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 thestrain planes is not incompatible with the author's 45 cuto€s for carbon/epoxy laminates in the tension-com￾pression quadrants. Both are close approximations, not precise answers. The 45 cuto€ would still exist at the ®bre level for both composite materials, but would sim￾ply 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 cuto€s for ®bre failures by brittle fracture, which is a constant-stress phenomenon because crack-tip stress intensities are una€ected by stresses parallel to the crack (transverse to the ®bre), and compressive instability, which is also a constant￾stress phenomenon,* are superimposed locally on the basic shear-failure envelopes. These three possible fail￾ure mechanisms are all that are considered for ®bre failures in the author's analyses of in-plane loads. A di€erence between the longitudinal tensile and compressive strengths of unidirectional laminae should be interpreted as implying that at least one of the fail￾ures 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 di€erent 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 fail￾ures is by shear, (as is quite likely for E-glass ®bres), one could perform a shear test on a ‹45 laminate, to gen￾erate 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 repre￾sented 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 bon￾ded 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 equiva￾lent 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 trans￾verse 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 indivi￾dual layer. Therefore, once the sti€ness 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 adjus￾ted accordingly; even the nonlinearities can be repli￾cated. 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 sti€ness of each equivalent ply would be the same as for a real unidirec￾tional 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 a€ect the longitudinal sti€ness but would not a€ect the transverse sti€ness within each tow of ®bres. (There would be a minor e€ect because of the in-plane separation of the tows of ®bres which would be ®lled with a di€erent 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 sti€ness by whatever degree of crimping was introduced by the weaving process and for the di€erence between tensile and compressive strengths which is exa￾cerbated by this same crimping. Some readers of earlier articles on the generalization of the maximum-shear-stress failure criterion to non￾isotropic homogeneous materials have expressed di- culty in accepting the concept of a 45-sloping constant￾shear-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 di€erences between principal stres￾ses 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 di€erence 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 una€ected by the simultaneous appli￾cation of transverse stresses. The laminate stress at which this would happen would vary with the ®bre pattern. The short￾ening 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 con￾stant-stress cut-o€ for the lamina automatically accounts for this e€ect. 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