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Formulation of a cohesive zone model for a Mode III crack
Engineering Fracture Mechanics 72 (2005) 1818–1829 www.elsevier.com/locate/engfracmech
Formulation of a cohesive zone model for a Mode III crack Wei Zhang, Xiaomin Deng
*
Department of Mechanical Engineering, University of South Carolina, Columbia, SC 29208, USA Received 13 July 2004; received in revised form 5 November 2004; accepted 15 November 2004 Available online 2 March 2005
Abstract Cohesive zone models have been proven effective in modeling crack initiation and propagation phenomena. In this work, a possible form for a Mode III cohesive zone model is formulated from elastic stress and displacement fields around a crack with a cohesive zone ahead of the crack tip. A traction–separation relation for the model is derived as a direct consequence of the formulation, which establishes some intrinsic connections between properties of the cohesive zone and those of the bulk material. Interestingly, this model states that the von Mises effective stress in the cohesive zone is constant, which may be related to the bulk materialÕs yield stress and is consistent with the assumption made in conventional strip-yield elastic–plastic solutions. Ó 2005 Elsevier Ltd. All rights reserved. Keywords: Crack; Cohesive zone model; Traction–separation relation; Mode III
1. Introduction Linear elastic fracture mechanics (LEFM) [1] predicts that the stress state at the tip of a crack in a brittle material is singular and infinite, which is known to be physically unrealistic. To overcome this difficulty in an elasticity-based treatment, Dugdale [2] and Barenblatt [3] argued that a cohesive zone exists ahead of the crack tip, which extends from the tip to some distance along the crack line and limits the magnitude of stress at the crack tip to physically meaningful levels. Today, the cohesive zone concept has become an effective way of describing and simulating material decohesion and debonding in monolithic and composite materials with and without a crack [4–9]. Under anti-plane shear conditions, the cohesive zone concept has been utilized to obtain elastic–plastic field solutions (the strip yield solutions) for a Mode III crack containing a line plastic zone ahead of the *
Corresponding author. Tel.: +1 803 777 7144; fax: +1 803 777 0106. E-mail address: [email protected] (X. Deng).
0013-7944/$ - see front matter Ó 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.engfracmech.2004.11.008
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crack tip. In particular, Unger [10] developed a finite-width Dugdale zone model to describe the Mode III crack-tip stress field, and Yi et al. [11] applied a crack-line field analysis method to the Mode III problem in an elastic–perfectly plastic plate and obtained the stress field near the crack line. In these two studies and in more general cohesive zone model studies, the cohesive zones are applied as zero-thickness material regions obeying phenomenological traction–separation laws. Many model variants exist in the literature and their parameters can significantly differ from one another [12]. In this paper, the authors will try to shed light on the formulation of cohesive zone models from admissible elastic stress and displacement fields around the tip of a Mode III crack containing a cohesive zone ahead of the crack tip. As demonstrated in this paper, the resulting elastic field solutions with the consideration of a cohesive zone are identical to those in a strip-yield model. However, unlike the approach used in the strip-yield model, a specific traction–separation law for the cohesive zone is not specified in advance. Instead, a traction–separation relation is extracted from the field solutions, which provide a possible form for a cohesive zone model and establishes explicit connections between the properties of the cohesive zone and those of the bulk material. In particular, a consequence of this model is that the von Mises effective stress in the cohesive zone is constant, which may be taken to be the bulk materialÕs yield stress and thus is consistent with the assumption made in the strip-yield model. This paper is arranged as follows. Section 2 presents the elastic solutions of the stress and displacement fields for a Mode III crack, which allow for a cohesive zone ahead of the crack tip; Section 3 derives an explicit traction–separation relation from the elastic field solutions; Section 4 discusses some issues concerning the cohesive zone model and its parameters; and Section 5 concludes with the main findings of the paper.
2. Stress and displacement fields Fig. 1 shows a schematic of a Mode III crack containing a cohesive zone ahead of the crack tip. The cohesive zone is defined as a transition region in which the surface traction smoothly changes from zero at the crack tip to a certain magnitude at the cohesive zone tip, where the displacement discontinuity across the zone disappears. The cohesive zone is a mathematical extension of the crack and can be viewed physically as the fracture process zone. To more effectively account for the surface boundaries introduced by a cohesive zone, we make use of the elliptic cylindrical coordinates u, v, w [13], which are related to the Cartesian coordinates x, y, z by 8 x ¼ c coshðuÞ cosðvÞ > < y ¼ c sinhðuÞ sinðvÞ ðu P 0; p 6 v 6 pÞ; ð1Þ > : z¼w
Fig. 1. A schematic of a Mode III crack containing a cohesive zone ahead of the crack tip.
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Fig. 2. (a) A 2D plot of the elliptic coordinate system with a family of solid lines (u = const) and dashed lines (v = const); (b) Locations of key points and lines in terms of elliptic coordinates, with the cohesive zone being represented by two line segments (u = 0, p 6 v 6 0 for the lower cohesive surface and u = 0, 0 6 v 6 p for the upper cohesive surface).
