Constitutive modelling of magnetic shape memory alloys with discrete and continuous symmetries

Abstract

A free energy-based constitutive formulation is considered for magnetic shape memory alloys. Internal state variables are introduced whose evolution describes the transition from reference state to the deformed and transformed one. We impose material symmetry restrictions on the Gibbs free energy and on the evolution equations of the internal state variables. Discrete symmetry is considered for single crystals, whereas continuous symmetry is considered for polycrystalline materials.

1. Introduction

The macroscopically observable large deformations in magnetic shape memory alloys (MSMAs) are caused by the microstructural reorientation of martensitic variants [1], field-induced phase transformation (FIPT) [2] or by the combination of the two mechanisms. In the variant reorientation mechanism, the variants have different preferred directions of magnetization, and the magnetic field is applied to select specific variants, which results in the macroscopic shape change. There are two major modelling approaches for such mechanisms. In microstructural-based models, the resulting macroscopic strain and magnetization response are predicted by minimizing a free energy functional. This functional includes terms related with the magnetostatic energy, the elastic energy and the magnetic anisotropy energy within the martensitic twin variant. Details of the microstructural-based modelling approach can be found in references [3,4]. The second approach to study the material behaviour is thermodynamics-based phenomenological modelling. The hysteretic behaviour of such a material system is taken into account through the evolution of internal state variables that account for the presence of the different martensitic variants within the magnetic alloy [5,6]. More detailed description of internal state variable-based approach for conventional shape memory alloys can also be found in references [7–9]. We proceed with the second approach to present a general modelling framework to capture both FIPT and variant reorientation.

The coupled MSMA behaviours can be modelled by considering the material as an electromagnetic continuum. Extensive work on different electromagnetic formulations had been proposed in the literature [10–14] based on different notions of breaking up long-range and short-range forces. A continuum theory for deformable ferromagnetic material is proposed in a recent work [15,16]. A nonlinear theory of magnetoelasticity for magnetosensitive elastomers is derived in [17,18]. A study of electrostatic forces on large deformations of polarizable material is presented in [19,20]. A theory for the equilibrium response of magnetoelastic membranes can be found in [21,22]. A continuum theory for the evolution of magnetization and temperature in a rigid magnetic body for ferro/paramagnetic transition is formulated in [23]. The variational formulations for general magneto-mechanical materials have been proposed by many authors [24–28].

In this work, we consider MSMA as electromagnetically active dissipative material systems. Because most of the experiments are conducted on single-crystal specimen, our major aim was to obtain the integrity basis for the Gibbs free energy and then derive constitutive equations of such a material system by considering finite symmetry. The symmetry restrictions on the evolution equations of the lower symmetric phases are also investigated. Finally, we consider continuous symmetry for polycrystalline materials to take into account anisotropy by introducing structural tensors. An evolution of a structural tensor is proposed to capture the changes in texturing owing to changes in the microstructure during phase transformation and reorientation.

2. Thermodynamic framework

We denote the reference configuration by Ω₀ and the current configuration by Ω. The spatial position in the deformed configuration is denoted by x=χ(X,t), where X∈Ω₀, and the deformation gradient is defined by F=∂x/∂X [29].

In the deformed configuration Ω, we denote the magnetic induction by b, the magnetic field h and the magnetization vector m. The free current density of the body is neglected in this study. The magnetization vector m is related through the following constitutive relation

$$\mathbf{\text{m}}=\frac{\mathbf{\text{b}}}{{\mu }_0}-\mathbf{\text{h}}.$$ 2.1

The local form of the Maxwell equations are given by

$$\mathrm{\nabla }\cdot \mathbf{\text{b}}=0,\mathrm{\nabla }\times \mathbf{\text{h}}=0,\mathrm{\nabla }\times \widetilde{\mathbf{\text{e}}}=-\stackrel{\ast }{\mathbf{\text{b}}},$$ 2.2

where, $\tilde{\mathbf{e}}=\dot{\mathbf{x}}\times b$. The asterisk (*) denotes the convective time derivative of the vector A

$$\stackrel{\ast }{\mathbf{A}}=\dot{\mathbf{A}}-\mathbf{L}\mathbf{A}+\mathbf{A}\hspace{0.17em}\mathbf{\text{tr}}\mathbf{(}\mathbf{L}\mathbf{)},$$

and $\mathbf{L}=\mathrm{\nabla }\otimes \dot{\mathbf{x}}$ is the velocity gradient. The material time derivative is denoted by d/dt or by a ‘dot’ and the spatial time derivative is denoted by ∂/∂t [30,31].

The conservation laws of mass, linear momentum and angular momentum are

$$\begin{array}{rl}\dot{\rho }+\rho \mathrm{\nabla }\cdot \dot{\mathbf{x}}& =0,\end{array}$$ 2.3a

$$\begin{array}{rl}\mathrm{\nabla }\cdot \mathit{\sigma }+\rho {\mathbf{f}}^b& =\rho \ddot{\mathbf{x}}\end{array}$$ 2.3b

and

$$\begin{array}{rl}\mathrm{skw}(\mathit{\sigma })& =0.\end{array}$$ 2.3c

Here, ρ is the mass density, σ is the total stress [10,11] generated owing to combined magneto-mechanical effect, f^b is the non-magnetic body force density. The local form of the energy balance can be obtained from the following global balance law

$$\begin{array}{rl}& \frac{\mathrm{d}}{\mathrm{d}t}{\int }_{\mathrm{\Omega }}[\rho u+\frac{1}{2}\rho \dot{\mathbf{x}}\cdot \dot{\mathbf{x}}+\frac{1}{2{\mu }_0}b\cdot b]\mathrm{d}v\\ & ={\int }_{\mathrm{\partial }\mathrm{\Omega }}[{\mathit{\sigma }}^T\dot{\mathbf{x}}-\mathbf{q}\mathbf{-}\mathbf{(}\mathbf{e}\times \stackrel{\mathbf{~}}{\mathbf{h}}\mathbf{)}\mathbf{]}\cdot \mathbf{n}\hspace{0.17em}\mathrm{d}\mathbf{a}+{\int }_{\mathit{\Omega }}\mathbf{[}\rho {\mathbf{r}}^{\mathrm{h}}+\rho \dot{\mathbf{x}}\cdot {\mathbf{f}}^b\mathbf{]}\hspace{0.17em}\mathrm{d}\mathbf{v},\end{array}$$

where u is the specific internal energy, $\frac{1}{2}\dot{\mathbf{x}}\cdot \dot{\mathbf{x}}$ is the specific kinetic energy, (1/2μ₀)b⋅b is the electromagnetic energy density of the free space, ${\mathit{\sigma }}^T\dot{\mathbf{x}}$ is the stress power, q is the heat flux, $(\tilde{\mathbf{e}}\times \mathbf{h})$ is the electro-magnetic energy flux (Poynting vector), r^h is the heat supply owing to external source and $\dot{\mathbf{x}}\cdot {\mathbf{f}}^b$ is the specific power consumed by the non-magnetic body force. The local form can be expressed as [32]

$$\rho \dot{u}+\mathrm{\nabla }\cdot \mathbf{q}\mathbf{-}{\mathit{\sigma }}^{{\mathbf{L}}^{\prime }}\mathbf{:}\mathbf{L}+\mathbf{m}\cdot \dot{\mathbf{b}}\mathbf{-}\rho {\mathbf{r}}^{\mathrm{h}}=\mathbf{0.}$$ 2.4

The work conjugate of the velocity gradient L is σ^L′, which is

$${\mathit{\sigma }}^{L^{\prime }}=\mathit{\sigma }-{\mathit{\sigma }}^{M^{\prime }},$$

where

$${\mathit{\sigma }}^{M^{\prime }}=(\frac{\mathbf{b}\otimes \mathbf{b}}{{\mu }_0}-\mathbf{m}\otimes \mathbf{b})-(\frac{1}{2}\frac{\mathbf{b}\cdot \mathbf{b}}{{\mu }_0}-\mathbf{m}\cdot \mathbf{b})\mathbf{I}\mathbf{.}$$

We consider that u(F,b,s,{ζ}), where s is the entropy. The set {ζ} represents the collection of tensor, vector and scalar internal state variables. We perform step-by-step partial Legendre transformations to change the variable space of u to Gibbs free energy G. Outline of the steps is shown as

$$u(\mathbf{F},b,s,\{\zeta \})\stackrel{\hspace{0.17em}}{\to }{\stackrel{\hspace{0.17em}}{\psi }}_1(\mathbf{F},\mathbf{h},\mathbf{T},\{\zeta \}\mathbf{)}\stackrel{\mathit{\Omega }\to {\mathit{\Omega }}_0}{⟶}{\stackrel{\mathbf{~}}{\psi }\hspace{0.17em}}_1\mathbf{(}\mathbf{E},\mathbf{H},\mathbf{T},\{\mathbb{Z}\}\mathbf{)}\to \mathbf{G}\mathbf{(}{\mathbf{S}}^E,\mathbf{H},\mathbf{T},\{\mathbb{Z}\}\mathbf{)},$$

where H=F^Th, $\mathbf{E}=\frac{1}{2}\mathbf{(}\mathbf{C}\mathbf{-}\mathbf{I}\mathbf{)}$, and C=F^TF. The material stress S^E is the work conjugate of the Green strain E and can be expressed as

$${\mathbf{S}}^E=\mathbf{(}det\mathbf{F}\mathbf{)}{\mathbf{F}}^{\mathbf{-}1}\mathbf{[}{\mathit{\sigma }}^E\mathbf{]}{\mathbf{F}}^{\mathbf{-}\mathbf{T}},$$ 2.5

where σ^E=σ−σ^h and σ^h=μ₀h⊗h−(μ₀/2)(h⋅h)I. Elaborate derivations can be found in [33]. Moreover, $\{\mathbb{Z}\}$ is the set of internal variables in the reference configuration. We obtain the following constitutive equations with the help of Coleman–Noll maximum entropy principle:

$$\begin{array}{rl}\mathbf{E}& =-{\rho }_{0}G{,}_{{\mathbf{S}}^E},\end{array}$$ 2.6a

$$\begin{array}{rl}{\mu }_0\mathbf{M}& =-{\rho }_{0}G{,}_{\mathbf{H}}\end{array}$$ 2.6b

and

$$\begin{array}{rl}s& =-G{,}_T,\end{array}$$ 2.6c

and the second law takes the following form

$$-\rho G{,}_{{\mathbb{Z}}_i}\cdot {\dot{\mathbb{Z}}}_i\ge 0.$$ 2.7

The Gibbs free energy for MSMAs with the set of external state variables {S^E,H,T} and internal state variables $\{\mathbb{Z}\}=\{{\mathbf{E}}^I,{\mathbf{M}}^I,{\xi }_{\mathbf{i}},\mathbf{g}\}$ can be written as

$$G=G({\mathbf{S}}^E,\mathbf{H},T,{\mathbf{E}}^I,{\mathbf{M}}^I,{\xi }_{\mathbf{i}},\mathbf{g}\mathbf{)}\mathbf{.}$$ 2.8

We assume the inelastic strain tensor E^I and internal magnetization vector M^I. The internal magnetization accounts for the rotation of magnetization vector and motion of magnetic domain walls. ξ_i (i=1,…,N) denotes the volume fraction of transforming material from austenite to martensite and also owing to reorientation. g is an internal state variable associated with the interaction energy during transformation and reorientation and so a scalar.

We denote stress-favoured, field-favoured and twinned martensitic variants by M₁, M₂ and M₃, respectively (figure 1). The austenitic phase is denoted by A. The volume fractions of M₁, M₂ and M₃, produced during phase transformation from A, are denoted by ξ₁, ξ₂ and ξ₃. We denote the reorientation volume fraction of M₂ from M₁ by ξ₄, of M₂ from M₃ by ξ₅ and of M₁ from M₃ by ξ₆. The production of volume fractions ξ_i from different phases is schematically presented in figure 1. The three phase transformations $A\leftrightarrow M_1$, $A\leftrightarrow M_2$, $A\leftrightarrow M_3$ and the field-induced variant reorientation $M_1\leftrightarrow M_2$ can be generated both directions. However, reorientation from stress-favoured variants M₁ and field-favoured variants M₂ to twinned martensitic variant M₃ is not energetically favourable and ${\dot{\xi }}_5$, ${\dot{\xi }}_6$ may occur in only one direction. We select the set of six scalar internal variables ξ_i (i=1,…,6), to describe the phase state of the material. The total volume fractions of M₁, M₂, M₃ are denoted by c₁, c₂, c₃ and the volume fraction of the austenitic phase (A) is denoted by c₄. The rate ${\dot{c}}_i$ of each volume fraction is obtained by summing up the reaction rates ${\dot{\xi }}_j$ [34], i.e.

$${\dot{c}}_i={\nu }_{ij}{\dot{\xi }}_j,$$ 2.9

where

$${\nu }_{ij}=\left[\begin{array}{cccccc}1& 0& 0& -1& 0& 1\\ 0& 1& 0& 1& 1& 0\\ 0& 0& 1& 0& -1& -1\\ -1& -1& -1& 0& 0& 0\end{array}\right].$$ 2.10

The total volume fractions are subjected to the following constraints

$$\sum _i^4{\dot{c}}_i=0,\sum _i^4{c}_i=1\text{and}0\le c_i\le 1.$$

If the initial volume fractions of M₁, M₂, M₃, and A are c₀₁, c₀₂, c₀₃, and c₀₄, respectively, then we can write

$$c_i=c_{0i}+{\nu }_{ij}{\xi }_j.$$ 2.11

We assume that the inelastic strain rate ${\dot{\mathbf{E}}}^I$ obeys the following flow rule

$${\dot{\mathbf{E}}}^I=\sum _{i=1}^3{\mathit{\Lambda }}_i^t\dot{{\xi }_i}+\sum _{i=4}^6{\mathit{\Lambda }}_i^r\dot{{\xi }_i}.$$ 2.12

The tensors ${\mathit{\Lambda }}_i^t$ and ${\mathit{\Lambda }}_i^r$ describe the direction and magnitude of the strain generated during phase transformation and variant reorientation, respectively.

