On the fundamental equations of unsteady anisothermal viscoelastic piezoelectromagnetism

European Journal of Mechanics / A Solids 78 (2019) 103848

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On the fundamental equations of unsteady anisothermal viscoelastic piezoelectromagnetism L.M.B.C. Campos a, F. Moleiro a, *, M.J.S. Silva b a b

LAETA, IDMEC, CCTAE, Instituto Superior T�ecnico, Universidade de Lisboa, Av. Rovisco Pais 1, 1049-001 Lisboa, Portugal Instituto Superior T�ecnico, Universidade de Lisboa, Av. Rovisco Pais 1, 1049-001 Lisboa, Portugal

A R T I C L E I N F O

A B S T R A C T

Keywords: Piezoelectromagnetism Thermal stresses Unsteady motions

The derivation of the piezoelectric equations requires only the Maxwell equations of electrostatics and the momentum equation with electric forces, together with the constitutive equations derived from the first principle of thermodynamics. In the non-isothermal case, the inclusion of thermal conduction requires also the second principle of thermodynamics and then the equation of energy becomes necessary. Furthermore, in the visco­ elastic case, the viscous stresses modify the momentum and energy equations. The sequence outlined above provides the simplest derivation of the fundamental equation of anisothermal viscoelastic piezoelectricity including thermal and viscous dissipation. The equations of piezomagnetism are similar to those of piezoelec­ tricity, replacing the electric by magnetic fields. The coupling of the unsteady electric and magnetic fields through electromagnetic waves interacts with both piezoelectricity and piezomagnetism leading to piezo­ electromagnetism. In the general case of unsteady anisothermal piezoelectromagnetism, the energy, momentum and Maxwell equations specify the temperature, displacement vector, electric field and magnetic field. The co­ efficients involve constitutive and diffusion tensors specifying the properties of matter, generally anisotropic, such as crystals and orthotropic plates. They can also specify the properties of amorphous substances corre­ sponding to the simplest isotropic case.

1. Introduction There is an extensive literature about adaptive structures using piezoelectric and piezomagnetic actuators for active control. Piezocer­ amics, electrostrictive materials, magnetostrictive materials and shape memory alloys are actuators that are currently being investigated to be applied on material systems for controlling and altering the response of structural components. They can be used for precision pointing, vibra­ tion suppression/isolation, solid-state motors, damage detection, and dimensional stability control (Carman and Mitrovic, 1995). Most of the literature considers either piezoelectric or piezomagnetic materials separately. A natural extension is to consider both simultaneously leading to piezoelectromagnetism, as stated in the title of the paper. Electricity and magnetism are coupled in the unsteady case through electromagnetic waves. These are thus included in piezo­ electromagnetism together with thermal, elastic and viscoelastic effects. The following account on the state-of-the-art focuses on piezoelectricity and piezomagnetism separately as an introduction (Section 1) to their

combination in the core of the paper (Sections 2 to 6). For instance, magnetostrictive materials are starting to become handy in many functional materials and therefore they are receiving more attention in the academic and industry worlds. They can convert magnetic energy into mechanical energy and vice-versa because they exhibit mechanical deformation when they are subjected to magnetic fields and exhibit magnetization when they are subjected to mechanical stresses. That means there is a coupling between magnetization and magnetostriction. Consequently, they are useful for actuation and sensing capabilities in many applications. These materials have many advantages because they can provide large output stresses, respond quickly, have high Curie temperature and have a large magneto­ mechanical coupling coefficient (Xu et al., 2013). With these charac­ teristics, magnetostrictive materials have a wide range of applications and they are mainly used in active vibration damping systems, high-precision linear motor, giant magnetostrictive micro-position de­ vices and electrical machines to reduce noise and vibrations (Kar­ unanidhia and Singaperumal, 2010; Sun et al., 2011; Park et al., 2014). To improve the control of the magnetostrictive material devices, it is

* Corresponding author. E-mail addresses: [email protected] (L.M.B.C. Campos), [email protected] (F. Moleiro), manuel.jose.dos.santos.silva@tecnico. ulisboa.pt (M.J.S. Silva). https://doi.org/10.1016/j.euromechsol.2019.103848 Received 30 July 2018; Received in revised form 1 August 2019; Accepted 3 September 2019 Available online 7 September 2019 0997-7538/© 2019 Elsevier Masson SAS. All rights reserved.

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European Journal of Mechanics / A Solids 78 (2019) 103848

Nomenclature

α αij β βij δij �_ ϵij

η λ D Si V

μ μij ν ω

Tij ! x

∂D ∂i Φ

ρ σ σij τ θ �~ Yijkl

υijkl ε

εij ϱ b S

Thermal expansion scalar Thermal expansion tensor Electromagnetic coupling scalar Electromagnetic coupling tensor Kronecker delta Derivative of the variable � with regard to time t Strain tensor Relaxation time Second elastic Lam�e moduli Domain of a system Area of the boundary of the domain D Volume of the domain D Magnetic permeability scalar Magnetic permeability tensor First elastic Lam�e moduli Frequency Total stress tensor

b ij T ξ ζ Ai Bi c Cv Di Ei eijk F gi Gi Hi hi Ji k kij p pijk Q q qijk S t Tij U ui W

Position vector Closed regular boundary of D Gradient operator Electrostatic potential Mass density Ohmic electric conductivity scalar Ohmic electric conductivity tensor Temporal quantity Temperature Variable � represented in the frequency domain Elastic tensor Viscosity tensor Dielectric permittivity scalar

fundamental to know an accurate constitutive relation that describes mathematically a relationship between several variables such as stress, strain, magnetic field and temperature, and to predict accurate re­ sponses if we know the configurations of mechanical loading and mag­ netic inputs. The constitutive model must predict, in an accurate way, non-linear, hysteric and dissipative magnetostriction to optimize the design of magneto-mechanical devices since there is an increase of in­ terest in magnetostrictive materials caused by the commercial avail­ ability of giant magnetostrictive materials. However, developing constitutive relations can be difficult because some experimental results (Moffet et al., 1991) show that the constitu­ tive relations of magnetostrictive materials are non-linear and the magnetization and magnetostrictive curves are seriously affected by several conditions, such as the amount of pre-stress applied to the ma­ terial and the external temperature. For instance, experiments per­ formed on Terfenol-D rod (Moffet et al., 1991; Liang and Zheng, 2007; Gao et al., 2008) showed that the increase of magnetization and magnetostrictive strain with the magnetic field slowed down in low and medium magnetic field regions, while the maximum magnetostricitive strain was higher in the high magnetic field region (where the saturation magnetization was achieved) when compressive pre-stresses were applied along axial direction of the rod. Experiments done on other magnetostrictive materials like Metglas (Spano et al., 1982) and Galfe­ nol (Mahadevan et al., 2010) showed the same behaviour of the magnetostrictive and magnetization curves with the magnetic field (the curves increased and then tended to saturation). Even for the negative magnetostrictive materials like Ni6 (Zhang et al., 2015), the effects of compressive pre-stresses on magnetostrictive and magnetization curves were similar to the effect of tensile forces on positive magnetostrictive

