## Abstract

When a crack tip impinges upon a bi-material interface, the order of the stress singularity will be equal to, less than or greater than one-half. The generalized stress intensity factors have already been determined for some such configurations, including when a finite-length crack is perpendicular to the interface. However, for these non-square-root singular stresses, the determination of the conditions for crack growth are not well established. In this investigation, the critical value of the generalized stress intensity factor for tensile loading is related to the work of adhesion by using a cohesive zone model in an asymptotic analysis of the separation near the crack tip. It is found that the critical value of the generalized stress intensity factor depends upon the maximum stress of the cohesive zone model, as well as on the Dundurs parameters (*α* and *β*). As expected this dependence on the cohesive stress vanishes as the material contrast is reduced, in which case the order of the singularity approaches one-half.

## 1. Introduction

The stress field in the immediate vicinity of a crack in a homogeneous material is characterized by the stress intensity factors, *K*_{I}, *K*_{II} and *K*_{III}, in modes I, II and III, respectively. In mode I, in which the crack is subjected to a tensile loading, the normal stress acting on the symmetry plane is given by (e.g. [1])
*K*_{I}≥*K*_{IC} the crack will propagate; otherwise it will not. The quantity *K*_{IC} is known as the critical stress intensity factor, and characterizes the resistance of the material to fracture.

The stress intensity factor (*K*_{I}) is related to the energy release rate (*G*) and the *J*-integral according to
*E**=*E* for plane stress and *E**=*E*/(1−*ν*^{2}) for plane strain, *E* is Young’s modulus and *ν* is Poisson’s ratio. Crack extension will occur when the energy release rate is equal to the work of adhesion (*w*), i.e.
*w*=2*γ* and *γ* is the surface energy. Thus, crack growth will occur when *J* or equivalently *K*_{I} reaches a critical value, i.e. *J*=*w* or *K*_{IC} is also known as the fracture toughness.

Now consider a crack with its tip terminating at an interface between two different materials as shown in figure 1*a*. The stress field is singular but the order of the singularity is no longer square-root (e.g. [2]). The normal stress acting on the symmetry plane can be written as
*Q* is referred to as the *generalized stress intensity factor* (e.g. [3]). The order of the singularity (λ) depends upon the material combination and its value can be less than, equal to or greater than one-half. It is noted that *Q* as defined by equation (1.4), differs from *K*_{I} by a factor of

Non-square-root singularities occur in certain adhesion problems. In [4], Adams considers the problem of a semi-infinite elastic strip in plane and axisymmetric geometries being pulled from an elastic half-space. An analytical procedure based on Mellin transforms is used to relate the critical value of the generalized stress intensity factor to the fracture toughness. The resulting pull-off force depends not only on the material combination but also on the maximum stress of the cohesive zone model.

The issue of non-square-root stress singularities also occurs in sharp finite-angle notches. A fracture criterion for sharp-notched samples in mode I was given by Gómez & Elices [5]. They related the critical value of the generalized stress intensity factor to that of a crack using a finite-element analysis. Very good agreement with experimental results for a variety of materials was obtained. In [6], Mohammed & Liechti used cohesive zone modelling to determine the initiation of failure over a range of bi-material corners angles between epoxy and aluminium. After calibration, the model results agreed well with the measurements. Adams & Hills [7] also related the critical value of the generalized stress intensity factor to the fracture toughness (critical value of *K*_{IC}) using an analytical procedure based on Mellin transforms similar to [4]. The results agreed very well (within 1%) with the finite-element results of Gómez & Elices [5] which, in turn, agreed very well with experiments.

In the current investigation, a crack which is perpendicular to, and whose tip impinges upon, a bi-material interface is analysed. The analytical procedure uses a cohesive zone model and Mellin transforms as in [4] and [7] in order to determine the critical value of the generalized stress intensity factor. In §2, an asymptotic analysis of the crack tip is presented using the method of Williams [8]. The analysis of the crack using the cohesive zone model and Mellin transforms is presented in §3. The results and discussion are given in §4 which also includes a sub-section in which the results of §3 are combined with existing results in the literature to analyse the propagation of a finite-length crack. Finally, the conclusions are presented in §5.

