## Abstract

Implicit constitutive relations that characterize the response of elastic bodies have greatly enhanced the arsenal available at the disposal of the analyst working in the field of elasticity. This class of models were recently extended to describe electroelastic bodies by the present authors. In this paper, we extend the development of implicit constitutive relations to describe the behaviour of elastic bodies that respond to magnetic stimuli. The models that are developed provide a rational way to describe phenomena that have hitherto not been adequately described by the classical models that are in place. After developing implicit constitutive relations for magnetoelastic bodies undergoing large deformations, we consider the linearization of the models within the context of small displacement gradients. We then use the linearized model to describe experimentally observed phenomena which the classical linearized magnetoelastic models are incapable of doing. We also solve several boundary value problems within the context of the models that are developed: extension and shear of a slab, and radial inflation and extension of a cylinder.

## 1. Introduction

Recently, there has been considerable interest in the development of implicit constitutive relations to describe the response of both fluids and solids. While implicit rate type models have been used to describe the response of materials, both for viscoelastic fluids and inelastic solids, purely algebraic implicit relations [1–5] have been introduced to study the response of elastic and electroelastic bodies. Such models are better suited to explain a plethora of phenomena such as the stresses and strains adjacent to crack tips and at the edge of notches [6,7] and several other problems wherein the traditional models of elasticity predict unphysical singularities. Moreover, the class of implicit models also allows one to obtain rational approximations wherein the linearized strain bears a nonlinear relationship to the stress. Such features, when generalized are very useful in describing the response that has been observed in magnetoelastic materials. Even a cursory glance of the experimental literature that is pertinent to the response of magnetoelastic bodies makes it evident that the classical models that are available are incapable of describing them. For instance, it is seen the strain, even when it is very small to be in what would be justifiably called the ‘small strain’ range, exhibits a nonlinear relationship to the stress and the other electromagnetic variables. Linearization (in the sense that the gradients of displacement are small) of the models that are used cannot lead to a model that can describe these experiments. Our experience with implicit constitutive relations emboldens us to develop such models within the context of magnetoelasticity, which when linearized would be capable of describing the experiments that are available. This is not the main reason for developing implicit constitutive relations for the response of magnetoelastic materials. Such models greatly enhance the ability of the modeller to describe the response of materials and more importantly implicit constitutive relations have the correct philosophical underpinnings. We shall not discuss these issues here but refer the reader to the papers mentioned earlier, where they are discussed in detail.

Continua that respond to stimulation by magnetic fields have been the object of continued interest for several decades. This response of the continua to magnetic fields has relevance to many applications (e.g. [8–16]). There is a considerable amount of observations and experimental data concerning such response, and of particular relevance to this study are the interesting results concerning the nonlinear response of such bodies in a variety of areas such as the response of ferromagnetic materials [17–20], and some relatively new classes of materials, namely magneto-active elastomers [21–30]. In particular in [17,18], one can find the documentation of numerous experiments which indicate that even within the context of small displacement gradients and hence small strains, the behaviour of many magneto-active materials can be highly nonlinear.

One of the most authoritative works on nonlinear magnetoelasticity is the monograph by Brown [31], which has influenced directly and indirectly many of the later researchers in the area. A more recent work that deserves mention is the book by Maugin [32] (a short discussion of the contributions of both these authors is presented in §3). In much of the literature, we see the use of explicit constitutive equations, where in particular the stresses are assumed to be functions of the strains and one of the magnetic variables (see §3). This assumption has been also used in all of the recent works in this area (e.g. [33–46]). Mention must also be made of the treatise by Truesdell & Toupin [47], their chapter dedicated to the response of electromagnetic continua has moulded the thinking of the more theoretically inclined modellers in continuum mechanics (see [47], §312).

In this study, we develop a more general approach than those that have been followed hitherto. Building on the recent generalization for the response of elastic bodies [1–3,48–54], we advocate the use of implicit constitutive relations to describe the response of magnetoelastic bodies.

The organization of the paper is as follows. In §2, we present the basic equations of kinematics and magnetostatics, which is followed in §3 with a detailed account of earlier works on magnetoelasticity, reviewing in particular some of the early experimental results on the behaviour of ferromagnetic materials, which needs to be taken into account in modelling the response of such materials. We then review the numerous theoretical works on magnetoelasticity, in particular the development of constitutive relations in the previous studies, to highlight the relevance and the novelty of the work that is carried out in this paper. In §4, we propose a general form for the implicit relations that can capture the behaviour exhibited by magneto-active elastic bodies. In §4*a–c*, several interesting special cases are derived from the general implicit relations that are developed. In §5, we study some simple boundary value problems within the context of the implicit constitutive relations that have been developed in the paper.

## 2. Basic equations

### (a) Kinematics

Let *t*. It is assumed that the mapping ** χ** which assigns the position

**x**at time

*t*, i.e.

**x**=

**(**

*χ***X**,

*t*) is sufficiently smooth to make all the derivatives that are taken, meaningful. The displacement

**u**, the deformation gradient

**F**, the Cauchy–Green tensors

**b**and

**c**, and the linearized strain tensor

**are defined through**

*ε*More details concerning kinematics can be found, for example, in [47,55]. In this work, we are only concerned with quasi-static problems and the above definitions suffice.

### (b) Equations of magnetostatics

The deformed configuration **H**, **B** and **M**, respectively.

