l184 L.J. Hart-Smith shear-failure cut-offs, the validity of the earlier 45deg approximation for carbon/epoxy laminates is clearly d (11) confirmed. Conversely, the earlier appro be significantly conservative for glass as the author had suspected without actually ble to for round fibres in precisely quantify the effect until now. (These expressions are derived from the solutions for Point(4)in Fig. 3 follows from point (3), the zero the fibre volume fraction as a function of each array axial stress point for the fibre, by retaining the same Setting the value of the array coefficient K at unity, axial strain and multiplying the transverse strain by Re eqn(2)would then predict the following strain-amplifi- from eqn(2). The line(14)in Fig 3 then defines the cation factors for the composite materials used in failure locus of shear failures in the fibres in terms of strains the lamina. It will be apparent that point (4) lies off the (0%)T300/914C carbon/epoxy, Re=1. 517 zero longitudinal stress line for the lamina, being asso- (0%)E-glass/LY556-epoxy, Re=5. 257 ciated with an effective transverse poisson ratio of (0%)E-glass/MY750-epoxy, RE= 5.159 UTL =VTL/Re (12) (0%)AS4/3501-6 carbon/epoxy, Re= 1. 488. Given that these amplification factors are effectively instead of the unrelated VlamtL for the laminate as a reduced in the ratio (UnT/vLT), or roughly 0.2/0.3 for whole. The reason for this is that, while the fibres have carbon/epoxy, in establishing the final slope of these no axial stress at point(4), the matrix does d Square fibres in squre array d Circular fibres in square array Circular fibres in hexagonal array
and d 4 Vf r 1128pVf 11 for round ®bres in square arrays. (These expressions are derived from the solutions for the ®bre volume fraction as a function of each array.) Setting the value of the array coecient K at unity, eqn (2) would then predict the following strain-ampli®- cation factors for the composite materials used in failure exercise.4 (0) T300/914C carbon/epoxy, R" 1517 (0) E-glass/LY556-epoxy, R" 5257 (0) E-glass/ MY750-epoxy, R" 5159 (0) AS4/3501-6 carbon/epoxy, R" 1488. Given that these ampli®cation factors are eectively reduced in the ratio (fLT=LT), or roughly 0.2/0.3 for carbon/epoxy, in establishing the ®nal slope of these shear-failure cut-os, the validity of the earlier 45deg; approximation for carbon/epoxy laminates is clearly con®rmed. Conversely, the earlier approximation would be signi®cantly conservative for glass ®bres, as the author had suspected without actually being able to precisely quantify the eect until now. Point (4) in Fig. 3 follows from point (3), the zero axial stress point for the ®bre, by retaining the same axial strain and multiplying the transverse strain by R" from eqn (2). The line (1)±(4) in Fig. 3 then de®nes the locus of shear failures in the ®bres in terms of strains in the lamina. It will be apparent that point (4) lies o the zero longitudinal stress line for the lamina, being associated with an eective transverse Poisson ratio of 0 TL f TL=R" 12 instead of the unrelated lamTL for the laminate as a whole. The reason for this is that, while the ®bres have no axial stress at point (4), the matrix does. Fig. 4. Fibre volumes for various arrays. 1184 L. J. Hart-Smith
Predictions of a generalized maximum-shear-stress failure criterion 1185 The strains on the lamina strain plane at which the by the organizers. There would be a small increase in fibre would fail by shear under the application of purely slope for carbon/epoxy laminae and a large increase in transverse tension or compression are consequently slope for glass-fibre- reinforced epoxies. The original given by maximum-strain model would be almost as good a presentati laminates as the + VTi runcated maximum-strain model is for carbon/epoxy (13) laminates, as will become evident from comparing the 7=干R(+ worked examples later in this paper with the corre- sponding solutions in Ref. 7. This similarity would be particularly strong if it were known that the actual fail- (The fibre strain Ef would be replaced by ef if the latter ures under uniaxial tension were by brittle fracture were numerically greater. The line(1)(4)in Fig 3 thus rather than by shear because, then, the shear cutoff line defines the shear-failure locus of the fibre in terms of should start from beyond the measured uniaxial strain lamina strains. It is not exactly at a slope of 45, something to failure. If a matrix were so soft in comparison with the author had previously adopted as what seemed to be the transverse stiffness of the fibres that it could exert no a legitimate simplifying assumption, at least for