《复合材料 Composites》课程教学资源（学习资料）第五章 陶瓷基复合材料_fibrous molithic-34

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 modulus

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 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

l180 L.J. Hart-Smith SEPARATE CHARACTERIZATIONS ARE NEEDED FOR EACH FAILURE MECHANISM IN EACH CONSTITUENT OF THE COMPOSITE OF MAT INTERACTIONS BETWEEN STRESSES AFFECTING THE SAME FAILURE MODE IN THE SAME CONSTITUENT OF THE COMPOSITE ARE PERMITTED INTERACTIONS BETWEEN DIFFERENT FAILURE MODES ARE SCIENTIFICALLY INCORRECT TYPICAL FAILURE MECHANISMS FOR FIBRE-POLY MER COMPOSITES FRACTURE OF FIBRES AT FLAWS AND DEFECTS, UNDER LONGITUDINAL TENSION FAILURE OF FIBRES REMOTE FROM ANY FLAWS OR DEFECTS, UNDER TENSILE LOADS MICRO-INSTABILITY, OR KINKING, OF FIBRES UNDER COMPRESSIVE LOADS SHEAR FAILURE OF WELL-STABILIZED FIBRES UNDER COMPRESSIVE LOADS DUCTILE FAILURE OF MATRIX, WITHOUT CRACKING, UNDER IN-PLANE LOADS CRACKING OF MATRIX BETWEEN THE FIBRES UNDER TRANSVERSE-TENSION LOADS, VHICH INVOLVES BOTH A MATERIAL PROPERTY AND A GEOMETRIC PARAMETER INTERFACIAL FAILURE BETWEEN THE FIBRES AND THE MATRIX INTERLAMINAR FAILURE OF MATRIX AT EDGES AND DISCONTINUITIES DELAMINATIONS BETWEEN THE PLIES UNDER IMPACT OR TRANSVERSE SHEAR LOADS DELAMINATIONS BETWEEN THICK PLIES INITIATING AT THROUGH-THICKNESS MATRIX RACKS WITHIN A TRANSVERSE PLY FATIGUE FAILURES IN THIN PLIES CAUSED BY THROUGH-THICKNESS TRANSVERSE CRACKS IN ADJACENT THICK PLIES EACH OF THESE POSSIBILITIES REQUIRES ITS OWN EQUATION, EVEN THOUGH NOT EVERY MODE CAN OCCUR FOR EVERY FIBRE-POLY MER COMBINATION AND EVEN THOUGH SOME MODES CAN BE SUPPRESSED BY SKILLFUL SELECTION OF THE STACKING SEQUENCE Fig. 1. Specification for fibre/polymer composite failure criteria. (under predominantly shear and transverse-compression obviously necessary re-assessment of some of the best loads) or by brittle fracture whenever the transverse- known composite failure models tension stress between the fibres is sufficient to cause In any event, the author's composite failure model microcracks in the resin matrix which are parallel to the when first formulated on the strain plane several years fibres to fast fracture Matrix failures are also influenced ago, accounted for all three possible fibre-failure od /aditiona of which is automatically precluded by terize possible failures of the matrix. The reason for this by residual thermal stresses within each lamina-the mechanisms cited above, but made no attempt to charac the traditional false assumption of homogeneity within was that the author worked with carbon/epoxy compo each lamina sites in the aerospace industry and it was not difficult to Cracking of the matrix between the fibres, which is at establish simple design rules which would ensure that the core of all progressive-failure and ply-discounting the strength of the fibres which carried most of the load models, is particularly difficult to cope with analytically would not be undercut by premature failures of the because, in contrast with the basic premise of laminated matrix which was there to stabilize the fibres, not to composite strength predictions-that each and every ply carry significant load itself. Consequently, the author's can be assessed independently of all others-matrix forays into real matrix failures, as contrasted with those cracking is influenced by the adjacent plies and cannot predicted to occur by so many interactive theories but be analyzed the same way. There must be a geometric which actually do not occur at either the stress levels or factor in the analysis as well, just as is the case for all densities calculated, have lagged behind his efforts in fracture-mechanics analyses of cracks in homogeneous regard to fibre failures and publicizing the need for materials. To assume otherwise is to imply that boron/ mechanistic failure models. Other investigators, most epoxy crack-patching of metallic aircraft could not notably Puck have worked more with glass-fibre-rein possibly extend the life of aircraft by retarding the forced plastics and have been unable to avoid the need growth of the cracks. Had the design of boron/epoxy to characterize these added failure mechanisms. It was rack patches relied on traditional composite stress Puck who first formulated a mechanics-based composite analysis techniques for laminated structures, it is clear strength-prediction theory in which failure of the fibres that the concept would never have been initiated since, and matrix were covered by separate equations. It is not without exception, these theories predict that no benefit at all surprising that, given his focus on glass-fibre- could possibly be obtained. Unfortunately, the fact that reinforced plastics, he has developed a far more com- such benefits have been demonstrated, many times(for prehensive model for matrix failures than the authors example by the pioneering work of Baker and his col- and could rely on the simpler maximum-strain fibre- leagues as in Ref 5), does not seem to have caused the failure model for those elements of 'his'composites