The magnetic field and magnetic induction satisfy the simplified form of Maxwell equations in the absence of electric interactions [56]

The magnetization field is defined in terms of **H** and **B** through the equation
*μ*_{0} is the magnetic permeability in free space. In vacuum and for non-magnetizable materials, we have **M**=**0** and (2.3) becomes

Across a surface of discontinuity in the body or the boundary **H** and **B** must satisfy the continuity conditions
**n** is the unit outward normal to **A**, the double bracket is defined as [**A**]=**A**^{o}−**A**^{i}, where **A**^{o} and **A**^{i} correspond to the field evaluated near

### (c) Balance equations

As mentioned in [57,58], there are different ways to write the equations of equilibrium when dealing with electromagnetic interactions (see [59] for the electroelastic case). The non-uniqueness in the manner in which the stress is defined, is due to the fact that it is not possible to separate the effects due to the different loads (due to electromagnetic fields and mechanical traction) inside a body [58].

In the case, for example, when we consider the magnetization **M** as the ‘independent variable’, the equations of equilibrium in the current configuration take the form (e.g. [36–39])
** σ** is in general a non-symmetric tensor,

*ρ*the mass density and

**f**would be the body forces of non-magnetic origin.

One of the simplest formulations, equivalent to (2.6) is based on the concept of the total stress tensor ** τ**, where the magnetic body force

**, obtaining an alternative definition for the stress that is symmetric and that satisfies the following equations of equilibrium (e.g. [36–39])**

*σ*The continuity condition across a surface of discontinuity in the current configuration is of the form [37–39]
*τ*_{m} is the Maxwell stress due to the magnetic field outside the material near the boundary of the body [56]

## 3. Constitutive relations in classical linear and nonlinear magnetoelasticity theories

In this section, we review some of the earlier works on electromagnetic fields and their interactions with continua with a view towards providing the state of the art of modelling such materials as it stands now, so that the reader can appreciate the different perspective provided in this paper with regard to the modelling the response of magnetoelastic bodies and assess its novelty and efficacy. We document and discuss both experimental results and the theoretical modelling efforts to highlight the fact that there is considerable experimental work, which is not adequately described by the models that are available. We also wish to show how the models developed earlier fit into the class of models developed here as very simple, in fact trivial approximations.

The effects of a magnetic field have fascinated the curious scientist from the very beginnings of science, a good brief historical account on electric and magnetic interactions can be found in www.magnet.fsu.edu/education/tutorials/index.html and also [60]. Special mention must be made of the early contributions of Coulomb [61,62], Poisson [63], Green [64], Joule [65], Thomson [66], Voigt [67,68] and Volterra [69]. Coulomb carried out some experiments to determine the forces of attraction and repulsion due to magnetic fields in magnetized bodies ([61], pp. 578–611). He assumed that such forces were created by interaction with a sort of magnetic fluid, and that idea was used by many later researchers [63,64]. Coulomb also noted that when divided into two parts, each part of a permanent magnet continues behaving in the same manner as the original body, in the sense that each piece has north and south poles (see [62], §XXV). Regarding the works of Poisson, he not only performed experimental research, for example, on the effect of temperature on the magnetic properties of some materials ([63], pp. 258), but also proposed some of the earliest mathematical models to compute the forces on a body due to magnetic fields (see pp. 259, Eqn. 2 of pp. 265 and Eqns. (c), (d) of pp. 299 of [63]). Green, in addition to calculating the magnetic forces within the context of the potential theory that he developed ([64], pp. 83–115), also addressed the issue of the magnetostriction effect on a wire, although he did not offer any experimental result to support his thesis ([64], pp. 17–18).

Credit goes to Joule and Thomson as the scientists that carried out experiments on magnetism, magnetostriction, thermomagnetism and other phenomena in a reasonably systematic manner. Joule was among the earliest researchers to investigate the change in the length of iron bars due to the application of a magnetic field (e.g. [65], pp. 46–53), wherein Joule credits Mr F. D. Arstall for the discovery of that phenomenon. More results are presented in [65], pp. 49 and pp. 235–264, see in particular Table 1 and pp. 239, 240, and the remark in pp. 245 where he stated that ‘the elongation is proportional, in a given bar, to the square of the magnetic intensity’. Regarding the works by Thomson, in [66], he was concerned with the effects of temperature on the magnetic and electric properties of some metals (see the articles in pp. 76–80 and §207 (pp. 313–314) of Part VII of [66]). In [70], we find more results on the influence of magnetic fields concerning the thermo-electric properties of different metals ([70], pp. 267–292), and we also find interesting results concerning the influence of the stresses on magnetization ([70], pp. 332–395 and §214–§222). Also worth noting is the observation that if the tension *σ* in a bar increases, the magnetization would diminish and vice-versa ([70], §182 pp. 332–353). We cannot tell directly from his experimental results if the effect would be linear or nonlinear. More results are presented in pp. 358–395, for an alloy of iron, nickel and cobalt. An interesting result that was established by him was that for *σ*>0, *M* decreased, but that was the case when the magnetization force was above a critical value, and if such a force was below that value, then the effect of increasing the tensile stress *σ* was to increase the magnetization.^{1}

We discuss the works by Voigt & Volterra [67–69] later on when reviewing the classical constitutive equations used in magnetoelasticity.

