## Abstract

Structural integrity of composite materials is governed by failure mechanisms that initiate at the scale of the microstructure. The local stress fields evolve with the progression of the failure mechanisms. Within the full span from initiation to criticality of the failure mechanisms, the governing length scales in a fibre-reinforced composite change from the fibre size to the characteristic fibre-architecture sizes, and eventually to a structural size, depending on the composite configuration and structural geometry as well as the imposed loading environment. Thus, a physical modelling of failure in composites must necessarily be of multi-scale nature, although not always with the same hierarchy for each failure mode. With this background, the paper examines the currently available main composite failure theories to assess their ability to capture the essential features of failure. A case is made for an alternative in the form of physical modelling and its skeleton is constructed based on physical observations and systematic analysis of the basic failure modes and associated stress fields and energy balances.

This article is part of the themed issue ‘Multiscale modelling of the structural integrity of composite materials’.

## 1. Introduction

The potential of composite materials in providing high stiffness, strength and toughness in lightweight structural applications is accompanied by the richness in the mechanisms underlying those properties. Inadequate account of those mechanisms in design procedures can lead to wasteful use of material and unreasonable reliance on testing. Historically, however, many analyses of composite materials were inspired and motivated by experience with metals, which led to the use of homogenization procedures that did not allow explicit account of failure mechanisms at length scales that were obliterated by the homogenization. While the homogenized models accurately predict the deformational response of composites prior to initiation of failure mechanisms, such models do not handle well the progression of the mechanisms and the criticality they induce.

The high and increasing computational power in recent years has opened up possibilities of simulating microstructure-level failure events by numerical stress analysis. However, accurate prediction of failure response requires use of appropriate physics of failure in the simulation procedures. Even here, metals-based modelling concepts are likely to result in inaccurate predictions, as will be discussed later.

The exposition to follow will first give a brief account of the homogenized models proposed over the years to describe failure of composites. The inherent inadequacy of the models to incorporate the physics of composite failure will be pointed out. Efforts to remedy the lack of predictive capability of the models, while keeping the single scale of homogenized composites, will be scrutinized. A case will be made to develop physics-based structural integrity models for composites on their own without relying on metals-based concepts of multi-scale modelling. A schematic failure analysis process for composite structures will then be outlined.

## 2. Failure theories for homogenized composites

The first notable effort to describe failure of a homogenized unidirectional (UD) composite appeared in 1965 by Azzi & Tsai [1], which became known as the Tsai–Hill theory. Hill [2] in 1948 proposed a generalization of the yield criterion for isotropic metals to orthotropic metals by assuming six independent yield stresses: three for normal stresses in the three symmetry directions and three for shear stresses in the three planes of symmetry. It is noteworthy that while the von Mises criterion expressed in stresses is derivable from the distortional energy density, this is not the case for the Hill criterion. Thus, Hill’s generalization of the von Mises criterion is simply a mathematical generalization. Still, one can argue that it is relevant to yielding of metals. Hill, in fact, motivated his criterion from the need to describe yielding in metals that develop texture from sheet forming by rolling, resulting in orthotropic symmetry. The Azzi–Tsai adaptation of the Hill criterion to UD composites was motivated by assumed similarity of the rolling direction in metal sheets and the reinforcing direction in UD composites. The final equation, referred to as the Tsai–Hill criterion, was derived by equating the six yield stresses in the Hill criterion to the corresponding ‘strengths’ of a UD composite, and further assuming transverse isotropy in the composite cross-sectional plane. Because the UD composites used as layers in laminates are usually thin, the criterion is commonly used in its two-dimensional version, given by
2.1where *σ*_{1} and *σ*_{2} are the normal stresses in the fibre and transverse directions, respectively, *σ*_{12} is the in-plane shear stress, and *X*, *Y* and *S* are the strength values of *σ*_{1}, *σ*_{2} and *σ*_{12}, respectively. For unequal strengths in tension and compression, a variation of equation (2.1) is easily derived.

Although initial verification of equation (2.1) was indicated for the case of UD composites loaded in tension at various angles to the fibre direction, many cases later did not support the criterion.

The next noteworthy development of failure criteria for UD composites appeared in a paper by Tsai & Wu [3] in 1971 based on a formulation proposed by Goldenblat & Kopnov [4] in 1965. The proposed formulation is a scalar function expressed as a polynomial of all stress tensor components. The coefficients of the polynomial terms represent the strength constants. The Tsai–Wu modification of this function was its simplification to a quadratic expression that allowed an interpretation as a quadric surface using analytical geometry. Thus, the ellipsoidal surface of ‘failure’ of a UD composite under in-plane stresses took the form
2.2where indices *p* and *q* take values 1, 2 and 6, and the summation convention for repeated indices is implied. The index 6 on the stress components stands for shear, and is the same as 12 in equation (2.1). The coefficient terms in the equation represent strength values. Coefficients of the linear terms are non-zero when the strengths for positive and negative stress components differ. Thus, *F*_{6} is always zero, and for the same reason, *F*_{16} and *F*_{26} vanish. Of the remaining coefficients, *F*_{1}, *F*_{2}, *F*_{11}, *F*_{22} and *F*_{66} can be expressed in terms of the normal strength values in the fibre and transverse directions and the shear strength in the plane of the composite. Finally, the coefficient *F*_{12} can, in principle, be obtained by a biaxial test, but the non-uniqueness of this procedure is a source of ambiguity in the criterion. It can be argued that because the Tsai–Wu criterion, equation (2.2), is not connected to the specifics of the failure in composites, its applicability will always be limited by the uncertainty of determining the *F*_{12} value.

