## Abstract

We consider the role of interfacial slippage in the deformation and stress fields near the tip of a plane interface crack occurring between a compressible hyperelastic material and a rigid substrate. Specifically, we draw comparisons between the two limiting cases of ‘no-slip’ (infinitely high friction) and ‘frictionless’ (zero friction) surfaces by performing corresponding asymptotic analyses in the crack-tip region. Our results indicate that for the no-slip case, when the body is subjected to far-field loading, the crack deforms to a wedge-like shape consistent with experimental observations reported in the literature. Moreover, in this case, the wedge angle is shown to be directly related to ratios of various Cauchy stress components on the bonded surface in the near-tip region. Finite-element simulations reveal that the wedge angle also depends on material compressibility and the far-field loading conditions. By contrast, the analysis of the frictionless case reveals that the crack consistently opens into a smooth parabolic shape with a right wedge angle and near-tip stress field dominated by the normal stress at the surface. The results established here can be used as a basis for the understanding of the role of varying degrees of slippage on interfacial cracks.

## 1. Introduction

Soft elastomers are widely used in coatings and non-structural adhesives such as pressure sensitive adhesives. The overall bonding strength between a soft elastomer layer and a relatively rigid substrate is primarily determined by the intrinsic work of adhesion defined as the amount of work required to separate a unit area of contacting surfaces. Studies in recent years have shown that the strength of soft adhesives can be significantly influenced by other factors, including surface roughness [1–3], viscoelasticity [4,5] and interfacial slippage [6–8]. The effect of interfacial slippage is twofold. First, when the adhesive layer is being detached from a substrate, partial slippage can occur owing to insufficient friction leading to energy dissipation on the interface [7,8]. The second aspect was based on the experimental observations of Newby *et al.* [6] with the conclusion that slippage can also affect the viscoelastic dissipation in the bulk adhesive material during detachment. Newby *et al.* [6] conducted peel-tests of adhesive tapes from silicon surfaces coated with organic molecules to alter the surface property (figure 1*a*). It was found that the peel force was larger on surfaces with lower work of adhesion, but higher slip resistance. To understand this result, they found that during detachment on the surfaces with low slippage the adhesive material behind the debonding edge flipped forward and formed a wedge at the tip of the interfacial crack (as shown in figure 1*b*). Based on scaling arguments, the wedge angle *ϕ* was shown to be related to slippage and the viscous dissipation in the adhesive material: the lower the slippage, the smaller the angle *ϕ*, which led to larger dissipation and hence a larger peel force. Specifically, the relation between the wedge angle *ϕ* and the interfacial shear traction owing to slippage was based on a crude model and was applicable only to small *ϕ*. From the solid mechanics perspective, slippage depends on the shear traction on the interface, and thus can be viewed as a boundary condition for the deformation of the soft adhesive. The focus of this paper was to provide more detailed and rigorous analysis on how slippage affects the local deformation of the adhesive material near the debonding edge.

To model the detachment process, we consider a plane–strain interface crack between a soft material and a rigid substrate (figure 2). Detachment occurs when the interface crack opens up and propagates forward under far-field loads. Following Newby *et al.* [6], the soft material is assumed to be elastic even though most soft adhesives are inherently viscoelastic. We are concerned with the local deformation field near the debonding edge (or tip of the interface crack). Consequently, the crack length is taken to be infinite. There are multiple volumes of literature concerned with the near-tip fields of interface cracks between dissimilar elastic solids. Earlier works in this area adopted linear elasticity to describe deformation of the solids [9–12], but these reported oscillatory singularities for the near-tip displacement and stress fields which can lead to interpenetration of the crack surfaces. These oscillations were considered to be unphysical [13]. For the special case of plane–strain interface cracks between an elastic material and a rigid substrate, the oscillatory behaviour of the near-tip displacement and stress fields disappears if the elastic material is incompressible but persists otherwise [14]. On the other hand, Hui *et al.* [15] showed that cracks in soft materials can become blunted before propagation owing to the capability of soft materials to undergo large deformation without fracture which requires that geometrical and material nonlinearities be taken into account. Efforts have been devoted to analysing the near-tip solutions within the context of finite strain elastostatics for either cracks in homogeneous solids [16–23] or interface cracks between dissimilar materials [24–28]. Among these works, Hermann [25,26] showed that for interface cracks between two compressible hyperelastic materials the crack-tip displacement and stress fields are characterized by smooth singularities without oscillation, whereas the oscillatory behaviour can still exist if the materials are incompressible as suggested by the finite-element simulations of Krishnan & Hui [29]. Both these results are contrary to the solutions of linear elasticity. In this paper, we adopt a class of compressible hyperelastic material models to account for potentially large deformation of the soft adhesives.

