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The influence of plasticity mismatch on the growth and coalescence of spheroidal voids on the bimaterial interface
International Journal of Plasticity 18 (2002) 249±279 www.elsevier.com/locate/ijplas
The in¯uence of plasticity mismatch on the growth and coalescence of spheroidal voids on the bimaterial interface Zhenhuan Li a,*, Wanlin Guo b,c a Department of Mechanics, Huazhong University of Science and Technology, Wuhan 430074, PR China The National Key Laboratory of Mechanical Structural Strength and Vibration, Xi'an Jiaotong University, Xi'an 710049, PR China c Department of Aircraft Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, PR China b
Received in ®nal revised form 28 June 2000
Abstract To investigate the macro-mechanical response and micro-mechanism of damage by void growth and coalescence on the interface in a bimaterial system, detailed ®nite element computations of a representative cylindrical cell containing a spherical void are performed. By comparison with the response of a homogeneous material cell model, signi®cant eects of the matrix plasticity mismatches due to the yield stress and the strain hardening exponent on the void growth and coalescence are revealed: (1) The growth rate of the void on the bimaterial interface is much faster than that in the homogeneous material, and the critical coalescence strain of the void on the interface is only about half of that in homogeneous material. (2) Due to the dierence in the deformation resistance of the matrix materials in the bimaterial system, all computations indicate that deformed voids are seriously distorted and the linking of adjacent voids takes place in the softer matrix material. Comparison of the computational results with the classical Rice±Tracey (R-T) model shows that the R-T model cannot make good prediction for the growth of the void on the bimaterial interface. On the basis of large numbers of numerical simulations, a correction coecient is introduced to improve the R-T model. # 2002 Elsevier Science Ltd. All rights reserved. Keywords: A. Voids and inclusions; B. Porous material; C. Finite element; Plasticity mismatch; Interface
* Corresponding author. E-mail address: [email protected] (Zhenhuan Li). 0749-6419/02/$ - see front matter # 2002 Elsevier Science Ltd. All rights reserved. PII: S0749-6419(00)00078-4
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1. Introduction The mechanics and micro-mechanisms of fracture on a bimaterial interface are the topics of considerable interests in the design of a vast number of structural components, such as welding structure, coating systems and advanced composites. Increasing applications of functionally graded materials (FGM) also require a basic understanding of the eect of material combination on fracture resistance. Similar to the processes in homogeneous materials, ductile fracture in a bimaterial system takes place as a successive processes of nucleation, growth and coalescence of micro-voids on the bimaterial interface. When the bimaterial system is loaded, voids are ®rst nucleated on the interface by local interface decohesion, then plastic straining of the surrounding ductile matrix material will control the growth process until a local internal necking and plastic collapse of the intervoid ligament occur. The latter event marks the incipient ductile fracture. During the last thirty years, there was a history of eorts directed at developing predictive tools for the growth and coalescence of voids in homogeneous materials. The growth of an isolated cylindrical void was investigated ®rst by McClintock (1968), then an analytical model (R-T model) for the spherical void embedded in an in®nite perfectly plastic solid subjected to remote triaxial stresses was presented by Rice and Tracey (1969). Although these pioneering works are only valid for lower porosity volume fraction, they revealed an essential feature that the void growth rate depends exponentially on the stress triaxiality. On the basis of the R-T model, Gurson (1977) further considered a cell containing a spherical void and suggested a plastic potential function for porosity materials. Since this model has the advantages of simulating damage and fracture in ductile materials, considerable attentions have been paid to it. However, the original Gurson model was derived from an approximate solution for rigid-perfectly plastic material cell containing a centered spherical void. To extend the Gurson model to periodic arrays of cylindrical and spherical voids in hardening materials, Tvergaard (1981, 1982) introduced two adjustable parameters q1 and q2 . However, recent investigations show that the q-values depend on both the strain hardening (n) and the ratio of the yield strength (ys ) to Young's modulus (E) of the matrix materials (Faleskog et al., 1998). The Gurson model also ignores voids interaction so that it cannot predict the coalescence of voids by localized internal necking of the intervoid matrix. Therefore, Tvergaard and Needleman (1984) made further modi®cation to the Gurson model to describe the drop-o of stress carrying capacity in materials after voids coalescence. Their constitutive relation has been referred to as GTN model and various of its extensions have been used to predicate ductile damage and fracture in metal materials (Xia and Shih, 1995a,b, 1996; Ruggieri et al., 1996; Faleskog et al., 1998; Gao et al., 1998a,b). On the other hand, numerical simulations have provided us better understanding of void growth and coalescence mechanisms. To our knowledge, original works to simulate the interaction of voids by ®nite element method (FEM) were made by Needleman (1972) and Tvergaard (1981), who analyzed two-dimensional square arrays of cylindrical holes under plane strain condition. However, compared with the real situation of spherical voids in three dimensions, such a two dimensional
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(2D) model cannot describe the stress triaxiality properly and always overestimates the interaction of voids. Further ®nite element computations have been carried out to investigate the three-dimensional eects by the aid of the axisymmetrical cylindrical unit cell containing a spherical void under various triaxial loading conditions (Andersson, 1977; Tvergaard, 1982; Bourcier et al., 1986; Koplik and Needleman, 1988; Brocks et al., 1995; Steglich et al., 1998; Sùvik and Thaulow, 1997; Li et al., 2000). However, such a cylindrical unit cell does not allow for the adjustment of arbitrary stress ratios in all three directions. The three-dimensional (3D) cell models subjected to stresses parallel to the principal geometrical axes are developed (McMeeking and Hom, 1990; Worswick and Pick, 1990; Nagaki et al., 1993; Richelsen and Tvergaard, 1994; Ruggieri et al., 1996; Zhang and Zheng, 1997; Zhang et al., 1999). Other cell structures, such as cubic primitive model and body centered cubic model, have been analyzed by Kuna and Sun (1996). It is shown that the axisymmetrical cell model leads to larger deformation and a lower failure stress than 3D cell models. This means that the hexagonal arrangement with axisymmetric unit cell simulation gives a lower bound solution for material failure by voids. This is encouraged from the viewpoint of structural safety design. Compared with the mass of researches into the mechanics and micro-mechanisms of ductile fracture by void growth and coalescence in the homogeneous materials, little attention has been devoted to the micro-mechanisms of ductile fracture on the interface in an elastic±plastic bimaterial system under triaxial stress conditions. As is well known, in the bimaterial system, the voids on the interface are surrounded by two kinds of matrix materials possessing dierent plastic properties. Recent investigation (Faleskog et al., 1998) has shown, for the homogeneous material, that over a range of constraint and initial void volume fraction typically encountered in applications, the growth and coalescence of voids rely strongly on the strain hardening exponent (n) and the strength ratio (ys =E) of the matrix material. Therefore, for the bimaterial system, we can roughly speculate that the plasticity mismatch of the matrix materials, which have no elastic mismatch, will have signi®cant eects on the growth and coalescence mechanisms of voids on the bimaterial interface. However, to our knowledge, no systematic analyses on the void growth and coalescence on the elastic± plastic bimaterial interface have been made in the existing literature. It is also susceptible whether the existing classical micro-mechanical model driven from the homogeneous porosity material can predict rationally the combination eects of the plasticity mismatch on the growth and coalescence of voids on the bimaterial interface. In order to predict ductile fracture on the interface in the bimaterial system, through knowledge of all the three basic stages relating to void nucleation, growth and coalescence are required. Due to the complexity of void nucleation process, a great deal of works remains to be done to obtain a better understanding of the relating phenomena. Therefore, the present work will consider only the growth and coalescence mechanisms of the voids on the interface, and the emphasis is focussed on the eect of the matrix plasticity mismatch. In particular, the two kinds of matrix materials composing the bimaterial system have similar elastic properties but dierent plastic properties.
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2. Cell model 2.1. Model description Because the cell model can describe the relation between the macroscopic mechanical responses and the microstructures of materials, it has widely been used to simulate and study the behavior of porous solids (Koplik and Needleman, 1988). The problems considered here are the growth and coalescence of the voids on the interface in the elastic-plastic bimaterial system subjected to far triaxial stresses, as illustrated in Fig. 1. The bimaterial system comprises two materials with dierent strain hardening exponents or yield stresses, and the intervoid bimaterial interface is bonded perfectly and is dicult to decohere during plastic deformation. In consistency with a common damage mechanics approach, the cylindrical cell can be regarded as a representative volume element when a periodic array of identical voids on the bimaterial interface is assumed. Fig. 1 shows the geometry of the cylindrical cell containing one spherical void on the bimaterial interface. For convenience, the rectangular coordinates (x, y, z) are adopted in this paper (see Fig. 1). The origin of the coordinate system is located at the midpoint of the bottom of the cell. Obviously, the initial void volume fraction can be written as
Fig. 1. Illustration of the problem and the 2D cell model.
