Nonlinear Static Analysis of a Bi-directional Functionally Graded Microbeam Based on a Nonlinear Elastic Foundation Using Modified Couple Stress Theory

In the present paper, a size-dependent Euler-Bernoulli beam model has been developed for nonlinear static analysis of a bidirectional functionally graded (BFG) microbeam based on a nonlinear elastic foundation according to the modified couple stress theory. In order to eliminate stretching and bending coupling caused by the non-symmetrical material variation along with the thickness, the problem is formulated with regard to the physical middle surface. By using Hamilton's principle, the underlying equations of motion, as well as the relevant boundary conditions of the problem, were deduced. The generalized differential quadrature method (GDQM) has also been utilized for solving the underlying equations for pinned-pinned (PP) and clamped-clamped (CC) boundary conditions to obtain the natural frequencies of the BFG microbeam. The precision of the present solution is evaluated through comparing the nonlinear static deflection provided by the proposed approach with the results available from previous studies. The results show that the average error of the proposed approach with the results available from previous studies is 1.3%. In addition, a parametric study has been performed to explore the impacts of the gradient indices, material length scale parameter, end supports, and the stiffness coefficients of the nonlinear foundation on the nonlinear static deflection of the BFG microbeam. The results indicate that the increase in K-NL has increased the value of W-NL/W-L, and the effect of K-NL on the increase in nonlinear deflection is more than that of linear deflection. Furthermore, the results illustrate that incrementing dimensionless length scale parameter l(0) increases the nonlinear deflection ratio W-NL/W-L so that increasing l(0) from 0.25 to 1 increases the nonlinear deflection ratio for K-P = 10 by about 4%, while for K-P = 100, it is about 1%.