carbon- stress on them, the failure envelope would shrink to the epoxy composites. The worked examples here will points A and C in Fig. 2 at the ends of the two radial show that the correct solution is quite close to that slope lines characterizing pure longitudinal loading on the for carbon-epoxy composites, but much closer to the fibre- and lamina-strain planes and be a simple square 90 slope of the maximum-strain model for glass-fibre- maximum-stress box on the laminate-strain plane. This reinforced laminates is referred to as netting theory(see Refs 10 and l1) This same kind of modification to the effective lor It should be noted that pucks maximum-strain chara itudinal strain under a transverse stress must also be terization of glass-fibre failures on the lamina or laminate applied to the transition from a pure uniaxial load to the equal-biaxial-strain point. At the equal biaxial strain points, as defined above, the actual transverse strain in 2(%) ne fibre will now be less than in the lamina, per eqn (12)and the equal-biaxial-strain point for the lamina will now be even closer to matching the uniaxial strain E1(%) value than was the case for earlier present-ations of thi theory in which the transition from uniaxial to biaxial loads was assumed to be governed by the measured lamina transverse Poisson ratio VTL. The equal-biaxial-strain point B" in Fig 3 has co-ordinates slightly less than the EL of the truncated maximum-strain failure model, at eL =E=Ei-VrvLT' 1+ ≈E[1-(1+um)h where vti is defined in eqn(12). It is quite distinct from the measured (or computed) transverse Poisson ratio VTL of the lamina. Points A, B and F in Fig. 3 refer to the corresponding points in Fig. 2 As regards the change in slope of the 45 line for fibre ailures on the strain plane, the net effect of accounting for the differences in stiffness via eqns(1)and(13)is shown in Fig. 5, using the material properties supplied *This improvement in the model is partly the consequence of responding to earlier unsubstantiated criticism in America that, even if it were conceded that the 45 slope were valid for isolated fibres(as it obviously can be for isotropic glass fibres) It would be replaced by nearly vertical lines at the lamina lev even for carbon/epoxy laminates, because the matrix is so much softer than the fibres. The new analysis is also in Fig. 5. Fibre shear failure cut-offs on lamina strain plane. tinuous lines for AS4/3501-6 carbon/epoxy lamina))Con- response to the challenge presented by the organizers in evaluat-(Dashed lines for E-glass/MY750 epoxy lamina and ing glass-fibre laminates, beyond the authors prior experience
The strains on the lamina strain plane at which the ®bre would fail by shear under the application of purely transverse tension or compression are consequently given by "L f TL 1 f LT 1 f TL "t L; "T R" 1 f LT 1 f TL "t L 13 (The ®bre strain "t L would be replaced by "c L if the latter were numerically greater.) The line (1)±(4) in Fig. 3 thus de®nes the shear-failure locus of the ®bre in terms of lamina strains. It is not exactly at a slope of 45, something the author had previously adopted as what seemed to be a legitimate simplifying assumption, at least for carbonepoxy composites.* The worked examples here will show that the correct solution is quite close to that slope for carbon-epoxy composites, but much closer to the 90 slope of the maximum-strain model for glass-®brereinforced laminates. This same kind of modi®cation to the eective longitudinal strain under a transverse stress must also be applied to the transition from a pure uniaxial load to the equal-biaxial-strain point. At the equal biaxial strain points, as de®ned above, the actual transverse strain in the ®bre will now be less than in the lamina, per eqn (12)Ðand the equal-biaxial-strain point for the lamina will now be even closer to matching the uniaxial strain value than was the case for earlier present-ations of this theory in which the transition from uniaxial to biaxial loads was assumed to be governed by the measured lamina transverse Poisson ratio TL. The equal-biaxial-strain point B'' in Fig. 3 has co-ordinates slightly less than the "t L of the truncated maximum-strain failure model, at "L "T "t L 1 ÿ 0 TLLT 1 0 TL "t L 1 ÿ 1 LT 0 TL 14 where 0 TL is de®ned in eqn (12). It is quite distinct from the measured (or computed) transverse Poisson ratio TL of the lamina. Points A, B and F in Fig. 3 refer to the corresponding points in Fig. 2. As regards the change in slope of the 45 line for ®bre failures on the strain plane, the net eect of accounting for the dierences in stiness via eqns (1) and (13) is shown in Fig. 5, using