(under predominantly shear and transverse-compression loads) or by brittle fracture whenever the transverse￾tension stress between the ®bres is sucient to cause microcracks in the resin matrix which are parallel to the ®bres to fast fracture. Matrix failures are also in¯uenced by residual thermal stresses within each laminaÐthe consideration of which is automatically precluded by the traditional false assumption of homogeneity within each lamina. Cracking of the matrix between the ®bres, which is at the core of all progressive-failure and ply-discounting models, is particularly dicult to cope with analytically because, in contrast with the basic premise of laminated composite strength predictionsÐthat each and every ply can be assessed independently of all othersÐmatrix cracking is in¯uenced by the adjacent plies and cannot be analyzed the same way. There must be a geometric factor in the analysis as well, just as is the case for all fracture-mechanics analyses of cracks in homogeneous materials. To assume otherwise is to imply that boron/ epoxy crack-patching of metallic aircraft could not possibly extend the life of aircraft by retarding the growth of the cracks. Had the design of boron/epoxy crack patches relied on traditional composite stress analysis techniques for laminated structures, it is clear that the concept would never have been initiated since, without exception, these theories predict that no bene®t could possibly be obtained. Unfortunately, the fact that such bene®ts have been demonstrated, many times (for example by the pioneering work of Baker and his col￾leagues as in Ref. 5), does not seem to have caused the obviously necessary re-assessment of some of the best￾known composite failure models. In any event, the author's composite failure model, when ®rst formulated on the strain plane several years ago, accounted for all three possible ®bre-failure mechanisms cited above, but made no attempt to charac￾terize possible failures of the matrix. The reason for this was that the author worked with carbon/epoxy compo￾sites in the aerospace industry and it was not dicult to establish simple design rules which would ensure that the strength of the ®bres which carried most of the load would not be undercut by premature failures of the matrix which was there to stabilize the ®bres, not to carry signi®cant load itself. Consequently, the author's forays into real matrix failures, as contrasted with those predicted to occur by so many interactive theories but which actually do not occur at either the stress levels or densities calculated, have lagged behind his e€orts in regard to ®bre failures and publicizing the need for mechanistic failure models. Other investigators, most notably Puck6 have worked more with glass-®bre-rein￾forced plastics and have been unable to avoid the need to characterize these added failure mechanisms. It was Puck who ®rst formulated a mechanics-based composite strength-prediction theory in which failure of the ®bres and matrix were covered by separate equations. It is not at all surprising that, given his focus on glass-®bre￾reinforced plastics, he has developed a far more com￾prehensive model for matrix failures than the author's and could rely on the simpler maximum-strain ®bre￾failure model for those elements of `his' composites Fig. 1. Speci®cation for ®bre/polymer composite failure criteria. 1180 L. J. Hart-Smith

