Numerical energy relaxation to model ... - Wiley Online Library

GAMM-Mitt. 38, No. 1, 171 – 196 (2015) / DOI 10.1002/gamm.201510009

Numerical energy relaxation to model microstructure evolution in functional magnetic materials Björn Kiefer1∗ , Karsten Buckmann1 , and Thorsten Bartel1 1

Institute of Mechanics, TU Dortmund, Leonhard-Euler-Str. 5, 44227 Dortmund, Germany

Received 2 October 2014, revised 25 October 2014, accepted 22 November 2014 Published online 11 March 2015 Key words Magnetic shape memory, magnetostriction, constitutive modeling Subject classification 74A60, 74D10, 74F15, 74N15, 74Q15, 82D40 This paper proposes energy relaxation-based approaches for the modeling of magnetostriction, with a particular focus on single crystalline magnetic shape memory alloy response. The theoretical development relies on concepts of energy relaxation in the context of nonconvex free energy landscapes whose wells define preferred states of spontaneous straining and magnetization. The constrained theory of magnetoelasticity developed by DeSimone and James [1] represents the point of departure for the model development, and its capabilities, but also limitations, are demonstrated by means of representative numerical examples. The key features that characterize the extended approach are (i) the incorporation of elastic deformations, whose distribution among the individual phases occurs in an energy minimizing fashion, (ii) a finite magnetocrystalline anisotropy energy, that allows magnetization rotations away from easy axes, and (iii) dissipative effects, that are accounted for in an incremental variational setting for standard dissipative materials. In the context of introducing elastic strain energy, two different relaxation concepts, the convexification approach and the rank-one relaxation with respect to first-order laminates, are considered. In this manner, important additional response features, e.g. the hysteretic nature, the linear magnetization response in the pre-variant reorientation regime, and the stress dependence of the maximum field induced strain, can be captured, which are prohibited by the inherent assumptions of the constrained theory. The enhanced modeling capabilities of the extended approach are demonstrated by several representative response simulations and comparison to experimental results taken from literature. These examples particularly focus on the response of single crystals under cyclic magnetic field loading at constant stress and cyclic mechanical loading at constant magnetic field. c 2015 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

1

Introduction

Functional materials, by definition, are materials whose intrinsic properties provide additional engineering functionality (e.g. sensing, actuation, energy harvesting, self-healing) beyond the usual structural requirements. An important subclass, that has widely been used for sensing, actuation, transduction, and information storage applications [2, 3], are magnetic functional ∗

Corresponding author

E-mail: [email protected], Phone: +49 231 755 5729, Fax: +49 231 755 2688

c 2015 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

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B. Kiefer, K. Buckmann, and T. Bartel: Numerical relaxation for magnetic materials

materials, in particular magnetostrictive materials. The recent resurgence of interest in magnetostrictives is primarily explained by the commercial availability of giant magnetostrictive materials, such as Terfenol-D [4] and Galfenol [5], that produce field-induced strains about two orders of magnitude above those that occur in naturally-magnetostrictive elements, e.g. iron for moderate magnetic fields. Moreover, multiferroics, i.e. materials that exhibit multiple ferroic (ferroelastic, ferromagnetic, ferroelectric) phase transitions, have recently drawn much research attention [6] due to their virtually unlimited potential to spawn novel functional material applications, for instance by providing magnetoelectric coupling properties that significantly outperform those of magnetoelectric materials undergoing single ferroic phase transitions. Multiferroics mostly occur in composite form, where a magnetostrictive material such as cobalt ferrite (CoFe2 O4 ) is used as one of the active phases. Magnetic shape memory alloys (MSMA), such as Ni2 MnGa, however, are intrinsically multiferroic materials, that undergo both thermally-induced austenite-martensite and paramagnetic-ferromagnetic phase transitions. These physical mechanisms give rise to response features—magnetic field induced strains of up to 10%, contactless actuation, fast response times, large bandwidth (up to 2 kHz), and conventional shape memory behavior—that in their combination are unique to MSMAs. This complexity of their constitutive behavior also makes them an ideal material class to study the influence of phase transitions on the macroscopic response of multiferroic (here coupling of ferromagnetism and ferroelasticity) materials. The quest of understanding the complex magneto-mechanically-coupled, anisotropic, nonlinear, and hysteretic behavior of MSMA can be supported by constitutive modeling and simulation efforts. Moreover, the availability of sufficiently accurate and numerically efficient and robust simulation tools are indispensable for the conceptualization and design of novel MSMA applications. In addition to now established MSMA modeling approaches—e.g. energy minimization, continuum thermodynamics, and phase-field concepts—see for instance the overviews given in [7–10], recent activity in this area aimed at improving the predictive capabilities of MSMA models, see, e.g., [11, 12], including extensions to fully 3-dimensional response [13], numerical implementation [14–16], and rigorous investigations of the mathematical underpinnings of such approaches [17]. The reader is also referred to related approaches for the modeling of macroscopic magnetostriction, e.g. [18–23], and references therein. While we make use of the same general continuum thermomechanics framework as in previous work [8, 10, 15, 24], in which the influence of crystallographic and magnetic microstructure evolution on the effective material properties is captured via dissipative and non-dissipative internal state variables, the MSMA modeling approach presented here is fundamentally different. It relies on concepts of energy relaxation in the context of non-convex free energy landscapes, whose wells define preferred states of straining and magnetization. The so-called constrained theory of magnetoelasticity developed by DeSimone and James [1], and particularly its application to MSMA modeling [25, 26], can be regarded as a point of departure for our model development. Their modeling framework essentially combines the Ball and James theory of microstructure formation [27] with classical micromagnetics approaches [28, 29]. It has successfully been demonstrated, that this physically well-motivated and mathematically rigorous theory can be applied to the modeling of MSMA and is able to predict important features of the magnetic shape memory effect (MSME), cf. [25, 26]. On the other hand, key response characteristics—e.g. the hysteretic nature, elastic effects, the linear magnetization response in the pre-variant reorientation regime, and the stress dependence of the maximum field induced strain, both of which are directly related to the rotation

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of magnetization vectors away from easy axes—are prohibited by the inherent assumptions of the constrained theory. To improve the energy relaxation-based modeling of MSMA, the following extensions are made: (i) elastic deformations are allowed and their distribution to the individual phases occurs in an energy minimizing fashion, (ii) the high, in fact infinite, magnetocrystalline anisotropy energy limit of the constrained theory is alleviated to allow for magnetization rotations away from easy axes, and (iii) dissipative effects are accounted for in an incremental variational setting for standard dissipative materials. The paper is structured as follows. First, a brief review of the constrained theory of magnetoelasticity is presented in Chapter 2. This theoretical framework is taken as a basis for the subsequent development of the extended modeling approach. Its predictive capabilities, but also limitations, are demonstrated with application to MSMA modeling by means of representative numerical examples. Chapter 3 addresses the first key component of the proposed extended model, the incorporation of non-energy-well states. This formally necessitates the introduction of additional internal degrees of freedom, whose evolution under loading is determined through energy minimization. Chapter 4 builds on the notion of standard dissipative materials to account for hysteretic effects in MSMA response. This results in the definition of an incremental potential through a variational minimization principle that governs the evolution of dissipative internal state variables. Chapter 5 presents selected results for the prediction of single crystal MSMA sample response under cyclic magnetic field loading at constant stress, as well as compressive stress loading at fixed magnetic field. These results, which are analyzed, compared to experiments, and discussed in detail, demonstrate the enhanced predictive capabilities of the proposed model. Chapter 6 is concerned with the utilization of rank-one convexification, which assumes an underlying (first-order) laminated microstructure and identically satisfies strain compatibility at twin boundaries. To the knowledge of the authors, [30] is the only publication in which such a laminate-based relaxation approach to model magnetostriction has previously been attempted. Simulation results for the modified relaxation approach are compared to those previously presented for the convexification scheme in Chapter 5, for which interface compatibility had not been taken into consideration. The paper concludes with a summary and an outlook on future work.

