Small scale effect on the free vibration of orthotropic arbitrary straight-sided quadrilateral nanoplates

As a first endeavor, the free vibration of orthotropic arbitrary straight-sided quadrilateral nanoplates is investigated using the nonlocal elasticity theory. The formulation is derived based on the first order shear deformation theory (FSDT). The solution procedure is based on the transformation of the governing equations from physical domain to computational domain and then discretization of the spatial derivatives by employing the differential quadrature method (DQM) as an efficient and accurate numerical tool. The formulation and the method of the solution are firstly validated by carrying out the comparison studies for the isotropic and orthotropic rectangular plates against existing results in literature. Then, the effects of nonlocal parameter in combination with the geometrical shape parameters, thickness-to-length ratio and the boundary conditions on the frequency parameters of the nanoplates are investigated.     

Nonlocal free vibration of orthotropic non-prismatic skew nanoplates

The free vibration of orthotropic non-prismatic skew nanoplate based on the first-order shear deformation theory (FSDT) in conjunction with Eringen’s nonlocal elasticity theory is presented. As a simple, accurate and low computational effort numerical method, the differential quadrature method (DQM) is employed to solve the related differential equations. For this purpose, after deriving the equations of motion and the related boundary conditions, they are transformed from skewed physical domain to rectangular computational domain of DQM and accordingly discretized. After validating the formulation and method of solution, the effects of nonlocal parameter in combination with geometrical parameters and boundary conditions on the natural frequencies of the orthotropic skew nanoplates are investigated.     

An efficient numerical method for analyzing the thermal effects on the vibration of embedded single-walled carbon nanotubes based on the nonlocal shell model

Employing the variational differential quadrature (VDQ) method, the effects of initial thermal loading on the vibrational behavior of embedded single-walled carbon nanotubes (SWCNTs) based on the nonlocal shell model are studied. According to the first-order shear deformation theory and considering Eringen's nonlocal elasticity theory, the energy functionality of the system is presented and discretized using the VDQ method. The effects of thermal loading and elastic foundation are simultaneously taken into account. The use of the numerical discretization technique in the context of variational formulation reduces the order of differentiation in the governing equations and consequently improves the convergence rate. The accuracy of the present model is first checked by comparison with molecular dynamics simulation results and those of other methods. The effects of involved parameters are then investigated on the fundamental frequencies of thermally preloaded embedded SWCNTs. The results imply that the thermal loading has a significant effect on the vibration analysis of embedded SWCNTs.     

A peridynamic Reissner-Mindlin shell theory

In this work, we have developed a state-based peridynamics theory for nonlinear Reissener-Mindlin shells to model and predict large deformation of shell structures with thick wall. The nonlocal peridynamic theory of solids offers an integral formulation that is an alternative to traditional local continuum mechanics models based on partial differential equations. This formulation is applicable for solving the material failure problems involved in discontinuous displacement fields. The governing equations of the state-based peridynamic shell theory are derived based on the nonlocal balance laws by adopting the kinematic assumption of the Reissner and Mindlin plate and shell theories. In the numerical calculations, the stress points are employed to ensure the numerical stability. Several numerical examples are conducted to validate the nonlocal structure mechanics model and to verify the accuracy as well as the convergence of the proposed shell theory.     

A VDQ-based multifield approach to the 2D compressible nonlinear elasticity

A numerical multifield methodology is developed to address the large deformation problems of hyperelastic solids based on the 2D nonlinear elasticity in the compressible and nearly incompressible regimes. The governing equations are derived using the Hu-Washizu principle, considering displacement, displacement gradient, and the first Piola-Kirchhoff stress tensor as independent unknowns. In the formulation, the tensor form of equations is replaced by a novel matrix-vector format for computational purposes. In the solution strategy, based on the variational differential quadrature (VDQ) technique and a transformation procedure, a new numerical approach is proposed by which the discretized governing equations are directly obtained through introducing derivative and integral matrix operators. The present method can be regarded as a viable alternative to mixed finite element methods because it is locking free and does not involve complexities related to considering several DOFs for each element in the finite element exterior calculus. Simple implementation is another advantage of this VDQ-based approach. Some well-known examples are solved to demonstrate the reliability and effectiveness of the approach. The results reveal that it has good performance in the large deformation problems of hyperelastic solids in compressible and nearly incompressible regimes.     

Decoupling the nonlocal elasticity equations for three dimensional vibration analysis of nano-plates

In this paper, a three dimensional vibration analysis of nano-plates is studied by decoupling the field equations of Eringen theory. Considering the small scale effect, the three dimensional equations of nonlocal elasticity are obtained. At first, three decoupled equations in terms of displacement components and three decoupled equations in terms of rotation components are obtained. In order to find the solution for a nano-plate based on the presented formulation, one of the three equations in terms of displacement components and corresponding rotation equation should be solved independently. Using some relations, the other two displacement components can be obtained in terms of the mentioned displacement and rotation component. A Navier-type method for finding the exact three dimensional solution of a nano-plate is presented using the Fourier series technique. Exact natural frequencies of nano-plates are presented and compared with the results of nonlocal first order and third order shear deformation theories.     

