1. Introduction

Strength of materials is one of the oldest fields of study in solid mechanics. An early text on the subject by Morley [1] in 1920 defines strength of materials as “...study of the distribution of internal forces, the stability and deformation of various elements of machines and structures subjected to straining actions.” In his classical review of the history of strength of materials, Timoshenko [2] describes the early strength theories for selecting “working stresses for the various cases of combined stresses”. At that time, the concern was mostly for metals that displayed an elastic limit called “yield- point stress”, or simply, “yield strength”, which was regarded as a material property. Theories were formulated based on considerations of stress tensor invariants or strain energy density and were cast as criteria in terms of that material property. Strictly speaking, these theories were hypotheses to be tested against test data. The graphical representation of the criteria, e.g., the von Mises ellipse in the two-dimensional principal stress space, which became known as a yield envelope, is taught in foundational courses in strength of materials. As the field of materials has grown from isotropic metals to anisotropic solids such as fiber reinforced composites, various generalizations of the original yield criteria have been proposed. At the same time, our understanding of what limits the load-carrying capacity of a material has improved due to advances in observational tools and in computational techniques. We can now reveal mechanisms of “first” failure initiation at the scale of a few microns by high-resolution X-ray computed tomography (X-CT). The strength theories have, however, not kept pace with this knowledge and the gap between mechanics descriptions of “working stresses” and physics of failure mechanisms remains wide.

There are significant advances being made for crystalline materials to translate the knowledge at the atomistic level to the measured yield strength by analyzing slip and dislocation processes [3]. This material property can be different in tension and compression and can be anisotropic. Among the first treatments of anisotropy in yield strength was that by Hill [4] who noted that at large plastic strains and in processes such as rolling, drawing and extrusion, metals develop a preferred orientation due to crystalline plane alignment. A metal sheet can then be viewed as having three orthogonal planes of symmetry as illustrated in Fig. 1. Hill assumed six values of the yield strength in this case: three corresponding to normal stresses in the three symmetry directions and three corresponding to shear stresses in the three symmetry planes. It is important to note that all six values are manifestations of the same physical mechanism of yielding. Hill [3] generalized the von Mises yield criterion for isotropic metals to the orthotropic metals by a straightforward mathematical generalization as follows.

Fig. 1.
figure 1

Illustration of planes of symmetry in a metal plate drawn in the rolling direction aligned with the x -axis.

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The von Mises criterion expressed in stress components is given by,

$${\left({\sigma }_{xx}-{\sigma }_{yy}\right)}^{2}+{\left({\sigma }_{yy}-{\sigma }_{zz}\right)}^{2}+{\left({\sigma }_{zz}-{\sigma }_{xx}\right)}^{2}+6\left({\tau }_{yz}^{2}+{\tau }_{zx}^{2}+{\tau }_{xy}^{2}\right)=2{A}^{2},$$
(1)

where the constant A is the yield strength in any direction. A mathematical generalization of this polynomial gives

$${F\left({\sigma }_{yy}-{\sigma }_{zz}\right)}^{2}+{G\left({\sigma }_{zz}-{\sigma }_{xx}\right)}^{2}+{H\left({\sigma }_{xx}-{\sigma }_{yy}\right)}^{2}+2\left({L\tau }_{yz}^{2}+{M\tau }_{zx}^{2}+{N\tau }_{xy}^{2}\right)=1.$$
(2)

The six constants F , G, H , L, M , N can be related to the six values of the yield strength: X , Y , Z in the x, y, z symmetry directions, respectively, and R, S, T relations [4]. in the yz, xz, xy symmetry planes, respectively, by the following

$$\begin{array}{c}\frac{1}{{X}^{2}}=G+H, \frac{1}{{Y}^{2}}=H+F, \frac{1}{{Z}^{2}}=F+G, \\ 2F=\frac{1}{{Y}^{2}}+\frac{1}{{Z}^{2}}+\frac{1}{{X}^{2}}, 2G=\frac{1}{{Z}^{2}}+\frac{1}{{X}^{2}}+\frac{1}{{Y}^{2}}, 2H=\frac{1}{{X}^{2}}+\frac{1}{{Y}^{2}}+\frac{1}{{Z}^{2}},\end{array}$$
(3)
$$2L=\frac{1}{{R}^{2}}, 2M \frac{1}{{S}^{2}}, 2N\frac{1}{{T}^{2}}.$$
(4)

It is noted that the isotropic yield strength A is not related to the six realizations of the yield strength for the orthotropic case. It is conceivable that this connection can be established via atomistic modeling, but this author is not aware of such work.

In a fiber reinforced composite material where all fibers are straight and parallel, three orthogonal planes of symmetry exist like in the metal sheet in Fig. 1. Motivated by this observation, the Hill formulation of yielding in orthotropic plates described above was adapted to the strength of unidirectional composite materials [5], commonly referred to as the Tsai–Hill strength criterion.

This criterion is expressed as

$$\frac{{\sigma }_{xx}^{2}}{{X}^{2}}+\frac{{\sigma }_{yy}^{2}}{{Y}^{2}}-\frac{{\sigma }_{xx}{\sigma }_{yy}}{{X}^{2}}+\frac{{\tau }_{xy}^{2}}{{T}^{2}}=1,$$
(5)

where only the stress components in the xy plane are considered and the condition of rotational symmetry about the x axis, i.e., isotropy in the yz plane, is assumed.

Notwithstanding the inappropriate labeling of the theory as Tsai–Hill, it did not show good agreement with test results. This example illustrates the inherent inadequacy in the phenomenological failure theories when the physics underlying the failure process is either not accounted for or is incorrectly addressed as in the Tsai–Hill case.