As a first step to understand the role of slippage, we assume two limiting cases (defined as ‘no-slip’ and ‘frictionless’) and perform asymptotic analysis for the corresponding crack-tip fields. The ‘no-slip’ condition means that friction can be infinitely large, so that no relative slippage can occur on the interface. With the ‘frictionless’ condition, it is assumed that the soft elastomer layer can freely slide on the substrate while remaining attached to the bonded surface. A comparison of these two cases can illustrate the effect of slippage on the near-tip fields of the interface crack.

The paper is organized as follows. In §2, we first review finite strain elastostatics for plane–strain geometries and then set up the governing equations and boundary conditions for the interface crack. The asymptotic solutions are presented in §3 for both no-slip and frictionless conditions. These solutions, together with results of finite-element simulations, are discussed in §4. We then conclude the paper in §5 with suggestions for future research.

## 2. Near-tip field of an interface crack

### (a) Plane strain elastostatics

We begin by describing the kinematics of finite deformation following standard notations [30,31]. The undeformed configuration of a solid is used as the reference configuration, and the position vector of a material point in the deformed configuration **x** (with basis *i*=1,2,3) is the sum of its counterpart in the undeformed configuration **X** (with basis *A*=1,2,3) and the resultant displacement vector **u**:
*V* is the deformed material volume and *V* _{0} is the original undeformed material volume. For an isotropic hyperelastic solid, the first Piola–Kirchoff stress tensor is given by
*ψ* is the strain energy density of the solid, which is a function of the three principal invariants *I*_{i} (*i*=1,2,3) of the right Cauchy–Green tensor **u** and hence the deformed position **x** are functions only of the referential planar components *X*_{1} and *X*_{2} and the out-of-plane displacement *u*_{3} is assumed to be zero [17]. As a result, the components of the right Cauchy–Green tensor *I*_{i} (*i*=1,2,3) of *I* and *J* defined in equations (2.8) and (2.3), respectively, as [17]
*ψ* is a function of *I* and *J* where
*i*=1,2 and *A*=1,2):
*ε*_{ij} is the planar Levi–Civita alternator [31,30] defined as
** σ** can be obtained using [31,30]

*i*,

*j*=1,2)

*I*and

*J*[17]:

*I*and

*J*and the latter ensures that the material volume does not vanish in the deformed configuration.

### (b) Interface crack: governing equations and boundary conditions

We are interested in the deformation and stress fields near the tip of a plane–strain interface crack between a soft elastic material and a rigid substrate. The undeformed geometry is shown in figure 2 with the elastic material shown as the grey portion above the hatched area which represents the rigid substrate. For the purpose of asymptotic analysis, we have assumed the interface crack to be infinitely long and to coincide with the negative *X*_{1} axis designated in figure 2 by a dashed line, with the crack tip situated at the origin. The elastic material is assumed to be of infinite extent occupying the entire half-plane above the *X*_{1} axis.

When a certain far-field loading is applied to the elastic material, the interface crack will open. To solve for the deformation field in the vicinity of the crack tip, it is convenient to use a set of polar coordinates *r* and *θ* (instead of *X*_{1} and *X*_{2}) also centred at the crack tip. This has the transformation (*A*=1,2):
*r* and *θ*, the planar deformation gradient components can be expressed as (*i*=1,2 and *A*=1,2)
*I* and *J* become (*i*=1,2)
*θ*=*π*) and the interface ahead of the crack tip (*θ*=0). The crack surface is assumed to be traction free as in previous analyses [17,18,25], which leads to
*N* *m*^{−1} and Young's modulus of the order of 1 MPa, one can derive a characteristic elasto-capillary length to be about 10 nm, defined as the ratio between surface tension and Young's modulus. This length is several orders of magnitude smaller than typical scale of crack deformation observed in experiments [6], meaning that surface tension is negligible in our problem and crack deformation is primarily controlled by far field loading and bulk elasticity.

