Microstructure-based stochastic finite element methods for elastostatics and elastodynamics

Consideration of global boundary value problems in elastostatics of materials with random elastic microstructures leads to the identification of three length scales: macro (global), meso (finite element size), and micro (microstructural heterogeneity). We first provide a formulation of the RVE and mesoscale moduli from the standpoint of elasticity of random media. This is followed by a discussion of basic random field models. These models provide inputs to two finite element schemes - based, respectively, on minimum potential and complementary energy principles - for bounding the global response. While in the classical case of a homogeneous material, these bounds are convergent with the finite elements becoming infinitesimal, the presence of microstructure prevents such a convergence. In the ensemble sense, we obtain micromechanically based stochastic finite elements of elastostatics. Turning to elastodynamics, we first recognize that the dynamics of any structure involves wave propagation in all the members of the structure, and so, a structural dynamics analysis should be based on elastodynamic equations. This is accomplished by a spectral approach where the response of each structural member is described by a stiffness matrix in the frequency space: a spectral finite element. Its stochastic versions are determined for materials described by random Fourier series.