The objective of this paper is to review key features of the failure processes in UD composites to reveal the nature of the complexities involved. Following this, arguments will be put forward to question the concept of “strength” as a representation of the failure criticality in composite materials. A mechanisms-based scheme for characterization of failure in terms of the first failure event, the subsequent failure events, and the final limiting event of load-carrying capacity will be proposed. The traditional strength of materials approaches that are legacies of metal yielding will be argued as not being the right direction for progress in the composites field.

2. Failure in Unidirectional (UD) Composites

The technological composites that have huge impact on aircraft and spacecraft structures, wind turbines, automotives, and more, consist of polymers reinforced with fibers of glass and carbon in one direction and stacking of these UD composites in different orientations to create laminates. The failure of single layers within a laminate sets limits to the “working stresses” in structures subjected to general loading, which can be combinations of normal forces, bending moments, and torsion. The problem of structural failure is cast as the “strength” of UD composites under combined local normal and shear stresses imposed on the boundary of a representative volume. In cases where the UD composite layers are thin, called plies, plane stress assumption allows one formulating failure criteria using three components of stresses, namely, normal stress along fibers, normal stress transverse to fibers, and in-plane shear stress. In the coordinate system shown in Fig. 1, these stresses are σxx, σyy and τxy. While the sign of the shear stress is of no consequence (except for kink band formation, to be discussed later), failure is highly sensitive to the sign of normal stresses in the longitudinal ( x ) and transverse ( y ) directions of fibers. In the following, we shall review the failure observations in each of the five possible single component cases and describe how the underlying failure mechanisms are different in each case. We shall then discuss the hitherto taken approaches and examine these in the light of the failure mechanisms observed. Finally, a mechanisms-based modeling approach will be advocated.

2.1. Tension along the fibers (σxx > 0)

In the early years, observations of failure in longitudinal tension were mostly on fracture surfaces of broken specimens because the faces of specimens did not reveal any useful information. The tensile strength values showed a large scatter and trying to find a cause of that was the focus of most studies. One study compared the scatter in the strength of dry carbon fiber bundles with that for composites made with the same fibers in an epoxy polymer cured in two conditions, resulting in two interfacial bond qualities [6]. The scatter was least in dry bundles and most in well-bonded composites. The authors found that in the well-bonded case a cluster of fibers that they described as “sub-bundles” failed before the final failure which occurred by interconnection of the sub-bundles along sheared longitudinal planes. Another study [7] examined fracture surfaces of broken UD composites and confirmed the findings of [6] while noting that the small zones of fiber failures formed independently of one another. These observations suggested the local nature of the failure process and gave rise to the concept of “local load sharing” in composites versus “global load sharing” in dry fiber bundles. It was thought that in the local load sharing the first broken fiber induced stress concentration on the neighboring fibers increasing their probability of failure. Several models for local load sharing were proposed along with statistical models of fiber failure to predict the probability of failure of a composite, e.g., [8].

In recent years, high-resolution techniques such as X-CT have revealed more details of the tensile failure process. In [9], UD composite model containing 125 quartz fibers in an epoxy matrix was used and reported direct observations of the fiber failure sequence. Figure 2 taken from [9] shows the formation of single fiber failures, called singlets, at low stresses followed by failure of two neighboring fibers (doublets), and multiple fiber failures ( k -plets) leading to a critical cluster of fiber failures that grows unstably to failure.

Fig. 2.
figure 2

X-ray radiograph of a quartz fiber reinforced epoxy composite at increasing tensile stress showing the sequence of fiber failures [9].

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In [9], the authors introduced the notion of a core region of a composite that is like a penny-shaped crack whose unstable growth causes final failure. Then, they estimated the size of the core by using a brittle fracture criterion. The predicted number of fibers in the core was in the ballpark of the observed number. While the model in [9] is quite approximate because of the assumption of a planar crack, it points to an important aspect of the failure process, which is that discrete fibers in a critical cluster have to be interconnected for unstable growth of the cluster. This realization suggests accounting for the failure in the matrix and at the fiber/matrix interface in analyzing the formation and growth of the broken fiber cluster. Observing the matrix cracks and interfacial debonds is not easy in the high-resolution X-CT techniques and the interpretation of the observed images has to be aided by computational simulations to get a fuller picture of the failure events. To summarize, while prediction of failure in UD composites under longitudinal tension is still under development, the failure mechanisms can be identified as initial random fiber failures, followed by an interactive failure process of random fiber cluster formation, whose random growth to a critical cluster size causes the final failure. Under the applied remote stress σxx the stress value at which final failure occurs, described as tensile “strength”, is a quantity of significant scatter.

2.2. Compression along the fibers (σxx > 0)

When a UD composite is subjected to longitudinal compression its failure displays drastically different mechanisms compared to the tensile loading case described above. The mean values of the two failure stresses are generally significantly different and the scatter in the failure stress is found to be much lower in compression than in tension. These observations intrigued early researchers who in the absence of knowledge of underlying mechanisms produced speculative models that did not predict the compressive failure stress well. The earliest most familiar model in this respect is Rosen’s model [10] in 1965 who contemplated two deformation modes under longitudinal compression as displayed in Fig. 3. The fibers deform out-of-phase in the extensional mode, while they deform in-phase in the shear mode. These modes of initially straight fibers give critical stresses to micro-buckling; the lower of the two according to Rosen’s model is equal to the shear modulus of the UD composite. This model prediction was found to overpredict the failure stress and the search for another modeling concept was set in motion.

Fig. 3.
figure 3

Extensional (out-of-phase) and shear (in-phase) micro-buckling modes postulated by Rosen [10].