At this point, it can be said that the two criteria discussed above are severely handicapped by their inherent limitations: the Tsai–Hill criterion is motivated by an incorrect failure mechanism (yielding, which is not the failure mechanism in UD composites with a non-metallic matrix) and the Tsai–Wu criterion has no specifics of a failure mechanism, rendering it simply a quadratic curve-fitting framework.

Numerous efforts to modify the type of failure criteria exemplified by the two criteria discussed above, including applying these to the case of fatigue failure, have not produced consistent improvements [5,6]. Other ways to approach formulation of failure criteria, while still keeping the composite material homogenized, were initiated by Hashin [7] in 1980. These will be briefly summarized next.

Hashin [7] pointed out that the single ellipsoid represented by equation (2.2) led to physically unacceptable interactions between stress components in some cases. He suggested to introduce piecewise smooth surfaces to describe critical states and additionally to recognize that composite failure involving fibre breakage was governed by stresses differently from when the failure occurs in the matrix only. Assuming the two failure modes to be independent, Hashin proceeded to formulate criteria for them separately. Thus, for example, the fibre failure mode when the fibre axial stress is tensile was assumed to be given by 2.3Thus, the transverse normal stress was assumed not to affect the fibre failure mode, and the quadratic form of the failure criterion was retained.

For the matrix failure mode, Hashin [7] proposed the notion of a failure plane not intersecting fibres in a UD composite. Assuming the fibre direction stress not to have influence on this plane, Hashin proposed that the other stress components would determine the inclination of the plane. He outlined the difficulties involved in determining the angle of this inclination and proposed that some (unknown) extremum principle would govern it. Notwithstanding Hashin’s word of caution, Puck and co-workers [8,9] launched an elaborate procedure to develop failure criteria incorporating the matrix failure plane idea. The resulting so-called Puck failure criteria for a UD composite require determination of seven material constants.

Without conducting a detailed and exhaustive scrutiny of all proposed failure theories that operate on the homogenized description of composites, one can reasonably put forth the argument that the inherent limitation in such theories would still be the inability to incorporate the conditions for initiation of the first failure event and for the sequential development of subsequent failure events. These failure events necessarily occur at scales that have been obliterated in the composite homogenization. The final failure event associated with criticality (e.g. unstable progression of failure events or transition to another failure mode) cannot generally be analysed without delineation of the subcritical path of the prior failure events. These arguments point to an alternative approach to failure assessment of composites where resort is made to the size scales of the composite constituents and the configurations in which the constituents have been put together. The multi-scale approach to structural integrity of composites will be discussed next.

## 3. Physical modelling of failure: mechanisms

The failure theories for homogenized composites, discussed above, served their purpose when understanding of the physical mechanisms was not adequate and the computer power needed to simulate the initiation and progression of failure events in different failure modes was not available. Today, there is sufficient basis for conducting failure analysis with a multi-scale approach that connects the failure process at the governing scales with the structural scale response characteristics needed for design. What remains to do is to develop efficient models that capture the physics of failure without overly complicating the failure analysis to be used by designers. The first step in this process is to have a clear understanding of the underlying failure mechanisms. Because the failure mechanisms differ significantly depending on whether fibre failures are involved and whether the driving forces for failure are tensile, compressive or shear, the discussion to follow will be arranged to account for these aspects.

### (a) Fibre failure mode in tension

Fibre failure within a UD composite under imposed overall axial tension has been a subject of study for many years owing to the importance of this failure mode to the ultimate composite failure. Direct and *in situ* observations of the failure mechanisms have been made difficult by the limited applicability of microscopy. Early observations by scanning electron microscopy combined with ultrasonic measurements [10] and by fractography [11], both on carbon–epoxy UD composites, suggested a local failure process in clusters of fibres. More recent work using high-resolution computed tomography [12] provided more details of the fibre failures and reported statistics of the broken fibre clusters (figure 1). They reported that at low load levels, fibre breaks appear singly and in clusters of two neighbouring fibres, while close to final failure, clusters of four or more fibre breaks were found. They did not see larger clusters forming from accumulation of smaller clusters, as the load was increased, but instead found them forming independently at random locations.

