Functionally Graded Magneto-electro-elastic Research Articles

The adaptive hygrothermo–magneto–electro–elastic coupling improved enriched finite element method for functionally graded magneto–electro–elastic structures

In this study, the hygrothermo–magneto–electro–elastic coupling improved enriched finite element method (HC-IEFEM) is proposed to analyze functionally graded magneto–electro–elastic (FG-MEE) structures. The static behavior of FG-MEE structures in the hygrothermal environment is studied. In the enriched finite element method (EFEM) enhanced by interpolation cover functions, the improved shape functions are added to solve the rank defect problem. Quadrilateral elements are used in the analysis of FG-MEE structures in this study. This improved method has proven to be relatively more efficient and resistant to mesh distortion than the traditional finite element method (FEM). Besides, the adaptive mesh refinement (AMR) scheme is used to refine the mesh in local areas to improve numerical calculation efficiency. Numerical examples show that the adaptive HC-IEFEM can achieve relatively high accuracy for analyzing FG-MEE structures. The proposed HC-IEFEM can obtain high accuracy in the hygrothermal environment using a relatively coarse mesh through the improved shape function and AMR scheme. Therefore, the significant potential is demonstrated in analyzing FG-MEE structures in the hygrothermal environment using the proposed adaptive HC-IEFEM.

Isogeometric analysis of magneto-electro-elastic functionally graded Mindlin microplates

In this study, an accurate numerical method for the static and dynamic response analysis of magneto-electro-elastic functionally graded (MEEFG) microplates with complex geometries is proposed with the application of isogeometric analysis (IGA). Leveraging Hamilton's principle and the extended modified couple stress theory, the weak form of motion equations is derived. By performing convergence analysis, the accuracy of the proposed numerical method is verified. To show the applicability of the new method, the influences of the microstructure effect and gradient index on the static and dynamic behaviors of both the MEEFG square and elliptical microplates with various boundary conditions are discussed in detail. Results illustrate that the microstructure effect dominated by the couple stress effect has an obvious impact on both the mechanical and electromagnetic behavior of structures with microscopic scales which will cause hardening of microstructural stiffness, whereas the macrostructures are minimally affected by the couple stress effect. Besides, microstructures with lower stiffness exhibit a more pronounced microstructure effect and a greater degree of stiffness hardening. The proposed IGA method could serve as a basis for the design and optimization of microelectromechanical devices with complex shapes.

Transient analysis of a cracked functionally graded magneto‐electro‐elastic rectangular finite plane under anti‐plane mechanical and in‐plane electric and magnetic impacts

AbstractThe transient anti‐plane problem of a functionally graded magneto‐electro‐elastic (FGMEE) rectangular plane weakened by several embedded and edge cracks under various boundary conditions is solved. First, the solutions to the dynamic magneto‐electro‐elastic dislocation in the finite rectangular domain are derived by employing Laplace and finite Fourier cosine transforms. The solutions can be utilized to construct the singular integral equations in Laplace transform domain for FGMEE rectangular plane with multiple embedded and edge cracks. The integral equations arising from dislocation solutions have Cauchy‐type singularities with unknown dislocation density functions. These unknown functions may be obtained by reducing the integral equations into a system of linear algebraic equations, hereby, the dynamic stress intensity factors (DSIFs) for several arbitrarily embedded and edge cracks are obtained. The overshoots of dynamic stress intensity factors are shown to be intensely affected by the rectangular plane dimensions, length and position of the cracks, loading parameter, and negative or positive gradient index.

Numerical modeling of geometrically nonlinear responses of smart magneto-electro-elastic functionally graded double curved shallow shells based on improved FSDT

