Dynamic Magnetic Shape Memory Alloys Responses: Eddy Current Effect and Joule Heating Krishnendu Haldar1∗and Dimitris C. Lagoudas2 1 Department
of Aerospace Engineering, Indian Institute of Technology Bombay, Mumbai 400076, India. 2 Department
of Aerospace Engineering & Department of Materials Science and Engineering Texas A&M University, College Station, TX 77843, USA.
Key Words: MSMA, variant reorientation, dynamic responses, eddy current, Joule heating ABSTRACT Generating high actuation frequency (∼ 1.0kHz) is one of the potential applications of Magnetic Shape Memory Alloys (MSMAs). In this work, dynamic responses of single crystal MSMAs due to variant reorientation are investigated. Time dependent part of the Maxwell equations becomes significant for a high frequency regime. Generation of an electric field and magnetic flux linkage due to the motion of the material points during deformation create a complex electro-magneto-mechanical coupling mechanism. We perform a thermodynamically consistent study to capture the variation of electromagnetic fields due to the deformation in the presence of fluctuating magnetic field, mainly focusing on eddy current and Joule heating. A comparison of MSMA responses with a typical ferromagnet/magnetostrictive material responses is discussed.
1
Introduction
MSMAs are best known for their unique ability to produce Magnetic Field Induced Strains (MFIS) up to 10% under a magnetic field [1–4]. Some of the commonly used MSMA material systems are NiMnGa [5–10], FePd [11–15] ∗
Corresponding author. Email:
[email protected]
1
and NiMnX, where X = In, Sn, Sb [16–18].Among them, NiMnGa alloys are widely investigated and will be the main focus of this study. Martensitic transformations in Ni2 MnGa alloys were first reported by Webster et al. [19]. Zasimchuk et al. [20] and Martynov and Kokorin [21] performed detailed studies on the crystal structure of martensite in the Ni2 MnGa alloys. Ullakko et al. [22, 23] are credited with first suggesting the possibility of a magnetic field-controlled shape memory effect in these materials. The unique magnetomechanical coupling makes MSMAs promising materials for multifunctional structures, actuator and sensor applications [24–28]. The coupled MSMA behaviors can be modeled by considering the material as an electromagnetic continuum. Extensive work on different electromagnetic formulations had been proposed in the literature [29–33] based on different notion of breaking up long range and short range forces. A continuum theory for deformable ferromagnetic materials and nonlinear magnetoelasticity for magneto sensitive elastomers are discussed in [34–37]. On the other hand, a study of electrostatic forces on large deformations of polarizable materials is discussed in[38, 39]. A theory for the equilibrium response of magneto-elastic membranes is formulated by [40, 41] and a continuum theory for the evolution of magnetization and temperature in a rigid magnetic body for ferro/paramagnetic transition could be found in [42]. The variational formulations for general magneto-mechanical materials have been proposed by many authors [43–47]. The macroscopically observable MFIS in MSMAs is caused by the microstructural reorientation of martensitic variants [2, 48], field induced phase transformation [17, 49–52] or a combination of the two mechanisms. In this work, we will focus on variant reorientation. In the variant reorientation mechanism, the variants have different preferred directions of magnetization and the magnetic field is applied to select certain variants among others, which results in the macroscopic shape change. There are two major modeling approaches for variant reorientation mechanism. In microstructural based models, the resulting macroscopic strain and magnetization response are predicted by minimizing a free energy functional. Details on the microstructural based modeling approach can be found in [1, 11, 53–56]. The second approach to study the material behavior is through thermodynamics based phenomenological modeling [57–61]. Most recent model development of variant reorientation in MSMAs and magnetomechanical loading analysis is reported in [62–64]. Moreover, a detailed anisotropic consideration due to single crystal discrete symmetry is recently presented in [65, 66]. A brief modeling approach for bulk polycrystalline materials is also mentioned in [65]. In this study, we systematically consider the time dependent Maxwell equations and the rate forms of the mechanical and 2
magnetization constitutive equations that include single crystal anisotropy [66]. Voltage generation (∼ 100 [V]) in MSMA due to mechanical impulse (∼ 1.4 [m/s]) inside a biased magnetic induction (∼ 1.5 [T]) is reported in [67]. A similar study of stress induced variant reorientation under a fixed biased magnetic field (1.6 [T]) is discussed in [28]. They found maximum voltage of 280 [mV] with the strain range of 4.9% at 10 [Hz]. Other relevant studies regarding voltage generation of MSMAs as sensors are reported in [68, 69]. Best of our knowledge, effect of eddy current and corresponding Joule heating are not so far reported for high-frequency actuation condition. Our main contributions in this article are to predict the above mentioned quantities, based on experimentally verified quasistatic constitutive equations, and to analyze other field variable responses under dynamic conditions. We also compare eddy current and Joule heating due to MSMA actuation with the 430-steel (rigid ferromagnet) and Terfenol-D (magnetostrictive) material responses.