where c is selected as a characteristic length of the cohesive zone. Fig. 2 shows a 2D plot of the elliptic coordinate system with a family of solid lines (u = const) and dashed lines (v = const) illustrating the curvilinear coordinates. Two coincident lines, namely v = p and v = p, are used to represent the traction-free surfaces of the crack, respectively. The origin of the Cartesian coordinates is at u = 0 and v = ±p/2, the crack tip is at x = c and y = 0 (u = 0, v = ±p), and the two coincident line segments with length 2c (u = 0,p 6 v 6 p) are the cohesive surfaces as indicated in Fig. 2. The derivatives of the Cartesian coordinates (x, y) with respect to the elliptic coordinates (u, v) are 2 3 oxðu; vÞ oxðu; vÞ " # sinhðuÞ cosðvÞ coshðuÞ sinðvÞ 6 ou 7 ov 6 7 ð2Þ 4 oyðu; vÞ oyðu; vÞ 5 ¼ c coshðuÞ sinðvÞ sinhðuÞ cosðvÞ : ou ov The Jacobian inverse of the above matrix yields the derivatives of the elliptic coordinates (u, v) with respect to the Cartesian coordinates (x, y): 2 3 ouðx; yÞ ouðx; yÞ " # 6 ox cosðvÞðe2u 1Þ sinðvÞðe2u þ 1Þ oy 7 6 7 ; ð3Þ 6 7¼A 4 ovðx; yÞ ovðx; yÞ 5 sinðvÞðe2u þ 1Þ cosðvÞðe2u 1Þ ox
oy
where A¼
2eu c½4 sinðvÞ2 e2u
þ ðe2u 1Þ2
:
ð4Þ
For a Mode III crack, the deformation is of the anti-plane shear type, so that ux = uy = 0 and uz = uz(x,y) in the Cartesian coordinate system. In the absence of any body forces, the equation of equilibrium can be written by the non-trivial out-of-plane stress components as rzx;x þ rzy;y ¼ 0:
ð5Þ
To find elastic solutions of the anti-plane shear problem, the complex variable method can be used. In the Cartesian coordinate system, we define the complex variable f ¼ x þ iy:
ð6Þ
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In the elliptic coordinate system, we define another complex variable x ¼ u þ iv:
ð7Þ
The coordinate transformation in Eq. (1) leads to the following conformal mapping between the two complex variables in Eqs. (6) and (7) f ¼ c coshðxÞ:
ð8Þ
An analytic elastic solution for the Mode III crack problem will now be obtained for an infinite plate. To focus on obtaining a solution in the near-tip region that includes the cohesive zone, a solution possessing the following features is sought: (a) there are no tractions on the free surfaces of the crack; (b) stresses are finite everywhere due to the existence of a cohesive zone; (c) tractions are continuous across the cohesive surfaces; (d) there is a finite displacement jump across the cohesive surfaces; and (e) stresses tend to zero far away from the near-tip region. There are otherwise no pre-determined conditions or cohesive laws prescribed along the two coincident cohesive zone surface boundaries. By selecting an appropriate complex variable function, we can obtain the following stress and displacement fields in terms of the elliptic coordinates (Appendix A) rzx ¼ smax eu=2 sinðv=2Þ;
ð9Þ
rzy ¼ smax eu=2 cosðv=2Þ;
ð10Þ
uz ¼
dtip u=2 ½3e sinðv=2Þ e3u=2 sinð3v=2Þ 8
ð11Þ
16 bc; 3
ð12Þ
with smax ¼ 2lb;
dtip ¼
where l is the shear modulus and b is a non-dimensional real constant. Hence, once b and c are determined, the complete stress and displacement fields, Eqs. (9)–(11), are determined. It is easy to verify that Eqs. (9) and (10) satisfy the stress equilibrium equation (Eq. (5)) when Eq. (3) is utilized. Using Eqs. (1) and (9)– (11), the normalized stress and displacement components are plotted against the normalized Cartesian coordinates in Figs. 3(a)–(c), respectively. The stress components in the cohesive zone (when u = 0) are rzx ¼ smax sinðv=2Þ;
ð13Þ
rzy ¼ smax cosðv=2Þ:
ð14Þ
The shear stresses reach their maximum value smax at the crack tip (rzx at u = 0, v = p) and at the cohesive zone tip (rzy at u = 0, v = 0), respectively. The displacement jump across the cohesive zone surfaces (upper surface minus lower surface) is computed from Eq. (11) as dz ¼ 2uz ¼ dtip sinðv=2Þ
3
ð0 6 v 6 pÞ;
ð15Þ
where the maximum separation in the cohesive zone occurs at the crack tip (v = p).
3. Traction–separation relation In the cohesive zone, the relationship between the cohesive stress (traction) acting on the cohesive surfaces and the displacement jump (separation) across the cohesive zone defines a cohesive zone model. The
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Fig. 3. A plot of full-field distributions of the normalized stress and displacement fields in the vicinity of the Mode III crack tip: (a) rzx/smax; (b) rzy/smax; and (c) uz/dtip.
combination of Eqs. (14) and (15), both of which are parametric functions of the variable v (0 6 v 6 p), leads to the following explicit traction–separation relation: rzy ¼ smax ½1 ðdz =dtip Þ
2=3 1=2
ð0 6 dz 6 dtip Þ;
ð16Þ
which can be viewed as a possible form of a Mode III cohesive zone model. Interestingly, a traction–separation relation of the same form has been derived for a Mode I crack with a different approach [14]. A normalized plot of the traction–separation relation in Eq. (16) is given in Fig. 4. It reveals that the cohesive traction decreases monotonically with the separation, and the maximum traction value occurs at the cohesive zone tip, where the separation is zero. It should be emphasized that the traction–separation relation is extendable to cases involving negative tractions and/or negative displacement jumps, even though only the case with a positive traction and a positive displacement jump is plotted in Fig. 4.
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Fig. 4. The traction–separation relation (in normalized form) for the cohesive zone model.
4. Discussions 4.1. Connection with a strip yield model It is seen in Fig. 3 that all stress components are finite everywhere and the maximum value of the cohesive traction, rzy, occurs at the cohesive zone tip (x = c, y = 0 or u = 0, v = 0). For the Mode III crack, the von Mises effective stress in the cohesive zone is qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi re ¼ 3ðr2zx þ r2zy Þ ¼ 3smax : ð17Þ Interestingly, re is a constant and is independent of the surface separation dz. To this end, it is worth noting that Dugdale [2] used the assumption of a uniform stress in the cohesive zone to formulate the Mode I strip-yield problem. Eq. (17) reveals that a form of the cohesive zone model with a uniform von Mises effective stress distribution in the cohesive zone of a Mode III crack is derivable from the overall formulation of the elastic field solutions. Furthermore, Eqs. (9)–(11) represent the elastic–plastic solution of the finite-width Dugdale zone model [10,15] when the plastic zone reduces to a line segment, in which the general Mode III solution was obtained by assuming an ellipse-shaped perfectly plastic zone ahead of the crack tip. An interpretation of Eq. (17) based on the Dugdale zone mode is that the von Mises effective stress in the cohesive zone of a Mode III crack equals the yield stress of the surrounding bulk material, even though Eq. (17) is derived without the a priori assumption that the cohesive zone is an elastic–plastic zone with an elastic–perfectly plastic behavior. Another point can also be drawn from the above discussions: According to Eq. (12), the non-dimensional parameter b now has the physical meaning of being equal to one half of the ratio between the yield stress in pure shear and the shear modulus. As noted by Unger [15], the Mode III stress and displacement fields, Eqs. (9)–(11), can be derived from a plastic-strip model by Cherepanov [16], in which a plastic zone occupies the location of the cohesive zone. The advantage of the cohesive zone model is that it clearly defines the separation of the cohesive surfaces, while the plastic-strip model lacks the ability to explain the displacement jump across the plastic zone. 4.2. Fracture energy In a cohesive zone model, the specific fracture energy that is dissipated when new fracture surfaces are created in a material corresponds to the area under the traction–separation curve. For the particular form of the cohesive zone model described by Eq. (16) (see also Fig. 4), this fracture energy is given by
