Size-dependent nonlinear forced oscillation of self-assembled nanotubules based on the nonlocal strain gradient beam model

Abstract

Self-assembled protein micro/nanotubule has been aroused considerable interest as a self-assembled supramolecular structure with bioactive properties. The prime aim of this study is to predict size dependency in the nonlinear forced oscillation of the self-assembled nanotubules embedded in an elastic biomedium. To accomplish this end, the nonlocal strain gradient elasticity theory including both softening and stiffening features of size effect is applied to the refined hyperbolic shear deformable beam model. By using the principle of Hamilton, unconventional governing differential equations of motion have been extracted. Subsequently, generalized differential quadrature method in conjunction with the Galerkin technique is employed to solve the nonclassical problem numerically. The nonlocal strain gradient frequency response and amplitude response relevant to the primary resonance of the self-assembled nanotubules are obtained corresponding to different types of boundary conditions. It is anticipated that the nonlocal size effect causes to decrease the excitation amplitudes associated with both bifurcation points, but its effect on the first one is more considerable. However, the strain gradient size dependency has an opposite influence and leads to increase them. Furthermore, it is found that by changing the end supports from simply supported to clamped one, the influence of the nonlocality on the excitation amplitude associated with the bifurcation points increases, but the influence of the strain gradient size dependency decreases.