2

The Constrained Theory of Magnetoelasticity

In this section, the constrained theory of magnetoelasticity developed by DeSimone and James [1] is briefly reviewed, with a particular focus on its application to the modeling of MSMA, cf. [25,26]. Figure 1 schematically visualizes the crystal structures of Ni2 MnGa undergoing a temperature-induced phase transformation from the paramagnetic cubic austenite parent phase to the ferromagnetic (5M) martensite product phase, which, due to the reduction in lattice symmetry, exhibits three (nearly) tetragonal variants. The magnetic easy axis in each variant is aligned with the respective short c-axis, thereby allowing for two possible magnetization directions. In view of an energetic modeling concept, the undeformed crystal structure of the austenite phase is associated with the unique minimum of a convex free energy. In the ferromagnetic martensite at lower temperature, however, all six combinations of spontaneous lattice deformations and spontaneous magnetizations shown in Figure 1, henceforth referred to as configurations, are energetically equivalent in the absence of external magnetomechanical loading. The state of the material is then associated with a non-convex multi-well-type energy



174


landscape, whose minima correspond to the elastically undeformed martensite variants with magnetizations along the respective easy axes. a0

a0

a0

εtr 1 c

a

m1

c m2

a

e3 e2 e1

α1

a

α2

a

εtr 2 a

a m3

α3

c

m4

m5

α5

α4

εtr 3

m6

α6

Fig. 1 (online colour at: www.gamm-mitteilungen.org) Schematic illustration of the cubic austenite parent phase (lattice parameter a0 ) and the (nearly) tetragonal variants (lattice parameters a and c, transformation strains εtri , easy axis magnetizations mi , and configuration volume fractions αi ) in the ferromagnetic martensite product phase.

The difficulty of applying classical micro-magnetic theory [28, 29, 31] to the modeling of magnetostrictive or MSMA response is that it generally requires expensive computations on very small time and length scales. This makes numerical response simulations on the macroscopic or component scale almost impossible and motivated the development of the constrained theory by DeSimone and James [1]. This theory yields a reduction in computational effort by two central assumptions. The first is the large body limit, in which the gradient type exchange energy that favors the alignment of neighboring magnetic moments is neglected. This is justified by the idea that not all details of the underlying microstructure, e.g. domain sizes and domain wall thicknesses, must be fully resolved to predict macroscopic response features [32]. The second key assumption is the high anisotropy limit, which may be applied to the modeling of materials whose energies grow very rapidly away from the energetic minima. Under this condition all possible material states are comprised of energetically favorable mixtures of the energy well configurations. For the modeling of MSMA this means that elastic deformations and magnetization rotations away from the easy axes are assumed to be negligible. One may consequently define the set of energy wells K through the combination of allowable spontaneous strain and magnetization states as K = {[εtr1 , m1 ], [εtr1 , m2 ], . . . , [εtr3 , m6 ]} .

(1)

Here we have deviated slightly from the original notation used, e.g., in [26] for consistency with our extended model formulation introduced below. It has been shown, see [1, 26], that if the energy wells satisfy the conditions of pairwise magnetoelastic compatibility ˆ ij ]sym , εtri − εtrj = [aij ⊗ n ˆ kl = 0 , (mk − ml ) · n

i, j ∈ {1, 2, 3} ,

(2a)

k, l ∈ {1, . . . , 6} ,

(2b)

where no summation over repeated indices is implied in this case, for all combinations of ˆ ij ). The set of admissible macroscopic states is then given by the convex the pairs (aij , n www.gamm-mitteilungen.org


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hull of K. Relation (2a) represents a small deformation version of the Hadamard condition for strain compatibility [33, 34], which implies a rank-one connection of the (symmetrized) deformation gradients in the neighboring regions, or, more precisely, the continuity in their ˆ however, tangential projections. The corresponding projections in the unit normal direction n, may suffer the jump a. Condition (2b) on the other hand guarantees that no magnetostatic energy builds up at the respective interfaces. The effective magnetization m and effective strains ε can then completely be constructed through the convex combination of well states, cf. [25, 26], where each state is weighted by the corresponding volume fraction αi , i.e. m =

6

αi mi = [α1 − α2 ] m1 + [α3 − α4 ] m3 + [α5 − α6 ] m5 ,

(3a)

i=1

ε = [α1 + α2 ] εtr1 + [α3 + α4 ] εtr2 + [α5 + α6 ] εtr3 .

(3b)

The complexity of the free-energy expression then consequently reduces significantly within the constrained theory. By normalizing each energy well state to have zero energy and assuming all states to be combinations of them, the elastic strain energy and the magnetic anisotropy term, which generally assume finite values in micromagnetics-based energy expressions, see also the extended model formulation proposed below, vanish identically. What remains, are the magnetostatic energy of the demagnetization field and the external field contributions. In consequence, the modeling of MSMA response on the basis of the DeSimone and James theory can be reduced to the constrained quadratic minimization problem [26] , min Π(α)

α∈A

with

with A := {α | αi ≥ 0,

6

αi = 1} ,

(4)

i=1

Π(α) = VBs

1 μ0 m(α) · D ·m(α) − μ0 h ·m(α) − σ : ε(α) . 2

(5)

Here, the external loading is prescribed in terms of the external stresses σ and magnetic fields h. It has further been assumed that an ellipsoidal sample body of volume VBs with spatially homogeneous strain and magnetization fields is considered. Alternatively, an analysis in which all field quantities are volume averaged over a sample of arbitrary geometry may also be conducted. In such cases, the self-field can be computed via the (average) demagnetization tensor D, which is the magnetostatic equivalent of tensorial Eshelby shape factors used in micromechanics. The demagnetization tensor is either known for ellipsoidal sample geometries [35, 36] or calculated via magnetostatic FE-analysis, see also the discussion in [14]. For the interpretation of microstructural quantities it is often convenient to introduce an alternative parametrization of volume fractions. Noting that in (3) all transformation strain and magnetization contributions are weighted with the sum, respectively the difference, of two volume fractions, one may define the alternative weighting factors, cf. [25], ξ1 = α1 + α2 , ξ2 = α3 + α4 , ξ3 = α5 + α6 , η1 = α1 − α2 , η2 = α3 − α4 , η3 = α5 − α6 .


(6)


176


The ξi can be interpreted as the variant volume fractions and the ηi , at least in this case where the magnetization vectors are constrained to their respective easy axes, as the net magnetizations of each variant. Another reason is that we later want to associate the evolution of the variant volume fractions ξi with dissipation, whereas changes in the net magnetizations ηi will be treated as fully energetic, i.e. non-dissipative. The physical motivation for this will be discussed in Section 4. With this reparametrization, the set of admissible ranges for the volume fractions α(ξi , ηi ) ∈ A may alternatively be expressed as 6 i

αi =

3

ξj = 1 , 0 ≤ ξi ≤ 1 , |ηi | ≤ 1 ,

j

0 ≤ α1 = 0.5 [ξ1 + η1 ] ≤ 1 ,

0 ≤ α2 = 0.5 [ξ1 − η1 ] ≤ 1 ,

0 ≤ α3 = 0.5 [ξ2 + η2 ] ≤ 1 ,

0 ≤ α4 = 0.5 [ξ2 − η2 ] ≤ 1 ,

0 ≤ α5 = 0.5 [ξ3 + η3 ] ≤ 1 ,

0 ≤ α6 = 0.5 [ξ3 − η3 ] ≤ 1 .

