ournal 1. Am Ceram Soc, s1 [9) 2105-12(1998 Assessment of the Interfacial Properties of Ceramic-Matrix Composites Using Strain Partition under Load Jean-Marie Morvan and Stephane Baste Laboratoire de Mecanique Physique, Universite Bordeaux 1, CNRS UPRES A 5469, 33405 Talence, France The interfacial sliding stress is a key parameter in the glob- curate study of the fracture behavior requires identification of I behavior of ceramic-matrix composites. An ultrasonic the damage threshold(that is to say, the stress where the crack characterization, through the complete determination of begins ), the debonding stress, the sliding stress, and the he stiffness tensor along a tensile test, detects all the dam interfacial shear stress. Three basic parameters can be exhibited ge mechanisms: transverse matrix microcracking and the from the stress-strain curves the stress at the onset of matri existence of longitudinal cracks at the fiber/matrix inter- cracking, the stress at saturation, and the ultimate strength. face. It also allows a strain partition under load. A micro Identification of the interfacial shear stress, T, has given rise mechanical model is established and gives access to the to numerous works. It has even led to new measurement tech- value of the interfacial sliding stress during the entire ten niques such as microindentation. ,9 This technique allows sile test, whether the damage occurs at the mesostructural study of the interfacial properties by analyzing the stress- or at the microstructural level of the composite. displacement curves obtained from push-out or push- hrough tests of a single fiber or of a bundle of fibers. 1 Some works have also focused on the measurement of the pull-out length, 12 the use of the intercrack distance at saturation, or the T IS now commonly recognized that the macroscopic nonlin- information held in the hysteresis coming from cyclic loading Lear behavior of ceramic-matrix composites( CMCs)unde tensile loading is the result of the combination of three main analysis 4, I5 is also included in these methods. However, they amage mechanisms: I matrix microcracking normal to the ten- are not exempt from criticism. For example, the push-out test sile axis. deflection of these cracks at the fiber/matrix interface creates a Poisson effect opposite to the one created during and fiber fracture. Matrix microcracking induces a loss of stiff. monotonic loading, tests on microcomposites lack the structure ness. The fiber-matrix debonding leads to fiber-matrix slid Nevertheless, some basic trends have a tendency to emerge prevents the composite from failing prematurely When a fiber The interaction between the fiber and the matrix involves the breaks, the load sharing that results is done through the sliding esistance along the fiber. 2 Thus, the sliding resistance(T)is a he fibers and the matrix, as well as the interfacial properties key parameter in the global behavior The importance of the sliding phenomenon is related to the misfit strain, the fiber surface roughness, and the debond trix cracking, followed by interaction of these cracks with the length. o However, surface roughness can be considered as a fibers and interfaces. 3 Such interactions involve the debond radial compressive stress at the interface. 6 The main informa fracture energy and the sliding resistance within the debond. tion is that t varies with the sliding length. Actually, the pa- Therefore. the role of t rameters that govern friction are not all constant. Abrasion and Beyond the protection o the fibers during processing roughness effects are functions of the sliding distance. The he fiber coating can improve both the crack deflection and the fiber pullout decreases the friction force because the length over which the fiber slides also decreases vsteresis is de oughness. The interphase controls the fracture resistance, be endent upon both the friction force and the length over which cause it promotes the crack arrest by matrix crack deflection. It he fibers slide. 7 Thus, the volume of fibers is an important also limits the extension of the cracks by energy absorption parameter, because the area available for energy dissipation These mechanisms occur either at the matrix/interphase or in- ncreases with it. 7 The hysteresis is larger with lower values of terphase/fiber interfaces or in the interphase itself when it is thick enough. Although the response of the interface is com- creases through more-effective bridging. 7 Debris at the inter plex, it is usually idealized as two mechanisms: the initial faces can influence the friction coefficient during a test. 6 esistance to relative motion(usually, the debonding) and the Some work has also been conducted on the decrease of inter frictional sliding resistance. facial shear during fatigue tests. The observation is made that The stress-strain curves of CMCs clearly show that the glob. assumption of a constant shear stress and a constant number of cracks created in the longitudinal bundles due to interfacial of the constituent o he first cycles might be wrong. Further- al strain is the sum of an elastic and an inelastic strain. The broken fibers during hat results from the thermal mismatch Thus, the identification of the micromechanical parameters conceal the abrasion effect. In the same manner, the interface that govern these phenomena is of prime importance. An roughness has a tendency to counterbalance the Poisson effect From a quantitative standpoint, carbon or boron nitride coat ngs have a tendency to diminish the interfacial shear stren A G. Evans--contributing editor however. strength is still a function of the residual stresses Nevertheless. the results found in the literature exhibit large scatter. 2,3, 20 In a two-dimensional chemical-vapor-infiltrated Sic-SiC(2D CVI SiC-SiC) composite, for example, the in- Manuscript No 191124 Received March 19, 1997: approved December 23, 1997 terfacial sliding stress can vary within a range of 2-370 ME 2405
Assessment of the Interfacial Properties of Ceramic-Matrix Composites Using Strain Partition under Load Jean-Marie Morvan and Stéphane Baste Laboratoire de Mécanique Physique, Université Bordeaux 1, CNRS UPRES A 5469, 33405 Talence, France The interfacial sliding stress is a key parameter in the global behavior of ceramic-matrix composites. An ultrasonic characterization, through the complete determination of the stiffness tensor along a tensile test, detects all the damage mechanisms: transverse matrix microcracking and the existence of longitudinal cracks at the fiber/matrix interface. It also allows a strain partition under load. A micromechanical model is established and gives access to the value of the interfacial sliding stress during the entire tensile test, whether the damage occurs at the mesostructural or at the microstructural level of the composite. I. Introduction I T IS now commonly recognized that the macroscopic nonlinear behavior of ceramic-matrix composites (CMCs) under tensile loading is the result of the combination of three main damage mechanisms:1 matrix microcracking normal to the tensile axis, deflection of these