## 2. Asymptotic analysis at the crack tip touching the interface

Consider a crack which is oriented perpendicular to an interface and with one of its tips terminating at that interface as shown in figure 1*a*. It is noted that for later reference the same configuration with a cohesive zone is shown in figure 1*b*. Of particular interest is the mode I behaviour of this crack. This configuration (figure 1*a*) will be used to analyse the asymptotic behaviour of a finite-length crack (figure 2) in the immediate vicinity of the tip which touches the interface. The solution for the order of the singularity at the crack tip was determined by Bogy [2] using a method based on Mellin transforms. For completeness, an analysis of that configuration will also be presented here, but will use the simpler asymptotic approach of Williams [8]. The superscripts (1) and (2) are used to refer to the cracked and uncracked materials, respectively (figures 1 and 2).

The boundary conditions along the crack surface are given by
*θ*=*π*/2 interface are
*θ*=*π* require
*μ* is the shear modulus, *κ*=(3−*ν*)/(1+*ν*) for plane stress, whereas *κ*=3−4*ν* for plane strain and *ν* is Poisson’s ratio.

Following the procedure of Williams [8], the result of applying the boundary conditions (2.1)–(2.3) to equation (2.4) is a system of eight homogeneous linear algebraic equations, the determinant of which must vanish for non-trivial solutions of *A*^{(i)},*B*^{(i)},*C*^{(i)} and *D*^{(i)} (*i*=1,2). Setting the determinant to zero yields the result given by Bogy [2]
*α* and *β* are defined by Dundurs [10] as
*α* and *β*. As expected equation (2.5) shows that the order of the singularity (λ) depends only on *α* and *β* [10] and is valid for plane stress or plane strain. The values of *α* and *β* do, however, depend on *κ*_{1},*κ*_{2} and in that manner are affected by the condition of plane stress or plane strain.

The variation of the order of the singularity (λ) in the Dundurs parameter plane is shown in figure 3. It is noted that the limit *α*=1 corresponds to a rigid cracked material (1), whereas the limit *α*=−1 corresponds to a rigid and yet uncracked material (2). The special case of identical materials (*α*=*β*=0) leads to λ=1/2 and the problem is simply a crack in a homogeneous material. For each pair of values *α* and *β*, the numerical solution of equation (2.5) gives one real root in the range 0<λ<1 and no complex roots. Roots greater than unity are not allowed based upon energy considerations, whereas negative values of λ lead to bounded stresses as *r*→0 and consequently are not of interest in this study. From figure 3, it is seen that the order of the singularity increases monotonically from zero when *α*=−1 and *β*=−1/2 to unity when *α*=1.

## 3. Analysis of the configuration with a cohesive zone model

It is recognized that the asymptotic solution of the linear elastic problem demands that the stresses be singular as *r*→0, and in particular that the normal stress acting on the plane of symmetry (*θ*=*π*) becomes infinite in front of the crack tip. Because such infinite stresses cannot exist, a simple cohesive zone model is used along the plane of symmetry in which the normal stress in the separation region (figure 1*b*) is given by
*h*_{0} is such that the product *h*_{0}*σ*_{0} is equal to the work of adhesion (*w*). Note, however, that in a continuum model the separation is zero when the surfaces touch. Therefore, when the continuum separation is *h*_{0} the atomic planes are separated by a distance equal to the equilibrium spacing plus an additional distance *h*_{0}. The cohesive zone extends over the interval (0,*r*_{0}) where the location of the cohesive zone tip (*r*=*r*_{0}) is, at this point in the analysis, unknown. Note that the cohesive zone model eliminates the stress singularity at the crack tip while maintaining bounded stresses at *r*=*r*_{0}. The cohesive zone model is due to Barenblatt [11]; it was Dugdale [12] who proposed a mathematically similar model to deal with small-scale plastic yielding at a crack tip.