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 moduli

which 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

l182 L.J. Hart-Smith match. This is explained in Fig. 3 where, for the first point (3), as shown. It remains only to compute the time in the author's works, the relationship between the associated transverse strain in the lamina, at point (4) transverse strains in the fibres and lamina is derived Strictly, this is a complicated micromechanical problem The points (1),(2), (3), and (4)in Fig 3 refer to the steps However, with the same model as was employed by in creating the accurate cut-off 1-4, rather than the ear- Chamis to derive an expression relating the transverse lier cut-off passing through the measured point I at a strain in the matrix between the fibres to the average slope of 45 lamina strain, the author has derived the following sim e The first step in constructing the failure envelope for ple solution for the corresponding strain ratio between embedded. rather than isolated. fibre is to draw the lamina and the fibres The formula results from an radial lines from the origin at slopes defined by VgT and assessment of the compatibility of deformations along USTL for the fibres, and VLT for the lamina, as shown in transverse axis through the middle of a fibre Fig 3 Point(1)on the lamina shear-stress cutoff line is at the uniaxial longitudinal strain point m=√+(-xD)1一动)+m EL =EL, ET=-VLTEL R A vertical line is then drawn through the measured strain-to-failure of the fibres, El, which is assumed to be the same for both the lamina and the embedded fibre. a Here, V is the fibre volume fraction, Vm the single 45-sloping line, denoting constant shear strain, is then Poisson ratio for the resin matrix, VTL the minor Pois passed through the uniaxial-tension failure point(2)for son ratio for an isolated fibre, E is the modulus of the ne fibre, which can occur at a different transverse strain resin matrix, and err is the corresponding transverse than that for the lamina reinforced by unidirectional modulus of the individual fibres. The transverse strain. fibres, because the two major Poisson ratios need not be ET, is the strain in the lamina, not the matrix, and Efr is the same. This sloping line will cross the purely trans- the transverse strain in the fibre. The coefficient K is a verse-stress line for the fibre close to the vertical axis at function of the fibre array, being [(2 3)/] for circular B=arctan v a= arctan VrT c= arctan VILT 450° A450 450° Possible positions for isotropic glass fibres Transversely isotropic carbon fibres Fig. 2. Strain-based failure envelopes for glass and carbon fibres, according to a generalization of the classical maximum stress criterion

match. This is explained in Fig. 3 where, for the ®rst time in the author's works, the relationship between the transverse strains in the ®bres and lamina is derived. The points (1), (2), (3), and (4) in Fig. 3 refer to the steps in creating the accurate cut-o€ 1±4, rather than the ear￾lier cut-o€ passing through the measured point 1 at a slope of 45. The ®rst step in constructing the failure envelope for an embedded, rather than isolated, ®bre is to draw radial lines from the origin at slopes de®ned by fLT and f TL for the ®bres, and LT for the lamina, as shown in Fig. 3. Point (1) on the lamina shear-stress cuto€ line is at the uniaxial longitudinal strain point. "L ˆ "t L; "T ˆ ÿLT"L …1† A vertical line is then drawn through the measured strain-to-failure of the ®bres, "t L, which is assumed to be the same for both the lamina and the embedded ®bre. A 45-sloping line, denoting constant shear strain, is then passed through the uniaxial-tension failure point (2) for the ®bre, which can occur at a di€erent transverse strain than that for the lamina reinforced by unidirectional ®bres, because the two major Poisson ratios need not be the same. This sloping line will cross the purely trans￾verse-stress line for the ®bre close to the vertical axis at point (3), as shown. It remains only to compute the associated transverse strain in the lamina, at point (4). Strictly, this is a complicated micromechanical problem. However, with the same model as was employed by Chamis8 to derive an expression relating the transverse strain in the matrix between the ®bres to the average lamina strain, the author has derived the following sim￾ple solution for the corresponding strain ratio between the lamina and the ®bres. The formula results from an assessment of the compatibility of deformations along a transverse axis through the middle of a ®bre. "T "f T ˆ  KVf p ‡ 1 ÿ  KVf   p 1 ÿ 2 m ÿ
Ef T Em ‡ f TLm   ˆ R" …2† Here, Vf is the ®bre volume fraction, m the single Poisson ratio for the resin matrix, f TL the minor Pois￾son ratio for an isolated ®bre, Em is the modulus of the resin matrix, and Ef T is the corresponding transverse modulus of the individual ®bres. The transverse strain, "T, is the strain in the lamina, not the matrix, and "f T is the transverse strain in the ®bre. The coecient K is a function of the ®bre array, being [(2p3)/] for circular Fig. 2. Strain-based failure envelopes for glass and carbon ®bres, according to a generalization of the classical maximum stress criterion. 1182 L. J. Hart-Smith