the material properties supplied by the organizers.4 There would be a small increase in slope for carbon/epoxy laminae and a large increase in slope for glass-®bre-reinforced epoxies. The original maximum-strain model would be almost as good a representation of glass-®bre/epoxy laminates as the truncated maximum-strain model is for carbon/epoxy laminates, as will become evident from comparing the worked examples later in this paper with the corresponding solutions in Ref. 7. This similarity would be particularly strong if it were known that the actual failures under uniaxial tension were by brittle fracture rather than by shear because, then, the shear cuto line should start from beyond the measured uniaxial strain to failure. If a matrix were so soft in comparison with the transverse stiness of the ®bres that it could exert no stress on them, the failure envelope would shrink to the points A and C in Fig. 2 at the ends of the two radial lines characterizing pure longitudinal loading on the ®bre- and lamina-strain planes and be a simple square maximum-stress box on the laminate-strain plane. This is referred to as netting theory (see Refs 10 and 11). It should be noted that Puck's maximum-strain characterization of glass-®bre failures on the lamina or laminate Fig. 5. Fibre shear failure cut-os on lamina strain plane. (Dashed lines for E-glass/MY750 epoxy lamina and continuous lines for AS4/3501-6 carbon/epoxy lamina). *This improvement in the model is partly the consequence of responding to earlier unsubstantiated criticism in America that, even if it were conceded that the 45 slope were valid for isolated ®bres (as it obviously can be for isotropic glass ®bres), it would be replaced by nearly vertical lines at the lamina level even for carbon/epoxy laminates, because the matrix is so much softer than the ®bres. The new analysis is also in response to the challenge presented by the organizers in evaluating glass-®bre laminates, beyond the author's prior experience. Predictions of a generalized maximum-shear-stress failure criterion 1185
1186 L.J. Hart-Smith strain planes is not incompatible with the author's 45 requires that laminates made from bi-directional woven cutoffs for carbon epoxy laminates in the tension-com- fabric layers be treated as combinations of two equiva- pression quadrants. Both are close approximations, not lent unidirectional layers-at the same height within the precise answers. The 45 cutoff would still exist at the laminate if it is a plain-weave fabric, or one above the fibre level for both composite materials, but would sim- other if it is a satin-weave fabric or the like. The trans ply not be evident at the lamina and laminate levels for verse strains involved in Figs 2-6 are those associated with a unidirectional fibre not a mixture of those acting Figure 6(failure of fibres on the fibre-strain plane) on two orthogonal sets of fibres. A plain-weave cloth shows how possible cutoffs for fibre failures by brittle can be decomposed into its equivalent layers by using fracture, which is a constant-stress phenomenon lamination theory in reverse. The combination of 0 and because crack-tip stress intensities are unaffected by 90% fibres to produce a 0 /90 laminate results in a stiff tresses parallel to the crack(transverse to the fibre), ness of something close to 55% of that of each indivi- ind compressive instability, which is also a constant- dual layer. Therefore, once the stifness and strengths of stress phenomenon, are superimposed locally on thethe fabric layer have been measured, they can be basic shear-failure envelopes. These three possible fail- increased in the ratio 1/0.55=1.82 (or whatever more ure mechanisms are all that are considered for fibre precise value is calculated for a specific material). When failures in the author's analyses of in-plane loads. needed, the matrix-dominated properties can be adjus a difference between the longitudinal tensile and ted accordingly; even the nonlinearities can be repl impressive strengths of unidirectional laminae should cated. The process can either be performed using logic be interpreted as implying that at least one of the fail- alone or by scaling (inversely) relevant details of the ures cannot be by shear. The 450-sloping lines in Fig. 6 output from a complete analysis of a 0/90% laminate for would then be passed through the numerically greater of which it has been assumed that the in-plane-shear the two measured strengths, on the assumption that the properties would not be altered by the separation of the lower number denotes a premature failure by a different constituents and that the transverse stifness