Two-phase Elasticity Research Articles

On the local/nonlocal piezoelectric nanobeams: Vibration, buckling, and energy harvesting

Based on a paradox-free nonlocal theory—two-phase local/nonlocal elasticity—vibration, buckling, and energy harvesting of piezoelectric nanobeams are investigated for the first time. By the means of the differential form of two-phase elasticity and Hamilton's principle, governing equations and boundary conditions are obtained. The exact solution as well as a numerical solution, Generalized Differential Quadrature Method (GDQM), are presented to extract results. Also, for the sake of obtaining equations for the forced vibration and energy harvesting analysis, the Galerkin method is utilized to discretize the governing equation. Given the fact that the differential nonlocal elasticity is not able to apply the size dependency on uniform loads, for the first time, the size-dependent piezoelectric load is taken into account through the two-phase elasticity. Also, vibration and energy harvesting of a clamped free nanobeam – which is a really good case for harvesting energy and cannot be accurately studied by differential nonlocal – are investigated employing the two-phase elasticity. To validate the present formulation and solution procedures, several comparison studies are conducted. Comparison between the common differential nonlocal elasticity and two-phase theory reveals that differential nonlocal elasticity is incompetent to yield reliable results for studying the vibration and energy harvesting of piezoelectric-based materials. Therefore, to study the mechanics of piezoelectric nano structures, other nonlocal theories such as two-phase local/nonlocal elasticity should be used. This paper can be a useful basis to investigate the vibration, buckling, and energy harvesting of nano piezoelectric devices and to improve their design.

Dynamic stability and vibration of two-phase local/nonlocal VFGP nanobeams incorporating surface effects and different boundary conditions

In the present study, it is shown that the fully nonlocal elasticity is failed to analyze the dynamic stability of Timoshenko nanobeams subjected to an axial load and, to solve this problem, for the first time, the two-phase local/nonlocal elasticity, as a paradox free form of nonlocal elasticity, is utilized to study the size-dependent dynamic stability and damping vibration of Viscoelastic Functionally Graded Porous (VFGP) Timoshenko nanobeams incorporating surface effects. Firstly, the governing equations, in presence of the axial and transverse displacements, are obtained through the Hamilton's principle. Next, to investigate the vibration and dynamic stability of nanobeams with different boundary conditions, the Generalized Differential Quadrature Method (GDQM) as well as Bolotin's method are utilized. After validation of the present results and formulation, to examine the influences of different parameters such as local phase fraction factor, nonlocal parameter, damping factor, FG index, volume fraction of porosities and surface effects in different boundary conditions, various benchmark results are presented. Furthermore, it is indicated that, against the fully nonlocal theory, using two-phase elasticity makes it possible to study the size dependent vibration and stability for several boundary conditions of Timoshenko nanobeams which are subjected to axial load.

Vibration of two-phase local/nonlocal Timoshenko nanobeams with an efficient shear-locking-free finite-element model and exact solution

According to paradoxical behaviors of common differential nonlocal elasticity, employing the two-phase local/nonlocal elasticity, to consider the size effects of nanostructures, has recently attracted the attentions of nano-mechanics researchers. Now, due to more complexity of the two-phase elasticity problems than the differential nonlocal ones, it is essential to achieve efficient methods for studying the mechanical characteristics of two-phase nanostructures. Therefore, in this work, the exact solution corresponding to the vibrations of two-phase Timoshenko nanobeams is provided for the first time. Furthermore, the shear-locking problem is investigated in the case of two-phase finite-element method (FEM), and since the FE model of local/nonlocal nanobeam is more complex than the classic one, due to coupling of all elements together, one of the main aims of the present work is to create an efficient locking-free local/nonlocal FEM with a simple and efficient beam element. To extract the exact natural frequencies, the basic form of two-phase elasticity is replaced with the equal differential equation and the obtained higher-order governing equations are solved by satisfying additional constitutive boundary conditions. To construct the two-phase FE model, an efficient and simple shear-locking-free Timoshenko beam element is introduced, and next, basic form of two-phase elasticity including local and integral form of nonlocal elasticity is utilized. No need for satisfying higher-order boundary conditions, shear-locking-free, simple shape functions and well convergence are advantages of the present two-phase finite element model. Several convergence and comparison studies are conducted, and the reliability and locking-free properties of the present two-phase finite element model are confirmed. Also, the influences of two-phase elasticity on the natural frequencies of Timoshenko nanobeams with different thickness ratios are studied.

