Predictions of a generalized maximum-shear-stress failure criterion l181 which failed last. In the models developed by both inance. It cannot be defined by consideration of fibre authors, the comparison between competing failure failures alone modes must necessarily be effected at a common strain reference-in each lamina. Additional fibre- or matrix failure modes dded to either model by super- 2 THE GENERALIZED MAXIMUM-SHEAR- position, not by interaction. Each mechanism governs STRESS FAILURE MODEL FOR FIBRES throughout a limited range of stresses-and none inter- acts with any other, even though individual stress com- Given that carbon fibres are transversely isotropic, and ponents may interact within a single failure mechanism. that glass fibres are essentially completely isotropic, any Strength predictions by brittle fracture, from small shear-failure mechanism would have the same critical Ind large flaws, and ductile failures in the same metals conditions for both the longitudinal-transverse (L-T) have co-existed for decades, the choice being dictated by and longitudinal-normal (L-n) planes within the fibres the state of the applied stress and the degree of alloying It is possible that, since carbon fibres are orthotropic, and heat treatment of the metals. Why should carbon the critical shear strain needed to cause failure in the fibres be so unique as to be required not to behave transverse-normal (T-N) plane may not be the same as similarly? And, given that glass fibres are even isotropic, for the other two planes. For this reason, the T-N cut why should this most common mechanism of failure fTs shown in earlier presentations of the author's theory shear, have been excluded from fibrous composite fail- have been relocated, to a parallel but possibly offset ure analyses? The author has never wavered in his belief position beyond the original failure envelope. This is that it shouldn,t be. Progress in the development of this unlikely to have any effect on the in-plane strengths failure model over the years, coupled with objections, predicted for fibre/polymer composite laminates, and constructive criticism, and help from many other done only because doing so simplifies the application of researchers around the world have strengthened the the analysis to the present problems and because it authors belief that only mechanistic failure models are might be necessary for assessing the response of com- appropriate for predicting the strength of fibre/polymer posites to transverse shear or other out-of-plane loads. composites-or any other material, for that matter The simplified failure envelope for the fibres is shown Before summarizing his theory and demonstrating in Fig. 2, for glass fibres on the left and carbon(and ow it can be used to solve at least some of the prob- other transversely isotropic)fibres on the right, drawn lems of the failure exercise described in Ref. 4, the to scale, using data provided in Ref. 4. Since glass is author would like to take this opportunity to express his isotropic, the failure envelope has the same form as for appreciation of the invitation to participate in the com- ductile isotropic metals. The corresponding corner parison and his hope that their goals will be achieved. points are labelled, to show equivalences and to identify The efforts made by the many participants certainly the associated states of stress. The entire shear-failure merit a successful outcome envelope for glass fibres can be constructed from a sin The nature of this failure model is that most of its gle measured strength(or strain to failure)because the predictions must be bounded between those of the two failure mechanism is prescribed to be constant around theories covered in a companion paper/ involving the the entire perimeter. Other than this one reference original and truncated maximum-strain failure models. strength, the only other quantities needed to construct (There are some minor exceptions, associated with the failure envelopes are the Poisson ratios, VLT (E V12) changing from a constant-strain to constant-stress cut- and vTL ( v21), to define the slopes of the constant off for compressive loads parallel to the fibre. Even if stress lines. If it is assumed that there is only one critical his best guesses at some of the matrix -failures prove to shear-strain-to-failure for transversely isotropic(car- be wide of the mark, just trying to solve the problems bon-type) fibres as well, the same can be said for all has accelerated the authors own learning of the subject fibres. The diagram on the right of Fig. 2 shows addi- and exposed just how fortunate he has been to have tional cut-offs(line IJ and its mirror image)for the 2-3 worked exclusively in a world which did not require plane transverse to the fibre axis in the event that the ch a focus on the more complicated portions of this failure strains are unequal discipline which have been encountered in other indus- The next step of the analysis has relied upon a stan tries. Despite the risk of discrediting his fibre-based dard simplifying assumption-that plane sections composite failure theory by making predictions about remain plane and that, therefore, the transverse strain matrix-dominated failures under circumstances for developed in the fibre and the matrix is much the same which he has absolutely no prior experiences to guide as that developed in each lamina. This is a reasonably him, the author has included his assessments of matrix accurate approximation for carbon/epoxy laminates, failures in the belief that doing so would at least because the fibres are so highly orthotropic, but is better contribute to the technology by exposing those areas in regarded as a conservative design procedure for glass- which more work needs to be done. Particularly in the fibre-reinforced plastics. Strictly, since the 45. sloping case of the±s5° laminate, the failure envelope lines in Fig. 2 refer to the fibres, they cannot also refer to defined by alternating regimes of fibre and matrix dom the composite laminae-unless the