Including Surface Stress Effects Research Articles

Theoretical prediction of proteins network-induced nonlocal response in molecules-resonator biosensor with hydrogen bonds including van der waals interactions

The present study investigates the adsorption-induced resonance frequency shift of a biomolecule-resonator system, considering the shear distortion effect, distributed adatoms, and small-scale effects using nonlocal elasticity theory. The adsorption-induced energy is modeled using a distributional approach for both the bio-receptor and spike protein. The dynamic behavior model for a microbeam resonator is derived, incorporating surface stress. The functional microbeam approach and the localized biomolecule approach are employed, along with van der Waals (vdW) interactions using the Lennard-Jones (6–12) potential to calculate the influence of all applied conditions. Explicit inertia moment and shear force are determined based on the nonlocal Timoshenko beam equations, with residual stress applied as an additive axial load. Numerical results demonstrate that the computed frequency shift depends on the active surface parameters, adsorbed adatoms, as well as the localized receptor and spike. The evaluation of results indicates that interatomic phenomena make the microsystem softer, emphasizing the importance of considering it in computations. Thus, the derived model is suitable for investigating the dynamic behavior of the biomolecule-resonator, applicable for determining both mass and density of spike and virus in the presence of adatom bonds.

Large-amplitude oscillations of composite conical nanoshells with in-plane heterogeneity including surface stress effect

In the current investigation, a new formulation for nonlinear vibration behaviors of functionally graded (FG) composite conical nanoshells are constructed using Gurtin-Murdoch elasticity theory based on higher-order shear deformation shell theory (HSDFST) framework. Both Voigt and Reuss homogenization procedures are considered for the estimation of the mechanical characteristics of FG materials. Using generalized differential quadrature method (GDQM) together with Galerkin technique, the surface elastic-based nonlinear frequency-responses of FG composite conical nanoshell are obtained. It has been illustrated that the decrease of material property gradient index or transformation of boundary condition from full simply supported to full clamped, surface stress effect on the nonlinear frequency of a FG composite conical nanoshell reduces. Also, decreasing semi-vertex angle increases the frequency ratio of ωNL/ωL which reveals higher geometrical nonlinearity. However, it is seen that surface elasticity effect on the nonlinear vibration behavior of FG composite conical nanoshells is not significant.

Surface elastic shell model for nonlinear primary resonant dynamics of FG porous nanoshells incorporating modal interactions

The surface to volume ratio becomes significant at nanoscale, so the surface stress plays an essential role in the mechanical responses of nanostructures. The focus of the present study is on an accurate analytical nonlinear primary resonance solution of functionally graded (FG) porous nanoshells under an external soft harmonic excitation including surface stress effects. To this end, the constitutive differential equations are adopted within the framework of the Gurtin-Murdoch theory of elasticity. In order to capture more accurate results, the nonlinear modal interactions between the main oscillation mode and higher symmetric vibration modes are taken into account. In addition, the mechanical properties of the FG porous nanoshell are estimated on the basis of the closed-cell Gaussian-Random field scheme. Thereafter, with the aid of the multiple time-scales technique, analytical expressions are expressed for the nonlinear surface elastic-based frequency-responses and amplitude-responses associated with the nonlinear primary resonance of FG porous nanoshells corresponding to various nonlinear modal interactions. It is demonstrated that by taking the modal interaction between higher symmetric vibration modes with the main oscillation mode into account, the frequency-response of FG porous nanoshells changes from the hardening behavior to the softening one. As a consequence, the modal interaction causes to shift the peak of the nonlinear resonance from the excitation frequency range of Ω/ω > 1 to Ω/ω < 1.

