Mixed Lagrangian formulation for size‐dependent couple stress elastodynamic and natural frequency analyses

Deng, Guoqiang; Dargush, Gary F.

doi:10.1002/nme.5310

Cited by 17 publications

(4 citation statements)

References 31 publications

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“…It is hard to solve analytically the size-effect related problems, because the size-dependent continuum theories are much more complicated than the classical elasticity theory. The finite element method (FEM), which is generally regarded as the most efficient and popular numerical tool for modelling solids [22], provides a promising solution [23][24][25][26][27]. However, due to the presence of the second-order derivatives of displacements in these size-dependent continuum theories, the displacement-based FEM simulation requires C 1 continuity for the displacement interpolation.…”

Section: Introductionmentioning

confidence: 99%

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

Shang

Qian

Cen

et al. 2019

Numerical Meth Engineering

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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.

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Section: Introductionmentioning

confidence: 99%

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

Shang

Qian

Cen

et al. 2019

Numerical Meth Engineering

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“…To effectively circumvent the obstacle imposed by the requirement of the C 1 continuity, various methodologies have been successfully proposed, resulting in a diverse of element schemes, such as the mixed element formulations (Deng and Dargush, 2017;Lee 2017, 2018;Papanicolopulos et al, 2019), the C 0-1 elements that adopt both the displacements and displacement derivatives as the nodal degrees of freedom (DOFs) Chen 2013, 2014;Wang et al, 2016;Torabi et al, 2019), etc. Among these works, an effective one to keep the element formulation relatively simple is to introduce the independent nodal rotation DOFs into the element construction.…”

Section: Couple Stress Based Membrane Elements 1097mentioning

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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“…A two-field variational formulation can be established by strongly imposing the couple stress constraint. Focusing on the electro-elastodynamics of the conservative flexoelectric system, the internal and external forces together with the motion between times T 0 " 0 and T , can be determined from a Hamilton's principle, Deng and Dargush [2017]; Sudeep et al [2005]. To this effect, we introduce the Lagrangian L as Lpu, 9 u, ψq " Kp 9 uq´Π int pu, ψq´Π ext pu, ψq…”

Section: The Displacement-potential Variational Formulationmentioning

confidence: 99%

On a family of numerical models for couple stress based flexoelectricity for continua and beams

Poya

Gil

Ortigosa

et al. 2019

Journal of the Mechanics and Physics of Solids

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A family of numerical models for the phenomenological linear flexoelectric theory for continua and their particularisation to the case of three-dimensional beams based on a skew-symmetric couple stress theory is presented. In contrast to the standard strain gradient flexoelectric models which assume coupling between electric polarisation and strain gradients, we postulate an electric enthalpy in terms of linear invariants of curvature and electric field. This is achieved by introducing the axial (mean) curvature vector as a strain gradient measure. The physical implication of this assumption is many-fold. Firstly, analogous to the standard strain gradient models, for isotropic (non-piezoelectric) materials it allows constructing flexoelectric energies without breaking material's centrosymmetry. Secondly, unlike the standard strain gradient models, nonuniform distribution of volumetric part of strains (volumetric strain gradients) do not generate electric polarisation, as also confirmed by experimental evidence to be the case for some important classes of flexoelectric materials. Thirdly, a state of plane strain generates out of plane deformation through strain gradient effects. Finally, under this theory, extension and shear coupling modes cannot be characterised individually as they contribute to the generation of electric polarisation as a whole. As a first step, a detailed comparison of the developed couple stress based flexoelectric model with the standard strain gradient flexoelectric models is performed for the case of Barium Titanate where a myriad of simple analytical solutions are assumed in order to quantitatively describe the similarities and dissimilarities in effective electromechanical coupling under these two theories. From a physical point of view, the most notable insight gained is that, if the same experimental flexoelectric constants are fitted in to both theories, the presented theory in general, reports up to 200% stronger electromechanical conversion efficiency. From the formulation point of a view, the presented flexoelectric model is also competitively simpler as it eliminates the need for high order strain gradient and coupling tensors and can be characterised by a single flexoelectric coefficient. In addition, three distinct mixed flexoelectric variational principles are presented for both continuum and beam models that facilitate incorporation of strain gradient measures in to a standard finite element scheme while maintaining the C 0 continuity. Consequently, a series of low and high order mixed finite element schemes for couple stress based flexoelectricity are presented and thoroughly benchmarked against available closed form solutions in regards to electromechanical coupling efficiency. Finally, nanocompression of a complex flexoelectric conical pyramid for which analytical solution cannot be established is numerically studied where curvature induced necking of the specimen and vorticity around the frustum generate moderate electric polarisation.

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Mixed Lagrangian formulation for size‐dependent couple stress elastodynamic and natural frequency analyses

Cited by 17 publications

References 31 publications

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

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

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

On a family of numerical models for couple stress based flexoelectricity for continua and beams

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