Micromechanical Consideration of Tensile Crack Behavior

7/28/2019 Micromechanical Consideration of Tensile Crack Behavior

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Micromechanical consideration of tensile crack behaviorbased on virtual internal bond in contrast to cohesive stress

Z.N. Zhang *, X.R. Ge

School of Naval Architecture, Ocean and Civil Engineering, Shanghai Jiaotong University, Shanghai, 200240, PR China

Available online 23 May 2005

Abstract

A modified version of the virtual internal bond model (VIB) is presented. This involves the introduction of a R-bondrestricting the relative rotation freedom of pairwise mass particle. Such a modification allows the VIB model to considerarbitrary values of the Poisson ratio. A linear elastic cohesive law considering both the R-bond and L-bond areassumed. The constitutive relationship is derived using the CauchyBorn rules. The derived constitutive associatesthe bond stiffness with the Youngs modulus and Poisson ratio of materials. This gives the bond stiffness in terms ofthe Youngs modulus and Poisson ratio of materials.

The modified VIB model is then used to analyze the tensile crack behavior. In contrast to the cohesive stress method,

the deformation-governed concept will be used. The local materials failure is assumed to coincide with the reduction ofthe bond density due to the local deformation rather than by the local cohesive stress. A phenomenological relationshipbetween the bond density and the deformation is established. The criterion which is applied to determined crack initi-ation and propagation is built into the constitutive model. As an example, the method is used to study the crack ini-tiation and propagation behavior under tensile loading. 2005 Elsevier Ltd. All rights reserved.

Keywords: Micromechanical model; Virtual internal bond; Tensile crack behaviors; Isotropic solid

1. Introduction

To predict the crack behavior, many approaches have been proposed. The VIB model [1] was used tostudy the crack growth behavior. The method, however, is limited to a specific value of the Poisson s ratio.This limitation will be overcome in this work by considering both the R- and L-bond in addition to usingthe CauchyBorn rule [25] that connects the continuum approach to that of the cohesive bond. Path

0167-8442/$ - see front matter 2005 Elsevier Ltd. All rights reserved.doi:10.1016/j.tafmec.2005.03.005

* Corresponding author. Tel.: +86 21 54740526; fax: +86 21 62932194.E-mail addresses: [email protected], [email protected] (Z.N. Zhang).

Theoretical and Applied Fracture Mechanics 43 (2005) 342359

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independent integral of the type in [6] has also been used. Cohesive surface methods [711] have also beenadvanced to represent the crack initiation and propagation process. It is difficult to associate the cohesivemethods with the physics of cracking, particularly when different size scales are involved. The works in [12

15] have discussed multiscaling near the crack tip by using the concept of the strain energy density function.In addition, discussions were also made with reference to the use of continuum concept and that of the par-ticle. Molecular dynamics (MD) has also been used to analyze fracture behavior at the atomistic level [1621]. The conventional treatment of crack behavior based on fracture mechanics can be found in [22]. Theapproach can be regarded as a simplified version of the multiscale treatment in [1215]. It has recently beenpointed out [23] that the 1/r-type of singularity for the potential function of clusters of molecules is anintrinsic property that is independent of the material constitutive relationship. This holds when the domi-nant lattice interaction at the crack tip is of the mutual attraction type. These fundamental researches im-proved the understandings of fracture behavior of materials and provided the theoretical basis for solvingthe practical crack problems. The development of numerical methods have involved many different ap-proaches. Among them is the VIB model [1] that has been used to study crack growth in isotropic solids.The method was extended to the anisotropic materials [24]. The finite element implementation of VIB wasmade in [25].

The method integrates the discrete character at the microscopic level to the mechanical properties at thecontinuum level. By incorporating a cohesive law which governs the interactions between two discrete massparticles, the fracture criterion is directly built into the constitutive relationship. No external criterion isrequired when using VIB to simulate the crack growth. However, the Poissons ratio in [1] must be fixedat 0.25. A R-bond is thus added to restrict the relative rotation freedom. The modified VIB model is thenused to examine crack growth by fracture. The condition corresponds to a reduction of the bond densitycause by the local deformation rather than the cohesive stress used in the earlier models.