The use of functionally graded magneto-electro-elastic (FG-MEE) shells in smart composite structures like sensors, transducers, vibration monitoring, aeronautics, shape tracking, medical and sonar applications has a lot of promise. The accurate modeling approach for situations with several physics linked is quite difficult. In this paper, an improved first-order shear deformation hypothesis is used to inspect the magneto-electro-elastic coupling performance of thin-walled smart structures with laminate design presenting FGM composite as inactive material and using the magneto-piezoelectric patches as combined sensor and actuator features. The govern equations of motion, electric and magnetic distributions through the thickness direction of shell element are derived from the virtual of Hamilton principle. The simulations are carried out using a self-coded FE software. A numerical analysis of the nonlinear deflection of FG-MEE shells is performed under multiphysics loading. To ensure that the current MEE formulation is accurate, the findings were compared to existing solutions from the literature, and remarkable agreement was identified. A parametric analysis is conducted to demonstrate how the material composition affects the deflection throughout the thickness. Further, new numerical results, physical insights and finding have been provided in the present manuscript taking into account influences of several important parameters such as material graduation index and various boundary conditions. The novel MEE FE model suggested in this study is highly useful for the accurate development and design of FG-MEE structures.

Free vibration of porous FG magneto-electro-elastic microbeams in the hygrothermal environment based on differential transformation method

Free vibration of the porous functionally graded magneto-electro-elastic (FGMEE) microbeams in hygrothermal environment is investigated. Four kinds of distributions of porosities including uniform porosity, “O,”“X,” and “V” type porosities are assumed in this paper. For the functionally graded microbeam, the physical neutral surface is considered. The modified couple stress theory is adopted to capture the size effect. The equations of motion for the porous FGMEE microbeam are derived and solved by differential transformation method. In the numerical examples, the effect of electric and magnetic loadings, material length scale parameter, the temperature rise, the moisture concentration, material gradient index, and porosity volume fraction on the natural frequency are analyzed and discussed in detail. The results indicate that different porous distribution modes have different effects on natural frequency. Moisture concentration/temperature rise can reduce the natural frequency regardless of what the temperature and moisture modes is. These results will be useful for the design of the porous FGMEE structures.

Dynamic fracture behavior for hole-initiated cracks in functionally graded magneto-electro-elastic bi-materials

In this article, the dynamic fracture behavior of hole-initiated cracks in functionally graded magneto-electro-elastic (FGMEE) bi-materials with exponential variation is studied by Green’s function method, and SH-wave is considered as an external load acting on the bi-materials. The mechanical model of the cracks is constructed through interface-conjunction and crack-deviation techniques, thereby simplifying the crack problem to solving a series of the first kind of Fredholm’s integral equations, from which the dynamic stress intensity factor (DSIF) at the crack tips is expressed. Numerical results clarified the factors that affect the DSIF, including wave number, incident angle, the geometry of the defect, and the gradient of the bi-materials. The accuracy of the present methods is examined by comparing the obtained results with the reference solutions. Compared with previous traditional numerical and analytical methods, the methods proposed in this paper are relatively more effective and applicable, and provide a new perspective for studying the fracture problems of FGMEE materials with more complex defects in practical engineering.

Multi-physics zonal Galerkin free element method for static and dynamic responses of functionally graded magneto-electro-elastic structures

Functionally graded magneto-electro-elastic (FGMEE) structures have been extensively applied in engineering applications, which are often in the multi-field coupling environment. Driven by the desire to identify and understand the complexities of coupled fields, the multi-physics zonal Galerkin free element method (MZG-FREM), which belongs to a weak-form meshless method, is proposed for static and transient responses of magneto-electro-elastic structure, which material properties vary along the thickness direction. Based on the thermodynamic potential function and generalized Hamilton variational principle, the equations for the FGMEE medium are derived. Through the zone technique, MZG-FREM processes a robust capability to handle the model with complex geometry and the results obtained by MZG-FREM are insensitive to irregularly distributed nodes. Besides, Galerkin method is introduced to set up the equation for each node to guarantee the stability and accuracy of the results. The effect of exponential factors on the responses of FGMEE structures is investigated by MZG-FREM with a modified Newmark scheme. The convergence, accuracy, and robustness of MZG-FREM are verified through numerical examples including FGMEE structure with complex geometry and FGMEE energy harvester composed of various materials. The proposed method MZG-FREM and obtained results can be effectively incorporated for the accurate design of FGMEE smart devices.

