A perturbation method for mixed three-dimensional problems of the theory of elasticity with a complex line of boundary-condition separation

Abstract

A modification of the perturbation method is proposed, based on the utilization of variational formulas and enabling asymptotic expansions (AE) to be obtained for mixed three-dimensional problems of the theory of elasticity with a complex line of boundary-condition separation. Application of Lighthill's method enables these expansions to be transformed into uniformly suitable ones. The problem for an elastic body with a slit (crack) and the contact problem of the theory of elasticity are considered separately. For the body with a slit the variational formula determines the variation of the displacement of the slit surface caused by variation in the shape of the slit contour. The effectiveness of this formula for constructing AE in problems associated with a perturbation of the shape of the slit contour is shown. Cases of slits of complex shape in an infinite body that differ slightly from a circular slit are examined in detail. A scheme for constructing similar AE is mentioned for spatial contact problems of the theory of elasticity with a complex shape of the contact area. A review of the application of perturbation methods to mixed problems in the theory of elasticity is contained in /1, 2/. The solutions of mixed spatial problems in the theory of elasticity with a complex line of boundary condition separation, obtained by using other methods, are discussed in /3–8/. The behaviour of the solution of the boundary value problem for a pseudodifferential equation (in particular, crack theory) for variation of the domain was investigated in /9/.