Figure 1.

Schematic of phase transformation and reorientation in MSMA.

Similarly, we consider that the rate of magnetization vector ${\dot{\mathbf{M}}}^I$ can be expressed in the following way,

$${\dot{\mathbf{M}}}^I=\sum _{i=1}^3{\mathit{\gamma }}_i^t\dot{{\xi }_i}+\sum _{i=4}^6{\mathit{\gamma }}_i^r\dot{{\xi }_i},$$ 2.13

where the vectors ${\mathit{\gamma }}_i^t$ and ${\mathit{\gamma }}_i^r$ take into account the direction and magnitude of the internal magnetization owing to phase transformation and reorientation, respectively.

The evolution of $\dot{g}$ can be represented by the following flow rule

$$\dot{g}=\sum _{i=1}^3{f}_i^t\dot{{\xi }_i}+\sum _{i=4}^6{f}_i^r\dot{{\xi }_i},$$ 2.14

where f_is are the hardening functions. Restrictions on the evolution equations will be imposed through the second law of thermodynamics.

It should be noted that G is an objective scalar function, because the quantities S^E, H are frame indifferent and do not depend on a superimposed rigid rotation, i.e. S^E=S^E*, and H=H*, where ‘*’ is a rigidly rotated current configuration. We assume that in the reference configuration E^I=E^I* and M^I=M^I* and so their rates ${\dot{\mathbf{E}}}^I={\dot{\mathbf{E}}}^{I\ast }$ and ${\dot{\mathbf{M}}}^I={\dot{\mathbf{M}}}^{I\ast }$.

3. Material symmetry and constitutive equations for magnetic shape memory alloys

We want to construct a Gibbs free energy

$$G=G(S^E,\mathrm{H},\mathrm{T},{\mathrm{E}}^I,{\mathrm{M}}^I,{\xi }_i,g,\{\mathbb{S}\}),$$ 3.1

which is invariant under a magnetic point group$\{\mathcal{M}\}$ or continuous group $\{\mathcal{A}\}$ (see appendix for more details). The anisotropy for a continuous symmetry group is considered by introducing structural tensors $\{\mathbb{S}\}$. Here, S^E,E^I are the polar, second-order, symmetric, i-tensors and H,M^I are the axial c-vectors. MSMA single crystals exhibit discrete symmetries and belong to $\{\mathcal{M}\},$ whereas polycrystals may exhibit continuous symmetry and belong to $\{\mathcal{A}\}$.

Let Υ_ϕ be the integrity basis of (3.1) which may be split such that

$${\mathrm{{\rm Y}}}_{\varphi }={\mathrm{{\rm Y}}}_{\varphi I}\bigcup {\mathrm{{\rm Y}}}_{\varphi P}\text{and}{\mathrm{{\rm Y}}}_{\varphi I}\bigcap {\mathrm{{\rm Y}}}_{\varphi P}=\mathrm{\varnothing }.$$

Here, Υ_ϕI contains the elements of the integrity basis with at least one internal variable, and Υ_ϕP is the set of the elements of the integrity basis with S^E,H. Based on the above arguments, we propose

$$G({\mathrm{{\rm Y}}}_{\varphi },T,{\xi }_i,g)=\sum _{j=1}^4{c}_j({\xi }_i)G_{P_j}({\mathrm{{\rm Y}}}_{\varphi P},T)+G_{\mathrm{mix}}({\mathrm{{\rm Y}}}_{\varphi I},T,g),$$ 3.2

where G_P1, G_P2, G_P3, G_P4 and G_mix are the Gibbs free energy of the stress-favoured martensitic variants, field-favoured martensitic variants, twinned martensitic variants, austenitic phase and mixing of different phases, respectively. We denote the sets obtained by taking partial derivative of the elements of Υ_ϕ with respect to H, S^E, E^I and M^I by Υ_H, ${\mathrm{{\rm Y}}}_{{\mathbf{S}}^E}$, ${\mathrm{{\rm Y}}}_{{\mathbf{E}}^I}$ and ${\mathrm{{\rm Y}}}_{{\mathbf{M}}^I}$, respectively. In addition, for continuous symmetry, we denote ${\mathrm{{\rm Y}}}_{\{\mathbb{S}\}}$ as the partial derivative of Υ_ϕ with respect to $\{\mathbb{S}\}$.

Using equations (2.6a)–(2.6c), one can write

$$\begin{array}{rl}\mathbf{E}& =-{\rho }_{0}G{,}_{{\mathbf{S}}^E}=\sum _{{\mathbf{T}}_{\mathbf{a}}\in {\mathbf{{\rm Y}}}_{{\mathbf{S}}^{\mathbf{E}}}}{\alpha }_a({\mathrm{{\rm Y}}}_{\varphi }){\mathbf{T}}_{\mathbf{a}},\end{array}$$ 3.3a

$$\begin{array}{rl}\mathbf{M}& =-\frac{{\rho }_0}{{\mu }_0}G{,}_{\mathbf{H}}=\sum _{{\mathbf{v}}_b\in {\mathbf{{\rm Y}}}_{\mathbf{H}}}{\beta }_b({\mathrm{{\rm Y}}}_{\varphi }){\mathbf{v}}_b\end{array}$$ 3.3b

and

$$\begin{array}{rl}s& =-{\rho }_{0}G{,}_T.\end{array}$$ 3.3c

By denoting $G_e({\mathrm{{\rm Y}}}_{\varphi P},T,{\xi }_i)=\sum _{j=1}^4{c}_j({\xi }_i)G_{P_j}({\mathrm{{\rm Y}}}_{\varphi P},T)$ in (3.2), equation (3.3a) can be written as $\mathbf{E}={\mathbf{E}}^e+{\overline{\mathbf{E}}}^I$, where ${\mathbf{E}}^e=-{\rho }_0{G}_e{,}_{S^E}$ and ${\overline{\mathbf{E}}}^I=-{\rho }_0{G}_{\mathrm{mix}}{,}_{S^E}$. If we further restrict G_mix to be only a linear function of S^E and E^I, then one may obtain ${\overline{\mathbf{E}}}^I={\mathbf{E}}^I$. Under this condition, E^e can be identified as elastic strain. Similarly, we can decompose the total magnetization as $\mathbf{M}={\mathbf{M}}^e+{\overline{\mathbf{M}}}^I$, where M^e=−ρ₀G_e,_H and ${\overline{\mathbf{M}}}^I=-{\rho }_0{G}_{\mathrm{mix}}{,}_H$.

The second law (2.7) can then be written as

$${\mathit{\pi }}_{E^I}:{\dot{\mathbf{E}}}^I+{\mathit{\pi }}_{M^I}:{\dot{\mathbf{M}}}^I+{\pi }_{{\xi }_i}{\dot{\xi }}_i+{\pi }_g\dot{g}+{\mathit{\pi }}_{\{\mathbb{S}\}}:\dot{\{\mathbb{S}\}}\ge 0.$$ 3.4

where

$$\begin{array}{rl}{\mathit{\pi }}_{E^I}& =-{\rho }_{0}G{,}_{{\mathbf{E}}^I}=\sum _{{\mathbf{R}}_{\mathbf{a}}\in {\mathbf{{\rm Y}}}_{{\mathbf{E}}^{\mathbf{I}}}}{\chi }_a({\mathrm{{\rm Y}}}_{\varphi }){\mathbf{R}}_{\mathbf{a}}\mathbf{.}\\ {\mathit{\pi }}_{M^I}& =-{\rho }_{0}G{,}_{{\mathbf{M}}^I}=\sum _{{\mathbf{u}}_b\in {\mathbf{{\rm Y}}}_{{\mathbf{M}}^{\mathbf{I}}}}{\phi }_b({\mathrm{{\rm Y}}}_{\varphi }){\mathbf{u}}_b,\\ {\pi }_{{\xi }_i}& =-{\rho }_{0}G{,}_{{\xi }_i},\\ {\pi }_g& =-{\rho }_{0}G{,}_g\\ \mathrm{and}{\mathit{\pi }}_{\{\mathbb{S}\}}& =-{\rho }_{0}G{,}_{\{\mathbb{S}\}}=\sum _{{\mathbf{U}}_{\mathbf{c}}\in {\mathbf{{\rm Y}}}_{\{\mathbb{S}\}}}{\omega }_c({\mathrm{{\rm Y}}}_{\varphi }){\mathbf{U}}_{\mathbf{c}}\mathbf{.}\end{array}$$

The rate ${\dot{\mathbf{E}}}^{\mathit{I}}$, ${\dot{\mathbf{M}}}^{\mathit{I}}$ and $\dot{g}$ can be obtained from the evolution equations (2.12)–(2.14).

Considering the evolution equations for the inelastic strain (2.12), we assume that transformation and reorientation depends on stress, field, temperature and structural tensor (for continuous symmetry only). Then, ${\mathit{\Lambda }}_v^{\beta }$ can be represented as

$${({\mathit{\Lambda }}_{ij}^{\beta })}_v={({\mathit{\Lambda }}_{ij}^{\beta })}_v({\mathbf{S}}^E,\mathbf{H},\mathbf{T},\{\mathbb{S}\}\mathbf{)},$$

whereas for the evolution of the magnetization and mixing energy equation, we can write

$${({\gamma }_i^{\beta })}_v={({\gamma }_i^{\beta })}_v({\mathbf{S}}^E,\mathbf{H},\mathbf{T},\{\mathbb{S}\}\mathbf{)}$$

and

$$f_v^{\beta }=f_v^{\beta }({\mathbf{S}}^E,\mathbf{H},\mathbf{T},\{\mathbb{S}\}\mathbf{)}\mathbf{.}$$

Here, β=t for transformation, β=r for reorientation, and v may vary from 1 to 6. Following appendix C, we further write

$${({\mathit{\Lambda }}_{ij}^{\beta })}_v=\sum _{p=1}^m{c}_{p_v}(I_1,\dots ,I_r){({\mathcal{D}}_{ij}^p)}_v.$$ 3.5

Similarly, from the evolution equation for the internal magnetization (2.13), we can write for any generic ${\mathit{\gamma }}_v^{\beta }$,

$${({\gamma }_i^{\beta })}_v=\sum _{p=1}^m{c}_{p_v}^{\prime }(I_1,\dots ,I_r){({\mathcal{D}}_i^p)}_v,$$ 3.6

and for the hardening function

$${(f^{\beta })}_v=c_{p_v}^{″}(I_1,\dots ,I_r).$$ 3.7

We further focus on continuous symmetry for which one needs to know about the evolution of structural tensors. Because the direction of texturing, which is denoted by e₃, may change owing to the microstructural change during phase transformation and variant reorientation, the structural tensors evolve with the change in the direction of e₃. Let $\mathit{\Sigma }\in \{\mathbb{S}\}$ and

$$\mathit{\Sigma }=\mathit{\Sigma }({\mathbf{e}}_3\mathbf{)}\mathbf{.}$$

The rate of change of the structural tensor can then be represented by

$$\dot{\mathit{\Sigma }}=\mathbf{\Xi }({\mathbf{e}}_3,{\dot{\mathbf{e}}}_3\mathbf{)}\mathbf{.}$$ 3.8

We can express e₃ with respect to the directional cosines such that ${\mathbf{e}}_3=\mathbf{(}\cos {\alpha }_1,$ ${\cos {\alpha }_2,\cos {\alpha }_3)}^{\mathrm{T}}$ and

$${\dot{\mathbf{e}}}_3=-(\sin {\alpha }_1({\dot{\alpha }}_1),\sin {\alpha }_2({\dot{\alpha }}_2),\sin {\alpha }_3{({\dot{\alpha }}_3))}^{\mathrm{T}}.$$

The evolution of the α_j (j=1,3) may be related with the evolution of the volume fractions such that

$${\dot{\alpha }}_j=\sum _{i=1}^3{\mathrm{\Theta }}_{ij}^t\dot{{\xi }_i}+\sum _{i=3}^6{\mathrm{\Theta }}_{ij}^r\dot{{\xi }_i}.$$ 3.9

Here, ${\mathrm{\Theta }}_{ij}^{\beta }$ are scalars that take into account the change in α_i owing to changes in ξ_j.