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Gf ¼ smax
Z
dtip
h
1 ðdz =dtip Þ
2=3
i1=2
0
ddz ¼
3p smax dtip : 16
ð18Þ
If the path-independent J integral [17] is made to equal to the fracture energy, and if we let smax be the pure shear yield stress k, then the crack tip opening displacement in the Cherepanov plastic-strip model [16] can be recovered, which is dtip ¼
16J : 3pk
ð19Þ
Unger [15] obtained the same expression by comparing his Dugdale zone solution to the classical LEFM field equations. It must be pointed out that Eq. (19) offers one possible method to determine the parameters in the cohesive zone model. 4.3. Connection with LEFM solutions Classical LEFM solutions of the asymptotic stress and displacement fields at a crack tip are well documented for all three fracture modes. For convenience, the Mode III solution in terms of polar coordinates is given in Appendix B, which shows an inverse square-root singularity and an infinite stress at the crack tip. The cohesive zone model-based solution developed in this study limits the stress level at the crack tip to a finite value. In order to make a connection between the cohesive zone solution and the classical solution, one can suppose that the stress component rzy from these two solutions match in value at some distance ahead of the crack tip. This distance should be larger than some characteristic length scale (e.g. the cohesive zone size 2c) of the process zone but still sufficiently small so that the classical asymptotic singular field still holds. The following analysis shows what happens if this connection is made. For convenience, the crack tip is chosen at x = c and y = 0 in the Cartesian coordinates (see Fig. 2). Then the transformation between the polar and Cartesian coordinates is given by x ¼ r cosðhÞ c ðp 6 h 6 pÞ: ð20Þ y ¼ r sinðhÞ Now suppose that rzy from the cohesive zone stress solution, Eq. (10), and the classical asymptotic solution, Eq. (B.2), have the same value at some distance in front of the crack tip, say at point (x = mc, y = 0), where m P 1 and c is one half of the cohesive zone size, then pffiffiffiffiffiffiffiffi K III = 2pc ¼ smax hðmÞ; ð21Þ where pffiffiffiffiffiffiffiffiffiffiffiffi hðmÞ ¼ earc coshðmÞ=2 m þ 1:
ð22Þ
Note that the coordinate relations u = arccosh(x/c), v = 0 and r = x + c, h = 0 have been used in deriving Eqs. (21) and (22). Using Eq. (21), Eqs. (B.2) and (10) can be written in normalized forms as, respectively, 1 rczy =s0 ¼ pffiffiffiffiffiffiffi cosðh=2Þ; r=c
ð23Þ
rzy =s0 ¼ eu=2 cosðv=2Þ=hðmÞ;
ð24Þ
where pffiffiffiffiffiffiffiffi s0 ¼ K III = 2pc:
ð25Þ
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In the above, the superscript c signifies the classical solution. The corresponding normalized displacements, Eq. (B.3) and Eq. (11), respectively, are pffiffiffiffiffiffiffi ucz =u0 ¼ 2 r=c sinðh=2Þ; ð26Þ uz =u0 ¼ ½3eu=2 sinðv=2Þ e3u=2 sinð3v=2Þ=3hðmÞ; where u0 ¼
K III l
rffiffiffiffiffiffi c : 2p
In the above derivations, Eqs. (12) and (21) have been used, that is rffiffiffiffiffiffi 8smax 8K III c c¼ : dtip ¼ 3l 3hðmÞl 2p
ð27Þ
ð28Þ
ð29Þ
Comparisons of the normalized stress rzy ahead of the crack tip, and of the corresponding displacement behind cohesive zone tip, both along the lower surface of the crack (y = 0), are shown in Fig. 5 for several m values. It is seen that the far field stress agrees well with each other when m is sufficiently large (e.g., m = 9). 4.4. On the formulation approach The cohesive zone formulation approach described in this work can be viewed in two ways. In the first case, the current approach provides a way of generating possible forms of the traction–separation relationship that are consistent with the linear elastic stress and displacement fields around cracks with cohesive zones ahead of the crack tip. The particular simple form of the cohesive zone model described in Eq. (16) is established based on a one-term stress and displacement fields corresponding to k = 1/2, which appears to be a preferred choice based on several considerations. It is noted that if the eigenfunctions are viewed as terms in a ‘‘Fourier’’ series expansion, then the eigenvalue k will correspond to the ‘‘frequency’’ of a single eigenfunction. Thus, terms with the lowest ‘‘frequencies’’ (i.e. k = 1/2 and 1/2) should be the dominant ones in the series. Since
Fig. 5. Comparisons of present elastic solutions along y = 0 (the lower surface) with classical solutions for various value m: (a) rzy/s0 and (b) uz/u0.
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k = 1/2 results in infinite stresses in the far field, the preferred term is k = 1/2. Moreover, it can be shown that the high ‘‘frequency’’ terms (e.g., k = 3/2) have zero contribution to the fracture energy even though these terms may change the form of the traction–separation relation. Nevertheless, if terms corresponding to other eigenfunctions are chosen, or if combinations of various terms are employed, then different forms of the traction–separation relationship can be obtained. The second way to view the current approach is that, the infinite series solutions derived in this work provide a means of understanding and possibly quantifying the elastic stress and displacement fields around a crack containing a cohesive zone ahead of the crack tip, where a cohesive law governing the behavior of the cohesive zone may be given as part of a boundary-value problem. In principle, relevant terms of the near-field series solutions must work together in order to match the traction–separation relation in the cohesive zone as dictated by the given cohesive law, thus allowing the determination of the coefficients for these terms, while the coefficients of other terms in the series may be determined by matching with the far fields, which are expected to contain information about the geometry and loading conditions of a particular problem. However, this option is not explored in the current work.
5. Summary and concluding remarks In this work, linear elastic solutions for the stress and displacement fields around a Mode III crack containing a cohesive zone ahead of the crack tip have been obtained without a priori specification of a cohesive law for the cohesive zone. A possible form of a Mode III cohesive zone model is proposed, which is based on a particular choice of the stress and displacement fields derived in this work. The traction–separation relation in this model contains two independent parameters, b and c, where b is dimensionless and has the physical meaning of being equal to one half of the ratio between the yield stress in pure shear and the shear modulus, and c is the half length of the cohesive zone. The parameters b and c are also related to the specific fracture energy of the material. The stress field corresponding to this cohesive zone model is finite everywhere and the maximum cohesive stress (traction in the cohesive zone) occurs at the cohesive zone tip. The von Mises effective stress in the cohesive zone is constant and can be interpreted as the yield stress of the surrounding bulk material, which is consistent with the assumption in strip-yield models for elastic–perfectly plastic materials that a line plastic zone exists directly ahead of the crack tip. The developed approach for deriving near-field stress and displacement solutions and for formulating crack-tip cohesive zone models can be extended to Mode I and Mode II cracks. Results of this extension will be reported separately [18].