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f0
2a30 : 3R20 H0
253
1
where a0 is the initial radius of the void, R0 and H0 are the initial radius and half of the height of the cell, respectively. In this investigation, only f0 0:001 is considered. The current void volume fraction f is de®ned as the ratio of the total void volume to the cell volume. For elastic-plastic incompressible matrix materials, the void volume fraction can be computed by f1
1
f0 V0 Ve : V
2
where V0 and V are the initial and current volume of the cell, respectively, Ve is the increase in volume of the cylindrical cell due to elastic dilatation arising from the imposing hydrostatic stress which can be approximated by (Koplik and Needleman, 1988): 3 1 2 Ve V0 1 f0 h : 3 E where E and are Young's modulus and Poisson's ratio of the matrix material. h is the macroscopic hydrostatic stress. Obviously, for a porous bimaterial system with identical elastic property but dierent plastic property, Eq. (3) can also be used to describe reasonably the elastic volume dilatation of the cell. Considering the dierences in matrix material properties, half of the cell model (region x0 50, 04z42H0 ) is analyzed with FEM, since the problem has re¯ectional symmetry with respect to the axial x 0. The ®nite element meshes of the geometry are constructed with an eight-node quadratic iso-parametric element as shown in Fig. 1 and 22 Gauss integral is adopted. The axisymmetric FE cell model consists of 480 elements and 1535 nodes as shown Fig. 1. To avoid the eect of FE mesh on the computational results, identical FE meshes are adopted in all FE analyses for the bimaterial and homogeneous material cell models. For comparison with FE results, a modi®ed R-T model is also considered. The RT model predicts the void growth by means of the radius a of the spherical void. An approximation of the relation derived by Rice and Tracey can be written as: : a : 0:283exp R " pe : a
4
where "pe is the eective plastic strain, rate is a modi®cation coecient and R is the stress triaxiality de®ned as h 5 R : e where e is the macroscopic eective stress.
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Then the increasing rate of the void volume fraction can be written as : : f a : 3 0:849exp R "pe : a f
6
In the classical R-T model, is set as approximately 1:5. 2.2. Boundary conditions and loading method Due to the axial symmetry, the shear stresses on x, z-plane will vanish. Therefore, the axial, radial and tangential directions will be the principal directions of stresses. Furthermore, for axial symmetry case the tangential and radial components of the macroscopic stress and strain tensors subjected on the cell boundary will be of equal magnitude. The structure is subjected to a homogeneous elongation, Uz1 , in the axial direction and the radial displacement Ux1 is also kept to be homogeneous by constraint conditions. The boundary conditions for the axisymmetric region analyzed numerically can be described as: : : : uz 0; Fx 0; Fy 0; on z 0 : : : Fx Fy Fz 0; on x2 y2 z2 a2 : : : : uz Uz1 ; Fx 0; Fy 0; on z 2H : : Fy 0; Fz 0; on x R Here, a is the current radius of the void, R and H are the current radius and half : of the height of the cell, respectively; uz is the axial displacement rate; Fi (i=x, y, z) is the traction in the ith direction. The prescribed radial concentrated load Fx1 is determined from the condition that the average macroscopic true stresses acting on the cell follow the proportional history : x x : : z z
7
with as a prescribed constant determined by the controlled stress triaxiality R and sx sy sz
Fx1 : 2 R0 Ux 2H0 Uz
Fz1 : R0 Ux 2
8 9
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255
where i (i x; y; z) is the macroscopic principal stress in the ith direction, Fz1 is the axial concentrated load which can be obtained by a spring attached to the node at the axis of symmetry. For the axisymmetric case, the corresponding macroscopic eective stress and hydrostatic stress can be given by: e jz
1 x j; h z 2x : 3
10
and the stress triaxiality R can be written as R
h 1 2 : e 3 1
11
The macroscopic principle strains and eective strain can be expressed by: R H 2 "x "y ln ; "z ln ; "e "x "y : R0 H0 3
12
In order to control the macroscopic stress triaxiality R throughout the loading history, has to remain constant whereas the ratio of the prescribed strains "x ="z will vary with time. To satisfy this requirement, a spring element, which measures the axial load Fz1 arising from the prescribed axial displacement Uz1 , is introduced. The radial concentrated load Fx1 depending on R is then applied. This loading process can be achieved automatically in the FE computation by the equilibrium iteration technology (Sùvik and Thaulow, 1997). 2.3. Matrix material description In this work, the Prandtl±Ruess's constitutive equation is considered to describe elastic-plastic stress vs. strain responses of the matrix materials. In the case of large deformation, the Prandtl±Ruess's constitutive equation can be written as: 2 3 7 0ij 0kl 7 13 D kl : 2 2 2 1 5 1 h 3 e 3 E where D ij is the rate of deformation tensor, and D ij 1=2 @vi =@xj @vj =@xi , :vi is the rate of material point, xi is the instantaneous coordinate of material point; Jij is the Jaumann rate of Cauchy stress ij ; ij is the Kronecker delta; h is the strain hardening parameter; 0ij is the deviatoric stress tensor de®ned by 0ij ij 13 ij kk ; e is the eective stress of the matrix; l is the loading coecient determined by the loading criteria: ( 0 0ij D ij < 0 l : 14 1 0 D ij 50 : Jij
E 6 6ik jl ij kl 1 4 1 2
ij
l
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The true stress vs. true strain responses of the matrix materials under uniaxial tension is assumed to be in the linear elastic±power hardening plastic form: 0 ys ; E"; "4 E B 15 @ E"n ys ys : ; "> "ys E where and " are the uniaxial stress and strain respectively, ys and "ys are the uniaxial yield stress and yield strain, respectively, n is the strain hardening exponent. In this investigation, the emphases will be focused on the in¯uences of plasticity mismatch on void growth and coalescence on the interface. Considering the strong dependences of void growth and coalescence on the strain hardening exponent (n) and yield strength (ys =E) of the matrix material, dierent combinations of n and ys =E between the matrix materials are considered: 1. The plasticity mismatch due to the matrix material hardening exponent Ð Material 1: n1 0:1, ys =E 0:0025, ys 500 MPa Material 2: n2 0:2, ys =E 0:0025, ys 500 MPa 2. The plasticity mismatch due to the matrix material yield stress Ð Material 1: n 0:1, ys =E1 0:00125, ys 250 MPa Material 2: n 0:1, ys =E 2 0:0025, ys 500 MPa For reference, the growth and coalescence of voids in the homogeneous media, which may be hard or soft, will also be analyzed. 3. Analyses of the computational results In this investigation, we will explore the dependences of the void growth and coalescence on the macroscopic stress triaxiality and the combination of plasticity mismatch. A broad spectrum of stress triaxiality 0:754R 43:0 will be investigated, which covers the range from necking smooth bars to sharp crack bodies (Koplik and Needleman, 1988). As is well known, for lower triaxiality level, the shear mode of void growth and coalescence will be dominant, which is clearly dierent from the simpler internal necking mode of voids considered here in higher triaxial stress ®elds. This goes beyond the scope considered in this paper. In the present paper, the macroscopic response of the cell model in the homogeneous material is ®rst analyzed in Section 3.1 to validate the FE method adopted in this paper. The eects of plasticity mismatches due to dierent strain hardening exponents and yield stresses on void growth and coalescence are studied in Sections 3.2 and 3.3. Finally, a modi®cation of the R-T model is made to describe the growth of voids on the interface reasonably.
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3.1. The macroscopic response of cell model For variations of stress triaxiality from R 1:0 toR 3:0, Fig. 2 shows the computing eective mechanical response of the cell model for the homogeneous material with n 0:1 and ys =E 0:0025. Fig. 2(a) displays the macroscopic eective stress- eective strain curves, which show the competition between the matrix material strain hardening and porosity induced softening. The abrupt fall in the strain-stress curve corresponds to void coalescence. This event is characterized by sudden shift to a uniaxial straining mode where any further plastic deformation is limited to the intervoid radial ligament. This macroscopic response of cell model substantively re¯ects the break down of the load carrying capability of the cell caused by void growth and internal necking. Fig. 2(c) gives the change in cell radius against the eective strain. It is shown clearly that the plastic collapse is eventually reached, at which the eective strain increases while the radial strain remains approximately constant. This implies that ¯ow localization takes place in the ligament between radial adjacent voids. As can be seen in Fig. 2(b), the void volume fraction increases steeply at this point and this event is associated with the sudden load drop in Fig. 2(a). Therefore, it is reasonable to de®ne the point, where the macroscopic radial strain turns to be constant with increasing macroscopic eective strain, as the inception of void coalescence (Koplik and Needleman,1988; Sùvik and Thaulow, 1997). To validate the present FE computations, Table 1 gives out the dependence of the critical eective strain "e c on the stress triaxiality R , which shows an excellent agreement with the prediction by K-N (Koplik and Needleman, 1988). Due to lower initial void volume fraction f0 , the present critical eective strain is slightly higher than that predicated by B-S-H and L-P-S, however, the error is less than 5%. So it can be concluded that the precision of the present FE computation is high enough to analyze the growth and coalescence of voids in the bimaterial system. 3.2. In¯uence of the mismatch in strain hardening exponents In this section, the bimaterial system with dierent strain hardening exponents, or n1 0:1 and n2 0:2, will be considered. Fig. 3 shows the macroscopic stress-strain responses of the homogeneous material cell as well as the bimaterial cell at dierent stress triaxiality levels. For two homogeneous materials with dierent hardening exponents, the critical coalescence strains are dierent. The critical strains in the material with higher strain hardening exponent are slightly higher for all triaxiality levels considered. For the bimaterial system, as expected, until the attainment of the critical constant radial strain, the response curve of the bimaterial cell lies between the curves of the two kinds of homogeneous material cells. However, it is surprising that the critical coalescence eective strain of the bimaterial system is remarkably lower than that of both homogeneous matrix materials. At ®rst glance, this result seems to be unexpected, since the critical coalescence strain of the bimaterial system seems to be supposed to lie between the critical eective strains of the two kinds homogeneous materials. Again, the actual explanation lies in the earlier ¯ow localization in the softer materials.
