## Abstract

We carried out a detailed analysis of the asymptotic stress and deformation fields at the tip of a mode III crack in a hyperelastic solid described by Gent's model. This model accounts for finite chain extensibility so that the deformation everywhere in the solid, including the crack tip, is bounded. We also present an exact solution for the ‘small-scale yielding’ problem where the region of large deformation is small compared with specimen dimensions. Our result shows that the crack tip stress field is non-separable. In addition, an *infinite* number of parameters are needed to completely specify the angular variation of crack tip stress field.

## 1. Introduction

Understanding the mechanical behaviour of soft elastic materials has many important applications. For example, much of what we have learned about the roles of mechanical forces in directing cellular behaviours such as adhesion, migration and cell differentiation has been derived from observations of cells plated on hydrogel substrates with tunable stiffness (Pelham & Wang 1997; Engler *et al.* 2006; Isenberg *et al.* 2009). Gels are also used as a scaffold for the cultivation of tissues (Drury & Mooney 2003). The ability to deform reversibly under various stimuli (e.g. temperature, chemicals and electric/magnetic field) has made soft elastic material an attractive candidate for actuators and sensors (Chaterji *et al.* 2007; Dong & Jiang 2007). Also, hydrogel has the potential to replace natural tissues such as cartilage because of its biocompatibility and high water content, which allows them to provide a high degree of lubrication under large normal stresses (Yasuda *et al.* 2005). The last example is particularly relevant to this work since these gels must also have a high resistance to fracture.

Some of these applications have stimulated interest in understanding the fracture mechanics of soft materials (e.g., Baumberger *et al.* 2006; Bouchbinder *et al.* 2008; Krishnan *et al.* 2008; Seitz *et al.* 2009; Cristiano *et al*. 2010; Livne *et al.* 2010). However, in comparison with the vast literature on fracture of stiff solids, such as ceramics and metals, fracture of highly compliant soft elastic solids received much less attention and is not well understood. One of the difficulties is that the elastic moduli of elastomers are typically of the order of 1 MPa, whereas the moduli of gels can be as low as 1 KPa. As a result, large deformation can arise at very low applied stresses. Because of the large deformation, linear elasticity, which works well for fracture of stiff solids, is not expected to capture the correct behaviour of the stress and deformation fields near the crack tip in soft materials.

The deformation of elastomers and elastic gels can be reasonably well described by hyperelasticity—where the work *W* to deform a unit volume of material to a given deformation state (characterized by **F**, the deformation gradient tensor) is independent of the loading path (see Holzapfel (2000) for basic theory of hyperelasticity). Briefly, a hyperelastic solid has a work function or strain energy density function *W* that only depends on the deformation gradient tensor **F**. In addition, most elastomers and gels are incompressible, i.e. the volume of every material element is conserved or det(**F**)=1. For incompressible hyperelastic materials, the first Piola–Kirchhoff stress tensor **S** is related to the deformation gradient tensor **F** through the constitutive relation
1.1
where *P* is a Lagrange multiplier to enforce the incompressibility constraint; it is usually interpreted as an isotropic pressure and can be determined by boundary conditions.

A work function *W* that has universal appeal is the neo-Hookean model
1.2
where *μ* is the small strain shear modulus, *I* is the invariant defined by and *λ*_{i} are the principal stretch ratios. A limitation with this model is that it underestimates the amount of strain hardening as deformation becomes large. More importantly, this model and many other models in the literature predict that the deformation (e.g. the stretch ratio in a tension test) can increase without bound. This feature is not consistent with the finite extensibility of the polymer chains that form the molecular network of many soft materials (see a review article by Horgan & Saccomandi (2006) for detailed discussions). Of course, these constitutive models are not intended to be used to describe the intense deformation near the tip of a crack. A more realistic way to study the crack tip stress fields in soft materials is to incorporate finite extensibility of polymer chains into the constitutive model.

