A nodal discontinuous Galerkin finite element method (DG-FEM) to solve the linear and nonlinear elastic wave equation in heterogeneous media with arbitrary high order accuracy in space on unstructured triangular or quadrilateral meshes is presented. This DG-FEM method combines the geometrical flexibility of the finite element method, and the high parallelization potentiality and strongly nonlinear wave phenomena simulation capability of the finite volume method, required for nonlinear elastodynamics simulations. In order to facilitate the implementation based on a numerical scheme developed for electromagnetic applications, the equations of nonlinear elastodynamics have been written in a conservative form. The adopted formalism allows the introduction of different kinds of elastic nonlinearities, such as the classical quadratic and cubic nonlinearities, or the quadratic hysteretic nonlinearities. Absorbing layers perfectly matched to the calculation domain of the nearly perfectly matched layers type have been introduced to simulate, when needed, semi-infinite or infinite media. The developed DG-FEM scheme has been verified by means of a comparison with analytical solutions and numerical results already published in the literature for simple geometrical configurations: Lamb's problem and plane wave nonlinear propagation.

The possibility of tuning the nonlinear effective response of composite materials and structures is of great importance for developing new concepts such as soft metamaterials, acoustic diodes, nonlinear waveguides and phononic crystals. In this paper we develop a homogenization technique for dispersions of nonlinear particles in a soft matrix able to take account of second and third order elastic nonlinearities. Based on this method, we prove the possibility to strongly amplify a given particles nonlinearity (either the second or the third one) under specific conditions concerning the linear response of the two constituents (particles and matrix). We finally give a realistic example based on a population of porous polymer particles embedded in a PDMS matrix.