A nonlinear strain gradient beam formulation

Kahrobaiyan, M. H.; Asghari, Mohsen; Rahaeifard, M.; Ahmadian, Mohammad Taghi

doi:10.1016/j.ijengsci.2011.01.006

Cited by 168 publications

(56 citation statements)

References 40 publications

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“…In another work, Ma et al In the current work, by employing strain gradient theory and Hamilton's principle, a large-deformation size-dependent Timoshenko microbeam model is presented. The present model accommodates some previously published beam models including the linear couple stress [Park and Gao 2006;Kong et al 2008;Ma et al 2008], linear strain gradient [Kaneko 1975;Kahrobaiyan et al 2011], nonlinear couple stress [Asghari et al 2010b], nonlinear strain gradient [Kahrobaiyan et al 2011] theories. Then, utilizing this model, the nonlinear size-dependent free vibration behavior of simply supported microbeams is described using the harmonic balance method.…”

Section: Introductionsupporting

confidence: 66%

“…Moreover, the governing equations and boundary conditions of a nonlinear Timoshenko beam modeled via the modified couple stress theory can be achieved if only l 0 = l 1 = 0 and l 2 = l (see [Asghari et al 2010b]). Additionally, by neglecting the shear deformation, governing equations and boundary conditions corresponding to the microscale Euler-Bernoulli beam model based on the strain gradient elasticity theory will be attained (see [Kahrobaiyan et al 2011]). …”

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confidence: 99%

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2012

J. Mech. Mater. Struct.

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A generalized plane strain micromechanical model is developed to predict the stress and strain fields and overall elastic properties of a unidirectional fiber-reinforced composite subjected to various axial and transverse normal loading conditions using a least-squares-based differential quadrature element method (DQEM). The representative volume element (RVE) of the composite consists of a quarter of the fiber surrounded by matrix to represent the real composite with a repeating square array of fibers. The cubic serendipity shape functions are used to convert the solution domain to a proper rectangular domain and the new versions of the governing equations and boundary conditions are also derived. The fully bonded fiber-matrix interface condition is considered and the displacement continuity and traction reciprocity are imposed on the fiber-matrix interface. Application of DQEM to the problem leads to an overdetermined system of linear equations mainly due to the particular periodic boundary conditions of the RVE. A least-squares differential quadrature element method is used to obtain solutions for the governing partial differential equations of the problem. The numerical results are in excellent agreement with the available analytical and finite element studies. Moreover, the results of this study reveal that the presented model can provide highly accurate results with a very small number of elements and grid points within each element. In addition, the model shows advantages over conventional analytical models for fewer simplifying assumptions related to the geometry of the RVE.

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Section: Introductionsupporting

confidence: 66%

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confidence: 99%

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2012

J. Mech. Mater. Struct.

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“…Wang et al [2010], on the basis of strain gradient elasticity theory, developed a microscale Timoshenko beam model in which the Poisson effect was incorporated and also investigated the static bending and free vibration of a simply supported microscale Timoshenko beam to illustrate this model. Kahrobaiyan et al [2011] developed a nonlinear size-dependent Euler-Bernoulli beam model based on strain gradient theory.…”

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confidence: 99%

A nonlinear Timoshenko beam formulation based on strain gradient theory

Ansari

Gholami

Darabi

2012

J. Mech. Mater. Struct.

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Developed herein is a comprehensive geometrically nonlinear size-dependent microscale Timoshenko beam model based on strain gradient and von Kármán theories. The nonlinear governing equations and the corresponding boundary conditions are derived from employing Hamilton's principle. A simply supported microbeam is considered to delineate the nonlinear size-dependent free vibration behavior of the presented model. Utilizing the harmonic balance method, the solution for free vibration is presented analytically. The influence of the geometric parameters, Poisson's ratio, and material length-scale parameters on the linear frequency and nonlinear frequency ratio are thoroughly investigated. The results obtained from the present model are compared, in special cases, with those of the linear strain gradient theory, linear and nonlinear modified couple stress theory, and linear and nonlinear classical models; excellent agreement is found. It is concluded that the nonlinear natural frequency and nonlinear frequency ratio predicted by strain gradient theory are more precise than those from the other theories mentioned, especially for shorter beams.

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“…Using the gradient elasticity theory, a higher-order Euler-Bernoulli beam model was developed by Papargyri-Beskou et al (2003) and Kahrobaiyan et al (2011). Papargyri-Beskou and Beskos (2008) presented a Kirchhoff microplate and conducted the static, stability and dynamic analysis of gradient elastic flexural plates.…”

Section: Introductionmentioning

confidence: 99%

Size-Dependent Bending, Buckling and Free Vibration Analyses of Microscale Functionally Graded Mindlin Plates Based on the Strain Gradient Elasticity Theory

Ansari

Hasrati

Shojaei

et al. 2016

Lat. Am. j. solids struct.

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In this paper, a size-dependent microscale plate model is developed to describe the bending, buckling and free vibration behaviors of microplates made of functionally graded materials (FGMs). The size effects are captured based on the modified strain gradient theory (MSGT), and the formulation of the paper is on the basis of Mindlin plate theory. The presented model accommodates the models based upon the classical theory (CT) and the modified couple stress theory (MCST) if all or two scale parameters are set to zero, respectively. By using Hamilton's principle, the governing equations and related boundary conditions are derived. The bending, buckling and free vibration problems are considered and are solved through the generalized differential quadrature (GDQ) method. A detailed parametric and comparative study is conducted to evaluate the effects of length scale parameter, material gradient index and aspect ratio predicted by the CT, MCST and MSGT on the deflection, critical buckling load and first natural frequency of the microplate. The numerical results indicate that the model developed herein is significantly sizedependent when the thickness of the microplate is on the order of the material scale parameters.

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A nonlinear strain gradient beam formulation

Cited by 168 publications

References 40 publications

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A nonlinear Timoshenko beam formulation based on strain gradient theory

Size-Dependent Bending, Buckling and Free Vibration Analyses of Microscale Functionally Graded Mindlin Plates Based on the Strain Gradient Elasticity Theory

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