There are many hyperelastic material models in the literature that exhibit finite extensibility behaviour (see Horgan & Saccomandi (2006) for more details). Among these models, two representative work functions are the one proposed by Arruda & Boyce (1993) and a much simpler one proposed later by Gent (1996). The work function of Gent's model has the following form:
1.3
where *J*_{m} is a material parameter controlling the degree of strain hardening. For small deformation, i.e. (*I*−3)/*J*_{m}≪1, both models reduce to the neo-Hookean model given in equation (1.2). As shown in Boyce & Arruda (2000), the work functions of Arruda & Boyce (1993) and Gent (1996) produce very similar theoretical predictions. In particular, they have the same asymptotic behaviour when . Therefore, the first-order behaviour of the stress field near the crack tip is identical for both models.

A difficulty with finite extensibility models is that the stress becomes infinite at *finite stretch ratios*. Because of this feature, it is extremely difficult to analyse the stress and deformation field near a crack tip using numerical methods. While there have been many crack tip analyses using different hyperelastic models (e.g. Knowles & Sternberg 1973, 1983; Stephenson 1982; Geubelle & Knauss 1994), very little work has been devoted entirely to the effect of finite extensibility models on the crack tip stress fields. This limitation motivates us to study the crack tip field using material models with finite extensibility. For simplicity, we focus on anti-plane shear or mode III cracks and use Gent's model as an example. However, the method of analysis can be applied to other stable hyperelastic solids (e.g. a generalized neo-Hookean solid, see Knowles (1977)), and we expect that qualitative features of the crack tip field are valid for other fracture modes as well. An example of mode III crack geometry is shown in figure 1.

This paper is organized as follows: in §2, we summarize the governing equations for anti-plane shear cracks. The general solution of the governing equations is presented in §3. In §4, we give the exact solution for the ‘small-scale yielding’ (SSY) problem where the nonlinear effect of large deformation is confined in a small region enclosing the crack tip. The asymptotic near-tip fields under the SSY condition are also shown in this section. The crack tip fields for general loading conditions are presented in §5. In §6, we highlight the significance of our results.

## 2. Governing equations for anti-plane shear cracks

Detailed derivation of the governing equations for finite anti-plane shear deformation around a crack can be found in Knowles (1977). Here, we briefly summarize the results. Let *x*_{i} denote the coordinates of a material point in the reference undeformed configuration (figure 2) and *y*_{i} its deformed coordinates, i.e.
2.1
where *u*_{i} is the displacement of the material point. In anti-plane shear deformation,
2.2
and the out-of-plane displacement *u*_{3} is a function of the in-plane material coordinates, i.e.
2.3

The non-zero off-diagonal entries in the deformation gradient tensor *F*_{ij}=∂*y*_{i}/∂*x*_{j} are defined to be the shear strain *γ*_{α},
2.4
In this paper, Greek indexes range from 1 to 2 and *w*,_{α}≡∂*w*/∂*x*_{α}. We consider isotropic incompressible hyperelastic materials with work functions that depend only on the invariant
2.5
such as Gent's model. Using the traction-free boundary conditions on the crack faces, Knowles (1977) showed that the only non-trivial components of the first Piola–Kirchhoff stress tensor **S** are *S*_{3α} and *S*_{α3}. The shear stresses are related to the work function *W* by (Knowles 1977)
2.6
In anti-plane shear, the shear components of the first Piola–Kirchhoff stresses and the Cauchy stresses are identical. Equation (2.6) implies that
2.7
where
2.8
For simple shear where *w*=*γx*_{2} or *γ*_{2}=*γ*, there is only one non-trivial stress component *S*_{32}=*τ*. The relation between *τ* and *γ* for a solid that obeys Gent's model is given by
2.9
Using equations (2.4) and (2.7) and enforcing equilibrium, the displacement field *w*(*x*_{1},*x*_{2}) satisfies the following equation (Knowles 1977):
2.10
The traction-free condition on the crack surfaces implies
2.11

