Abstract:
Taking into account local scale refinements in nonlinear structural mechanics can be achieved either by keeping the same model and by working on the discretisation strategy, or by changing the continuous model through kinematic enrichment.
In the first case, one must introduce a nonconforming matching strategy between the finite element meshes of a coarse global model and of local small scale models, and use efficient domain decomposition algorithms. Due to the presence of small scale structures, one can prove that the matching of interface rigid body motions is sufficient to ensure stability and convergence. One can also introduce optimal Dirichlet Neumann domain decomposition algorithms which reduce the full problem to the cost of severally coarse global problems and to the parallel solution of local fine grid subproblems. Such techniques will be proved to be efficient when dealing with relatively soft materials at the subgrid model.
But they cannot handle problems such as fiber reinforced sheet where the fibers are very stiff compared to the filling material. In such situations, introducing a richer kinematic of Timoshenko type at the fiber level yields a better representation of the local behavior of the constitutive material, in particular with respect to microbuckling. The resulting problem will present a very large stiffness ratio between shear, elongation, flexion and local deformation of the filling material. This must be handled by mixed finite elements which can be largely inspired from shell theory.
The purpose of the talk is to give an overview of the different numerical techniques handling local scale refinements in nonlinear structural mechanics, and to review the corresponding convergence results.
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