《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