Dielectric permittivity tensor Electric charge density Viscous entropy

Viscous stresses Shear viscosity Bulk viscosity Magnetic potential vector Magnetic induction Speed of the light in vacuum Specific heat Electric displacement Electric field Permutation symbol Free energy Pyroelectric vector Electric current Magnetic field Pyromagnetic vector Heat flux Thermal conductivity scalar Thermal conductivity tensor Piezoelectric scalar Piezoelectric tensor Heat Piezomagnetic scalar Piezomagnetic tensor Elastic entropy Time Elastic stresses Internal energy Displacement vector Work

materials previously mentioned. To summarize, in the positive magne­ tostriction, compressive mechanical stress decreases the magnetization of materials expanding under the external magnetic field, while tensile stress does the opposite. On the other hand, the materials with negative magnetostriction contracting in the direction of the magnetic field with compressive mechanical stresses exhibit the magnetization property increased (Cullity and Graham, 2009; Sablik et al., 1987). While non-linearities pose modelling problems, they could also be beneficial for control in the response of a structural system, such as vibration control. They can induce not only elongations but also significant stiff­ ness changes (Clark, 1980) and both can be used to alter vibration characteristics. Since non-linear properties are attractive but they are difficult to model in practical applications, some works develop a non-linear material model to overcome this obstacle. The experimental results can provide the fundamental features of magnetostrictive constitutive models because we can conclude that prestress and temperature have similar effects for several kinds of these materials and therefore, the experimental data can help to develop a general constitutive relation for these materials. Heretofore, modelling approaches for magnetostrictive materials relied on the theoretical de­ velopments constructed for linear piezoelectric material systems (IEEE, 1971). These approaches can provide an analytical framework for approximating the response in specific regimes. However, they neglect coupling terms that must be included to establish accurate predictive capabilities required in structural applications. Some non-linear coupling terms which must be included are the effect of pre-load, magnetic field intensity and temperature dependence on the mechani­ cal response (Clark, 1993). Several constitutive models have been built to describe the features above leading to non-linear coupled 2

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European Journal of Mechanics / A Solids 78 (2019) 103848

magnetostrictive relations (because of their behaviour, it is not feasible to describe these relations with linear coupled constitutive coefficients). Some approaches to the development of non-linear constitutive relations focused on electromechanical systems (PartonKudryavtsev, 1988; Ikeda, 1990) using thermodynamics and energy methods. Furthermore, Joshi (1992) presented a non-linear constitutive relation for piezoelectric materials explicitly defining symmetry relations for the non-linear terms. Concerning magnetostrictive materials, Clark (1980) describes phenomenological approaches to develop constitutive relations for these materials and other analytical approaches have focused on hysteresis predictions for electromechanical materials (Sreeram et al., 1993). The Zheng-Liu model (Zheng and Liu, 2005) can adequately simulate the magnetostrictive strain under low, medium and high magnetic fields, and moreover can simulate the saturation phenomenon of magneto­ strictive strain (this model divides the elastic strain produced by pre-stress into non-linear part associated with the domain rotation and linear part that is not related with the domain rotation) although it does not consider the variations of temperature. Regarding the influence of the temperature on magnetostrictive curves, several constitutive models can be built to describe the magneto-thermo-mechanical coupling (Zheng and Sun, 2007; Wang and Zhou, 2010). Some modelling works about magneto-mechanical effects follow the phenomenological description as in the Preisach model (Adly et al., 1991a) or utilize the physically based energy aspects as in the Jiles-Atherton model (Sablik and Jiles, 1993). Otherwise, another class of constitutive models makes use of the thermodynamic principles whose goal is to describe the magnetostrictive material response through the definition of a specific free energy function in the Taylor series form (Carman and Mitrovic, 1995; Wan et al., 2003) or through the introduction of internal state variables (Zheng and Sun, 2007; Lin­ nemann et al., 2009). Some of them are based on the Gibbs free energy (Zhang et al., 2015). In (Zhang et al., 2015), the free energy was expanded into a polynomial form using Taylor series. The relationship between stress and strain and the relationship between magnetization and magnetic field with the polynomial form were obtained with the thermodynamic relations. A compact magneto-mechanical coupled constitutive model was successfully established by introducing a non-linear function to describe non-linear strain and magnetization caused by domain rotations. The numerically simulated results of the magnetostrictive strain and magnetization curves and the effect of pre-stress on them show a good agreement with the non-linear coupled constitutive model based on Gibbs free energy in the regions of low, moderate and high magnetic fields, but in (Zhang et al., 2015) it was not considered variations of temperature. Therefore, the comparison demonstrated that building constitutive relations may show the non-linear magneto-mechanical coupling characteristics, and the model has therefore many applications for magnetostrictive materials. The work (Carman and Mitrovic, 1995) intends to propose a non-linear constitutive relation for magnetostrictive materials. That includes non-linear coupling effects arising between temperature or pre-stress and magnetic field strengths where the relations were derived from thermodynamic equations using Gibbs free energy expanded in a Taylor series. The expanding form solely have the material constants that can be determined from experimental data available in the litera­ ture. Relations between the non-linear material coefficients and linear coefficients present in the literature are derived by considering that the magnetostrictive material is placed in a biased magnetic field and operated at a small permuted field strength. By comparing experimental results obtained on a Terfenol-D rod operating under both magnetic field and stress external stimuli with theoretical values, the accuracy of the non-linear constitutive relation is evaluated. Results indicate that the model predicts appropriately the non-linear relations between strain and magnetic field in specific regimes (low and medium fields). The article (Ho, 2016) also presents a constitutive model for the field-induced magnetostriction that is mainly attributed to the

