Macroscopic behaviour of magneticshape-memory polycrystals and polymer

composites

May 16, 2006

S. Contia∗, M. Lenzb, M. Rumpfba Fachbereich Mathematik, Universitat

Duisburg-Essen, Lotharstr. 65, 47057 Duisburg,Germany

b Institut fur Numerische Simulation, UniversitatBonn, Nussallee 15, 53115 Bonn, Germany

Single crystals of magnetic shape-memory materialsdisplay a large spontaneous deformation in responseto an applied magnetic field. In polycrystalline ma-terial samples the effect is frequently inhibited by in-compatibilities at grain boundaries. This motivatestechnological interest in textured polycrystals andcomposites of single-crystal magnetic shape-memoryparticles embedded in a soft polymer matrix. We usea continuum model based on elasticity and micro-magnetism to study the induced macroscopic mate-rial behavior in dependence of such mesoscopic struc-tures via numerical simulation in two dimensions.

Keywords: Magnetic shape-memory, composites,polycrystals, homogenization

1 Introduction

Ferromagnetic shape-memory materials exhibit com-parably large strains in response to an applied mag-netic field. For single crystals one can achieve strainsof order of magnitude 10% [1, 2, 3, 4]. In the searchfor applicable devices, polycrystal samples have beenexplored, mainly due to their much easier production.In polycrystals the effectivity drops significantly, asa consequence of the rigidity of interacting grains.The same difficulty is present in shape-memory ma-terials, such as InTl or NiTi. In many cases, theshape-memory effect disappears due to blocking atgrain boundaries, this typically being the case whena small number of variants is present. In magneticshape-memory (MSM) materials the number of vari-

*Corresponding author: Sergio Conti, Tel. +49-(0)203-379-2696; Fax +49-(0)203-379-2117; Email sergio.conti@uni-due.de

ants is typically very low. In particular, NiMnGa hasa cubic-to-tetragonal transformation, with only threevariants. Furthermore, compatibility conditions forthe magnetization at grain boundaries are also rele-vant. The role of texturing has been, to some extent,investigated experimentally [5, 6], showing that ori-enting the grains has a strong impact on the macro-scopic MSM properties. Here we investigate the roleof misalignment numerically and quantify its impacton the effective spontaneous strain.

A recently-proposed alternative for shape mem-ory devices is to embed small single-crystal shape-memory particles in a soft polymer matrix [7, 8]. Thisapproach gives a large freedom in the material de-velopment, which includes the type of polymer, thedensity of particles, their shape, and their orientation[9]. In the case of classical magnetostrictive materi-als, such as Terfenol-D, experiments have shown thatusing elongated particles, and orienting them via abias field during solidification of the polymer, leadsto a much larger magnetostriction of the composite[10, 11]. We exemplarily demonstrate the potentialof this approach and show that a significant percent-age of the single-crystal spontaneous strain can berealized in polymer composites.

2 Micromagnetic-elastic model

We briefly present our model in a geometrically linearsetting; for more details and a full nonlinear discus-sion see [12]. We work on a domain Ω ⊂ R2 occupiedby a polycrystal, which consists of grains ωi, eachof which is a single crystal. In the case of polymercomposites, the ωi denote the particles in a polymermatrix Ω \ ω. In both cases, ω = ∪ωi.

Kinematics. Let u : Ω → R2 be the elastic dis-placement, p : ω → 1, 2 be the phase index, whichis supposed to be constant on each grain (each par-ticle, in the case of composites), and M : R2 → R2

the magnetization. Notice that v and p are definedon the reference configuration, whereas M is definedon the deformed configuration.

Elasticity. We assume the elastic deformationv(x) = x + u(x) to be injective on Ω. We use linearelasticity throughout. The particles have two energy-minimizing phases, which are distinguished by thephase parameter p, and are elastically anisotropic.

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Figure 1: Top line: Reference configuration (left) anddeformed configuration (right) for a periodic poly-crystal, containing grains with seven different orien-tations. Bottom line: the same for a composite. Forillustrative purposes, a finite portion of a periodicsample is plotted, with the computational cell in thecenter. The checkerboard pattern in the grains (par-ticles) illustrates the lattice orientation (it is not re-lated to a numerical mesh), and the shading (coloringin the case of polymer composite) shows the elasticenergy density.

Their elastic energy is

Eelast =∫

ωW ((∇v(x))Q(x)− εp(x)) dx . (2.1)

Here Q : ω → SO(2) represents the crystal latticeorientation in the reference configuration, εp is theeigenstrain of phase p, say,

ε1 =(−ε0 00 ε0

), ε2 =

(ε0 00 −ε0

), (2.2)

and the energy density W (F ) penalizes deviationsfrom a preferred, phase-dependent, strain. Weparametrize it with the cubic elastic constants C11,C12 and C44. Notice that we only linearize the de-formation, and that the lattice rotation is insertedexplicitly and nonlinearly through Q(x).