cracks at the fiber/matrix interface, and fiber fracture. Matrix microcracking induces a loss of stiffness. The fiber–matrix debonding leads to fiber–matrix sliding, with friction depending on the nature of the interface, and prevents the composite from failing prematurely. When a fiber breaks, the load sharing that results is done through the sliding resistance along the fiber.2 Thus, the sliding resistance (t) is a key parameter in the global behavior. Thus, the fracture properties of CMCs are governed by matrix cracking, followed by interaction of these cracks with the fibers and interfaces.3 Such interactions involve the debond fracture energy and the sliding resistance within the debond. Therefore, the role of the interphase is of first importance. Beyond the protection provided to the fibers during processing, the fiber coating can improve both the crack deflection and the stress transfer by modifying, among other things, the surface roughness.4 The interphase controls the fracture resistance, because it promotes the crack arrest by matrix crack deflection. It also limits the extension of the cracks by energy absorption. These mechanisms occur either at the matrix/interphase or interphase/fiber interfaces or in the interphase itself when it is thick enough. Although the response of the interface is complex, it is usually idealized as two mechanisms: the initial resistance to relative motion (usually, the debonding) and the frictional sliding resistance.5 The stress–strain curves of CMCs clearly show that the global strain is the sum of an elastic and an inelastic strain. The inelastic strain is controlled by the opening of the transverse cracks created in the longitudinal bundles due to interfacial debonding and fiber–matrix sliding.6 Thus, the identification of the micromechanical parameters that govern these phenomena is of prime importance. An accurate study of the fracture behavior requires identification of the damage threshold (that is to say, the stress where the cracking begins), the debonding stress, the sliding stress, and the interfacial shear stress. Three basic parameters can be exhibited from the stress–strain curves: the stress at the onset of matrix cracking, the stress at saturation, and the ultimate strength.7 Identification of the interfacial shear stress, t, has given rise to numerous works.2 It has even led to new measurement techniques such as microindentation.8,9 This technique allows study of the interfacial properties by analyzing the stress– displacement curves10 obtained from push-out4 or pushthrough tests of a single fiber or of a bundle of fibers.11 Some works have also focused on the measurement of the pull-out length,12 the use of the intercrack distance at saturation,3 or the information held in the hysteresis coming from cyclic loading of either microcomposites or real composites.13 Finite-element analysis14,15 is also included in these methods. However, they are not exempt from criticism. For example, the push-out test creates a Poisson effect opposite to the one created during monotonic loading; tests on microcomposites lack the structure effect.11 Nevertheless, some basic trends have a tendency to emerge. The interaction between the fiber and the matrix involves the misfit strain induced by the difference of properties between the fibers and the matrix, as well as the interfacial properties.7 The importance of the sliding phenomenon is related to the misfit strain, the fiber surface roughness, and the debond length.10 However, surface roughness can be considered as a radial compressive stress at the interface.16 The main information is that t varies with the sliding length. Actually, the parameters that govern friction are not all constant. Abrasion and roughness effects are functions of the sliding distance. The fiber pullout decreases the friction force because the length over which the fiber slides also decreases.17 Hysteresis is dependent upon both the friction force and the length over which the fibers slide.17 Thus, the volume of fibers is an important parameter, because the area available for energy dissipation increases with it.17 The hysteresis is larger with lower values of frictional strength; however, the opening of the cracks decreases through more-effective bridging.17 Debris at the interfaces can influence the friction coefficient during a test.16 Some work has also been conducted on the decrease of interfacial shear during fatigue tests.18 The observation is made that assumption of a constant shear stress and a constant number of broken fibers during the first cycles might be wrong. Furthermore, the compression that results from the thermal mismatch of the constituents, especially if the thermal expansion coefficient of the matrix is greater than that of the fiber, might conceal the abrasion effect. In the same manner, the interface roughness has a tendency to counterbalance the Poisson effect during pullout.19 From a quantitative standpoint, carbon or boron nitride coatings have a tendency to diminish the interfacial shear strength; however, strength is still a function of the residual stresses.3 Nevertheless, the results found in the literature exhibit large scatter.2,3,20 In a two-dimensional chemical-vapor-infiltrated SiC–SiC (2D CVI SiC–SiC) composite, for example, the interfacial sliding stress can vary within a range of 2–370 MPa.7 A. G. Evans—contributing editor Manuscript No. 191124. Received March 19, 1997; approved December 23, 1997. J. Am. Ceram. Soc., 81 [9] 2405–12 (1998) Journal 2405
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<<1
What 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 roughness 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 requires 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 interface. 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 saturation 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 effective 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, respectively, of the cracked material. S° is the compliance tensor of the uncracked material, L a fourth-order tensor (the coefficients 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 transverse 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 between 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
