Magneto-electro-thermo-elastic Research Articles

Nonlinear postbuckling analysis of magneto-electro-thermo-elastic laminated microbeams based on modified couple stress theory

In this article, the nonlinear postbuckling behavior of the magneto-electro-thermo-elastic (METE) laminated microbeams is presented. According to the modified couple stress theory (MCST) and Reddy's three-order shear deformation theory (RTSDT), in conjunctions with the von Karman geometric nonlinearity, the nonlinear static model of METE laminated microbeam is established. The nonlinear governing equations and the corresponding boundary conditions are derived by using the principle of virtual work principle. Afterwards, using an analytical method, the nonlinear postbuckling behavior of METE laminated microbeam with simple supported and clamped boundary conditions at the both ends is described. Moreover, numerical examples are exhibited to reveal the effects of the material length scale parameter, slenderness ratio, temperature rise, magneto-electric potential on the critical buckling load and the nonlinear postbuckling response. Also the distribution law of magneto-electric potential through the thickness direction of the microbeam with various lay-up modes is discussed.

A nonlocal strain gradient shell model with the surface effect for buckling analysis of a magneto-electro-thermo-elastic cylindrical nanoshell subjected to axial load.

This paper develops a novel size-dependent magneto-electro-thermo-elastic (METE) cylindrical nanoshell which is made of BaTiO3-CoFe2O4 materials. To illustrate the newly developed model, the buckling problem of the METE cylindrical nanoshell subjected to temperature changes, initial magnetic and electric potentials, and axial load is analytically solved on the basis of Kirchhoff-Love theory. To model the size dependency effects, nonlocal strain gradient theory (NSGT) and surface elasticity theory are considered simultaneously. In the process, governing differential equations of the shell are derived using Hamilton's principle. Bifurcation conditions for buckling of the METE cylindrical nanoshell are obtained using Navier's method. The influences of the scale parameter, structure parameter, surface effect, temperature change, initial magnetic potential and initial electric potential on buckling behavior are examined in detail. The present model can be used as a basic model in the study of the effects of temperature changes, initial magnetic and electric potentials, and the axial load on the buckling behavior of METE cylindrical nanoshells. The results provide insights for future experimental research and show that METE cylindrical nanoshells are potential candidates for nanocomponents.

Size-dependent free vibration and buckling analysis of magneto-electro-thermo-elastic nanoplates based on the third-order shear deformable nonlocal plate model

In this paper, a size-dependent nanoplate model is developed to describe the free vibration and buckling behaviors of magneto-electro-thermo-elastic (METE) rectangular nanoplates. It is assumed that the METE nanoplate is subjected to external electric voltage, external magnetic potential, and uniform temperature rise. The nonlocal elasticity theory along with the third-order shear deformation plate theory is employed for the size-dependent mathematical modeling of nanoplates. The presented model has two advantages over available models: (1) the need for the correction factor is bypassed and (2) it can be successfully applied to thick nanoplates. The governing equations and the corresponding boundary conditions are derived using the Hamilton’s principle which are then discretized on the space domain based on the generalized differential quadrature (GDQ) method. Afterward, an efficient numerical Galerkin procedure is adopted to reduce the discretized equations into Duffing-type ordinary differential equations. Numerical results are presented to examine the influences of nonlocal parameter, length-to-thickness ratio, temperature rise, external electric potential, external magnetic potential and type of boundary condition on the free vibration, and buckling behaviors of METE nanoplates. It is revealed that frequency and critical buckling load of nanoplates are dependent on magneto-electro-mechanical loadings, whereas they are less dependent on thermal loading.

