theoretical and applied fracture mechanIcs ELSEVIER Theoretical and Applied Fracture Mechanics 32(1999)15-25 www.elsevier.com/locate/tafmec Determination of mechanical properties of fiber-matrix interface from pushout test . Ye..K. Kaw Mechanical Engineering Department, Unirersity of South Florida, Tampa, FL 33620-5350, USA Abstract For ceramic matrix composites, the pushout test is the most widely used test for finding the two mechanical properties of the fiber-matrix interface-(1)the coefficient of friction and (2) the residual radial stress. Experimental measurements from the pushout test do not directly give the values of these two mechanical properties of the fiber- natrix interface, but need to be regressed to theoretical models. Currently, approximate theoretical models based on shear-lag analysis are used for regression. In this paper, the adequacy of the shear-lag analysis model in accurately inding the mechanical properties of the fiber-matrix interface is discussed. An elasticity solution of the pushout test based on boundary element method is developed. Regressing one set of available experimental data from a pushout test to both shear-lag analysis and boundary element method models gives values differing by 15% for the coefficient of friction but similar values for the residual radial stress Parametric studies were also conducted to show the difference between the shear-lag analysis and boundary element method results for factors such as fiber to matrix elastic moduli ratios, coefficient of friction and fiber volume fractions. 1999 Elsevier Science Ltd. All rights reserved 1. Introduction Since the early 1980s, several experimental tests, such as the pushout test, have been developed to Fracture toughness of ceramic matrix compos find the two mechanical properties of the fiber- ites(CMCs)is dependent and sensitive to the two matrix interface in ceramic matrix composites mechanical properties of fiber-matrix interface [1].(CMCs). In the pushout test, a composite is sliced namely, the coefficient of friction and the residual normal to the fiber direction and the specimen is radial stress. Hence, there is a need to find these placed on a platform with a hole as shown in properties accurately. Only if these two properties Fig. 1. A micro-indentor with a radius that is 60- are found accurately, can we have reliable quan- 90% of the radius of the fiber pushes on the fiber. tification of the correlation of the microstructure Generally, the pushout force on the fiber and the and the fracture toughness, and then only can we displacement on the surface of the fiber below the have better control of the composite behavior for matrix surface due to interfacial slip are measured onducting optimum and reliable design of ce- to construct a pushout force vs. displacement ramic matrix composite structures curve. The pushout force-displacement curve from the test is then regressed to a theoretical model of the test for determining the two mechanical properties of the fiber-matrix interface. So far, the Corresponding author. Tel. +1-813-974-5626: fax: +1-813. theoretical models used for regression are limited 974-3539: e-mail: kaw deng. usf.edu to shear-lag analysis (SLA). In this paper, we 0167-8442/99/. see front matter @1999 Elsevier Science Ltd. All rights reserved PI:S0167-8442(99)00022
Determination of mechanical properties of ®ber±matrix interface from pushout test J. Ye, A.K. Kaw * Mechanical Engineering Department, University of South Florida, Tampa, FL 33620-5350, USA Abstract For ceramic matrix composites, the pushout test is the most widely used test for ®nding the two mechanical properties of the ®ber±matrix interface ± (1) the coecient of friction and (2) the residual radial stress. Experimental measurements from the pushout test do not directly give the values of these two mechanical properties of the ®ber± matrix interface, but need to be regressed to theoretical models. Currently, approximate theoretical models based on shear±lag analysis are used for regression. In this paper, the adequacy of the shear±lag analysis model in accurately ®nding the mechanical properties of the ®ber±matrix interface is discussed. An elasticity solution of the pushout test based on boundary element method is developed. Regressing one set of available experimental data from a pushout test to both shear±lag analysis and boundary element method models gives values diering by 15% for the coecient of friction but similar values for the residual radial stress. Parametric studies were also conducted to show the dierence between the shear±lag analysis and boundary element method results for factors such as ®ber to matrix elastic moduli ratios, coecient of friction and ®ber volume fractions. Ó 1999 Elsevier Science Ltd. All rights reserved. 1. Introduction Fracture toughness of ceramic matrix composites (CMCs) is dependent and sensitive to the two mechanical properties of ®ber±matrix interface [1], namely, the coecient of friction and the residual radial stress. Hence, there is a need to ®nd these properties accurately. Only if these two properties are found accurately, can we have reliable quanti®cation of the correlation of the microstructure and the fracture toughness, and then only can we have better control of the composite behavior for conducting optimum and reliable design of ceramic matrix composite structures. Since the early 1980s, several experimental tests, such as the pushout test, have been developed to ®nd the two mechanical properties of the ®ber± matrix interface in ceramic matrix composites (CMCs). In the pushout test, a composite is sliced normal to the ®ber direction and the specimen is placed on a platform with a hole as shown in Fig. 1. A micro-indentor with a radius that is 60± 90% of the radius of the ®ber pushes on the ®ber. Generally, the pushout force on the ®ber and the displacement on the surface of the ®ber below the matrix surface due to interfacial slip are measured to construct a pushout force vs. displacement curve. The pushout force±displacement curve from the test is then regressed to a theoretical model of the test for determining the two mechanical properties of the ®ber±matrix interface. So far, the theoretical models used for regression are limited to shear±lag analysis (SLA). In this paper, we www.elsevier.com/locate/tafmec Theoretical and Applied Fracture Mechanics 32 (1999) 15±25 * Corresponding author. Tel.: +1-813-974-5626; fax: +1-813- 974-3539; e-mail: kaw@eng.usf.edu 0167-8442/99/$ - see front matter Ó 1999 Elsevier Science Ltd. All rights reserved. PII: S 0 1 6 7 - 8 4 4 2 ( 9 9 ) 0 0 022-1