September 1998 Assessment of the Interfacial Properties of CMCs Using Strain Partition under load 2407 micromechanical equilibrium of the composite using a shear- lag analysis gives access to interfacial sliding stress. The fol (e)+() (10) lowing assumptions are made: (i) isostrain conditions prevail in the cross-sectional area of the sample at the end of the sliding length, (ii) a constant interfacial friction stress exists within the sliding length, and (iii)the stress in the matrix is completely transferred to the fiber in the crack plane f the poisson ratio of the fiber and the matrix are assumed to be the same, the far-field stress of the composite(oo)is In the same manner, the stress in the matrix om() must de- related to the far-field fiber stress(o and the far-field matrix crease from the far-field stress o at the end of the cohesive stress(om)by the relation length to zero in the crack plane in : 12. Thus, σn+V,σ (12) where Ve and Vm represent the volume fractions of the fibers and the matrix, respectively Therefore, the strain variation in the matrix, 8(), is also The isostrain condition is expressed this way G Ooo of om (13) where the indices c, f, and m refer to the composite, fiber, and is given br acement of the free crack surface of the matrix, 8ms The elementary cell that was used in the model is depicted in [((-a (14) size on each side of the crack is given by in2. This distance is TH itself divided into two parts: the debonding length upon which the sliding occurs(a), and the length upon which the interface remains cohesive(/). R is the fiber radius () (15) Assuming that a constant sliding friction stress T opposes the between the fiber and matrix displacements in the crack plane 2+8 The fiber strain e=), the derivative of the fiber displacement along the tensile axis(8 )) increases the same way: By using Eq(7). d64(-) Using Eqs. (5)and(17), the measured inelastic strain(e inelastic) In the cohesive zone, only elasticity occurs. Within the can be related to the sliding friction stress lebon, the fiber displacement in the z-direction(see Fig. 2) vill be maximum in the crack plane; that is to say, 2= 2 Therefore we have (18) Is to say, aE8-21/+RE (19) Because east= oo/Ee, we finally obtain the relation Debonding crack inelastic elastic 22 2u- Crack opening Transverse he interfacial sliding stress is directly dependent upon Matrix Crack both the elastic and inelastic strains as well as the transverse crack density and the area upon which the sliding occurs, hrough a, Ie, and I IIL E accurately study the damage evolution in composites, experimental device that couples an ultrasonic immersion tanl Fig. 2. Schematic depiction of an elementary cell containing an open to a tensile machine and an extensometer has been developed. 29 This apparatus allows the study of the loss of stiffness of the
micromechanical equilibrium of the composite using a shearlag analysis gives access to interfacial sliding stress. The following assumptions are made: (i) isostrain conditions prevail in the cross-sectional area of the sample at the end of the sliding length, (ii) a constant interfacial friction stress exists within the sliding length, and (iii) the stress in the matrix is completely transferred to the fiber in the crack plane. If the Poisson ratio of the fiber and the matrix are assumed to be the same, the far-field stress of the composite (s`) is related to the far-field fiber stress (sf ) and the far-field matrix stress (sm) by the relation s` = Vmsm + Vfsf (6) where Vf and Vm represent the volume fractions of the fibers and the matrix, respectively. The isostrain condition is expressed this way: s` Ec = sf Ef = sm Em (7) where the indices c, f, and m refer to the composite, fiber, and matrix, respectively, for the Young’s moduli (E) and the stresses (s). The elementary cell that was used in the model is depicted in Fig. 2. Because the intercrack distance is given as l, the cell size on each side of the crack is given by l/2. This distance is itself divided into two parts: the debonding length upon which the sliding occurs (ld), and the length upon which the interface remains cohesive (lc). R is the fiber radius. Assuming that a constant sliding friction stress t opposes the relative sliding motion of the fiber and the matrix and that the fiber stress (sf (z)) increases linearly from the far-field fiber stress sf at the end of the cohesive length, we can write sf~z! = sf + S 2t R Dz (8) The fiber strain «f (z), the derivative of the fiber displacement along the tensile axis (df (z)), increases the same way: «f~z! = ddf~z! dz = sf Ef + S 2t REf Dz (9) In the cohesive zone, only elasticity occurs. Within the debond, the fiber displacement in the z-direction (see Fig. 2) will be maximum in the crack plane; that is to say, z 4 l/2. Therefore, we have df = FS sf Ef DzGU 0 lc + FS sf Ef Dz + S t REf Dz 2 GU l c l/2 (10) Thus, df = S sf Ef D l 2 + t REf FS l 2D 2 − lc 2 G (11) In the same manner, the stress in the matrix sm(z) must decrease from the far-field stress sm at the end of the cohesive length to zero in the crack plane in z 4 l/2. Thus, sm~z! = smS1 − z l D (12) Therefore, the strain variation in the matrix, «m(z), is also linear: «m~z! = sm Em S1 − z l D (13) and the displacement of the free crack surface of the matrix, dm, is given by dm = FS sm Em DzGU 0 lc + F sm Em Sz − z 2 2l DGU l c l/2 (14) Thus, dm = sm Em S lc 2 2l + 3l 8 D (15) Because of the symmetry of the representative cell, one-half the crack-opening displacement, U, is simply the difference between the fiber and matrix displacements in the crack plane: U = df − dm = sf Ef S l 2D + t REf FS l 2D 2 − lc 2 G − sm Em S lc 2 2l + 3l 8 D (16) By using Eq. (7), U = s` Ec S l 8 − lc 2 2l D + t REf FS l 2D 2 − lc 2 G (17) Using Eqs. (5) and (17), the measured inelastic strain («inelastic) can be related to the sliding friction stress: «inelastic = bTS U a D (18) This is to say, «inelastic = bT a H s` Ec S l 8 − lc 2 2l D + t REf FS l 2D 2 − lc 2 GJ (19) Because «elastic 4 s`/Ec, we finally obtain the relation t = REf FS l 2D 2 − lc 2 G FS a bT D«inelastic − S l 8 − lc 2 2l D«elasticG (20) Finally, the interfacial sliding stress is directly dependent upon both the elastic and inelastic strains as well as the transverse crack density and the area upon which the sliding occurs, through a, lc, and l. III. Experimental Results To accurately study the damage evolution in composites, an experimental device that couples an ultrasonic immersion tank to a tensile machine and an extensometer has been developed.29 This apparatus allows the study of the loss of stiffness of the Fig. 2. Schematic depiction of an elementary cell containing an open crack. September 1998 Assessment of the Interfacial Properties of CMCs Using Strain Partition under Load 2407
2408 Journal of the American Ceramic Society-Monan and Baste Vol 81. No 9 Table L. Initial Compliances of a 2D Sic-SiC Composite tensor coefficient interval(x 10GPa)") Sir Extensometer ±0.5 Immersed Transducer 104 s品品 ±04 0.01 ±0.2 Contact Transducer by propagation in the nonprincipal plane(1-45%). How when the material exhibits tetragonal symmetry, plane (1-45) becomes a principal plane. Thus, it is impossible to indepen- dently measure the two stiffnesses C23 and C44. The value of Fig 3. Schematic of a sample instrumented for strain partition under the in-plane shear modulus Ca4 is obtained from the phase velocity of a shear wave generated with a pair of contact trans- verse transducers. This value is used together with plane (1 45)data to simplify and improve the identification of C? manufactured from preforms built up for Sic-Sic composite The composite studied is a woven 2D material under tensile stress because it makes it possible to tic-tensor variation under load. Becauseary to identify the elas- silicon carbide by Societe Europeenne de Propulsion(SEP by complete stiffness tensor, it is possible to know which coeffi- using CVI. Although the material possesses a significant po- cients-and, therefore, which engineering constants--are af- rosity the signal frequency used, which is equal to 2.25 MHz, fected during the damage process. 