Nonlinear vibration analysis of two-phase local/nonlocal nanobeams with size-dependent nonlinearity by using Galerkin method

It has been proved that using pure nonlocal elasticity, especially in differential form, leads to inconsistent and unreliable results. Therefore, to obviate these weaknesses, Eringen’s two-phase local/nonlocal elasticity has been recently used by researchers to consider the nonlocal size dependency of nanostructures. Given this, for the first time, the size-dependent nonlinear free vibration of nanobeams is investigated in this article within the framework of two-phase elasticity by using the Galerkin method. Contrary to differential nonlocal elasticity, the size dependency of the axial tension force, due to von Karman nonlinearity, is considered, and its effect on the nonlinear vibration is examined. The correct procedure of using the Galerkin method for studying the nonlinear vibration of two-phase nanobeams is introduced. It is shown that, although it is possible to extract the linear mode shapes of two-phase nanobeams using its equal differential equation, integral form of two-phase elasticity should be considered in the Galerkin method. Furthermore, it is observed that in two-phase elasticity, applying classic mode shapes in the Galerkin method leads to significant errors, especially in higher nonlocal parameters and lower values of local phase fraction and amplitude ratios.

Vibration analysis of mass nanosensors with considering the axial-flexural coupling based on the two-phase local/nonlocal elasticity

The influences of considering coupled axial-flexural vibrations on efficiency of cantilever mass nanosensors, modeled by Rayleigh and Timoshenko beam theories, are investigated in the frame work of a paradox-free nonlocal theory, i.e. two-phase local/nonlocal elasticity. However, the effects of axial-flexural coupling due to eccentricity of attached mass have been ignored in previous studies on mass nanosensors. Governing equations, boundary conditions, and corresponding compatibility conditions are derived by means of Hamilton’s principle and differential law of two-phase elasticity. Next, The Generalized Differential Quadrature Method (GDQM) as well as Generalized Integral Quadrature Method (GIQM) are utilized to attain the discretized two-phase formulation and coupled vibration characteristics of mass nanosensor. Several analogical studies are performed in detail to confirm the credibility of the present formulation and results. Comparisons between the uncoupled and coupled vibrations disclose that there are significant errors in obtaining the natural frequencies of corresponding mass value when the axial and transverse vibrations are considered separately. Therefore, the coupling of axial-lateral displacements resulted from the added mass eccentricity must be considered. This work can be a useful step forward to examine the coupling effects on vibration behavior of mass nanosensors, and it can be helpful to design the nanosensors with higher accuracy.

Thermal vibration and buckling analysis of two-phase nanobeams embedded in size dependent elastic medium

In this paper, vibration and buckling of two-phase nanobeams embedded in size dependent elastic medium and under thermal load are analyzed. Due to paradoxes of common differential nonlocal elasticity, such as neglecting the size effect of uniform loads, failure to satisfy the constitutive boundary conditions, associated to transformation of integral nonlocal equation to differential one, and incompatibility between the results of differential nonlocal with those of integral nonlocal, the size dependent effects of nanobeam, elastic medium and thermal load are taken into account simultaneously by using two-phase local/nonlocal Eringen's elasticity, for the first time. Governing equations and corresponding boundary conditions are derived using Hamilton's principle. To obtain natural vibration frequencies as well as critical buckling temperature, three different methods of solution are presented, i.e. exact solution, Generalized Differential Quadrature Method (GDQM) and Finite Element Method (FEM) which is based on the integral form of two-phase elasticity. Several comparison studies are conducted to examine the validity of the present formulation and results. The effects of applying two-phase elasticity on elastic medium and thermal load in vibration and buckling of nanobeams with different boundary conditions are investigated in details. Differences appeared in present results, especially in the cases with higher temperature and nonlocality as well as stiffer elastic medium, reveal that the size dependency of elastic medium and uniform thermal load, which is neglected by differential nonlocal, must be considered by employing two-phase elasticity.

Axial and Torsional Free Vibrations of Elastic Nano-Beams by Stress-Driven Two-Phase Elasticity

Size-dependent longitudinal and torsional vibrations of nano-beams are examined by two-phase mixture integral elasticity. A new and efficient elastodynamic model is conceived by convexly combining the local phase with strain- and stress-driven purely nonlocal phases. The proposed stress-driven nonlocal integral mixture leads to well-posed structural problems for any value of the scale parameter. Effectiveness of stress-driven mixture is illustrated by analyzing axial and torsional free vibrations of cantilever and doubly clamped nano-beams. The local/nonlocal integral mixture is conveniently replaced with an equivalent differential law equipped with higher-order constitutive boundary conditions. Exact solutions of fundamental natural frequencies associated with strain- and stress-driven mixtures are evaluated and compared with counterpart results obtained by strain gradient elasticity theory. The provided new numerical benchmarks can be effectively employed for modelling and design of Nano-Electro-Mechanical-Systems (NEMS).