On the Constitutive Equations of Magnetoelectroelastic Solids

In this chapter we present a widely useful composite model for the calculation of magnetoelectric coupling and all other properties of a two-phase multiferroic composite consisting of aligned piezomagnetic (or piezoelectric) spheroidal inclusions in a piezoelectric (or piezomagnetic) matrix. Both perfect and imperfect interface conditions are considered. Among the many features of the properties reported is the intriguing magnetoelectric coupling that signifies the "0 þ 0 ! 1" product effect of the multiferroic composite. It is also reported that, due to the piezoelectric-piezomagnetic interaction, the elastic C 44 of the composite can be substantially higher than that of either of the two phases. We have used the theory to calculate the 17 independent material constants: 5 elastic, 3 piezoelectric, 3 piezomagnetic, 2 dielectric, 2 magnetic, and 2 magnetoelectric coefficients of a transversely isotropic BaTiO 3 -CoFe 2 O 4 composite, and show how these magneto-electro-elastic constants depend on the volume concentration, aspect ratio of inclusions, and the interface condition. We conclude by pointing out that a weak interface model is often required to capture the experimentally measured data of a bulk multiferroic composite. IntroductionA multiferroic composite generally refers to a two-phase composite involving two ferroic phases. Most notable among them is the piezoelectric-piezomagnetic composite. The piezoelectric phase, such as barium titanate (BaTiO 3 , or BTO), lead zirconium titanate (PZT), and lead magnesium titanate-lead titanate (PMT-PT), is generally transversely isotropic, and so is the piezomagnetic phase, such as cobalt ferrite (CoFe 2 O 4 , or CFO), Terfenol-D, lanthanum strontium manganite (LSMO), and nickel ferrite (NFO). In such composites, strong coupling tends to develop Y. Wang • G.J. Weng (*) Department of Mechanical and Aerospace Engineering, Rutgers University, New Brunswick, NJ 08903, USA e-mail: yang.wang@rutgers.edu; weng@jove.rutgers.edu © Springer International Publishing Switzerland 2016 S.A. Meguid (ed.), Advances in Nanocomposites, DOI 10.1007/978-3-319-31662-8_8 189 among the elastic, piezoelectric, dielectric, piezomagnetic, and magnetic properties of the constituent phases. These coupling effects can lead to some unique overall properties for the composite. Among the most intriguing features of the coupled effects is the generation of the magnetoelectric coupling coefficients, α 33 and α 11 , which are absent in either the piezoelectric or the piezomagnetic phase alone. This property enables the composite to exhibit magnetization under an electric field, and electric polarization under a magnetic field. It is the so-called "0 þ 0 ! 1" product effect that makes this class of composite so appealing. For instance, in information storage this coupling allows the data to be written electronically and read magnetically; it does not require the generation of a strong local magnetic field to write. In addition, the strong coupling can also make the elastic constant, C 44 , of...

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“…This leads to eight different variants of the magneto-electro-elastic constitutive equations. Following Soh and Liu (2005), we write…”

Section: Appendix 1: the Eight Variants Of The Coupled Constitutive Ementioning

confidence: 99%

Magnetoelectric Coupling and Overall Properties of a Class of Multiferroic Composites

Wang

Weng

2016

Advances in Nanocomposites

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“…According to [134], eight sets of multi-coupled constitutive equations can be defined, depending on the choice of the independent state variables. In this work for both EI, NEI, from (23) and the two remaining Maxwell equations transformed in (33), the independent variables are taken as…”

Section: Compatibility and Constitutive Equations For Eimentioning

confidence: 99%

“…For an FEM amenable implementation, a Legendre transformation to exchange E by P , H by M and T e by s is applied to (4): to the total thermodynamic energy e T , other energies produced by the exchanged variables must be subtracted to obtain the enthalpy result Π = e T − E e · P − µ 0 H e · M − T e s. A similar procedure, although with different terms, is developed in [134], [141], [47], [88], [16]. In this work, we obtain…”