Acknowledgments The authors greatly appreciate the support of a grant from DEPSCoR/ONR (Grant # N00014-03-10807) and helpful discussions with Dr. J. Zuo and Dr. A. Reynolds.
Appendix A. Elastic solution for a Mode III crack Keeping in mind the relations between the Cartesian and the elliptic coordinate systems (Eq. (1)), we can define uz ðxðu; vÞ; yðu; vÞÞ ¼ uw ðuðx; yÞ; vðx; yÞÞ:
ðA:1Þ
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For anti-plane shear problems, 2 ou ! ! ezx u 6 z;x 1 1 ox ¼ ¼ 6 2 2 4 ou ezy uz;y oy
the only non-zero strain components are 1 30 ov ouz B C ox 7 7B ou C: @ 5 ov ouz A oy ov
1827
ðA:2Þ
Denoting l the shear modulus, the non-zero stress components are related to the strains through the HookeÕs law, so that rzx ezx ¼ 2l : ðA:3Þ rzy ezy Eqs. (5), (A.2) and (A.3) then lead to the Laplace equation r2 uz ðx; yÞ ¼ r2 uw ðu; vÞ ¼ 0;
ðA:4Þ
2
where $ is the Laplacian operator. In the elliptic coordinate system, Eq. (A.4) can be written as 2 1 o uw ðu; vÞ o2 uw ðu; vÞ þ r2 uw ðu; vÞ ¼ ¼ 0: ou2 ov2 c2 ½cosh2 ðuÞ cos2 ðvÞ
ðA:5Þ
The most important task is to find a function for uz = uw that satisfies Eq. (A.4) and at the same time yields stresses that satisfy the traction-free boundary conditions at the crack surfaces, i.e., rzy = 0 for u > 0 and v = ±p. Different from the LEFM, a solution that leads to a finite stress at the crack tip is sought here, since a cohesive zone is expected ahead of the crack tip to remove the stress singularity there. Let f(f) = p(x,y) + iq(x,y) be a holomorphic (analytic) function of the complex variable f, then both p(x,y) and q(x,y) satisfy the harmonic equation r2 pðx; yÞ ¼ 0; r2 qðx; yÞ ¼ 0:
ðA:6Þ
Therefore, the solution of Eq. (A.4) can be written as uz ¼
2 1 Re½f ðfÞ ¼ ½f ðfÞ þ f ðfÞ: l l
Substituting Eq. (A.7) into Eq. (A.2), we have ( 0 1 ezx ¼ 2l ½f 0 ðfÞ þ f ðfÞ : 0 ezy ¼ 2li ½f 0 ðfÞ f ðfÞ
ðA:7Þ
ðA:8Þ
Combining Eq. (A.3) and Eq. (A.8), we obtain rzx irzy ¼ 2f 0 ðfÞ:
ðA:9Þ
Now, letÕs consider the following analytic (holomorphic) function f 0 ðfÞ ¼ gðxÞ ¼ lBekx
ðB ¼ a ibÞ;
ðA:10Þ
where a, b and k are real constants to be determined by the boundary conditions. Then Eq. (A.9) becomes rzx irzy ¼ 2lða ibÞekðuþivÞ : Separating the real and imaginary parts, we have ( rzx ¼ 2leku ½b sinðkvÞ þ a cosðkvÞ : rzy ¼ 2leku ½b cosðkvÞ a sinðkvÞ
ðA:11Þ
ðA:12Þ
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Substituting boundary conditions rzy = 0 for v = ±p results in b cosðkpÞ a sinðkpÞ ¼ 0 : b cosðkpÞ þ a sinðkpÞ ¼ 0
ðA:13Þ
The existence of a non-trivial solution (i.e., B 5 0) requires that the determinant of the coefficient matrix of the preceding equation be zero, that is, sinð2kpÞ ¼ 0:
ðA:14Þ
The above equation in turn requires that the constant k must satisfy k ¼ n=2 ðn is an integerÞ:
ðA:15Þ
If we take the view that stresses must go to zero far away from the crack tip (when u ! 1), k has to be negative according to Eq. (A.12). Of all the eigenfunctions in the form of Eq. (A.12), the function with the eigenvalue k = 1/2 for which a = 0 contributes the most to the stress field far away from the crack tip. If the function with k = 1/2 is chosen to construct the solution, Eq. (A.10) becomes df ðfÞ dx df ðfÞ=dx ¼ ¼ ilbex=2 : dx df c sinhðxÞ
ðA:16Þ
Integrating the above ordinary differential equation gives f ðfÞ ¼
ilbc x=2 ilbc ðuþivÞ=2 ð3e þ e3x=2 Þ ¼ ½3e þ e3ðuþivÞ=2 ; 3 3
ðA:17Þ
where the integration constant has been taken to be zero, since at v = 0 the displacement (real part) must be zero. The stress components can be obtained from Eq. (A.12) as rzx ¼ 2lbeu=2 sinðv=2Þ;
ðA:18Þ
rzy ¼ 2lbeu=2 cosðv=2Þ:
ðA:19Þ
Then displacement can be derived by using Eqs. (A.7) and (A.17), uz ¼
2bc u=2 ½3e sinðv=2Þ e3u=2 sinð3v=2Þ: 3
ðA:20Þ
Note that the procedure for constructing the solution can be written more succinctly if we start directly from a general analytic function, which was not done in order to reveal details about the choice of the general eigenfunction (Eq. (A.10)). Also note that k = ±1 seems to cause singularity problems in Eq. (A.7) since factor (k2 1) will appear in the denominator. However, this can be avoided by replacing B with B = B 0 (k2 1) in Eq. (A.10). Hence the solution Eq. (A.15) is actually valid for all integers.
Appendix B. Classical solution of Mode III crack The asymptotic singular crack-tip stress and displacement fields from classical LEFM can be conveniently expressed in terms of polar coordinates as K III rzx ¼ pffiffiffiffiffiffiffi sinðh=2Þ; 2pr
ðB:1Þ
K III rzy ¼ pffiffiffiffiffiffiffi cosðh=2Þ; 2pr
ðB:2Þ
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uz ¼
2K III r !1=2 sinðh=2Þ; 2p l
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ðB:3Þ
where KIII is the stress intensity factor.