## 3. General solution for Gent's model

### (a) Governing equation and hodograph transform

The nonlinear partial differential equation (PDE) given by equation (2.10) can be transformed to a *linear* PDE using the hodograph transform introduced by Neuber (1961). Detailed discussion on applying this technique to anti-plane shear crack problems can be found in Rice (1967). The basic idea is to use the strains *γ*_{α} instead of the physical coordinates *x*_{α} as the independent variables. Specifically, the physical coordinates (*x*_{1},*x*_{2}) can be written as the gradient of a potential function *ψ*(*γ*_{1},*γ*_{2}) in the strain plane (*γ*_{1},*γ*_{2}), i.e.
3.1
It is convenient to introduce the polar coordinate system (*γ*,*ϕ*) in the strain plane (*γ*_{1},*γ*_{2}), i.e.
3.2
where *ϕ* is the angle measured positive counterclockwise from the positive *γ*_{2} axis. With respect to the polar strain coordinates (*γ*, *ϕ*), equation (3.1) is
3.3
Rice (1967) has shown that the *nonlinear* PDE in equation (2.10) is equivalent to the following *linear* PDE governing the potential in the strain plane:
3.4
where
3.5
This transformation allows one to circumvent the complicated nonlinear PDE in equation (2.10) by solving the linear PDE (3.4) in the strain plane. Once the potential function *ψ*(*γ*_{1}, *γ*_{2}) is determined, the shear strains *γ*_{α} can be determined using equation (3.1).

For Gent's model, a straightforward calculation using equations (1.3), (2.5) and (2.8) shows that
3.6
Substituting equation (3.6) into (3.4), we obtain
3.7
For convenience, we introduce a new strain variable
3.8
Let us normalize *x*_{i} by *a*, where *a* is a length scale in the problem of interest (e.g. crack length), i.e. *X*_{i}≡*x*_{i}/*a*. Using this normalization, equation (3.3) becomes
3.9
and
3.10
where is the normalized potential function.

With respect to these new variables, the governing equation (3.7) becomes 3.11

### (b) Boundary conditions

Since we are interested in the asymptotic crack tip fields, the crack can be taken to be semi-infinite with its tip located at the origin *X*_{1}=*X*_{2}=0. The crack surfaces occupy *X*_{2}=0,*X*_{1}<0 (figure 2). The traction-free boundary condition on the crack surfaces is *τ*_{2}(*X*_{1}<0,*X*_{2}=0)=0. Using equations (3.9) and (3.10), this condition is equivalent to
3.12

### (c) Analysis of crack tip fields

We anticipate that the material elements near the crack tip deform to the finite extensibility limit, owing to the significant stress concentration at the crack tip. In other words, as the crack tip is approached, , or . Therefore, we change the variable from *ρ* to
3.13
Physically, *η* represents the deviation of the effective strain *γ* from its maximum value of . With this change of coordinates in the strain plane, the crack tip corresponds to *η*=0 and equation (3.11) becomes
3.14
where
3.15
Following Rice (1967), we look for separable solutions of the form
3.16
Substituting equation (3.16) into (3.14) results in
3.17
The general solutions of equation (3.17) subjected to the traction boundary-free condition (3.12) are
3.18
and
3.19
where
3.20
3.21
and *c*_{αk} are arbitrary constants. The coefficients *a*_{mk} for *m*≥1 in equation (3.20) and *b*_{mk} for *m*≥3 in equation (3.21) can be obtained from the recursive relations shown in the electronic supplementary material, note S1. We note that the radius of convergence of these power series is 1, so the solution is valid for any finite region containing the crack tip.

