NUMERICAL ANALYSES OF DUCTILE FRACTURE BEHAVIOR IN 2D PLANE STRAIN AND AXISYMMETRIC MODELS USING THE COMPLETE GURSON MODEL

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1 DRAFT Proceedings of PVP ASME Pressure Vessels and Piping Division Conference July 26-30, 2009, Prague, Czech Republic PVP NUMERICAL ANALYSES OF DUCTILE FRACTURE BEHAVIOR IN 2D PLANE STRAIN AND AXISYMMETRIC MODELS USING THE COMPLETE GURSON MODEL Jie Xu a,b Department of Structural Engineering a Norwegian University of Science and Technology b Beijing University of Science and Technology jie.xu@sintef.no Zhiliang Zhang Department of Structural Engineering Norwegian University of Science and Technology Trondheim, Norway zhiliang.zhang@ntnu.no Erling Østby SINTEF, Materials and Chemistry Trondheim, Norway Erling.Ostby@sintef.no Bård Nyhus SINTEF, Materials and Chemistry Trondheim, Norway Bard.Nyhus@sintef.no Dongbai Sun Beijing University of Science and Technology, Beijing, China dbsun@mater.ustb.edu.cn ABSTRACT Ductile crack growth plays an important role in the analysis of fracture behavior of structures. A strong geometry dependence of ductile crack growth resistance emerges under large scale yielding conditions. This geometry dependence is associated with different levels of crack tip constraint. However, an independent relationship between the fracture resistance and crack tip constraint has also been observed in experimental studies for selected specimen geometries. To verify these results, crack growth resistance curves for plane strain, mode I crack growth under large scale yielding have been computed using the complete Gurson model. Single edge notched bend (SENB) and tension (SENT) specimens with three different crack geometries have been selected for the numerical analyses. Specimen size effect on ductile crack growth behavior has also been studied. SENT specimen appears as an alternative to conventional fracture specimens to characterize fracture toughness of circumferentially cracked pipes due to its similar geometry constraint ahead of the crack tip with that of cracks in pipes. The application of SENT specimen into the axisymmetric model so as to form a pipe segment with a long internal circumferential crack under large scale yielding conditions is examined. The effect of biaxial load conditions (axial tension combined with internal pressure) on the resistance curves is considered. INTRODUCTION Constraint has been an important consideration in fracture mechanics. The purpose of studying constraint is to find a proper parameter to characterize the crack tip stress-strain fields, so that results from one test geometry can be transferred to another. This transferability of material toughness remains a key issue in applications of fracture mechanics to assess the integrity of structural components. Ductile crack growth plays an important role in the analysis of the fracture behavior of structures. The loading-carrying ligament will be reduced with further crack extension and therefore the capacity of the structures will be decreased. Crack tip constraint can well explain the dependence of fracture toughness on specimen geometry, crack size, loading mode, yield strength mismatch as well as residual stress [1-9]. Experimental results consistently reveal the strong effects of specimen geometry and loading mode (tension vs. bending), associated with different levels of crack tip constraint, on measured resistance curves (for typical examples, see Brocks and Schmitt [6], Hancock et al. [5], O Dowd and Shih [10] and Joyce and Link [7]). However, an independent relationship between the fracture toughness and crack tip constraint for SENT (single edge notched tension) specimens with different initial crack lengths has also been observed in experimental studies [11], as shown in Fig Copyright 2009 by ASME

2 Fig. 1. CTOD-R curves for SENT specimens with different crack lengths compared with a standardized SENB specimen [11]. To further understand the relation between the constraint level and ductile fracture toughness, a numerical study has been carried out to investigate the effect of crack tip constraint on ductile crack growth behavior. The complete Gurson model developed and implemented by the authors [12] has been utilized in this study. In addition, it has been known that the standard SENB specimens with crack depth of a/w= have a much higher geometry constraint than circumferential cracks in pipes, which therefore lead to considerably conservative decisions for the fracture mechanics testing in engineering critical assessment of pipelines. For circumferential