On the gradient strain elasticity theory of plates

Lazopoulos, K.A.

doi:10.1016/j.euromechsol.2004.04.005

Cited by 96 publications

(43 citation statements)

References 16 publications

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“…REZA ANSARI, RAHEB GHOLAMI AND MOHAMMAD ALI DARABI vibration behaviors of microstructures [Lazopoulos 2004;Park and Gao 2006;Kong et al 2008;Ma et al 2008;Asghari et al 2010b;2010a;Lazopoulos and Lazopoulos 2010]. Lazopoulos [2004] was apparently the first to employ the von Kármán kinematic relations in nonclassical analyses of beams and plates.…”

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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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mentioning

confidence: 99%

A nonlinear Timoshenko beam formulation based on strain gradient theory

Ansari

Gholami

Darabi

2012

J. Mech. Mater. Struct.

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“…vibration behaviors of microstructures [Lazopoulos 2004;Park and Gao 2006;Kong et al 2008;Ma et al 2008;2011;Asghari et al 2010b;2010a;Lazopoulos and Lazopoulos 2010]. Lazopoulos [2004] was apparently the first to employ the von Kármán kinematic relations in nonclassical analyses of beams and plates.…”

Section: Introductionmentioning

confidence: 99%

“…Lazopoulos [2004] was apparently the first to employ the von Kármán kinematic relations in nonclassical analyses of beams and plates. Mindlin [1965] proposed a higher-order gradient theory for elastic materials by considering the first and second derivatives of the strain tensor effective on the strain energy density.…”

Section: Introductionmentioning

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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“…Altan and Aifantis (1992) suggested a simplified strain gradient model including a single strain gradient coefficient of length squared dimension. Based on this model, Lazopoulos (2004) investigated the buckling behavior of a long rectangular plate subjected to uniaxial compression and small lateral load. Lam et al (2003) reduced five scale constants in strain gradient theory to three ones and presented the modified strain gradient theory.…”

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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On the gradient strain elasticity theory of plates

Cited by 96 publications

References 16 publications

A nonlinear Timoshenko beam formulation based on strain gradient theory

A nonlinear Timoshenko beam formulation based on strain gradient theory

Untitled

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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