Journal of the American Ceramic Societ and Baste Vol 81. No 9 What is clear is that friction effects will be more important in are functions of the elastic properties of both systems that have residual compressive thermal stress acting the uncracked material and the material that surrounds the ess can increase the sliding resistance. This resistance is also prone to increase when some displacement mismatch results from variations in the radius of the fiber along its length when S=S+-oA the fiber slides All this information shows that a rigorous evaluation of t where S and C are the compliance and stiffness tensors, re- implies that the crack-opening displacement must be identified spectively, of the cracked material. S is the compliance tensor s a function of the applied stress. Crack growth usually re- of the uncracked material, Aa fourth-order tensor( the coeffi- quires some sliding of the matrix over the fibers cients of which are dependent upon both the crack geometry The use of an experimental device that couples an ultrasonic and the mechanical properties of the medium that surrounds immersion tank to a tensile machine and an extensometer not them), and o the crack density only allows study of the loss of stiffness of the material under For the transverse crack system, normal to the tensile axis, tensile stress, because it gives access to the complete stiffne only three components of A, noted A, are different from tensor variation, but also allows one to know precisely which zero: 28 A33, Ai4, and Ass. After every compliance is known, it rtion of the global strain is either elastic or inelastic 22A evious study s has shown that the inelastic strains measured This parameter represents the density of cracks in a square of nd the density of the transverse cracks. "78 both the oper during monotonous loading are governed side dimension 2a. 27 Actually, only the matrix and the trans- e results can verse bundles are cracked in the material. The definition of the used to evaluate the interfacial shear stress of the composite elementary cell must consider this to accurately identify the during monotonic load density of cracks in a representative element of the sample(that The following work has been obtained from a shear-lag is to say, an element that contains both cracked and uncracked analysis.9 ,24,25 Only normal axial stresses are assumed to exist sub-elements). This element includes the fully cracked square ithin the fiber. The shear stress is concentrated at the inter- of side dimension 2a and the adjacent uncracked sub-elements face. The radial stress-the summation of the residual stress of extensions(2x,-2a)and(2x2-2c)in directions I and 2, hat results from thermal expansion coefficient mismatch and respectively( see Fig. 1). Thus, the effective crack density, B the poisson effect---will not onsidered. because it can be is related to o through the equation xpressed in a Coulomb formulation of the interfacial shear ress. We will not make any distinction between the sliding istance and the debond length either, although sliding can he case where the sliding length is correlated to the length of Actually, Br is also related to the intercrack distance () the Mode II longitudinal crack. In the cohesive zone, elasticity prevails, beyond this point, sliding is dominant. 6 Crack satu ration occurs when the crack spacing has become sufficient small that the cracks interact with each other. When slip zones from neighboring cracks overlap, debonding is complete and Therefore in an extensometer of length L the number of cracks no further cracking occurs. 6 The load is then shared only by given by the fibers n=L (4 IL. Formulation of the model Because the inelastic strain comes from the crack-opening The model is based on the analytical expressions of the displacements(2U) due to fiber/matrix sliding in the tensile elastic properties of a fibrous composite that contains cracks. direction, it can be expressed by The model also is based on a shear-lag analysis of the crack closure traction applied by the bridging fibers The variation of the elastic properties of a cracked material e inelastic Anelastic n(2U)m(2b) L can be deduced by replacing the cracked medium with an ef- fective equivalent medium. 