A size-dependent elastic theory for magneto-electro-elastic materials

https://doi.org/10.1016/j.euromechsol.2020.104198Get rights and content

Highlights

  • Conventional MEE parameters are size-dependence.

  • Formulation of the boundary value problems for MEE materials.

  • General solutions and specific solutions for one-dimensional beam models.

Abstract

A size-dependent elastic theory for magneto-electro-elastic (MEE) nano-materials is proposed. The theory features not only the inclusion of the classical parameters such as piezoelectric and piezomagnetic constants, the magneto-electro, dielectric and magnetic permeability coefficients, but also the nonlocal and strain gradient parameters and their induced high-order MEE parameters. The governing equations and the boundary conditions are derived with the aid of the variational principle. To illustrate the theory, the general solutions of the complete boundary value problems of a one-dimensional beam problem is formulated. It is found that the equation of motion of the present beam model is two orders higher than that of the classical Euler—Bernoulli model. Therefore, it indicates that the general solutions presented in this paper may be served as benchmark theoretical results for the future study.

Introduction

With the rapid development of nanotechnology, nanostructures are widely used as key components in micro-electro-mechanical systems and nano-electro-mechanical systems. Therefore, it is of great importance to study their mechanical behaviors to facilitate their potential applications in nanotechnology. However, conventional elastic continuum theories cannot characterize the size effects observed both in the experiments and in molecular simulations. As a result, modelling such size-dependent properties of nanostructures motivates one to introduce one or several material length parameters in the constitutive equations (Askes and Aifantis, 2011). Earlier attempts of establishing such a theory include but not limit to the strain gradient elasticity theory (Mindlin and Eshel, 1968), surface elasticity theory (Gurtin and Ian Murdoch, 1975), nonlocal elasticity theory (Eringen, 1983) and modified couple stress theory (Yang et al., 2002). Recently, these elastic theories have now been widely used as useful tools in modelling the size-dependent structures in the form of rods, beams, plates and shells.

Based on the assumption that the strain energy of small scaled materials are contributed by the usual strain energy and higher-order strain energy induced by the higher-order stresses work-conjugate to strain gradients, Mindlin (1965) formulated a strain gradient elasticity for elastic materials. Later, a modified strain gradient elasticity theory was established by Lam et al. (2003) who introduced a new equilibrium equation characterizing the behaviors of higher-order stresses and moments of couples. Since then, this theory is widely applied to the studies of size-dependent materials and structures. For the beam models, one may refer to the Euler–Bernoulli beam models (Barretta et al., 2017; Čanađija et al., 2016; Civalek et al., 2020; Ebrahimi et al., 2020; Lazopoulos, 2012; Murmu and Adhikari, 2012; Xu and Deng, 2016) and the higher-order shear beam models (Akgöz and Civalek, 2017; Challamel, 2013). For plate models, one can refer to the Kirchhoff plate models (Jafari et al., 2016; Niiranen and Niemi, 2017; Wang et al., 2016; Zhang et al., 2015), the Mindlin plate models (Ansari et al., 2015) and the higher-order shear deformation plate models (Ansari et al., 2015; Sahmani and Ansari, 2013). For the shell models, one may mention the relevant works (Daneshmand et al., 2013; Mohandes and Ghasemi, 2019; Zeighampour and Tadi Beni, 2014). It is concluded from these works that the strain gradient elasticity theory predicts a ‘stiffening effect’, which is only in part consistent with the experiment results (Fleck et al., 1994; Yang et al., 2002). Nevertheless, the introduction of extra material length parameters in the constitutive equations increases the order of partial differential equations of motion of the engineering structures. As a result, the analytical solutions of the corresponding boundary value problems are challenging from the mathematical point of view. For instance, Akgöz and Civalek (2013) investigated the effect of different higher-order boundary conditions on the buckling of strain gradient Euler–Bernoulli beams. The effect of different higher-order boundary conditions on the dynamic behaviors of thin beams was systematically studied by Xu and Deng (2016). Along this line, Jafari et al. (2016), Wang et al. (2016) and Niiranen and Niemi (2017) discussed the boundary value problems of Kirchhoff plate models.

