8‐node hexahedral unsymmetric element with rotation degrees of freedom for modified couple stress elasticity

Abstract

A new C 0 8-node 48-DOF hexahedral element is developed for analysis of size-dependent problems in the context of the modified couple stress theory by extending the methodology proposed in our recent work (Shang et al., Int J Numer Methods Eng 119(9): 807-825, 2019) to the three-dimensional (3D) cases.There are two major innovations in the present formulation. First, the independent nodal rotation degrees of freedom (DOFs) are employed to enhance the standard 3D isoparametric interpolation for obtaining the displacement and strain test functions, as well as to approximatively design the physical rotation field for deriving the curvature test function. Second, the equilibrium stress functions instead of the analytical functions are used to formulate the stress trial function whilst the couple stress trial function is directly obtained from the curvature test function by using the constitutive relationship. Besides, the penalty function is introduced into the virtual work principle for enforcing the C 1 continuity condition in weak sense. Several benchmark examples are examined and the numerical results demonstrate that the element can simulate the size-dependent mechanical behaviors well, exhibiting satisfactory accuracy and low susceptibility to mesh distortion. K E Y W O R D Shexahedral element, modified couple stress theory, rotation degree of freedom, size-dependent, unsymmetric FEM INTRODUCTIONMany experimental observations have shown that the micro/nano structures, which have been widely used in modern engineering applications, may experience size-dependent mechanical behaviors. Various non-classical higher-order continuum theories, such as the strain gradient theory (SGT) and the couple stress theory (CST), have been developed for describing such size-dependent phenomena. 1-7 However, as these higher-order theories are more complex than the classical one, the analytical or semi-analytical solutions are available only for restricted problems with simple geometries and certain boundary conditions. Therefore, the numerical approaches with high accuracy and efficiency are clearly required for the solution procedure, in which the well-accepted finite element method (FEM) is usually recognized as a very efficient choice. In recent year, the isogeometric method 8-11 and the meshless method, 12,13 which can formulate highly continuous basis functions, have also been applied to the size-dependent problems. However, the computation expenses Int J Numer Methods Eng. 2020;121:2683-2700. wileyonlinelibrary.com/journal/nme

Comments on “4‐node unsymmetric quadrilateral membrane element with drilling DOFs insensitive to severe mesh‐distortion” by Yan Shang and Wengen Ouyang, International Journal for Numerical Methods in Engineering 2018; 113: 1589–1606

“…In fact, it is equivalent to the hybrid stress element based on the Hellinger-Reissner variational principle. The other papers by Shang et al [2][3][4] have the same problem. The unsymmetric element method first proposed by Rajendran and Liew 5 has the unsymmetric stiffness matrix as its obvious characteristic while also the major disadvantage due to the need for more computer memory and computations.…”

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

“…Therefore, the displacement u in the previous equations can be determined using C 0 continuous interpolation functions. The commonly used C 0 20-node isoparametric interpolation appears to be a good choice for the present 20-node element, but can be further enriched by the nodal rotation DOFs through the 3D link interpolation scheme, 38 as follows:…”

“…The unsymmetric FEM is essentially characteristic of independently or partially independently designing the element's test function and trial function. [37][38][39][40] In the present development, the C 1 requirement is satisfied in a weak sense by introducing an independent rotation field into the virtual work principle and using the penalty function method. 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.…”

An efficient 4‐node facet shell element for the modified couple stress elasticity

“…Over the past decades, tremendous efforts have been made to develop finite elements with concise formulation and satisfactory performance. But unfortunately, those existing works primarily focus on the membrane elements, [25][26][27][28][29][30][31] solid elements, [32][33][34][35][36] and plate bending elements. [37][38][39][40] The literatures regarding the shell elements, especially the ones which can be used for analysis of shells with complex geometries, are very limited.…”

“…The y-direction deflections at the load point A of the pinched cylinder (μm) The reference values are obtained using 200,000 hexahedral solid element 35. …”