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-tensionabsence 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 homo￾geneous nonisotropic materials, however, as the equa￾tions in Jones's work9 make clear. Consider, for example, uniform heating of a transversely isotropic solid. If the coecients of thermal expansion in the principal axes di€er, 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 sti€er load path and impose di€erent strains-to-failure, which are cov￾ered by the present analysis for ®bres. In transverse compression, the failures of unidirec￾tional 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 lami￾nate with the layers of ®bres in the di€erent directions well interspersed, the ®bres would be better stabilized to resist transverse compression loadsÐjust as they are similarly able to withstand higher longitudinal com￾pressive stressesÐand this cuto€ would be moved out￾ward, possibly becoming totally ine€ective. 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