The observations reported have focused on fibre failures, but have not sufficiently clarified the roles of matrix and the fibre–matrix interface in transferring load from the broken to intact fibres, except in trying to find evidence to support the proposed models for the load transfer.

### (b) Fibre failure mode in compression

Although the strength of a UD composite under axial compression is characterized as a fibre failure mode, as by Hashin [7], and subsequently by many, the failure mechanisms involved in this case depend significantly on the matrix behaviour. As described by Jelf & Fleck [13], the fibre failure in compression may be categorized as elastic microbuckling or plastic microbuckling, depending on whether the matrix stress–strain behaviour is linear or nonlinear. For the latter case, the compressive strength is predicted well by Budiansky’s kink band model [14], which has the fibre misalignment and matrix shear yield stress as parameters. Kyriakidis *et al.* [15] further verified the dependence of the compressive strength on fibre misalignment angle and generalized it to fibre waviness as the microstructure (defect) scale. The microstructural features of the kink bands and mechanisms of their formation were clarified by these authors. Their analysis found that in the presence of fibre waviness, localization of shear deformation occurs in the matrix. This forms bands and the flow of the matrix in the bands results in bending of fibres and eventual breakage, which results in the formation of the observed kink bands. A fairly comprehensive review of the role of various parameters on the compressive strength can be found in Waas & Shultheisz [16]. Lee & Waas [17] conducted experimental and numerical studies of the influence of a wide range of parameters such as fibre stiffness, fibre volume fraction and fibre misalignment on the compressive strength.

Berbinau *et al.* [18] analysed the role of the fibre failure itself in the criticality of failure in compression and showed that the earlier contention that the fibres fail by tension in bending within the kink band was not plausible.

### (c) Matrix and fibre–matrix interface failure in transverse tension

The cracks observed on loading a UD composite under tension normal to fibres have commonly been referred to as ‘transverse cracks’. The appearance of these cracks is typified by the images shown in figure 2 [19]. Owing to the brittle nature of such cracks (i.e. unstable growth), it is common to study these in a constrained environment, e.g. in a laminate, often a cross ply laminate, in order to have them stopped at the lamina/lamina interfaces. Furthermore, by applying cyclic loading of appropriate level, the crack formation can be slowed down for closer examination. The cracking images in figure 2 were obtained in cross ply laminates under cyclic transverse tension.

In the early work on composite failure, as described in common textbooks on introduction to composite materials [20,21], the failure initiation in transverse tension is attributed to fibre–matrix de-bonding. While observations seem to imply this, a closer analysis suggests that a failure mechanism in the matrix could be a precursor of the observed de-bonding. This will be discussed further in §4.

### (d) Matrix failure in transverse compression

Studies of failure in transverse compression of thin UD composites (lamina) are few owing to difficulties of *in situ* observations. The earliest observations of transverse compression failure can be found reported in Agarwal *et al.* [20] on thick UD composites that show failure on planes parallel to fibres but otherwise inclined to the loading direction. This seems to suggest an influence of the shear stress in the failure process. More recent observations of this failure mode have been reported in Gonzalez & Llorca [22], reproduced in figure 3. As seen in figure 3, the failure plane is inclined (in this case at 56°), confirming earlier observations. Their observations of the fracture surface showed the presence of hackles, supporting previous suggestions that local shear in the matrix plays a role in this failure mode.

### (e) Matrix failure in in-plane shear

Applying in-plane shear to a thin and flat UD composite causes difficulties of producing a uniform stress state owing to the influence from gripping ends. The preferred specimen is therefore a thin-walled tube in torsion with the fibres running in the circumferential (hoop) direction. Quaresimin & Carraro [23] and Carraro & Quaresimin [24] used this specimen and loading mode. Their microscopic observations of failure induced by cyclic in-plane shear confirmed earlier observations by Redon [25] (figure 4) and by Plumtree & Shi [26]. As seen in figure 4, microcracks develop in the matrix along planes inclined to the fibres in a UD composite subjected to in-plane shear. These cracks tend to turn and grow in the fibre direction, merging together to form ‘axial’ cracks.

### (f) Failure modes in combined loading

The individual failure initiation mechanisms described in the previous sections are governed by the appropriate local conditions. The criticality of the local conditions in each case will be affected when the loading modes are combined.

For fibre failure in tension, presence of the transverse stress (tension or compression) or the in-plane shear stress, or both, will aid or abate the formation of failed fibre clusters depending on how the local load transfer from a failed fibre to its neighbouring fibres is affected. Systematic experimental observations of these effects do not seem to be available.

In the case of axial compression in the presence of in-plane shear, Jelf & Fleck [27] reported that for a carbon–epoxy UD composite, produced as pultruded thin tubes, superposition of torque on axial compression did not change the mechanism of plastic microbuckling resulting in kink band formation. Although applying torque alone caused splitting parallel to fibres, when it was combined with axial compression, failure initiated as plastic microbuckling. Vogler *et al.* [28] conducted tests on a UD carbon–polyetheretherketone produced as flat specimens subjected to axial compression and in-plane shear. Their experimental observations indicated kink band formation to be the governing mechanism in compression, the effect of in-plane shear being rotation and bending of the fibres within the band. The cause of fibre breakage was found to be the axial compression.