2. Constitutive model for linear elastic materials

2.1. Physical consideration of material behavior

Atoms within the material are known to interact and their en-mass effects can be incorporated into acontinuum approach. It is important to study the macroscopic response of materials in relation to themicrostructure. In the VIB model, the solid is considered to consist of numerous discrete mass particlesat the microscopic scale. These mass particles are connected with a virtual internal bond, which is embed-ded with assumed cohesive law. These bonds only restrict the radial displacement of a two-coupled particlesas shown as Fig. 1. For a two-coupled particles, besides the radial displacement, there is also relative rota-tion freedom as shown in Fig. 2. To restrict the rotation freedom, a R-bond is introduced into the VIB

Fig. 1. Virtual internal bond model.

Z.N. Zhang, X.R. Ge / Theoretical and Applied Fracture Mechanics 43 (2005) 342359 343


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model, Fig. 3. Here, the radial bond is referred as the L-bond. Each L-bond is accompanied with three R-bonds, which restrict the L-bond rotation towards the three coordinate axes respectively.

2.2. Constitutive relationship

The interaction laws between the particles presumably would determine the macroscopic properties ofthe material. The interaction in the VIB model is nonlinear. However, in what follows, the interaction isassumed to be linear in that both the L-bond and R-bond are linearly elastic. The energy potential storedin a single L-bond can be expressed as:

UL 12k2 1

where k is the L-bond stiffness coefficient; is the stretch of the L-bond.

The energy potential stored in the R-bond is:

UR1 12

rb21

UR2 12

rb22

UR3 12

rb23

8>: 2

where r is the R-bond stiffness coefficient; URi(i= 1,2,3) is the energy potential of the R-bond correspond-ing to the coordinate-xi such that bi is the rotation angles of L-bond towards coordinate- xi. Hence, theenergy potential within a two-coupled particles i j could be expressed as:

Uij UL UR1 UR2 UR3 3

Fig. 3. The L-bond and the R-bond.

Fig. 2. Decomposition of bond deformation (b: rotation angle; I: radial displacement; I0: the original length of L-bond).

344 Z.N. Zhang, X.R. Ge / Theoretical and Applied Fracture Mechanics 43 (2005) 342359


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According to Eqs. (1)(3), the pairwise particle potential is associated with the bond stiffness and theparticles position which must be determined. The position of the atoms may be related to that of the con-tinuum field through the local deformation gradient. This is the so-called CauchyBorn rule [25] as shown

in Fig. 4 [5].The theoretical basis of quasi-continuum method is to make a continuum volume element mechanicallyequivalent to a discrete microstructure. The equivalence principle is the strain energy density equivalence.In an arbitrary representative element with volume V, the total strain energy stored in L-bond could beexpressed as:

UL

ZL2L1

Z2p0

Zp0

ULDL0; h;/ sinh dh d/ d0 4

where DL(0,h,/) is the spatial distribution density of 0-length bond; DL0; h;/ sinhdh d/ means thenumber of0-length bonds per unit volume between the bond angle (h,h + dh) and (/,/ + d/) in sphericalcoordinate; L1, L2 is the minimum and maximum original length of L-bond respectively. The total strainenergy stored in R-bond can thus be expressed as:

UR

ZL2L1

Z2p0

Zp0

UR1DR10; h;/ sinhdh d/ d0

ZL2L1

Z2p0

Zp0

UR2DR20; h;/ sinhdh d/ d0

ZL2L1

Z2p0

Zp0

UR3DR30; h;/ sinhdh d/ d0 5

where DRi(0,h,/)(i= 1,2,3) is the distribution function of R-bond related to the L-bond with originallength 0. URi is the corresponding R-bond energy. The total strain energy stored in bond is:

U UL UR 6

The L-bond stretch in a certain direction n is:

0nieijnj 7

where n = (sin h cos /, sin h sin /, cos h) is the unit orientation vector of L-bond in the sphere coordinatesystem, shown as Fig. 5; eij is the strain tensor; 0 is the original length of L-bond. The rotation anglesof L-bond with orientation n towards three coordinate axes are given by

b1 nieijg0

j

b2 nieijg00

j

b3 nieijg000

j

8>: 8

Fig. 4. The CauchyBorn rule: an undistorted lattice with vectors A1 is mapped to the deformed configuration by application of thelocal deformation gradient F (from [5]).