27 The cracks--2a deep, 2b thick This crack opening comes from the elasticity mismatch be- and 2c wide--are modeled as elliptic cylinders with an aspect tween the fiber and the matrix, as well as from the relative ratio(8= 2b/2a) that approaches zero(Fig. 1). The effective sliding at the interface between them. The study of the state of Matrix crack Fiber matrix Fig. 1. Crack geometry in the case of a transverse crack with cla > I and an aspect ratio 8<<1What is clear is that friction effects will be more important in systems that have residual compressive thermal stress acting across the fiber/matrix interface. In these systems, the rough￾ness can increase the sliding resistance. This resistance is also prone to increase when some displacement mismatch results from variations in the radius of the fiber along its length when the fiber slides.21 All this information shows that a rigorous evaluation of t implies that the crack-opening displacement must be identified as a function of the applied stress. Crack growth usually re￾quires some sliding of the matrix over the fibers.3 The use of an experimental device that couples an ultrasonic immersion tank to a tensile machine and an extensometer not only allows study of the loss of stiffness of the material under tensile stress, because it gives access to the complete stiffness tensor variation, but also allows one to know precisely which proportion of the global strain is either elastic or inelastic.22 A previous study23 has shown that the inelastic strains measured during monotonous loading are governed by both the opening and the density of the transverse cracks.23 These results can be used to evaluate the interfacial shear stress of the composite during monotonic loading. The following work has been obtained from a shear-lag analysis.9,24,25 Only normal axial stresses are assumed to exist within the fiber. The shear stress is concentrated at the inter￾face. The radial stress—the summation of the residual stress that results from thermal expansion coefficient mismatch and the Poisson effect—will not be considered, because it can be expressed in a Coulomb formulation of the interfacial shear stress.25 We will not make any distinction between the sliding distance and the debond length either, although sliding can increase the debond. Furthermore, we will limit our analysis to the case where the sliding length is correlated to the length of the Mode II longitudinal crack. In the cohesive zone, elasticity prevails; beyond this point, sliding is dominant.26 Crack satu￾ration occurs when the crack spacing has become sufficiently small that the cracks interact with each other. When slip zones from neighboring cracks overlap, debonding is complete and no further cracking occurs.26 The load is then shared only by the fibers. II. Formulation of the Model The model is based on the analytical expressions of the elastic properties of a fibrous composite that contains cracks. The model also is based on a shear-lag analysis of the crack closure traction applied by the bridging fibers. The variation of the elastic properties of a cracked material can be deduced by replacing the cracked medium with an ef￾fective equivalent medium.27 The cracks—2a deep, 2b thick, and 2c wide—are modeled as elliptic cylinders with an aspect ratio (d 4 2b/2a) that approaches zero (Fig. 1). The effective elastic properties are functions of the elastic properties of both the uncracked material and the material that surrounds the cracks: S = S° + p 4 vL (1) where S and C are the compliance and stiffness tensors, re￾spectively, of the cracked material. S° is the compliance tensor of the uncracked material, L a fourth-order tensor (the coeffi￾cients of which are dependent upon both the crack geometry and the mechanical properties of the medium that surrounds them), and v the crack density. For the transverse crack system, normal to the tensile axis, only three components of L, noted LT, are different from zero:28 LT 33, LT 44, and LT 55. After every compliance is known, it is then easy to identify the crack-density parameter v. This parameter represents the density of cracks in a square of side dimension 2a. 27 Actually, only the matrix and the trans￾verse bundles are cracked in the material. The definition of the elementary cell must consider this to accurately identify the density of cracks in a representative element of the sample (that is to say, an element that contains both cracked and uncracked sub-elements). This element includes the fully cracked square of side dimension 2a and the adjacent uncracked sub-elements of extensions (2x1 − 2a) and (2x2 − 2c) in directions 1 and 2, respectively (see Fig. 1). Thus, the effective crack density, bT, is related to v through the equation bT = vS 2x1 2a DS2x2 2c D (2) Actually, bT is also related to the intercrack distance (l):27 l = 2a bT (3) Therefore, in an extensometer of length L, the number of cracks n is given by the equation n = LS bT 2aD (4) Because the inelastic strain comes from the crack-opening displacements (2U) due to fiber/matrix sliding in the tensile direction, it can be expressed by «inelastic = DLinelastic L = n~2U! L = n~2b! L = dbT (5) This crack opening comes from the elasticity mismatch be￾tween the fiber and the matrix, as well as from the relative sliding at the interface between them. The study of the state of Fig. 1. Crack geometry in the case of a transverse crack with c/a >> 1 and an aspect ratio d << 1. 2406 Journal of the American Ceramic Society—Morvan and Baste Vol. 81, No. 9