4. Integrity basis for finite symmetry

Our next objective is to find out the integrity basis for the Gibbs free energy G(Υ_ϕ) and material tensors ${\mathit{\Lambda }}_i^t$, ${\mathit{\Lambda }}_i^r$, ${\mathit{\gamma }}_i^t$ and ${\mathit{\gamma }}_i^r$ for MSMA material systems which belong to a specific class of symmetry group. For finite symmetry, we consider FIPT and variant reorientation in a single-crystal specimen. FIPT takes place from a higher symmetric austenitic phase to a lower symmetric martensitic phase, and we consider NiMnCoIn single crystal which has a ferromagnetic austenitic phase and antiferromagnetic martensitic phase. Field-induced variant reorientation is observed in the lower symmetric martensitic phase owing to reorientation of martensitic variants. The most widely used material for this mechanism is Ni₂MnGa. Now, we consider the symmetries associated with such material systems to generate the integrity basis.

(a). Ferromagnetic cubic austenitic phase

The integrity basis for the Gibbs free energy is derived for the parent austenitic phase. It is a well-known fact that cubic symmetry does not support ferromagnetism [35–37]. For example, the symmetry of bcc α-iron is often thought to be cubic, but is tetragonal owing to the axially symmetric magnetic moment [38]. Similarly, a reduction in the symmetry of the Ni crystal structure occurs from fcc to trigonal ($3\underset{\_}{m}$) owing to the alignment of the magnetic moment along the [111] direction. The Cu₂MnAl Heusler alloy has the L2₁ chemical structure and belongs to Fm3m space group even though the magnetic point group of this compound, such as Ni, is $3\underset{\_}{m}$ [35].

In the present case, the NiMnCoIn crystal exhibits L2₁-type Heusler structure with the Fm3m space group [39]. The magnetic point group has not been reported to date. Because the crystal structure and space group of NiMnCoIn resembles Cu₂MnAl Heusler alloy, we consider that the ferromagnetic austenitic phase belongs to $3\underset{\_}{m}$ magnetic point group. The irreducible representation for $3\underset{\_}{m}$ can be found in [40]. The threefold rotations are along the z-axis which is perpendicular to the plane of the paper and obey the right-hand rule. The decompositions of axial c-vectors and polar i-tensors are given in table 1.

Table 1.

Decomposition of magneto-mechanical quantities of $3\underset{\_}{m}$ magnetic point group.

type	decomposition
axial c-vectors	Γ(1)⊕Γ(3)
polar i-tensors	2Γ(1)⊕2Γ(3)

The Gibbs free energy of the austenitic phase is represented by G_P4(Υ_ϕP,T), and we want to calculate the elements of integrity basis Υ_ϕP. The basic quantities for the variables (S^E,H,T) are

$$\begin{array}{rl}{\mathit{\Gamma }}^{(1)}:\{u_1^{(1)},u_2^{(1)},u_3^{(1)}\}& =\{H_3,S_{11}^E+S_{22}^E,S_{33}^E\},\\ {\mathit{\Gamma }}^{(3)}:\{{\mathbf{u}}_1^{(3)},{\mathbf{u}}_2^{(3)},{\mathbf{u}}_3^{(3)}\}& =\{\left[\begin{array}{c}{H}_1\\ H_2\end{array}\right],\left[\begin{array}{c}{S}_{13}^E\\ S_{23}^E\end{array}\right],\left[\begin{array}{c}2S_{12}^E\\ S_{11}^E-S_{22}^E\end{array}\right]\}.\end{array}$$

The elements of the integrity bases, in terms of the basic quantities, are given by

$$\text{Degree 1:}{u}_i^{(1)}(i=1,3)$$

and

$$\text{Degree 2:}{\mathbf{u}}_i^{(3)}\cdot {\mathbf{u}}_j^{(3)}(i,j=1,3).$$

(b). Antiferromagnetic monoclinic martensitic phase

Here, we derive the integrity basis of the Gibbs free energy first and then of material tensors associated with the evolution equations of the internal state variables.

(i). Integrity basis for the Gibbs free energy

The martensitic phase is 14M monoclonic [2,39] and belongs to 2/m (C_2h) classical point group. The three magnetic point groups of 2/m are $\underset{\_}{2}/\underset{\_}{m}$, $2/\underset{\_}{m}$, NiMnCoIn martensitic phase has not been reported in the literature so far.

The integrity basis for each magnetic point group is different. We need to identify the group which is closest to the observed material response. For example, any material belonging to group $\underset{\_}{2}/\underset{\_}{m}$ is a ferromagnetic material [37,41]. Thus, we eliminate this group for the antiferromagnetic martensitic phase. Now, both $2/\underset{\_}{m},\hspace{0.17em}\underset{\_}{2}/m$ are antiferromagnetic, and it can be shown that the integrity bases for magneto-mechanical coupling up to second order are the same for $2/\underset{\_}{m}$ and $\underset{\_}{2}/m$ [40]. We select $\underset{\_}{2}/m$ to proceed. Another example of a material system that belongs to 2/m class is the martensitic phase of Ni–Ti [42], which is widely used in shape memory alloys. The irreducible representation for $\underset{\_}{2}/m$ can be obtained from [40]. The decompositions of axial c-vectors and polar i-tensors are presented in table 2.

Table 2.

Decomposition of magneto-mechanical quantities of $\underset{\_}{2}/m$ magnetic point group.

type	decomposition
axial c-vectors	2Γ(2)⊕Γ(3)
polar i-tensors	4Γ(1)⊕2Γ(4)

Because the nucleating phase is martensite during phase transformation, we impose symmetry restrictions of $\underset{\_}{2}/m$ on G_mix to calculate Υ_ϕI. The basic quantities are given by

$$\begin{array}{rl}{\mathit{\Gamma }}^{(1)}:& \{u_1^{(1)},u_2^{(1)},u_3^{(1)},u_4^{(1)}\}=\{S_{11}^E,S_{22}^E,S_{33}^E,S_{12}^E\},\\ & \{u_5^{(1)},u_6^{(1)},u_7^{(1)},u_8^{(1)}\}=\{E_{11}^I,E_{22}^I,E_{33}^I,E_{12}^I\},\\ {\mathit{\Gamma }}^{(2)}:& \{u_1^{(2)},u_2^{(2)},u_3^{(2)},u_4^{(2)}\}=\{H_1,H_2,M_1^I,M_2^I\},\\ {\mathit{\Gamma }}^{(3)}:& \{u_1^{(3)},u_2^{(3)}\}=\{H_3,M_3^I\}\\ \mathrm{and}{\mathit{\Gamma }}^{(4)}:& \{u_1^{(4)},u_2^{(4)},u_3^{(4)},u_4^{(4)}\}=\{S_{23}^E,S_{31}^E,E_{23}^I,E_{31}^I\}.\end{array}$$

The elements of the integrity basis, in terms of basic quantities, are

$$\text{Degree 1:}{u}_i^{(1)}(i=1,8)$$

and

$$\text{Degree 2:}{u}_j^{(2)}{u}_k^{(2)},u_l^{(3)}{u}_m^{(3)},u_r^{(4)}{u}_s^{(4)}(j,k=1,4;\hspace{0.17em}l,m=1,2;\hspace{0.17em}r,s=1,4).$$

The integrity basis Υ_ϕP can be obtained by considering E^I=0 and M^I=0.

(ii). Integrity basis for material tensors

The stress-favoured martensitic variant is nucleated from the austenitic phase under high stress and low magnetic field, because austenitic phase is stable only at high field. We consider that only the stress-favoured single-crystal martensitic variant exists. Nucleation of the new phase causes inelastic deformation and the changes in strain and magnetization are taken into account through evolution of E^I and M^I which are related to that of ξ₁ through transformation tensor ${\mathit{\Lambda }}_1^t$ and transformation vector ${\mathit{\gamma }}_1^t$, respectively. We consider the strain evolution equation in the following form

$${\dot{E}}_{ij}^I={({\mathit{\Lambda }}_{ij}^t)}_1({\mathbf{S}}^E){\dot{\xi }}_1.$$ 4.1

As described in appendix C, we construct V =V (t,S^E), and the basic quantities are given by

$$\begin{array}{rl}{\mathit{\Gamma }}^{(1)}:& \{u_1^{(1)},u_2^{(1)},u_3^{(1)},u_4^{(1)}\}=\{S_{11}^E,S_{22}^E,S_{33}^E,S_{12}^E\},\\ & \{u_5^{(1)},u_6^{(1)},u_7^{(1)},u_8^{(1)}\}=\{t_{11},t_{22},t_{33},t_{12}\}\\ \mathrm{and}{\mathit{\Gamma }}^{(4)}:& \{u_1^{(4)},u_2^{(4)},u_3^{(4)},u_4^{(4)}\}=\{S_{23}^E,S_{31}^E,t_{23},t_{31}\}.\end{array}$$

Elements of the integrity bases are

$$\text{Degree 1:}{u}_1^{(1)},u_2^{(1)},u_3^{(1)},u_4^{(1)},u_5^{(1)},u_6^{(1)},u_7^{(1)},u_8^{(1)}$$

and

$$\begin{array}{rl}\text{Degree 2:}& {(u_1^{(4)})}^2,{(u_2^{(4)})}^2,{(u_3^{(4)})}^2,{(u_4^{(4)})}^2,u_1^{(4)}{u}_2^{(4)},u_1^{(4)}{u}_3^{(4)},u_1^{(4)}{u}_4^{(4)},\\ & u_2^{(4)}{u}_3^{(4)},u_2^{(4)}{u}_4^{(4)},u_3^{(4)}{u}_4^{(4)}.\end{array}$$

Following appendix C, we find that $\{I\}=\{u_1^{(1)},u_2^{(1)},u_3^{(1)},u_4^{(1)},{(u_1^{(4)})}^2,{(u_2^{(4)})}^2,u_1^{(4)}{u}_2^{(4)}\}$ are independent of t and $\{L\}=\{u_5^{(1)},u_6^{(1)},u_7^{(1)},u_8^{(1)},u_1^{(4)}{u}_3^{(4)},u_1^{(4)}{u}_4^{(4)},u_2^{(4)}{u}_3^{(4)},u_2^{(4)}{u}_4^{(4)}\}$ are linear in t, and we can write

$${({\mathit{\Lambda }}_{ij}^t)}_1=\sum _{p=1}^8{c}_{p_1}(\{I\}){({\mathcal{D}}_{ij}^p)}_1,$$ 4.2

where the elements of the set $\{{\mathcal{D}}_{ij}\}=\mathrm{\partial }\{L\}/\mathrm{\partial }{t}_{ij}$.

Similarly, for the internal magnetization, the evolution equation is

$${\dot{M}}_i^I={({\gamma }_i^t)}_1({\mathbf{S}}^{\mathit{E}}\mathbf{)}{\dot{\xi }}_1,$$ 4.3

and, as mentioned earlier, we construct V =V (r,S^E). The basic quantities for this case

$$\begin{array}{rl}{\mathit{\Gamma }}^{(1)}:& \{u_1^{(1)},u_2^{(1)},u_3^{(1)},u_4^{(1)}\}=\{S_{11}^E,S_{22}^E,S_{33}^E,S_{12}^E\},\\ {\mathit{\Gamma }}^{(2)}:& \{u_1^{(2)},u_2^{(2)}\}=\{r_1,r_2\},\\ {\mathit{\Gamma }}^{(3)}:& \{u_1^{(3)}\}=\{r_3\}\\ \mathrm{and}{\mathit{\Gamma }}^{(4)}:& \{u_1^{(4)},u_2^{(4)}\}=\{S_{32}^E,S_{31}^E\}.\end{array}$$

Among the elements of the integrity basis, only $\{L\}=\{u_1^{(2)},u_2^{(2)},u_1^{(3)}\}$ are linear in r. We write

$${({\gamma }_i^t)}_1=\sum _{p=1}^3{c}_{p_1}^{\prime }(\{I\}){({\mathcal{D}}_i^p)}_1,$$ 4.4

where the elements of $\{{\mathcal{D}}_i\}=\mathrm{\partial }\{L\}/\mathrm{\partial }{r}_i$ and {I} is the same as described for ${\mathit{\Lambda }}_1^t$.

