SHRINKAGE, RESIDUAL STRESS, AND CRACKING IN HETEROGENEOUS MATERIALS. A Thesis. Submitted to the Faculty. Purdue University.

Transcription

1 SHRINKAGE, RESIDUAL STRESS, AND CRACKING IN HETEROGENEOUS MATERIALS A Thesis Submitted to the Faculty of Purdue University by Jae Heum Moon In Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy May 2006 Purdue University West Lafayette, Indiana

2 Dedicated to my family for their love and sacrifice ii

3 iii ACKNOWLEDGEMENTS I would like to express my deepest gratitude to Professor Jason Weiss for his countless guidance, support, advice, and patience throughout the course of my research. Without his supervision and advice, this research would not have been possible. I would also like to thank Professors Menashi Cohen, John Haddock, and Chin- Teh Sun for serving on my committee. The valuable suggestions and encouragement given by them has helped me immensely. I gratefully acknowledge the support received from the National Science Foundation and Purdue University while pursuing this research work. This research was conducted in the Material Sensing Laboratory and as such I gratefully acknowledge the support that has made this work possible. Thanks are also due to Ms. Janet Lovell and Mr. Mark Baker for the constant supports and help rendered by them in resolving various issues and problems regarding laboratory works. Very special thanks to Farshad Rajabipour, Bradly Pease, Jon Couch. This work would not have been completed without them. Any number of words would be insufficient to express my appreciation to all my friends at Purdue, Aleksandra Radlinska, Yogini Deshpande, Sulapha Peethamparan, and all my fellow students in Materials Area of School of Civil Engineering for their help and company.

4 iv Finally, I would like to express my sincere gratitude and love to my family members for their constant support, encouragement and love, which has served as an anchor for me. I will always treasure the sacrifices done by my parents for providing me with every possible opportunity to excel, and the love and care of my sisters.

5 v TABLE OF CONTENTS Page LIST OF TABLES... ix LIST OF FIGURES....x ABSTRACT... xvii CHAPTER 1: INTRODUCTION Background Proposed Research Organization of Contents... 7 CHAPTER 2: AUTOGENOUS AND DRYING SHRINKAGE OF CONCRETE AND THE RESIDUAL STRESS DEVELOPMENT IN THE RESTRAINED RING SPECIMENS Introduction Research Significance Analytical Approach Moisture Gradients and Their Relationship with Shrinkage Analytical Solution Stress Development in the Concrete Ring due to Differential Shrinkage: σθθ, diff. shr.(r, γ) Stress Development in the Concrete Ring due to Steel Ring Constraint: ( ) σ θθ, rest ring. r Superposition of Differential Shrinkage and Steel Ring Restraint... 23

6 vi Page...Page 2.6 Finite Element Analysis (FEA) Comparison of the Analytical Approach with FEA and Application to an Experiment Comparison with FEA Application to Experiments Conclusions CHAPTER 3: DEGREE OF RESTRAINT Introduction Ring Specimen Geometry Quantifying the Degree of Restraint under Uniform Shrinkage (Analytical Approach) Quantifying the Degree of Restraint in the Presence of Moisture Gradient A Practical Example of Geometry Selection for the Restrained Ring Test Summary and Conclusions CHAPTER 4: INTERNAL STRESS DEVELOPMENT IN HETEROGENEOUS SYSTEMS Introduction Effective Material Property Calculations Single Aggregate Prism Systems Multiple Aggregate Systems Summary CHAPTER 5: ANALYSIS PROCEDURES FOR ASSESSING THE SHRINKAGE AND CRACKING IN A CEMENTITIOUS COMPOSITE ON THE MESO-SCALE Introduction Overview of the Approach Analysis Procedures Preprocessing: Acquiring the Image from a Realistic Specimens Specimen Preparation

7 vii Page...Page Scanning Finite Element Simulations General Approach Failure Criterion Material Properties OOF Simulation (Performing Trials) Post-Processing: Analysis of Results Results and Discussions Conventional Approach Using Effective Medium Assumption OOF Simulation (Results) Summary CHAPTER 6: PARAMETRIC STUDY OF SHRINKAGE CRACKING IN A CONCRETE COMPOSITE Introduction Importance of This Research One Directionally Restrained Boundary Condition Volume fraction of aggregate Size Distribution of Aggregate Bond Condition Free Boundary Condition Volume fraction of aggregate Discussion CHAPTER 7: SUMMARY Introduction Quantification of the Residual Stress Development in Concrete Due to Non-Uniform Shrinkage and the Degree of Restraint Exhibited by the Restrained Ring Test When Different Specimen Geometries are Used

8 viii Page...Page 7.3 Discussion of the Comparison between OOF Simulation Results in a Heterogeneous Material and Experimental Measurements Conclusion LIST OF REFERENCES APPENDIX VITA

9 ix LIST OF TABLES Table Page 5.1 Material properties of cement paste Material properties of aggregate Information on the volume of aggregate in each composite shown in Figure

10 x LIST OF FIGURES Figure Page 1.1 The influence of aggregate on shrinkage and microcracking Conceptual illustration of the restraint components and a sample of stress gradient in the concrete ring Relative humidity (RH) gradients calculated using Eq. 2.3 (RH I =100%, RH S =50%, thickness=75mm) Stress gradient due to self-restraint in the concrete ring (i.e., without the steel ring (Eq. 2.10)) (R OC =0.225m, R IC =0.150m, E con =21GPa, ε SH-const = -100 µε) Conceptual illustration of the shrink-fit approach Stress gradient due to steel ring constraint in the concrete ring (Eq. 2.12) (R OC =0.225m, R IC =R OS =0.150m, R IS =0.1406m, E con =21GPa, E steel =200GPa, ε SH-const = -100 µε) Stress gradient in the concrete ring exposed to circumferential drying with steel ring (Eq. 2.13) (R OC =0.225m, R IC =R OS =0.150m, R IS =0.1406m, E con =21GPa, E steel =200GPa, ε SH-const = -100 µε) Comparison of stress gradients due to self-restraint in the concrete ring without steel ring (FEA and Eq. 2.10) (R OC =0.225m, R IC =0.150m, E con =21GPa, ε SH-const = -100 µε) Comparison of stress gradients due to self-restraint in the concrete ring without steel ring (FEA and Eq. 2.10) (γ=0.002) (R OC =0.225m, R IC =0.150m, E con =21GPa, ε SH-const = -100 µε).. 29

11 xi Figure Page 2.9 Comparison of stress gradients due to self-restraint in the concrete ring without steel ring (FEA and Eq. 2.10) (γ=0.02) (R OC =0.225m, R IC =0.150m, E con =21GPa, ε SH-const = -100 µε) Comparison of σ Actual-Max due to self-restraint in the concrete ring without steel ring (R OC =0.225m, R IC =0.150m, R IS =0.1406m, E con =21GPa, ε SHconst= -100 µε) Comparison of stress gradients due to steel ring restraint in the concrete ring (FEA and Eq. 2.12) (R OC =0.225m, R IC =R OS =0.150m, R IS =0.1406m, E con =21GPa, E steel =200GPa, ε SH-const = -100 µε) Comparison of stress gradients due to circumferential drying in a concrete ring with steel ring (R OC =0.225m, R IC =R OS =0.150m, R IS =0.1406m, E con =21GPa, E steel =200GPa, ε SH-const = -100 µε) Stress gradients due to circumferential drying in the concrete ring with steel ring (calculated using Eq and real test data) (0.50 W/C, 50% Sand, R OC =0.225m, R IC =R OS =0.150m, R IS =0.1406m) Predicted depth of microcracking in the concrete ring (0.50 W/C, 50% sand, R OC =0.225m, R IC =R OS =0.150m, R IS =0.1406m) Drying direction and the resulting stress development of restrained ring A conceptual illustration of the degree of restraint The influence of specimen geometry on the degree of restraint and the ratio εst/ε SH Comparison between the analytical solution and the experimentally determined degree of restraint before the through cracking (top and bottom drying) The simulated humidity profiles corresponding to different γ values Degree of restraint as drying progresses; obtained from finite element simulations (top and bottom drying) Influence of shrinkage condition on the deformation of concrete and steel rings.. 58

12 xii Figure Page 3.8 The effect of non-uniform deformation of the steel ring (simulations results) Comparison on the degree of restraint obtained from the analytical procedure (Eq. 6, uniform drying) and the values obtained from finite element simulations (non-uniform drying) Graphical representation of Eq. 3.5 and 3.6 in terms of possible combinations of geometry parameters for a desired degree of restraint Shrinkage exponent n as a function of the aggregate to paste stiffness ratio Acoustic activity in mortars at early ages (50% aggregate volume) An illustration of the two geometries simulated Stress development in an (a) unrestrained single aggregate specimen and b) the externally restrained single aggregate specimen Stress localization for different boundary conditions Stress development along the Y-axis of a restrained prism specimen over an aggregate (restrained boundary condition) Experimental data from a restrained specimen containing an Instrumented Aggregate Multiple aggregate system composed of unit cell matrix Comparisons of equivalent elastic modulus of composites Maximum stress development in an externally unrestrained composite Maximum stress development in a composite externally restrained in one direction Cracking tendency of a composite (horizontally restrained boundary conditions) Flow chart of OOF simulation on the meso-scale Specimen preparation (surface treatments). 107

13 xiii Figure Page 5.3 Flat-bed color scanner and concrete samples (saw-cut, polished, and stained) Image analysis (re-colorization, part of the scanned image V agg = 55.7 %) Meshed sample Failure criteria Color scaled image of stress distribution in a composite (Paste-1, Horizontally restrained, ε paste = -140 µε) Cracked image (paste-1, horizontally-restrained, ε paste =140 µε) Equivalent elastic modulus of a composite (OOF simulation vs. Hansen s model) Equivalent stress vs. shrinkage strain of paste Shrinkage strain of paste vs. load (horizontally-restrained boundary condition) Cracks in a composite (one directionally restrained boundary condition) Cracks in a composite (free boundary condition) Color contour image of stress distribution in a composite (One directionally restrained boundary condition) Cumulative area fraction vs. stress Strain energy density versus area fraction Strain energy density level versus normalized stored energy Strain energy density level vs. cumulative values of normalized stored energy Shrinkage strain of paste vs. cumulation of normalized stored energy 142

14 xiv Figure Page 5.20 Relationships among the percentage of the released energy, area fraction of cracked zones, and the magnitude of shrinkage strain of paste (free (free boundary condition) Relationship between the area fraction of cracked zones and the percentage of the released energy (free boundary condition) Equivalent elastic modulus of a composite with the volume fraction of aggregate Equivalent shrinkage strain of a concrete as a function of the volume fraction of aggregate Equivalent stress of a concrete (ε paste = 225 µε) Series law for the calculation of the tensile strength of a concrete composite Strength and equivalent stress of a composite with the variation of the volume fraction of aggregate (paste 1 in Table 5.1) Cracking potential of composites based on the volume fraction of aggregate and material properties Images with different volume fraction of aggregate Images of composites with cracks (one directionally restrained boundary condition) Average sectional stresses of composites vs. (a) shrinkage strain of mortar and (b) shrinkage strain of paste (one directionally restrained boundary condition) a Stored strain energy vs. shrinkage strain of mortar b Stored strain energy vs. shrinkage strain of paste c Energy release at failure vs. volume fraction of aggregate d Area fraction of cracked zone vs. percent-released energy e Acoustic energy development as a function of time (Chariton et al., 2002) 171

15 xv Figure Page 6.11 Composites with different number of aggregates maintaining the volume fraction of aggregate Stress development in composites Sectional stress development in a composite under the restrained boundary condition The sectional ratio of the total chord length of the aggregates to the height of the concrete Images of cracked composites at failure (one directionally restrained boundary condition) Cracking behavior of a mortar composite (V agg = 55.7 %) with different bond conditions (one directionally restrained boundary condition) Image of the cracked mortar composite (V agg = 55.7 %) at failure (ε paste = -190 µε) (one directionally restrained boundary condition with perfectly plastic failure criterion for bond phase) Cracking behavior of a mortar composite with different failure criterion on the bond phase (one directionally restrained boundary condition Images of composites with cracks (free boundary condition - ε paste = -250 µε) Relationship between the area fraction of cracked zones and the shrinkage strain of paste (free boundary condition) Cracking behavior of mortar composites (free boundary condition) Relationship between the area fraction of cracked zones and the energy release (Free boundary condition) Images of cracked composites at ε paste = -250 µε (free boundary condition) Volume fraction of cracked zones depending on the ratio of bond material properties to paste material properties Cracking behavior of a mortar composite (V agg = 55.7 %) with different bond conditions (free boundary condition).. 203

16 xvi Figure Page 6.26 Relationship between the area fraction of cracked zones and the percent released energy (bond phase, free boundary condition) Cracked image of a mortar composite (V agg = 55.7 %) (ε paste = -250 µε) (free boundary condition with perfectly plastic failure criterion for bond phase) Area fraction of cracked zones vs. shrinkage strain of paste Cracking behavior of a mortar composite with different failure criterion on the bond phase (free boundary condition) Relationship between the area fraction of cracked zones and the percent released energy A comparison of the shrinkage strain of mortar and the shrinkage strain of paste as (OOF simulation, elastic) Relationship between the shrinkage strain of mortar and the shrinkage strain of paste (OOF simulation, free boundary condition) Energy release and Area fraction of cracked zones versus shrinkage strain of mortar and paste phase (OOF simulation, free boundary condition) Restrained (a) and free (b) shrinkage specimen geometries with locations of acoustic emission sensors (Pease et al., 2004) Autogenous shrinkage of a paste specimen and a mortar specimen, and the dynamic elastic modulus of paste (w/c = 0.3) Cumulative acoustic energies of mortar specimens (free boundary condition).. 226

17 xvii ABSTRACT Moon, Jae Heum, Ph.D., Purdue University, May Shrinkage, Restraint, Residual Stress, and Cracking in Heterogeneous Materials. Major Professor: W. Jason Weiss. Concrete experiences volumetric changes as a result of material formation (cement hydration), thermal variations, or moisture losses. If these volume changes are prevented by the structure surrounding the concrete or volumetrically stable phases inside the concrete, residual stresses can develop. In many cases, these residual stresses may be large enough to result in cracking. While the premature cracking of concrete is significant enough to be considered in the design of concrete facilities, existing test methods and design methodologies need to be updated to better quantify the cracking potential especially when the concrete experiences non-uniform deformations. Most design and testing approaches assume that concrete behaves like a homogeneous material. However, concrete is a composite that consists of cement paste and aggregates that have dissimilar material properties. When concrete changes its volume, localized internal residual stresses develop due to heterogeneity (i.e., paste and aggregates) that could lead to microcracking and premature cracking of concrete. Therefore, to properly evaluate the cracking behavior of concrete, an analytical tool considering the heterogeneous nature of concrete should be developed.

18 xviii This research begins with quantifying the impact of non-uniform shrinkage on the residual stress development in the restrained ring test considering that the concrete experiences non-uniform moisture loss (drying). The role of the boundary conditions and the degree of restraint on residual stress development is also discussed. Next, the internal residual stresses that develop in a multi-phase composite system are examined by varying the series of parameters (material properties of each-phase, volume fraction of aggregate, and bond conditions between the phases). Analytical modeling is used to assess the microcracking and cracking behavior of concrete composite systems using the objectoriented finite element code. It is the hypothesis of this research that the cracking behavior of concrete can be properly evaluated by assuming that concrete is a heterogeneous multi-phase composite with the assistance of a recently developed object-oriented finite element code that enables meshing a complex material meso-structure using optical mesostructural images. Although this research focuses on shrinkage in cementitious composites only, this software could have numerous applications in determining damage development in other composites as well.

19 1 CHAPTER 1: INTRODUCTION 1.1 Background Cementitious materials change volume in response to moisture variations, temperature variations, and chemical reactions. If these volume changes are restrained, residual stresses can develop and could lead to microcracking and cracking. Several models exist to estimate shrinkage in mature concrete sections (Bazant and Panula, 1979; Gardner and Lockman, 2002; and ACI 209 under revision). While these models may be useful for predicting long-term maximum shrinkage for design codes, they do not work well in predicting the rates of shrinkage that are important for performance models, especially when these models are focused on describing early-age behavior. Recently developed theoretical models describe the risk of cracking associated with the restraint of autogenous, drying, and thermal shrinkage (Weiss et al., 1999; Yang et al., 1999), however, these models are based on macroscopic properties and require the direct input of measured shrinkage values. Although many improvements have been made in concrete technology over the last several decades, it is still common to observe cracking in concrete soon after it is placed (Weiss et al., 1997; Burrows, 1999; Weiss and Shah, 2002; RILEM, 2002). This premature microcracking and cracking is mainly caused by excessive volume changes at an early age when the concrete does not have sufficient strength to resist these changes.

20 2 The heterogeneity of concrete results in internal residual stresses even if the net section force is low. Few approaches exist to explicitly quantify the role of each of the constituent phases (paste, transition zone or bond, and aggregate) in contributing directly to shrinkage and cracking. If such an approach exists it could be useful for performance prediction and mixture optimization models. Historically, the contribution of the aggregate phase has only been considered through a series of approximate equations that result in the shrinkage of concrete being considered as the product of the free shrinkage of a pure paste phase and the volume fraction of the paste phase raised to an exponent (usually between 1.2 and 1.7) (Pickett, 1956; Hobbs, 1974). While this type of approach generally works well to describe long-term shrinkage, it does not describe local stresses and microcracking that may develop inside the concrete. The approximate method for predicting shrinkage described above does not account for changes in the relative stiffness between the paste and the aggregates, aggregate size or shape, or damage development (microcracking), which of itself is ultimately important for use in transport models or through-section cracking predictions. In other words, previous models provide averaged (equivalent) residual stress calculations using the average material properties of a composite that would have higher internal residual stresses due to the stress concentration induced by heterogeneity. For example, Figure 1.1 (Pease et al., 2003b) illustrates the shrinkage strains and the passive acoustic energy measured in paste and mortar systems. It can be seen that the shrinkage strain of a mortar specimen with 45 percent aggregate (by volume) has lower shrinkage strain than a paste specimen (Figure 1.1 (a)). It can also be seen in Figure 1.1

21 3 (b) that, with closely spaced aggregates (45% by volume), the microcracking is more constant over time (i.e., a smooth curve). However, when the aggregates are spaced further apart (15 percent by volume), discrete acoustic activities (sudden jumps in acoustic energy) are observed, which is consistent with the idea that cracks propagate through the paste between the aggregates. However, it has been observed that the cumulative acoustic energy does not linearly increase by increasing the volume fraction of aggregate, regardless of the boundary condition as shown in Figure 1.1 (c). This result implies that each phase (paste, aggregate, and bond) has an important role in the microcracking and cracking behaviors of concrete. Free Shrinkage (µε) Free Shrinkage Paste 45% Agg Age of Specimen (Hours) Cumulative A c o u s Acoustic t i c E n e r g Energy y ( µ V s ) (nvs) % Aggregate 45% Aggregate Age of Age of Specimen (Hours) Cumulative Acoustic 24 Hours (nvs) Restrained Free Aggregate Content (%) (a) (b) (c) Figure 1.1 The influence of aggregate on shrinkage and microcracking: (a) shrinkage strain versus age, (b) cumulative acoustic energy versus age (restrained boundary condition), and (c) cumulative acoustic energy (i.e., microcracking of damage) versus volume fraction of aggregate

22 4 1.2 Proposed Research To resolve the previously mentioned problems the restrained ring test will be further investigated in this research. This research begins by considering the uniform (autogenous) and non-uniform (drying) shrinkage behavior of concrete. This work begins considering the concrete as a homogeneous material so that the effects of uniform and non-uniform shrinkage can be assumed as it relates to the residual stress development in the restrained ring test. Analytical solutions are developed for the calculation of the residual stresses that are induced by non-uniform shrinkage. A method will also be provided to quantifying the degree of restraint in the restrained ring test for specimens with different geometries. To better understand how mixture proportions influence the potential for premature cracking in concrete, this work investigates the use of new simulation techniques. A model was used that enabled the concrete to be considered as a multi phase composite systems. This enabled the paste and aggregate phases to be explicitly considered thereby enabling the stress concentrations around the aggregates to be quantified. A recently developed object-oriented finite element code (OOF) (Fuller et al. 2003) is used in this research. OOF provides a user interface which enables the use of actual 2-D images of concrete to be rapidly input in a finite element model. A procedure was developed for taking an image of the cross-section of the concrete, developing a finite element model for this composite computing the residual stresses that develop as a result of volume change, and implementing a cracking criterion to describe damage development.

23 5 This thesis describes methodologies to quantify residual stress development in concrete when it experiences uniform and non-uniform shrinkage. In addition to assuming concrete to be a homogeneous material, this research shows how optical microscopy could be combined with analytical tools (FEM) for predicting the stress fields in a heterogeneous medium.

24 6 1.3 Objectives of the Study The objectives of this thesis are to: Provide a theoretical solution for quantifying the equivalent residual stress development in concrete when it experiences non-uniform shrinkage in a ring geometry due to moisture gradient. Provide a theoretical solution to combine the equivalent residual stress development for the specimen with non-uniform shrinkage and the residual stress that develops from the boundary condition (degree of restraint) for the restrained ring geometry. Investigate the importance of heterogeneity on meso-structure scale as it relates to the development of residual stress concentrations, and cracking in concrete composite systems. Evaluate the role of each parameter (volume fraction of each phase, material properties of each phase, geometry of each phase, and bond condition on stress development and cracking potential in concrete composites.

25 7 1.4 Organization of Contents Chapter 2 provides a solution for the calculation of the equivalent residual stress in a restrained ring test when it experiences non-uniform shrinkage due to moisture gradients. Chapter 3 provides a solution for the calculation of the degree of restraint in a restrained ring test. This solution enables the degree of restraint to be quantified and used in the selection of a test specimen geometry that best simulates a field structure. Chapter 4 describes the concept of stress localization and concentration in a composite system due to the existences of material phases that have different material properties. This chapter describes the fundamental behavior of the composite system. Chapter 5 describes the procedure of using new simulation techniques, to assess the complex geometries that may be expected in an actual composite phase distribution. This chapter also discusses the analysis procedures for OOF simulation results. Chapter 6 discusses the parameters that affect the microcracking and cracking in a composite. The role of each parameter (volume fraction of each phase, material properties of each phase, size and shape of the aggregates in the aggregate phases, and the bonding condition between the aggregate and paste will be discussed. Chapter 7 contains the summary and conclusions of this research.

26 8 CHAPTER 2: AUTOGENOUS AND DRYING SHRINKAGE OF CONCRETE AND THE RESIDUAL STRESS DEVELOPMENT IN THE RESTRAINED RING SPECIMENS 2.1 Introduction Cementitious materials change volume in response to moisture variation, temperature variation, and chemical reaction. If restrained, these volumetric changes result in residual stress development that can lead to cracking. The restrained ring test has recently become a popular method to assess a mixture s susceptibility to restrained shrinkage cracking (Krause et al., 1995; AASHTO PP 34-99; Shah et al., 1992; Berke et al., 1997; Grzybowski 1989; Lim et al., 1999). Over the last 80 years, researchers have considered various adaptations of the ring test. For example, early work by Carlson and Reading (1988) used the ring to qualitatively compare the shrinkage cracking potentials of various cement compositions. Malhotra and Zoldners (1967) proposed that a pressurized ring test could be performed as a potential method for assessing the tensile strength of concrete. Swamy and Stavrides (1979) suggested a ring test where strain could be measured for assessing the behavior of fiber reinforced concrete. Grzybowski and Shah (1989) used the ring test to investigate strain development in plain and fiber reinforced concrete using strain gages on the concrete surface for the calibration of a modeling approach. Kovler et al. (1993) combined the passive restraint from the classic ring test with the active approach advocated by Malhotra for the development of a test with an inner core ring that was made from a material with a higher thermal expansion

27 9 coefficient. In this test Kovler used the ring to apply passive restraint until a specific time at which a tensile load was applied to the concrete ring by introducing a temperature rise in the specimen assembly. Kovler s approach enabled the testing time to be shortened as the temperature rise that was required to cause fracture was related to the potential additional stress capacity of the material. Weiss and Shah (2002) used the ring geometry to demonstrate the benefit of a fracture mechanics approach for predicting the geometry-dependent failure of the restrained concrete ring. More recently, in an effort to increase the severity of the test, He et al. (2004) proposed an elliptical ring to increase the stress concentration provided by the ring. While the aforementioned research shows a wide range of applications for the ring approach, this thesis will build on recent work to use measurements from an instrumented ring test to make the test more quantitative. Attiogbe et al. developed expressions based on a thin ring (See et al., 2003a; Attiogbe et al., 2003; See et al., 2003b), while Weiss et al. (Weiss and Ferguson 2001; Hossain and Weiss 2003) proposed a solution for a thick-walled concrete ring under uniform radial drying. This work enabled the strain measured in a steel ring to be used to determine the maximum residual tensile stress in the concrete ring using Eq R + R R OS R IS σ Actual Max = ε Steel (t) E s Eq. 2.1 R R 2R 2 OS 2 OC where R OS is the outer radius of steel ring, R IS is the inner radius of steel ring, R OC is the outer radius of concrete ring, and E S is the elastic modulus of the steel ring. It should be noted, however, that although steel was used in this expression, the restraining ring can be made of any material provided it remains linear and elastic during the test. It has been 2 OC 2 OS 2 OS

28 10 subsequently shown that this solution converges with the thin-ring solution for a sufficiently small specimen (Attiogbe et al., 2004). Additionally, it was shown that the combination of an elastic solution with Eq. 2.1 can provide an assessment of creep in the ring (Hossain et al., 2003). While Eq. 2.1 is appropriate for the case of uniform drying along the radial direction, Figure 2.1 illustrates how a moisture gradient develops due to circumferential drying, which can significantly influence the stress distribution (Weiss and Shah, 2002; Bazant and Chern, 1985; Kim and Weiss, 2003; Weiss and Shah, 2001). Stress gradient Drying from the outer circumferential surface Stress gradient due to humidity gradient Stress gradient due to steel ring restraint (a) Total Stress Development (b) Self Restraint (c) Ring Restraint Figure 2.1 Conceptual illustration of the restraint components and a sample of stress gradient in the concrete ring Moisture gradients were described as early as the 1930s by Carlson, who began looking at the role of specimen thickness on drying shrinkage (Carlson, 1937). Bazant and Najjar (1971) suggested that moisture diffusion analysis could be made more accurate with the use of a diffusion coefficient that was non-linear with moisture content.