Next, we shall discuss some of the early experimental results concerning the behaviour of ferromagnetic materials. Since magnetic fields might affect different materials in different ways, for brevity we restrict our discussion to only ferromagnetism, and we review only some of the fundamental experimental works in this area. *We wish to emphasize that there is an abundance of experimental phenomena for which the implicit constitutive theories presented in §4 can be more suitable for the modelling than the theories that are currently available*. Of the numerous experimental results that are available the books by Becker & Döring [17], Bozorth [18] and the article by Lee [71] deserve special recognition. Other useful contributions are the article due to Kneller [20], the book by Chikazumi & Charap [72], the articles by Mason [73] and Carr [19]. In [17], we can find plenty of experimental data concerning the saturation phenomenon (e.g. Figs. 62, 72, 77 and 91 therein) where the magnetization *M* shows a nonlinear behaviour in terms of the magnetic field *H* as depicted in^{2} figure 1. From the results provided in [17] as well as the experimental information presented in other works, it is possible to observe the strong dependence of the magnetic properties, in this case, for example, the magnetization *M*, on the temperature of the material. Although the effect of temperature is very important, in our present communication we do not consider the influence of it, leaving that for future work study.

In [17], ch. III, interesting results are presented for the magnetic field *H* and other magnetic variables in terms of the stress, see, for example, Fig. 116 (tensile stress *σ* in a bar), Fig. 119 for the magnetic behaviour as function of the shear stress *τ* and normal stress *σ* due to torsion and tension in a bar. Also see Fig. 125 for the behaviour of *H* as a function of *σ* and the diameter of the bar, and Fig. 135 for the behaviour of the magnetic susceptibility *χ* and *H* versus *σ*. Finally in [17], ch. IV, results for the magnetostriction, or longitudinal deformation *ε*=*δl*/*l* (using the notation of that work) are presented as a function of *M* (see [17], Figs. 188, 189, 192 and 193), for *ε* versus *H* for different external tensile stresses *σ* (see Fig. 195) and for the percentage of change in the volume of a bar versus *H* (see [17], Figs. 200–202). *In all these results, we can see a strong nonlinear dependence of the deformation on H (or M) and in σ, and for all these results ε is very small, its maximum is of order 10^{−5}.*

In [18], we can also find interesting experimental data on the magnetic and elastic properties of iron and some iron alloys. In Fig. 1–11, some results are presented for *ε* versus *H*, where for some materials the bars get longer as *H* increases, while for others they get shorter for increasing *H*. In [18], Figs. 4-23 and 5-96, results for *B* versus *H* and *M* versus *B* are presented, while in ch. 13 of that work we can find several plots for the magnetic behaviour as a function of the stresses (e.g. Fig. 13-1). In [18], Fig. 13-49 and Fig. 13-60, more results are presented for *ε* versus *H* and *σ*. Finally, we mention Fig. 13-79 for λ (the elongation of a bar) versus *H* and *σ* where *σ*>0 (tension) and also in some cases when *σ*<0 (compression), and Fig. 13-123 where we find a plot for *ε* versus *σ*. In all these plots, we can observe strong nonlinear behaviour with regard to the different variables, within the context of small strains as with the results presented in [17].

In the review article by Lee [71], in §1.2, a discussion is presented concerning early experimental works, in particular we mention Figs. 1.1, 1.2, where results for *ε* versus *M* (and *σ*) are presented, in which, like in [17,18], we can observe a highly nonlinear behaviour for *ε* (the strains remain small, of the order of 10^{−5}). See also [71], Fig. 1.6 for results for *M* versus *H*, for the change in volume of a bar versus *H*, and *ε* versus *M*, where we can clearly also observe nonlinear behaviour^{3} for the magnetization, the volumetric strain and the elongation of a bar, in terms of the magnetic field *H*. In [71], Fig. 5.1, results are presented for single crystals of iron, for *ε* versus the magnetization; we notice that for *M* below a certain critical value we have *ε*>0, whereas if *M* is above such critical value it is observed that *ε*<0.

*We have presented a short review of some selected experimental works especially with regard to ferromagnetism, as such experimental evidence cries out for a theory other than those that are in place that when approximated to the small strain range can describe the nonlinear response that is observed*. This is precisely what the models developed in this paper can provide and thus such experimental evidence provides a strong motivation, rationale and justification of the work that is carried out. We now turn to a discussion of the classical constitutive theories which have been used to model the behaviour of magneto-active materials that highlights their inadequacy in describing a great deal of available experimental data. As in the discussion of the earlier experimental investigations, here we do not review all the works on magnetoelasticity (and magnetoplasticity), but rather we consider some selected relevant studies in that area.

The most influential work on magnetoelastic interactions is the monograph by Brown [31]. The articles by Tiersten [74,75], Pao and co-workers [57,76], the books by Maugin [32], Eringen & Maugin [77] and by Hutter *et al*. [58] all provide useful and important information. Early systematic treatments of the mathematical modelling of magnetoelastic materials can be found in the works by Voigt [67,68], the papers by Volterra [69] and by Mason [73]. Many of these works have been influenced by the different theories on electroelasticity, which were developed earlier (see [78] and the review in §3(a) in [4]). In all these works as well as several others that we will cite later, the main assumption has been to express the stress tensor (either the total stress tensor or other definitions for it [37,38,58]), and one of the magnetic variables as functions (or functionals in the case of bodies with memory) of the strains,^{4} i.e. expressions of the form:
**F** and **M** for bodies with memory, where **F** can be replaced by either the right Cauchy–Green deformation tensor, the Green–Saint Venant strain tensor, or the linearized strain tensor (in the case of small gradient of the displacement field), and where the magnetization **M** can be interchanged with **B** or **H** (if that field is considered to be the independent variable).

