On size-dependent mechanics of Mindlin plates made of polymer networks

The recent advances of solid mechanics of polymer networks are that they can be well-modelled by a physically-based size-dependent constitutive relation via a simplified strain gradient elasticity theory. However, boundary value problems of plate models composed of polymer networks have not been reported, which limit wide applications of the models in the engineering science. In this paper, we systematically established a variationally consistent boundary value problems of Mindlin plate models for polymer networks leading to the framework of a simplified strain gradient elasticity. This study considers the strain energy produced by the strain gradient in the thickness direction and proposes a well-posed boundary value problem for a Mindlin plate with arbitrary boundaries, discussing possible boundary conditions, especially higher-order nonconventional ones. The senses of stress resultants and double stresses acting on the face of a volume element are firstly explained. Surprisingly, it is found that unexpected corner condition related to normal derivatives of shear force, bending moment, and twisting moment exists for plates with irregular boundaries—contradicting conventional mechanics notions of plates. For illustrative purpose, static bending analyses of a simply supported rectangular plate subjected to a uniformly distributed loading and a concentrated loading are provided. The effective Young's modulus predicted by this approach agrees well with reported result in the open literature. This work may be helpful in developing efficient numerical methods and offers new insights into the existence of corner condition in Mindlin plates within the context of a simplified strain gradient elasticity theory.