29 It also allows one to pre put us close to the limit of the scales of homogeneity require cisely measure which portion of the global strain is either elastic or inelastic The characterization of anisotropic materials using an ultra- sonic methods gives access to the purely elastic portion of 33 their behavior. The complete determination of the stiffness ten- sor of a composite with orthorhombic symmetry allows, by simply inverting the tensor, a description in terms of compli nces,which, in turn, allows reconstruction of the elastic por tion of the hardening curve. Wave-speed measurements are performed by using ultrasonic pulses that are refracted through a plate sample immersed in water. Because of the signal dis- tortion that is caused by the propagation in a porous medium a special signal-processing method has been developed; this method leads to the correct measurement of the phase velocity 查 of the pulses through the porous sample tress(MPa) c The main principles of the ultrasonic evaluation of the elastic oefficients of anisotropic materials have been given by Roux. To identify the nine elastic constants Cu that fully describe the elastic behavior of an orthotropic material, the wave-propagation velocities are collected in the two accessible principal planes(planes(1-2)and(1-3), Fig. 3)and in a non- principal plane (plane (1-45), described by the bisectrix of axes 2 and 3, Fig. 3). The identification in plane (1-2) gives four elastic constants: Cu, C22, C66, and C12. Three other elas tic coefficients are obtained in plane(1-3): C33, Css, and C13 6 The two remaining coefficients-C23 and Carare identified tress(MP 0 100150200250 Strain (%t (B 6 Variation of the compliance tensor coefficients( fa 2D SiC-SiC composite(and their 90% relative Fig. 4. Stress-strain curve of a 2D SiC-SiC composite. versus stress applied in direction 3 (see Fig. 3)
material under tensile stress because it makes it possible to perform the angular investigation necessary to identify the elastic-tensor variation under load. Because it gives access to the complete stiffness tensor, it is possible to know which coefficients—and, therefore, which engineering constants—are affected during the damage process.29 It also allows one to precisely measure which portion of the global strain is either elastic or inelastic.22 The characterization of anisotropic materials using an ultrasonic method30 gives access to the purely elastic portion of their behavior. The complete determination of the stiffness tensor of a composite with orthorhombic symmetry allows, by simply inverting the tensor, a description in terms of compliances, which, in turn, allows reconstruction of the elastic portion of the hardening curve. Wave-speed measurements are performed by using ultrasonic pulses that are refracted through a plate sample immersed in water. Because of the signal distortion that is caused by the propagation in a porous medium, a special signal-processing method has been developed; this method leads to the correct measurement of the phase velocity of the pulses through the porous sample.31 The main principles of the ultrasonic evaluation of the elastic coefficients of anisotropic materials have been given by Roux.32 To identify the nine elastic constants Cij that fully describe the elastic behavior of an orthotropic material, the wave-propagation velocities are collected in the two accessible principal planes (planes (1–2) and (1–3), Fig. 3) and in a nonprincipal plane (plane (1–45°), described by the bisectrix of axes 2 and 3, Fig. 3). The identification in plane (1–2) gives four elastic constants: C11, C22, C66, and C12. Three other elastic coefficients are obtained in plane (1–3): C33, C55, and C13. The two remaining coefficients—C23 and C44—are identified by propagation in the nonprincipal plane (1–45°). However, when the material exhibits tetragonal symmetry, plane (1–45°) becomes a principal plane. Thus, it is impossible to independently measure the two stiffnesses C23 and C44. The value of the in-plane shear modulus C44 is obtained from the phase velocity of a shear wave generated with a pair of contact transverse transducers. This value is used together with plane (1– 45°) data to simplify and improve the identification33 of C23. The composite studied is a woven 2D SiC–SiC composite manufactured from preforms built up from multiple layers of silicon carbide by Socie´te´ Europe´enne de Propulsion (SEP, Saint-Me´dard-En-Jalles, France). The SiC matrix was added by using CVI. Although the material possesses a significant porosity, the signal frequency used, which is equal to 2.25 MHz, put us close to the limit of the scales of homogeneity required Fig. 3. Schematic of a sample instrumented for strain partition under load. Fig. 4. Stress–strain curve of a 2D SiC–SiC composite. Table I. Initial Compliances of a 2D SiC–SiC Composite Compliance tensor coefficient Compliance, S ((× 103 GPa)−1) Relative confidence interval ((× 103 GPa)−1) S° 11 8.5 ±0.2 S° 22 3.5 ±0.4 S° 33 4.6 ±0.5 S° 44 10.4 ±0.7 S° 55 17.5 ±0.2 S° 66 18.4 ±0.4 S° 12 −1.5 ±0.2 S° 13 0.01 ±0.2 S° 23 −2.2 ±0.3 Fig. 5. Variation of the compliance tensor coefficients (A) S33 and (B) S22 of a 2D SiC–SiC composite (and their 90% relative confidence intervals) versus stress applied in direction 3 (see Fig. 3). 2408 Journal of the American Ceramic Society—Morvan and Baste Vol. 81, No. 9
September 1998 Assessment of the Interfacial Properties of CMCs Using Strain Partition under load 2409 250 elastic ￡fsE 6. Under-load strain partition of a 2D SiC-SiC composite to interpret wave propagation. 3I Under tensile stress, this com- matrix cracking. This damage mechanism starts at a stress of posite exhibits a nonlinear behavior related to the matrix mi--80 MPa and saturates at a stress of-120 MPa. Then, the cocracking, because the matrix has a lower failure than variation of S33 becomes smaller, which indicates that another the fibers. The specimen, which has a flat dogbone shape, a mechanism is occurring. This mechanism is the matrix micro- thickness of 3 mm, and a density of 2.7 g/cm, was submitted cracking that spreads inside the bundles. The same comments to tensile stress in direction 3, parallel to the direction of one of can be made on the variation of S22, although there is less the bundles. The stress-strain curve of the sample is shown in global variation. This variation, which is due to the fiber- Fig. 4 latrix debonding, occurs with a 10 MPa delay, compared The loading was applied over eighteen steps of stress until the transverse cracking the sample failed. These steps were necessary for the ultrasonic When unloading/reloading cycles are performed, the sliding evaluation. During the test, the total strain, a, was measured that occurs at the fiber/matrix interface is subordinate to the with an extensometer specially configured to work under im- direction of the load. As the stress decreases during unloading, mersion. The total