With regard to the studies of Voigt, it is necessary to recognize that his studies on magnetoelasticity were influenced by his deeper works on piezoelectricity (see the 1910 edn. of [68]), where he assumed linear constitutive equations for the stress and one of the electric variables (e.g. [4], §3(a)). In [67] in order to study magneto-optic interactions (assuming rigid bodies), Voigt postulated the use of a very simple relation of the form
*χ* is a constant. In [68] (ch. VI, Part V) apart from giving a good review of previous works on magnetostriction (applied to crystals), in §237 (Eqn. 292) and §240 (Eqn. 311) Voigt proposed extensions of the previous model (3.2) of the form (in Cartesian coordinates index notation):
*χ*_{ij}, *μ*_{ij} are constants, which are defined as the magnetic susceptibility and magnetic permeability, respectively. Voigt indicated that such expressions (3.3) were taken from Thomson's works. In (3.3), no deformation of the body was considered. The linearized model (3.3) is not suitable to study the magnetic behaviour near the saturation point (figure 1), as recognized by Voigt himself ([68], §251), therefore, in §258 (Eqns. 369, 370) of the same work, he proposed extensions of (3.3) considering power series expansions in terms of *H*_{j}. In [68], ch. VII, Part I, he considered the deformation of the body, but decoupled from the magnetic field.

Volterra [69] (see also [47], §309) considered an extension of the simple models due to Voigt (3.2) and (3.3), proposing a relation for **B** in terms of **H** (incorporating the history of **H**) of the form:
*μ* is a constant.

In the review article [73] (see in particular §II therein), Mason decided to work with an ‘elastic enthalpy’ function (see Eqn. (1) [73]), which depends on the stress tensor and one of the magnetic variables, obtaining in Eqn. (3)(1) an expression for the strain as a derivative of that function in the stress (see also [73], Eqns. (4) and (8)) for some polynomial approximations for that enthalpy function. The purpose of Mason was to apply such constitutive equations in the modelling of ferromagnetic materials, which as mentioned earlier, undergo small strains but exhibit several nonlinear phenomena. In [73], §III and §IV, such equations are used to fit some experimental data, in particular for the behaviour of shear and longitudinal waves in bars (see Figs. 5–10 therein). Mason did not indicate why he chose to consider the stress as one of the independent variables, but in [79] he used the same kind of relations where the strains are expressed as functions of the stresses for the modelling of piezoelectric crystals (see [4], §3(a)). Interestingly in [73], Appendix II, for some particular polynomial expansions for the elastic enthalpy function, he inverted the expressions for the strain in terms of the stress, in order to obtain the classical equations for the stress in this case as function of the strain.

Similar analysis considering the stress as the independent variable, but only for one-dimensional problems, has been carried out by Lee [71], see in particular §9 and Eqns. (9.1) and (9.2) therein for some specific expressions for the linearized strain as function of the tension.

Some further comments regarding the work of Brown, Tiersten, Pao, Maugin, Eringen and Hutter is warranted. The monograph by Brown [31] provides, in addition to new and original work, a compilation of results of his previous work (e.g. [80–82]), where he provides a critical assessment of several earlier attempts at modelling the response of ferromagnetic materials (see [81], §1.2 [31], §1.1 [81] and pp. 349). Brown is right on the mark with his assessment of the previous works when he identifies the errors (e.g. [19,83,84]), with regard to the assumption that the energy of the body can be decomposed into a part that represented purely as the elastic energy with no magnetic contribution, and a part that captures the magnetic energy assuming the body as rigid. Brown assumed the existence of an energy function depending on the deformation gradient (i.e. he started with a general formulation for large deformations) and the magnetization (see [31], §6.3 and Eqn. 6.21), obtaining explicit expressions for the stress tensor and the magnetic field in terms of derivatives of such an energy function (see [31], Eqns. 6.25 and 6.24, respectively), i.e. expressions similar to (3.1). The use of such an energy function (depending only on the deformation gradient and the magnetization) was also suggested on a study on nonlinear dielectrics by Toupin [78], as was acknowledged by Brown (see Preface of [31] and the third paragraph in pp. 995 of [31]). In [31], §9, starting from the general expressions for ** τ** and

**H**, Brown considered the special case |∇

**u**|∼

*O*(

*δ*),

*δ*≪1, which is a very relevant approximation for the response of most ferromagnetic materials, leading to the classical expressions of the linearized model

Tiersten [74,75] considered the nonlinear response of magnetoelastic bodies and obtained the governing field equations by assuming that the body comprised two conjoint continua, a deformable continuum which he endowed with a lattice and an electric spin continua which responds to the magnetic field. In [74], §9 and [75], §6, Tiersten considered standard nonlinear constitutive equations (3.1) as the starting point for his subsequent studies, considering dissipation [74] and interaction with thermal loads [75].