Selection of size dependency theory effects on the wave's dispersions of magneto-electro-thermo-elastic nano-beam resting on visco-elastic foundation

Current research develops a comprehensive wave propagation analysis of a magneto-electro-thermo-elastic (METE) nano-beam (NB) resting on the visco-elastic medium. To model the size dependency effects, modified couple stress (MCS) and Eringen's nonlocal (ENL) theories are employed to analyze and describe the wave propagation behaviors for those nano-beams. These theories were the most used in the literature due to the inclusion of one additional size-dependent length scale parameter. Those theories are compared side by side in this investigation and their impacts/differences on the wave propagation of specific materials are explored. Sinusoidal shear deformation beam model with Hamilton's principle is adopted to develop the governing equations of motion. Then, an analytical solution is executed to extract numerical results for transverse wave propagation in both elastic and METE configurations of the nano-beam. The effects of size-dependent length scale of both theories, thickness of NB, Winkler-Pasternak coefficients, thermal gradient, and magnetic potential and external electric voltage are illustrated and discussed in details. It is concluded that wave frequency decreases with increment of nonlocal parameter for ENL model. On the other hand, a stiffening effect takes place for the wave frequency when the MCS model is considered. Hence, the results indicate that there is a significant difference between ENL and MCS theories in the estimation of the behavior of wave propagation in small-scale structures. One of the main results of this investigation indicates that the MCS theory has similar nonlocal effects as the Eringen's theory for specific conditions.

Transient responses of functionally graded magneto-electro-elastic structures with holes in thermal environment using stabilized node-based smoothed radial point interpolation method

In the present work, a stabilized node-based smoothed radial point interpolation method (SNS-RPIM) is proposed for analysing the behaviour of functionally graded magneto-electro-elastic (FGMEE) structures with holes in thermal environment. By introducing the stable item related to the gradient variance of field variables, a close-to-exact SNS-RPIM model is constructed, and the temporal instability of traditional node-based smoothed radial point interpolation method (NS-RPIM) is cured. Several numerical examples validate that SNS-RPIM performs well in efficiency and convergence than finite element method (FEM) and NS-RPIM and could solve magneto-electro-thermo-elastic (METE) multi-physics coupling problems more accurately. It is also found that the position of the holes has a greater influence on the displacement and magnetic potential of the structure and has less influence on the electric potential. This provides a reference for the design and application of FGMEE structures with holes working in thermal environment in practical engineering.

Surface energy layers investigation of intelligent magnetoelectrothermoelastic nanoplates through a vibration analysis

Intelligent materials such as magnetoelectrothermoelastic (METE) nanoplates offer great potential to supply more efficient energy harvesting devices, transducers, sensors, and actuators. In this study, the influences of slanted angle of the METE nanoplate and orthotropic angle of Pasternak foundation on the magnitude of surface energy layers are investigated through a vibrational analysis. The nanoplate is exposed to outer thermal, electric, magnetic and in-plane loadings. For modeling the plate’s nonlocal behavior and surface stresses, the governing equations are constructed based on Hamilton’s principle. Galerkin method is implemented to solve the equilibrium motions, and also for the authenticity of the solution, the equations are resolved by the Navier’s method. Obtained results are deemed useful for the mechanical analysis and design of nano-/microelectromechanical system nanostructures constructed from the METE materials. The numerical examples indicated that after a particular value of slanted angle of the nanoplate, α ≥ 80, the magnitude of surface energy layers in the vibration behavior becomes opposite to the case when the slanted angles are 0° < α < 80°. In addition, the value of the slanted angle of the nanoplate can cause the harmonic responses of the vibration due to the orthotropic angle of Pasternak foundation to shift in harmony with the slanted angle degree.

The multi-physic cell-based smoothed finite element method for dynamic characterization of magneto-electro-elastic structures under thermal conditions

This study was aimed to computationally investigate the thermal load combining the mechanical load effect on dynamic characteristics of intelligent composite structures. For this purpose, the cell-based smoothed finite element method (CS-FEM) was applied to the multi-physic coupling problem, and the coupled multi-physic CS-FEM (CPCS-FEM) integrating the magneto-electro-thermo-elastic (METE) coupling effect was put forward. Compared with standard FEM, this method with higher accuracy, lower mesh restriction and much less calculation for stochastic issues was more capable in handling severe mesh distortion and deformation. The convergence, accuracy and efficiency of CPCS-FEM were validated with three numerical cases. With CPCS-FEM integrating the modified Newmark scheme, the thermal effects on generalized displacement (x- and z-direction displacement components, electric and magnetic potentials) of magneto-electro-elastic sensors were explored. This study presents an effective approach to model the complicated multi-physic problem, and the simulation findings contribute to the design of intelligent structures in service under thermal conditions.