(7)

The volume fractions ξi and ηi fully describe the microstructural state of the material. Since we will restrict our attention to two-dimensional considerations in the following examples, although this is not a limitation of the modeling approach in general, the internal state variable vector is in this case specifically given by p = [ξ1 , ξ2 , η1 , η2 ]t . Experimental investigations in MSMA typically focus on magnetic field induced variant reorientation at constant stress and stress-induced variant reorientation at constant magnetic field. The mechanical load and magnetic field directions are usually perpendicular and aligned with the edges of the single crystalline samples. The specimen are typically of prismatic (rectangular cuboid) shape and are carefully cut to align with specific, e.g.100aust, crystallographic directions. In particular, we consider the slightly off stoichiometric Heusler-type alloy Ni49.7 Mn29.1 Ga21.2 , for which the lattice constants a0 = 0.584 nm (austenite) and a = 0.595 nm, c = 0.561 nm (five-layered modulated martensite) have been reported [37]. The Bain strain tensors for the two martensite variants are thus assumed to be of the form εtr1 = 2 e1 ⊗ e1 + 1 e2 ⊗ e2 + 1 e3 ⊗ e3 , εtr2 = 1 e1 ⊗ e1 + 2 e2 ⊗ e2 + 1 e3 ⊗ e3 ,

(8)

with lattice strain constants 1 = (a−a0 )/a0 = 0.0188 and 2 = (c−a0 )/a0 = −0.0394, where {e1 , e2 , e3 } denotes an orthonormal frame aligned with the crystal axes, and, in this case, also sample edges. For these cubic-to-tetragonal phase transformation strain tensors it is easily shown that the compatibility conditions (2) can be satisfied. More specifically, considering a two-dimensional setting, see also Figure 3 below, a 180◦ domain wall between m1 = ms e1 and m2 = −ms e1 , where ms represents the saturation magnetization constant, implies a ˆ 12 = e2 . Crystallographic compatibility is trivially satisfied in this case, unit normal of n since both magnetization wells correspond to the same variant. A 90◦ domain√wall between ˆ 13 = 1/ 2 (e1 + e2 ), m1 = ms e1 and m3 = ms e2 on the other hand implies the normal n ◦ i.e. a 45√ inclination of the interface normal. Strain compatibility in this case is achieved for a13 = 2 [(2 − 1 )e1 + (1 − 2 )e2 ]. A more general discussion of the implications of compatibility in the presence of elastic deformations and magnetization rotations is given in Section 6 in the context of rank-one-convexification.



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The influence of the sample geometry on the magnetic field experienced by the material is captured through the demagnetization tensor concept. Since tr[D] = 1 must hold inside the magnetized body and the considered samples possess planar symmetry, it is sufficient to know one of its components, i.e. D = D11 e1 ⊗e1 +

1 − D11 (e2 ⊗ e2 + e3 ⊗ e3 ) . 2

(9)

The specific choice of model parameters used in the following numerical examples is summarized in Table 1. Table 1 Parameter set: constrained theory based model. Model parameter Saturation magnetization Transformation strain coefficient Transformation strain coefficient Demagnetization factor: sphere (1:1:1) Demagnetization factor: short square cyl. (9:5:5) Demagnetization factor: long square cyl. (5:1:1)

Symbol

Value

Unit

Reference

ms 1 2 D11 D11 D11

514 0.0188 −0.0394 0.3333 0.2154 0.0872

kA/m [-] [-] [-] [-] [-]

[9, 38] [37] [37] -

Figure 2 shows MSMA response curves that have been predicted with the described constrained theory based model. The configuration diagrams in Figure 2a) and b) indicate the state of the microstructure for given magnetomechanical load level and sample geometry. Note that single phase states as well as phase-mixtures involving combinations of all, or just a subset of, variant-magnetization configurations may generally occur. Furthermore, the influence of the sample geometry is clearly visible in these results, which emphasizes the fact that these simulations capture a system response. This correlation stems from the geometry dependence of the demagnetization field and its influence on the magnetostatic energy storage. A subtle point that is often overlooked is that this sample shape dependence is usually either not addressed or not reported in magnetic measurements found in the literature, so that sample responses are interpreted as true constitutive responses [14]. Figure 2c) and d) depict magnetization and strain response curves associated with magnetic field loading at constant compressive stress. In these response plots, relevant states have been numbered throughout the loading sequence. The lack of dissipation in the system causes the loading and unloading states to coincide. It is further observed, that a complete switching from the compressive stress favored configurations 1 and 2 (variant 1) to the field favored configurations 3 and 4 (variant 2) is always possible, provided the magnetic field is large enough, regardless of the stress level. Both these latter observations, however, contradict experimental findings [39].

3

Incorporation of Non-Energy-Well States

The response simulations of the previous section have demonstrated that the constrained theory is capable of predicting many key features of MSMA response. It also has has the advantage of combining strong physical motivation with mathematical rigor, although the short summary presented above is certainly not suitable to do the latter point justice (see [39] and [1]).



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B. Kiefer, K. Buckmann, and T. Bartel: Numerical relaxation for magnetic materials 0.0

0.6 3

2

-0.3

σ 11 [MPa]

μ0 h2 [T]

0.0

-0.6

1

sphere short cyl. long cyl. 4

-1.0

-0.5

-2.0 -3.0 −1.05 MPa −1.75 MPa −3.00 MPa

-4.0 0.0

μ0 h1 [T]

0.5

-5.0

1.0

-1.0

-0.5

a)

1.0

1 2

0.0

μ0 h2 [T]

0.5

5

0.0

ε11 [%]

4 6

2

1

-0.5

−1.05 MPa −1.75 MPa −3.00 MPa

-0.5

0.0

μ0 h2 [T]

0.5

3 5

4.0 3.0 2.0 −1.05 MPa −1.75 MPa −3.00 MPa

1.0 0.0

-1.0 -1.0

4

5.0

0.5

1.0

-1.0

c)

1.0

b)

6.0 3

m2 /ms [-]

3

4

-1.0

0.3

-0.5

2 6 1

0.0

μ0 h2 [T]

0.5

1.0

d)

Fig. 2 (online colour at: www.gamm-mitteilungen.org) Prediction of MSMA response based on the constrained theory of magnetoelasticity. Configuration (phase) diagrams a) in magnetic field and b) in magnetic field-stress space for three demagnetization factors (cf. Table 1). Response curves for magnetic field loading at constant compressive stress: c) magnetization and d) strain curves.

