Composites Science and Technology 58(1998 c 1998 Published by Elsevier Science Ltd. All rig Printed in G PII:s0266-3538(97)00193-0 0266-353898s PREDICTIONS OF A GENERALIZED MAXIMUM-SHEAR- STRESS FAILURE CRITERION FOR CERTAIN FIBROUS COMPOSITE LAMINATES L.J. Hart-Smith Douglas Products Division, Boeing Commercial Airplane Group, Long Beach, California, USA (Received 6 November 1995; revised 8 April 1996: accepted 23 September 1997) Abstract between these strains) for carbon/ epoxy laminates and The use of the author's generalization of the maximum- shown that the difference does need to be accounted for shear-stress failure criterion for fibre/polymer composites with glass-fibre laminates is illustrated by sample solutions of specific problems The theory did not evolve instantaneously. Indeed, it provided by the organizers of the world-wide failure passed through a phase in which it was expressed on the exercise. New refinements of the theory justify an earlier stress plane, like so many other composite failure the approximation of it for use with carbon/epoxy laminates and ories, before the benefits of expressing it on the strain remove a degree of conservatism when the original theory plane instead became apparent. As the development as applied to glass-fibre-reinforced polymer composites progressed, it became clear that there were fundamental The intent of this exercise is to compare the independent irrecoverable errors in the many published and coded predictions for these same problems made by several origi- interactive failure theories for composites, and an added nators of composite failure models and, simultaneously, to goal has been to lay scientific foundations for future compare the predictions with test data. C 1998 Published failure models of all inevitably heterogeneous composite by Elsevier Science Ltd. All rights reserved materials by emphasizing mechanistic models and shunning the abstract mathematical models developed Keywords: composite laminate strength, lamina failure on the false assumption that composites of materials criteria. fibre shear failures could be regarded as homogeneous anisotropic solids This simplification is appropriate for computing stifi- nesses, but not for strengths. In a fibre/polymer compo- 1 INTRODUCTION site, for example, only a fibre, the matrix, or an interface can fail-and separate characterizations are required for The origin of the author's generalization of the classical each of these mechanisms, as indicated in Fig. 1. Indeed maximum-shear-stress yield or failure criterion for multiple characterizations are sometimes required for metals to fibre/polymer composites can be traced back each constituent of the composite, because more than to his recognition in 1983 that the highest measure- one mechanism of failure can occur(depending on the ments of the fibre-dominated in-plane shear strength of state of stress) and a separate characterization is a+45 T-300/N5208 carbon/epoxy laminate were required for each of these, also Fibres can fail by shear, almost precisely half of the uniaxial tension or com- as is indicated by the same longitudinal tensile or com pression strength of the corresponding 00/90 laminates. pressive strengths, by compressive instability, or by No composite failure theory of the day predicted this. brittle fracture. The matrix can fail by ductile shear* Indeed. no other one does so even now. Yet the shear strength of ductile metals has been known for centuries What is apparently ductile shear at the macroscopic level is to be close to half the tension or compression strength actually better characterized at the microscopic level as linearly The authors composite failure model is simply an elastic behaviour at the lower stiffness remaining when many attempt to develop an equivalent analysis method for the matrix. What is actually transmitting the shear load from fibre to fibrous composite laminates fibre is a series of discrete ligaments of matrix. These cracks occur described in several references (e.g. Refs 2 and 3). It is 4 to th fibre axes, and are stable once a saturation en the summarized here because significant improvements were load is removed. The author is indebted to professor made while solving the problems posed by the organi- for explaining this to him. There is no permanent set of the kind zers of the failure exercise. 