An inhomogeneous stabilized node-based smoothed radial point interpolation method for the multi-physics coupling responses of functionally graded magneto-electro-elastic structures

In this paper, an inhomogeneous stabilized node-based smoothed radial point interpolation method (ISNS-RPIM) is developed for analyzing functionally graded magneto-electro-elastic (FGMEE) structures in hygrothermal environment. The change of material gradient follows the exponential function distribution. Combining with the gradient smoothing technique (GST) and the coupling constitutive relationship, and considering the hygrothermo-magneto-electro-elastic (HMEE) coupling effects of FGMEE material, the equations for ISNS-RPIM are deduced. The generalized displacement of the FGMEE structures is obtained, and the error in total energy norm, CPU time and efficiency of ISNS-RPIM, NS-RPIM and FEM are researched. Numerical examples are provided to explore the influence of the following parameters in FGMEE structures including exponential factor, hygrothermal loading and empirical constants, and demonstrate the correctness, high effectivity and insensitivity to mesh distortion of ISNS-RPIM. The present method has shown great potential in solving practical complex problems.

Nonlinear damping of auxetic sandwich plates with functionally graded magneto-electro-elastic facings under multiphysics loads and electromagnetic circuits

This work addresses the damped transient response of smart sandwich plates subjected to coupled/multiphysics loads. The plate is made of an auxetic core with magneto-electro-elastic functionally graded (MEEFG) facesheets. The well known active constrained layer damping (ACLD) technique making use of the piezoelectric (PE) patch also known as1-3 PZC layer embedded on the viscoelastic (VE) layer is adopted to attenuate the vibrations. A finite element (FE) based approach has been adopted in order to derive the governing equations. Further, the solutions to the considered problem are obtained by adopting the direct iterative method. The primary focus of this work is to evaluate the effects of the auxetic core's rib-length ratio, inclination angle on the damped structural response, which has been carried out for the first time in the literature. Also, parametric studies on the geometric and material properties such as thickness configuration of core and facesheets, ACLD patch positions, PE fiber orientation angle, control gain, functional gradation pattern, and gradient index have been carried out. Emphasize has been made to understand the impact of applying open and closed circuits associated with the rest of the parameters, on the controlled nonlinear behaviour of smart sandwich plates. The numerical results of this work make some major revelation about the implementation of auxetic cores in the smart structures, which make this work interesting.

Static and dynamic analysis of functionally graded magneto-electro-elastic plates and shells

Functionally graded magneto-electro-elastic (FG-MEE) plates and shells have great application potential in smart structures for vibration control, shape control and health monitoring. The precise modeling technique for multi-physics coupled problems is a big challenge. This paper develops a finite element (FE) model coupled with magneto-electro-elastic fields for static and dynamic analysis of FG-MEE plates and shells. The FE model is derived by using the first-order shear deformation hypothesis with consideration of linear multi-physics coupled constitutive equations. Eight-node quadrilateral plate/shell elements are proposed for FG-MEE structures, including five mechanical DOFs at each node, one electric and magnetic DOF at each MEE layer of elements. The model is first validated by MEE laminated plates and shells, later applied to parametric study of FG-MEE structures with functionally graded electric and magnetic properties.

Extended isogeometric analysis for fracture in functionally graded magneto-electro-elastic material

In this work, we present an extended isogeometric analysis (XIGA) for cracked functionally graded magneto-electro-elastic (FGMEE) material. The material properties inside the FGMEE domain vary exponentially. An electrically and magnetically impermeable crack in the FGMEE domain is modeled by adding appropriate enrichment functions into the isogeometric analysis (IGA) approximation. Heaviside function is utilized to model the displacement, electric, and magnetic potential along the crack face. In contrast, conventional four-field crack tip enrichment functions are used to mimic the crack tip singularity characteristics. The generalized intensity factors (IFs), i.e., the stress intensity factors (SIFs), electric displacement intensity factor (EDIF) and magnetic induction intensity factor (MIIF), are assessed using magneto-electro-mechanical interaction integral. A series of numerical studies over the cracked homogeneous and FGMEE material are presented. The accuracy of XIGA is examined by comparing the obtained results with the reference solutions. Various parameters such as volume fraction, crack length, loading combination parameter (both electrical and magnetic), material non-homogeneity parameter and crack orientation are taken to examine their effects on the generalized IFs.