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Formulation of a cohesive zone model for a Mode III crack Wei Zhang, Xiaomin Deng
*
Department of Mechanical Engineering, University of South Carolina, Columbia, SC 29208, USA Received 13 July 2004; received in revised form 5 November 2004; accepted 15 November 2004 Available online 2 March 2005
Abstract Cohesive zone models have been proven effective in modeling crack initiation and propagation phenomena. In this work, a possible form for a Mode III cohesive zone model is formulated from elastic stress and displacement fields around a crack with a cohesive zone ahead of the crack tip. A traction–separation relation for the model is derived as a direct consequence of the formulation, which establishes some intrinsic connections between properties of the cohesive zone and those of the bulk material. Interestingly, this model states that the von Mises effective stress in the cohesive zone is constant, which may be related to the bulk materialÕs yield stress and is consistent with the assumption made in conventional strip-yield elastic–plastic solutions. Ó 2005 Elsevier Ltd. All rights reserved. Keywords: Crack; Cohesive zone model; Traction–separation relation; Mode III
1. Introduction Linear elastic fracture mechanics (LEFM) [1] predicts that the stress state at the tip of a crack in a brittle material is singular and infinite, which is known to be physically unrealistic. To overcome this difficulty in an elasticity-based treatment, Dugdale [2] and Barenblatt [3] argued that a cohesive zone exists ahead of the crack tip, which extends from the tip to some distance along the crack line and limits the magnitude of stress at the crack tip to physically meaningful levels. Today, the cohesive zone concept has become an effective way of describing and simulating material decohesion and debonding in monolithic and composite materials with and without a crack [4–9]. Under anti-plane shear conditions, the cohesive zone concept has been utilized to obtain elastic–plastic field solutions (the strip yield solutions) for a Mode III crack containing a line plastic zone ahead of the *
Corresponding author. Tel.: +1 803 777 7144; fax: +1 803 777 0106. E-mail address: [email protected] (X. Deng).
0013-7944/$ - see front matter Ó 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.engfracmech.2004.11.008
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crack tip. In particular, Unger [10] developed a finite-width Dugdale zone model to describe the Mode III crack-tip stress field, and Yi et al. [11] applied a crack-line field analysis method to the Mode III problem in an elastic–perfectly plastic plate and obtained the stress field near the crack line. In these two studies and in more general cohesive zone model studies, the cohesive zones are applied as zero-thickness material regions obeying phenomenological traction–separation laws. Many model variants exist in the literature and their parameters can significantly differ from one another [12]. In this paper, the authors will try to shed light on the formulation of cohesive zone models from admissible elastic stress and displacement fields around the tip of a Mode III crack containing a cohesive zone ahead of the crack tip. As demonstrated in this paper, the resulting elastic field solutions with the consideration of a cohesive zone are identical to those in a strip-yield model. However, unlike the approach used in the strip-yield model, a specific traction–separation law for the cohesive zone is not specified in advance. Instead, a traction–separation relation is extracted from the field solutions, which provide a possible form for a cohesive zone model and establishes explicit connections between the properties of the cohesive zone and those of the bulk material. In particular, a consequence of this model is that the von Mises effective stress in the cohesive zone is constant, which may be taken to be the bulk materialÕs yield stress and thus is consistent with the assumption made in the strip-yield model. This paper is arranged as follows. Section 2 presents the elastic solutions of the stress and displacement fields for a Mode III crack, which allow for a cohesive zone ahead of the crack tip; Section 3 derives an explicit traction–separation relation from the elastic field solutions; Section 4 discusses some issues concerning the cohesive zone model and its parameters; and Section 5 concludes with the main findings of the paper.
2. Stress and displacement fields Fig. 1 shows a schematic of a Mode III crack containing a cohesive zone ahead of the crack tip. The cohesive zone is defined as a transition region in which the surface traction smoothly changes from zero at the crack tip to a certain magnitude at the cohesive zone tip, where the displacement discontinuity across the zone disappears. The cohesive zone is a mathematical extension of the crack and can be viewed physically as the fracture process zone. To more effectively account for the surface boundaries introduced by a cohesive zone, we make use of the elliptic cylindrical coordinates u, v, w [13], which are related to the Cartesian coordinates x, y, z by 8 x ¼ c coshðuÞ cosðvÞ > < y ¼ c sinhðuÞ sinðvÞ ðu P 0; p 6 v 6 pÞ; ð1Þ > : z¼w
Fig. 1. A schematic of a Mode III crack containing a cohesive zone ahead of the crack tip.
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Fig. 2. (a) A 2D plot of the elliptic coordinate system with a family of solid lines (u = const) and dashed lines (v = const); (b) Locations of key points and lines in terms of elliptic coordinates, with the cohesive zone being represented by two line segments (u = 0, p 6 v 6 0 for the lower cohesive surface and u = 0, 0 6 v 6 p for the upper cohesive surface).
