Variational formulations and general boundary conditions for sixth-order boundary value problems of gradient-elastic Kirchhoff plates

Sixth-order boundary value problems for gradient-elastic Kirchhoff plate bending models are formulated in a variational form within an H³ Sobolev space setting. Existence and uniqueness of the weak solutions are then established by proving the continuity and coercivity of the associated symmetric bilinear forms. Complete sets of boundary conditions, including both the essential and the natural conditions, are derived accordingly. In particular, the gradient-elastic Kirchhoff plate models feature two different types of clamped and simply supported boundary conditions in contrast to the classical Kirchhoff plate model. These new types of boundary conditions are given additional attributes singly and doubly; referring to free and prescribed normal curvature, respectively. The formulations and results of the analyzed strain gradient plate models are compared to two other generalized Kirchhoff plate models which follow a modified strain gradient elasticity theory and a modified couple stress theory. It is clarified that the results are extendable to these model variants as well. Finally, the relationship of the natural boundary conditions to external loadings is analyzed.