reorientation of magnetic moments associated with applied magnetic fields (Cullity and Graham, 2009). The constitutive model can account for the hysteretic properties of magnetization and magnetostriction and follows the first and second laws of thermodynamics. The internal state variables were introduced to characterize irreversible thermodynamics and consequently, a decomposition of the magnetic flux density and the strain was made to account the reversible and irreversible parts. This constitutive model was able to describe the magneto-mechanical behaviour by comparing simulation results with the experimental data. Again, it was concluded that the slope of the magnetic hysteresis decreases with an increase of the compressive pre-stresses and the magnitude of the magnetic field increases with an increase of the pre-load to reach the same magnetostrictive strain. Otherwise, there are some materials like lead zirconate titanate (PZT) that can be considered as ferroelectric materials because they can exhibit non-linear coupling between mechanical and non-mechanical responses in addition to the time-dependent response that depend on the magnitude of the external effects such as mechanical, electrical and thermal. Under these stimulations, they have been used in structural health-monitoring systems, actuation and sensing applications. The article (Muliana, 2011) focus on the application of ferroelectric materials as actuators where high electric fields are applied to produce desired deformations. Non-linear behaviours, depicted in electro-mechanical hysteresis responses of PZT subjected to cyclic electric fields, with amplitudes above the coercive electric field, due to polarization switching, were shown in (Schmidt, 1981; Chan and Hagood, 1994). The effect of compressive stresses that are applied parallel to the polling axis on the electro-mechanical hysteresis behav­ iours of PZT-based ceramics have also been studied (Arndt et al., 1984; Hwang et al., 1995) and it was concluded that compressive stress ac­ celerates depolarization in the ferroelectric ceramics. Experimental studies on PZTs at various isothermal temperatures (Hooker, 1998) show that the hysteresis response depends also on their ambient tem­ peratures. Furthermore, electrical and mechanical responses of ferro­ electric materials are time-dependent (Fett and Thun, 1998; Hall, 2001). The effects of electric field and mechanical stress are important to predict non-linear and hysteresis behaviours of ferroelectric materials and establish constitutive models. They can be classified as purely phenomenological models, derived from classical mechanics and ther­ modynamics relations principles, or as models related to micro­ mechanics that incorporate changes in the polycrystalline structure under external excitations. About the phenomenological models for piezoelectric materials, the electromechanical hysteresis response is derived from the thermodynamic potential that is decomposed into reversible and irreversible parts and the last part is associated with the residual electric polarization (Bassiouny et al., 1988a, 1988b; Bassiouny and Maugin, 1989). Other models present equations for describing a slow hysteresis behaviour between electric and elastic responses where the material irreversibility is associated with the residual response be­ tween polarization and electric field during polarization switching (Huang and Tiersten, 1998). On the other side, a hysteresis model be­ tween the electric field and polarization based on free-energy of a single crystal structure can also be considered and then a stochastic homoge­ nization approach is utilized to obtain macroscopic behaviours of polycrystal piezoceramics (Smith et al., 2003). For describing polariza­ tion switching in ferroelectric materials, there are other micromechanics based constitutive models that can be found on several references (Fan et al., 1999; Shilo et al., 2007). Further discussion on constitutive models developed for ferroelectric ceramics can be found in Smith (2005). There have been several experimental and theoretical studies on understanding polarization switching behaviour in piezoelectric mate­ rials. The article (Muliana, 2011) focus on understanding the electro-mechanical response and polarization switching behaviour of ferroelectric ceramics that depend strongly upon the external electric fields and ambient temperatures and is time and rate-dependent within a 3

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European Journal of Mechanics / A Solids 78 (2019) 103848

context of dielectric and piezoelectric relaxation. It presents a non-linear time-dependent polarization model based on a single integral function and the total polarization is additively decomposed into the time-dependent and residual components that are produced by the po­ larization switching in the ferroelectric materials. The development of the constitutive model for magnetostrictive material is similar to those present in the literature for piezoelectric and electrostrictive material systems. The similarity between the models suggests that a unified generic constitutive relation providing the response of a wide range of electro-magneto-thermal-mechanical ma­ terials is possible. Magnetostrictive alloys, that show an inherent coupling between magnetic field and deformation, and ferroelectric ceramics, that have a coupling between electric field and deformation, have many applica­ tions for actuators and sensors and both materials show non-linear behaviour. There are several works, not only for ferroelectric ceramics (Chen and Lynch, 2001; Hwang et al., 1995) but also for magnetostrictive materials (Calkins et al., 2000) that present microscopically motivated models which approximate the switching process, modelled by energy criterion, for every single crystallite, and then by averaging over numerous union domains, the macroscopic behaviour is obtained. Consequently, they generally lead to a large number of internal variables that increase the numerical effort. On the other hand, macroscopic constitutive models are based on phenomenological approaches of the material behaviour. Some of them are purely phenomenological and not thermodynamically consistent (Adly et al., 1991b; Pasco and Berry, 2004). Another class of macro­ scopic constitutive models about the principles of thermodynamics is formulated. The definition of a specific free energy function determines the material behaviour. For piezoelectric materials, the strains, the electric field, and the temperature are often used as independent vari­ ables. For magnetostrictive alloys, the magnetization is used instead of the electric field. In the article (Linnemann et al., 2009), a thermodynamic consistent constitutive model is presented which accounts for hysteresis effects in ferroelectric ceramics and magnetostrictive materials. The electric and magnetic fields and the strains are split. The irreversible fields and the irreversible strains serve as internal variables describing the domain state of the material where the irreversible field strength determines, with a one-to-one relation, the irreversible strains which arise from domain switching processes. The formulation is based on a free energy function and a switching criterion and the constitutive relations are able to approximate the hysteresis curves and the related butterfly hysteresis curves of the ferromagnetic and ferroelectric materials. The constitutive models are therefore very useful to predict the behaviour of piezoelectric and piezomagnetic materials and there are several ways to establish them, some of them using the concept of free energy and the thermodynamic principles. The relations are very similar and they are derived from the same fundamental principles. That means that it is possible to develop universal constitutive relations, coupling the equations of piezoelectricity and piezomagnetism, which apply not only to elastic materials but also to viscoelastic materials, as it will be noticed in this paper. The main goal of this work is to couple the un­ steady electric and magnetic fields leading to piezoelectromagnetism. The basic (i) unsteady piezoelectricity combining elastic stresses with electrostatic field (Section 2) may be extended to: (ii) nonisothermal conditions including thermal conduction (Section 3); (iii) viscoelastic materials with viscosity tensor reducible or not to a relax­ ation time (Section 4). The sequence (i) to (iii) provides the simplest derivation of the fundamental equations of anisothermal viscoelastic piezoelectricity and can be extended to include as well as piezomag­ netism and piezoelectromagnetism (Section 5). The number of inde­ pendent constitutive and diffusive properties is reduced by material symmetries (Appendix A-Appendix C). The starting point (Section 2) is the first principle of thermodynamics

