Page 1
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
Page 2
Outline Introduction Literature Review Objectives Mathematical Modeling Transducer Design Actuator Results References
Outline
1 Introduction
2 Literature Review
3 Objectives
4 Mathematical Modeling
5 Applications of Transducer Design
6 Rod Actuator Characteristics
Manik Kumar, Sajan Wahi, Dr. Sushma SantapuriCoupled Magnetomechanical Modeling of Magnetostrictive Materials 2 / 31
Page 3
Outline Introduction Literature Review Objectives Mathematical Modeling Transducer Design Actuator Results References
Outline
1 Introduction
2 Literature Review
3 Objectives
4 Mathematical Modeling
5 Applications of Transducer Design
6 Rod Actuator Characteristics
Manik Kumar, Sajan Wahi, Dr. Sushma SantapuriCoupled Magnetomechanical Modeling of Magnetostrictive Materials 2 / 31
Page 4
Outline Introduction Literature Review Objectives Mathematical Modeling Transducer Design Actuator Results References
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.
Manik Kumar, Sajan Wahi, Dr. Sushma SantapuriCoupled Magnetomechanical Modeling of Magnetostrictive Materials 3 / 31
Page 5
Outline Introduction Literature Review Objectives Mathematical Modeling Transducer Design Actuator Results References
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.
Manik Kumar, Sajan Wahi, Dr. Sushma SantapuriCoupled Magnetomechanical Modeling of Magnetostrictive Materials 4 / 31
Page 6
Outline Introduction Literature Review Objectives Mathematical Modeling Transducer Design Actuator Results References
Potential Applications of Magnetostrictive Material
Figure 2: Applications of magnetostrictive material [Source: Olabi and Grunwald(2008)]
and many more applications . . .
Manik Kumar, Sajan Wahi, Dr. Sushma SantapuriCoupled Magnetomechanical Modeling of Magnetostrictive Materials 5 / 31
Page 7
Outline Introduction Literature Review Objectives Mathematical Modeling Transducer Design Actuator Results References
Outline
1 Introduction
2 Literature Review
3 Objectives
4 Mathematical Modeling
5 Applications of Transducer Design
6 Rod Actuator Characteristics
Manik Kumar, Sajan Wahi, Dr. Sushma SantapuriCoupled Magnetomechanical Modeling of Magnetostrictive Materials 5 / 31
Page 8
Outline Introduction Literature Review Objectives Mathematical Modeling Transducer Design Actuator Results References
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.
Manik Kumar, Sajan Wahi, Dr. Sushma SantapuriCoupled Magnetomechanical Modeling of Magnetostrictive Materials 6 / 31
Page 9
Outline Introduction Literature Review Objectives Mathematical Modeling Transducer Design Actuator Results References
Outline
1 Introduction
2 Literature Review
3 Objectives
4 Mathematical Modeling
5 Applications of Transducer Design
6 Rod Actuator Characteristics
Manik Kumar, Sajan Wahi, Dr. Sushma SantapuriCoupled Magnetomechanical Modeling of Magnetostrictive Materials 6 / 31
Page 10
Outline Introduction Literature Review Objectives Mathematical Modeling Transducer Design Actuator Results References
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.
Manik Kumar, Sajan Wahi, Dr. Sushma SantapuriCoupled Magnetomechanical Modeling of Magnetostrictive Materials 7 / 31
Page 11
Outline Introduction Literature Review Objectives Mathematical Modeling Transducer Design Actuator Results References
Outline
1 Introduction
2 Literature Review
3 Objectives
4 Mathematical Modeling
5 Applications of Transducer Design
6 Rod Actuator Characteristics
Manik Kumar, Sajan Wahi, Dr. Sushma SantapuriCoupled Magnetomechanical Modeling of Magnetostrictive Materials 7 / 31
Page 12
Outline Introduction Literature Review Objectives Mathematical Modeling Transducer Design Actuator Results References
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
).
Manik Kumar, Sajan Wahi, Dr. Sushma SantapuriCoupled Magnetomechanical Modeling of Magnetostrictive Materials 8 / 31
Page 13
Outline Introduction Literature Review Objectives Mathematical Modeling Transducer Design Actuator Results References
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]
Manik Kumar, Sajan Wahi, Dr. Sushma SantapuriCoupled Magnetomechanical Modeling of Magnetostrictive Materials 9 / 31
Page 14
Outline Introduction Literature Review Objectives Mathematical Modeling Transducer Design Actuator Results References
Locally Linearized Constitutive Model (cont.)
