Mixed-mode cohesive laws and the use of linear-elastic fracture mechanics

Small-scale cohesive-zone models based on potential functions are expected to be consistent with the important features of linear-elastic fracture mechanics (LEFM). These include an inverse-square-root 𝐾 -field ahead of a crack, with the normal and shear stresses being pro- portional to the mode-I and mode-II stress-intensity factors, 𝐾 𝐼 and 𝐾 𝐼𝐼 , the work done against crack-tip tractions being equal to ( 𝐾 2 𝐼 + 𝐾 2 𝐼𝐼 ) ∕ ̄𝐸 , where ̄𝐸 is the appropriate modulus, and failure being controlled by the toughness. The use of an LEFM model also implicitly implies that the partition of the crack-tip work into shear and normal components is given by a phase angle defined as 𝜓 𝐾 = tan −1 ( 𝐾 𝐼𝐼 ∕ 𝐾 𝐼 ) . In this paper, we show that the partition of crack-tip work in a cohesive-zone model is consistent with LEFM if the normal and shear deformations across an interface are uncoupled. However, we also show that this is not the case for coupled cohesive laws, even if these are derived from a potential function. For coupled laws, LEFM cannot be used to predict the partition of work at the crack tip even when the small-scale requirements for LEFM conditions being met; furthermore, the partition of the work may depend on the loading path. This implies that LEFM cannot be used to predict mixed-mode fracture for interfaces that are described by coupled cohesive laws, and that have a phase-angle-dependent toughness.


Introduction
Cohesive-zone models, originating from the work of Hillerborg et al. [1] and Needleman [2], are widely used to simulate the initiation and growth of cracks in problems ranging from the materials scale [2][3][4][5] to the structural scale, such as adhesive joints [6][7][8] and wind turbine blades [9]. In these models, the fracture process is described by a traction-separation relationship, known as a cohesive law, that comprises both a strength (peak traction) and a fracture energy (area under the traction-separation curve) [10,11]. The use of cohesive laws allows a transition between the strength-based approach to fracture of [12] and the energy-based method of [13] that underpins linear-elastic fracture mechanics (LEFM) [14][15][16].
Since [17] and [18] generalized cohesive laws to include shear tractions, cohesive-zone modelling has been extended to mixedmode fracture, with many different cohesive laws being developed. These cohesive laws can be divided into several fundamentally different groups [19]. First, there are those derived from potential functions, and those that are not derivable from potential functions. Second, there are what are termed as ''uncoupled'' and ''coupled'' mixed-mode laws.
Potential-based cohesive laws are independent of the loading history. The normal and shear tractions depend only on the values of the normal and tangential openings; they do not depend on the path by which those openings are reached. As an example, micromechanical modelling can be used to show that cross-over fibre-bridging gives coupled laws for which a potential function exists [20]. For cohesive laws that cannot be derived from a potential function, the cohesive tractions and the work of the cohesive tractions depend on the loading path. Such laws can be used to model fracture processes that include history-dependent phenomena such as plasticity or frictional sliding [21]. However, attention is focused in this paper on conditions that might be consistent with LEFM, so only potential-based cohesive laws are considered in the present work; history-dependent mechanisms are excluded.
In ''uncoupled'' mixed-mode cohesive laws, the normal tractions depend only on the normal openings, and the shear tractions depend only on the tangential (shear) openings. However, despite the terminology, coupling between the two modes of deformation is inherently introduced through the failure criterion [22]. This coupling generally manifests itself as a relationship between the critical normal and shear displacements. In particular, shear decreases the critical opening-displacement, and opening decreases the critical shear displacement. More details are given in Appendix. In coupled cohesive laws, the normal and shear tractions each depend on both the normal and tangential openings. It is not necessary to describe an additional mixed-mode failure criterion with such coupled laws, but the coupling should be consistent with any observed mixed-mode failure criterion.
Under small-scale conditions, the driving force for crack growth in an elastic body is the gradient of total potential energy of the system with respect to the length of the traction-free portion of the crack [23]. In linear-elastic fracture mechanics (LEFM), this is designated by the energy-release rate,  [13], which is identical to the value of the -integral taken around the crack tip [23]. Fracture occurs when  = , which is identified as the toughness, and is considered to be a material property. Mixed-mode fracture in an LEFM framework is described in terms of the mode-I and mode-II stress-intensity factors, and : the amplitudes of the singular normal and shear stresses in the -dominant region near the crack tip. A phase angle describes the ratio between these two parameters as = tan −1 ( ∕ ) , and the toughness is assumed to be a unique function of the phase angle, ( ) [24,25]. Crack growth occurs when  = ( ), where the phase angle describes the ratio ∕ at fracture. Two implicit assumptions of LEFM are that the work at the crack tip, and its partition into shear and normal components are both path-independent i.e., independent of whether and are applied proportionally (simultaneously) or non-proportionally (e.g. sequentially). The use of LEFM is predicated on the assumption that any portion of a body not described as a continuum elastic medium is limited to a very small region near the crack tip, and that the macroscopic response of the body is linear-elastic. The use of LEFM as a powerful quantitative tool that is ubiquitous in engineering design is not predicated on singular stresses actually existing at the crack tip, but rather on the fact that the fracture process at the crack tip is dependent only on a macroscopic description of the -field [26]. In other words, the crack tip (and its partition) are uniquely defined by and , and independent of the cohesive length, provided this latter parameter is small enough. The implication of this is that any loading-path dependence that might exist for deformation of the crack tip potentially is inconsistent with the assumptions that underpin the use of LEFM.
In the present study, we investigate this specific issue within the broad framework of small-scale fracture that is generally taken to correspond to LEFM conditions. It is emphasized again that for a fracture problem to be described by LEFM merely requires a smallscale cohesive zone. It does not require singular stresses to actually exist at the crack tip. This has been demonstrated by appropriate small-scale cohesive-zone analyses [16,27,28]. In this paper, we use small-scale cohesive-zone models with potential-based cohesive laws to satisfy one obvious requirement of path-independence, and examine whether there are additional constraints on tractionseparation laws for them to provide path-independent, mixed-mode behaviour. In particular, we are interested in whether there may be limitations on when an LEFM framework might be valid to describe small-scale fracture with uncoupled and coupled, potential-based, mixed-mode cohesive laws.