strain to failure reached 0.8%. The behavior sliding opposite to that created by an increase in stress car occur. The transverse cracks that have been created during the ycles performed are hysteresis loops that exhibit many monotonic loading are then prone to close. Thus, the inelastic changes in slope, relevant to the interfacial debonding exte strain varies during the cycles. Therefore, identifying the in- and emphasize the presence of sliding with friction that need elastic strains as the residual strains at zero stress leads to ar a threshold of stress to occur. Even after unloading, the residual underestimate of their contribution to the total strain. To sepa train remains far from negligible. It is noteworthy that, until rate and accurately identify the effects of the initiation and 150 MPa, the stress-strain curve of the 2D SiC-SiC composite growth of matrix microcracks under tensile loading, as well as that th the he effec acks w clic load steps, which are performed at constant stress is necessary to perform a strain partition under load The accuracy and the reliability of the complete determina- Complete determination of the stiffness tensor, together with tion of the stiffness tensor, together with its variation during the its variation during the test, allows one to assess the elastic test, allow, by inverting this tensor, one to obtain the tensor of portion of the material behavior. The elastic strain is obtained elastic compliances and its own variation. Table I shows the initial elastic compliances with their confidence intervals. They tensile axis is then simply S33, depicted in Fig. 5, multiplied by have been recovered from the experimental measurements of the applied stress the stiffness of the material Figure 5 shows the compliances that were most affected by (21) the transverse and longitudinal crack patterns, with their con- fidence intervals. As observed in the stress-strain curve. the Because the extensometer gives access to the total strain, we compliance variations exhibit three domains. Until a stress of 0 MPa(which is the damage threshold of this composite)is elastic attained, the various compliances remain unchanged. After this int, matrix microcracking begins. The compliance variation The results of the strain partition are plotted in Fig. 6. Simi- along the tensile axis is very large; the value of S33 increases by lar to the compliance variations, this plot exhibits three zones more than 300%, which is representative of a large interbundle Until the damage threshold, the behavior is linear elastic. Then, Inter- Bundle Crack Density Intra-Bundle Crack Density -t- Stress(MPa) Fig. 7. Constitutive laws of the 2D SiC-SiC composite, depicting the crack-density variations interbundle scale an intrabundle scale
to interpret wave propagation.31 Under tensile stress, this composite exhibits a nonlinear behavior related to the matrix microcracking, because the matrix has a lower failure strain than the fibers. The specimen, which has a flat dogbone shape, a thickness of 3 mm, and a density of 2.7 g/cm3 , was submitted to tensile stress in direction 3, parallel to the direction of one of the bundles. The stress–strain curve of the sample is shown in Fig. 4. The loading was applied over eighteen steps of stress until the sample failed. These steps were necessary for the ultrasonic evaluation. During the test, the total strain, «, was measured with an extensometer specially configured to work under immersion. The total strain to failure reached 0.8%. The behavior remained elastic until a load of ∼80 MPa was attained. The two cycles performed are hysteresis loops that exhibit many changes in slope, relevant to the interfacial debonding extent, and emphasize the presence of sliding with friction that needs a threshold of stress to occur. Even after unloading, the residual strain remains far from negligible. It is noteworthy that, until 150 MPa, the stress–strain curve of the 2D SiC–SiC composite shows that the strain increases during the ultrasonic evaluation steps, which are performed at constant stress. The accuracy and the reliability of the complete determination of the stiffness tensor, together with its variation during the test, allow, by inverting this tensor, one to obtain the tensor of elastic compliances and its own variation. Table I shows the initial elastic compliances with their confidence intervals. They have been recovered from the experimental measurements of the stiffness of the material. Figure 5 shows the compliances that were most affected by the transverse and longitudinal crack patterns, with their confidence intervals. As observed in the stress–strain curve, the compliance variations exhibit three domains. Until a stress of 80 MPa (which is the damage threshold of this composite) is attained, the various compliances remain unchanged. After this point, matrix microcracking begins. The compliance variation along the tensile axis is very large; the value of S33 increases by more than 300%, which is representative of a large interbundle matrix cracking. This damage mechanism starts at a stress of ∼80 MPa and saturates at a stress of ∼120 MPa. Then, the variation of S33 becomes smaller, which indicates that another mechanism is occurring. This mechanism is the matrix microcracking that spreads inside the bundles. The same comments can be made on the variation of S22, although there is less global variation. This variation, which is due to the fiber– matrix debonding, occurs with a 10 MPa delay, compared to the transverse cracking. When unloading/reloading cycles are performed, the sliding that occurs at the fiber/matrix interface is subordinate to the direction of the load. As the stress decreases during unloading, sliding opposite to that created by an increase in stress can occur. The transverse cracks that have been created during the monotonic loading are then prone to close. Thus, the inelastic strain varies during the cycles. Therefore, identifying the inelastic strains as the residual strains at zero stress leads to an underestimate of their contribution to the total strain. To separate and accurately identify the effects of the initiation and growth of matrix microcracks under tensile loading, as well as the effect of these cracks when cyclic loading is performed, it is necessary to perform a strain partition under load. Complete determination of the stiffness tensor, together with its variation during the test, allows one to assess the elastic portion of the material behavior. The elastic strain is obtained from the generalized Hooke’s law. The elastic strain along the tensile axis is then simply S33, depicted in Fig. 5, multiplied by the applied stress: «elastic = S33s (21) Because the extensometer gives access to the total strain, we have «inelastic = «total − «elastic (22) The results of the strain partition are plotted in Fig. 6. Similar to the compliance variations, this plot exhibits three zones. Until the damage threshold, the behavior is linear elastic. Then, Fig. 6. Under-load strain partition of a 2D SiC–SiC composite. Fig. 7. Constitutive laws of the 2D SiC–SiC composite, depicting the crack-density variations on an interbundle scale and an intrabundle scale. September 1998 Assessment of the Interfacial Properties of CMCs Using Strain Partition under Load 2409