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A remarkable departure from the Rosen model of fiber micro-buckling was proposed by Argon [11] in 1971 who recognized the role of misaligned fibers in initiating compressive failure. Argon envisioned a misaligned fiber to rotate under compression causing a local shear in the surrounding region. With increasing compression, the shear deformation of the local region becomes excessive and unstable at some point leading to a shear collapse. At this point, the local region forms a band of rotated fibers analogous to a kink band in crystals. Then, Argon derived an approximate relationship between the remote compressive stress and the shear strength of the matrix at the point of initiation of the kink band. This relationship is simply,

$${\upsigma }_{A}=\frac{{\tau }_{y}}{\phi }.$$
(5)

Kink bands became a focus of study for compression failure in fiber reinforced composites, and a prominent model proposed by Budiansky [12] was able to connect to the Rosen and Argon models. The Budiansky model addressed the kink band geometry depicted in Fig. 4 where an observed kink band [13] is also shown. As shown, a kink band is described by the width of the kink band W and two angles of rotation: the rotation of the kink band itself, given by the angle β, and the rotation of fibers within the kink band, angles ϕ and ϕs, where the angle ϕ is the initial fiber misalignment and the angle ϕs is the shear-induced rotation. Budiansky derived for an elastic composite with inextensible fibers the compressive stress to kink band formation as

$${\sigma }_{c}=G+{E}_{T} \mathrm{tan} \beta ,$$
(6)

where ET is the transverse Young’s modulus. For β = 0, the stress to kink band formation gives the Rosen prediction. However, the physical observations show non-zero kink band rotation angles.

Fig. 4.
figure 4

A kink band observed in a carbon-epoxy composite (left, Pimenta et al, [13]) and a sketch describing the geometry of the band (right).

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Based on post-buckling analysis of elastic composites, Budiansky [12] argued that the compressive strength was not sensitive to defects such as fiber misalignment. However, he showed that if the composite had significant plasticity in shear, then the initial fiber misalignment had a significant effect. Based on perfect plasticity in shear, he derived the ratio of the new (plasticity-based) failure stress, σs, to the elasticity-based failure stress σc , as

$$\frac{{\sigma }_{s}}{{\sigma }_{c}}=\frac{1}{\phi /{\phi }_{s}+1}.$$
(7)

Thus, if the misalignment angle ϕ is, say 2°, and the shear induced rotation, i.e., shear strain is γs = 0.002, then the compressive 5failure stress is predicted to be 1/18th of the elastic value σc.

Substituting σc from Eq. (6) into Eq. (7) and using G = τy/ϕs where τy is the shear yield stress, one obtains,

$${\sigma }_{s}=\frac{{\tau }_{y}+\left({E}_{T}/{\phi }_{s}\right)\mathrm{tan} \beta }{{\phi +\phi }_{s}}.$$
(8)

At β = 0, the compression failure stress becomes

At the point when the shear induced rotation is incipient, i.e., ϕs = 0, the elastic kink band formation stress becomes equal to the stress predicted by Argon’s model (see Eq. (5)). Note that the kink band rotation angle β appearing in Eq. (8), as been found by Budiansky and Fleck [14], hasn’t a large effect on the compression failure stress for most angles found in experimental observations. Also, the fiber misalignment angle is comparatively much more important. A recent review has summarized the large literature devoted to the study of fiber misalignment and waviness on compression failure [15].

For the purpose of this exposition, the point to note is that the mechanisms underlying failure in longitudinal tension and in longitudinal compression are widely different. The thrusts of the models concerned with prediction of the failure stresses in the two cases are also different. The statistical fiber failure models are the focus of tensile failure while fiber misalignment (waviness) is the parameter of consequence in compression failure. It is natural, therefore, that the interactions with other imposed stresses (transverse tension and compression, and in-plane shear) in the two cases will also be different. This aspect will be addressed next.

2.3. Tension along fibers combined with transverse normal and in-plane shear stresses

As described above, the dominant mechanism of failure under longitudinal tension consists of random single fiber failures followed by multiple fiber failures aided by local load sharing, progressing to clusters of fiber failures, culminating in a fiber failure cluster large enough to cause final failure by unstable growth. This failure process is not describable by deterministic rate equations and is dominated by the statistical nature of fiber failure. The stress that matters most in this process is the axial tensile stress in fibers which is governed by the applied tensile force in the fiber direction. Therefore, any influence of other applied loads, e.g., the transverse normal force (tensile or compressive) and the loading causing in-plane shear stress, must come by altering the axial tensile stress in the fibers.

To understand how superimposing a transverse tensile load affects the longitudinal tensile failure in UD composites by experimental observations is not easy. Under such biaxial loading the fiber/matrix interface tends to fail at low transverse tension resulting in axial splitting that separates a rectangular specimen in two or more sub-specimens of smaller widths. Using tubes with filament wound fibers in the hoop direction and applying internal pressure combined with axial tension leads to the same problem. Furthermore, the statistical nature of the fiber failure process and the localized stress concentration on fibers due to neighboring fiber breaks makes it difficult to track any influence of the transverse tension in the longitudinal fiber failure progression.

Superposition of a transverse compressive load on longitudinal tension has fewer practical difficulties than in the transverse tension case described above because the fiber/matrix interface can sustain a relatively high compressive stress. The transverse compressive stress induces inelastic deformation leading to failure of the matrix. The matrix deformation is widespread and is not restricted to the sites of fiber failures caused by longitudinal tension. The progression of mechanisms at the micro-level under combined longitudinal tension and transverse compression have not been observed with clarity, only the remote uniform stresses at the final failure event have been recorded. The longitudinal tensile strength, which has significant scatter, shows a slight reduction with transverse compression up to a point, beyond which the data suggest a change of the governing mechanism to the transverse compression induced failure.