Analytical buckling solution of magneto-electro-thermo-elastic cylindrical shells under multi-physics fields

A new analytical buckling solution of a cylindrical shell made of two-phase magneto-electro-thermo-elastic (METE) composites under multi-physical fields is obtained by a Hamiltonian-based approach. Two types of technologically important distribution model are considered: continuous fibers and laminates. Based on the Reissner’s shell theory, the exact solution expanded into symplectic series is rigorously obtained from governing equations under the Hamiltonian description which has four possible forms of explicit expressions. Accurate critical buckling loads and analytical buckling mode shapes for various boundary conditions are obtained. A comprehensive comparison is presented to verify the proposed solution and very good agreement is reported. Effects of geometrical parameters, boundary conditions, cases of eigenfunctions, volume fractions and external magneto-electro-thermal loadings on buckling behaviors of the shell are investigated also.

A stabilized node-based smoothed radial point interpolation method for functionally graded magneto-electro-elastic structures in thermal environment

It is well-known that the standard finite element method (FEM) has the disadvantage of ‘overly-stiff’ and could not offer reliable enough solutions. We proposed a novel stabilized node-based smoothed radial point interpolation method (SNS-RPIM) to analyze the free vibration and the behavior of functionally graded magneto-electro-elastic (FGMEE) devices under mechanical and thermal load, which cured the ‘overly-soft’ and temporally instability of traditional NS-RPIM. The gradient variance items associated with the field nodes are applied to construct the system stiffness matrix. The detailed numerical study has shown the superiority of the proposed method. SNS-RPIM performs well in solving static magneto-electro-thermo-elastic (METE) multi-physics coupling field problems, which validates the accuracy, convergence of the proposed method. Moreover, SNS-RPIM is less insensitive to mesh distortion than FEM and NS-RPIM. The advantages mentioned above make SNS-RPIM very helpful for the design of the actual intelligent devices.

Rate of surface energy changes on the wave propagation analysis of METE nanoplates based on couple stress small-scale and nonlocal strain gradient theories

The modified couple stress and nonlocal strain gradient small-scale models are manipulated to scrutinize generally the rate of surface energy layer changes on the wave propagation analysis of a rectangular magneto-electro-thermo-elastic (METE) nanoplate. The METE nanoplate is a compound of BiTiO3–CoFe2O4 nanomatters. The presented model considers concurrently modified couple stress, nonlocal, strain gradient, and surface energy hypotheses. Using the refined plate theory (RPT), which is a suitable plate theory in comparison with the other models, the nonclassical governing equilibrium equations are derived according to Hamilton’s principle. The RPT has been implemented for the first time, based on the authors’ knowledge, to derive the presented governing equilibrium equations through the modified couple stress hypothesis. With incorporating the impacts of diverse external in-plane loads, an analytical solution is utilized in order to compute the rate of surface energy variations upon the wave propagation of a rectangular METE nanoplate. For exactness of the anticipated model, the procured results are compared with those formerly mentioned outcomes in the literature. The results of the present study represented that for the wave propagation analysis of the METE nanoplate, the modified couple stress and strain gradient small-scale coefficients would have a reverse influence on the rate of surface layers changes in comparison with that of nonlocal one.