This notwithstanding, the high anisotropy limit, i.e. the constraining of strain and magnetization states to the energy wells, also imposes severe limitations, that in some cases completely suppress important physical effects. The most striking implication is perhaps that at infinite magnetocrystalline anisotropy there exists no mechanism to limit the variant reorientation process, so that the blocking stress phenomenon, or even the stress dependence of the fieldinduced strain magnitude, can not be captured. Motivated by these observations, this section is concerned with the extension of the modeling approach by introducing elastic strains, to allow deviations from spontaneous strain states, and magnetization rotations, to allow rotational deviations from spontaneous magnetization (easy axis) states. 3.1 Elastic strain energy in the context of convexification A question that naturally arises when introducing the notion of straining in the context of phase mixtures is how to relate the local phase strains εi to the macroscopic strain ε. Such considerations can be interpreted as homogenization assumptions. For the determination of the convex hull of an underlying non-convex energy landscape, a piecewise homogeneous strain state is assigned to each phase i via εi = ε + [[εi ]] , www.gamm-mitteilungen.org

(10)


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with [[εi ]] representing the respective strain jump. The elastic Helmholtz free energy of the phase mixture for a macroscopically prescribed strain state is then defined as ψ = ψ (ε, [[εi ]] , ξi ) =

nv

ξi ψi (ε, [[εi ]]) ,

(11)

i=1

where nv denotes the total number of phases, in this case martensitic variants. Assuming linear elastic response, and in the small deformation context an additive split of the infinitesimal tr strain tensor εel i := εi − εi , the elastic strain energy in phase i is introduced as 1 ψi = ψi (ε, [[εi ]]) = (ε + [[εi ]] − εtri ) : Ei : (ε + [[εi ]] − εtri ) . 2

(12)

The convex hull ψ C is then defined in terms of the constrained minimization problem n v ψ C = ψ C (ε, ξi ) = inf ψ (ε, [[εi ]] , ξi ) = inf ξi ψi (ε, [[εi ]]) , (13) [[εi ]]

[[εi ]]

i=1

subject to ([[εi ]] , ξi ) := − g=g

nv

ξi [[εi ]] =

i=1

nv

ξi (ε − εi ) = ε −

i=1

nv

ξi εi = 0 .

(14)

i=1

Through nvthe rewriting of the above constraint, for which (10) and the partition of unity property i=1 ξi = 1 were used, it is clear that (14) ensures ε to indeed represent the volume average strain of the phase mixture. The associated relation ε(ξi ) =

nv

ξi εi =

i=1

nv

tr ξi (εel i + εi )

(15)

i=1

is the extension of (3)2 when accounting for elastic deformations. The constitutive variational problem (13), subject to (14), yields the equality of stresses in each phase as a necessary condition. This is consistent with the statement that the partial relaxation of the effective elastic strain energy is interpretable as enforcing a Reuss/Sachs homogenization assumption, see [40]. In this sense, the convexification is known to yield the theoretically lowest possible energy bound. In some cases, particularly for isotropic small strain elasticity, a closed-form expression of the convex hull may be derived. To this end, exploiting the equality of stresses in phases i and j and assuming Ei = Ej = E, the strain jump in phase i may be expressed as Ei : (ε + [[εi ]] − εtri ) = Ej : (ε + [[εj ]] − εtrj ) , [[εi ]] = [[εj ]] − εtrj + εtri .

(16)

The substitution of (16) into constraint (14) then yields g=−

nv

ξi [[εi ]] = −

i=1


nv

ξi ([[εj ]] − εtrj + εtri ) = 0 ,

(17)

i=1


180


so that an explicit expression for the energy minimizing strain jump in phase j is found in terms of the known transformation strains and martensitic volume fractions [[εj ]] =

εtrj

−

nv

ξi εtri .

(18)

i=1

Consequently, the convex hull in the considered case takes the form ψ C (ε, ξi ) =

nv

ξi ψi (ε, [[εi ]])

i=1

nv

1

= ξi ε + [[εi ]] − εtri : E : ε + [[εi ]] − εtri 2 i=1 =

nv nv

1 ξj εtrj . ε − ξj εtrj : E : ε − 2 j=1 j=1

(19)

It is easily that the relaxed energy (19) indeed attains its minimum value of zero at nvobserved, ε = j=1 ξj εtrj , which is the assumption of the constrained theory (3)2 . In the context of MSMA modeling, where experiments are typically conducted at constant prescribed stress and variable magnetic field or at constant field and prescribed variable stress, it is much more convenient to work in terms of applied macro-stresses σ, rather than macrostrains. Under the assumption of linear elastic behavior in each single variant, an alternative expression of the convex hull ψ C (σ, ξi ) may be derived in closed form by inverting the linear relation deduced from (19) nv

⇔ ξj εtrj σ = ∂ε ψ C = E : ε − j=1

ε = E−1 : σ +

nv

ξi εtri

(20)

i=1

and substituting the resulting strain expression into (19), which yields the simpler expression 1 ψ C (σ) = σ : E−1 : σ . 2

(21)

Note that, due to the fact that the convexification results in homogeneous stresses and by having assumed isotropy, this energy density does not explicitly depend on the variant volume fractions. The explicit form of the compliance tensor for isotropic linear elasticity, considering a standard isochoric-volumetric decomposition, reads S := E−1 =

1 dev 1 1 ⊗1 + I , 9κ 2μ

(22)

where κ and μ are the compression and shear modulus, respectively, 1 the second-order identity tensor, and Idev the symmetric fourth-order deviatoric projection tensor with Cartesian components [Idev ]ijkl = 21 (δik δjl + δil δjk ) − 13 δij δkl . www.gamm-mitteilungen.org


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3.2 Finite magnetocrystalline anisotropy energy The second step of the model extension is concerned with allowing magnetization rotations away from their respective easy axes. Figure 3 illustrates the four rotation angles θi , with |θi | ≤ π/2, that are introduced to parametrize these rotations in a two-dimensional setting. This reduction to two dimensions is not a general limitation of the approach, but rather a matter of convenience, and is considered to be absolutely sufficient for the magnetomechanical experiments to which the response simulations will be compared. A parametrization of the rotations in three dimensions, e.g. through Euler angles, Rodrigues parameters, or quaternions, would naturally be more involved, but conceptually straightforward. (2) m1 (1)

θ1

m3

m2

θ3

θ2

(2)

(4) θ4

(1)

m4

(3)

(2) εtr 1 , ξ1 , η1

(4) e2 , [010]

εtr 2 , ξ2 , η2

(3)

e1 , [100]

(4)

Fig. 3 (online colour at: www.gamm-mitteilungen.org) Parametrization of the microstructure: martensitic variants (volume fractions ξi , transformation strains εtri ), magnetic domains (net magnetizations ηi ), and magnetization vectors (rotations θi ).

In compliance with the introduced rotation angles, the domain magnetization vectors are now defined as, see Figure 3, m1 := ms (cos(θ1 ) e1 + sin(θ1 ) e2 ),

m2 := −ms (cos(θ2 ) e1 + sin(θ2 ) e2 ) ,

m3 := ms (− sin(θ3 ) e1 + cos(θ3 ) e2 ), m4 := ms (sin(θ4 ) e1 − cos(θ4 ) e2 ) . (23) The effective magnetization of the phase mixture is then defined in terms of the mixture rule m(ξj , ηj , θi ) =

4

αi (ξj , ηj ) mi (θi ) .

(24)

i=1

which, considering (23), is an extension of (3)1 when accounting for magnetization rotations. Note that the term net magnetization for the variables ηi is strictly speaking only appropriate for θi = 0, but will henceforth still be used in a generalized sense. Though deviations of the magnetization in each of the variant/domain configurations from the spontaneous magnetization states are now generally possible, they are strongly penalized in high anisotropy (hard magnetic) materials such as Ni-Mn-Ga. Considering uniaxial symmetry for the tetragonal martensitic variants, the magnetocrystalline anisotropy energy in each



182


phase may be expressed as, see [36, 41] ψian = ψian (θi ) =

N

Kn sin2n (θi ) ,

(25)

n=1

where the Kn are magnetocrystalline anisotropy constants. Typically only the first term of the series expansion is used to capture the behavior of Ni2 MnGa, cf. [8, 42–44]. By applying a standard mixture rule, the magnetocrystalline anisotropy energy of the phase mixture is defined as follows ψ an = ψ an (ξj , ηj , θi ) =

4

αi (ξj , ηj ) ψian (θi ) =

i=1

4

αi (ξj , ηj )K1 sin2 (θi ) .

i=1

(26)

With (25), it is easily observed that the effective anisotropy energy (26) is indeed zero when the magnetization vectors in each domain are aligned with their respective easy axis (θi = 0). Finally, with all of the additional free energy contributions of the extended model defined in (19)–(26), the total potential of the system under stress-magnetic field loading (σ, h) takes the explicit form Π(σ, h, ξi , ηi , θi ) 1 = VBs ψ rel + ψ an (ξi , ηi , θi ) + μ0 m(ξi , ηi , θi ) · D ·m(ξi , ηi , θi ) 2 − μ0 h ·m(ξi , ηi , θi ) − σ : ε(σ, ξi ) = VBs

−

(27)

4 1 σ : E−1 : σ + αi (ξj , ηj )K1 sin2 (θi ) 2 i=1

4 4

1 μ0 αi (ξj , ηj ) mi (θi ) · D · αi (ξj , ηj ) mi (θi ) 2 i=1 i=1 4 2

− μ0 h · αi (ξj , ηj ) mi (θi ) − ξi σ : εtri .