4 This refinement distin associated with ductile yielding of metallic alloys. Regardless of guishing between the transverse strain in each lamina the physics of the situation, all that needs to be noted for macro level analyses is that, after the first few load cycles, the in-plane and that in the fibres, has confirmed the validity of the shear stiffness is more accurately given by the secant modulus at original formulation (in which there was no distinction failure than by the initial tangent modulusPREDICTIONS OF A GENERALIZED MAXIMUM-SHEAR￾STRESS FAILURE CRITERION FOR CERTAIN FIBROUS COMPOSITE LAMINATES L. J. Hart-Smith Douglas Products Division, Boeing Commercial Airplane Group, Long Beach, California, USA (Received 6 November 1995; revised 8 April 1996; accepted 23 September 1997) Abstract The use of the author's generalization of the maximum￾shear-stress failure criterion for ®bre/polymer composites is illustrated by sample solutions of speci®c problems provided by the organizers of the world-wide failure exercise. New re®nements of the theory justify an earlier approximation of it for use with carbon/epoxy laminates and remove a degree of conservatism when the original theory was applied to glass-®bre-reinforced polymer composites. The intent of this exercise is to compare the independent predictions for these same problems made by several origi￾nators of composite failure models and, simultaneously, to compare the predictions with test data. # 1998 Published by Elsevier Science Ltd. All rights reserved Keywords: composite laminate strength, lamina failure criteria, ®bre shear failures 1 INTRODUCTION The origin of the author's generalization of the classical maximum-shear-stress yield or failure criterion for metals to ®bre/polymer composites can be traced back to his recognition in 19831 that the highest measure￾ments of the ®bre-dominated in-plane shear strength of a ‹45 T-300/N5208 carbon/epoxy laminate were almost precisely half of the uniaxial tension or com￾pression strength of the corresponding 0/90 laminates. No composite failure theory of the day predicted this. Indeed, no other one does so even now. Yet the shear strength of ductile metals has been known for centuries to be close to half the tension or compression strength. The author's composite failure model is simply an attempt to develop an equivalent analysis method for ®brous composite laminates. The author's ®bre-dominated theory has already been described in several references (e.g. Refs 2 and 3). It is summarized here because signi®cant improvements were made while solving the problems posed by the organi￾zers of the failure exercise.4 This re®nement, distin￾guishing between the transverse strain in each lamina and that in the ®bres, has con®rmed the validity of the original formulation (in which there was no distinction between these strains) for carbon/epoxy laminates and shown that the di€erence does need to be accounted for with glass-®bre laminates. The theory did not evolve instantaneously. Indeed, it passed through a phase in which it was expressed on the stress plane, like so many other composite failure the￾ories, before the bene®ts of expressing it on the strain plane instead became apparent. As the development progressed, it became clear that there were fundamental irrecoverable errors in the many published and coded interactive failure theories for composites, and an added goal has been to lay scienti®c foundations for future failure models of all inevitably heterogeneous composite materials by emphasizing mechanistic models and shunning the abstract mathematical models developed on the false assumption that composites of materials could be regarded as homogeneous anisotropic solids. This simpli®cation is appropriate for computing sti€- nesses, but not for strengths. In a ®bre/polymer compo￾site, for example, only a ®bre, the matrix, or an interface can failÐand separate characterizations are required for each of these mechanisms, as indicated in Fig. 1. Indeed, multiple characterizations are sometimes required for each constituent of the composite, because more than one mechanism of failure can occur (depending on the state of stress) and a separate characterization is required for each of these, also. Fibres can fail by shear, as is indicated by the same longitudinal tensile or com￾pressive strengths, by compressive instability, or by brittle fracture. The matrix can fail by ductile shear* Composites Science and Technology 58 (1998) 1179±1208 # 1998 Published by Elsevier Science Ltd. All rights reserved Printed in Great Britain PII: S0266-3538(97)00193-0 0266-3538/98 $Ðsee front matter 1179 *What is apparently ductile shear at the macroscopic level is actually better characterized at the microscopic level as linearly elastic behaviour at the lower sti€ness remaining when many inclined microcracks have spread from ®bre to ®bre throughout the matrix. What is actually transmitting the shear load from ®bre to ®bre is a series of discrete ligaments of matrix. These cracks occur under the resolved tensile component of the applied shear load, at 45 to the ®bre axes, and are stable once a saturation density has been established. A virtually full elastic recovery is made when the load is removed. The author is indebted to Professor Alfred Puck for explaining this to him. There is no permanent set of the kind associated with ductile yielding of metallic alloys. Regardless of the physics of the situation, all that needs to be noted for macro level analyses is that, after the ®rst few load cycles, the in-plane shear sti€ness is more accurately given by the secant modulus at failure than by the initial tangent modulus