3D LATTICE MODELING OF TENSILE TEST ON SIMULATED CEMENT PASTE AT MICROSCALE

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1 3D LATTICE MODELING OF TENSILE TEST ON SIMULATED CEMENT PASTE AT MICROSCALE Zhiwei Qian (1), Guang Ye (1, 2), E. Schlangen (1) and K. van Breugel (1) (1) Microlab, Faculty of Civil Engineering and Geosciences, Delft University of Technology, the Netherlands (2) Magnel Laboratory for Concrete Research, Department of Structural Engineering, Ghent University, Technologiepark-Zwijnaarde 904 B-9052, Ghent (Zwijnaarde), Belgium Abstract Looking into materials is always a dream for scientists and engineers in the field of materials science. Plenty of research activities have been performed to approach this dream, including numerical simulations and lab experiments. In this paper, a tensile test on cement paste is to be simulated virtually on computer. But it is different from previous attempts because we are working at a lower scale, i.e. the micro scale. Although there is a rich set of modeling techniques available to fulfill the task, it is decided to apply 3D lattice model rather than continuum finite element model to carry out the simulation because lattice is a discretized model and it would be more preferable to simulate fracture process at micro scale. The cement paste used in the numerical experiment is virtually simulated by the cement hydration and microstructure formation model HYMOSTRUC3D. The simulation results reflect the ability of lattice modeling techniques and the addressed principles can be used potentially to produce a complete multiscale simulation. 1. INTRODUCTION In the field of material science, it is believed that the microstructure determines its global performance. Scientists keep on exploring the fundamentals of materials throughout the history. Plenty of models have been proposed to help us look into materials at different scales, namely macro, meso, micro and even nano scales. This paper attempts to simulate a tensile test on cement paste at microscale using the 3D lattice model. The concept "lattice" was first introduced into the filed of fracture mechanics in [1] and the corresponding model was named Lattice Fracture Model. However, it was restricted to 2D analysis of concrete at meso scale, although the principles must also hold for other configurations. Afterwards the model was expanded into 3D domain successfully [2, 3]. In this paper, the 3D lattice model is applied at a lower scale, namely micro scale. The relationship of mechanical properties between cement hydration product CSH (Calcium 1161

2 Silicate Hydrate) gel and cement paste is defined, more specifically, the tensile strength and Young's modulus of cement paste can be calculated based on its microstructure and mechanical properties of the corresponding components, such as CSH outer product, CSH inner product and unhydrated cement paste. 2. HYDRATION AND MICROSTRUCTURE FORMATION The starting point of this research is the microstructure information of cement paste, which can be collected via CT scan and/or computer simulation. As this is a computationally driven research project, it is decided to adopt third party tools to simulate the hydration and microstructure formation process of cement paste. The HYMOSTRUC3D (Hydration, Morphology and Structure, 3D) model [4, 5] and the CEMHYD3D model [6] are the most popular hydration models all over the world. The HYMOSTRUC3D model is developed by the Microlab in Delft, the Netherlands, and the CEMHYD3D is the product of NIST (National Institute of Standards and Technology, US). Among others, one difference of these two models is that HYMOSTRUC3D is a particle-based model while CEMHYD3D is voxelbased. It is convenient for this research to adopt HYMOSTRUC3D model because clear relationship can be created between particle structure and lattice structure. In the HYMOSTRUC3D model, it is assumed that only CSH (Calcium Silicate Hydrates) gel and CH (Calcium Hydroxides) gel are produced during the hydration process, furthermore the CH product is transferred to volume equivalent CSH product for simplicity. Hence, the hydrating cement particle may consist of three layers in general, namely unhydrated cement, inner product and outer product. As the hydrating cement particle is assumed to be in the shape of sphere, the location of which is expressed by sphere center coordinates and the size by diameters. Hereby, an example is given to illustrate the necessary input parameters for the HYMOSTRUC3D model and the corresponding outputs which indicate the microstructure of cement paste at a certain degree of hydration. In this example, the cement paste is in the shape of a cube with the dimension of µ m. The Blaine value of cement is 420 m / kg and the water/cement ratio is 0.4. The environment temperature is 20 C. The mineralogical composition of the Portland cement used in this study is given in percentage of weight content in Table 1. Table 1: Mineralogical composition of the cement CEM I 32.5R [5] C 3 S C 2 S C 3 A C 4 AF 63% 13% 8% 9% Figure 1 illustrates the relationship between the degree of hydration and curing age, which is one of the outcomes from the HYMOSTRUC3D model. 1162