A Numerical Investigation into the Primary Resonant Dynamics of Magneto-Electro-Thermo-Elastic Plates

The geometrically nonlinear forced vibration response of magneto-electro-thermo-elastic (METE) rectangular plates is analyzed herein using a numerical approach. The shear deformation effect is taken into account based on the first-order shear deformation plate theory. The geometrical nonlinearity is also considered using the von Karman hypothesis. Based upon a variational approach, the energy functional of problem is obtained and represented in matrix form. Then, the variational differential quadrature technique is utilized to directly discretize that functional. For the solution in the time domain, the time periodic discretization method is employed. Finally, the pseudo arc-length continuation technique is used to find the frequency-response curves of METE plates under various boundary conditions. The influences of electric voltage, magnetic potential and temperature difference on the primary resonant dynamics of METE plates are investigated.

A comprehensive investigation into the impact of nonlocal strain gradient and modified couple stress models on the rates of surface energy layers of BiTiO3–CoFe2O4 nanoplates: a vibration analysis

According to the nonlocal strain gradient and modified couple stress theories, a unified size-dependent plate model is proposed for the vibration analysis of rectangular magneto-electro-thermo-elastic (METE) nanoplates to analyze the rates of its surface energy layers. The METE nanoplate is composed of BiTiO3–CoFe2O4 materilas. The developed model can simultaneously catch nonlocal, strain gradient, and modified couple stress models, in addition to surface energy effects. Using the refined plate theory, we have developed the non-classical governing equations based on the principle of minimum potential energy. The refined plate theory is employed for the first time to get the governing equations through the modified couple stress theory. Analytical solutions are obtained for the natural frequency of rectangular METE nanoplates under the influence of various external in-plane loadings. Verification of the proposed model is done based on the comparison of the degenerated results with those previously accessible in the literature. It is demonstrated that the nonlocal scale parameter exerts an opposite impact on the degree of surface layers, as compared with those of the strain gradient and modified couple stress scale parameters. So, by enhancing the nonlocal scale parameter, it would be possible to increase the effects of surface energy layers on the natural frequencies ratio of METE nanoplate, while the influence of the surface energy layers could be reduced with the enhancement of the strain gradient and modified couple stress scale parameters.

Effect of BaTiO3/CoFe2O4 micro-topological textures on the coupled static behaviour of magneto-electro-thermo-elastic beams in different thermal environment

The use of composite materials with multifunctional capabilities is an increasing requirement for structures or components where the sensory function is accompanied by the diagnosis and the actuation functions, such as autonomic, adaptive or self-sustaining systems. In this context, the present study aims to characterize the coupled response of magneto-electro-thermo-elastic (METE) beams made from Barium Titanate (BaTiO3) and Cobalt Ferric Oxide (CoFe2O4) composite having various micro-topological textures, as well as their static response when submitted to different temperature distribution profiles. To this purpose, a three-dimensional finite element accounting for the coupling between the multiple physical fields in presence, is developed and implemented. The spatial heterogeneous BaTiO3/CoFe2O4 microstructure is also assessed by considering typical Body Centered Cubic (BCC), Face Centered Cubic (FCC) and Simple Cubic (SC) spatial packing arrangements. A special attention is paid to the influence of these micro-topological structures on the pyroeffects and its contribution towards the direct and derived quantities of the METE beam. The results obtained, suggest that the composite heterogeneous microstructure have a relevant influence on the static response of the METE beam in thermal environment.

Buckling analysis of skew magneto-electro-thermo-elastic nanoplates considering surface energy layers and utilizing the Galerkin method

In the present study, buckling analysis of skew magneto-electro-thermo-elastic (METE) nanoplates is carried out by the surface layers and nonlocal small-scale hypotheses. The equilibrium governing skew METE nanoplates is attained via the refined plate hypothesis; then, the equations of motion are obtained by using the principle of virtual work. The Galerkin method is considered to solve these equations. The governing equations are tested using the Navier’s solution to verify the exactness of the Galerkin method. The monumental elements, which consist of electric potential, magnetic potential, nonlocal parameter, nanoplate thickness, skew angle, and temperature change, are investigated with regard to the rate of the surface layers of the buckling behavior of a skew METE nanoplate. Based on the numerical results, there is a good similarity between the results of the present work and some accessible cases already mentioned in the literature. It is also shown that by raising the skew angle, the influence of the surface layers on the buckling of the METE nanoplate is decreased.