Nonlinear resonance responses of geometrically imperfect shear deformable nanobeams including surface stress effects

In this work, a thorough investigation is presented into the nonlinear resonant dynamics of geometrically imperfect shear deformable nanobeams subjected to harmonic external excitation force in the transverse direction. To this end, the Gurtin–Murdoch surface elasticity theory together with Reddy’s third-order shear deformation beam theory is utilized to take into account the size-dependent behavior of nanobeams and the effects of transverse shear deformation and rotary inertia, respectively. The kinematic nonlinearity is considered using the von Kármán kinematic hypothesis. The geometric imperfection as a slight curvature is assumed as the mode shape associated with the first vibration mode. The weak form of geometrically nonlinear governing equations of motion is derived using the variational differential quadrature (VDQ) technique and Lagrange equations. Then, a multistep numerical scheme is employed to solve the obtained governing equations in order to study the nonlinear frequency–response and force–response curves of nanobeams. Comprehensive studies into the effects of initial imperfection and boundary condition as well as geometric parameters on the nonlinear dynamic characteristics of imperfect shear deformable nanobeams are carried out through numerical results. Finally, the importance of incorporating the surface stress effects via the Gurtin–Murdoch elasticity theory, is emphasized by comparing the nonlinear dynamic responses of the nanobeams with different thicknesses.

Geometrically nonlinear free vibration and instability of fluid-conveying nanoscale pipes including surface stress effects

This paper is aimed to examine the geometrically nonlinear vibration and stability of nanoscale pipe conveying fluid incorporating surface stress effect. To approach this, the von-Karman hypothesis and Timoshenko beam theory are used to model the nanoscale pipe as a nonlinear Timoshenko nanobeam. Then, Hamilton’s principle and the Gurtin–Murdoch continuum elasticity are used to derive the governing equations of motion and associated boundary conditions incorporating the surface stress effect. Afterward, by the generalized differential quadrature method and harmonic balance method, the obtained nonlinear differential equations are discretized and simplified, before solving numerically through the Newton–Raphson method. The effects of the surface stress parameters on the stability and imaginary and real parts of frequency of nanopipes are discussed. Results are performed for nanopipes with different end supports made of silicon (Si) and aluminum (Al).

On the bending and stability of nanowire using various HSDTs

In this article, various higher-order shear deformation theories (HSDTs) are developed for bending and buckling behaviors of nanowires including surface stress effects. The most important assumption used in different proposed beam theories is that the deflection consists of bending and shear components and thus the theories have the potential to be utilized for modeling of the surface stress influences on nanowires problems. Numerical results are illustrated to prove the difference between the response of the nanowires predicted by the classical and non-classical solutions which depends on the magnitudes of the surface elastic constants.

Surface stress effects on the nonlinear postbuckling characteristics of geometrically imperfect cylindrical nanoshells subjected to axial compression

For structures at nanoscale, the surface effects can be important due to the high ratio of surface area to volume. In the current investigation, the nonlinear axial postbuckling behavior of geometrically imperfect cylindrical nanoshells is studied including surface stress effects. For this purpose, Gurtin–Murdoch continuum elasticity theory in conjunction with von Karman–Donnell-type geometric nonlinearity is implemented into the classical shell theory. By the developed size-dependent shell model, the surface effects which include surface elasticity and residual surface stress are taken into account. In order to satisfy balance conditions on the surfaces of nanoshell, a linear variation through the thickness is considered for the normal stress component of the bulk. Based on the variational approach using virtual work's principle, the non-classical governing differential equations are derived. In order to solve the nonlinear problem, a boundary layer theory is employed which contains simultaneously the nonlinear prebuckling deformations, initial geometric imperfections and large deflections corresponding to the postbuckling domain. Subsequently, a two-stepped singular perturbation methodology is utilized to predict the nonlinear critical buckling loads as well as the postbuckling equilibrium paths. It is observed that by taking surface stress effects into account, the both critical buckling load and critical end-shortening of a cylindrical nanoshell made of Silicon increase.

Surface stress effects on the nonlinear postbuckling characteristics of geometrically imperfect cylindrical nanoshells subjected to torsional load

The prime aim of the current investigation is to predict the nonlinear torsional buckling and postbuckling behavior of geometrically imperfect cylindrical nanoshells including surface stress effects. To this end, the size-dependent governing differential equations of cylindrical nanoshell based on von Karman–Donnell-type of kinematic nonlinearity are derived using a combination of Gurtin–Murdoch elasticity theory and the classical shell theory, and employing the principle of virtual work. Subsequently, by considering the transverse displacement and Airy stress function as independent variables, a boundary layer theory is put to use which takes surface stress effects into account in conjunction with the nonlinear prebuckling deformations, large postbuckling deflections and initial geometric imperfection. Finally, an efficient solution methodology based on a two-stepped singular perturbation technique is conducted to obtain the size-dependent postbuckling equilibrium paths of nanoshells. It is revealed that after the minimum point of postbuckling load, by increasing the value of applied torsional load, the difference between postbuckling load-deflection curves corresponding to the perfect and imperfect nanoshells tends to decrease.

