Stress trajectories element method for stress determination from discrete data on principal directions

This paper presents a Trefftz-element numerical method for the reconstruction of stress trajectories and the determination of full stress tensors in two-dimensional elastic bodies from discrete data on principal directions. The conventional techniques cannot be used because neither displacements nor tractions are specified on the boundary. The proposed approach involves the subdivision of the domain into smaller subdomains and the introduction of the Cauchy integrals with unknown densities on element boundaries in order to approximate complex potentials within the elements. For polynomial approximations of the densities, this leads to piecewise polynomial approximations for the complex potentials within the entire domain and, therefore, all elasticity equations are automatically satisfied as in the Trefftz method. Continuity of the complex potentials is forced at the collocation points, which forms the first group of equations. The second group is formed by satisfying the data on principal directions known in some locations. All these equations are homogeneous; therefore, it is assumed that the average value of the maximum shear stresses at data points is unity. This guarantees the existence of a non-trivial solution of the system; however it addresses the non-uniqueness of the reconstruction of the full stress tensor. The technique is validated by reconstructing stress trajectories and determining maximum shear stresses from synthetic and photoelasticity data. It has been applied to reconstruction of tectonic stresses in the Australian region and the results were compared with previous approaches.