including the heat and work of elastic stresses and electric forces (Sub­ section 2.1); omitting temperature changes in the first instance, it leads to the constitutive equations for the elastic stresses and electric displacement, as functions of the strains and electric field. The co­ efficients are the elastic, dielectric and piezoelectric tensors (Subsection 2.2). Substitution in the Maxwell equations of electrostatics and mo­ mentum equation specifies the coupled equations for the displacement vector and electrostatic potential (Subsection 2.3). In non-adiabatic conditions (Section 3), the heat must be included in the first principle of thermodynamics. The entropy, as the dependent variable, and the temperature, as the independent variable, are added to the constitutive equations and the coefficients include the specific heat, the pyroelectric vector and the thermal expansion tensor (Subsection 3.1). In the presence of heat conduction, the second principle of ther­ modynamics specifies entropy production in the equation of energy (Subsection 3.2) The three equations of momentum, electrostatics and energy couple the displacement vector, electrostatic potential and temperature (Subsection 3.3). In the case of viscoelastic material (Section 4), the viscous stresses are added to the elastic stresses, modifying one constitutive equation, and hence the momentum equation (Subsection 4.1); the viscous dissi­ pation also adds to entropy production and modifies the equation of energy (Subsection 4.2). The set of three equations of momentum, electrostatics and energy (Subsection 4.3) has several simplifications, e. g. (i) with viscosity specified by a relaxation time, (ii) in the linear case or (iii) in the steady case. Considering piezomagnetism, in addition to piezoelectricity, the magnetic energy in the first principle of thermodynamics will be necessary (Subsection 5.1), implying another constitutive relation for the magnetic induction. The piezomagnetism also implies the addition of the magnetic field as the independent variable. New coefficients will appear such as the magnetic permeability, electromagnetic coupling tensor, piezomagnetic tensor and pyromagnetic vector. If the electric currents associated with the magnetic field are due to Ohmic electrical conduction, there is entropy production by the Joule effect appearing in the energy equation (Subsection 5.2). The magnetic force appears in the momentum equation and there is another Maxwell equation for the magnetic field, thus totalling four fundamental equations (Subsection 5.3). Since the unsteady electric and magnetic fields are coupled in electromagnetic waves, they interact with piezoelectricity and piezo­ magnetism leading in general to piezoelectromagnetism (Section 5). The maximum number of components of the constitutive and diffu­ sion tensors applies to anisotropic materials without any symmetries (Appendix A). In the case of crystallographic systems or classes and orthotropic plates, the number of independent material properties de­ creases as more symmetries are included (Appendix B). The number of independent material properties is minimum for isotropic materials, simplifying the fundamental equations (Appendix C). 2. Two fundamental equations of adiabatic piezoelectricity The two fundamental equations of piezoelectricity are the mo­ mentum equation and equation of electrostatics (Subsection 2.3), involving the elastic, dielectric and piezoelectric tensors in the consti­ tutive relations (Subsection 2.2) derived from the first principle of thermodynamics involving the work of elastic stresses and electric forces (Subsection 2.1). 2.1. Work of the elastic stresses and electric forces The first principle of thermodynamics (Guggenheim, 1949; Callen, 1960) states that the internal energy is the sum of the heat θdS where θ is the temperature and S the entropy with the work dW, dU ¼ θdS þ dW;

4

(1a)

L.M.B.C. Campos et al.

dW ¼ Tij dϵij

European Journal of Mechanics / A Solids 78 (2019) 103848

(1b)

Di dEi ;

(iii) the piezoelectric tensor

where is considered the work of: (i) the elastic stresses Tij on the strains ϵij ; (ii) the electric displacement Di on the electric field Ei . The exact � differentials in (1a) and (1b) show that the internal energy U S; ϵij ; Ei is a function of state depending on the entropy, strains and electric field. The entropy S is replaced by the temperature θ that is easier to measure. As in (Muliana, 2011), the constitutive model for piezoelectric material is based on thermodynamics and classical mechanics approaches regarding continuous bodies and in this case the model is described in terms of stress, strain, electric field and electric displacement, as usually it is made for piezoelectric materials. Thermal effect is also considered and so the temperature and entropy must be incorporated. We can develop different constitutive relations depending upon the choice of independent variables (Carman and Mitrovic, 1995). In this work, the independent variables are temperature, magnetic field and strain and the thermodynamic function is expressed in terms of these variables. The free energy is introduced as a Legendre transform of the internal energy exchanging entropy for temperature, F¼U

pijk �

Tij ¼ ϒ ijkl ϵkl

Di dEi

Sdθ;

Di ¼

S¼

∂F ; ∂ϵij ∂F ; ∂Ei ∂F : ∂θ

The first fundamental equation (Stratton, 1941; Campos, 2011) is the Maxwell equation for the electric charge density,

(3)

(4c)

ρu€i

ϱEi ¼ ∂j Tij

¼ ϒ ijkl ∂j ϵkl

pijk ∂j Ek ;

(11a) (11b)

where ρ is the mass density, dot denotes derivative with regard to time, for instance, u_i �

∂ui ; ∂t

(11c)

and (9a) was used in (11b). For a steady magnetic induction, (12a)

B_ i ¼ 0; the electrostatic field is irrotational (Campos, 2011),

(5b)

∂i Ej ¼ ∂j Ei ;

(5c)

and derives from an electrostatic potential (Campos, 2011), Ei ¼

(12b)

(12c)

∂i Φ;

also, the strains are related to the displacement vector through the next equation, as stated in (Carman and Mitrovic, 1995):

(i) the elastic tensor

2ϵij ¼ ∂i uj þ ∂j ui :

(6)

(12d)

Substitution of (12c) and (12d) in (10b) and (11b) leads to the fundamental equations of electricity (13) and momentum (14) for the particle displacement and electrostatic potential:

(ii) the dielectric tensor

∂D ∂2 F ∂2 F εij � i ¼ ¼ ¼ εji ; ∂Ej ∂Ej ∂Ei ∂Ei ∂Ej

(10c)

(4b)

The coefficients specified by the partial derivatives, according to (Muliana, 2011), are:

∂Tij ∂2 F ∂2 F ¼ ¼ ¼ ϒ klij ; ∂ϵkl ∂ϵkl ∂ϵij ∂ϵij ∂ϵkl

(10b)

that may be replaced by the covariant derivative if curvilinear co­ ordinates are used. The second fundamental equation is the momentum equation (11a) balancing the inertia force against the divergence of the elastic stresses and the electric force,

the stress tensor (5b) and electric displacement (5c) depend only on the strains ϵij and electric field Ei (Joshi, 1992),

ϒ ijkl �

¼ εij ∂i Ej þ pjki ∂i ϵjk ;

∂i � ∂=∂xi :

(5a)

dDi ¼ εij dEj þ pjki dϵjk :

(10a)

(4a)

Assuming isothermal conditions,

pijk dEk ;

ϱ ¼ ∂i Di

where the electric displacement (9b) may be replaced in the previous equation, leading to (10b), and noting that ∂i denotes the gradient operator,

2.2. Elastic, dielectric and piezoelectric tensors

dTij ¼ ϒ ijkl dϵkl

(9b)

2.3. Particle displacement and electrostatic potential

as stated in (Muliana, 2011). From (4a-4c), it follows that the stresses Tij , electric displacement Di and entropy S each depend on the strains ϵij , electric field Ei and temperature θ.

θ ¼ const:;

(9a)

that is, the constitutive relations (5b) and (5c) become respectively (9a) and (9b), noticed also in (Joshi, 1992).

shows that the free energy depends on the strains ϵij , electric field Ei , and temperature θ. The corresponding partial derivatives specify respec­ tively the stresses (4a), minus the electric displacement (4b) and minus the entropy (4c), Tij ¼

pijk Ek ;

Di ¼ εij Ej þ pjki ϵjk ;

its exact differential, dF ¼ Tij dϵij

(8)

that appear both in (5b) and (5c). If the constitutive tensors ϒ ijkl , εij and pijk are constant, integrating equations (5b) and (5c), we arrive at

(2)

θS;

∂Tij ∂2 F ∂2 F ∂Dk ¼ ¼ ¼ ¼ pjik ∂Ek ∂Ek ∂ϵij ∂ϵij ∂Ek ∂ϵij

ϱ ¼ εij ∂i ∂j Φ

(7)

5

� 1 pjki ∂i ∂j uk þ ∂i ∂k uj ; 2

(13)

L.M.B.C. Campos et al.