As in Rice (1967), we assume completeness of eigenfunctions *h*_{k}(*η*)*g*_{k}(*ϕ*), and the general solution of *Ψ* is
3.22
Using equations (3.9), (3.10) and (3.22), the normalized radial distance from the crack tip, *R*≡*r*/*a*, is related to the potential *Ψ* by
3.23
The dominant asymptotic behaviour of *Ψ* as the crack tip is approached can be determined by neglecting higher order terms of *η*. Substituting equation (3.22) into (3.23) and keeping the lowest order terms in *η* gives
3.24
We impose the condition that for all *ϕ* when (i.e. there is a one-to-one correspondence between the crack tip in the physical plane and the *η* in the strain plane). This condition and equation (3.24) imply that *c*_{2k}=0 for all *k* in equation (3.22). Therefore, the normalized potential function is
3.25
We emphasize that equation (3.25) is an *exact* solution for an anti-plane shear crack. The undetermined constants *c*_{1k} can be used to enforce boundary conditions of a mode III crack in a finite domain. For example, Rice (1967) has used this technique to obtain a closed form solution for a finite edge crack in a semi-infinite power-law hardening material.

Equation (3.25) implies that the dominant asymptotic behaviour of the normalized potential is
3.26
Since the coefficients *c*_{1k} cannot be determined by local asymptotic analysis, it is not possible to sum the series in equation (3.26) without specifying the external load and the specimen geometry. Therefore, *q*(*ϕ*) *cannot* be determined by asymptotic analysis. This result is significant, since this is the only example to the best of our knowledge that the near-tip field of cracks cannot be characterized by a finite number of loading parameters. Indeed, this result shows that an *infinite number* of parameters are needed to completely specify the crack tip field. Note that *q is an odd function* because it is a linear combination of sine functions.

To find the stress distribution near the crack tip, we need to transform back to the physical plane. This is accomplished by expressing equations (3.9) and (3.10) in terms of *η* using *ρ*=1−*η*, i.e.
3.27
Substituting equation (3.26) into (3.27), we found, for small *η*,
3.28
We keep the *η*^{2} order term in equation (3.28) since it may be the dominant term in *X*_{1} when *ϕ*=0. Note that , where −*π*≤*θ*≤*π* is the polar angle in the physical plane (figure 2). Near the crack tip, equation (3.28) provides a relation between the angle *ϕ* in the strain plane and the polar angle *θ* in the physical plane, i.e.
3.29
where *f*(*ϕ*)≡*q*′/*q*, and we have used the identity that .

Assuming that *q*(*ϕ*) is known, for a given (*ρ*,*ϕ*) in the strain plane, equations (3.29) and (3.23) provide two equations to solve for (*R*,*θ*), which allows the strain fields (*ρ*,*ϕ*) to be expressed in terms of *R* and *θ*. The stresses can be obtained using the constitutive model (2.7) and (2.8) with *W* given by equation (1.3). This procedure is simple in principle, but is quite complicated to carry out analytically. In particular, *q*(*ϕ*) cannot be determined unless the external loading condition and sample geometry are specified. Thus, the near-tip fields are no longer determined solely by a finite number of loading parameters (such as stress intensity factors) but by an unknown *function* which depends on boundary conditions. To better understand the nature of these fields, we proceed with the solution of the SSY problem before determining the general form of the asymptotic stress fields near the crack tip.

## 4. Exact solution: ‘small-scale yielding’

A problem of central importance in fracture mechanics is the ‘SSY’ problem where the effect of nonlinearity is confined to a small region near the crack tip. In our case, the SSY condition is satisfied by applying a very small load to a hyperelastic specimen, e.g. *w*_{0}/*t*≪1 in figure 1. Because the applied displacement is small, material points everywhere except very close to the crack tip are subjected to small deformation and Gent's model reduces to linear Hooke's law, where *τ*_{α}=*μγ*_{α}. In the mathematical formulation of SSY, the local fields near the crack tip are determined by replacing the actual crack by a semi-infinite crack in an infinite solid with asymptotic boundary condition such that, at large *r*, the field approaches that of the standard linear elastic singularity with a strength given by the mode III stress intensity factor *K*_{III} (see Hui & Ruina (1995) for a detailed review of SSY). Specifically, the asymptotic boundary condition for our mode III problem is
4.1
In the strain plane, Rice (1967) has shown that equation (4.1) becomes
4.2
Since *K*^{2}_{III}/(2*πμ*^{2}) is the only length scale in the problem, we choose our length scale *a* (see discussions associated with equations (3.9) and (3.10)) to be
4.3
so that
4.4
Equation (4.4) satisfies (3.11) as and the traction-free boundary condition (3.12). Equation (4.4) suggests a solution of the form
4.5
Substituting equation (4.5) into (3.11), we find that the exact solution of the SSY problem is
4.6
See the electronic supplementary material, note S2, for details of the derivation. We emphasize that equation (4.6) is a special case of the general solution given by equation (3.25) with *c*_{10}=−2 and *c*_{1k}=0 for *k*≥1.

