1872 K Goda/Composites Science and Technology 59(1999)1871-1879 (a) If the interface is weak, a shear stress concentra- bond layer and the shear modulus, similar to the for- tion parallel to the fiber/matrix interface often mulation taken for a 2-node line element. A global causes interfacial shear debonding along the stiffness matrix is constituted from the three-element fiber -axis stiffness matrices. and therefore the whole structural (b)However, if the interface has a strong bond, a analysis can be carried out following an ordinary finite rack initiates at the fiber break and extends into element procedure. In this study a relatively brittle the matrix perpendicular to the fiber axis. material such as epoxy is used as a matrix, so that the (c) If the matrix consists of a ductile material, it yields effect of (3)was not taken into account. Thus, it is and the yield zone spreads along the broken fiber assumed that the matrix and interface elements as well as the fiber element behave as a linear elastic body, The shear-lag model [ 13] is widely used in estimating respectively, and are statically fractured when the local axial fiber stress distributions around fiber break points stress satisfies a fracture criterion. Namely, the Youngs in a composite, simulating its axial fracture process and modulus of a fiber element is changed to zero if the so on. However, the effect of (2)is not contained in the normal stress of the fiber element achieves its tensile shear-lag model. Therefore, in the present study a finite strength. The shear modulus of an interface element is element method is applied for modeling interfacial changed to zero if the shear stress of the matrix element debonding and matrix cracking. The present finite-ele- achieves the so-called interfacial shear strength. For a ment model is based on the model of a monolayer matrix element, the Von Mises criterion is applied, in composite suggested by Mandel et al. [14]. Fig. I shows which the elastic modulus of this element is changed to the model and mesh, in which a 2-node line element zero if the equivalent stress of this element achieves its representing a fiber element is incorporated into the tensile strength. In the remainder of this article, we call nodes along the y axis of a 4-node isoparametric ele- their fractures "damages, and individually we call ment based on a plane stress condition. This plane ele- them fiber break, interfacial debonding and matrix ment represents a matrix element and takes into account cracking, respectively a multi-axial stress state of tensile and shear stresses The composite model used in this study is a boron around a fiber break epoxy monolayer, and 10 fibers are placed in the finite Furthermore, a shear spring element representing an element mesh, as shown in Fig. 1. Prior to the present interfacial bond (referred to as"interface element") simulation, the effect of the division number per fiber connects the fiber and matrix elements. Deformation was preliminarily investigated in the cases of 10, 20 and resistance of the interface element is determined by the 30 elements per fiber. The calculation results of 20 and spring constant and the relative displacement of the 30 elements showed almost the same stress distributions fiber and matrix elements. The stiffness matrix of a around a broken fiber. around which the most drastic shear spring element is determined by the size of the change in stress occurs. Therefore 20 elements per fiber were selected for the actual simulation. according to this meshing, the number of nodes is 462, and the numbers ▲▲▲▲▲▲▲▲▲▲▲▲ Fiber elemet of fiber. matrix and interface elements are 200. 190 and Interface 220, respectively Matrix element 2. 2. Simulation Occurrences of fiber breaks, matrix cracking and interfacial debonding would cause complicated stress distributions throughout method for estimating reasonably what type of damage occurs in each element, should be incorporated within the simulation procedure. In order to achieve such an estimation, an Imin method [15]is employed in this study, which was originally used in searching for yielding △22 in a metal with an elasto-plastic finite element method. According to this method. a ratio of the dif- ference between strength and stress to the stress incre- ment is calculated by each element, and the element giving the minimum ratio causes one of the damages, i.e Fig. 1. Finite-element model and mesh. Reprinted with permission the fiber break, the matrix cracking and the interfacial from Trans JSME 1997; 63A: 445-452. C 1999 The Japan Society of debonding. The following is the present simulation Mechanical Engineers [16- procedure:(a) If the interface is weak, a shear stress concentra￾tion parallel to the ®ber/matrix interface often causes interfacial shear debonding along the ®ber-axis. (b) However, if the interface has a strong bond, a crack initiates at the ®ber break and extends into the matrix perpendicular to the ®ber axis. (c) If the matrix consists of a ductile material, it yields and the yield zone spreads along the broken ®ber. The shear-lag model [13] is widely used in estimating axial ®ber stress distributions around ®ber break points in a composite, simulating its axial fracture process and so on. However, the e€ect of (2) is not contained in the shear-lag model. Therefore, in the present study a ®nite element method is applied for modeling interfacial debonding and matrix cracking. The present ®nite-ele￾ment model is based on the model of a monolayer composite suggested by Mandel et al. [14]. Fig. 1 shows the model and mesh, in which a 2-node line element representing a ®ber element is incorporated into the nodes along the y axis of a 4-node isoparametric ele￾ment based on a plane stress condition. This plane ele￾ment represents a matrix element and takes into account a multi-axial stress state of tensile and shear stresses around a ®ber break. Furthermore, a shear spring element representing an interfacial bond (referred to as ``interface element'') connects the ®ber and matrix elements. Deformation resistance of the interface element is determined by the spring constant and the relative displacement of the ®ber and matrix elements. The sti€ness matrix of a shear spring element is determined by the size of the bond layer and the shear modulus, similar to the for￾mulation taken for a 2-node line element. A global sti€ness matrix is constituted from the three-element sti€ness matrices, and therefore the whole structural analysis can be carried out following an ordinary ®nite element procedure. In this study a relatively brittle material such as epoxy is used as a matrix, so that the e€ect of (3) was not taken into account. Thus, it is assumed that the matrix and interface elements as well as the ®ber element behave as a linear elastic body, respectively, and are statically fractured when the local stress satis®es a fracture criterion. Namely, the Young's modulus of a ®ber element is changed to zero if the normal stress of the ®ber element achieves its tensile strength. The shear modulus of an interface element is changed to zero if the shear stress of the matrix element achieves the so-called interfacial shear strength. For a matrix element, the Von Mises criterion is applied, in which the elastic modulus of this element is changed to zero if the equivalent stress of this element achieves its tensile strength. In the remainder of this article, we call their fractures ``damages'', and individually we call them ®ber break, interfacial debonding and matrix cracking, respectively. The composite model used in this study is a boron/ epoxy monolayer, and 10 ®bers are placed in the ®nite element mesh, as shown in Fig. 1. Prior to the present simulation, the e€ect of the division number per ®ber was preliminarily investigated in the cases of 10, 20 and 30 elements per ®ber. The calculation results of 20 and 30 elements showed almost the same stress distributions around a broken ®ber, around which the most drastic change in stress occurs. Therefore 20 elements per ®ber were selected for the actual simulation. According to this meshing, the number of nodes is 462, and the numbers of ®ber, matrix and interface elements are 200, 190 and 220, respectively. 2.2. Simulation procedure Occurrences of ®ber breaks, matrix cracking and interfacial debonding would cause complicated stress distributions throughout a composite. Therefore, a method for estimating reasonably what type of damage occurs in each element, should be incorporated within the simulation procedure. In order to achieve such an estimation, an rmin method [15] is employed in this study, which was originally used in searching for yielding regions in a metal with an elasto±plastic ®nite element method. According to this method, a ratio of the dif￾ference between strength and stress to the stress incre￾ment is calculated by each element, and the element giving the minimum ratio causes one of the damages, i.e. the ®ber break, the matrix cracking and the interfacial debonding. The following is the present simulation procedure: Fig. 1. Finite-element model and mesh. Reprinted with permission from Trans JSME 1997;63A:445±452. # 1999 The Japan Society of Mechanical Engineers [16]. 1872 K. Goda / Composites Science and Technology 59 (1999) 1871±1879