MIXED FORMULATION OF FINITE ELEMENT METHOD WITHIN STRAIN GRADIENT ELASTICITY
DOI:
https://doi.org/10.15407/dopovidi2025.03.061Keywords:
gradient theory of elasticity, stress and strain gradients, variational equations, finite element method, mixed approximation, convergenceAbstract
One of the generalized continuum theories related to the microstructure scale is the Toupin-Mindlin gradient theory of elasticity, which allows one to take into account structural inhomogeneities and material damage at the micro level. The peculiarity of solving variational equations in gradient theory is the consideration of the first partial derivatives of the components of the small strain tensor. A necessary condition for the convergence of a solution based on the finite element method is the property of the approximation functions that ensures the continuity of displacements and their first derivatives at the boundary between elements. This study considers an alternative approach, according to which the solution of boundary value problems in gradient elasticity theory is based on the use of a variational formulation of displacements, deformations, stresses, and their gradients. This formulation greatly simplifies the choice of approximation functions, since there is no need to use finite elements that ensure the continuity of the first derivatives of displacements between elements. To analyze the correctness of the mixed approximation, the variational equations of the mixed method are transformed into an equivalent form with respect to displacements, strains, and their gradients. Based on the obtained a priori estimates, conditions that ensure the convergence of approximate solutions based on the mixed approximation are determined.
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