Predictions of a generalized maximum-shear-stress failure criterion l183 fibres in a hexagonal array, 4/I for circular fibres in a along the transverse axis through the middle of each square array, and unity for square fibres in a square fibre, of 'diameter'd are array. In other words, it has a value close to unity regardless of the stacking array. The effect of the value Elam=Emr(1-d)+errd (3) of K on Re is not large, being greater for typical com- posites when K is least, particularly when Re is much where the subscripts lam, m, and f refer to the lamina eater than unity. This equation satisfies obvious sanity matrix, and fibre, respectively. The axial strain in the checks that the strains are equal when the two stiffnesses fibre for the particular state in which it has zero axial match, regardless of the fibre content, and that the ratio stress is, by definition, -y/TLEST, at point(3)in Fig. 3 is infinite for zero matrix stiffnes The matrix is prescribed to undergo the same strain The derivation of eqn(2)is as follows, using the ter- along the axis of the fibres. The transverse stress in each minology in Fig. 4. It is not necessary to assume that the constituent of the composite would then follow from transverse stress is uniform throughout the thickness of standard equations, of the type given in the standard each lamina, only that it is constant along the datum text by Jonesas through the middle of the fibres (Obviously, this stress will be less on other strata where the matrix makes up (E2+2E1) where=1-u221.(4) more of the total composite of materials, being lowest for any load path passing entirely through the matrix and totally missing the stiffer fibres. The lamina strains With el defined to match the state of zero axial stress in the fibres, as above, and o2 taken as constant through out both fibre and matrix on this particular plane, it follows that [EmT +VmE1]=[EST+VLTEI (5) enc Em Er EsT Em whence u2 Substitution of eqn()into eqn(3)then yields =d+(-d E5+mz1(8) from which eqn(2)follows directly, once the fibre 'dia meter'd is related to the fibre volume fraction V as a function of the fibre array. (Equation( 8)also satisfies the obvious sanity checks for equal fibre and matrix a =arctan properties and for zero matrix stiffness. The effective fibre diameter is related to the form of ratan the array, as is explained in Fig. 4 rotan y d=vv for a square fibre in a square array. B=B+/R 6 ≠ arctan √3 Fig. 3. Conversion of 45 slope for fibre failure on fibre strain plane to corresponding line on lamina strain plane for round fibres in a hexagonal array

