- Email: [email protected]
Multiscale shear fracture of heterogeneous material using the virtual internal bond
Theoretical and Applied Fracture Mechanics 47 (2007) 185–191 www.elsevier.com/locate/tafmec
Multiscale shear fracture of heterogeneous material using the virtual internal bond Z.N. Zhang
a,b,* ,
X.R. Ge
c,d
a
c
Shanghai Institute of Applied Mathematics and Mechanics, Shanghai University, Shanghai 200072, China b Department of Civil Engineering, Shanghai University, Shanghai 200072, China Key Laboratory of Rock and Soil Mechanics, Institute of Rock and Soil Mechanics, Chinese Academy of Sciences, Wuhan 430071, China d Institute of Geotechnical Engineering, Shanghai Jiaotong University, Shanghai 200030, China Available online 6 March 2007
Abstract Material behavior entails multiscaling. The macro response of materials is affected by the microstructure. The method of VMIB (virtual multidimensional internal bond) developed from the VIB (virtual internal bond) for the continuum consists of discrete microscopic mass particles. These mass particles are connected with virtual bond possessing both normal and shear stiffness. The macro constitutive relation is derived in terms of bond stiffness. Shear fracture simulation of heterogeneous material involves introducing heterogeneity into the constitutive relation by two ways. First, heterogeneous material may consist of micro components with different stiffness. This character represented by the bond stiffness is assumed to be randomly distributed. Second, the micro components composed of the heterogeneity may possess the same stiffness, but their critical deformation, say the strain strength, may be different. This character represented by the strain strength is assumed to be randomly distributed. For a given statistical distribution function, the bond stiffness and the strain strength can be generated in numerical simulation implementation. Using VMIB, the macro response of materials is determined by the evolution of bond at the microscopic scale. To reflect micro mechanism of shear fracture, a phenomenological bond evolution function is proposed such that the shear fracture process of heterogeneous material is simulated. The results demonstrate that the present method can represent the shear fracture process and the basic characteristics of shear fracture of heterogeneous material. 2007 Elsevier Ltd. All rights reserved. Keywords: Heterogeneity; Multiscale numerical method; Shear fracture; Virtual multidimensional internal bond
1. Introduction Simulation of shear fracture of the heterogeneous material is important. The prevailing methods can *
Corresponding author. Address: Department of Civil Engineering, Shanghai University, Shanghai 200072, China. E-mail addresses: [email protected], [email protected] (Z.N. Zhang).
be divided into two categories. One is the continuum-mechanics-based method where the heterogeneity is introduced into the numerical model by assigning element mechanical properties with the statistical method [1,2]. The other is the discrete method where the continuous solid is represented mechanically by a discrete network such that each element represents a microelement of the heterogeneous solid [3–7]. The mechanical properties of
0167-8442/$ - see front matter 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.tafmec.2007.01.001
186
Z.N. Zhang, X.R. Ge / Theoretical and Applied Fracture Mechanics 47 (2007) 185–191
the heterogeneous elements are assumed to be randomly distributed. Both methods possess advantages and disadvantages. They can compensate each other. In other words, the continuum-mechanics-based method reduces the internal freedom though inadequate for treating the problems of fracture. The discrete method on the other hand is suited for simulating the fracture behaviors of solids, but it usually involves with a large number of freedom. It would be ideal to integrate the two approaches. The Virtual Internal bond (VIB) model [8,9] suggests this possibility. From the view of VIB, the continuous solid is considered to consist of microscopic material particles. Accordingly, the macro constitutive relation is thus derived from the cohesive law between particles. VIB spans two scales to consider the solid properties. The work in [10–12] developed VIB by introducing a shear bond into the original VIB or assigning the normal bond with a shear stiffness [13]. The model is known as the VMIB (Virtual Multidimensional Internal Bonds) model. It can present the diversity of Poisson ratio and applied to more extensive engineering materials. In [14], a modified VIB, i.e. MVIB is also proposed to address the fixed Poisson ratio problem of VIB by respectively considering the energy contribution of the spherical and deviatoric deformation. Both the VMIB and the MVIB are consistent with the classic linear elasticity theory. They are multiscale in character and consider the response of solids at a higher level by mechanisms at a lower scale. The multiscale analysis method can reveal the essence of macro response more comprehensively. Recent works in [15–18] have discussed multiscaling near the crack tip by using the concept of the strain energy density function. In [19], the dual scale (micro and macro) features were analyzed for periodic heterogeneous solids although the scheme applies to any two scales
such that the models can be connected and extended to multiscale. The micro mechanism of shear fracture is different from that of tensile fracture. It is more complex than that of tensile especially for heterogeneous materials. The fluctuations of mechanical properties can seriously affect the macroscopic behaviors of fracture. For example, the fracture always initiate simultaneously at different locations with a zigzag pattern. This work attempts to simulate the shear fracture process of heterogeneous materials using VMIB. The motivation here is to develop a multiscale method to model the heterogeneous materials. 2. Modeling of heterogeneous materials According to VIB and VMIB, the continuous solid is mechanically represented by a microscopic network. The macro constitutive relation at the continuum level is derived from the interactions between material particles. The methodology of VIB and VMIB can be graphically shown in Fig. 1. For homogeneous material, the constitutive relation is simple, which has been well analyzed by VIB and VMIB. However, the constitutive relation of heterogeneous materials is much more complex. The mechanical properties varies with the location. It is very difficult to accurately describe the properties at a certain location. Therefore, the statistical method is employed to describe the distribution of heterogeneity. To model the heterogeneous materials, the following two methods are employed. 2.1. Stiffness-controlled method For some heterogeneous materials, the micro component may have different stiffness. The microscopic fluctuation of stiffness may also affect the macroscopic fracture behavior. To begin with, let
Fig. 1. Heterogeneous solid, its representation with VMIB and the unit cell.