For matrix failure under combined action of transverse tension and in-plane shear, the failure initiation mechanism to occur first will depend on whether the local stress field causes the critical conditions for fibre–matrix de-bonding or for producing cracks on planes inclined to the fibres. Observations by Quaresimin & Carraro [23] on glass–epoxy tubes with fibres in the circumferential direction subjected to simultaneous cyclic axial tension and torsion indicated that two regimes of interaction existed: for low-to-moderate values of in-plane shear, the dominating mechanism was fibre–matrix de-bonding, whereas for higher values of shear, the cracks forming along the fibres resulted from growth and merger of the cracks illustrated in figure 2. Plumtree & Shi [26] reported similar observations on flat UD coupons of carbon–epoxy loaded in off-axis tension.

Observations on UD composite failure under transverse compression were noted above. Combining this loading mode with in-plane shear has experimental challenges. There seems to be no data reported for this case.

## 4. Physical modelling of failure: multi-scale approaches

A general review of multi-scale approaches to composite materials can be found in [29]. Our focus here is to use the multi-scale methodology to support physical modelling, by which is meant to capture understanding of the observed failure mechanisms into models that have at the minimum two scales: a scale where the failure events occur and a scale where description of the structural failure could be given. In this context, the following subsections describe multi-scale approaches with respect to the individual failure modes. Section 4a describes an overall strategy for failure assessment.

### (a) Fibre failure mode in tension

The observed failure process in this case, as described in §3a, suggests that the first scale at which a failure event occurs is the fibre diameter. A broken fibre either exists from the manufacturing process or it results from loading as breakage at the weakest point. The next event is failure of fibres in the neighbourhood of a broken fibre caused by the stress enhancement induced by the first broken fibre. Observations indicate localized failure in clusters of fibres, one or more of which grow unstably, triggering ultimate failure of the UD composite. A refinement in the failure process is multiple failed fibre clusters interconnecting to a larger failed plane growing unstably to failure.

Most literature has focused on analysing conditions for formation of a cluster of broken fibres. The scale at which the stress and failure analysis is conducted consists of a representative region containing sufficient fibres to simulate a fibre-cluster failure. An example of this is in Nedele & Wisnom [30] where a representative region is modelled as an assembly of five concentric cylinders. This axisymmetrical model is illustrated in figure 5.

The Nedele–Wisnom analysis calculated the stress enhancement in fibres neighbouring the central broken fibre and the probability of failure in these fibres accounting for the statistical fibre strength distribution. The matrix in the model was assumed as either elastic or perfectly plastic and consideration was given to a frictional stress transfer from the de-bonded length of the broken fibre. No account was taken of cracking of the matrix in the interfibre region. This has been done in an unpublished work [31]. Leaving the details of the stress and failure analysis to [31], the main results are as follows. The broken fibre placed at the centre of the axisymmetrical model initiates matrix cracking or fibre–matrix de-bonding depending on whether the broken fibre exists before loading is applied or if it breaks under loading. The subsequent failure events consist of failure of fibres neighbouring the existing broken fibre. The stress enhancement in the neighbouring fibres is affected by the matrix crack emanating from the broken fibre end or as a result of kinking out of the de-bond crack. The stress enhancement on the neighbouring fibres extends over an axial distance and its intensity depends on the extent to which the matrix crack has grown, and if the broken central fibre has de-bonded from the matrix, then the extent of the de-bonding matters.

### (b) Fibre failure mode in compression

As described in §3b, the critical mechanism for this mode is kink-band formation, as observed in most glass–epoxy and carbon–epoxy composites. Budiansky & Fleck [32] review analytical treatments of kink bands, where the initial fibre misalignment angle appears as the microstructural parameter. Kyriakidis *et al.* [15] modelled the UD composite as alternating layers of fibres and matrix and introduced the axial misalignment as waviness, which for long wavelength approximates to the misalignment. This microstructural model clarified the roles of matrix properties as well as the sensitivity of the compressive strength (limit load) to the initial imperfections.

### (c) Matrix and fibre–matrix interface failure in transverse tension

It is important to recognize that in the absence of a crack, or a weak plane that can potentially form a crack, failure in a given material will be a ‘point process’, i.e. it will initiate at a material point when critical conditions for the particular failure mode are met at that point. Subsequent events will depend on the region in which the failed material point exists. If the material point lies in a brittle region, which has little capacity to deform inelastically (i.e. irreversibly), then the consequence of failure can be formation of a brittle crack, if appropriate conditions exist, such as a critical release of energy along a preferred plane. On the other hand, if the region is a polymer that deforms inelastically, then crazes or shear bands, or both, can form [33], leading eventually to ductile failure.