In composites we additionally have the elastic en-ergy of the polymer matrix, which is assumed to bean isotropic, linearly elastic material.

Micromagnetism. Let M = Msm be the magne-tization, where Ms denotes the saturation magneti-zation. The vector field m : R2 → R2 has unit lengthon the deformed domain v(ω), i.e., on the grains (par-ticles), and vanishes elsewhere. The magnetic energy

is given by the coupling to the external field Hext,the demagnetization term, and the phase-dependentanisotropy term:

Em =∫

R2

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M2s

µ0|Hd|2 −

Ms

µ0Hext ·m dy

+Ku

∫v(ω)

ϕp(v−1(y))

((R∇vv−1Q(y))T m

)dy .

The field Hd : R2 → R2 is the projection of m ontocurl-free fields. Furthermore, ϕ2(m) = m2

1|m|2 , ϕ1(m) =

m22

|m|2 are the two magnetic anisotropy functions re-flecting the phase-dependent easy axis, and Ku is theuniaxial anisotropy constant. The matrix R∇v(x) ∈SO(2) is the rotation associated with the elastic de-formation v at a reference point x. Explicitly, weconsider R ≈ Id + 1

2(∇v − (∇v)T ).

Homogenization and numerical methods. Inthe spirit of the theory of homogenization [13], westudy periodic configurations, where each periodiccell contains a small number of particles, as illus-trated in Figure 1. We use boundary elements [14]to express both the elastic and the magnetic problemin the full space in terms of the deformation and themagnetization on the boundary of each grain (parti-cle). Details are given in [12].

Material parameters. We use parameters forNiMnGa, precisely: Ms

µ0' 0.50 MPa

T [15], Ku '0.13 M Pa [4], ε0 ∼ 0.058, C11 = 160 G Pa, C44 =40 G Pa, C11−C12 = 4 GPa [16] For the polymer wetake λ ∼ 20 G Pa, µ ∼ 1 G Pa.

3 Polycrystals

We investigate magnetostriction in a periodic poly-crystal. For better comparison, we keep the graingeometry fixed, and vary only the lattice orientationin the grains. This means that the ωi are fixed, andthe Qi are varied. Figure 2 shows the results for threedifferent choices of the orientation, ranging from al-most oriented to completely random. Whereas anorientation mismatch of about 2 does not substan-tially reduce the magnetostriction with respect to asingle crystal, already with an average mismatch of8 the effect is reduced by almost 30%. In the un-structured case only one-quarter of the single-crystalspontaneous strain survives.

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Figure 2: Spontaneous deformation of a polycrystalwith different texturing. Only one unit cell is plot-ted. With fixed grain geometry, we vary the amountof misorientation. The average orientation mismatchat grain boundaries is 2 (first line) and 8 (secondline). The third line depicts a configuration withcompletely random lattice orientation (i.e. averagemismatch close to 22.5). The first column showsthe reference configuration, the second one the de-formed state at the spontaneous deformation, withan applied horizontal external field Hext = 1T . Thespontaneous strain is 5.6%, 4.2%, and 1.6% in thethree cases. For comparison, the value for a singlecrystal is 5.8%.

4 Composites

As a second case we consider polymer composites.Again, we keep the shape and size of the particles

Figure 3: Spontaneous deformation of a compositewith different texturing. The volume fraction of theparticles is about 50%. Only one unit cell is plotted.With fixed particle geometry, we vary the amountof misorientation between the particles. The aver-age orientation mismatch is 2 (first line), 8 (secondline), and completely random (third line). Again, thefirst column shows the reference configuration, thesecond one the deformed state at the spontaneous de-formation, with an applied horizontal external fieldHext = 1T . The spontaneous strain is 3.7%, 3.9%and 2.9% in the three cases, compared to the 5.8% incase of a single crystal.

fixed, and vary their orientation (and consequentlytheir spatial arrangement). We assume that the vol-ume fraction of the active material is 50%. Fig-ure 3 shows that the dependence of the macroscopicmagnetostriction on the orientation is much smaller

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than for polycrystals. In particular, misorientationsof 2 and 8 give almost exactly the same magne-tostriction, which in turn is comparable to the oneof the polycrystal at 8 misorientation. (Indeed, inthis range the spontaneous strain also depends sig-nificantly on the details of the geometry. For a sym-metric arrangement of particles one actually observesmonotonicity of the spontaneous strain with respectto the lattice alignment. Here, we refer to Fig. 12 in[12].) Even in the non-oriented case the compositerecovers almost one-half of the single-crystal magne-tostriction.

Acknowledgements. This work was par-tially supported by Deutsche Forschungsgemein-schaft through the Priority Program 1095 Analysis,Modeling and Simulation of Multiscale Problems.

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