of each mechanism (A more precise slope could be used on the equivalent ply would be the same as for a real unidirec lamina-strain plane when appropriate, as shown in tional lamina made from the same fibres and resin The Fig 5). In the event that it is known by fractographic justification for this second assumption is that any inspection of the broken fibres that neither of the fail- crimping of the fibres in a real fabric would affect the ures is by shear, (as is quite likely for E-glass fibres), one longitudinal stiffness but would not affect the transverse could perform a shear test on a+45 laminate, to gen- stiffness within each tow of fibres. (There would be a erate data near the middle of the sloping line, far away minor effect because of the in-plane separation of the from any failures by other mechanisms. Unfortunately, tows of fibres which would be filled with a different based on past experie ence wi ith carbon/epoxy laminates, combination of resin and fibres than within the tows. at least, such a test is likely to result in a premature This cross-plying technique has already been used to failure, giving a cut-off more severe than that based on generate more reliable measurements of unidirectional the higher of the two measured axial strengths. This lamina strength than are usually obtained by direct should be physically impossible if the test really repre- measurement of the lamina strengths, as discussed in ented the true material strength devoid of any influence Refs 13 and 14. The process automatically accounts for of the geometry of the test specimen. The highest known the loss of stifness by whatever degree of crimping was test results have been obtained using the Douglas bon- introduced by the weaving process and for the difference ded tapered rail shear coupon described in Ref. 12 between tensile and compressive strengths which is exa- PP It is necessary to note that the formulation of the cerated by this same crimping neralized maximum-shear-stress failure theory Some readers of earlier articles on the generalization of the maximum-shear-stress failure criterion to non- *In all his earlier works on this subject, the author had sotropic homogeneous materials have expressed diffi described this cut-off as a constant-strain line. since the fibres culty in accepting the concept of a 45-sloping constant would buckle once they had reached a critical shortening shear-strain line representing a constant critical stress strain which would be unaffected by the simultaneous appli- criterion for anything other than an isotropic solid, like ation of transverse stresses The laminate stress at which thi a glass fibre. (The confusion seems to arise from the would happen would vary with the fibre pattern. The short- obviously dissimilar differences between principal stres ening strain ould not. However. he had overlooked the ses in the L-t plane for fibres subjected to axial tension changed the reference point for the buckling process. He is on the one hand and transverse compression on the indebted to the editorial review for pointing this out. A con- other )Reference 15 includes an attempt by the author stant-stress cut-off for the lamina automatically accounts for to explain this apparent contradiction, in terms of the this effect. Ironically, with the distinction derived above difference between isotropic and nonisotropic materials between transverse fibre and lamina strains, the new position Briefly, while isotropic homogeneous materials can of this cut-off, for both carbon and glass fibres, is almost undergo strains in the absence of stresses, as the result coincident with the constant-longitudinal-strain line. of uniform heating for example, or stresses in the
strain planes is not incompatible with the author's 45 cutos for carbon/epoxy laminates in the tension-compression quadrants. Both are close approximations, not precise answers. The 45 cuto would still exist at the ®bre level for both composite materials, but would simply not be evident at the lamina and laminate levels for glass-®bre/epoxies. Figure 6 (failure of ®bres on the ®bre-strain plane) shows how possible cutos for ®bre failures by brittle fracture, which is a constant-stress phenomenon because crack-tip stress intensities are unaected by stresses parallel to the crack (transverse to the ®bre), and compressive instability, which is also a constantstress phenomenon,* are superimposed locally on the basic shear-failure envelopes. These three possible failure mechanisms are all that are considered for ®bre failures in the author's analyses of in-plane loads. A dierence between the longitudinal tensile and compressive