where c is selected as a characteristic length of the cohesive zone. Fig. 2 shows a 2D plot of the elliptic coordinate system with a family of solid lines (u = const) and dashed lines (v = const) illustrating the curvilinear coordinates. Two coincident lines, namely v = p and v = p, are used to represent the traction-free surfaces of the crack, respectively. The origin of the Cartesian coordinates is at u = 0 and v = ±p/2, the crack tip is at x = c and y = 0 (u = 0, v = ±p), and the two coincident line segments with length 2c (u = 0,p 6 v 6 p) are the cohesive surfaces as indicated in Fig. 2. The derivatives of the Cartesian coordinates (x, y) with respect to the elliptic coordinates (u, v) are 2 3 oxðu; vÞ oxðu; vÞ " # sinhðuÞ cosðvÞ coshðuÞ sinðvÞ 6 ou 7 ov 6 7 ð2Þ 4 oyðu; vÞ oyðu; vÞ 5 ¼ c coshðuÞ sinðvÞ sinhðuÞ cosðvÞ : ou ov The Jacobian inverse of the above matrix yields the derivatives of the elliptic coordinates (u, v) with respect to the Cartesian coordinates (x, y): 2 3 ouðx; yÞ ouðx; yÞ " # 6 ox cosðvÞðe2u 1Þ sinðvÞðe2u þ 1Þ oy 7 6 7 ; ð3Þ 6 7¼A 4 ovðx; yÞ ovðx; yÞ 5 sinðvÞðe2u þ 1Þ cosðvÞðe2u 1Þ ox
oy
where A¼
2eu c½4 sinðvÞ2 e2u
þ ðe2u 1Þ2
:
ð4Þ
For a Mode III crack, the deformation is of the anti-plane shear type, so that ux = uy = 0 and uz = uz(x,y) in the Cartesian coordinate system. In the absence of any body forces, the equation of equilibrium can be written by the non-trivial out-of-plane stress components as rzx;x þ rzy;y ¼ 0:
ð5Þ
To find elastic solutions of the anti-plane shear problem, the complex variable method can be used. In the Cartesian coordinate system, we define the complex variable f ¼ x þ iy:
ð6Þ
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In the elliptic coordinate system, we define another complex variable x ¼ u þ iv:
ð7Þ
The coordinate transformation in Eq. (1) leads to the following conformal mapping between the two complex variables in Eqs. (6) and (7) f ¼ c coshðxÞ:
ð8Þ
An analytic elastic solution for the Mode III crack problem will now be obtained for an infinite plate. To focus on obtaining a solution in the near-tip region that includes the cohesive zone, a solution possessing the following features is sought: (a) there are no tractions on the free surfaces of the crack; (b) stresses are finite everywhere due to the existence of a cohesive zone; (c) tractions are continuous across the cohesive surfaces; (d) there is a finite displacement jump across the cohesive surfaces; and (e) stresses tend to zero far away from the near-tip region. There are otherwise no pre-determined conditions or cohesive laws prescribed along the two coincident cohesive zone surface boundaries. By selecting an appropriate complex variable function, we can obtain the following stress and displacement fields in terms of the elliptic coordinates (Appendix A) rzx ¼ smax eu=2 sinðv=2Þ;
ð9Þ
rzy ¼ smax eu=2 cosðv=2Þ;
ð10Þ
uz ¼
dtip u=2 ½3e sinðv=2Þ e3u=2 sinð3v=2Þ 8
ð11Þ
16 bc; 3
ð12Þ
with smax ¼ 2lb;
dtip ¼
where l is the shear modulus and b is a non-dimensional real constant. Hence, once b and c are determined, the complete stress and displacement fields, Eqs. (9)–(11), are determined. It is easy to verify that Eqs. (9) and (10) satisfy the stress equilibrium equation (Eq. (5)) when Eq. (3) is utilized. Using Eqs. (1) and (9)– (11), the normalized stress and displacement components are plotted against the normalized Cartesian coordinates in Figs. 3(a)–(c), respectively. The stress components in the cohesive zone (when u = 0) are rzx ¼ smax sinðv=2Þ;
ð13Þ
rzy ¼ smax cosðv=2Þ:
ð14Þ
The shear stresses reach their maximum value smax at the crack tip (rzx at u = 0, v = p) and at the cohesive zone tip (rzy at u = 0, v = 0), respectively. The displacement jump across the cohesive zone surfaces (upper surface minus lower surface) is computed from Eq. (11) as dz ¼ 2uz ¼ dtip sinðv=2Þ
3
ð0 6 v 6 pÞ;
ð15Þ
where the maximum separation in the cohesive zone occurs at the crack tip (v = p).
3. Traction–separation relation In the cohesive zone, the relationship between the cohesive stress (traction) acting on the cohesive surfaces and the displacement jump (separation) across the cohesive zone defines a cohesive zone model. The
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Fig. 3. A plot of full-field distributions of the normalized stress and displacement fields in the vicinity of the Mode III crack tip: (a) rzx/smax; (b) rzy/smax; and (c) uz/dtip.
combination of Eqs. (14) and (15), both of which are parametric functions of the variable v (0 6 v 6 p), leads to the following explicit traction–separation relation: rzy ¼ smax ½1 ðdz =dtip Þ
2=3 1=2
ð0 6 dz 6 dtip Þ;
ð16Þ
which can be viewed as a possible form of a Mode III cohesive zone model. Interestingly, a traction–separation relation of the same form has been derived for a Mode I crack with a different approach [14]. A normalized plot of the traction–separation relation in Eq. (16) is given in Fig. 4. It reveals that the cohesive traction decreases monotonically with the separation, and the maximum traction value occurs at the cohesive zone tip, where the separation is zero. It should be emphasized that the traction–separation relation is extendable to cases involving negative tractions and/or negative displacement jumps, even though only the case with a positive traction and a positive displacement jump is plotted in Fig. 4.
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Fig. 4. The traction–separation relation (in normalized form) for the cohesive zone model.
4. Discussions 4.1. Connection with a strip yield model It is seen in Fig. 3 that all stress components are finite everywhere and the maximum value of the cohesive traction, rzy, occurs at the cohesive zone tip (x = c, y = 0 or u = 0, v = 0). For the Mode III crack, the von Mises effective stress in the cohesive zone is qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi re ¼ 3ðr2zx þ r2zy Þ ¼ 3smax : ð17Þ Interestingly, re is a constant and is independent of the surface separation dz. To this end, it is worth noting that Dugdale [2] used the assumption of a uniform stress in the cohesive zone to formulate the Mode I strip-yield problem. Eq. (17) reveals that a form of the cohesive zone model with a uniform von Mises effective stress distribution in the cohesive zone of a Mode III crack is derivable from the overall formulation of the elastic field solutions. Furthermore, Eqs. (9)–(11) represent the elastic–plastic solution of the finite-width Dugdale zone model [10,15] when the plastic zone reduces to a line segment, in which the general Mode III solution was obtained by assuming an ellipse-shaped perfectly plastic zone ahead of the crack tip. An interpretation of Eq. (17) based on the Dugdale zone mode is that the von Mises effective stress in the cohesive zone of a Mode III crack equals the yield stress of the surrounding bulk material, even though Eq. (17) is derived without the a priori assumption that the cohesive zone is an elastic–plastic zone with an elastic–perfectly plastic behavior. Another point can also be drawn from the above discussions: According to Eq. (12), the non-dimensional parameter b now has the physical meaning of being equal to one half of the ratio between the yield stress in pure shear and the shear modulus. As noted by Unger [15], the Mode III stress and displacement fields, Eqs. (9)–(11), can be derived from a plastic-strip model by Cherepanov [16], in which a plastic zone occupies the location of the cohesive zone. The advantage of the cohesive zone model is that it clearly defines the separation of the cohesive surfaces, while the plastic-strip model lacks the ability to explain the displacement jump across the plastic zone. 4.2. Fracture energy In a cohesive zone model, the specific fracture energy that is dissipated when new fracture surfaces are created in a material corresponds to the area under the traction–separation curve. For the particular form of the cohesive zone model described by Eq. (16) (see also Fig. 4), this fracture energy is given by