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where g is the unit vector perpendicular to n, g0 n x*1 n; g00 n x*2 n; g000 n x*3 n; x*i isthe unit orientation vector of coordinate-xi(for details, refer to Appendix A). The concrete expression ofg is:

g0 sin2hsin2/ cos2h sin2h cos / sin /; sin h cos / cos h

g00 sin2h cos / sin /; cos2h sin2hcos2/; sin h sin / cos h

g000 sin h cos / cos h; sin h sin / cos h; sin2h

8>:

Using Eqs. (1)(8) and equating the strain energy function at the continuum level to the potential energystored in the cohesive bonds due to an imposed deformation, the stress tensor can be expressed as the rela-tions used in the hyper-elastic theory [26,27]:

rij

oU=V

oeij

1

VZ

L2

L1

Z2p

0Z

p

0 k2

0nmemnnnninjDL0; h;/ sinh dh d/ d0

1

V

ZL2L1

Z2p0

Zp0

rb1nig0

jDR10; h;/ sinhdh d/ d0

1

V

ZL2L1

Z2p0

Zp0

rb1nig00

jDR20; h;/ sinhdh d/ d0

1

V

ZL2L1

Z2p0

Zp0

rb1nig000

j DR30; h;/ sinhdh d/ d0 9

where V is the representative element volume.The elastic modulus can be expressed as:

Cijmn o

2U=V

oeijoemn

1

V

ZL2L1

Z2p0

Zp0

k20ninjnmnnDL0; h;/ sinhdh d/ d0

1

V

ZL2L1

Z2p0

Zp0

rnig0

jnmg0nDR10; h;/ sinhdh d/ d0

1

V

ZL1L2

Z2p0

Zp0

rnig00

j nmg00nDR20; h;/ sinhdh d/ d0

1

V

ZL2L1

Z2p0

Zp0

rnig000

j ximg000n DR30; h;/ sinhdh d/ d0 10

Fig. 5. Indication of the L-bond orientation in sphere coordinate system.

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For the simplification purpose, let the original length of L-bond distribute uniformly with original length0. Therefore, Eqs. (9), (10), becomes the following form:

rij

1

VZ2p

0

Zp0 k

2

0nmemnnnninjDLh;/ sinhdh d/

1

VZ2p

0

Zp0 rb1n

ig0

jDR1h;/ sinhdh d/

1

V

Z2p0

Zp0

rb1nig00

jDR2h;/ sinhdh d/ 1

V

Z2p0

Zp0

rb1nig000

j DR3h;/ sinhdh d/ 11

Cijmn 1

V

Z2p0

Zp0

k20ninjnmnnDLh;/ sinhdh d/ 1

V

Z2p0

Zp0

rnig0

jnmg0nDR1h;/ sinhdh d/

1

V

Z2p0

Zp0

rnig00

j nmg00nDR2h;/ sinhdh d/

1

V

Z2p0

Zp0

rnig000

j nmg000n DR3h;/ sinhdh d/

12

2.3. Relationship between the bond stiffness and the material constants

For the isotropic material, the bond density distributed uniformly in spatial. So, both the L-bond densityand the R-bond density could be taken as constant and DL(h,/) = DR1(h,/) = DR2(h,/) = DR3 (h,/). Con-sider

R2p0

Rp0

Dh;/ sin hdh d/ 1, such that the bond density can be written as:

DLh;/ DR1h;/ DR2h;/ DR3h;/ 1

4p

Using the FEM analysis, the elastic tensor Cijmn may take the matrix form as

X

C1111 C1122 C113312

C1112 C112112

C1132 C112312

C1113 C1131

C2211 C2222 C22331

2 C2212 C2221

1

2 C2232 C2223

1

2 C2213 C2231C3311 C3322 C3333

12

C3312 C332112

C3332 C332312

C3313 C3331

C1211 C1222 C123312

C1212 C122112

C1232 C122312

C1213 C1231

C2311 C2322 C233312

C2312 C232112

C2332 C232312

C2313 C2331

C1311 C1322 C133312

C1312 C132112

C1332 C132312

C1313 C1331

2666666664

3777777775

13

where X can be fitted into the relationship:

r X e 14

with r = [r11,r22,r33,r12,r23,r13]T and e = [e11,e22,e33,2e12,2e23,2e13]

T. Note that rij and eij are the compo-nents of stress and strain tensor, respectively. According to Eq. (13), the elastic matrix corresponding toEq. (12) becomes

X 1

15V

3k20 2r k20 r k

20 r 0 0 0

3k20 2r k20 r 0 0 0

3k20 2r 0 0 0

k20 1.5r 0 0

symmetry k20 1.5r 0

k20 1.5r

2666666664

3777777775

15

In Eq. (15), the relationship between k, r and the macro material constants, i.e. the Youngs modulus Eand the Poisson ratio m could be obtained:



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k 3EV

1 2m20

r 3EV1 4m

1 m1 2m

8>>>>>:

