Bending and vibration analysis of generalized gradient elastic plates

Xu, Xiaojian; Deng, Zichen; Zhang, Kai

doi:10.1007/s00707-014-1142-0

Cited by 22 publications

(5 citation statements)

References 39 publications

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“…Yaghoubi et al [154] developed a continuum model for medium-thick nanobeams based on this theory, and derived the governing equations, initial conditions, and boundary conditions using the variational principle. A mechanical model of static buckling and vibration in nano-Kirchhoff and Mindlin plates has been developed by Xu et al [155] using the Mindlin velocity gradient theory. It was found that the natural frequency ratio decreased with increasing gradient parameter, and when the dimensionless gradient parameter is small, the fundamental frequency ratio is close to 1, indicating that the gradient parameter has no significant effect on large-size plates.…”

Section: Velocity Gradientmentioning

confidence: 99%

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A review on the size-dependent bulking, vibration and, wave propagation of nanostructures

Wang

Zhao

et al. 2023

J. Phys.: Condens. Matter

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Size effect is a typical characteristic of micro-/nano-materials, which can contribute to a variety of size-dependent behaviors, phenomena, and properties, such as stiffness softening, deformation springback, etc. The intrinsic causes of size effects are micro-structural properties of materials, and the sensitivity of microstructural properties of materials is closely related to the smallest structural unit of the crystal, crystal defects and geometric dimensions, and is heavily influenced by the material’s field conditions. The modeling method based on nonlocal theory and gradient theory in the model is not only consistent with experimental and molecular dynamics simulation results, but also provides a solid explanation for the size effect underlying ”softening” and ”hardening” behaviors. Taking this as a basic point, this paper further considers the real working environment of materials, and systematically reviews the static and dynamic mechanical behavior cases of various nano-structures, mainly involving bulking, vibration and wave propagation of micro-beams and plates under different theories. A description and discussion of the differences in mechanical properties resulting from size effects under various theoretical frameworks and three key bottleneck problems are provided: the selection of kernel functions, the determination of size parameters, and the physical meaning of boundary conditions at higher orders. A summary is provided of the possible avenues and potentials for size effect models in future research. Many studies have shown that size parameters have a significant impact on the mechanical behavior of micro-/nano-structures, and these effects will increase as the size of the structure decreases. Nevertheless, different theories have varying scopes of application and size effects, and further research is needed to develop a unified size-dependent theory with universal applicability. A major focus of this paper is on the size effect of micro-/nano-structures, as well as provides the necessary data support to resolve the bottleneck problem associated with the size effect in the processing and manufacturing industries, and realizes the design and optimization of micro-scale parts based on their size.

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Section: Velocity Gradientmentioning

confidence: 99%

“…Based on the gradient theory with higher-order inertia effects proposed by Polizzotto [153], Xu et al [155] developed a nano-Euler beam model, which gives closed solutions for the free vibration frequencies of the beam under the conditions of clamped, simply-supported and cantilever boundary conditions.…”

Section: Velocity Gradientmentioning

confidence: 99%

A review on the size-dependent bulking, vibration and, wave propagation of nanostructures

Wang

Zhao

et al. 2023

J. Phys.: Condens. Matter

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“…For instance, the deflection of gradient elastic Kirchhoff plates [28,29] and Kirchhoff-Love cylindrical shells [33,34] requires C 2 -continuity. The deflection and rotations of gradient elastic Mindlin plates [31,49] require C 1 -continuity. Moreover, gradient elastic Reddy plates [32] require both the C 1 -continuity of rotation and the C 2 -continuity of deflection.…”

Section: Introductionmentioning

confidence: 99%

Size-Dependent Free Vibration of Non-Rectangular Gradient Elastic Thick Microplates

Zhang

et al. 2022

Symmetry

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The free vibration of isotropic gradient elastic thick non-rectangular microplates is analyzed in this paper. To capture the microstructure-dependent effects of microplates, a negative second-order gradient elastic theory with symmetry is utilized. The related equations of motion and boundary conditions are obtained using the energy variational principle. A closed-form solution is presented for simply supported free-vibrational rectangular microplates with four edges. A C1-type differential quadrature finite element (DQFE) is applied to solve the free vibration of thick microplates. The DQ rule is extended to the straight-sided quadrilateral domain through a coordinate transformation between the natural and Cartesian coordinate systems. The Gauss–Lobato quadrature rule and DQ rule are jointly used to discretize the strain and kinetic energies of a generic straight-sided quadrilateral plate element. Selective numerical examples are validated against those available in the literature. Finally, the impact of various parameters on the free vibration characteristics of annular sectorial and triangular microplates is shown. It indicates that the strain gradient and inertia gradient effects can result in distinct changes in both vibration frequencies and mode shapes.

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“…The existing two-dimensional non-local models for thin elastic plates are usually based on the above-mentioned differential constitutive relations (e.g. [20][21][22][23][24][25][26][27][28]). In these models, threedimensional → two-dimensional reduction is carried out using ad-hoc assumptions neglecting the variation of non-local properties across the thickness.…”

Section: Introductionmentioning

confidence: 99%

“…The existing 2D nonlocal models for thin elastic plates are usually based on the above mentioned differential constitutive relations, e.g., see Lu et al (2007), Duan and Wang (2007), Aghababaei and Reddy (2009), Pradhan and Phadikar (2009), Malekzadeh et al (2011), Xu et al (2014), Thai et al (2014), Jung and Han (2014), and Mousavi et al (2017). In these models, 3D → 2D reduction is carried out using ad-hoc assumptions neglecting the variation of nonlocal properties across the thickness.…”

Section: Introductionmentioning

confidence: 99%

A non-local asymptotic theory for thin elastic plates

2017

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The 3D dynamic nonlocal elasticity equations for a thin plate are subject to asymptotic analysis assuming the plate thickness to be much greater than a typical microscale size. The integral constitutive relations, incorporating the variation of an exponential nonlocal kernel across the thickness, are adopted. Long-wave low-frequency approximations are derived for both bending and extensional motions. Boundary layers specific for nonlocal behaviour are revealed near the plate faces. It is established that the effect of the boundary layers leads to first-order corrections to the bending and extensional stiffness in the classical 2D plate equations.

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Bending and vibration analysis of generalized gradient elastic plates

Cited by 22 publications

References 39 publications

A review on the size-dependent bulking, vibration and, wave propagation of nanostructures

A review on the size-dependent bulking, vibration and, wave propagation of nanostructures

Size-Dependent Free Vibration of Non-Rectangular Gradient Elastic Thick Microplates

A non-local asymptotic theory for thin elastic plates

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