2
Electro-magneto-mechanical field equations for dynamic system
We denote the reference configuration B0 , which is assumed to be free from any externally applied stimuli, and the current configuration is denoted by Bt . The body consists of material points X ∈ B0 . The spatial position in the deformed configuration is denoted by x = ϕ(X, t), with t representing time, so that the deformation gradient can be defined as F = ∇X ϕ with J = det(F ) > 0. In the deformed configuration Bt , we denote the magnetic induction by b(x, t), the magnetic field by h(x, t), the magnetization vector by m(x, t), and the electric displacement by d(x, t) with respect to a rest frame. We consider that the body is only magnetizable and there are no polarization and free charge. Magnetic and electric constitutive relations are written as m = b/µ0 − h, d = ε0 e.
(1)
Here µ0 is the magnetic permeability, and ε0 is the electric permittivity of the free space. An electric field will be present due to the rate of change of magnetic flux (Faraday’s law). Let a generic point P ∈ Ωt move with a velocity x˙ during deformation. The motion of a magnetic body under applied ˙ magnetic field will further produce motional electromotive force (emf), x×b. The total electric field then becomes e e = e + x˙ × b. At a moving point, the
3
e e f} are written as remaining electromagnetic variables {e j f , h, b, m e jf = jf ,
e b = b − c−2 (x˙ × e) ≈ b,
f = m, m
e = h − x˙ × d h
with respect to the rest frame variables {j f , h, b, m}, where j f is the free f are decoupled with the velocity current. The moving frame fields e j f and m x˙ as there are no free charge qf and electrical polarization p, respectively. The local form of the Maxwell equations are: ∇x · d = 0,
∇x · b = 0,
∗
e=e ∇x × h j f + d,
∗
∇x × e e = −b,
(2) ∗
where, for a general vector α, convective time derivative is denoted by α = ˙ − Lα + α tr (L) with α ˙ as total time derivative. In a moving frame Ohms α e law is given by j f = Ωe e = j f , where Ω is the conductivity tensor and positive definite. We further write ∗
∇x × e e = −b ⇒ ∇x × e = −∂t b, ∗ e=e ∇x × h j f + d ⇒ Ω−1 ∇x × h = e e + 0 Ω−1 ∂t e ⇒ ∇x × h = Ωe e. Note that 0 ∼ 10−12 and for a conductor |Ω| ∼ 107 . This means the product 0 Ω−1 ∼ 10−19 and thus negligible. The conservation laws of mass, linear momentum, angular momentum and energy are: ˙ ˙ ∇x · σ + ρf b = ρg, skw (σ) = skw (ρg ⊗ x), 0 f · b˙ − ρrh = 0. ρu˙ + ∇ · q − σ L : L − e jf · e e+m
ρ˙ + ρ∇x · x˙ = 0,
(3) (4)
The generalized momentum density is given by g = x˙ +
0 0 e×b=v+ e×b ρ ρ
(5)
and the total stress is decomposed as 0
σ = σL + σM 0
0
(6)
where, σ L is the work conjugate of the velocity gradient L and the general0 ized Maxwell stress σ M is given by, b · b 0 e · e b⊗b M0 σ = − m ⊗ b + 0 e ⊗ e + 0 e × b ⊗ x˙ − + − m · b I. (7) µ0 2µ0 2
4
2.1
Thermodynamic framework and constitutive equations
The constitutive responses with respect to a potential ψ(F , b, s, {ζ}), where s is the entropy and the set {ζ} represents the collection of internal state variables, are obtained through Coleman and Noll procedure [70], 0
σ L = ρ∂F ψF T ,
µ0 m = −ρ∂b ψ,
s = −∂T ψ,
−∂ζi ψ · ζ˙i > 0.
(8)
After performing Legendre transformations [65, 71], with respect to a Gibbs free energy G(S E , H, T, {Z}) the following constitutive equations are obtained E = −ρ0 ∂SE G,
µ0 M = −ρ0 ∂H G,
s = −∂T G,
−ρ∂Zi G · Z˙ i > 0.