(c). Tetragonal ferromagnetic martensitic phase

We consider tetragonal symmetry for variant reorientation. The integrity basis of the Gibbs free energy is derived first, followed by the integrity basis of the material tensors.

(i). Integrity basis for the Gibbs free energy

The martensitic phase has 10 M structure and belongs to I4/mmm space group [43]. The classical point group is 4/mmm (D_4h). The five magnetic point groups are $\underset{\_}{4}/\underset{\_}{m}\underset{\_}{m}m,4/\underset{\_}{m}mm,\underset{\_}{4}/mm\underset{\_}{m},4/\underset{\_}{m}\underset{\_}{m}\underset{\_}{m}$ and $4/m\underset{\_}{m}\underset{\_}{m}$. Among them only $4/m\underset{\_}{m}\underset{\_}{m}$ is ferromagnetic and rest of the members are antiferromagnetic [37]. So, we consider $4/m\underset{\_}{m}\underset{\_}{m}$ to develop the integrity basis.

There are three possible variants for tetragonal martensite. We denote variant-3 to be that which has its shorter length (c) along the z-direction (also the axis of the fourfold discrete symmetric rotations). The x- and y-axes are parallel to the longer side with length a. The irreducible representation can be obtained from [40]. Co–Ni–Al, for example, is another material system of MSMAs that also belongs to the 4/mmm class [44]. The magneto-mechanical decompositions are presented in table 3, and the basic quantities are given by

$$\begin{array}{rl}{\mathit{\Gamma }}^{(1)}:& \{u_1^{(1)},u_2^{(1)},u_3^{(1)},u_4^{(1)},u_5^{(1)},u_6^{(1)}\}=\{H_3,M_3^I,S_{11}^E+S_{22}^E,S_{33}^E,E_{11}^I+E_{22}^I,E_{33}^I\},\\ {\mathit{\Gamma }}^{(3)}:& \{u_1^{(3)},u_2^{(3)}\}=\{S_{12}^E,E_{12}^I\},\\ {\mathit{\Gamma }}^{(4)}:& \{u_1^{(4)},u_2^{(4)}\}=\{S_{11}^E-S_{22}^E,E_{11}^I-E_{22}^I\}\\ \mathrm{and}{\mathit{\Gamma }}^{(5)}:& \{{\mathbf{u}}_1^{(5)},{\mathbf{u}}_2^{(5)},{\mathbf{u}}_3^{(5)},{\mathbf{u}}_4^{(5)}\}=\{\left[\begin{array}{c}{H}_2\\ -H_1\end{array}\right],\left[\begin{array}{c}{M}_2^I\\ -M_1^I\end{array}\right],\left[\begin{array}{c}{S}_{23}^E\\ -S_{31}^E\end{array}\right],\left[\begin{array}{c}{E}_{23}^I\\ -E_{31}^I\end{array}\right]\}.\end{array}$$

The components of the basic quantities for Υ_ϕI are presented with respect to the orientation of the crystal with crystallographic c-axis along z. The elements of the integrity basis are given by

$$\begin{array}{rl}\text{Degree 1:}& u_i^{(1)}(i=1,6)\\ \text{and}\text{Degree 2:}& u_l^{(3)}{u}_m^{(3)},u_r^{(4)}{u}_s^{(4)},{\mathbf{u}}_i^{(5)}\cdot {\mathbf{u}}_j^{(5)}(l,m,r,s=1,2;\hspace{0.17em}i,j=1,4).\end{array}\}$$ 4.5

The integrity basis for Υ_ϕP can be obtained from with E^I=0 and M^I=0.

Table 3.

Decomposition of magneto-mechanical quantities of $4/m\underset{\_}{m}\underset{\_}{m}$ magnetic point group.

type	decomposition
axial c-vectors	Γ(1)⊕Γ(5)
polar i-tensors	2Γ(1)⊕Γ(3)⊕Γ(4)⊕Γ(5)

(ii). Integrity basis for material tensors

We assume that the initial phase of the single-crystal MSMA is the stress-favoured variant-1 and with the application of magnetic field, the field-favoured variant-2 nucleates. The internal strain and magnetization thus generated are taken into account by considering the evolution of internal variables E^I and M^I. We write the strain evolution equation for nucleation of variant-2 as

$${\dot{E}}_{ij}^I={({\mathit{\Lambda }}_{ij}^r)}_4({\mathbf{S}}^{\mathbf{E}}\mathbf{)}{\dot{\xi }}_4\mathbf{.}$$

As described in appendix C, we construct V =V (t,S^E) and the basic quantities are given by

$$\begin{array}{rl}{\mathit{\Gamma }}^{(1)}:& \{u_1^{(1)},u_2^{(1)},u_3^{(1)},u_4^{(1)}\}=\{S_{11}^E+S_{22}^E,S_{33}^E,t_{11}+t_{22},t_{33}\},\\ {\mathit{\Gamma }}^{(3)}:& \{u_1^{(3)},u_2^{(3)}\}=\{S_{12}^E,t_{12}\},\\ {\mathit{\Gamma }}^{(4)}:& \{u_1^{(4)},u_2^{(4)}\}=\{S_{11}^E-S_2^E,t_{11}-t_{22}\}\\ \mathrm{and}{\mathit{\Gamma }}^{(5)}:& \{{\mathbf{u}}_1^{(5)},{\mathbf{u}}_2^{(5)}\}=\{\left[\begin{array}{c}{S}_{23}^E\\ -S_{31}^E\end{array}\right],\left[\begin{array}{c}{t}_{23}\\ -t_{31}\end{array}\right]\}.\end{array}$$

Here, $\{I\}=\{u_1^{(1)},u_2^{(1)},u_1^{(3)},u_1^{(4)}{\mathbf{u}}_1^{(5)}\cdot {\mathbf{u}}_1^{(5)}\}$ is independent of t and $\{L\}=\{u_3^{(1)},u_4^{(1)},u_1^{(3)}{u}_2^{(3)},$ $u_1^{(4)}{u}_2^{(4)},{\mathbf{u}}_1^{(5)}\cdot {\mathbf{u}}_2^{(5)}\}$ are linear in t. We write

$${({\mathit{\Lambda }}_{ij}^r)}_4=\sum _{p=1}^5{c}_{p_4}(\{I\}){({\mathcal{D}}_{ij}^p)}_4.$$ 4.6

where the elements of the set $\{{\mathcal{D}}_{ij}\}=\mathrm{\partial }\{L\}/\mathrm{\partial }{t}_{ij}$.

Similarly, the magnetization evolution equation for nucleation of variant-2 is given by

$${\dot{M}}_i^I={({\gamma }_i^r)}_4({\mathbf{S}}^{\mathit{E}}\mathbf{)}{\dot{\xi }}_4\mathbf{.}$$ 4.7

We construct V =V (r,S^E), and the basic quantities are given by

$${\mathit{\Gamma }}^{(1)}:\{u_1^{(1)},u_2^{(1)},u_3^{(1)}\}=\{r_3,S_{11}^E+S_{22}^E,S_{33}^E\}$$

and

$$\begin{array}{rl}{\mathit{\Gamma }}^{(5)}:& \{{\mathbf{u}}_1^{(5)},{\mathbf{u}}_2^{(5)}\}=\{\left[\begin{array}{c}{r}_2\\ -r_1\end{array}\right],\left[\begin{array}{c}{S}_{23}^E\\ -S_{31}^E\end{array}\right]\}.\end{array}$$

For this case, $\{I\}=\{u_2^{(1)},u_3^{(1)},{\mathbf{u}}_2^{(5)}\cdot {\mathbf{u}}_2^{(5)}\}$ is independent of r, and $\{L\}=\{u_1^{(1)},{\mathbf{u}}_1^{(5)}\cdot {\mathbf{u}}_2^{(5)}\}$ is linear in r. This implies

$${({\gamma }_i^r)}_4=\sum _{p=1}^2{c}_{p_4}^{\prime }(\{I\}){({\mathcal{D}}_i^p)}_4,$$ 4.8

where $\{{\mathcal{D}}_i\}=\mathrm{\partial }\{L\}/\mathrm{\partial }{r}_i$.

5. Integrity basis of the Gibbs free energy for continuous symmetry

We have already discussed anisotropy for a single crystal, single variant MSMAs by considering finite symmetry. However, anisotropy may exist in a specimen owing to the polycrystalline nature of the presence of multiple variants of the martensitic phase. The variants may have some preferred directions of anisotropy.

We confine our analysis by considering transverse isotropy which belongs to $D_{\mathrm{\infty }h}$ group for which the structural tensor has the form e₃⊗e₃, where e₃ is the preferred unit direction of texturing (appendix B). We consider a⊗a as the mechanical structural tensor and f⊗f as the magnetic structural tensor such that $\{\mathbb{S}\}=\{\mathbf{a}\otimes \mathbf{a},\mathbf{f}\otimes \mathbf{f}\}$. The unit vectors a and f are the direction of mechanical and magnetic anisotropy, respectively. Moreover, a⊗a and f⊗f may evolve during loading owing to microstructural change. The Gibbs free energy for continuous symmetry can be expressed as

$$G=G({\mathbf{S}}^E,\mathbf{H},\mathbf{T},{\mathbf{E}}^{\mathit{I}},{\mathbf{M}}^{\mathit{I}},{\xi }_{\mathbf{i}},g,\mathbf{a}\otimes \mathbf{a},\mathbf{f}\otimes \mathbf{f}\mathbf{)},$$ 5.1

We assume that the mechanical transverse anisotropy is predominant in the directional tensor of the inelastic strain. Thus, ${\mathit{\Lambda }}_i^{\beta }={\mathit{\Lambda }}_i^{\beta }({\mathbf{S}}^E,\mathbf{a}\otimes \mathbf{a},\mathbf{H}\mathbf{)}$. Similarly, the directional vector for magnetization evolution ${\mathit{\gamma }}_i^{\beta }({\mathbf{S}}^E,\mathbf{f}\otimes \mathbf{f},\mathbf{H})$ is dominated by the magnetic transverse anisotropy. We consider $\mathbf{a}={\mathbf{(}\cos {\alpha }_1^m,\cos {\alpha }_2^m,\cos {\alpha }_3^m\mathbf{)}}^{\mathrm{T}}$ and $\mathbf{f}={\mathbf{(}\cos {\alpha }_1^{\mathbf{f}},\cos {\alpha }_2^{\mathbf{f}},\cos {\alpha }_3^{\mathbf{f}}\mathbf{)}}^{\mathrm{T}}$, where (α₁,α₂,α₃) are the angles made by the unit directional vector with the global axes. Denoting r as either a or f, we can write

$${[\mathbf{r}\otimes \mathbf{r}\mathbf{]}}_{ij}=\left[\begin{array}{ccc}{\cos }^2{\alpha }_1& \cos {\alpha }_1\cos {\alpha }_2& \cos {\alpha }_1\cos {\alpha }_3\\ \cos {\alpha }_1\cos {\alpha }_2& {\cos }^2{\alpha }_2& \cos {\alpha }_2\cos {\alpha }_3\\ \cos {\alpha }_1\cos {\alpha }_3& \cos {\alpha }_2\cos {\alpha }_3& {\cos }^2{\alpha }_3\end{array}\right]\mathbf{.}$$ 5.2

By taking the time derivative of (5.2), one can write

$$\dot{(\mathbf{a}\otimes \mathbf{a}\mathbf{)}}={\mathbf{L}}^m\mathbf{(}{\alpha }_{\mathbf{i}}^m,{\dot{\alpha }}_{\mathbf{i}}^m\mathbf{)}$$

and

$$\dot{(\mathbf{f}\otimes \mathbf{f}\mathbf{)}}={\mathbf{L}}^f\mathbf{(}{\alpha }_{\mathbf{i}}^{\mathit{f}},{\dot{\alpha }}_{\mathbf{i}}^{\mathit{f}}\mathbf{)},$$

and the evolution equations for α are defined through equation (3.9)

$${\dot{\alpha }}_j^m=\sum _{i=1}^3{\mathrm{\Theta }}_{ij}^{tm}\dot{{\xi }_i}+\sum _{i=4}^6{\mathrm{\Theta }}_{ij}^{rm}\dot{{\xi }_i}$$ 5.3a

and

$${\dot{\alpha }}_j^f=\sum _{i=1}^3{\mathrm{\Theta }}_{ij}^{tf}\dot{{\xi }_i}+\sum _{i=4}^6{\mathrm{\Theta }}_{ij}^{rf}\dot{{\xi }_i}.$$ 5.3b

Here, ${\mathrm{\Theta }}_{ij}^{\beta m}$ (β=t for transformation, β=r for reorientation) are scalars that take into account the change in α_i owing to change in ξ_j. Moreover, we assume ${\mathrm{\Theta }}_{ij}^{\beta m}({\mathbf{S}}^E,\mathbf{H},\mathbf{a}\otimes \mathbf{a},\mathbf{f}\otimes \mathbf{f}\mathbf{)}$, ${\mathrm{\Theta }}_{ij}^{\beta f}({\mathbf{S}}^E,\mathbf{H},\mathbf{a}\otimes \mathbf{a},\mathbf{f}\otimes \mathbf{f}\mathbf{)}$ and hardening function $f_i^{\beta }({\mathbf{S}}^E,\mathbf{H},\mathbf{a}\otimes \mathbf{a},\mathbf{f}\otimes \mathbf{f}\mathbf{)}$ depend on both mechanical and magnetic anisotropy. The isotropic scalar invariants are well established in the literature [45,46] and are not explicitly discussed here.