### (a) Crack tip stress fields of the small-scale yielding problem

We use the procedure outlined at the end of §3 to derive the stress and strain fields near the crack tip. Substituting equation (4.6) into equations (3.9) and (3.10), the physical coordinates are
4.7
4.8
and
4.9
The relation between *ϕ* and *θ* can be determined by taking the ratio of equations (4.7) and (4.8), i.e.
4.10
We are interested in the behaviour as . Let *η*=1−*ρ* so that as . Equation (4.10) becomes
4.11
One must be careful when taking the limit of , that is, to neglect the second term in equation (4.11). Note that if this term can be neglected, then *ϕ* depends *only* on *θ* for sufficiently small *η*. However, this should not be done since at *ϕ*=0 (this corresponds to *γ*_{1}=0,*γ*_{2}≠0), and the second term becomes unbounded as long as *η*≠0. Since the first term in equation (4.11) vanishes at *ϕ*=0, no matter how small *η* is, there exists a small boundary layer around *ϕ*=0 where the second term is important. The size of this layer is of the order of *η*. This result implies that the near-tip field is not separable and hence cannot be self-similar—a characteristic feature of many crack tip solutions.

For small *η*, the solution of equation (4.11) is
4.12
and
4.13
Using equations (4.12) and (4.13), we plot the relation between *ϕ* and *θ* in figure 3 for different values of *η*. Figure 3 shows that, for small *η*>0, *θ* is a continuous function of *ϕ* that changes rapidly near *ϕ*=0. The width of the transition regions or boundary layers where rapid change of *θ* occurs is of order *η*. These boundary layers become infinitesimally thin as (see the *η*=0.001 curve in figure 3). This feature motivates us to define the discontinuous function
4.14
which can be used as a first-order approximation for the function *ϕ*(*θ*,*η*) as . A schematic of the three regions in equation (4.14) is shown in figure 4.

If we replace *ρ* by (1−*η*) in equation (4.9) and neglect terms with order higher than *η*^{4}, equation (4.9) becomes
4.15
Note that it is necessary to keep the *η*^{4} term since *ϕ*≈0 in region II. Using the approximation *ϕ*≈*ϕ*_{o}, the leading order behaviour of equation (4.15) in region I is
4.16
The effective shear stress *τ* is
4.17
The two shear stress components are
4.18
In region II, the leading order behaviour of equation (4.15) is
4.19
The effective shear stress is
4.20
The two shear stress components are
4.21
In region III, the two shear stress components are
4.22
Recall *R*=*r*/*a*, where *a*=*K*^{2}_{III}/(2*πμ*^{2}*J*_{m}) (equation (4.3)); thus, the amplitude of the near-tip fields is directly proportional to *K*^{2}_{III}/2*μ*, which is the applied energy release rate *J* (see below). Equation (4.21) states that the material ahead of the crack tip (region II) is in a state of pure shear.