(9)
Here, H = F T h is the magnetic field, E = 21 (F T F − I) is the Green 0 strain, S E = JF −1 σ E F −t is the work conjugate of E, where σ E = σ L + µ0 h ⊗ m + µ20 (m · m)I and {Z} is the set of internal variables in B0 . The Gibbs free energy for MSMA consists with the set of internal state variables {Z} = {E r , M r , ξ, g}. In a typical experimental condition, the MSMA sample, initially at austenitic phase, is subjected to a constant compressive mechanical load. A phase transformation into the martensitic variant takes place by cooling down the temperature. This particular martensitic variant, among the other three, is called stress favored variant or variant-1. Then a magnetic field is applied along the perpendicular to the loading direction. After a critical value, field induced variant (or variant-2) nucleates and both variants coexist by forming a twinned microstructure. Complete variant reorientation to field induced variant takes place at a sufficiently high applied magnetic field. By switching from variant-1 to variant-2 with magnetic field thus generates high actuation strain or reorientation strain with hysteresis. Variant-1 in the initial configuration only contains 180o domain walls as there is no remnant magnetization before applying any magnetic field. When a magnetic field is applied, the magnetization vectors start rotating in each domain from the easy axis to the hard axis. The associated energy is known as Magneto-crystalline Anisotropy Energy (MAE). Once the critical field for the variant reorientation has been reached, the field favored variant nucleates, and the structural rearrangement takes place to form an intermediate twin structure forming 900 domain walls. When MAE is sufficient to overcome the energy required for twin boundary motion, magnetic field favored martensitic variant (variant-2) grows, and field induced macroscopic shape change is observed. The applied field for the completion of the reorientation process 5
is around 1.5 [T]. In MSMA, however, the magnetic domain wall motion appears to be associated with a a minimal amount of dissipation. More detail could be found in [7, 72]. We denote the reorientation strain tensor by E r and internal magnetization vector due to reorientation by M r . ξ denotes the volume fraction of the reoriented field-favored martensitic variant (variant-2) and the scalar g is an internal state variable associated with the interaction (mixing) energy ˙ r, during reorientation [65]. We assume that the reorientation strain rate E r ˙ and the rate of mixing energy the rate of internal magnetization vector M g obey the following flow rules ˙ ˙ r = Λr ξ, E
˙ ˙ r = γ r ξ, M
˙ g˙ = f r ξ.
(10)
The tensors Λr describes the direction and magnitude of the strain generated during variant reorientation, γ r takes into account the direction and magnitude of the internal magnetization due to reorientation respectively and f r is the hardening function. Expanding the entropy inequality (9d), we get ˙ r + πM r · M ˙ r + πξ ξ˙ + πg g˙ ≥ 0. π Er : E
(11)
The thermodynamic driving forces are denoted by π Er = −ρ0 ∂Er G
π M r = −ρ0 ∂M r G
πξ = −ρ0 ∂ξ G
πg = −ρ0 ∂g G.
If the above relations are substituted in to (11), we get π Er : Λr ξ˙ + π M r · γ r ξ˙ + πξ ξ˙ + πg f r ξ˙ ≥ 0,
⇒ π r ξ˙ ≥ 0,
where the total thermodynamic driving force π r due to reorientation is given by π r = π E r : Λ r + π M r · γ r + πξ + πg f r . The following reorientation function, Φ r , is then introduced, ( π r − Y r , ξ˙ > 0 r r Φ := , Φ ≤ 0, r r −π − Y , ξ˙ < 0
(12)
(13)
where Y r is a positive scalar associated with the internal dissipation during reorientation and can be found from calibration. The proposed transformation function is similar to the transformation function used with conventional shape memory behavior [73, 74]. It is assumed that the constraints of the 6
reorientation process follows the principle of maximum dissipation and can be expressed in terms of the Kuhn Tucker type conditions [75] r
Φ ≤ 0,
r Φ ξ˙ = 0 .
(14)
The martensitic phase of Ni2 MnGa has 10M structure with I4/mmm space group [76]. The classical point group is 4/mmm (D4h ). The discrete symmetry restrictions for a specific Gibbs free energy potential, constitutive equations and evolution equations are elaborately discussed in [65, 66]. The incremental constitutive equations for single crystal anisotropy can be written in the following forms: ˙ = [L]bEc ˙ + [K]H, ˙ bSc ˙ + [K0 ]H. ˙ = [L0 ]bEc ˙ M
(15) (16)
Here [L]6×6 is a mechanical tangent stiffness matrix, [K]6×3 and [L0 ]3×6 are magneto-mechanical stiffness matrices, and [K0 ]3×3 is a magnetic stiffness matrix. In a general way, half-vectorization of a n × n symmetric matrix A is denoted by vech(A) = [A11 , ..., An1 , A22 , ..., An2 , ..., A(n−1)(n−1) , A(n−1)n , Ann ]T and we write for simplicity vech(A) = bAc.