29 11 Bazant and Chern (1985) developed a general procedure for considering moisture gradients, stress development, and microcracking. Grzybowski and Shah (1989a and 1989b) developed a model for restrained shrinkage cracking that considered moisture gradients in the ring analysis that used a linear specimen approximation and a continuum damage model, while Weiss and Shah (2001) used the moisture profile to illustrate the specimen size dependency on the age of cracking. They described how the moisture gradient resulted in complex stress fields that change shape over time (Weiss and Shah, 2001; Weiss, 1999). To quantify the moisture field that develops, Schieβl et al. (2000) suggested a procedure based on electrical impedance. Grasley and Lange (2004) demonstrated a novel sensor which can be used to measure the humidity profile directly. Once the humidity profile is known, it can be used to compute the internal residual stress in a specimen (Weiss and Shah, 2002b; Grasley and Lange, 2004; Bazant, 1986). The existence of moisture gradients has been shown to result in the development of surface microcracking based on analytical modeling (Bazant and Chern, 1985; Granger et al., 1997), acoustic emission measurements (Weiss and Shah, 2001c), and optical microscopy (Bisschop and Van Mier, 2001). While this research has pointed toward the importance of moisture gradients in shrinkage and cracking studies, little has been developed in understanding how moisture gradients influence the results that are obtained from the ring test. The residual stress development and cracking in the restrained ring depends on the drying conditions. Stresses in the thick wall under uniform drying are highest at the inner radius and the stresses decay as a function of r 2 (Dally and Riley, 1991), while the ring that dries from the outer circumference shows that the maximum actual residual

30 12 stress occurs at the outer radius (at least for short drying times) (Weiss and Shah, 2001c). This difference in the shape of the stress field is important since it changes how the crack initiates, develops, and propagates. Acoustic emission has been used to show that in rings that dry uniformly along the radius, the cracking begins at the inner circumference of the concrete rings while in specimens that dry from the outer circumference of the rings, the cracks begin at the outer edge and propagate toward the center (Kim and Weiss, 2003; Hossain and Weiss, 2005). Analytical methods are needed to better characterize the stress field for modeling failure in the ring. This thesis outlines a unique approach for determining this stress field by considering the moisture gradient and the steel ring restraint in the analysis of rings that dry from the outer circumference. 2.2 Research Significance This chapter will propose a solution for determining the residual stress using the restrained ring test geometry in which the residual stress that develops in the ring is thought to be due to a combination of the self-restraint that develops due to non-uniform drying conditions and the restraint that is provided by the steel ring. Based on this approach, an analytical equation is presented for computing the residual stress field that develops in the restrained ring. This expression can be used to describe stress gradients that develop in a concrete ring. The development of this equation can enable the use of a simple economic test (which has been proposed by AASHTO (AASHTO PP 34-99) and, more recently, by ASTM (ASTM C )), to provide information about the role of moisture gradients in the shrinkage cracking behavior of concrete. In addition to use for standard tests, this approach has implications in predicting the cracking behavior of slabs

31 13 and bridge decks in which the stress development is due to the concurrent effects of the self-restraint and the restraint by the boundary conditions. It is fully anticipated that the approach presented can be extended for general use in quantifying the cracking potential of concrete under non-uniform drying. 2.3 Analytical Approach It can be argued that the stress that develops in the restrained ring due to circumferential drying can be considered as a combination of two components as shown in Figure 2.1 (i.e., self-restraint due to non-uniform drying and restraint that comes from external restraint supplied by the ring). Therefore, if the residual stress development can be computed for each restraint component, all of the stress fields can be obtained. An example of how the moisture gradient varies in concrete as a function of drying time is shown in Figure 2.2 (γ is the product of the moisture diffusion coefficient and the drying time and is explained in detail in the following section) (Moon et al., 2004). It can be noticed that for short drying times (i.e., low values of γ), the moisture gradient is very severe, with the majority of the drying taking place along the outer edge of the specimen. As drying time increases (i.e., γ increases), the depth of the drying front progresses further into the concrete and the moisture distribution is more gradual along the radial direction.

32 RH (%) Concrete Distance from the outer surface (m) γ =0.002 =0.004 =0.008 =0.02 =0.04 =0.08 =0.2 =0.5 =100 Figure 2.2 Relative humidity (RH) gradients calculated using Eq. 2.3 (RH I =100%, RH S =50%, thickness=75mm) It should be noted that the loss of moisture causes the specimen to shrink. If the concrete ring is unrestrained (i.e., no steel ring), it will shrink in the radial direction. However, when the steel ring is present, the steel ring limits this shrinkage and residual stresses develop (σ θθ,rest.-ring ). This restraint causes a pressure to develop at the interface between the steel and concrete which can be calculated directly since the strain at the inner surface of the steel ring (i.e., strain ε steel ) is directly measured (Hossain and Weiss, 2003a; Shah and Weiss, 2003). In addition to the restraint from the steel, residual stress develops due to self-restraint caused by circumferential drying (σ θθ,diff.-shr. ). The total

33 15 stress development can be expressed as the sum of the stress due to restraint and the differential shrinkage. σ Eq. 2.2 θθ ( r, γ) = σθθ,rest. ring + σθθ,diff. shr. The analytical solution of each component of this expression is further described in section 2.5, however in order to enable a better understanding of the foundation of the differential shrinkage behavior due to the moisture gradient, it is first necessary to describe the moisture gradient and its relationship to drying shrinkage in Section Moisture Gradients and Their Relationship with Shrinkage The analysis described in this thesis considers a ring that is exposed to drying from the outer circumference. The surface that is exposed to drying shrinks more rapidly than the internal concrete, which loses water more slowly. Early work on moisture gradients by Carlson in the 1930s considered the moisture gradient to be explained with a diffusion-controlled process similar to that used in thermal analysis (Carlson, 1937). Bazant and Najjar (1971) proposed that moisture loss in concrete can be better expressed using a nonlinear form of Fick s second law. They proposed that in addition to being age-dependent the diffusion coefficient was non-linear with respect to moisture (i.e., pore pressure, relative humidity, or degree of saturation). While the non-linear approach is more accurate, especially for long term drying, this work will utilize a linear diffusion function (in the form of Eq. 2.3), which was fit to physical tests to estimate the humidity profiles as shown in Figure 2.2 (Moon et al., 2004). The linear diffusion approach is preferred here since it enables the use of the

34 16 closed-form solution for determining the relative humidity (RH) distribution which is amenable to hand calculation. It should be noted, however, that a more general solution will be pursued in future work. As a result, a simple moisture gradient can be determined using Eq RH(x, t) x = RH I (RHI RHS ) erfc Eq D t where, RH(x,t) is the relative humidity at a depth x from the drying surface, t is the drying time, erfc is the complementary error function, RH S is the relative humidity at the surface of the specimen, RH I is the internal relative humidity, and D is the aging moisture diffusion coefficient of concrete. To simplify the combined treatment of the diffusion coefficient and drying time here, a single parameter (γ) was introduced as twice the square root of the product of the diffusion coefficient and time ( γ = 2 Dt ). The internal relative humidity (i.e., pore pressure) could be used to predict the induced drying shrinkage strain (Bazant, 1986). The relationship between shrinkage and relative humidity can be approximated for the case of high relative humidity (i.e., RH > 50%) as linearly proportional (Weiss and Shah, 2002c). As a result, the drying shrinkage strain can be expressed by a constant free shrinkage coefficient (ε SH-Const ) and the change of relative humidity RH as illustrated in Eq. 4: ε t) = ε RH Eq. 2.4 ( SH const The constant free shrinkage coefficient (ε SH-Const ) can be considered as the slope of the shrinkage versus the change in the relative humidity relationship. The change of

35 17 relative humidity ( RH) represents the difference between 100% RH and the internal RH of a concrete specimen with time. 2.5 Analytical Solution As previously described, the stress development in the concrete ring is mainly due to two components; self-restraint due to differential shrinkage caused by moisture gradients and external restraint due to the effect of the steel ring. In this section, an analytical solution for calculating each component is presented. It should be noted that in its initial form of this solution it is assumed that the concrete ring specimen exhibits linear creep (relaxation) without microcracking (Bazant and Chern, 1985; Kim and Weiss, 2003; Bisschop and Van Mier, 2001; Granger et al., 1997) Stress Development in the Concrete Ring due to Differential Shrinkage: σθθ, diff. shr.(r, γ) The equation for calculating stress development in the concrete ring due to differential shrinkage can be obtained by using an approach that is similar to the approach used for differential thermal effects (Boley and Weiner, 1988). It should be remembered that the analysis of the effect of self-restraint does not consider external restraint by the steel ring. Plane stress is assumed in the derivation and, due to the axi-symmetric geometry, the shearing stress at the interface is zero. Using Hooke s law and the drying shrinkage strain from Eq. 4, the stress components (i.e., stress in radial direction: σ rr and stress in circumferential direction: σ θθ ) can be expressed in radial components as:

36 18 E con du u σ rr = + ν (1 + ν) εsh const RH 1 ν Eq dr r σ θθ E con = 1 ν 2 u + ν r du dr (1 + ν) ε SH const RH Eq. 2.6 where, u is the radial displacement ( ε rr = du / dr, = u / r ), r is the radial distance in the cylindrical coordinates system, and E con is the effective elastic modulus of concrete (note: the effective modulus can be used to consider the effects of creep). Substituting Eq. 2.5 and 2.6 into the stress equilibrium equation ( σ r + ( σ σθθ )/ r 0) results in the following expression; d dr 1 d(r u) = ε r dr SH ε θθ rr / rr = d( RH) const (1 + ν) Eq. 2.7 dr This expression can be solved to yield the general solution for the hollow ring geometry: (1 + ν) ε r SH const C 2 u(r) = RH rdr + C1 r + r R IC r Eq. 2.8 where C 1 and C 2 are constants of integration. Application of the boundary conditions (i.e., no traction on the inner or outer surface of the ring) enables the constants of integration (C 1 and C 2 ) to be obtained. Combining Eq. 2.5, 2.6 and 2.8 enables the residual stress ( σ (r, θθ, diff. shr. γ) ) in the concrete ring caused by differential shrinkage to be computed using Eq. 2.9 : 2 2 ε + R OC r SH const E r R con IC, diff. shr. = erfc(a) r dr + erfc(a) r dr erfc(a) r r R RIC RIC OC R IC σθθ Eq. 2.9 (a) where, A is given as; 2

37 19 A = (R OC r) / γ Eq. 2.9 (b) While Eq. 2.9 (a) provides an expression for stress which can be integrated to yield Eq as a closed-form expression for the residual stress caused by different shrinkage. σ θθ where, f (r) 2 2 ε + SH const E r R con IC, diff. shr. = [f (R OC ) f (R IC )] + f (r) f (R IC ) erfc(a) r r R OC R IC R OC 2 A π erf (A ) R OC = γ erfc (A ) A + erfc (A ) A γ π A A 2 4 e 8 2 e 2 Eq (a) γ Eq (b) Figure 2.3 shows the example of resulting stress gradients in a concrete ring using Eq for the case of a ring without steel restraint (R OC =225mm, R IC =150mm, ε SHconst= -100 µε). The stresses that are developed due to self-restraint have steep gradients near the outer surface when drying begins (i.e., low values of γ). With increasing drying time (i.e., increasing γ), the gradient flattens and the shrinkage becomes more uniform. Decreasing the severity of the humidity profile implies that the strain (or stress) developed by self-restraint will be decreased.

38 20 3 Outer Surface Of the Conc. Ring Inner Surface Of the Conc. Ring Stress (MPa) 2 1 γ =0.002 =0.004 =0.008 =0.02 =0.04 =0.08 =0.2 =0.5 = Distance from the outer surface (m) Figure 2.3 Stress gradient due to self-restraint in the concrete ring (i.e., without the steel ring (Eq. 2.10)) (R OC =0.225m, R IC =0.150m, E con =21GPa, ε SH-const = -100 µε) Stress Development in the Concrete Ring due to Steel Ring Constraint: σ ( ) θθ, rest ring. r The stress development due to the external restraint from the steel ring must also be considered. Previous research has shown that the restraint from the steel ring can be simulated by separating the steel and concrete ring and treating the problem as a shrinkfit problem. The concrete ring is permitted to shrink some amount ( U SH ) that is equal to that caused by drying and autogenous shrinkage. The composite cylinder can be considered to have a fictitious pressure that is applied on the outer surface of the steel ring that is equal to the pressure on the internal surface of the concrete ring (Weiss and Shah, 2002; Weiss and Ferguson, 2001; Hossain and Weiss, 2003a; Hossain and Weiss,

39 ). The pressure is adjusted until the steel ring is compressed ( U Steel ) and the concrete ring is expanded ( U conc ) to compensate for the shrinkage as shown in Figure 2.4 (further details are provided in (Hossain and Weiss, 2003a)). Free Shrinkage (No Steel Ring) R OC R IS R OS (R IC ) U SH U conc U SH U steel (a) Before Shrinkage Occurs (b) Free Shrinkage (No Steel Ring) (c) Shrinkage with Steel Ring Figure 2.4. Conceptual illustration of the shrink-fit approach The fictitious external pressure could be determined from Eq since the strain in the steel can be obtained experimentally using the strain gage on the inner surface of the steel ring (Hossain and Weiss, 2003a). p i 2 R OS R = εsteel (t) ES Eq R 2 IS 2 OS The fictitious pressure that can be thought to act on the steel ring could be related to an internal fictitious pressure that acts on the concrete ring and as a result the stress distribution in the concrete ring can be determined as shown in Eq (Hossain and Weiss, 2003a).

40 22 σ 2 2 R OS R + OC ( r) = p i Eq R OC R OS r θθ,rest. ring 2 Figure 2.5 shows an example of the stress gradient in the concrete ring calculated using Eq for the case of the steel ring restraint (R OC =225mm, R OS =R IC =150mm, R IS =140.6mm, E steel =200GPa, ε SH-const = -100 µε) where ε steel is the strain that would be measured in the steel ring. The maximum stress develops in the circumferential direction along the interface between concrete and steel. It can be noticed that Eq is also able to be used in the case of autogenous shrinkage because it can consider a uniform shrinkage of the matrix. Stress (MPa) γ =0.002 =0.004 =0.008 =0.02 =0.04 =0.08 =0.2 =0.5 =100 Inner Surface of the Conc. Ring Distance from the outer surface(m) Figure 2.5 Stress gradient due to steel ring constraint in the concrete ring (Eq. 2.12) (R OC =0.225m, R IC =R OS =0.150m, R IS =0.1406m, E con =21GPa, E steel =200GPa, ε SH-const = -100 µε)

41 Superposition of Differential Shrinkage and Steel Ring Restraint By combining Eq. 2.10, 2.11, and 2.12, the stress development in the concrete ring can be expressed as the sum of the stress caused by the self-restraint (i.e., differential shrinkage in Section 2.5.1) and the external restraint (i.e., stress caused by the steel ring in Section 2.5.2). Eq enables the stress ( σ θθ ) at any point in the concrete ring to be calculated. σ θθ ( r, γ) = σ θθ,rest. ring + σ θθ,diff. shr R OS R IS = ε R steel ( t) E S (R OC R OS ) r 2 2 ε r + R SH const con IC 2 (f (R OC ) f (R IC )) + f (r) f (R IC ) erfc(a) r r R OC R IC OC + E Eq Figure 2.6 shows the total stress gradient in the concrete calculated using Eq where the concrete ring is restrained by a steel ring (R OC =225mm, R OS =R IC =150mm, R IS =140.6mm, E steel =200GPa, E con =21GPa, ε SH-const = -100 µε). As expected, the stress development is dependent on the extent of drying time (i.e., γ). It can be noticed that at short drying times the maximum stress occurs on the outer circumference. As drying continues, the position of σ Actual-Max changes from the outer circumference of the concrete ring to the inner surface after γ exceeds approximately 0.2 (i.e., longer drying time depending on the elastic modulus of the concrete) (Figure 2.6).

42 24 Stress (MPa) γ =0.002 =0.004 =0.008 =0.02 =0.04 =0.08 =0.2 =0.5 =100 Inner Surface of the Conc. Ring Steel Ring 0 E con / E steel = Distance from the outer surface (m) Figure 2.6 Stress gradient in the concrete ring exposed to circumferential drying with steel ring (Eq. 2.13) (R OC =0.225m, R IC =R OS =0.150m, R IS =0.1406m, E con =21GPa, E steel =200GPa, ε SH-const = -100 µε) This result can be explained by the fact that, at initial drying times γ, the stress development in a concrete ring is mainly governed by self restraint, while it is mainly governed by the external steel ring restraint for longer drying periods. To verify the applicability of the superposition of stresses obtained from Eq and 2.12, each case of restraint (self restraint: Eq and steel ring restraint: Eq. 2.12) will be compared with the results of finite element analysis separately, and Eq will be finally compared with the finite element analysis in the following section.

43 Finite Element Analysis (FEA) A series of finite element analyses were simulated using finite element analysis (FEA) in ANSYS to verify the appropriateness of the developed approach. The ring specimen was considered as an axi-symmetric geometry where the drying shrinkage gradient occurs along a radial section of the ring specimen. Quadrilateral eight-node elements were used for the simulations. The geometry of the concrete specimens simulated had a depth of 75 mm, an outer radius of concrete 225 mm, and an inner radius of concrete 150 mm. The geometry and material properties of the steel ring were fixed (thickness = 9.4 mm, E s = 200 GPa, and ν s = 0.3). Poisson s ratio and the effective elastic modulus of concrete were selected to be 0.18 and 21 GPa, respectively. To verify the effect of self-restraint and steel ring restraint, two types of analyses were performed: with the steel ring and without the steel ring. For this analysis, it was assumed that the concrete ring dries only from the outer circumference of the ring. Eq. 2.3 was used to determine the relative humidity profile while RH I and RH S were assumed as 100% RH and 50% RH, respectively. A free shrinkage strain constant (ε SH-const ) was assumed to have a value of -100 µε. Finite element analyses were performed by varying the severity of the drying γ ( γ=0.001 ~ 100). Theoretically, an infinite value of γ corresponds to a concrete ring, which dries uniformly along the radius (as described in Eq. 2.1), however a γ value of 100 was found to be essentially equivalent to higher values, and, as such, it is the largest γ simulated. Because the FEA program does not support the drying shrinkage loading directly, shrinkage was introduced using a temperature load variable substitution. The

44 26 temperature distribution was assumed to vary as an error function (erf) (the change in temperature in this thesis is analogous to a change in relative humidity) and a thermal expansion coefficient was the input for the concrete (this can be thought of as being analogous to the free shrinkage constant) to obtain the same effect as drying shrinkage. The maximum stress was selected along a plane perpendicular to the depth direction at midheight in the ring specimen to obtain the stress values that were reported. The contact condition between steel and concrete for the FE analysis was chosen to simulate an unbonded condition by using a pseudo-bar that essentially permitted a friction-free condition along the vertical direction of the ring, which is compatible with the results previously presented in the literature (Moon et al., 2004). 2.7 Comparison of the Analytical Approach with FEA and Application to an Experiment Comparison with FEA To verify the appropriateness of the equations that were developed, the stress values calculated by Eq. 2.10, 2.12, and 2.13 were compared with those obtained from the finite element analyses. Figure 2.7 shows the stress gradients in the concrete ring without the steel ring restraint as compared to the solution presented in Eq A generally good agreement was found resulting in minor differences (below 5%) between the finite element analysis and the analytical solution (Eq. 2.10) especially for low γ values (i.e., short drying times) along the outer circumference of the ring.

45 27 2 FEA γ =0.002 FEA γ=0.02 FEA γ=0.2 FEA γ =100 Eq. 10 Stress (MPa) Distance from the outer surface (m) Figure 2.7 Comparison of stress gradients due to self-restraint in the concrete ring without steel ring (FEA and Eq. 2.10) (R OC =0.225m, R IC =0.150m, E con =21GPa, ε SH-const = -100 µε) To explain some of the differences along the outer circumference of the ring, it should be remembered that the analytical solutions (Eq. 2.10, 2.12, and 2.13) were developed considering plane stress conditions with a uniform stress field in the z (depth) direction. Some of these differences occur due to the plane stress assumption which is not especially well suited for very short drying times. To verify this, additional finite element simulations were performed using both plane stress and plane strain conditions. Figures 2.8 and 2.9 show the stress gradients in the concrete ring obtained from the plane stress and strain analysis assuming a uniform stress field in the z direction. As expected,

46 28 the residual stress (σ Actual-Max ) that was obtained directly from finite element analysis of the actual geometry is located between the plane strain and plane stress approximations (Figure 2.10). Since Eq was developed from the plane stress condition, the values from Eq matched well with those from the plane stress condition finite element analyses (Figure 2.10). The results converge toward the plane stress solution as γ increases (Figure 2.10). It is believed that for a first approximation, this approach is reasonably accurate since it should be remembered that there is a high degree of uncertainty associated with the actual ring specimen tests in which micro-cracking occurs along the outer circumference.