Before examining whether the failure ensued is to be brittle or ductile, it is appropriate to look at the stress field under which failure initiates. It is a common error to assume that a polymer that shows extensive inelastic deformation under a uniaxial test in the laboratory also behaves in-elastically within a composite with stiff reinforcement. Asp *et al.* [34] found that the behaviour of an epoxy polymer subjected to the so-called composite-like stress state was brittle, in spite of extensive ductility displayed under uniaxial stress. This gave a plausible explanation of the observed low strain to failure under transverse tension of UD composites and the insensitivity of this strain to modifications of the polymer morphology.

Asp *et al.* [35,36] pursued the line of thinking in [34] and developed a criterion for cavitation-induced brittle failure in the matrix polymer within composites. According to their work, in regions of matrix close to the stiff fibres in a UD composite, the stress state developed under remote transverse tension is triaxial with nearly equal principal stresses. Independent tests conducted on an epoxy polymer produced a critical value of the dilatational energy density, which when used for the same epoxy within the composite, predicted the composite failure stress well. The observed mechanism in the unreinforced polymer under hydrostatic tension was cavitation, and it was reasonable to assume that the expansion of the cavity formed becomes unstable at a material-specific value associated with the initiation of the instability.

A remarkable implication of the Asp *et al.* theory [35,36] is that what is commonly assumed to be fibre–matrix de-bonding may actually be a consequence of the brittle failure induced by the cavity ‘burst’ when the critical dilatational energy density is reached. Because this happens close to the fibre surface (at specific locations), where the largest dilatational energy density exists, the consequent fibre–matrix interface breakage appears as de-bonding.

The fibre–matrix de-bonding mechanism is also possible without the matrix cavitation failure as a precursor. This will be the case if favourable conditions for cavitation do not exist or if the fibre–matrix interface is sufficiently weak or is sufficiently weakened by defects. In that case, the radial tensile stress, and possibly this stress combined with the shear stress on the fibre surface, will break the fibre–matrix interface bonds, initiating de-bonding. For modelling purposes, it is difficult to know what interface failure properties to use, as these depend on the actual quality of the bond formed during the composite manufacturing process, which is not the same for model composites with one or few filaments that are used for experimental determination of the bond characteristics. The interface bond strength cannot be determined accurately by theoretical means. Several experimental methods have therefore been devised (see Zhandarov & Mader [37] for a review). These are either stress-based (strength) or energy-based (toughness) methods. As noted above, the characteristics of the fibre–matrix interface obtained by the experimental methods are commonly under simple conditions (e.g. single-fibre tests). Using the material constants thus obtained for interfaces that are under constrained conditions within composites would require caution. Applying the interface toughness criterion for evaluating initiation of fibre–matrix de-bonding also faces difficulties owing to the uncertainty of knowing the flaw size and its variability.

For ductile failure to occur under transverse tension of a UD composite, conditions must exist for inelastic deformation of the polymer matrix within the composite. Because the inelastic deformation requires shear stress to drive it, the distortional part of the strain energy density at a point must be sufficient to initiate what may be called ‘yielding’ (although this term originates from crystalline metal behaviour). For isotropic metals, the initiation of yielding is satisfactorily given by the critical value of the distortional energy density obtained experimentally for the given metal. Equivalently, the yield criterion for metals can be expressed in terms of the second invariant of the deviatoric stress tensor, as is the case for the von Mises criterion. The initiation of yielding in polymers is governed by molecular phenomena that differ significantly from the dislocation motion underlying yielding of crystalline metals. Still, it is common to describe the onset of inelastic deformation in polymers by the approaches used for metal yielding. In contrast to metals, the inelastic response of glassy polymers displays pressure sensitivity, as discussed by Rottler & Robbins [38]. This has prompted modifying the metal yield criteria by including the hydrostatic stress, e.g. by adding to the threshold of the octahedral shear stress *τ*_{0} a constant *α* times the mean pressure *p*, as
4.1where *p* is the average of the three principal stresses, *p*=−(*σ*_{1}+*σ*_{2}+*σ*_{3})/3.

Equation (4.1) is the modified von Mises yield criterion, which in energy terms states that the dilatational energy density contributes as well to the shear-driven onset of inelastic deformation in polymers. Additionally, temperature and strain rate are also found to affect the inelastic deformation [39].

In a polymer matrix within a UD composite, the stress triaxiality is generally high except in resin-rich regions. The inelastic deformation will thus tend to occur away from the fibre–matrix interfaces. Once initiated, the inelastic deformation can lead to shear banding before crack formation. Estevez *et al.* [40] have studied the crack formation process in glassy polymers by considering the competition between shear banding and crazing. Based on their study, it can be stated that the role of the distortional part of the strain energy density at a point is to localize inelastic deformation in shear bands, whereas the dilatational component is responsible for cavitation leading to craze formation, craze widening and breakdown of craze fibrils. The mix of the two energy components determines the ease or difficulty of ductile crack formation in the matrix within the composite.