3
Small deformation approximation and quasistatic responses
We reduce the model infinitesimal strain approximation for h by considering i T 1 E which E ≈ ε = 2 ∇u + (∇u) , S ≈ σ E , H ≈ h and M ≈ m. Under these conditions, we have the following system of equations: ∇·b ∇×e σ˙ ˙ m
= = = =
0, ∇ · d = 0, ˙ −∂t b, ∇ · σ = ρv, ˙ + [K]h˙ [L](ε) ˙ ˙ + [K0 ]h. [L0 ](ε)
∇ × h = Ωe e skw (σ) = 0
We further write from (6) n o µ0 0 0 0 σ = σ L + σ M = σ E − µ0 h ⊗ m − (m · m)I + σ M = σ E + σ h , 2 7
(17) (18) (19) (20)
where, modified Maxwell stress σ h takes the following form µ0 0 σ h = σ M − µ0 h ⊗ m − (m · m)I 2 i h h i µ0 0 = µ0 h ⊗ h − (h · h)I + 0 e ⊗ e − (e · e)I . 2 2 The rate form of the MSMA constitutive relations ((19),(20)) and the components of [L], [K], [L0 ] and [K0 ] can be found in [65, 66]. In 2-D 0 −1 1 ν 0 max E , [K] = E∆Mε ν 1 0 + µ0 hy 0 1 ,(21) [L] = 2 1−ν r(1 + ν) 1 0 0 2 (1 + ν) 0 0 0 0 0 [L ] = , 0 0 0 0
0 a1 ∆M/r [K ] = . 0 −b1 ∆M/r 0
(22)
The material parameters a1 = µ0 MxC , b1 = µ0 ∆M = µ0 (M sat − MyC ), where MxC is the x-component of the magnetization response at forward reorientation start and ∆M = (M sat − MyC ) + µ0 hy is the difference between the saturation magnetization and the y-component of the magnetization at forward reorientation start . The parameter r = ∂Φ , is an explicit function ∂ξ of the internal variable ξ, which depends on the magnetic field hy . The functional form r(ξ(hy )) depends on the selection of the hardening function. Thermodynamic force calculation and selection of hardening function are briefly presented in Appendix A.
3.1
Quasistatic conditions and constitutive responses
A half space MSMA conductor is considered for 1-D approach (Fig. 1). Although the geometry is idealized, its simplicity allows relatively simple semianalytic solutions to gain a useful insight into the processes. The x = 0 plane is kept fixed, while a biasing compressive stress σ ∗ is applied at x = L along the negative x direction. A magnetic field h0y (t) is then applied in the y-direction such that it produces actuation due to reorientation along the ˆ y (x, t) is a perturbation field due to the presence of the electric field x-axis. h ˆ y (x, t) and e˜z are zero. We consider e˜z . However, for a quasistatic case, both h a linearly applied field in the form h0y (t) = c1 t + c2 . Considering µ0 hy = HsM2 at t = ts and µ0 hy = HfM2 at t = tf , we can write HfM2 − HsM2 , µ 0 c1 = tf − ts
HsM2 tf − HfM2 ts µ0 c2 = . tf − ts 8
(23)
Figure 1: Schematic representation of the geometry and assumptions on the field variables. Infinitely extended y and z planes are considered to avoid electromagnetic edge effects.
The reduced 1-D constitutive equations become (from (19) and (20)):
(a)
(b)
Figure 2: Predictions of (a) magnetization and (b) strain responses for variant reorientation mechanism (after experimental data correction [77]) with an incremental method.
9
∆Mεmax ˙ hy , 0 = ε˙xx + µ0 r ˙by = µ0 1 − µ0 [∆M ][∆M] h˙ y . r
(24) (25)
We assume that total stress (component σxx ) inside the specimen is constant and is maintained at σ ∗ = −2 MPa to trigger field induced variant reorientation and so σ˙ xx = 0. Equation (24) reads Z εmax ∆M ˙ εmax ∆M ε˙xx = ∂x v = −µ0 hy , ⇒ εxx = − µ0 dhy (26) r r and from (25) we write Z [∆M ][∆M] dhy . by = µ 0 1 − µ 0 r
(27)
We present the quasistatic responses for magnetization and strain, obtained in an incremental way, in Fig. 2. Comparisons of predicted responses with experimental data and data corrections due to demagnetization effect are discussed in [77].
4
Dynamic effect on MSMA actuation
In this section, we consider the dynamic MSMA responses. The geometry of the body is same as presented in Fig. 1. We further assume that inertia effect is negligible and conservation of linear momentum is not taken into account. A motional emf vby is generated when high actuation is considered. ˆ y (x, t) is created due to the presence of eddy Consequently, a magnetic field h current and is added up with the existing magnetic field. The total field becomes ˆ y (x, t) = c1 t + c2 + h ˆ y (x, t). hy (x, t) = h0y (t) + h In this context, Faraday’s law and Ampere’s law reduce to ∂x ez = ∂t by [V /m2 ], ∂x hy = Ωe ez [A/m2 ].