J. Ye, AK Kaw Theoretical and Applied Fracture Mechanics 32(1999)15-25 sliding friction stress can be overestimated if the Indent transverse expansion of the fiber is not taken into account. The Poisson expansion of the fibers under the compressive loads and the consequent of the normal stress across the interface lead to a nonlinear variation of the frictional shear stress Matrix along the embedded fiber length is most often used for modeling the pushout test Base and for the evaluation of the fiber-matrix interface ig. 1. Schematic of a single fiber pushout test. However, SLA models have assumptions such as-approximate shear stress distribution on the interface. and axial stresses in the fiber and matrix focus our attention on the adequacy of using SLa are independent of radial direction. So how ade models in extracting the mechanical properties of quate is the SLA model for extracting the two the fiber-matrix interface mechanical properties of the fiber-matrix inter- The first pushout test was conducted in [2] face? To answer this question, an elasticity model Measured was the force necessary to slip a fiber of the pushout test based on boundary element along part or all of its length by pushing on the method(BEM)was developed. The BEM model is end with an indentor on a flat-end probe. It was then used to extract the two fiber-matrix interface assumed that the fiber and matrix are bonded only properties. The results from the BEM model are frictionally in these composites. In [3, 4] use was compared with the SLA model. Further for a made of a variation of the pushout technique [2] complete study, parametric studies are conducted for measuring the interfacial friction stress. In to compare BEM and SLA results for several these experiments, the force F required to push out rameters such as coefficient of friction, fiber to ort fibers from thin composite specimen was matrix elastic moduli ratios, and fiber volume measured and the friction stress was calculated fractions from a simple force balance analysis, F= 2Trr La where rr is the fiber radius, L the fiber length and or is the shear stress along the fiber-matrix in- 2. Shear-lag analysis of pushout test terface. This assumes that the interfacial shear stress over the embedded length that supports the Fig. 2 is the schematic of the pushout test in the external force is constant. However. a constant shear-lag analysis. It shows a composite geometry shear stress approximation may only be reason- of length L and a uniform pressure loading p on of friction is small. Thi ry short or the the fiber to simulate the indentor load Coulomb neglects the variation of the radial stress normal to the interface due to the poissons effect under pushout loading. Actually, in a pushout test the xternal applied force is compressive that expand the fiber in the transverse direction, thereby in creasing the normal stress and hence the frictional stress at the interface A shear-lag analysis(SLA)[5] was proposed for modeling the pushout test. An exponential de- crease was predicted for the interfacial shear stress along the fiber length. The results showed that the Fig. 2. Schematic of the pushout test for shear-lag analysis
focus our attention on the adequacy of using SLA models in extracting the mechanical properties of the ®ber±matrix interface. The ®rst pushout test was conducted in [2]. Measured was the force necessary to slip a ®ber along part or all of its length by pushing on the end with an indentor on a ¯at-end probe. It was assumed that the ®ber and matrix are bonded only frictionally in these composites. In [3,4] use was made of a variation of the pushout technique [2] for measuring the interfacial friction stress. In these experiments, the force F required to push out short ®bers from thin composite specimen was measured and the friction stress was calculated from a simple force balance analysis, F 2prfLrrz, where rf is the ®ber radius, L the ®ber length and rrz is the shear stress along the ®ber±matrix interface. This assumes that the interfacial shear stress over the embedded length that supports the external force is constant. However, a constant shear stress approximation may only be reasonable if the embedded length is very short or the coecient of friction is small. This approximation neglects the variation of the radial stress normal to the interface due to the PoissonÕs eect under pushout loading. Actually, in a pushout test the external applied force is compressive that expands the ®ber in the transverse direction, thereby increasing the normal stress and hence the frictional stress at the interface. A shear±lag analysis (SLA) [5] was proposed for modeling the pushout test. An exponential decrease was predicted for the interfacial shear stress along the ®ber length. The results showed that the sliding friction stress can be overestimated if the transverse expansion of the ®ber is not taken into account. The Poisson expansion of the ®bers under the compressive loads and the consequent increase of the normal stress across the interface lead to a nonlinear variation of the frictional shear stress along the embedded ®ber length. Shear±lag analysis such as that discussed in [5] is most often used for modeling the pushout test and for the evaluation of the ®ber±matrix interface properties [6]. However, SLA models have assumptions such as ± approximate shear stress distribution on the interface, and axial stresses in the ®ber and matrix are independent of radial direction. So how adequate is the SLA model for extracting the two mechanical properties of the ®ber±matrix interface? To answer this question, an elasticity model of the pushout test based on boundary element method (BEM) was developed. The BEM model is then used to extract the two ®ber±matrix interface properties. The results from the BEM model are compared with the SLA model. Further for a complete study, parametric studies are conducted to compare BEM and SLA results for several parameters such as coecient of friction, ®ber to matrix elastic moduli ratios, and ®ber volume fractions. 2. Shear±lag analysis of pushout test Fig. 2 is the schematic of the pushout test in the shear±lag analysis. It shows a composite geometry of length L and a uniform pressure loading p on the ®ber to simulate the indentor load. Coulomb friction law is assumed at the interface, and Fig. 2. Schematic of the pushout test for shear±lag analysis. Fig. 1. Schematic of a single-®ber pushout test. 16 J. Ye, A.K. Kaw / Theoretical and Applied Fracture Mechanics 32 (1999) 15±25
J. Ye, AK Kaw Theoretical and Applied Fracture Mechanics 32(1999)15-25 residual radial compression due to thermal ex- This type of measurement also has its limitations pansion mismatch between the fibers and matrix is The measurement of slip length is difficult to make assumed. Using SLA [6,7], analytical models are and in many cases the fiber slips at both ends, developed for the pushout test. Three expressions hence making it difficult to relate the slip length (depending on type of experimental data collected) one measurement. In addition, accounting for wo mecheressing experimental data to find the more than one slip zone in a theoretical model is anical properties of the interface are Imost intractable due to the nonlinear nature of given below the frictional interface 2. 