Nonlinear Vibration Analysis of Viscoelastic Smart Sandwich Plates Through the use of Fractional Derivative Zener Model

In this paper, the nonlinear vibrational behavior of a sandwich plate with embedded viscoelastic material is studied through the use of constitutive equations with fractional derivatives. The studied sandwich structure is consisted of a viscoelastic core that is located between the faces of functionally graded magneto-electro-elastic (FG-MEE). In order to determine the frequency-dependent feature of the viscoelastic layer, four-parameter fractional derivative model is utilized. The material properties of FG-MEE face sheets have been distributed considering the power law scheme along the thickness. In addition, for derivation of the governing equations on the sandwich plate, first-order shear deformation plate theory along with von Karman-type of kinematic nonlinearity are implemented. The derived partial differential equations (PDEs) have been transformed to the ordinary differential equations (ODEs) through the Galerkin method. After that, the nonlinear vibration equations for the sandwich plate have been solved by multiple time scale perturbation technique. Moreover, for evaluating the effect of different parameters such as electric and magnetic fields, fractional order, the ratio of the core-to-face thickness and the power low index on the nonlinear vibration characteristics of sandwich plates with FG-MEE face sheets, the parametric analysis has been performed. The obtained results revealed the enhanced nonlinear natural frequency through an increment in the fractional order. Furthermore, the prominent influence of fractional order on the nonlinear frequency of sandwich plate was declared at the negative electric potential and positive magnetic potential.

Magneto-electro-mechanical vibration of porous functionally graded smart sandwich plates with viscoelastic core

In this paper, the vibrations of a sandwich plate with functionally graded magneto-electro-elastic (FG-MEE) face sheets and porous and viscoelastic core were investigated. Power-law rule modified by two types of porosity distributions was used to model the FG-MEE plates. The plate with even porosity distribution was considered as the FG-MEE-I model, while the one with uneven porosity distribution was labeled as the FG-MEE-II model. The normal and shear stresses were considered in the core layer, and the interlayer was modeled by the standard linear solid scheme. First-order shear deformation plate theory was used to derive the governing equation of the sandwich panel including the FG-MEE plate and viscoelastic core interaction. The governing equations were solved by the Navier method. A detailed parametric analysis was conducted to assess the effects of electric and magnetic fields, core-to-face sheet thickness ratio, and power-law index on the linear vibration characteristics of sandwich plates with functionally graded MEE face sheets. It is observed that for the FG-MEE-I model, an increase in the porosity coefficient led to a reduction in the frequency of the FG-MEE sandwich plate. On the contrary, for the FG-MEE-II model, an increment in the porosity coefficient enhanced the natural frequency.

Size‐dependent continuum‐based model of a flexoelectric functionally graded cylindrical nanoshells

Flexoelectricity is dependent on the strain gradient, which is high in microscale and nanoscale, leading to its considerable effect on the electromechanical behavior of structures in microscale and nanoscale. Using the Love's theory for thin shells and the modified flexoelectricity theory, governing coupled equations of the functionally graded magneto‐electro‐elastic (FGMEE) cylindrical nanoshells were formulated along with their boundary conditions using the Hamilton's principle and variation method. It is worth noting that the governing coupled equations of FGMEE cylindrical nanoshell with an electric potential‐independent polarization parameter, where the polarization is independent of the electric potential, are considered the major novelty of this study. To show the ability and uniqueness of the formulation, free vibrations of the shell were examined in a specific state, regardless of the magnetic field effect under the clamped–clamped boundary condition. The effects of different parameters including geometry, size effect, and external voltage on the vibration frequency of the nanoshell were investigated. According to the results, the vibration behavior of the flexoelectric nanoshell is largely dependent on the size effect. Given the application of this structure in sensors and actuators, the external voltage has a considerable effect on the electromechanical behavior of the structure in the presence of the flexoelectric effect.