The continuum models on the basis of the Eringen's nonlocal elasticity theory can be used to predict reasonable well with the results of molecular simulations. Therefore, these models have been extensively applied in nanostructures such as carbon nanotubes and graphene sheets (Rafii-Tabar et al., 2016; Thai et al., 2017). Most of these cited work show that the nonlocal elasticity theory predicts a ‘softening effect’. However, a counterintuitive ‘stiffening effect’ that the vibration frequencies of beams increase with increasing the nonlocal material length parameter was reported for the dynamic behaviors of nanostructures (Eltaher et al., 2016; Hosseini-Hashemi et al., 2013; Wang et al., 2007). With a view toward obtaining the expected ‘softening effect’ for nonlocal elasticity theory, several interesting works have been conducted to address this issue. Challamel (2013) derived the variationally consistent boundary conditions for various nonlocal beam models. Xu et al. (2016) formulated the variationally consistent boundary value problems of nonlocal Euler–Bernoulli and Timoshenko beam models by utilizing the weighted residual approaches (WRAs). From the numerical scheme, the Symplectic method capturing the ‘softening effect’ for nonlocal Timoshenko beams was conducted (Zhang et al., 2019). As anticipated, these works can capture the expected ‘softening effect’ for nonlocal cantilever beams. Recently, Tuna and Kirca (2016), Fernández-Sáez et al. (2016) and Romano et al. (2017) also captured the ‘softening effect’ for Euler–Bernoulli beams by using the original integral form of nonlocal constitutive equation. In addition, the ill-posedness of the Eringen model has been studied in terms of nonlocal elastic beams (Romano et al., 2017). Again, the ‘softening effect’ for Euler–Bernoulli beams is observed. Motivated by the Eringen integral model, the stress-driven nonlocal integral model was proposed for free vibration problems of thin elastic beams (Apuzzo et al., 2017).

From the above-mentioned literature review, it is found that the strain gradient elasticity theory and the nonlocal elasticity theory are the two distinct size-dependent continuum theories. As a result, combination of the above two elasticity theories into an elasticity theory is straightforward to provide a plausible explanation for the results of experiments and molecular simulations. Challamel and Wang (2008) established a consistent nonlocal strain gradient theory based on a novel idea of combining the local and the non-local curvatures in the constitutive elastic relation. Askes and Aifantis (2011) proposed a combined constitutive model by taking into account both the strain gradient and velocity gradient. Based on the nonlocal effects of the strain field and its first gradient strain field, Lim et al. (2015) formulated a nonlocal strain gradient elasticity theory which includes higher-order stress gradients and strain gradient nonlocality. Recently, the buckling (Farajpour et al., 2016; Xu et al., 2017a; Yu et al., 2019), vibration (Li et al., 2016; Şimşek, 2016; Xu et al., 2017b; Zhu and Li, 2017) and wave propagation characteristics (Ebrahimi et al., 2016; Lim et al., 2015) based on the nonlocal elasticity theory have been developed in order to evaluate the effect of two material length parameters on the static and dynamic behaviors of engineering structures. It is stated that the conventional results will be obtained when the two material length parameters are the same for simply supported boundary conditions. However, we shall illustrate in this paper that it is not correct for nonlocal strain gradient beams for the reason that there may exist two alternative options for the higher-order boundary conditions (see Section 5.2). From the mathematical point of view, the additional boundary conditions should be added to constitute well-posed boundary value problems of strain gradient elastic models. As a result, the variationally consistent boundary conditions of strain gradient beam models should be correctly derived to avoid the possible counterintuitive ‘stiffening effect’ reported in the literature. To this end, we derive the boundary conditions of nonlocal strain gradient beams in combined with the nonlocal geometric relations by using the WRAs. It is worth mentioning that the thickness effect should be considered in developing the beam models using the strain gradient elasticity. This effect involves the contribution of strain gradient in the z direction. The representative works dealing with several different kind of beam models will be found in the literature (Chen et al., 2019a, 2019b; Tang et al., 2019a, 2019b), and the references therein.

Magneto-electro-elastic (MEE) materials are a class of smart composite materials, which are able to convert the energy of the system from the mechanical energy to magnetic or electric energies and vice vasa. With the rapid development of nanotechnology, MEE materials have been widely found in micro-/nano engineering science. Huang and Yu (2006) proposed a constitutive relation for piezoelectric materials by taking into the surface elasticity. Shen and Hu (2010) established an elastic theory of flexoelectricity for dielectrics involving surface effect. The constitutive relations within the framework of nonlocal elasticity for piezoelectric materials have also been developed by several researchers (Ke et al., 2012). Although these works show that the size effect may be prominent for structures at micro-/nano scale, they obviously fail to take into account the effects of high-order MEE constants. For MEE materials at micro-/nano scale, it is anticipated not only the surface elasticity and nonlocal elasticity, but also the strain gradient induced higher-order MEE constants will become surprisingly significant on the mechanical properties of MEE materials. In other words, ignoring the above effect for modelling the MEE materials may not be adequate accuracy. The reason is that these newly constants are coupled with notable strain gradients and/or stress gradients. Therefore, accurately capturing the mechanical properties of MEE materials from mechanical testing is of great importance in the engineering applications, such as energy harvesting and piezoelectric sensors. Developing an elastic theory that takes into account the strain/stress gradients as well as the induced higher-order MEE constants is the topic of current interest.