Work of cohesive tractions
The local work done (per unit area) against cohesive tractions across a small element of the interface, , can be decomposed into the local work done against normal tractions (designated as mode-I),  , and the local work done against shear tractions (designated as mode-II),  : where and are the normal and shear tractions, and are the normal and shear displacements. Under pure mode-I conditions ( = 0), local failure of the interface occurs when = c , where c is the normal displacement at failure. This corresponds to  = , which is defined as the mode-I toughness. Under pure mode-II conditions ( = 0), local failure of the interface occurs when = c , where c is the shear displacement at failure. This corresponds to  = , which is defined as the mode-II toughness.
Of particular interest in fracture mechanics is the work done against the tractions at a cohesive crack tip (defined as the point at which the active cohesive zone ends, 1 = 0 in Fig. 1). The normal and shear displacements at the cohesive crack tip are designated by and , and the two terms for the work done against the corresponding tractions at this location are designated by  and  . When the -integral [29] is evaluated along the cohesive zone out to a region where  = 0, its value is given by [30]: where ( , ) is the potential function used for the traction-separation law. In this paper, the concept of an instantaneous cohesive length at the tip of the cohesive crack [16,28] is used. This can be defined for a homogeneous system in modes I and II as wherē= ∕(1 − 2 ) in plane strain,̄= in plane stress, and and are Young's modulus and Poisson's ratio. These are slightly different from similar quantities defined in terms of the failure parameters [1,16,31]. They have the advantages that they can be used to describe the state of the cohesive zone at any stage of loading, and they can be defined for coupled cohesive laws. The cohesive lengths can be normalized by a characteristic dimension of the geometry, such as a layer thickness, ℎ, so that = ∕ℎ. If̃is very small, one is in a small-scale regime, and the principles of LEFM are expected to apply. In particular, this means that there will be a -dominant region ahead of the cohesive crack tip, where the stresses across the interface follow an inverse square-root relationship with respect to distance from the tip, 1 . In the absence of a modulus mismatch across the interface, the normal tractions and shear tractions along 2 will be described in this region by: where and are the mode-I and mode-II stress-intensity factors. Close to the crack tip, the stresses will deviate from this relationship, with the details of the stress field being dependent on the cohesive law. Beyond the -dominant region, the stresses will deviate from this relationship, following the non-singular, elastic, stress field of the structure. The region over which the -field describes the stresses may be very small; however such a region will exist if̃is small enough. Again, we emphasize that a central tenet of LEFM is that it can be used to describe fracture if̃is small, it does not have to be zero. It is for this reason that cohesive-zone models can be used to describe LEFM under small-scale conditions [16,27,28].
Under LEFM conditions, an evaluation of the -integral in the -dominant region gives [29]: Owing to the path-independency of the -integral [29], is equal to (Eq. (2)), so that Irwin's virtual crack closure relation holds in LEFM:  = | | 2 ∕ . So, a consistent connection between LEFM models and CZM models will be that  =  , if̃is small enough for LEFM assumptions to be valid.