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Fig. 2. Finite element results for the homogeneous porosity material under dierent stress triaxialities R =1.0, R =2.0 and R=3.0: (a) macroscopic eective stress±eective strain response; (b) void volume fraction vs. macroscopic eective strain; (c) radial strain vs. macroscopic eective strain.
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259
Fig. 2. (continued) Table 1 The critical eective strain "e c in comparison to the results by Koplik and Needleman (1988) (K-N), Brocks, Sun and HoÈnig (1995) (B-S-H) and Leblond, Perrin and Suquet (1994) (L-P-S) R
Present results (f0 0:001)
K-N (f0 0:0013)
B-S-H (f0 0:002)
L-P-S (f0 0:002)
1.0 2.0 3.0
0.82 0.22 0.08
0.83 0.22 0.08
0.79 0.20 0.08
0.79 0.21 0.07
Fig. 4 compares the void growth in the homogeneous materials with that on the bimaterial interface for dierent stress triaxiality levels. The predictions by the R-T model are also shown in the ®gure. It is easy to ®nd that, for the homogeneous materials with dierent strain hardening exponents, the growth rate of voids in the softer material is slightly faster than that in the harder materials. The R-T model gives good predictions of the growth rate of voids when 1:0 < R < 2:0, but overestimates the void growth for lower or higher stress triaxiality. In addition, by comparisons with the growth rate of voids in homogeneous materials, the volume fractions of voids on the bimaterial interface increase much more rapidly. As shown in Fig. 4, for the stress triaxiality levels considered in this study, the R-T model, which was obtained by solving the boundary problem of the homogeneous in®nite prefect plastic material containing void, can not predict the void growth on the
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Fig. 3. Macroscopic eective stress vs. eective strain curves of the bimaterial system with dierent hardening exponents under dierent stress triaxialities: (a) R=0.75, (b) R=1.0, (c) R =2.0, (d) R=3.0.
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Fig. 3. (continued)
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Fig. 4. Void volume fraction vs. eective strain curve of the bimaterial system and homogeneous materials with dierent hardening exponents under dierent stress triaxialities: (a) R=0.75, (b) R=1.0, (c) R=2.0, (d) R=3.0.
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Fig. 4. (continued)
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interface correctly. Therefore, it is necessary to modify the original R-T model for bimaterial systems. Fig. 5 shows the variations of the critical coalescence eective strain "e c with the stress triaxiality R for the homogeneous materials and the bimaterial system. It can be found that "e c of the bimaterial is about 50% lower than that of the homogeneous materials. Correspondingly, Fig. 6 plots the critical void volume fraction fc as a function of R . Although the growth of voids in the bimaterial system is signi®cantly rapider than that in the homogeneous materials, fc of the bimaterial system is much lower. It is worthy to notice that fc is sensitive to the stress triaxialty level for both the bimaterial system and homogeneous materials. It is also susceptible whether fc can be regarded as a material constant to predict fracture in specimens with dierent geometrical con®gurations. Fig. 7 shows the initial and deformed shapes of the cell model under dierent stress triaxiality level. It is striking to ®nd that the material plasticity mismatch and stress triaxiality cause such signi®cant dierences in the deformed void shape. For homogeneous materials, it can be seen that the void expands in the main tension direction and becomes prolate ellipsoidal when the stress triaxiality level is lower. On the contrary, the void grows in radial direction and become oblate ellipsoidal in the higher triaxial stress ®elds. For voids on the bimaterials interface, the growth of the semi-ellipsoidal void in the harder material is very dicult; however, the semiellipsoidal void in the softer material grows rapidly and expands along radial direction.
Fig. 5. The critical eective strain vs. stress triaxiality for the bimaterial system and the homogeneous materials with dierent hardening exponents.
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Fig. 6. The critical void volume fraction vs. stress triaxiality for the bimaterial system and the homogeneous materials with dierent hardening exponents.
As shown in Fig. 7, due to higher constraint from the harder material, radial expanding of the semi-ellipsoidal void in the softer material doesn't take place on the interface but a small distance from the interface so that the shape of void is distorted badly. Similar to the shape change of void in homogeneous materials, the pro®le of void on the interface is slightly elongated in the main tension direction at lower stress triaxiality levels; on the contrary, the void are relatively elongated in the radial direction at higher stress triaxiality levels. Since the R-T model does not take account of void shape changes, it of course has no capability of predicting of the growth rate of voids on the bimaterial interface and in homogeneous materials. 3.3. In¯uence of the mismatch in yield stress In this section, the attention is restricted to the situation where the bimaterial system comprises two kinds of elastic-plastic materials possessing identical elastic property and strain hardening exponent but dierent yield stress. For convenience, a representative case with ys 2 2 ys 1 will be investigated. As shown by Fig. 1, the stronger material carries the same axial load as the softer material; as a result the strain in the softer material is much higher than that in the harder material. Therefore, the plastic ¯ow localization is expected to take place earlier in the softer material.
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Fig. 8 shows the curve of the macroscopic eective stress vs. eective strain of the cell for the bimaterial system. For comparison, the macroscopic responses of the two corresponding homogeneous materials are also displayed in the ®gure. As expected,
Fig. 7. Deformed cells when ee=(ee)c for (a) the bimaterial system with dierent hardening exponents, and (b) the homogeneous material under dierent stress triaxialities.
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267
Fig. 8. The macroscopic eective stress vs. eective strain curves of the bimaterial system with dierent yield stresses under dierent stress triaxialities: (a) R=0.75, (b) R=1.0, (c) R=2.0, (d) R=3.0.
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Fig. 8. (continued)
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for the homogeneous materials, the dierence in yield stress leads to signi®cant differences in load carrying capability. Due to the identical strain-hardening exponent, the eect of yield stress on the critical eective strain "e c associated with the void coalescence is very weak. However, for the bimaterial system, it is interesting to ®nd that "e c is much lower than that of the homogeneous materials in the whole range of triaxiality considered. This unexpected phenomenon mainly comes from the earlier plastic ¯ow localization in the softer material of the bimaterial system. Fig. 9 illustrates the eect of yield stress mismatch on the growth rate of void on the interface at dierent stress triaxiality levels. For comparison, the FE results of the two corresponding homogeneous materials and the predictions of the classical R-T model are also displayed in Fig. 9. For homogeneous materials, the eects of yield stress on void growth and coalescence can be ignored. However, in the bimaterial system, the mismatch in yield stress can increases the growth rate of voids on the interface remarkably. This means that the damage on the interface is much more dangerous than that in homogeneous materials. It is well known that the classical R-T model is derived from a homogeneous porosity material. From Fig. 9, it can be seen clearly that the classical R-T model does not give a good prediction for the growth rate of voids on the interface between materials with dierent yield stresses. Figs. 10 and 11 show variations of "e c and fc with stress triaxiality R for two homogeneous materials and the corresponding bimaterial system. It can be found that, for homogeneous materials, the dierence of "e c caused by change in yield stress is very small, and the "e c R curves of the two homogeneous materials are almost coincident. However, for the bimaterial system, "e c is about 50% lower at the same triaxiality level. For the dierent stress triaxiality levels considered, Fig. 11 indicates that the critical void volume fraction fc of the bimaterial system is much lower than that in the two homogeneous materials. It can be noticed further that fc has strong dependence on R , and it is not a monotonic function of R , fc increases in the lower stress triaxiality range to its peak value fc max , then decreases gradually with increasing R . Fig. 12 displays the initial and deformed shapes of the cell under dierent stress triaxialities. By comparison with the shape changes of voids in homogeneous materials as shown in Fig. 7(b), it can be found that the deformed void shapes are seriously misshapen in the bimaterial system with yield stress mismatch. In the bimaterial system, the growth of semi-ellipsoidal void in the softer material is easier than that in the harder material. As depicted in Fig. 12, radial expanding of the semi-ellipsoidal void in the softer material takes precedence over that in the harder material; as a result the deformed void shape in the bimaterial interface becomes misshapen. 3.4. Modi®cation of the R-T model As aforementioned, since the classical R-T model was obtained from the homogeneous perfect plastic material containing an ideal spherical void and ignored the void shape changes in dierent triaxial stress ®elds, this model has no capability of predicting the growth of voids on the bimaterial interface. One possible way of improving this model is to introduce a modi®cation coecient as a function of stress triaxiality, which re¯ects indirectly the eect of the void shape, as shown by Eq. (4).
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Fig. 9. The void volume fraction vs. the macroscopic eective strain for the bimaterial system and homogeneous materials with dierent yield stresses under dierent stress triaxialities: (a) R=0.75, (b) R=1.0, (c) R=2.0, (d) R=3.0.