47 A FEA γ =0.002 Plane Strain Plane Stress Eq. 10 Stress (MPa) Distance from the outer surface (m) Figure 2.8 Comparison of stress gradients due to self-restraint in the concrete ring without steel ring (FEA and Eq. 2.10) (γ=0.002) (R OC =0.225m, R IC =0.150m, E con =21GPa, ε SH-const = -100 µε)

48 30 2 B FEA γ =0.02 Plane Strain Plane Stress Eq.10 Stress (MPa) Distance from the outer surface (m) Figure 2.9 Comparison of stress gradients due to self-restraint in the concrete ring without steel ring (FEA and Eq. 2.10) (γ=0.02) (R OC =0.225m, R IC =0.150m, E con =21GPa, ε SH-const = -100 µε)

49 FEA (Exact Geometry) FEA (Plane Strain) FEA (Plane Stress) Eq. 10 Stress (MPa) γ Figure 2.10 Comparison of σ Actual-Max due to self-restraint in the concrete ring without steel ring (R OC =0.225m, R IC =0.150m, R IS =0.1406m, E con =21GPa, ε SH-const = -100 µε) Figure 2.11 shows the stress gradients that developed as a result of the steel ring restraint. The strain values (ε steel (t)) obtained from the finite element analyses were used for the calculations of stress using Eq. 2.12, and it can be seen that the stress gradients calculated by Eq corresponded well with those of the finite element analyses. Since each component of stress development (self-restraint and steel ring restraint) from the analytical solution was separately evaluated, the stress gradients calculated using Eq (the sum of Eq. 2.10, self restraint and Eq. 2.12, steel ring restraint) were compared with the finite element analyses and a reasonably good agreement was also observed (Figure 2.12). As a result, it appears appropriate to utilize

50 32 the analytical approach described here, if the linear diffusion, linear creep, and no microcracking assumptions are used γ FEA =0.002 FEA γ=0.02 FEA γ=0.2 Eq. 12 Stress (MPa) Distance from the outer surface (m) Figure 2.11 Comparison of stress gradients due to steel ring restraint in the concrete ring (FEA and Eq. 2.12) (R OC =0.225m, R IC =R OS =0.150m, R IS =0.1406m, E con =21GPa, E steel =200GPa, ε SH-const = -100 µε)

51 33 2 FEA γ=0.002 FEA γ=0.02 FEA γ=0.2 FEA γ=100 Eq. 13 Stress (MPa) Distance from the outer surface (m) Figure 2.12 Comparison of stress gradients due to circumferential drying in a concrete ring with steel ring (R OC =0.225m, R IC =R OS =0.150m, R IS =0.1406m, E con =21GPa, E steel =200GPa, ε SH-const = -100 µε) Application to Experiments To assess whether the values predicted by this approach are logical and of the right order of magnitude, Eq was used to describe a series of experiments that were performed on a mortar with a water to cement ratio of 0.50 and an aggregate volume of 50% (Hossain and Weiss, 2005). The ring specimen had an outer radius of the concrete of 225 mm, an inner radius of the concrete of 150 mm, and an inner radius of the steel ring which was mm (ν s =0.3, E s =200 GPa). The age-dependent diffusion

52 34 coefficient was determined for this mixture as described in Eq using electrical conductivity measurements (Moon et al., 2004; Rajabipour et al., 2004). d (t) = d + 1 t ( m 2 /sec) Eq D 2 where, d 1 had a value of approximately 9,0 d 2 had a value of , and t is the drying time, which is given in days. The age-dependent splitting tensile strength (f sp ) was measured and found to be described by Eq (Hossain and Weiss 2003a). f sp C4(t t0) (t) = f Eq C (t t ) 4 0 where, C 4 was the rate constant with a value of 2.45 day -1, t o was the setting time of 0.25 days, and f was the long-term strength, which had a value of 4.7 MPa. Before the maximum residual stress σ Actual-Max was calculated using Eq. 2.13, two additional material properties needed to be estimated. First, the free shrinkage strain constant was estimated from unrestrained free shrinkage tests. It has been previously shown (Hossain and Weiss 2003a) that the free shrinkage displacement ( U SH ) can be considered as the combination of the autogenous shrinkage displacement in a sealed specimen ( U AUTO ) and the drying shrinkage displacement ( U DRY ) (Eq. 2.16). U = U + U Eq SH AUTO DRY Based on the experimental results, Eq was found to describe the autogenous deformation ( U AUTO ) in an unrestrained sealed specimen as a function of time U Eq C 2 AUTO (t) = R IC εauto (t) = R IC C1 (t t0) where the coefficients C 1 =-52.83, C 2 =0.59, and t 0 =0.25 (mortar, 0.5 w/c, 50% sand) correspond well to measurements in a sealed concrete (Weiss and Ferguson, 2001b). The

53 35 drying shrinkage deformation ( U DRY ) can be obtained using Eq which was derived by integrating Eq. 2.8 (Hossain and Weiss, 2005). 2 ε R DRY const R IC U DRY(t) = erfc(a) r dr 2 2 Eq R R R IC OC OC IC It should be noted that the free shrinkage strain constant ( ε SH const renamed in Eq as the effective drying shrinkage strain constant ( ) in Eq. 2.8 is ε DRY const denote that it only considers the drying shrinkage component since the autogenous ) to shrinkage component is already considered using Eq The effective drying shrinkage constant of -1,200 µε was estimated for this specific concrete, and the deformations predicted by Eq and 2.18 were well correlated to physical measurements. Second, the effective elastic modulus was estimated using displacement compatibility as illustrated in Figure 2.4 (Attiogbe et al. 2004) E con (t) p (t) R 2 2 [(1 + ν )R + (1 ν ) R ] i IC = Eq C OC C IC Uconc(t) (ROC R IC) where, p i (t) and U conc (t) can be calculated by Eq and Eq With all the material properties now known, Eq was used to compute the stress gradients along the radius of the specimen as shown in Figure It can be seen in Figure 2.13 that almost immediately upon drying the stress that develops on the outer circumference of the concrete ring is higher than the tensile strength. It can be assumed that in the region where the maximum stress exceeds the tensile strength microcracking would occur (Bazant and Chern, 1985; Granger et al., 1997; Grasley et al., 2003; Weiss,

54 ). It can also be seen that the depth of the concrete in which the maximum stress exceeds the tensile strength increases with time Day of Drying 2 days 7 days 12 days Strength Stress (MPa) days 7 days 12 days Distance from the outer surface (m) Figure 2.13 Stress gradients due to circumferential drying in the concrete ring with steel ring (calculated using Eq and real test data) (0.50 W/C, 50% Sand, R OC =0.225m, R IC =R OS =0.150m, R IS =0.1406m) Figure 2.14 shows the progression of stable cracking (i.e., the depth where the residual stress exceeds the tensile strength) in a concrete ring with time, thereby suggesting that the crack has propagated at least 8 mm from the surface and this depth of the stable crack is likely greater since these stresses can be expected to be redistributed across the cross-section due to cracking. It should be noted that the laboratory specimen developed a visible crack during the 12 th day of drying (Hossain and Weiss, 2005), at

55 37 which time the magnitude of the average far-field stress is on the order of 2.5 to 3.0 MPa. If it is assumed that these specimens have properties similar to other mortars (K IC ~ 0.5 to 1.0 MPa mm ½ ) (Shah and Jenq, 1989) this far field stress and crack depth could be consistent with the level of stress that is required to cause through-cracking. Depth from the circumferce where σ 1 (t) reaches f sp (t) (m) Age of Drying (days) Visible Crack Figure 2.14 Predicted depth of microcracking in the concrete ring (0.50 W/C, 50% sand, R OC =0.225m, R IC =R OS =0.150m, R IS =0.1406m) While this approach appears reasonable, further work is necessary to validate and calibrate this procedure. Specifically, additional work is being performed to enable a description of a wider range of moisture gradients (Bazant, 1986; Rajabipour et al., 2004; Grasley and Lange, 2004). Acoustic emission testing is being refined to assess the extent

56 38 of the stable crack that can be expected to develop from the outer surface of the concrete ring (Neithalath et al., 2005) and to document how these cracks develop over time. In addition, further work is being performed to describe how these microcracks coalesce to form a visible crack (Bisschop and Van Mier, 2001; Hossain, 2003d; Hossain and Weiss, 2005). 2.8 Conclusions This chapter has described the development of an analytical approach for calculating stress development in a restrained concrete ring test under circumferential drying. An equation was developed that considered the stress that develops due to restraint by separating the problem into two components: 1) self-restraint due to moisture gradients (Eq. 2.10) and 2) external restraint due to the steel ring (Eq. 2.12). The analytical residual stress equation (Eq. 2.13) was compared with the results from finite element analysis and a reasonable agreement was shown. It was also verified that the restrained ring that dries from the circumference has a high stress at the outer radius for short drying time (i.e., low γ) and a higher potential to crack from the outer circumference. The developed solutions were applied to properties that correspond to an actual test to show a potential application for the basic approach, which resulted in reasonable stress fields. However, further work is needed to demonstrate the application to a wider range of concretes and to improve how relaxation and microcracking are considered.

57 39 CHAPTER 3: DEGREE OF RESTRAINT 3.1 Introduction Cementitious materials change volume as a result of autogenous, drying, or thermal shrinkage. When these volume changes are prevented by the structure surrounding concrete, residual tensile stresses can develop inside the material. If these residual stresses exceed the tensile strength of concrete, cracking may occur. Over the years, engineers have sought to develop simple tests to assess how susceptible a given concrete mixture may be to shrinkage cracking. While tests like ASTM C (standard test method for the length change of hardened hydraulic-cement mortar and concrete) are frequently used to measure the free shrinkage of a concrete mixture, free shrinkage by itself is not sufficient to predict whether cracking will occur. Rather, the potential for cracking is dependent on the interaction of several factors, including the magnitude of free shrinkage, rate of shrinkage, elastic modulus, degree of restraint, creep/stress relaxation, and fracture toughness (Weiss et al., 2000). Several researchers have developed experimental procedures to study how material properties influence the potential for shrinkage cracking (Swamy and Stavrides, 1979; Carlson and Reading, 1988; Weigrink et al., 1996). Several of these test procedures have been suggested to evaluate residual stresses that develop in concrete when shrinkage is restrained. Examples of such tests include the restrained ring test (Grzybowski and Shah, 1990; Hossain and Weiss, 2004), the passive linear restraint tests

58 40 (Springenschmidt et al., 1985; Weiss et al., 1998), and the active linear restraint tests (Kovler, 1994; Toma et al., 1999; Altoubat and Lange, 2002;). While a review of these test methods can be found elsewhere (Weiss, 1999), this paper focuses on assessing the degree of restraint and the measurable strain in the restrained ring test. Due to its simplicity and low cost, the restrained ring test has been used by numerous researchers to assess the potential for shrinkage cracking in concrete mixtures (Carlson and Reading, 1988; Grzybowski and Shah, 1990; Kovler et al., 1993; Weiss et al., 1999; Attiogbe et al., 2004). The ring test consists of a concrete annulus that is cast around a steel ring. As the concrete ring dries, it attempts to shrink. The steel ring prevents this shrinkage causing tensile stress to develop in the concrete. If these stresses are large enough, cracking may occur. Various geometries of the ring test have been used by researchers. AASHTO PP34-99 (AASHTO standard practice for cracking tendency using a ring specimen) was developed as a provisional standard test method for assessing the cracking potential of a concrete mixture when it is restrained. AASHTO PP34-99 uses a 12.5 mm thick steel ring, a 75 mm thick concrete ring, and allows the concrete to dry from the outer circumference. While this method has been used in several studies, it has been reported that the low degree of restraint provided by the steel ring results in a fairly long time before the first visible cracking is observed (Attiogbe et al., 2003). As a result, an alternative test geometry was developed for inclusion as an ASTM standard (ASTM C : standard test method for determining age at cracking and induced tensile stress characteristics of mortar and concrete under restrained shrinkage). ASTM C uses a 12.5 mm thick steel ring, however, the concrete wall thickness was reduced to 37.5 mm to encourage cracking to develop at an earlier age. This

59 41 restriction in specimen size, however, makes it difficult to test concrete with a larger aggregate or fiber reinforcement. While the development of these standards is a positive step forward, it should be noted that the restrained ring test is not intended to measure a fundamental material property; rather, the ring test measures the material s response to a specific stimulus (e.g., drying at a constant temperature and relative humidity) under specific boundary conditions (e.g., specimen size, drying direction, and degree of restraint). No one specimen geometry or external stimulus can be used to simulate all the possible conditions that may be encountered in the field. It is therefore important that procedures are developed to enable the ring test to be interpreted for a variety of applications. Specifically, this may include applications where the degree of restraint provided by the structure may vary over a wide range (15-90%) (NCHRP Project 12.37, 1995) or when the surface to volume ratio (this is analogous to varying thickness) of the concrete may vary from structure to structure. For example, the developer of a repair material may want to test their material under a high degree of restraint with a rapid rate of shrinkage to simulate a thin fully-bonded repair. Alternatively, someone developing a bridge deck mixture may be more interested in testing a thicker section (i.e., slower overall shrinkage rate) with a lower degree of restraint. This thesis will specifically look at how various geometric factors influence the results of the ring test. An analytical solution will be presented that describes the degree of restraint as a function of test geometry and material properties. Finite element simulations will be performed to quantify how material properties, geometry, and drying direction influence the degree of restraint. It is intended that the results of these

60 42 simulations will improve how the restrained ring test is used since it helps tailoring the ring geometry for specific applications. 3.2 Ring Specimen Geometry Figure 3.1 shows the typical geometry of the restrained ring test. The focus here primarily will be on assessing two aspects of the restrained ring test: the influence of the drying direction and the influence of the steel ring thickness. Figure 3.1 Drying direction and the resulting stress development of restrained ring

61 43 First, the influence of the drying direction will be considered. The simplest case which encountered consists of uniform shrinkage throughout the concrete in both the radial and height directions. This may be observed in the cases of autogenous shrinkage and in many of the basic analytical modeling approximations. The second drying condition considered is drying from the top and bottom of the concrete ring. This geometry may be preferred compared to circumferential drying since it allows the concrete ring to be sufficiently thick to enable large aggregate and fiber reinforced concrete to be tested (Hossain and Weiss, 2006). Top and bottom drying results in uniform shrinkage along the radial direction but not along the height direction. Since both the uniform shrinkage and top and bottom drying conditions have uniform shrinkage along the radial direction, the residual stresses that develop in these cases are the highest at the inner radius of the concrete ring and decrease as a function of 1/r 2 (where r is radial distance) through the concrete wall thickness (Hossain and Weiss, 2004). The third drying condition considered in this paper is the case of circumferential drying as suggested by the standard testing procedures ASTM C and AASHHTO PP In specimens that dry from the outer circumference, shrinkage is uniform throughout the height of the specimen, but not through the radial direction. Since the specimen loses the majority of water at the circumference (i.e., drying face), the stresses are highest at the drying face (Weiss and Shah, 2001; Moon et al., 2004; Moon and Weiss, 2006), which results in a complicated stress distribution that changes shape over time. In addition to considering the drying direction, this thesis investigates the influence of steel ring thickness. A thicker steel ring will provide a higher degree of restraint, resulting in higher stress development in the concrete. On the other hand, the

62 44 increase of the steel ring thickness decreases the magnitude of strain that can be measured at the inner surface of the steel ring, which is needed for stress calculations inside concrete (Hossain and Weiss, 2004). Therefore, it is necessary to quantify how the thickness of the steel ring changes the degree of restraint, as well as the stress measurements. 3.3 Quantifying the Degree of Restraint under Uniform Shrinkage (Analytical Approach) Figure 3.2 can be used to provide a basic illustration of the concept of the degree of restraint (Weiss, 1999; See et al., 2003). In this conceptual example, a concrete prism is thought to be axially restrained by a steel prism (Figure 3.2.(a)). If the concrete shrinks and there is no connection between the concrete and the steel (free boundary condition), a difference in the length of the specimens will exist as illustrated in Figure 3.2.(b). This difference is what one would expect to measure in a free shrinkage test like ASTM C If the concrete is restrained by the steel prism (assuming bending is prevented), the concrete and steel will deform together (Figure 2.(c)).

63 45 Figure 3.2 A conceptual illustration of the degree of restraint This will result in some of the shrinkage of the concrete being prevented, resulting in a smaller length change than for a specimen under the free shrinkage condition. The degree of restraint ( ψ ) can be defined as; L L free restrained restrained ψ = Eq. 3.1 (a) L free L = 1 L free where L free is the displacement of concrete due to free shrinkage, and L restrained is the displacement of concrete under restraint as shown in Figure 3.2. The degree of restraint can be estimated for linear specimens using Eq. 3.1 (b) (ACI 207.2R, 1995): C C ψ = Eq. 3.1 (b) E C E A C A + E S A S

64 46 where A C and A S are the cross-sectional areas, and E C and E S are the elastic moduli of the concrete and steel elements, respectively. In the case of a restrained ring test, the degree of restraint can be written as Eq. 3.1 (c): U (t) U (t) SH R S R IC OS ψ = Eq. 3.1 (c) U (t) SH R IC where U SH RIC (t) is the free shrinkage displacement of the concrete ring (i.e., if no steel ring was present) measured at the inner surface of the concrete ring (R IC ) and U S Ros (t) is the actual displacement of the concrete ring under the restraint provided by the steel ring. The degree of restraint in Eq. 3.1 (c) is defined at the concrete-steel interface (R IC =R OS ; Figure 3.1) where the restraint is maximum. The degree of restraint can vary between zero (corresponding to no restraint; U SH RIC = U S Ros ) and one (corresponding to perfect restraint; U S Ros = 0). For the case of uniform shrinkage (i.e., no moisture gradient), the free shrinkage displacement, U SH RIC (t) can be calculated using Eq. 3.2.a based on the linear free shrinkage of the concrete, ε SH (t) (Weiss et al., 1999). U SH R IC (t) = ε (t) R Eq. 3.2 (a) SH IC The actual displacement of the restrained concrete ring at R OS, U S Ros (t), can be calculated using the strain that is measured on the inner surface of the steel ring (ε st (t)) as shown in Eq. 3.2.b. U [ R + R ν (R R )] 1 (t) = εst (t) S IS Eq (b) 2R OS IS OS S R OS OS

65 47 where ν S is the Poisson s ratio of steel. By substituting 3.2.a and 3.2.b into 3.1 (c), the degree of restraint can be written as follows: 2 1 ε st (t) R IS ψ = 1 (1 + ν ) + (1 ν ) 2 S S 2 εsh (t) R OS Eq. 3.3 Meanwhile, the ratio ε st (t)/ε SH (t) can be related to the specimen geometry and the material properties of the concrete and steel by considering the interfacial pressure that develops at the concrete-steel interface. The magnitude of the interfacial pressure can be calculated using either Eq. 3.4 (a), which is based on the measured steel strain from an experiment or Eq. 3.4 (b), which is based on fundamental computation using a shrinkfit approximation that incorporates the free shrinkage strain of the concrete and elastic properties of the concrete and steel (Hossain et al. 2003; Hossain and Weiss 2004): p R OS R IS = εst (t) E S Eq. 3.4 (a) 2R 2 OS p 0 ' εsh (t) E C = Eq. 3.4 (b) ' E C [(1 + νs )R IS + (1 νs )R OS] [(1 ν C )R OS + (1 + ν C )R OC ] E (R R ) (R R ) S OS IS OC OS In these equations, E S is the elastic modulus of steel, E C is the effective elastic modulus of concrete (considering the creep effect), and ν C is the Poisson s ratio of concrete. Substitution of Eq. 3.4 (a) into 3.4 (b) enables the ratio ε st (t)/ε SH (t) to be determined as:

66 48 ε ε st SH (t) (t) = E E ' C S 2 R 1 R IS OS 2 E E ' C S (1 + ν S R ) R IS OS R 1 R 2 IS OS + (1 ν 2 S 1 ) (1 + ν C R ) R OC OS R 1 R 2 OC OS + (1 ν 2 C ) Eq. 3.5 Eq. 3.5 can now be substituted into Eq. 3.3, which results in an expression for the degree of restraint as only a function of the geometry and material properties (namely the stiffness and the Poisson s ratio) of the concrete and steel. ψ = 1 ' E C 1 Eq. 3.6 E 2 2 S R R IS OC 1 (1 + ν + ν C ) (1 C ) ' E R R OS C OS 2 2 E S R OC R IS 1 (1 + ν ) + ν S (1 S ) R OS R OS This enables one to determine the degree of restraint prior to conducting the experiment and provides the opportunity to adjust the test geometry to satisfy the desired degree of restraint that best approximates the actual field conditions. In addition to determining the degree of restraint, an important parameter that needs to be considered is the magnitude of the strain that will be measured from the inner surface of the steel ring (ε st (t)). From a practical point of view, resolving the magnitude of the residual stress in a typical concrete with an accuracy of better than 10 psi requires a minimum steel strain value of 50 µε before the specimen fails, which can be ensured by determining the minimum value of ε st /ε SH at failure for a given ultimate free shrinkage of concrete (ε SH ). For example, for concrete with an ultimate free shrinkage of ε SH = 400 µε, a minimum value of at least ε st /ε SH = would be desired to obtain a strain in the

67 49 steel of 50 µε (ε st = 50 µε). Eq. 3.5 can now be used to check whether the test geometry satisfies this requirement. It is now possible to evaluate the effect of specimen geometry on the degree of restraint (Ψ) and ε st /ε SH. Figure 3.3 (a) illustrates the influence of steel ring thicknesses by varying R IS /R OS between 0 (a solid cylinder) and 1 (an infinitely thin ring). As expected, the degree of restraint decreases with decreasing the thickness of the steel ring. In addition, a stiffer concrete (higher effective elastic modulus) results in a lower degree of restraint. It is shown that in all cases when R IS /R OS decreases below 0.6, only a slight increase (less than 10%) in the degree of restraint occurs by further increasing the steel ring thickness. Similarly, Figure 3.3 (b) shows how an increase in the concrete ring thickness (R OC /R IC ) reduces the degree of restraint.

68 50 (a) (b) Figure 3.3 The influence of specimen geometry on the degree of restraint and the ratio εst/ε SH : (a) influence of the concrete stiffness and (b) influence of the concrete ring thickness (E C = 21 GPa, E S = 200 GPa, uniform shrinkage)

69 51 Eq. 3.5 and 3.6 provide an analytical approach to determine the degree of restraint and to ensure that an accurate strain measurement can be obtained from the steel ring so that the stress computation can be performed. To illustrate the applicability of this technique for predicting the degree of restraint prior to testing, the degree of restraint determined using Eq. 3.6 was compared with experimentally-measured values (using Eq. 3.3), as illustrated in Figure 3.4. Figure 3.4 Comparison between the analytical solution and the experimentally determined degree of restraint before the through-cracking (top and bottom drying) The experimental data used here was obtained from a series of mortar ring tests with a w/c of 0.30 and 50% fine aggregate by volume (Hossain and Weiss 2004). Three

70 52 different restrained rings were used with varying steel thicknesses (3.2, 9.4, and 19 mm). The geometry parameters were as follows: R OC = 225 mm, R IC = R OS = 150 mm. The mortar rings were exposed to drying from the top and bottom in a 50 % relative humidity environment. The effective elastic modulus of concrete was estimated to be 60% of the actual measured elastic modulus to account for creep effects (Bazant and Wittmann 1982). The free shrinkage strain (ε SH (t)) and the steel ring strain (ε st (t)) were measured experimentally and used in Eq. 3.3 to calculate the degree of restraint and these results were compared with the values obtained from the analytical solution (Eq. 3.6). It must be noted that the analytical approach discussed earlier is initially developed for a uniform shrinkage condition. However, the results of a series of finite element simulations (as described in the next section) suggest that the analytical approach can be used regardless of the drying condition of the concrete ring. Figure 3.4 shows that there is reasonable agreement between the analytical approach and the experimentally measured degree of restraint. It is also evident that the degree of restraint decreases over time as the stiffness of concrete increases. 3.4 Quantifying the Degree of Restraint in the Presence of Moisture Gradient To consider how the degree of restraint is influenced by the presence of a moisture gradient inside concrete (caused by preferential drying from exposed surfaces), a series of finite element simulations were performed using the commercial analysis code ANSYS. The ring specimen was considered to be axi-symmetric and was modeled using quadrilateral eight-node elements. Both rings (concrete and steel rings) were considered

71 53 to have a height of 75 mm. A fixed value of R IC =R OS = 150 mm was used in the simulations. Also, a base value of R OC = 225 mm and R IS = mm were used, however for geometrical analysis, the values of R OC and R IS were varied to provide an R IS /R OS ranging from 0.5 to 1 and an R OC /R IC ranging from 1.25 to 2. Other parameters used in the simulations were: E C = 21 GPa, E S = 200 GPa, ν c = 0.18, and ν s = No creep non-linearity or microcracking effects were considered in the finite element analysis. A perfectly unbonded condition between the steel and concrete rings was assumed (i.e., no transfer of tensile and shear stresses at the interface). This interface condition can be approached experimentally by using a form release agent or a thin plastic sheet between the concrete and steel (Hossain et al. 2003). Three different cases of drying were considered: a) uniform drying, b) top and bottom drying, and c) circumferential drying. In cases (b) and (c), a moisture gradient develops inside the concrete as it dries, preferentially from the exposed surfaces. To incorporate moisture gradients in the model, a linear moisture diffusion function was used to describe the relative humidity at each point inside the concrete as a function of the drying time (t), as described by Eq. 3.7 (Moon et al. 2004). RH(x, t) = RH I (RH I RH S ) erfc 2 x Dt Eq. 3.7 where, RH (x, t) is the relative humidity at a depth x from the drying surface, erfc is the complementary error function, RH S is the relative humidity at the surface of the specimen (considered to be 50% in the simulations), RH I is the internal humidity if the specimen was completely sealed ( RH was assumed here to be 100% and, as such, the I effect of autogenous shrinkage was neglected), and D is the aging moisture diffusion

72 54 coefficient of concrete (m 2 /sec). It should be noted that although non-linear diffusion coefficients are commonly suggested for use in concrete (Bazant and Najjar 1971), and while these equations may be more scientifically correct, the application of a linear coefficient here has enormous computational benefits with no loss in the generalities of the conclusions of this thesis. By considering D to be only dependent on specimen age, a single parameter γ can be introduced as γ = 2 Dt to represent the time and diffusion coefficient variations in Eq It has been shown that this approach can be used to reasonably describe the moisture gradient that develops inside mortar due to drying (Moon et al. 2004). Figure 3.5 shows an example of relative humidity profiles that correspond to different values of γ. A low γ value (i.e., γ < 0.01) refers to a very steep moisture gradient, while the severity of the gradient decreases as γ increases. As γ approaches infinity (practically this corresponds to a γ = 100), the entire cross-section of the concrete has come to equilibrium with the ambient humidity.

73 55 Figure 3.5 The simulated humidity profiles corresponding to different γ values The free shrinkage strain (ε SH ) at each point inside the concrete can be obtained from the relative humidity using an approach similar to that suggested by Bazant and Wittmann (1982). The relationship between shrinkage and relative humidity can be assumed to be linear for high relative humidities (i.e., RH > 50%) (Weiss and Shah 2001). In this case, the free shrinkage strain can be estimated as: ε SH (t) = ε SH (100% RH(x, t)) Eq. 3.8 (100% RH ) S where ε SH- is the ultimate free shrinkage of the concrete at a specific ambient relative humidity, RH S. For the simulations in this paper, a value of ε SH- = -100 µε was assumed

74 56 at a RH S = 50%, however, the simulation results can be simply scaled to consider other shrinkage values. The degree of restraint was determined using Eq. 3.1 (c) with values of U SH RIC and U S ROS that were determined at the mid-height of the ring from the finite element simulations. It was observed that in the case of uniform shrinkage the simulation results match very well with the analytical procedure (Eq. 3.6). For the cases in which a moisture gradient develops inside the concrete (i.e., non-uniform drying), the degree of restraint was determined for different values of the drying time parameter, γ. Figure 3.6 shows the results for the case of top and bottom drying. It can be seen that a small variation in the degree of restraint occurs as drying progresses. These variations are less than 7 % over the entire duration of drying. Similar results were obtained for concrete experiencing circumferential drying. These results indicate that the degree of restraint is (to a reasonable extent) independent of the moisture profile inside the concrete and as such, the analytical procedure developed for a uniform shrinkage condition (Eq. 3.6) can be employed for other cases of drying to estimate the degree of restraint.

75 57 Figure 3.6 Degree of restraint as drying progresses; obtained from finite element simulations (top and bottom drying) The small variation in the degree of restraint in the cases of non-uniform drying is primarily caused by a non-uniform deformation of the steel ring throughout the height. Figure 3.7 provides a conceptual illustration of the deformations of the steel ring and the pressure gradient that develops at the concrete-steel interface. When the concrete shrinks uniformly, the interfacial pressure is (almost) uniform across the height (Figure 3.7 (a)). However, in the cases of non-uniform drying, a pressure gradient is observed along the height due to the non-uniform shrinkage inside the concrete (Figure 3.7 (b) and 3.7 (c)).