Z.N. Zhang, X.R. Ge / Theoretical and Applied Fracture Mechanics 47 (2007) 185–191
187
the microscopic bond stiffness be assumed to be randomly distributed such that the stiffness k, r can be expressed as
To simplify, the distribution density D(h, /) can be taken as unity for the isotropic solid. According to [10,11], the mean value of k; r can be determined by
k ¼ kk
3E 4pð1 2mÞ 3ð1 4mÞE r ¼ 4pð1 þ mÞð1 2mÞ
ð1Þ
r ¼ kr
where k, r are respectively the normal and shear stiffness of bond. Note that k; r are the corresponding mean value of k, r while k is a statistical variable and the mean value k ¼ 1 is held. In [4,6,7], each bond (or spring) is treated as an element to describe the heterogeneous materials at the microscopic level. This method can simulate the fracture behavior more precisely. However, this treatment often involves with a large number of freedom. To reduce the freedom, the number of bonds in the unit cell is assumed to be randomly distributed and the mechanical properties are considered at the unit cell level rather than the bond level, shown as Fig. 1. According to [8,10,11], the potential of this unit cell with an imposed deformation can be expressed as Z 2p Z p U¼ U L Dðh; /Þ sinðhÞdh d/ 0
þ
0
Z
2p
Z
0
ðU R1 þ U R2 þ U R3 ÞDðh; /Þ sinðhÞ dhd/
0
where UL, URi (i = 1, 2, 3) are respectively the potential contributions of normal stiffness and shear stiff2 2 ness of bond, U L ¼ 12 kkðni eij nj Þ ; U R1 ¼ 12 krðni eij g0j Þ ; 1 00 2 1 000 2 U R2 ¼ 2 krðni eij gj Þ , and U R3 ¼ 2 krðni eij gj Þ Here, n is the unit orientation vector of bond, n = (sin h cos /, sin h sin /, cos h) in the spherical coordinate system while g is the orientation vector perpendicular * * to n, g0 ¼ n ðx1 nÞ, g00 ¼ n ðx2 nÞ, and * g000 ¼ n ðx3 nÞ. The spatial distribution bond density is D(h, /). Within the framework of hypo-elastic theory [20,21], the elastic tensor of the unit cell can be derived as
¼
oU oeij oemn Z 2p Z p 0
þ
Z 0
ð4Þ
where E, m are respectively the mean Young’s modulus and Poisson ratio of heterogeneous materials. For two-dimensional cases, the elastic tensor expression is reduced to Z p C ijmn ¼ ðkkni nj nm nn þ krni g0j nm g0n 0
þ krni g00j nm g00n ÞDðhÞ dh
ð5Þ
where g 0 = [cos h, sin h]T; g00 = [ cos h, sin h]T; n = [sin h, cos h]T. The corresponding mean value of bond stiffness k; r can be calculated by 2E pð1 mÞ ð1 3mÞE r ¼ pð1 m2 Þ k ¼
ð6Þ
for the plane-stress problem and
p
ð2Þ
C ijmn ¼
k ¼
kkni nj nm nn Dðh; /Þ sinðhÞ dh d/
0 2p Z p
ð3Þ
ð7Þ
for the plane-strain problems. Numerically, the sample value of k can be generated by statistical methods, such as Monte Carlo and others, if a certain probability distribution law is given. 2.2. Strain strength-controlled method For a heterogeneous material, the micro components may have different strain strength even though the stiffness may be the same. The fluctuation of strain strength can also affect the macroscopic fracture behavior which can be demonstrated by assigning each bond with different strain strength, i.e. the critical deformation over which the bond is assumed to break. Again, the strain strength is assumed to be randomly distributed. For each bond, it is assumed that eb ¼ keb
krðni g0j nm g0n þ ni g00j nm g00n
0
000 þ ni g000 j nm gn ÞDðh; /Þ sinðhÞ dh d/
2E pð1 þ mÞð1 2mÞ ð1 4mÞE r ¼ pð1 þ mÞð1 2mÞ k ¼
ð8Þ
where eb is the strain strength and eb is its mean value.
