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 ˆ f TL 1 ‡ f LT 1 ‡ f TL  "t L; "T ˆ 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 carbon￾epoxy 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-®bre￾reinforced laminates. This same kind of modi®cation to the e€ective long￾itudinal 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 ˆ "T ˆ "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 e€ect of accounting for the di€erences in sti€ness 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 corre￾sponding solutions in Ref. 7. This similarity would be particularly strong if it were known that the actual fail￾ures 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 sti€ness 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 charac￾terization of glass-®bre failures on the lamina or laminate Fig. 5. Fibre shear failure cut-o€s on lamina strain plane. (Dashed lines for E-glass/MY750 epoxy lamina and con￾tinuous 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 evaluat￾ing glass-®bre laminates, beyond the author's prior experience. Predictions of a generalized maximum-shear-stress failure criterion 1185