76 58 Figure 3.7 Influence of shrinkage condition on the deformation of concrete and steel rings: (a) uniform shrinkage, (b) top and bottom drying, and (c) circumferential drying To further investigate this behavior, the interfacial pressure gradient was obtained using finite element simulations for a concrete ring experiencing top and bottom drying. The geometrical parameters considered in this case were as follow: R OS = R IC = 150 mm, R OC = 225 mm, steel ring thickness = 9.38 mm, and height = 75 mm. Figure 3.8 (a) shows the simulation results in terms of pressure profiles as drying progresses. A higher pressure is always observed in the middle region of the steel ring. Initially, the pressure at the top and bottom regions is zero, which suggests that the top and bottom concrete is not in contact with steel during this period. However, the pressure increases as drying progresses (i.e., increasing γ) and approaches a more uniform distribution at higher values of γ. The pressure gradient causes a non-uniform strain profile at the inner surface

77 59 of the steel ring (Figure 3.8 (b)). Due to stress redistribution within the thickness of the steel, the strain profile does not exhibit a significant gradient across the height and as such, has a minor effect on the degree of restraint. Further investigation is required to study the effect of bending for thin and tall steel rings for which bending is expected to be more significant.

78 60 (a) (b) Figure 3.8 The effect of non-uniform deformation of the steel ring (simulations results): (a) pressure gradients at R OS and (b) steel ring strain at R IS

79 61 Finally, Figure 3.9 provides a comparison of the degree of restraint calculated from the analytical procedure (Eq. 3.6, uniform drying) and the results obtained from finite element simulations for the cases of non-uniform drying. The results provided here correspond to a ring with R IC =R OS = 150 mm, R OC = 225 mm, height = 75 mm, E C = 21 GPa, and E s = 200 GPa. The simulations were performed with different values of γ ranging from to 100 to determine the degree of restraint as the drying progresses. A good agreement is exhibited between the results obtained from the analytical procedure and the finite element simulations, suggesting that the analytical procedure is capable of estimating the degree of restraint regardless of the direction and uniformity of drying.

80 62 Figure 3.9 Comparison on the degree of restraint obtained from the analytical procedure (Eq. 6, uniform drying) and the values obtained from finite element simulations (nonuniform drying) 3.5 A Practical Example of Geometry Selection for the Restrained Ring Test To illustrate the usefulness of the approach described here, an example is provided in which the geometry of a restrained ring is determined to correspond with an actual field application and to enable reasonable stress measurement. In this example, it is assumed that the field concrete is a floor slab that is six inches (152.4 mm) thick and the slab experiences drying from the top and bottom surfaces. The concrete has an ultimate free shrinkage of approximately 400 µε at 50% relative humidity (i.e., the shrinkage was measured using ASTM C ). The actual elastic modulus is assumed

81 63 to be 25 GPa, however, the effective elastic modulus can be reduced to 15 GPa (60%) to account for the creep effect. Further, the restraint that is observed in a typical floor slab can be calculated to be approximately on the order of 60 %. A reasonable restrained ring test geometry could consist of a concrete ring that is six inches tall and is allowed to dry from the top and bottom surfaces to ensure similar moisture gradients and shrinkage rates as those in the floor. Also, a concrete ring with circumferential drying could be used, in which case the concrete ring thickness should be three inches (i.e., half of the depth as drying occurs from only one side). In this example, the ring with top and bottom drying is considered (however, the same procedure can be applied to determine the desired geometry of the ring with circumferential drying). From the analytical procedure described earlier (Eq. 3.5 and 3.6), a reasonable target ring geometry can be determined to approximate the condition in the floor. Figure 3.10 is a graphical representation of Eq. 3.5 and 3.6 and can be used to illustrate different combinations of geometry parameters that satisfy a desired degree of restraint while meeting a minimum value of the ratio ε st /ε SH to ensure proper stress calculations. To obtain a ring geometry, first the thickness of the concrete ring should be determined by considering the maximum size of the aggregates. In this case, if a 25 mm aggregate is used, the concrete ring thickness can be chosen to be 125 mm (five times the maximum aggregate size). By assuming R IC = R OS = mm (six inches), the value of R OC will be mm. From a practical point of view, a standard mm (11 inches) radius cardboard tube form can be used to provide the outer mold for the concrete ring. This gives an R OC = mm, which yields an R OC /R IC = From Figure 3.10 (a), to achieve a 60 % degree of restraint, R IS /R OS = is obtained, which gives a steel ring

82 64 thickness of 8.08 mm. A commercially available steel ring with 9.5 mm (3/8 inch) thickness can be chosen (outer radius is six inches). This corresponds to an R IS /R OS = and a degree of restraint of ψ = 63.8 %, which is quite similar to 60 % and is therefore acceptable for practical purposes (as shown as a solid point in Figure 10 (a)). The next step is to check the value of ε st /ε SH to ensure proper stress calculations. Using the ultimate free shrinkage of concrete (ε SH = 400 µε) a minimum acceptable value of ε st /ε SH = 50/400 = is desired. From Figure 3.10 (b), the solid point corresponding to R OC /R IC = 1.83, and R IS /R OS = lies above the ε st /ε SH = curve and, as such, the strain ratio requirement is also satisfied.

83 65 (a) (b) Figure 3.10 Graphical representation of Eq. 3.5 and 3.6 in terms of possible combinations of geometry parameters for a desired degree of restraint (a) and an acceptable strain ratio (b) (assumed material properties: E C = 15 GPa, E S = 200 GPa, ν C = 0.18, and ν S = 0.3)

84 66 Finally, the actual maximum residual stress that develops inside the concrete ring can be calculated using the method suggested by Hossain and Weiss (2004); Eq R IC + R OC R IC R IS σ = ε Actual max st ( t)e Eq. 3.9 S R OC R IC 2R IC It should be noted that this equation is applicable only when shrinkage is uniform in the radial direction (i.e., uniform drying or top and bottom drying), and an alternative formula (as described elsewhere; Moon and Weiss, 2006) needs to be used for the case of circumferential drying. To summarize, the final geometry of the restrained ring test can be selected as follows; concrete ring with mm (11 inches) outer radius (R OC ) and steel ring with mm (6 inches) outer radius (R OS ) and 9.5 mm (3/8 inch) thickness. Both rings are 6 inches high. A standard mm (11 inches) radius cardboard tube form is used as the outer mold for the concrete ring. 3.6 Summary and Conclusions The following conclusions can be drawn in this chapter: To utilize the restrained ring test method to simulate an actual field condition requires that an appropriate specimen geometry be determined prior to performing the experiment. The geometry of the ring test that should be selected for a particular application must satisfy a desired degree of restraint, match the actual drying conditions in the field, and ensure proper strain measurements to enable accurate stress calculations. An analytical procedure was presented to determine the degree of restraint in the restrained ring test for concrete under uniform shrinkage (Eq. 3.6). It was shown that the

85 67 degree of restraint depends only on the geometry of the rings and the material properties (namely the elastic modulus and the Poisson s ratio) of the concrete and the steel. The degree of restraint increases with increasing the thickness and elastic modulus of the restraining ring (steel) and decreases with increasing the thickness of the concrete ring. In addition, it was shown that to ensure proper stress calculations inside the concrete ring, a minimum acceptable strain ratio ε st /ε SH must be met. This imposes an additional requirement on the restrained ring geometry (Eq. 3.5). A series of finite element simulations were performed to examine the applicability of the analytical solution to the cases of non-uniform (i.e., top and bottom or circumferential) drying, in which a moisture gradient develops inside the concrete ring. It was observed that the degree of restraint shows minor changes as the drying progresses and is not significantly influenced by the presence of a moisture gradient inside the concrete ring. This suggests that the analytical solution that was initially developed for uniform shrinkage is also applicable to estimate the degree of restraint in the cases of non-uniform drying. It should, however, be noted that the drying direction can significantly influence the distribution of the residual stresses inside the concrete ring (Moon and Weiss, 2006) and must not be neglected during stress calculations.

86 68 CHAPTER 4: INTERNAL STRESS DEVELOPMENT IN HETEROGENEOUS SYSTEMS 4.1 Introduction This chapter provides an overview of a research program that is currently being conducted to develop procedures to enable information from sensors placed in a concrete structure or pavement to be utilized efficiently as real-time feedback for updating performance simulation models (Weiss, 2001a). While this project has many facets, one aspect of this work is to better understand how an embedded stress sensor may be used to quantify residual stresses in concrete. Toward this end, a series of preliminary computations have been performed to better understand the residual stresses that develop in a heterogeneous composite system when one phase (the paste phase) shrinks and the other phase (the aggregate phase) does not. Subsequent computations have been performed to investigate stresses that develop when both phases experience differential movement, though they will not be discussed here. The overall research program is being conducted to try to resolve a few key questions that need to be answered to enable residual stress sensors to be used more efficiently in concrete. Numerous researchers and engineers frequently simplify concrete by assuming it behaves like a homogeneous material by using effective material properties. While this is commonly done, the impact of simplifications made to implement effective material

87 69 properties remain unclear, especially as these simplifications relate to micro-cracking, which influences system compliance and durability. While residual stress sensors are being developed, research is needed to address how the external boundary conditions influence the stress fields that develop around an aggregate with respect to directionality and magnitude. Numerous studies are being conducted to assess how the mixture proportions of concrete can be optimized. The majority of these studies utilize parameters that can be easily measured like strength or free shrinkage to optimize mixture proportions. This study takes a more fundamental look at the influence of aggregate volume, shape, bond, and stiffness on the resulting stress development, micro-cracking, and through-cracking. It is the goal of this research that through further studies, improved guidelines may be available to suggest how mixtures can be designed more efficiently to improve field performance. This chapter is divided into four main components. The first portion of this chapter describes the overview and motivation for the research. The second section describes the typical computations that are used for determining the effective material properties of a composite by using the properties of each constituent. The following two sections of this chapter (sections three and four) discuss preliminary investigations aimed at understanding how external restraint influences the stresses around an aggregate and how micro-cracking develops in these systems. Section three discusses simulations and experiments that were performed in which a single aggregate is considered in a shrinking matrix. Section four considers a collection of hexagonal unit cells, each consisting of an

88 70 aggregate in a paste matrix. Finally, section five provides an overview of the preliminary observations that can be made from this work. It should be noted that the majority of the calculations in this paper assume the paste and aggregate to behave as non-aging, linear-elastic materials. This approximation enables the models to remain relatively simple. As the models are refined over time, age and time-dependent rheological material properties will be incorporated into these models. 4.2 Effective Material Property Calculations Concrete shrinks in response to drying, self-desiccation, and chemical reactions. The paste component is generally recognized as the component responsible for the shrinkage, while the aggregate component is frequently thought of as an inert-filler that reduces the overall shrinkage of the concrete system. Several researchers in the 1950 s discussed how the shrinkage of a concrete (ε Concrete ) could be described as a fraction of the shrinkage of the paste (Carlson, 1937; Pickett, 1956; L Hermite, 1960). Pickett developed an expression that is frequently used to describe the relationship between the shrinkage of the paste and the shrinkage of the concrete (Eq. 4.1). ε Eq. 4.1 n Concrete = ε Paste ( 1 VAgg ) where ε Paste is the shrinkage of the paste, V Agg is the volume fraction of the aggregate, and n is a constant for a particular system. To arrive at this formulation, Pickett considered the effect of spherical aggregate particles in a concrete composite. Pickett computed the restraining effect of aggregates by assuming that the aggregate and the

89 71 concrete act elastically. Pickett assumed that the exponent n would be related to the relative elastic properties of the aggregate and the concrete. He developed a theoretical expression that showed that the value of n could range from 0 to approximately 2. This theoretical expression is somewhat difficult to use appropriately, however, as it requires a measure of the elastic modulus of the concrete composite. Pickett experimentally determined that the exponent n would have a value of 1.7 for mortars made using Ottawa sand (ASTM C-190). L Hermite (1960) wrote an extensive summary of research on shrinkage and summarized the work of Dutron (Dutron, 1934) where Eq. 4.1 had been written in a slightly different form. In re-analyzing the results of Dutron using Eq. 4.1, L Hermite found that the exponent n ranged between 1.2 and 1.7 for normal strength mortars and concretes made using different aggregate. Recently, simulations were performed (using the planar geometry as described in Section 4.4) where the exponent, n was related to the elastic modulus of the aggregates (Figure 4.1). A hyperbolic equation (Eq. 4.2) was fit to the data shown in Figure 4.1 n 1 = n E Eq. 4.2 Paste 1 + C1 E Agg where n refers to the value of n for an infinitely stiff aggregate, which can be taken as 1.405, and C 1 is a constant that can be taken as E Paste and E Agg are the elastic moduli of paste and aggregate, respectively. While Eq. 4.2 may need to be modified to account for low paste moduli, different aggregate geometries, or differences in the Poisson s ratio between the two materials, it appears possible to use this expression to

90 72 estimate the free shrinkage of a concrete composite. It should be noted, however, that this simulation was performed for a planar geometry (assuming plane stress) and some differences may exist for a truly three-dimensional composite geometry. This suggests that the overall shrinkage of a concrete can be reasonably estimated using Eq Neville (1996) points out that the aggregate restrains the shrinkage and this restraint causes residual stress to develop in the paste. However, while many authors use Eq. 4.1 (Mindess et al., 1996; Mehta and Monteiro, 1986) to approximate the overall shrinkage, others (Swazye, 1960) have cautioned against relying on such approximations. An objection that has been raised to the use of Eq. 1 may be attributed in part to the role of residual stresses that result in non-linearities associated with creep and microcracking in the system. To measure the residual stresses that develop in a composite system, Dela and Stang (Dela and Stang, 2000) introduced an aggregate sensor to measure the pressure on a spherical embedded inclusion inside a cement paste. They concluded that the effects of stress relaxation were substantial at very early ages while the subsequent stress development could lead to cracking. It has also previously been suggested that the region around an aggregate may be susceptible to cracking as somehow attempted to measure the extent of microcracking in cement mortars using acoustic emission measurements (Pease et al., 2003b, 2004). Figure 4.2 shows the results of early-age measurements in terms of cumulative acoustic activity in externally-unrestrained sealed mortar samples with different water-tocement ratios (w/c) (Pease et al. 2003b). Substantial acoustic activity occurs in the specimens with the lower w/c at early ages. The specimens with a w/c of 0.30 showed

91 73 the greatest number of acoustic events, followed by the specimens with a w/c of 0.35, 0.40, and 0.50, respectively. It was hypothesized that the lower w/c mixtures undergo more autogenous shrinkage and, as a result, they are more likely to experience higher residual stresses generated by the internal restraint from the aggregates. This implies that concrete made using a lower w/c would be more prone to microcracking due to the internal restraint of aggregates against autogenous shrinkage. The current paper builds on these observations and describes a series of experiments (Pease et al., 2003b) in which the role of aggregate inclusions is investigated by using different model systems. To understand how micro-cracking occurs in low w/c pastes, Pease et al. (2003b) and Moon et al., (2004) used a low w/c cylindrical paste specimen. A steel rod was placed in the center of the paste specimen. This system was used to simulate the shrinkage of paste around an aggregate. Specimens were prepared with different pasteto-aggregate diameter ratios. An elastic shrink-fit theory (Timoshenko and Goodier, 1970) was used to try to interpret the results from the test. It was concluded that the maximum residual stress that develops in the paste could be computed using Eq. 4.3 (after correcting the elastic modulus of paste to account for creep):

92 Shrinkage Exponent, n Ratio of Aggregate and Paste Stiffnesses (E Agg /E Paste ) Figure 4.1 Shrinkage exponent n as a function of the aggregate to paste stiffness ratio Cumulative Events (hits) W/C 0.35 W/C 0.40 W/C 0.50 W/C Age of Specimen (hrs) Figure 4.2 Acoustic activity in mortars at early ages (50% aggregate volume)

93 75 R Agg 2 OP 2 OP + R 2 OA 2 OA R R σ = ε Paste E Paste ( t) Eq. 4.3 ( ) E Paste ( t) R + R 1 ν Agg + ν Paste + E R R 2 OP 2 OP 2 OA 2 OA where ν corresponds to Poisson s ratios of paste (denoted with a subscript Paste) and aggregate (denoted with a subscript Agg), and R corresponds to the outer radius of the aggregate (denoted with a subscript OA) and the paste cylinder (denoted with a subscript OP), respectively. Eq. 4.3 was used to estimate the maximum stress level that could develop around the aggregate and to compare the cracking behavior of different specimens. As the aggregate volume increased (i.e., lower R OP /R OA ) there was an increase in both the average residual stress level and the acoustic activity. As the paste radius decreased, the potential for through-cracking increased. Through-cracking was observed to correspond to a sudden rise in acoustic energy in these specimens. Previous research assumed that acoustic activity is synonymous with the development of microcracking. To measure this more directly, Lura et al. (2005) are investigating the cracking around an idealized aggregate using gallium impregnation. The main benefit of gallium impregnation is that it could be performed without inducing additional cracking into the system; thereby, cracks are imaged as they were in the unperturbed specimen, before damage from preparation could occur. As mentioned earlier, a composite system can be described by equivalent material properties, such as the equivalent elastic modulus. Hansen (1965) proposed an equation

94 76 for calculating the equivalent elastic modulus of a spherical composite, which contains a spherical inclusion at the center of a sphere of the paste matrix (Eq. 4.4). E composite (1 VAgg ) E Paste + (1 + VAgg ) E Agg = E Paste Eq. 4.4 (1 + VAgg ) E Paste + (1 VAgg ) E Agg To provide a simplified solution, Hansen assumed that the Poisson s ratio was the same for each phase (0.20). Such equivalent material properties can be used directly to estimate the residual stress that would develop when concrete is completely restrained from shrinking freely by multiplying Eq. 4.1 and Eq This will be discussed in greater detail later. 4.3 Single Aggregate Prism Systems As previously discussed, there are questions as to how the external boundary conditions influence the residual stress fields that develop around an aggregate when it is suspended in a paste matrix. This section simulated a geometry in which a single aggregate was placed inside a prism of paste. The paste was allowed to shrink and the stresses were computed for an unrestrained and a horizontally-restrained specimen. The single aggregate prism specimens used in this investigation are illustrated in Figure 4.3. The length (L) of the specimen is five times its width (H). The specimens contain one aggregate in the center, with a diameter (D=2R OA ) that was varied from 0.1 to 0.2, 0.4, 0.5, 0.6, 0.8, and 0.9 times of the height (H) of the prism. Two external

95 77 boundary conditions were considered in this analysis: a) unrestrained and b) horizontallyrestrained, where the ends of the specimen were completely fixed so as to not permit any movement in the x-direction, though movement in the y-direction was permitted. L L H D H Single Aggregate - Unrestrained Single Aggregate - Restrained (a) (b) Figure 4.3 An illustration of the two geometries simulated: a) an unrestrained single aggregate system and b) a restrained single aggregate system These specimens were simulated using finite element analysis (FEA) in the commercial program ANSYS. The model systems were meshed using quadratic rectangular eight-node elements and analyzed using plane-stress approximations. The effect of autogenous shrinkage was simulated using a temperature substitution analogy (Moon et al., 2004). A uniform temperature load was applied, while the thermal expansion coefficient of the cement paste was set to µε/µε/ºc and the thermal expansion coefficient of the aggregate was set to zero. The autogenous shrinkage strain was assumed to be -100 microstrain (µε which was applied using a temperature load ( T = -100 ºC). It should be noted that since linear elasticity was assumed, the stresses can be scaled proportionally to reflect an increase or decrease in autogenous shrinkage. The

96 78 paste was assumed to have a modulus of 20 GPa and a Poisson s ratio of 0.20, while the elastic properties of the aggregate were 200 GPa and 0.30, respectively. It should be noted that this solution does not consider the effect of creep or changing the elastic properties of the paste. A perfect-bond between the aggregate and the paste is assumed in the simulations unless it is specifically noted as otherwise. The preliminary results showed that the bond condition has a minor effect on the system with unrestrained boundary conditions, while it has a substantial effect with restrained boundary conditions. A further discussion with respect to the bonding effect will be presented later. Figure 4.4 shows stresses that developed along the x-axis and along the y-axis for two different external boundary conditions when the diameter of the aggregate (2R OA ) is 0.1H. In the case of the unrestrained specimens, the residual stress level decreases as the distance from the aggregate increases (Figure 4.4 (a)). This solution is similar to the stresses that develop in the ring test (Pease et al., 2003b; Weiss, 1999) although a slight correction is needed in the case of the larger aggregates to account for the top and bottom edges of the prism specimen. Very similar stresses develop in the x and y directions. For this particular geometry, the bond between the aggregate and the matrix has little influence on the developed stresses. Numerical simulations have shown that the difference between the bonded and unbonded case is less than 7%. In pure cement paste specimens (i.e., no aggregate), the specimen will not experience any internal stress development under the unrestrained boundary conditions. However, when aggregates are present, internal stresses that develop at the interface between the two phases could result in internal microcracking.

97 79 In the externally restrained specimen (fully restrained in one direction) a stress gradient develops along the x-axis (i.e., for element A in Figure 4.4) that is similar to the stress gradient in the unrestrained specimen. However, the externally restrained specimen develops a stress gradient along the y-axis (i.e., for element B in Figure 4.4) that is much greater than the stresses that develop in the unrestrained specimen. This is not unexpected; however, if a pure paste specimen were restrained, the residual stress in elements along the y-axis would be constant and equal to 2 MPa. As a result, the maximum stresses that develop in Figure 4.4 (b) are approximately 14% higher than the specimen without aggregate at a small distance from the aggregate surface and similar to the specimen without aggregate at a sufficient distance away from the aggregate. The results indicate that the inclusion of aggregate may increase the cracking potential of a specimen by combining the influence of internal restraint with the stresses that develop from the external restraint. In the case of the unrestrained boundary conditions, the stress level reduces quickly by moving away from the aggregate. However, in the case of the specimen with externally restrained boundary conditions, a high stress level is maintained even when moving away from the aggregate (in the y- direction). It can be imagined that if a small crack develops in the unrestrained specimen, this crack could reduce a substantial amount of the stored residual stress. As a result, a microcrack could initiate, yet there may not be enough stored energy to cause this crack to propagate across the specimen. However, the externally restrained specimen stores substantial energy, which may not substantially decrease as the crack develops. Therefore, the externally restrained specimen will have the higher potential of throughcracking when the crack initiates.

98 80 3 Unrestrained Single Aggregate Specimen 3 Restrained Single Aggregate Specimen 2 2 Stress (MPa) 1 0 A B B Stress (MPa) 1 0 A B B -1 Y D OA A -1 Y D OA A -2 X H -2 X H Distance from an aggregate (mm) (a) Distance from an aggregate (mm) (b) Figure 4.4 Stress development in an (a) unrestrained single aggregate specimen and b) the externally restrained single aggregate specimen As previously discussed, the bond between the aggregate and paste has an important role on the stress development in a composite system when the composite is externally restrained. Figure 4 (b) shows that the maximum stress that develops along the y-axis (element B) is not at the aggregate-paste interface. This may be explained by the fact that the actual maximum stress develops along a diagonal (approximately 55 o from the y-axis for the geometry shown) when the aggregate is perfectly bonded with cement paste (Figure 4.5 (a)), whereas it develops along the y-axis when the aggregate is

99 81 perfectly unbonded (Figure 4.5 (b)). Assuming that the aggregate and paste are initially perfectly bonded, it could be expected that the tensile stresses that develop at the pasteaggregate interface could lead to microcracking (debonding). Therefore, it can be expected that the initial debonding initiates at the interface at approximately 55º (for the geometry shown) from the y-axis and grows along the interface between the paste and aggregate until the aggregate becomes fully debonded from the paste. Externally Restrained Boundary Condition Externally Unrestrained Boundary Condition Max. Stress Development Agg. Agg. Agg. (a) (b) (c) Figure 4.5 Stress localization for different boundary conditions: (a) perfectly bonded aggregate for the externally restrained specimen, (b) perfect unbonded aggregate for the externally restrained specimen, and (c) perfectly bonded/unbonded aggregate for the externally unrestrained specimen This debonding causes a redistribution of stresses. Figure 4.6 shows the residual stresses along the y-axis for different bond conditions (externally restrained specimen in horizontal direction). In a perfectly bonded condition, a lower stress is observed at the aggregate-paste interface, which can be attributed to the fact that the aggregate is

100 82 participating in transferring stress through the system. The local variations in stress arise due to the differences in the elastic properties of the aggregate and the paste. When the aggregate is unbonded, tensile stresses can no longer be transferred through the aggregate. This results in the development of a maximum stress at the aggregate-paste interface (element B). Figure 4.6 also shows the stress development around a spherical air void, which can be considered as an aggregate with zero stiffness. The main difference between an air void and an unbonded aggregate is the ability of the unbonded aggregate to transfer compressive stress. In a system containing an unbonded aggregate, a radial pressure develops at the top and bottom of the aggregate as the paste shrinks. This effectively implies that the aggregate wedges the void open. In the case of an air void, no such stresses develop since the deformation of the paste in the y-axis direction is not restrained by the presence of aggregate. This explains the slight difference between the stress development in the paste around an air void and around an unbonded aggregate inclusion (Figure 4.6).