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3. Bond evolution function for shear fracture Tensile fracture is considered in [10] in which the macro response is attributed to the evolution of microscopic bond. Different bond evolution function leads to different macro response. In [22], a simple version of the bond evolution function for uniaxial compressive load was proposed, but it cannot account for the effects of the integrated deformation, i.e. combination of the normal deformation and rotation. A comprehensive description of bond deformation can be made by employing the integrated deformation bond index, which is similar to that in [23,24]
d¼
‘ b þ eb 0:5ð1 þ mÞeb
ð9Þ
Note that d is the integrated deformation index of bond, ‘ ¼ jni edij nj j; b ¼ jni eij gj j; Here, edij is the corresponding deviatoric tensor of strain tensor eij and eb is the uniaxial strain strength. According to the criterion in [23], the discrete element is linear under the condition d < 1 while the discrete element is assumed to break under the condition d P 1. The discrete element experience two phases, namely the linear deformation and the breaking state. To account for the pre-peak strengthening effect and the post-peak softening effect, the following bond evolution function is proposed: k k0 ¼ ð10Þ expðc dn Þ r r0 where c, n are model parameters, which governs different characters of macro response; k 0 and r0 are the initial bond stiffness, which can be determined respectively by Eqs. (4), (6) and (7). The effect of c, n on macro response is shown in Fig. 2. Fig. 2a shows that c mainly governs the peak stress, i.e. the uniaxial strength. The bigger the c, the lower is the peak stress. According to Fig. 2b, the n governs the ‘brittleness’ of the material. The bigger the n, the more brittle is the material. 4. Simulation example and discussion 4.1. Simulation strategy
Fig. 2. Effect of model parameters on stress–strain relationship: (a) effect of c, n = 40; (b) effect of n, c = 0.15.
To examine the performance of the method, the shear fracture process of a specimen under uniaxial compressive load is simulated. The dimensions and boundary conditions of the specimen are shown in Fig. 3. The 3-node plane triangular element is adopted in the FEM implementation. The node number is 435 and the element number is 784. The Gauss point number for computing elastic tensor, say Eq. (5), is 24, hence the sample volume of k is 24 · 784 = 18,816. As an example, let the statistical variable k obey the normal distribution function with the mean value k ¼ 1. Different heterogeneous materials require different statistical distribution function. The choice of probability distribution function will not be discussed. Before applying the FEM, the sample values of k are generated by statistical method. The distribution of k is shown graphically
Z.N. Zhang, X.R. Ge / Theoretical and Applied Fracture Mechanics 47 (2007) 185–191
189
Table 1 Material parameters and the corresponding model parameters Material parameters
Model parameters
E (GPa)
m
eb · 103
c
n
23.6
0.15
2.0
0.15
4.0
4.2. Discussion
Fig. 3. Simulation object dimension and its boundary conditions: (a) dimensions; (b) boundary conditions.
Fig. 4. The value distribution of k.
in Fig. 4. The material parameters and the model parameters are listed in Table 1. The displacement-controlled loading method is adopted. The simulation is performed in three schemes as the following: Scheme I: Keep the strain strength eb constant and treat the bond stiffness as a statistic variable. This scheme refers to the stiffness-controlled method mentioned in Section 2.1. Scheme II: Keep the bond stiffness constant and treat the strain strength eb as a statistic variable. This scheme refers to the strain strengthcontrolled method mentioned in Section 2.2. Scheme III: Keep both the bond stiffness and strain strength constant. This corresponds to the case of treating the material as a homogeneous one.
