Surface effects on nonlinear vibration of embedded functionally graded nanoplates via higher order shear deformation plate theory

In this paper, free vibration behavior of functionally nanoplate resting on a Pasternak linear elastic foundation is investigated. The study is based on third-order shear deformation plate theory with small scale effects and von Karman nonlinearity, in conjunction with Gurtin–Murdoch surface continuum theory. It is assumed that functionally graded (FG) material distribution varies continuously in the thickness direction as a power law function and the effective material properties are calculated by the use of Mori–Tanaka homogenization scheme. The governing and boundary equations, derived using Hamilton's principle are solved through extending the generalized differential quadrature method. Finally, the effects of power-law distribution, nonlocal parameter, nondimensional thickness, aspect of the plate, and surface parameters on the natural frequencies of FG rectangular nanoplates for different boundary conditions are investigated.     

On the forced vibration analysis of Timoshenko nanobeams based on the surface stress elasticity theory

This paper aims to present nonlinear forced vibration characteristics of nanobeams including surface stress effect. By considering the local geometrical nonlinearity based on von Karman relation, a new formulation of the Timoshenko beam model is developed through the Gurtin–Murdoch elasticity theory in which the effect of surface stress is incorporated. By using a variational approach on the basis of Hamilton’s principle, the size-dependent equations of motion and associated boundary conditions are obtained. The generalized differential quadrature (GDQ) method is employed to discretize the non-classical governing differential equations over the spatial domain by using the shifted Chebyshev–Gauss–Lobatto grid points. Subsequently, a Galerkin-based numerical approach is put to use in order to reduce the set of nonlinear equations into a time-varying set of ordinary differential equations of Duffing-type. In the next step, the time domain is discretized via spectral differentiation matrix operators which are defined based on the derivatives of a periodic base function. Finally, the pseudo arc-length method is employed to solve the resulting nonlinear parameterized algebraic equations. The frequency–response curves for forced vibration behavior of nanobeams including the effect of surface stress are predicted corresponding to various values of beam thickness, length to thickness ratio and surface elastic constants. It is revealed that by incorporating the surface stress effect, the maximum amplitude occurs at lower excitation frequencies and the wide of region of the response tends to decrease.     

Postbuckling characteristics of nanobeams based on the surface elasticity theory

In the present paper, an attempt is made to numerically investigate the postbuckling response of nanobeams with the consideration of the surface stress effect. To accomplish this, the Gurtin–Murdoch elasticity theory is exploited to incorporate surface stress effect into the classical Euler–Bernoulli beam theory. The size-dependent governing differential equations are derived and discretized along with various end supports by employing the principle of virtual work and the generalized differential quadrature (GDQ) method. Newton’s method is applied to solve the discretized nonlinear equations with the aid of an auxiliary normalizing equation. After solving the governing equations linearly, to obtain each eigenpair in the nonlinear model, the linear response is used as the initial value in Newton’s method. Selected numerical results are given to show the surface stress effect on the postbuckling characteristics of nanobeams. It is found that by increasing the thickness of nanobeams, the postbuckling equilibrium path obtained by the developed non-classical beam model tends to the one predicted by the classical beam theory and this anticipation is the same for all selected boundary conditions.     

Surface and nonlocal effects on the nonlinear free vibration of non-uniform nanobeams

The surface and nonlocal effects on the nonlinear flexural free vibrations of elastically supported non-uniform cross section nanobeams are studied simultaneously. The formulations are derived based on both Euler–Bernoulli beam theory (EBT) and Timoshenko beam theory (TBT) independently using Hamilton’s principle in conjunction with Eringen’s nonlocal elasticity theory. Green’s strain tensor together with von Kármán assumptions are employed to model the geometrical nonlinearity. The differential quadrature method (DQM) as an efficient and accurate numerical tool in conjunction with a direct iterative method is adopted to obtain the nonlinear vibration frequencies of nanobeams subjected to different boundary conditions. After demonstrating the fast rate of convergence of the method, it is shown that the results are in excellent agreement with the previous studies in the limit cases. The influences of surface free energy, nonlocal parameter, length of non-uniform nanobeams, variation of nanobeam width and elastic medium parameters on the nonlinear free vibrations are investigated.     