Definitions of mode-mixedness
There are several definitions of mode-mixedness in the cohesive-zone literature (Fig. 2). The one we will focus on in this paper has a direct connection with the concept of a phase angle in LEFM. It is defined in terms of the ratio of the work done against each mode of deformation, so that, at any point along the interface, the local phase angle is As 1 approaches zero, this tends to the crack-tip phase angle, which is defined as [22,32] = tan −1 The distance over which ( 1 ) is equal to decreases with decreasing cohesive length, [16,27,28]. For the special case of uncoupled cohesive laws, no modulus mismatch across the interface, and a very small value of̃, the mode-I and mode-II work done against crack-tip tractions can be identified with and through Eq. (6) as 1 The phase angle used in LEFM is defined as: Therefore, as has been shown to be the case [16,27], is expected to equal under these conditions. Furthermore, if there is a modulus mismatch across the interface, , scales with the elastic properties and cohesive length as predicted by LEFM [16,27,33].
It is noted that an alternative measure of mode-mixedness ( Fig. 2), based on the ratio of the two tractions: can vary with the choice of cohesive law. It does not have the potential advantage of , in linking crack-tip deformation to macroscopic conditions under LEFM conditions. Under LEFM conditions, the magnitude of the stresses within the field region are dictated by the stress-intensity factors. So, ( 1 ) = in this region. However, it is axiomatic to LEFM that fracture is controlled by the deformation at the crack tip, and that the -field controls this deformation through . Therefore, it would seem to be an unnecessary restriction on modelling mixed-mode fracture to impose an additional constraint on cohesive laws that the crack-tip stresses in the entire cohesive zone should be in the same ratio as the stress-intensity factors [34]. In conclusion, one expects ( 1 ) = in the field, but expects to depend on the choice of cohesive law at the crack tip. Conversely, one expects ( 1 ) to equal close to the crack tip, but for there to be no connection between ( 1 ) and in the -field. The observation that = has already been shown to be valid if the cohesive laws are uncoupled [16,27]. However, a consideration of Eq. (2) for the case when the cohesive-laws are coupled indicates that the ratio between the two quantities ( and  ) may depend on the loading path, as discussed in Ref. [35]. In such a case there may not be a unique relationship between and . This could have implications for the use of LEFM to predict the failure of interfaces if the fracture-process mechanism behaves in accordance with a coupled traction-separation law. Mixed-mode LEFM models are all predicated on an assumption that deformation at the crack tip, where fracture takes place, is uniquely defined by and , with no path dependence. If coupled laws give path-dependent deformation at the crack tip, then it would imply that the use of LEFM may implicitly require the assumption of uncoupled cohesive laws. It is the purpose of this paper to explore this point.
In this context it should be emphasized that we are exploring the effects of using coupled and uncoupled laws, wheñis small enough for the problem to be in the LEFM limit. It has already been shown that, in this limit, uncoupled laws result in being equal to , provided sufficient care is taken to ensure that in finite element modelling the mesh size is small enough to observe the plateau in ( 1 ). We are interested in whether the same conclusion can be made for uncoupled laws, given the same care about mesh size and limitations oñ.
This focus is in contrast to that of earlier work [27,32,36], which explored the crack-tip phase angle and mixed-mode fracture at large cohesive lengths, well away from the LEFM limit. This body of work indicates that, for large cohesive lengths, the crack-tip phase angle tends to move away from the values controlled by the local K-field to values controlled by the macroscopic loads and geometries, as suggested by Charalambides et al. [37]. For example, the paper by Conroy et al. [32] explores values of cohesive lengths that range from values slightly bigger than ones for which LEFM should unambiguously be valid to much larger values. At  the lower end, the phase angle for an uncoupled law approaches the LEFM value, while the phase angle for a coupled law shows a larger discrepancy. In this paper, we explore in detail the difference between coupled and uncoupled laws, while ensuring that we are unambiguously within the range where LEFM is valid.

