Nonlinear axisymmetric bending analysis of strain gradient thin circular plate

Abstract

A size-dependent nonlinear bending theory for axisymmetric thin circular plate is proposed by using the principle of minimum potential energy. The formulation is based on the strain gradient theory of Zhou et al. and the von Kármán geometric nonlinearity. The governing equations and boundary conditions are obtained and further reduced to that based on the couple stress theory, modified couple stress theory and even classical theory by neglecting some or all strain gradient components, respectively. Besides, the corresponding linear theory is also obtained by excluding the nonlinear terms from the present theory. The bending problems for both simply supported and fully clamped circular plate subjected to uniformly distributed load are solved by using the differential quadrature method (DQM) and iteration method. The comparison between theoretical and numerical results of linear bending deflection shows good agreement. The numerical results of nonlinear bending deflection based on these different theories reveal the size-dependency of circular plate bending rigidity. The effect of strain gradients enhances the bending rigidity of circular plate, in which rotation gradient plays a dominant role in controlling the stiffening effect of bending rigidity. When the thickness of circular plate is close to the higher-order material constant, the strain gradient effects are comparable or even dominant in comparison with the traditional bending rigidity. When the thickness of circular plate is much greater than the higher-order material constant, all strain gradient effects can be ignorable and the differences of deflections among these theories are negligible.