+

i=1

(28)

i=1

Note that the relaxed elastic strain energy density was left general in Equation (27) to allow the consideration of other energy relaxation concepts, e.g. rank-one convexification, while it was specified to ψ rel = ψ C (σ) for the current convexification approach in (28). The total potential (28) again defines the thermodynamically reversible energy storage contribution. The implications of dissipative constitutive effects are addressed in the following chapter. Again it is pointed out that (28) reduces to the equivalent constrained theory expression (5) for εi ≡ εtri , εel i ≡ 0 , and θi ≡ 0, i.e. when the strain and magnetization states take values in the respective energy wells. It is further observed that for the special case of experiments under constant stress and varying magnetic field, the mechanical contribution to the total potential 2 1 C −1 ψ (σ) − σ : ε(σ, ξi ) = − σ : E : σ − ξi σ : εtri 2 i=1


(29)


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in fact differs from the constrained theory only in form of the additive constant given by the first term in (29). This implication would no longer hold, however, if different elastic properties were assigned to each variant under the consideration of tetragonal anisotropy.

4

Incremental Energy Minimization

From experimental observations [39, 45] it is clear that the magnetic shape memory response in Ni2 MnGa is strongly hysteretic. On the other hand, easy and hard axis magnetization curves of single crystalline Ni2 MnGa, in which the variant switching is suppressed through the application of compressive stresses above the blocking level, reveal that magnetization processes associated with pure magnetic domain wall motion (easy axis) or pure magnetization rotation (hard axis), respectively, are essentially hysteresis free [43]. Physical observation thus suggests that in Ni2 MnGa single crystals only the mechanism of variant reorientation, i.e. twin boundary motion, is in fact associated with dissipation. From a modeling point of view this can be captured by making a distinction between energetic and dissipative internal state variables that effectively describe the evolution of microstructure. To incorporate dissipative effects within an energy minimization setting, we adopt the concept of standard dissipative materials in the sense of Halphen and Nguyen [46], see also [47, 48]. In this framework the evolution of the internal state variables has an underlying extremum property associated with the minimization problem ˙ p) dv . where D := ζ(p, (30) inf Π˙ + D , p˙

Bs

Here, Π˙ is the rate of energy storage in the body, and D the volume integral over the dissipa˙ p). The Euler-Lagrange equation of this variational problem reads tion potential ζ(p, 0 ∈ ∂p Π + ∂p˙ D ,

(31)

which is known as (an integral form of) the Biot equation and describes the optimal evolution of the internal state variables in standard dissipative materials. A time discrete approximation of (30) in the time interval [tn , tn+1 ], also accounting for the physical constraints on p, takes the algorithmic form [48] red In+1 := inf In+1 pn+1

s.t. pn+1 ∈ P ,

(32)

where P denotes the set of admissible internal state variable values, see also Table 2, based on the incremental potential In+1 := Πn+1 − Πn + Δt D([pn+1 − pn ]/Δt, pn+1 ) .

(33)

In this case, a rate-independent form of the dissipation potential is chosen with ˙ p) = ζ(p,

nv

Yξ (ξ)|ξ˙i | ,

(34)

i=1



184


where Yξ (ξ) represents the, possibly loading history dependent, critical driving force, or threshold, for the evolution of the variant volume fractions. It is again emphasized that, although formally introduced as internal state variables, the evolution of the net magnetizations ηi and the magnetization rotations θi are treated as non-dissipative, cf. [24]. As discussed, this is motivated by experimental observations [43], which show that pure magnetic domain wall motion and magnetization rotation in Ni2 MnGa single crystals are not associated with hysteresis. It would, however, should the application of the modeling approach to other magnetostrictive materials for example demand this, be straightforward to associate the other mechanisms with hysteresis by assigning non-zero threshold values for the evolution of the respective internal state variables. With (34), the incremental dissipative work term takes the specific form Δt D([pn+1 − pn ]/Δt, pn+1 ) = VBs

nv

Yξ (ξi,n+1 )|ξi,n+1 − ξi,n | .

(35)

i=1

In the following examples we restrict our attention to two-dimensional loading cases for which experimental data is also available. In this case, the internal state variable vector reduces to p = [ξ1 , ξ2 , η1 , η2 , θ1 , θ2 , θ3 , θ4 ]t . Furthermore, to capture hardening behavior during variant switching, the following linear dependence of the threshold value on the variant volume fraction is assumed Y0 + ξ1,n+1 ΔY , if ξ1,n+1 > ξ1,n (36) Yξ (ξ1,n+1 ) = Y0 + (1 − ξ1,n+1 )ΔY , if ξ1,n+1 < ξ1,n , where Y0 and ΔY are additional material parameters. Certainly nonlinear hardening functions could also be introduced here, see, e.g. [8, 23, 24], but in this initial investigation of the interplay of the mechanisms of microstructure evolution within an energy relaxation framework the added level of complexity seemed unnecessary. In the constitutive variational problem (32), the minimization with respect to the energetic internal state variables {ηi , θi } on the one-hand and the dissipative variables ξi on the other, could generally be executed in a sequential manner, i.e. the minimization could be performed for an incremental free energy that has already been relaxed with respect to the purely energetic degrees of freedom. Our implementation, however, determines the solutions for all current internal state variables in a single minimization step. The details of the algorithmic setting of the proposed energy relaxation based model for the simulation of single crystalline MSMA response is summarized in Table 2.

5

Numerical Examples

In this section, the advantages of the extended modeling approach are demonstrated by comparing corresponding MSMA response predictions to those previously presented for the constrained theory based model at the end of Section 2. For further model validation, the extended model predictions are also directly compared to experimental measurements taken from the literature. All Ni49.7 Mn29.1 Ga21.2 and sample geometry parameters required for the following simulations are specified in Table 3.