Finite-element modelling
The problem was modelled by finite-element (FE) simulations, using the commercial code ABAQUS. The finite-element domain (of radius ) and the mesh for the mixed-mode -field is shown in Fig. 3. A crack extends along the plane 2 = 0, from 1 = − to 1 = 0. The traction-separation relationships used to model the cohesive zone were specified along the crack plane from 1 = 0 to 1 = (Fig. 3).
Quadratic plane-strain elements were used for the elastic solid, and quadratic cohesive elements of non-zero thickness were used in the cohesive zone. As can be seen from Fig. 3, a combination of quadrilateral and triangular elements allows for a structured increase in the size of the plane-strain elements as one moves away from the vicinity of the crack tip.
The mixed-mode cohesive laws were implemented as user-defined elements. The cohesive elements had a length 2 equal to 5 × 10 −7 in the range 0 ⩽ 1 ∕ ⩽ 8.2 × 10 −3 . The mesh was then gradually increased to 5 × 10 −2 at 1 ∕ = 1. The height of the cohesive elements was equal to 5 × 10 −8 along the cohesive interface. Only positive values of were studied, so there was no issue with interpenetration.
Several potential-based mixed-mode cohesive laws were tested. Particular results are presented for the laws shown schematically in Fig. 4, and discussed in more detail in Appendix: an uncoupled trapezoidal law of [22], an uncoupled linear law [16,28], and the coupled Park-Paulino-Roesler (PPR) law [38,39].

Boundary conditions
The displacement components, 1 and 2 , are related to the singular field of Fig. 3 by [40]: where is the shear modulus,̄= in plane stress and̄= ∕(1 − ) in plane strain, is Poisson's ratio, and the magnitude of the stress intensity factors is | | = √ 2 + 2 . These displacement components are prescribed remotely on the boundary at = by means of a user-defined ABAQUS subroutine. The magnitude of | | is varied through incremental changes in 1 and 2 , such that is kept at the desired value.
The application of displacements that match those expected in an LEFM field does not, by itself, ensure that a -controlled stress field will be established. This requires an additional condition that both ∕ and ∕ are small enough. Although the geometry of Fig. 3 is the conventional one used to describe -fields in infinite bodies with semi-infinite cracks, it must be remembered that introduces an arbitrary length scale that will determine if the cohesive zone satisfies the small-scale conditions or not.