(28) 1
(29) (30)
Units: It is always helpful to keep track of the units. We have 0 = 8.85 · 10−12 F/m, µ0 = 4π · 10−7 N/A2 and c = 3 · 108 m/s. Note that F=A.s/V and Volt can further be written as V =N.m/A.s. Electric field e is V/m and magnetic induction b is T=N/A.m. Magnetization m and magnetic field h have unit A/m. The unit for current density j is A/m2 . The conductivity Ω is S/m, where Simens S=A/V. 1
10
Denoting β1 = c1 /µ0 [A/m.s], ∆Mεmax β2 = −µ0 [m/A], r [∆M ][∆M] µr = µ0 1 − µ0 [N/A2 ], r we solve for ez with a given ∂t by (equation (25)) from the following equation ∂x ez = ∂t by = µr ∂t hy ,
(31)
ˆ y . The velocity due to actuation is obtained from (26) where ∂t hy = c1 + ∂t h as ∂x v = β2 ∂t hy .
(32)
ˆy Knowing the velocity we then compute eez = ez + vby , from which we find h from (30) ˆ y = Ωe ∂x h ez .
(33)
We solve all the above equations numerically in a staggered way, and the outline is presented in Tab. 2. Material constants are given in Table. 1. The µ0 HsM2 = 0.6 [T], µ0 HfM2 = 0.9 [T], µ0 HsM1 = 0.75 [T], µ0 HfM1 = 0.17 [T] M sat =742 [kA/m], K =0.5 [T]−1 , H C = HsM2 , εmax = 5.65%, E=2.0 [GPa] n1 = 6.6, n2 = 1.4, n3 = 0.8, n4 = 0.3, (Hardening exponents, Appendix A) Ω = 1.43 · 107 [S/m], σ ∗ = −2.0 [MPa], ρ = 1.206 · 104 [kg/m3 ] Table 1: Used material constants in the analysis. The values of the magnetic parameters can be found in [66, 77]. The value for conductivity is taken from [78, 79]. initial-boundary values are ˆ y (x, 0) = 0 IC : h BC : ez (0, t) = 0,
x ∈ [0, L] vx (0, t) = 0, 11
ˆ y (0, t) = 0 h
t ∈ [0, +∞) (34)
1. Given external applied magnetic field h0y = c1 t + c2 . (1,n)
ˆ (1,n) ˆ (1,n) = 0, h = 0, ∆h = 0. y y
2. Initialize v (1,n) = 0, ez 3. For i = 1 : M
(space discretization, along length)
4. For n = 1 : N , (i,n)
5. Find: hy
(time discretization) 0, (i,n)
= hy
(i,n)
ˆy +h
(i,n)
,
∆hy
ˆ (i,n) = c1 ∆t + ∆h . y
ˆ (i,n) 6. Find: [∂t hy ](i,n) = c1 + ∆h /∆t. y (i,n)
7. Compute: ξ (i,n) , r(i,n) , β1 (i,n)
8. Compute: eez
(i,n)
= ez
(i,n+1)
(i,n)
, β2
(i,n)
+ v (i,n) by
(i,n)
(i,n)
, µr
(i,n)
(i,n)
. (all are function of hy
)
. (i,n)
9. Compute: by = by + µr ∆hy , (i,n+1) (i,n) (i,n) (i,n) εxx = εxx + β2 ∆hy . (i,n)
10. From (32), find: v (i+1,n) = v (i,n) + [β2 (i+1,n)
11. From (31), find: ez
(i,n)
= ez
(i,n)
+ [µr
[∂t hy ](i,n) ]∆x.
[∂t hy ](i,n) ]]∆x.
(i,n) ˆ (i+1,n) ˆ (i,n) 12. From (33), find h =h + Ωe ez ∆x. y y
ˆ (i+1,n+1) = h ˆ (i+1,n) 13. Update: v (i+1,n+1) = v (i+1,n) , h (i+1,n+1) (i+1,n) ez = ez .
Table 2: Numerical scheme to solve the coupled electro-magneto-mechanical problem in a staggered way. Next, we predict some simulations for dynamic conditions. Our magnetic range of interest is between reorientation start and finish. Time at HsM2 is considered as t = 0 and at HfM2 , t = ts−f . The time dependent applied field rate is considered in the form of h˙ 0y = c(t) such that h0y
Z =
c(t)dt + c2 .
