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=[ECij ˆ 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 funda￾mental solutions around the axis of rotational symmetry, three-dimensional axisymmetric prob￾lems are transformed to a one-dimensional prob￾lem. The axisymmetric form of boundary integral equation then is: Crr…p† Crz…p† Crz…p† Czz…p†   ur…p† uz…p†   ˆ 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† ˆ ÿ0:5n…1 ÿ n†; U2…n† ˆ …1 ‡ n†…1 ÿ n†; U3…n† ˆ ÿ0:5n…1 ‡ n†: …19† Thus, the coordinates of any point on the ele￾ment can be expressed in terms of nodal coordi￾nate as follows: r…n† ˆ X 3 cˆ1 Uc…n†rc; z…n† ˆ X 3 cˆ1 Uc…n†zc; …20† where rc and zc are the coordinates of the nodes in r and z direction, respectively. Similarly, the vari￾ations of the displacement and traction over the element can also be expressed as: ur…n† ˆ X 3 cˆ1 Uc…n†…ur†c; uz…n† ˆ X 3 cˆ1 Uc…n†…uz†c; tr…n† ˆ X 3 cˆ1 Uc…n†…tr†c; tz…n† ˆ X 3 cˆ1 Uc…n†…tz†c: …21† The transformation from actual variable C to intrinsic variable n can be expressed as: dC ˆ  …dr† 2 ‡ …dz† 2 q ˆ  dr dn  2 ‡ dz dn  2 s dn ˆ J…n†dn; …22† where J…n† ˆ  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: ‰CŠ ur…p† uz…p†   ˆ ÿ2p XM mˆ1 Z ‡1 ÿ1 ‰T Š‰UŠ J rq  m dnfug ‡ 2p XM mˆ1 Z ‡1 ÿ1 ‰UŠ‰UŠ J rq  m dnftg: …23† The integration on the right-hand side is per￾formed 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 ma￾trix form of the equation is: ‰HŠ‰uŠ ˆ ‰GŠ‰tŠ; …24† where the matrices [H] and [G] contain the inte￾grals of the traction and displacement kernels, respectively. Before solving Eq. (24), the boundary condi￾tions are applied. The boundary conditions are known displacements or tractions, or a relation￾ship between them. Eq. (23) then can be rear￾ranged 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: ‰AŠ‰xŠ ˆ ‰BŠ‰yŠ ˆ ‰EŠ; …25† J. Ye, A.K. Kaw / Theoretical and Applied Fracture Mechanics 32 (1999) 15±25 19