Mixed Formulation of Finite Element Method Within Toupin–Mindlin Gradient Elasticity Theory

The mixed formulation of the finite element method for the problems of the Toupin–Mindlin gradient theory of elasticity is justified. This theory permits accounting for scale effects stemming from the material microstructure dimensions, particularly in problems with the limitations of classical elasticity. The variational formulation of the boundary problem is examined where strains, stresses, and their gradients enter into the variational equations along with displacements as equivalent arguments. The key feature of these equations is that they involve only the first-order partial derivatives of displacements, in contrast to the differential equations of the classical problem formulation that involve the derivatives of displacements to the fourth order inclusive, and the Lagrange variational equation in displacements, which incorporates their double differentiation. Their solution based on the mixed finite element method greatly simplifies the choice of approximation functions since there is no need to use finite elements that ensure the continuity of the first displacement derivatives at the element boundaries. This formulation based on a separate approximation of displacements, strains, stresses, and their gradients, is applied to solving the boundary problems of elasticity theory, where the strain gradient is included. For the mixed-method variational equations, the condition is formulated so as to ensure the uniqueness of the solution and stability of the mixed approximation for gradient elasticity theory problems. This condition is defined with the orthogonal projection operator that establishes a one-to-one correspondence between the classical and mixed approximation of strain distributions. A well-suited formulation for the practical application of variational equations for displacements and strains is proposed with the weakest requirements for mixed approximation stability.