An energy error-based method for the resolution of the Cauchy problem in 3D linear elasticity

A new method is described for the problem of expanding known displacement fields at the boundary of a solid together with the surface tractions on it, towards the solid interior up to the inaccessible part of the boundary. The solid is supposedly linearly elastic with known elastic moduli (but not necessarily homogeneous nor isotropic). The problem is the Cauchy problem for the Lamé Operator. A new form of this Cauchy problem suited for applications associated with surface tangential fields’ measurements is also stated and studied. The method is based on the splitting of the elastic fields into two separate solutions of well-posed problems; the gap between these fields is subsequently minimised with respect to the unknown boundary data in order to produce the desired expanded elastic fields. The gap used here is an energy error associated with the elastic energy of the system. Various 3D applications are given, including non-linear boundary conditions on the unreachable boundary.