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 <z<Ls d (12) Im (r1,z)=i(r,2) (rr, z)l=uor (rr, z)l, Lo are for the slip zone with (rr, 3)<0, L0 <z<L (13) Fig. 3. Schematic of the pushout test for boundary element To be satisfied in the stick zone are dr(rr, z)=om(e, z), Ls <z<L, dr(rm, z)=om(r, z), Ls<z<L l(r,0)=ln,0≤r≤a (rr, z)=ur(r, 2), Ls <z<L, The traction free conditions for the fibers are u!(r1,z)=2(r1,z),L、<z<L (0)=0,a<r≤r such tha o(,0)=0,0≤r≤r (6)an(,2)<0,L<z<L (15) and for the matrix are lor(r, z)l< uor(r, z)l, Ls <z<L, d2(r,0)=0,r≤r≤rm, (7) where Lo is the length of the open zone and ls-l dm(,0)=0,r≤r≤rm is the length of the slip zone. In the slip zone, the condition of positive dis- At the bottom(z=L)of the composite, only the sipation has also to be met, that is, the slippage is matrix is constrained in axial direction beyond the in the same direction as the shear stress as the pushout hole of radius ro, that is pushout force is increased [91 2(r,L)=0,0≤F≤rm dm(r,D)=0,r≤r≤rm 4. Analysis by boundary element method (r,L)=0,r≤r≤10<r Based on linear elasticity, BEM formulation [10-12] is applied to the fiber and matrix region eparately. The governing three-dimensional (,L)=0,0≤r≤r, (9) boundary, integral equation for linearly elastic o(r,L)=0,0≤r≤r single body for a source point p is The fiber-matrix interface at r=r is modeled ac Ty (p, q)u, a)ds(q) cording to the Coulomb friction law as: (,2)=m(r,z)=0.0≤z<L + Uy(p, q)t (a)ds(a), Ty and Uy are three-dimensional fundam for the open zone where the crack is open, solutions for traction and displacement, respec (r,z)-u(r,z)>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 < rf: 5 The traction free conditions for the ®bers are: rf z r; 0 0; a < r 6 rf; rf rz r; 0 0; 0 6 r 6 rf; 6 and for the matrix are: rm z r; 0 0; rf 6 r 6 rm; rm rz r; 0 0; rf 6 r 6 rm: 7 At the bottom (z L) of the composite, only the matrix is constrained in axial direction beyond the pushout hole of radius r0, that is, um z r; L 0; r0 6 r 6 rm; rm rz r; L 0; rf 6 r 6 rm; rm z r; L 0; rf 6 r 6 r0 < rm: 8 and rf z r; L 0; 0 6 r 6 rf; rf rz r; L 0; 0 6 r 6 rf: 9 The ®ber±matrix interface at r rf is modeled according to the Coulomb friction law as: rf rr rf;z rm rr rf;z 0; 0 6 z < L0; rf rz rf;z rm rz rf;z 0; 0 6 z < L0; 10 for the open zone where the crack is open, um r rf;z ÿ uf r rf;z > 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