Z-M. Huang / Compute Structures80(2002)l591170 (10.1), is valid in most cases. On the other hand, the 0.5, sensitive to the specific fiber packing geometry. To ac- The most important feature is that Eq(9)should be count for this sensitivity, two bridging parameters are valid until rupture of the composite. This is because the incorporated into the corresponding independent bridg fiber packing geometry does not change or only varies ing elements, i.e.[71 very little when the composite deforms from an elastic region to an inelastic one. Thus. if the fiber material is a2=B+(1-) El, 0≤B≤1, inearly elastic until rupture and the matrix is elastic- plastic, we should have a3=x+(1-2)x,0≤x≤1 an=Em/E (10.1) The bridging parameters B and a can be adjusted by a2=0.5(1+Ean/E2 comparing the predicted effective transvers a3=0.5(1+Gm/G12) (10.3) plane shear moduli, Ezz and G12 of the composite, 1.e (+ mau(+man) where(refer to Fig. 5) (Vr+man)(,2+a22/m S2)+VVm(S2-Sm2)a12 E ∫E,when≤ 理, when a (104) G vr/Gn2+Ima33/Gm (12.2) Gn={05/1+四,when≤哩 > with measured ones. The so calibrated bridging meters can be used in(ll. 1)and (11. 2)for later inelastic ym is matrix Poissons ratio and analysis. If no other information is available. B=a=0.5 can be employe 啁=V(G)2+(璺)2-())+3(唱)2(106 is the matrix von misses effective stress. when the 4. Thermal load effect problem under consideration is fully three-dimensional the corresponding bridging matrix, A, is given in Ap- Suppose that the working temperature of the UD pendⅸxA composite, Ti is different from a reference temperature In reality, the composite longitudinal property hardly To at which the internal stresses of the fiber and the depends on the fiber packing geometry. The matrix are already known. Because of mismatch be- sponding independent bridging element formula, Eq. tween the coefficients of thermal expansion of the fiber Oy=yield strength E=tan(aYoung s modulus ng modulus E Fig. 5. An elastic-plastic stress-strain curve with definition of material parameters.k11 ¼ 1; k21 ¼ k31 ¼ 0:5; and all the other kij ¼ 0: ð9Þ The most important feature is that Eq. (9) should be valid until rupture of the composite. This is because the fiber packing geometry does not change or only varies very little when the composite deforms from an elastic region to an inelastic one. Thus, if the fiber material is linearly elastic until rupture and the matrix is elastic– plastic, we should have a11 ¼ Em=Ef 11; ð10:1Þ a22 ¼ 0:5ð1 þ Em=Ef 22Þ; ð10:2Þ a33 ¼ 0:5ð1 þ Gm=Gf 12Þ; ð10:3Þ where (refer to Fig. 5) Em ¼ Em; when rm e 6 rm Y Em T ; when rm e > rm Y;  ð10:4Þ Gm ¼ 0:5Em=ð1 þ mmÞ; when rm e 6 rm Y Em T =3; when rm e > rm Y:  ð10:5Þ mm is matrix Poisson’s ratio and rm e ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðrm 11Þ 2 þ ðrm 22Þ 2 ðrm 11Þðrm 22Þ þ 3ðrm 12Þ 2 q ð10:6Þ is the matrix von Misses effective stress. When the problem under consideration is fully three-dimensional, the corresponding bridging matrix, ½A, is given in Ap￾pendix A. In reality, the composite longitudinal property hardly depends on the fiber packing geometry. The corre￾sponding independent bridging element formula, Eq. (10.1), is valid in most cases. On the other hand, the composite transverse and in-plane shear responses are sensitive to the specific fiber packing geometry. To ac￾count for this sensitivity, two bridging parameters are incorporated into the corresponding independent bridg￾ing elements, i.e. [7] a22 ¼ b þ ð1 bÞ Em Ef 22 ; 0 6 b 6 1; ð11:1Þ a33 ¼ a þ ð1 aÞ Gm Gf 12 ; 0 6 a 6 1; ð11:2Þ The bridging parameters b and a can be adjusted by comparing the predicted effective transverse and in￾plane shear moduli, E22 and G12 of the composite, i.e. E22 ¼ ðVf þ Vma11ÞðVf þ Vma22Þ ðVf þ Vma11ÞðVfSf 22 þ a22VmSm 22Þ þ VfVmðSm 12 Sf 12Þa12 ; ð12:1Þ G12 ¼ Vf þ Vma33 Vf=Gf 12 þ Vma33=Gm ; ð12:2Þ with measured ones. The so calibrated bridging para￾meters can be used in (11.1) and (11.2) for later inelastic analysis. If no other information is available, b ¼ a ¼ 0:5 can be employed. 4. Thermal load effect Suppose that the working temperature of the UD composite, T1 is different from a reference temperature, T0 at which the internal stresses of the fiber and the matrix are already known. Because of mismatch be￾tween the coefficients of thermal expansion of the fiber Fig. 5. An elastic–plastic stress–strain curve with definition of material parameters. Z.-M. Huang / Computers andStructures 80 (2002) 1159–1176 1163