3 Blaine Degree of hydration (%) Curing age (hours) Figure 1: Relationship between the degree of hydration and curing age (water/cement = 0.4, Blaine = 420m 2 /kg, temperature = 20 C) Figure 2 shows an image of the microstructure of the cement paste at the curing age 794 hours, and the corresponding degree of hydration is 85%. Figure 2: Image of the cement paste microstructure (water/cement = 0.4, Blaine = 420m 2 /kg, degree of hydration = 85%) 1163

4 3. 3D LATTICE MODELLING OF TENSILE TEST In the lattice analysis module, it is assumed that the lattice structure consists of beam elements with circular cross-section. Hence, it is required to determine the location of nodes, the geometric parameters (e.g. length, radius of cross-section) and mechanical properties (e.g. Young s and shear modulus, tensile strength) of elements. A lattice structure can be generated on the basis of the particle structure from HYMOSTRUC3D model through the following three stages. The first stage is to determine the geometry of the lattice structure. It is reasonable to assume that one node represents one hydrating cement particle, hence, the node coordinates are exactly equal to the coordinates of hydrating cement particle center. Moreover, it is assumed that one element is generated between two hydrating cement particles if they have contact volume as shown in Figure 3 [3]. The length of the element is equal to the distance of the two nodes, the radius of cross-section is determined by the size of contact volume. h h l Figure 3: Contact hydrating cement particles and beam element [3] The second stage is to determine the Young s modulus and shear modulus ( E, G ) of an element. They are calculated via two averaging steps as shown in Figure 4(a) and 4(b). First of all, the modulus of hydrating cement particle ( Ep, G p) is defined as a weighted average value of the modulus of unhydrated cement, inner product and outer product. Then the modulus of beam element can be calculated by averaging particle modulus by weight. 1164

5 R o R 1 R 2 R i R u l (a) Determination of particle modulus (b) Determination of element modulus Figure 4: Determination of Young's modulus and shear modulus of an element [3] The last stage is to determine the tensile strength of an element. Actually this is the trickiest one in the lattice model. For simplicity, it is assumed that the tensile strength is proportional to the Young s modulus of the element. The coefficient is assumed to be Based on the above principles and the microstructure information, a lattice structure is generated, as shown in Figure 5(a). As a tensile test is to be simulated, the force applied is a uniform surface load in the z -direction, all the other surfaces are free to expand and/or shrink, as shown in Figure 5(b). (a) Initial mesh of the system (b) Applied load and boundary conditions Figure 5: 3D lattice structure of the hydrating cement paste (water/cement = 0.4, degree of hydration = 85%) 1165