Now consider the crack shown in figure 1 for which the generalized stress intensity *Q* has been calculated (e.g. [13]). Provided that the magnitude of *Q* is insufficient to cause the cohesive zone opening at the crack tip to be equal to *h*_{0}, i.e. *Q* will make the separation exceed *h*_{0} which will cause the crack to propagate. This configuration includes a cohesive zone extending in front of the crack tip as shown in figure 1*b*. It can be represented by the sum of two problems—the same problem without the cohesive zone (i.e. continuity along the entire interface) and a residual problem with the cohesive zone which is also subjected to the opening stresses

The corresponding boundary conditions on the free surfaces of this residual problem are given by equations (2.1) and (2.2). On the plane of symmetry

The Mellin transform of a suitably regular function *f*(*r*) on *s* is a complex transform parameter. Before applying the Mellin transform, we note that the stress and displacement fields can be found from the Airy stress function which is a solution of the biharmonic equation [16], i.e.
*ϕ*(*r*,*θ*),*r*^{2}*τ*_{rr}(*r*,*θ*),*r*^{2}*τ*_{θθ}(*r*,*θ*),*r*^{2}*τ*_{rθ}(*r*,*θ*),*ru*_{r}(*r*,*θ*),*ru*_{θ}(*r*,*θ*), respectively. The application of the Mellin transform to equation (3.6) results in [15]
*a*(*s*),*b*(*s*),*c*(*s*) and *d*(*s*) are unknown complex functions to be determined from the boundary conditions. The transforms of the relevant stresses and displacements given by equation (3.7) are

The unknown displacement discontinuity in the cohesive zone *r*_{0}) according to
*r*=0 and at *r*=*r*_{0}. The values of *M* and *N* will be chosen for convergence. The Mellin transform of the displacement discontinuity ^{r} [17].

The normal stress acting on the symmetry plane can now be found using the inverse Mellin transform of *c* is taken such that the integral of equation (3.15) in the complex plane exists, e.g. *c*=−1. Consequently, a new integration variable *p* is defined such that *s*=−1+i*p*.

The evaluation of equation (3.15) results in
*f*_{m}(*m*=1,2,…*M*) and *g*_{n} (*n*=1,2,…*N*). These unknown constants are determined by applying the first of the mixed conditions of equation (3.3) to the normal stress in equation (3.16). The result is

Equation (3.18) has the additional unknown quantity *Q*^{′} which can be used to determine the length of the cohesive zone (*r*_{0}). A numerical solution of equation (3.18) is obtained by satisfying this set of equations at a discrete number of suitably chosen collocation points (*r*_{i}, *i*=1,2,…*M*+*N*+1) resulting in *M*+*N*+1 linear equations with *M*+*N*+1 dimensionless unknowns (*f*′_{m},*g*′_{n},*Q*′). The length of the cohesive zone is then given by
*Q*^{′} and *f*′_{1} are determined numerically for a given material combination as described earlier in this section. From the form given by equation (3.20), it is seen that the critical value of the generalized stress intensity factor (*Q*_{C}) can be expressed in terms of the work of adhesion (*w*) and the cohesive zone stress (*σ*_{0}). However, for the special case of a crack in a homogeneous material (*α*=0,*β*=0), for which λ=1/2 and the dependency on *σ*_{0} vanishes.

## 4. Results and discussion

It is convenient to rewrite equation (3.20) in dimensionless form as
*R*(*α*,*β*) versus the composite material parameters (*α*,*β*) are shown in figure 4. It is noted that *R*(*α*,*β*) represents the dimensionless critical value of the generalized stress intensity factor for *α* and *β* through equation (2.5). A curve-fit to these results is given by
*C*_{i}) given in table 1. The use of equation (4.2) leads to a maximum error of less than 0.8%. It is noted that a plot of *R*(*α*,*β*) versus λ shows that *R*(*α*,*β*) does depend on both *α* and *β* and not simply on λ.