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Fig. 9. (continued)
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In order to evaluate such a modi®cation factor , void growth predictions by the classical R-T model are ®tted to the FE results of the cell model. The computational results for the dependence of on R are shown in Fig. 13. Even for the homogeneous materials, it can be seen that is not a constant 1.5 as in the classical R-T model but has close relation with the stress triaxiality. When 1:0 < R < 2, is close to the constant 1.5 but for lower or higher R , will be less than 1.5. It is also shown that the dierence in the plastic properties such as strain-hardening exponent and yield stress only have a weak eect on in homogeneous materials. However, in a bimaterial system, the situation is completely dierent. It is very interesting to notice that not only strongly depends on the stress triaxality but also has close relation with the plasticity mismatch of the bimaterial system. For the two kinds of plasticity mismatches discussed above, is a monotonic decreasing function of R within the stress triaxiality range considered. By ®tting the FE results of the bimaterial cell model, can be estimated by exp lR :
16
where 2:67, l 0:24 for the hardening exponent mismatch and 2:97, l 0:22 for the yield stress mismatch considered in the present investigation.
Fig. 10. The critical eective strain vs. stress triaxiality for the bimaterial system and the homogeneous materials with dierent yield stresses.
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Fig. 14 displays the comparison of the FE results of the cell with the predictions by the modi®ed R-T model for the void growth on the bimaterial interface. In the ®gure, the void volume fraction fis plotted as a function of the eective strain "e . As expected, for dierent stress triaxiality and the two plasticity mismatches considered, the predictions by the modi®ed R-T model give satisfying agreement with the FE results of cell model for the bimaterial system. 4. Discussion The stress triaxialty concerned in the paper is the macroscopic stress triaxiality which is average or global quantity. In fact, due to the localized plastic strain, the microscopic stress triaxiality in the cell consisting of homogeneous matrix material cannot be uniform. The local microscopic strain mode and stress triaxiality have obvious eect on the micro-mechanism of the void growth and coalescence (Li et al., 2000). For bimaterial system, there are two semi-ellipsoidal voids, one in the soft material and other in the hard material. The axial forces are identical in both cells but the lateral forces are dierent, leading to two dierent stress ratios. For two constant macroscopic stress triaxialities R 1=3 and R 1, Fig. 15 gives the changes of the local stress triaxialty in the upper soft material cell and the lower
Fig. 11. The critical void volume fraction vs. stress triaxiality for the bimaterial system and the homogeneous materials with dierent yield stresses.
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hard material cell with the macroscopic eective strain. It can easily be found that, though the macroscopic stress triaxiality of the bimaterial cell remains a constant during deformation, the stress triaxiality of soft materials cell is higher than that of hard materials cell. Corresponding to void coalescence, the stress triaxiality of the soft cell becomes higher and higher but that of the hard cell become lower and lower with increasing eective strain. It is worthy of noting that the stress triaxiality in the upper can reach 0.75 but that in the lower hard cell can only reach 0.1 corresponding to uniaxial tension R 1=3. This means that though the macroscopic stress triaxiality of bimaterial cell is lower, the local stress triaxiality of soft material cell can still reach enough high to drive the cavity growth and coalescence. Due to earlier localized plastic strain and higher local stress triaxiality in the soft material cell, the growth of semi-ellipsoidal void in the soft material is faster than that in the hard material. This means that the void coalescence in the bimaterial system is easier than that in the homogeneous material. Due to dierent void growth rates in the soft and the hard material cell, it seems doubtful that the total porosity is a good measure. How to de®ne the porosity is very important to understand the micro-mechanism of void growth and coalescence. However, due to interfering and shielding of two semiellipsoidal voids in the bimaterial, the law for cavity growth in the interface will be very complicate and dierent from that in the corresponding homogeneous material. Further researches relating these issues are to be appear in another paper.
Fig. 12. Deformed cells when ee=(ee)c for the bimaterial system with dierent yield stresses under dierent stress triaxialities.
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Fig. 13. The correction coecient vs. the stress triaxiality R for the homogeneous materials and the bimaterial systems with dierent (a) hardening exponents, and (b) yield stresses.
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Fig. 14. Comparisons of FE results with the predictions of the modi®ed R-T model for the growth of the voids. Results are shown for the bimaterial systems with dierent (a) hardening exponents, and (b) yield stresses.
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Fig. 15. The local stress triaxiality in the upper cell and lower cell for the macroscopic stress triaxiality R 1=3 and R 1:0.
5. Conclusions In the present paper, the eects of plasticity mismatch on the void growth and coalescence on the bimaterial interface have been investigated carefully by use of a axisymmetric cell model with a spherical void on the interface of two elastic±plastic materials. By controlling constant macroscopic stress triaxiality throughout the multiaxial loading history, the macroscopic responses of cell model are investigated in details by the aid of the large deformation elastic±plastic FE calculations. From the numerical results the following conclusions relating to the plasticity mismatch can be drawn: . The plasticity mismatch has signi®cant eects on the growth rate and coalescence strain of voids on the bimaterial interface. Compared with the behavior of voids in homogeneous materials, the growth rate of voids on the interface are much higher, and the critical macroscopic eective strains are less than about 50% for the two kinds of plasticity mismatches considered. So the damage by void growth on the interface is much more dangerous than that in homogeneous materials. . Unlike the elliptical shapes of deformed voids in the homogeneous materials, the deformed shapes of voids on the bimaterial interface are seriously misshapen. Due to earlier plastic ¯ow localization and higher local stress triaxiality
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in the softer material; the semi-spherical void in the softer material will expand mainly along the radial direction. However, due to higher deformation resistance in the harder material, the semi-spherical void in the harder material is nearly unchanged throughout the loading history. As a result, the linking of adjacent voids on the interface takes place in the softer material and with a small distance from the interface. . The classical R-T model coming from the homogeneous porosity materials cannot predict the void growth on the bimaterial interface. A simple modi®cation of the classical R-T model has been suggested. . The critical void volume fraction fc has strong dependence on the macroscopic stress triaxiality R , and is not a monotonic function of R . It is also susceptible whether fc can be regarded as a material constant to predict the damage by the voids on the bimaterial interface in dierent triaxial stress ®elds. Acknowledgements The authors thank the anonymous reviewer for valuable comments. References Andersson, H., 1977. Analysis of a model for void growth and coalescence ahead of a moving crack tip. Journal of the Mechanics and Physics of Solids 25, 217. Bourcier, R.J., Koss, D.A., Smelser, R.E., Richmond, O., 1986. The in¯uence of porosity on the deformation and fracture of alloys. Acta Metallurgica 34, 2443. Brocks, W., Sun, D.Z., HoÈnig, A., 1995. Veri®cation of the transferability of micromechanical parameters by cell model calculations with visco-plastic materials. International Journal of Plasticity 11, 971±989. Faleskog, J., Gao, X., Shih, C.F., 1998. Cell model for nonlinear fracture analysis- I. Micromechanics calibration. International Journal of Fracture 89, 355±373. Gao, X., Faleskog, J., Shih, C.F., 1998a. Cell model for nonlinear fracture analysis- II. Fracture-process calibration and veri®cation. International Journal of Fracture 89, 375±398. Gao, X., Faleskog, J., Shih, C.F., 1998b. Ductile tearing in part-through crack, experiments and cell model prediction. Engineering Fracture Mechanics 59, 761±777. Gurson, A.L., 1977. Continuum theory of ductile rupture by void nucleation and growth: part I Ð yield criteria and ¯ow rules for porous ductile media. Journal of Engineering Materials and Technology 99, 2±15. Koplik, J., Needleman, A., 1988. Void growth and coalescence in porous plastic solids. International Journal of Solids and Structures 24, 835±853. Kuna, M., Sun, D.Z., 1996. Three-dimensional cell model analyses of void growth in ductile materials. International Journal of Fracture 81, 235±258. Leblond, J.B., Perrin, G., Suquet, P., 1994. Exact results and approximate model for porous viscoplastic solid. International Journal of Plasticity 10, 213±235. Li, G.C., Ling, X.W., Shen, H., 2000. On the mechanism of void growth and the eect of straining mode in ductile materials. International Journal of Plasticity 16, 39±58. McClintock, F.A., 1968. A criterion of ductile fracture by growth of holes. ASME, Journal of Applied Mechanics 35, 363. McMeeking, R.M., Hom, C.L., 1990. Finite element analysis of void growth in elastic±plastic materials. International Journal of Fracture 42, 1±9.