As an extension work of Lim et al. (2015), the objectives of this paper are to develop a size-dependent elastic theory for MEE nano-materials, and in addition, to present the general solutions of nonclassical boundary value problems. The layout of this work is as follows. Section 2 presents the thermodynamic framework of size-dependent elastic theory for MEE nano-materials, in which the newly complete constitutive equations are presented. Section 3 formulates the equilibrium equations and complete boundary conditions by using the usual variational principle. To illustrate the proposed theory, the variationally consistent boundary conditions are derived with the help of WRAs and discussed for different selections of higher-order boundary conditions in Section 4. In Section 6, the numerical results of the analytical solutions derived in Section 5 illustrate that the degenerated nonlocal model can capture the expected ‘softening effect’ for beams subject to different boundary conditions. Importantly, the ‘softening effect’ observed in this paper is consistent with the results reported in the literature (Tuna and Kirca, 2016; Xu et al., 2016).

Section snippets

Thermodynamic framework

A higher-order nonlocal strain gradient theory featuring the higher-order stress gradient and strain gradient nonlocality was recently established by Lim et al. (2015). Within the framework of this nonlocal strain gradient theory, the strain energy of an elastic body occupying volume V readsUn=12εijVK1(y,x,le1)cijklεkly(y)dV+12lm2εij,mVK2(y,x,le2)cijklεkl,my(y)dV.where εij and εkly are the strain tensors at point x and point y, cijkl is the classical elastic tensor, le1 and le2 are the

Variational principle

In this section, the variational principle for MEE materials using the nonlocal strain gradient elasticity is presented. Let us consider an elastic body occupying a finite domain V of boundary surface Ω.

Eq. (2) can be rewritten in a compact form by substituting Eqs. (13), (14) into Eq. (2) asU=12V(σijεij+σijm(1)εij,mDkEkBkHkDkm(1)Ek,mBkm(1)Hk,m)dV.

For the first two integrations of Eq. (15), the detailed derivation can be found elsewhere (Lim et al., 2015; Mindlin and Eshel, 1968;

MEE beam model

A beam model is simply presented in this section to illustrate the proposed theory of MEE materials. For convenience of the illustration, the above integral-differential equations are here converted by the corresponding differential ones by properly choosing a special kernel function (Lim et al., 2015; Romano et al., 2017). It is also assumed that the two nonlocal parameters have the same value, i.e., le1 = le2 = le. For preliminaries, we first present in Sections 4.1 Boundary conditions, 4.2

Analytical frequency solutions

In Section 4, the Euler-Bernoulli beam theory has been generalized to the establishment of the nonlocal strain gradient beam theory that features two material length parameters. In order to illustrate the size effect on the free vibration of MEE beams, the boundary value problems of the proposed model are solved analytically. Boundary value problems (55) and (56) are sixth order for w and fourth order for u. Therefore, one needs to know five boundary conditions for each end of a beam. In order

Numerical results and discussion

In this section, the free vibration of nonlocal strain gradient MEE beams subject to different types of boundary conditions are studied in detail. It is known that the strain gradient parameter lm and the nonlocal parameter le have the ability for increasing and decreasing the effective bending rigidities of the material. In addition, when these parameters are small compared to the characteristic sizes of the material, the size effects can be neglected. In order to illustrate these size

Concluding remarks

The main contribution of the present work are as follows.

  • The nonlocal strain gradient elastic theory is here extended for MEE materials by introducing the nonlocal effect and strain gradient effect into the strain energy from Eq. (2).

  • The thermodynamic framework and variational principle are developed from the theoretical aspect.

  • The general solutions are presented for equation of motion of a MEE beam, and the analytical solutions are given for beams subject to different selections of

Declaration of competing interest

The authors declare that there is no conflict of interest reported in this paper.