Surface stress effects on the postbuckling behavior of geometrically imperfect cylindrical nanoshells subjected to combined axial and radial compressions

Because of surface free energy effects at nanoscale, study of the mechanical behavior of nanostructures including surface stress effects is a topic of substantial interest. Herein, the nonlinear buckling and postbuckling characteristics of geometrically imperfect cylindrical nanoshells under combined axial and radial compressive loads are investigated in the presence of surface stress effects. An efficient size-dependent shell model is proposed based on Gurtin–Murdoch elasticity theory and von Karman–Donnell-type of kinematic nonlinearity. On the basis of variational approach using the principle of virtual work, the non-classical governing differential equations are derived. Afterwards, a boundary layer theory is employed incorporating surface stress effects in conjunction with nonlinear prebuckling deformations, initial geometric imperfections and large postbuckling deflections. Then a two-stepped singular perturbation methodology is put to use in order to solve the size-dependent nonlinear problem corresponding to axial dominated and radial dominated loading cases. It is shown that in the case of radial dominated loading, the combination of hydrostatic pressure with axial compression causes to decrease approximately the effect of surface stress compared to the absence of axial load. However, for the axial dominated loading case, in comparison with no applied radial load, the surface stress effects is more significant in the presence of hydrostatic pressure.

On the postbuckling behavior of geometrically imperfect cylindrical nanoshells subjected to radial compression including surface stress effects

The main objective of the present study is to investigate the effect of surface stress on the nonlinear buckling and postbuckling behavior of cylindrical nanoshells with initial geometric imperfection subjected to radial compressive load. Gurtin–Murdoch elasticity theory is implemented into the classical shell theory to develop a size-dependent shell model which is capable to capture surface stress effects efficiently. In order to satisfy balance conditions on the surfaces of nanoshell, a linear variation through the thickness is considered for the normal stress component of the bulk. The principle of virtual work is put to use in order to formulate the non-classical governing differential equations. Afterwards, a boundary layer theory is employed including the nonlinear prebuckling deformations, initial geometric imperfection and large postbuckling deflections. Finally, a two-stepped singular perturbation methodology is utilized to obtain the size-dependent critical buckling loads and the postbuckling equilibrium paths of imperfect nanoshells corresponding to both lateral and hydrostatic pressure loading cases. It is found that for the positive and negative values of surface elastic constants, the both critical buckling load and critical end-shortening of nanoshell increase and decrease, respectively.

Nonlocal plate model for dynamic pull-in instability analysis of circular higher-order shear deformable nanoplates including surface stress effect

In the present paper, dynamic pull-in instability and free vibration characteristics of circular higher-order shear deformable nanoplates subjected to hydrostatic and electrostatic forces are studied including surface stress effect. For this purpose, Eringen’s nonlocal elasticity continuum in conjunction with the Gurtin-Murdoch elasticity theory is incorporated into the classical higher-order shear deformation plate theory to develop size-dependent plate model able to consider both of small scale and surface stress effects. The non-classical governing differential equations are then discretized along with simply supported and clamped edge supports by employing generalized differential quadrature (GDQ) method. To evaluate the size-dependent pull-in voltage of nanoplates, the hydrostatic-electrostatic actuation is assumed to be calculated by neglecting the fringing field effects and utilizing the parallel plate approximation. It is demonstrated that the pull-in instability occurs at lower voltages for nanoplates with higher values of nonlocal parameters. Moreover, it is found that surface stress effect can increase or decrease the pull-in voltage of nanoplates which depends on the sign of surface elastic constants.