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Table 2 Algorithmic setting. 0. Initialization: Set counter n = 1, specify set of constraints 4 αi = 1 , |θi | ≤ π/2 , P := i=1

0 ≤ α1 = 0.5 [ξ1 + η1 ] ≤ 1 , 0 ≤ α3 = 0.5 [ξ2 + η2 ] ≤ 1 ,

0 ≤ α2 = 0.5 [ξ1 − η1 ] ≤ 1 , 0 ≤ α4 = 0.5 [ξ2 − η2 ] ≤ 1 ,

and choose admissible set p1 = [ξ1 , ξ2 , η1 , η2 , θ1 , θ2 , θ3 , θ4 ]t1 ∈ P. 1. Load stepping: Specify magnetomechanical loads hn+1 , σ n+1 , and set initial values for internal state variables p0n+1 = pn . 2. Update of internal state variables: Minimize the incremental potential pn+1 = arg inf {In+1 (σ n+1 , hn+1 , pn+1 )} , pn+1

with

In+1 = Πn+1 − Πn + VBs

2 i=1

s.t. pn+1 ∈ P

Yξ (ξi,n+1 )|ξi,n+1 − ξi,n | ,

Π specified in (28), and Yξ (ξi ) given in (36), using a constrained global minimizer, e.g. M ATLAB function GlobalSearch with active set strategy for constraints. 3. Postprocessing: Calculate dependent state variables 2 ξi,n+1 εtri , εn+1 = E−1 : σn+1 + i=1 4 mn+1 = αi (ξj,n+1 , ηj,n+1 ) mi (θi,n+1 ) i=1

with mi (θi ) defined in (23). Save history data pn+1 (global variables), step counter n ⇐ n + 1, and return to 1.

Table 3 Model parameters: extended modeling approach. Model parameter

Symbol

Value

Unit

Reference

Young’s modulus Poisson’s ratio Saturation magnetization Anisotropy constant Initial threshold Delta threshold Strain coefficient Strain coefficient Demagn. coeff. short square cyl. (9:5:5)

E ν ms K Y0 ΔY 1 2 D11

5000 0.3 514 0.167 0.03 0.003 0.0188 −0.0394 0.2154

MPa [-] kA/m MJ/m3 MJ/m3 MJ/m3 [-] [-] [-]

[9, 38] [38, 39] [37] [37] -




1.0

0.5

0.5

m2 /ms [-]

m2 /ms [-]

186

0.0 -0.5

−0.60 MPa −1.40 MPa −3.00 MPa

-1.0

-0.5

-0.5

0.0

μ0 h2 [T]

0.5

1.0

-1.0

5.0

5.0

ε11 [%]

6.0

4.0 3.0 2.0 −0.60 MPa −1.40 MPa −3.00 MPa

1.0 -1.0

-0.5

μ0 h2 [T]

0.5

c)

0.5

1.0

b)

3.0 2.0

0.0

1.0

0.0

μ0 h2 [T]

4.0

−1.05 MPa −1.75 MPa −3.00 MPa

1.0 0.0

-0.5

a)

6.0

0.0

−1.05 MPa −1.75 MPa −3.00 MPa

-1.0 -1.0

ε11 [%]

0.0

-1.0

-0.5

0.0

μ0 h2 [T]

0.5

1.0

d)

Fig. 4 (online colour at: www.gamm-mitteilungen.org) Magnetic-field induced variant reorientation at constant stress; experiments [39] (left) and simulations (right). a) Measured and b) computed magnetization response. c) Measured and d) computed strain response.

The first example considers the magnetic field induced magnetization and strain responses at constant stress. This stress is applied along the 1-direction of the sample, while the cyclic magnetic field is applied along the 2-direction. The initial configuration for the computation is chosen to be ξ1 = 1. This is consistent with experiments, in which a sufficiently large compressive stress is typically applied prior to the magnetic field application, to guarantee this initial stress-favored single variant configuration that is characterized by the alignment of the short crystallographic c-axis along the compression direction. Figures 4b) and 4d) show the response predictions based on the proposed model. Comparison to the corresponding experimental data reported by Heczko [39], see Figures 4a) and 4c), demonstrates that the extended model clearly captures all of the desired features of macroscopic Ni2 MnGa response.1 It is particularly observed in Figure 4b), that in contrast to the behavior predicted in Figure 2c), 1 It shall be emphasized here that special care must be taken when comparing simulation results to reported response data. In single crystalline MSMA experiments it is usually convenient to measure straining with respect to an initial single variant configuration. However, these initial configurations typically vary for tests at constant stress and variable magnetic field (initial variant 1, short axis in compression direction) and tests at constant field and variable mechanical load (initial variant 2, long axis in compression direction). Moreover, both of these initial configurations differ from the austenite reference configuration assumed in the theoretical strain definition. Hence, a strain conversion procedure is required. To this end, one may define the theoretical axial strain component ε11 = (l − a0 )/a0 , see also (8), where l represents the current (unit cell) length in the 1-direction. Analogously, the axial strains measured by Heczko [39] under cyclic magnetic field and compressive stress loading are defined as H,σ εH,h 11 = (l−c)/c and ε11 = (l−a)/a, respectively. Rearranging these relations, comparable strain measures may



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initial linear magnetic field-dependencies of the magnetization response curves occur, which can be attributed to magnetization rotations in each of the domains, cf. illustrations in Figure 5 for the low and medium stress levels, that are prohibited in the constrained theory. Comparison to the experimental data of Figure 4a), however, reveals that this effect is clearly physical. Then, upon reaching a critical magnetic field value, the martensite phase starts to switch from the initial stress-favored to the magnetic field-favored variant for the low (black) and mid (red) stress-levels. See again the corresponding schematic illustrations in Figure 5. As a result, the slope of the magnetization curve increases significantly and becomes field dependent. Thus, below the saturation level, two mechanisms, namely field induced variant reorientation and rotation of the magnetization, occur simultaneously. The variant switching also causes the magnetic field induced straining observed in Figure 4d). At high magnetic fields all (black), or most (red), of the martensite phase has switched to the magnetic fieldfavored variant. At the compression stress level of −3.0 MPa (green), however, the variant reorientation is completely suppressed and consequently the magnetization response remains linear until reaching saturation. The stress level at which this effect first occurs is known as the blocking stress, cf. [45]. As observed, the maximum magnetic field induced strain is highly stress level-dependent, ranging from zero (green) to full reorientation (black). This phenomenon is properly captured by the extended model, which introduces the full rotation of magnetization in the stress-favored variant as a limiting mechanisms that is lacking in the constrained theory model. Upon magnetic unloading, the martensitic phase reorients back, either partially (black) or completely (red), to the stress-favored variant. The point at which variant reorientation is initiated for unloading significantly differs from the termination point of the reorientation process in the loading regime due to the hysteretic behavior of the MSME. Note that while the computations predict vanishing macroscopic magnetization for the fully removed magnetic field at all stress levels, the underlying mechanisms that are activated to reach this point actually differ significantly, see Figure 5. Magnetization rotation towards the respective easy axes initially occurs only for the stress levels having undergone partial (red) or no (green) switching. At some point variant reorientation becomes active as a second mechanism (red and black curves), which can be verified by comparison to the associated strain response. Since only partial recovery of the stress-favored variant occurs at the low stress level (black), a third mechanism must be activated, namely formation of domains in the field favored variant, to reduce the net magnetization, and consequently the magnetostatic energy, to zero at zero applied field. At the medium and high stress levels, the magnetization curves are pointsymmetric with respect to the origin, while the strain curves are symmetric with respect to the vertical axis, since the configuration at zero magnetic field is identical to the initial configuration in both cases. It is also observed, that at low stress level the strain and magnetization response curves differ significantly between the first and second half of the loading cycle, due to a difference in starting configurations. This phenomenon has been termed the first cycle effect, cf. [24], and is also properly captured by the model predictions. The response of all be calculated as εH,h 11 = ε11

a0 a0 + −1, c c

and

εH,σ 11 = ε11

a0 a0 + −1. a a

(37)

In addition to this conversion, field induced strain curves are typically shifted to start at zero strain for all stress levels, which eliminates initial elastic offsets.