Manik Kumar, Sajan Wahi, Dr. Sushma SantapuriCoupled Magnetomechanical Modeling of Magnetostrictive Materials 10 / 31
Page 15
Outline Introduction Literature Review Objectives Mathematical Modeling Transducer Design Actuator Results References
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.
Manik Kumar, Sajan Wahi, Dr. Sushma SantapuriCoupled Magnetomechanical Modeling of Magnetostrictive Materials 11 / 31
Page 16
Outline Introduction Literature Review Objectives Mathematical Modeling Transducer Design Actuator Results References
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.
Manik Kumar, Sajan Wahi, Dr. Sushma SantapuriCoupled Magnetomechanical Modeling of Magnetostrictive Materials 12 / 31
Page 17
Outline Introduction Literature Review Objectives Mathematical Modeling Transducer Design Actuator Results References
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
Manik Kumar, Sajan Wahi, Dr. Sushma SantapuriCoupled Magnetomechanical Modeling of Magnetostrictive Materials 13 / 31
Page 18
Outline Introduction Literature Review Objectives Mathematical Modeling Transducer Design Actuator Results References
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
Manik Kumar, Sajan Wahi, Dr. Sushma SantapuriCoupled Magnetomechanical Modeling of Magnetostrictive Materials 14 / 31
Page 19
Outline Introduction Literature Review Objectives Mathematical Modeling Transducer Design Actuator Results References
Effect of stress
Figure 8: Effect of stress (σ) on magnetization and magnetostriction
Manik Kumar, Sajan Wahi, Dr. Sushma SantapuriCoupled Magnetomechanical Modeling of Magnetostrictive Materials 15 / 31
Page 20
Outline Introduction Literature Review Objectives Mathematical Modeling Transducer Design Actuator Results References
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)
Manik Kumar, Sajan Wahi, Dr. Sushma SantapuriCoupled Magnetomechanical Modeling of Magnetostrictive Materials 16 / 31
Page 21
Outline Introduction Literature Review Objectives Mathematical Modeling Transducer Design Actuator Results References
Outline
1 Introduction
2 Literature Review
3 Objectives
4 Mathematical Modeling
5 Applications of Transducer Design
6 Rod Actuator Characteristics
Manik Kumar, Sajan Wahi, Dr. Sushma SantapuriCoupled Magnetomechanical Modeling of Magnetostrictive Materials 16 / 31
Page 22
Outline Introduction Literature Review Objectives Mathematical Modeling Transducer Design Actuator Results References
Applications of Transducer Design
� Applications of transducer
1 2D axisymmetric rod actuator.
2 Composite unimorph bending actuator.
Manik Kumar, Sajan Wahi, Dr. Sushma SantapuriCoupled Magnetomechanical Modeling of Magnetostrictive Materials 17 / 31
Page 23
Outline Introduction Literature Review Objectives Mathematical Modeling Transducer Design Actuator Results References
2D Axisymmetric Transducer Design
Schematic view of Galfenol rod actuator
Manik Kumar, Sajan Wahi, Dr. Sushma SantapuriCoupled Magnetomechanical Modeling of Magnetostrictive Materials 18 / 31
Page 24
Outline Introduction Literature Review Objectives Mathematical Modeling Transducer Design Actuator Results References
Outline
1 Introduction
2 Literature Review
3 Objectives
4 Mathematical Modeling
5 Applications of Transducer Design
6 Rod Actuator Characteristics
Manik Kumar, Sajan Wahi, Dr. Sushma SantapuriCoupled Magnetomechanical Modeling of Magnetostrictive Materials 18 / 31
Page 25
Outline Introduction Literature Review Objectives Mathematical Modeling Transducer Design Actuator Results References
Rod Actuator Characteristics
A. Magnetic Flux Distribution
Figure 9: 3D of the norm of magnetic flux density
Manik Kumar, Sajan Wahi, Dr. Sushma SantapuriCoupled Magnetomechanical Modeling of Magnetostrictive Materials 19 / 31
Page 26
Outline Introduction Literature Review Objectives Mathematical Modeling Transducer Design Actuator Results References
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
Manik Kumar, Sajan Wahi, Dr. Sushma SantapuriCoupled Magnetomechanical Modeling of Magnetostrictive Materials 20 / 31
Page 27
Outline Introduction Literature Review Objectives Mathematical Modeling Transducer Design Actuator Results References
Rod Actuator Characteristics (cont.)
