13th International Conference on Fracture June 16–21, 2013, Beijing, China -5- 3.1. Governing Equations We examine the class of axisymmetric problems where fracture extension in brittle elastic media satisfies Hooke’s Law and the corresponding Navier equations: i.e. 2 , 2 ; ( ) 0 ij kk ij ij i k ki u u σ λε δ με μ λ μ = + ∇ + + = (6) and λand μ are Lamé’s constants and 2∇ is Laplace’s operator. The boundary integral equation governing axisymmetric deformations of the geomaterial region can be written as { } * * 0 lk k lk k lk k i r c u P u u P d r Γ + + Γ= ∫ (7) where Γ is the boundary of the domain; ku and kP are, respectively, the displacements and tractions on Γand * ik u and * ikP are the fundamental solutions [45,46]. In (7), lk c is a constant, which can take values of either zero (within the domain), /2 ijδ (if the point is located at a smooth boundary) or is a function of the discontinuity at a corner and of Poisson’s ratio. For axial symmetry, the displacement fundamental solutions take the forms 2 2 2 4 4 * 1 3 1 4(1 )( ) (7 8 ) ( ) ( ) ( ) 2 4 rr z R e z u C K m E m rR r rR m ν ρ ρ ν ⎧ ⎫ ⎧ ⎫ − + − − − ⎪ ⎪ ⎪ ⎪ = − − ⎨ ⎬ ⎨ ⎬ ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ ⎩ ⎭ (8) 2 2 * 1 3 1 ( ) 1 ( ) ( ) 2 2 rz e z u C z E m K m R R m ⎧ ⎫ ⎪ + ⎪ = − ⎨ ⎬ ⎪ ⎪ ⎩ ⎭ (9) …etc., where 2 2 2 1 2 2 2 2 2 2 1 2 ( ); ( ); ( ); 1 4 1 ( ); ; ; 4 (1 ) i i i i i z z z r r r r r m m rr e r r R r z m C R ρ πμ ν = − = + = + = − = − = + = = − (10) and ( ) K m and ( ) E m are complete elliptic integrals of the first and second-kind and ( , ) r z and ( , ) i i r z correspond to the coordinates of the field and source points respectively. The relevant fundamental solutions for * lkP can be obtained by manipulating results of the types (8) and (9). Upon discretization of the boundary Γ, the integral equation can be expressed in the form of a boundary element matrix equation [ ]{ } [ ]{ } = D U T P (11) where [ ]D and [ ]T are obtained, respectively, by integration of the displacement and traction fundamental solutions. When considering the discretization of the boundary Γof the domain, quadratic elements can be employed quite effectively; the variations of the displacements and tractions within an element can be described by 3 0 i n n i n u a P ζ = ⎫ ⎬ = ⎭ ∑ (12) where ζ is the local coordinate.. Then modeling cracks that occur at the boundaries or within the interior of the elastic geomaterial, it is necessary to modify (12) to take into consideration the 1/ ζ type locally two-dimensional stress singularity at the crack tip. In contrast to finite element approaches that use quarter-point elements, here we utilize the singular traction quarter-point boundary elements [40] where the tractions can be expressed in the form
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