1. Fiber displacement 2.3. Maximum pi In the first case, the pushout force, P on the fiber is measured as a function of the displacement The third type of experimental here the maximum pushout force, Pmax, is mea Au, on the surface of the fiber below the matrix sured as a function of specimen thickness, L. The surface due to interfacial slip. The SLA equation Is SLA model equation [5] is same as Eq (3), except given by [5] L=L, and hence is given by Ag=(1-2vk)o∫P kEf 1,(00+k(P/xr2) (1) This type of measurement does require several samples of different thickness as opposed to only one in the previous two cases. But, the slip length is predetermined; that is, it is equal to the length of Er(1+Vm)+Em(1-ve) (2) the specimen. Also, as will be shown later that this type of measurement is not influenced by indentor Youngs modulus of matrix are Er and Em, re- radius and type. Since this model seems to have the pectively while L is the specimen length. The least limitations of the three sets of possible ex Poisson's ratio of fiber and matrix are v, Vm, re- perimental data, this SLa model will be the used in spectively with u being the coefficient of friction. this study for comparison with the BEM model Residual radial stress at the interface is o and re stands for the fiber radius However, in an analytical study of [8]. the 3. Boundary element method problem statement above pushout force vs. displacement curve(eq (1)) was found to be highly influenced by many An axisymmetric model is composed of a solid extrinsic factors of the test such as type and radius cylindrical fiber of radius, rr, and length L, and a of indentor This observation cannot be ignored in hollow cylinder of matrix of internal radius, rf, analyzing experimental data and outer radius, Im, and length, L. The load due to a hard flat indentor is simulated by a uniform 2.2. Pushout force displacement of the fiber over 60-90% of the fiber radius, and the corresponding force on the fiber is The second type of experimental measurements is where the pushout force, P, on the fiber is calculated by integrating the axial stresses on the measured as a function of the length of the inter fiber over the loading area The boundary conditions for the BEM model facial slip zone, Ls. The SLA equation [5] is given (Fig. 3)are as follows. At the top surface(==0)of the composite, the fiber is subjected to a uniform vertical displacement, la, over the indentor radius therwise it is tract
residual radial compression due to thermal expansion mismatch between the ®bers and matrix is assumed. Using SLA [6,7], analytical models are developed for the pushout test. Three expressions (depending on type of experimental data collected) used in regressing experimental data to ®nd the two mechanical properties of the interface are given below. 2.1. Fiber displacement In the ®rst case, the pushout force, P on the ®ber is measured as a function of the displacement, Duz on the surface of the ®ber below the matrix surface due to interfacial slip. The SLA equation is given by [5] Duz rf 1 ÿ 2mfkr0 kEf P 2pr2 f lr0 ÿ 1 2lk ln r0 k P=pr2 f r0 ; 1 where k Emmf Ef 1 mm Em 1 ÿ mf : 2 YoungÕs modulus of matrix are Ef and Em, respectively while L is the specimen length. The PoissonÕs ratio of ®ber and matrix are mf; mm, respectively with l being the coecient of friction. Residual radial stress at the interface is r0 and rf stands for the ®ber radius. However, in an analytical study of [8], the above pushout force vs. displacement curve (Eq. (1)) was found to be highly in¯uenced by many extrinsic factors of the test such as type and radius of indentor. This observation cannot be ignored in analyzing experimental data. 2.2. Pushout force The second type of experimental measurements is where the pushout force, P, on the ®ber is measured as a function of the length of the interfacial slip zone, Ls. The SLA equation [5] is given by P pr2 f r0 k e 2lkLs=rf ÿ 1 : 3 This type of measurement also has its limitations. The measurement of slip length is dicult to make and in many cases the ®ber slips at both ends, hence making it dicult to relate the slip length as one measurement. In addition, accounting for more than one slip zone in a theoretical model is almost intractable due to the nonlinear nature of the frictional interface. 2.3. Maximum pushout force The third type of experimental measurements is where the maximum pushout force, Pmax, is measured as a function of specimen thickness, L. The SLA model equation [5] is same as Eq. (3), except L Ls, and hence is given by Pmax pr2 f r0 k e 2lkL=rf ÿ 1 : 4 This type of measurement does require several samples of dierent thickness as opposed to only one in the previous two cases. But, the slip length is predetermined; that is, it is equal to the length of the specimen. Also, as will be shown later that this type of measurement is not in¯uenced by indentor radius and type. Since this model seems to have the least limitations of the three sets of possible experimental data, this SLA model will be the used in this study for comparison with the BEM model. 3. Boundary element method problem statement An axisymmetric model is composed of a solid cylindrical ®ber of radius, rf, and length L, and a hollow cylinder of matrix of internal radius, rf, and outer radius, rm, and length, L. The load due to a hard ¯at indentor is simulated by a uniform displacement of the ®ber over 60±90% of the ®ber radius, and the corresponding force on the ®ber is calculated by integrating the axial stresses on the ®ber over the loading area. The boundary conditions for the BEM model (Fig. 3) are as follows. At the top surface (z 0) of the composite, the ®ber is subjected to a uniform vertical displacement, ua, over the indentor radius a; otherwise it is traction free, that is, J. Ye, A.K. Kaw / Theoretical and Applied Fracture Mechanics 32 (1999) 15±25 17
J. Ye, AK Kaw Theoretical and Applied Fracture Mechanics 32(1999)15-25 The conditions. or(rm, z)=om(rr, z), Lo 0.0≤z<L tively, u and t, are boundary displacement and (11) traction, respectively
uf z r; 0 ua; 0 6 r 6 a 0; 0 6 z < L0: 11 The conditions: rf rr rf;z rm rr rf;z; L0 < z < Ls; rf rz rf;z rm rz rf;z; L0 < z < Ls; uf r rf;z um r rf;z; L0 < z < Ls; jrf rz rf;zj ljrf rr rf;zj; L0 < z < Ls; 12 are for the slip zone with rm rr rf;z < 0; L0 < z < Ls: 13 To be satis®ed in the stick zone are: rf rr rf;z rm rr rf;z; Ls < z < L; rf rz rf;z rm rz rf;z; Ls < z < L; uf r rf;z um r rf;z; Ls < z < L; uf z rf;z um z rf;z; Ls < z < L; 14 such that: rf rr rf;z < 0; Ls < z < L; jrf rz rf;zj < ljrf rr rf;zj; Ls < z < L; 15 where L0 is the length of the open zone and Ls ÿ L0 is the length of the slip zone. In the slip zone, the condition of positive dissipation has also to be met, that is, the slippage is in the same direction as the shear stress as the pushout force is increased [9]. 4. Analysis by boundary element method Based on linear elasticity, BEM formulation [10±12] is applied to the ®ber and matrix region separately. The governing three-dimensional boundary integral equation for linearly elastic single body for a source point p is Cijui p ÿ Z S Tij p; quj qdS q Z S Uij p; qtj qdS q; 16 Tij and Uij are three-dimensional fundamental solutions for traction and displacement, respectively; uj and tj are boundary displacement and traction, respectively, Fig. 3. Schematic of the pushout test for boundary element method. 18 J. Ye, A.K. Kaw / Theoretical and Applied Fracture Mechanics 32 (1999) 15±25