European Journal of Mechanics / A Solids 78 (2019) 103848

� 1 ϒ ijkl ∂j ∂k ul þ ∂j ∂l uk 2

ρu€i

pijk ∂j ∂k Φ ¼ ϱEi ;

where kij is the thermal conductivity tensor. The second principle of thermodynamics specifies the entropy production, � θS_ ¼ Ji ∂i θ ¼ kij ð∂i θÞ ∂j θ > 0; (20a)

(14)

the r.h.s. of (14) is non-linear on account of (13), and may be omitted in the linear approximation.

(20b)

kij ¼ kji ;

3. Three fundamental equations of anisothermal piezoelectricity

that must be positive implying that the thermal conductivity tensor is symmetric (20b) and positive-definite (20a). The equation of energy,

In non-adiabatic conditions, the heat appears in the internal energy (1a) and the free energy (3) has the temperature as an independent variable. The temperature appears in the constitutive relations for the elastic stresses and electric displacement as well as in a third constitutive equation for the entropy (Subsection 3.1); the specific heat, the pyro­ electric vector and the thermal expansion tensor are added to the constitutive equations as coefficients. In anisothermal conditions, heat conduction leads to entropy production in the energy equation (Sub­ section 3.2). The three fundamental equations of momentum, electro­ statics and energy couple the displacement vector, electrostatic potential and temperature (Subsection 3.3).

∂Q þ ∂i Ji ¼ 0; ∂t

(21)

considering the substitution of (16) and the heat flux equation, becomes θ

∂S ¼ kij ∂i ∂j θ: ∂t

(22)

This is the third fundamental equation that adds to the electrostatic (10a) and momentum (11a) equations. 3.3. Displacement vector, electrostatic potential and temperature

3.1. Specific heat, pyroelectric vector and thermal expansion tensor

The equations (10b) and (11b) are no longer valid, since the constitutive relations (15a-15c) should be substituted in electrostatic (10a), momentum (11a) and energy (22) equations leading respectively to (23-25):

In non-adiabatic conditions, the inclusion of the heat in the internal energy (1a) implies that the free energy (3) depends on the temperature, and thus so do the elastic stresses (15a), electric displacement (15b) and entropy (15c). The constitutive equations, according to (Muliana, 2011), become:

ϱ ¼ εij ∂i Ej þ pjki ∂i ϵjk þ gi ∂i θ;

(23)

(15a)

ρu€

(24)

dDi ¼ εij dEj þ pjki dϵjk þ gi dθ;

(15b)

kij ∂i ∂j θ ¼ Cv θ_

dθ θ

(15c)

dTij ¼ ϒ ijkl dϵkl

dS ¼

pijk dEk þ αij dθ;

αij dϵij þ gi dEi þ Cv :

(25)

θαij ϵ_ ij þ θgi E_ i :

(16)

�

1 2

(26)

εij ∂i ∂j Φ þ pjki ∂i ∂j uk þ ∂i ∂k uj þ gi ∂i θ;

ϱ¼

� 1 ϒ ijkl ∂j ∂k ul þ ∂j ∂l uk 2

ρu€i Cv θ_

(ii) the pyroelectric vector

∂Di ∂2 F ∂2 F ∂S gi � ¼ ; ¼ ¼ ∂θ ∂θ∂Ei ∂Ei ∂θ ∂Ei

pijk ∂j Ek þ αij ∂j θ;

For a steady magnetic induction, that is, when the equations (12a) and (12b) hold, the electrostatic potential (12c) may be used together with the displacement vector (12d) leading to its coupling with the temperature on the three fundamental equations of electricity (26), momentum (27) and energy (28):

The new constitutive coefficients in the three previous equations in addition to those already presented in (5b) and (5c) are, regarding (Muliana, 2011): (i) the specific heat at constant strain and electric field � � � � ∂S ∂Q Cv � θ ¼ ; ∂θ ϵij ; Ej ∂θ ϵij ; Ej

ϱEi ¼ ϒ ijkl ∂j ϵkl

� 1 θαij ∂i u_j þ ∂j u_i 2

pijk ∂j ∂k Φ

αij ∂j θ ¼

ϱ∂i Φ;

_ ¼ kij ∂i ∂j θ: θgi ∂i Φ

(27) (28)

In the steady case, (17)

∂t ¼ 0;

(29a)

the energy equation decouples (29b) and specifies the temperature, (iii) the thermal expansion tensor 2

αij �

2

∂Tij ∂F ∂F ∂S ¼ ¼ αji : ¼ ¼ ∂θ ∂θ∂ϵij ∂ϵij ∂θ ∂ϵij

kij ∂i ∂j θ ¼ 0:

The temperature, as a solution of (29b), is substituted in the elec­ trostatic (26) and momentum (27) equations, the latter without the inertia force. The electric force on the r.h.s. of (27) can be omitted in the linear case.

(18)

If the constitutive coefficients are all constant, the differentials may be omitted from (15a-15c).

4. Anisothermal viscoelastic piezoelectricity In the case of a viscoelastic material, the viscous stresses (Batchelor, 1967; Campos, 2012) must be included in the momentum equation (Subsection 4.1); the viscous dissipation adds to entropy production in the energy equation (Subsection 4.2) and also appears in the diffusive energy flux. The assumption that the viscous tensor is related to the elastic tensor through a relaxation time is more readily implemented in the frequency domain (Subsection 4.3).

3.2. Thermal conductivity and energy equation In the anisothermal case, the temperature gradients lead to the ex­ istence of a heat flux specified by the Fourier law (Carman and Mitrovic, 1995), Ji ¼

kij ∂j θ;

(29b)

(19)

6

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European Journal of Mechanics / A Solids 78 (2019) 103848

4.1. Elastic and viscous stress tensors

be taken as proportional to the elastic tensor multiplied by minus a constant relaxation time and then we have

In the case of a viscoelastic material, the total stress tensor consists of the elastic stresses (15a) plus the viscous stresses proportional to the rate-of-strain ϵ_ kl through a viscosity tensor:

regarding equation (15a). Changing to the frequency domain,

The electrostatic equation (26) is not affected, but the momentum equation (11a) changes to

ρu€

ϱEi ¼ ∂j T ij ¼ ∂j Tij þ υijkl ∂j ϵ_ kl :

Zþ∞ T~ij ð! x ; ωÞ �

(31)

∞

~εkl ð! x ; ωÞ �

¼

eiωt εkl ð! x ; tÞ dt; ∞

Zþ∞

ρu€i

1 υijkl 2

eiωt T ij ð! x ; tÞ dt; Zþ∞

The substitution of the constitutive relation (15a) and electrostatic potential (12c) in (31) leads to the momentum equation � 1 ϒ ijkl ∂j ∂k ul ∂j ∂l uk 2 � ∂j ∂k u_l ∂j ∂l u_k pijk ∂j ∂k Φ αij ∂j θ

(38)

ηϵ_ kl ;

¼ ϒ ijkl ϵkl

(30)

b ij : T ij ¼ Tij þ υijkl ϵ_ kl ¼ Tij þ T

α�ij θ ¼ ϒ ijkl ϵ�kl þ υijkl ϵ_ kl

T ij þ pijk Ek

~ θð! x ; ωÞ �

(32)

(39)

eiωt θð! x ; tÞ dt; ∞

the relaxation time may depend on frequency:

ϱ∂i Φ;

(40)

and comparing with (27), a new viscous term was added.