Figs. 5–7 show respectively the simulation results of Schemes I, II and III. Fig. 8 shows the mechanical response of the three Schemes of simulation. From Fig. 5, it can be concluded that no fracture initiates in the 15th step while fracture begin to initiate simultaneously at several locations in the 16th step. In the 17th and 18th step, the fracture propagates and finally goes though the specimen. From the inclination of fracture, it can be concluded that it is a shear fracture. This agrees with the observation by experiment. The zigzag of fracture demonstrates the effect of heterogeneity on macro fracture. Fig. 6 shows the fracture process of strain strength-controlled scheme. The fracture mode in Fig. 6 is similar to that in Fig. 5. But the fracture in Fig. 6 initiates earlier and propagates slower than that of Fig. 5. The fracture is also zigzaging, which demonstrates the effect of heterogeneity on macro fracture. Fig. 7 shows the shear fracture model of the corresponding homogeneous material. The fracture model in homogeneous material is obviously different that in heterogeneous material in that the fracture trace approaches a straight line and no zigzag fracture is observed. Comparing Figs. 5–7, the effect of heterogeneity on fracture behavior follows. Fig. 8 shows the mechanical response of the three Schemes of simulation. From Fig. 8, it can be seen that the peak-value corresponds to the homogeneous material is the highest while the two corresponding to the heterogeneous materials are lower. Though some component in the heterogeneous material can be
Fig. 5. Fracture process of heterogeneous materials (stiffnesscontrolled, Scheme I; each step = 1.2 · 102 mm): (a) Step = 15; (b) Step = 16; (c) Step = 17; (d) Step = 18.
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Z.N. Zhang, X.R. Ge / Theoretical and Applied Fracture Mechanics 47 (2007) 185–191
Fig. 6. Fracture process of heterogeneous materials (strain strength-controlled, Scheme II) (each step = 1.2 · 102 mm): (a) Step = 13; (b) Step = 14; (c) Step = 15; (d) Step = 16; (e) Step = 17.
Fig. 7. Fracture process of homogeneous materials (Scheme III, each step = 1.2 · 102mm): (a) Step = 15; (b) Step = 16; (c) Step = 17; (d) Step = 18; (e) Step = 19; (f) Step = 20.
using VMIB. The heterogeneity is represented in two ways. One is to make the stiffness of bond randomly distributed and another is to make the strain strength randomly distributed. To reflect the shear fracture mechanism, a bond evolution function is developed. The simulation results demonstrate that the present method can simulate the shear fracture process and can basically reflect the effect of heterogeneity on the fracture behavior. Acknowledgements
Fig. 8. Relationship of stress–strain (Hete. 1 – stiffness-controlled; Hete. 2 – strain strength-controlled; Homo – homogenous case).
stronger than the homogeneous material, they do not strengthen the integrated material. This demonstrates that the fluctuation might weaken the material strength to some degree. It can be attributed to the effect of heterogeneity on micro fracture. 5. Conclusion remarks This paper developed a multiscale method to simulate the shear fracture of heterogeneous materials
Project supported by National Natural Science Foundation of China (No. 50609013); supported by the Open Research Fund of the Key Laboratory of Rock and Soil Mechanics, Chinese Academy of Sciences under grant NO. Z110511. References [1] C.A. Tang, H. Liu, P.K.K. Lee, Y. Tsui, L.G. Tham, Numerical studies of the influence of microstructure on rock failure in uniaxial compression – Part I: effect of heterogeneity, International Journal of Rock Mechanics and Mining Sciences 37 (4) (2000) 555–569. [2] C.A. Tang, R.H.C. Wong, K.T. Chau, P. Lin, Modeling of compression-induced splitting failure in heterogeneous brittle porous solids, Engineering Fracture Mechanics 72 (4) (2005) 597–615.
Z.N. Zhang, X.R. Ge / Theoretical and Applied Fracture Mechanics 47 (2007) 185–191 [3] P.K.V.V. Nukala, S. Zapperi, S. Simunovic, Statistical properties of fracture in a random spring model, Physical Review E 71 (2005), 066106(1–11). [4] M. Sahimi, J.D. Goddard, Elastic percolation models for cohesive mechanical failure in heterogeneous systems, Physical Review B 33 (11) (1986) 7848–7851. [5] M. Sahimi, S. Arbabi, Scaling laws for fracture of heterogeneous materials and rock, Physical Review Letters 77 (17) (1996) 3689–3692. [6] M. Sahimi, S. Arbabi, Percolation and fracture in disordered solids and granular media: approach to a fixed point, Physical Review Letters 68 (5) (1992) 608–611. [7] M. Sahimi, S. Arbabi, Mechanics of disordered solids. III. Fracture properties, Physical Review B 47 (2) (1993) 713– 722. [8] H.J. Gao, P. Klein, Numerical simulation of crack growth in an isotropic solid with randomized internal cohesive bond, Journal of the Mechanics and Physics of Solids 46 (2) (1998) 187–218. [9] P. Klein, H. Gao, Crack nucleation and growth as strain localization in a virtual-bond continuum, Engineering Fracture Mechanics 61 (1) (1998) 21–48. [10] Z.N. Zhang, X.R. Ge, Micromechanical consideration of tensile crack behavior based on virtual internal bond in contrast to cohesive stress, Theoretical and Applied Fracture Mechanics 43 (3) (2005) 342–359. [11] Z.N. Zhang, X.R. Ge, Micromechanical modelling of elastic continuum with virtual multi-dimensional internal bonds, International Journal of Numerical Methods in Engineering 65 (2006) 135–146. [12] Z.N. Zhang, X.R. Ge, Y.H. Li, A multiscale mechanical model of materials based on Virtual Internal Bond theory, Acta Mechanica Solida Sinica 19 (3) (2006) 196–202. [13] Z.N. Zhang, X.R. Ge, A new quasi-continuum constitutive model for crack growth in an isotropic solid, European Journal of Mechanics – A/Solids 24 (2) (2005) 243–252.