The dimensionless stress intensity factor, *α* for different values of *β* are shown in figure 5 for *β* the value of λ increases with increasing *α*. Because *R*(*α*,*β*) for small *α* (for which λ<1/2 from figure 3) but greater than *R*(*α*,*β*) for greater values of *α* (for which λ>1/2 from figure 3). This also causes the trend of *β* to be opposite to that of *R*(*α*,*β*) with *β*. From figure 6, for which the value of

The corresponding dimensionless critical length (*r*_{0C}) of the cohesive zone is given by equations (3.19) and (3.20) as
*α* and for various values of *β*. The most discernible trend is an increase in this value as *β* decreases putting the material pair in the lower left corner of the composite material plane for which λ decreases toward zero. Thus, as the singularity tends to disappear the length of the cohesive zone increases as is expected.

### (a) Results for a finite-length crack

Consider now the example of a finite-length crack (figure 2) for which the values of the generalized stress intensity factors were determined by Chen [13]. The notation used by Chen differs somewhat from that used here. In particular, in [13], 1−λ rather than λ was used to represent the order of the singularity, *K* rather than *Q* was used to represent the generalized stress intensity factor, and the internal pressure in the crack was *σ*_{0}, whereas *p* is used here. Let *T*(*α*,*β*) represent the results given in table 3 of [13] for loading by an internal pressure *p*. Then
*Q* in equation (4.4) equal to *Q*_{C} in equation (4.1), the critical value of the internal pressure for crack propagation becomes
*α*)*R*(*α*,*β*)/(1+*α*)*T*(*α*,*β*) as a function of the composite material parameter *α* and for various values of *β*. In figures 9 and 10 are shown the variation of *α* and for various values of *β* with *α* increases (i.e. material 1 becomes stiff compared with material 2) the critical value of

It is noted that the use of a cohesive zone model implies some limitations on the values of certain parameters. For example, Jin & Sun [18] investigated the use of a cohesive zone model in a power-law hardening material under mode III loading. They found that the ratio of the peak cohesive stress to the yield stress needed to be greater than about two, which also implied that the cohesive zone width was small compared to the crack length. Such an analysis in the context of the current configuration, in which a crack impinges upon a bi-material interface, is well beyond the scope of this work. However, the ratio of the cohesive zone length to the crack length can be readily determined. In terms of the dimensionless parameters used
*r*_{0}/2*a* in the results presented here, and from figure 7, *r*_{0}/2*a*<0.09 and the use of a cohesive zone appears to be justified.

## 5. Conclusion

The propagation of a crack tip impinging upon a bi-material interface has been investigated. It is known that the order of the stress singularity can be equal to, less than or greater than one-half. The critical value of the generalized stress intensity factor for tensile loading has been related to the work of adhesion by using a cohesive zone model in an asymptotic analysis of the separation near the crack tip. It has been found that the critical value of the generalized stress intensity factor depends upon the maximum stress of the cohesive zone model, as well as on the Dundurs parameters (*α* and *β*). As expected, this dependence on the cohesive zone stress vanishes as the material contrast is reduced, in which case the order of the singularity approaches one-half.

For a finite-length crack subject to an internal pressure, the generalized stress intensity factor has been calculated for a range of material combinations and has been presented previously in the literature. Those results have been combined with those given in the current investigation to determine the critical internal pressures for propagation of a finite-length crack.

## Data accessibility

The only data are those generated by solving the equations described in the paper and have been presented in the graphs of figures 3–10.

## Authors' contributions

G.G.A. is the sole author of this paper. He did not receive advice or input from anyone else.

## Competing interests

There are no conflicts or competing interests in this work.

## Funding

This work was done without funding from any source.

- Received August 17, 2015.
- Accepted October 27, 2015.

- © 2015 The Author(s)