Application of the reciprocal theorem to some problems for the elastic half-space

Abstract

BEl-rt's reciprocal theorem is used to determine the normal displacements due to a normal point force at the plane surface ofa generally-anisotropic half-space or a transversely-isotropic general elastic foundation, using a state of plane sinusoidal loading as auxiliary solution. The method is also extended to give a dual integral equation formulation of the axisymmetric contact problem for an inhomogeneous hail-space in which the elastic moduli are arbitrary functions of depth.

Surface Displacements due to a Steadily Moving Point Force

“…The same method could also be used to establish a similar relation between the point and line load problems for the elastostatic problem for a generally anisotropic material. This was proved by a different (and less straightforward) method by Barber and Sturla (1992). Notice that the surface displacement is proportional to r -1, as we deduced from selfsimilarity considerations in the Introduction.…”

“…The more challenging transient problem in which the half-space is initially quiescent and the force moves for a finite time was investigated by Payton (1964), who gave results for the special case where Poisson's ratio u = ¼. A more convenient solution to the steady-state problem for general Poisson's ratio was given by Churilov (1977), who also extended the argument to give simple solutions to some elastodynamic contact problems (Churilov, 1978).The steady-state problem is self-similar and we can therefore use equilibrium and dimensional arguments to demonstrate that the stresses must decay with R-2 and the displacements with R -l, where R is the distance from the moving force (see, for example, Willis, 1967;Barber and Sturla, 1992). The problem therefore reduces to the determination of the q%dependence of the normal surface displacements in a cylindrical polar coordinate system (r, th, z) moving with the force.…”

Inverse extraction of interfacial tractions from elastic and elasto-plastic far-fields by nonlinear field projection

“…The Maxwell-Betti reciprocal theorem is a particularly useful tool for extracting information concerning solutions to a linearly elastic boundary value problem without obtaining a complete solution for the stress and displacement fields (Barber and Sturla, 1992). It forms the basis for a computational method in linear elasticity called the boundary element method.…”

Adhesive Frictionless Contact Between an Elastic Isotropic Half-Space and a Rigid Axi-Symmetric Punch

“…For example, in the context of boundary element methods, BRT is used to construct an integral equation in terms of the problem unknowns, which can then be numerically solved (see, e.g., [2]). Examples of application of BRT to contact problems can be found in [8,31], and [9], among others.…”

“…Betti's reciprocity theorem has been used to obtain the solution of several contact problems, such as in [8,9,31]. In particular, Shield [31] also selected Boussineq's solution to recover Sneddon's theorem in the circular contact region case.…”