Investigation of the effect of BaTiO3/CoFe2O4 particle arrangement on the static response of magneto-electro-thermo-elastic plates

In this article, a framework based on finite element (FE) methods is proposed for predicting the influence of spatial arrangement of two phase Barium Titanate (BaTiO3) and Cobalt Ferric Oxide (CoFe2O4) particulate composites on the static response of magneto-electro-thermo-elastic (METE) plates. The coupled material properties such as piezoelectric, piezomagnetic, dielectric, magnetic permeability, thermal expansion and pyro co-efficients vary significantly with the spatial arrangement of BaTiO3/CoFe2O4 particulates. The coupled FE governing equations accounting the effect of particle arrangement is presented by incorporating linear coupled constitutive equations of METE composites. Through the condensation technique, the governing equations of METE plates are solved to obtain direct (thermal displacements, electric and magnetic potentials) and derived quantities (stresses, electric displacements and magnetic flux densities). A special attention has been placed on evaluating the pyro-electric and pyro-magnetic coupling effects for different packing arrangement considered namely, Body Centered Cubic (BCC), Face Centered Cubic (FCC) and Simple Cubic (SC) METE particulate composites. Further, parametric studies are carried out to analyse the influence of boundary conditions and aspect ratio. The present study reveals that the multiphysics response of METE plates changes significantly with the packing arrangements of BaTiO3/CoFe2O4 particulates and geometrical parameters. It is believed that the obtained solutions would provide insights into design aspects of METE structures.

The establishment of coupled magneto-electro-thermo-elastic theory with the consideration of surface and non-local effects and its application in laminated nano-devices

A fully coupled linear two-dimensional (2D) magneto-electro-thermo-elastic (METE) laminated plate theory considering non-local and surface effects is established for the first time based on the principle of virtual work and Mindlin's theory. This theory is suitable in macro and micro scales, and these equations obtained are general and can be reduced to some cases at particular conditions. After theoretical and numerical validation, the performance of a nano-scaled energy harvester made of piezoelectric and piezomagnetic layers is studied by using of the 2D METE theory as a specific example. A critical thickness is obtained and numerical results show that surface effect plays key role on the size-dependent behavior, including the resonant frequency, output voltage, output power, and magneto-electric coupling coefficient. The present study may be useful to multi-field coupling analysis of size-dependent nanostructures.

Size-dependent bending, buckling and vibration of higher-order shear deformable magneto-electro-thermo-elastic rectangular nanoplates

The aim of the present study is to propose a unified size-dependent higher-order shear deformable plate model for magneto-electro-thermo-elastic (METE) rectangular nanoplates by adopting the nonlocal elasticity theory to capture the size effect, and by utilizing a generalized shape function to consider the effects of transverse shear deformation and rotary inertia. By considering various shape functions, the proposed plate model can be reduced to the nonlocal plate model based upon the Kirchhoff, Mindlin and Reddy plate theories, as well as the parabolic, trigonometric, hyperbolic and exponential shear deformation plate theories. The governing equations of motion and corresponding boundary conditions of METE nanoplates subjected to external in-plane, transverse loads as well as magnetic, electric and thermal loadings, are obtained using Hamilton’s principle. Then, as in some case studies, the static bending, buckling, and free vibration characteristics of simply-supported METE rectangular nanoplates are investigated based upon the Navier solution approach. Numerical results are provided in order to investigate the influences of various parameters including the nondimensional nonlocal parameter, type of transverse loading, temperature change, applied voltage, and external magnetic potential on the mechanical behaviors of METE nanoplates. Furthermore, comparisons are made between the results predicted by different nonlocal plate models by utilizing the developed unified nonlocal plate model and selecting the associated shape functions. It is illustrated that by using the presented unified nonlocal plate model, the development of a nonlocal plate model based upon any existing higher-order shear deformable plate theory is a simple task.

