A Stochastic Nonlocal Model for Materials with Multiscale Behavior

Integral-type nonlocal mechanics is employed to model the macroscale behavior of multiscale materials, with the associated nonlocal kernel representing the interactions between mesoscale features. The nonlocal model is enhanced by explicitly considering the spatial variability of subscale features as stochastic contributions resulting in a stochastic characterization of the kernel. By appropriately representing the boundary conditions, the nonlocal boundary value problem (BVP) of the macroscale behavior is transformed into a system of equations consisting of a classical BVP together with two Fredholm integral equations. The associated integration kernels can be calibrated using either experimental measurements or micromechanical analysis. An efficient and computationally expedient representation of a resulting stochastic kernel is achieved through its polynomial chaos decomposition. The coefficients in this decomposition are evaluated from statistical samples of the disturbance field associated with a random distribution of microcracks. The new model is shown to be capable of predicting nonlocal features, such as the size effect and boundary effect, of the behavior of materials with random microstructures.