16

The case ofr = 0 is equivalent to the case of not considering rotation in the energy potential In [1], therotation energy potential is not considered and hence the Poisson ratio of m = 0.25 was obtained. For thetwo dimensional cases, the bond construction is shown in Fig. 6. Eq. (12) reduces to the form:

Cijmn 1

V

Zp0

k20ninjnmnn rnig0

jnmg0n rnig

00j nmg

00n

1

pdh 17

where n = [sinh, cosh]T

; g0 = [cosh, sinh]T

; g00 = [cosh, sinh]T

; and 1/p the bond density. The elastic ma-trix corresponding to Eq. (17) is

X 1

8V

3k20 2r k20 2r 0

k20 2r 3k20 2r 0

0 0 k20 2r

264

375 18

For plane-strain, the relationship between k, r and E, m could be obtained by using Eq. (18):

k 2EV

201 2m1 m

r

EV1 4m

1 2m1 m

8>>>>>:

19

3. Extension to the nonlinear materials

3.1. Micromechanism of material weakening due to the deformation

The material weakening behavior is assumed to be deformation-governed in that the local material fail-ure is considered as the result of bond density reduction induced by the local deformation, not induced bythe local cohesive stress. For the usual material, its failure process usually undergoes the linear elastic

Fig. 6. Virtual bond in two-dimensional cases.



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phase, strengthening phase and the strain softening phase, shown as Fig. 7. In the linear elastic phase, themodulus keeps constant. However, in the strengthening and the strain-softening phase, the modulus de-creases with increasing deformation. This is because some flaws and dislocations initiate in the material

as the deformation increases. Additional increase in deformation leads to the coalescence of these flawsand dislocations. This is then followed by the initiation of micro-cracks as shown in Fig. 8. Nonlinear weak-ening behavior of the material finally takes place.

Assumed in this work is that the reduction of material modulus could be considered as the result of thebond density reduction. In the linear elastic phase, both the L-bond and the R-bond distribution density arekept constant. As the deformation exceed the linear elastic limit, the bond density reduces due to theappearance of flaws, and dislocations. This is shown in Fig. 9. As the deformation further increases, thebond density reduces more seriously and tends to zero.

The bond density reduction with the deformation can be qualitatively shown in Fig. 10. To quantita-tively describe the density reduction with the deformation increasing, the bond density can be phenomeno-logically expressed. With reference to the L-bond density, it can be shown that

DLh;/ 1

4pif l0 6 ec

1

4pexpkl0 ec

2=e2c if l

0 > ec

8>: 20

where l0 = jni eijnjj. The symbol j j means the absolute value and k is a coefficient used to adjust the reductionratio of bond density while ec is the critical value of strain, the determination of which will be described inSection 3.2. The R-bond density can be expressed as:

Fig. 7. Three phases of the whole process stressstrain relationship for the uniaxial tension test (rp: the propotional strength limit; ep:the proportional strain limit; rb: the tension strength of material; eb: the strain corresponding to rb).

Fig. 8. IIlustration of materials weakening with deformation increasing.



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DRibi

1

4p

if b0i 6 bc

1

4pexpkb0i bc

2=b2c if b

0i > bc

8>: 21

where b0i is the absolute value of the L-bond rotation angle towards the coordinate-xi; i.e. b0i jbij (Eq. (8));

bc is the critical value of L-bond rotation angle. Refer to Section 3.2. For the two-dimensional cases, the L-bond density could be expressed as:

DLh;/

1

pif l0 6 ec

1

pexpkl0 ec

2=e2c if l0 > ec

8>: 22

Fig. 10. Bond density varying tendency with the deformation increasing (D0: the initial bond density).

Fig. 9. Bond density variation with the deformation increasing.



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and the R-bond density can be expressed as:

DRibi

1

pif b0i 6 bc

1p

expkb0 bc2=b2c if b

0i > bc

8>: 23

where l0 = jnieijnjj; n = (sinh, cosh) in the plane coordinate, shown as Fig. 6; b01 jnieijg

0jj; b

02 jnieijg

00j j;

g0 cos h; sin h; g00 cos h; sin h.

3.2. Parameter determination

In Eqs. (20)(23), there are totally three parameters, namely ec, bc and k , as discussed in Section 3.1.When the deformation is within the linear elastic limit, i.e. e 6 ep, both the L-bond and the R-bond density

are kept constant. As the deformation exceeds the linear elastic limit ep, i.e. e > ep, shown in Fig. 7, the bonddensity begins to reduce. Therefore, the ec should approximately take the value of ep.At the onset of the linear elastic limit, shown in Fig. 11, the strain is: e22 = ep, e11 = me22 = mep,

e12 = e21 = 0. The rotation angle of L-bond with orientation n = (sinh, cosh) is b nieijg0

j epsinh cosh

Fig. 11. IIlustration of uniaxial tension test and bond model in two-dimension.