The approximate asymptotic solution given by equations (4.18), (4.21) and (4.22) is a discontinuous function of *θ*, whereas the actual asymptotic solution is continuous. At distances close to the crack tip, one expects that the approximate solution should approach the exact solution except in the vicinity of the boundary layer at *θ*=±*π*/2. To verify this hypothesis, we compared the approximate asymptotic solution with the exact asymptotic solution obtained by numerically solving equations (4.9) and (4.10). Figure 5 plots the effective stress *τ* versus *θ* along two different circular paths with normalized radius *R* of 0.005 and 0.001, respectively. The solid lines are the approximate solution whereas the dashed lines are the exact asymptotic solution. It is clear that the approximate asymptotic solution captures the behaviour of the near-tip field, with the exception of a boundary layer at *θ*=±*π*/2.

### (b) J-integral

A simple way to check the validity of our asymptotic result is to evaluate the path-independent *J*-integral (Eshelby 1969, Knowles & Sternberg 1972) using the crack tip field obtained above. In anti-plane shear, the *J*-integral is
4.23
where *Γ* is any contour enclosing the crack tip that starts from the lower crack surface and ends at the upper crack surface, and **n** is the unit normal vector of *Γ*. The path independence of the *J*-integral allows us to choose a circular path of radius *r*=*εa* or *R*=*ε* with its centre at the crack tip. We can always choose *ε* to be sufficiently small so that the crack tip field described by equations (4.16)–(4.22) is valid—in other words, the artificial discontinuities introduced by the approximation *ϕ*≈*ϕ*_{0} do not affect the calculation of the *J*-integral. Since the asymptotic fields are divided into three regions, we have
4.24

In region I, the work function is
4.25
Equation (4.25) implies
4.26
Therefore, using equations (4.18) and (4.26), the first integral in equation (4.24) as is
4.27
In the same manner, we can show that the second integral in equation (4.24) is zero as and the third integral equals *μJ*_{m}*πa*/2. Therefore, the *J*-integral is
4.28
where *a*=*K*^{2}_{III}/(2*πμ*^{2}*J*_{m}) (equation (4.3)). Substituting equation (4.3) into (4.28) recovers the well-known result
4.29
An interesting result is that only stress fields in regions I and III, i.e. in the sectors *π*/2<*θ*≤*π* and −*π*≤*θ*<−*π*/2, contribute to the energy release rate. This is because *τ*_{2} in region II approaches as while the perimeter of the integration path is directly proportional to *r*. As shown in figure 4, material points in regions I and III lie *behind* the crack tip.

## 5. Asymptotic fields under arbitrary mode III loading

We are now in a position to analyse the mode III crack tip stress fields under general loading conditions. The primary difficulty is that the *function* *q*(*ϕ*) in equation (3.26) cannot be determined by a local analysis. Without *q*(*ϕ*), it is not possible to determine the full asymptotic field since the relation between the strain angle *ϕ* and the physical angle *θ* depends on the solution of equation (3.29). Nevertheless, it is possible to show that (see electronic supplementary material, note S3, for details) the solution of equation (3.29) in the limit of small *η* is independent of *q*(*ϕ*) and is still given by *ϕ*_{0}(*θ*) (equation (4.14)) except for boundary layers at *θ*=±*π*/2. This result allows us to determine an approximate asymptotic stress field similar to the SSY problem. Using exactly the same procedure as the SSY problem, we found the following crack tip stress field.

In region I, where *π*/2≤*θ*≤*π*,
5.1
In region II, where −*π*/2<*θ*≤*π*/2
5.2
Finally, in region III, where −*π*≤*θ*<−*π*/2,
5.3
Even with the approximation *ϕ*≈*ϕ*_{0}(*θ*), the asymptotic fields contain an *unknown* *function* *q* which describes the angular variation of the crack tip fields. Fortunately, because of the path independence of the *J*-integral, the dependence of the *dominant* crack tip stress field on the radial distance *r* is independent of the external loading and specimen geometry, as long as one avoids the boundary layers at *θ*=±*π*/2.

