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

摘要

In the present paper, a size-dependent Euler–Bernoulli beam model has been developed for nonlinear static analysis of abidirectional functionally graded (BFG) microbeam based on a nonlinear elastic foundation according to the modified couplestress theory. In order to eliminate stretching and bending coupling caused by the non-symmetrical material variation alongwith the thickness, the problem is formulated with regard to the physical middle surface. By using Hamilton’s principle, theunderlying equations of motion, as well as the relevant boundary conditions of the problem, were deduced. The generalizeddifferential 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 ofthe present solution is evaluated through comparing the nonlinear static deflection provided by the proposed approach withthe results available from previous studies. The results show that the average error of the proposed approach with the resultsavailable from previous studies is 1.3%. In addition, a parametric study has been performed to explore the impacts of thegradient indices, material length scale parameter, end supports, and the stiffness coefficients of the nonlinear foundation onthe nonlinear static deflection of the BFG microbeam. The results indicate that the increase in K_(NL) has increased the value ofW_(NL)/W_L, and the effect of K_(NL) on the increase in nonlinear deflection is more than that of linear deflection. Furthermore, theresults illustrate that incrementing dimensionless length scale parameter l_0 increases the nonlinear deflection ratio W_(NL)/W_Lso that increasing l0 from 0.25 to 1 increases the nonlinear deflection ratio for K_P = 10 by about 4%, while for K_P = 100, itis about 1%.