Fig. 12. Effect ofk on the uniaxial stressstrain relationship, ec = ep bc = bp.



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mep cos h sin h, where g0 = (cosh, sinh). The maximum ofb is: bp 12

ep mep. As discussed in Section 3.1,when the L-bond rotation angle is smaller than bp, the R-bond density does not reduce with increasingdeformation. Therefore, the value ofbc should be no less than bp, i.e. bcP bp. Here, k is a coefficient used

to adjust the bond reduction ratio with deformation. For different materials, their actual strength rb may bedifferent though their proportional strength limit rp and initial Youngs modulus are identical.Figs. 1214 show the effects of the three parameters on the uniaxial tension complete stressstrain rela-

tionship. The model parameters for these figures are: E= 40 GPa, m = 0.2, ep = 0.0006, bp 12

ep mep 0.00036, V= 1, l0 = 1. Fig. 12 shows the effect of parameters k on the complete stressstrain relationship.From Fig. 12, it can be found that the peak-stress, i.e. rb, decreases with k increasing. This is because thatthe bigger the k is, the faster the density decreases with deformation increasing according to Eqs. (20)(23).The ratio of the bond density reduction dominates the material strength. Fig. 13 shows the effect ofbc onthe complete stressstrain relationship. From Fig. 13, it can be found that the bc dominate the post-peakbehavior of material. As bc varies from bp to 5bp, the post-peak behavior changes considerably while thepeak stress changes very little. The bigger the bc is, the slower the post-peak stress declines. Fig. 14 shows

Fig. 13. Effect of bc on the uniaxial stressstrain relationship, k = 0.1 ec = ep.

Fig. 14. Effect of ec on the uniaxial stressstrain relationship, k = 0.1 bc = bp.



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the effect of ec on the stressstrain relationship. From Fig. 14, it can be found that the value of ec has aconsiderable effect on the peak-stress value, but little on the varying ratio of the post-peak stress. The smal-ler the value ofec is, the smaller the peak-stress. This is because that the smaller the value of ec, the earlier

the L-bond density reduction begins.

4. Application to the crack initiation and propagation

4.1. Mechanism of simulating crack initiation and propagation

When the deformation exceeds certain value, material modulus would reduce. This simulates crack ini-tiation and propagation as shown in Fig. 15. As the deformation of the element at the fracture tip exceedcertain value, the corresponding element would have negligible contribution because its stiffness tends tozero. This is equivalent to the state of fracture propagation. By the same reason, as the local deformationexceed certain value, crack would initiate as shown in Fig. 15(b).

4.2. Simulation example

To simulate crack initiation and propagation, a typical example is presented. Consider a rectangularplate with an around hole at the center, whose dimensions are shown in Fig. 16(a). A uniform tension stress

Fig. 15. Mechanism of the presented model simulating crack initiation and propagation. (a) Fracture propagation (from [1]). (b) Crackinitiation.



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rp is applied at the edges. Due to symmetry, only one quarter of the geometry need to be considered. Theboundary conditions are shown in Fig. 16(b). The model parameters are listed in Table 1. The FEM mesh

scheme is shown as Fig. 17. The total number of elements is 870. The number of nodes along coordinate-x1is 30, and the number of nodes along the quarter arc of the hole is 31. The element type is 4-node planeisoparameter element. In [1], the representative element volume is unit. In [25], the original bond lengthis taken as unity. The same will be done in this simulation. Figs. 1821 show the results of the simulation.

Fig. 16. Simulation object and its boundary conditions.

Table 1Model parameters for numerical simulation

E m eb ep ec bc k

40 GPa 0.2 0.002 0.0006 0.0006 0.000036 0.1

Fig. 17. Mesh scheme of Fig. 16(b) for FEM implementation.



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Fig. 18. Relationship between applied stress and the normal stress at point-1 (Fig. 16(b)).

Fig. 19. Normal stress along the coordinate-x1 under different applied stress.

Fig. 20. The final deformed mesh (post-peak: rp = 0.005 MPa).