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Nagaki, S., Goya, M., Sowerby, R., 1993. The in¯uence of void distribution on the yielding of an elastic± plastic porous solid. International Journal of Plasticity 9, 199±212. Needleman, A., 1972. Void growth in an elastic±plastic medium. ASME, Journal Applied Mechanics 39, 964. Rice, J.R., Tracey, D.M., 1969. On the ductile enlargement of voids in triaxial stress ®elds. Journal of the Mechanics and Physics of Solids 17, 201±207. Richelsen, A.B., Tvergaard, V., 1994. Dilatant plasticity on upper bound estimates for porous ductile solid. Acta Metallurgica 42, 2561±2577. Ruggieri, C., Panontin, T.L., Dodds, R.H., 1996. Numerical modeling of ductile crack growth in 3-D using computational cell elements. International Journal of Fracture 82, 67±96. Steglich, D., Brocks, W., 1998. Micromechanical modeling of damage and fracture of ductile materials. Fatigue & Fracture of Engineering Materials and Structures 21, 1175±1188. Sùvik, O., Thaulow, C., 1997. Growth of spherical voids in elastic±plastic solids. Fatigue and Fracture of Engineering Materials and Structures 20, 1731±1744. Tvergaard, V., 1981. In¯uence of voids on shear band instabilities under plane strain conditions. International Journal of Fracture 17, 389±407. Tvergaard, V., 1982. On localization in ductile materials containing voids. International Journal of Fracture 18, 237±252. Tvergaard, V., Needleman, A., 1984. Analysis of the cup-cone fracture in a round bar. Acta Metallurgica 32, 157±169. Worswick, M.J., Pick, R.J., 1990. Void growth and constitutive softening in a periodically voided solid. Journal of the Mechanics and Physics of Solids 38, 601±625. Xia, L., Shih, C.F., 1995a. Ductile crack growth Ð I. A numerical study using computational cells with microstructurally based length scales. Journal of the Mechanics and Physics of Solids 43, 233±259. Xia, L., Shih, C.F., 1995b. Ductile crack growth Ð II. Void nucleation and geometry eects on macroscopic fracture behavior. Journal of the Mechanics and Physics of Solids 43, 1953±1981. Xia, L., Shih, C.F., 1996. Ductile crack growth Ð III. Transition to cleavage fracture incorporating statistics. Journal of the Mechanics and Physics of Solids 44, 603±639. Zhang, K.S., Zheng, C.Q., 1997. 3D analysis of spherical void contained cell under dierent triaxial stress state. In: New Progress of Solid Mechanics. Tsinghua University Press. Zhang, K.S., Bai, J.B., Francois, D., 1999. Ductile fracture of materials with high void volume fraction. International Journal of Solids and Structures 36, 3407±3425.
The in¯uence of plasticity mismatch on the growth and coalescence of spheroidal voids on the bimaterial interface Zhenhuan Li a,*, Wanlin Guo b,c a Department of Mechanics, Huazhong University of Science and Technology, Wuhan 430074, PR China The National Key Laboratory of Mechanical Structural Strength and Vibration, Xi'an Jiaotong University, Xi'an 710049, PR China c Department of Aircraft Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, PR China b
Received in ®nal revised form 28 June 2000
Abstract To investigate the macro-mechanical response and micro-mechanism of damage by void growth and coalescence on the interface in a bimaterial system, detailed ®nite element computations of a representative cylindrical cell containing a spherical void are performed. By comparison with the response of a homogeneous material cell model, signi®cant eects of the matrix plasticity mismatches due to the yield stress and the strain hardening exponent on the void growth and coalescence are revealed: (1) The growth rate of the void on the bimaterial interface is much faster than that in the homogeneous material, and the critical coalescence strain of the void on the interface is only about half of that in homogeneous material. (2) Due to the dierence in the deformation resistance of the matrix materials in the bimaterial system, all computations indicate that deformed voids are seriously distorted and the linking of adjacent voids takes place in the softer matrix material. Comparison of the computational results with the classical Rice±Tracey (R-T) model shows that the R-T model cannot make good prediction for the growth of the void on the bimaterial interface. On the basis of large numbers of numerical simulations, a correction coecient is introduced to improve the R-T model. # 2002 Elsevier Science Ltd. All rights reserved. Keywords: A. Voids and inclusions; B. Porous material; C. Finite element; Plasticity mismatch; Interface
* Corresponding author. E-mail address: [email protected] (Zhenhuan Li). 0749-6419/02/$ - see front matter # 2002 Elsevier Science Ltd. All rights reserved. PII: S0749-6419(00)00078-4
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1. Introduction The mechanics and micro-mechanisms of fracture on a bimaterial interface are the topics of considerable interests in the design of a vast number of structural components, such as welding structure, coating systems and advanced composites. Increasing applications of functionally graded materials (FGM) also require a basic understanding of the eect of material combination on fracture resistance. Similar to the processes in homogeneous materials, ductile fracture in a bimaterial system takes place as a successive processes of nucleation, growth and coalescence of micro-voids on the bimaterial interface. When the bimaterial system is loaded, voids are ®rst nucleated on the interface by local interface decohesion, then plastic straining of the surrounding ductile matrix material will control the growth process until a local internal necking and plastic collapse of the intervoid ligament occur. The latter event marks the incipient ductile fracture. During the last thirty years, there was a history of eorts directed at developing predictive tools for the growth and coalescence of voids in homogeneous materials. The growth of an isolated cylindrical void was investigated ®rst by McClintock (1968), then an analytical model (R-T model) for the spherical void embedded in an in®nite perfectly plastic solid subjected to remote triaxial stresses was presented by Rice and Tracey (1969). Although these pioneering works are only valid for lower porosity volume fraction, they revealed an essential feature that the void growth rate depends exponentially on the stress triaxiality. On the basis of the R-T model, Gurson (1977) further considered a cell containing a spherical void and suggested a plastic potential function for porosity materials. Since this model has the advantages of simulating damage and fracture in ductile materials, considerable attentions have been paid to it. However, the original Gurson model was derived from an approximate solution for rigid-perfectly plastic material cell containing a centered spherical void. To extend the Gurson model to periodic arrays of cylindrical and spherical voids in hardening materials, Tvergaard (1981, 1982) introduced two adjustable parameters q1 and q2 . However, recent investigations show that the q-values depend on both the strain hardening (n) and the ratio of the yield strength (ys ) to Young's modulus (E) of the matrix materials (Faleskog et al., 1998). The Gurson model also ignores voids interaction so that it cannot predict the coalescence of voids by localized internal necking of the intervoid matrix. Therefore, Tvergaard and Needleman (1984) made further modi®cation to the Gurson model to describe the drop-o of stress carrying capacity in materials after voids coalescence. Their constitutive relation has been referred to as GTN model and various of its extensions have been used to predicate ductile damage and fracture in metal materials (Xia and Shih, 1995a,b, 1996; Ruggieri et al., 1996; Faleskog et al., 1998; Gao et al., 1998a,b). On the other hand, numerical simulations have provided us better understanding of void growth and coalescence mechanisms. To our knowledge, original works to simulate the interaction of voids by ®nite element method (FEM) were made by Needleman (1972) and Tvergaard (1981), who analyzed two-dimensional square arrays of cylindrical holes under plane strain condition. However, compared with the real situation of spherical voids in three dimensions, such a two dimensional
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(2D) model cannot describe the stress triaxiality properly and always overestimates the interaction of voids. Further ®nite element computations have been carried out to investigate the three-dimensional eects by the aid of the axisymmetrical cylindrical unit cell containing a spherical void under various triaxial loading conditions (Andersson, 1977; Tvergaard, 1982; Bourcier et al., 1986; Koplik and Needleman, 1988; Brocks et al., 1995; Steglich et al., 1998; Sùvik and Thaulow, 1997; Li et al., 2000). However, such a cylindrical unit cell does not allow for the adjustment of arbitrary stress ratios in all three directions. The three-dimensional (3D) cell models subjected to stresses parallel to the principal geometrical axes are developed (McMeeking and Hom, 1990; Worswick and Pick, 1990; Nagaki et al., 1993; Richelsen and Tvergaard, 1994; Ruggieri et al., 1996; Zhang and Zheng, 1997; Zhang et al., 1999). Other cell structures, such as cubic primitive model and body centered cubic model, have been analyzed by Kuna and Sun (1996). It is shown that the axisymmetrical cell model leads to larger deformation and a lower failure stress than 3D cell models. This means that the hexagonal arrangement with axisymmetric unit cell simulation gives a lower bound solution for material failure by voids. This is encouraged from the viewpoint of structural safety design. Compared with the mass of researches into the mechanics and micro-mechanisms of ductile fracture by void growth and coalescence in the homogeneous materials, little attention has been devoted to the micro-mechanisms of ductile fracture on the interface in an elastic±plastic bimaterial system under triaxial stress conditions. As is well known, in the bimaterial system, the voids on the interface are surrounded by two kinds of matrix materials possessing dierent plastic properties. Recent investigation (Faleskog et al., 1998) has shown, for the homogeneous material, that over a range of constraint and initial void volume fraction typically encountered in applications, the growth and coalescence of voids rely strongly on the strain hardening exponent (n) and the strength ratio (ys =E) of the matrix material. Therefore, for the bimaterial system, we can roughly speculate that the plasticity mismatch of the matrix materials, which have no elastic mismatch, will have signi®cant eects on the growth and coalescence mechanisms of voids on the bimaterial interface. However, to our knowledge, no systematic analyses on the void growth and coalescence on the elastic± plastic bimaterial interface have been made in the existing literature. It is also susceptible whether the existing classical micro-mechanical model driven from the homogeneous porosity material can predict rationally the combination eects of the plasticity mismatch on the growth and coalescence of voids on the bimaterial interface. In order to predict ductile fracture on the interface in the bimaterial system, through knowledge of all the three basic stages relating to void nucleation, growth and coalescence are required. Due to the complexity of void nucleation process, a great deal of works remains to be done to obtain a better understanding of the relating phenomena. Therefore, the present work will consider only the growth and coalescence mechanisms of the voids on the interface, and the emphasis is focussed on the eect of the matrix plasticity mismatch. In particular, the two kinds of matrix materials composing the bimaterial system have similar elastic properties but dierent plastic properties.