Pao and co-workers (see [76], §2) also considered very simple models of the form (3.1) for rigid magnetizable bodies, whereas in §4 (see Eqns. (4.1)–(4.3)) the usual assumption of an energy function that depends on the deformation gradient and the magnetization (3.1) is used. In [76], §5 and §6, the above equations are linearized with regard to the strains (see also [85]). In the articles [57,86], an extension of the above models is presented (e.g. Eqns. (3.4) and (3.7)), where we have expressions of the form

Maugin has carried out extensive studies in the field of electromagnetics starting with his PhD dissertation on the subject in 1970 [87]. We shall cite a few of these studies and refer the reader to the numerous references to his own work and others that can be found in them [32,87–89]. Maugin also considered constitutive assumptions of the form (3.1) interchanging however the roles of **H** and **M** and considering as the dependent magnetic variable,^{5} see [32], §2.5, §2.9, §2.10, §3.1, Eqn. (3.1.3) and §3.5. In ch. 6 of his book [32], constitutive equations are presented for the special case of elastic ferromagnets using the constitutive assumptions that was discussed earlier (see §6.4 therein), and these constitutive relations are then linearized in Part C (Eqns. 6.4.36 and 6.4.47).

It is interesting to note that in ch. 5 of the book by Eringen & Maugin [77], after providing the list of variables that are relevant for the problem of electromagnetic interactions with continua (Eqn. (5.3.1) therein), in Eqn. (5.3.3) Eringen and Maugin mention implicit constitutive theories in order to determine the stresses, strains, electric field, polarization, magnetic induction and magnetization (among other variables for the problem); they also mention in [77], pp. 133, that with the use of such implicit relations ‘there is no need to distinguish between dependent or independent variables’. Despite the above remarks, shortly thereafter Eringen and Maugin abandon the use of implicit constitutive relations and work with explicit constitutive equations, considering the stress tensor, the polarization and the magnetization among other quantities as dependent variables (e.g. [77], Eqn. (5.4.6) in pp. 135 §8 and §5.1.3), and for the particular case of magnetoelastic interactions, see [77], ch. 8 (Eqns. 8.2.17–8.2.22), and some other special cases, see [90], Part II of ch. 9. The same use of explicit constitutive equations has been considered in previous works by Eringen and co-workers (e.g. [91–93]).

In electromagnetodynamics of continuous media, there are different ways to express the field equations, and in particular there are different possible definitions for the stress tensor (e.g. [31], pp. 82 and [57]). These differences appear among other reasons, because of the impossibility of separating (from the point of view of the experimental observations) the different loads inside an electro- and magneto-active body, where the body deforms simultaneously due to the action of mechanical stresses and body loads due to electromagnetic fields. In [58], Hutter, van de Ven and Ursescu review all the different formulations that were used until that time (among them some of their own work such as [94–96]), demonstrating that all such formulations are equivalent, if the comparison is presented in terms of ‘physically measurable quantities’ (see [58], §3.1).^{6} A key ingredient in this work [58] was the consideration of different forms of the field equations, and expressions for the constitutive relations, which following the traditional treatment, divided the different variables of the problems into dependent variables (such as the stress tensor), and independent variables (such as the deformation gradient), see [58], §2.6 and Eqn. (2.6.5). The different theories are discussed in §3.3–3.6 (among them, the two-dipole model, the Lorentz formulation, etc.), considering in each case the use of explicit constitutive equations (see [58], Eqns. (3.3.23), (3.3.56), (3.4.28), (3.5.10) and (3.6.16)).

There are many other investigators that have considered the use of explicit constitutive equations of the form (3.1) or some expressions that are similar in form in the study of magnetoelastic interactions; we mention: [97], where in Eqn. 3.6 they assume the stress as a nonlinear function of the linearized strain and the magnetization; [98] that is a study of magnetizable fluids; [99] a study of rigid thermally and electrically active conductors; [100] for a simplified theory where decoupled constitutive relations between the stresses and deformations, and the electromagnetic fields is considered; [101–103] for some models concerning memory effects (see [101], §4 and Eqn. (4.3); [103], Eqn. (3.3) and [102], §3); the series of works by Rivlin and co-workers [104–108] for a number of models with regard to rigid electromagnetic bodies, and deformable electromagnetic bodies as well; and finally [109,110] for some very simplified linear constitutive equations and [111,112] for other references with regard to studies on magnetoelasticity within the context of large nonlinear deformations.

In all these theoretical works reviewed above (with the exception of [71,73]), the main assumption has been to express the stress tensor and one of the magnetic variables (among other quantities of interest) as explicit functions of the strains, and one of the independent magnetic variables. However, in order to study appropriately the multitude of nonlinear phenomena observed, for example, in ferromagnetic materials [17] (among many other applications), more general constitutive relations are needed, the development of which is the intent of this study.

## 4. Implicit relations for nonlinear magnetoelastic bodies

As in the case of the response of electroelastic bodies [4], we propose the following two implicit relations between the total stress tensor ** τ**, the left Cauchy–Green strain tensor

**b**, the magnetic induction

**B**and the magnetic field

**H**:

Before turning our attention to study some special cases that arise from the constitutive relations (4.1), it is important to introduce the concept of ‘reference’ or ‘characteristic’ values for the stresses, which we denote by *τ*_{0}, for the magnetic field, which we denoted by *H*_{0}, and for the magnetic induction, which we denote by *B*_{0}, following the same procedure presented in [4]. The reason for introducing these quantities is in order to define proper non-dimensional variables, so that we can derive a number of special cases from the general implicit relations, by allowing some of these dimensionless variables to be sufficiently small. We define the non-dimensional stress tensor, magnetic field and magnetic induction, respectively, through
** τ**,

**H**and

**B**.

In this work, we do not consider (4.1) directly, but we study two simpler cases. In the first one, we assume that **H** and **b**, while in §4*c* we assume that the second relation depends on **b** but not in ** τ**.