Vibration analysis of orthotropic graphene sheets using nonlocal elasticity theory and differential quadrature method

The small scale effect on the vibration analysis of orthotropic single layered graphene sheets (SLGS) is studied. Elastic theory of the graphene sheets is reformulated using the nonlocal differential constitutive relations of Eringen. The equations of motion of the nonlocal theories are derived for the graphene sheets. Differential quadrature method (DQM) is employed to solve the governing differential equations for various boundary conditions. Nonlocal theories are employed to bring out the small scale effect of the nonlocal parameter on the natural frequencies of the orthotropic graphene sheets. Further, effects of (i) nonlocal parameter, (ii) size of the graphene sheets, (iii) material properties and (iv) boundary conditions on nondimensional vibration frequencies are investigated.     

Nonlocal static analysis of a functionally graded material curved nanobeam

This article is concerned with the analytical solution for a curved nanobeam based on nonlocal elasticity. The structure is made of functionally graded (FG) material, and its property varies in accordance with a power law function through the thickness. To obtain the displacement function, the static differential equations for a curved FG beam are combined with the nonlocal Eringen stress equations. By using the direct method for solving the nonlocal force–strain and moment–curvature relations covering the distributed loads, the explicit expressions of nonlocal strains are achieved. The strain-displacement relations are also employed to find displacement field. Numerical examples with different types of boundary conditions are carried out in order to investigate the effects of nonlocal parameters, the nonhomogeneity index, and geometric characteristics.     

Nonlocal,strain gradient and surface effects on vibration and instability of nanotubes conveying nanoflow

ABSTRACT

In this paper, the size-dependent vibration and instability of nanoflow-conveying nanotubes with surface effects using nonlocal strain gradient theory (NSGT) are examined. Hence, based on Gurtin-Murdoch theory, the nonclassical governing equations are derived by extended Hamilton's principle. To study the small-size effects on the flow field, the Knudsen number is applied. Applying Galerkin's approach, the partial differential equations converted to ordinary differential equations. The effects of the main parameters like nonlocal and strain gradient parameters, length to diameter ratio, thickness, surface effects, Knudsen number and different boundary conditions on the eigenvalue and critical fluid velocity of the nanotube are explained.     

Vibration characteristics of embedded multi-layered graphene sheets with different boundary conditions via nonlocal elasticity

A nonlocal elastic plate model accounting for the small scale effects is developed to investigate the vibrational behavior of multi-layered graphene sheets under various boundary conditions. Based upon the constitutive equations of nonlocal elasticity, derived are the Reissner–Mindlin-type field equations which include the interaction of van der Waals forces between adjacent and non-adjacent layers and the reaction from the surrounding media. The set of coupled governing equations of motion for the multi-layered graphene sheets are then numerically solved by the generalized differential quadrature method. The present analysis provides the possibility of considering different combinations of layerwise boundary conditions in a multi-layered graphene sheet. Based on exact solution, explicit expressions for the nonlocal frequencies of a double-layered graphene sheet with all edges simply supported are also obtained. The results from the present numerical solution, where possible, are indicated to be in excellent agreement with the existing data from the literature.     

A mixed finite element for the nonlinear analysis of in-plane loaded masonry walls

A mixed membrane eight-node quadrilateral finite element for the analysis of masonry walls is presented. Assuming that a nonlinear and history-dependent 2D stress-strain constitutive law is used to model masonry material, the element derivation is based on a Hu-Washizu variational statement, involving displacement, strain, and stress fields as primary variables. As the behavior of masonry structures is often characterized by strain localization phenomena, due to strain softening at material level, a discontinuous, piecewise constant interpolation of the strain field is considered at element level, to capture highly nonlinear strain spatial distributions also within finite elements. Newton's method of solution is adopted for the element state determination problem. For avoiding pathological sensitivity to the finite element mesh, a novel algorithm is proposed to perform an integral-type nonlocal regularization of the constitutive equations in the present mixed formulation. By the comparison with competing serendipity displacement-based formulation, numerical simulations prove high performances of the proposed finite element, especially when coarse meshes are adopted.     

Small scale effect on the buckling of single-layered graphene sheets under biaxial compression via nonlocal continuum mechanics

In this article, the small scale effect on the buckling analysis of biaxially compressed single-layered graphene sheets (SLGS) is studied using nonlocal continuum mechanics. The nonlocal mechanics accounts for the small size effects when dealing with nano size elements such as graphene sheets. Using the principle of virtual work the governing equations are derived for rectangular nanoplates. Solutions for buckling loads are computed using differential quadrature method (DQM). It is shown that the nonlocal effect is quite significant in graphene sheets and has a decreasing effect on the buckling loads. When compared with uniaxially compressed graphene, the biaxially compressed one show lower influence of nonlocal effects for the case of smaller side lengths and larger nonlocal parameter values. This difference in behavior between uniaxial and biaxial compressions decreases as the size of the graphene sheets increases.     