Acknowledgements

The work was supported by the Fundamental Research Funds for the Central Universities, CHD (No. 300102219315), the National Natural Science Foundation of China (No. 11902045), and Natural Science Basic Research Plan in Shaanxi Province of China (Nos. 2020JQ-337 and 2020JQ-340).

References (70)

  • M.A. Eltaher et al.

    A review on nonlocal elastic models for bending, buckling, vibrations, and wave propagation of nanoscale beams

    Appl. Math. Model.

    (2016)
  • J. Fernández-Sáez et al.

    Bending of Euler–Bernoulli beams using Eringen's integral formulation: a paradox resolved

    Int. J. Eng. Sci.

    (2016)
  • N.A. Fleck et al.

    Strain gradient plasticity: theory and experiment

    Acta Metall. Mater.

    (1994)
  • S. Hosseini-Hashemi et al.

    An exact analytical approach for free vibration of Mindlin rectangular nano-plates via nonlocal elasticity

    Compos. Struct.

    (2013)
  • A. Jafari et al.

    Size dependency in vibration analysis of nano plates; one problem, different answers

    Eur. J. Mech. Solid.

    (2016)
  • M.H. Kahrobaiyan et al.

    A nonlinear strain gradient beam formulation

    Int. J. Eng. Sci.

    (2011)
  • S.M.H. Karparvarfard et al.

    A geometrically nonlinear beam model based on the second strain gradient theory

    Int. J. Eng. Sci.

    (2015)
  • L.-L. Ke et al.

    Nonlinear vibration of the piezoelectric nanobeams based on the nonlocal theory

    Compos. Struct.

    (2012)
  • D.C.C. Lam et al.

    Experiments and theory in strain gradient elasticity

    J. Mech. Phys. Solid.

    (2003)
  • A.K. Lazopoulos

    Dynamic response of thin strain gradient elastic beams

    Int. J. Mech. Sci.

    (2012)
  • L. Li et al.

    Buckling analysis of size-dependent nonlinear beams based on a nonlocal strain gradient theory

    Int. J. Eng. Sci.

    (2015)
  • L. Li et al.

    Nonlinear bending and free vibration analyses of nonlocal strain gradient beams made of functionally graded material

    Int. J. Eng. Sci.

    (2016)
  • L. Li et al.

    Free vibration analysis of nonlocal strain gradient beams made of functionally graded material

    Int. J. Eng. Sci.

    (2016)
  • C.W. Lim et al.

    A higher-order nonlocal elasticity and strain gradient theory and its applications in wave propagation

    J. Mech. Phys. Solid.

    (2015)
  • R.D. Mindlin

    Second gradient of strain and surface-tension in linear elasticity

    Int. J. Solid Struct.

    (1965)
  • R.D. Mindlin et al.

    On first strain-gradient theories in linear elasticity

    Int. J. Solid Struct.

    (1968)
  • M. Mohandes et al.

    A new approach to reinforce the fiber of nanocomposite reinforced by CNTs to analyze free vibration of hybrid laminated cylindrical shell using beam modal function method

    Eur. J. Mech. Solid.

    (2019)
  • T. Murmu et al.

    Nonlocal elasticity based vibration of initially pre-stressed coupled nanobeam systems

    Eur. J. Mech. Solid.

    (2012)
  • J. Niiranen et al.

    Variational formulations and general boundary conditions for sixth-order boundary value problems of gradient-elastic Kirchhoff plates

    Eur. J. Mech. Solid.

    (2017)
  • C. Polizzotto

    Nonlocal elasticity and related variational principles

    Int. J. Solid Struct.

    (2001)
  • C. Polizzotto

    A gradient elasticity theory for second-grade materials and higher order inertia

    Int. J. Solid Struct.

    (2012)
  • C. Polizzotto

    A second strain gradient elasticity theory with second velocity gradient inertia – Part I: constitutive equations and quasi-static behavior

    Int. J. Solid Struct.

    (2013)
  • H. Rafii-Tabar et al.

    Nonlocal continuum-based modeling of mechanical characteristics of nanoscopic structures

    Phys. Rep.

    (2016)
  • G. Romano et al.

    Constitutive boundary conditions and paradoxes in nonlocal elastic nanobeams

    Int. J. Mech. Sci.

    (2017)
  • S. Sahmani et al.

    On the free vibration response of functionally graded higher-order shear deformable microplates based on the strain gradient elasticity theory

    Compos. Struct.

    (2013)
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