A model for magnetoelastic interaction with special reference to - - PowerPoint PPT Presentation

A model for magnetoelastic interaction with special reference to giant magnetostrictive behaviour

Anouar Belahcen1, Reijo Kouhia2, Paavo Rasilo3, and Matti Ristinmaa4

1Aalto University, Electrical Engineering, Espoo, Finland 2 Tampere University of Technology, Civil Engineering, Tampere, Finland 3Tampere University of Technology, Electrical Energy Engineering, Tampere, Finland 4Lund University, Division of Solid Mechanics, Lund, Sweden Glasgow, UK, 11-15 June, 2018

1 Introduction 2 Constitutive eq 3 Galfenol 4 Results 5 Conclusions

Content

1 Introduction 2 Constitutive equations 3 Application to galfenol 4 Some results 5 Concluding remarks and future work

Magnetoelastic interaction – RK ECCM 2018 2/14

1 Introduction 2 Constitutive eq 3 Galfenol 4 Results 5 Conclusions

1 Introduction 2 Constitutive equations 3 Application to galfenol 4 Some results 5 Concluding remarks and future work

Magnetoelastic interaction – RK ECCM 2018 3/14

1 Introduction 2 Constitutive eq 3 Galfenol 4 Results 5 Conclusions

Introduction

Lagrangian approach Formulation based on the theory by Dorfmann, Ogden and Bustamante Magnetic behaviour can depend strongly on deformation

http://www.tdvib.com/wp-content/uploads/2015/09/Galfenol-Stress-Annealed Tested-Under-Compression-media.pdf Magnetoelastic interaction – RK ECCM 2018 4/14

1 Introduction 2 Constitutive eq 3 Galfenol 4 Results 5 Conclusions

1 Introduction 2 Constitutive equations 3 Application to galfenol 4 Some results 5 Concluding remarks and future work

Magnetoelastic interaction – RK ECCM 2018 5/14

1 Introduction 2 Constitutive eq 3 Galfenol 4 Results 5 Conclusions

Constitutive equations

Lagrangian forms of the primary magnetic fields are H L ≡ F T H , and BL ≡ JF −1B Formulation based on the complementary form Ω∗(F, H L) of the total energy function Ω(F, BL) and thus τ = J−1F ∂Ω∗ ∂F , B = −J−1F ∂Ω∗ ∂H L Ω∗(F, H L) = Ω(F, BL) − H L·BL The total energy function Ω is related to the Helmholtz free energy per unit mass ψ as Ω ≡ ρ0Φ+ 1

2µ−1 0 JB · B,

where Φ(F, BL) ≡ ψ(F, J−1FBL) ρ0, µ0 are the density in the ref. configuration and the magnetic permeability in vacuum, respectively.

Magnetoelastic interaction – RK ECCM 2018 6/14

1 Introduction 2 Constitutive eq 3 Galfenol 4 Results 5 Conclusions

Constitutive equations

Integrity basis I1 = tr C, I2 = 1

2[(tr C)2 − tr C 2], I3 = det C,

I4 = H L·H L, I5 = H L·CH L, I6 = H L·C 2H L where C = F TF. Coleman-Noll procedure results in constitutive equations τ =J−1[2bΩ∗

1 + 2(I1b − b2)Ω∗ 2 + 2I3I Ω∗ 3 + 2bH ⊗ bH Ω∗ 5

+ 2(bH ⊗ b2H + b2H ⊗ bH )Ω∗

6],

B = − J−1(2bH Ω∗

4 + 2b2H Ω∗ 5 + 2b3H Ω∗ 6),

b = FF T, Ω∗

i = ∂Ω∗/∂Ii, and ⊗ is the standard tensor product.

Magnetoelastic interaction – RK ECCM 2018 7/14

1 Introduction 2 Constitutive eq 3 Galfenol 4 Results 5 Conclusions

Complementary energy function

Elastic part (J = det F = √I3) Ω∗e = 1 2K 1 2(J2 − 1) − ln J

• + 1

2G(tr ¯ C − 3), ¯ C = J−2/3C coupled magneto-elastic part Ω∗m =Ω∗m(I1, I3, I4, I5, I6) = − 1

2µ0I4 − g(4)(I1, I3, h4) − g(5)(I1, I3, h5) − g(6)(I1, I3, h6)

hi = √Ii, i = 4, 5, 6.

Magnetoelastic interaction – RK ECCM 2018 8/14

1 Introduction 2 Constitutive eq 3 Galfenol 4 Results 5 Conclusions

1 Introduction 2 Constitutive equations 3 Application to galfenol 4 Some results 5 Concluding remarks and future work

Magnetoelastic interaction – RK ECCM 2018 9/14

1 Introduction 2 Constitutive eq 3 Galfenol 4 Results 5 Conclusions

Application to galfenol

Functional form of the g(i)-functions g(4)(I1, I3, I4) = µ0α4 f ′(h4; Γ) ln [cosh (f(h4; Γ))] , where Γ = Γ(I1, I3) = I1I−1/3

3

− 3 f(h4; Γ) = ξ(4)

0 (Γ)h4/h0 + n

• i=1

ξ(4)

i

(Γ)h4/h0 − η(4)

i

(Γ), and • denotes the Macaulay brackets.

Magnetoelastic interaction – RK ECCM 2018 10/14

1 Introduction 2 Constitutive eq 3 Galfenol 4 Results 5 Conclusions

1 Introduction 2 Constitutive equations 3 Application to galfenol 4 Some results 5 Concluding remarks and future work

Magnetoelastic interaction – RK ECCM 2018 11/14

1 Introduction 2 Constitutive eq 3 Galfenol 4 Results 5 Conclusions

Some results

Some parameters: α4 = Msat = 1370 kA/m, χm = 400, ⇒ h0 = Msat/χm It also appears that when n = 1: ξ(4) ∼ exp(−Γ), η(4)

1

∼ CΓ, ξ(4)

1

≈ 0.65 (constant)

Model with solid lines and experimental data with dashed lines Magnetoelastic interaction – RK ECCM 2018 12/14

1 Introduction 2 Constitutive eq 3 Galfenol 4 Results 5 Conclusions

1 Introduction 2 Constitutive equations 3 Application to galfenol 4 Some results 5 Concluding remarks and future work

Magnetoelastic interaction – RK ECCM 2018 13/14

1 Introduction 2 Constitutive eq 3 Galfenol 4 Results 5 Conclusions

Concluding remarks and future work

Investigation of the deformation dependency of magnetic properties Improvement of the model Parameter estimation Micromechanical models.

Glasgow Window by Barbara Rae, 1986

Thank you for your attention!

Magnetoelastic interaction – RK ECCM 2018 14/14