A comparative study on the vibration of functionally graded Magneto-Electro-Elastic rectangular plate rested on a Pasternak foundation based on exponential and first-order shear deformation theories

In this paper, the in-plane and out-of-plane free vibrations of the functionally graded magneto-electro-elastic (FG-MEE) rectangular plate on a pasternak foundation were evaluated based on exponential shear deformation theory (ESDT) and first order shear deformation theory (FSDT). The material properties of FG-MEE varied along the thickness according to a power-law distribution model. It was assumed that the FG-MEE plate is subjected to initial external electrical and magnetic potentials; mreover, simply-supported boundary conditions were considered for all the edges of the FG-MEE plate. Firstly, the corresponding partial differential equations (PDEs) were derived using Hamilton’s principle, then, the natural frequencies were determined by solving the obtained equations through Navier’s approach according to the assumed boundary conditions. The results revealed that the natural frequencies of the plate decrease/increase with the increase of the electric/magnetic potentials. Moreover, the results showed a 0.03%, difference between the natural frequencies of the plate with a thickness-to-length ratio of 0.1 based on FSDT and ESDT; when the aspect ratio was increased to 0.2 and 0.3 this difference rose to 0.2% and 0.5%, respectively.

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.

Multiphysical time-dependent creep response of FGMEE hollow cylinder in thermal and humid environment

In this paper, we investigate the history of radial displacement, stresses, electric potential, and magnetic potential of a functionally graded magneto-electro-elastic (FGMEE) hollow cylinder subjected to an axisymmetric hygro-thermo-magneto-electro-mechanical loading for the plane strain condition. The material properties are taken as a power-law function of radius. Using stress-displacement relations, equations of equilibrium, electrostatic and magnetostatic equations, we find a differential equation including creep strains. Initially, eliminating creep strains, we obtain an analytical solution for the primitive stresses and electric and magnetic potential. In the next step, considering creep strains, we find the creep stress rates by applying the Norton law and Prandtl–Reuss equations for steady-state hygrothermal boundary condition. Finally, using an iterative method, we find the time-dependent creep stresses, radial displacement, and magnetic and potential field redistributions at any time. In numerical section, are comprehensively investigate the effects of grading index, hygrothermal environmental conditions, rotating speed, and temperature- and moisture-dependency of elastic constant of FGMEE.

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.

Coupling Magneto-Electro-Elastic node-based smoothed radial point interpolation method for free vibration and transient analysis of Functionally Graded Magneto-Electro-Elastic structures

A Coupling Magneto-Electro-Elastic (MEE) Node-based Smoothed Radial Point Interpolation Method (CM-NS-RPIM) was proposed to solve the free vibration and transient responses of Functionally Graded Magneto-Electro-Elastic (FGMEE) structures. By introducing the modified Newmark method, the displacement, electrical potential and magnetic potential of the structures under transient mechanical loading were obtained. Based on G space theory and the weakened weak (W2) formulation, the equations of the multi-physics coupling problems were derived. Using triangular background elements, the free vibration and transient responses of three numerical examples were studied. Results proved that CM-NS-RPIM performed better than the standard FEM by reducing the overly-stiff of structures. Moreover, CM-NS-RPIM could reduce the number of nodes while guaranteeing the accuracy. Besides, triangular elements could be generated automatically even for complex geometries. Therefore, the effectiveness and validity of CM-NS-RPIM were demonstrated, which were valuable for the design of intelligence devices, such as energy harvesters and sensors.

Analytical Solution for the Creep Problem of a Rotating Functionally Graded Magneto–Electro–Elastic Hollow Cylinder in Thermal Environment

In this paper, an analytical method is presented for the problem of the time-dependent response of a functionally graded magneto–electro–elastic (FGMEE) rotating hollow cylinder in thermal environment. The material properties of FGMEE are supposed to be power-law functions of radius. Applying the equations of equilibrium and electrostatic and magnetostatic equations, a differential equation which includes creep strains is achieved. At first, an exact solution for the primitive stresses, electric and magnetic potentials are achieved by eliminating creep strains in the mentioned differential equation. Then, Prandtl–Reuss equations, as well as Norton’s law, are employed for the FGMEE. Now, creep stress rates can be achieved by considering only creep strains in the mentioned differential equation. As a final step, time-dependent creep stress, electric potential and magnetic potential redistributions at any time can be achieved using an iterative method. Numerical examples are presented to disclose the influence of creep evolution, thermal loading, angular velocity and grading index on the primitive and creep response of the FGMEE hollow cylinder. Results show that the enhancement in tensile hoop stress during the creep evolution must be considered in designing. So, the creep analysis is vital to have more reliable and accurate aerospace smart structures.