188


6.0 5.0

0.5

6

ε11 [%]

m2 /ms [-]

4

5

4

3

5

0.0 2

6

-0.5

4.0 3.0

3

2.0 1.0

1

0.0

-1.0 -1.0

-0.5

0.0

μ0 h2 [T]

0.5

1

-1.0

1.0

-0.5

a)

1.0

2

0.0

1.0

0.5

μ0 h2 [T]

b)

5.0

4

4.0

0.5 5

ε11 [%]

m2 /ms [-]

4

3

0.0 6

2

-0.5

5

3.0 3

2.0 1.0

1

1

0.0

-1.0 -1.0

-0.5

0.0

μ0 h2 [T]

0.5

1.0

-1.0

c)

-0.5

0.0

6

μ0 h2 [T]

2

0.5

1.0

d)

Fig. 5 (online colour at: www.gamm-mitteilungen.org) Evolution of the microstructure of simulations presented in Figure 4. a) Magnetization and b) strain response at low stress. c) Magnetization and d) strain response at medium stress.

further loading cycles is then (point) symmetric and starts and ends with the configuration at the end of the first cycle. The second example deals with stress-induced variant reorientation at constant magnetic field. In this case, a compressive stress is cyclically applied along the 1-direction of the sample, while the constant magnetic field is applied along the 2-direction. The initial state of the microstructure for the computation is now set to ξ2 = 1. Figures 6a) and 6c) depict the experimental magnetization and strain response curves for this loading case as reported in [39]. Figures 6b) and 6d) show the corresponding simulation results and Figure 7 shows sketches of the associated microstructure evolution. In the initial configuration, the applied magnetic field is aligned with the easy axis of variant 2, which results in a state of saturation magnetization at zero stress for medium (green) and high (red) magnetic field cases. On the other hand, the macroscopic magnetization is zero during the entire loading cycle when no magnetic field is applied (black), since vanishing net magnetization is energetically favorable in this case. Variant reorientation from the magnetic field-favored to the stress-favored variant is observed in all response curves, but the onset of switching is delayed with increasing magnetic field. While the macroscopic magnetization in absence of an applied field remains zero (black), it remains at saturation level for 1.1 T, since this field is large enough to align the magnetization in variant 1 with the external field (rotation of 90◦ from easy axes). Since only partial alignment of field and magnetization occurs in variant 1 at the intermediate field level (green),



189

1.0

1.0

0.8

0.8

m2 /ms [-]

m2 /ms [-]

GAMM-Mitt. 38, No. 1 (2015)

0.6 0.4 0.2

2.0

4.0

6.0

|σ 11 | [MPa]

8.0

0.0 T 0.4 T 1.1 T

0.0 0.0

10.0

2.0

a)

0.0

4.0

6.0

|σ 11 | [MPa]

8.0

10.0

b)

0.0 0.0 T 0.4 T 1.1 T

-1.0 -2.0 -3.0 -4.0

-2.0 -3.0 -4.0

-5.0

-5.0

-6.0

-6.0

0.0

2.0

4.0

6.0

|σ 11 | [MPa]

8.0

0.0 T 0.4 T 1.1 T

-1.0

ε11 [%]

ε11 [%]

0.4 0.2

0.0 T 0.4 T 1.1 T

0.0 0.0

0.6

10.0

0.0

c)

2.0

4.0

6.0

|σ 11 | [MPa]

8.0

10.0

d)

Fig. 6 (online colour at: www.gamm-mitteilungen.org) Compressive stress induced variant reorientation at constant magnetic field; experiments [39] (left) and simulations (right). a) Measured and b) computed magnetization response. c) Measured and d) computed strain response.

a significant change in magnetization is observed during the mechanically induced variant switching. Both simulations with non-zero magnetic field show a complete reorientation back into the magnetic field-favored variant at low stresses. This effect has been termed magnetic field-biased pseudoelasticity or magnetoelasticity [24]. At zero applied field no reorientation back to the field-favored variant occurs and consequentially a maximum remanent strain is observed. Although not pictured here, partial recovery of the initial variant configuration occurs at intermediate magnetic induction levels of, in this case, 0.0–0.4 T. This phenomenon of partial magnetic field-biased variant reorientation is often, somewhat misleadingly, referred to as pseudoplastic behavior [49–51]. Again, model predictions show excellent qualitative and reasonable quantitative agreement with experimental results. The stress-induced variant reorientation response predictions also reveal the importance of including elastic strain energy in the model formulation, although the effect is not very pronounced at the particular stress levels considered here.

6

Rank-One Convexification

In Section 3.1 a convexification procedure was considered through which the lowest energy bound, the convex hull, is computed by a partial energy relaxation with respect to the phase strain jumps. This approach, however, gives no consideration to fulfilling crystallographic



190

B. Kiefer, K. Buckmann, and T. Bartel: Numerical relaxation for magnetic materials 0.0 1

0.8 0.6

1

-1.0

ε11 [%]

m2 /ms [-]

1.0

2

0.4

2

-2.0 -3.0 4

-4.0 -5.0

0.2 4

0.0 0.0

2.0

3

4.0

6.0

|σ 11 | [MPa]

3

-6.0 8.0

10.0

0.0

2.0

a)

4.0

6.0

|σ 11 | [MPa]

8.0

10.0

b)

Fig. 7 (online colour at: www.gamm-mitteilungen.org) Evolution of the microstructure for the response simulations at 0.4 T. a) Magnetization and b) strain response.

compatibility requirements at the respective interfaces, in this case twin boundaries, and, in fact, does not even acknowledge the existence of interfaces. On the other hand, compatibility requirements on laminated energy minimizing microstructures were discussed in Section 2. That discussion, however, due to the assumptions of the constrained theory, was limited to energy minimization in the context of convex combinations of well states. In what follows, we make use of a more general notion of rank-one convexification with respect to first-order laminates to enforce crystallographic compatibility in the presence of elastic deformations and magnetization rotations, i.e. while allowing the occurrence of non-energy-well states. Only some basic elements of the concept of lamination can be reviewed here, but this approach has widely been used in the literature, see, e.g., [52–58] for applications to the modeling of martensitic phase transformations or [59–63] in the context of crystal plasticity. The numerical application of sequential lamination for the modeling of SMA behavior, with a particular focus on the mathematical foundations of this theory, e.g. in terms of the relation to gradient Young measures, was pointed out in [64–67], see also the review article of [68]. The incorporation of dissipative effects for such theories are elaborated in [69–72]. In this context we also refer to recent work presented in [73] and references therein. Again having a laminated martensite microstructure, such as the one sketched in Figure 3, in mind, strain compatibility may be expressed in terms of the jump condition, see also (2a), ˆ sym , [[ε]] := ε1 − ε2 = [a ⊗ n]

(38)

where, however, we demand compatibility with respect to the total strains, not the respective transformation strains. Recall that in the convexification approach the phase strain jumps followed entirely from energy minimization, disregarding compatibility requirements. Again using the definition of the volume average strain of the phase mixture (15) tr el tr ε = ξ1 ε1 + ξ2 ε2 = ξ1 (εel 1 + ε1 ) + ξ2 (ε2 + ε2 ) ,

(39)

the following closed form expressions for the elastic strains in the two variants can be derived ˆ ]sym − εtr1 , εel 1 = ε + ξ2 [a ⊗ n

ˆ ]sym − εtr2 . εel 2 = ε − ξ1 [a ⊗ n

The partially (rank-one) relaxed energy density is then defined as ˆ ξi ) , ˆ ξi ) = inf ψ (ε, a, n, ψ R1 = ψ R1 (ε, n, a


(40)

(41)


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191

where, with (40) and the constraint ξ1 + ξ2 = 1, the unrelaxed energy density is given by ˆ , ξi ) = ψ (ε, a, n