strengths of unidirectional laminae should be interpreted as implying that at least one of the failures cannot be by shear. The 45-sloping lines in Fig. 6 would then be passed through the numerically greater of the two measured strengths, on the assumption that the lower number denotes a premature failure by a dierent mechanism. (A more precise slope could be used on the lamina-strain plane when appropriate, as shown in Fig. 5). In the event that it is known by fractographic inspection of the broken ®bres that neither of the failures is by shear, (as is quite likely for E-glass ®bres), one could perform a shear test on a 45 laminate, to generate data near the middle of the sloping line, far away from any failures by other mechanisms. Unfortunately, based on past experience with carbon/epoxy laminates, at least, such a test is likely to result in a premature failure, giving a cut-o more severe than that based on the higher of the two measured axial strengths. This should be physically impossible if the test really represented the true material strength devoid of any in¯uence of the geometry of the test specimen. The highest known test results have been obtained using the Douglas bonded tapered rail shear coupon described in Ref. 12. It is necessary to note that the formulation of the generalized maximum-shear-stress failure theory requires that laminates made from bi-directional woven fabric layers be treated as combinations of two equivalent unidirectional layersÐat the same height within the laminate if it is a plain-weave fabric, or one above the other if it is a satin-weave fabric or the like. The transverse strains involved in Figs 2±6 are those associated with a unidirectional ®bre, not a mixture of those acting on two orthogonal sets of ®bres. A plain-weave cloth can be decomposed into its equivalent layers by using lamination theory in reverse. The combination of 0 and 90 ®bres to produce a 0/90 laminate results in a sti- ness of something close to 55% of that of each individual layer. Therefore, once the stiness and strengths of the fabric layer have been measured, they can be increased in the ratio 1/0.55=1.82 (or whatever more precise value is calculated for a speci®c material). When needed, the matrix-dominated properties can be adjusted accordingly; even the nonlinearities can be replicated. The process can either be performed using logic alone or by scaling (inversely) relevant details of the output from a complete analysis of a 0/90 laminate for which it has been assumed that the in-plane-shear properties would not be altered by the separation of the constituents and that the transverse stiness of each equivalent ply would be the same as for a real unidirectional lamina made from the same ®bres and resin. The justi®cation for this second assumption is that any crimping of the ®bres in a real fabric would aect the longitudinal stiness but would not aect the transverse stiness within each tow of ®bres. (There would be a minor eect because of the in-plane separation of the tows of ®bres which would be ®lled with a dierent combination of resin and ®bres than within the tows.) This cross-plying technique has already been used to generate more reliable measurements of unidirectional lamina strength than are usually obtained by direct measurement of the lamina strengths, as discussed in Refs 13 and 14. The process automatically accounts for the loss of stiness by whatever degree of crimping was introduced by the weaving process and for the dierence between tensile and compressive strengths which is exacerbated by this same crimping. Some readers of earlier articles on the generalization of the maximum-shear-stress failure criterion to nonisotropic homogeneous materials have expressed di- culty in accepting the concept of a 45-sloping constantshear-strain line representing a constant critical stress criterion for anything other than an isotropic solid, like a glass ®bre. (The confusion seems to arise from the obviously dissimilar dierences between principal stresses in the L±T plane for ®bres subjected to axial tension on the one hand and transverse compression on the other.) Reference 15 includes an attempt by the author to explain this apparent contradiction, in terms of the dierence between isotropic and nonisotropic materials. Brie¯y, while isotropic homogeneous materials can undergo strains in the absence of stresses, as the result of uniform heating for example, or stresses in the *In all his earlier works on this subject, the author had described this cut-o as a constant-strain line, since the ®bres would buckle once they had reached a critical shortening strain which would