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2. Cell model 2.1. Model description Because the cell model can describe the relation between the macroscopic mechanical responses and the microstructures of materials, it has widely been used to simulate and study the behavior of porous solids (Koplik and Needleman, 1988). The problems considered here are the growth and coalescence of the voids on the interface in the elastic-plastic bimaterial system subjected to far triaxial stresses, as illustrated in Fig. 1. The bimaterial system comprises two materials with dierent strain hardening exponents or yield stresses, and the intervoid bimaterial interface is bonded perfectly and is dicult to decohere during plastic deformation. In consistency with a common damage mechanics approach, the cylindrical cell can be regarded as a representative volume element when a periodic array of identical voids on the bimaterial interface is assumed. Fig. 1 shows the geometry of the cylindrical cell containing one spherical void on the bimaterial interface. For convenience, the rectangular coordinates (x, y, z) are adopted in this paper (see Fig. 1). The origin of the coordinate system is located at the midpoint of the bottom of the cell. Obviously, the initial void volume fraction can be written as
Fig. 1. Illustration of the problem and the 2D cell model.
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f0
2a30 : 3R20 H0
253
1
where a0 is the initial radius of the void, R0 and H0 are the initial radius and half of the height of the cell, respectively. In this investigation, only f0 0:001 is considered. The current void volume fraction f is de®ned as the ratio of the total void volume to the cell volume. For elastic-plastic incompressible matrix materials, the void volume fraction can be computed by f1
1
f0 V0 Ve : V
2
where V0 and V are the initial and current volume of the cell, respectively, Ve is the increase in volume of the cylindrical cell due to elastic dilatation arising from the imposing hydrostatic stress which can be approximated by (Koplik and Needleman, 1988): 3 1 2 Ve V0 1 f0 h : 3 E where E and are Young's modulus and Poisson's ratio of the matrix material. h is the macroscopic hydrostatic stress. Obviously, for a porous bimaterial system with identical elastic property but dierent plastic property, Eq. (3) can also be used to describe reasonably the elastic volume dilatation of the cell. Considering the dierences in matrix material properties, half of the cell model (region x0 50, 04z42H0 ) is analyzed with FEM, since the problem has re¯ectional symmetry with respect to the axial x 0. The ®nite element meshes of the geometry are constructed with an eight-node quadratic iso-parametric element as shown in Fig. 1 and 22 Gauss integral is adopted. The axisymmetric FE cell model consists of 480 elements and 1535 nodes as shown Fig. 1. To avoid the eect of FE mesh on the computational results, identical FE meshes are adopted in all FE analyses for the bimaterial and homogeneous material cell models. For comparison with FE results, a modi®ed R-T model is also considered. The RT model predicts the void growth by means of the radius a of the spherical void. An approximation of the relation derived by Rice and Tracey can be written as: : a : 0:283exp R " pe : a
4
where "pe is the eective plastic strain, rate is a modi®cation coecient and R is the stress triaxiality de®ned as h 5 R : e where e is the macroscopic eective stress.
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Then the increasing rate of the void volume fraction can be written as : : f a : 3 0:849exp R "pe : a f
6
In the classical R-T model, is set as approximately 1:5. 2.2. Boundary conditions and loading method Due to the axial symmetry, the shear stresses on x, z-plane will vanish. Therefore, the axial, radial and tangential directions will be the principal directions of stresses. Furthermore, for axial symmetry case the tangential and radial components of the macroscopic stress and strain tensors subjected on the cell boundary will be of equal magnitude. The structure is subjected to a homogeneous elongation, Uz1 , in the axial direction and the radial displacement Ux1 is also kept to be homogeneous by constraint conditions. The boundary conditions for the axisymmetric region analyzed numerically can be described as: : : : uz 0; Fx 0; Fy 0; on z 0 : : : Fx Fy Fz 0; on x2 y2 z2 a2 : : : : uz Uz1 ; Fx 0; Fy 0; on z 2H : : Fy 0; Fz 0; on x R Here, a is the current radius of the void, R and H are the current radius and half : of the height of the cell, respectively; uz is the axial displacement rate; Fi (i=x, y, z) is the traction in the ith direction. The prescribed radial concentrated load Fx1 is determined from the condition that the average macroscopic true stresses acting on the cell follow the proportional history : x x : : z z
7
with as a prescribed constant determined by the controlled stress triaxiality R and sx sy sz
Fx1 : 2 R0 Ux 2H0 Uz
Fz1 : R0 Ux 2
8 9
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255
where i (i x; y; z) is the macroscopic principal stress in the ith direction, Fz1 is the axial concentrated load which can be obtained by a spring attached to the node at the axis of symmetry. For the axisymmetric case, the corresponding macroscopic eective stress and hydrostatic stress can be given by: e jz
1 x j; h z 2x : 3
10
and the stress triaxiality R can be written as R
h 1 2 : e 3 1
11
The macroscopic principle strains and eective strain can be expressed by: R H 2 "x "y ln ; "z ln ; "e "x "y : R0 H0 3
12
In order to control the macroscopic stress triaxiality R throughout the loading history, has to remain constant whereas the ratio of the prescribed strains "x ="z will vary with time. To satisfy this requirement, a spring element, which measures the axial load Fz1 arising from the prescribed axial displacement Uz1 , is introduced. The radial concentrated load Fx1 depending on R is then applied. This loading process can be achieved automatically in the FE computation by the equilibrium iteration technology (Sùvik and Thaulow, 1997). 2.3. Matrix material description In this work, the Prandtl±Ruess's constitutive equation is considered to describe elastic-plastic stress vs. strain responses of the matrix materials. In the case of large deformation, the Prandtl±Ruess's constitutive equation can be written as: 2 3 7 0ij 0kl 7 13 D kl : 2 2 2 1 5 1 h 3 e 3 E where D ij is the rate of deformation tensor, and D ij 1=2 @vi =@xj @vj =@xi , :vi is the rate of material point, xi is the instantaneous coordinate of material point; Jij is the Jaumann rate of Cauchy stress ij ; ij is the Kronecker delta; h is the strain hardening parameter; 0ij is the deviatoric stress tensor de®ned by 0ij ij 13 ij kk ; e is the eective stress of the matrix; l is the loading coecient determined by the loading criteria: ( 0 0ij D ij < 0 l : 14 1 0 D ij 50 : Jij
E 6 6ik jl ij kl 1 4 1 2
ij
l
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The true stress vs. true strain responses of the matrix materials under uniaxial tension is assumed to be in the linear elastic±power hardening plastic form: 0 ys ; E"; "4 E B 15 @ E"n ys ys : ; "> "ys E where and " are the uniaxial stress and strain respectively, ys and "ys are the uniaxial yield stress and yield strain, respectively, n is the strain hardening exponent. In this investigation, the emphases will be focused on the in¯uences of plasticity mismatch on void growth and coalescence on the interface. Considering the strong dependences of void growth and coalescence on the strain hardening exponent (n) and yield strength (ys =E) of the matrix material, dierent combinations of n and ys =E between the matrix materials are considered: 1. The plasticity mismatch due to the matrix material hardening exponent Ð Material 1: n1 0:1, ys =E 0:0025, ys 500 MPa Material 2: n2 0:2, ys =E 0:0025, ys 500 MPa 2. The plasticity mismatch due to the matrix material yield stress Ð Material 1: n 0:1, ys =E1 0:00125, ys 250 MPa Material 2: n 0:1, ys =E 2 0:0025, ys 500 MPa For reference, the growth and coalescence of voids in the homogeneous media, which may be hard or soft, will also be analyzed. 3. Analyses of the computational results In this investigation, we will explore the dependences of the void growth and coalescence on the macroscopic stress triaxiality and the combination of plasticity mismatch. A broad spectrum of stress triaxiality 0:754R 43:0 will be investigated, which covers the range from necking smooth bars to sharp crack bodies (Koplik and Needleman, 1988). As is well known, for lower triaxiality level, the shear mode of void growth and coalescence will be dominant, which is clearly dierent from the simpler internal necking mode of voids considered here in higher triaxial stress ®elds. This goes beyond the scope considered in this paper. In the present paper, the macroscopic response of the cell model in the homogeneous material is ®rst analyzed in Section 3.1 to validate the FE method adopted in this paper. The eects of plasticity mismatches due to dierent strain hardening exponents and yield stresses on void growth and coalescence are studied in Sections 3.2 and 3.3. Finally, a modi®cation of the R-T model is made to describe the growth of voids on the interface reasonably.