Under an in-plane shear stress superimposed on longitudinal tension the testing issues are also severe but less troublesome than when transverse tension is superimposed. The shear stress on the axial plane parallel to fibers causes the fiber/matrix interface to fail. This stress, called the axial shear (versus the transverse shear stress), can affect the local stress transfer from a broken fiber to the neighboring fibers. The details of this effect are not easy to observe and are instead explored by numerical stress analysis involving some failure related assumptions. Experimental work is mostly limited to observations of the failure surfaces and the combined load level at failure. An example is shown in Fig. 5 where the axial failure stress is plotted against the shear stress [16]. The authors used tubular specimens of carbon/epoxy cross ply laminates to obtain the failure stresses at different ratios of axial to shear stresses. They used a neural network methodology to identify failure modes. As seen in the first quadrant of the plot in Fig. 5, there appears to be a small effect of the shear stress on the tensile failure mode, which as described above is fiber dominated. The other failure modes will be discussed in later sections.

Fig. 5.
figure 5

The experimental data of axial strength plotted against the axial shear strength for a carbon/epoxy composite [16]. The plot also shows two loading paths, BCT and AT, that produce strength values V and U, respectively, indicating insignificant loading path effect.

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2.4. Compression along fibers combined with transverse normal and in-plane shear stresses

The failure mechanisms under axial compression were described in Section 2.2. Accordingly, the axial stress to kink band formation is of main interest. Superposition of a transverse tensile stress on the longitudinal compressive stress gives similar testing difficulties as in the case of the longitudinal tensile stress described above. It is conceivable that the transverse tensile stress affects matrix yielding and thereby influences the fiber rotation leading to kink band formation. However, a direct observation of this effect has not been possible. The case of superimposing transverse compression on longitudinal compression has been possible to implement by laterally confining a circular cylindrical specimen as done in [17]. The improvement in the axial compressive strength with proportional transverse compression can be attributed partly to the prevention of axial splitting which tends to happen due to weak fiber/matrix interfaces such as in the E-glass/vinylester composite used in [17]. Superposition of in-plane shear stress on axial compression in UD composites was studied experimentally in [18].

Figure 6 taken from this work shows an example to a pre-failure kink band formed under axial compression. On applying in-plane shear, the fibers in the band rotate further and form the type of kink band illustrated in Fig. 4. This “interaction” between axial compression and in-plane shear results in reducing the axial compressive strength. The strength data plotted in the left quadrant of Fig. 5 show this combined loading effect. It was suggested by [18] and later by [19] that reversing the sign of the shear stress could improve the axial compressive strength by reducing the fiber rotation within a kink band.

Fig. 6.
figure 6

A kink band formed under axial compression before final failure [18].

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2.5. Tension transverse to fibers (σyy > 0)

When a typical laboratory specimen of a UD composite is loaded in tension normal to fibers, it fails abruptly at low σyy values. This prevents the detailed observation of the failure process. However, UD composites are the building blocks of a multidirectional laminate, and it is therefore important to know this failure process since it corresponds to the first failure of a lamina within a laminate, also known as “first ply failure”. Observations have shown that the first ply failure does not lead to the failure of a laminate. Instead, this failure takes the form of a transverse crack in that ply which on further loading develops more parallel cracks. In a well-designed laminate, the multiple transverse cracks in a ply can have limited effect on the load-carrying capacity of the laminate. However, the intense stress field ahead of these cracks leads to local separation of two plies at their interface, called delamination, which triggers further failure events culminating in the total failure of a laminate [20]. The problem of composite laminate failure is rich in complexity and has been an active field of research for decades [20]. Here, our interest is to examine the failure process that leads to the formation of transverse cracks under the stress σyy > 0.

The observed failure of a UD composite under σyy > 0 occurs at a strain that is much lower than the failure strain of the polymer matrix. This intriguing fact was treated in [21] and was found to be due to the local triaxiality of the stress in the inter-fiber region. It was shown that under a hydrostatic tension a glassy polymer such as epoxy suffers cavitation caused by volumetric expansion. The subsequent work formulated a criterion and successfully predicted the observed failure strains [22, 23]. Stress field computations showed that the favorable stress state for cavitation existed at points in the matrix near the fiber/matrix interface. The scale of cavities formed is, however, too small to allow an in-situ observation, so it is reasonably assumed that on unstable growth of a cavity will result a local debond at the fiber/matrix interface. Numerical studies have been conducted to predict cavitation induced failure [24, 25] that have also investigated the effects on this failure of the nonuniformity of fiber distribution in the UD composite cross-section caused by manufacturing processes. Taking as a starting point the presence of a debond crack, the studies have examined the link-up of such cracks to form a transverse crack [26, 27] in compliance with observation data shown in Fig. 7 [28].

Fig. 7.
figure 7

The transverse crack formation process, (a) merger of debond cracks, and (b) resulting transverse crack [28].

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2.6. Tension transverse to fibers combined with axial normal and in-plane shear stresses

Considering the transverse crack formation as the main mechanism, the tensile stress in fibers is not expected to have a significant effect on it. The same can be said about the compressive stress in fibers unless the fibers are misaligned and are about to form a kink band. The kink band formation, as described above, is a highly localized mechanism and it can enhance matrix yielding around misaligned fibers. This can promote ductile crack formation in the matrix. However, this effect, if present, will be difficult to observe due to the problems associated with biaxial testing of UD composites.

