An improved quadrilateral finite element for nonlinear second-order strain gradient elastic Kirchhoff plates

Babu, Bishweshwar; Patel, B.P.

doi:10.1007/s11012-019-01087-z

Cited by 9 publications

(3 citation statements)

References 44 publications

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“…Equations (19)(20)(21)(22) are written in matrix form as w h e re Z = [w, θ x , V x , M x ] T i s re fe r re d t o a s t h e s t at e ve c t or ; H = F G Q −F T , w i t h…”

Section: Governing Equation For Bending Buckling and Free Vibration Of Nanoplates In The Hamiltonian Systemmentioning

confidence: 99%

“…reported finite element formulations for nonlocal elastic Euler-Bernoulli beam and Kirchhoff plate, and analyzed bending, vibration, and buckling of simply supported nonlocal plates. Bahu and Patel 21 developed an improved quadrilateral finite element for nonlinear second-order strain gradient elastic Kirchhoff plates based on the nonlocal theory. Necira 22 developed the hierarchical finite element method for size-dependent free vibration analysis of Mindlin nano-plates with curvilinear plan-forms.…”

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confidence: 99%

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New analytic bending, buckling, and free vibration solutions of rectangular nanoplates by the symplectic superposition method

Zheng

Huang

et al. 2021

Sci Rep

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New analytic bending, buckling, and free vibration solutions of rectangular nanoplates with combinations of clamped and simply supported edges are obtained by an up-to-date symplectic superposition method. The problems are reformulated in the Hamiltonian system and symplectic space, where the mathematical solution framework involves the construction of symplectic eigenvalue problems and symplectic eigen expansion. The analytic symplectic solutions are derived for several elaborated fundamental subproblems, the superposition of which yields the final analytic solutions. Besides Lévy-type solutions, non-Lévy-type solutions are also obtained, which cannot be achieved by conventional analytic methods. Comprehensive numerical results can provide benchmarks for other solution methods.

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“…Equations (19)(20)(21)(22) are written in matrix form as w h e re Z = [w, θ x , V x , M x ] T i s re fe r re d t o a s t h e s t at e ve c t or ; H = F G Q −F T , w i t h…”

Section: Governing Equation For Bending Buckling and Free Vibration Of Nanoplates In The Hamiltonian Systemmentioning

confidence: 99%

mentioning

confidence: 99%

New analytic bending, buckling, and free vibration solutions of rectangular nanoplates by the symplectic superposition method

Zheng

Huang

et al. 2021

Sci Rep

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“…Recently, there is considerable effort regarding the development of finite element formulations for size-dependent theories using Kirchhoff plates. Babu and Patel [50,51] developed a quadrilateral Kirchhoff plate element for second-order strain gradient theory proposed by Ru and Aifantis [52,53]. Beheshti [54] used the same formulation to develop four types of rectangular elements with different nodal degree of freedoms.…”

Section: Introductionmentioning

confidence: 99%

Finite Element Analyses of the Modified Strain Gradient Theory Based Kirchhoff Microplates

Kandaz

Dal

2021

Surfaces

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In this contribution, the variational problem for the Kirchhoff plate based on the modified strain gradient theory (MSGT) is derived, and the Euler-Lagrange equations governing the equation of motion are obtained. The Galerkin-type weak form, upon which the finite element method is constructed, is derived from the variational problem. The shape functions which satisfy the governing homogeneous partial differential equation are derived as extensions of Adini-Clough-Melosh (ACM) and Bogner-Fox-Schmit (BFS) plate element formulations by introducing additional curvature degrees of freedom (DOF) on each node. Based on the proposed set of shape functions, 20-, 24-, 28- and 32- DOF modified strain gradient theory-based higher-order Kirchhoff microplate element are proposed. The performance of the elements are demonstrated in terms of various tests and representative boundary value problems. Length scale parameters for gold are also proposed based on experiments reported in literature.

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Higher-order finite strip method (H-FSM) with nonlocal strain gradient theory for analyzing bending and free vibration of orthotropic nanoplates

Tanzadeh,

Amoushahi

2024

Acta Mech

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An improved quadrilateral finite element for nonlinear second-order strain gradient elastic Kirchhoff plates

Cited by 9 publications

References 44 publications

New analytic bending, buckling, and free vibration solutions of rectangular nanoplates by the symplectic superposition method

New analytic bending, buckling, and free vibration solutions of rectangular nanoplates by the symplectic superposition method

Finite Element Analyses of the Modified Strain Gradient Theory Based Kirchhoff Microplates

Higher-order finite strip method (H-FSM) with nonlocal strain gradient theory for analyzing bending and free vibration of orthotropic nanoplates

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