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3.1. The macroscopic response of cell model For variations of stress triaxiality from R 1:0 toR 3:0, Fig. 2 shows the computing eective mechanical response of the cell model for the homogeneous material with n 0:1 and ys =E 0:0025. Fig. 2(a) displays the macroscopic eective stress- eective strain curves, which show the competition between the matrix material strain hardening and porosity induced softening. The abrupt fall in the strain-stress curve corresponds to void coalescence. This event is characterized by sudden shift to a uniaxial straining mode where any further plastic deformation is limited to the intervoid radial ligament. This macroscopic response of cell model substantively re¯ects the break down of the load carrying capability of the cell caused by void growth and internal necking. Fig. 2(c) gives the change in cell radius against the eective strain. It is shown clearly that the plastic collapse is eventually reached, at which the eective strain increases while the radial strain remains approximately constant. This implies that ¯ow localization takes place in the ligament between radial adjacent voids. As can be seen in Fig. 2(b), the void volume fraction increases steeply at this point and this event is associated with the sudden load drop in Fig. 2(a). Therefore, it is reasonable to de®ne the point, where the macroscopic radial strain turns to be constant with increasing macroscopic eective strain, as the inception of void coalescence (Koplik and Needleman,1988; Sùvik and Thaulow, 1997). To validate the present FE computations, Table 1 gives out the dependence of the critical eective strain "e c on the stress triaxiality R , which shows an excellent agreement with the prediction by K-N (Koplik and Needleman, 1988). Due to lower initial void volume fraction f0 , the present critical eective strain is slightly higher than that predicated by B-S-H and L-P-S, however, the error is less than 5%. So it can be concluded that the precision of the present FE computation is high enough to analyze the growth and coalescence of voids in the bimaterial system. 3.2. In¯uence of the mismatch in strain hardening exponents In this section, the bimaterial system with dierent strain hardening exponents, or n1 0:1 and n2 0:2, will be considered. Fig. 3 shows the macroscopic stress-strain responses of the homogeneous material cell as well as the bimaterial cell at dierent stress triaxiality levels. For two homogeneous materials with dierent hardening exponents, the critical coalescence strains are dierent. The critical strains in the material with higher strain hardening exponent are slightly higher for all triaxiality levels considered. For the bimaterial system, as expected, until the attainment of the critical constant radial strain, the response curve of the bimaterial cell lies between the curves of the two kinds of homogeneous material cells. However, it is surprising that the critical coalescence eective strain of the bimaterial system is remarkably lower than that of both homogeneous matrix materials. At ®rst glance, this result seems to be unexpected, since the critical coalescence strain of the bimaterial system seems to be supposed to lie between the critical eective strains of the two kinds homogeneous materials. Again, the actual explanation lies in the earlier ¯ow localization in the softer materials.
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Fig. 2. Finite element results for the homogeneous porosity material under dierent stress triaxialities R =1.0, R =2.0 and R=3.0: (a) macroscopic eective stress±eective strain response; (b) void volume fraction vs. macroscopic eective strain; (c) radial strain vs. macroscopic eective strain.
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Fig. 2. (continued) Table 1 The critical eective strain "e c in comparison to the results by Koplik and Needleman (1988) (K-N), Brocks, Sun and HoÈnig (1995) (B-S-H) and Leblond, Perrin and Suquet (1994) (L-P-S) R
Present results (f0 0:001)
K-N (f0 0:0013)
B-S-H (f0 0:002)
L-P-S (f0 0:002)
1.0 2.0 3.0
0.82 0.22 0.08
0.83 0.22 0.08
0.79 0.20 0.08
0.79 0.21 0.07
Fig. 4 compares the void growth in the homogeneous materials with that on the bimaterial interface for dierent stress triaxiality levels. The predictions by the R-T model are also shown in the ®gure. It is easy to ®nd that, for the homogeneous materials with dierent strain hardening exponents, the growth rate of voids in the softer material is slightly faster than that in the harder materials. The R-T model gives good predictions of the growth rate of voids when 1:0 < R < 2:0, but overestimates the void growth for lower or higher stress triaxiality. In addition, by comparisons with the growth rate of voids in homogeneous materials, the volume fractions of voids on the bimaterial interface increase much more rapidly. As shown in Fig. 4, for the stress triaxiality levels considered in this study, the R-T model, which was obtained by solving the boundary problem of the homogeneous in®nite prefect plastic material containing void, can not predict the void growth on the
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Fig. 3. Macroscopic eective stress vs. eective strain curves of the bimaterial system with dierent hardening exponents under dierent stress triaxialities: (a) R=0.75, (b) R=1.0, (c) R =2.0, (d) R=3.0.
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Fig. 3. (continued)
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Fig. 4. Void volume fraction vs. eective strain curve of the bimaterial system and homogeneous materials with dierent hardening exponents under dierent stress triaxialities: (a) R=0.75, (b) R=1.0, (c) R=2.0, (d) R=3.0.
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Fig. 4. (continued)
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interface correctly. Therefore, it is necessary to modify the original R-T model for bimaterial systems. Fig. 5 shows the variations of the critical coalescence eective strain "e c with the stress triaxiality R for the homogeneous materials and the bimaterial system. It can be found that "e c of the bimaterial is about 50% lower than that of the homogeneous materials. Correspondingly, Fig. 6 plots the critical void volume fraction fc as a function of R . Although the growth of voids in the bimaterial system is signi®cantly rapider than that in the homogeneous materials, fc of the bimaterial system is much lower. It is worthy to notice that fc is sensitive to the stress triaxialty level for both the bimaterial system and homogeneous materials. It is also susceptible whether fc can be regarded as a material constant to predict fracture in specimens with dierent geometrical con®gurations. Fig. 7 shows the initial and deformed shapes of the cell model under dierent stress triaxiality level. It is striking to ®nd that the material plasticity mismatch and stress triaxiality cause such signi®cant dierences in the deformed void shape. For homogeneous materials, it can be seen that the void expands in the main tension direction and becomes prolate ellipsoidal when the stress triaxiality level is lower. On the contrary, the void grows in radial direction and become oblate ellipsoidal in the higher triaxial stress ®elds. For voids on the bimaterials interface, the growth of the semi-ellipsoidal void in the harder material is very dicult; however, the semiellipsoidal void in the softer material grows rapidly and expands along radial direction.
Fig. 5. The critical eective strain vs. stress triaxiality for the bimaterial system and the homogeneous materials with dierent hardening exponents.
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Fig. 6. The critical void volume fraction vs. stress triaxiality for the bimaterial system and the homogeneous materials with dierent hardening exponents.
As shown in Fig. 7, due to higher constraint from the harder material, radial expanding of the semi-ellipsoidal void in the softer material doesn't take place on the interface but a small distance from the interface so that the shape of void is distorted badly. Similar to the shape change of void in homogeneous materials, the pro®le of void on the interface is slightly elongated in the main tension direction at lower stress triaxiality levels; on the contrary, the void are relatively elongated in the radial direction at higher stress triaxiality levels. Since the R-T model does not take account of void shape changes, it of course has no capability of predicting of the growth rate of voids on the bimaterial interface and in homogeneous materials. 3.3. In¯uence of the mismatch in yield stress In this section, the attention is restricted to the situation where the bimaterial system comprises two kinds of elastic-plastic materials possessing identical elastic property and strain hardening exponent but dierent yield stress. For convenience, a representative case with ys 2 2 ys 1 will be investigated. As shown by Fig. 1, the stronger material carries the same axial load as the softer material; as a result the strain in the softer material is much higher than that in the harder material. Therefore, the plastic ¯ow localization is expected to take place earlier in the softer material.
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Fig. 8 shows the curve of the macroscopic eective stress vs. eective strain of the cell for the bimaterial system. For comparison, the macroscopic responses of the two corresponding homogeneous materials are also displayed in the ®gure. As expected,
Fig. 7. Deformed cells when ee=(ee)c for (a) the bimaterial system with dierent hardening exponents, and (b) the homogeneous material under dierent stress triaxialities.
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Fig. 8. The macroscopic eective stress vs. eective strain curves of the bimaterial system with dierent yield stresses under dierent stress triaxialities: (a) R=0.75, (b) R=1.0, (c) R=2.0, (d) R=3.0.
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Fig. 8. (continued)
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for the homogeneous materials, the dierence in yield stress leads to signi®cant differences in load carrying capability. Due to the identical strain-hardening exponent, the eect of yield stress on the critical eective strain "e c associated with the void coalescence is very weak. However, for the bimaterial system, it is interesting to ®nd that "e c is much lower than that of the homogeneous materials in the whole range of triaxiality considered. This unexpected phenomenon mainly comes from the earlier plastic ¯ow localization in the softer material of the bimaterial system. Fig. 9 illustrates the eect of yield stress mismatch on the growth rate of void on the interface at dierent stress triaxiality levels. For comparison, the FE results of the two corresponding homogeneous materials and the predictions of the classical R-T model are also displayed in Fig. 9. For homogeneous materials, the eects of yield stress on void growth and coalescence can be ignored. However, in the bimaterial system, the mismatch in yield stress can increases the growth rate of voids on the interface remarkably. This means that the damage on the interface is much more dangerous than that in homogeneous materials. It is well known that the classical R-T model is derived from a homogeneous porosity material. From Fig. 9, it can be seen clearly that the classical R-T model does not give a good prediction for the growth rate of voids on the interface between materials with dierent yield stresses. Figs. 10 and 11 show variations of "e c and fc with stress triaxiality R for two homogeneous materials and the corresponding bimaterial system. It can be found that, for homogeneous materials, the dierence of "e c caused by change in yield stress is very small, and the "e c R curves of the two homogeneous materials are almost coincident. However, for the bimaterial system, "e c is about 50% lower at the same triaxiality level. For the dierent stress triaxiality levels considered, Fig. 11 indicates that the critical void volume fraction fc of the bimaterial system is much lower than that in the two homogeneous materials. It can be noticed further that fc has strong dependence on R , and it is not a monotonic function of R , fc increases in the lower stress triaxiality range to its peak value fc max , then decreases gradually with increasing R . Fig. 12 displays the initial and deformed shapes of the cell under dierent stress triaxialities. By comparison with the shape changes of voids in homogeneous materials as shown in Fig. 7(b), it can be found that the deformed void shapes are seriously misshapen in the bimaterial system with yield stress mismatch. In the bimaterial system, the growth of semi-ellipsoidal void in the softer material is easier than that in the harder material. As depicted in Fig. 12, radial expanding of the semi-ellipsoidal void in the softer material takes precedence over that in the harder material; as a result the deformed void shape in the bimaterial interface becomes misshapen. 3.4. Modi®cation of the R-T model As aforementioned, since the classical R-T model was obtained from the homogeneous perfect plastic material containing an ideal spherical void and ignored the void shape changes in dierent triaxial stress ®elds, this model has no capability of predicting the growth of voids on the bimaterial interface. One possible way of improving this model is to introduce a modi®cation coecient as a function of stress triaxiality, which re¯ects indirectly the eect of the void shape, as shown by Eq. (4).