101 83 Stress (MPa) Perfectly Bonded Perfectly Unbonded Air Void No Agg Distance from the Agg. (mm) Figure 4.6 Stress development along the Y-axis of a restrained prism specimen over an aggregate (restrained boundary condition) To provide a physical model that is similar to the simulations, a prism specimen was restrained in one direction with a single aggregate. It is believed that this physical model will be able to be used to calibrate material parameters for future modeling developments. The physical model consisted of a prismatic specimen that had a length of 250 mm, a width of 50 mm, and a height of 25 mm. This specimen geometry is similar to the passive dog-bone restraint frame described in the literature (Chariton and Weiss, 2002). A cement paste specimen was placed in the frame with a w/c of 0.3. Specific details on the cement and the free shrinkage of this cement paste are described in the literature (Pease et al., 2003b). In addition to the cement paste, the specimen had an

102 84 instrumented aggregate that was placed in the center of the prism. The instrumented aggregate consisted of a thin-walled copper cylinder with an outer diameter of 15.8 mm, a height of 25 mm, and a wall thickness of 0.6 mm. Two strain gages were placed on the inner surface of the copper cylinder at 90 from one another to measure the strain that develops on the aggregate in different directions, as illustrated in Figure 4.7. The specimen was sealed to prevent drying for the duration of the experiment. As illustrated by Figure 4.7, the instrumented aggregate (copper ring) recorded a strain development caused by the autogenous shrinkage of the cement paste. These strains were measured by the strain gages and recorded by a Strainsmart acquisition system every five minutes. The results show an initial compressive strain recorded by both strain gages. Since the paste is still in a plastic phase, it is highly unlikely that these strains are attributed to a stress development in the system. Rather these strains could be thought to be due to a slight temperature rise in the system (approximately 2 C). After the time of setting (approximately seven hours) a tensile strain is measured in the direction of the y- axis (B) and a compression strain in the direction of the x-axis (A) is observed. This can be attributed to the development of a radial pressure at point B accompanied by aggregate-paste debonding at point A. The strain increased over time due to the increase in autogenous shrinkage. At an age of 27 hours, a visible crack was observed from the aggregate to one of the specimen edges. It should be noted that the strain in the copper ring does not drop to zero after cracking, presumably due to the bond between the ring and paste and continued stress transfer from the mortar to the copper ring.

103 85 Strain (µε) B A Position A Position B Copper Ring (with strain gages) Time (Hours) (a) Cummulative Acoustic Emission Energy (nvs) Copper Ring (with strain gages) Jump in Acoustic Activity to 10.3 nvs Time (Hours) (b) Figure 4.7 Experimental data from a restrained specimen containing an Instrumented Aggregate : a) strain development and b) acoustic activity

104 86 In addition to measuring strain development, acoustic emission was used to estimate the extent of cracking that occurred in the specimen. Acoustic emission describes a class of testing that relies on the use of piezo-electric transducers to measure the vibration (or acoustic activity) that occurs when a disturbance (i.e., crack) occurs in a material. This disturbance results in the release of energy and the propagation of an elastic wave. For the sake of brevity, the reader is referred to existing literature on the specific details of the acoustic emission equipment and the testing approach employed herein (Kim and Weiss, 2003). Figure 4.7 (b) shows the cumulative acoustic emission energy recorded during this experiment. The first acoustic event was recorded shortly after setting. During the initial period after setting, microcracking is expected at the aggregate-paste interface which corresponds to a continuous release of acoustic energy (Pease et al., 2003b). When the visible through-crack propagated, a rapid increase of the acoustic energy was observed. 4.4 Multiple Aggregate Systems A model was developed using a combination of hexagonal unit cells (Figure 4.8 (a)), each consisting of a cylindrical aggregate particle surrounded by a hexagon of the paste matrix. This model system was used to begin to assess the behavior of a multiaggregate system. The model system consisted of approximately 16 unit cells. Simulation results for residual stress were acquired from the center hexagonal unit cell to avoid boundary effects at the edges of the specimen. The length of each side of the hexagon (l Hex ) was fixed as 16.7 % of the length of the system while the size of the

105 87 aggregates was varied to investigate the effect of aggregate volume fraction on stress development. For this purpose, simulations were performed by varying the aggregate radius (R OA ) from 0.1R OP, to 0.2R OP, 0.4R OP, 0.5R OP, 0.6R OP, 0.8R OP, and 0.9R OP where R OP is the equivalent radius of hexagon corresponding to the same cross-sectional area as a cylinder (Figure 4.8 (a)) (Pease et al. 2003b). The elastic modulus of the cement paste was assumed to be 20 GPa while the elastic modulus of aggregate was varied to observe the effect of aggregate stiffness on stress development. The Poisson s ratio of the paste and aggregate was assumed to be 0.2 and 0.3 respectively. Two external boundary conditions, unrestrained and horizontally restrained (Figure 4.8 (b)) were applied in the simulations. First, a series of simulations were performed to obtain the equivalent elastic modulus of the composite system. A pseudo displacement load of -100 µε was applied to the horizontal direction and the reaction force was obtained. The overall average section stress and average section strain were used, along with Hooke s law, to determine the equivalent elastic modulus of the composite. Figure 4.9 show the results of the simulations for different aggregate volume fractions. Figure 4.9 (a) provides a comparison between the simulation results and the results obtained by using a series and a parallel composite model, as well as the model used by Hansen (1965) (Eq. 4.4). The results shown correspond to a case when the ratio of E agg to E paste is 10. Figure 4.9 (b) provides a comparison between the simulation results (shown as data points) and the results obtained from the Hansen s model (shown as lines) for different E agg /E paste values. It can be observed in both Figures that the simulation results are in good agreement with

106 88 the results from the Hansen s model despite the differences in the Poisson s ratio and the differences in the 2d versus 3d model assumptions. y Hexagonal Unit Cell Model Unrestrained Agg. x l Hex R OA R OP Single Unit Cell Equivalent Cylinder Restrained (a) (b) Figure 4.8 Multiple aggregate system composed of unit cell matrix

107 89 Equivalent E composite (GPa) E Agg / E Paste = 10 Parallel Model Series Model Hansen's Model Simulation Vol. of Agg. (%) (a) Equivalent E composite ( GPa) E agg /E Paste Simulation Hansen's Model Vol. of Agg. (%) (b) Figure 4.9 Comparisons of equivalent elastic modulus of composites

108 90 A second series of simulations were performed to obtain the internal stress development inside the composite system as the cement paste shrinks around the aggregates. The autogenous shrinkage of the cement paste was assumed to be -100 microstrain (µε). Figure 4.10 shows the maximum stresses that develop in an externally unrestrained system as a function of the aggregate volume fraction. Figure 4.10 (a) shows the maximum principal stress in the cement paste for different values of E agg /E paste, ranging from 0.1 to 10. A higher level of stress is observed as the volume fraction and the stiffness of aggregate increase. Figure 4.10 (b) provides a comparison between the simulation results and the maximum residual stress of the paste determined using Eq A good agreement between the two is observed for composites with an aggregate volume less than 60 percent. However, at higher volume fractions, the two methods predict stress levels that are slightly different, which is mainly due to the fact that Eq. 3 is derived for a single aggregate system (a cylindrical paste shrinking around a cylindrical aggregate). As the volume fraction of aggregate increases, the distance between the aggregates becomes smaller and the aggregates begin influencing one another.

109 91 Max. Principal Stress(MPa) E Agg /E Paste (Simulation) Externally Unrestrained Vol. of Agg. (%) (a) Externally Unrestrained Stress (MPa) E Agg 10 E Paste 5 2 Simulation Eq Vol. of Agg. (%) (b) Figure 4.10 Maximum stress development in an externally unrestrained composite

110 92 Figure 4.11 show the maximum residual stresses that develop in the composite system when it is horizontally restrained. In Figure 4.11 (a), the maximum principal stress that develops in the paste is shown as a function of the aggregate volume fraction. It is observed that for a value of E agg /E paste higher than 2, the maximum stress level is relatively independent of the aggregate volume fraction. However, for lower stiffness aggregates, the maximum stress initially decreases with increasing the volume fraction of aggregate and reaches a minimum around a 65% volume fraction. After this point, a slight increase in the stress level is observed as the volume of aggregates increases. Alternatively, the maximum residual stress in concrete can be computed by considering the system as a homogenous material with equivalent properties. In this case, the equivalent shrinkage strain can be determined by applying Eq. 4.1 and 4.2, while Eq. 4.4 could determine the equivalent elastic modulus of the composite. The equivalent maximum residual stress is determined by the application of Hooke s law. Figure 4.11 (b) shows the results for composites with different volume fractions of aggregate. As the aggregate volume increases, the equivalent shrinkage strain (calculated using Eq. 4.1 and 4.2) decreases, which leads to a decrease in the equivalent maximum residual stress. However, such analysis does not consider the residual stresses that develop due to the internal restraint provided by the aggregates. For comparison, the maximum principal stress in the composite obtained from the FEA simulations is also included in Figure 4.11(b). These results, on the other hand, do not suggest that the residual stress level in the composite decreases significantly with increasing the volume fraction of aggregate.

111 93 Max. Principal Stress (MPa) Stress (MPa) Externally Restrained (Simulation) E Agg / E Paste Vol. of Agg. (%) (a) Externally Restrained Internal Equivalent E Agg 10 E Paste Vol. of Agg. (%) (b) Figure 4.11 Maximum stress development in a composite externally restrained in one direction

112 94 While this may appear to contradict many field observations, such as those by Darwin et al. (2004), who noted that visible cracking in the field was substantially reduced in materials with a lower paste volume, it should be noted that these observations do not consider the potential contribution of microcracking or interfacial cracking that can be expected to occur in a heterogeneous composite system. Up to this point, this chapter has focused on assessing the residual stress development in the paste matrix (and aggregate). It is also necessary to consider the effect of the bond condition between aggregate and cement paste to better understand the potential for microcracking at the interface between aggregate and cement paste. As previously discussed, the bond condition has little effect on the internal stress development in a composite system for the unrestrained boundary condition because the matrix shrinks around the aggregates, subjecting them to an almost uniform compressive pressure along the interface. However, in the case of the restrained boundary condition, the bond condition and the bond strength become much more significant parameters. The influence of bond strength was investigated using finite element analysis. A composite system consisting of three phases (cement paste, aggregate, and bond elements) was prepared with an aggregate volume fraction of 15% (Figure 4.12). The time-dependent values of autogenous strain, elastic modulus, and fracture energy of cement paste were used in these simulations. For the sake of brevity, the exact time dependent functions are not provided, however, they describe the behavior of a paste with a w/c of 0.30, which is similar to that measured by Pease et al. (2004). For the trials described in this paper, the bond elements were assigned stiffness and fracture energy that were 10% of the values for cement paste. Further work is needed to characterize the bond stiffness and strength

113 95 more thoroughly for further simulations. To determine whether interfacial cracking occurred, a fracture energy analysis was performed to determine whether the stored strain energy in each element exceeded the fracture energy. When the stored strain energy exceeded the fracture energy, the stiffness and fracture energy of those elements was decreased to 10% of the previous values and the simulation was re-run until a stable geometric configuration was obtained. This simulation technique is typically used when a model has high non-linearity, such as cracking. The elastic modulus of aggregate that was used was 100 GPa and the cracking of the aggregate was not considered in this study in an effort to focus on observing the cracking tendency of cement paste with low bond strength. For the analysis shown in Figures 4.12 and 4.13, the shrinkage strain was applied incrementally and the stress development and cracking was monitored. Figure 4.12 (a) shows the initial status of the microcracking, where crack coalescence can be observed during loading as the black-colored regions. These debonded regions developed along the edges of the aggregates under the horizontally restrained boundary condition. The debonding between the aggregate and the paste increased until the debonding reached the y-axis position (i.e., the top and bottom of the aggregate as shown in Figure 4.12 (b)) at which time a vertical through-crack developed (Figure 4.12 (b)).

114 96 (a) (b) Figure 4.12 Cracking tendency of a composite (horizontally restrained boundary conditions) It was previously discussed that the distance between the aggregates has an important role in the internal stress development and cracking tendencies. However, it is impossible to think that the simple hexagonal unit cell composite model can completely represent a complex system like a concrete composite. In real cementitious composites, the shape of the aggregate, the size distribution of the aggregate, and the spacing between the aggregates would influence the internal stress development (Jones and Kaplan 1957). Therefore, additional modeling approaches are needed.

115 Summary This chapter describes the effect of internal and external restraint on residual stress development in a concrete composite. Three model systems were evaluated including 1) a single aggregate, 2) a composition of hexagonal unit cells, and 3) a real composite image system. The following observations can be made: The increase in aggregate volume reduces the overall free shrinkage of a cementitious composite. The addition of aggregate results in the generation of internal residual stresses. The bond condition between the aggregate and the cement paste has a small role on stress development when a composite is not externally restrained. The bond condition becomes more critical for describing the behavior of the system when an external restraint is provided. In the case of the externally unrestrained boundary condition, it was observed that higher internal stresses develop with a higher volume fraction of aggregate and higher stiffness of aggregate. In the case of the externally restrained (in one direction) boundary condition, the maximum residual stress level is not observed to vary significantly with increasing the volume fraction of aggregate. The local residual stresses that develop in a composite are higher than the equivalent stresses that are computed using the equivalent shrinkage strain and equivalent elastic modulus of a composite. This is due to the fact that the

116 98 latter technique does not consider the stresses generated by the internal restraint of the aggregates, and indicates that it is possible to underestimate the microcracking and cracking potential of concrete if estimation is performed only using equivalent parameters.

117 99 CHAPTER 5: ANALYSIS PROCEDURES FOR ASSESSING THE SHRINKAGE AND CRACKING IN A CEMENTITIOUS COMPOSITE ON THE MESO-SCALE 5.1 Introduction Concrete changes volume in response to chemical reactions, moisture changes, and temperature variations. Many documents highlight the importance of considering the influence of volume change on the potential for cracking (ACI 209R-92; Kosmatka et al., 2003), however, the majority of these documents and simulation models (Bernard et. al., 2002; Foster, 2000; Weiss et al., 1998) treat the concrete as a homogeneous material and focus only on the development of large, visible cracks. Currently, few engineers explicitly consider the heterogeneous nature of concrete in designing for shrinkage cracking. Researchers (Grzybowski and Shah, 1990; Bisschop and van Mier, 2001; Kim and Weiss, 2003; Moon et al., 2005) have shown that the combination of the external restraint from the structure surrounding the concrete and the internal restraint provided by the aggregate can lead to the development of microcracking even when visible cracking is not predicted or observed. These microcracks may play a key role in relaxing stresses (Pease et al., 2005), increasing fluid transport (Yang et al., 2005) and serving as an initiation point for visible crack formation (Chariton et al., 2002). Chapter 4 showed that the use of the conventional analysis approach of assuming that concrete acts as a homogeneous material may not provide sufficient information about the microcracking that would be experienced in the composite. Chapter 4 introduced the concept of

118 100 assessing concrete as a heterogeneous system. Finite element simulations were proposed that consider idealized meso-scale composite systems consisting of hexagonal unit cells. While these models begin to illustrate the stress distribution in a composite due to the stress concentration provided by the material heterogeneity, they do not provide enough information to completely analyze the microcracking behavior of real heterogeneous composite systems because aggregates do not have a single shape and are not distributed uniformly. This chapter describes an initial research effort to utilize a simulation technique to assess the potential for microcrack development in concrete. The behavior of realistic two-dimensional concrete composites system was simulated using an Object-Oriented Finite Element (OOF ver ) code which was developed by the National Institute of Standards and Technology (NIST, Langer et al., 2001). OOF is compatible with UNIX or Linux operating systems and it provides an easy to use interface which enables users to define each phase in a composite and to mesh elements of a composite system from an optical scan of an surface of the material. The OOF program uses portable pixmap image files that can be obtained by scanning the surface of real composites (actually, the PPM2OOF code (ver ) which is the complementary code of OOF, provides this interface). This approach is only applicable when each phase in a composite has a different color (or color intensity) because the process of defining each phase is performed based on the color information of each phase. Therefore, the color images of a concrete (or mortar) need to be obtained where the concrete has paste and aggregates that are different colors. Under these conditions, the OOF program can be used to perform finite element simulations on multiple-phase meso-scale structures rather easily.

119 101 This chapter describes the simulation procedure in three sections. The first section describes the preprocessing step, indicating how the image files are obtained from an actual concrete sample, how the colors of the paste and aggregate are used to define each phase, and how this information can be used to develop a finite element mesh for each sample. The second section of this chapter describes the finite element simulation process, and the third section describes the postprocessing procedure indicating how image analysis is performed for data analysis using color contour images of stress or energy. The results and discussion sections of this chapter discuss how the results of the simulations can be used for assessing the shrinkage cracking behavior of a heterogeneous system. 5.2 Overview of the Approach The importance of performing a meso-scale analysis to describe the cracking behavior of concrete composites is discussed in this chapter. A realistic two-dimensional composite system of aggregates in a cement paste matrix is simulated using OOF code. The sequence of the meso-scale OOF simulation procedures is described in the flowing chart (Figure 5.1), and the section thereafter describes the preprocessing, finite element simulation, and postprocessing in detail.

120 102 OOF SIMULATION ON THE MESO-SCALE (Object-Oriented Finite Element Simulation) 1. PREPROCESSING 1) Acquisition of concrete specimen 2) Surface preparation (saw-cut and polish) 3) Chemical treatment (phenolphthalein) 4) Acquisition of 2-D image (scanning) 5) Re-coloring 6) Meshing 2. FINITE ELEMENT SIMULATION 1) Modeling of shrinkage strain of paste 2) Determination of fracture criterion 3) Boundary condition 4) Load 3. POSTPROCESSING 1) Acquisition of images of stress/energy distributions 2) Image analysis DATA ANALYSIS Figure 5.1 Flow chart of OOF simulation on the meso-scale 5.3 Analysis Procedures The analysis procedures of OOF simulation consists of three sections: preprocessing, finite element simulation, and postprocessing. While a more detailed description of each section will be described later in this chapter, a brief description of each process is provided below.

121 103 Preprocessing: The preprocessing procedure described in this chapter was used to obtain an image of the concrete section that will be used for finite element simulation. The preprocessing procedure consisted of obtaining a high-resolution color image. The color in the image was used to define the phases (i.e., the aggregate and paste matrix) of the composite. To obtain a quality image, the concrete or mortar specimen was saw-cut and polished. The polished surface was stained using phenolphthalein to change the color of the paste phase to pink, thereby enabling each phase to be more clearly identified. The stained surface was then scanned using a flat-bed type scanner. The scanned image was saved as a TIFF format file that records the initial color of each pixel. The saved image file was re-colored using a color histogram to identify similar phase areas (i.e., areas of similar color) that can be converted to a grayscale image (i.e., black, gray, and white) to represent any phase (aggregate, interfacial zone, or paste) with the assistance of Adobe Photoshop. In this research, each phase was defined as having the same material properties throughout the phase. The re-colored image was then saved as a portable pixmap image file and transferred to PPM2OOF code (complementary program of OOF code for pre-processing) for meshing. The mesh was successively refined until the boundaries between the phases had a shape that was reasonably similar to the original image. The refinement of the mesh at the boundaries was performed using the PPM2OOF code (Carter et al. 2000). In summary, the preprocessing consisted of four components; a) concrete specimen preparation, b) concrete specimen scanning, c) image analysis (re-coloring) of the scanned image, and d) meshing.

122 104 Finite Element Simulation: This section begins by describing how the shrinkage of concrete was simulated using the technique of thermal expansion substitution, which is previously mentioned in Chapter 2. For a composite system, a coefficient of thermal expansion of zero is applied to the aggregate and a coefficient of thermal expansion of 10-6 ε/ o C is applied to the paste phase. A negative temperature is applied to a meshed composite system, which introduce a shrinkage strain in the paste phase (here, a change in temperature of -1 o C represents a -1 µε shrinkage strain). Two different boundary conditions were considered in this study. The unrestrained specimen corresponds to a free shrinkage specimen while the restrained specimen considers restraint in one direction (i.e., the x-direction) similar to what may be expected in a bridge deck or pavement. The magnitude of the shrinkage strain of paste was successively increased until a meshed composite failed (externally restrained boundary condition) or up to a designated value (free boundary condition). For cracking analysis, a brittle failure criterion was used. Further discussion on the failure criterion is provided later in this chapter. Postprocessing: This section describes the developing methods used to assess the results of the finite element analyses. In addition to using the stresses obtained directly at the nodes from the finite element analysis, image analysis was performed using the color contour images obtained from the OOF simulations, where each color represents a specific level of stress or strain energy density. The area fraction of each color level was obtained using Image-Pro Plus TM. This section discusses

123 105 how the information that was obtained from the results of the OOF simulations were analyzed. At the end of this chapter, the differences between the conventional analysis approach which assumes that the concrete behaves as a homogeneous material and the meso-scale approach (OOF simulation) which considers concrete as heterogeneous material will be discussed. The application of the meso-scale approach for cracking analysis of composites will also be illustrated. 5.4 Preprocessing: Acquiring the Image from a Realistic Specimens Specimen Preparation The simulations that were performed in this research used two-dimensional images. To obtain a realistic cementitious composite, actual two-dimensional corss-sectional surfaces of concrete (mortar) were prepared. The cross-sectional surface of the concrete was obtained by saw-cutting and polishing the specimen. To avoid the destruction (raveling) of the paste phase during cutting, the mortar specimen was allowed to hydrate until a sufficient strength was achieved at the time of cutting. The saw-cut surface was polished after cutting to obtain a flat and clean surface, which provided a clearly defined composite image (Figure 5.2 (a)). Polishing was performed using equipments composed of a flat-turntable with fine grits ranging from 6 µm to 0.25 µm. A general description of polishing procedures are provided by Allman and Lawrence (1972) and John et al. (1998). When the polishing process was finished, the polished surface was washed with

124 106 water and the specimen surface was dried using compressed air. After the specimen surface was dried, phenolphthalein was applied on the polished surface using a cottontipped stick (Figure 5.2 (b)). The application of phenolphthalein changed the color of the cement paste phase to pink, which helped to better define the paste phase during the image analysis (re-coloring) procedure. One alternative to the application of phenolphthalein is to use white cement with dark color aggregates, which provide better color-defined phases.

125 cm 2.54 cm (a) Cross section of a sample (saw-cut) (b) Polished surface treated with phenolphthalein Figure 5.2 Specimen preparation (surface treatments)

126 Scanning A computer image file was obtained by scanning an actual concrete surface using a computer scanner (Figure 5.3), which provided a two-dimensional color image scan with a high resolution (600 dpi is used in this research). Images with high resolution were needed for the meshing procedure to provide clear boundaries among phases because meshing is performed based on the size of each pixel. The resolution of an image also becomes important when the bond phase, between the paste and aggregate phases, needs to be considered because the bond phase will have very small thickness (µm scale). The mechanical properties of the bond phase are typically reported to be different (lower strength and stiffness) than those of bulk cement paste due to increased porosity (Neville, 1996). Scrivener and Gariner (1988) investigated the increase of porosity at the interfacial zone and found that the porosity at the interface is three times higher than that of the bulk paste zone (between 0 to 50 µm from the boundary surfaces of aggregates). The scanning resolution used in this research was 600 dpi (dot(pixel) per inch) which provides 42 µm per pixel. The size (thickness) of the bond phase was determined by image processing that was based on a pixel unit. Therefore, a one-pixel thick layer that has 42 µm thickness could be artificially placed around the aggregate phases, which is compatible with the theoretical bond phase (interfacial zone). The details of ordering the bond phase in a scanned image are described in the next section. While a higher resolution would be better suited for the finer refinement at the bond phases, a high-end computer (faster CPU (higher than Intel P4-3GHz) and higher RAM (random access memory) capacity of more than 1GB) would be required for the preprocessing phase

127 109 because a higher resolution image requires more memory. The scanned image file is saved as a TIFF format file that records the initial color of each pixel. Some file formats (such as JPEG) compress the size of the original image file, resulting in the loss of the initial pixels of data.

128 Figure 5.3 Flat-bed color scanner and concrete samples (saw-cut, polished, and stained) 110

129 Image Analysis (Re-Coloring) The first step in the image analysis process was to distinguish between the two phases of the specimen, namely the aggregates and the paste. This was done based on the color difference between the aggregates and paste in the scanned image. Natural aggregates come in several different colors and the paste does not have uniform gray color which makes it difficult to differentiate between the two phases simply by using image analysis techniques. As a result, it was decided to stain the paste by applying phenolphthalein before scanning (this gives paste relatively uniform pink color due to the high alkalinity of the paste). A two phase image was generated from the scanned image using the commercial image analysis program, Adobe Photoshop (Figure 5.4). First, the (pink) cement paste phase was selected and reassigned uniformly as white. The rest of the image that represents the aggregates was reassigned the color gray. In this research, the air voids were not considered in the simulation. This implies that the areas of air voids were assumed to belong to the paste phase. Using the image analysis program, it was also possible to assign a third phase (i.e., bond phase) as the paste-aggregate interface, which can have properties that are different than the bulk paste. By using the expand or contract options in the image analysis program, a third phase was generated at the perimeter of each aggregate with a single pixel thickness. At the end of the re-coloring process, the file format was converted to PPM so that it could be used for the OOF simulation (PPM2OOF and OOF are compatible with PPM format images).