Models for ductile crack formation in polymers are complex and require many material constants for implementation. Huang & Talreja [41] have shown that the Rice–Tracey ductile fracture model [42] is capable of predicting ductile cracking in polymers with high accuracy. The Rice–Tracey model assumes that pre-existing microvoids (from defects or inclusions) grow within plastically deformed material to a critical size, at which point neighbouring microvoids coalesce and cause fracture. The microvoid growth is controlled by the stress triaxiality and equivalent plastic strain, given by
4.2where *σ*_{m} and *σ*_{p} are the mean stress and the von Mises equivalent stress, respectively, *ε*_{p} is the equivalent plastic strain, and *R* and *R*_{0} are the current and initial void radii, respectively. The lower limit of integration is often taken as zero for simplicity. In implementation, the damage variable is used instead of the void size. When *D* reaches a critical value *D*_{c} at the considered material point, a crack is assumed to occur or to pass through that point by void coalescence. Rice & Tracey [42] calculated the constant *α* to be 0.283 under assumptions of spherical voids and an infinite medium. The critical void growth ratio *D*_{c} is believed to be a material constant that does not change with geometry or loading conditions. However, it cannot be directly measured. Instead, a calibration procedure is carried out to obtain its value by matching the simulation-predicted results with experimental measurements for either a smooth or pre-cracked specimen under selected loading conditions. The value of *D*_{c} thus obtained is then used to predict the ductile fracture of the same material under other geometries and loading conditions.

### (d) Matrix failure in transverse compression

As noted in §3d, observed failure of a UD composite under transverse compression indicates that failure occurs on a plane inclined to the loading direction. This prompts the suggestion that the failure mode is a manifestation of the matrix yielding in compression governed by cohesion and friction acting on the plane. The criterion for this type of yielding is the classical Mohr–Coulomb criterion, expressed by
4.3where *τ* is the shear stress at yielding acting on the critical plane, *σ* is the normal stress on that plane, and *c* and *ϕ* are material constants representing cohesion and friction angle, respectively. This criterion is similar to equation (4.1), if *c* is considered yield stress in pure shear and *ϕ* is viewed as the effect of hydrostatic stress. The angle made by the inclined failure plane with respect to the normal to the compression loading direction is given by
4.4In a UD composite, when compression is applied normal to the fibres, the inclination of the failure plane will be altered by the triaxility of the local stress state [22,43].

The formation of the failure plane in a UD composite involves progression from yielding at the most favourable point to connectivity of the yield (failure) planes at the subsequent points of yielding. This connectivity is supposedly by bands in which ductile deformation and failure mechanisms take place. These aspects cannot be treated in a homogenized composite, but require representation of the microstructure for detailed local stress field calculations. Gonzalez & Llorca [22] have presented an approach of this nature. They include in their analysis also the effect of fibre–matrix interface failure via an assumed cohesive zone model.

### (e) Matrix failure in in-plane shear

The stress–strain behaviour of UD composites under pure in-plane shear is found to be nonlinear. Part of this nonlinearity is attributed to the inelastic deformation of the matrix and the other part to the formation of the type of cracks shown in figure 4. Unless defects in the matrix lie with potential weak planes inclined to the fibre direction, assuming these cracks to form owing to the tensile stress acting normal to those planes would not be reasonable. Puck & Shurmann [44] offered, however, the explanation that these cracks form at 45° to the fibre direction under the action of the tensile principal stress, assuming a pure shear stress in the matrix between the fibres. Actual experimental evidence, such as in figure 4, shows these cracks to be not at 45° and not straight. These cracks are curved (‘sigmoidal’, as described in [26]), multiple, and tend to link up near the fibre surfaces, eventually forming a wavy crack (described in observations as a transverse crack) with its plane running overall parallel to the fibres. Along the lines of Puck & Shurmann [44], Carraro & Quaresimin [24] assumed the shear-induced crack to be formed by the largest local (tensile) principal stress, which they calculated numerically. The direction of the local maximum principal stress differed from 45° to the fibre axis, assumed in [44] because of the triaxiality of the local stress field.

The assumption of a nucleation (fracture) plane in polymers warrants closer examination. Contrary to metals, where specific crystalline planes exist, and can be expected to nucleate fracture planes under appropriate stress conditions, in polymers no such planes exist. To form a fracture plane, the molecules will need to be stretched beyond their breaking point in a manner that breakage points are roughly aligned along a plane. Because the observed shear-induced cracks do not appear to be aligned with the plane of maximum shear, assuming a ‘stretch and break’ process, governed by tension, is reasonable.

### (f) Failure modes in combined loading

Having considered the individual failure modes in single loads, i.e. tension or compression in fibre or transverse directions, or shear in the plane of a UD composite, the next area of inquiry is concerning failure when these loads are combined. The combined effect is perceived to be an ‘interaction’, meaning mutually enhanced degradation (or, exceptionally, strengthening).