Nonlinear Vibrational Analysis of Nanobeams Embedded in an Elastic Medium including Surface Stress Effects

Due to size-dependent behavior of nanostructures, the classical continuum models are not applicable for the analyses at this submicron size. Surface stress effect is one of the most important matters which make the nanoscale structures have different properties compared to the conventional structures due to high surface to volume ratio. In the present study, nonlinear free vibrational characteristics of embedded nanobeams are investigated including surface stress effects. To this end, a thin surface layer is assumed on the upper and lower surfaces of the cross section to separate the surface and bulk of nanobeams with their own different material properties. Based on harmonic balance method, closed-form analytical solution is conducted for nonlinear vibrations to obtain natural frequencies of embedded nanobeams with and without considerations of surface elasticity and residual surface tension effects corresponding to the various values of nondimensional amplitude, elastic foundation modulus, and geometrical variables of the system. Selected numerical results are given to indicate the influence of each one in detail.

A geometrically non-linear plate model including surface stress effect for the pull-in instability analysis of rectangular nanoplates under hydrostatic and electrostatic actuations

Presented herein is a comprehensive investigation on the size-dependent pull-in instability of geometrically non-linear rectangular nanoplates including surface stress effects undergoing hydrostatic and electrostatic actuations. To this end, based on the Gurtin–Murdoch theory, a non-classical continuum plate model capable of incorporating size-effects is developed; then, by means of the principle of virtual work, the governing equations of the actuated nanoplate are obtained. Subsequently, the generalized differential quadrature (GDQ) method is used to discretize the governing equations and associated boundary conditions, before solving numerically by the pseudo arc-length algorithm. Finally, the influences of important parameters including the geometrical non-linearity, thickness of the nanoplate, surface elastic modulus, residual surface stress and boundary conditions on the pull-in behavior of the actuated nanoplate are thoroughly studied. In addition, the effect of the material on the pull-in voltage and pressure is investigated by comparing the results obtained from nanoplates made of two different materials including aluminum (Al) and silicon (Si).

Surface stress effect on the pull-in instability of circular nanoplates

The prime aim of the current study is to investigate the pull-in instability characteristics of hydrostatically and electrostatically actuated circular nanoplates including surface stress effect. Gurtin–Murdoch elasticity theory is incorporated into the classical plate theory to develop a non-classical plate model considering surface stress effect. The governing size-dependent differential equations of a hydrostatically and electrostatically actuated circular nanoplate are discretized along with simply-supported and clamped boundary conditions using the generalized differential quadrature (GDQ) method. Selected numerical results are given to indicate the capability of the present size-dependent plate model for predicting the normalized pull-in voltage and pull-in hydrostatic pressure of nanoplates with inclusion surface stress effect. It is found that the pull-in phenomenon of actuated nanoplate is strongly size-dependent. The results indicate that surface stress effect plays a more significant role in the pull-in phenomenon of nanoplates of lower thicknesses. This investigation might be helpful to evaluate the mechanical characteristics of electrostatical actuators.

Surface stress effect on the postbuckling and free vibrations of axisymmetric circular Mindlin nanoplates subject to various edge supports

Postbuckling behavior of circular nanoplates and their free vibration characteristics in the vicinity of postbuckling domain are investigated with the consideration of surface stress effect. To this end, Mindlin’s plate theory in conjunction with the Gurtin–Murdoch elasticity theory is utilized to derive nonlinear equations of motion incorporating geometric nonlinearity and surface stress effect. On the basis of generalized differential quadrature (GDQ) method, the non-classical governing differential equations are discretized along simply-supported and clamped boundary conditions and are then parameterized and solved using the pseudo arc-length continuation method. The postbuckling configurations of axisymmetric circular nanoplates are obtained as a function of applied axial compressive load based on the static analysis. Afterward, on the basis of dynamic analysis, the natural frequencies and associated mode-shapes of circular nanoplates corresponding to both prebuckling and postbuckling domains are predicted including surface stress effect. It is revealed that by decreasing the magnitude of thickness, the surface stress effect on the postbuckling configurations of nanoplates becomes more prominent. Moreover, the effect of surface stress may shift the postbuckling domain to higher or lower applied axial loads, depending on the magnitude and sign of surface elastic constants. These anticipations are the same corresponding to various edge supports.

