Finite element analysis of plane strain solids in strain-gradient elasticity

Trefftz collocation method for two‐dimensional strain gradient elasticity

“…where the components of the displacement, traction, and double traction are defined by u = {u 1 , u 2 , 0}, t = {t 1 , t 2 , 0} and m = {m 1 , m 2 , 0}, respectively; the comma denotes differentiation with respect to the corresponding coordinates; and the rigid body displacements in the vertical direction are avoided in all variants introducing additional point constraint at x = y = 0: u 2 = 0. The difference between the given variants (29), (30), and (31) is in the high-order boundary conditions. In the Variant I, we have stress-free conditions with zero double-tractions on the whole surface of the region.…”

“…Definitions of tractions and double tractions components in (29), (30), and (31) for the corresponding boundaries of the rectangular region can be obtained by using (2),(3), and (4) and these are the following:…”

“…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.…”

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

“…In general, they are restricted to regular element geometry. Although the geometry requirement can be relaxed by using mapped meshes or smoothing the mesh, the former is restrictive and the latter requires special preprocessing treatment …”