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