130 cm 2.54 cm Figure 5.4 Image analysis (re-colorization, part of the scanned image V agg = 55.7 %)

131 Meshing Meshing was performed using the PPM2OOF code (complementary preprocessing program of OOF) (Carter et al., 2000). Before meshing, material properties were ordered for each phase (i.e., the cement paste phase, aggregate phase, and bond phase) which were previously grouped by color in the re-coloring process. In this research, it was assumed that all aggregates have the same material properties and the paste phases also have single values that describe their material properties. Three node triangular elements were used in this research. The meshing and refining processes were semi-automatically performed until the meshed boundaries among the phases become reasonably the same as the original boundary shapes. The details of the meshing and refining procedures can be found in the PPM2OOF Manual (Carter et al., 2000). Careful application of the meshing process was performed to optimize element refinement at the boundaries while avoiding excessive refinement which can lead to element distortion and thus divergences of numerical calculations. During the meshing procedures, the total number of meshed elements was maintained at less than 200,000, which was found to be consistent with the computer (RAM-1.5 GB) used for this research. The total number of meshed elements is dependent on the size of an image and the degree of refinement. Figure 5.5 shows an example of a meshed composite system that contains about 70,000 elements (the size of the real sample for the portion of this image was 2.54 cm * 1.27 cm).

132 cm 2.54 cm Figure 5.5 Meshed sample

133 Finite Element Simulations General Approach For finite element simulations, the boundary condition, the type of load, and the magnitude of the load need to be determined. In this study, two different boundary conditions were considered (i.e., the free boundary and one directionally restrained boundary conditions). The unrestrained specimen corresponded to a free shrinkage specimen while the restrained specimen considers restraint in a bridge deck or pavement, and plane stress approximation was used in the simulations. The shrinkage loading consisted of the application of a uniform volume change in the paste and no volume change in the aggregates using a temperature substitution analogy as described in Chapter 2 (Moon et al., 2005). The elements representing cement paste were ordered to have a non-zero thermal expansion coefficient (α = 1.0E-6 ε/ o C in this research) while the elements representing aggregates were determined as having a zero thermal expansion coefficient. Therefore, by applying a temperature load (a negative temperature in the case of the shrinkage of cement paste) onto the entire meshed elements, FE simulations were performed that simulated concrete shrinkage. For example, a -1 o C of temperature load will introduce the shrinkage strain of the cement paste phase as a magnitude of -1 µε when the thermal expansion coefficient of the paste is 1.0E-6 ε/ o C. This approach is compatible with the concept that the paste phase changes its volume due to autogenous or drying shrinkage while aggregates do not change their volumes. Here, it should be kept in mind that the thermal expansion coefficient and the temperatures used in this research

134 116 do not represent the real thermal expansion coefficient of paste nor the real temperature. The magnitude of the temperature drop (which is compatible with the magnitude of the shrinkage strain of cement paste) was successively changed (increased) to observe the stress development, failure stress (strength), and cracking behaviors of composite systems Failure Criterion Cracking analysis in finite element simulations can be performed by applying appropriate failure criterion, and the type of failure criterion can be divided in three categories: perfectly brittle, quasi-brittle, and perfectly plastic (Figure 5.6). Concrete is known to behave like a quasi-brittle material while each phase (cement paste and aggregate) in the concrete behaves more closely to a perfectly brittle material. Therefore, even though cement paste and aggregates are not perfectly brittle materials, for the simplicity of the application, a perfectly brittle failure criterion was used for each phase throughout the course of this research. The perfectly brittle failure criterion was applied using damisotropic function in the OOF code (Langer et al., 2001). The damisotropic function is composed of three parts: the basic material properties (elastic modulus, poisson s ratio, and thermal expansion coefficient), the maximum stresses (tensile and compressive), and the knockdown factor (0~1). The knockdown factor is used to reduce the elastic modulus of elements when the elements need to behave as cracked ones in simulations. The knockdown factor is automatically multiplied to the elastic modulus of an element if the stress in the element exceeds the strength value and the simulation is restarted at that stage of loading using the new material properties of elements until the

135 117 simulation satisfies the convergence. Therefore, the knockdown factor has a value between 0 to 1, and it needs to be determined considering the failure criterion of a material. To apply the perfectly brittle failure criterion, the knockdown factor should be small enough to drop the elastic modulus close to zero but it should be large enough to avoid the convergence error, which typically happens when the elastic modulus of elements is too close to zero value, compared to the elastic moduli of other elements. With the consideration of the range of the material properties (elastic modulus and tensile strength) of cement paste, the knockdown factor in this research was chosen to be 10-5 through trial and error. Stre s s Stre s s Stre s s Strain Strain Strain (a) Perfectly brittle (b) Quasi-brittle (c) Perfectly plastic Figure 5. 6 Failure criteria

136 Material Properties In this study, three different material properties of pastes were chosen and used for the simulations (Table 5.1). The typical fine aggregate used in Indiana is likely to be composed of many kinds of minerals, however, for this research, the average material properties of siliceous quartz aggregate (Table 5.2) were chosen as siliceous quartz is the representative material for typical fine aggregate (West, 1995). Table 5.1 Material properties of cement paste Paste-1 Paste-2 Paste-3 Elastic modulus (GPa) Poisson s ratio Tensile strength (MPa) Compressive strength (MPa) Table 5.2 Material properties of aggregate Fine aggregate Elastic modulus (GPa) 50 Poisson s ratio 0.2 Tensile strength (MPa) 20 Compressive strength (MPa) 200

137 OOF Simulation (Performing Trials) A single composite image was used for simulations in this research. A mortar specimen was made using sieved fine aggregates, which were retained between #16 and #8 sieves (the size distribution of the fine aggregate: 1.19~2.36 mm). The original size of the mortar specimen was 25.4 cm (10 inch) long with a 2.54 cm (1 inch) wide square cross-section. The specimen was cured for two days and preprocessing (saw-cutting, polishing, chemical treatment, scanning, re-coloring, and meshing) was performed following the procedures described in Section 5.4. The size of the image used for simulations was 2.54 cm (1 inch) wide and 1.27 cm (0.5 inch) high. The mixture had an aggregate volume of 55 % (the obtained image was observed to contain 55.7 % of fine aggregate) (Figure 5.4). The percentage of aggregates in an image was obtained using a commercial image program (ImagePro Plus TM ). Three different material properties of cement paste were used for the simulations (Table 5.1). The material properties of fine aggregates are shown in Table 5.2. For these simulations, it was assumed that cement paste and aggregates are perfectly bonded. Simulations were performed to observe the influence of autogenous shrinkage on microcracking and cracking by increasing the magnitude of uniform shrinkage strain of the paste phase up to the time of failure (in the case of the restrained boundary condition) and up to -300 µε (in the case of the free boundary condition). All simulation data files were saved and the postprocessing tasks were performed, which are described in the next session.

138 Post-Processing: Analysis of Results The post-processing portion of this work consisted of developing methods to assess the results of the finite element analyses. In addition to using the stresses obtained directly at the nodes from the finite element analysis, image analysis was performed using the color contour images obtained from the OOF simulations, where each color represented a specific level of stress or strain energy density. Numerical analysis using color contour images can be performed. The benefit of this approach is that the stress or energy development and distributions in a composite system can be easily quantified. For the quantification of the specific stress or energy levels in a composite, the area fraction of each color level was obtained using Image-Pro Plus TM program. To acquire color scaled images, consistent maximum and minimum limits of scale needed to be determined. This research illustrates only the tensile stresses as the minimum limit of the scale for the color contour of the stress distribution was fixed to zero. Therefore, compressive stress is represented along with the zero stress elements. After the determination of the maximum and minimum magnitude of scale, the type of color scale was chosen. In this research, the Tequila_Sunrise option in OOF was chosen (Figure 5.7). The Tequila_sunrise scale varies color from red (minimum magnitude of scale) to yellow (maximum magnitude of scale), changing the RGB (red, green, blue) values from 255, 0, 0 to 255, 221, 0. The value of red is always 255 and blue is always zero in the scale. Therefore, each color level can be defined by observing the intensity change of the green color value.

139 121 The total number of intervals in the color scale was selected. Theoretically, it is possible to order the total intervals up to 256, however, only 80 scale intervals were used in this research. The color contour images from the simulations were saved as portable pixmap image format (ppm) by OOF code (OOF saves images in ppm format) and the image files were converted to bitmap image files (bmp) for post processing. During this process, the image files were not compressed since this will cause information to be lost. Using Image-Pro Plus TM, the color contour image was analyzed regarding the area fraction of each color level, referring to the intensity change of the green color. As a result of image analysis, Image-Pro Plus TM provided a table of the total number of pixels for each color level. Yellow Figure 5. 7 Color scaled image of stress distribution in a composite (Paste-1, Horizontally restrained, ε paste = -140 µε)

140 122 The OOF code automatically changed the color of elements to black when the elements reached to the maximum stress (strength) values (Figure 5. 8). Therefore, from the image analysis using these images, the area fraction of cracked zones was determined. Figure 5. 8 Cracked image (paste-1, horizontally-restrained, ε paste =140 µε) 5.7 Results and Discussions In this section, two analytical approaches were discussed. The conventional approach for the calculation of residual stress development in concrete using equivalent parameters, which assumes the concrete composite to be a homogeneous effective medium, and a meso-scale OOF simulation approach, which considers the heterogeneity of the composite. First, using the conventional approaches, equivalent parameters are calculated, hence, the equivalent residual stresses are calculated as previously discussed in Chapter 4. The meso-scale approach will show the results of the OOF simulations, showing the stress and energy distributions and cracking behaviors in a heterogeneous

141 123 composite system. Discussions follow regarding the differences between the approaches and the importance of the meso-scale approach Conventional Approach Using Effective Medium Assumption Chapter 4 showed the conventional approach for the calculation of the residual stress using equivalent parameters (equivalent shrinkage strain and equivalent elastic modulus). An actual composite system was shown in Figure 5.4. The volume fraction of aggregate was determined from Figure 5.4 (55.7 %) and the material properties shown in Tables 5.1 and 5.2 were used to compute equivalent properties. The equivalent strain was calculated by Eq. 4.1 and Eq. 4.2, and the equivalent elastic modulus was calculated by Eq All of these calculated equivalent parameters were compared with the values directly obtained from the finite element simulations. It was observed that the difference in the equivalent elastic modulus between the OOF simulation and Hansen s model was less than 5 percent (Figure 5.9). Therefore, it can be said that the conventional equations for the equivalent strain and equivalent elastic modulus developed by Pickett (1956) and Hansen (1965) can be used not only for the idealized composite system (hexagonal unit cell composite), but also for the cases of real composite systems.

142 Equivalent elastic modulus (GPa) Equivalent Elastic Modulus (GPa) OOF Simulation Hansen's Model Paste number (Table 5.1) Paste number (Table 5.1) Figure 5.9 Equivalent elastic modulus of a composite (OOF simulation vs. Hansen s model) Figure 5.10 shows the equivalent stress values for each type of cement paste (Table 5.1). The equivalent stress was calculated by multiplying the equivalent strain with the equivalent elastic modulus obtained from Eqs. 4.1, 4.2, and 4.4. As can be seen, equivalent stress increases when the shrinkage strain of paste and elastic modulus of paste increases. Here, it can be noticed that the equivalent stress for any type of paste does not reach the tensile strength of paste (5 MPa) up to µε.

143 125 2 Equivalent stress (MPa) Paste-1 Paste-2 Paste Shrinkage strain of paste (µε) Figure 5.10 Equivalent stress vs. shrinkage strain of paste As previously discussed in Chapter 4, the conventional approach cannot account for the microcracking behavior in a composite system appropriately because the conventional approach transforms a composite system to an equivalent homogeneous system, which neglects stress localization. This aspect also becomes important when a concrete specimen is in the free boundary condition because the conventional approach assumes that concrete will not experience stress development in a specimen in the free boundary condition while stresses develop and stress localizations occur in a real concrete specimen due to the existence of aggregates, which restrain the volume change of the paste phase due to shrinkage. In the case of the meso-scale approach, the stress

144 126 development and localization in a composite can be observed, and the strengths of the different composites can be evaluated from simulations if the material properties of each phase are known OOF Simulation (Results) Three different material properties were used in the simulations to describe the properties of the cement paste phase (Table 5.1). OOF simulations were performed by systematically increasing the shrinkage strain of the paste phase. Two different boundary conditions (horizontally-restrained and free boundary conditions) were applied. In the case of the restrained boundary condition, simulations were performed until the reaction force of the composite at the boundaries decreased to nearly zero. In the case of the unrestrained boundary condition, simulations were performed with shrinkage in the paste phase of up to -300 µε. Figure 5.11 shows the reaction force that developed at the horizontal boundary of each composite with different paste properties in the case of the restrained boundary condition. By knowing the cross-sectional length (height) of the composite, which is used for the OOF simulation (in this study, 1.27 cm) and assuming that the composite has a unit thickness (1 m), the average cross-sectional stress was calculated as shown in Figure Here, the average sectional stress along the restrained direction at failure was determined to be 1.7 MPa, 0.9 MPa, and 0.5 MPa for paste-1, paste-2, and paste-3, respectively, for the restrained specimens. As can be seen in Figure 5.11, paste-1 developed the highest reaction force and failed when the shrinkage strain of cement paste

145 127 reached to -140 µε. The cases of paste-2 and paste-3 failed at -125 µε and -120 µε, respectively. Referring to the material properties of cement paste in Table 5.1, the maximum tensile strain of paste at failure is -250 µε (which can be calculated by dividing the tensile strength by the elastic modulus) while all three composites failed at a much lower shrinkage strain of the paste, which is due to the existence of stress concentration and localization in a composite system induced by the heterogeneity. Therefore, it can be said that the concrete would fail at a lower magnitude of shrinkage strain of the paste because of the stress concentration and localization while the equivalent (apparent) shrinkage strain of a composite is smaller than the maximum tensile strain of the paste phase. Reaction Force (N) Paste-1 (Elastic) Paste-2 (Elastic) Paste-3 (Elastic) Paste-1 (Cracked) Paste-2 (Cracked) Paste-3 (Cracked) Shrinkage Strain of Paste (µε) Average Cross Sectional Stress (MPa) Figure 5.11 Shrinkage strain of paste vs. load (horizontally-restrained boundary condition)

146 128 Figures 5.12 and 13 show the cracked images of each case. The results show that composites demonstrated localized through-cracking in the case of the restrained boundary condition while distributed cracking occurred in the case of the free boundary condition. It appears that the change of material properties also changed the paths of the cracking in the case of the restrained boundary condition (Figure 5.12) while the amount of microcracks varied with the change of the material properties of the paste in the case of the free boundary condition (Figure 5.13). The amount of microcracking will be further discussed later in this chapter.

147 129 (a) Paste-1 (ε paste = µε) (b) Paste-2 (ε paste = µε) (c) Paste-3 (ε paste = µε) Figure 5.12 Cracks in a composite (one directionally restrained boundary condition)

148 130 (a) Paste-1 (ε paste = µε) (b) Paste-2 (ε paste = µε) (c) Paste-3 (ε paste = µε) Figure 5.13 Cracks in a composite (free boundary condition)

149 131 Image analyses were performed using the color contour images of the stress and energy distributions. To observe the change of the stress and energy developments, two different simulations were performed: an elastic condition with no cracking and cracking simulations (Figure 5.14). It was observed in the cracking simulations that the level of stress successively increased by increasing the paste shrinkage and that the initial cracks occurred at the boundaries between the paste and the aggregates where the highest stresses developed due to stress concentration and localization. From Figure 5.14, it can be seen that the higher stresses are localized at the boundaries between the paste and the aggregates and these higher stress levels (brighter yellow color) developed in a composite decrease after cracking.

150 132 (Pa) 6E6 0 (a) Elastic condition with no cracking (ε paste = -140 µε) (b) Cracking simulation (ε paste = -140 µε) Figure 5.14 Color contour image of stress distribution in a composite (One directionally restrained boundary condition)

151 133 Figure 5.15 shows the cumulative area fraction of the composite that carries a certain level of residual stresses when the specimen is externally restrained (Figure 5.15 (a)) or unrestrained (Figure 5.15 (b)). For example, in Figure 5.15 (a), the elastic curve (hollow circles) shows that 40 percent of the composite surface area has a residual stress level that is higher than 2 MPa stress. By comparing the elastic and cracked curves, it is apparent that cracking releases stresses in the material. Cracking occurs where tensile stresses exceed the paste strength. This leads to through-cracking only after the stress in a small percentage (0.5 %) of the total area exceeded the strength when the specimen is externally restrained (Figure 5.15 (a)). However, the specimen without restraint (i.e., free boundary conditions) did not exhibit a through crack even though the stress in a considerable fraction of the area exceeded the tensile strength (Figure 5.15 (b)). In this case, dispersed microcracking occurred at areas with high residual stresses, without significant coalescence of microcracks (i.e., stable crack growth). A final observation can be made based on the results presented in Figure 5.15 (a). In the restrained sample, a through cracking splits the specimen into two parts. Each of these parts behaves independently and similarly to a specimen with free boundary condition (compare the stress level between the cracked and the free boundary in Figure 5.15 (a)).

152 134 Cumulative area fraction of stress level (%) Horizontally Restrained Boundary Paste-1 Elastic Cracked Free boundary ε paste = -140 µε Average Sectional Stress (Elastic, 1.7 MPa) Tensile Strength of paste-1 (5 MPa) Stress (MPa) (a) Restrained boundary condition (ε paste = -140 µε) Cumulative area fraction of stress level (%) Free Boundary Paste-1 Elastic Cracked ε paste = -250 µε Tensile Strength of paste-1 (5MPa) Strain Energy Density (N/m 2 ) (b) Free boundary condition (ε paste = -250 µε) Figure 5.15 Cumulative area fraction vs. stress

153 135 Strain energy was used for qualifying and quantifying the cracking behavior of the cement composite. The amount of strain energy that was stored and released before and after cracking was detected. Figure 5.16 shows the distribution of the energy density level in a composite system. As shown in Figure 5.16, the area fractions at lower levels of strain energy density increased (the area fractions at higher levels of strain energy density decreased) after cracking. In the case of the restrained boundary condition, the elastic curve exhibits two peaks, one at higher strain energy density (around 150 N/m 2 ) which corresponds to higher stress level, and the other at lower energy density (around 25 N/m 2 ) which corresponds to a lower stress level. Cracking causes release of energy in the high strain energy density (high stress) areas as shown by the elimination of the high energy peak in Figure 5.16 (a). In the case of free boundary condition, the occurrence of microcracking did not change the distribution of strain energy density considerably (Figure 5.16 (b)).

154 Paste - 1 Elastic Cracked Area Fraction (%) Strain Energy Density (N/m 2 ) (a) Restrained boundary condition (ε paste = -140 µε) 16 Area Fraction (%) Paste - 1 Elastic Cracked Strain Energy Density (N/m 2 ) (b) Free boundary condition (ε paste = -250 µε) Figure 5.16 Strain energy density versus area fraction

155 137 To clearly observe the magnitude of the stored energy for each energy density level, a normalized stored energy value was used. The normalized stored energy is an energy value which is calculated by multiplying each energy density level with its area fraction, which represents the magnitude of the stored energy in a composite for each energy density level per unit area. Figure 5.17 shows the change of the normalized stored energy for the energy density level in a composite due to cracking. As can be seen, portions of the higher strain energy density levels mainly decreased after cracking while portions of the lower energy density levels increased in the case of the restrained boundary condition (Figure 5.17 (a)). A similar tendency was also observed in the case of the free boundary condition (Figure 5.17 (b)). One interesting finding was that the curve had a bimodal shape. It was found in the simulations that the portion of the first peak represents the stored energy mainly in the aggregate phases while the portion of the second peak is the stored energy in the paste phase. Therefore, from Figure 5.17, it can be seen that the energy mainly decreased in the second peak, which means that the paste phase released energy due to cracking. The total area below each curve indicates the normalized total strain energy stored in a composite. Therefore, it can be expected that the composite after cracking had a smaller amount of stored energy due to the cracking and the difference of the total stored energy is the released energy.

156 138 Normalized Stored Energy (N*m) Paste-1 Elastic Cracked Strain Energy Density (N/m 2 ) (a) Restrained boundary condition (ε paste = -140 µε) 5 Normalized Stored Energy (N*m) Paste-1 Elastic Cracked Strain Energy Density (N/m 2 ) (b) Free boundary condition (ε paste = -250 µε) Figure 5.17 Strain energy density level versus normalized stored energy

157 139 Figure 5.18 shows the changes of the cumulative normalized stored energy for the elastic condition and the cracking condition. It can be clearly seen that the energies at higher levels of a strain energy density were mainly released by cracking. The difference between the elastic and the cracking conditions represents the released energy by cracking. When the composite made using paste-1 was restrained it released about 45 percent of its total stored energy at failure (ε paste = -140 µε) (Figure 5.18 (a)). In the case of the free boundary condition, initial cracking occurred when the shrinkage strain of the paste was -150 µε and the microcracks increased until the simulation was stopped at -300 µε. Figure 5.18 (b), free boundary condition, shows about 33 percent release of the total stored energy at -250 µε of paste shrinkage. Therefore, by comparing the total stored energy in a composite for both simulation cases (elastic and cracking), it is possible to observe and quantify the cracking related energy changes of the composite systems.

158 Cumulation of Normalized Stored Energy (N*m) Energy Release Paste-1 Elastic Cracked Strain Energy Density (N/m 2 ) (a) Restrained boundary condition (ε paste = -140 µε) 200 Cumulation of Normalized Stored Energy (N*m) Paste-1 Elastic Cracked Energy Release Strain Energy Density (N/m 2 ) (b) Free boundary condition (ε paste = -250 µε) Figure 5.18 Strain energy density level vs. cumulative values of normalized stored energy

159 141 Figure 5.19 shows the relationships between the shrinkage strain of the paste and the cumulative normalized stored energy for a composite with different paste properties. In the case of the restrained boundary condition, the composite model which was made using paste-1 (which has a higher elastic modulus and tensile strength) stored more energy and released more energy at failure due to cracking (Figure 5.19 (a)). It was also observed that the composite failed at a higher paste shrinkage load when the material properties of the paste were higher (-140 µε for paste-1, -125 µε for paste-2, and -120 µε for paste-3) where paste-1 had the highest elastic modulus and strength and paste-3 had the lowest values. In the case of the free boundary condition, cracks began to occur when the shrinkage strain of the paste reached -150 µε for paste-1, -140 µε for paste-2, and -140 µε for paste-3. The cumulative normalized stored energy of the unrestrained specimen increased as microcracking occurred at the external boundaries of the aggregate (until the shrinkage strain reached -220 µε for paste-1, -210 µε for paste-2, and -200 µε for paste- 3). As the shrinkage of the paste continued to increase, the microcracks began to coalesce, resulting in a decrease in the stored strain energy.

160 142 Cumulation of Normalized Stored Energy (N*m) One Directionally Restrained Boundary Paste-1 (Elastic) Paste-2 (Elastic) Paste-3 (Elastic) Paste-1 (Cracked) Paste-2 (Cracked) Paste-3 (Cracked) Shrinkage Strain of Paste (µε) (a) Restrained boundary condition Cumulation of Normalized Stored Energy (N*m) Free Boundary Paste-1 (Elastic) Paste-2 (Elastic) Paste-3 (Elastic) Paste-1 (Cracked) Paste-2 (Cracked) Paste-3 (Cracked) Shrinkage Strain of Paste (µε) (b) Free boundary condition Figure 5.19 Shrinkage strain of paste vs. cumulation of normalized stored energy

161 143 Although the analytical approach using the energy of a composite (which is stored and released) is informative and important for cracking analysis, this approach cannot be used directly for quantifying the number of microcracks that may have formed. For example, a composite with paste-3 released lower energy (Figure 5.19 (b)) than the cases of paste-1 and paste-2 while there was a large number of microcracks in the case of the free boundary condition (Figure 5.13). To observe the relationship between the number of microcracks and the energy approach, the percentage of the energy that was released due to cracking and the area fraction of cracked zones were plotted (Figure 5.20) in the case of free boundary condition. As shown in Figure 5.20 (a), lower percentage of stored energy was released in the case of paste-1 while the magnitude of the stored energy and the released energy is higher (Figure 5.19 (b)). A similar tendency was observed in the case of the area fraction of cracked zones as shown in Figure 5.20 (b), where a smaller amount of microcracks developed in the case of paste-1 at the same magnitude of paste shrinkage. This result indicates that a composite with a paste phase that has a higher elastic modulus and tensile strength will have a smaller amount of microcracks and release a lower percentage of its stored energy, although the magnitude of the stored energy and the released energy is higher. From this observation, it can be noticed that the percent released energy would have a relationship with the number of microcracks (area fraction of cracked zones).

162 144 Released Energy (%) Free Boundary Condition Paste-1 Paste-2 Paste Shrinkage Strain of Paste (µε) (a) Percent released energy versus paste strain Area Fraction of Cracked Zones (%) Free Boundary Condition Paste-1 Paste-2 Paste Shrinkage Strain of Paste (µε) (b) Area fraction of cracked zones versus paste strain Figure 5.20 Relationships among the percentage of the released energy, area fraction of cracked zones, and the magnitude of shrinkage strain of paste (free boundary condition)

163 145 Figure 5.21 shows the relationship between the area fraction of cracks and the percentage of the released energy. The area fraction of cracks can be seen to have a single relationship with the percent-released energy regardless of the material properties of the paste phase in a composite. This result indicates that the degree of microcracking (percentage of microcracking) can be determined regardless of the change in the paste properties if the percentage of the released energy and its relationship with the degree of microcracking are known. Area Fraction of Cracked Zones (%) Free Boundary Condition Paste-1 Paste-2 Paste Released Energy (%) Figure 5.21 Relationship between the area fraction of cracked zones and the percentage of the released energy (free boundary condition)

164 Summary This chapter presented the application of a new approach for assessing the potential of microcracking in a concrete composite undergoing uniform paste shrinkage. The use of the NIST object-oriented finite element code enabled scanned twodimensional concrete images to be used to develop a finite element mesh to analyze composite materials on the meso-scale. The simulation results were analyzed using color contour images of stress and stored energy. It was shown that the external boundary condition plays a significant role in governing the cracking behavior of a composite. Externally-restrained composites exhibited localized sudden through-cracking while unrestrained composites showed distributed cracks with stable growth. It was also found that a composite with a stiffer (higher elastic modulus) cement paste increased the brittleness of the composite, although it delayed the paste strain that was required for initial cracking. It was found, in the case of free boundary condition, that there exists a single relationship between the percentage of released energy and the area fraction of cracked zones regardless of the variation of the material properties. This result indicates the possible analytical approach for quantifying the degree of microcracking of a composite using the parameter of the percent-released energy at cracking. Further observations are required to observe the influence of the aggregate volume fraction, the aggregate shape, the aggregate size distribution, and the bond condition between the cement paste and the aggregates to clearly understand the cracking behavior of concrete composites.