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.
Manik Kumar, Sajan Wahi, Dr. Sushma SantapuriCoupled Magnetomechanical Modeling of Magnetostrictive Materials 21 / 31
Page 28
Outline Introduction Literature Review Objectives Mathematical Modeling Transducer Design Actuator Results References
Rod Actuator Characteristics (cont.)
2. Magnetic Induction (B)-Current Density (J0)
Figure 12: Magnetic induction (B) vs current density (J0) for anhystereticmodel at various pre-stress values.
Manik Kumar, Sajan Wahi, Dr. Sushma SantapuriCoupled Magnetomechanical Modeling of Magnetostrictive Materials 22 / 31
Page 29
Outline Introduction Literature Review Objectives Mathematical Modeling Transducer Design Actuator Results References
Composite Unimorph Transducer
Figure 13: Cantilevered composite magnetostrictive unimorph
Manik Kumar, Sajan Wahi, Dr. Sushma SantapuriCoupled Magnetomechanical Modeling of Magnetostrictive Materials 23 / 31
Page 30
Outline Introduction Literature Review Objectives Mathematical Modeling Transducer Design Actuator Results References
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
Manik Kumar, Sajan Wahi, Dr. Sushma SantapuriCoupled Magnetomechanical Modeling of Magnetostrictive Materials 24 / 31
Page 31
Outline Introduction Literature Review Objectives Mathematical Modeling Transducer Design Actuator Results References
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
Manik Kumar, Sajan Wahi, Dr. Sushma SantapuriCoupled Magnetomechanical Modeling of Magnetostrictive Materials 25 / 31
Page 32
Outline Introduction Literature Review Objectives Mathematical Modeling Transducer Design Actuator Results References
Bending Actuator Characteristics
Figure 15: Tip displacement of cantilevered Galfenol-Aluminium unimorph
Manik Kumar, Sajan Wahi, Dr. Sushma SantapuriCoupled Magnetomechanical Modeling of Magnetostrictive Materials 26 / 31
Page 33
Outline Introduction Literature Review Objectives Mathematical Modeling Transducer Design Actuator Results References
Bending Actuator Characteristics (cont.)
Figure 16: Free strain (εxx) on Galfenol surface
Manik Kumar, Sajan Wahi, Dr. Sushma SantapuriCoupled Magnetomechanical Modeling of Magnetostrictive Materials 27 / 31
Page 34
Outline Introduction Literature Review Objectives Mathematical Modeling Transducer Design Actuator Results References
Bending Actuator Characteristics (cont.)
Figure 17: Free strain (εxx) on aluminum surface
Manik Kumar, Sajan Wahi, Dr. Sushma SantapuriCoupled Magnetomechanical Modeling of Magnetostrictive Materials 28 / 31
Page 35
Outline Introduction Literature Review Objectives Mathematical Modeling Transducer Design Actuator Results References
Bending Actuator Characteristics (cont.)
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)
Manik Kumar, Sajan Wahi, Dr. Sushma SantapuriCoupled Magnetomechanical Modeling of Magnetostrictive Materials 29 / 31
Page 36
Outline Introduction Literature Review Objectives Mathematical Modeling Transducer Design Actuator Results References
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.
Manik Kumar, Sajan Wahi, Dr. Sushma SantapuriCoupled Magnetomechanical Modeling of Magnetostrictive Materials 29 / 31
Page 37
Outline Introduction Literature Review Objectives Mathematical Modeling Transducer Design Actuator Results References
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.
Manik Kumar, Sajan Wahi, Dr. Sushma SantapuriCoupled Magnetomechanical Modeling of Magnetostrictive Materials 30 / 31
Page 38
Outline Introduction Literature Review Objectives Mathematical Modeling Transducer Design Actuator Results References
THANK YOU FOR YOUR PATIENCEQUESTIONS ?
Manik Kumar, Sajan Wahi, Dr. Sushma SantapuriCoupled Magnetomechanical Modeling of Magnetostrictive Materials 31 / 31