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4.3. Discussion on numerical results

Fig. 18 shows the relationship between the applied stress rp and normal stress r22 at the Point-1, Fig.16(b). From Fig. 18, it can be found that the stress at Point-1 increases with the applied stress rpincreasing at first. As the applied stress reaches a certain value, the Point-1 stress begins to decline. Thisdecline indicates that the zone around Point-1 reaches the post-peak softening phases. With further in-crease of the applied stress, the stress of Point-1 becomes smaller and smaller due to the reduction ofstiffness. When the stiffness becomes small enough, the contribution of the corresponding element tothe whole structure is negligible, although these elements are still kept in the FEM iterative computation.This is equivalent to saying that a crack has initiated at Point-1. From Fig. 18, it also can be found thatas the applied stress reaches a certain value, it would never reverse. It decreases though it doesnt reach

the tension strength. This is because the net area along coordinate-x1 is smaller than the rectangularedge of the applied stress due to the existence of hole and the yielding area around Point-1. Althoughthe applied stress doesnt reach the tension strength, the elements along coordinate-x1 will have reachedtheir tension strength followed by entering the softening phases. Hence, the applied stress rp woulddecline.

Fig. 19 shows the stress along coordinate-x1 under different applied stress rp. It can be found that thestress at the hole edge is much bigger than that farther away from the edge when the applied stress issmall, e.g. rp = 28.8 MPa. This is the typical result of the stress intensity at the hole edge. Increasingthe applied stress, the domain near the hole edge enter the strain-softening phase. As a consequence,the stress declines. This indicates crack initiation. The crack propagates further with additional increaseof the applied stress. The same procedure will repeat. The peak-stress, like a wave, spread forward from

the hole edge to the farther end. Moreover, it also can be noticed that the stress intensity becomes weakerand weaker with the crack growth. The final deformation is shown in Fig. 20. It can be found that theelements at the upper part deform very little. However, the deformation of the elements along the coor-dinate-x1 is severe. This is equivalent to saying that the plate is intensely fractured along coordinate-x1 . Italso can be noticed that a neck-narrowing phenomenon occurs. This is indicated by the point (x1 = 3)moving to the left. The effects of the Poisson ratio on the crack behaviors are shown in Fig. 21. It canbe found that the Poisson ratio has considerable effect on the simulated results. The higher the Poissonratio is, the bigger the computed stress at the tip of crack. This is because for plane strain, the transversedeformation is restricted. The Poisson ratio is a parameter which characterize the property of transversedeformation. Under the condition of the transverse deformation being restricted, the magnitude of Pois-son ratio do have effect on the stress field.

Fig. 21. Effect of Poisson ratio on stress field at crack tip (Point-1).



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5. Concluding remarks

By adding a R-bond which is used to restrict the rotation freedom. This is in contrast to the VIB model

in [1]. The present model can consider Poissons ratio other than 0.25. The derived constitutive relationshipconnects the discrete structure characters at the microscopic, i.e. the bond stiffness, and the mechanical

properties at the macroscopic, i.e. the Poisson ratio and the Youngs modulus.Based on the modified VIB model, the local materials failure and the fracture behavior are analyzed. In

contrast to the cohesive stress method, the presented method attributes the local material failure to thebond density reduction induced by the local deformation rather than induced by the local cohesive stress.By incorporating a relationship between the bond density and the deformation, the criterion of fracture isdirectly built into the constitutive model. Through simulation, an example of crack initiation and propa-gation is presented and the results are discussed.

Acknowledgement

During the review of this paper, Prof. G.C. Sih provided some constructive advices and help. Theauthors would like to thank Prof. G.C. Sih for his valuable instruction, help and encouragement.

Appendix A

For the L-bond whose unit orientation vector is n, n = (sin h cos /, sin h sin /, cos h), its rotation angletowards xi-axes is equal to the angle of n towards g

i, which is perpendicular to n and located in the planeconstructed by n and x

*i where x

*i is the unit orientation vector of xi-axes. So, g

i n x*

i n, i= 1,2,3,shown as Fig. A.1.

g0 sin2hsin2/ cos2h sin2h cos / sin /; sin h cos / cos h

g00 sin2h cos / sin /; cos2h sin2hcos2/; sin h sin / cos h

g000 sin h cos / cos h; sin h sin / cos h; sin2h

8>: A:1

Fig. A.1. Indication of gi and n.



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The rotation angle of the L-bond towards xi-axes is:

bi nmemngin A:2

The rotation angles are shown in Fig. A.2, emn is the strain tensor at the Point o.

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Micromechanical Consideration of Tensile Crack Behavior

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