2410 Journal of the American Ceramic SocieryMorvan and Baste Vol 81. No 9 Inter-Bundle Inelastic Strains IntraBundle Inelastic Strains Fig 8. Evolution of the inelastic strains of a 2D SiC-SiC composite the interbundle matrix microcrack and with it, the value at saturation of the transverse crack pattern with respect inelastic strain appears. The inelastic strain exhibits two very to the two-scale effect different increases related to the scale at which they occur: an increment of strain at constant stress(AEhs), which starts at-80 MPa and stops at-160 MPa, and a strain that requires an increase in stress and begins at 120 MPa. Pr\S%2-S% Thus, Eq.(22) can be improved by using the following relation. elastic Ebs.8 where ebs represents the inelastic strain that grows at constant stress and es is the inelastic strain due to an increment of stress Iv. Identification of the model The experimental variation of the compliances gives access to the constitutive law of the transverse crack densities using (Fig. 7; in this figure, the symbols represent the values dentified from Eq (1)using the compliance variations and the traight lines represent the interpolated evolution laws ). Be cause of the two-scale effect of the microcracking, the identi fication is not straightforward. Actually, the values of the com- liances of the so-called uncracked material must be adjusted For the interbundle microcracking. the values of the compli ances at 80 MPa (the damage threshold) have been used whereas, for the intrabundle scale, the values at 120 MPa(th onset of intrabundle cracking) were considered. As a result both the interbundle and intrabundle transverse crack densities (A) can be described with a linear function. Up to 80 MPa, the crack density is equal to zero, because the damage has not begun. The interbundle crack density saturates at-120 MPa, at hich point the intrabundle crack density begins After the transverse crack densities are known, Eq (5)gives access to the inelastic strain variation at both scales of the 添 composite. These variations are depicted in Fig. 8, where the symbols represent the experimental data and the straight lines represent the predictions The two-scale effect in the 2D SiC-SiC composite must also be taken into account in Eq.(20). Because this relation has EE been written at the fiber scale. it must be modified at the bundle scale. Assuming that the bundles are elliptic, the radius of the fiber R in Eq. (20)must be replaced by the term [2m Mf/(m2+ MI where M and m are the major and minor semi-axe respectively, of the bundle. The Young's modulus of the fiber, Er must then become E, which is the Youngs modulus of the bundle 34 The compliance variations are sensitive not only to the pres- ence of cracks but also to their size. thus. the debonding length la can be estimated using the transverse crack density and the variation of S22(this variation is dependent upon Mode II ), acking). Assuming that la is zero when the material is ur cracked and is equal to one-half the intercrack distance when the matrix microcracking is at saturation, it becomes possible, using a simple rule of mixtures, to estimate la from the values of the compliances S22, of the uncracked material, and S%2, the scale(Aubard35)
the interbundle matrix microcracking begins, and with it, the inelastic strain appears. The inelastic strain exhibits two very different increases related to the scale at which they occur: an increment of strain at constant stress (D«bs), which starts at ∼80 MPa and stops at ∼160 MPa, and a strain that requires an increase in stress and begins at 120 MPa. Thus, Eq. (22) can be improved by using the following relation: «inelastic = «bs + «fs (23) where «bs represents the inelastic strain that grows at constant stress and «fs is the inelastic strain due to an increment of stress. IV. Identification of the Model The experimental variation of the compliances gives access to the constitutive law of the transverse crack densities using Eq. (1) (Fig. 7; in this figure, the symbols represent the values identified from Eq. (1) using the compliance variations and the straight lines represent the interpolated evolution laws). Because of the two-scale effect of the microcracking, the identification is not straightforward. Actually, the values of the compliances of the so-called uncracked material must be adjusted. For the interbundle microcracking, the values of the compliances at 80 MPa (the damage threshold) have been used, whereas, for the intrabundle scale, the values at 120 MPa (the onset of intrabundle cracking) were considered. As a result, both the interbundle and intrabundle transverse crack densities can be described with a linear function. Up to 80 MPa, the crack density is equal to zero, because the damage has not begun. The interbundle crack density saturates at ∼120 MPa, at which point the intrabundle crack density begins. After the transverse crack densities are known, Eq. (5) gives access to the inelastic strain variation at both scales of the composite. These variations are depicted in Fig. 8, where the symbols represent the experimental data and the straight lines represent the predictions. The two-scale effect in the 2D SiC–SiC composite must also be taken into account in Eq. (20). Because this relation has been written at the fiber scale, it must be modified at the bundle scale. Assuming that the bundles are elliptic, the radius of the fiber R in Eq. (20) must be replaced by the term [2m2 M2 /(m2 + M2 )]1/2, where M and m are the major and minor semi-axes, respectively, of the bundle. The Young’s modulus of the fiber, Ef , must then become Et , which is the Young’s modulus of the bundle.34 The compliance variations are sensitive not only to the presence of cracks but also to their size. Thus, the debonding length ld can be estimated using the transverse crack density and the variation of S22 (this variation is dependent upon Mode II cracking). Assuming that ld is zero when the material is uncracked and is equal to one-half the intercrack distance when the matrix microcracking is at saturation, it becomes possible, using a simple rule of mixtures, to estimate ld from the values of the compliances S° 22, of the uncracked material, and Ss 22, the value at saturation of the transverse crack pattern with respect to the two-scale effect: ld = a bT S S22 − S° 22 S22 s − S° 22D (24) Fig. 8. Evolution of the inelastic strains of a 2D SiC–SiC composite. Fig. 9. Micrographs of the transverse cracking of a 2D SiC–SiC composite ((A) interbundle scale (Guillaumat34) and (B) intrabundle scale (Aubard35)). 2410 Journal of the American Ceramic Society—Morvan and Baste Vol. 81, No. 9