The effect of combining in-plane shear with transverse tension has been studied by numerical simulation [29]. An illustrative example from that study is shown in Fig. 8 where the combined stress state at initiation of the first failure event is plotted. As can be seen, the effect of superimposing axial shear stress on the transverse tensile stress has negligible effect for shear stress less than 10 MPa. The simulation showed that in this range of the shear stress the first failure mechanism was cavitation induced brittle cracking. Then, the first failure mechanism shifted to matrix yielding at higher shear stresses. The reduction of the critical shear stress at matrix yielding by the transverse tension is significant and it occurs by altering the local stress state in the matrix.

Fig. 8.
figure 8

Simulated combined transverse tension and axial shear loading in a UD composite at initiation of the first failure event [29].

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2.7. Compression transverse to fibers (σyy > 0)

Under compression transverse to fibers in a UD composite, formation of a failure plane has observed in [30]. That evidence is shown in Fig. 9. The inclined plane of final separation shows evidence of ductile failure of the matrix regions between fibers. The inclination of the failure plane was found to vary in the range 50-56° measured from the direction normal to the applied compressive stress. This suggests that there are other possible effects than the shear stress which for a homogeneous solid will be maximum at 45°. The authors [30] attribute the additional angle to the effect of decohesion of the fiber/matrix interface. Another numerical simulation study in [31] indicated the failure plane angle to depend on the fiber volume fraction and interface failure properties. It is to be noted that the interface properties are not determined as unique, independently obtained properties but are calibrated based on the model used.

Fig. 9.
figure 9

The failure plane observed in a UD composite under transverse compression (indicated by arrows). The angle of the plane measured from the vertical (thickness) direction is indicated [30].

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2.8. Compression transverse to fibers combined with axial normal and in-plane shear stresses

Like the case of tension transverse to fibers, the effect of axial stress (tensile or compressive) on the transverse failure in compression has not been detected. The in-plane shear stress on the other hand is expected to affect the transverse compression failure via its effect on the matrix yielding. Experimental failure stress data have been reported in the literature in different combinations of transverse normal stresses and in-plane shear stresses. A recent paper [32] collected such data for four different UD composites and compared predictions by four different strength theories. This comparison is shown in Fig. 10. The details of the four theories are not important for our purpose here, only that the predictions differ significantly based on the assumptions made in these phenomenological theories. Here, attention is drawn to the second quadrants of the four plots where the transverse compression is denoted by the negative values of σ2 and the in-plane shear stress is denoted by σ12. Note that a continuous curve is predicted by the Hashin theory that does not involve the inclination of the failure plane, while the other three theories each produce two piece-wise continuous curves by using an additional fitting constant given by the failure plane consideration. This aspect will be discussed later.

Fig. 10.
figure 10

Failure envelopes described by four failure theories in combined transverse normal stress (σ2) and shear stress (τ21) compared with data for (a) E-glass/LY556, (b) AS4/55A, (c) T800/3900-2, and (d) IM7/8552 [32].

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2.9. In-plane shear stress, (τxy ≠ 0)

Inducing a uniform state of shear stress in a UD composite faces practical difficulties. Various methods were proposed, and strength values obtained by commonly used methods gave different results [33, 34]. One of the testing problems is failure initiating from grips. To avoid this, the Iosipescu test was developed, but it does not generate a pure shear stress state [35]. For producing pure shear, the preferred specimen is a thin-walled tube in torsion with the fibers running in the circumferential (hoop) direction. This specimen geometry was used in [36, 37] to perform microscopic observations of failure under cyclic in-plane shear. Their findings confirmed earlier observations reported in [38, 39]. In [38], shear stress was introduced directly at the fiber level by cutting fibers that retracted inducing equal and opposite forces locally (Fig. 11b).

Fig. 11.
figure 11

Image of fracture surface showing hackles caused by failure in shear [39] (a), and sigmoidal shape cracks in the matrix produced by axial shear between fibers [38] (b).

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In [39] shear failure was induced between fibers in a 10° off-axis tension fatigue. The key features of failure from in-plane shear are depicted in Fig. 11. As seen in Fig. 11a, the shear failure involves plastic deformation of matrix along planes inclined to the fiber direction resulting in shear cusps called as “hackles”. The cracks formed by the local maximum shear stress between the fibers grow and turn towards the fiber/matrix interfaces, forming sigmoidal shape cracks seen in Fig. 11b.

2.10. In-plane shear stress combined with axial and transverse stresses

The failure caused by the in-plane shear stress is primarily a ductile failure mechanism involving plastic deformation of the matrix. The fiber/matrix interface failure could result from a matrix crack formed by ductile failure of the matrix diverting into the interface, as in Fig. 11b. In any case, the final failure appears as cracks along fibers known as axial splits. If a tensile stress normal to fibers is superimposed on the in-plane shear stress, then any “interaction” must come from a change brought about by the superimposed stress in the local stress field induced by the in-plane shear stress. This change could alter the orientation of the ductile crack in the matrix that would still divert into the fiber/matrix interface, resulting in the axial splits. Depending on the fiber/matrix interface fracture resistance (“strength”), the superimposed tensile normal stress would cause failure of the interface, enhancing the axial split process. This failure process, if correct as described, would show a reduction in the shear strength with increasing superimposed tensile normal stress. This is in fact the case, as seen in the first quadrants of the four plots shown in Fig. 10.

If a transverse compressive stress is superimposed on the in-plane shear stress, then the ductile failure of the matrix in the inter-fiber regions will still occur, but the cracks formed will have an altered orientation. These cracks on growing will divert into the fiber/matrix interface, but their growth along the fibers will be dampened by the superimposed compressive normal stress. The final failure from axial splits will therefore be delayed, resulting in “strengthening” of the in-plane shear strength. This is indeed the case, as seen in the second quadrants of the plots in Fig. 10.