Size-Dependent Buckling and Postbuckling Analyses of First-Order Shear Deformable Magneto-Electro-Thermo Elastic Nanoplates Based on the Nonlocal Elasticity Theory

This paper presents a nonlocal nonlinear first-order shear deformable plate model for investigating the buckling and postbuckling of magneto-electro-thermo elastic (METE) nanoplates under magneto-electro-thermo-mechanical loadings. The nonlocal elasticity theory within the framework of the first-order shear deformation plate theory along with the von Kármán-type geometrical nonlinearity is used to derive the size-dependent nonlinear governing partial differential equations and associated boundary conditions, in which the effects of shear deformation, small scale parameter and thermo-electro-magneto-mechanical loadings are incorporated. The generalized differential quadrature (GDQ) method and pseudo arc-length continuation algorithm are used to determine the critical buckling loads and postbuckling equilibrium paths of the METE nanoplates with various boundary conditions. Finally, the influences of the nonlocal parameter, boundary conditions, temperature rise, external electric voltage and external magnetic potential on the critical buckling load and postbuckling response are studied.

Nonlocal nonlinear first-order shear deformable beam model for postbuckling analysis of magneto-electro-thermo elastic nanobeams

In this study, the size-dependent postbuckling behavior of magneto-electro-thermo-elastic (METE) nanobeams with different edge supports is investigated. Based on the nonlocal first-order shear deformation beam theory and considering the von Karman hypothesis, a size-dependent nonlinear METE nanobeam model is developed, in which the effects of small scale parameter and thermo-electro-magnetic-mechanical loadings are incorporated. A numerical solution procedure based on the generalized differential quadrature (GDQ) and pseudo arc-length continuation methods is utilized to describe the size-dependent postbuckling behavior of METE nanobeams under various boundary conditions. The effects of different parameters such as nonlocal parameter, external electric voltage, external magnetic potential and temperature rise on the postbuckling path of METE nanobeams are explored. The results indicate that increasing the non-dimensional nonlocal parameter, imposed positive voltage, negative magnetic potential and temperature rise decrease the critical buckling load and post-buckling load-carrying capacity of METE nanobeams, whilean increase in the negative voltage, positive magnetic potential lead to a considerable increase of critical buckling load as well as postbuckling strength of the METE nanobeams.

Nonlocal free vibration in the pre- and post-buckled states of magneto-electro-thermo elastic rectangular nanoplates with various edge conditions

Considering the small scale effect together with the influences of transverse shear deformation, rotary inertia and the magneto-electro-thermo-mechanical coupling, the linear free vibration of magneto-electro-thermo-elastic (METE) rectangular nanoplates with various edge supports in pre- and post-buckled states is investigated herein. It is assumed that the METE nanoplate is subjected to the external in-plane compressive loads in combination with magnetic, electric and thermal loads. The Mindlin plate theory, von Kármán hypothesis and the nonlocal theory are utilized to develop a size-dependent geometrically nonlinear plate model for describing the size-dependent linear and nonlinear mechanical characteristics of moderately thick METE rectangular nanoplates. The nonlinear governing equations and the corresponding boundary conditions are derived using Hamilton’s principle which are then discretized via the generalized differential quadrature method. The pseudo-arc length continuation approach is used to obtain the equilibrium postbuckling path of METE nanoplates. By the obtained postbuckling response, and taking a time-dependent small disturbance around the buckled configuration, and inserting them into the nonlinear governing equations, an eigenvalue problem is achieved from which the frequencies of pre- and post-buckled METE nanoplates can be calculated. The effects of nonlocal parameter, electric, magnetic and thermal loadings, length-to-thickness ratio and different boundary conditions on the free vibration response of METE rectangular nanoplates in the pre- and post-buckled states are highlighted.