September 1998 Assessment of the Interfacial Properties of CMCs Using Strain Partition under load 2411 600 250 Crack Free Zone 200 50 200 Fig. 10. Characteristic lengths of the transverse microcracking of a 2D SiC-SiC composite Using the elastic strains calculated from Eq.(21), the pre- tion, or failure, occurs. The model describes what can be called dicted linear variation of the inelastic strains that appear in Fig. a mixed bonding-that is to say, an initially strong bonding 8, the parameters identified from the micrographs in Fig.9 that promotes elasticity, becomes weak with increasing stress, (given in Table ID), together with the variations of the compli and then allows fiber-matrix sliding when delamination oc- ances that appear as plain lines in Fig. 5, it is possible to predict curs. Furthermore, the increase of T at both scales can be re- the characteristic lengths, intercrack distances(Eq.(3), and ated to the saturation of the sliding. The relative fiber-matrix debond lengths(Eq.(22) of the microcracking(Fig. 10), as sliding does not become more difficult because of a clamping well as the interfacial sliding stress at both scales of the com stress or debris that lock the sliding; more probably, the diffi- posite(Fig. 11). The average lengths of the elementary cell culty is caused by the decrease in the number of sliding sites, given in Table II take into account the waviness of the because of saturation. bundle remark must be made regarding the interbundle scale. considering a linear variation for the inelastic strains V. Conclusion stead of the values directly identified from the experiment ( Fig. 8)is equivalent to compensate the delay that results from The fundamental strain mecl of ceram the mode acking. In Fig. 10, as the intercrack distance posites(CMCs)are matrix mi king, which induces a loss diminishes, the sliding length increases; it increases up to one of stiffness. and fiber-matrix ing which leads to inter- half the intercrack distance and then stabilizes because of satu- facial frictional sliding. Thu ial sliding stress is a k ation. No further transverse cracking appears, and either the arameter in the global behavior. Accurate knowledge of the cracking begins at another scale or the sample breaks. In regard strain mechanisms is necessary to correctly identify the value to T(Fig. 11), the average value at the interbundle scale is of the interfacial sliding stress. Ultrasonic characterization different from the value at the fiber scale. This variation can be through the complete determination of the stiffness tensor explained by the difference that exists in the radii, as well as in along the entire test can detect all the damage mechanisms of the debond lengths. The lower value at the bundle scale is-200 CMCs: the transverse matrix microcracking. as well as the 1Pa, whereas it is -100 MPa at the fiber scale. This range is resence of longitudinal cracks at the fiber/matrix interface consistent with previous results found elsewhere. 34 The higher is of great help in the aniso- value of T at the bundle scale explains the progressive slidin tropic damage and it also allows one to conduct the strain whereas, at the fiber scale, the sliding occurs simultaneously to partition under load. Because the strain partition under loa the increment of stress. It is noteworthy that, whatever the separates the various mechanisms responsible for the nonlinear scale. the evaluation is not constant. It decreases to a minimum behavior of cmcs. it then seems clear that those two elemen- value and then increases to an even-higher value when satura- tary mechanisms occur at both scales of the material: at the tra bundle 300 H200 Stress(MPa) Fig. 11. Interfacial sliding stress of a 2D SiC-SiC composite
Using the elastic strains calculated from Eq. (21), the predicted linear variation of the inelastic strains that appear in Fig. 8, the parameters identified from the micrographs in Fig. 9 (given in Table II), together with the variations of the compliances that appear as plain lines in Fig. 5, it is possible to predict the characteristic lengths, intercrack distances (Eq. (3)), and debond lengths (Eq. (22)) of the microcracking (Fig. 10), as well as the interfacial sliding stress at both scales of the composite (Fig. 11). The average lengths of the elementary cell given in Table II take into account the waviness of the bundle.36 A remark must be made regarding the interbundle scale: considering a linear variation for the inelastic strains instead of the values directly identified from the experiment (Fig. 8) is equivalent to compensate the delay that results from the Mode II cracking. In Fig. 10, as the intercrack distance diminishes, the sliding length increases; it increases up to onehalf the intercrack distance and then stabilizes, because of saturation. No further transverse cracking appears, and either the cracking begins at another scale or the sample breaks. In regard to t (Fig. 11), the average value at the interbundle scale is different from the value at the fiber scale. This variation can be explained by the difference that exists in the radii, as well as in the debond lengths. The lower value at the bundle scale is ∼200 MPa, whereas it is ∼100 MPa at the fiber scale. This range is consistent with previous results found elsewhere.34 The higher value of t at the bundle scale explains the progressive sliding, whereas, at the fiber scale, the sliding occurs simultaneously to the increment of stress. It is noteworthy that, whatever the scale, the evaluation is not constant. It decreases to a minimum value and then increases to an even-higher value when saturation, or failure, occurs. The model describes what can be called a mixed bonding—that is to say, an initially strong bonding that promotes elasticity, becomes weak with increasing stress, and then allows fiber–matrix sliding when delamination occurs. Furthermore, the increase of t at both scales can be related to the saturation of the sliding. The relative fiber–matrix sliding does not become more difficult because of a clamping stress or debris that lock the sliding; more probably, the difficulty is caused by the decrease in the number of sliding sites, because of saturation. V. Conclusion The fundamental strain mechanisms of ceramic-matrix composites (CMCs) are matrix microcracking, which induces a loss of stiffness, and fiber–matrix debonding, which leads to interfacial frictional sliding. Thus, interfacial sliding stress is a key parameter in the global behavior. Accurate knowledge of the strain mechanisms is necessary to correctly identify the value of the interfacial sliding stress. Ultrasonic characterization through the complete determination of the stiffness tensor along the entire test can detect all the damage mechanisms of CMCs: the transverse matrix microcracking, as well as the presence of longitudinal cracks at the fiber/matrix interface. Thus, this technique is of great help in measuring the anisotropic damage, and it also allows one to conduct the strain partition under load. Because the strain partition under load separates the various mechanisms responsible for the nonlinear behavior of CMCs, it then seems clear that those two elementary mechanisms occur at both scales of the material: at the Fig. 10. Characteristic lengths of the transverse microcracking of a 2D SiC–SiC composite. Fig. 11. Interfacial sliding stress of a 2D SiC–SiC composite. September 1998 Assessment of the Interfacial Properties of CMCs Using Strain Partition under Load 2411