Pulsating fluid induced dynamic instability of visco-double-walled carbon nano-tubes based on sinusoidal strain gradient theory using DQM and Bolotin method

This research deals with the dynamic instability analysis of double-walled carbon nanotubes (DWCNTs) conveying pulsating fluid under 2D magnetic fields based on the sinusoidal shear deformation beam theory (SSDBT). In order to present a realistic model, the material properties of DWCNTs are assumed viscoelastic using Kelvin–Voigt model. Considering the strain gradient theory for small scale effects, a new formulation of the SSDBT is developed through the Gurtin–Murdoch elasticity theory in which the effects of surface stress are incorporated. The surrounding elastic medium is described by a visco-Pasternak foundation model, which accounts for normal, transverse shear and damping loads. The van der Waals interactions between the adjacent walls of the nanotubes are taken into account. The size dependent motion equations and corresponding boundary conditions are derived based on the Hamilton’s principle. The differential quadrature method in conjunction with Bolotin method is applied for obtaining the dynamic instability region. The detailed parametric study is conducted, focusing on the combined effects of the nonlocal parameter, magnetic field, visco-Pasternak foundation, Knudsen number, surface stress and fluid velocity on the dynamic instability of DWCNTs. The results depict that the surface stress effects on the dynamic instability of visco-DWCNTs are very significant. Numerical results of the present study are compared with available exact solutions in the literature. The results presented in this paper would be helpful in design and manufacturing of nano/micro mechanical systems in advanced biomechanics applications with magnetic field as a parametric controller.     

Influence of surface energy on the nanoindentation response of elastically-layered viscoelastic materials

In the context of integrated nonlinear viscoelastic contact mechanics, a nonlinear finite element model is developed to predict and analyze the quasistatic response of nanoindentation problems of an elastically-layered viscoelastic materials considering the surface elasticity effects. Effects of surface energy are accounted for by employing the Gurtin–Murdoch continuum model for surface elasticity. The linear viscoelastic response is modeled by the Schapery’s creep model with a Prony’s series to express the transient component in the creep compliance. The viscoelastic constitutive equations are cast into a recursive form that needs only the previous time increment rather than the entire strain history. To satisfy the contact constraints exactly, the Lagrange multiplier method is adopted to enforce the contact conditions into the system. The equilibrium indentation configuration is obtained through the Newton–Raphson iterative procedure. The developed model is verified then applied to investigate the quasistatic nanoindentation response of two different indentation problems with different geometry and loading conditions. Results show the significant effects of surface energy and viscoelasticity on the quasistatic nanoindentation response.     

Nonlocal nonlinear bending and free vibration analysis of a rotating laminated nano cantilever beam

In this article, we present the nonlocal, nonlinear finite element formulations for the case of nonuniform rotating laminated nano cantilever beams using the Timoshenko beam theory. The surface stress effects are also taken into consideration. Nonlocal stress resultants are obtained by employing Eringen’s nonlocal differential model. Geometric nonlinearity is taken into account by using the Green Lagrange strain tensor. Numerical solutions of nonlinear bending and free vibration are presented. Parametric studies have been carried out to understand the effect of nonlocal parameter and surface stresses on bending and vibration behavior of cantilever beams. Also, the effects of angular velocity and hub radius on the vibration behavior of the cantilever beam are studied.     

Analysis of nanocontact problems of layered viscoelastic solids with surface energy effects under different loading patterns

Surface stresses have a remarkable effect on nanocontact response of layered viscoelastic solids, especially under specific loading patterns. In the framework of nonlinear viscoelastic contact mechanics, a numerical model is developed to investigate the quasistatic nanocontact response of elastically layered viscoelastic solids under different loading patterns. The developed model accounts for surface energy effects by adopting the complete Gurtin–Murdoch surface elasticity model. The Schapery’s constitutive viscoelastic creep model is used for the stress, strain, and time relationships. The transient term in the creep compliance is expressed by Prony’s series. Frictionless contact condition is assumed throughout the contact interface. The equilibrium contact configuration, in which the contact constraints are exactly satisfied without any need for an appropriate value for the penalty parameter, is obtained by using the Lagrange multiplier method in the framework of the Newton–Raphson procedure. The developed model is applied to study and analyze the quasistatic nanocontact response of two different problems under different loading patterns. Results show the significant effect of the type of loading pattern and its rate on the nanocontact response of elastically layered viscoelastic solids.     