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[14] K.Y. Volokh, H. Gao, On the modified virtual internal bond method, Journal of Applied Mechanics 72 (6) (2005) 969– 971. [15] G.C. Sih, Implication of scaling hierarchy associated with nonequilibrium: field and particulate, Theoretical and Applied Fracture Mechanics 37 (2001) 335–369. [16] G.C. Sih, B. Liu, Mesofracture mechanics: a necessary link, Theoretical and Applied Fracture Mechanics 37 (2001) 371– 395. [17] G.C. Sih, X.S. Tang, Dual scaling damage model associated with weak singularity for macroscopic crack possessing a micro/mesoscopic notch tip, Theoretical and Applied Fracture Mechanics 42 (2004) 1–24. [18] G.C. Sih, X.S. Tang, Simultaneity of multiscaling for macromeso-micro damage model represented by strong singularities, Theoretical and Applied Fracture Mechanics 42 (3) (2004) 199–225. [19] K. Terada, I. Saiki, K. Matsui, Y. Yamakawa, Two-scale kinematics and linearization for simultaneous two-scale analysis of periodic heterogeneous solids at finite strain, Computer Methods in Applied Mechanics and Engineering 192 (31–32) (2003) 3531–3565. [20] J.E. Marsden, T.J.R. Hughes, Mathematical Foundations of Elasticity, Prentice-Hall, Englewood Cliffs, NJ, 1983. [21] R.W. Ogden, Non-linear Elastic Deformations, John Wiley and Sons, New York, 1984. [22] Z.N. Zhang, X.R. Ge, Y.H. Li, Application of VMIB (Virtual Multi-dimensional Internal Bond) to the uniaxial failure of rocklike materials, Journal of Geotechnical Engineering 28 (4) (2006) 504–509 (in Chinese). [23] C.S. Chang, T.K. Wang, L.J. Sluys, J.G.M. Van Mier, Fracture modelling using a micro-structural mechanics approach-I. Theory and formulation, Engineering Fracture Mechanics 69 (17) (2002) 1941–1958. [24] J.G.M. Van Mier, Fracture Processes of Concrete, CRC Press, Boca Raton, 1997.
Multiscale shear fracture of heterogeneous material using the virtual internal bond Z.N. Zhang
a,b,* ,
X.R. Ge
c,d
a
c
Shanghai Institute of Applied Mathematics and Mechanics, Shanghai University, Shanghai 200072, China b Department of Civil Engineering, Shanghai University, Shanghai 200072, China Key Laboratory of Rock and Soil Mechanics, Institute of Rock and Soil Mechanics, Chinese Academy of Sciences, Wuhan 430071, China d Institute of Geotechnical Engineering, Shanghai Jiaotong University, Shanghai 200030, China Available online 6 March 2007
Abstract Material behavior entails multiscaling. The macro response of materials is affected by the microstructure. The method of VMIB (virtual multidimensional internal bond) developed from the VIB (virtual internal bond) for the continuum consists of discrete microscopic mass particles. These mass particles are connected with virtual bond possessing both normal and shear stiffness. The macro constitutive relation is derived in terms of bond stiffness. Shear fracture simulation of heterogeneous material involves introducing heterogeneity into the constitutive relation by two ways. First, heterogeneous material may consist of micro components with different stiffness. This character represented by the bond stiffness is assumed to be randomly distributed. Second, the micro components composed of the heterogeneity may possess the same stiffness, but their critical deformation, say the strain strength, may be different. This character represented by the strain strength is assumed to be randomly distributed. For a given statistical distribution function, the bond stiffness and the strain strength can be generated in numerical simulation implementation. Using VMIB, the macro response of materials is determined by the evolution of bond at the microscopic scale. To reflect micro mechanism of shear fracture, a phenomenological bond evolution function is proposed such that the shear fracture process of heterogeneous material is simulated. The results demonstrate that the present method can represent the shear fracture process and the basic characteristics of shear fracture of heterogeneous material. 2007 Elsevier Ltd. All rights reserved. Keywords: Heterogeneity; Multiscale numerical method; Shear fracture; Virtual multidimensional internal bond
1. Introduction Simulation of shear fracture of the heterogeneous material is important. The prevailing methods can *
Corresponding author. Address: Department of Civil Engineering, Shanghai University, Shanghai 200072, China. E-mail addresses: [email protected], [email protected] (Z.N. Zhang).