Let us study the first case where the relations (4.1) simplify to

If _{1} leads to the following representation [113,114]:
*α*_{i}, *i*=0,2,3,…,15 are scalar functions that depend on the following list of invariants:
_{2} the vector relation becomes [113,114]:
*β*_{j}, *j*=0,1,…,6 depend on the invariants (4.5)_{1,2,3}, (4.6)_{5}, (4.7)_{1,2} and

The representation (4.4)–(4.10) are too general to be used to correlate with experimental data as there are so many material functions that depend on numerous invariants. It is left to the ingenuity of the modeller to come up with much simpler sub-classes that would be meaningful models.

### (a) Small gradient of the displacement field: ∣ ∇ u ∣ ∼ O ( δ ) , *δ*≪1

In this first special case, we study what happens if we assume *δ*≪1, when attention is focused on (4.4). From (2.1), we have that **b**≈2** ε**+

**I**. Let us consider the following approximations for the functions

*α*

_{i},

*i*=0,1,2,…,15, which are valid if

**b**≈2

**+**

*ε***I**:

_{ikl}=∂

*α*

_{i}/∂

*ε*

_{kl}|

_{(τ,0,B)}. Using (4.11) in (4.4) and appealing to the approximation

**b**≈2

**+**

*ε***I**, after some manipulations, which for brevity are not presented here, and neglecting terms of order

*δ*

^{2}or higher, we obtain an equation of the form (in Cartesian coordinates index notation)

**and**

*τ***B**. If the inverse of

**can be expressed as**

*ε*Relations (4.13) are important in their own right, since they present a rational manner of approximation of the general constitutive relation (4.3), of the behaviour of magneto-sensitive bodies, that in the small strain regime leads to a nonlinear relationship between the strain and the stresses, the magnetic field and the magnetic induction that need not be small. Of course, quantities such as stresses have dimension and so their smallness or largeness depends on the dimension being used. The point is that there are no restrictions on these quantities and other than the fact that the quantities under consideration are of the same order. The more important point is that the relationship in the linearized strain regime is nonlinear.

### (b) Small magnetic induction: ∣ H ∣ ∼ O ( δ ) , *δ*≪1

Starting from (4.9) in this section, we analyse the case *δ*≪1. From (4.9), let us consider the following approximation for the functions *β*_{i}, *i*=0,1,2,…,6:
_{ij}=∂*β*_{i}/∂*H*_{j}|_{(τ,B,0)}. The approximation (4.14) is valid if *δ*≪1.

Replacing (4.14) in (4.9), and neglecting the terms of order *δ*^{2} or higher, after some manipulations of (4.9) we obtain the relation
** τ**,

**B**. If

*δ*≪1, we would obtain from (4.3) that

In the case in which both conditions *δ*≪1 are satisfied, it is easy to prove from (4.13) that we obtain two constitutive equations for the linearized strain tensor ** ε** and the magnetic field

**H**of the form

### (c) Small total stress tensor: ∣ ∇ τ ∣ ∼ O ( δ ) , *δ*≪1

In this section, we will derive the classical constitutive equations (3.1) (in terms of **b** and **B** instead **M**) starting with the expression (4.1). In order to do so, let us consider the special cases where the implicit tensor relation ** τ**,

**b**and

**B**, and the implicit vector relation

**b**,

**B**and

**H**, i.e.:

Now, let us assume that *δ*≪1, and let us consider the following approximations;
*Υ*_{ikl}=∂*α*_{i}/∂*τ*_{kl}|_{(0,b,B)}, which is valid if *δ*≪1. Replacing (4.19) in (4.4), neglecting the terms of order *δ*^{2} or higher, and following the same methodology presented in the previous sections, it is easy to prove that the total stress tensor is given as a function of the left Cauchy–Green strain tensor and the magnetic induction, i.e.

## 5. Boundary value problems

We present two simple solutions to boundary value problems within the context of (4.4) and (4.9), first for the homogeneous distribution of stresses that are assumed to produce homogeneous distribution of strains, followed by the study of the problem of radial expansion and extension of a circular annulus, where the distribution of strains and stresses are in general inhomogeneous.

### (a) Extension of a slab

Let us consider a slab that in the reference unstressed and unstrained configuration is defined as
*L*_{i}>0, *i*=1,2,3 are constants. The slab is assumed to be under the effect of the stress distribution and magnetic induction fields
*τ*_{i}, *i*=1,2,3 and *B*_{o}≥0 are constants.

We assume that such stress and magnetic induction distributions produce the uniform deformation field^{7} :
_{2,3} we obtain

Regarding (4.9), first it is necessary to assume a possible expression for the magnetic field **H** that would be produced by the stress field and magnetic induction (5.2). Let us assume that the magnetic field is given by
*H*_{i}, *i*=1,2,3 are constant. Using (5.8) in (4.9) and considering (5.2)_{2}, we obtain the implicit relations
**H**. It is interesting to note that another possibility is the simpler expression
*β*_{j}, *j*=0,1,2,…,5 only depend on *τ*_{i}, *i*=1,2,3, *B*_{o} and *H*_{2}.

