Dynamic Fracture Mechanics With Crack Edges Contact Interaction

Abstract

This paper resents some elastodynamic contact problems with unilateral restrictions and friction for bodies with cracks. There are considered case of general dynamic loading and also relevant case of harmonic loading. The mathematical aspects of those problems are discussed briefly. The problems of a tension-compression plane harmonic wave interaction with one and two co-linear cracks of the finite length with allowance of unilateral contact interaction of the crack edges are solved. The influence of contact interaction of the cracks edges on a stress intensity factor is worked out. 1. Statement of problem Let we have an elastic body in three-dimensional Euclidean space R? that occupies the volume V. The boundary of the body 9V is a piece wisesmooth one and consists of the parts 9Vp and 9Vu, where the vectors of surface load p(x,t) and displacements u(x,t) are assigned respectively. There is an arbitrary oriented crack in the body, which is described by its surfaces Q^ and Q. The body may by affected upon by body forces b(x,t). Its stress-strain state is described by the equations of the linear elastodynamic in displacements Guz' and Zozulya [4] Agiiy+b,. = £>d] u, , VXEV , Vtei = [to,TJ (l) The operator Ay for an isotropic body has the form Transactions on Modelling and Simulation vol 20, © 1998 WIT Press, www.witpress.com, ISSN 1743-355X 24 Boundary Elements where 9= --and <9, = — are derivatives with respect to the space 0x. c% coordinate and time respectively, A, and JLI are the Lame constants, p is the density of the material. The summation convention applies to repeated indices. For the correct formulation of the elastodynamic problems it is necessary to assign the initial and boundary and conditions. We present these in the form u,(x,to) = u,o(x) , ^U((x,tJ=vXx), VxeV pXx,t) = (x,t)n/x) = Y/kt), VxE9V, , VteT (2) u,.(x,t) = cp,(x,t) , VxE6\, , VtET Let us formulate the conditions, that must be satisfied on the crack edges. For the contact forces of interaction and displacements discontinuity vectors the one sided restrictions with friction in the form of inequalities on the edges of the cracks must be satisfied Duvaut. and Lions [3], Guz' and Zozulya [4], Panagiotopoulos [7]