flaws in pipes, the SENT specimen has frequently been used because it has a geometry constraint that is similar to the cracks in pipes. Much work has already been carried out on the tensile testing for SENT specimens as alternative fracture mechanics specimens, and examples can be found in references [11, 13-17]. In studying the long circumferential cracks in pipes, the crack geometry, applied load and boundary conditions are symmetrical about the axis of revolution. These axisymmetric bodies can be represented by a typical radial plane containing the axis of rotational symmetry, therefore the analysis of threedimension can be reduced to two-dimension in the radial and axial directions. Axisymmetric model could be a good approximation in the analyses of pipes with long circumferential cracks. This work presents the application of axisymmetric model to study the ductile fracture behavior for pipes with long circumferential cracks under large scale yielding conditions as well. The effect of combined load (axial tension along with internal pressure) conditions on the fracture resistance curves is considered. NOMENCLATURE a: initial crack length Δa: crack growth W: specimen width 2L: specimen length t: pipe wall thickness r i : inner radius of the pipe D: outer diameter of the pipe E: Yong s modulus ν: Poisson s ratio n: strain hardening exponent σ 0 : yield stress σ f : flow stress ε 0 : yield strain ε p : equivalent plastic strain q: von Mises stress f : void volume fraction f 0 : initial void volume fraction f c : critical void volume fraction f F : void volume fraction at final failure SENT: single edge notched tension SENB: single edge notched bend CTOD: crack tip opening displacement FE MODELS OF SENB AND SENT SPECIMENS The stress-strain behavior of the model material in this study is characterized by the following power law hardening model: n p f 0 1 (1) 0 where σ f is the flow stress, σ 0 is the yield stress, ε p is the equivalent plastic strain, ε 0 is the yield strain and n is the strain hardening exponent. The finite element analyses consider material with a yield stress σ 0 =400MPa, Young s modulus E=200GPa and Poisson ratio ν=0.3. The effect of initial volume fraction, f 0, has also been investigated in the analyses. The dimensions of the specimens are schematically shown in Fig. 2 and. In all the analyses, the span of the specimens, S, is chosen to be 4 times the width, W, for SENB and L/W=5 for SENT specimens. In order to study the effect of crack tip constraint on ductile fracture behavior, three different ratios of initial crack length to specimen width,, 0.3, and, are selected for both SENB and SENT. As for the size effect, three different specimen sizes,, 30 and 50mm are used both for SENB and SENT specimens. Figure 2 (c) and (d) respectively shows the finite element mesh and crack tip mesh arrangement in the local region. Due to the symmetry, only one half of the specimen was modeled in the finite element analyses and 2D plane strain model with 4-node elements (ABAQUS type CPE4) has been used for the parameter study. A remote homogenous displacement controlled boundary condition (clamped) was applied for the SENT specimen. Large deformation effect is accounted for in all the analyses. The region with uniform element size extends to the width of the specimen and 3.0mm above the symmetrical interface is used to simulate the ductile crack growth. The element size is mm in this local 2 Copyright 2009 by ASME

3 region except for 0.1 5mm for a single row which representing the damage layer along the crack growth path. All the analyses have been performed with ABAQUS [18]. (c) (d) Fig. 2. Schematic plot of the specimen. SENB; SENT; (c) finite element mesh; (d) crack tip mesh arrangement. THE COMPLETE GURSON MODEL (CGM) FOR VOID NUCLEATION, GROWTH AND COALESCENCE Ductile crack growth in metals is a result of nucleation, growth and coalescence of microvoids. Great effects have been made in developing the constitutive models for elastic-plastic materials incorporating void mechanisms and the best model appears to be the one originally introduced by Gurson [19] and later modified by Tvergaard [20] and Needleman [21]. The yield function of the Gurson-Tvergaard-Needleman model has the following form: q 2 2 f 2q cosh q m f 2 f 1 2 q f 0 (2) 1 where q is the von Mises stress, σ f is the flow stress, f is the void volume fraction and σ m is the mean normal stress component. q 1 and q 2 are constants. Due to the incompressible nature of the matrix material, the growth of existing voids can be expressed as: p df (1 f )d : (3) growth where