relevant moduliwhich failed last. In the models developed by both authors, the comparison between competing failure modes must necessarily be e€ected at a common strain referenceÐin each lamina. Additional ®bre- or matrix￾failure modes are added to either model by super￾position, not by interaction. Each mechanism governs throughout a limited range of stressesÐand none inter￾acts with any other, even though individual stress com￾ponents may interact within a single failure mechanism. Strength predictions by brittle fracture, from small and large ¯aws, and ductile failures in the same metals have co-existed for decades, the choice being dictated by the state of the applied stress and the degree of alloying and heat treatment of the metals. Why should carbon ®bres be so unique as to be required not to behave similarly? And, given that glass ®bres are even isotropic, why should this most common mechanism of failure, shear, have been excluded from ®brous composite fail￾ure analyses? The author has never wavered in his belief that it shouldn't be. Progress in the development of this failure model over the years, coupled with objections, constructive criticism, and help from many other researchers around the world have strengthened the author's belief that only mechanistic failure models are appropriate for predicting the strength of ®bre/polymer compositesÐor any other material, for that matter. Before summarizing his theory and demonstrating how it can be used to solve at least some of the prob￾lems of the failure exercise described in Ref. 4, the author would like to take this opportunity to express his appreciation of the invitation to participate in the com￾parison and his hope that their goals will be achieved. The e€orts made by the many participants certainly merit a successful outcome. The nature of this failure model is that most of its predictions must be bounded between those of the two theories covered in a companion paper7 involving the original and truncated maximum-strain failure models. (There are some minor exceptions, associated with changing from a constant-strain to constant-stress cut￾o€ for compressive loads parallel to the ®bre.) Even if his best guesses at some of the matrix-failures prove to be wide of the mark, just trying to solve the problems has accelerated the author's own learning of the subject and exposed just how fortunate he has been to have worked exclusively in a world which did not require such a focus on the more complicated portions of this discipline which have been encountered in other indus￾tries. Despite the risk of discrediting his ®bre-based composite failure theory by making predictions about matrix-dominated failures under circumstances for which he has absolutely no prior experiences to guide him, the author has included his assessments of matrix failures in the belief that doing so would at least contribute to the technology by exposing those areas in which more work needs to be done. Particularly in the case of the ‹55 laminate, the failure envelope is de®ned by alternating reÂgimes of ®bre and matrix dom￾inance. It cannot be de®ned by consideration of ®bre failures alone. 2 THE GENERALIZED MAXIMUM-SHEAR￾STRESS FAILURE MODEL FOR FIBRES Given that carbon ®bres are transversely isotropic, and that glass ®bres are essentially completely isotropic, any shear-failure mechanism would have the same critical conditions for both the longitudinal±transverse (L±T) and longitudinal±normal (L±N) planes within the ®bres. It is possible that, since carbon ®bres are orthotropic, the critical shear strain needed to cause failure in the transverse±normal (T±N) plane may not be the same as for the other two planes. For this reason, the T±N cut￾o€s shown in earlier presentations of the author's theory have been relocated, to a parallel but possibly o€set position beyond the original failure envelope. This is unlikely to have any e€ect on the in-plane strengths predicted for ®bre/polymer composite laminates, and is done only because doing so simpli®es the application of the analysis to the present problems and because it might be necessary for assessing the response of com￾posites to transverse shear or other out-of-plane loads. The simpli®ed failure envelope for the ®bres is shown in Fig. 2, for glass ®bres on the left and carbon (and other transversely isotropic) ®bres on the right, drawn to scale, using data provided in Ref. 4. Since glass is isotropic, the failure envelope has the same form as for ductile isotropic metals. The corresponding corner points are labelled, to show equivalences and to identify the associated states of stress. The entire shear-failure envelope for glass ®bres can be constructed from a sin￾gle measured strength (or strain to failure) because the failure mechanism is prescribed to be constant around the entire perimeter. Other than this one reference strength, the only other quantities needed to construct the failure envelopes are the Poisson ratios, LT (ˆ 12) and TL (ˆ 21), to de®ne the slopes of the constant￾stress lines. If it is assumed that there is only one critical shear-strain-to-failure for transversely isotropic (car￾bon-type) ®bres as well, the same can be said for all ®bres. The diagram on the right of Fig. 2 shows addi￾tional cut-o€s (line IJ and its mirror image) for the 2±3 plane transverse to the ®bre axis in the event that the failure strains are unequal. The next step of the analysis has relied upon a stan￾dard simplifying assumptionÐthat plane sections remain plane and that, therefore, the transverse strain developed in the ®bre and the matrix is much the same as that developed in each lamina. This is a reasonably accurate approximation for carbon/epoxy laminates, because the ®bres are so highly orthotropic, but is better regarded as a conservative design procedure for glass- ®bre-reinforced plastics. Strictly, since the 45 sloping lines in Fig. 2 refer to the ®bres, they cannot also refer to the composite laminaeÐunless the relevant moduli Predictions of a generalized maximum-shear-stress failure criterion 1181