Buckling and free vibration of magnetoelectroelastic nanoplate based on nonlocal theory
Introduction
Nanostructures have increased considerable attention among the experimental and theoretical research communities. These nanostructures [1], [2], [3] are found to be possessing extraordinary mechanical, electrical, electronics and thermal properties as compared to the conventional structural materials. A vast area of novel applications of these nanostructures is foreseen in the coming years. These include aerospace, biomedical, bioelectrical, superfast microelectronics, etc. Understanding the accurate mechanical and physical properties of these nanostructures and their impacts on its performance and reliability is thus necessary for its productive applications. Therefore, microstructure-dependent size effects are often observed [4], [5], [6], [7], [8].
In the domain of materials science, some recent advances are the smart or intelligent materials where piezoelectric and piezomagnetic materials are involved. These materials called magnetoelectroelastic composites have the ability of converting energy from one form (among magnetic, electric, and mechanical energies) to the other. Furthermore, they exhibit magnetoelectric effect that is not present in single-phase piezoelectric or piezomagnetic materials [9], [10], [11], [12], [13], [14].
Recently, much attention has been paid to the structural analysis of the magnetoelectroelastic plate. Pan [15] presented an exact closed-form solution for the static deformation of the layered piezoelectric/piezomagnetic plate based on a new and simple formalism resembling the Stroh formalism, and the propagator matrix method was used to handle the multilayered case. Using the state vector method, Wang et al. [16] obtained an analytical solution for magneto-electro-elastic, simply supported and multilayered rectangular plates in the form of infinite series. The state-vector approach was proposed by Chen et al. [17] for the analysis of free vibration of magneto-electroelastic layered plates. Wang et al. [18] derive the analytical solution for a three-dimensional transversely isotropic axisymmetric multilayered magneto-electro-elastic (MEE) circular plate under simply supported boundary conditions. Liu and Chang [19] presented the closed form for the vibration problem of a transversely isotropic magneto-electro-elastic plate. A nonlinear large-deflection model for magnetoelectroelastic rectangular thin plates is proposed by Xue et al. [20]. The bending problem for a transversely isotropic MEE rectangular plate is analyzed by imposing the Kirchhoff thin plate hypothesis on the plate constituent. An equivalent single-layer model for the dynamic analysis of magnetoelectroelastic laminated plates is presented by Milazzo [21]. The electric and magnetic fields are assumed to be quasi-static and the first-order shear deformation theory is used.
Considered the nonhomogeneous magnetoelectroelastic solids, Bhangale and Ganesan [22] carried out static analysis of FGM magneto-electro-elastic plate by finite element method under mechanical and electrical loading. Wu et al. [23] extended the Pagano method for the three-dimensional plate problem to the analysis of a simply-supported, functionally graded rectangular plate under magneto-electro-mechanical loads.
To the best of authors’ knowledge, however, the buckling and free vibration of magnetoelectroelastic nanoplate resting on a Pasternak foundation has not been considered.
Based on the nonlocal theory, the buckling and free vibration analysis of a magnetoelectroelastic nanoplate resting on a Pasternak foundation is investigated. The in-plane electric and magnetic fields can be ignored for nanoplates. According to Maxwell equations and magnetoelectric boundary conditions, the variation of electric and magnetic potentials along the thickness direction of the nanoplate is determined. The governing equations of magnetoelectroelastic nanoplate are derived based on application of Hamilton’s principle. Numerical results reveal the effects of the electric and magnetic potentials, spring and shear coefficients of the Pasternak foundation on the buckling load and natural frequency.
Section snippets
Nonlocal theory of magnetoelectroelasticity
Nonlocal elastic theory assumes that the stress state at a reference point x in the body is regarded to be dependent not only on the strain state at x but also on the strain states at all other points x′ of the body. The most general form of the constitutive relation in the nonlocal elasticity type representation involves an integral over the entire region of interest. The integral contains a nonlocal kernel function, which describes the relative influences of the strains at various locations
Modeling of the problem
Consider a magnetoelectroelastic nanoplate with length l, width b and thickness h resting on a Pasternak foundation as depicted in Fig. 1. A Cartesian coordinate system (x, y, z) is used to describe the plate with z along the plate thickness direction and the x − y plane sitting on the midplane of the undeformed plate. The magnetoelectroelastic body is poled along z-direction and subjected to an electric potential V0 and a magnetic potential Ω0 between the upper and lower surfaces of the plate.
Numerical results
In this section, numerical examples of buckling and free vibration for magnetoelectroelastic nanoplate are investigated. The magnetoelectroelastic composite made of the piezoelectric material BaTiO3 as the inclusions and piezomagnetic material CoFe2O4 as the matrix is considered. The materials properties for the composite are [26], [27]:
Conclusions
In this paper, the buckling and free vibration analysis of magnetoelectroelastic nanoplate resting on Pasternak foundation is investigated based on nonlocal theory. The in-plane electric and magnetic fields can be ignored for nanoplates. According to Maxwell equations and magnetoelectric boundary conditions, the variation of electric and magnetic potentials along the thickness direction of the nanoplate is determined. From the numerical results, some conclusions can be drawn.
- (i)
For a
Acknowledgement
This work is supported by the Natural Science Foundation of Hebei Province, China (E2013402077).
References (27)
ZnO nanowire and nanobelt platform for nanotechnology
Mater Sci Eng R: Reports
(2009)- et al.
Thermo-mechanical vibration of a single-walled carbon nanotube embedded in an elastic medium based on nonlocal elasticity theory
Comput Mater Sci
(2009) - et al.
Nonlocal elasticity theory for vibration of nanoplates
J Sound Vib
(2009) - et al.
Nonlocal plate model for free vibrations of single-layered graphene sheets
Phys Lett A
(2010) - et al.
Small scale effect on the free vibration of orthotropic arbitrary straight-sided quadrilateral nanoplates
Compos Struct
(2011) - et al.
Nonlinear vibration of the piezoelectric nanobeams based on the nonlocal theory
Compos Struct
(2012) - et al.
Closed-form solutions for the magnetoelectric coupling coefficients in fibrous composites with piezoelectric and piezomagnetic phases
Int J Solids Struct
(2000) - et al.
State vector approach to analysis of multilayered magneto-electro-elastic plates
Int J Solids Struct
(2003) - et al.
Modal analysis of magneto-electro-elastic plates using the state-vector approach
J Sound Vib
(2007) - et al.
Large deflection of a rectangular magnetoelectroelastic thin plate
Mech Res Commun
(2011)
An equivalent single-layer model for magnetoelectroelastic multilayered plate dynamics
Compos Struct
Three-dimensional static behavior of function-ally graded magneto-electro-elastic plates using the modified Pagano method
Mech Res Commun
Dynamic fracture analysis of a penny-shaped crack in a magnetoelectroelastic layer
Int J Solids Struct
Cited by (151)
Effect of crack damage on size-dependent instability of graphene sheets
2024, Applied Mathematical ModellingAccurate buckling analysis of magneto-electro-elastic cylindrical shells subject to hygro-thermal environments
2023, Applied Mathematical ModellingSize-dependent nonlinear bending analysis of nonlocal magneto-electro-elastic laminated nanobeams resting on elastic foundation
2023, International Journal of Non-Linear MechanicsCitation Excerpt :The strain gradient theory also models the hardening effect [26] while the nonlocal strain gradient theory models both hardening and softening effects [27]. Li et al. [28] studied buckling load and vibration frequency of Mindlin nanoplates through nonlocal theory. Ebrahimi et al. [29–31] studied buckling behaviors of nonlocal FG-MEE nanobeams using different beam theories.