Dislocations and internal loading in a semi-infinite elastic medium with surface stresses

This paper presents analytical solutions for shear and opening dislocations in an elastic half-plane with surface stresses by using the Gurtin–Murdoch continuum theory of elastic material surfaces. The fundamental solutions corresponding to buried vertical and horizontal loads are also presented. Fourier integral transforms are used in the analysis. It is found that a characteristic length parameter that depends on the surface and bulk elastic moduli exists for this class of problems, and it represents the influence of surface stresses on the bulk elastic field. Selected numerical results are presented to demonstrate the influence of surface stresses on the bulk stress field. The fundamental solutions presented in this study can be used to develop boundary integral equation and other methods to analyze complicated fracture and boundary-value problems associated with nano-scale structures and soft elastic solids.     

The nonlinear vibration of an initially stressed laminated plate

Nonlinear partial differential equations of motion for a laminated plate in a general state of non-uniform initial stress are presented in various plate theories. This study uses Lo’s displacement field to derive the governing equations. The higher-order terms in Lo’s theory can be disregarded, to obtain the equations of simpler forms and even other theories for laminated plate. These nonlinear partial equations are transformed to ordinary nonlinear differential equations using the Galerkin method. The Runge–Kutta method is used to obtain the ratio of nonlinear frequency to linear frequency. The numerical solutions of an initially stressed laminate plate based on various plate theories obtained by the Galerkin and Runge–Kutta method are presented herein. Using these equations with various theories, the nonlinear vibration behavior of laminated plate is studied. The results show that apparent discrepancies exist among the various displacement fields, which indicates the transverse shear strain, normal strain and initial stress state have great effect on the vibration behavior of laminate plate under nonlinear vibration.     

An analytical study on the nonlinear free vibration of functionally graded nanobeams incorporating surface effects

Nonlinear free vibration of simply supported FG nanoscale beams with considering surface effects (surface elasticity, tension and density) and balance condition between the FG nanobeam bulk and its surfaces is investigated in this paper. The non-classical beam model is developed within the framework of Euler–Bernoulli beam theory including the von Kármán geometric nonlinearity. The component of the bulk stress, σ_zz, is assumed to vary cubically through the nanobeam thickness and satisfies the balance conditions between the FG nanobeam bulk and its surfaces. Accordingly, surface density is introduced into the governing equation of the nonlinear free vibration of FG nanobeams. The multiple scales method is employed as an analytical solution for the nonlinear governing equation to obtain the nonlinear natural frequencies of FG nanbeams. Several comparison studies are carried out to demonstrate the effect of considering the balance conditions on free nonlinear vibration of FG nanobeams. Lastly, the influences of the FG nanobeam length, volume fraction index, amplitude ratio, mode number and thickness ratio on the normalized nonlinear natural frequencies of the FG nanobeams are discussed in detail.     

Boundary element analysis of nanoinhomogeneities of arbitrary shapes with surface and interface effects

In this paper, a boundary element method (BEM) is proposed to analyze the stress field in nanoinhomogeneities with surface/interface effect. To consider this effect, the continuity conditions along the internal interfaces between the matrix and inhomogeneities are modeled by the well-known Gurtin–Murdoch constitutive relation. In the numerical analysis, the interface elastic moduli and the geometry of the nanoscale inhomogeneity are varied to show their influence on the induced stress field. The interaction between nanoscale inhomogeneities and the effect of different geometric shapes of inhomogeneities, including ellipse, triangle, and square are also investigated for different interface material parameters. It is shown that the elastic field can be greatly influenced by the interfacial energy and geometry of nanoscale inhomogeneities. The proposed BEM formulation is very general, including the complete Gurtin–Murdoch model and is further convenient for arbitrary shapes of inhomogeneity.     