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 analysis of Timoshenko nanobeams including surface stress effect

A modified continuum model is developed to predict the postbuckling deflection of nanobeams incorporating the effect of surface stress. To have this problem in view, the classical Timoshenko beam theory in conjunction with the Gurtin–Murdoch elasticity theory is utilized to propose non-classical beam model taking surface stress effect into account. The geometrical nonlinearity is considered in the analysis using the von Karman assumption. By employing the principle of virtual work, the size-dependent governing differential equations and related boundary conditions are derived. On the basis of the shifted Chebyshev–Gauss–Lobatto grid points, the generalized differential quadrature (GDQ) method is adopted as an accurate, simple and computational efficient numerical solution to discretize the non-classical governing differential equations along with various end supports. Selected numerical results are worked out to demonstrate the nonlinear equilibrium paths of the postbuckling behavior of nanobeams corresponding to different values of beam thickness, buckling mode number, surface elastic constants, and various types of boundary conditions.

Surface Stress Effect on the Vibrational Response of Circular Nanoplates With Various Edge Supports

The classical continuum theory cannot be directly used to describe the behavior of nanostructures because of their size-dependent attribute. Surface stress effect is one of the most important size dependencies of structures at this submicron size, which is due to the high surface to volume ratio of nanoscale domain. In the present study, the nonclassical governing differential equation together with corresponding boundary conditions are derived using Hamilton's principle, into which the surface energies are incorporated through the Gurtin-Murdoch elasticity theory. The model developed herein contains intrinsic length scales to take the size effect into account and is used to analyze the free vibration response of circular nanoplates including surface stress effect. The generalized differential quadrature (GDQ) method is employed to discretize the governing size-dependent differential equation along with simply supported and clamped boundary conditions. The classical and nonclassical frequencies of circular nanoplates with various edge supports and thicknesses are calculated and are compared to each other. It is found that the influence of surface stress can be different for various circumferential mode numbers, boundary conditions, plate thicknesses, and surface elastic constants.

A sixth-order compact finite difference method for non-classical vibration analysis of nanobeams including surface stress effects

A non-classical model for the free vibrations of nanobeams accounting for surface stress effects is developed in this study. Based on Gurtin–Murdoch elasticity theory, the influence of surface stress is incorporated into the Euler–Bernoulli beam theory. A compact finite difference method (CFDM) of sixth order is employed for discretizing the non-classical governing differential equation to obtain the natural frequencies of nanobeams subject to different boundary conditions. To check the validity of the present numerical solution, based on an exact solution, an explicit formula for the fundamental frequency of simply-supported nanobeams is obtained. Good agreement between the results of exact and numerical solutions is achieved, confirming the validity and accuracy of the present numerical scheme. The comparison between the results generated by CFDM with those obtained by the conventional finite difference method (FDM) further reveals the advantages of the compact method over its classical counterpart. The influences of beam thickness, surface density, surface residual stress, surface elastic constants, and boundary conditions on the natural frequencies of nanobeams are also investigated. It is indicated that the effect of surface stress on the vibrational response of a nanobeam is dependent on its aspect ratio and thickness.

Surface Stress Effect on the Pull-In Instability of Hydrostatically and Electrostatically Actuated Rectangular Nanoplates With Various Edge Supports

This paper is aimed to investigate the size-dependent pull-in behavior of hydrostatically and electrostatically actuated rectangular nanoplates including surface stress effects based on a modified continuum model. To this end, based on the Gurtin–Murdoch theory and Hamilton's principle, the governing equation and corresponding boundary conditions of an actuated nanoplate are derived; the step-by-step linearization scheme and the differential quadrature (GDQ) method are used to discretize the governing equation and associated boundary conditions. The effects of the thickness of the nanoplate, surface elastic modulus and residual surface stress on the pull-in instability of the nanoplate are investigated. Plates made of two different materials including aluminum (Al) and silicon (Si) are selected to explain the variation of the pull-in voltage and pressure with respect to plate thickness.