Ni-Mn-Ga single crystal Shape Memory Alloy magneto-thermo-mechanical modeling

In the frame of Thermodynamics of irreversible process, a mo del describing the thermomagneto-mechanical behavior of a single crystal of Ni-Mn-Ga is built. The choice of internal variables is linked to the physics of the p roblem (fraction of martensite variants, fraction of Weiss domains, magnetization an gle). The simulations permit to describe the paths in the space (stress, temperature, magne tic field) in agreement with experimental tests. A special attention will be devoted to the control laws required to use these functional materials as sensors or actuators.


Introduction
The interest of Magnetic Shape Memory Alloys (MSMAs) compared with the classical Shape Memory Alloys (SMAs) is their possible activation not only by stress and temperature actions but also by magnetic field. A model of rearrangement process of martensite platelets in a non-stoichiometric Ni 2 MnGa MSMA single crystal under magnetic field and (or) stress actions has been proposed by the same authors in [1]. The aim of this following paper is to extend these works purpose to the anisothermal behavior when including the process of martensite platelets reorientation and phase transformation austenite ⇋ martensite.
Firstly, MSMA present the same properties as classical shape memory alloys but with the addition of a magnetic field sensibility. Several models are then devoted to the variant reorientation process. Some relatively old models are based on simple energy function [2], energy minimization [3,4] or using a magnetic stress to disconnect mechanical and magnetic behaviours [5]. One of the first thermodynamic approaches was built in [6] and the addition of internal variables in the thermodynamics of irreversible processes was proposed in [7,8] A very interesting experimental study concerning the shape memory and martensitic deformation response of Ni 2 MnGa single crystals is performed by Callaway et al. [9]. Few papers relate the global modeling of MSMA including temperature, stress and magnetic field effects in the same formalism. The thermodynamical approach proposed in this paper is a relevant way to model the complex behaviour of MSMA in a global and macroscopic form. a e-mail: christian.lexcellent@univ-fcomte.fr 2 Gibbs free energy expression associated with a magneto-thermo-mechanical loading The Gibbs free energy G expression can be split into four parts : the chemical one G chem (generally associated with the latent heat of the phase transformation), the thermal one G therm (associated with the heat capacity), the mechanical one G mech and the magnetic one G mag .
where the state variables are: The coupling between mechanic and magnetism is made by the choice of internal variables, for instance z i , and not by the addition of an interaction term G mech−mag in the free energy. For a Ni-Mn-Ga single crystal presenting three variants of martensite and a mother phase (cubic austenite ⇋ tetragonal martensite), under thermal, magnetic loading and mechanical one, the Gibbs free energy expression can be: C p is the heat capacity, E the Young modulus, β a = a a 0 , β c = c a 0 . a 0 lattice parameter of A. a and c lattice parameters of M; A and K constant belonging to the mechanical part of free energy G. T c is the Curie temperature and m 0 the magnetisation at 0K. In order to simplify the model, m 0 and T c parameters are considered as the same for austenite and martensite. µ 0 , χ a and χ t are the vaccum magnetic permeabilty, the magnetic susceptibilities in the easy and transverse axis.

Clausius-Duhem inequality
In the classical frame of Thermodynamics of Irreversible Process, the total strain in y direction, the magnetization in x direction, and the entropy can be written as: A magneto-thermal effect takes place in the entropy expression due to the temperature dependence of m s . This effect will be neglected in the present paper. The thermodynamical forces associated with the progression of the Weiss domains widths α, α A and rotation angle θ are: The choice of the free energy expression confirms that the pure magnetic behavior is considered as reversible. Actually, the magnetization curves of the two martensite variants have no hysteresis on the figure 2 taken from [10]. Finally, the thermodynamical forces associated with the z i martensite and austenite fractions are: The mechanical behavior is highly irreversible, hence the Clausius-Duhem inequality has to be written: where dD is the dissipation increment. This expression can be reduced to:

kinetic equations (example with two variants)
In the previous inequality, three of the four variables are independent and the evolution depends on the configuration of the forces. For example, if the sample contains only two variants (z o = z 3 = 0) during the evolution at constant temperature, then z = z 1 = 1 − z 2 and the Clausius-Duhem inequality becomes: An external loop, e.g. a complete rearrangement from z = 0 to z = 1 (path a) and from z = 1 to z = 0 (path b), is reported on figure 3. Rearrangement begins when (π f 1 − π f 2 ) ≥ π cr (T ) for the path a and when (π f 1 − π f 2 ) ≤ −π cr (T ) for the path b. After the rearrangement start, the behavior is modeled according to the following kinetic equations: λ is considered as a constant value in the present paper. But, in [1], the value of λ was considered depending on the previous strain history. The concept of memorized point was introduced and a difference appears between partial loops and major loops. Moreover, π cr (T ) is a function of the temperature as it was shown in [11]. A linear dependence is used in this paper as π cr (T ) =    A numerical simulation was done with the help of the Matlab/Simulink software. The powerful of the modeling approach presented here is that either mechanical (pseudo-elasticity and martensite reorientation), thermal, and magnetic effects can be taken into account in the same numerical simulation. Some different cases will be considered in this section.

Conclusion
The purpose of this paper was to propose a full single crystal modeling of Magnetic Shape Memory alloys including the magneto-thermo-mechanical coupling. In the frame of the thermodynamics of irreversible processes, a model was proposed using a Gibbs free energy expression. As it can be seen in (2), the model is quite complicated and its use may be difficult. Nevertheless in practical applications and for specific conditions, some reductions can be made (isothermal process on 2D motions) and in such a case, the usefulness of the model is shown. For example, in the paper [13], an extension of the quasi-static isothermal model to the dynamical case was made, including the magnetic circuit creating the magnetic field and a dynamical load applied to the MSMA sample. A thermodynamic approach with hamiltonian modeling was used by the authors to design actuators [13] and new control laws [14]. Moreover, the thermodynamical approach proposed in this paper to model a specific MSMA can be easily extended to other materials, such as future MSMA monocristals and polycristals. Finally, this global model can be a basis to model MSMA in a Finite Element Analysis to design new mechanical structures for actuation and sensing applications.