The effects of fiber stresses — tensile or compressive — on the in-plane shear strength is not expected to be significant, unless the fibers are broken, in which case the enhancement of the plastic deformation in the vicinity of the broken fiber ends will induce matrix cracking. These effects are difficult to observe by experiments and must instead be clarified by numerical simulations.

3. Modeling Approaches to Failure in UD Composites

As described in the Introduction, the history of failure theories for UD composites began in 1965 with the Azzi-Tsai paper [5] that adapted the Hill theory for orthotropic metal yielding [4] to the composites case by equating the yield stress (“strength”) to the final failure stress of composites. A review of the key features of the failure mechanisms in UD composites described in Section 2 provides strong evidence that these mechanisms are widely different from the metal yielding mechanism. In fact, they depend heavily on whether the applied normal stress is in the fiber direction or in the direction transverse to it, or whether the normal stress is tensile or compressive, or whether the applied stress is shear in the plane of the composite. Furthermore, and even more importantly, the combined effects of the various stresses depend on which of the single-component mechanisms is operative in each combined stressing case. This fact is crucial as the success of any proposed failure theory depends on capturing the so-called “interaction” effects.

The next consequential development in failure theories for UD composites came in 1971 when Tsai and Wu [40] proposed as a failure criterion a quadratic polynomial expression adapted from a previously proposed general expression [41]. The Tsai–Wu expression for a 2D case with the three in-plane stress components could be depicted as an ellipse for which analytic geometry provided certain restrictions on the coefficients of the polynomial. These coefficients could be expressed in terms of the measured failure stresses (strengths) in the fiber direction (tensile and compressive), in the transverse direction (tensile and compressive), and in the in-plane shear, except an additional coefficient described as an interaction factor. The interaction factor does not have a physical interpretation but can be expressed in terms of strengths in two principal directions subjected to a restriction imposed by the requirement of convexity of the ellipse. However, the restriction only places a limit to the range in which the factor can vary, while its determination is not fixed. The interaction factor was a subject of intensive discussions in the literature and much controversy. Hashin in 1980 [42] pointed to a fundamental difficulty with the interaction factor and proposed a way to avoid it, as described next.

Hashin noted that if the interaction factor is found by a biaxial tension test, then it turns out to depend on the compressive strengths in the principal directions of the composite. To avoid this physically unacceptable result, Hashin proposed to separate the failure of the composite in fiber and matrix modes. Then, he suggested that the matrix failure mode be described using the failure plane concept à la the Coulomb–Mohr failure theory for soils. The inclination of the failure plane would then be an unknown for which some stationarity (optimization) principle could be used.

Hashin’s proposal of a failure plane, although not taken to fruition by him, was pursued by many in the literature. The prominent examples of such works are [43,44,45], where several assumptions were made to determine the inclination of the failure plane. In [43], an elaborate scheme was developed to implement the failure plane methodology, resulting in four “inclination parameters” in addition to the three conventional material strength values. As expected, other phenomenological assumptions by other authors have resulted in different material parameters. A detailed examination of the failure plane approach was presented in [46] where it was pointed out that the lack of explicit presence of fibers in this approach necessitates introducing remedial phenomenological parameters. In other words, the presence of fibers is only accounted for by restricting the failure plane to be parallel to the fiber direction, but how the fibers affect the tractions on the plane cannot be determined. This type of imaginary separation of fibers and matrix (called “inter-fiber region” in [43]) prevents calculating the actual stresses that cause failure of the assumed plane. The fact that the failure plane-based approaches show good agreement with measured failure stress data is of little comfort as this can be attributed to a typically large number of adjustable parameters, which differ from one approach to another and depend on the assumptions made in each approach. Another fundamental problem with the cited approaches that are based on a failure plane parallel to fibers lies in the fact that the local stress field in the inter-fiber region can cause failure on a plane inclined to fibers, e.g., in a resin-rich pocket. Such a stress field cannot, however, be computed when the entire matrix was homogenized as a single continuum and the failure plane orientation was restricted a priory.

A different line of approach to failure in UD composites was called “computational micromechanics” [47, 48]. This approach explicitly accounts for fibers by constructing a representative volume element (RVE) with bounding surfaces parallel to the chosen coordinate planes in which a large enough number of fibers are placed to conform to a selected representation criterion. The bounding surfaces are subjected to prescribed tractions or displacements according to the loading modes (σxx, σyy, τxy) in single or combined form. The local stress fields within the matrix (or at fiber/matrix interfaces) are calculated typically by a finite element method. Certain criteria for failure initiation are then applied on the local stress fields. This allows the failure analysis to be more physics-based than in the failure plane-based phenomenological approaches. A further advantage in this approach is that failure can be simulated under loading conditions that are difficult to implement in physical experiments, e.g., under combined transverse tension and out-of-plane shear [49]. This virtual testing can also help explore material selection without expensive physical tests [50].

A crucial step in the computational micromechanics approaches is the analysis of the failure process. Once the stress fields in the matrix are calculated, one must ask: How does the failure initiate, progress, and attain criticality? In the case of a matrix material like epoxy, one must resort to the physics of failure in this material that was studied and reported in the literature. For instance, the observed pressure sensitivity of yielding in epoxy can be accounted for by appropriately modifying the metal yield criterion [51]. What is often not accounted for is that yielding is totally suppressed in hydrostatic tension, i.e., when the three principal stresses are tensile and equal. This was first pointed out in [21] in trying to explain why the transverse failure strains in epoxy-based UD composites are much lower than in unreinforced epoxies. Figure 12 taken from that work shows this dramatic effect. The failure in equi-triaxial tension was found to occur by dilatation induced cavitation, a phenomenon explained in detail in [52]. The stored dilatational energy density at which this brittle cavitation occurs is much lower than the stored distortional energy density that causes yielding. Thus, initially when the epoxy matrix within the composite is deforming elastically, the brittle cavitation failure event can occur before yielding initiates if the condition of equi-triaxial tension exists. This was demonstrated first for transverse tension of UD composites containing uniform distribution of fibers in the cross-section [22, 23] and later for various nonuniform distribution cases [24, 25].