(35)
We consider a constant rate c(t) = c1 for which the values of c1 and c2 are the 12
same as given in (23). We present two cases where two different lengths are considered without altering the frequency (25 Hz, equivalent to ts−f =0.02 s)
4.1
Case-I: Small length and low frequency
(a)
(b)
Figure 3: (a) Volume fraction of the field induced martensitic variant (variant-2) and (b) velocity distribution due to actuation. In this case, the length of the bar is considered 25 mm and the frequency is 25Hz. The plot for the field induced volume fraction is presented in Fig. 3(a). The distribution along the length almost remains uniform for the entire time period. Note that, along the length, each material point has an electric field, ˆ y . The induced magnetic field thus which creates an induced magnetic field h nucleates more field induced variants and an increase in volume fraction is observed along the length. The velocity distribution along the time axis is increasing-decreasing by nature Fig. 3(b). This means, the velocity increases initially with the rate of nucleation of the new field induced variant and then gradually slows down at the end. Induced magnetic field due to the motion is presented in Fig. 4(a) and the Maxwell stress is given in Fig. 4(b). Both of them are significant. The maximum Maxwell stress turns out to be around 0.4 [MPa], 20% of the -2 [MPa] blocking stress. The eddy current density along the transverse direction is shown in Fig. 5(a). The predicted current density is large. The current flows along the width of the bar and the body is heated up due to the resistance. The Joule heating is given in Fig. 5(b) and is significant too. A comparison with the standard steel will be provided as the work progress (see Table. 3). This means the increased temperature 13
(a)
(b)
Figure 4: (a) Induced magnetic field due to induced electric field and (b) Maxwell stress in the x-direction.
(a)
(b)
Figure 5: (a) Eddy current density and (b) corresponding Joule heating. (heat) may facilitate to nucleate thermally induced phase transformation. However, we did not consider thermal coupling in this study.
4.2
Case-II: Large length and low frequency
We increase the length four times, i.e., 100 mm while keeping the frequency same. The volume fraction distribution becomes strongly nonuniform due 14
(a)
(b)
Figure 6: (a) Volume fraction of the field induced martensitic variant (variant-2) and (b) velocity distribution due to actuation. to the increase in length (Fig. 6a). The distribution indicates that complete reorientation (ξ = 1) takes place almost everywhere after half of the length. Once the complete reorientation takes place, actuation stops and a sharp instantaneous drop in velocity to zero is observed (Fig. 6b).
(a)
(b)
Figure 7: (a) Induced magnetic field due to induced electric field and (b) Maxwell stress in the x-direction. The induced magnetic field becomes high, and it reaches its maximum value around 0.8 [T] (Fig. 7a). Maxwell stress is given in Fig. 7(b). The eddy 15
current density (Fig. 8b) increases almost twice by increasing the length four times. The Joule heating also increases than the previous case (Fig. 8b).
(a)
(b)
Figure 8: (a) Eddy current density and (b) corresponding Joule heating. One should notice that a sawtooth is created around x=65 mm for both current and power responses. This is due to the fact that the velocity becomes zero after this point and so the motional emf drops. This is mainly the part Ωvx by that disappears due to zero velocity. However, the presence of applied field rate contributes in the increasing of jf in spite of zero velocity. The same argument holds true for the power P . ˆ y , no sharp drop is observed. As ∂ hˆ y = Ω(ez + vx by ), it Note that for h ∂x turns out that the contribution from Ωvx by is very small (in the order of 10−2 ) compared to the static part Ωez .
4.3
Case-III: Small length and high frequency
We observed in the previous cases that the length has a significant effect on the output variables. Now we keep the length of the body same as Case-I, 25 mm, but consider a frequency 500Hz. The volume fraction and velocity are plotted in Fig. (9). As the velocity becomes very high, we suspect that inertia may play an important role in this regime. Recall that we assumed conservation of linear momentum is trivially satisfied due to constant stress assumption and negligible inertial force. The linear momentum equation reads ∂x σxx = ρv˙ x = ρ[∂t vx + vx ∂x (vx )] 16
(a)
(b)
Figure 9: (a) Volume fraction of the field induced martensitic variant and (b) velocity distribution due to actuation. and from the known values of the right hand side, we calculate total stress distributions at 25 Hz and 500 Hz for L=25 mm (Fig. 10). Note that at
(a)
(b)
Figure 10: Total stress distribution due to the frequency (a) 25 Hz and (b) 500 Hz for L = 25 mm. 500 Hz, the maximum stress level (0.6 MPa) is nearly 30% of the assumed biasing stress (2 M P a). However, at low frequency this approximation holds good (Fig. 10a). We perform some parametric studies by varying length up to 100 mm 17
and frequency up to 0.5 KHz. The variation of the maximum stress response is presented in Fig. 11(a). The white dotted line represents the limit below which the stress level is less than 5% with respect to the blocking stress. Considering this range as an admissible domain of L and f , we project this line on the maximum velocity plot (Fig. 11b), eddy current and Joule heating plots (Fig. 12).