Section: Compatibility and Constitutive Equations For Eimentioning

confidence: 99%

Multiphysics and Thermodynamic Formulations for Equilibrium and Non-equilibrium Interactions: Non-linear Finite Elements Applied to Multi-coupled Active Materials

Pérez-Aparicio

Palma

Taylor

2015

Arch Computat Methods Eng

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Combining several theories this paper presents a general multiphysics framework applied to the study of coupled and active materials, considering mechanical, electric, magnetic and thermal fields. The framework is based on thermodynamic equilibrium and nonequilibrium interactions, both linked by a two-temperature model.The multi-coupled governing equations are obtained from energy, momentum and entropy balances; the total energy is the sum of thermal, mechanical and electromagnetic parts. The momentum balance considers mechanical plus electromagnetic balances; for the latter the Abraham representation using the Maxwell stress tensor is formulated. This tensor is manipulated to automatically fulfill the angular momentum balance. The entropy balance is formulated using the classical Gibbs equation for equilibrium interactions and non-equilibrium thermodynamics. For the non-linear finite element formulations, this equation requires the transformation of thermoelectric coupling and conductivities into tensorial form.The two-way thermoelastic Biot term introduces damping: thermomechanical, pyromagnetic and pyroelectric converse electromagnetic dynamic interactions. Ponderomotrix and electromagnetic forces are also considered.The governing equations are converted into a variational formulation with the resulting four-field, multicoupled formalism implemented and validated with two custom-made finite elements in the research code FEAP. Standard first-order isoparametric eight-node elements with seven degrees of freedom (dof) per node (three displacements, voltage and magnetic scalar potentials plus two temperatures) are used. Non-linearities and dynamics are solved with Newton-Raphson and Newmark-β algorithms, respectively.Results of thermoelectric, thermoelastic, thermomagnetic, piezoelectric, piezomagnetic, pyroelectric, pyromagnetic and galvanomagnetic interactions are presented, including non-linear dependency on temperature and some second-order interactions.

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“…In recent years, an area of increasing interest is the fracture mechanics of magnetoelectroelastic materials, which combine the ferromagnetic and ferroelectric phases, and most of the achievements have been made on anti-plane problems [1,2] and two-dimensional in-plane problems [3][4][5][6] under the static deformation assumption. The analysis of dynamic fracture problems of magnetoelectroelastic materials is very limited [7][8][9] , with all of the studies made on anti-plane problems.…”

Section: Introductionmentioning

confidence: 99%

A penny-shaped crack in a magnetoelectroelastic layer under radial shear impact loading

Feng

Hui

Han

2007

Acta Mech. Solida Sin.

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This paper analyzes the dynamic magnetoelectroelastic behavior induced by a pennyshaped crack in a magnetoelectroelastic layer. The crack surfaces are subjected to only radial shear impact loading. The Laplace and Hankel transform techniques are employed to reduce the problem to solving a Fredholm integral equation. The dynamic stress intensity factor is obtained and numerically calculated for different layer heights. And the corresponding static solution is given by simple analysis. It is seen that the dynamic stress intensity factor for cracks in a magnetoelectroelastic layer has the same expression as that in a purely elastic material. And the influences of layer height on both the dynamic and static stress intensity factors are insignificant as h/a > 2.

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On the Constitutive Equations of Magnetoelectroelastic Solids

Cited by 82 publications

References 38 publications

Magnetoelectric Coupling and Overall Properties of a Class of Multiferroic Composites

Magnetoelectric Coupling and Overall Properties of a Class of Multiferroic Composites

Multiphysics and Thermodynamic Formulations for Equilibrium and Non-equilibrium Interactions: Non-linear Finite Elements Applied to Multi-coupled Active Materials

A penny-shaped crack in a magnetoelectroelastic layer under radial shear impact loading

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