Directly ahead of the crack tip (where *θ*=0), the elastic material is bonded to the rigid substrate and we assume no normal separation along the interface after deformation. Consequently,
*et al.* [6,7], there may be relative slippage along the tangential direction of the interface. How the slippage affects the crack-tip deformation field is the focus of this paper. As discussed earlier, we consider two limiting cases: ‘no-slip’ where the interface friction can be infinitely large and relative slippage is prohibited, and ‘frictionless’ where the interface allows for free slippage. Certainly, a more realistic model would be somewhere in between these two limiting cases describing finite slippage. This will be the subject of a future paper. In the no-slip case, by definition we have

### (c) Material model

To solve the problem posed in §2*b*, it is necessary to specify the strain energy density function *ψ* by selecting an appropriate model for the material bonded to the substrate. There are many hyperelastic models available in the literature to describe the nonlinear stress–deformation response of soft elastomers, including the ones that capture the strain hardening effects at large deformation, for example, the Arruda–Boyce model [34], the Gent model [35] and the generalized neo-Hookean model proposed by Knowles [19] and later adopted by others for crack-tip field analyses [21,23,27,28]. Because our focus here is the effect of interface slippage, it is advantageous to use a simple model that well describes the elastic behaviour of soft elastomers. One option is the incompressible neo-Hookean model, since most elastomers have a Poisson's ratio *ν* close to 0.5. However, in a previous work [29], results of finite-element simulations reveal an oscillatory singularity for the stress fields near the tip of an axisymmetric crack between an incompressible neo-Hookean material and a rigid substrate assuming the no-slip condition. This was attributed to the combined effects of the no-slip condition and incompressibility constraints. In this case, analytical solutions of the crack tip fields are difficult to obtain.

To allow analytical solutions for crack tip fields and to highlight the effect of interface slippage, we relax the incompressibility constraint and consider a set of compressible material models. Insight was found through the work of Blatz & Ko [36], where the following strain energy density function was introduced for continuum or foam polyurethane rubber
*f* represents the degree of dilation from voids in the bulk material, *μ* is the material modulus of rigidity and *β* represents the degree of compressibility which is a function of the Poisson's ratio *ν*
*f*=1 and equation (2.27) reduces to
*β*, where *x*_{1} and *x*_{2},

## 3. Asymptotic solutions

In this section we solve for the asymptotic deformation and stress fields near the tip of the interface crack. Following the method used in similar crack-tip analyses [17,20,21,27,28], we attempt the following near-field solution for the deformed coordinates *x*_{i} for both frictionless and no-slip conditions
*i*=1,2) are constants representing the deformed coordinates of the crack tip, and the functions *v*_{i} are unknown. The exponent *m* must be real to ensure a smooth crack opening as opposed to the oscillatory behaviour encountered in previous linear elastic solutions. In addition, motivated by the anticipation that the stress and deformation gradient may become unbounded near the crack tip (*r*→0) but displacement fields should remain bounded, we further assume that 0<*m*<1 [17,20,25]. Next, we discuss the two cases of frictionless and no-slip conditions separately.

### (a) Frictionless condition

First, the boundary condition in equation (2.34) yields *J* are
*β*>0 and 0<*m*<1, the first term *r*→0 which yields
*m*<1, the only possible solution is *m*=1/2 and
*a* is an arbitrary constant. The fact that *v*_{1}(*θ*) vanishes for all *θ* between 0 and *π* means that we must include higher-order terms in equation (3.1)
*n*>1/2. As a result, the leading-order term of *J* becomes
*r*→0, so that
*w*_{1}(*π*). This contradiction indicates that the assumption in equation (3.10) is not valid and we are left with the following two cases
*r*→0, and *w*_{1}(*θ*) is governed by a complicated nonlinear ordinary differential equation. Numerically, we have confirmed that case I does not admit any solutions that satisfy the governing equation (2.31a) and the boundary equations (2.32a) and (2.34) for *i*=1,2)
*β*>0 it must then hold that 1/2<*n*<3/2. Within this range of *n*, the only possibility is *n*=1 and
*x*_{1} and *x*_{2} is
*J* is then
*x*_{1} in equation (3.16a) had the same order (*J*=1. As mentioned earlier, with the frictionless condition, the near-tip solution for an interface crack is the same as that for a mode I crack in a homogeneous material. In addition, for the plane–stress deformation of an incompressible neo-Hookean material, the out-of-plane principal stretch ratio is given by 1/*J*, where *J* is the determinant of the in-plane components of the deformation gradient *I* is the trace of the in-plane components of the right Cauchy–Green tensor *β*=1. Therefore, it is not surprising that we have obtained the same near-tip solution as in [21] and we have shown above that the solution is also valid for *β* other than 1.

