Free vibration analysis of pre-stretched hyperelastic micromorphic continua with arbitrary shapes

Abstract

A novel numerical strategy is developed in this article to study the free vibrations of hyperelastic micromorphic continua under bending load. In the proposed approach, the variational differential quadrature (VDQ) method and the idea of position transformation are used. The 3D micromorphic hyperelasticity is first formulated via vector-matrix relations which can be readily utilized in the coding process of numerical methods. The present numerical approach is able to address problems with irregular domains. To this end, the domain of elements is transformed into a regular one by the technique of mapping of position field based on the finite element shape functions. Being locking-free, simple implementation, computational efficiency and fast convergence rate are other features of the present variational approach. Three numerical examples including rectangular, sector and circular plates under bending load are considered whose free vibration behavior is analyzed. It is shown that the method is capable of predicting the relation between natural frequencies of micromorphic hyperelastic continua with load factor in an efficient way. The effects of internal length scale, scale-transition parameter and mode transition on the results are investigated.