2 i=1

=

ˆ , ξi ) = ξi ψi (ε, a, n

2 1 i=1

2

el ξi (εel i : E : εi )

1 ε : E : ε − ε : E : (ξ1 εtr1 + ξ2 εtr2 ) 2

1 ˆ sym : E : [a ⊗ n] ˆ sym + 2(εtr2 − εtr1 ) + ξ1 ξ2 [a ⊗ n] 2 1 1 + ξ1 εtr1 : E : εtr1 + ξ2 εtr2 : E : εtr2 . 2 2

(42)

Note that while the convexification approach is known to yield the lowest possible energy bound, the laminate-based rank-one relaxation scheme computes an upper bound to the desired quasiconvex hull. It should also be mentioned, that a caveat to the current approach is given in the fact that no effort has been made to enforce magnetic compatibility, see Equation (2b), at the twin and domain interfaces, —a condition which is only trivially satisfied for the constrained case where magnetization vectors are fixed to the easy axes, see discussion in Section 2. If crystallographic and magnetic interface compatibility were enforced simultaneously, the approach could be interpreted as a rank-one convexification with respect to first and second-order laminates, even though strain compatibility is always trivially satisfied at magnetic domain boundaries within the same variant. Moreover, the application of the proposed modeling concept to other magnetostrictive materials might even necessitate the introduction of higher-order sequential laminates. These considerations are the subject of on-going and future work. Enforcing the necessary condition for (41), namely

ˆ , ξi ) = ξ1 ξ2 (n ˆ · E · n) ˆ ·a+n ˆ · E : (εtr2 − εtr1 ) = 0 , ∂a ψ (ε, a, n

(43)

for this simplified case, then yields the closed form expression of the optimal strain jump

ˆ · E · n) ˆ −1 · n ˆ · E : (εtr2 − εtr1 ) . a∗ := −(n (44) Following the substitution a∗ = a, expression (42) can now alternatively be used as relaxed elastic strain energy density in the total system potential (27). To evaluate the capabilities of the proposed model when considering a partial rank-one relaxation, a semi-cyclic mechanical loading test at different magnetic field levels and variable compressive stress is again considered. For the following simulations, the model parameters listed in Table 3 have once again been used. Figure 8, which is the analog of Figure 6 in the convexification case, shows the resulting model predictions. Figure 9 depicts visualizations of the evolving laminated microstructure, that again reflect current values of the internal state variables, i.e. configuration volume fractions, magnetization rotation angles, and, since lamination has now explicitly been considered, the orientation of the twin interface. Interestingly, under the considered loading conditions, compatibility at the twin boundaries yields an interface orientation angle of 45◦ , see Figure 9, that is constant throughout the loading sequence and independent of the applied magnetic field level. Recall that the same value was obtained




0.0

0.8

-1.0

ε11 [%]

m2 /ms [-]

192

0.6 0.4 0.2 0.0 0.0

4.0

8.0

6.0

|σ 11 | [MPa]

-2.0 -3.0 -4.0 -5.0

0.0 T 0.4 T 1.1 T

2.0

0.0 T 0.4 T 1.1 T

-6.0 0.0

10.0

2.0

4.0

8.0

6.0

|σ 11 | [MPa]

a)

10.0

b)

Fig. 8 (online colour at: www.gamm-mitteilungen.org) Compressive stress induced variant reorientation at constant magnetic field. Computed a) magnetization, and b) strain response under partial rank-one relaxation. 1

1

1.0

2

2

m2 /ms [-]

0.8 3

3

0.6 4

0.4 6

4

5

5

0.2 6

0.0 0.0

2.0

4.0

6.0

8.0

10.0

|σ 11 | [MPa]

Fig. 9 (online colour at: www.gamm-mitteilungen.org) Evolution of the laminated microstructure for the response simulations at 0.4 T (left) and zoom-ins that illustrate the associated phase magnetization orientations (right).

in Section 2, where, in the context of the constrained theory, compatibility was enforced with respect to the transformation strains. The rank-one convexification approach presented here, ˆ via minimization of (41), or rather the system however, determines the interface normal n potential (27). The normal may thus generally evolve with loading, and is not prescribed or computed a priori. The fact that the predictions coincide in this case is attributed to the relatively low stress level and particular loading paths. Moreover, the microstructures sketched in Figure 9 can directly be interpreted as idealized representations of actual microstructures, as they can be observed in experiments [74–76], whereas the microstructural sketches of Figure 7 presented in Section 5 are merely cartoons that visualize current values of the configuration volume fractions and magnetization rotations. The macroscopic response curves respectively predicted by the convexification (Figure 6) and



GAMM-Mitt. 38, No. 1 (2015)

193

the rank-one convexification (Figure 8) schemes, are almost indistinguishable in this particular example. A careful investigation of how response predictions compare for these two approaches under more general loading conditions is the subject of on-going work.

7

Discussion and Future Work

In this work, an incremental energy relaxation-based model for nonlinear magnetostriction was established, with a particular application to the prediction of single-crystalline magnetic shape memory alloy response. The approach builds on the constrained theory of magnetoelasticity, but additionally accounts for elastic deformations, finite magnetocrystalline anisotropy energy, and dissipative effects. The presented model allows the simultaneous activation of the variant reorientation, domain wall motion, and magnetization rotation mechanisms. Changes in net magnetization and rotation of magnetization vectors away from easy axes are considered to be purely energetic, while the evolution of the variant volume fractions is treated as dissipative in nature. Instabilities related to the non-convex nature of the underlying energy landscape have been circumvented by incorporating effective, partially relaxed energy densities, either through convexification or laminate-based rank-one convexification, to describe elastic energy storage. Numerical simulations for two representative loading cases were considered, namely magnetic field cycling at constant stress and mechanical load cycling at constant applied field. Through comparison to experimental data taken from literature, it was demonstrated that the extended model captures all important features of the complex MSMA response for both loading scenarios. In particular, the model not only captures the magnetomechanical coupling (magnetic field-induced straining), nonlinearity, anisotropy, and sample geometry dependence of the response also predicted by the constrained theory, but accounts for additional key response features, such as hysteresis, elastic deformations, the first cycle effect, the stress level-dependence of the maximum field-induced strain, the linearity of the magnetization response at constant variant volume fraction under cyclic magnetic loading, as well as magnetic field-biased pseudoelastic/pseudoplastic responses under mechanical loading. It should be emphasized, that the model relies on a very reasonable number of material parameters, of which most have a very clear physical interpretation and can directly be determined for a given material system, e.g. the transformation Bain strains, the saturation magnetization, and the magnetocrystalline anisotropy energy constant. However, the parameters related to the initiation and evolution hardening of variant reorientation, which are essentially related to the twin boundary mobility, could be identified in a more rigorous fashion, i.e. through nonlinear optimization based [77] parameter identification, to further improve the quantitative accuracy of the model predictions. On-going work is concerned with generalizing the approach to a fully three-dimensional setting, which mostly hinges on adequate parametrizations of rotations (e.g. concerning interface and magnetization orientations), as well as the construction of appropriate compatible lamination sequences for the rank-one relaxation scheme, cf. [78]. Other important aspects that are currently addressed are the application of the established approach to other magnetostrictive material systems (e.g. cobalt ferrite, Terfenol-D or Galfenol) and the implementation of the developed constitutive models into fully magneto-mechanically-coupled finite element codes [23, 79].



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Acknowledgment The financial support by the German Research Foundation (DFG) through the Research Unit 1509: Ferroic Functional Materials: Multi-Scale Modeling and Experimental Characterization, project P7 (Ki 1392/4-1), is gratefully acknowledged.

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