J. Ye, AK Kaw Theoretical and Applied Fracture Mechanics 32(1999)15-25 is outside s The transformation from actual variable te di, i is inside s (17) intrinsic variable 5 can be expressed as dir, i is on smooth bor (r)2+(d)2 After integrating the three-dimensional fund mental solutions around the axis of rotational ymmetry, three-dimensional axisymmetric prob (卖)+() lems are transformed to a one-dimensional prob lem. The axisymmetric form of boundary integral J()d, equation then is where(5)=VJ,(2)2+J(5)is the Jacobian,and Cr(p)Cr(p) u(p) J, ($)and (S)are the components of the Jacobian (p)C=(p)]a(p) vector in the r and z directions, respecllcobio Tr(p, q) T2(p, q)ur(q) Since (u) and &t) are nodal displacements and Tr,q)Ta(p,q)u(q)∫ ra dr(g) tractions(known and unknown) and they are not functions of integral variable, so they can be taken +2xz/mq)U(q)1∫4(q) outside of the integral. Then the discretized of Uz(p, q) U(p, q)l(9/g dr(q. boundary integral equation is The numerical implementation of boundary lq{u=-2∑/{mJn}ndr integral equation can be carried out by discretizing the boundary into elements. Three nodal points define each element. The shape functions have the +2a∑/,{Jn母 following form 中1(3)=-0.5(1-) 中2(3)=(1+9)(1-), (19) The integration on the right-hand side is per- formed one element In at a time throughout the Thus, the coordinates of any point on the ele Eq(23)can be written for each node. There are ment can be expressed in terms of nodal coordi- two equations for each node, one in each direction nate as follows Then the integration are performed from the first element to the last element and added together to r(3) 2()r form a set of linear algebraic equations. The ma- trix form of the equ uation IS. (20 [H]= z(9)=∑中(2 where the matrices [H and [G] contain the inte- where r and z are the coordinates of the nodes in grals of the traction and displacement kernels r and z direction, respectively. Similarly, the vari- respectively ations of the displacement and traction over the Before solving Eq .(24), the boundary condi element can also be expressed as tions are applied. The boundary conditions are known displacements or tractions, or a relation ship between them. Eq.(23)then can be rear u()=∑…(9));u()=∑中5)n), ranged according to boundary conditions; that is unknown displacements and tractions are moved t() 中2(3)(t)e;t2(3)=更2()(1)2 to the left-hand side and all the known quantities are moved to the right-hand side as follows (21)4x=Bly=[E
Cij 0; i is outside S; dij; i is inside S; 1 2 dij; i is on smooth boundary: 8 < : 17 After integrating the three-dimensional fundamental solutions around the axis of rotational symmetry, three-dimensional axisymmetric problems are transformed to a one-dimensional problem. The axisymmetric form of boundary integral equation then is: Crr p Crz p Crz p Czz p ur p uz p ÿ2p Z C Trr p; q Trz p; q Tzr p; q Tzz p; q ur q uz q rq dC q 2p Z C Urr p; q Urz p; q Uzr p; q Uzz p; q tr q tz q rq dC q: 18 The numerical implementation of boundary integral equation can be carried out by discretizing the boundary into elements. Three nodal points de®ne each element. The shape functions have the following form: U1 n ÿ0:5n 1 ÿ n; U2 n 1 n 1 ÿ n; U3 n ÿ0:5n 1 n: 19 Thus, the coordinates of any point on the element can be expressed in terms of nodal coordinate as follows: r n X 3 c1 Uc nrc; z n X 3 c1 Uc nzc; 20 where rc and zc are the coordinates of the nodes in r and z direction, respectively. Similarly, the variations of the displacement and traction over the element can also be expressed as: ur n X 3 c1 Uc n urc; uz n X 3 c1 Uc n uzc; tr n X 3 c1 Uc n trc; tz n X 3 c1 Uc n tzc: 21 The transformation from actual variable C to intrinsic variable n can be expressed as: dC dr 2 dz 2 q dr dn 2 dz dn 2 s dn J ndn; 22 where J n Jr n 2 Jz n 2 q is the Jacobian, and Jr n and Jz n are the components of the Jacobian vector in the r and z directions, respectively. Since {u} and {t} are nodal displacements and tractions (known and unknown) and they are not functions of integral variable, so they can be taken outside of the integral. Then the discretized of boundary integral equation is: C ur p uz p ÿ2p XM m1 Z 1 ÿ1 T U J rq m dnfug 2p XM m1 Z 1 ÿ1 UU J rq m dnftg: 23 The integration on the right-hand side is performed one element Cm at a time throughout the boundary. Eq. (23) can be written for each node. There are two equations for each node, one in each direction. Then the integration are performed from the ®rst element to the last element and added together to form a set of linear algebraic equations. The matrix form of the equation is: Hu Gt; 24 where the matrices [H] and [G] contain the integrals of the traction and displacement kernels, respectively. Before solving Eq. (24), the boundary conditions are applied. The boundary conditions are known displacements or tractions, or a relationship between them. Eq. (23) then can be rearranged according to boundary conditions; that is, unknown displacements and tractions are moved to the left-hand side and all the known quantities are moved to the right-hand side as follows: Ax By E; 25 J. Ye, A.K. Kaw / Theoretical and Applied Fracture Mechanics 32 (1999) 15±25 19
J. Ye, AK Kaw Theoretical and Applied Fracture Mechanics 32(1999)15-25 where [x] and ly consist of unknown and known Since the results are limited to cases where the quantities, respectively, matrices [A] and [B] con- whole fiber has slipped, only the slip interface tain rearranged corresponding coefficients of [H conditions are discussed. The simulation of slip and [G], and [el consists of known coefficients interface is performed by iteration method. It ssumed that the two bo 4. 1. Temperature effects before any pushout force is being applied. So the first iteration is for perfect bond(continuity of the temperature effects. the radial and axial stresses and displacements at the composite consists of two sub-regions-fiber and interface)case. After the first iteration, the ratio of matrix For an elastic body with sub-regions un- local tangential stress to local normal stress is dergone uniform temperature change, if the coef- calculated to see if there is any violation of stress ficient of thermal expansion a is different for the state. If the ratio is greater than the coefficient of two sub-regions, then there would be strain and friction, A, then the originally assumed sticking stress at the interface of the sub-regions caused by condition is violated. and Coulomb's friction this temperature mismatch between two different condition o,=+uo, is applied instead. Since the materials. This mismatch results in additional unknowns in the matrix equations are in the global tractions at the surface. system, we need to represent the local tractions in terms of their corresponding global values using (q)r=(q)le+1=2△n(q (26) the following transformation equations where the subscript T and E stand for thermo- ∫4 cosO sin 8 elastic and elastic, respectively, E and v are the 0 e -coset Young's modulus and Poisson's ratio, respectively, where 0 is the contact angle between the unit AT is the temperature change, and n are th components of the unit outward normal at g. Then tangential vector and the positive radial axis, tt and tn are local tractions in tangential and normal directions, respectively. Because 0c is perpendicu- integral identity with uniform temperature change lar to the radial axis in the pushout model,the can be written as: absolute value of local tractions ft and In equal to Cui(p)=-/T, (p,q)u(q)ds(q) absolute value of global tractions ty and tr If 2/t>u, the slip occurs between the node Uu(p, q)a)ds(q) pair. So in the next iteration, the Coulomb slip condition is applied instead of stick condition aE△T That is for global coordinates (P,q)ny(q)dS(q).(27) In Eq.(26), I: should be against the relative axial 4.2. Interface conditions movement of node pairs. Then for matrix, if 0. the minus sign should be used in the The interface is modeled using the Coulomb relation of t, and t, that is, t'=-urh, and vice w. The element mesh along the interface is the versa. same for the two bodies. that is the two bodies or displacement between node pair a and b, have the same element length. Each node on fiber have following relationships in the global coordi and its corresponding node on matrix form a node nates pair a and b corresponding to interface node on For u-u>o, u=u, u=u+Aus,(30) fiber and matrix, respectively. Both nodes in a node pair have the same unit outward normal but where Aus is the amount of slip between node a with opposite sign and b
