Penalty C0 8-node quadrilateral and 20-node hexahedral elements for consistent couple stress elasticity based on the unsymmetric finite element method

Penalty hexahedral element formulation for flexoelectric solids based on consistent couple stress theory

“…It is worth to notice that, the stress solutions obtained from Equation (37) are only a subset of those of Equation (36), which gives rise to a certain approximation. But, it can be proved that the low-order stress solutions obtained in this way are exactly the solutions of Equation (36). As shown in Section 2, the stresses and couple stress are essentially the first and second derivatives of displacements, respectively.…”

“…That means, when the couple stresses take the constant values or vary linearly in space, the corresponding stresses vary linearly or quadratically. For the deformation state in which the couple stresses are constant or vary linearly, Equation ( 38) is satisfied automatically and Equation (36) degenerates exactly into Equation (37). Therefore, only the constant, linear and quadratic stress solutions of Equation (37) are used in this work, and it is still possible to ensure the convergence of the element based on the CST after introducing the above simplification.…”

“…Forty-eight groups of solutions to Equation (37), which can guarantee the quadratic completeness of Cartesian coordinates, have been proposed in the earlier work. 36 Note that the usage of 48 stress modes will lead to the subsequent matrix inversion of a 48 × 48 matrix which requires a considerable amount of computation cost. To improve the computational efficiency, we will further reduce these stress modes into 42 groups by considering the strain compatibility equations.…”

“…In this work, a 20‐node hexahedral element is proposed for size‐dependent electromechanical analysis of dielectric solids based on Hadjesfandiari's flexoelectricity model. The new element can be viewed as the extension and modification of the element model 36 developed for the CCST which is formulated within the framework of the penalty unsymmetric FEM. The unsymmetric FEM is essentially characteristic of independently or partially independently designing the element's test function and trial function 37–40 .…”

“…Then, the element's test functions of the displacement, mechanical rotation and electric potential are formulated using the quadratic serendipity isoparametric interpolation function, from which the strain, curvature and electric field can be further derived. The construction of the stress trial function follows the same strategy as Reference 36, that is, it is first built at the basis of polynomial stress functions and finally determined using the weighted residual method. Note that the stress functions used here can satisfy the equilibrium equations and deformation compatibility equations a priori, whilst those in Reference 36 can only satisfy the equilibrium equations.…”