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Gf ¼ smax
Z
dtip
h
1 ðdz =dtip Þ
2=3
i1=2
0
ddz ¼
3p smax dtip : 16
ð18Þ
If the path-independent J integral [17] is made to equal to the fracture energy, and if we let smax be the pure shear yield stress k, then the crack tip opening displacement in the Cherepanov plastic-strip model [16] can be recovered, which is dtip ¼
16J : 3pk
ð19Þ
Unger [15] obtained the same expression by comparing his Dugdale zone solution to the classical LEFM field equations. It must be pointed out that Eq. (19) offers one possible method to determine the parameters in the cohesive zone model. 4.3. Connection with LEFM solutions Classical LEFM solutions of the asymptotic stress and displacement fields at a crack tip are well documented for all three fracture modes. For convenience, the Mode III solution in terms of polar coordinates is given in Appendix B, which shows an inverse square-root singularity and an infinite stress at the crack tip. The cohesive zone model-based solution developed in this study limits the stress level at the crack tip to a finite value. In order to make a connection between the cohesive zone solution and the classical solution, one can suppose that the stress component rzy from these two solutions match in value at some distance ahead of the crack tip. This distance should be larger than some characteristic length scale (e.g. the cohesive zone size 2c) of the process zone but still sufficiently small so that the classical asymptotic singular field still holds. The following analysis shows what happens if this connection is made. For convenience, the crack tip is chosen at x = c and y = 0 in the Cartesian coordinates (see Fig. 2). Then the transformation between the polar and Cartesian coordinates is given by x ¼ r cosðhÞ c ðp 6 h 6 pÞ: ð20Þ y ¼ r sinðhÞ Now suppose that rzy from the cohesive zone stress solution, Eq. (10), and the classical asymptotic solution, Eq. (B.2), have the same value at some distance in front of the crack tip, say at point (x = mc, y = 0), where m P 1 and c is one half of the cohesive zone size, then pffiffiffiffiffiffiffiffi K III = 2pc ¼ smax hðmÞ; ð21Þ where pffiffiffiffiffiffiffiffiffiffiffiffi hðmÞ ¼ earc coshðmÞ=2 m þ 1:
ð22Þ
Note that the coordinate relations u = arccosh(x/c), v = 0 and r = x + c, h = 0 have been used in deriving Eqs. (21) and (22). Using Eq. (21), Eqs. (B.2) and (10) can be written in normalized forms as, respectively, 1 rczy =s0 ¼ pffiffiffiffiffiffiffi cosðh=2Þ; r=c
ð23Þ
rzy =s0 ¼ eu=2 cosðv=2Þ=hðmÞ;
ð24Þ
where pffiffiffiffiffiffiffiffi s0 ¼ K III = 2pc:
ð25Þ
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In the above, the superscript c signifies the classical solution. The corresponding normalized displacements, Eq. (B.3) and Eq. (11), respectively, are pffiffiffiffiffiffiffi ucz =u0 ¼ 2 r=c sinðh=2Þ; ð26Þ uz =u0 ¼ ½3eu=2 sinðv=2Þ e3u=2 sinð3v=2Þ=3hðmÞ; where u0 ¼
K III l
rffiffiffiffiffiffi c : 2p
In the above derivations, Eqs. (12) and (21) have been used, that is rffiffiffiffiffiffi 8smax 8K III c c¼ : dtip ¼ 3l 3hðmÞl 2p
ð27Þ
ð28Þ
ð29Þ
Comparisons of the normalized stress rzy ahead of the crack tip, and of the corresponding displacement behind cohesive zone tip, both along the lower surface of the crack (y = 0), are shown in Fig. 5 for several m values. It is seen that the far field stress agrees well with each other when m is sufficiently large (e.g., m = 9). 4.4. On the formulation approach The cohesive zone formulation approach described in this work can be viewed in two ways. In the first case, the current approach provides a way of generating possible forms of the traction–separation relationship that are consistent with the linear elastic stress and displacement fields around cracks with cohesive zones ahead of the crack tip. The particular simple form of the cohesive zone model described in Eq. (16) is established based on a one-term stress and displacement fields corresponding to k = 1/2, which appears to be a preferred choice based on several considerations. It is noted that if the eigenfunctions are viewed as terms in a ‘‘Fourier’’ series expansion, then the eigenvalue k will correspond to the ‘‘frequency’’ of a single eigenfunction. Thus, terms with the lowest ‘‘frequencies’’ (i.e. k = 1/2 and 1/2) should be the dominant ones in the series. Since
Fig. 5. Comparisons of present elastic solutions along y = 0 (the lower surface) with classical solutions for various value m: (a) rzy/s0 and (b) uz/u0.
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k = 1/2 results in infinite stresses in the far field, the preferred term is k = 1/2. Moreover, it can be shown that the high ‘‘frequency’’ terms (e.g., k = 3/2) have zero contribution to the fracture energy even though these terms may change the form of the traction–separation relation. Nevertheless, if terms corresponding to other eigenfunctions are chosen, or if combinations of various terms are employed, then different forms of the traction–separation relationship can be obtained. The second way to view the current approach is that, the infinite series solutions derived in this work provide a means of understanding and possibly quantifying the elastic stress and displacement fields around a crack containing a cohesive zone ahead of the crack tip, where a cohesive law governing the behavior of the cohesive zone may be given as part of a boundary-value problem. In principle, relevant terms of the near-field series solutions must work together in order to match the traction–separation relation in the cohesive zone as dictated by the given cohesive law, thus allowing the determination of the coefficients for these terms, while the coefficients of other terms in the series may be determined by matching with the far fields, which are expected to contain information about the geometry and loading conditions of a particular problem. However, this option is not explored in the current work.
5. Summary and concluding remarks In this work, linear elastic solutions for the stress and displacement fields around a Mode III crack containing a cohesive zone ahead of the crack tip have been obtained without a priori specification of a cohesive law for the cohesive zone. A possible form of a Mode III cohesive zone model is proposed, which is based on a particular choice of the stress and displacement fields derived in this work. The traction–separation relation in this model contains two independent parameters, b and c, where b is dimensionless and has the physical meaning of being equal to one half of the ratio between the yield stress in pure shear and the shear modulus, and c is the half length of the cohesive zone. The parameters b and c are also related to the specific fracture energy of the material. The stress field corresponding to this cohesive zone model is finite everywhere and the maximum cohesive stress (traction in the cohesive zone) occurs at the cohesive zone tip. The von Mises effective stress in the cohesive zone is constant and can be interpreted as the yield stress of the surrounding bulk material, which is consistent with the assumption in strip-yield models for elastic–perfectly plastic materials that a line plastic zone exists directly ahead of the crack tip. The developed approach for deriving near-field stress and displacement solutions and for formulating crack-tip cohesive zone models can be extended to Mode I and Mode II cracks. Results of this extension will be reported separately [18].
