Stress-driven local/nonlocal mixture model for buckling and free vibration of FG sandwich Timoshenko beams resting on a nonlocal elastic foundation

Abstract

We study the buckling and free vibration of functionally graded (FG) sandwich Timoshenko beams resting on an elastic foundation. In contrast to the majority of the literature on this subject, the behaviors of both the beam and elastic foundation are considered as nonlocal by applying the stress-driven strategy equipped with a bi-Helmholtz kernel. We find that in the presence of a nonlocal elastic foundation, the local/nonlocal mixture should be adopted in order to obtain a well-posed formulation of the problem at hand. The equations of motion and the standard boundary conditions are obtained by invoking Hamilton's principle. Each integro-differential constitutive law is transformed into an equivalently differential form equipped with four non-standard constitutive boundary conditions. The generalized differential quadrature method (GDQM) is then utilized to solve the corresponding eigenvalue problems numerically. Several comparative studies are conducted to validate the effectiveness of our solution. The numerical simulation results present the size-effect on the critical buckling loads and natural frequency of the beams with various boundary types, providing a new benchmark for further study of modeling small-scale beam structures using the bi-Helmholtz kernel-based stress-driven mixture model. Moreover, the influence of considering the size-dependency of the elastic foundation is also investigated.