where [x] and [y] consist of unknown and known quantities, respectively, matrices [A] and [B] contain rearranged corresponding coecients of [H] and [G], and [E] consists of known coecients. 4.1. Temperature eects For incorporating the temperature eects, the composite consists of two sub-regions ± ®ber and matrix. For an elastic body with sub-regions undergone uniform temperature change, if the coef- ®cient of thermal expansion a is dierent for the two sub-regions, then there would be strain and stress at the interface of the sub-regions caused by this temperature mismatch between two dierent materials. This mismatch results in additional tractions at the surface: tj qT tj qE aE 1 ÿ 2m DT nj q; 26 where the subscript T and E stand for thermoelastic and elastic, respectively, E and m are the YoungÕs modulus and PoissonÕs ratio, respectively, DT is the temperature change, and nj are the components of the unit outward normal at q. Then from Eq. (13), the three-dimensional boundary integral identity with uniform temperature change can be written as: Cijui p ÿ Z S Tij p; quj qdS q Z S Uij p; qtj qdS q aEDT 1 ÿ 2m Z S Uij p; qnj qdS q: 27 4.2. Interface conditions The interface is modeled using the Coulomb law. The element mesh along the interface is the same for the two bodies, that is, the two bodies have the same element length. Each node on ®ber and its corresponding node on matrix form a node pair a and b corresponding to interface node on ®ber and matrix, respectively. Both nodes in a node pair have the same unit outward normal but with opposite sign. Since, the results are limited to cases where the whole ®ber has slipped, only the slip interface conditions are discussed. The simulation of slip interface is performed by iteration method. It is assumed that the two bodies are sticking together before any pushout force is being applied. So the ®rst iteration is for perfect bond (continuity of radial and axial stresses and displacements at the interface) case. After the ®rst iteration, the ratio of local tangential stress to local normal stress is calculated to see if there is any violation of stress state. If the ratio is greater than the coecient of friction, l, then the originally assumed sticking condition is violated, and CoulombÕs friction condition rt lrr is applied instead. Since the unknowns in the matrix equations are in the global system, we need to represent the local tractions in terms of their corresponding global values using the following transformation equations: tt tn coshc sinhc sinhc ÿ coshc tr tz ; 28 where hc is the contact angle between the unit tangential vector and the positive radial axis, tt and tn are local tractions in tangential and normal directions, respectively. Because hc is perpendicular to the radial axis in the pushout model, the absolute value of local tractions tt and tn equal to absolute value of global tractions tz and tr; If jtz=trj P l, the slip occurs between the node pair. So in the next iteration, the Coulomb slip condition is applied instead of stick condition. That is for global coordinates: tz ltr: 29 In Eq. (26), tz should be against the relative axial movement of node pairs. Then for matrix, if ua z ÿ ub z > 0, the minus sign should be used in the relation of t b r and t b z , that is, t b z ÿlt b r , and vice versa. For displacement between node pair a and b, we have following relationships in the global coordinates: For ua z ÿ ub z > 0; ua r ub r ; ua z ub z Dus; 30 where Dus is the amount of slip between node a and b. 20 J. Ye, A.K. Kaw / Theoretical and Applied Fracture Mechanics 32 (1999) 15±25
J. Ye, AK Kaw Theoretical and Applied Fracture Mechanics 32(1999)15-25 In the matrix equation, uf, and u, are solved done because the computational time required in independently, and their solutions would yield the BEM is prohibitive not for individual runs but amount of slip because for the number of runs required to extract (31) the mechanical properties of the interface. To In summary, for Coulomb slip interface condition total of 30 data points provided in[14](used in[13] the following relationship apply in the range of specimen thickness of up to 2045 u=u t=-tt um is used (32) multiple regressio lysis [5] of the So u can be expressed in terms of u,, u, and u, can experimental data from Table 2 to Eq(4)gave the be solved independently and all the unknown residual radial stress oo=-75 MPa and coefficient tractions in the slip region can be expressed in of friction u=0. 12 terms of tr. Therefore, total unknowns in the slip region are up, u, u and r. There are four boundary 5.2. Extraction of interfacial properties using bour integral equations and four continuity equations dary element method model for each node pair, then the problem can be solved Since BEM is a numerical technique, its results are single data points of maximum pushout force 5. Results for a particular specimen length, friction and residual clamping stress (material 5.1. Extraction of interfacial properties using shear system and fiber volume fraction are fixed in the ig analysis model experiment [13]. Hence, the regression analysis involved is unlike the analytical SLa as discussed By measuring the maximum pushout force, in Section 5.1. An indirect method has to be used Pmax as a function of specimen thickness, L, Eq (4) by assuming a value for the regression parameters, be used to extract the mechanical properties of Oo and u, and calculating the maximum pushout the fiber-matrix interface, Go and H. The experi- force using BEM for the 10 diferent specimen mental data in [13] was used. The material prop lengths in Table 2. Then the sum of the square of erties and geometry are listed in Table I and ar the residuals, S, of the maximum pushout force is used in this paper, unless noted otherwise calculated as follows: Only 10 individual pushout test data points S,=>I[(Pmax)BEMI -[(Pmax)) (33) (Table 2) was used for this study. This was mainly where(Pmax )BEM is the maximum pushout force Table from BEM and(Pmax )EXPT is the maximum push Specimen dimensions and material properties of silicon carbide/ out force from experiment aluminosilicate glass composite sample [131 The regression parameters are changed until Fiber Matrix one obtains the minimum value of the sum of the Diameter, D(um) square of the residuals, S,. The SLA results of oung's modulus, E(GPa) Fo=-75MPa and u=0. 12 were used as a starting Poissons ratio. y 0.24 point. The value of the regression parameters ob Coefficient of thermal expansion, tained via BEM were Go=-74 MPa and u=0.14 x(10-6K-) This shows about 16% difference for the coefficient Table 2 Maximum pushout force(N)vS specimen thickness(um) in a pushout test for specimen given in Table 1[14 Maximum pushout force(M) a30936354313513233415412