The notion of interaction in combined loading has its history in yielding of metals. It is conceivable that each loading mode applied separately, or two or more loading modes applied together, may contribute to energy needed to drive the mechanisms underlying yielding. These mechanisms are basically related to dislocation mobility, and therefore the relevant strain energy density involved is distortional. This is a key reason for the von Mises criterion’s ability to successfully describe the combined effect of the normal and shear stresses. The ensuing quadratic expression of the criterion indicates the nature of the combined action of the individual stress components in initiating yielding.

For combined action of the stress components in contributing to the failure of a composite, there is no fundamental justification similar to that for metal yielding. First of all, there is no single failure mode in composites such as metal yielding to warrant a quadratic combination of the partial effect of each stress component. The conditions governing each failure mode are different, as discussed above. Therefore, the only reasonable course of action in describing the combined effects is to consider each failure mode separately and analyse how the individual effects combine for that particular failure mode. This will be attempted next.

A general remark before considering the combined effects of loading modes is in order. Because each loading mode generally induces a separate failure mechanism, it would be convenient and appropriate to consider a given failure mode as being driven primarily by one loading mode, denoted as the ‘dominant’ loading mode, and any other loading mode that is acting simultaneously with this loading mode as the ‘modifying’ loading mode. Thus, for example, the remotely applied tension is the dominant loading mode for tensile failure of fibres and an in-plane shear would be the modifying loading mode for this failure mode.

Let us begin with the fibre failure mode in tension, discussed in §§3a and 4a. The essential feature of this failure mode is the local load sharing from one failed fibre to its neighbouring intact fibres. The properties of the matrix and the fibre–matrix interface as well as the microstructural parameters such as the interfibre spacing play roles in this load transfer process. If remotely applied transverse tension (or compression) and in-plane shear stress are combined with the axial tension, then these additional loads will affect the local load transfer by changing the local stress field. Without fully analysing these modifying effects, it is commonly assumed that the transverse load (tension or compression) has no effect and that the in-plane shear contributes to the fibre failure mode in tension in the form of a quadratic interaction, as in equation (2.3). In fact, this equation also suggests that if the in-plane shear is the dominant loading mode, then the mechanisms of shear cracks leading to transverse crack formation, discussed in §§3e and 4e, will be modified by the imposed axial tension in the same way. Obviously, this assertion is not supported by any analysis at the microstructural level.

Consider next the fibre failure mode in compression, discussed in §§3b and 4b. Although different mechanisms underlying this failure mode are possible, in the common case of glass–epoxy and carbon–epoxy, the mechanism involved is plastic microbuckling leading to kink-band formation. The roles of the local shear stress and fibre misalignment have been identified as the important factors governing this mechanism. If transverse normal stress (tensile or compressive) and the in-plane shear stress are applied in addition to the axial tension, then their effects must be considered in terms of any changes induced in the local shear stress and in the fibre misalignment. Experimental studies have focused on combining the axial compression with the in-plane shear [27,28]. In those experiments, both these studies found that the fibre failure mode in compression remained unchanged by the additional application of the in-plane shear. If this is the case, then the contribution of the in-plane shear can be analysed in terms of the modification induced in the local shear associated with the kink-band formation. It must be realized, however, that if the applied axial compression does not trigger plastic microbuckling, then increasing the in-plane shear would likely result in shear cracks followed by transverse cracks, the mechanisms that come into play when the in-plane shear alone is applied. The role of the axial compression in these mechanisms is as yet not clear.

As far as failure in the matrix polymer is concerned, all combinations of loads applied—tension and compression normal to fibres and in-plane shear—induce mechanisms of yielding and crack formation. In going from the unreinforced polymer to the same polymer within a UD composite, care must be exercised in accounting for the stress triaxiality induced by the presence of fibres and the potential failure surfaces at the fibre–matrix interfaces. In theories developed for homogenized composites, such as by Puck & Schurmann [8], these aspects cannot be accounted for directly. Various assumptions and curve-fitting parameters must be introduced related to the fracture planes within the matrix between the fibres. Although such theories facilitate simple procedures for design, certain fundamental objections to their validity remain. For instance, what triggers the formation of the fracture planes within composites where the local stress fields are essentially triaxial? In other words, if no specific weak planes exist beforehand in the matrix, then what is the mechanism involved in getting to the flat planes of fracture from discrete failures at points? It is true that fracture planes are observed in experiments, but closer examination of the planes often suggests some precursor mechanism(s). For instance, in the case of compression normal to fibres, the observed hackles near the fibres on the inclined fracture planes indicate the role of the local stresses in initiating discrete failures.