## 6. Conclusion and discussion

One of the central premises of elastic fracture mechanics is that there is a small number of loading parameters that completely determine the behaviour of the crack tip fields. For example, for cracks in a homogeneous, linearly elastic and isotropic solid, these parameters are the modes I, II and III stress intensity factors. The existence of these parameters depends on the *similarity* of the near-tip crack tip stress distribution for all crack configurations. Our analysis shows that the crack tip field of Gent's model is non-separable. In addition, the angular variation of the crack tip stress distribution is dependent on the external loading and crack configuration. In other words, an infinite number of parameters are needed to completely specify the crack tip stress and deformation field. Such a scenario may also occur for materials with zero hardening, i.e. the stresses approach finite values while the deformation can increase without bound (e.g. *n*=1/2 in generalized neo-Hookean material, see Knowles (1977)). Also, we emphasize that the method of asymptotic analysis used in this paper can be readily applied to other hyperelastic materials. For example, we analysed the mode III crack tip field in generalized neo-Hookean materials for *n*>1/2 and obtained identical leading order asymptotic solutions to those in Knowles (1977). Note that the solutions in Knowles (1977) were based on the assumption that crack tip fields are in separable forms of (*r*, *θ*), while no such assumption is made in our method.

We have obtained an exact solution for the SSY problem. Our solution shows that, in the sector |*θ*|≤*π*/2, one component of the shear stress field dominates over the other, indicating that the material element ahead of the crack tip is under simple shear deformation. The dominant stress component in the sector of |*θ*|≤*π*/2 has a classical inverse square root singularity and is independent of the polar angle *θ*. Surprisingly, the stresses are more singular behind the crack tip, in the sector *π*≥|*θ*|>*π*/2. Indeed, since the strains are bounded, these stresses must vary as 1/*r*1/*r* owing to the path independence of the *J*-integral.

In an earlier work, Horgan & Saccomandi (2001) studied the general behaviour of hyperelastic materials loaded in anti-plane shear. As a particular example, they examined the asymptotic stress and deformation field at the tip of a crack in a Gent solid where the work function is given by equation (1.3). In their analysis, they assumed that the displacement field *w* near the crack tip has the separable form
6.1
With this assumption, they found that the asymptotic stresses *τ*_{3i} as one approaches the crack tip (; figure 2) are *bounded* and *constant*. In particular, the shear stresses have the following form:
6.2
where *A* is a numerical constant that cannot be determined from a local analysis. It can be readily shown that their solution satisfies all the field equations and the traction-free condition on the crack face and is therefore an asymptotic solution of the crack problem. This result assumes that the finite extensibility limit is not reached near the crack tip, and thus predicts a bounded stress field. This is quite different from our solution, where we expect the material elements to deform to the extensibility limit near the crack tip because of the severe stress concentration there. This assumption leads to a singular crack tip stress field, similar to the case of simple shear deformation where the shear stress becomes infinite as the material deforms to its extensibility limit. Indeed, the shear stress *τ* in simple shear is related to the shear strain *γ* by equation (2.9) (Horgan & Saccomandi 2001). Note that as . More importantly, the energy release rate or *J*-integral (Eshelby 1969; Knowles & Sternberg 1972) in almost any elastic crack problem is a *positive* number. For example, the *J*-integral for the geometry shown in figure 1 is easily computed to be *W*_{0}*t*, where
6.3
On the other hand, the path independence of *J* implies that one can compute *J* by evaluating it on any arbitrarily small circle centred at the crack tip. Since the perimeter along this path can be arbitrarily small, *J*=0 if both the stresses and deformation are bounded at the crack tip, contradicting the fact that *J*=*W*_{0}*t*>0. In this work, we show that there exists a different solution of the crack tip field that is singular and also satisfies the condition of the non-zero energy release rate to the crack tip.

## Acknowledgements

The authors acknowledge support from the Materials and Surface Engineering programme, CMMI and the National Science Foundation (grant no. CMMI-0900586).

- Received April 10, 2011.
- Accepted May 17, 2011.

- This journal is © 2011 The Royal Society