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 criterionmatch. 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