Dynamic instability analysis of doubly clamped cylindrical nanowires in the presence of Casimir attraction and surface effects using modified couple stress theory

The dynamic instability of free-standing size-dependent nanowires by considering the Casimir forceand surface effects is investigated in the following research work. The study is carried out for nanosystemswith circular cross section and cylinder–plate geometry for which the governing equation of motion is derivedbased on the Gurtin–Murdoch model and modified couple stress theory. Two methods including the proximityforce approximation for small separations and Dirichlet asymptotic approximation for large separations areutilized to formulate the Casimir attraction of a free-standing cylinder–plate geometry. To solve the complexnonlinear problem faced in this work, a stepwise numerical procedure is developed and the effects of lengthscale parameter, surface energy and vacuum fluctuations on the dynamic instability and adhesion time ofnanowires are studied. Based on the obtained results, the phase portrait of Casimir-induced nanowires showsperiodic and homoclinic orbits.     

Free vibration analysis of double-bonded isotropic piezoelectric Timoshenko microbeam based on strain gradient and surface stress elasticity theories under initial stress using differential quadrature method

The size-dependent effect on free vibration of double-bonded isotropic piezoelectric Timoshenko microbeams using strain gradient and surface stress elasticity theories under initial stress is presented. This article is developed for isotropic piezoelectric material. Due to the high surface-to-volume ratio, surface stress has an important role with micro- and nanoscale materials. Thus, the Gurtin–Murdoch continuum mechanic approach is used. Governing equations of motion are derived by Hamilton's principle and solved by the differential quadrature method. The effects of pre-stress load, surface residual stress, surface mass density, surface piezoelectrics, Young's modulus of surface layers, three material length scale parameters, thickness to material length scale parameter ratios, various boundary conditions, and two elastic foundation coefficients are investigated. It is concluded that the effect of pre-stress load in greater modes is negligible for higher aspect ratios and this effect is similar to lower aspect ratios. Also, the size-dependent effect on the dimensionless natural frequency for strain gradient theory is higher than that for modified couple stress theory and classical theory, which is due to increasing stiffness of the Timoshenko microbeam model. Moreover, the results show that dimensionless natural frequency affects more by considering the material length scale parameters with respect to surface effect. The results are compared with the obtained results from the literature and show good agreement between them. It is concluded that the amplitude of the transverse displacements (w₀) for a microbeam (MB) is more than the transverse displacements (w₁) for a piezoelectric microbeam (PMB). On the other hand, using a piezoelectric layer for PMB, the amplitude of the transverse displacements (w₁) reduces considerably with respect to MB, in which this effect leads to increase the stiffness of the microbeam and stability of microstructures. With considering the piezoelectric layer, the obtained results can be used to control the amplitude and vibration of microstructures, prevent the resonance phenomenon, design smart structures, and can be employed for micro-electro-mechanical systems and nano-electro-mechanical systems.     

Crack-like grain-boundary sliding wedges in nano-films

Time-dependent grain-boundary (GB) sliding in a nano-film on a substrate is studied. The Gurtin–Murdoch surface elasticity is incorporated onto the GB by ignoring the residual surface tension, and the GB is allowed to slide by the diffusion-controlled mechanism. The sliding viscosity can be homogeneous or heterogeneous along the GB. By using the Green's function method, the original boundary value problem is reduced to a Cauchy singular integro-differential equation of the first order which can be further changed into state-space equations by means of an adapted collocation method. The state-space equations can be solved through a rigorous eigenmode analysis. We demonstrate that such a sliding process leads to the formation of a crack-like GB wedge which causes the shear stress along the GB to decay exponentially with time. Furthermore, the stress field exhibits the weak logarithmic singularity at the tip of the sliding wedge due to the incorporation of surface elasticity. Detailed numerical results are obtained for eigenvalues, eigenfunctions, the transient shear stress along the GB, the opening displacement of the wedge and the strength of the logarithmic singularity at the wedge tip.     

Multiple circular nano-inhomogeneities and/or nano-pores in one of two joined isotropic elastic half-planes

