Mixed Fem Formulation Within the Toupin-Mindlin Gradient Elasticity Theory: Validity Analysis

The Toupin-Mindlin deformation gradient elasticity theory is one of the generalized continuum theories related to the microstructure dimension, which permits accounting for scale effects in the material. The peculiarity of solving the variational equations of the gradient theory is to take into account the first partial derivatives of the components of the infinitesimal strain tensor. A necessary condition for the convergence of approximate solutions of the problem based on the finite element method is the property of the approximation functions to ensure the continuity of displacements and their first derivatives at the boundary between elements. This leads to significant complications in the structure of finite element spaces and mathematical and computational difficulties. In this paper, we consider an alternative approach in which a mixed variational formulation with respect to displacements-deformations-stresses and their gradients is used to solve boundary value problems of the deformation gradient elasticity theory. The use of the mixed formulation greatly simplifies the choice of approximating functions since there is no need to use finite elements to ensure the continuity of the first derivatives of displacements between elements. To substantiate the correctness of the mixed approximation, the variational equations of the mixed method are transformed to an equivalent form in terms of displacements and strains. A condition is formulated that ensures the existence, uniqueness, and stability of approximate solutions based on the mixed method. The convergence conditions of the mixed approximation of displacements and strains to the exact solution of the variational problem are investigated. An inequality for estimating the total approximation error for deformations and their gradients in the energy norm is obtained. The estimates of the mixed approximation errors for strains, displacements, and potential energy with respect to the deformation gradient are proved. Estimates of the closeness of solutions for deformations using the classical and deformation gradient elasticity theory are given. Based on the obtained a priori estimates, the conditions that ensure the convergence of approximate solutions based on the mixed approximation are determined. It is shown that the superconvergence of the first derivatives of displacements on uniform and quasi-uniform partitions improves the accuracy of the calculation of deformations and their gradients, which makes it possible to justify the convergence of the mixed method even for simpler linear finite elements used to approximate displacements and strains. Sufficient conditions for the convergence of the mixed method are the continuous approximation of the displacement fields by finite elements of the second or higher order of approximation and the continuous approximation of the strains by finite elements of at least the first order of approximation, and these approximations are interconnected by the condition of solving a finite-dimensional problem.