~k T~ij þ pijk E

4.2. Viscous dissipation and energy flux

The constitutive relation (40) is more general than (38) because, returning to the time domain, it involves a convolution:

The work of the viscous stresses on the rate-of-strain adds to the entropy production (20a) by thermal conduction: � b ij ϵ_ ij ¼ υijkl ϵ_ kl ϵ_ ij : θS_ kij ð∂i θÞ ∂j θ ¼ T (33)

D

Zþ∞

D

ϱ ~¼

∂D

Z u_j ρu€i

ϱEi

(43) (44)

The unsteady magnetic induction is coupled to the unsteady electric field generalizing piezoelectricity and piezomagnetism (Cady, 1946; Landau and Lifshitz, 1959) to piezoelectromagnetism. This generaliza­ tion involves: (i) a fourth constitutive relation for the magnetic induc­ tion and a fourth term in every constitutive relation due to the magnetic field (Subsection 5.1); (ii) the Ohmic electric current in the Maxwell equations and associated resistive dissipation by the Joule effect in the energy equation (Subsection 5.2); (iii) a set of four fundamental equa­ tions of electricity, magnetism, momentum and energy including the scalar electric potential, vector magnetic potential, temperature and displacement vector as variables (Subsection 5.3).

where on the r.h.s.: (i) the second term is non-dissipative; (ii) the first term specifies the viscous energy flux which must be added to the thermal conduction (19), leading to (36)

The energy equation (21) including viscous dissipation and consid­ ering the equations (33) and (36) becomes:

Cv θ_

αij ∂j ~θ ¼ 0;

(42)

5. Unsteady piezoelectromagnetism

D

� � 1 υijkl ∂i u_j ð∂k u_l þ ∂l u_k Þ ¼ 2 � 1 _ θgi ∂i Φ: θαij ∂i u_j þ ∂j u_i 2

�

The non-linear terms have been omitted.

�

kij ∂i ∂j θ

ul þ ∂j ∂l ~ uk iωηðωÞ� ∂j ∂k ~

kij ∂i ∂j ~ θ ¼ iωCv ~ θ:

∂j Tij dV ;

b ij u_j ¼ υijkl u_j ϵ_ kl : Ji þ kij ∂j θ ¼ T

1 ϒ ijkl ½1 2

~ þpijk ∂j ∂k Φ

(35)

i

(41)

�

1 2

~ þ pjki ∂i ∂j ~ uk þ ∂i ∂k ~ uj þ gi ∂i ~θ; εij ∂i ∂j Φ

ρω2 u~i

Z b ij u_j dS T

τÞ dτ:

It is simpler to write the fundamental equations of electricity (26), momentum (32) and energy (37) in the frequency domain, leading respectively to (42-44):

D

¼

ϵ_ kl ð! x ; τÞηðt ∞

taking into account the symmetry of the viscous stresses in ði; jÞ. An integration by parts leads to Z Z � � �� b ij dV u_j ∂j T θb S_ dV ¼ ∂i Tb ij u_j D

iωηðωÞ �

αij θ ¼ ϒ ijkl ϵkl þ ϒ ijkl

T ij þ pijk Ek

Considering a domain D with a closed regular boundary ∂D , the entropy production due to viscous dissipation is given by � � Z Z Z b ij ϵ_ ij dV ¼ T b ij ∂i u_j dV ; θb S_ dV ¼ T (34) D

αij ~θ ¼ ϒ ijkl~εkl ½1

(37)

5.1. Magnetic permeability and electromagnetic coupling tensors and pyromagnetic vector

The three fundamental equations of electrostatics (26), momentum (32) and energy (37) couple the particle displacement, electrostatic potential and temperature. In the linear approximation, the r.h.s. of (32) can be omitted. Additionally, in the steady case (29a), the energy equation (29b) decouples the temperature.

Adding the magnetic energy to (1a) and (1b), the internal energy leads to dU ¼ Tij dϵij

Di dEi

Hi dBi þ θdS

(45)

where the magnetic field Hi and magnetic induction Bi also appear in the free energy (repeated here for convenience),

4.3. Relaxation time in the frequency domain In the viscoelastic stress-strain relation (30), the viscous tensor may 7

L.M.B.C. Campos et al.

F¼U

European Journal of Mechanics / A Solids 78 (2019) 103848

(Campos, 2011),

(46)

θS;

dF ¼ Tij dϵij

Di dEi

Hi dBi

Sdθ:

(47a) θS_

The constitutive coefficients are indicated in (4a-4c) and in the next equation:

∂F : ∂Bi

Hi ¼

qkij dBk þ αij dθ;

(54b)

and entropy production requires that the Ohmic conductivity tensor is positive definite (54a) and implies that it is symmetric (54b). The Joule dissipation appears in the energy equation (21) that becomes

It follows that ðTij ; Di ; Hi ; SÞ depend on ðϵkl ; Ek ; Bk ; θÞ leading to the constitutive relations pijk dEk

The Joule dissipation is added in (20a) due to thermal conduction, � kij ð∂i θÞ ∂j θ ¼ Gi Ei ¼ σ ij Ei Ej > 0; (54a)

σ ij ¼ σ ji ;

(47b)

dTij ¼ ϒ ijkl dϵkl

(53c)

Gi ¼ σij Ej :

and whose its differential leads in this case to

(55)

θS_ ¼ kij ∂i ∂j θ þ σ ij Ei Ej :

(48a)

dDi ¼ εij dEj þ pjki dϵjk þ βij dBj þ gi dθ;

(48b)

In the momentum equation (11a), the magnetic force (56a) must be included,

dHi ¼ μij 1 dBj þ qijk dϵjk þ βji dEj þ hi dθ;

(48c)

ρu€i

dS ¼ Cv

dθ θ

αij dϵij þ gi dEi þ hi dBi :

(48d)

The constitutive relations (48a-48d) may be substituted in the fundamental equations of energy (55), momentum (56), electricity (10a) and magnetism (53b) with thermal and electric dissipation, leading respectively to:

(49)

(ii) the inverse magnetic permeability tensor

μij

(56b)

5.3. Scalar electric and vector magnetic potentials

¼ qikj ;

1

(56a)

where (53c) was used in (56b).