ε p is the plastic strain tensor and I is the second-order unit tensor. There are two different coalescence criterions used in the literatures, the so-called critical void volume fraction criterion and Thomason s plastic limit load model [22]. The former one was empirically used in the Gurson model, which assumes that the void coalescence to occur when a critical void volume fraction, f c has been reached. The void coalescence ends when the void volume fraction reaches another value-the void volume fraction at final failure, f F. Tvergaard and Needleman [20] introduced a function that amplifies void growth to simulate the void coalescence process: f for f f f f f c * * u c fc f fc for f fc ff fc where f u * =1/q 1. When f > f c, f * replaces f in Eq. 2. In deriving the Gurson model, only homogeneous deformation has been considered. Thomason later suggested that the localized deformation mode of void coalescence should be treated differently. There is a competition between these two deformation modes in Thomason s plastic limit load mode. At the beginning, small voids could deform independently with each other. With further development of plastic deformation, the stress for localized deformation decreases and therefore the localized deformation mode will eventually become dominant. By combining the Gurson model and Thomason s plastic limit load model, a complete Gurson model was obtained by the authors previous work [12], in which the whole ductile fracture process, including void nucleation, growth and coalescence can be simulated. The complete Gurson model was implemented into ABAQUS using a user material subroutine UMAT developed by Zhang [23-25]. FE MODELING FOR CRACK TIP CONSTRAINT ANALYSES Effect of initial crack length and loading mode Figures 3-8 present the calculated CTOD-resistance curves for both SENB and SENT specimens. In these cases, the strain hardening exponent for the model material considered is n=5, and initial void volume fraction is f 0 =05. (4) 3 Copyright 2009 by ASME

4 a/w= SENB n=5 f 0 =05 a/w= SENT n=5 f 0 = Fig. 3. CTOD- Δa curves., 0.3 and. n=5, f 0 =05. SENB, mm. Fig. 6. CTOD- Δa curves., 0.3 and. n=5, f 0 =05. SENT, mm. a/w= SENB n=5 f 0 =05 a/w= SENT n=5 f o = Fig. 4. CTOD- Δa curves., 0.3 and. n=5, f 0 =05. SENB, mm. Fig. 7. CTOD- Δa curves., 0.3 and. n=5, f 0 =05. SENT, mm. a/w= SENB n=5 f 0 =05 a/w= SENT n=5 f o = Fig. 5. CTOD- Δa curves., 0.3 and, n=5, f 0 =05. SENB, mm. Fig. 8. CTOD- Δa curves., 0.3 and. n=5, f 0 =05. SENT, mm. 4 Copyright 2009 by ASME

5 It can be seen that, for both SENB and SENT specimens, the CTOD-R curves are quite similar at the very beginning for all the models. This is consistent with the observations obtained in the previous works [5, 26], in which an insensitive relation between the crack length and the fracture resistance at the onset of crack growth has been found. With further crack growth, a higher resistance curve was obtained for the shallow cracked specimens loaded in both bending and tension. Furthermore, the difference for the resistance curves among different specimen geometries becomes gradually larger with the specimen width increasing from 10 to 50mm. Figure 9 presents the CTOD values for different specimens at the same crack growth, Δa =mm. For the models with mm, fracture toughness as denoted by CTOD shows less dependence on the specimen geometries for SENT specimens than it does for SENB. This result agrees well with the experimental observations by Nyhus et al. [11], which indicates that the fracture toughness from SENT specimens is not sensitive to the initial crack lengths. While with the increase of specimen sizes, and 50mm, the initial crack length starts to show evident influence on the fracture toughness for both SENB and SENT specimens. Similar experimental results for larger specimens can also be found from [7, 27], in which evident difference between the resistance curves has been obtained for SENB specimens with 3~5, mm SENT-0.1 a = mm n=5 f 0 =05 SENT-0.3 SENT- SENB-0.1 SENB-0.3 SENB- Fig. 9. CTOD vs. specimens for Δa =mm., 0.3 and. n=5, f 0 =05., 30 and 50mm for both SENB and SENT. For further understanding this phenomenon, the Q- parameter [3, 4] has been selected to quantify the crack tip constraint for each case. In the following, the concept and definition of Q-parameter used in this study are briefly introduced. It is now well understood that crack tip constraints affect the distribution of stresses around a crack and consequently preclude the use of a single parameter characterization of the crack tip stress field. A second parameter based