July 2001 Multiple Cracking and Tensile Behavior 1569 Existing Cracks Hypothetical New Crack (4 Stiffness Change Due to Matrix Cracks in the Longitudinal fiber bundle A partial fiber/matrix interfacial debond model has been ana E10° lyzed through use of a concentric cylinder with fiber radius, R, as shown in Fig 9.The problem can be solved by two different Er shear-lag analysis. In this paper, both of the theories were applied The average applied stress, o,u, in the longitudinal fiber bundles under a composite stress, o, was approximated by E LL+LL where A is the load-partitioning factor Fig. 8. Schematic drawing of transverse crack calculations for cross-ply (11) ated composites using the shear-lag model. Here, E and a denote the modulus and the CTE of the plies, with subscripts I and 2 indicating and E and Elu are the elastic moduli of the composite and the 0° plies and90° plies, respectively longitudinal unidirectional plies in each laminate, respectively. For the composite, the resulting values are A= 1. 44 below 50 MPa, A=2.68 at omc(180 MPa), and A= 2.75 above o(300 MPa).A =E11+E2)(21EE2 d approximation can be obtained through linear interpolation ψ=-2k/t(a2-a1)△T-{1+2(E1)}l (A) Hutchinson and Jensen Theory: Hutchinson and Jensen found a solution for an axisymmetric cylindrical model with a where AT is the change in temperature, o, is the average stress single matrix crack using the Lame problem. The axial stress, of (equivalent stress)in the laminate, K is an effective shear stiffnes radial stress, o, and axial strain, Ef, for a fiber in the bonded of the bonding layer, and /, is the transverse crack spacing. The egion( denoted by a superscript + under an average applied dimensionless parameter S is the ratio between the shear and axial stress,Gu in a longitudinal fiber bundle are given by the stiffnesses of the shear-lag model. The average stress, Olu in 0. following plies along the x-axis is given by the following o=a-a,EmE (12a) 1+l2 L,Ex cosh(2Ex'1r sh(El/t,) cosh(2Er'l12) E,+e (ax2-a)△71 cosh(E//t,) where a, to as are parameters provied by Hutchinson and Jensen The first and second terms are due to the applied load and thermal and listed in Table I for isotopic materials. e is the mismatch load, respectively. Thus, the elastic modulus, E, for each laminate strain between fiber and matrix, due to thermal stresses. The stress is obtained using the following equation crack tip are given by A d△er=Er El1E1(E11+E22) the following equation is obtained with Ao as the free variable in (1 1+12LEIEI+ RE] tanh(EI/i2) the solution △Er=b2△o/Em ( Matrix Crack Density in the Longitudinal Fiber bundles is difficult to estimate the matrix crack density within 0 plies Full contact over the debonded region occurs with a constan using a theoretical approach. Therefore, an empirical equation sliding stress, T, and the fiber stress has a maximum value, on/p, roposed by Evans et al. has been directly applied in order to at the matrix crack surface account for the 3-D composite matrix crack der 7≈7G/m-1)/(G/。-1) where p is the fiber volume fraction and I is the matrix crack where omc, o and s are the matrix crack onset stress, the crack spacing. The total strain is then given by aturation stress, and the saturated crack spacing, respectively. The olid line shown in Fig. 5 obtained using omc 180 MPa,o 300 MPa, and 1s =45.4 mm agrees well with the experimental (15) Matrix Crack Debonding Region R Fibe o Fig. 9. Partial interfacial debonding model for a unidirectional compositej 5 ÎKt2~E1t1 1 E2t2!/~2t1E1E2! (5) c 5 22K/t2@~a2 2 a1!DT 2 $~t1 1 t2!/~E1t1!