As an example, the magnetic energy is independent of crystallographic orientation for NiMnCoIn polycrystals and provides an opportunity to use polycrystals for actuator application [47]. Modelling of such material responses can be approached by following the continuous symmetry method, as discussed in this section.

6. Applications of the theory for magnetic shape memory alloys

Here, we propose a specific form of the Gibbs free energy and explicit expressions of the magneto-mechanical constitutive equations are derived. A specific loading path is selected to further reduce the constitutive equations to a simpler form. We consider two examples to demonstrate the impact of considering symmetry restrictions in the modelling. Variant reorientation for a single-crystal NiMnGa will be considered followed by an example of phase transformation in a Fe–Mn–C polycrystalline MSMA.

(a). Discrete symmetry and field-induced variant reorientation

We consider the stress-favoured martensitic variant reorients to the field-favoured variant, for which ξ₁=ξ₂=ξ₃=ξ₅=ξ₆=0 (figure 1) and c₃=c₄=0. The reorientation process begins with a stress-favoured variant (M₁). Variant-1 is selected by applying traction on the single crystal. When the magnetic field intensity is high enough along the perpendicular direction of the traction, variant-2 becomes preferred.

Variant-1 (shorter axis is along the X₁-direction) is selected by applying traction on the single crystal along X₁. The variant-2 has its shorter length along the X₂-direction. When the magnetic field intensity is high enough along the direction of spontaneous magnetization (X₂), variant-2 becomes preferred. Here, X_i(i=1,2,3) are the global coordinate axes in the reference configuration. We assume that these two structural phases are magnetoelastic. Because the orientations of variant-1 and variant-2 are different than variant-3, the components of the basic quantities are also different. Changes in the local coordinate systems for the variant-1 and variant-2 can be taken into account by changing the indices such that

$$\left[\begin{array}{c}1\to 2\\ 2\to 3\\ 3\to 1\end{array}\right]\text{and}\left[\begin{array}{c}1\to 3\\ 2\to 1\\ 3\to 2\end{array}\right],$$ 6.1

respectively. The left columns are for the local index (x,y,z), and the right columns indicate the global index (X₁,X₂,X₃). The integrity basis can then be directly derived from (4.5).

(i). Stress-favoured variant

The elements of the integrity basis Υ_ϕP for G_P1 are obtained from (4.5) by using (6.1)a. We write

$$\begin{array}{rl}{I}_1& =H_1,I_2=H_2^2+H_3^2,I_3=S_{22}^E+S_{33}^E,I_4=S_{11}^E\\ I_5& ={[S_{31}^E]}^2+{[S_{12}^E]}^2,I_6={[S_{23}^E]}^2,I_7=S_{22}^E{S}_{33}^E\mathrm{and}{I}_8=H_2{S}_{12}^E+H_3{S}_{31}^E.\end{array}$$

We consider elastic and magnetic energies with quadratic dependence on stress and field, respectively, whereas only terms of first degree in stress and field are considered for the magneto-mechanical coupling energy. Under these assumptions, G_P1 can be expanded as

$$\begin{array}{rl}{G}_{P_1}(I_1,I_2,I_3,I_4,I_5,I_6,I_7,I_8)& =G_0^1-\frac{1}{{\rho }_0}(a_1{I}_1+a_2{I}_1^2+a_3{I}_2+a_4{I}_3^2+a_5{I}_4^2+a_6{I}_5\\ & +a_7{I}_6+a_8{I}_7+a_9{I}_8+a_{10}{I}_1{I}_3+a_{11}{I}_1{I}_4+a_{12}{I}_3{I}_4),\end{array}$$ 6.2

where −1/ρ₀ is a normalizing factor. With this definition, we return to (3.3a) and (3.3b) and write¹

$${\mathbf{E}}_1={\zeta }_1^1\mathbf{i}\otimes \mathbf{i}+{\zeta }_1^2\hspace{0.17em}\mathrm{\text{Sym}}\mathbf{[}\mathbf{i}\otimes \mathbf{j}\mathbf{]}+{\zeta }_1^3\mathbf{j}\otimes \mathbf{j}+{\zeta }_1^4\hspace{0.17em}\mathrm{\text{Sym}}\mathbf{[}\mathbf{j}\otimes \mathbf{k}\mathbf{]}+{\zeta }_1^5\hspace{0.17em}\mathrm{\text{Sym}}\mathbf{[}\mathbf{k}\otimes \mathbf{i}\mathbf{]}+{\zeta }_1^6\mathbf{k}\otimes \mathbf{k}$$ 6.3

and

$${\mu }_0{\mathbf{M}}_1={\eta }_1^1\mathbf{i}+{\eta }_1^2\mathbf{j}+{\eta }_1^3\mathbf{k}\mathbf{.}$$ 6.4

Here, ${\zeta }_1^1$, ${\zeta }_1^2$, ${\zeta }_1^3$, ${\zeta }_1^4$, ${\zeta }_1^5$ and ${\zeta }_1^6$ have the linear combination of the following quantities $(I_1,I_3,I_4),(S_{12}^E,H_2),(I_1,I_3,I_4,S_{33}^E),(S_{23}^E),(S_{33}^E,H_3)$ and $(I_1,I_3,I_4,S_{22}^E),$ respectively. Similarly, ${\eta }_1^1$, ${\eta }_1^2$ and ${\eta }_1^3$ are linear function of $(I_1,I_3,I_4),(H_2,S_{12}^E)$ and $(H_3,S_{31}^E),$ respectively. The unit vectors along the global X₁,X₂,X₃-axes are denoted by i,j,k, respectively.

(ii). Field-favoured variant

Like stress-favoured variant, Υ_ϕP for field-favoured variant are obtained from (4.5) by using (6.1)b and we write

$$\begin{array}{rl}{J}_1& =H_2,J_2=H_3^2+H_1^2,J_3=S_{33}^E+S_{11}^E,J_4=S_{22}^E,\\ J_5& ={[S_{12}^E]}^2+{[S_{23}^E]}^2,J_6={[S_{31}^E]}^2,J_7=S_{33}^E{S}_{11}^E\mathrm{and}{J}_8=H_3{S}_{23}^E+H_1{S}_{12}^E\end{array}$$

It should be noted that the elements of the integrity basis of variant-2 are different than variant-1 owing to different orientation. Considering similar assumptions of magento-mechanical energy for variant-1, G_P2 can be expanded as

$$\begin{array}{rl}{G}_{P_2}(J_1,J_2,J_3,J_4,J_5,J_6,J_7,J_8)& =G_0^2-\frac{1}{{\rho }_0}(b_1{J}_1+b_2{J}_1^2+b_3{J}_2+b_4{J}_3^2+b_5{J}_4^2+b_6{J}_5\\ & +b_7{J}_6+b_8{J}_7+b_9{J}_8+b_{10}{J}_1{J}_3+b_{11}{J}_1{J}_4+b_{12}{J}_3{J}_4).\end{array}$$ 6.5

Like variant-1, we can write

$${\mathbf{E}}_2={\zeta }_2^1\mathbf{i}\otimes \mathbf{i}+{\zeta }_2^2\hspace{0.17em}\mathrm{\text{Sym}}\mathbf{[}\mathbf{i}\otimes \mathbf{j}\mathbf{]}+{\zeta }_2^3\mathbf{j}\otimes \mathbf{j}+{\zeta }_2^4\hspace{0.17em}\mathrm{\text{Sym}}\mathbf{[}\mathbf{j}\otimes \mathbf{k}\mathbf{]}+{\zeta }_2^5\hspace{0.17em}\mathrm{\text{Sym}}\mathbf{[}\mathbf{k}\otimes \mathbf{i}\mathbf{]}+{\zeta }_2^6\mathbf{k}\otimes \mathbf{k}$$ 6.6

and

$${\mu }_0{\mathbf{M}}_2={\eta }_2^1\mathbf{i}+{\eta }_2^2\mathbf{j}+{\eta }_2^3\mathbf{k}\mathbf{.}$$ 6.7

Here, ${\zeta }_2^1$, ${\zeta }_2^2$, ${\zeta }_2^3$, ${\zeta }_2^4$, ${\zeta }_2^5$ and ${\zeta }_2^6$ have the linear combination of the follow quantities $(J_1,J_3,J_4,S_{33}^E)$, $(S_{12}^E,H_1)$, (J₁,J₃,J₄), $(S_{23}^E,H_3)$, $(S_{31}^E)$ and $(J_1,J_3,J_4,S_{11}^E),$ respectively. Similarly, ${\eta }_1^1$, ${\eta }_1^2$ and ${\eta }_1^3$ are linear function of $(H_1,S_{12}^E)$, (J₁,J₃,J₄), and $(H_3,S_{23}^E),$ respectively.

(iii). Mixture of both variants

Because variant-2 is nucleating, symmetry restrictions associated with variant-2 are considered. Using (4.5) and (6.1)b, the elements of the set Υ_ϕI can be expressed as

$$\begin{array}{rlrlrl}{K}_1& =S_{33}^E+S_{11}^E& & K_2=S_{22}^E& & K_3=E_{33}^I+E_{11}^I\\ K_4& =E_{22}^I& & K_5=H_2& & K_6=M_2^I\\ K_7& ={[S_{12}^E]}^2+{[S_{23}^E]}^2& & K_8={[S_{31}^E]}^2& & K_9=S_{33}^E{S}_{11}^E\\ K_{10}& ={[E_{12}^I]}^2+{[E_{23}^I]}^2& & K_{11}={[E_{31}^I]}^2& & K_{12}=E_{33}^I{S}_{11}^E\\ K_{13}& =S_{31}^E{E}_{31}^I& & K_{14}=[S_{33}^E-S_{11}^E][E_{33}^I-E_{11}^I]\\ K_{15}& =E_{12}^I{S}_{12}^E+E_{23}^I{S}_{23}^E& & K_{16}=H_3^2+H_1^2\\ \mathrm{and}{K}_{17}& ={[M_3^I]}^2+{[M_1^I]}^2& & K_{18}=H_3{M}_3^I+H_1{M}_1^I\end{array}$$

Further considering first-order coupling between stress and inelastic strain and between field and internal magnetization, the expanded form of the Gibbs free energy can be written as

$$\begin{array}{rl}& G_{\mathrm{mix}}(K_1,K_2,K_3,K_4,K_5,K_6,K_{12},K_{13},K_{14},K_{15},K_{18},g)\\ & =G_0^I-\frac{1}{{\rho }_0}(c_1{K}_1{K}_3+c_2{K}_1{K}_4+c_3{K}_2{K}_3+c_4{K}_2{K}_4+c_5{K}_5{K}_6\\ & +c_6{K}_{12}+c_7{K}_{13}+c_8{K}_{14}+c_9{K}_{15}+c_{10}{K}_{18})-\frac{1}{{\rho }_0}g.\end{array}$$ 6.8

The constitutive equations are written as

$$\begin{array}{rl}{\overline{\mathbf{E}}}^I& =[c_1(E_{33}^I+E_{11}^I)+c_2{E}_{22}^I-c_8(E_{33}^I-E_{11}^I)+c_6{E}_{33}^I]\mathbf{i}\otimes \mathbf{i}+2c_9{E}_{12}^I\hspace{0.17em}\text{Sym}[\mathbf{i}\otimes \mathbf{j}]\\ & +[c_3(E_{33}^I+E_{11}^I)+c_4{E}_{22}^I]\mathbf{j}\otimes \mathbf{j}+2c_9{E}_{23}^I\hspace{0.17em}\text{Sym}[\mathbf{j}\otimes \mathbf{k}]+2c_7{E}_{31}^I\hspace{0.17em}\text{Sym}[\mathbf{k}\otimes \mathbf{i}]\\ & +[c_1(E_{33}^I+E_{11}^I)+c_2{E}_{22}^I+c_8(E_{33}^I-E_{11}^I)]\mathbf{k}\otimes \mathbf{k}\end{array}$$ 6.9

and

$${\mu }_0{\overline{\mathbf{M}}}^I=c_5{M}_2^I\mathbf{j}+c_{10}(M_1^I\mathbf{i}+M_3^I\mathbf{k}).$$ 6.10