In the matrix equation, ua z and ub z are solved independently, and their solutions would yield the amount of slip Dus ua z ÿ ub z : 31 In summary, for Coulomb slip interface condition the following relationship apply: ub r ua r ; t b r ÿt a r : 32 So ub r can be expressed in terms of ua r , ua z and ub z can be solved independently and all the unknown tractions in the slip region can be expressed in terms of t a r . Therefore, total unknowns in the slip region are ua r , ua z , ub z and t a r . There are four boundary integral equations and four continuity equations for each node pair, then the problem can be solved. 5. Results 5.1. Extraction of interfacial properties using shear± lag analysis model By measuring the maximum pushout force, Pmax as a function of specimen thickness, L, Eq. (4) can be used to extract the mechanical properties of the ®ber±matrix interface, r0 and l. The experimental data in [13] was used. The material properties and geometry are listed in Table 1 and are used in this paper, unless noted otherwise. Only 10 individual pushout test data points (Table 2) was used for this study. This was mainly done because the computational time required in BEM is prohibitive not for individual runs but because for the number of runs required to extract the mechanical properties of the interface. To eliminate any bias, every third data point of the total of 30 data points provided in [14] (used in [13] in the range of specimen thickness of up to 2045 lm is used. Using multiple regression analysis [5] of the experimental data from Table 2 to Eq. (4) gave the residual radial stress r0 ÿ75 MPa and coecient of friction l 0.12. 5.2. Extraction of interfacial properties using boundary element method model Since BEM is a numerical technique, its results are single data points of maximum pushout force for a particular specimen length, coecient of friction and residual clamping stress (material system and ®ber volume fraction are ®xed in the experiment [13]. Hence, the regression analysis involved is unlike the analytical SLA as discussed in Section 5.1. An indirect method has to be used by assuming a value for the regression parameters, r0 and l, and calculating the maximum pushout force using BEM for the 10 dierent specimen lengths in Table 2. Then the sum of the square of the residuals, Sr of the maximum pushout force is calculated as follows: Sr Xf PmaxBEM i ÿ PmaxEXPT i g2 ; 33 where (Pmax)BEM is the maximum pushout force from BEM and (Pmax)EXPT is the maximum pushout force from experiment. The regression parameters are changed until one obtains the minimum value of the sum of the square of the residuals, Sr. The SLA results of r0 ÿ75MPa and l 0.12 were used as a starting point. The value of the regression parameters obtained via BEM were r0 ÿ74 MPa and l 0.14. This shows about 16% dierence for the coecient Table 1 Specimen dimensions and material properties of silicon carbide/ aluminosilicate glass composite sample [13] Fiber Matrix Diameter, D (lm) 137 5500 YoungÕs modulus, E (GPa) 413 87 PoissonÕs ratio, m 0.19 0.24 Coecient of thermal expansion, a (10ÿ6 Kÿ1) 4.3 4.61 Table 2 Maximum pushout force (N) vs. specimen thickness (lm) in a pushout test for specimen given in Table 1 [14] Specimen thickness (lm) 218 374 691 800 1014 1191 1333 1489 1604 2045 Maximum pushout force (N) 0.3100 0.9990 2.745 2.355 4.301 6.185 7.328 5.511 5.921 7.865 J. Ye, A.K. Kaw / Theoretical and Applied Fracture Mechanics 32 (1999) 15±25 21
J. Ye, AK Kaw Theoretical and Applied Fracture Mechanics 32(1999)15-25 of friction and less than 2% difference for the re- the two major sources -thermal expansion mis- sidual radial stress. Fig. 4 shows the regressio match and roughness are given by: curves obtained using sla and bem models. For interfacial residual radial stresses. note that (oo)hema=k(x-xm)△T th he input to the BEM model is the temperature (oo)rouch=ky (34) change and not directly the residual stress at the fiber-matrix interface as used in the SLA model. where(oo)thermal is the thermal exe sig acial re- nsion mismatch How is the appropriate temperature change cal induced interfacial residual radial stress and culated for use in the BEM model? A residual (oo)rough is the roughness induced stress is first calculated for a temperature change of sidual radial stress. In Eq.(34), AT=-l K for the problem of a perfectly bonded composite cylinder [15] same properties and ge- k1 EmEt ometry(except the cylinders are infinitely long) Er(1+vm)+Em(1-ve (35) were used as given in Table 1. Then after calcula- The coefficient of thermal expansion of fiber and tion of the residual stress for AT=-l K, and since matrix, are given by af and om, respectively. The there is a linear relationship between AT and oo, temperature change is AT and Us is the fiber-ma- the appropriate AT can be found to develop the trix gap due to amplitude of roughness of surfaces required interfacial residual stress. For example, no distinction will be made between the two for Go=-75 MPa in the present this calc residual stresses. The total residual radial stress is lated temperature change is AT=-3263 K. simulated through a temperature change because But this temperature change AT=-3263 K is this study is focussed on addressing the adequacy different from the processing temperatures change of using SLa in extracting mechanical properties △T==700K K [14]. Then how could a of the fiber-matrix interface, and not on hot temperature change of only -3263 K justified? mechanical behavior of fiber-matrix interface First but secondary, the Sic fibers are coated with should be modeled. a carbon layer of 3 um thickness that affects the Figs. 5-8 show trends of fiber, matrix and in- thermal stresses [6] facial stresses in the composite specimen. Fig. 5 Second but more importantly [14] the roughness shows the distribution of the axial stress along the induced interfacial residual stresses is higher than radial coordinate for different distances from the the thermal expansion mismatch induced interfa loading end for a short fiber length (Ls=L cial residual stress. The approximate residual ra- 218 um). The axial stress changes drastically at the dial stress calculated in SLA [13, 16] where from edge of the indentor on the top of the fiber(==0) z=213 mm · Experimental da Shear-lag SLA亿z=0mm) Specimen thickness(mm) Fig. 5. Axial stress in fiber along the interfac Inction of Fig. 4. Comparison of extraction of interface properties from radius with distance from the loaded end(up=10 um, Ip=67.5 experimental data using SLA and BEM models m,L=L=218pm,H=0.12,00=-75MPa,△T=-3263K)