on the elastic T-stress has been proposed by Betegon and Hancock [2] to describe the crack tip stress filed. For elastic-plastic problems, O Dowd and Shih [3, 4] proposed a J-Q formulation to characterize the crack tip stress field and quantify constraint levels for various geometry and loading configurations in elastic-plastic materials, where J sets the deformation level and Q is a stress triaxiality parameter which is a direct measure of the stress field that is related to a reference field usually described by the HRR field. Because of its simplicity, the J-Q theory is widely used in engineering fracture mechanics analysis. The Q-parameter was originally defined as Re f ( ) Q, at x /( J / 0) 2, θ = 0 (5) 0 where σ θθ is the opening stress component, x is the distance from the crack tip along the crack plane (θ = 0). Because of the use of CTOD as the crack driving force, the following definition of Q has been used in this study: Re f ( ) Q, at x/ctod = 4, θ = 0 (6) 0 A modified boundary layer (MBL) model solution with T=0 is adopted to represent the reference stress field σ θθ Ref. Figure 10 presents the calculated CTOD-Q relation for all the specimens at Δa =mm SENB-0.1 SENB-0.1 SENB-0.1 SENB SENB- SENB-0.3 SENB-0.3 SENB- SENB- 0.1 SENB a = n=5 f 0 = SENT-0.1 SENT-0.3 SENT- SENT-0.1 SENT-0.3 SENT SENT a = n=5 f 0 = Q SENT-0.3 SENT- SENT- Q Fig. 10. CTOD-Q relation. n=5, f 0 =05, Δa =mm., 0.3 and., 30 and 50mm. SENB; SENT. 5 Copyright 2009 by ASME

6 For small SENT specimens, mm in Fig. 10, the corresponding Q values have no significant change for cases with different initial crack lengths at Δa =mm. That indicates the crack tip constraint is rather similar for small SENT specimens with different crack lengths, which therefore has no significant effect on the resistance curves as have been observed in Fig. 6. As for small SENB specimens, small decrease of Q-value with the decrease of initial crack length can be seen in Fig. 10, which well explains the slight influence of crack tip constraint on the resistance curves as have been shown in Fig. 3. As for the larger specimens, and 50mm loaded in both bending and tension, a significant decrease of CTOD with the increase of Q-parameter can be observed in Fig. 10 and. Therefore, crack tip constraint exhibits distinct influence on the fracture resistance curves for the larger specimens with material considered herein. Effect of specimen size In the following, further discussion will be focused on the specimen size effect on the crack growth resistance curves. Figures show the calculated CTOD-R curves for shallow () and deep (a/w=) cracked specimens loaded in both bending and tension. Three different specimen sizes are examined,, 30 and 50mm respectively. For the shallow cracked specimens,, it can be seen from Fig that there is less dependence on specimen size for both bending and tension specimens. Therefore, the fracture toughness parameter can be directly transferred from small specimens to the larger ones for crack growth no more than 2.5mm for SENB and 2.8mm for SENT specimens in this study. As for deep cracked cases, for example a/w=, however, a more pronounced specimen size effect on crack growth resistance curves can be observed for both SENB and SENT specimens as can be seen from Fig The crack growth resistance increases with the decrease of specimen size. This increase in toughness is usually associated with a loss of constraint as the specimen size is reduced, which can be seen from Fig. 10. Thus, the direct transferability of fracture toughness parameter from small specimens to the large ones does not exist anymore as for specimens with deep cracks. Similar findings for the effect of specimen size on the resistance curves which was denoted by J-Δa have also been reported by the authors [28] for specimens loaded in bending. SENB a/w= n=5 f 0 =05 SENB n=5 f 0 = SENT a/w= n=5 f 0 =05 SENT n=5 f 0 = Fig. 11. CTOD- Δa curves., 30 and 50mm. n=5, f 0 =05.. SENB; SENT Fig. 12. CTOD- Δa curves., 30 and 50mm. n=5, f 0 =05. a/w=. SENB; SENT. Effect of initial void volume fraction The effects of specimen geometry and size may depend on the material toughness level. Under this consideration, 6 Copyright 2009 by ASME

