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 matricesloads, 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