Results
The results presented in this section are divided into two main classes. In the first set of results, the loading is done in such a way that remains constant throughout the loading procedure. This is described as proportional loading. In the second set of results, the loading is done in such a way that changes during the loading procedure. This is described as non-proportional loading. . The value of̃∕ for this plot is equal to 0.01375, which satisfies small-scale conditions. The excellent agreement between the numerical results and the asymptotic field can be seen from this plot for both the opening and shear tractions. The field under these conditions extends to within about 0.01 from the crack tip, with the relationship between the cohesive length and the extent of the singular field being visible from Fig. 5(b). Fig. 5 provides what might be considered to be a classic understanding of LEFM: an inverse square root relationship between stress and distance from the crack tip, with a magnitude given by and , but which breaks down near the crack tip. 3 This verifies the ability of a cohesive-zone model to describe LEFM under appropriate small-scale conditions.
The variation of the phase angle, ( 1 ), with 1 , is illustrated in Fig. 6 with the same three cohesive laws (with different ∕ ratios) as the plots in Fig. 5, but with three different phase angles. As expected, tends to close to the crack tip in all cases, but generally deviates from this equality in the field. The exception is the special case of ∕ = 1, for which ( 1 ) equals for all values of 1 ∕ . This is because the ratio of the two stresses is equal to the square root of the ratio of the two modes of work in this law. Therefore, there is a special case agreement between ( 1 ) and within the field where the stresses must also scale with . For the other cohesive laws, the same agreement between the stresses and applies, but now the ratio of the stresses is not the same as the ratio of the square root of the work.
This point is emphasized in Fig. 7, which shows how the traction ratios, represented by the phase angle , vary with 1 . For ∕ = 1, the traction ratio is identical to the square root of the work ratios for a linear cohesive law. Therefore, = , both near the crack tip and in the field. For other values of ∕ , = only in the field. However, it should be remembered that it is at the crack tip where fracture occurs, and where one needs a measure of mode-mixedness that can be linked to LEFM. As discussed earlier, such a measure is provided by , which equals for an uncoupled law.  Similar conclusions can be drawn from calculations conducted using uncoupled laws with different shapes. Although this has been demonstrated before for beam-like geometries [16], here we show the results for a -field geometry using a trapezoidal law (described in Appendix A.1) for two values of mode-mixedness. Fig. 8 shows how the stress field evolves near the crack tip for = 45 o , and a peak traction ratiô∕̂= 2. The length of the fracture process zone is less than 1% and, in the -field zone ( 1 ∕ >10 −2 ), the normal and shear tractions are identical to the asymptotic field. In this case, the tractions of the uncoupled cohesive law are at their maximum values, established by their cohesive strengths, all the way to the cohesive crack tip, because neither law has entered the softening regime under the conditions for which the plot has been made. It should be emphasized that is small enough for LEFM to be valid, as can be seen from the stress field of Fig. 8. Fig. 9 shows how the phase angle ( 1 ) varies with 1 for = 45 o and 60 o . As before, it can be seen that the crack-tip phase angle, tends to . Away from the crack-tip region, there is no particular significance to this partition of work. However, it should be noted that, for these calculations, much of the -field is associated with the initial, linear portion of the traction-separation law. This means that, for the two cases with identical mode-I and mode-II cohesive laws, the laws look like linear laws with equal cohesive lengths. As discussed in connection with Fig. 6, this means that in the -field region the special case of ( 1 ) = is met.

Coupled cohesive laws
The results for the coupled cohesive law developed by Park et al. [38], which we refer to as the PPR law, are described in this section. Fig. 10 shows the normal and shear tractions ahead of the crack tip with = 45 o , and with values of and cohesive strengths corresponding to those used for Fig. 8. In this case, the cohesive-zone is fully developed, so the stresses at the crack tip are approximately zero. As with the uncoupled law, the cohesive-length scale, , is so small that the stresses are described by the asymptotic -field at distances greater than about 0.01 from the crack tip. Again, this confirms the ability of a cohesive-zone model to describe LEFM under appropriate conditions. The phase angle, , is plotted in Fig. 11 for three peak-traction ratios,̂∕̂, and two values of . These plots illustrate the effect of different parameters for the PPR cohesive law. A key difference between the results for this form of a coupled law, and the results for uncoupled cohesive laws, is that, in general, ≠ . The only situation in which = is the special case of =45 o , when the shear and normal laws are identical. A similar result that, in general, ≠ for coupled mixed-mode laws was found when several other coupled cohesive laws were explored, including those of Xu and Needleman [18], and Sørensen and Goutianos [41].    The reason for the discrepancy between and can be seen by a simple examination of the form of the equations. If, in general, = ( , ) and = ( , ), then the crack-tip phase angle, which from Eq. (2) is given by will generally depend on how varies with , and it is going to be path dependent. In particular, there is no reason why should be related to .

Non-proportional loading
In the previous section, we showed that = for uncoupled laws and proportional loading; but this identity was valid only for very special forms of coupled laws. In this section, we explore the effect of non-proportional loading on this relationship. Specifically, we do this by determining the evolution of the phase angle as the geometry is loaded to the same final conditions ( = 45 o ), but following two different loading paths, 1 and 2 , illustrated schematically in Fig. 12. Fig. 13 shows how the phase angle ( 1 ) varies with 1 for an uncoupled trapezoidal law at four discrete points along the two non-proportional loading paths, 1 and 2 identified in Fig. 12. It can be seen that the crack-tip phase angle, , always matches the applied value of , at all points during loading. Similar results were obtained for all the other paths and cohesive parameters that were explored.
The corresponding results for the PPR cohesive law are shown in Fig. 14. In this case, it will be remembered that was not equal to for proportional loading. Here the two parameters are in closer agreement for a trajectory that starts off dominated by mode-I. However, the two are even more divergent for the mode-II dominated trajectory than for the proportional trajectory, indicating clear evidence of path-dependence for the crack-tip phase angle. This path dependence of was confirmed as being S. Goutianos et al.  a general result for other paths and cohesive parameters for coupled laws. It should be emphasized that, in all cases, the total work at the crack tip remained the same. There was no path dependence to this quantity, as expected for potential-based laws. The path-dependency was only related to how the crack-tip work was partitioned between the two modes.