165 147 CHAPTER 6: PARAMETRIC STUDY OF SHRINKAGE CRACKING IN A CONCRETE COMPOSITE 6.1 Introduction The previous chapter demonstrated the applicability of performing a meso-scale finite element simulation using the OOF approach developed by NIST since it enables the use of actual 2-D images to generate the finite element mesh for the composite. It can be expected that the mixture proportions (namely the volume fraction of the aggregate, the size distribution of the aggregates, and the shape of the aggregate (shape of aggregate is not investigated in this study)) will influence the behavior of concrete composites. This chapter investigates the role of aggregate volume fraction, aggregate size, and bond between the aggregate and paste using the OOF simulation approach. The simulation results will be compared to both analytical approaches and experimental results. A discussion will be provided to describe how the OOF simulation technique could be used to estimate microcracking and cracking of concrete.

166 Importance of This Research No design tool currently exists which clearly considers the influence of material heterogeneity on the cracking behavior of concrete. Conventional design and analysis tools assume that the concrete behaves as a homogeneous material considering only the continuum response of concrete while ignoring microcracking. If the microcracking is not considered, the long-term durability of the concrete structure may not able to be accurately described. The OOF simulation technique was performed by considering the material at the meso-scale (i.e., mm) to enable the investigation of the microcracking in concrete. In order to develop a better design tool for evaluating the behavior of concrete, the key parameters need to be evaluated. In this research, five different parameters were considered including: 1) the external boundary condition, 2) the material properties (elastic modulus, strength), 3) the volume fraction of aggregate, 4) the size distribution of the aggregate, and 5) the bond condition between the paste and the aggregates. While previous chapter dealt with the material properties (paste property changes), this chapter will discuss the other parameters (the volume fraction of the aggregate, the size distribution of the aggregate, and the bond condition between the paste and the aggregates) considering two different boundary conditions (one directionally restrained boundary condition and free boundary condition).

167 One Directionally Restrained Boundary Condition Volume fraction of aggregate Theoretical Observation (Conventional Approach) The material properties of a composite are governed by the properties of the constituent materials. In the case of concrete (or mortar), tensile strength is mainly governed by the paste phase, which typically has lower tensile strength than that of the aggregates. Chapter 4 showed the conventional approach for calculating the equivalent elastic modulus of a composite using the Hansen s model (Eq. 4.4). Figure 6.1 shows the equivalent elastic modulus (as determined using Hansen s model) for a mortar that is composed of paste -1 and siliceous quartz aggregate as described in Tables 5.1 and 5.2.

168 150 Equivalent Elastic Modulus (GPa) E Agg. (50 GPa) E paste (20 GPa) Volume Fraction of Aggregate (%) Figure 6.1 Equivalent elastic modulus of a composite with the volume fraction of aggregate The equivalent shrinkage strain of a concrete composite can be estimated using Pickett s equation (Eq. 4.1 and 4.2). Figure 6.2 shows the equivalent shrinkage strains that may be experienced in a concrete when the uniform shrinkage strain of paste is -225 µε. For the calculations, the material properties of the mortar consisted of paste 1 and siliceous quartz aggregates as described in Tables 5.1 and 5.2.

169 Equivalent Strain (µε) ε paste (225 µε) Volume Fraction of Aggregate (%) Figure 6.2 Equivalent shrinkage strain of a concrete as a function of the volume fraction of aggregate Therefore, if the concrete specimen is restrained in one direction, the average sectional stress (equivalent stress) experienced in the concrete can be calculated by multiplying the equivalent elastic modulus by the equivalent shrinkage strain as illustrated in Figure 6.3 (assuming no creep). It can be seen that the equivalent stress that develops in the concrete (the product of strain and elastic modulus) decreases with increasing aggregate volume fraction.

170 152 5 Equivalent Stress (MPa) Volume Fraction of Aggregate (%) Figure 6.3 Equivalent stress of a concrete (ε paste = 225 µε) While this information is useful for evaluating the possible stress development in a concrete specimen, it does not provide information about the cracking potential of concrete because the strength of concrete also varies as a factor of the volume fraction of the aggregate. The tensile strength of a composite can be approximately evaluated using the series law (Reuss model) as illustrated in Figure 6.4 (a) (Daniel, 1994). If a composite is uniformly layered and the layers are perfectly bonded, the tensile strength of the composite can be calculated by Eq f ' t composite d E paste = Econcrete ε s E Agg paste ult Eq. 6.1

171 153 where, E concrete is the equivalent elastic modulus of concrete (Eq. 4.4) and ε paste-ult is the shrinkage strain of the paste phase at the failure stress. The parameters d and s are indicated in Figure 6.1 (a). If the aggregates are assumed to have a single size (diameter is d) and are to be placed following the square array rule (Figure 6.4 (b)), Eq. 6.1 can be modified, replacing d/s term by the volume fraction of aggregate (Eq. 6.2) f ' t composite 4 V Agg E paste = E concrete 1+ 1 ε π E Agg paste ult Eq. 6.2 Eq. 6.2 is valid when the volume fraction of aggregate is lower than 78.5 % because the maximum volume fraction of aggregate in the square array is 78.5 %. Pas te Aggre gate d d s s d (a) (b) Figure 6.4 Series law for the calculation of the tensile strength of a concrete composite: (a) layer array (b) square array (particle array)

172 154 Figure 6.5 shows an example of the variations of the strength and the equivalent stress of a concrete composite depending on the volume fraction of aggregate. As shown in Figure 6.5, the strength of a composite decreased when the volume fraction of aggregate increased. It should be noted, however, that the strength does not decrease as rapidly as residual stress. The equivalent stress and strength curves meet at approximately 9 percent of the volume fraction of aggregate when the paste strain was about -225 µε, as shown in Figure 6.5. From this investigation, theoretically, it can be said that a volume fraction of aggregate at a converging point is a critical volume fraction of aggregate with which concrete will have a higher cracking potential. 5 Equivalent Stress (MPa) Strength (Composite) ε paste = 225 µε ε paste = 150 µε ε paste = 100 µε Volume Fraction of Aggregate (%) Figure 6.5 Strength and equivalent stress of a composite with the variation of the volume fraction of aggregate (paste 1 in Table 5.1)

173 155 Therefore, it can be expected that the cracking potential of a composite will decrease with a higher or lower volume fraction of aggregate than the critical volume fraction of aggregate. In a practical point of view, higher volume fraction of aggregate is recommendable because the cement content for a concrete mixture can de reduced by increasing aggregate volume (economical). While Figure 6.5 shows the converging point at about 9 percent, it was also observed that the volume fraction of aggregate at converging point increases with lower material properties (especially the elastic modulus) of paste phase. Further investigations were performed using different material properties of the paste phase (paste 2 and paste 3 in Table 5.1) as shown in Figure 6.6. It can be seen that each composite had a converging point at different paste shrinkage (-225 µε for paste-1, -200 µε for paste-2, and -180 µε for paste-3). The composite with paste-1 has the highest paste strain which corresponds to the highest strength of a composite. This is mainly due to the fact that paste-1 has the highest paste strength than the others resulting in the increase of the composite strength. It was previously observed that higher elastic modulus of paste phase will increase the stress leading to the increase of cracking potential. Therefore, if the strengths of all paste phases are same, a concrete with higher elastic modulus will have higher cracking potential (cracking at lower paste shrinkage) while Figure 6.6 shows reverse results due to the change of strength. While this theoretical approach does not consider the stress concentration and localization in a composite system, it does illustrate that increasing the volume of aggregate reduces the potential for cracking.

174 Difference Between Strength and Stress (MPa) Paste - 1 at ε paste = 225 µε Paste - 2 at ε paste = 201 µε Paste - 3 at ε paste = 180 µε Volume Fraction of Aggregate (%) Figure 6.6 Cracking potential of composites based on the volume fraction of aggregate and material properties

175 OOF Simulations Seven different mortar images were prepared and used for this investigation. It should be noted that two images are artificially created images (Figures 6.7 (c) and (d)) where aggregates have been removed from an actual mortar image using photo editing tools to vary the volume fraction of aggregate. Each image has a different volume fraction of aggregate. Figure 6.7 shows the images used for this study. The first three images were obtained from mortar specimens that contained sieved fine aggregates. Figure 6.7 (a) is a pre-processed image from a mortar specimen, which consisted of fine aggregates that were larger than 4.75 mm. Figure 6.7 (b) is a mortar image with 3.35 mm ~ 4 mm fine aggregates, and Figure 6.7 (c) is a mortar image with 1.18 ~2.36 mm fine aggregates. These three mortar specimens had an aggregate volume of 55% (the images were observed to contain 61.8, 59.3, and 55.7 % of aggregates, respectively). Figure 6.7 (d) and (e) are artificial images obtained by deleting aggregate particles in the image of Figure 6.7 (c). Figures 6.7 (d) and (e) were prepared to observe the cracking behavior at a low volume fraction of aggregate. Figures 6.7 (f) and (g) are the preprocessed images from mortar specimens with 45% and 15% volume fraction of fine aggregates (the images were observed to contain 42.6 and 23.4 % of aggregates, respectively). The original sizes of images are 5 cm wide and 2.5 cm high for Figure 6.7 (a) and (b), and 2.54 cm side and 1.27 cm high for the other images. Simulations were performed increasing the shrinkage strain of paste under the horizontally restrained boundary condition. The material properties of paste-1 (Table 5.1) and the fine aggregate (Table 5.2) were used for the paste phase and aggregate phase, respectively.

176 cm 5 cm (a) V agg = 61.8 %, original size of image (5 cm x 2.5 cm) 2.5 cm 5 cm (b) V agg = 59.3 %, original size of image (5 cm x 2.5 cm) Figure 6.7 Images with different volume fraction of aggregate (continued)

177 cm 2.54 cm (c) V agg = 55.7 %, original size of image (2.5 cm x 1.27 cm) 1.27 cm 2.54 cm (d) V agg = 26.6 %, original size of image (2.5 cm x 1.27 cm) Figure 6.7 Images with different volume fraction of aggregate (continued)

178 cm 2.54 cm (e) V agg = 13.0 %, original size of image (2.5 cm x 1.27 cm) 1.27 cm 2.54 cm (f) V agg = 42.6 %, original size of image (2.5 cm x 1.27 cm) Figure 6.7 Images with different volume fraction of aggregate (continued)

179 cm 2.54 cm (g) V agg = 23.4 %, original size of image (2.5 cm x 1.27 cm) Figure 6.7 Images with different volume fraction of aggregate As a first step in performing the OOF simulations, a perfect-bond condition was assumed to exist between the aggregate and paste phase (with which bond phase has the same material properties as paste phase) and a brittle failure criterion applied to both the paste and aggregate. The details of OOF simulation procedures can be found in Chapter 5. Figure 6.8 shows the image of each mortar composite obtained from OOF simulation with the cracks that would exist in the material at the time of failure. Each mortar composite failed at a different magnitude of paste shrinkage strain.

180 162 (a) V agg = 61.8 % (failed at ε paste = µε) (b) V agg = 59.3 % (failed at ε paste = µε) Figure 6.8 Images of composites with cracks (one directionally restrained boundary condition continued)

181 163 (b) V agg = 55.7 % (failed at ε paste = µε) (d) V agg = 26.6 % (failed at ε paste = µε) Figure 6.8 Images of composites with cracks (one directionally restrained boundary condition continued)

182 164 (e) V agg = 13.0 % (failed at ε paste = µε) (f) V agg = 42.6 % (failed at ε paste = µε) Figure 6.8 Images of composites with cracks (one directionally restrained boundary condition continued)

183 165 (g) V agg = 23.4 % (failed at ε paste = µε) Figure 6.8 Images of composites with cracks (one directionally restrained boundary condition) Results of simulations performed on perfectly restrained specimens are shown in Figure 6.9. Figure 6.9 (a) shows that the composite with a higher aggregate volume had a greater increase of the average section stress, and failed at a lower mortar strain (lower equivalent strain of a composite). While each composite shows the noticeable change of the mortar shrinkage at failure, the shrinkage strain of the paste at failure did not change significantly (-120 ~ -145 µε) as a function of the volume fraction of aggregate (13.0 ~ 61.8 %) (Figure 6.9 (b)). This result indicates that, if the cracking potential of a composite is evaluated by the equivalent values (such as equivalent elastic modulus and strain of a composite), it seems a composite with a higher aggregate volume to have higher cracking potential. It is believed, however, that the potential for shrinkage cracking in actual concrete decreases by increasing the volume fraction of aggregate. This may be explained by the fact that, in many field applications, the concrete behavior

184 166 is considered on the homogeneous scale. As such, a concrete with a higher aggregate volume would exhibit less overall length change of the composite although the shrinkage of the paste may be the same as shown in Figure 6.9 (b).

185 167 Average Sectional Stress (MPa) Vagg = 61.8 % Vagg = 59.3 % Vagg = 55.7 % Vagg = 26.6 % Vagg = 13.0 % Vagg = 42.6 % Vagg = 23.4 % Shrinkage Strain of Mortar (µε) (a) Average Sectional Stress (MPa) Vagg = 61.8 % Vagg = 59.3 % Vagg = 55.7 % Vagg = 26.6 % Vagg = 13.0 % Vagg = 42.6 % Vagg = 23.4 % Shrinkage Strain of Paste (µε) (b) Figure 6.9 Average sectional stresses of composites vs. (a) shrinkage strain of mortar and (b) shrinkage strain of paste (one directionally restrained boundary condition)

186 168 In addition to observing the strain in the paste at failure, the change of stored energy in each composite was also recorded. Figure 6.10 (a) and (b) shows the stored strain energy of the composites as a function of the mortar shrinkage and as a function of the paste shrinkage. It can be observed that, as the aggregate volume increases, higher energy is stored in a composite with lower mortar shrinkage (Figure 6.10 (a)). However, it can be seen that, as the aggregate volume increases, the stored energy is reduced with the same paste strain (Figure 6.10 (b)). Once the shrinkage of the paste reaches a critical value, for each mixture, a sudden decrease in stored energy is observed. This sudden decrease in energy is consistent with the observation of a through cracking. It can be noted that composites did not release all of their stored energy upon failure (i.e., through cracking) (Figure 6.10 (a) and (b)). As discussed in Chapter 5, after the specimen fails, the internal restraint provided by the aggregate still exists while the energy that is stored due to the external restraint is released by cracking. Figure 6.10 (c) shows that, as a higher volume fraction of aggregate is used, the amount energy released by the development of a crack would be reduced. Figure 6.10 (d) shows the relationship between the percent released energy (i.e., the ratio of the energy released on cracking and the energy in the specimen without permitting cracking) and the area fraction of cracks after failure (i.e., area fraction of cracked zones). The area fraction of the cracked zone is primarily a measure of the area of the localized through cracks. Some distributed microcracks may also be included. It should be noted, however, that the area of microcracks was extremely small for these simulations. Therefore, we can think that the area fraction of cracked zone is similar to the width of the localized crack times the length of the localized crack. A linear

187 169 relationship exists between the percent energy released and the area of the cracks. This implies that a brittle specimen (i.e., a specimen which releases the majority of its energy during the formation of a localized crack) would have a larger crack width. This can be observed in experimental observations as well (Shah et al., 2004; Kim and Weiss 2003). From this result, it can be expected that a concrete composite with a higher volume fraction of aggregate releases less energy when through cracking occurrs. This is similar to the observation of Chariton et al. (2002) (Figure 6.10 (e)), where the energy released due to distributed cracking due to moisture loss and cracking around the aggregates needed to be seperated from the energy consumed during localized cracking. Stored Strain Energy (N*m) Vagg = 61.8 % Vagg = 59.3 % Vagg = 55.7 % Vagg = 26.6 % Vagg = 13.0 % Vagg = 42.6 % Vagg = 23.4 % Shrinkage Strain of Mortar (µε) Figure 6.10 (a) stored strain energy vs. shrinkage strain of mortar

188 170 Stored Strain Energy (N*m) Vagg = 61.8 % Vagg = 59.3 % Vagg = 55.7 % Vagg = 26.6 % Vagg = 13.0 % Vagg = 42.6 % Vagg = 23.4 % Shrinkage Strain of Paste (µε) Figure 6.10 (b) stored strain energy vs. shrinkage strain of paste Released Energy Due to Cracking (N*m) (e) (g) (d) (f) (a) Vagg = 61.8 % (b) Vagg = 59.3 % (c) Vagg = 55.7 % (d) Vagg = 26.6 % (e) Vagg = 13.0 % (f ) Vagg = 42.6 % (g) Vagg = 23.4 % (b) Volume Fraction of Aggregate (%) (c) (a) Figure 6.10 (c) energy release at failure vs. volume fraction of aggregate

189 171 Area Fraction of Cracked Zone (%) (a) Vagg = 61.8 % (b) Vagg = 59.3 % (c) Vagg = 55.7 % (d) Vagg = 26.6 % (e) Vagg = 13.0 % (f ) Vagg = 42.6 % (g) Vagg = 23.4 % (c) (b) (a) Released Energy (%) (f) (d) (g) (e) Figure 6.10 (d) area fraction of cracked zone vs. percent-released energy Figure 6.10 (e) Acoustic energy development as a function of time (Chariton et al., 2002)

190 Size Distribution of Aggregate It was observed from the simulations that the specimen (a), (b), and (c) in Figure 6.7 (V agg = 61.8, 59.3, and 55.7 percent, respectively) showed similar mechanical behavior (Figures 6.9 and 6.10). It should be remembered that specimen (a) consisted of aggregates larger than 4.75 mm, specimen (b) consisted of aggregates between 3.35 mm and 4.00 mm, and specimen (c) consisted of aggregates between 1.18 mm and 2.36 mm. Therefore, it could be said that the size of aggregate does not appear to have a significant influence on the cracking behavior of concrete. However, it was observed from the specimen with a well graded aggregates distribution (i.e., specimen (f) and (g) in Figure 6.7) (Vagg = 42.6 and 23.4 percent) that they showed through-cracking near the larger aggregates (Figures 6.8 (f) and (g)). This result indicates that there exists an effect of aggregate distribution on the cracking behavior of a composite. To better understand this behavior, additional simulations were performed. These simulations consisted of aggregates placed in the center of a sample. Five different composites were simulated and each sample contained a different number of aggregates although the volume fraction of the aggregate was maintained constant (Figure 6.11). Table 6.1 and Figure 6.11 describes the geometry of the aggregates that were used in each composite. The material properties used in these simulations are the same as those described in Table 5.1 and 5.2 (paste 1 and the aggregate (siliceous quartz)). Each composite was horizontally restrained and the simulations were performed by increasing the shrinkage strain of paste until the composites failed.

191 173 Table 6.1 Information on the volume of aggregate in each composite shown in Figure 6.11 Composite Agg. Agg. Total Agg. Radius (m) Area (m 2 ) Area (m 2 ) 1EA EA EA EA EA EA 2EA 3EA 4EA 5EA Figure 6.11 Composites with different number of aggregates maintaining the volume fraction of aggregate The simulations showed that the size of the aggregate did not have a significant role on the stresses that were developed or on the cracking potential. The shrinkage strain of the paste at failure was observed to decrease in a composite with smaller

192 174 aggregates (Figure 6.12 (a)). This, however, may be due to the influence of the size and number of the aggregates, where a composite with five small size aggregates has the largest aggregate chord length along the center cross section (Figure 6.12 (b)).

193 EA 2EA 3EA 4EA 5EA Stress (MPa) Shrinkage Strain of Paste (µε) (a) 1.6 Aggregate Chord Length at the Critical Section (m) EA 2 EA 3EA 4EA 5EA Specimen (b) Figure 6.12 Stress development in composites: (a) stress development and (b) aggregate chord length

194 176 However, these results do not explain why the concrete composites with distributed aggregates particles would show through crackings around the larger aggregate (as shown in Figures 6.8 (f) and (g)). To better explain this observation, it is necessary to understand how the stresses develop in a composite system when it is restrained from shrinking freely. Figure 6.13 illustrates the sectional stresses that develop in a composite system when the paste shrinks. It should be noted here that the (average) sectional stress should be the same for any vertical section that would be taken in the composite. Due to the existence of aggregates, more stress will develop locally (at the surface of the aggregate) in the paste phase as discussed in Chapter 4. As such, it can be imagined that in regions of the specimen, where coarser (larger) aggregate exists, a greater potential exists for larger stress concentration. As such, sections with a greater aggregate sizes would have a high potential to become the critical section where failure would occur. Figure 6.13 Sectional stress development in a composite under the restrained boundary condition

195 177 In an effort to determine the volume fraction of aggregate at each section, ImagePro Plus TM software was used. Figure 6.14 shows the ratio of the total chord length of the aggregates (L agg ) to the total length of the specimen height (Length) (i.e., sectional ratio) along the entire specimens in the case of mortars with well graded aggregates (V agg = 42.6 and 23.4 %). As shown in Figure 6.14, the major of throughcracking occurred near the larger aggregate where the sectional ratio of the aggregate chord length was high (even though it was not always the highest). It should be noted that further work may be needed to determine whether this is driven by the aggregate size or solely the clump of aggregates.

196 Lagg/Length Dist ance (cm) (a) V agg = 42.6 % Figure 6.14 The sectional ratio of the total chord length of the aggregates to the height of the concrete (continued)

197 179 Lagg/Length Distance (cm) (b) V agg = 23.4 % Figure 6.14 The sectional ratio of the total chord length of the aggregates to the height of the concrete

198 180 While larger aggregates would seem to increase the cracking potential of a concrete, the use of larger aggregates are frequently preferred in the field due to the potential benefit for decreasing the paste content of the concrete. This work does not contradict that observation, rather it points to the need for uniformity in the aggregate distribution as clumping of larger aggregates appears to result in stress localization, Bond Condition A cracking analysis was performed assuming that the paste and aggregates were perfectly bonded. The perfect bond condition consists of a specimen where the material properties of the bond zone were the same as those of the bulk paste. It is typically believed, however, that a bond phase exists (interfacial transition zone (ITZ)) which contains higher porosity, resulting in a lower elastic modulus and lower strength than those of bulk paste. Scrivener and Gariner (1988) observed that the mechanical properties, such as strength and stiffness, decrease about three times at the ITZ, and the ITZ has a width of about 50 µm. As described in Chapter 5, the scanned image had a 600 dpi resolution so one pixel is 42 µm wide. Therefore, by ordering an additional one-pixel phase around the aggregate phases through the pre-processing (Chapter 5), simulations considering the effect of bond phase can be performed. To investigate the effect of the bond phase on the cracking behavior, four different material properties (25%, 50%, 75%, and 100% of the elastic modulus and tensile strength of bulk paste) were used to simulate the properties of the bond phase. Poisson s ratio was assumed to be constant (0.18) for these simulation. For this

199 181 investigation, a single mortar image (V agg = 55.7 %) was used and the material properties of the bulk paste and aggregates were not changed (for bulk paste, paste-1 in Table 5.1 and for aggregate, in Table 5.2). Figure 6.15 shows the images of the cracked composites at failure when the bond condition was varied in the case of one directionally restrained boundary condition. The composite with the lower material properties of the bond phase failed at a lower magnitude of paste shrinkage strain (the 25 % bond case failed at ε paste = -85 µε, the 50% bond case at ε paste = -100 µε, the 75% bond case at ε paste = -120 µε, and the 100 % bond case at ε paste = -140 µε).

200 182 (a) material properties of bond phase (25% of paste phase) (failed at ε paste = -85 µε) (b) material properties of bond phase (50% of paste phase) (failed at ε paste = -100 µε) Figure 6.15 Images of cracked composites at failure (one directionally restrained boundary condition - continued)

201 183 (c) material properties of bond phase (75% of paste phase) (failed at ε paste = -120 µε) (d) material properties of bond phase (100% of paste phase) (failed at ε paste = -140 µε) Figure 6.15 Images of cracked composites at failure (one directionally restrained boundary condition) Figure 6.16 (a) shows the stored strain energy in a composite with different bond conditions. As expected, the case with the bond phase which has 25 % of paste properties shows the lowest stored energy at the same magnitude of paste shrinkage strain. The specimen with a weaker bond failed at a much lower paste shrinkage strain

202 184 than the specimen with stronger bond. It was also observed that more energy was released from a concrete which has the higher material properties of the bond phase (Figure 6.16 (b)).