6 The next step is to perform the 3D lattice analysis to simulate the fracture process of the cement paste system. The basic idea of lattice analysis is that imposing a prescribed displacement on the frame structure, finding the critical element which has the highest stress/strength ratio, removing it from the system. This procedure is repeated until the system fails. Roughly speaking, lattice analysis is a set of linear analysis on frame structures using Finite Element Method. This implies that the fundamental of lattice analysis is nothing else but the conventional structural analysis. As a result, the steps required for lattice analysis are quite similar to the standard finite element analysis for frame structure, except that the critical element is removed and the analysis is repeated until the system fails. For the example given in the previous section, 1755 analysis steps are performed until the system fails. The final load-displacement diagram is presented in Figure 6. The tensile strength of the specimen can be calculated on basis of the peak load in the load-displacement diagram, which is 2.14MPa. The Young s modulus of the specimen is the slope of the curve 4 in the linear stage in the load-displacement diagram, which is equal to MPa Step 1012 Load (mn) Step Displacement (µm) Figure 6: Load-displacement diagram (simulation of tensile test on cement paste) During the lattice analysis, the sequence of broken elements is recorded. It is assumed that one broken element represents one microcrack in the material. In 3D configuration, the microcrack is assumed to be in the shape of cylinder, the length of which is the elongation of broken element and the cross-section of which is equal to the cross-section of broken element. Figure 7(a) and 7(b) show the microcracks at Step 1012 and Step 1745 respectively. 1166

7 (a) 3D microcracks at Step 1012 (b) 3D microcracks at Step 1745 Figure 7: 3D microcracks at peak load point and failure state It is observed that the microcracks appear everywhere within the cube, and not just lie in some specific regions. The reason for this phenomenon is obvious, because the external force applied is a uniform tensile surface load in the vertical direction ( z -direction) and no notch is made on the specimen, which results in a uniform tensile stress state at every point in the cube. Similar results can be found in [8] while using another approach. In [8], the microstructure of cement paste is simulated by the NIST s 3D model (CEMHYD3D) and solid element is adopted in the fracture analysis. Furthermore, most of the microcracks concentrate at a certain region, because stress concentration occurs when the first few microcracks have formed. These microcracks make up the main crack which might be observed in a lab experiment. 4. CONCLUSIONS AND FURTHER RESEARCH The 3D lattice model is applied to simulate a tensile test on simulated cement paste at micro scale in this paper. The study shows the ability of lattice modeling and its validity for micro scale application. Compared with 2D lattice modeling, 3D analysis requires much more computational resources in terms of computing time and computer memory. It is always a challenge for algorithm designers to reduce these high demands. Some progress is made in this paper, for instance, element-by-element scheme and preconditioning techniques are incorporated into the linear algebraic solver. The modeling principles addressed in this paper should also hold for higher scale, which corresponds with mortar and concrete. A complete multiscale computational system can be built up with this solid fundamental. Furthermore, temperature change induced fracture and time dependent behavior such as shrinkage and relaxation can also be simulated using the 3D lattice analysis. REFERENCES [1] Schlangen, E. Experimental and Numerical Analysis of Fracture Processes in Concrete. Ph.D. Thesis, Delft: Delft University Press, 1993,

8 [2] Lilliu, G. 3D Analysis of Fracture Processes in Concrete. Ph.D. Thesis, Delft: Delft University of Technology, 2007, 151. [3] Qian, Z. 3D Lattice Analysis of Cement Paste. MSc Thesis, Delft: Delft University of Technology, 2008, 40. [4] van Breugel, K. Simulation of hydration and formation of structure in hardening cement-based materials. Ph.D. Thesis (2nd Edition), Delft: Delft University Press, 1997, 305. [5] Ye, G. Experimental Study & Numerical Simulation of the Development of the Microstructure and Permeability of Cementitious Materials. Ph.D. Thesis, Delft: Delft University Press, 2003, 186. [6] Bentz, D. 'CEMHYD3D: A Three-Dimensional Cement Hydration and Microstructure Development Modeling Package (Version 3.0)', National Institute of Standards and Technology Interagency Report, Technology Administration, U.S. Department of Commerce, NISTIR 7232, June 2005 [7] Schlangen, E. and Garboczi, E.J. 'Fracture simulations of concrete using lattice models: Computational aspects'. Engineering Fracture Mechanics 57(2/3)(1997) [8] Bernard, F., Kamali-Bernard, S. and Prince W. '3D multi-scale modeling of mechanical behaviour of sound and leached mortar'. Cement and Concrete Research 38(2008)