be unaected by the simultaneous application of transverse stresses. The laminate stress at which this would happen would vary with the ®bre pattern. The shortening strain would not. However, he had overlooked the Poisson-induced axial strains caused by those stresses, which changed the reference point for the buckling process. He is indebted to the editorial review for pointing this out. A constant-stress cut-o for the lamina automatically accounts for this eect. Ironically, with the distinction derived above between transverse ®bre and lamina strains, the new position of this cut-o, for both carbon and glass ®bres, is almost coincident with the constant-longitudinal-strain line. 1186 L. J. Hart-Smith
Predictions of a generalized maximum-shear-stress failure criterion 1187 y>(1+4)E1 7=(1+ar)Et ae arctan ttle fracture L-N plane critical Shear failure L-T plane critical Fig. 6. Superposition of additional fibre failure modes on basic maximum-shear-stress failure criterion absence of strains, as for hydrostatic compression of an 3 CUTOFFS IMPOSED BY MATRIX SHEAR incompressible material, there is a one-to-one relation FAILURES between stress and strain for isotropic materials most of the time. This is the exception to the rule for homo- The failure envelopes shown in Figs 2, 3, 5 and 6 lack a geneous nonisotropic materials, however, as the equa- roof to define any limits imposed by the in-plane shear tions in Jones's work. make clear. Consider, for strength of the matrix between the fibres. Since being example, uniform heating of a transversely isotropic formulated on the strain plane, the authors model has solid. If the coefficients of thermal expansion in the always included a non-interactive horizontal plateau, principal axes differ, it is inevitable that shear strains will located by the shear-strain-to-failure, as shown by the develop between axes inclined at +45 to the material lamina failure model in Fig. 7. This refers to shear with axes, even though there are no stresses anywhere in the respect to fibres in the 0 and 90 directions. Most of solid. However, only those components of stress and any such load would be reacted by fibres at # 45, if any strain for which there is a matching strain or stress were present. They would provide a far stiffer load path contribute to the distortional energy of deformation. and impose different strains-to-failure, which are cov Therefore, the criterion should not be applied to shear ered by the present analysis for fibres stresses deduced from Mohr circles, for example, but In transverse compression, the failures of unidirec only to the shear stress associated with the shear strain. tional tape laminae are akin to the collapse of too large Even for transversely isotropic solids, there are three a pile of stacked logs and little influenced by additional increments of stress for each strain, and vice versa. stress components other than transverse shear, which is Obviously, if the shear strain is constant along some not considered here. Naturally, in a well-designed lami- certain lines, the associated shear stress must also be nate with the layers of fibres in the different directions constant. The remaining increments of shear stress, at well interspersed, the fibres would be better stabilized to the Mohr circle level, have been shown in Ref. 15 to resist transverse compression loads-just as they are have no matching shear strains. This is the explanation similarly able to withstand higher longitudinal com- of the apparent inconsistency; isotropic behaviour can pressive stresses-and this cutoff would be moved out- be inferred from that for nonisotropic materials, but no ward, possibly becoming totally ineffective. Like matrix vIce versa cracking fibres under transverse-tension
absence of strains, as for hydrostatic compression of an incompressible material, there is a one-to-one relation between stress and strain for isotropic materials most of the time. This is the exception to the rule for homogeneous nonisotropic materials, however, as the equations in Jones's work9 make clear. Consider, for example, uniform heating of a transversely isotropic solid. If the coecients of thermal expansion in the principal axes dier, it is inevitable that shear strains will develop between axes inclined at 45 to the material axes, even though there are no stresses anywhere in the solid. However, only those components of stress and strain for which there is a matching strain or stress contribute to the distortional energy of deformation. Therefore, the criterion should not be applied to shear stresses deduced from Mohr circles, for example, but only to the shear stress associated with the shear strain. Even for transversely isotropic solids, there are three increments of stress for each strain, and vice versa. Obviously, if the shear strain is constant along some certain lines, the associated shear stress must also be constant. The remaining increments of shear stress, at the Mohr circle level, have been shown in Ref. 15 to have no matching shear strains. This is the explanation of the apparent inconsistency; isotropic behaviour can be inferred from that for nonisotropic materials, but not vice versa. 