(a)
(b)
Figure 11: (a) Variation of maximum total stress and (b) maximum velocity at different lengths and frequencies. The dotted lines are the threshold above which the data start to accumulate spurious values due to constant stress assumption and lack of satisfying conservation of linear momentum. It is evident from the above study that we need to consider full coupling of the conservation of linear momentum equation, coupled magneto-mechanical constitutive equations and Maxwell equations for accurate predictions in a wide range of length-frequency combinations. At smaller time scale inertia will undoubtedly play a crucial role in generating stress waves. Since the MFIS strongly depends on the stress intensity, a highly nonuniform strain field may be produced. A proper wave propagation study is required to conclude about the inertia effect, which is not investigated in this study. However, the presented analysis gives feasible results only at a specific zone of interest, as mentioned inFig. 11. Finally, we present some quantitative data for mechanical waves, electromagnetic waves and skin depth for MSMA conductors, operating under fluctuating field. The propagation speed of longitudinal q mechanical waves along an elastic MSMA rod (1-D) is given by vm = Eρ = 407.23 m/s. On the other hand, if we assume linear magnetic MSMA material for which 18
(a)
(b)
Figure 12: (a) Maximum eddy current density and (b) Joule heating at different lengths and frequencies. µ = max {µr (hy )} ≈ 3.5µ0 , then the plane wave propagation speed is given by sr r ω Ω 2 0 µr ue = , where κ=ω 1+( ) + 1. κ 2 0 ω The values at 1 MHz and 1 kHz frequencies are 1.13 · 103 m/s and 35.88 m/s, respectively. The skin depth, δ = 1/κ, increases from 0.18 mm to 5.7 mm due to the decrease in frequency. Further decreasing the frequency, e.g. at 100 Hz, ue = 11.34 m/s and δ = 18.1 mm.
4.4
Comparison with rigid and magnetostrictive ferromagnets
In this section, we compare the eddy current and Joule heating due to MSMA actuation with the conventional rigid magnet and magnetostrictive materials. We select ‘430 steel’ as a rigid ferromagnet which has saturation magnetization M sat = 955kA/m. Such a soft ferromagnetic material reaches saturation when the applied field is around 2.5 [mT] [80]. This value is significantly small compared to the operating field range of forward variant reorientation (0.60.9 [T]) for MSMA. We can thus easily assume that magnetization remains fixed in this range and there is no rate of change of magnetization. On the other hand, for a magnetostrictive material, such as Terfenol-D, magnetization and strain saturation take place around 0.25 [T] [81]. Almost same 19
electrical conductivity (Ω = 1.66 · 106 [S/m]) of Terfenol-D [82] with the 430 steel suggests that both of them will behave almost same in the considered range of the magnetic field (0.6-0.9 [T]) .
jz [A/mm2 ]
8
jz [A/mm2 ]
2.5 2 1.5 1 0.5
6 4 2
0 100
0 100
1
50
x [mm]
x [mm]
0.01
50 0 0.02
t [s]
2 0
(a)
t [s]
×10-3
(b)
Figure 13: Induced eddy current at (a) 25 [Hz] and (b) 200 [Hz] for a considered length of 100 [mm]. The absence of rate of change in magnetization and magnetostriction gives ∂t by = µ0 hy and eez = ez . We write from equations (29) and (30) ∂x2 jz =
1 ∂t jz , γ
where γ =
1 . Ωµ0
(36)
We solve the above equation with the same initial-boundary values, used for the MSMA problem. The analytic solution of the above equation is given by ( 0 (x, t) = (0, 0) jz (x, t) = . (37) A erf (ζ) + B (x, t) ∈ (0, L] × (0, +∞) √ Here ζ = x/2 γt and A, B are arbitrary constants. From jz (0, t) = 0 we get B = 0. Considering the fact that µ0 h(0, 0+ ) = HsM2 and µ0 h0 (0, tf ) = HfM2 , we further write from the relation Ω1 ∂x jz (0, t) = µ0 ∂t hy (0, t) Z µ0
M2
Hf
/µ0
M2 /µ0
Hs
A dh(0, t) = √ Ω πγ
Z 0
tf
√ Ω(HfM2 − HsM2 ) πγ dt √ =⇒ A = . (38) √ 2 tf t 20
The plots are given in Fig. 13 for the eddy current at two different finish time tf = 0.02 [s] and tf = 2.5 · 10−3 [s], which correspond to 25 [Hz] and 200 [Hz], respectively. At low frequency, saturation of eddy current reaches at the end of the length. However, saturation reaches much earlier for the high frequency. A similar trend is also observed for the Joule heating. Maximum values of the eddy current and the Joule heating and a comparison with the MSMA responses are given in Table 3.