We assume the strain evolution depends on the deviatoric stress S^′E and write the strain evolution equation as

$${\dot{\mathbf{E}}}^I={\mathit{\Lambda }}_4^r({\mathbf{S}}^{\mathrm{\prime }E}\mathbf{)}{\dot{\xi }}_4\mathbf{.}$$ 6.11

For the present case, we could write $\{I\}=\{S_{33}^{\mathrm{\prime }E}+S_{11}^{\mathrm{\prime }E},S_{22}^{\mathrm{\prime }E},{(S_{31}^{\mathrm{\prime }E})}^2,{(S_{33}^{\mathrm{\prime }E}-S_{11}^{\mathrm{\prime }E})}^2,{(S_{12}^{\mathrm{\prime }E})}^2+{(S_{23}^{\mathrm{\prime }E})}^2\}$. The elements of the set $\{\mathcal{D}\}$ are given by

$${\mathcal{D}}^1=\mathbf{i}\otimes \mathbf{i}+\mathbf{k}\otimes \mathbf{k},{\mathcal{D}}^2=\mathbf{j}\otimes \mathbf{j},{\mathcal{D}}^3=2S_{31}^{\mathrm{\prime }E}\hspace{0.17em}\text{Sym}[\mathbf{i}\otimes \mathbf{k}]$$

and

$${\mathcal{D}}^4=(S_{33}^{\mathrm{\prime }E}-S_{11}^{\mathrm{\prime }E})(\mathbf{k}\otimes \mathbf{k}-\mathbf{i}\otimes \mathbf{i}),{\mathcal{D}}^5=2S_{12}^{\mathrm{\prime }E}\hspace{0.17em}\text{Sym}[\mathbf{i}\otimes \mathbf{j}]+2S_{23}^{\mathrm{\prime }E}\hspace{0.17em}\text{Sym}[\mathbf{j}\otimes \mathbf{k}].$$

Spanning ${\mathit{\Lambda }}_4^r$ in terms of the elements of $\{\mathcal{D}\}$, we write

$$\begin{array}{rl}{\mathit{\Lambda }}_4^r& =c_{14}(\mathbf{i}\otimes \mathbf{i}+\mathbf{k}\otimes \mathbf{k})+c_{24}\mathbf{j}\otimes \mathbf{j}+2c_{34}{S}_{31}^{\mathrm{\prime }E}\hspace{0.17em}\text{Sym}[\mathbf{i}\otimes \mathbf{k}]\\ & +c_{44}(S_{33}^{\mathrm{\prime }E}-S_{11}^{\mathrm{\prime }E})(\mathbf{k}\otimes \mathbf{k}-\mathbf{i}\otimes \mathbf{i})+2c_{54}(S_{12}^{\mathrm{\prime }E}\hspace{0.17em}\text{Sym}[\mathbf{i}\otimes \mathbf{j}]+S_{23}^{\mathrm{\prime }E}\hspace{0.17em}\text{Sym}[\mathbf{j}\otimes \mathbf{k}]).\end{array}$$

Similarly, the magnetization evolution equation is given by

$${\dot{\mathbf{M}}}^I={\mathit{\gamma }}_4^r({\mathbf{S}}^{\mathrm{\prime }E}\mathbf{)}{\dot{\xi }}_4$$

for which $\{I\}=\{S_{33}^{\mathrm{\prime }E}+S_{11}^{\mathrm{\prime }E},{(S_{12}^{\mathrm{\prime }E})}^2+{(S_{23}^{\mathrm{\prime }E})}^2\}$ and

$${\mathcal{D}}^1=\mathbf{j},{\mathcal{D}}^2=S_{12}^{\mathrm{\prime }E}\mathbf{i}+S_{23}^{\mathrm{\prime }E}\mathbf{k}.$$

The expression for ${\mathit{\gamma }}_4^r$ is then given by

$${\mathit{\gamma }}_4^r=c_{14}^{\prime }\mathbf{j}+c_{24}^{\prime }(S_{12}^{\mathrm{\prime }E}\mathbf{i}+S_{23}^{\mathrm{\prime }E}\mathbf{k}).$$

(iv). A specific magneto-mechanical loading path

We consider a single variant (variant-1) is under axial traction along the X₁-direction and a magnetic field is applied along the X₂-direction. Under these magneto-mechanical loading conditions, we assume ${\mathbf{S}}^{\mathit{E}}={\mathbf{S}}_{11}^{\mathbf{E}}\mathbf{i}\otimes \mathbf{i}$ and H=H₂j and the strain constitutive equations may be reduced to

$$\begin{array}{rl}{\mathbf{E}}_1& =2a_5{S}_{11}^E\mathbf{i}\otimes \mathbf{i}+2a_9{H}_2\hspace{0.17em}\text{Sym}[\mathbf{i}\otimes \mathbf{j}]+a_{12}{S}_{11}^E\mathbf{j}\otimes \mathbf{j}+a_{12}{S}_{11}^E\mathbf{k}\otimes \mathbf{k},\\ {\mathbf{E}}_2& =(2b_4{S}_{11}^E+b_{10}{H}_2)\mathbf{i}\otimes \mathbf{i}+(b_{12}{S}_{11}^E+b_{11}{H}_2)\mathbf{j}\otimes \mathbf{j}+((2b_4+b_8)S_{11}^E+b_{10}{H}_2)\mathbf{k}\otimes \mathbf{k}\\ \mathrm{and}{\mathit{\Lambda }}_4^r& =(c_{14}+c_{44}{S}_{11}^{\mathrm{\prime }E})\mathbf{i}\otimes \mathbf{i}+c_{24}\mathbf{j}\otimes \mathbf{j}+(c_{14}-c_{44}{S}_{11}^{\mathrm{\prime }E})\mathbf{k}\otimes \mathbf{k}.\end{array}$$

We have the following remarks on the strain constitutive equations:

• — Variant-1 has shear strain components owing to the presence of magnetic field.

• — The remaining strain components of variant-1 are independent of the field.

• — Variant-2 does not have any shear components.

• — All the diagonal components of strain of variant-2 depend on stress and magnetic field.

• — ${\mathit{\Lambda }}_4^r$ does not contain any off diagonal components.

Next, considering the magnetization constitutive response, the reduced form can be written as

$$\begin{array}{rl}{\mathbf{M}}_1& =(a_1+a_{11}{S}_{11}^E)\mathbf{i}+2a_3{H}_2\mathbf{j},\\ {\mathbf{M}}_2& =(b_1+2b_2{H}_2+b_{10}{S}_{11}^E)\mathbf{j}\\ \mathrm{and}{\mathit{\gamma }}_4^r& =c_{14}^{\prime }\mathbf{j}.\end{array}$$

Further assuming c′₁₄ is constant, the internal magnetization can be written as

$${\overline{\mathbf{M}}}^I=c_{14}^{\prime }{\xi }_4\mathbf{j}.$$

Thus, being consistent with the symmetry restrictions, we have the following remarks on the simplified magnetic constitutive equation:

• — Only the X₁ magnetization component of variant-1 depends on stress.

• — There is no X₁ magnetization component in the variant-2.

• — ${\mathit{\gamma }}_4^r$ is restricted to have no X₁ component.

(b). Continuous symmetry and field-induced phase transformation

We consider FIPT in an Fe–Mn–C polycrystalline MSMA where the austenitic phase is paramagnetic and martensitic phase is ferromagnetic [48]. Initially, the specimen is at a high temperature under compressive loading without any magnetic field and completely austenitic. The initial state is denoted by point-1 (figure 2) from where the temperature decreases to point-2 under zero magnetic field. A magnetic field H=f(T) is then applied, and the martensitic transformation ends at point-3 where field-induced martensitic variant (M₂) is present owing to the high magnetic field. Between point-2 and point-3, both stress-favoured and field-favoured variants nucleate, whereas stress-favoured variants reorientate to the field-favoured ones. The direction of the texturing of the stress-favoured variant at the beginning (point-2) is denoted by a_i, and the texturing direction at point-3 is denoted by a_f. As a result of the transformation and reorientation of this process (from point 2 to 3), the direction of texturing continuously changes from a_i to a_f. We introduce a structural tensor a⊗a in the Gibbs free energy to take into account the directionality of the magneto-mechanical responses along a. The polycrystalline austenitic phase is assumed to be isotropic. We simplify the analysis by considering no reorientation from M₁ to M₂, i.e. ξ₄=0. Moreover, ξ₃=ξ₅=ξ₆=0 (figure 1).

Figure 2.

Schematic of a stress–field–temperature phase diagram with the projections of the martensitic start (M_s) and martensitic finish (M_f) surfaces on the σ−T and H−T planes and a magneto-thermal loading path on the H−T plane at stress level σ*. (Online version in colour.)

We denote the Gibbs free energy of the austenitic and the martensitic phases by G_P4 and G_P1=G_P2=G_Pm, respectively. The Gibbs free energy of the transforming phase is denoted by $G_{P_4\to P_{\mathrm{m}}}$. Thus, from (3.2), we write

$$G({\mathrm{{\rm Y}}}_{\varphi },T,\xi ,g)=\xi G_{P_{\mathrm{m}}}({\mathrm{{\rm Y}}}_{\varphi P},T)+(1-\xi )G_{P_4}({\mathrm{{\rm Y}}}_{\varphi P},T)+G_{\mathrm{mix}}({\mathrm{{\rm Y}}}_{\varphi I},g).$$

We consider the following assumptions on the integrity basis for this study.

• 1. The martensitic and austenitic phases are linear thermoelastic. Therefore, G has a second-order dependence on S^E. Moreover, G only depends on first-order coupling between S^E and T.

• 2. G depends only on the first-order coupling of E^I and S^E. We assume that the inelastic deformation is an isochoric process and generation of transformation strain E^I is proportional to the deviatoric stress. This means tr(E^I)=0. We also assume that G depends only on the first-order coupling of M^I and H.

• 3. In general, magnetostriction in MSMAs is not observed. Quadratic coupling of the magnetic field H with the S^E and E^I is therefore neglected.

Under these assumptions, we consider Υ_ϕ to be composed of the following set of nine invariants

$$\begin{array}{rl}{I}_1& =\mathbf{H}\cdot \mathbf{H},I_2=\text{tr}({\mathbf{S}}^E),I_3=\text{tr}({\mathbf{S}}^{E^2}),\\ I_4& ={(\mathbf{H}\cdot \mathbf{a})}^2,I_5=\mathbf{a}\cdot {\mathbf{S}}^E\mathbf{a},I_6=\mathbf{a}\cdot {{\mathbf{S}}^E}^2\mathbf{a}\\ \text{and}{I}_7& =\text{tr}({\mathbf{S}}^E{E}^{\mathit{I}}),I_8=\text{tr}({\mathbf{S}}^E(\mathbf{a}\otimes \mathbf{a})E^{\mathit{I}}),I_9={\mathbf{M}}^I\cdot \mathbf{H}.\end{array}\}$$ 6.12

