A Timoshenko beam element based on the modified couple stress theory

Kahrobaiyan, M. H.; Asghari, Mohsen; Ahmadian, M.T.

doi:10.1016/j.ijmecsci.2013.11.014

Cited by 100 publications

(42 citation statements)

References 40 publications

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“…First equation and system of the rest two equations can be solved independently. We have to mention that system (68) coincide with the one presented in [47,50,51] up to notation. Therefore analysis and verification presented in [47,50] take place in considered here case.…”

Section: Timoshenko's Couple Stress Theory Of the Curved Rodsmentioning

confidence: 80%

“…In most publications the considered models are based on Euler-Bernoulli and Timoshenko's hypothesis. Among the many articles on couple stress theories of beams based on Euler-Bernoulli hypothesis we mention here [43][44][45][46] and on Timoshenko's hypothesis we mention here [46][47][48][49][50][51][52], curved rods along with others have been considered in [4,16,42,[53][54][55], for more information see the extensive review paper [56].…”

Section: Introductionmentioning

confidence: 99%

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Couple stress theory of curved rods. 2-D, high order, Timoshenko’s and Euler-Bernoulli models

Zozulya

2017

Curved and Layered Structures

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New models for plane curved rods based on linear couple stress theory of elasticity have been developed. 2-D theory is developed from general 2-D equations of linear couple stress elasticity using a special curvilinear system of coordinates related to the middle line of the rod as well as special hypothesis based on assumptions that take into account the fact that the rod is thin. High order theory is based on the expansion of the equations of the theory of elasticity into Fourier series in terms of Legendre polynomials. First, stress and strain tensors, vectors of displacements and rotation along with body forces have been expanded into Fourier series in terms of Legendre polynomials with respect to a thickness coordinate. Thereby, all equations of elasticity including Hooke's law have been transformed to the corresponding equations for Fourier coefficients. Then, in the same way as in the theory of elasticity, a system of differential equations in terms of displacements and boundary conditions for Fourier coefficients have been obtained. Timoshenko's and EulerBernoulli theories are based on the classical hypothesis and the 2-D equations of linear couple stress theory of elasticity in a special curvilinear system. The obtained equations can be used to calculate stress-strain and to model thin walled structures in macro, micro and nano scales when taking into account couple stress and rotation effects.

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Section: Timoshenko's Couple Stress Theory Of the Curved Rodsmentioning

confidence: 80%

Section: Introductionmentioning

confidence: 99%

Couple stress theory of curved rods. 2-D, high order, Timoshenko’s and Euler-Bernoulli models

Zozulya

2017

Curved and Layered Structures

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“…Besides, the developments of structural beam/plate elements based on size‐dependent continuum theories have also received a considerable attention from scholars. For example, Kahrobaiyan et al developed Timoshenko beam elements and applied them to the MEMS; Ansari et al constructed Euler‐Bernoulli and Timoshenko beam elements based on Mindlin's strain gradient theory; Reddy et al performed the nonlinear finite element analysis of the functionally graded circular plates with the modified couple stress theory; Ansari et al proposed Mindlin plate elements based on the most general form of the strain gradient theory.…”

Section: Introductionmentioning

confidence: 99%

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

Shang

Jia

2020

Numerical Meth Engineering

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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

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“…At microscales and nanoscales, the mechanical response has been indicated to be size-dependent [21][22][23][24][25][26][27][28][29][30][31] and thus classical models of elasticity must be modified so as to include size influences [32][33][34][35][36][37][38][39][40][41][42]. A number of modification procedures based on the nonlocal elasticity [43][44][45][46], couple stress model [47][48][49][50][51][52], and strain gradient theory [53,54] have been proposed. More recently, a significant number of size-dependent models with microstructure-dependent deformational and nonlocal stress influences have been proposed [55,56].…”

Section: Introductionmentioning

confidence: 99%

Asymmetric Oscillations of AFG Microscale Nonuniform Deformable Timoshenko Beams

2019

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A nonlinear vibration analysis is conducted on the mechanical behavior of axially functionally graded (AFG) microscale Timoshenko nonuniform beams. Asymmetry is due to both the nonuniform material mixture and geometric nonuniformity. Using the Timoshenko beam theory, the continuous models for translation/rotation are developed via an energy balance. Size-dependence is incorporated via the modified couple stress theory and the rotation via the Timoshenko beam theory. Galerkin’s method of discretization is applied and numerical simulations are conducted for a size-dependent vibration of the AFG microscale beam. Effects of material gradient index and axial change in the cross-sectional area on the force and frequency diagrams are investigated.

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A Timoshenko beam element based on the modified couple stress theory

Cited by 100 publications

References 40 publications

Couple stress theory of curved rods. 2-D, high order, Timoshenko’s and Euler-Bernoulli models

Couple stress theory of curved rods. 2-D, high order, Timoshenko’s and Euler-Bernoulli models

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

Asymmetric Oscillations of AFG Microscale Nonuniform Deformable Timoshenko Beams

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