3 CUTOFFS IMPOSED BY MATRIX SHEAR FAILURES The failure envelopes shown in Figs 2, 3, 5 and 6 lack a roof to de®ne any limits imposed by the in-plane shear strength of the matrix between the ®bres. Since being formulated on the strain plane, the author's model has always included a non-interactive horizontal plateau, located by the shear-strain-to-failure, as shown by the lamina failure model in Fig. 7. This refers to shear with respect to ®bres in the 0 and 90 directions. Most of any such load would be reacted by ®bres at 45, if any were present. They would provide a far stier load path and impose dierent strains-to-failure, which are covered by the present analysis for ®bres. In transverse compression, the failures of unidirectional tape laminae are akin to the collapse of too large a pile of stacked logs and little in¯uenced by additional stress components other than transverse shear, which is not considered here. Naturally, in a well-designed laminate with the layers of ®bres in the dierent directions well interspersed, the ®bres would be better stabilized to resist transverse compression loadsÐjust as they are similarly able to withstand higher longitudinal compressive stressesÐand this cuto would be moved outward, possibly becoming totally ineective. Like matrix cracking between the ®bres under transverse-tension Fig. 6. Superposition of additional ®bre failure modes on basic maximum-shear-stress failure criterion. Predictions of a generalized maximum-shear-stress failure criterion 1187
1188 L.J. Hart-Smith 4 CUTOFFS IMPOSED BY TRANSVERSE CRACKS IN THE MATRIX BETWEEN THE FIBRES The authors assessment of cracking of the matrix under ransverse-tension loads is explained in Ref. 16. This is a fracture-mechanics problem, with the failure stress varying from fibre pattern to fibre pattern. There is no universal characterization equivalent to that in Fig. 7 which can be formulated on the lamina strain plane. regardless of whether the assessment is in terms of stress or strain. The nominal failing stress in the lamina, when such cracking occurs, is a function of the orientation and thickness of adjacent plies, as well as the thickness of the ply under consideration. The transverse-tension strength measured on an isolated all-90o lamina applies 712--2LT only to that isolated lamina. It is neither an upper-nor a lower-bound estimate of strength for the very same Fig. 7. Matrix shear failure cut-off for fibre/polymer laminae. lamina when it is embedded in a multidirectional struc- tural laminate. This laminate strength cannot be pre- dicted using the traditional ply-by-ply decomposition loads, this potential failure mechanism can be charac- used for other failure modes in composite laminates terized properly only at the laminate level Nevertheless. once the laminate has been defined. and Although he has developed a formula for ductile the operating environment specified, the influence of matrix failures which interact not only the stresses dis- biaxial stresses on matrix cracking can be depicted as cussed above, but also the compatible matrix stress shown in Fig. 8-in the form of a constant-transverse- developed in the matrix parallel to the fibres, the author tension cut-off. The only difficulty, at the macro level of tends to assess the stresses discussed above separatel analysis used here, is that the line can be located only by (i.e. noninteractively) because his matrix-failure theory has yet to properly account for the residual thermal stresses in the matrix caused by curing at elevated tem- peratures. These stresses, which are customarily exclu- ded from consideration by the standard assumption of omogenizing fibres and matrix to create one composite Constant transverse material, are typically very much greater than those tress components which are retained in most ana- lyses-at least at the macro level. The author's thoi on what is needed to properly characterize matrix ures in fibre/polymer composites can be found in Ref. 16. The author's empirical equation for ductile matrix failures (not cracking) under a combination of inter- active stresses. extracted