### (b) No-slip condition

Again, we start with the form of solution given in equation (3.1), but the boundary condition for *x*_{1} at *θ*=0 is given in equation (2.33) such that the material remains perfectly fixed to the bonded surface and *v*_{1}(*θ*) does not vanish because of the no-slip boundary condition in equation (2.33). This solution implies that *J*=𝒪 (*r*^{−1}) and to determine the asymptotic behaviour of *J* we must pursue higher-order terms in *x*_{1} and *x*_{2}. Similar to equation (3.8) where *x*_{1}(*r*,0)=*r*, we impose the further constraint *n*≤1, hence 1/2<*n*≤1. Proceeding in a manner similar to the frictionless condition, we find *i*=1,2)
*n*=1 and
*r*→0 for the deformed coordinates *x*_{1} and *x*_{2} is given by
*J* is found to be

### (c) Asymptotic behaviour of *J* at *θ*=*π*

As shown in equations (3.17) and (3.25), the first-order asymptotic solutions for *J* for the frictionless and no-slip conditions are both order *θ*=*π*), the first-order term vanishes, and the asymptotic behaviour of *J* is determined by even higher-order terms 𝒪 (*r*^{−1/2}). This is important, because our derivation relies on the assumption that the term associated with *J* in the traction-free boundary condition (see equation (2.32) for *θ*=*π*) was less dominant than the other. This assumption is questionable if the asymptotic behaviour of *J* at *θ*=*π* is unknown.

The same issue was encountered by Knowles & Sternberg [24], who solved the near-tip fields for a plane–stress crack between two sheets of different incompressible neo-Hookean materials. In their work, it was found that *r*→0 except at *θ*=*π*, where it was determined that *J* at *θ*=*π* was determined by establishing a relation between *I* and *J*. Using the same method, we have determined that for both the frictionless and no-slip cases near *r*→0
*J* is indeed less dominant for *β*>0. Therefore, the solutions obtained in equations (3.16) and (3.24) are valid for any value of *β*>0. As demonstrated in Knowles & Sternberg [24], it is possible to derive a solution for *J* that is uniformly valid for all *θ* including a boundary layer near *θ*=*π* within which the asymptotic behaviour of *J* changes from *θ*=0.

## 4. Results and discussion

### (a) Crack-tip deformation and stress fields

For the frictionless condition, we see that the crack-opening shape from equation (3.16) at *θ*=*π* has the approximation
*a*. After loading, the frictionless case shows an opening that resembles a parabolic shape with a vertical slope at the crack tip which is the same as the upper half of that of a loaded mode I crack [20,21,24]. The deformation gradient components are found to be
*σ*_{22} has the most severe singularity and dominates the other stress components. Therefore, near the crack tip, the material is expected to be under a nearly uniaxial tensile stress state in the *x*_{2}-direction. On the bonded interface (*θ*=0), *σ*_{12}=0 and *σ*_{22}≫*σ*_{11}, so the bonding strength of the interface is primarily dependent on normal traction in the *x*_{2}-direction. It must also be noted that the stress components (for either the frictionless or the no-slip case) are seemingly unbounded at *θ*=*π*. This is because when deriving the Cauchy stress with equation (2.15), we kept only the first-order term of *J* which vanishes at *θ*=*π*. A higher-order term of J should be used at *θ*=*π* (see §3*c*) which would result in bounded Cauchy stress. This was not included in the Cauchy stress results, because we are primarily interested in the stresses on the bonded interface at *θ*=0.

For the no-slip case, equations (3.24a) and (3.24b) on the free edge (*θ*=0) lead to the following crack-opening shape
*b* shows a plot of the normalized crack-opening shape for the no-slip condition and we now see the emergence of the wedge-like shape observed by Newby *et al.* [6]. In this plot, the normalized wedge angle will always be *π*/4; however, from equations (3.24a) and (3.24b), this angle *ϕ* is related to the ratio of the two coefficients *a*_{1} and *a*_{2} by

The near-tip stress field for the no-slip case is significantly different from the frictionless case, because all stress components have the same first-order dependence on *r* and *θ* with only different coefficients. The ratios between different stress components are related to the wedge angle *ϕ* of the opened crack surface by
*θ*=0), as the wedge angle *ϕ*→*π*/2 then *σ*_{22}≫*σ*_{12}≫*σ*_{11}. This trend along the bonded edge is reversed when *ϕ*→0 and *σ*_{11}≫*σ*_{12}≫*σ*_{22}. The change in the crack tip stress state has important implications on the crack propagation behaviour on the bonded surface. For example, a small wedge angle *ϕ* means a normal stress *σ*_{22} much smaller than the other stress components. This means the interface can be held together with weaker adhesion if slippage is suppressed and a small wedge is achieved, consistent with the experimental observations of Newby *et al.* [6].