LEFM assumptions
Mixed-mode loading in an LEFM framework is completely described by the energy-release rate, , and the phase angle, . It is assumed that any small-scale deformation at the crack tip is uniquely described by these two parameters, which are both independent of the loading history. Therefore, in corresponding cohesive zone modelling, both the magnitude of the work done at the crack tip   Table A. 1, witĥ=2̂. during this deformation and the partition of this work into normal and shear components can be deduced uniquely from the two parameters. 4 Mixed-mode failure criteria used in LEFM analyses are all predicated on this concept of path-independence.
The use of potential-based traction-separation laws within a cohesive-zone framework, ensures the same total work is done against crack-tip tractions for any loading path under mixed-mode loading. Therefore, this class of cohesive law results in an agreement with one LEFM assumption: the energy-release rate does not depend on the loading history. However, not all potentialbased cohesive laws match the second assumption: the LEFM partition of the work at the crack tip can be described only in terms of and . The present paper confirms the earlier results of [16,27,28] that the LEFM assumption about the prediction of work is satisfied for uncoupled cohesive laws if̃≠ 0. 5 However, it is shown here that coupled cohesive laws generally result in a different partition of crack-tip work from that assumed by LEFM. Furthermore, while the total work is path independent, this partition of crack-tip work can be path dependent. This conclusion has been illustrated by the results presented in this paper, but it was also confirmed by testing other potential-based coupled cohesive laws from the literature [18,38,41], with proportional and non-proportional loading paths.

Implications for LEFM mixed-mode failure criteria
The implicit assumption behind LEFM mixed-mode failure criteria is that an interface separates when the energy-release rate, , exceeds a critical value, , which is identified as the toughness, and is a function of the phase angle:  ≥ ( ). This functional 4 This is rigorously correct only when the second Dundurs parameter,̃, is equal to zero. Wheñ≠ 0, LEFM cannot be used to partition the work done in deforming the crack-tip region into shear and normal components, although the total work is still given by the energy-release rate [33,42]. 5 LEFM requires an additional length parameter to represent the behaviour of uncoupled potential-based cohesive laws wheñ≠ 0 [16,27].  Table A. 1, witĥ=2̂. dependence of toughness on phase angle can take any form, including non-monotonic forms. However, in LEFM, the phase angle is defined only in terms of the geometry and the loads, and is path independent. Therefore, the toughness of an interface is implicitly assumed to be path independent.
In practice, most LEFM mixed-mode fracture tests are conducted under proportional loading, so that is constant throughout a test. An envelope of toughness is developed as a function of through a series of separate tests, each one exploring a different value of . With this approach, it would not matter if the actual crack-tip phase angle, , of the fracture process was incorrectly described by , a unique mixed-mode failure envelope would always be developed that described the experimental results. This failure envelope could then be used predictively in design, under the same assumptions of LEFM and proportional loading.
It would be relatively easy to develop a cohesive-law that describes such limited data. Both coupled and uncoupled laws could work; indeed, even a law not based on a potential function could work, if the issue of path dependence is not explored. However, only the uncoupled law would be consistent with LEFM assumptions. More detailed experimental studies might reveal path-dependence, violating LEFM, in which case coupled laws derived from a potential function or cohesive laws not derived from a potential function might be more appropriate.