®bres in a hexagonal array, 4= for circular ®bres in a square array, and unity for square ®bres in a square array. In other words, it has a value close to unity regardless of the stacking array. The e€ect of the value of K on R" is not large, being greater for typical com￾posites when K is least, particularly when R" is much greater than unity. This equation satis®es obvious sanity checks that the strains are equal when the two sti€nesses match, regardless of the ®bre content, and that the ratio is in®nite for zero matrix sti€ness. The derivation of eqn (2) is as follows, using the ter￾minology in Fig. 4. It is not necessary to assume that the transverse stress is uniform throughout the thickness of each lamina, only that it is constant along the datum through the middle of the ®bres. (Obviously, this stress will be less on other strata where the matrix makes up more of the total composite of materials, being lowest for any load path passing entirely through the matrix and totally missing the sti€er ®bres.) The lamina strains along the transverse axis through the middle of each ®bre, of `diameter' d are "lam ˆ "mT… †‡ 1 ÿ d "f Td …3† where the subscripts lam, m, and f refer to the lamina, matrix, and ®bre, respectively. The axial strain in the ®bre for the particular state in which it has zero axial stress is, by de®nition, ÿf TL"f T, at point (3) in Fig. 3. The matrix is prescribed to undergo the same strain along the axis of the ®bres. The transverse stress in each constituent of the composite would then follow from standard equations, of the type given in the standard text by Jones9 as 2 ˆ E2 l … † "2 ‡ 12"1 where l ˆ 1 ÿ 1221: …4† With "1 de®ned to match the state of zero axial stress in the ®bres, as above, and 2 taken as constant through￾out both ®bre and matrix on this particular plane, it follows that 2 ˆ Em lm ‰"mT ‡ m"1Š ˆ Ef T lf "f T ‡ f LT"1 ; where "1 ˆ ÿf TL"f T: …5† Hence, Em lm "mT ˆ Ef T lf "f T ÿ f LT Ef T lf ÿ m Em lm   fTL"fT ÿ
…6† whence "mT "f T ˆ 1 ÿ 2 m ÿ
Ef T Em ‡ mfTL …7† Substitution of eqn (7) into eqn (3) then yields "lam "f T ˆ d ‡ … †… 1 ÿ d 1 ÿ 2 m
Ef T Em ‡ mf TLŠ …8† from which eqn (2) follows directly, once the ®bre `dia￾meter' d is related to the ®bre volume fraction Vf as a function of the ®bre array. (Equation (8) also satis®es the obvious sanity checks for equal ®bre and matrix properties and for zero matrix sti€ness.) The e€ective ®bre diameter is related to the form of the array, as is explained in Fig. 4. d ˆ  Vf p for a square fibre in a square array; …9† d ˆ  2 p3  Vf r ˆ 1
050pVf ; …10† for round ®bres in a hexagonal array Fig. 3. Conversion of 45 slope for ®bre failure on ®bre strain plane to corresponding line on lamina strain plane. Predictions of a generalized maximum-shear-stress failure criterion 1183

l184 L.J. Hart-Smith shear-failure cut-offs, the validity of the earlier 45deg approximation for carbon/epoxy laminates is clearly d (11) confirmed. Conversely, the earlier appro be significantly conservative for glass as the author had suspected without actually ble to for round fibres in precisely quantify the effect until now. (These expressions are derived from the solutions for Point(4)in Fig. 3 follows from point (3), the zero the fibre volume fraction as a function of each array axial stress point for the fibre, by retaining the same Setting the value of the array coefficient K at unity, axial strain and multiplying the transverse strain by Re eqn(2)would then predict the following strain-amplifi- from eqn(2). The line(14)in Fig 3 then defines the cation factors for the composite materials used in failure locus of shear failures in the fibres in terms of strains the lamina. It will be apparent that point (4) lies off the (0%)T300/914C carbon/epoxy, Re=1. 517 zero longitudinal stress line for the lamina, being asso- (0%)E-glass/LY556-epoxy, Re=5. 257 ciated with an effective transverse poisson ratio of (0%)E-glass/MY750-epoxy, RE= 5.159 UTL =VTL/Re (12) (0%)AS4/3501-6 carbon/epoxy, Re= 1. 488. Given that these amplification factors are effectively instead of the unrelated VlamtL for the laminate as a reduced in the ratio (UnT/vLT), or roughly 0.2/0.3 for whole. The reason for this is that, while the fibres have carbon/epoxy, in establishing the final slope of these no axial stress at point(4), the matrix does d Square fibres in squre array d Circular fibres in square array Circular fibres in hexagonal array

