应变梯度 Mindlin 板边值问题的讨论

It is reported from both the experiments and molecular dynamics that the materials and structures will exhibit a remarkable size-effect when their characteristic sizes shrink down to the micro- to nano-scales. Therefore, it is a hot topic of current interest in the research communities that whether is it possible to establish an accurate continuum model that is capable of predicting the mechanical behaviors of materials and structures. Although numerous studies have been carried out on the mechanical behaviors of Mindlin plates, their variationally consistent boundary value problems and the related issues have not been well addressed in the open literature. Firstly, the strain energy of an isotropic Mindlin plate within the context of the simplified strain gradient elasticity is given. Then, the variationally consistent boundary value problems of the Mindlin plate model and the corresponding corner conditions in terms of displacement derivations are derived using the variational principle and tensor analysis. It is verified that the present boundary value problems of the Mindlin plate model recover to the corresponding boundary value problems of the Timoshenko beam model and Kirchhoff plate model. It is found that the governing equation of the transverse behavior of the present Mindlin plate model involves a set of a decoupled twelfth-order partial differential equation, and therefore, should enforce six boundary conditions on each plate side, to constitute a well-posed boundary value problem. The possible boundary conditions of a circular plate and a rectangular plate are discussed. The corner conditions, produced by the double stresses, are closely related to normal gradients of classical components of the shear force, the bending moment and the twisting moment. The corner conditions, existed in the present Mindlin plate model, are firstly clarified. The present work is expected to be a useful tool for developing effective numerical methods, such as the finite element method and the Garlerkin method.