Since the stress field (5.2)_{1}, the magnetic induction (5.2)_{2} and the magnetic field (5.8) are assumed to be constant, then the simplified forms of Maxwell equations (2.2) and the equilibrium equation (2.7) (without mechanical body forces) are satisfied automatically. Regarding the continuity conditions (2.5), (2.8), two special cases are analysed separately.

#### (i) A long slab under uniaxial tension

In this first case, we assume *L*_{1}≪*L*_{2} and *L*_{3}≪*L*_{2}, i.e. the slab is very long in the reference configuration in the 2 direction. In addition, let us assume that the slab is under the effect of an external mechanical traction only on the surfaces *X*_{2}=±*L*_{2}/2, while on the surfaces *X*_{1}=±*L*_{1}/2 and *X*_{3}=±*L*_{3}/2 the slab is in contact with free space and is free of mechanical traction. Finally, let us work with the second possible expression (5.12) for **H**.

Since *L*_{1}≪*L*_{2} and *L*_{3}≪*L*_{2} as an approximation the continuity conditions are considered only for the surfaces *X*_{1}=±*L*_{1}/2 and *X*_{3}=±*L*_{3}/2. From (5.2)_{2} and (2.5)_{1} far from *X*_{2}=±*L*_{1}/2, we find that there is no restriction on the form of the magnetic induction in vacuum, while from (5.12) and (2.5)_{2}, again far from *X*_{2}=±*L*_{1}/2, we have that the magnetic field in vacuum should be of the form

From (5.13), (2.4) and (2.9), the non-zero components of the Maxwell stress tensor are

Given *B*_{o}, we need to find *H*_{2} by considering (4.10) and (5.10); then using (5.15) and (5.5)–(5.7) we can find λ_{i}, *i*=1,2,3 produced by such a load and the magnetic induction.

#### (ii) A thin plate under biaxial tension

In this second variant of the problem, we assume that *L*_{2}≪*L*_{1} and *L*_{2}≪*L*_{3}, i.e. we work with a thin plate. The mechanical load is assumed to be applied on the surfaces *X*_{1}=±*L*_{1}/2, *X*_{3}=±*L*_{2}/2, while the surfaces *X*_{2}=±*L*_{2}/2 are in contact with vacuum. In addition, only expression (5.12) is considered.

Since *L*_{2}≪*L*_{1} and *L*_{2}≪*L*_{3} as an approximation the continuity conditions (2.5) are considered only for the surfaces *X*_{2}=±*L*_{2}/2. From (5.2)_{2} and (2.5)_{2}, far from *X*_{1}=±*L*_{1}/2, *X*_{3}=±*L*_{2}/2, the magnetic induction in vacuum is approximately of the form
*X*_{1}=±*L*_{1}/2, *X*_{3}=±*L*_{2}/2 the magnetic effect has been incorporated in the definition of the external loads

### (b) Shear of a slab

In this second problem, we consider the same slab defined in (5.1), assuming *L*_{2}≪*L*_{1} and *L*_{2}≪*L*_{3}, i.e. we work with a thin plate. The external mechanical traction is applied on the surfaces *X*_{2}=±*L*_{2}/2, while on the surfaces *X*_{1}=±*L*_{1}/2, *X*_{3}=±*L*_{3}/2 the plate is adjacent to vacuum.

We assume the body is under the effect of the stress distribution and magnetic induction of the form:
*τ*_{i}, *i*=1,2,3, *τ*_{12} and *B*_{o} are constant.

It is assumed that the above stress field and magnetic induction produces the deformation
*κ* is a constant. Using (5.20) in (2.1)_{2,3}, the deformation gradient and the left Cauchy–Green strain tensors are

It follows from (5.19), (5.21)_{2} and (4.4) the following implicit relations hold:

We assume that (5.19) will produce a constant magnetic field of the form

In this problem in general **H**=*H*_{2}**e**_{2} may not be a solution of the problem (as in the case of the problem discussed in §5*a*). If one assumes that *H*_{2}=*H*_{3}=0 from (5.27) to (5.29) we still have three relations for one only unknown *H*_{2}, and unless some compatibility conditions (which for brevity we do not discuss here) are imposed, the problem cannot be solved in general.

Regarding the continuity conditions (2.5), since *L*_{2}≪*L*_{1} and *L*_{2}≪*L*_{3}, as an approximation the only surfaces where it would be important to check such conditions are *X*_{2}=±*L*_{2}/2. It follows from (5.19)_{2} and(2.5)_{2} that the external magnetic induction must be of the form
_{1} we infer that the external magnetic field should be of the form

On the surfaces *X*_{2}=±*L*_{2}/2, the external traction is assumed to incorporate in its definition the effect of the magnetic fields, therefore, it is not necessary to calculate the Maxwell stresses for such surfaces.

Given, for example, *B*_{o} and *τ*_{12}, from (5.22) to (5.25) we can obtain *κ* and *τ*_{i}, *i*=1,2,3 such that the deformation (5.20) can be produced, while from (5.27) to (5.29) using *B*_{o} and the above values for *τ*_{12}, *τ*_{i}, *i*=1,2,3, it is possible to obtain *H*_{i}, *i*=1,2,3, and finally from (5.32) we have the expression for the external magnetic induction that is necessary in order for (5.19), (5.26) to be a solution of the boundary value problem.