and d ˆ  4  Vf r ˆ 1
128pVf …11† for round ®bres in square arrays. (These expressions are derived from the solutions for the ®bre volume fraction as a function of each array.) Setting the value of the array coecient K at unity, eqn (2) would then predict the following strain-ampli®- cation factors for the composite materials used in failure exercise.4 (0) T300/914C carbon/epoxy, R" ˆ 1
517 (0) E-glass/LY556-epoxy, R" ˆ 5
257 (0) E-glass/ MY750-epoxy, R" ˆ 5
159 (0) AS4/3501-6 carbon/epoxy, R" ˆ 1
488. Given that these ampli®cation factors are e€ectively reduced in the ratio (fLT=LT), or roughly 0.2/0.3 for carbon/epoxy, in establishing the ®nal slope of these shear-failure cut-o€s, the validity of the earlier 45deg; approximation for carbon/epoxy laminates is clearly con®rmed. Conversely, the earlier approximation would be signi®cantly conservative for glass ®bres, as the author had suspected without actually being able to precisely quantify the e€ect until now. Point (4) in Fig. 3 follows from point (3), the zero axial stress point for the ®bre, by retaining the same axial strain and multiplying the transverse strain by R" from eqn (2). The line (1)±(4) in Fig. 3 then de®nes the locus of shear failures in the ®bres in terms of strains in the lamina. It will be apparent that point (4) lies o€ the zero longitudinal stress line for the lamina, being asso￾ciated with an e€ective transverse Poisson ratio of 0 TL ˆ f TL=R" …12† instead of the unrelated lamTL for the laminate as a whole. The reason for this is that, while the ®bres have no axial stress at point (4), the matrix does. Fig. 4. Fibre volumes for various arrays. 1184 L. J. Hart-Smith

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

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 the

strain 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

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-tension

absence 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

1188 L.J. Hart-Smith 4 CUTOFFS IMPOSED BY TRANSVERSE CRACKS IN THE MATRIX BETWEEN THE FIBRES The authors assessment of cracking of the matrix under ransverse-tension loads is explained in Ref. 16. This is a fracture-mechanics problem, with the failure stress varying from fibre pattern to fibre pattern. There is no universal characterization equivalent to that in Fig. 7 which can be formulated on the lamina strain plane. regardless of whether the assessment is in terms of stress or strain. The nominal failing stress in the lamina, when such cracking occurs, is a function of the orientation and thickness of adjacent plies, as well as the thickness of the ply under consideration. The transverse-tension strength measured on an isolated all-90o lamina applies 712--2LT only to that isolated lamina. It is neither an upper-nor a lower-bound estimate of strength for the very same Fig. 7. Matrix shear failure cut-off for fibre/polymer laminae. lamina when it is embedded in a multidirectional struc- tural laminate. This laminate strength cannot be pre- dicted using the traditional ply-by-ply decomposition loads, this potential failure mechanism can be charac- used for other failure modes in composite laminates terized properly only at the laminate level Nevertheless. once the laminate has been defined. and Although he has developed a formula for ductile the operating environment specified, the influence of matrix failures which interact not only the stresses dis- biaxial stresses on matrix cracking can be depicted as cussed above, but also the compatible matrix stress shown in Fig. 8-in the form of a constant-transverse- developed in the matrix parallel to the fibres, the author tension cut-off. The only difficulty, at the macro level of tends to assess the stresses discussed above separatel analysis used here, is that the line can be located only by (i.e. noninteractively) because his matrix-failure theory has yet to properly account for the residual thermal stresses in the matrix caused by curing at elevated tem- peratures. These stresses, which are customarily exclu- ded from consideration by the standard assumption of omogenizing fibres and matrix to create one composite Constant transverse material, are typically very much greater than those tress components which are retained in most ana- lyses-at least at the macro level. The author's thoi on what is needed to properly characterize matrix ures in fibre/polymer composites can be found in Ref. 16. The author's empirical equation for ductile matrix failures (not cracking) under a combination of inter- active stresses. extracted from Ref. 16. is Constant longitudinal stress line G(会)+()+)=1 in which Em is the modulus of the resin matrix, El the longitudinal modulus of the unidirectional lamina, and s the measured in-plane shear strength of the lamina The os and t represent the obvious in-plane stress com- ponents in the lamina, the direction I being along the length of the fibres. This is an entirely empirical expres ion; setting the direct reference strengths at twice the =ARCTAN V shear strength is based only on Mohr circles, not curve a =ARCTAN fits to data. A further reason for excluding the first and third terms from all but isolated unidirectional laminae Fig. 8. Characterization of intralaminar matrix cracking in is that the interactions between stresses often do not fibre/polymer composites. (1) Transverse tension cracking of matrix in isolated unidirectional lamina, (2)arbitrary desigr become significant below the strain limits imposed by limit imposed for cracking in brittle matrix, (3)inoperable cut the fibres off for unattainable matrix failures in ductile matrices

