A three‐dimensional phenomenological model for Magnetic Shape Memory Alloys

Abstract

We present a three-dimensional thermodynamically-consistent phenomenological model for the magneto-mechanical behavior of magnetic shape memory materials featuring a cubic-totetragonal martensitic transformation. Existence of energetic solutions for both the constitutive relation problem and the three-dimensional quasi-static evolution problem are proved. The proposed model reduces to some former one via parameter asymptotics by means of a rigorous Γ-convergence argument. Magnetic Shape Memory Alloys (MSMAs) present the usual superelastic and shape memory effects of SMAs combined with an impressive magnetostrictive behavior up to strains of 5-8%. This is the effect of the ferromagnetic nature of martensites in MSMAs which present the classical magnetic domain structure (domains with uniform magnetization). Local magnetization tends to align with external magnetic fields. Generally, this alignment corresponds to two mechanisms: the magnetic domain wall motion and the magnetization vector rotation. In MSMAs, a third phenomenon arises for martensitic variants rearrange in order to minimize the magnetic energy.Our focus here is on the phenomenological description of MSMA behavior in three dimensions. We concentrate our attention on so-called magnetically uniaxial materials, namely those presenting a single easy axis (preferred magnetization axis) at the martensitic crystal level. This is the case for MSMAs featuring a cubic-to-tetragonal martensitic transformation as Ni 2 MnGa, FePd, FePt. By applying a magnetic field, one variant may be favored with respect to the others. This results in a variant reorientation and a macroscopic strain.Some alternative phenomenological modeling of MSMAs are proposed in [4] and [5]. The referred papers, albeit taking their origin from the same basic principles, differ from the present approach as they are essentially restricted to two dimensions (or two martensitic variants), and they feature a scalar internal variable (local proportion of one martensitic variant with respect to the others) and a different choice of the relevant potentials. On the contrary, the present model is three-dimensional and tensorial in nature.

Existence and time-discretization for the finite-strain Souza–Auricchio constitutive model for shape-memory alloys

“…Results in the non-isothermal case have also been obtained els2,Krejci-Stefanelli09,Krejci-Stefanelli10,mpp,mpp2 [18,27,28,40,37]. Moreover, extensions of the original model to residual plasticity c3,auricchio07b,els [7,8,17], more realistic non-symmetric behaviors and transformation-dependent material parameters ars2 [9], and the ferromagnetic shape-memory effect paperalpha,gamm,bs,bks,hmm [1,2,11,12,47] are also available.…”

The Souza-Auricchio model for shape-memory alloys

“…When the body is thin in at least one direction and mechanical cycles have a suitably low frequency one can assume that the heat produced in the specimen is immediately transferred to the surrounding environment acting as a heat bath. Hence, by assuming the temperature to agree with a given external temperature t → θ(t) = θ ext (t) for all times, one is interested in considering a quasistatic evolution problem (7). This has been done in [90] and [87] in the polycrystal and single-crystal case, respectively.…”

“…The SA model has been extended to include the ferromagnetism of the martensitic phases in [6,7,8,23]. Within the single-crystal setting, we assume magnetic uniaxiality within a cubic-tetragonal system.…”

Experimental characterization and modeling of a three-variant magnetic shape memory alloy

“…Modeling the magneto-mechanical responses of MSMAs has been undertaken by several groups such as Hirsinger and Lexcellent (2003), Chernenko et al (2004), Kiefer and Lagoudas (2005), Kiefer (2006), Kiang and Tong (2007), Li et al (2007), Ma and Li (2007),Sarawate and Dapino (2007,2009,2010), Zhu and Dui (2007) Kiefer andLagoudas (2008, 2009), Li and Ma (2008), Morrison et al (2008), Auricchio et al (2011), Stefanelli and Bessoud (2011), Zhu and Yu (2013), LaMaster et al (2013aLaMaster et al ( , 2013bLaMaster et al ( , 2014, Chen et al (2014) among others. These models have been validated with experimental results from two-dimensional magneto-mechanical tests carried out on samples trained to accomodate only two martensite variants.…”