Twenty‐four–DOF four‐node quadrilateral elements for gradient elasticity

Abstract

Computational analysis of gradient elasticity often requires the trial solution to be C 1 , yet constructing simple C 1 finite elements is not trivial. In this paper, three four-node 24-DOF quadrilateral elements for gradient elasticity analysis are devised by generalizing some of the advanced element formulations for thin-plate analysis. These include the discrete Kirchhoff method, a relaxed hybrid-stress method, and the hybrid-stress method with equilibrating internal force modes. The first two methods start with the derivation of a C 0 displacement, which is quadratic complete in the Cartesian coordinates. In the first method, at the midside points are derived and interpolated together with those at the nodes. Strain is derived from the displacement interpolation yet the second-order displacement derivatives are derived from the displacement-gradient interpolation. In the second method, only the assumed constant double-stress modes are employed to enforce the continuity of the normal derivative of the displacement. In the third method, the equilibrating internal force modes require the C 1 displacement to be defined only along the element boundary and the domain interpolation can be avoided. Patch test involving linear stress and constant double stress as well as other tests are presented to validate the proposed elements. KEYWORDSC 1 displacement, discrete Kirchhoff, finite element, gradient elasticity, hybrid formulation INTRODUCTIONConventional elasticity theories employ strain as the only deformation measure and do not take the internal length scale of the material into account. They work well while the characteristic length of the deformation is considerably larger than the internal length scale. The requirement is met in conventional engineering applications of compact materials but not necessarily in, eg, natural materials, artificial cellular/foam materials, and micro/nano-devices. 1-6 A bottleneck for many gradient-enhanced theories to gain wider acceptance is probably the C 1 requirement on the basis functions. As meshless and isogeometric methods can construct highly continuous basis functions, 2,7-11 the integration of the weak form and the enforcement of boundary conditions in these method are not as straightforward as the finite element method. Furthermore, the support of the highly continuous basis functions is larger than those of the finite element model, leading to less sparse system matrices and long computing time. To date, the finite element method remains to be a well-accepted tool for engineering analyses. Unfortunately, the existing C 1 finite elements pose strict requirement on the mesh topology, 128 possess second-/third-order derivatives of the interpolated variable as the nodal degrees of freedom (DOFs), and/or employ heterosis nodes which are different from other element nodes in the nodal DOFs. 12 For instance, Agyris's three-node and six-node triangles use (u, u, x , u, y , u, xx , u, xy , u, yy ) as the parameters at the corner nodes and u, n , the displacement gradient along th...

Trefftz collocation method for two‐dimensional strain gradient elasticity

“…One of the possible variant is to introduce some assessments for the field variables based on the approximate solution for the maximum traction t 0 and displacements u 0 that arise in the considered problem. Thus, we can introduce the normalizing constants in (27) as follows:…”

“…where d is the characteristic geometric dimension of the considered region; and the value of the length scale parameter l in the definitions of R 4 and R 6 is inessential since it will be canceled in (27).…”

“…This means that the edges of the region are the main source of the nonclassical gradient effects under pure bending within the considered simplified theory (5) and variant I of the boundary conditions (in more general SGET formulations, the gradient effects under pure bending have a more complex nature 6 ). The normalizing constants in the error functions (27) for the pure bending problem are defined based on the corresponding classical solution as follows:…”

“…12,13 The pioneering work on the development of numerical methods in SGET was undertaken by Oden et al in their paper in 1970, 14 where they adopted the finite element method (FEM). Today, a lot of advanced numerical approaches and methods have been proposed in SGET that can be divided into the mixed FEM with C 0 -continuous interpolation and its variants, [15][16][17][18][19][20][21][22][23][24][25][26][27] FEM with C 1 -continuous interpolation, [28][29][30][31] isogeometric analysis, [32][33][34][35][36][37] singular boundary element methods (BEM) 38,39 and meshless Galerkin-type methods. [40][41][42] Among recent works, we can also mention the penalty three-dimensional elements approach 43 and the nonlocal operator method.…”

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

“…In past decades, great efforts have been made to develop robust plane elements for higher‐order continuum theories. For instance, Zervos et al formulated different C 1 triangular and quadrilateral elements for elastoplasticity strain gradient problems; Beheshti developed 4‐node quadrilateral elements based on the Hermite shape functions for the strain‐gradient elasticity; Papanicolopulos et al proposed a general framework for developing mixed finite elements for strain‐gradient boundary‐value problems using either Lagrange multiplier or penalty methods; Choi and Lee extended the smoothed FEM to the modified couple stress theory; Kwon and Lee proposed a mixed element formulation using the Lagrange multiplier method and the convergence criteria; Garg and Han developed penalty plane and axisymmetric elements for the couple stress elasticity in which the independent nodal drilling DOFs are introduced; Wang et al developed the quasi‐conforming C 0‐1 elements in which both nodal displacements and nodal displacement derivatives are adopted as DOFs for the modified couple stress theory; Chen and his coauthors also proposed similar models for the strain gradient/couple stress theories using the refined nonconforming element technique; Phunpeng and Baiz constructed a mixed element for strain‐gradient elasticity problems using the FEniCS environment; Sze and Wu formulated three 4‐node 24‐DOF quadrilateral elements for the gradient elasticity analysis by generalizing different thin plate element models.…”