(i) the piezomagnetic tensor

∂Hi ∂2 F ∂2 F ∂Hj � ¼ ¼ ¼ ∂Bj ∂Bj ∂Bi ∂Bi ∂Bj ∂Bi

1 ϱEi ¼ eijk Gj Bk c

1 ¼ eijk σjl El Bk ; c

The constitutive coefficients that appear from (48a-48d), besides (68) and (16-18), regarding (Carman and Mitrovic, 1995), are:

∂Hi ∂2 F ∂2 F ∂Tjk qijk � ¼ ¼ ¼ ∂ϵjk ∂ϵjk ∂Bi ∂Bi ∂ϵjk ∂Bi

∂j Tij

kij ∂i ∂j θ þ σij Ei Ej ¼ Cv θ_ (50)

ρu€

¼ μji 1 ;

ϒ ijkl ∂j ϵkl þ pijk ∂j Ek þ qkij ∂j Bk

1 c

αij ∂j θ ¼ ϱEi þ eijk Bk σjl El ;

∂Di ∂2 F ∂2 F ∂Hj βij � ¼ ¼ ¼ ∂Bj ∂Bj ∂Ei ∂Ei ∂Bj ∂Ei

σij Ej þ εij E_ j þ pijk ϵ_ jk þ βij B_ j þ gi θ_ ¼ c eijk μkl1 ∂j Bk þ qkln ∂j ϵln þ βlk ∂j El þ hk ∂j θ (51)

�

(61a)

Bi ¼ eijk ∂j Ak ; (iv) the pyromagnetic vector hi �

2

∂Hi ∂F ∂F ∂S ¼ : ¼ ¼ ∂θ ∂θ∂Bi ∂Bi ∂θ ∂Bi

(60)

The magnetic induction (53a) is specified by the vector magnetic potential (Campos, 2011):

¼ βji ;

2

(58) (59)

ϱ ¼ εij ∂i Ej þ pjki ∂i ϵjk þ βij ∂i Bj þ gi ∂i θ;

(iii) the electromagnetic coupling tensor

(57)

θαij ϵ_ ij þ θgi E_ i þ θhi B_ i ;

the unsteady electric field satisfies (52)

eijk ∂j Ek ¼

1_ Bi c

(61b)

leading to the scalar electric potential Φ in If all constitutive coefficients are constant, the differentials in (48a48d) may be omitted.

Ei ¼

The pair of Maxwell magnetic equations (Campos, 2011) is (53a) and (53b) where c is the speed of light in vacuum: (53a)

1 eijk ∂j Hk ¼ ðGi þ D_ i Þ; c

(53b)

1_ Ai : c

(61c)

Substitution of (12d), (61a) and (61c) in equations (57)–(60) spec­ ifies the particle displacement, temperature, scalar electric and vector magnetic potential, through the coupled equation of energy (57), mo­ mentum (58), electricity (59) and magnetism (60), valid for an elastic medium in the presence of unsteady electric and magnetic fields and thermal and electrical conduction; these are the equations of unsteady, dissipative, anisothermal piezoelectromagnetism coupling elastic and electromagnetic damped waves. The extension to viscoelastic materials can be made as before (Section 4).

5.2. Ohmic electrical resistance and Joule dissipation

∂i Bi ¼ 0;

∂i Φ

6. Conclusion

the electric current, in the case of Ohmic conduction (53c), is propor­ tional to the electric field through the electric conductivity tensor

The fundamental equations of piezoelectricity were obtained in the 8

L.M.B.C. Campos et al.

European Journal of Mechanics / A Solids 78 (2019) 103848

first instance (i) in the adiabatic case (Section 2), and then extended to include anisothermal conditions, heat conduction (Section 3) and viscoelastic materials (Section 4). In all cases, the Maxwell equation (61b) shows that, for a steady magnetic induction (12a), the electrostatic field is decoupled leading to piezoelectricity associated with electric charges (10a). Similarly, the Maxwell equation (53b) shows that, if the electric displacement is steady,

ϱ_ ¼ ∂i D_ i ¼

the magnetic field is decoupled, eijk ∂j Hk ¼

Gi ; c

(62b)

and noting that

∂i Bi ¼ 0;

(62c)

is specified by the magnetic potential Ak , Bi ¼ eijk ∂j Ak ;

c eijk ∂i ∂j Hk ¼

∂i Gi :

(63)

The coupling of the unsteady electric and magnetic fields leads to electromagnetic waves, that can be neglected in piezoelectricity and piezomagnetism if the period is small compared with other time scales; since the speed of light is very large, for example c ¼ 3 � 1010 cm s 1 in vacuum, the propagation time is short for moderate length scales. When this approximation is not valid, the extension to piezoelectromagnetism (Section 5) includes both piezoelectricity and piezomagnetism and their coupling through electromagnetic waves. Thus piezoelectromagnetism may be expected to supersede separate piezoelectricity and piezomag­ netism for high frequencies when coupling through electromagnetic waves cannot be neglected. The most general anisotropic conditions have been considered in (i) the fundamental equations of energy (57), momentum (58), electricity (59) and magnetism (60) involving various constitutive and diffusion tensors (Appendix A) that simplify for crystals and materials with symmetries (Appendix B) leading to simpler funda­ mental equations (Appendix C).

(62a)

D_ i ¼ 0;

∂i Gi

(62d)

Acknowledgements

leading to piezomagnetism associated with electric currents (62b). In the unsteady case, the electric and magnetic fields are coupled, as can be seen: (i) from the Maxwell equations (53b) and (61a); (ii) from the scalar and vector potentials (61a) and (61c); (iii) from the conservation of electric charges and currents that follows from (10a) and (53b),

This work was supported by FCT (Foundation of Science and Tech­ nology) through IDMEC (Institute of Mechanical Engineering), under LAETA Pest – OE/EME/LA0022.

Appendix A. Constitutive and diffusion tensors of matter The properties of the material are represented by the constitutive tensors in (48a-48d) and diffusion tensors in (19, 36, 55), that include scalars, vectors and tensors with two, three or four indices (Schouten, 1954; Campos, 2014). The scalar constitutive property is the specific heat (16) that has only one component: (A.1a)

#Cv ¼ 1: The constitutive vectors specify the pyroelectric (17) and pyromagnetic (52) effects and have 3 components: #gi ¼ 3;

(A.1b)

#hi ¼ 3:

(A.1c)

The tensors with two indices are all symmetric and thus have 6 independent components. These tensors are: (i) constitutive tensors such as the thermal expansion (18), dielectric permittivity (7), magnetic permeability (50) and electromagnetic coupling (51), #αij ¼ 6;

(A.2a)

#εij ¼ 6;

(A.2b)

#μij ¼ 6;

(A.2c)

#βij ¼ 6;

(A.2d)

(ii) diffusion tensors such as the thermal conductivity (20b) and Ohmic electrical conductivity Eqn. (54b), #kij ¼ 6;

(A.2e)

#σij ¼ 6:

(A.2f)

The tensors with three indices are both constitutive, namely the piezoelectric (8) and piezomagnetic (49) tensors, that are symmetric in two indices, and thus have 18 independent components: #pijk ¼ 18;

(A.3a)

#qijk ¼ 18:

(A.3b) 9

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European Journal of Mechanics / A Solids 78 (2019) 103848