be divided into two categories. One is the continuum-mechanics-based method where the heterogeneity is introduced into the numerical model by assigning element mechanical properties with the statistical method [1,2]. The other is the discrete method where the continuous solid is represented mechanically by a discrete network such that each element represents a microelement of the heterogeneous solid [3–7]. The mechanical properties of
0167-8442/$ - see front matter 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.tafmec.2007.01.001
186
Z.N. Zhang, X.R. Ge / Theoretical and Applied Fracture Mechanics 47 (2007) 185–191
the heterogeneous elements are assumed to be randomly distributed. Both methods possess advantages and disadvantages. They can compensate each other. In other words, the continuum-mechanics-based method reduces the internal freedom though inadequate for treating the problems of fracture. The discrete method on the other hand is suited for simulating the fracture behaviors of solids, but it usually involves with a large number of freedom. It would be ideal to integrate the two approaches. The Virtual Internal bond (VIB) model [8,9] suggests this possibility. From the view of VIB, the continuous solid is considered to consist of microscopic material particles. Accordingly, the macro constitutive relation is thus derived from the cohesive law between particles. VIB spans two scales to consider the solid properties. The work in [10–12] developed VIB by introducing a shear bond into the original VIB or assigning the normal bond with a shear stiffness [13]. The model is known as the VMIB (Virtual Multidimensional Internal Bonds) model. It can present the diversity of Poisson ratio and applied to more extensive engineering materials. In [14], a modified VIB, i.e. MVIB is also proposed to address the fixed Poisson ratio problem of VIB by respectively considering the energy contribution of the spherical and deviatoric deformation. Both the VMIB and the MVIB are consistent with the classic linear elasticity theory. They are multiscale in character and consider the response of solids at a higher level by mechanisms at a lower scale. The multiscale analysis method can reveal the essence of macro response more comprehensively. Recent works in [15–18] have discussed multiscaling near the crack tip by using the concept of the strain energy density function. In [19], the dual scale (micro and macro) features were analyzed for periodic heterogeneous solids although the scheme applies to any two scales
such that the models can be connected and extended to multiscale. The micro mechanism of shear fracture is different from that of tensile fracture. It is more complex than that of tensile especially for heterogeneous materials. The fluctuations of mechanical properties can seriously affect the macroscopic behaviors of fracture. For example, the fracture always initiate simultaneously at different locations with a zigzag pattern. This work attempts to simulate the shear fracture process of heterogeneous materials using VMIB. The motivation here is to develop a multiscale method to model the heterogeneous materials. 2. Modeling of heterogeneous materials According to VIB and VMIB, the continuous solid is mechanically represented by a microscopic network. The macro constitutive relation at the continuum level is derived from the interactions between material particles. The methodology of VIB and VMIB can be graphically shown in Fig. 1. For homogeneous material, the constitutive relation is simple, which has been well analyzed by VIB and VMIB. However, the constitutive relation of heterogeneous materials is much more complex. The mechanical properties varies with the location. It is very difficult to accurately describe the properties at a certain location. Therefore, the statistical method is employed to describe the distribution of heterogeneity. To model the heterogeneous materials, the following two methods are employed. 2.1. Stiffness-controlled method For some heterogeneous materials, the micro component may have different stiffness. The microscopic fluctuation of stiffness may also affect the macroscopic fracture behavior. To begin with, let
Fig. 1. Heterogeneous solid, its representation with VMIB and the unit cell.