Fig. 12.
figure 12

Stress-strain response of an epoxy under uniaxial and equi-triaxial tension [21].

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It should be noted that the condition for cavitation is favorable for the loading mode σyy > 0 described in Section 2.5 but it is no longer favorable if shear is superimposed in sufficient measure, as illustrated in Fig. 8. As seen there, the transverse failure stress remains unaffected by the shear stress less than 10 MPa. For higher shear stress values, the first failure event becomes yielding governed by the critical distortional energy density for epoxy, as analyzed in [29]. This example is illustrative of the presence of two independent mechanisms, each governed by its own criticality condition. When the loading modes are combined, both mechanisms compete to attain their respective criticality. The “interaction” of multiple failure mechanisms is discussed next.

3.1. Multiple failure mechanisms

The review of the failure mechanisms in Section 2 brings out a fact that is crucial to proper modeling of failure in UD composites. It is that there are two independent failure mechanisms of fiber failure, in tension and in compression, and two independent failure mechanisms in matrix, namely, brittle cavitation and ductile failure. Since the brittle cavitation occurs near the fiber/matrix interface, it can also be viewed as an interface debonding mechanism. One can argue that a tensile stress normal to that interface can also cause debonding. In any case, each of these failure mechanisms consists of a sequence of failure events that culminate in the final failure event commonly represented by a stress termed as “strength”. Thus, the five measured strength values in the in-plane loading modes are: axial tensile strength, axial compressive strength, transverse tensile strength, transverse compressive strength, and in-plane shear strength. The aim of the failure theories discussed above is to predict strength in any combination of the five basic loading modes in terms of the five measured strength values.

It would not be an understatement to say that the failure prediction problem in combined loading is complex. However, it is possible to identify those approaches that do not have the potential to properly address this problem. The phenomenological approaches such as Tsai-Hill [5] and Tsai-Wu [40] fall in this category. The approaches that do not account for the explicit presence of fibers cannot calculate stresses in the matrix perturbed by fibers and therefore do not have the ability to treat any of the failure events occurring in the matrix. The approach initiated by Hashin [42] and followed by Christensen [53] is of this type. In fact, the authors in [32] show that Christensen’s criterion for matrix failure can be deduced from the phenomenological Tsai-Wu criterion [40]. In any case, this scrutiny leaves us with the computational micromechanics approach as the type of approach that could meet the challenge of combined loading failure in UD composites. However, there is one problem with the way most authors have used this approach, as discussed next.

3.2. Failure at a point vs failure on a plane

In a pristine material with no pre-existing defects, failure at a point initiates when a failure event that requires the least energy among all potential failure events takes place. In a heterogeneous material, the local stress state under a remote uniform stress is generally triaxial and varies spatially. The problem of failure initiation cannot therefore be cast in terms of the remote stresses unless those are related to the local stresses. Relating the remote stresses to the local stresses requires knowing the geometry of the phases (fibers and matrix in composites), the properties of the phases, and the distribution of the discrete phase (fibers) into the continuous phase (matrix). The effect of fiber geometry (circular vs non-circular) is a secondary effect and is often neglected in calculating the local stresses. Fibers are also generally approximated as having axial stresses only. However, the distribution of fibers in the composite cross-section is found to have significant effect on the local stresses in the matrix. For this reason, the approaches for constructing a representative volume element (RVE), accounting for the degree of nonuniformity, were developed [24, 25]. When a remotely applied stress, e.g., on the RVE bounding surfaces, is increased from zero, linear elastic response of fibers and matrix can be assumed initially until a departure from elasticity takes place. While for fibers a linear elastic response until failure is a good approximation, matrix can be assumed to be linear elastic until yielding initiates at any point in the matrix. As noted above, a pre-yielding failure event of brittle cavitation in an epoxy matrix can occur if the local triaxial stress state is favorable. Depending on whether brittle cavitation or yielding occurs first, the subsequent stress analysis has to be account for this occurrence appropriately. The occurrence of brittle cavitation can be assumed to result in a small fiber/matrix debonding since the favorable conditions for cavitation are near the fiber/matrix interface. Studies that have accounted for the presence of one or more debond cracks in analysis of subsequent failure events are, e.g., [26, 27], where the growth and kink-out of the debond cracks have been analyzed by fracture mechanics methods. The culmination of these failure events is formation of transverse cracks as shown in experimental observations such in Fig. 7. The plane of final failure is then what results from an unstable growth of the transverse crack. Studies such as [26, 27] have examined the failure process (debond crack growth, kink-out and merger) only under imposed transverse tension. The study in [29] has clarified the initiation of brittle cavitation vs yielding in the matrix under a combined transverse tension and axial shear. Work is needed to carry this further to examining the subsequent failure events that follow either the fiber/matrix debonding or matrix yielding to final failure. It should be mentioned that a study [54] examined the failure events after a debond crack has formed when a transverse compression is imposed. The growth and kink-out of the debond crack were studied carefully but while the kink-out angle could be determined, the formation of a failure plane was not determined.