2412 Journal of the American Ceramic SocieryMorvan and Baste Vol 8l. No 9 Table Il. Parameters Used to Assess Interfacial posites s and Relationships to Constituent Properties, J. Am. Ce vans, J.-M. Domergue, and E. Vagaggini, ""Methodology for Re- Interbundle lating the Tensile Constitutive Behavior of Ceramic-Matrix Composites to Con- Radius of the fiber, R(um) stituent Properties,J.Am. Ce Push-Out(Indentation) Tests Minor semi-axis of the bundle in Ceramic-Matrix Composites, J. Major semi-axis of the bundle 1500 in Fiber-Reinforced Brittle Matrix Composites: Basic Problems, "Mater. Sci. Youngs modulus of the fiber, E(GPa) l80 eR. J. Kerans, T. A. Parthasarathy, P, D. Jero, A. Chatterjee, and N. J. Pa- Youngs modulus of the bundle, ino. ' Fracture and Sliding in the Fibre/Matrix Interface and Failure Processes 200 Crack depth, 2a(m) of Hysteresis Observe During Fatigue of Ceramic-Matrix Composites, J Am. Ceram. Soc., 73 Crack width, 2c(um Depth of elementary cell 550 1879-83(1990) "D. Rouby and P. Reynaud, ""Fatigue Behaviour Related to Interface Modi- width of elementary cell, 2x,(um) 3500 fication during Load Cycling in Ceramic-Matrix Composites, Compos. Sci IT. A. Parthasarathy, D. B. Marshall, and R J. Kerans, ""Analysis of the Effect of Interfacial Roughness on Fiber Debonding and Sliding in Brittle Ma- interbundle matrix, and at the microstructure level, which con- Hysteresis Measurements and the Constituent Properties of onships between mesostructure level, constituted by both the bundles and the Ceramic Matrix tal Studies on Unidirectional Materials, J. Am. Ce chanical model used with these accurate measurements of the ram,w. Hutchonna-ndh95) various components of the total strain gives access to the value Pullout in Brittle Composites with Fnit of the interfacial sliding stress during the entire tensile test. In 2S. Baste and J-M van,""Unde the model, the interfacial sliding stress is dependent upon both Matrix Composite Using an Ultrasonic Method,E the elastic and inelastic strains but also upon the transverse crack density and the area upon which the sliding occurs. As a M. Morvan and S. Baste. Effects of Two-Scale transverse result, according to the scale of the composite, the interfacial sliding stress exhibits a different value, because of the nature of Danchaivijit and D K. Shetty,""Matrix Cracking in Ceramic Matrix 2C.-H. Hsueh, operties of Fiber-Reinforced Ceramic Compo sing a Mechanical Properties Microprobe, J. Am. Ce ZL. s. Sigl and A G. Evans, Effects of Residual Stress and Frictional Sliding on Cracking and Pull Out in Brittle Matrix Composites, ' Mech. Mater Cracking in Brittle-Matrix Fiber Composites, ""Acta Metall, 33 [11]2013-21 8,1-12(1989) 27N. Laws, G J. Dvorak, and M. Hejazi, ""Stiffness Changes in Unidirec- keaaoree Evans and F w. Zok, " The Physics and Mechanics of Fibre tional Composites Caused by Crack Systems, Mech. Mater, 2, 123-37(1983) d Brittle Matrix Composites, J. Mater. Sci., 29, 3857-96(1994) Evans, F. W. Zok, and J. Davis, "The Role of Interfaces in Fibe isotropic Solids."Philos. Mag, 36,367-zg es sociated with Cracks in Reinforced Brittle Matrix Composites, Compos. Sci. Technol, 42, 3-24 9B. Audoin and S. Baste. "Ultrasonic Evaluation of Stiffness Tensor Roughness, and Fiber Coatings on Interfacial Properties in Ceramic Compos- Roux, B. Hosten, B. Castagnede, and M. Deschamps, ""Caracterisation Pifabram. Soc., 79[1\13-1 1996) Mecanique des Solides par Spectro-Interferometrie Ultrasonore, Rev. Phys. els of Fiber-Matrix Interfacial Debonding, " JAm. Ceram.Soc,756]1694-96(1992) Audoin and J. Roux, "An Innovative Application of the Hilbert Trans- form to Time Delay Estimation of Overlapped Ultrasonic due to Matrix Cracking in Unidirectional Fiber-Reinforced Composites, ""Mech. oux,"Elastic Wave Propagation in Anisotropic I Lamon, "Interface and Interfacial Mechanics: Intluence on the Mechani 73 in Proceedings of IEEE 1990 Ultrasonics Symposia M2210 cal Behavior of Ceramic Matrix Composites(CMC), ""J Phys. IV, 3, 1607-16 Engineers, Piscataway, NJ, 1990 Voy. Institute of Electronic and D. B. Marshall. " An Indentation Method for Measuring Matrix-Fiber Fri B Hosten, ""Stiffness Matrix Invariants late the Characterization of Composite Materials with Ultrasonic Methods, Ultrasonics, 30 16] 365-71 C-260(1984 "D. B. Marshall and W. C. Oliver. ""Measurement of Interfacial Mechanical L. Guillaumat, Microfissuration des CMCs: Relation avec la Microstruc- rties in Fiber-Reinforced Ceramic Composites, "J. Am. Ceram. Soc. ortement Mecanique""(Microcracking in CMCs: Relation be 8J542-48(1987 PC-H. Hsueh and M. K. Ferber. "Evaluations of Residual Axial Stresses tur d Mechanical Behavior ): Ph. D. Thesis. University of and Interfacial Friction in Nicalon Fibre orced Macro- Defect-Free Cement 3SX. Aubard. Modelisation et Identification du ce Composites, J Mater. Sci., 28, 2551-56(1993). des Materiaux Composites 2D SiC-SiC"(Modeling and Identification of the Fernandez, and M. J. Purdy, ""Determination Composite Materials); Ph. D. Thesis. Un of Interfacial Properties Using a Single- Fiber Microcomposite Test, J. Am ersity of Paris VI, Paris, France, 1992 Ceram.Soc,79]1083-91(1996) =F. E. Heredia, S M. Spearing, A. G. Evans, P. Mosher, and w.A. Curtin, Woven C/SiC Composite under Mechanical Loading. Part 1: Mechanical Char- Mechanical Properties of Continuous-Fiber-Reinforced Carbon Matrix Co acterization, ""Compos. Sci. Techno, 56, 1363-72(1996)
mesostructure level, constituted by both the bundles and the interbundle matrix, and at the microstructure level, which consists of both the fibers and the intrabundle matrix. A micromechanical model used with these accurate measurements of the various components of the total strain gives access to the value of the interfacial sliding stress during the entire tensile test. In the model, the interfacial sliding stress is dependent upon both the elastic and inelastic strains but also upon the transverse crack density and the area upon which the sliding occurs. As a result, according to the scale of the composite, the interfacial sliding stress exhibits a different value, because of the nature of the bonding. References 1 D. B. Marshall, B. N. Cox, and A. G. Evans, ‘‘The Mechanics of Matrix Cracking in Brittle-Matrix Fiber Composites,’’ Acta Metall., 33 [11] 2013–21 (1985). 2 A. G. Evans and F. W. Zok, ‘‘The Physics and Mechanics of FibreReinforced Brittle Matrix Composites,’’ J. Mater. 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Parameters Used to Assess Interfacial Sliding Stress Parameter Interbundle Intrabundle Radius of the fiber, R (mm) 7.5 Minor semi-axis of the bundle, m (mm) 150 Major semi-axis of the bundle, M (mm) 1500 Young’s modulus of the fiber, Ef (GPa) 180 Young’s modulus of the bundle, Et (GPa) 200 Crack depth, 2a (mm) 150 15 Crack width, 2c (mm) 2000 15 Depth of elementary cell, 2x1 (mm) 200 30 Width of elementary cell, 2x2 (mm) 3500 30 2412 Journal of the American Ceramic Society—Morvan and Baste Vol. 81, No. 9