We assume G_P4=G_P4(I₁,I₂,I₃,T) for the isotropic austenitic phase. The Gibbs free energy of the transversely isotropic martensitic phase is denoted by G_Pm=G_Pm(I₁,I₂,I₃,I₄,I₅,I₆,T). We assume that the energy associated with inelastic transformation is given by G_mix=G_mix(Υ_ϕI,g)=G_mix(I₇,I₈,I₉,g). The Gibbs free energies for the austenitic and martensitic phases can be expanded up to second degree of the elements of the integrity basis. We consider a general expression

$$\begin{array}{rl}& G_{P\alpha }(I_1,I_2,I_3,I_4,I_5,I_6,T)\\ & =G_0^{\alpha }-\frac{1}{{\rho }_0}(a_1^{\alpha }{I}_1+a_2^{\alpha }{I}_1^2)-\frac{1}{{\rho }_0}(a_3^{\alpha }{I}_2+a_4^{\alpha }{I}_2^2)\\ & -\frac{1}{{\rho }_0}(a_5^{\alpha }{I}_3+a_6^{\alpha }{I}_3^2)-\frac{1}{{\rho }_0}(a_7^{\alpha }{I}_4+a_8^{\alpha }{I}_4^2)-\frac{1}{{\rho }_0}(a_9^{\alpha }{I}_5+a_{10}^{\alpha }{I}_5^2)\\ & -\frac{1}{{\rho }_0}(a_{11}^{\alpha }{I}_6+a_{12}^{\alpha }{I}_6^2)-\frac{1}{{\rho }_0}(a_{13}^{\alpha }(\mathrm{\Delta }T)+a_{14}^{\alpha }{(\mathrm{\Delta }T)}^2)-\frac{1}{{\rho }_0}(a_{15}^{\alpha }{I}_1{I}_2+a_{16}^{\alpha }{I}_1{I}_3\\ & +a_{17}^{\alpha }{I}_1{I}_4+a_{18}^{\alpha }{I}_1{I}_5+a_{19}^{\alpha }{I}_1{I}_6+a_{20}^{\alpha }{I}_1\mathrm{\Delta }T)-\frac{1}{{\rho }_0}(a_{21}^{\alpha }{I}_2{I}_3+a_{22}^{\alpha }{I}_2{I}_4\\ & +a_{23}^{\alpha }{I}_2{I}_5+a_{24}^{\alpha }{I}_2{I}_6+a_{25}^{\alpha }{I}_2\mathrm{\Delta }T)-\frac{1}{{\rho }_0}(a_{26}^{\alpha }{I}_3{I}_4+a_{27}^{\alpha }{I}_3{I}_5+a_{28}^{\alpha }{I}_3{I}_6\\ & +a_{29}^{\alpha }{I}_3\mathrm{\Delta }T)-\frac{1}{{\rho }_0}(a_{30}^{\alpha }{I}_4{I}_5+a_{31}^{\alpha }{I}_4{I}_6+a_{32}^{\alpha }{I}_4\mathrm{\Delta }T)\\ & -\frac{1}{{\rho }_0}(a_{33}^{\alpha }{I}_5{I}_6+a_{34}^{\alpha }{I}_5\mathrm{\Delta }T)-\frac{1}{{\rho }_0}{a}_{35}^{\alpha }{I}_6\mathrm{\Delta }T.\end{array}$$ 6.13

where −1/ρ₀ is a normalizing factor, T₀ is a reference temperature and ΔT=T−T₀, the coefficients are assumed scalar constants. The austenitic phase and the martensitic phase are denoted by α=4 and α=m, respectively.

The inelastic energy G_mix can be expanded up to second degree of the elements of the integrity basis as

$$\begin{array}{rl}{G}_{\mathrm{mix}}(I_7,I_8,I_9,g)& =G_0^I-\frac{1}{{\rho }_0}(b_1{I}_7+b_2{I}_7^2)-\frac{1}{{\rho }_0}(b_3{I}_8+b_4{I}_8^2)\\ & -\frac{1}{{\rho }_0}(b_5{I}_9+b_6{I}_9^2)-\frac{1}{{\rho }_0}(b_7{I}_7{I}_8+b_8{I}_7{I}_9+b_9{I}_8{I}_9)-\frac{1}{{\rho }_0}g.\end{array}$$ 6.14

Because we assume that the elastic energy functions for the austenitic and martensitic phases depend only on the quadratic power of the stress, we neglect $I_2,I_3^2,I_5,I_6^2$, I₂I₃,I₂I₆, I₃I₅,I₃I₆, I₅I₆. Moreover, we consider the magneto-mechanical coupling energy where the order of the stress components is one, and so I₁I₃,I₁I₆, I₃I₄,I₄I₆ are neglected. We write

$${\mathbf{E}}^{\alpha }={\zeta }_1^{\alpha }\mathbf{I}+{\zeta }_2^{\alpha }{\mathbf{S}}^{\mathit{E}}+{\zeta }_3^{\alpha }\mathbf{(}\mathbf{a}\otimes \mathbf{a}\mathbf{)}+{\zeta }_4^{\alpha }\mathbf{[}{\mathbf{S}}^{\mathit{E}}\mathbf{(}\mathbf{a}\otimes \mathbf{a}\mathbf{)}+\mathbf{(}\mathbf{a}\otimes \mathbf{a}\mathbf{)}{\mathbf{S}}^{\mathit{E}}\mathbf{]}$$ 6.15

and

$${\mu }_0{\mathbf{M}}^{\alpha }={\eta }_1^{\alpha }\mathbf{H}+{\eta }_2^{\alpha }\mathbf{(}\mathbf{H}\cdot \mathbf{a}\mathbf{)}\mathbf{a}\mathbf{.}$$ 6.16

where, ${\zeta }_1^{\alpha }$, ${\zeta }_3^{\alpha }$, ${\eta }_1^{\alpha }$, ${\eta }_2^{\alpha }$ are linear function of I₁,I₂,I₄,I₅,ΔT and ${\zeta }_2^{\alpha }$, ${\zeta }_4^{\alpha }$ are linear function of ΔT. It should be noted that for the austenitic phase (α=4), I₄,I₅,I₆ and a are zero.

In the transforming state, we consider only first-order coupling between stress and transformation strain and between magnetic field and internal magnetization. The strain response of the mixing phase, from (6.14), is given by

$${\overline{\mathbf{E}}}^I=b_1{\mathbf{E}}^{\mathbf{I}}+b_3\mathrm{\text{Sym}}\mathbf{[}\mathbf{(}\mathbf{a}\otimes \mathbf{a}\mathbf{)}{\mathbf{E}}^{\mathbf{I}}\mathbf{]}\mathbf{.}$$ 6.17

We consider ${\mathit{\Lambda }}_1^t={\mathit{\Lambda }}_2^t={\mathit{\Lambda }}^t$ such that ${\dot{\mathbf{E}}}^I={\mathit{\Lambda }}^t\dot{\xi }$. Assuming that transformation strain depends on deviatoric stress, i.e. Λ^t(S^′E,a⊗a) has a linear dependence in stress, we can write

$${\mathit{\Lambda }}^t=t_1\mathbf{I}+{\mathit{t}}_2{{\mathbf{S}}^{\prime }}^E+{\mathit{t}}_3\mathbf{(}\mathbf{a}\otimes \mathbf{a}\mathbf{)}+{\mathit{t}}_4\mathbf{(}{{\mathbf{S}}^{\prime }}^E\mathbf{(}\mathbf{a}\otimes \mathbf{a}\mathbf{)}+\mathbf{(}\mathbf{a}\otimes \mathbf{a}\mathbf{)}{{\mathbf{S}}^{\prime }}^E\mathbf{)}+{\mathit{t}}_5\mathbf{(}\mathbf{a}\otimes \mathbf{a}\mathbf{)}{{\mathbf{S}}^{\prime }}^E\mathbf{(}\mathbf{a}\otimes \mathbf{a}\mathbf{)},$$ 6.18

where t₁,…,t₅ are assumed to be constants. In a similar way, for magnetic response, we can write

$${\overline{\mathbf{M}}}^I=c_3{\mathbf{M}}^{\mathit{t}},$$

where, $c_3=b_{001}^I$. We consider ${\mathit{\gamma }}_1^t={\mathit{\gamma }}_2^t={\mathit{\gamma }}^t$ so that ${\dot{\mathbf{M}}}^I={\mathit{\gamma }}^t\dot{\xi }$. Assuming γ^t(S^′E,H,a⊗a) has a linear dependence in stress, we can write

$${\mathit{\gamma }}^t=s_1\mathbf{H}+{\mathbf{s}}_2{{\mathbf{S}}^{\prime }}^E\mathbf{H}+{\mathbf{s}}_3\mathbf{(}\mathbf{a}\otimes \mathbf{a}\mathbf{)}\mathbf{H}+{\mathbf{s}}_4{{\mathbf{S}}^{\prime }}^E\mathbf{(}\mathbf{a}\otimes \mathbf{a}\mathbf{)}\mathbf{H}+{\mathbf{s}}_5\mathbf{(}\mathbf{a}\otimes \mathbf{a}\mathbf{)}{{\mathbf{S}}^{\prime }}^E\mathbf{H},$$ 6.19

where s₁,…,s₅ are assumed to be constants.

(i). A specific magneto-mechanical loading path

We consider a specimen that is initially entirely in the austenitic phase and under axial traction along the X₁ direction with a magnetic field applied along the X₂-direction. Under these loading conditions, ${\mathbf{S}}^{\mathit{E}}={\mathbf{S}}_{11}^{\mathbf{E}}\mathbf{i}\otimes \mathbf{i}$ and H=H₂j. At the beginning when the field is low, only the stress-favoured variant is nucleated with the decrease in temperature. The direction of the transverse anisotropy is then along the unit direction a_i=(1,0,0)^T at the initial condition. At high field, the direction changes to a_f=(0,1,0)^T owing to the presence of field-favoured variants.

Our main focus in this subsection is on the evolution of the structural tensor. We assume that $\mathbf{a}=\mathbf{(}\cos \beta ,\sin \beta ,0\mathbf{)}$, where β is the angle with the (1,0,0)-direction. The structural tensor may be then written as

$${[\mathbf{a}\otimes \mathbf{a}\mathbf{]}}_{\mathbf{i}\mathbf{j}}=\left[\begin{array}{ccc}{\cos }^2\beta & \cos \beta \sin \beta & 0\\ \cos \beta \sin \beta & {\sin }^2\beta & 0\\ 0& 0& 0\end{array}\right],$$

and the time derivative as

$${\dot{[\mathbf{a}\otimes \mathbf{a}\mathbf{]}}}_{ij}=\left[\begin{array}{ccc}-\sin 2\beta & \cos 2\beta & 0\\ \cos 2\beta & \sin 2\beta & 0\\ 0& 0& 0\end{array}\right]\dot{\beta }=[D_c]\dot{\beta }.$$

The evolution of the angle β can be written from (5.3) in a simple form

$$\dot{\beta }=\mathrm{\Theta }\dot{\xi }.$$

and so

$${\dot{[\mathbf{a}\otimes \mathbf{a}\mathbf{]}}}_{ij}=[D_c]\mathrm{\Theta }\dot{\xi }.$$

Moreover, the evolution of g is related to the evolution of ξ by

$$\dot{g}=f^t\dot{\xi },$$ 6.20

where f^t is a hardening function. Substituting back all the evolution equations in (6.21), we obtain

$${\mathit{\pi }}_{E^I}:{\mathit{\Lambda }}^t\dot{\xi }+{\mathit{\pi }}_{M^I}:{\mathit{\gamma }}^t\dot{\xi }+{\pi }_{\xi }\dot{\xi }+{\pi }_g{f}^t\dot{\xi }+{\mathit{\pi }}_{\mathbf{a}\otimes \mathbf{a}}:[D_c]\mathrm{\Theta }\dot{\xi }\ge 0.$$ 6.21

or,

$${\pi }^t\dot{\xi }\ge 0,$$

where the total thermodynamic driving force π^t owing to phase transformation is given by

$${\pi }^t={\mathit{\pi }}_{E^I}:{\mathit{\Lambda }}^t+{\mathit{\pi }}_{M^I}:{\mathit{\gamma }}^t+{\pi }_{\xi }+{\pi }_g{f}^t+{\mathit{\pi }}_{\mathbf{a}\otimes \mathbf{a}}:[D_c]\mathrm{\Theta }\ge 0.$$ 6.22

The following transformation function, Φ^t, is then introduced,

$${\mathrm{\Phi }}^t:=\{\begin{array}{ll}{\pi }^t-Y^t& \dot{\xi }>0,\\ -{\pi }^t-Y^t& \dot{\xi }<0,\end{array}{\mathrm{\Phi }}^t\le 0,$$ 6.23

where Y ^t is a positive scalar associated with the internal dissipation during phase transformation. It is assumed that the constraints of the transformation process follows the principle of maximum dissipation and can be expressed in terms of the Kuhn–Tucker-type conditions

$${\mathrm{\Phi }}^t\le 0,{\mathrm{\Phi }}^t\dot{\xi }=0.$$ 6.24

At the end of this section, we have the following remarks:

• — γ^t can not only be a function of S^E. The stress is always coupled with the magnetic field (6.19).

• — The intensity of multi-field coupling may be high. The influence of magnetic field on stress has been reported to be more than 15% compared with the stress level under a no field condition by solving a simplified magneto-mechanical boundary value problem for MSMA [49].

Before presenting some numerical examples, we would like to point out that Landis [50] used a phase-field approach to show that under strong enough combinations of competing field and stress, non-favourable variants can be created where the magnetization is not aligned with the minimum strain martensitic direction. The present modelling approach could incorporate such non-standard behaviours by allowing such high-energy variants, in addition to the stress-favoured and field-favoured variants which are considered in this work (figure 1).

7. Numerical examples

We will consider two examples to demonstrate the capability of the proposed generalized model for single-crystal field-induced variant reorientation (FIVR) and FIPT. The available experimental data for the above-mentioned mechanisms are mostly based on single-crystal specimens, and so we present some results considering discrete symmetry.