loads, this potential failure mechanism can be charac￾terized properly only at the laminate level. Although he has developed a formula16 for ductile matrix failures which interact not only the stresses dis￾cussed above, but also the compatible matrix stress developed in the matrix parallel to the ®bres, the author tends to assess the stresses discussed above separately (i.e. noninteractively) because his matrix-failure theory has yet to properly account for the residual thermal stresses in the matrix caused by curing at elevated tem￾peratures. These stresses, which are customarily exclu￾ded from consideration by the standard assumption of homogenizing ®bres and matrix to create one composite material, are typically very much greater than those stress components which are retained in most ana￾lysesÐat least at the macro level. The author's thoughts on what is needed to properly characterize matrix fail￾ures in ®bre/polymer composites can be found in Ref. 16. The author's empirical equation for ductile matrix failures (not cracking) under a combination of inter￾active stresses, extracted from Ref. 16, is 1 2S   Em EL     2 ‡ 12 S  2 ‡ 2 2S  2  1 …15† in which Em is the modulus of the resin matrix, EL the longitudinal modulus of the unidirectional lamina, and S the measured in-plane shear strength of the lamina. The s and  represent the obvious in-plane stress com￾ponents in the lamina, the direction 1 being along the length of the ®bres. This is an entirely empirical expres￾sion; setting the direct reference strengths at twice the shear strength is based only on Mohr circles, not curve ®ts to data. A further reason for excluding the ®rst and third terms from all but isolated unidirectional laminae is that the interactions between stresses often do not become signi®cant below the strain limits imposed by the ®bres. 4 CUTOFFS IMPOSED BY TRANSVERSE CRACKS IN THE MATRIX BETWEEN THE FIBRES The author's assessment of cracking of the matrix under transverse-tension loads is explained in Ref. 16. This is a fracture-mechanics problem, with the failure stress varying from ®bre pattern to ®bre pattern. There is no universal characterization equivalent to that in Fig. 7 which can be formulated on the lamina strain plane, regardless of whether the assessment is in terms of stress or strain. The nominal failing stress in the lamina, when such cracking occurs, is a function of the orientation and thickness of adjacent plies, as well as the thickness of the ply under consideration. The transverse-tension strength measured on an isolated all-90 lamina applies only to that isolated lamina. It is neither an upper- nor a lower-bound estimate of strength for the very same lamina when it is embedded in a multidirectional struc￾tural laminate. This laminate strength cannot be pre￾dicted using the traditional ply-by-ply decomposition used for other failure modes in composite laminates. Nevertheless, once the laminate has been de®ned, and the operating environment speci®ed, the in¯uence of biaxial stresses on matrix cracking can be depicted as shown in Fig. 8Ðin the form of a constant-transverse￾tension cut-o€. The only diculty, at the macro level of analysis used here, is that the line can be located only by Fig. 7. Matrix shear failure cut-o€ for ®bre/polymer laminae. Fig. 8. Characterization of intralaminar matrix cracking in ®bre/polymer composites. (1) Transverse tension cracking of matrix in isolated unidirectional lamina, (2) arbitrary design limit imposed for cracking in brittle matrix, (3) inoperable cut￾o€ for unattainable matrix failures in ductile matrices. 1188 L. J. Hart-Smith