Coupled Magnetomechanical Modeling of …...Manik Kumar, Sajan Wahi, Dr. Sushma SantapuriCoupled...

Outline Introduction Literature Review Objectives Mathematical Modeling Transducer Design Actuator Results References

Coupled Magnetomechanical Modeling ofMagnetostrictive Materials with Application to

Transducer Design

Manik Kumar, Sajan Wahi, Dr. Sushma Santapuri

Department of Applied MechanicsIndian Institute of Technology Delhi

Manik Kumar, Sajan Wahi, Dr. Sushma SantapuriCoupled Magnetomechanical Modeling of Magnetostrictive Materials 1 / 31


Outline

1 Introduction

2 Literature Review

3 Objectives

4 Mathematical Modeling

5 Applications of Transducer Design

6 Rod Actuator Characteristics



Introduction

Figure 1: Phenomenon of magnetostrictive materials

Magnetostrictive Materials are a class of smart materials thatexhibits coupling between magnetic and mechanical domains.

They undergo change in shape when subjected to externalmagnetic field.



Why their is need for Rare Earth Materials

� Ferromagnets.

Low magnetostriction ∼100 ppm.

� Rare earth materials(Terbium, Dysprosium).

Low curie point at room temperature.

� Rare earth alloysTerfenol-D (TbxDy1−xFe2)

Maximum magnetostriction ∼1250 ppm.Brittle in nature.

Galfenol (FexGa1−x)

Maximum magnetostriction ∼250 ppm.High tensile strength.



Potential Applications of Magnetostrictive Material

Figure 2: Applications of magnetostrictive material [Source: Olabi and Grunwald(2008)]

and many more applications . . .



Outline

1 Introduction

2 Literature Review

3 Objectives






Magnetostrictive bending actuator

1 Mudivarthi et al. (2008)

Developed 3D Bidirectional Magnetoelastic Model (BCMEM).Computationally expensive model.

2 Graham et al. (2009)

Developed 2D Bidirectional Magnetoelastic Model (BCMEM).Compuatationally efficient as compared to Mudivarthi et al.(2008).

3 Cao et al. (2015)

Incorporates nonlinear magnetomechanical model.Computationally expensive model.



Outline

1 Introduction

2 Literature Review

3 Objectives






Objectives

a Computationally efficient modeling of magnetostrictive material.

b Design of magnetostrictive actuator.

Computational framework of 2D magnetostrictive rod actuator.

Finite element framework for 1D unimoprh bending actuator.



Outline

1 Introduction

2 Literature Review

3 Objectives






Energies in Magnetostrictive material

ψ(ε,m) = ψanisotropy + ψmagnetoelastic + ψzeeman

where m, ε are the magnetic moment and elastic strain.

Figure 3: Magnetic domains

ψ(ε,mk, ξk) = Σkξk(ψk

anisotropy + ψkelastic + ψk

zeeman)

where ξk is domain volume fraction given byexp(−ψk

ω)

r∑k=1

exp(−ψkω

).



Locally Linearized Constitutive Model

Constrained Locally Linearized Constitutive Model:In nonlinear model, a single energy expression is used for anyparticle orientation whereas in this case a local energy expression isanalytically calculated about each easy axis ck = [c1, c2, c3],

ψcons(m1,m2,m3, L) =ψanisotropy + ψzeeman + ψmagnetoelastic+

L(m21 +m2

2 +m23 − 1)

Using Taylor series expansion upto second order differential.

∂ψkcons∂mk

i

=∂ψkcons∂mk

i

∣∣∣ck

+∂2ψkcons∂mk

imkj

∣∣∣ck(mk

j − ckj ) = 0

[Kk][mk − ck] = [Bk]



Locally Linearized Constitutive Model (cont.)



Effect of prestress (σ) on nature of B-H curves

Figure 4: Comparison of magnetic induction (B) vs magnetic field (H) between thenonlinear and locally linearized model along [100] at various prestress values.



Effect of prestress (σ) on nature of λ-H curves

Figure 5: Comparison of magnetostriction (λ) vs magnetic field (H) between thenonlinear and locally linearized model along [100] at various prestress values.



Effect of magnetic field (H) on nature of B-σ curves

Figure 6: Magnetic Induction (B) vs stress (σ) along [1 0 0] at variousprestress values



Effect of magnetic field (H) on nature of ε-σ curves

Figure 7: Strain (ε) vs stress (σ) along [1 0 0] at various prestress values



Effect of stress

Figure 8: Effect of stress (σ) on magnetization and magnetostriction



3D Magnetomechanical Governing Equations

General constitutive modelling of magnetostrictive materialsinvolves coupling between the magnetic and mechanical BVPs.