Single-crystal NiMnGa is consider for FIVR. This ferromagnetic material system has $4/m\underset{\_}{m}\underset{\_}{m}$ symmetry group. The proposed Gibbs free energy and the general constitutive equations for FIVR are described in §6a. Experiments, related to these material systems, are mostly performed under two-dimensional magneto-mechanical loading conditions (see [51,52]) and motivated by experiments, a reduced form of the constitutive equations is presented in §6a(iv). The calibration of material parameters, a₅,b₄,b₁₀, c₁₄,a₁,a₁₁,a₃, b₁,b₂, c′₁₄ for this simple loading conditions can be found in [33,53]. A simulation and a prediction is presented in figure 3. A complex loading path is considered in figure 4 to predict magnetic field-induced strain. Initially, the stress-favoured variant was present at zero field and −2 [MPa]. Then a quadratic stress-dependent magnetic field is applied up to 1 [T] for the reorientation to a field-favoured variant, while a bi-quadratic stress-dependent field is considered during unloading. The maximum strain is assumed stress-dependent and a quadratic fit is considered from the available maximum strain data [54]. The problem is solved incrementally and the simulation result is presented in figure 4b.

Figure 3.

(a) Model simulation of strain-field response at −1.4 MPa and (b) model prediction of magnetization response at −1.4 MPa. Experimental data are taken from [54]. (Online version in colour.)

Figure 4.

(a) A magneto-mechanical loading path where stress and field are simultaneously applied, and (b) model prediction of the field-induced strain. (Online version in colour.)

Single-crystal NiMnCoIn, which belongs to $3\underset{\_}{m}$ and 2/m point groups in the cubic ferromagnetic austenitic phase and monoclinic antiferromagnetic martensitic phase, respectively, is considered for FIPT. In an typical experiment for FIPT [2,47], the specimen is held under compressive stress, and a magnetic field is applied coaxially with the mechanical load. Initially, the specimen is in antiferromagnetic martensitic phase. After a critical applied magnetic field is reached, the ferromagnetic austenitic phase nucleates, and phase transformation completes with a further increase in magnetic field. The specimen returns to the martensitic phase again when the magnetic field decreases below a critical value, characteristic of the material. The Gibbs free energy for such a mechanism, similar to FIVR, is considered by a quadratic expansion of the corresponding integrity basis. The detailed derivations of constitutive equations, model calibrations, model simulations and predictions can be found in [55]. Here, we present only a couple of results, which are simplified, one-dimensional versions of this generalized modelling framework where the traction and magnetic field are applied along the same axial direction. Figure 5 represents the model predictions of the field-induced strain and magnetization [55].

Figure 5.

(a) Field-induced strain prediction at T=200 K at σ_M= −60 MPa. (b) Model predictions of magnetization at 230 K and σ_M=−100 MPa. Experimental data are taken from [55]. (Online version in colour.)

Finally, we want to comment on the following fact. For the purpose of understanding the complex and coupled physical behaviours of MSMAs in a continuum scale, we consider a full quadratic expansion of the Gibbs free energy of each phase. Thus, the model parameters, appearing in this study for a fully coupled anisotropic three-dimensional case, are a large number. However, the non-zero parameters that are important for real material applications are a smaller set and can be found by appropriate experiments. This study provides the framework to include the largest possible set of material parameters and, approximations will be based on available experimental data and loading paths that may not require the full three-dimensional implementation.

8. Conclusion

A generalized modelling approach for magnetic shape memory alloys is introduced in this work. The integrity basis for the Gibbs free energy is derived by considering material symmetry. The evolution equations of the internal variables are restricted by group symmetry operations. Finite symmetry is considered for single-crystal MSMA. For polycrystalline MSMAs, continuous symmetry is considered, and anisotropy is taken into account by introducing structural tensors in the Gibbs free energy and evolution equations. Selected results are presented for FIPT and variant reorientation as special cases of the general theory. Considering symmetry restrictions in the modelling not only provides insights to construct an energy potential and evolution equations of the internal variables, but also systematically captures cross-coupling between multiple fields.

Appendix A. Finite symmetry for magneto-crystalline material

The point groups for crystals belongs to classical point groups. We denote the classical group by $\mathcal{G}$. Because a reversal of time changes the sign of the current and hence reverses the direction of the magnetic moment vector in magnetic crystals, we need an additional time-inversion operation τ. When this operator acts on a classical point group, it is possible to find a new group, known as magnetic point groups [37]. In this group, half of the elements of the ordinary point group $\mathcal{G}$ are multiplied by the time-inversion operator τ. The other half forms a subgroup, $\mathcal{H}$, of $\mathcal{G}$. The magnetic point group, $\mathcal{M}$, can be written as

$$\mathcal{M}=\mathcal{H}+\tau (\mathcal{G}-\mathcal{H}).$$

We denote the magnetic point group by

$$\mathcal{M}=\{{\mathcal{M}}^{\alpha }\}$$

for α=1,…,n, where n is the order of the group. The representation of the group is thus given by

$$T(\mathcal{M})=\{T({\mathcal{M}}^{\alpha })\}.$$

If we denote the fully reduced representation of T by Γ, then

$$\mathit{\Gamma }=\underset{i=1}{\overset{r}{⨁}}{n}_i{\mathit{\Gamma }}^{(i)},$$ A 1

where Γ⁽¹⁾,…Γ^(r) are the irreducible representation of $\mathcal{M}$. We use the orthogonality of characters to obtain the coefficients n_i of equation (A 1) [40].

(a) Determination of polynomial integrity basis

Let W(X,Y,…) be a scalar-valued tensor function so that it is invariant under $\mathcal{M}$ [56,57]. If A∈{X,Y,…} and $\mathbf{Q}\in T(\mathcal{M}),$ then the following transformation holds true

$$A_{ijk\dots n}^{\prime }={(-1)}^p(det\hspace{0.17em}\mathbf{Q})Q_{ip}{Q}_{jq}\dots Q_{nu}{A}_{pqr\dots u}.$$

In the above transformation, p=1 for c-tensors and p=0 for i-tensors. $det\hspace{0.17em}\mathbf{Q}=1$ for polar tensors and $det\hspace{0.17em}\mathbf{Q}=-1$ for axial tensors. Polar tensors do not change sign under improper rotation, whereas axial tensors do. Tensors of any order that are invariant under time inversion are known as i-tensors and tensors whose components change sign with time reversal are known as c-tensors.

Given a magnetic point group, we now determine the basic quantities of X,Y,… Let

$$\mathbf{u}={[u_1,\dots .,u_m]}^{\mathrm{T}}={[X_1,\dots X_p,Y_1,\dots ,Y_q,\dots ]}^{\mathrm{T}}$$

denote the column vector whose entries are the independent components of X,Y,… So, the restrictions imposed on the scalar function are

$$W(\mathbf{u})=W({\mathcal{T}}_k\mathbf{u})(k=1,\dots ,n),$$

where $\{{\mathcal{T}}_k\}$ are m-dimensional matrix representation of $T(\mathcal{M})$. The representation $\{{\mathcal{T}}_k\}$ can be decomposed into irreducible representations associated with the group $\{\mathcal{M}\}$. We denote these representations by $\{{\mathit{\Gamma }}_k^{(1)},{\mathit{\Gamma }}_k^{(2)}\dots \}$. Thus, we are looking for a similarity transformation with a non-singular m×m matrix R such that

$$\mathbf{R}{\mathcal{T}}_k{\mathbf{R}}^{-1}=\underset{i=1}{\overset{r}{⨁}}{n}_i{\mathit{\Gamma }}_k^{(i)}(k=1,\dots ,n),$$

where

$$\mathbf{R}\mathbf{u}=\left[\begin{array}{c}{\mathbf{u}}^{(1)}\\ {\mathbf{u}}^{(2)}\\ ⋮\\ {\mathbf{u}}^{(r)}\end{array}\right],{\mathbf{u}}^{(1)}=\left[\begin{array}{c}{u}_1^{(1)}\\ u_2^{(1)}\\ ⋮\\ u_{n_1}^{(1)}\end{array}\right],{\mathbf{u}}^{(2)}=\left[\begin{array}{c}{u}_1^{(2)}\\ u_2^{(2)}\\ ⋮\\ u_{n_2}^{(2)}\end{array}\right],\dots ,{\mathbf{u}}^{(r)}=\left[\begin{array}{c}{u}_1^{(r)}\\ u_2^{(r)}\\ ⋮\\ u_{n_r}^{(r)}\end{array}\right].$$

We can then express the scalar function as

$$W(\mathbf{u})=P({\mathbf{u}}^{(1)},{\mathbf{u}}^{(2)},\dots ,{\mathbf{u}}^{(r)})=P({\Gamma }_k^{(1)}{\mathbf{u}}^{(1)},{\Gamma }_k^{(2)}{\mathbf{u}}^{(2)},\dots ,{\Gamma }_k^{(r)}{\mathbf{u}}^{(r)}).$$

The set {u⁽¹⁾,u⁽²⁾,…,u^(r)}, associated with $\{{\mathit{\Gamma }}_k^{(1)},{\mathit{\Gamma }}_k^{(2)},\dots ,{\mathit{\Gamma }}_k^{(r)}\}$, forms the carrier space for the irreducible representation and the elements of the set are known as basic quantities.

Let I₁,…,I_s be the polynomials in the basic quantities {u⁽¹⁾,u⁽²⁾,…,u^(r)} such that I₁,…,I_s are each invariant under $\mathcal{M}$ and such that every function P(u⁽¹⁾,u⁽²⁾,…,u^(r)), which is invariant under $\mathcal{M}$, can be expressed as functions of I₁,…,I_s. The I₁,…,I_s form an integrity basis, invariant under $\mathcal{M}$ [56].

Appendix B. Continuous symmetry for magneto-noncrystalline material

A group containing an infinite number of continuous elements is called a continuous group. Integrity basis for a scalar-valued tensor function W(X,Y,…) can be obtained by following Rivlin & Spensor [45,46] for O(3) and SO(3). In this study, we consider only transverse isotropy, whose symmetry can be classified into five groups: $\mathcal{A}=\{C_{\mathrm{\infty }},C_{\mathrm{\infty }v},C_{\mathrm{\infty }h},D_{\mathrm{\infty }},D_{\mathrm{\infty }h}\}$.

We consider $D_{\mathrm{\infty }h}$ in this study. The structural tensor for $D_{\mathrm{\infty }h}$ is given by $\{\mathbb{S}\}=\{{\mathbf{e}}_3\otimes {\mathbf{e}}_3\}$ [58], where e₃ is the unit direction of transverse isotropy.

Appendix C. Symmetry restrictions for general constitutive relations

So far, we have considered a tensor-valued scalar function of the form W(X,Y,…). Now, we focus on the most general constitutive form

$$T_{i_1\dots i_n}=T_{i_1\dots i_n}(\mathbf{A},\mathbf{B},\dots \mathbf{)},$$

which is invariant under $\{\mathcal{M}\}$ or $\{\mathcal{A}\}$. We convert the tensor-valued function by introducing an arbitrary tensor t, which has same order and symmetry properties as T. We define a scalar V such that

$$V=t_{i_1\dots i_n}{T}_{i_1\dots i_n},$$

and V =V (t,A,B,…) is linear in t. Now one can find the integrity basis for V with the arguments t,A,B,… as described in the previous subsection. Let the set {I}, with elements I₁,…,I_r of the integrity basis, be independent of t and the set {L}, with elements L₁,…,L_m, be linear in t and the rest of the elements are of higher order in t. Because V is linear in t it may be represented by

$$V=\sum _{p=1}^m{c}_p(I_1,\dots ,I_r)L_p,$$

where c_ps are scalar polynomials. Then, the generalized form of the constitutive relation, which is invariant under $\{\mathcal{M}\}$, can be written as

$$\begin{array}{rl}{T}_{i_1\dots i_n}=\frac{\mathrm{\partial }V}{\mathrm{\partial }{t}_{i_1\dots i_n}}& =\sum _{p=1}^m{c}_p(I_1,\dots ,I_r)\frac{\mathrm{\partial }{L}_p}{\mathrm{\partial }{t}_{i_1\dots i_n}}\\ & =\sum _{p=1}^m{c}_p(I_1,\dots ,I_r){\mathcal{D}}_{i_1\dots i_n}^p,\end{array}$$ C 1

where

$${\mathcal{D}}_{i_1\dots i_n}^p=\frac{\mathrm{\partial }{L}_p}{\mathrm{\partial }{t}_{i_1\dots i_n}}.$$

We denote the set $\{\mathcal{D}\}$ by {∂L₁/∂t_i1…in,∂L₂/∂t_i1…in…∂L_m/∂t_i1…in}.

Funding statement

The authors acknowledge the financial support of the Army Research Office, grant no. W911NF-06-1-0319 for the initial stages of this work, NSF-IIMEC (International Institute for Multifunctional Materials for Energy Conversion) under grant no. DMR-0844082 and NSF-NIRT, grant no. CMMI: 0709283 for the support of the first.