### (c) Inflation and extension of a cylindrical annulus

In this section, we study the problem of inflation and axial extension of a cylindrical tube. We do not obtain closed form solutions for the boundary value problem, considering that in order to do that we would need explicit expressions for the functions *α*_{i}, *i*=0,1,…,15 and *β*_{j}, *j*=0,1,…,6 from (4.4), (4.9), which is practically impossible to specify based on experimental data. In fact, numerous sets of values for these material constants could fit the data equally well (in general, the situation is worse as they are function and any function is equivalent to specifying infinite set of constants).

Consider the annulus, described in cylindrical coordinates, in the reference configuration
*H*_{zo} is constant.

In the case of the equilibrium equations (2.7) (in the absence of mechanical body forces), we infer from (5.34)_{1} that
_{2,3} it is easy to see that they are satisfied automatically.

Let us assume now that a such stress tensor field and magnetic induction produce the following deformation in the tube (described in cylindrical coordinates):
*f*(*R*) is a function to be determined. The deformation gradient and left Cauchy–Green deformation tensors (2.1)_{2,3} are, respectively:
*f*′=d*f*/d*R*.

It follows from (5.34), (5.37)_{2}, (4.4) and (4.9) that
*α*_{i}, *i*=0,1,…,15 depend on the invariants
*β*_{j}, *j*=0,1,…,6 depend on the invariants (5.42)_{1,2,3}, (5.45)_{1,2,3} and (4.10):

Equation (5.35) is an ordinary differential equation, while (5.38)–(5.41) are algebraic equations (in general nonlinear), we thus have a differential algebraic system to solve. These five equations can be used to find the functions *τ*_{r}(*r*), *τ*_{θ}(*r*), *τ*_{z}(*r*), *B*_{z}(*r*) and *f*(*R*).

From (5.36)_{1}, the differential equation (5.35) can be re-written as
_{1} is invertible so that there exists *f*^{−1} such that *r*=*f*^{−1}(*R*), we need to solve (5.47), (5.38)–(5.41) to find *τ*_{r}(*R*), *τ*_{θ}(*R*), *τ*_{z}(*R*), *B*_{z}(*R*) and *f*(*R*). To find the solutions, we need specific forms for the functions *α*_{i}, *i*=0,1,…,15 and *β*_{j}, *j*=0,1,…,6.

About possible boundary and continuity conditions, let us assume *L*≪*R*_{o}, therefore we study the continuity conditions (2.5) only across the boundaries *r*=*r*_{i} and *r*=*r*_{o}, where *r*_{i}=*f*(*R*_{i}) and *r*_{o}=*f*(*R*_{o}). It follows from (5.34)_{2} that equation (2.5)_{2} is satisfied automatically, whereas from (5.34)_{3} we infer that equation (2.5)_{1} is satisfied if the magnetic field outside the tube (in vacuum) **H**^{o} is of the form

Let us suppose that we apply a mechanical radial traction *P* on the inner surface of the annulus *r*=*r*_{i}, and that the outer surface of the annulus *r*=*r*_{o} is free of mechanical loads. It then follows that the above expressions for the components of *τ*_{m} namely (5.49)_{1}, and (2.8), that
*τ*_{mrr} along with *P* explicitly in the external load.

## Data accessibility

Our paper has no data.

## Author contributions

Both authors R.B. and K.R.R. worked on the theoretical formulation of the new classes of implicit relations. R.B. solved the boundary value problems, while K.R.R. and R.B. reviewed the different works in order to write the extended review of mathematical models in nonlinear magnetoelasticity. K.R.R. revised the whole article critically before submission.

## Funding statement

R.B. would like to express his gratitude for the financial support provided by FONDECYT (Chile) under grant no. 1120011. K.R.R. thanks the National Science Foundation and the Office of Naval Research for support of this work.

## Conflict of interests

We have no competing interests.

## Acknowledgements

R.B. and K.R.R. would like to thank the reviewers of this communication for their constructive comments.

## Footnotes

↵1 It is interesting to note that with regard to the results on magnetostriction presented by Thomson, he considered the tension on the bar as the independent variable, and the deformation due to the magnetic field and the tension was considered as an effect or consequence of the application of such loads.

↵2 See as well [20], Fig. 5; [9], Fig. 11.10; and [32], §1.6 and Fig. 1.6.6.

↵3 In §7.3 [71], mentioned that the Curie temperature can be affected by the pressure applied on the material.

↵4 The notable exceptions being the paper [73], the beginning of [77], ch. 5 and the article [71].

↵5 Though using the assumption that the stresses are expressed as functions of the strains and the magnetic field, at the beginning of [32], §2.5 pp. 92, Maugin says that ‘forces applied to a body cause it to undergo motion, and the motions caused differs depending on the nature of the body’, which should have led him to use express the effects in terms of the causes and not vice-versa.

↵6 Regarding the electric and magnetic variables, they stated that ‘regarding electromagnetic field quantities, we take the position that they are not measurable except in vacuo’ [58].

↵7 It is possible that other solutions for the displacement field different from (5.3) could be also compatible with (5.2), (4.4), (4.9), (2.2) and (2.7). In this paper, we do not study the possibility of such non-uniqueness to the solutions for (2.7) and (2.2).

- Received December 12, 2014.
- Accepted January 8, 2015.

- © 2015 The Author(s) Published by the Royal Society. All rights reserved.