C0 and C1 theories and test functions for FEM patch test in microstructures

Abstract

Among many theories and categories in microstructures, rotation-displacement used as "independent" or "dependent" variables, is a noticeable topic. In FEM, it is called C 0 and C 1 theory. The convergence criteria of finite elements for microstructures are less mature than those for the conventional thin plate bending problem. In this paper, the patch test functions for assessing convergence of the C 0 and C 1 finite elements in microstructures is established based on the enhanced patch test theory. The author has further explored the C 0 and C 1 finite element theories and investigated the difference and correlation between their finite element formulations. Newly proposed finite element theories for microstructures are as follows: (1) the displacement-rotation dependent C 1 element that requires the element function satisfying both C 0 and C 1 continuity; (2) the displacement-rotation independent C 0 element which requires new convergence criteria, such as non-zero constant shear stress patch test and zero constant shear stress patch test for approximating C 1 element. couple stress theory, displacement-rotation dependent theory, displacement-rotation independent theory, finite element method for microstructures, test function for patch test PACS: 02.70.Dh, 61.72.-y, 81.30.-tSince the 1960s, experiments have shown that material has localized deformation due to impurities, scaled defects, crystal lattice mismatch and micro cracks at micro or nano scales. These phenomena are also known as shear band, rupture and stress concentration, etc. As conventional continuum theory can not explain or solve the problem of localization at micro-scale, theories for microstructures need to be developed.Theory for microstructures includes couple stress and strain gradient theories. The couple stress theory can be seen as a special format of strain gradient theory which uses rotation as a variable to describe curvature, while the strain gradient theory uses strain as variable to describe curvature. Though both theories can describe the localized deformation at micro-scale, the couple stress theory contains fewer rotation variables than the strain gradient theory does the strain variables. In the couple stress theory, the variables related to micro-scale impurities or defects are formulated into rotation equilibrium equations. In the strain gradient theory, these variables are formulated into higher order strain terms in geometric equations. In both cases, new parameters which describe the material scale characteristics are introduced as higher order term (4th order) into the partial differential governing equation. Yet, in conventional continuum mechanics, this partial differential governing equation is a 2nd order equation.A series of research studies in the couple stress and strain gradient theories have been made. For example, in the 1960s, Toupin [1], Koiter [2] and Mindlin [3] proposed strain gradient theory by introducing strain gradient into constitutive equation [3]. Between the 80-90s, Aifantis [4], Fleck and Hutchinson ...

Strain gradient differential quadrature finite element for moderately thick micro‐plates

“…Based on References 26, 67 and Equations (57a)‐(57c), three test functions are assumed as where S n are undetermined coefficients.…”

“…In comparison to the conventional plate finite elements, the convergence criteria for gradient-enhanced plate finite elements are still inadequate. Here, we employ the enhanced patch test theory 26,67 for the constrained couple stress plane element and the classical Mindlin plate element to assess the convergence of our element. In our present analysis, we select an element patch with six quadrilateral elements, as shown in Figure 3.…”

“…Papanicolopulos et al 25 developed a C 1 ‐type hexahedral element to deal with three‐dimensional (3D) gradient elasticity problems. Using the enhanced patch test theory, Chen 26 presented the patch test functions to assess the convergence of C 0 ‐ and C 1 ‐continuous couple stress‐based finite elements. Zhao et al 27 developed a 24‐degree of freedom (DOF) quadrilateral element for couple stress/strain gradient elasticity using the refined Kirchhoff plate element RDKQ with weak C 1 ‐continuity, as well as the nonconforming element CQ12 with weak C 0 ‐continuity.…”

A weak continuity condition of FEM for axisymmetric couple stress theory and an 18-DOF triangular axisymmetric element

“…In fact, it is difficult to satisfy this kind of patch test. The behavior of the C 0 type couple stress elements is simlar to that of the Mindlin plate elements [20,21]. For C 1 thin plate elements, the displacement test function of the patch test can be expressed as…”

A polygonal element for couple stress/strain gradient elasticity based on SBFEM and spline interpolation

“…偶应力/应变梯度有限元中应变同时出现了位移的 一阶和二阶导数, 要求节点参数包括位移的函数值 和导数值. 此外, Soh和陈万吉 [12,13] 的研究表明, 偶 应力/应变梯度理论的分片检验函数应是满足体力 为零的平衡方程的2次位移函数, 这还要求单元对 位移函数满足2次完备性和C 0 连续, 目前用于求解 偶应力/应变梯度理论的绝大部分单元都不能通过 新提出的分片检验. 随后, 借助分片检验的理论分 析, 多个满足分片检验条件的三角形和四边形偶应 力/应变梯度理论单元被陆续提出 [12,[14][15][16][17][18] .…”