A 3‐node C 0 triangular element for the modified couple stress theory based on the smoothed finite element method

SummaryIn this work, the recently proposed unsymmetric 4‐node 12‐DOF (degree‐of‐freedom) membrane element (Shang and Ouyang, Int J Numer Methods Eng 113(10): 1589‐1606, 2018), which has demonstrated excellent performance for the classical elastic problems, is further extended for the modified couple stress theory, to account for the size effect of materials. This is achieved via two formulation developments. Firstly, by using the penalty function method, the kinematic relations between the element's nodal drilling DOFs and the true physical rotations are enforced. Consequently, the continuity requirement for the modified couple stress theory is satisfied in weak sense, and the symmetric curvature test function can be easily derived from the gradients of the drilling DOFs. Secondly, the couple stress field that satisfies a priori the related equilibrium equations is adopted as the energy conjugate trial function to formulate the element for the modified couple stress theory. As demonstrated by a series of benchmark tests, the new element can efficiently capture the size‐dependent responses of materials and is robust to mesh distortions. Moreover, as the new element uses only three conventional DOFs per node, it can be readily incorporated into the standard finite element program framework and commonly available finite element programs.

Section: Two-dimensional Problemmentioning

confidence: 99%

Section: The Cantilever Thin Beammentioning

confidence: 99%

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A simple unsymmetric 4‐node 12‐DOF membrane element for the modified couple stress theory

Shang

Qian

Cen

et al. 2019

“…However, since the CST element exhibits a very stiff behavior, it is difficult to obtain a solution appropriately for the membrane dominant problem. In the proposed shell element, a membrane element developed by the current authors is used. This subsection summarizes the formulation of the membrane element.…”

Section: Smoothed Flat Shell Elementmentioning

confidence: 99%

Rotation‐free triangular shell element using node‐based smoothed finite element method

Choi

Lee

2018

Self Cite

In this paper, a triangular thin flat shell element without rotation degrees of freedom is proposed. In the Kirchhoff hypothesis, the first derivative of the displacement must be continuous because there are second-order differential terms of the displacement in the weak form of the governing equations. The displacement is expressed as a linear function and the nodal rotation is defined using node-based smoothed finite element method. The rotation field is approximated using the nodal rotation and linear shape functions. This rotation field is linear in an element and continuous between elements. The curvature is defined by differentiating the rotation field, and the stiffness is calculated from the curvature. A hybrid stress triangular membrane element was used to construct the shell element. The penalty technique was used to apply the rotation boundary conditions. The proposed element was verified through several numerical examples. KEYWORDS flat shell, NS-FEM, rotation-free, thin-shell element Int J Numer Methods Eng. 2018;116:359-379.wileyonlinelibrary.com/journal/nme

“…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.…”

Section: Introductionmentioning

confidence: 99%

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

Shang

Jia

2020

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