Acknowledgments The authors greatly appreciate the support of a grant from DEPSCoR/ONR (Grant # N00014-03-10807) and helpful discussions with Dr. J. Zuo and Dr. A. Reynolds.
Appendix A. Elastic solution for a Mode III crack Keeping in mind the relations between the Cartesian and the elliptic coordinate systems (Eq. (1)), we can define uz ðxðu; vÞ; yðu; vÞÞ ¼ uw ðuðx; yÞ; vðx; yÞÞ:
ðA:1Þ
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For anti-plane shear problems, 2 ou ! ! ezx u 6 z;x 1 1 ox ¼ ¼ 6 2 2 4 ou ezy uz;y oy
the only non-zero strain components are 1 30 ov ouz B C ox 7 7B ou C: @ 5 ov ouz A oy ov
1827
ðA:2Þ
Denoting l the shear modulus, the non-zero stress components are related to the strains through the HookeÕs law, so that rzx ezx ¼ 2l : ðA:3Þ rzy ezy Eqs. (5), (A.2) and (A.3) then lead to the Laplace equation r2 uz ðx; yÞ ¼ r2 uw ðu; vÞ ¼ 0;
ðA:4Þ
2
where $ is the Laplacian operator. In the elliptic coordinate system, Eq. (A.4) can be written as 2 1 o uw ðu; vÞ o2 uw ðu; vÞ þ r2 uw ðu; vÞ ¼ ¼ 0: ou2 ov2 c2 ½cosh2 ðuÞ cos2 ðvÞ
ðA:5Þ
The most important task is to find a function for uz = uw that satisfies Eq. (A.4) and at the same time yields stresses that satisfy the traction-free boundary conditions at the crack surfaces, i.e., rzy = 0 for u > 0 and v = ±p. Different from the LEFM, a solution that leads to a finite stress at the crack tip is sought here, since a cohesive zone is expected ahead of the crack tip to remove the stress singularity there. Let f(f) = p(x,y) + iq(x,y) be a holomorphic (analytic) function of the complex variable f, then both p(x,y) and q(x,y) satisfy the harmonic equation r2 pðx; yÞ ¼ 0; r2 qðx; yÞ ¼ 0:
ðA:6Þ
Therefore, the solution of Eq. (A.4) can be written as uz ¼
2 1 Re½f ðfÞ ¼ ½f ðfÞ þ f ðfÞ: l l
Substituting Eq. (A.7) into Eq. (A.2), we have ( 0 1 ezx ¼ 2l ½f 0 ðfÞ þ f ðfÞ : 0 ezy ¼ 2li ½f 0 ðfÞ f ðfÞ
ðA:7Þ
ðA:8Þ
Combining Eq. (A.3) and Eq. (A.8), we obtain rzx irzy ¼ 2f 0 ðfÞ:
ðA:9Þ
Now, letÕs consider the following analytic (holomorphic) function f 0 ðfÞ ¼ gðxÞ ¼ lBekx
ðB ¼ a ibÞ;
ðA:10Þ
where a, b and k are real constants to be determined by the boundary conditions. Then Eq. (A.9) becomes rzx irzy ¼ 2lða ibÞekðuþivÞ : Separating the real and imaginary parts, we have ( rzx ¼ 2leku ½b sinðkvÞ þ a cosðkvÞ : rzy ¼ 2leku ½b cosðkvÞ a sinðkvÞ
ðA:11Þ
ðA:12Þ
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Substituting boundary conditions rzy = 0 for v = ±p results in b cosðkpÞ a sinðkpÞ ¼ 0 : b cosðkpÞ þ a sinðkpÞ ¼ 0
ðA:13Þ
The existence of a non-trivial solution (i.e., B 5 0) requires that the determinant of the coefficient matrix of the preceding equation be zero, that is, sinð2kpÞ ¼ 0:
ðA:14Þ
The above equation in turn requires that the constant k must satisfy k ¼ n=2 ðn is an integerÞ:
ðA:15Þ
If we take the view that stresses must go to zero far away from the crack tip (when u ! 1), k has to be negative according to Eq. (A.12). Of all the eigenfunctions in the form of Eq. (A.12), the function with the eigenvalue k = 1/2 for which a = 0 contributes the most to the stress field far away from the crack tip. If the function with k = 1/2 is chosen to construct the solution, Eq. (A.10) becomes df ðfÞ dx df ðfÞ=dx ¼ ¼ ilbex=2 : dx df c sinhðxÞ
ðA:16Þ
Integrating the above ordinary differential equation gives f ðfÞ ¼
ilbc x=2 ilbc ðuþivÞ=2 ð3e þ e3x=2 Þ ¼ ½3e þ e3ðuþivÞ=2 ; 3 3
ðA:17Þ
where the integration constant has been taken to be zero, since at v = 0 the displacement (real part) must be zero. The stress components can be obtained from Eq. (A.12) as rzx ¼ 2lbeu=2 sinðv=2Þ;
ðA:18Þ
rzy ¼ 2lbeu=2 cosðv=2Þ:
ðA:19Þ
Then displacement can be derived by using Eqs. (A.7) and (A.17), uz ¼
2bc u=2 ½3e sinðv=2Þ e3u=2 sinð3v=2Þ: 3
ðA:20Þ
Note that the procedure for constructing the solution can be written more succinctly if we start directly from a general analytic function, which was not done in order to reveal details about the choice of the general eigenfunction (Eq. (A.10)). Also note that k = ±1 seems to cause singularity problems in Eq. (A.7) since factor (k2 1) will appear in the denominator. However, this can be avoided by replacing B with B = B 0 (k2 1) in Eq. (A.10). Hence the solution Eq. (A.15) is actually valid for all integers.
Appendix B. Classical solution of Mode III crack The asymptotic singular crack-tip stress and displacement fields from classical LEFM can be conveniently expressed in terms of polar coordinates as K III rzx ¼ pffiffiffiffiffiffiffi sinðh=2Þ; 2pr
ðB:1Þ
K III rzy ¼ pffiffiffiffiffiffiffi cosðh=2Þ; 2pr
ðB:2Þ
W. Zhang, X. Deng / Engineering Fracture Mechanics 72 (2005) 1818–1829
uz ¼
2K III r !1=2 sinðh=2Þ; 2p l
1829
ðB:3Þ
where KIII is the stress intensity factor.
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