Variational Principle for the Temperature Problem of the Elasticity Theory for Stresses

The variational formulation of a problem is frequently used in strength analysis of parts and structural elements. The authors present a variational principle whose Euler's equations are the differential equations of thermoelasticity for stresses.     

Unified finite element analysis

A special matrix is introduced, the elements of which are zero or first-order differential operators. This matrix is used to define a boundary value problem which covers a wide class of engineering applications. An equivalent variational formulation is found using an extension of the Gauss theorem. From this variational principle the equations of the elements are derived. The unified formulae presented here can be useful for educational purposes and for the design of a finite element system for total analysis.     

A unified nonlocal formulation for bending,buckling and free vibration analysis of nanobeams

Abstract

A unified nonlocal formulation is developed for the bending, buckling, and vibration analysis of nanobeams. Theoretical formulations of eighteen nonlocal beam theories are presented by using unified formulation. Small scale effect is considered based on the nonlocal differential constitutive relations of Eringen. The governing equations of motion and associated boundary conditions of the nanobeam are derived using Hamilton's principle. Closed form solutions are presented for a simply supported boundary condition using Navier's solution technique. Numerical results for axial and transverse shear stress are first time presented in this study which will serve as a benchmark for the future research.     

Small scale effect on the thermal buckling of orthotropic arbitrary straight-sided quadrilateral nanoplates embedded in an elastic medium

As a first endeavor, the small scale effect on the thermal buckling characteristic of orthotropic arbitrary straight-sided quadrilateral nanoplates embedded in an elastic medium is investigated. The surrounding elastic medium is modeled as the two-parameter elastic foundation. The formulation is derived using the classical plate theory (CPT) in conjunction with the nonlocal elasticity theory. The solution procedure is based on the transformation of the governing equations from physical domain to computational domain and then discretization of the spatial derivatives by employing the differential quadrature method (DQM) as an efficient and accurate numerical tool. The fast rate of convergence of the method is shown and the results are compared against existing results in literature. Then, the influence of small scale parameter in combination with the elastic medium parameters, geometrical shape and the boundary conditions on the thermal buckling load of the nanoplates is investigated.     

On nonlocal residuals and fluctuations

The nonlocal residual is a novel physical quantity introduced in the nonlocal field theory of mechanics. In this paper, the nonlocal residual and some related problems are discussed. Firstly, a representative theorem of nonlocal residual is proved, in which the relation between the nonlocal residual and the spatial distributed fluctuation of the interaction among microstructures in materials is established. The existence of nonlocal residuals of body force, body moment and energy is investigated in detail based on the objectivity of the balance equations. To meet the requirements in physics, an eigen-scale parameter is introduced into the nonlocal kernel. And the properties of nonlocal kernel are then discussed. Finally, the nonlocal hyperelastic constitutive equation is deduced through the representation of the nonlocal residual of energy. Results show that the nonlocality of hyperelastic constitutive equation comes directly from the interaction potential among microstructures within materials.     

Thermo-mechanical vibration of a single-walled carbon nanotube embedded in an elastic medium based on nonlocal elasticity theory

A single-elastic beam model has been developed to analyze the thermal vibration of single-walled carbon nanotubes (SWCNT) based on thermal elasticity mechanics, and nonlocal elasticity theory. The nonlocal elasticity takes into account the effect of small size into the formulation. Further, the SWCNT is assumed to be embedded in an elastic medium. A Winkler-type elastic foundation is employed to model the interaction of the SWCNT and the surrounding elastic medium. Differential quadrature method is being utilized and numerical solutions for thermal-vibration response of SWCNT is obtained. Influence of nonlocal small scale effects, temperature change, Winkler constant and vibration modes of the CNT on the frequency are investigated. The present study shows that for low temperature changes, the difference between local frequency and nonlocal frequency is comparatively high. With embedded CNT, for soft elastic medium and larger scale coefficients (e₀a) the nonlocal frequencies are comparatively lower. The nonlocal model-frequencies are always found smaller than the local model-frequencies at all temperature changes considered.     

A size-dependent nonlinear Timoshenko microbeam model based on the strain gradient theory

The geometrically nonlinear governing differential equations of motion and the corresponding boundary conditions are derived for the mechanical analysis of Timoshenko microbeams with large deflections, based on the strain gradient theory. The variational approach is employed to achieve the formulation. Hinged-hinged beams are considered as an important practical case, and their nonlinear static and free-vibration behaviors are investigated based on the derived formulation.