## 5. Beyond unidirectional composite failure

UD composites form the building blocks in many composite structures, e.g. in stacked configurations in multidirectional laminates. Failure ensued in a single layer is often the beginning and not the end of the structural failure. In going beyond UD composite failure, two aspects are noteworthy: one, that the UD composite failure within a layer is necessarily under combined loading, and two, that this failure is constrained by the presence of the other differently oriented layers. When failure in a UD composite lying within a laminate occurs, it is not an ultimate failure in the sense of breaking apart as separate pieces, but rather as initiation of a crack that has its plane aligned with the fibre direction in the composite layer. This failure, described in the early literature as ‘first-ply failure’, has a progression consisting of multiple parallel cracks, which increase in number, reducing their mutual spacing, until a saturation state is reached. Failure in other plies progresses similarly, in a sequence determined by the criticality of their stress states.

Although investigated thoroughly over the years, and familiar to most composite researchers, the UD composite cracking process in a laminate was briefly outlined above to put in perspective the theories and models discussed in previous sections concerning UD composite failure. Going beyond these theories to treating composite laminate failure has additional challenges. Figure 6 illustrates the nature of these challenges by displaying the range of failure modes, within plies and between plies. The images in figure 6 pertain to a cross-ply laminate tested in tension–tension fatigue to the point when all of its ‘subcritical’ failure events have taken place, leaving the axial plies to fail in what was described above as fibre failure mode in tension. The subcritical failures are marked in the X-ray radiograph in the figure and for clarity are illustrated in a three-dimensional sketch on the side. Images at the bottom of the figure show the multiple ply cracks (left) and their diversion into the ply interfaces is focused on in the image to the right.

Until significant coupling of the transverse cracks through the interfaces occurs, the individual plies retain their load-bearing capacity in spite of sustaining a multitude of cracks. The displacement of the crack surfaces under loading changes the average deformational response of the laminate, however. This response has been treated fairly generally in Talreja [46,47], and since then by many researchers. Our focus here is failure in the sense of loss of load-bearing capacity, which for laminates degrades significantly when delamination of the plies disrupts the initial load sharing among the plies with perfectly bonded ply–ply interfaces. Assessment of this requires a fracture mechanics methodology for analysing advancement of multiply connected cracks. Such a methodology is as yet unavailable. Current efforts using cohesive zones face uncertainty of determining the properties (parameters characterizing the de-cohesion process) by experimental means.

## 6. A laminate failure assessment methodology

While the physical modelling of failure in composites has still challenges to meet, a roadmap to reaching the goal can be outlined. Such a map is laid out in figure 7. As indicated there, the first step is to characterize the microstructure of the composite as it results from the manufacturing process used. The characterization of the microstructure is to be at a scale needed for the failure analysis. Thus, depending on which failure mode is being analysed, e.g. a fibre failure mode in a UD composite, the choice of the microstructural region will be accordingly. In each case, the microstructural region must also include the appropriate manufacturing defects (e.g. fibre misalignment) within the region. This region, often called the representative volume element (RVE), is denoted here as real initial material state (RIMS) to capture the fact that it represents the ‘real’ material state as produced at the end of the manufacturing process, i.e. ‘initially’, when the composite is placed in the service environment. The realization of RIMS is labelled as RVE-1 and RVE-2 at the lamina and the laminate levels, respectively. The stress and failure analysis of the appropriate RVE will generate the conditions (criteria) for the particular failure mode. The failure conditions for lamina failure modes will enter the RVE-based analyses of the subsequent failure modes in a laminate.

## 7. Concluding remarks

A scrutiny of the fundamentals of the common failure theories advanced over the past many years reveals the inherent deficiency of the theories to treat the range of failure mechanisms observed. This deficiency is rooted in homogenizing the composite microstructure and thereby losing the ability to explicitly analyse the initiation and progression of the failure events involved. Current capabilities of observing the composite microstructure before, during and after failure, aided by the computer power to simulate the failure process, calls for a different approach. Such an approach has been discussed here. It has been motivated by the detailed physical understanding of the individual failure modes operating in UD composites under different imposed elementary loading modes and their combinations. The models to treat the failure modes have also been discussed.

The current state of composite failure assessment has advanced to the stage that UD composite failure can be simulated well, although some improvements can still be made. Greater challenges lie in dealing with failure in composite laminates, and subsequently in composite structures. Hurdles are mainly in treating intralaminar cracks that have diverted into the lamina interfaces, and the interconnection between these cracks that results from growth in the interfaces. The load-bearing capacity of a laminate with this type of damage is yet to be determined reliably. Current trends in treating this problem by the cohesive zone models are subject to uncertainty owing to the non-uniqueness of the material parameters (strength and fracture toughness) entering such models.

## Competing interests

The author declares that there are no competing interests.

## Funding

I received no funding for this study.

## Footnotes

One contribution of 22 to a Theo Murphy meeting issue ‘Multiscale modelling of the structural integrity of composite materials’.

- Accepted March 4, 2016.

- © 2016 The Author(s)

Published by the Royal Society. All rights reserved.