While systematic studies are yet to be done that follow the failure events from the first until the last, i.e., which causes final failure, from the studies so far nothing suggests that the failure involves separating two sides of a “failure plane” in a decohesion process. In fact, the likely scenario is initiation of failure at a point (brittle cavitation or yielding), subsequent crack formation (fiber/matrix debonding or ductile cracking of matrix), and crack growth until instability. Prior to the unstable crack growth, it is likely that the crack path is not straight because of the changing stress field in a heterogeneous medium. The suggestion in 1980 by Hashin [42] of a flat failure plane parallel to fibers may have been expedient then but has little justification now with the computational power available. Unfortunately, continuing to use the failure plane concept has led to a flurry of failure theories, each with adjustable parameters and none working satisfactorily in every situation of combined loading.

4. Way Forward in Modeling of Failure in UD Composites

The two issues discussed in Sections 3.1 and 3.2 need to be addressed for a physically justifiable failure theory to be developed. The approaches so far have treated failure under combined loadings with “interaction” criteria implying that a single underlying mechanism exists driven by combined effects of multiple loading modes. A common approach is to use quadratic failure criteria for the combined effects. Hashin [42] set this trend in motion by proposing the following criteria for combined transverse stress σ2 (same as σyy) and in-plane shear stress σ6 (same as τxy) for σ2>0

$${\left(\frac{{\upsigma }_{2}}{Y}\right)}^{2}+{\left(\frac{{\upsigma }_{6}}{Y}\right)}^{2}=1$$
(10)

and σ2>0

$${\left(\frac{{\upsigma }_{2}}{{2T}^{\mathrm{^{\prime}}}}\right)}^{2}+\left[{\left(\frac{{Y}^{\mathrm{^{\prime}}}}{{2T}^{\mathrm{^{\prime}}}}\right)}^{2}-1\right]\frac{{\upsigma }_{2}}{Y}+{\left(\frac{{\upsigma }_{6}}{Y}\right)}^{2}=1,$$
(11)

where Y and Y′ are the values of the tensile and compressive strength normal to fibers, respectively; T and T′ are the values of shear strength in a plane across fibers (“transverse shear strength”) and parallel to fibers (“axial shear strength”), respectively. For modifications to these expressions and other quadratic forms, see [32].

The failure mechanisms reviewed in Section 2 suggest strongly that several independent mechanisms exist, each governed differently by the local stresses. Take for instance the fiber failure mechanism in tension and the matrix failure mechanism in axial shear. These two failure mechanisms were reviewed in Sections 2.1 and 2.9, respectively. Under combined application of these stresses, the failure stresses are shown in Fig. 5 for a carbon/epoxy composite. The authors of that study [16] found two separate failure modes, each still governed by their respective stress, modified only by the superposition of the other stress. The transition between the two failure modes is a mixed effect. Another example is the combined loading with transverse tension and axial shear shown in Fig. 8 [29]. Here, the output of the failure analysis of the first failure event using computational micromechanics is plotted. The two separate modes are clearly seen; one due to transverse loading where the brittle cavitation is operative, and the other under axial shear stress that initiates yielding in the matrix. The brittle cavitation mechanism is insensitive to the superposition of the shear stress, while the initiation of yielding is affected by the application of the transverse stress, which adds to the distortional energy density induced by the axial shear stress. Yet another example is provided by the data in Fig. 10 [32] (second quadrant) showing the outcome of failure under combined transverse compression and axial shear. A closer examination of the data indicates presence of two distinct mechanisms, one due to transverse compression and the other caused by axial shear. Figure 13 shows how the two distinct mechanisms operate in their own ranges of governance, while a failure criterion with single expression will tend to curve-fit the data. The failure mechanism labeled as Mechanism 1 in the figure is due to transverse compression and, as described in Section 2.7, is essentially ductile failure of the matrix, while Mechanism 2, described in Section 2.9, represents cracking in the matrix along fibers caused by the axial shear. The strengthening effect on the axial shear stress to failure caused by superposition of the transverse compression can be attributed to the resistance to frictional sliding between the surfaces of the axial cracks induced by the closing pressure given by the transverse compression. While matrix yielding is involved in both mechanisms, their progression to final failure is along different paths. Therefore, a single curve will not be able to produce the failure envelope. The approach taken to treat the shear strengthening effect separately by failure plane methods, e.g., in [43,44,45], produce two piecewise continuous curves in the failure envelope at the cost of adjustable parameters.

Fig. 13.
figure 13

A schematic of failure stress representation under combined transverse compression σ2 and axial shear τ21. Mechanism 1 refers to yielding in the matrix and Mechanism 2 is axial cracking parallel to fibers.

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5. Concluding Remarks

The classical strength of materials approach is to predict “working stresses” for the purpose of design against failure under combined loading modes. Such stresses are expressed in terms of failure stresses described as “strengths” measured on laboratory specimens under uniform tension, compression, and shear stresses. This approach has worked well for metals where yielding is the operative mechanism providing the elastic limit (yield strength). The concept of strength (final failure stress) in fiber reinforced composites is complex in the light of a multitude of operating failure mechanisms (see [55] for a detailed treatment). Therefore, the classical strength of materials approach has not been successful for composites. Here, we have reviewed some key features of the failure mechanisms under elementary stresses and have argued why using failure criteria is not likely to succeed in predicting failure under combined loading. We have examined the descriptions of “interactions” by typical quadratic polynomials and pointed to the fundamental flaws in such descriptions. The approach of describing failure of composites by failure plane concepts has also been scrutinized and argued to be without basis in the failure mechanisms. The way forward is advocated to be where the local stress fields within a composite are computed, and energy-based criteria are used to determine the first (least energy requiring) failure event. Then, the subsequent failure events are similarly determined, and the analysis is carried out to the final failure. This failure analysis approach recognizes the individual and independent mechanisms operating at the local (micromechanical) level. This approach is a departure from the strength of materials approach and is likely to provide reliable “working stresses” for design against failure.