Generalized conforming Trefftz element for size-dependent analysis of thin microplates based on the modified couple stress theory

Shang, Yan; Mao, Yu-Hao; Cen, Song; Li, Chenfeng

doi:10.1016/j.enganabound.2021.01.007

Cited by 12 publications

(4 citation statements)

References 45 publications

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“…In our previous works, 39,52 it has been proved that the above set of 14 conditions can effectively meet the C 1 compatibility requirement of the Kirchhoff thin plate in weak sense. In general, the requirement is satisfied more strictly as the mesh being gradually refined.…”

Section: The Interpolations Related To the Bending Deformationsmentioning

confidence: 94%

“…Since the out-of-plane displacement of the mid-surface ŵm (x, ŷ) is related only to the bending deformations, it can be initially assumed as the linear combination of the Trefftz functions of the micro thin plate bending problem 39 in the context of the modified couple stress theory:…”

Section: The Interpolations Related To the Bending Deformationsmentioning

confidence: 99%

“…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. 41 The isogeometric analysis (IGA) proposed by Hughes et al 42 which adopts the CAD basis functions as the shape functions is recently regards as an effective method to overcome the obstacle of high-order continuity.…”

Section: Introductionmentioning

confidence: 99%

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

Shang

Cen

et al. 2021

Numerical Meth Engineering

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This work proposes a simple but robust 4-node 24-DOF facet shell element for static analysis of small-scale thin shell structures. To accommodate the size effects, the modified couple stress theory is employed as the theoretical basis.The element is constructed via two main innovations. First, the trial functions that can a priori satisfy related governing differential equations are adopted as the basic functions for formulating the element interpolations. Second, the generalized conforming theory and the penalty function method are employed to meet the C 1 continuity requirement in weak sense for ensuring the computation convergence property. Several benchmarks of shells with different geometries are tested to assess the new facet shell element's capability. The numerical results reveal that the element can effectively simulate the size-dependent mechanical behaviors of small-scale thin shells, exhibiting satisfactory numerical accuracy and low susceptibility to mesh distortion. Moreover, as the shell element uses only six degrees of freedom per node, it can be incorporated into the commonly available finite element programs very readily.

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Section: The Interpolations Related To the Bending Deformationsmentioning

confidence: 94%

Section: The Interpolations Related To the Bending Deformationsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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

Shang

Cen

et al. 2021

Numerical Meth Engineering

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“…Numerous experimental observations have shown that the mechanical behaviors of the micro/nano structures are size dependent (Fleck et al, 1994;Nix and Gao, 1998;Lam et al, 2003). To effectively capture such size effects, in past decades, substantial efforts have been devoted to develop different reliable numerical methods based on the higher-order continuum theories that contain additional internal material length scale parameters instead of the classical continuum theory, such as the finite element method (FEM) (Papanicolopulos et al, 2019;Pedgaonkar et al, 2021;Shang et al, 2021), the boundary element method (Rodopoulos et al, 2021), the isogeometric method (Yu et al, 2019;Fan et al, 2020;Thai et al, 2020) and the meshless method (Roque et al 2013a, b).…”

Section: Introductionmentioning

confidence: 99%

Couple stress-based unsymmetric 8-node planar membrane elements with good tolerances to mesh distortion

Shang

2021

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PurposeThe paper aims to propose two new 8-node quadrilateral membrane elements with good distortion tolerance for the modified couple stress elasticity based on the unsymmetric finite element method (FEM).Design/methodology/approachThe nodal rotation degrees of freedom (DOFs) are introduced into the virtual work principle and constrained by the penalty function for approximating the test functions of the physical rotation and curvature. Therefore, only the C0 continuity instead of C1 continuity is required for the displacement during the element construction. The first unsymmetric element assumes the test functions of the displacement and strain using the standard 8-node isoparametric interpolations, while these test functions in the second model are further enhanced by the nodal rotation DOFs. Besides, the trial functions in these two elements are constructed based on the stress functions that can a priori satisfy related governing equations.FindingsThe benchmark tests show that both the two elements can efficiently simulate the size-dependent plane problems, exhibiting good numerical accuracies and high distortion tolerances. In particular, they can still exactly reproduce the constant couple stress state when the element shape deteriorates severely into the degenerated triangle. Moreover, it can also be observed that the second element model, in which the linked interpolation technique is used, has better performance than the first one, especially in capturing the steep gradients of the physical rotations.Originality/valueAs the proposed new elements use only three DOFs per node, they can be readily incorporated into the existing finite element (FE) programs. Thus, they are of great benefit to analysis of size-dependent membrane behaviors of micro/nano structures.

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An efficient 3-node triangular plate element for static and dynamic analyses of microplates based on modified couple stress theory with micro-inertia

Mao

Shang

Cen

et al. 2022

Engineering with Computers

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Generalized conforming Trefftz element for size-dependent analysis of thin microplates based on the modified couple stress theory

Cited by 12 publications

References 45 publications

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

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

Couple stress-based unsymmetric 8-node planar membrane elements with good tolerances to mesh distortion

An efficient 3-node triangular plate element for static and dynamic analyses of microplates based on modified couple stress theory with micro-inertia

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