from Ref. 16. is Constant longitudinal stress line G(会)+()+)=1 in which Em is the modulus of the resin matrix, El the longitudinal modulus of the unidirectional lamina, and s the measured in-plane shear strength of the lamina The os and t represent the obvious in-plane stress com- ponents in the lamina, the direction I being along the length of the fibres. This is an entirely empirical expres ion; setting the direct reference strengths at twice the =ARCTAN V shear strength is based only on Mohr circles, not curve a =ARCTAN fits to data. A further reason for excluding the first and third terms from all but isolated unidirectional laminae Fig. 8. Characterization of intralaminar matrix cracking in is that the interactions between stresses often do not fibre/polymer composites. (1) Transverse tension cracking of matrix in isolated unidirectional lamina, (2)arbitrary desigr become significant below the strain limits imposed by limit imposed for cracking in brittle matrix, (3)inoperable cut the fibres off for unattainable matrix failures in ductile matrices
loads, this potential failure mechanism can be characterized properly only at the laminate level. Although he has developed a formula16 for ductile matrix failures which interact not only the stresses discussed above, but also the compatible matrix stress developed in the matrix parallel to the ®bres, the author tends to assess the stresses discussed above separately (i.e. noninteractively) because his matrix-failure theory has yet to properly account for the residual thermal stresses in the matrix caused by curing at elevated temperatures. These stresses, which are customarily excluded from consideration by the standard assumption of homogenizing ®bres and matrix to create one composite material, are typically very much greater than those stress components which are retained in most analysesÐat least at the macro level. The author's thoughts on what is needed to properly characterize matrix failures in ®bre/polymer composites can be found in Ref. 16. The author's empirical equation for ductile matrix failures (not cracking) under a combination of interactive stresses, extracted from Ref. 16, is 1 2S Em EL 2 12 S 2 2 2S 2 1 15 in which Em is the modulus of the resin matrix, EL the longitudinal modulus of the unidirectional lamina, and S the measured in-plane shear strength of the lamina. The s and represent the obvious in-plane stress components in the lamina, the direction 1 being along the length of the ®bres. This is an entirely empirical expression; setting the direct reference strengths at twice the shear strength is based only on Mohr circles, not curve ®ts to data. A further reason for excluding the ®rst and third terms from all but isolated unidirectional laminae is that the interactions between stresses often do not become signi®cant below the strain limits imposed by the ®bres. 4 CUTOFFS IMPOSED BY TRANSVERSE CRACKS IN THE MATRIX BETWEEN THE FIBRES The author's assessment of cracking of the matrix under transverse-tension loads is explained in Ref. 16. This is a fracture-mechanics problem, with the failure stress varying from ®bre pattern to ®bre pattern. There is no universal characterization equivalent to that in Fig. 7 which can be formulated on the lamina strain plane, regardless of whether the assessment is in terms of stress or strain. The nominal failing stress in the lamina, when such cracking occurs, is a function of the orientation and thickness of adjacent plies, as well as the thickness of the ply under consideration. The transverse-tension strength measured on an isolated all-90 lamina applies only to that isolated lamina. It is neither an upper- nor a lower-bound estimate of strength for the very same lamina when it is embedded in a multidirectional structural laminate. This laminate strength cannot be predicted using the traditional ply-by-ply decomposition used for other failure modes in composite laminates. Nevertheless, once the laminate has been de®ned, and the operating environment speci®ed, the in¯uence of biaxial stresses on matrix cracking can be depicted as shown in Fig. 8Ðin the form of a constant-transversetension cut-o. The only diculty, at the macro level of analysis used here, is that the line can be located only by Fig. 7. Matrix shear failure cut-o for ®bre/polymer laminae. Fig. 8. Characterization of intralaminar matrix cracking in ®bre/polymer composites. (1) Transverse tension cracking of matrix in isolated unidirectional lamina, (2) arbitrary design limit imposed for cracking in brittle matrix, (3) inoperable cuto for unattainable matrix failures in ductile matrices. 1188 L. J. Hart-Smith