%s#l # (6) where DT is the change in temperature, s#l is the average stress (equivalent stress) in the laminate, K is an effective shear stiffness of the bonding layer, and lL is the transverse crack spacing. The dimensionless parameter j is the ratio between the shear and axial stiffnesses of the shear-lag model. The average stress, s#lu, in 0° plies along the x-axis is given by the following: s# lu 5 t1 1 t2 E1t1 1 E2t2 SE1 1 t2E2 t1 cosh ~2jx9/t2! cosh ~jlL/t2! Ds#l 1 t2E1E2 E1t1 1 E2t2 ~a2 2 a1!DTF1 2 cosh ~2jx9/t2! cosh ~jlL/t2! G (7) The first and second terms are due to the applied load and thermal load, respectively. Thus, the elastic modulus, El , for each laminate (E, F, G) is obtained using the following equation: El 5 jlLt1E1~E1t1 1 E2t2! ~t1 1 t2!@lLE1jt1 1 t2 2 E2 tanh ~jlL/t2!# (8) (3) Matrix Crack Density in the Longitudinal Fiber Bundles It is difficult to estimate the matrix crack density within 0° plies using a theoretical approach. Therefore, an empirical equation proposed by Evans et al. 7 has been directly applied in order to account for the 3-D composite matrix crack density. l # < l # s~s# s/s# mc 2 1!/~s#/s# mc 2 1! (9) where s# mc, s# s, and l # s are the matrix crack onset stress, the crack saturation stress, and the saturated crack spacing, respectively. The solid line shown in Fig. 5 obtained using s# mc 5 180 MPa, s# s 5 300 MPa, and l # s 5 45.4 mm agrees well with the experimental data. (4) Stiffness Change Due to Matrix Cracks in the Longitudinal Fiber Bundle A partial fiber/matrix interfacial debond model has been ana￾lyzed through use of a concentric cylinder with fiber radius, R, as shown in Fig. 9.9,13 The problem can be solved by two different approaches: (1) elastic analysis of the Lame problem, and (2) shear-lag analysis. In this paper, both of the theories were applied. The average applied stress, s#lu, in the longitudinal fiber bundles under a composite stress, s#, was approximated by s# lu 5 ls# (10) where l is the load-partitioning factor l 5 Elu/E (11) and E and Elu are the elastic moduli of the composite and the longitudinal unidirectional plies in each laminate, respectively. For the composite, the resulting values are l 5 1.44 below 50 MPa, l 5 2.68 at s# mc (180 MPa), and l 5 2.75 above s# s (300 MPa). A good approximation can be obtained through linear interpolation between s# mc and s# s. 12 (A) Hutchinson and Jensen Theory: Hutchinson and Jensen found a solution for an axisymmetric cylindrical model with a single matrix crack using the Lame problem.9 The axial stress, sf 1, radial stress, sr 1, and axial strain, εf 1, for a fiber in the bonded region (denoted by a superscript 1) under an average applied stress, s#lu, in a longitudinal fiber bundle are given by the following: sf 1 5 a1s#lu 2 a2EmεT (12a) sr 1 5 a3 2 a4EmεT (sr at r 5 R) (12b) εf 1 5 a5~s#lu/Em! 1 a6εT (12c) where a1 to a6 are parameters provied by Hutchinson and Jensen9 and listed in Table I for isotopic materials. εT is the mismatch strain between fiber and matrix, due to thermal stresses. The stress and strain differences in the fiber above and below the debond crack tip are given by Dsf 5 sf 2 sf 1 and Dεf 5 εf 2 εf 1. Thus, the following equation is obtained with Dsf as the free variable in the solution: Dεf 5 b2Dsf/Em (13) Full contact over the debonded region occurs with a constant sliding stress, t, and the fiber stress has a maximum value, s#lu/r, at the matrix crack surface. sf 5 s#lu/r 2 2t~l/2 2 x!/R (14) where r is the fiber volume fraction and l is the matrix crack spacing. The total strain is then given by εcu 5 εf 1 1 2 l E l/22ld l/ 2 Dε dx (15) Fig. 8. Schematic drawing of transverse crack calculations for cross-ply laminated composites using the shear-lag model. Here, E and a denote the elastic modulus and the CTE of the plies, with subscripts 1 and 2 indicating 0° plies and 90° plies, respectively. Fig. 9. Partial interfacial debonding model for a unidirectional composite. July 2001 Multiple Cracking and Tensile Behavior 1569