of friction and less than 2% dierence for the residual radial stress. Fig. 4 shows the regression curves obtained using SLA and BEM models. For interfacial residual radial stresses, note that the input to the BEM model is the temperature change and not directly the residual stress at the ®ber±matrix interface as used in the SLA model. How is the appropriate temperature change calculated for use in the BEM model? A residual stress is ®rst calculated for a temperature change of DT ÿ1 K for the problem of a perfectly bonded composite cylinder [15] same properties and geometry (except the cylinders are in®nitely long) were used as given in Table 1. Then after calculation of the residual stress for DT ÿ1 K, and since there is a linear relationship between DT and r0, the appropriate DT can be found to develop the required interfacial residual stress. For example, for r0 ÿ75 MPa in the present case, this calculated temperature change is DT ÿ3263 K. But this temperature change DT ÿ3263 K is dierent from the processing temperatures change of only DT@ ÿ700 K [14]. Then how could a temperature change of only ÿ3263 K justi®ed? First but secondary, the SiC ®bers are coated with a carbon layer of 3 lm thickness that aects the thermal stresses [6]. Second but more importantly [14] the roughness induced interfacial residual stresses is higher than the thermal expansion mismatch induced interfacial residual stress. The approximate residual radial stress calculated in SLA [13,16] where from the two major sources ± thermal expansion mismatch and roughness are given by: r0thermal k1 af ÿ amDT ; r0rough k1 Us rf ; 34 where (r0)thermal is the thermal expansion mismatch induced interfacial residual radial stress and (r0)rough is the roughness induced interfacial residual radial stress. In Eq. (34), k1 EmEf Ef 1 mm Em 1 ÿ mf : 35 The coecient of thermal expansion of ®ber and matrix, are given by af and am, respectively. The temperature change is DT and Us is the ®ber±matrix gap due to amplitude of roughness of surfaces. No distinction will be made between the two residual stresses. The total residual radial stress is simulated through a temperature change because this study is focussed on addressing the adequacy of using SLA in extracting mechanical properties of the ®ber±matrix interface, and not on how mechanical behavior of ®ber±matrix interface should be modeled. Figs. 5±8 show trends of ®ber, matrix and interfacial stresses in the composite specimen. Fig. 5 shows the distribution of the axial stress along the radial coordinate for dierent distances from the loading end for a short ®ber length (Ls L 218 lm). The axial stress changes drastically at the edge of the indentor on the top of the ®ber (z 0). Fig. 4. Comparison of extraction of interface properties from experimental data using SLA and BEM models. Fig. 5. Axial stress in ®ber along the interface as a function of radius with distance from the loaded end (up 10 lm, rp 67.5 lm, Ls L 218 lm, l 0.12, r0 ÿ75 MPa, DT ÿ3263 K). 22 J. Ye, A.K. Kaw / Theoretical and Applied Fracture Mechanics 32 (1999) 15±25
J. Ye, AK Kaw Theoretical and Applied Fracture Mechanics 32(1999)15-25 axial locations close to ends of the fiber. Note that BEM in SlA. axial stresses in the fiber and ma assumed to vary only along the length of the specimen and are independent of the radial loca -B0 tion Figs. 6 and 7 show the interfacial radial stresses and shear stresses along the length of the interface respectively. From Figs. 6 and 7, the shear stresses Location along the interface Z(mm) and radial stresses by SLa are almost constant Fig. 6. Radial stress along the fiber-matrix interface from SLa rough out the interface. The difference between and BEM models(up=10 un 675m,L=L=218pn results of BEM and Sla is about 20% away from u=0.12,00=-75MPa,△T=-3263K the two ends of the interface. The differences in the shear and radial stresses from sla and bem 16 models near the ends of the interface are due to Shear-lag two reasons. First, different loading conditions are used in the two analyses In bem, a uniform dis placement was used while in SLA a uniform pressure loading was used. Second, in SLA, the residual stress is assumed constant throughout the interface. but the residual radial stress is nonuni 200 form as found in bem models Location along the interface Z(mm) 5.3. Parametric studies to show difference between Fig. 7. Shear stress along the fiber-matri and BEM models(up =10 um, Ip=67.5 um, L=L=218 um bem and sla results μ=0.12,00=-75MPa,T=-3263K) Fig. 8 shows the maximum pushout force as a function of the radius of the indentor The maxi- mum pushout force does not change significantly L=2045 In fact, the largest difference is less than 0.01% This shows that the indentor radius does not affect =1014 the maximum pushout force vs specimen thick ness data. and hence the extraction of the two mechanical properties of the fiber-matrix inter face. Also, the radius of the hole has a negligible 0 0.2 0. 4 0.6 0.8 1 effect on the results of the test [8]. Since these two Loading radius /fiber radius a /r, parameters are not accounted for in the shear-la model, it is important for the experimentalist to Fig.8. Maximum pushout force as a function of loading radius know that these parameters do not influence the ratios for different specimen thicknesses using BEM(Go -75MPa,H=0.12,△T=-3263K) maximum pushout force vs specimen length data Fig.9 shows the maximum pushout force as a function of coefficient of friction of the interface Away from the loading end, the axial stress varies for oo=-75 MPa, L=218 um for SLA and BEM little along the radial coordinate. At the axial lo models. The difference between the results of bem cations close to the free end of the fiber, the axial and Sla is approximately 20%. Similar magni stress starts to increase from approximately half of tudes of difference were observed for longer spec- the fiber radius. a similar trend is also observed in imens as well for longer specimen lengths, but the axial stress Fig 10 shows the maximum pushout force as a changes less drastically along the fiber radius at function of Young's modulus ratio(Er/Em) with
Away from the loading end, the axial stress varies little along the radial coordinate. At the axial locations close to the free end of the ®ber, the axial stress starts to increase from approximately half of the ®ber radius. A similar trend is also observed in for longer specimen lengths, but the axial stress changes less drastically along the ®ber radius at axial locations close to ends of the ®ber. Note that in SLA, axial stresses in the ®ber and matrix are assumed to vary only along the length of the specimen and are independent of the radial location. Figs. 6 and 7 show the interfacial radial stresses and shear stresses along the length of the interface, respectively. From Figs. 6 and 7, the shear stresses and radial stresses by SLA are almost constant through out the interface. The dierence between results of BEM and SLA is about 20% away from the two ends of the interface. The dierences in the shear and radial stresses from SLA and BEM models near the ends of the interface are due to two reasons. First, dierent loading conditions are used in the two analyses. In BEM, a uniform displacement was used while in SLA a uniform pressure loading was used. Second, in SLA, the residual stress is assumed constant throughout the interface, but the residual radial stress is nonuniform as found in BEM models. 