Development of the Stress Trajectories Element Method

This project is aimed at development of theory and numerical techniques for determination of stresses acting within a solid body (statically determined) caused by applied forces or contacts with other solids. The key innovation of this proposal is the investigation of consistency between constitutive equations (rheology) and stress trajectories. In general, these cannot be given simultaneously and either of them could be used with equations of equilibrium to form a closed system of partial differential equations. We address three essentially different cases when stress trajectories are known (i) everywhere inside the domain; (ii) at discrete points inside the domain; (iii) on the boundary. However in all these cases the stress trajectory concept is used as a unified approach; therefore the method proposed in this application is referred to as the stress trajectories element method, STEM. Case 1. Stress trajectories are known everywhere in the domain. If rheology is unknown then the problem of stress determination can be reduced to a boundary value problem for hyperbolic equations (classical or non-classical types). If rheology is known then the main concern is whether this rheology is consistent with the given stress trajectory pattern. If they are consistent, stress field in not unique and the degree of non-uniqueness has to be investigated. In the case of inconsistence, no solutions for the complete stress tensor exist. Numerical approach utilises elements formed by trajectories of different families.Case 2. Stress orientations are known at discrete points. To solve this problem one needs information regarding rheology. Then the problem is classified as an optimisation problem rather that a boundary value problem; therefore its solution requires minimisation of a certain functional and, if this procedure is unstable, application of regularisation methods. Numerical scheme assumes the use of prescribed or adaptive mesh with homogeneous trajectories within elements.Case 3. Trajectories are known on boundary. In this case the rheology should be given together with an extra boundary condition. The problem is ill-posed and its solvability depends upon the index that can be determined from the analysis of principal orientations of the stress tensor on the boundary. Solution is non-unique or does not exist in this case. Numerical method can be based on adaptive mesh with elements developing from the boundary.The research programme includes thorough analysis of all these situations and development of a variant of the STEM is the expected outcome of this project. The STEM method is novel and innovative; from the best of our knowledge, this approach has not been applied for different rheologies. We intend to detail and develop a methodology of stress field identification that can be applied on different scale levels. Application of the STEM to the problem of stress field determination in stable blocks of lithosphere is among the tasks of this proposal.