Conclusions
Different types of mixed-mode, potential-based cohesive laws under small-scale conditions have been used to explore how the behaviour of the crack-tip region compares to the assumptions that underpin linear-elastic fracture mechanics (LEFM). It has been shown that the fundamental assumptions of LEFM are fully consistent with uncoupled, potential-based laws. For these types of law, not only is the work done against crack-tip tractions independent of the loading path and equal to the value of the -integral, but the partition of this work into the two orthogonal modes is also in agreement with LEFM assumptions. The crack-tip phase angle is equal to the phase angle of the surrounding -field if small-scale conditions are met.
Coupled, potential-based cohesive laws result in the work done against the crack-tip tractions being path-independent and equal to that given by the -integral (consistent with LEFM). However, the partition of this work into normal and shear components does not necessarily agree with that indicated by the surrounding -field, even under small-scale conditions. In particular, the crack-tip phase angle can be path dependent.
These results have implications for the interpretation of mixed-mode fracture experiments and design based on LEFM concepts. LEFM assumes that deformation at a crack tip is uniquely described by the -field. It also assumes that the local conditions for mixedmode crack advance are controlled by and , and, hence, mixed-mode failure is independent of the loading path. However, if the normal and shear deformation processes at the crack tip are coupled, these assumptions would generally be violated to some degree.
A full understanding of mixed-mode failure criteria requires path dependence to be explored. In the absence of any significant path dependence being observed experimentally, uncoupled, potential-based cohesive laws with suitable empirical mixed-mode failure criteria would seem to be adequate, and, perhaps, the easiest to implement numerically. In addition, the use of path-dependent functions or coupled laws would need to be validated to ensure they did not introduce stronger path-dependence than merited by the experimental results. Only if significant path dependence that needs to be modelled is observed experimentally, would it seem to be imperative to use a coupled law, or path-dependent cohesive laws.

Declaration of competing interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

A.1. Uncoupled cohesive laws
Two forms of mixed-mode uncoupled cohesive laws are used in this study. The first is a special case of linear laws, for which the tractions are linearly dependent on the displacements until failure. Mathematically, these are described by where and are the stiffnesses of the two modes, which need not be identical. The peak tractions were set high enough that fracture did not occur in this study. However, they can easily be added if fracture needs to be modelled explicitly. It should be noted that a physical manifestation of a linear-elastic cohesive law could be an interface bonded by compliant brittle elastic springs. However, from a modelling perspective, linear laws have the unique feature that the instantaneous cohesive-lengths [28], do not vary during loading. Furthermore, the simplicity of linear cohesive-laws means that the results of all calculations performed with them can be expressed in terms of only four non-dimensional parameters: where the first two terms describe the cohesive laws, and the second two terms describe the remote loading. and therefore the mixed mode linear uncoupled laws are based on a potential function. The second form of uncoupled law used in this study are trapezoidal laws [22]. The tractions for these laws increase linearly with displacement until the normal and tangential displacements are 1 and 1 ; at which point the peak tractions arêand̂, respectively. The tractions remain at these levels while the relevant displacements remain less than 2 and 2 , at which point they drop linearly to zero at = c and = c . These laws can be expressed in the range −90 o ≤ ≤ 90 o as where ⟨⋯⟩ are Macaulay brackets [43]. Macaulay brackets of the form ⟨ − ⟩ 1 are interpreted as being equal to 0 when < , or equal to ( − ) when ≥ . The non-dimensional presentation of results for these trapezoidal laws is slightly more complicated than for the linear laws, because of the additional parameters required to describe the laws. The problem is completely described by ten non-dimensional groups. There are the two loading parameters, | |∕(̄√ ) and , and eight parameters that describe the cohesive laws. The values of these eight parameters that are used in this paper are given in Table A.1.
Finally, it should be noted that what are termed as ''uncoupled'' mixed-mode cohesive laws are actually coupled through a failure criterion of the general form The non-dimensional exponents and are given by: .
(A.18) where 1 and 1 are the normal and tangential openings corresponding to the pure mode-I and pure mode-II peak tractions, respectively.
The problem of this paper is completely described by ten non-dimensional groups, including the two loading parameters, | |∕(̄√ ) and . The eight parameters used in this paper that describe the cohesive laws are given in Table A.2. The parameters are chosen in such a way so as to ensure that the shape of pure mode-I and mode-II cohesive laws are similar to the shape of the corresponding mode-I and mode-II uncoupled cohesive laws. They have identical peak tractions, critical openings and fracture energies as the corresponding uncoupled laws (Table A.1).