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Fig. 9. The void volume fraction vs. the macroscopic eective strain for the bimaterial system and homogeneous materials with dierent yield stresses under dierent stress triaxialities: (a) R=0.75, (b) R=1.0, (c) R=2.0, (d) R=3.0.
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Fig. 9. (continued)
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In order to evaluate such a modi®cation factor , void growth predictions by the classical R-T model are ®tted to the FE results of the cell model. The computational results for the dependence of on R are shown in Fig. 13. Even for the homogeneous materials, it can be seen that is not a constant 1.5 as in the classical R-T model but has close relation with the stress triaxiality. When 1:0 < R < 2, is close to the constant 1.5 but for lower or higher R , will be less than 1.5. It is also shown that the dierence in the plastic properties such as strain-hardening exponent and yield stress only have a weak eect on in homogeneous materials. However, in a bimaterial system, the situation is completely dierent. It is very interesting to notice that not only strongly depends on the stress triaxality but also has close relation with the plasticity mismatch of the bimaterial system. For the two kinds of plasticity mismatches discussed above, is a monotonic decreasing function of R within the stress triaxiality range considered. By ®tting the FE results of the bimaterial cell model, can be estimated by exp lR :
16
where 2:67, l 0:24 for the hardening exponent mismatch and 2:97, l 0:22 for the yield stress mismatch considered in the present investigation.
Fig. 10. The critical eective strain vs. stress triaxiality for the bimaterial system and the homogeneous materials with dierent yield stresses.
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Fig. 14 displays the comparison of the FE results of the cell with the predictions by the modi®ed R-T model for the void growth on the bimaterial interface. In the ®gure, the void volume fraction fis plotted as a function of the eective strain "e . As expected, for dierent stress triaxiality and the two plasticity mismatches considered, the predictions by the modi®ed R-T model give satisfying agreement with the FE results of cell model for the bimaterial system. 4. Discussion The stress triaxialty concerned in the paper is the macroscopic stress triaxiality which is average or global quantity. In fact, due to the localized plastic strain, the microscopic stress triaxiality in the cell consisting of homogeneous matrix material cannot be uniform. The local microscopic strain mode and stress triaxiality have obvious eect on the micro-mechanism of the void growth and coalescence (Li et al., 2000). For bimaterial system, there are two semi-ellipsoidal voids, one in the soft material and other in the hard material. The axial forces are identical in both cells but the lateral forces are dierent, leading to two dierent stress ratios. For two constant macroscopic stress triaxialities R 1=3 and R 1, Fig. 15 gives the changes of the local stress triaxialty in the upper soft material cell and the lower
Fig. 11. The critical void volume fraction vs. stress triaxiality for the bimaterial system and the homogeneous materials with dierent yield stresses.
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hard material cell with the macroscopic eective strain. It can easily be found that, though the macroscopic stress triaxiality of the bimaterial cell remains a constant during deformation, the stress triaxiality of soft materials cell is higher than that of hard materials cell. Corresponding to void coalescence, the stress triaxiality of the soft cell becomes higher and higher but that of the hard cell become lower and lower with increasing eective strain. It is worthy of noting that the stress triaxiality in the upper can reach 0.75 but that in the lower hard cell can only reach 0.1 corresponding to uniaxial tension R 1=3. This means that though the macroscopic stress triaxiality of bimaterial cell is lower, the local stress triaxiality of soft material cell can still reach enough high to drive the cavity growth and coalescence. Due to earlier localized plastic strain and higher local stress triaxiality in the soft material cell, the growth of semi-ellipsoidal void in the soft material is faster than that in the hard material. This means that the void coalescence in the bimaterial system is easier than that in the homogeneous material. Due to dierent void growth rates in the soft and the hard material cell, it seems doubtful that the total porosity is a good measure. How to de®ne the porosity is very important to understand the micro-mechanism of void growth and coalescence. However, due to interfering and shielding of two semiellipsoidal voids in the bimaterial, the law for cavity growth in the interface will be very complicate and dierent from that in the corresponding homogeneous material. Further researches relating these issues are to be appear in another paper.
Fig. 12. Deformed cells when ee=(ee)c for the bimaterial system with dierent yield stresses under dierent stress triaxialities.
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Fig. 13. The correction coecient vs. the stress triaxiality R for the homogeneous materials and the bimaterial systems with dierent (a) hardening exponents, and (b) yield stresses.
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Fig. 14. Comparisons of FE results with the predictions of the modi®ed R-T model for the growth of the voids. Results are shown for the bimaterial systems with dierent (a) hardening exponents, and (b) yield stresses.
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Fig. 15. The local stress triaxiality in the upper cell and lower cell for the macroscopic stress triaxiality R 1=3 and R 1:0.
5. Conclusions In the present paper, the eects of plasticity mismatch on the void growth and coalescence on the bimaterial interface have been investigated carefully by use of a axisymmetric cell model with a spherical void on the interface of two elastic±plastic materials. By controlling constant macroscopic stress triaxiality throughout the multiaxial loading history, the macroscopic responses of cell model are investigated in details by the aid of the large deformation elastic±plastic FE calculations. From the numerical results the following conclusions relating to the plasticity mismatch can be drawn: . The plasticity mismatch has signi®cant eects on the growth rate and coalescence strain of voids on the bimaterial interface. Compared with the behavior of voids in homogeneous materials, the growth rate of voids on the interface are much higher, and the critical macroscopic eective strains are less than about 50% for the two kinds of plasticity mismatches considered. So the damage by void growth on the interface is much more dangerous than that in homogeneous materials. . Unlike the elliptical shapes of deformed voids in the homogeneous materials, the deformed shapes of voids on the bimaterial interface are seriously misshapen. Due to earlier plastic ¯ow localization and higher local stress triaxiality
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in the softer material; the semi-spherical void in the softer material will expand mainly along the radial direction. However, due to higher deformation resistance in the harder material, the semi-spherical void in the harder material is nearly unchanged throughout the loading history. As a result, the linking of adjacent voids on the interface takes place in the softer material and with a small distance from the interface. . The classical R-T model coming from the homogeneous porosity materials cannot predict the void growth on the bimaterial interface. A simple modi®cation of the classical R-T model has been suggested. . The critical void volume fraction fc has strong dependence on the macroscopic stress triaxiality R , and is not a monotonic function of R . It is also susceptible whether fc can be regarded as a material constant to predict the damage by the voids on the bimaterial interface in dierent triaxial stress ®elds. Acknowledgements The authors thank the anonymous reviewer for valuable comments. References Andersson, H., 1977. Analysis of a model for void growth and coalescence ahead of a moving crack tip. Journal of the Mechanics and Physics of Solids 25, 217. Bourcier, R.J., Koss, D.A., Smelser, R.E., Richmond, O., 1986. The in¯uence of porosity on the deformation and fracture of alloys. Acta Metallurgica 34, 2443. Brocks, W., Sun, D.Z., HoÈnig, A., 1995. Veri®cation of the transferability of micromechanical parameters by cell model calculations with visco-plastic materials. International Journal of Plasticity 11, 971±989. Faleskog, J., Gao, X., Shih, C.F., 1998. Cell model for nonlinear fracture analysis- I. Micromechanics calibration. International Journal of Fracture 89, 355±373. Gao, X., Faleskog, J., Shih, C.F., 1998a. Cell model for nonlinear fracture analysis- II. Fracture-process calibration and veri®cation. International Journal of Fracture 89, 375±398. Gao, X., Faleskog, J., Shih, C.F., 1998b. Ductile tearing in part-through crack, experiments and cell model prediction. Engineering Fracture Mechanics 59, 761±777. Gurson, A.L., 1977. Continuum theory of ductile rupture by void nucleation and growth: part I Ð yield criteria and ¯ow rules for porous ductile media. Journal of Engineering Materials and Technology 99, 2±15. Koplik, J., Needleman, A., 1988. Void growth and coalescence in porous plastic solids. International Journal of Solids and Structures 24, 835±853. Kuna, M., Sun, D.Z., 1996. Three-dimensional cell model analyses of void growth in ductile materials. International Journal of Fracture 81, 235±258. Leblond, J.B., Perrin, G., Suquet, P., 1994. Exact results and approximate model for porous viscoplastic solid. International Journal of Plasticity 10, 213±235. Li, G.C., Ling, X.W., Shen, H., 2000. On the mechanism of void growth and the eect of straining mode in ductile materials. International Journal of Plasticity 16, 39±58. McClintock, F.A., 1968. A criterion of ductile fracture by growth of holes. ASME, Journal of Applied Mechanics 35, 363. McMeeking, R.M., Hom, C.L., 1990. Finite element analysis of void growth in elastic±plastic materials. International Journal of Fracture 42, 1±9.
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