5.3. Parametric studies to show dierence between BEM and SLA results Fig. 8 shows the maximum pushout force as a function of the radius of the indentor. The maximum pushout force does not change signi®cantly. In fact, the largest dierence is less than 0.01%. This shows that the indentor radius does not aect the maximum pushout force vs. specimen thickness data, and hence the extraction of the two mechanical properties of the ®ber±matrix interface. Also, the radius of the hole has a negligible eect on the results of the test [8]. Since these two parameters are not accounted for in the shear±lag model, it is important for the experimentalist to know that these parameters do not in¯uence the maximum pushout force vs. specimen length data. Fig. 9 shows the maximum pushout force as a function of coecient of friction of the interface for r0 ÿ75 MPa, L 218 lm for SLA and BEM models. The dierence between the results of BEM and SLA is approximately 20%. Similar magnitudes of dierence were observed for longer specimens as well. Fig. 10 shows the maximum pushout force as a function of YoungÕs modulus ratio (Ef/Em) with Fig. 6. Radial stress along the ®ber±matrix interface from SLA and BEM models (up 10 lm, rp 67.5 lm, Ls L 218 lm, l 0.12, r0 ÿ75 MPa, DT ÿ3263 K). Fig. 7. Shear stress along the ®ber±matrix interface from SLA and BEM models (up 10 lm, rp 67.5 lm, Ls L 218 lm, l 0.12, r0 ÿ75 MPa, T ÿ3263 K). Fig. 8. Maximum pushout force as a function of loading radius ratios for dierent specimen thicknesses using BEM (r0 ÿ75 MPa, l 0.12, DT ÿ3263 K). J. Ye, A.K. Kaw / Theoretical and Applied Fracture Mechanics 32 (1999) 15±25 23
J. Ye, AK Kaw Theoretical and Applied Fracture Mechanics 32(1999)15-25 20 24 0.6 E ■BEM 羹言14 L EL=2 12 0.16 Coefficient of frictio ≡60.2 Fig. 9. Maximum pushout force as a function of coefficient of 0.0平14 friction for SLA and BEM models (Go=-75 MPa. AT=-3263 Fiber volume fraction Ve Fig. ll. Maximum pushout force as a function of fiber volume fraction for BEM and SLA models(oo=-75 MPa, H=0.12. L=L=218um,△T=-3263K) 0.8 4- BEM translate into less or more differences in the ex- tracted values of the two mechanical properties of the fiber-matrix interface. Based on the data ob Youngs modulus ratio E/E tained for Fig. 4, the difference between the bem and sla results r trix moduli ratio for sla and bem models (on -75 MPa. maximum pushout force as opposed to about 20% u=0.12,L=L=218um) for the extracted value of the coefficient of friction 00=-75 MPa, u=0. 12 for L=218 um for SLA 6 Conclusions and BEM models. Except for the Youngs modu- lus of the fiber, all other properties assumed are The adequacy of using SLA models to extract same as given in Table 1. The residual radial stress the two mechanical properties of the fiber-matrix in all the cases was assumed to be -75 MPa; interface is studied The following are the conclu hence, the temperature changes were recalculated sions of this study for BEM model. Again. there is about 20% dif-. The coefficient of friction of the fiber-matrix in- ference between beM and sla results. Similar terface extracted by using BEM and Sla differs magnitudes of difference were observed for longer by 15%. However, the difference is only 1% for specimens as well. the interfacial residual radial stress Fig. 1l shows the maximum pushout force as a Like the SLA model results, the interface function of fiber volume fraction vr with go stresses in bEM remain constant through the 75 MPa, u=0.12 for L=218 um. The temp thickness of the specimen except at the top ture changes for all fiber volume fractions were and the bottom surfaces of the specimen kept the same(-3263 K) and interfacial residual The maximum pushout force is independent of stresses for use in the sla model were calculated the indentor radius and type, and push from the infinite composite cylinder problem [15]as through-hole radius. This is important observa discussed in the beginning of this section. Again, tion for the experimentalist in conducting the there is about 20% difference between the results of pushout test. Also, the SLa model does not BEM and SLA. Similar magnitudes of difference need to account for these extrinsic factors were observed for longer specimens as well The parametric studies show that there is ap In Figs. 9-1l, although there is a 20%differenc proximately 20% difference between the maxi- between the maximum pushout force values cal- mum pushout force obtained by BEM and culated using Sla and BEM models, does it SLA when coefficient of friction fiber to matrix
r0 ÿ75 MPa, l 0.12 for L 218 lm for SLA and BEM models. Except for the YoungÕs modulus of the ®ber, all other properties assumed are same as given in Table 1. The residual radial stress in all the cases was assumed to be ÿ75 MPa; hence, the temperature changes were recalculated for BEM model. Again, there is about 20% difference between BEM and SLA results. Similar magnitudes of dierence were observed for longer specimens as well. Fig. 11 shows the maximum pushout force as a function of ®ber volume fraction Vf with r0 ÿ75 MPa, l 0.12 for L 218 lm. The temperature changes for all ®ber volume fractions were kept the same (ÿ3263 K) and interfacial residual stresses for use in the SLA model were calculated from the in®nite composite cylinder problem [15] as discussed in the beginning of this section. Again, there is about 20% dierence between the results of BEM and SLA. Similar magnitudes of dierence were observed for longer specimens as well. In Figs. 9±11, although there is a 20% dierence between the maximum pushout force values calculated using SLA and BEM models, does it translate into less or more dierences in the extracted values of the two mechanical properties of the ®ber±matrix interface. Based on the data obtained for Fig. 4, the dierence between the BEM and SLA results averaged about 16% for the maximum pushout force as opposed to about 20% for the extracted value of the coecient of friction. 6. Conclusions The adequacy of using SLA models to extract the two mechanical properties of the ®ber±matrix interface is studied. The following are the conclusions of this study: · The coecient of friction of the ®ber±matrix interface extracted by using BEM and SLA diers by 15%. However, the dierence is only 1% for the interfacial residual radial stress. · Like the SLA model results, the interfacial stresses in BEM remain constant through the thickness of the specimen except at the top and the bottom surfaces of the specimen. · The maximum pushout force is independent of the indentor radius and type, and pushthrough-hole radius. This is important observation for the experimentalist in conducting the pushout test. Also, the SLA model does not need to account for these extrinsic factors. · The parametric studies show that there is approximately 20% dierence between the maximum pushout force obtained by BEM and SLA when coecient of friction, ®ber to matrix Fig. 9. Maximum pushout force as a function of coecient of friction for SLA and BEM models (r0 ÿ75 MPa, DT ÿ3263 K). Fig. 10. Maximum pushout force as a function of ®ber to matrix moduli ratio for SLA and BEM models (r0 ÿ75 MPa, l 0.12, Ls L 218 lm). Fig. 11. Maximum pushout force as a function of ®ber volume fraction for BEM and SLA models (r0 ÿ75 MPa, l 0.12, Ls L 218 lm, DT ÿ3263 K). 24 J. Ye, A.K. Kaw / Theoretical and Applied Fracture Mechanics 32 (1999) 15±25