The constant coefficients in our asymptotic solutions, specifically *a* and *b* for the frictionless case and *a*_{1} and *a*_{2} for the no-slip case, describe the intensity of crack-tip deformation and stress fields. They can be related to the energy release rate by evaluating the *J*-integral generalized to finite strain elastostatics by Knowles & Sternberg [37]
*i*,*A*=1 or 2, *Γ* is a contour enclosing the crack tip in the reference configuration and *N*_{A} is component of the outward unit normal vector **N** of the contour *Γ*. Because the *J*-integral is path-independent, we selected a small circle enclosing the crack tip for *Γ* and let the radius of the circle approach zero. In this way, we evaluated the *J*-integral using our asymptotic solution and found the following results:

Finally, it should be noted that the material constant *β* (which describes material compressibility) does not appear explicitly in the first-order asymptotic solutions above for either the frictionless or no-slip cases. It may appear in higher-order terms describing these solutions. This, however, is a subject for further investigation. Another possibility is that the parameter *β* can affect the coefficients such as *a*_{1} and *a*_{2} for the no-slip case, as suggested by the finite-element results presented in the following section.

### (b) Finite-element analysis

To validate the asymptotic solutions, a plane–strain finite-element analysis (FEA) model was created for both the frictionless and no-slip cases using ABAQUS (version 6.12, Dassault Systems Simulia Corp., Providence, RI). The Blatz–Ko model adopted in equation (2.29) is a special case of the ABAQUS built-in hyperfoam model with no thermal expansion effects. The undeformed geometry of the FEA model is shown in figure 4 with point O designated as the crack tip (note the similarities to figure 2). The boundary AO is the crack surface and is traction free (see the boundary conditions in equation (2.32)). The boundary OD is the bonded interface with zero vertical displacement (*u*_{2}=0 and *x*_{2}=0). The frictionless case requires the shear traction on OD to be zero and this edge follows the boundary condition in equation (2.34). For the no-slip case, the horizontal displacement is also zero (*u*_{1}=0) on OD and follows the boundary condition in equation (2.33). A uniform displacement Δ was applied on the upper boundary CB to deform the crack resembling a far-field load in model I condition. Here, the far-field strain is defined as *ϵ*≡Δ/*h*, where *h* is the vertical dimension of the body (in this case, *h*=2*L*, where *L* is the dimensional length of the model). To overcome numerical difficulties, the two side boundaries BA and DC were constrained with zero displacement in the horizontal direction but allowed to slide freely in the vertical direction (zero shear traction). Typical meshes consisted of 0.05*L*-sized quadrilateral elements around the exterior and transitioning to a small portion of triangular elements near the crack tip at point O of size 0.0001*L* in order to resolve the local deformation and stress fields there.

We first compare the crack-opening shapes of the FEA results and those predicted by the asymptotic solutions in equations (3.16) and (3.24) for the frictionless and no-slip cases, respectively. Examples for *β*=4.5 with a far-field strain *ϵ*=100% are shown in figure 5. In these results, the asymptotic solutions closely match the FEA results near the crack tip; however, significant deviation occurs away from the crack tip as expected. The coefficients in the asymptotic solutions (namely *a*, *b* for the frictionless case, and *a*_{1}, *a*_{2} for the no-slip case) were determined by fitting equations (3.16) and (3.24) to the FEA results in the near-tip region. In the no-slip case in figure 5*b*, the magnitudes of the coefficients were determined to be *a*_{1}≃0.5918*L*^{1/2} and *a*_{2}≃2.2545*L*^{1/2}.