203 185 Stored Strain Energy (N*m) Bond Phase 25 % 50 % 75 % 100 % Shrinkage Strain of Paste (µε) Energy Release Due to Cracking (N*m) (a) Ratio of bond material properties to paste material properties (%) (b) Figure 6.16 Cracking behavior of a mortar composite (V agg = 55.7 %) with different bond conditions (one directionally restrained boundary condition): (a) stored strain energy vs. shrinkage strain of paste and (b) energy release due to cracking vs. bond condition

204 186 From this investigation, it can be said that the bond condition has an important role on the cracking potential of a composite in the case of the restrained boundary condition. As such future research is needed to better quantify the properties of bond that should be used in the simulations. All of the simulations were performed assuming that each phase (paste, aggregate, and bond phases) act as a perfectly brittle material. However, cement paste is a viscoelastic material. It is expected that the visco-elastic behavior at the interface between the paste and the aggregates will alter the behavior of the composite. To observe how this cracking behavior changes, a perfectly plastic failure criterion was applied to the bond phase and additional simulations were performed. Similar to the trials described in previous session, the mortar composite with a 55.7 % volume fraction of aggregate was used for the simulations and the material properties of paste-1 and fine aggregate were used (in Tables 5.1 and 5.2, respectively). The bond phase had the same material properties of those of the paste phase up to the maximum tensile stress, but the perfectly plastic condition was applied when the stress exceeded the maximum tensile stress. Figure 6.17 shows the cracked mortar composite at failure (which failed when ε paste = -190 µε) in the case of one directionally restrained boundary condition. Compared to the perfectly brittle case (Figure 6.15 (d)), which failed at -140 µε of paste shrinkage strain, failure occurred at a higher paste shrinkage strain and more distributed cracks (microcracks) were observed. It can be expected that, if lower material properties are applied to the bond phase with the perfectly plastic failure criterion, there will be more distributed cracks (microcracks) at the interfaces.

205 187 In the case of the perfectly plastic failure criterion for the bond phase, distributed crackings occurred at the phase boundary, and through-cracking followed when the elements in the bulk paste phase exceeded the maximum tensile stress. Compared to the perfectly brittle case (Figure 6.15 (d)), the through-cracking zones appear to be wider (area fraction of cracked zones: 3.37 % (c.f. perfectly brittle: 1.89%)), but it is because of the perfectly plastic failure condition at the bond phase, which does not release energy at the cracking points, resulting in a transfer of the excessive stresses to the nearby paste phases and eventual failure. Figure 6.17 Image of the cracked mortar composite (V agg = 55.7 %) at failure (ε paste = -190 µε) (one directionally restrained boundary condition with perfectly plastic failure criterion for bond phase)

206 188 Figure 6.18 (a) shows the stored energy in a composite with different failure criteria for the bond phase. The specimen with the perfectly plastic failure criterion for the bond phase failed at a higher shrinkage strain of paste than the perfectly brittle failure criterion. Figure 6.18 (b) shows the energy release due to cracking. Even though the magnitude of the energy release for the case of the perfectly plastic failure criterion for the bond phase is not appreciable, successive increases of energy release were observed before failure (Figure 6.18 (c)). These release of energy can be attributed to more cracking at the interface between aggregate and paste. 200 Stored Strain Energy (N*m) Bond Phase Perfectly Plastic Perfectly Brittle Shrinkage Strain of Paste (µε) (a) stored strain energy vs. shrinkage strain of paste Figure 6.18 Cracking behavior of a mortar composite with different failure criterion on the bond phase (one directionally restrained boundary condition - contiuned)

207 189 Energy release due to cracking (N*m) Bond Phase Perfectly Plastic Perfectly Brittle Shrinkage Strain of Paste (µε) (b) energy release due to cracking vs. volume fraction of aggregate Energy release due to cracking (N*m) Bond Phase Perfectly Plastic Perfectly Brittle Shrinkage Strain of Paste (µε) (c) energy release due to cracking vs. volume fraction of aggregate (range up to 5 N. m) Figure 6.18 Cracking behavior of a mortar composite with different failure criterion on the bond phase (one directionally restrained boundary condition)

208 Free Boundary Condition Volume fraction of aggregate Figure 6.19 shows the images of each mortar after the application of -250 µε of paste shrinkage strain for the free boundary condition simulations (paste 1 and the aggregates shown in Tables 5.1 and 5.2). Mortar with a lower volume fraction of aggregate can be seen to exhibit less cracking. In the case of the free boundary condition, the aggregates provide internal restraint against paste shrinkage. (a) V agg = 61.8 % (at ε paste = -250 µε) Figure 6.19 Images of composites with cracks (free boundary condition, ε paste = -250 µε - continued)

209 191 (b) V agg = 59.3 % (at ε paste = -250 µε) (c) V agg = 55.7 % (at ε paste = -250 µε) Figure 6.19 Images of composites with cracks (free boundary condition, ε paste = -250 µε - continued)

210 192 (d) V agg = 42.6 % (at ε paste = -250 µε) (e) V agg = 23.4 % (at ε paste = -250 µε) Figure 6.19 Images of composites with cracks (free boundary condition, ε paste = -250 µε) Figures 6.20 (a) and (b) show the relationship between the area fraction of the cracked zones and the paste strain. A composite with a higher volume fraction of aggregate would have more microcracks due to the higher number of restraint points (aggregates) and the higher bond area. With the increase of paste strain, the microcracks begin at the bond regions and grow into the bulk paste.

211 193 Figure 6.21 shows the cracking behavior in concrete under free shrinkage using the energy approach. Figure 6.21 shows that a composite with higher volume fraction of aggregate had higher stored energy at the same mortar shrinkage and started microcracking at lower mortar shrinkage. Figure 6.21 (b) shows the changes of the stored energy in composites with increasing the shrinkage strain of paste, and Figure 6.21 (c) illustrates that more energy was released with a higher volume fraction of aggregate.

212 194 Area Fraction of Cracked Zones (%) Area Fraction of Cracked Zones (%) Vagg = 61.8 % Vagg = 59.3 % Vagg = 55.7 % Vagg = 42.6 % Vagg = 23.4 % Shrinkage Strain of Paste (µε) (a) Vagg = 61.8 % Vagg = 59.3 % Vagg = 55.7 % Vagg = 42.6 % Vagg = 23.4 % Shrinkage Strain of Paste (µε) (b) Figure 6.20 Relationship between the area fraction of cracked zones and the shrinkage strain of paste (free boundary condition): (a) total range up to 300 µε and (b) range up to 250 µε

213 195 Stored Strain Energy (N*m) Vagg = 61.8 % Vagg = 59.3 % Vagg = 55.7 % Vagg = 42.6 % Vagg = 23.4 % Shrinkage Strain of Mortar (µε) (a) stored strain energy vs. shrinkage strain of mortar Stored Strain Energy (N*m) Vagg = 61.8 % Vagg = 59.3 % Vagg = 55.7 % Vagg = 42.6 % Vagg = 23.4 % Shrinkage Strain of Paste (µε) (b) stored strain energy vs. shrinkage strain of paste Figure 6.21 Cracking behavior of mortar composites (free boundary condition cont.)

214 196 Energy release due to cracking (N*m) Vagg = 61.8 % Vagg = 59.3 % Vagg = 55.7 % Vagg = 42.6 % Vagg = 23.4 % Shrinkage Strain of Paste (µε) (c) energy release due to cracking vs. volume fraction of aggregate Figure 6.21 Cracking behavior of mortar composites (free boundary condition) Figure 6.22 (a) shows the relationship between the area fraction of cracked zones and the energy that is released. Figure 6.22 (b) shows the relationship between the area fraction of cracked zones and the percent-released energy. As previously observed in Chapter 5, the relationship between the area fraction of cracked zones and the percentreleased energy does not change when the material properties of the phases were varied. Therefore, it can be said that the change of the relationship between the area fraction of cracked zones and the percent-released energy for each composite in Figure 6.22 (b) is mainly dependent on the volume fraction of aggregate.

215 197 Area Fraction of Cracked Zones (%) Area Fraction of Cracked Zones (%) Vagg = 61.8 % Vagg = 59.3 % Vagg = 55.7 % Vagg = 42.6 % Vagg = 23.4 % Released Energy (N*m) (a) Vagg = 61.8 % Vagg = 59.3 % Vagg = 55.7 % Vagg = 42.6 % Vagg = 23.4 % Released Energy (%) (b) Figure 6.22 Relationship between the area fraction of cracked zones and the energy release (Free boundary condition): (a) released energy and (b) percent-released energy

216 198 From Figure 6.20, it can be said that a concrete with higher volume fraction of aggregate will have more microcracks in the free shrinkage condition. While the cracking potential or the magnitude of the microcracking of a composite will increase with a higher volume fraction of aggregate, it should be noticed that the equivalent shrinkage strain (global strain) of a composite will also vary with the volume fraction of aggregate. It was observed in that the equivalent strain of a composite decreases with increasing the volume fraction of aggregate, which is desirable for the stability of a structure. Figure 6.19 (d) and (e) clearly showed that microcracks developed primarily where coarser (larger) aggregates are located, since larger aggregates provide higher restraint to the paste deformation in the free boundary condition.

217 Bond Condition Figure 6.23 shows the images of the mortar when a free shrinkage of the paste (ε paste = -250 µε) is applied with a different bond conditions. A single composite image (V agg = 55.7 %) was used and four different material properties were applied to the bond phase as previously described in Section The simulations with bond phase which had lower elastic modulus and strength showed a larger area fraction of cracked zones (Figure 6.24) because of the lower strength of the bond phase resulting in the increased cracking at the interfaces between the paste and the aggregates in a composite system.

218 200 (a) material properties of bond phase (25% of paste phase) (at ε paste = -250 µε) (b) material properties of bond phase (50% of paste phase) (at ε paste = -250 µε) Figure 6.23 Images of cracked composites at ε paste = -250 µε (free boundary condition - continued)

219 201 (c) material properties of bond phase (75% of paste phase) (at ε paste = -250 µε) (d) material properties of bond phase (100% of paste phase) (at ε paste = -250 µε) Figure 6.23 Images of cracked composites at ε paste = -250 µε (free boundary condition)

220 202 Area Fraction of Cracked Zones (%) Bond Phase Properties 25 % 50 % 75 % 100 % ( = paste - 1) Shrinkage Strain of Paste (µε) Figure 6.24 Volume fraction of cracked zones depending on the ratio of bond material properties to paste material properties Figure 6.25 (a) shows the stored energy in a composite with different bond conditions. The composites with lower bond properties can be seen to store less energy as cracked at lower paste shrinkage strain. From Figure 6.25 (b), it can be seen that a concrete with lower bond properties began microcracking at a lower paste shrinkage. However, the rate of energy release (slope in Figure 6.25 (b)) was lower for the specimen with poor bond. Therefore, it can be seen that, at lower values of paste shrinkage, higher energy was released from a concrete with lower bond material properties but more energy was released from a concrete with higher bond material properties at larger paste shrinkage.

221 203 Stored Strain Energy (N*m) Bond Phase 25 % 50 % 75 % 100 % Shrinkage Strain of Paste (µε) (a) stored strain energy vs. shrinkage strain of paste Energy release due to cracking (N*m) Bond phase 25 % 50 % 75 % 100 % Shrinkage Strain of Paste (µε) (b) energy release due to cracking vs. bond condition Figure Cracking behavior of a mortar composite (V agg = 55.7 %) with different bond conditions (free boundary condition)

222 204 Figure 6.24 illustrates that the relationship between the area fraction of cracked zones and the percent of the stored energy that was released does not change significantly as the material properties of the bond phase are varied. Area Fraction of Cracked Zones (%) Bond Phase 25 % 50 % 75 % 100 % Released Energy (%) Figure 6.26 Relationship between the area fraction of cracked zones and the percent released energy (bond phase, free boundary condition) Figure 6.27 shows the response of the composite with the two different failure criteria (i.e., the perfectly brittle failure criterion and perfectly plastic failure criterion) for the bond phase. A smaller amount of cracking was observed for the material with the brittle failure criterion (area fraction of cracked zones: perfectly plastic at 2.19 %, and perfectly brittle at 3.14 % at -250 µε of paste strain) at lower paste strain, but the

223 205 difference decreased with higher paste shrinkage (Figure 6. 28). It should be noted that, while the bond properties significantly influence the restrained system (Figure 6.15, 6.16, 6.17), they are less significant in the free shrinkage specimens. Figure 6.27 Cracked image of a mortar composite (V agg = 55.7 %) (ε paste = -250 µε) (free boundary condition with perfectly plastic failure criterion for bond phase)

224 206 Area Fraction of Cracked Zones (%) Failure Criterion (Bond Phase) Perfectly Plastic Perfectly Brittle Shrinkage Strain of Paste (µε) Figure 6.28 Area fraction of cracked zones vs. shrinkage strain of paste Figure 6.29 shows the change in the stored energy and the energy release due to cracking for the perfectly plastic failure criterion and the perfectly brittle failure criterion. As can be seen, by applying the perfectly plastic failure criterion to the bond phase, a smaller amount of energy is released because the toughness of the composite increased and more stresses were transferred through the bond phase noticeably.

225 207 Stored Strain Energy (N*m) Failure Criterion (Bond Phase) Perfectly Plastic Perfectly Brittle Shrinkage Strain of Paste (µε) (a) stored strain energy vs. shrinkage strain of paste Energy release due to cracking (N*m) Failure Criterion (Bond Phase) ITZ: Perfectly Plastic ITZ: Perfectly Brittle Shrinkage Strain of Paste (µε) (b) energy release due to cracking vs. volume fraction of aggregate Figure 6.29 Cracking behavior of a mortar composite with different failure criterion on the bond phase (free boundary condition)

226 208 Figure 6.30 shows the relationship between the area fraction of cracked zones and the percent-released energy. This relationship, as discussed, is independent of the material properties of the phases in the perfectly brittle failure criterion as illustrated in Figures 5.26 and However, in Figure 6.30, this relationship can be seen to change depending on the failure criterion. Therefore, it can be concluded that the relationship between the area fraction of cracked zones and the percent-released energy is dependent not only on the volume fraction of aggregate but also on the failure criterion of the phases. Area Fraction of Cracked Zones (%) Failure Criterion (Bond Phase) Perfectly Plastic Perfectly Brittle Released Energy (%) Figure 6.30 Relationship between the area fraction of cracked zones and the percent released energy

227 209 From this investigation, it can be concluded that the cracking behavior and cracking potential of a composite will be highly affected by the condition of the bond phase. Knowledge about the mechanical properties of the bond phase are still limited though, so further research is required to apply the proper material properties to the bond phase to get more realistic results from simulations.

228 Discussion By performing the series of OOF simulations, the cracking behavior of concrete composites was observed for two different boundary conditions. In the case of a restrained boundary condition, it was discussed that, theoretically, there exists a critical volume fraction of aggregate that has the highest cracking potential under the shrinkage condition. From a practical point of view, a large enough volume fraction of aggregate was recommended to reduce the potential for cracking. It was found that the role of aggregate size is less important. However, cracks form at the coarser (larger) aggregates especially when they clump together as such proper aggregate distribution would be beneficial to reduce the through-cracking potential. In the case of the free boundary condition, it was shown from the simulations that a concrete with a higher volume fraction of aggregate has more distributed cracks (microcracks). It was observed that the bond condition between aggregate and paste has an important role on the cracking potential and the behavior of a composite for the externally restrained boundary conditions. Further research is expected using proper material properties for the bond phase. While this chapter provided a discussion of the cracking behavior of concrete, it should be noted that all of the simulations performed were based on linear elastic material behaviors. The material properties of cement paste change with time due to cement hydration (aging), and the paste experiences creep under loading time. The magnitude of the shrinkage strain of the paste is a function of time. Therefore, to improve the simulation of concrete composites, time should be included. To incorporate creep to the

229 211 meso-scale OOF simulations, a creep function is needed in which the microcracking and bond effects are separated as the conventional creep models assume concrete to be a homogeneous material.

230 212 CHAPTER 7: SUMMARY 7.1 Introduction This study was composed of two parts. First, the study focused on developing a methodology that would enable data from the restrained ring test to be used to quantify the internal stress development in a concrete specimen when non-uniform shrinkage occurs due to moisture gradients. Chapter 2 focused on the development of an analytical procedure that could be used to compute the complex residual stress fields that develop in the case of non-uniform drying due to time-dependent moisture diffusion. Chapter 3 provided an approach to quantify the degree of restraint that is experienced in restrained ring specimens as the geometry of the concrete and steel ring are varied. The second part of this study focused on adapting a finite element simulation technique for the evaluation of microcracking and cracking behavior that is caused by the heterogeneous nature of concrete when the paste and aggregate are considered as two separate phases. It was shown that the cracking behavior was highly dependent on both the external restraint provided by the boundary condition and the internal restraint that was provided by the aggregates. Chapter 4 illustrated that, by assuming that concrete behaves as a homogeneous material, areas of high residual stress that increase the cracking potential in concrete may not be considered. A meso-scale finite element simulation technique was implemented to enable actual images of concrete to be directly used for simulations as

231 213 described in Chapter 5. Chapter 6 described the influence of many material properties on cracking from the results of the series of OOF simulations. 7.2 Quantification of the Residual Stress Development in Concrete Due to Non-Uniform Shrinkage and the Degree of Restraint Exhibited by the Restrained Ring Test When Different Specimen Geometries are Used Chapter 2 described the development of an analytical approach for calculating the residual stresses that develop in a restrained concrete ring under circumferential drying. Equation 7.1 (or Eq. 2.13) provides the ability to compute the stress that develops as a function of two components: 1) external restraint due to the steel ring and 2) self-restraint due to moisture gradients. σ θθ ( r, γ ) = σ θθ, rest. ring + σ θθ, diff. shr ROS RIS = R ε steel ( t) ES ( ROC ROS ) r OC + r R + R 2 2 ε SH const Econ IC 2 ( f ( R ) f ( R )) + f ( r) f ( R ) erfc( A) r OC IC IC r OC RIC Eq. 7.1 where, R OC is the outer radius of the concrete, R IC is the outer radius of the steel (inner radius of the concrete), E con is the effective elastic moduli of concrete, E Steel is the elastic modulus of the steel, ε Steel is the strain of steel ring (which can be measured from the attached strain gages), ε SH-const is the free shrinkage of the concrete, r is the location in the

232 214 concrete ring where the stresses are being calculated, and erfc is the complementary error function, f ( r ) = 1 erfc ( A) A 2 R OC + erfc ( A) γ A + 2 A π A 4 e + π erf ( A) R e 2 2 OC γ 2 2 A γ Eq. 7.2 where A = (R OC r) / γ, and = 2 D t γ where D is the diffusion coefficient of concrete and t is the time of drying that has been experienced when the calculation is performed. To utilize the restrained ring test method for the simulation of an actual field condition, it is essential that an appropriate ring geometry should be determined prior to performing the experiment (ASTM C recommends a single geometry which typically develops about 70 % of the degree of restraint to the concrete ring). The geometry of the ring test that should be selected for a particular application should be designed to satisfy the desired degree of restraint, match the actual drying conditions in the field, and ensure proper strain measurements to enable accurate stress calculations. An analytical procedure was presented in Chapter 3 to determine the degree of restraint in the restrained ring test for concrete under uniform shrinkage (Eq. 3.6 or 7.3). It was shown that the degree of restraint depends only on the geometry of the rings and the material properties (namely, the elastic modulus and the Poisson s ratio (ν)) of the concrete and the steel, regardless of the drying condition. ψ = 1 ' E C 1 Eq. 7.3 E 2 2 S R R IS OC 1 (1 + ν + ν C ) (1 C ) ' E R R OS C OS 2 2 E S R OC R IS 1 (1 + ν ) + ν S (1 S ) R OS R OS

233 215 The degree of restraint increases when the thickness and elastic modulus of the restraining ring (steel) are increased and the degree of restraint decreases with increasing the thickness of the concrete ring. In this study, the analytical solutions were compared with the results from finite element analysis and reasonable agreements were shown. The use of the developed solutions will enable the restrained ring test to be used for quantifying the potential development of the residual stresses induced by non-uniform shrinkage (drying) in concrete structures with the consideration of the degree of restraint. Further research should be directed at studying the effect of ring height as a ring with an increased height would induce bending to thin and tall steel rings (resulting in the non-uniform deformation of concrete along the height direction) for which bending is expected to be more significant.

234 Discussion of the Comparison between OOF Simulation Results in a Heterogeneous Material and Experimental Measurements To illustrate some of the potentially interesting research findings that can be obtained from the use of the OOF modeling approach, some of the modeling observations are presented and compared with experimental behavior of mortar and paste specimens. For comparison, the OOF simulation data in Section was used, and these simulations were performed using the material properties (paste 1 and aggregate in Tables 5.1 and 5.2). First, the simulation results were compared with the theoretical solution provided by Pickett (1956) and Hansen (1965). Figure 7.1 shows the relationship between the shrinkage strain of the paste and shrinkage strain of mortar obtained from the OOF simulations (assuming elastic-non-cracking condition) and the analytical equation based on Pickett s model (Eq. 4.1 and 4.2). As can be seen, the results of Pickett s model match well with the OOF simulation results (i.e., the results always within 6 % of the Pickett s model).

235 217 Shrinkage Strain of Mortar (µε) Pickett (Theory) Elastic (Simulation, Vagg = 61.8 %) Elastic (Simulation, Vagg = 42.6 %) Elastic (Simulation, Vagg = 23.4 %) Shrinkage Strain of Paste (µε) Figure 7.1 A comparison of the shrinkage strain of mortar and the shrinkage strain of paste as (OOF simulation, elastic) Figure 7.2 shows the free shrinkage strain of mortars considering both the elastic response and the response if cracking is permitted. The mortars begin to develop microcracking at a free shrinkage strain in the paste of approximately -150 µε. These microcracks cause the shrinkage of mortar composites to begin to plateau even though the shrinkage in the paste continues to increase.

236 218 Shrinkage Strain of Mortar (µε) Elastic (Simulation) Cracked (Simulation, Vagg = 61.8 %) Cracked (Simulation, Vagg = 42.6 %) Cracked (Simulation, Vagg = 23.4 %) Microcracking Shrinkage Strain of Paste (µε) Figure 7.2 Relationship between the shrinkage strain of mortar and the shrinkage strain of paste (OOF simulation, free boundary condition) Figure 7.3 illustrates how the area of cracking increases (a) and how the energy is released (b) in the mortar specimens as a function of the shrinkage of the mortar (i.e., composite) and paste (i.e., driving force). It shows that, at the beginning of the simulation, the shrinkage of mortar increased without cracking. It was observed that limited cracking began to occur when the paste shrinkage reached -150 µε (Figure 7.3 (a) and (c)). However, it should be noted that, after the mortar developed a cracked area of at least 0.2%, cracking and shrinkage can occur without a substantial increase in the shrinkage of the composite (Figure 7.3 (b) and (d)). This illustrates that a mortar or concrete could not continue to shrink or even expand when microcracks develop as

237 219 shown in Figures 7.2 and 7.3 (mortar with V agg = 61.8 %). It should be noted, however, that further work is needed to confirm this with experimental measurements. Energy Release Due to Cracking (N*m) Vagg = 61.8 % Vagg = 42.6 % Vagg = 23.4 % Shrinkage Strain of Mortar (µε) (a) energy release vs. shrinkage strain of mortar Figure 7.3 Energy release and Area fraction of cracked zones versus shrinkage strain of mortar and paste phase (OOF simulation, free boundary condition - continued)

238 220 Energy Release Due to Cracking (N*m) Vagg = 61.8 % Vagg = 42.6 % Vagg = 23.4 % Shrinkage Strain of Paste (µε) (b) energy release vs. shrinkage strain of paste phase Area Fraction of Cracked Zones (%) % Vagg = 61.8 % Vagg = 42.6 % Vagg = 23.4 % Shrinkage Strain of Mortar (µε) (c) area fraction of cracked zones vs. shrinkage strain of mortar Figure 7.3 Energy release and Area fraction of cracked zones versus shrinkage strain of mortar and paste phase (OOF simulation, free boundary condition - continued)

239 221 Area Fraction of Cracked Zones (%) Vagg = 61.8 % Vagg = 42.6 % Vagg = 23.4 % Shrinkage Strain of Paste (µε) (d) area fraction of cracked zones vs. shrinkage strain of paste Figure 7.3 Energy release and Area fraction of cracked zones versus shrinkage strain of mortar and paste phase (OOF simulation, free boundary condition) To compare the simulation results with the experimental behavior of mortar and paste specimens three different water to cement ratio materials were used (0.30, 0.35, and 0.40). The mortar mixtures had 55% fine aggregate (by volume). The specimens were prepared using the procedure described in (Pease, 2005) with a 2.5 cm wide, 2.5 cm high, and 30 cm long specimen (Figure 7.4). Shrinkage measurements were performed using a non-contact laser on paste and mortar specimens from the time of initial set to 24 hours after placing (Pease et al., 2005). In addition to measuring the length changes, the internal damage was detected using acoustic emission. Puri (2003) showed that the acoustic energy has a linear relationship with the fracture energy density (i.e., the energy release due to cracking). To

240 222 detect the internal damage (i.e., microcracking) in mortar specimens caused by autogenous shrinkage, passive acoustic emission testing was used for both a restrained shrinkage test and a free shrinkage test (Figure 7.4, Pease et al., 2004). The acoustic emission sensors were placed on the specimen at the time of initial setting (3 hours for w/c = 0.30, 4 hours for w/c = 0.35, and 5 hours for w/c = 0.40). Further details about the passive acoustic emission testing can be found in (Chariton and Weiss, 2002; Kim and Weiss, 2003; Pease, 2005). Figure 7.4 Restrained (a) and free (b) shrinkage specimen geometries with locations of acoustic emission sensors (Pease et al., 2004) Figure 7.5 (a) shows the autogenous shrinkage strain measured in a paste specimen with a w/c of 0.30 and the shrinkage strain measured in a mortar specimen after