25 [mm] jzmax P max 100 [mm] jzmax P max
MSMA 25 [Hz] 200 [Hz] 3.38 19.12 5.15 165.85 7.95 28.46
50.11 1.13·103
430 Steel/ Terfenol-D 25 [Hz] 200 [Hz] 1.25 6.11 0.92 22.51 2.20 2.81
6.11 22.51
Table 3: Comparison of MSMA and ferromagnetic/magnetostrictive responses at different frequencies and lengths. Comparing the responses in Fig. 13(a) and Fig. 8(a), we observe that eddy current generation in MSMA is higher than the conventional ferromagnetic/magnetostrictive materials (see also Table 3). Presence of rate of change of magnetization and velocity of the material points during variant reorientation make MSMA responses significant different. However, for a non-MSMA materials, responses relax quickly with the evolution of time. Power loss also follows the similar trend as of the eddy current. Before conclusion, let us have a quick estimation of temperature change due to the Joule heating by solving a 1-D steady-state heat equation. Consider Fig. 1 and assume that the surface x = 0 is insulated. The temperature is uniform, and the system is below the martensitic finish temperature Mf . If g0 is the energy generation and is assumed to be homogeneous for simplicity, then the heat conduction equation is written as 1 ∂xx T + g0 = 0 k
(39)
where k is the thermal conductivity. Thermal analysis with ambient heat convection could be found in [83]. Let us assume the temperature at x = L is maintained at Mf and the other boundary condition is ∂x T = 0 at x = 0. g0 2 The temperature distribution is then given by T (x) = − 2k [x − L2 ] + Mf , where L is the length of the slab. From Table. 3, let us consider the case L = 25 [mm] and 200 [Hz] for which g0 = P max = 19.12 [mW/mm3 ]. The 21
thermal conductivity of Nickel based alloy is nearly equal to that of Nickel, i.e., k = 91 W/m·K. For the given thermal conductivity, energy generation, and geometry the temperature at the adiabatic surface becomes T (0) = 98◦ C and quadratically decreases to T (L) = Mf = 32◦ C (Ni50 Mn28 Ga22 composition, [84]). The austenitic finish temperature for the same chemical composition is Af = 52◦ C. We are now looking for a range of x for which g0 2 [x − L2 ] + Mf ≥ Af and the solution for the above mentioned T (x) = − 2k material parameters is x ≤20.6 [mm]. This implies that more than 80% of the length will be fully transformed to the austenitic phase and the MFIS will be decreased due to the presence of austenite [83]. However, this is a gross estimation, and one needs to solve a coupled magneto-thermo-mechanical system for more precise information.
5
Conclusion
We found that the time dependent part of the Maxwell equations, which generally omitted in the quasi-static MSMA analysis, is significant for high frequency analysis. The high actuation strain rate generated due to variant reorientation cuts the applied magnetic field and produces motional electromotive force (emf). The total induced emf due to the rate of change of magnetic field and motional emf generates an eddy current. Additional magnetic field due to eddy current adds up with the existing static magnetic field and helps in variant reorientation. We solved numerically an Initial Boundary Value Problem by using the existing MSMA constitutive equations in a staggered way to capture electro-magneto-mechanical coupling responses of different field variables. We found the eddy current and Joule heating are significant. A comparison of eddy current and Joule heating between MSMA and rigid-ferromagnetic/magnetostrictive materials reveals that the differences are significantly high in MSMA. These could be the two essential parameters for shape and material optimization in device level MSMA applications.
Acknowledgments The authors would like to 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 author during his graduate 22
study at Texas A&M University.
APPENDIX A
Thermodynamic driving force
The four critical magnetic fields are: the start of forward reorientation, HsM2 , the end of forward reorientation, HfM2 , the start of reverse reorientation, HsM1 , and the end of reverse reorientation, HfM1 . The reduced form of the thermodynamic force (12) is given by π r = (σ ∗ +
µ0 2 cur H )ε + µ0 ∆M2 H2 + f r . 2 y
The hardening function is chosen as [85] ( A − 2 (1 + ξ n1 − (1 − ξ)n2 ) + B, r f := − C2 (1 + ξ n3 − (1 − ξ)n4 ) + D,
ξ˙ > 0 , ξ˙ < 0 ,
(A-1)
(A-2)
where, n1 , n2 , n3 and n4 are some given exponents. We need to know the parameters A, B, C, D, Y r . From the Kuhn Tucker condition (14) we obtain two conditions at the beginning and two conditions at the finish of the forward reorientation. They are for ξ˙ > 0, for ξ˙ > 0,
π r (σ ∗ , HsM2 ) − Y r = 0, π r (σ ∗ , HfM2 ) − Y r = 0,
at ξ = 0 at ξ = 1
(A-3a) (A-3b)
Similarly, for reverse reorientation we get two more equations, π r (σ ∗ , HsM1 ) + Y r = 0, π r (σ ∗ , HfM1 ) + Y r = 0,
for for
ξ˙ < 0, ξ˙ < 0,
at ξ = 1 at ξ = 0
(A-4a) (A-4b)
The constant stress level is denoted by σ ∗ . The continuity of the hardening function [73] gives us Z 1 Z 1 r f dξ = f r dξ. (A-5) 0
˙ ξ>0
0
˙ ξ