We are also interested in the Cauchy stresses along the bonded interface (*θ*=0). The asymptotic solutions predict for the frictionless case that *r*→0 (see equation (4.4)), whereas for the no-slip case that all Cauchy stress components have the same asymptotic order *β*=4.5 and a far-field strain of *ϵ*=100%. To further validate the near-tip stresses in the no-slip case (figure 6*b*), the stresses on the bonded edge (*σ*_{11}, *σ*_{12} and *σ*_{22} where *θ*=0) were also calculated using the coefficients (*a*_{1} and *a*_{2}) found from the data in figure 5*b* were within 10% of the FEA data. Agreement between our asymptotic solutions with FEA results was also confirmed using *β*=1.

It was shown in equation (4.10) that for the no-slip case the wedge angle *ϕ* can indicate the relative comparison of different Cauchy stress components in the near-tip region. From the FEA results, the wedge angle *ϕ* was found to depend on both the far-field strain *ϵ* and the material compressibility parameter *β*. Both effects are plotted in figure 7 using various values of *β* and *ϵ*. Figure 7*a* is a plot of the wedge angle *ϕ* with increasing far-field strain *ϵ* and various values of the bulk modulus *β*. For *β*=1, the wedge angle *ϕ* decreases as the far-field strain is increased and approaches a value of *ϕ*≃1.4321 radians (slightly less than *π*/2). For larger *β*, the material is less compressible and the wedge angle *ϕ* first decreases (until *ϵ*≃10%) and then increases. The increase of *ϕ* at large strains is likely due to the loading geometry of our FEA model, in particular, the horizontal constraints on the side boundaries BA and DC shown in figure 4. Figure 7*b* shows the dependence of *ϕ* on the compressibility parameter *β* (only far-field strains *ϵ*≥10% are shown). It can be seen here that the wedge angle *ϕ* is smaller for less compressible materials (larger *β*), and saturates as the incompressibility limit is approached. The level where *ϕ* saturates is dependent on the applied far-field strain *ϵ*.

## 5. Conclusion

In this work, we have shown that the deformation and stress fields near the tip of an interface crack are sensitive to the slippage on the bonded interface. Specifically, for the no-slip case (or infinitely high friction), the crack deforms to a wedge-like shape upon far-field loading and is consistent with the experimental observation of Newby *et al.* [6]. More interestingly, the wedge angle *ϕ* is related to ratios of different Cauchy stress components in the near-tip region. If the angle *ϕ*≃0, the crack-tip stress field is dominated by *σ*_{11}, the normal stress component along a direction tangent to the bonded interface. If the angle *ϕ* is increased to *π*/2, the normal stress component *σ*_{22} which is perpendicular to the bonded interface becomes dominant. The wedge angle *ϕ* can depend on material compressibility *β* and far-field loading conditions, as illustrated by our FEA results. By contrast, the frictionless case showed the crack always opens into a smooth parabolic shape with a wedge angle *ϕ*=*π*/2 and the near-tip stress field is always dominated by the normal stress *σ*_{22}. The results in this work can be used to understand how slippage affects the propagation of interface cracks.

A more realistic model should allow finite slippage on the interface which is the intermediate case between the two limiting cases considered in this paper. This finite slippage condition could be accommodated by prescribing an appropriate shear traction boundary condition on the interface directly ahead of the crack tip, as governed by certain cohesive zone models along the shear direction. In this case, it is speculated that the wedge angle *ϕ* should depend on the slippage as shown in the rough model proposed by Newby *et al.* [6]. This will be the subject of a future paper.

## Funding statement

This work was supported by the National Sciences and Engineering Research Council of Canada through a Discovery Grant and start-up funds from the University of Alberta.

## Acknowledgements

The authors thank the anonymous reviewers whose suggestions helped us improve the quality of the paper.

## Appendix A. Derivation of asymptotic behaviour of *J* at *θ*=*π*

Following the method used in Knowles & Sternberg [24], we derive the asymptotic behaviour of *J* on the crack surface (*θ*=*π*), specifically equation (3.26). From equation (2.11)
**I** is the identity tensor and **I**, respectively. For plane–strain deformations, equation (A 1) implies
*θ*=*π*), where the boundary condition in equation (2.32) dictates that *I* at *θ*=*π* is given by
*r*→0, *J*^{2(β+1)} dominates over *J*^{−2β} as *r*→0 and *J*^{2(β+1)}≃*I* and therefore

- Received June 24, 2014.
- Accepted July 14, 2014.

- © 2014 The Author(s) Published by the Royal Society. All rights reserved.