Navier’s equation in weak form is given by∫V

[T · δ S + ρ

∂2u

∂t2· δu + c

∂u

∂t· δu

]dV =∫

∂Vt · δu d∂V +

∫V

fB · δu dV

Also, the magnetostatic governing equation in weak form valid inthe magnetic material medium and the surrounding free space isgiven by ∫

Egradδφ · B dV = 0 (1)



Outline

1 Introduction

2 Literature Review

3 Objectives






Applications of Transducer Design

� Applications of transducer

1 2D axisymmetric rod actuator.

2 Composite unimorph bending actuator.



2D Axisymmetric Transducer Design

Schematic view of Galfenol rod actuator



Outline

1 Introduction

2 Literature Review

3 Objectives






Rod Actuator Characteristics

A. Magnetic Flux Distribution

Figure 9: 3D of the norm of magnetic flux density



Rod Actuator Characteristics (cont.)

B. Strain Distribution

Figure 10: Axial strain distribution at various prestress values in Galfenolrod (a) 0 MPa, (b) 15 MPa, (c) 30 MPa, (d) 45 MPa, (e) 60 MPa




1. Magnetostriction (λ)-Current Density (J0)

-9000 -6000 -3000 0 3000 6000 90000

50

100

150

200

250

Mag

neto

stric

tion

(ppm

)

Current Density (kA/m2)

-J

Figure 11: Magnetostriction (λ) vs current density (J0) for anhystereticmodel at various pre-stress values.




2. Magnetic Induction (B)-Current Density (J0)

Figure 12: Magnetic induction (B) vs current density (J0) for anhystereticmodel at various pre-stress values.



Composite Unimorph Transducer

Figure 13: Cantilevered composite magnetostrictive unimorph



Composite Unimorph Transducer (cont.)

Figure 14: Beam cross-section

Weak form expression of 1D Euler-Bernoulli Beam∫ L

0(duxdx

N +du2

y

dx2M)dx = 0



Composite Unimorph Transducer (cont.)

{NM

}=

[EeffectiveAeffective b(Eal

t222 − Eg

t212 )

b(Ealt222 − Eg

t212 ) EeffectiveIeffective

]{ε0

κ

}+

{−EgAgλEgb

t212 λ

}

Eeffective =(EgAg+EalAal)

(Ag+Aal),

Ieffective = Ig + Ial +Ag(t12 )

2 +Aal(t22 )

2, and Aeffective = Ag +Aal



Bending Actuator Characteristics

Figure 15: Tip displacement of cantilevered Galfenol-Aluminium unimorph



Bending Actuator Characteristics (cont.)

Figure 16: Free strain (εxx) on Galfenol surface




Figure 17: Free strain (εxx) on aluminum surface




Thickness Ratio (tr)

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

No

rma

lize

dT

ipD

isp

lace

men

t

100

110

120

130

140

150

160

170

Our Model

Datta Model

Figure 18: Comparison of normalized tip displacement as predicted by ourmodel and Datta et al. (2008)



References

Armstrong, W. D. (1997). Burst magnetostriction in tb 0.3 dy 0.7 fe 1.9.Journal of applied physics, 81(8):3548–3554.

Atulasimha, J. and Flatau, A. B. (2011). A review of magnetostrictiveiron–gallium alloys. Smart Materials and Structures, 20(4):043001.

Cao, Q., Chen, D., Lu, Q., Tang, G., Yan, J., Zhu, Z., Xu, B., Zhao, R.,and Zhang, X. (2015). Modeling and experiments of a laminatedmagnetostrictive cantilever beam. Advances in MechanicalEngineering, 7(4):1687814015573761.

Datta, S., Atulasimha, J., Mudivarthi, C., and Flatau, A. (2008). Themodeling of magnetomechanical sensors in laminated structures.Smart Materials and Structures, 17(2):025010.

Evans, P. and Dapino, M. (2010). Efficient magnetic hysteresis model forfield and stress application in magnetostrictive galfenol. Journal ofapplied physics, 107(6):063906.



References (cont.)

Graham, F., Mudivarthi, C., Datta, S., and Flatau, A. (2009). Modelingof a galfenol transducer using the bidirectionally coupledmagnetoelastic model. Smart Materials and Structures, 18(10):104013.

Mudivarthi, C., Datta, S., Atulasimha, J., and Flatau, A. (2008). Abidirectionally coupled magnetoelastic model and its validation using agalfenol unimorph sensor. Smart Materials and Structures,17(3):035005.

Olabi, A.-G. and Grunwald, A. (2008). Design and application ofmagnetostrictive materials. Materials & Design, 29(2):469–483.



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