Z.N. Zhang, X.R. Ge / Theoretical and Applied Fracture Mechanics 47 (2007) 185–191
187
the microscopic bond stiffness be assumed to be randomly distributed such that the stiffness k, r can be expressed as
To simplify, the distribution density D(h, /) can be taken as unity for the isotropic solid. According to [10,11], the mean value of k; r can be determined by
k ¼ kk
3E 4pð1 2mÞ 3ð1 4mÞE r ¼ 4pð1 þ mÞð1 2mÞ
ð1Þ
r ¼ kr
where k, r are respectively the normal and shear stiffness of bond. Note that k; r are the corresponding mean value of k, r while k is a statistical variable and the mean value k ¼ 1 is held. In [4,6,7], each bond (or spring) is treated as an element to describe the heterogeneous materials at the microscopic level. This method can simulate the fracture behavior more precisely. However, this treatment often involves with a large number of freedom. To reduce the freedom, the number of bonds in the unit cell is assumed to be randomly distributed and the mechanical properties are considered at the unit cell level rather than the bond level, shown as Fig. 1. According to [8,10,11], the potential of this unit cell with an imposed deformation can be expressed as Z 2p Z p U¼ U L Dðh; /Þ sinðhÞdh d/ 0
þ
0
Z
2p
Z
0
ðU R1 þ U R2 þ U R3 ÞDðh; /Þ sinðhÞ dhd/
0
where UL, URi (i = 1, 2, 3) are respectively the potential contributions of normal stiffness and shear stiff2 2 ness of bond, U L ¼ 12 kkðni eij nj Þ ; U R1 ¼ 12 krðni eij g0j Þ ; 1 00 2 1 000 2 U R2 ¼ 2 krðni eij gj Þ , and U R3 ¼ 2 krðni eij gj Þ Here, n is the unit orientation vector of bond, n = (sin h cos /, sin h sin /, cos h) in the spherical coordinate system while g is the orientation vector perpendicular * * to n, g0 ¼ n ðx1 nÞ, g00 ¼ n ðx2 nÞ, and * g000 ¼ n ðx3 nÞ. The spatial distribution bond density is D(h, /). Within the framework of hypo-elastic theory [20,21], the elastic tensor of the unit cell can be derived as
¼
oU oeij oemn Z 2p Z p 0
þ
Z 0
ð4Þ
where E, m are respectively the mean Young’s modulus and Poisson ratio of heterogeneous materials. For two-dimensional cases, the elastic tensor expression is reduced to Z p C ijmn ¼ ðkkni nj nm nn þ krni g0j nm g0n 0
þ krni g00j nm g00n ÞDðhÞ dh
ð5Þ
where g 0 = [cos h, sin h]T; g00 = [ cos h, sin h]T; n = [sin h, cos h]T. The corresponding mean value of bond stiffness k; r can be calculated by 2E pð1 mÞ ð1 3mÞE r ¼ pð1 m2 Þ k ¼
ð6Þ
for the plane-stress problem and
p
ð2Þ
C ijmn ¼
k ¼
kkni nj nm nn Dðh; /Þ sinðhÞ dh d/
0 2p Z p
ð3Þ
ð7Þ
for the plane-strain problems. Numerically, the sample value of k can be generated by statistical methods, such as Monte Carlo and others, if a certain probability distribution law is given. 2.2. Strain strength-controlled method For a heterogeneous material, the micro components may have different strain strength even though the stiffness may be the same. The fluctuation of strain strength can also affect the macroscopic fracture behavior which can be demonstrated by assigning each bond with different strain strength, i.e. the critical deformation over which the bond is assumed to break. Again, the strain strength is assumed to be randomly distributed. For each bond, it is assumed that eb ¼ keb
krðni g0j nm g0n þ ni g00j nm g00n
0
000 þ ni g000 j nm gn ÞDðh; /Þ sinðhÞ dh d/
2E pð1 þ mÞð1 2mÞ ð1 4mÞE r ¼ pð1 þ mÞð1 2mÞ k ¼
ð8Þ
where eb is the strain strength and eb is its mean value.
188
Z.N. Zhang, X.R. Ge / Theoretical and Applied Fracture Mechanics 47 (2007) 185–191
3. Bond evolution function for shear fracture Tensile fracture is considered in [10] in which the macro response is attributed to the evolution of microscopic bond. Different bond evolution function leads to different macro response. In [22], a simple version of the bond evolution function for uniaxial compressive load was proposed, but it cannot account for the effects of the integrated deformation, i.e. combination of the normal deformation and rotation. A comprehensive description of bond deformation can be made by employing the integrated deformation bond index, which is similar to that in [23,24]
d¼
‘ b þ eb 0:5ð1 þ mÞeb
ð9Þ
Note that d is the integrated deformation index of bond, ‘ ¼ jni edij nj j; b ¼ jni eij gj j; Here, edij is the corresponding deviatoric tensor of strain tensor eij and eb is the uniaxial strain strength. According to the criterion in [23], the discrete element is linear under the condition d < 1 while the discrete element is assumed to break under the condition d P 1. The discrete element experience two phases, namely the linear deformation and the breaking state. To account for the pre-peak strengthening effect and the post-peak softening effect, the following bond evolution function is proposed: k k0 ¼ ð10Þ expðc dn Þ r r0 where c, n are model parameters, which governs different characters of macro response; k 0 and r0 are the initial bond stiffness, which can be determined respectively by Eqs. (4), (6) and (7). The effect of c, n on macro response is shown in Fig. 2. Fig. 2a shows that c mainly governs the peak stress, i.e. the uniaxial strength. The bigger the c, the lower is the peak stress. According to Fig. 2b, the n governs the ‘brittleness’ of the material. The bigger the n, the more brittle is the material. 4. Simulation example and discussion 4.1. Simulation strategy
Fig. 2. Effect of model parameters on stress–strain relationship: (a) effect of c, n = 40; (b) effect of n, c = 0.15.