The paper considers the problem of multiple interacting circular nano-inhomogeneities or/and nano-pores located in one of two joined, dissimilar isotropic elastic half-planes. The analysis is based on the solutions of the elastostatic problems for (i) the bulk material of two bonded, dissimilar elastic half-planes and (ii) the bulk material of a circular disc. These solutions are coupled with the Gurtin and Murdoch model of material surfaces [Gurtin ME, Murdoch AI. A continuum theory of elastic material surfaces. Arch Ration Mech Anal 1975;57:291–323; Gurtin ME, Murdoch AI. Surface stress in solids. Int J Solids Struct 1978;14:431–40.]. Each elastostatic problem is solved with the use of complex Somigliana traction identity [Mogilevskaya SG, Linkov AM. Complex fundamental solutions and complex variables boundary element method in elasticity. Comput Mech 1998;22:88–92]. The complex boundary displacements and tractions at each circular boundary are approximated by a truncated complex Fourier series, and the unknown Fourier coefficients are found from a system of linear algebraic equations obtained by using a Taylor series expansion. The resulting semi-analytical method allows one to calculate the elastic fields everywhere in the half-planes and inside the nano-inhomogeneities. Numerical examples demonstrate that (i) the method is effective in solving the problems with multiple nano-inhomogeneities, and (ii) the elastic response of a composite system is profoundly influenced by the sizes of the nano-features.     

Nonlinear vibration and dynamic response of shear deformable laminated plates in hygrothermal environments

This paper deals with hygrothermal effects on the nonlinear vibration and dynamic response of shear deformable laminated plates. The temperature field considered is assumed to be a uniform distribution over the plate surface and through the plate thickness. The material properties of the composite are affected by the variation of temperature and moisture, and based on a micro-mechanical model. The formulations are based on higher-order shear deformation plate theory and general von Kármán-type equation of motion, which includes hygrothermal effects. The equations of motion are solved by an improved perturbation technique to determine nonlinear frequencies and dynamic responses of shear deformable antisymmetric angle-ply and unsymmetric cross-ply laminated plates. The numerical illustrations concern the nonlinear vibration and dynamic response of the shear deformable laminated plates under different sets of hygrothermal environmental conditions. Effects of temperature rise, the degree of moisture concentration, and fiber volume fraction on natural frequencies, nonlinear to linear frequency ratios and dynamic responses are studied.     

Nonlinear analysis of forced vibration of nonlocal third-order shear deformable beam model of magneto–electro–thermo elastic nanobeams

This paper deals with the forced vibration behavior of nonlocal third-order shear deformable beam model of magneto–electro–thermo elastic (METE) nanobeams based on the nonlocal elasticity theory in conjunction with the von Kármán geometric nonlinearity. The METE nanobeam is assumed to be subjected to the external electric potential, magnetic potential and constant temperature rise. Based on the Hamilton principle, the nonlinear governing equations and corresponding boundary conditions are established and discretized using the generalized differential quadrature (GDQ) method. Thereafter, using a Galerkin-based numerical technique, the set of nonlinear governing equations is reduced into a time-varying set of ordinary differential equations of Duffing type. The pseudo-arc length continuum scheme is then adopted to solve the vectorized form of nonlinear parameterized equations. Finally, a comprehensive study is conducted to get an insight into the effects of different parameters such as nonlocal parameter, slenderness ratio, initial electric potential, initial external magnetic potential, temperature rise and type of boundary conditions on the natural frequency and forced vibration characteristics of METE nanobeams.     

Saint-Venant torsion of a circular bar with radial cracks incorporating arbitrarily varied surface elasticity

We analytically investigate the contribution of arbitrarily varied surface elasticity to the Saint-Venant torsion problem of a circular cylinder containing a radial crack. The varied surface elasticity is incorporated by using a modified version of the continuum-based surface/interface model of Gurtin and Murdoch. In our discussion, the surface shear modulus is assumed to be arbitrarily varied along the crack surfaces. Both internal and edge cracks are studied. By using Green's function method, the boundary value problem is reduced to the Cauchy singular integro-differential equation of first order, which can be numerically solved by using the Gauss–Chebyshev integration formula, the Chebyshev polynomials, and the collocation method. The torsion problem of a cylinder containing two symmetric collinear radial cracks of equal length with symmetrically varied surface elasticity is also solved by using a similar method. Our numerical results indicate that the variation of the surface elasticity exerts a significant influence on the strengths of the logarithmic stress singularity at the crack tips, the torsional rigidity, and the jump in warping function.     

热环境下功能梯度材料板的自由振动和动力响应

基于Reddy高阶剪切变形理论和广义Kármán型方程,用双重Fourier级数展开法求得了四边简支功能梯度材料板的自由振动及动力响应的解析解,分析中考虑了热传导以及组分材料热物参数对温度变化的依赖性,讨论了材料组分指数和热环境对固有频率及动力响应的影响。