The tensors with four indices are the constitutive elastic tensor (6) and diffusive viscous tensor (30) and both are doubly symmetric, that is: (i) symmetric in the first two ði; jÞ and last two ðk; lÞ indices; (ii) symmetric in the two pairs. Thus, both have 21 independent components like a symmetric 6 � 6 matrix: 6�7 ¼ 21; 2

(A.4a)

6�7 ¼ 21: 2

(A.4b)

#ϒ ijkl ¼ #υijkl ¼

The number of independent components indicated above in (A.1a-A.1c), in (A.2a-A.2f), in (A.3a-A.3b) and in (A.4a-A.4b) is the maximum possible for a material without any symmetry. The symmetric materials like crystals or orthotropic plates have a smaller number of independent constitutive and diffusion parameters, with the minimum for isotropic substances (Appendix B). Appendix B. Crystal classes and anisotropic and isotropic materials The maximum number of constitutive and diffusion parameters applies to materials without any symmetries and the number reduces for the 7 crystallographic systems and 21 crystal classes as they become more symmetric (Lewis, 1899). Likewise, symmetries that leave materials invariant, like for orthotropic plates (Howell et al., 2009), reduce the number of elastic moduli, and the number of independent components of constitutive and diffusive tensors (Schouten, 1954; Campos, 2003, 2005, 2014; Lewis, 1899; Howell et al., 2009). The smallest number of material properties applies to an isotropic or amorphous substance, for which all directions are equivalent, and the constitutive and diffusion tensors can involve only: (a) scalars; (b) the identity matrix that is symmetric, (B.1a)

δij ¼ δji ; (c) the permutation symbol with three indices, as many as the dimensions of space, that is completely skew-symmetric, eijk ¼ ejki ¼ ekij ¼

eikj ¼

ejik ¼

(B.1b)

ekji :

They may appear in products, sums or combinations as shown next. The specific heat (A.1a) is a scalar and exists for all materials (B.2a). The pyroelectric (A.1b) and pyromagnetic (A.1c) vectors cannot be isotropic unless they are zero as stated in (B.2b) and (B.2c) and thus these effects cannot exist in amorphous substances, although they occur in some crystals: isotropic :

(B.2a)

Cv 6¼ 0;

gi ¼ 0;

(B.2b)

hi ¼ 0:

(B.2c)

The symmetric constitutive tensors from (A.2a-A.2c) and diffusion tensors from (A.2e-A.2f) all reduce to a scalar for isotropic materials, namely the thermal expansion (B.3a), dielectric permittivity (B.3b), magnetic permeability (B.3c), thermal conductivity (B.3d) and Ohmic conductivity (B.3e):

αij ¼ αδij ;

(B.3a)

εij ¼ εδij ;

(B.3b)

μij ¼ μδij ;

(B.3c)

kij ¼ kδij ;

(B.3d)

σ ij ¼ σ δij :

(B.3e)

The exception is the electromagnetic coupling (A.2d) that does not exist in an isotropic material (B.4c), because in (51): (i) the electric displacement is a polar vector; (ii) the magnetic induction is an axial vector; (iii) thus βij is a pseudo-tensor (B.4a) whereas the identity matrix (B.1a) is an absolute tensor, so (B.4b) implies (B.4c): βij ¼ βδij :

(B.4a)

β¼0

(B.4b)

⇒βij ¼ 0:

(B.4c)

The piezoelectric (A.3a) and piezomagnetic (A.3b) effects cannot exist, p ¼ 0;

(B.5a)

q ¼ 0;

(B.5b)

in an isotropic medium, because the permutation symbol is skew-symmetric (B.1b) in contrast with the symmetric pair of indices in (8) and (49): pijk ¼ peijk ;

(B.5c)

qijk ¼ qeijk ;

(B.5d)

The elastic (A.4a) and viscous (A.4b) tensors in an isotropic substance involve two elastic Lam�e moduli (B.6a) and the two shear and bulk 10

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European Journal of Mechanics / A Solids 78 (2019) 103848

viscosities (B.6b): ϒ ijkl ¼ λδik δjl þ νδij δkl ;

(B.6a)

υijkl ¼ ζδik δjl þ ξδij δkl :

(B.6b)

Since constitutive and diffusive tensors have a smaller number of independent components in an isotropic medium, the fundamental equations are simplified (Appendix C). Appendix C. Fundamental equations for isotropic materials In the case of an isotropic material, equations (B.3d) and (B.3e) imply that the heat flux (19) is anti-parallel to the temperature gradient through the thermal conductivity, Ji ¼

(C.1a)

k∂i θ;

and the Ohmic electric current (53c) is parallel to the electric field through the electric conductivity, (C.1b)

Gi ¼ θEi :

The elastic (5b) and viscous (31) stresses in an isotropic material relate to the strains (12d) and rates of strain (30) respectively through the Lam�e elastic moduli (B.6a), (C.2a)

Tij ¼ λϵij þ νϵkk δij ; and shear and bulk viscosities (B.6b), � � � � � b ij ¼ 2ζ ϵ_ ij 1ϵ_ kk δij þ ξ_ϵkk δij ¼ ζ ∂i u_j þ ∂j u_i þ ξ T 3

� 2ζ ∂k u_k δij : 3

(C.2b)

Considering only the elastic stresses, for an isotropic material, the constitutive relations from (48a-48d) state that: (i) the electric displacement and field are parallel and related through the dielectric permittivity, (C.3a)

Di ¼ εEi ; (ii) the magnetic induction and field are parallel and related through the magnetic permeability,

(C.3b)

Bi ¼ μHi ;

(iii-iv) the elastic stresses (C.3d) and entropy (C.3c) involve only the temperature and strains, dS ¼ Cv

dθ θ

(C.3c)

αdϵii ;

(C.3d)

Tij ¼ λϵij þ νϵkk δij þ αθδij :

It follows that the Maxwell equations (10a), (53a), (53b), (53c) and (61b) for an isotropic medium specify the coupled electric and magnetic fields, namely (C.4a)

ε∂i Ei ¼ ϱ; μ_ Hi ;

eijk ∂j Ek ¼

(C.4b)

c

(C.4c)

∂i Hi ¼ 0; σ

ε

(C.4d)

eijk ∂j Hk ¼ Ei þ E_ i ; c c

and they can be solved separately; the solutions can be substituted in the equations of energy (57) and momentum (58) that, for an isotropic material, simplify respectively to (C.5) and (C.6): (C.5)

Cv θ_ ¼ k∂i ∂i θ þ σ Ei Ei þ αθϵ_ ii ;

ρu€

λ∂j ϵij

ν∂i ϵjj þ α∂i θ ¼ ϱEi þ

μσ c

(C.6)

eijk Ej Hk :

The equations of the electromagnetic field (C.4a-C.4d), energy (C.5) and momentum (C.6) involve the temperature θ, displacement vector ui in the strains (12d), the scalar potential Φ and vector potential Ai in the electric (61c) and magnetic (61a) fields: (C.7)

μHi ¼ eijk ∂j Ak :

11

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European Journal of Mechanics / A Solids 78 (2019) 103848

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