To examine the performance of the method, the shear fracture process of a specimen under uniaxial compressive load is simulated. The dimensions and boundary conditions of the specimen are shown in Fig. 3. The 3-node plane triangular element is adopted in the FEM implementation. The node number is 435 and the element number is 784. The Gauss point number for computing elastic tensor, say Eq. (5), is 24, hence the sample volume of k is 24 · 784 = 18,816. As an example, let the statistical variable k obey the normal distribution function with the mean value k ¼ 1. Different heterogeneous materials require different statistical distribution function. The choice of probability distribution function will not be discussed. Before applying the FEM, the sample values of k are generated by statistical method. The distribution of k is shown graphically
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Table 1 Material parameters and the corresponding model parameters Material parameters
Model parameters
E (GPa)
m
eb · 103
c
n
23.6
0.15
2.0
0.15
4.0
4.2. Discussion
Fig. 3. Simulation object dimension and its boundary conditions: (a) dimensions; (b) boundary conditions.
Fig. 4. The value distribution of k.
in Fig. 4. The material parameters and the model parameters are listed in Table 1. The displacement-controlled loading method is adopted. The simulation is performed in three schemes as the following: Scheme I: Keep the strain strength eb constant and treat the bond stiffness as a statistic variable. This scheme refers to the stiffness-controlled method mentioned in Section 2.1. Scheme II: Keep the bond stiffness constant and treat the strain strength eb as a statistic variable. This scheme refers to the strain strengthcontrolled method mentioned in Section 2.2. Scheme III: Keep both the bond stiffness and strain strength constant. This corresponds to the case of treating the material as a homogeneous one.
Figs. 5–7 show respectively the simulation results of Schemes I, II and III. Fig. 8 shows the mechanical response of the three Schemes of simulation. From Fig. 5, it can be concluded that no fracture initiates in the 15th step while fracture begin to initiate simultaneously at several locations in the 16th step. In the 17th and 18th step, the fracture propagates and finally goes though the specimen. From the inclination of fracture, it can be concluded that it is a shear fracture. This agrees with the observation by experiment. The zigzag of fracture demonstrates the effect of heterogeneity on macro fracture. Fig. 6 shows the fracture process of strain strength-controlled scheme. The fracture mode in Fig. 6 is similar to that in Fig. 5. But the fracture in Fig. 6 initiates earlier and propagates slower than that of Fig. 5. The fracture is also zigzaging, which demonstrates the effect of heterogeneity on macro fracture. Fig. 7 shows the shear fracture model of the corresponding homogeneous material. The fracture model in homogeneous material is obviously different that in heterogeneous material in that the fracture trace approaches a straight line and no zigzag fracture is observed. Comparing Figs. 5–7, the effect of heterogeneity on fracture behavior follows. Fig. 8 shows the mechanical response of the three Schemes of simulation. From Fig. 8, it can be seen that the peak-value corresponds to the homogeneous material is the highest while the two corresponding to the heterogeneous materials are lower. Though some component in the heterogeneous material can be
Fig. 5. Fracture process of heterogeneous materials (stiffnesscontrolled, Scheme I; each step = 1.2 · 102 mm): (a) Step = 15; (b) Step = 16; (c) Step = 17; (d) Step = 18.
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Fig. 6. Fracture process of heterogeneous materials (strain strength-controlled, Scheme II) (each step = 1.2 · 102 mm): (a) Step = 13; (b) Step = 14; (c) Step = 15; (d) Step = 16; (e) Step = 17.
Fig. 7. Fracture process of homogeneous materials (Scheme III, each step = 1.2 · 102mm): (a) Step = 15; (b) Step = 16; (c) Step = 17; (d) Step = 18; (e) Step = 19; (f) Step = 20.
using VMIB. The heterogeneity is represented in two ways. One is to make the stiffness of bond randomly distributed and another is to make the strain strength randomly distributed. To reflect the shear fracture mechanism, a bond evolution function is developed. The simulation results demonstrate that the present method can simulate the shear fracture process and can basically reflect the effect of heterogeneity on the fracture behavior. Acknowledgements
Fig. 8. Relationship of stress–strain (Hete. 1 – stiffness-controlled; Hete. 2 – strain strength-controlled; Homo – homogenous case).
stronger than the homogeneous material, they do not strengthen the integrated material. This demonstrates that the fluctuation might weaken the material strength to some degree. It can be attributed to the effect of heterogeneity on micro fracture. 5. Conclusion remarks This paper developed a multiscale method to simulate the shear fracture of heterogeneous materials
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