7 model material with a lower initial void volume fraction, for example 10% of the previous value used, which representing higher material toughness, has been analyzed. Figure 13 presents the results of resistance curves for both SENB and SENT specimens with different initial crack lengths at mm. It can be seen that, for model material with a lower initial void volume fraction f 0 =005, significantly higher resistance curves are obtained compared with that of f 0 =05. Moreover, the effect of initial crack length on the resistance curves becomes smaller, especially for SENB specimens with relative higher crack tip constraint levels, and a/w=. Similar trends can also be obtained for other specimens with smaller sizes, and 30mm. Therefore, the influence of crack tip constraint on resistance curves is strongly related to the material toughness. A weaker dependence of crack growth resistance on crack tip constraint can be expected for material with higher fracture toughness. Furthermore, the effect of specimen sizes on resistance curves becomes smaller for both SENB and SENT SENB a/w= n=5 f 0 =005 SENB n=5 f 0 =005 a/w= SENT n=5 f 0 = a/w= Fig. 13. CTOD- Δa curves., 0.3 and, mm, n=5, f 0 =005. SENB; SENT. In addition, the results of size effect for specimens with a lower initial void volume fraction, f 0 =005, a/w= are displayed in Fig. 14. It can be observed that, for material with f 0 =005, other parameters are fixed, remarkably higher crack growth resistance than the cases with f 0 =05 can be found. SENT a/w= n=5 f 0 = Fig. 14. CTOD- Δa curves., 30 and 50mm. a/w=, n=5, f 0 =005. SENB; SENT. FE MODELS FOR LONG CIRCUMFERENTIALLY CRACKED PIPES As have been observed that the SENT specimen can be used to represent a pipe section. Figure 15 shows how the SENT specimen rotates around the symmetric axis. Only 180 of the geometry of the pipe with an internal circumferential crack for visual purpose is displayed in Fig. 15. The configuration and finite element mesh of the SENT specimen as have been used in the previous study can be seen in Fig. 2. Two-dimensional 4-node axisymmetric elements (CAX4) are used for the numerical study. Figure 15 gives the schematic illustration of axisymmetric model, in which the width (W) of the SENT specimen corresponds to the thickness of the pipe (as denoted by t hereafter), namely W=t, r i is the inner radius of the pipe, D is the outer diameter. In all the analyses, pipe wall thickness is kept constant, t=30mm, the pipe length is chosen to be ten times of the pipe wall thickness, and fixed ratio of D/t=22 are used in this study. 7 Copyright 2009 by ASME

8 Fig. 15. Sketch of geometry for pipe with an internal circumferential crack; 2D axisymmetric FE model for pipe with an internal circumferential crack. Due to the axisymmetric conditions are assumed, thus, model corresponds to a pipe loaded in tension with a crack all around the circumference. In the case of biaxial loading, the internal pressure can be applied to the model by specifying the inner surface of the pipe (which was represented by the left side of the SENT specimen as can be seen in Fig. 15 ) a uniform pressure as a constant load and then the remote controlled displacement for the tensile loading is prescribed gradually. If the crack is located on the side on which the pressure is applied, the crack surface is also loaded with internal pressure. Figure 16 shows how the internal pressure was applied on a model with an internal crack. The pressure on the crack surface can be seen in Fig. 16. a t D/2 r i a w FE ANALYSES OF COMBINED LOADING EFFECT ON DUCTILE CRACK GROWTH RESISTANCE FOR PIPES WITH LONG INTERNAL CIRCUMFERENTIAL CRACKS Figure 17 presents the calculated resistance curves for pipes with D/t=22, t=30mm, submitted to different internal pressure. The results of corresponding SENT (with the same initial crack length) and standard SENB specimens from 2D plane strain models are plotted together as well. CT0D, mm =0 t=30 a/t=0.1 D/t=22 h = n=5 f 0 =05 =0 0 = SENT() SENB(a/W=) 2.0 =0 = =0 0 = SENT() SENB(a/W=) t=30 a/t=0.3 D/t=22 n=5 f 0 = Crack tip =0 t=30 a/t= D/t=22 h = n=5 f 0 =05 =0 0 = SENT(a/W=) SENB(a/W=) (c) Fig. 16. Internal pressure in the model with an internal crack; Internal pressure ahead of the crack tip. The boundary conditions in the axisymmetric model are quite similar to those in the 2D plane strain model, except there is no constraint in x-direction in the axisymmetric model. Due to the symmetry, only half of the pipe was modelled in the finite element analyses as can be seen from Fig. 15. Large deformation effect is accounted for all the analyses Fig. 17. CTOD- Δa curves for pipes, SENB and SENT specimens. σ h /σ 0 =0, 0.25, 0 and n=5, f 0 =05, D/t=22, t=30. a/t=0.1; a/t=0.3; (c) a/t=. Three different ratios of hoop stress (σ h =P r i /t, σ h is the hoop stress, P is the magnitude of internal pressure, r i is the 8 Copyright 2009 by ASME

9 inner radius, t is the wall thickness of the pipe) produced circumferentially by internal pressure to material yield stress, σ h /σ 0 =0.25, 0 and 0.75 are considered. In addition, pipes with three different initial crack lengths, a/t=0.1, 0.3 and respectively, are employed respectively. For the shallow cracked pipes, a/t=0.1 in Fig. 17, small effect of internal pressure on the resistance curves can be seen. Similar behaviour for pipes with a/t=0.3 can also be observed in Fig. 17. As for deeply cracked pipes, a/t= in Fig. 17 (c), almost no effect of internal pressure on the resistance curves has been obtained. This indicates that the resistance curve obtained from uniaxial loading can be applied to biaxial situations. In addition, the result from SENB specimen is significantly conservative than that of SENT for both shallow and deep cracked cases. The present observations are in agreement well with the large-scale experimental results reported by Østby et al. [29], in which two different internal pressure were tested in the pipe for a/t=0.2. It has been demonstrated in [29] that the crack growth resistance curves measured in the pipes are in accordance with the results from the SENT testing. Moreover, the measured crack growth resistance curves for the pipes with different internal pressure and the resistance curves from SENT testing are quite similar. This work provides an additional support to the previous findings using a 2D axisymmetric model that a small effect of biaxiality on resistance curves can be expected. In addition, similar results from other 2D and 3D numerical simulations can also be found in [11, 16, 30, 31]. CONCLUSIONS 2D plane strain FE analyses have been carried out to study the effect of crack tip constraint on ductile crack growth behavior under large scale yielding conditions. Single edge notched bend (SENB) and tension (SENT) specimens with shallow and deep cracks, which consists of, 0.3, and, are selected for this purpose. Specimen size effect on ductile crack growth behavior has also been investigated. The complete Gurson model has been utilized to predict ductile crack growth resistance curves. Crack tip constraint has no significant effect on the crack initiation toughness for specimens loaded in both bending and tension. With further crack growth, it has been found for the small specimens, mm in this study, the initial crack length has no significant effect on the CTOD resistance curves for SENT specimens, which agrees well with the experimental observations. However, with the increase of specimen size, crack tip constraint will significantly influence the crack growth resistance for both SENB and SENT specimens. With regard to the specimen size effect on the ductile crack growth resistance, it has been observed that CTOD resistance curves display less dependence on the specimen size for shallow cracked specimens,, loaded in both bending and tension. While a more pronounced size effect can be seen with further crack growth for the deeply cracked specimens, a/w=, loaded in both bending and tension. However, both the effects of specimen geometry and size are strongly related to the material toughness level, as denoted by the initial void volume fraction in this study. For material with comparatively lower fracture toughness, the effects of specimen geometry and size are rather significant. On the contrary, both effects are strongly reduced when the material possesses a relatively higher toughness level. In addition, a 2D axisymmetric model has been carried out to investigate the biaxial loading effect on material resistance for pipes with long internal circumferential cracks. A small influence of internal pressure on crack growth resistance curves has been observed for pipes with different initial crack length. Moreover, a more conservative result from SENB than that from SENT specimen has been obtained compared with pipes, which therefore validates the use of SENT specimen in fracture characterization of circumferentially cracked pipes as well when the pipe submitted to biaxial loading conditions. ACKNOWLEDGMENTS The financial support from the China Scholarship Council and the industry partners and the Research Council of Norway through the Arctic Materials Project are gratefully acknowledged. REFERENCES [1] Larsson SG, Carlsson AJ. Influence of non-singular stress terms and specimen geometry on small-scale yielding at crack tip in elastic-plastic materials. J Mech Phys Solids 1973; 21: [2] Betegon C, Hancock JW. Two-parameter characterization of elastic-plastic crack-tip fields. J Appl Mech 1991; 58: [3] O Dowd NP, Shih CF. 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