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 Belahcen 1 , Reijo Kouhia 2 , Paavo Rasilo 3 , and Matti Ristinmaa 4 1 Aalto University, Electrical Engineering, Espoo, Finland 2 Tampere University of Technology, Civil Engineering, Tampere, Finland 3 Tampere University of Technology, Electrical Energy Engineering, Tampere, Finland 4 Lund University, Division of Solid Mechanics, Lund, Sweden Glasgow, UK, 11-15 June, 2018

1 Introduction 2 Constitutive eq Content 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 2/14

1 Introduction 1 Introduction 2 Constitutive eq 3 Galfenol 4 Results 5 Conclusions 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 Introduction 3 Galfenol 4 Results 5 Conclusions 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 1 Introduction 2 Constitutive eq 3 Galfenol 4 Results 5 Conclusions 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 Constitutive equations 3 Galfenol 4 Results 5 Conclusions Lagrangian forms of the primary magnetic fields are H L ≡ F T H , B L ≡ J F − 1 B and Formulation based on the complementary form Ω ∗ ( F , H L ) of the total energy function Ω( F , B L ) and thus τ = J − 1 F ∂ Ω ∗ B = − J − 1 F ∂ Ω ∗ ∂ F , ∂ H L Ω ∗ ( F , H L ) = Ω( F , B L ) − H L · B L The total energy function Ω is related to the Helmholtz free energy per unit mass ψ as Ω ≡ ρ 0 Φ+ 1 2 µ − 1 Φ( F , B L ) ≡ ψ ( F , J − 1 FB L ) 0 J B · B , where ρ 0 , µ 0 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 Constitutive equations 3 Galfenol 4 Results 5 Conclusions Integrity basis 2 [(tr C ) 2 − tr C 2 ] , I 3 = det C , I 1 = tr C , I 2 = 1 I 4 = H L · H L , I 5 = H L · CH L , I 6 = H L · C 2 H L where C = F T F . Coleman-Noll procedure results in constitutive equations τ = J − 1 [2 b Ω ∗ 1 + 2( I 1 b − b 2 )Ω ∗ 2 + 2 I 3 I Ω ∗ 3 + 2 bH ⊗ bH Ω ∗ 5 + 2( bH ⊗ b 2 H + b 2 H ⊗ bH )Ω ∗ 6 ] , B = − J − 1 (2 bH Ω ∗ 4 + 2 b 2 H Ω ∗ 5 + 2 b 3 H Ω ∗ 6 ) , b = FF T , Ω ∗ i = ∂ Ω ∗ /∂I i , and ⊗ is the standard tensor product. Magnetoelastic interaction – RK ECCM 2018 7/14

1 Introduction 2 Constitutive eq Complementary energy function 3 Galfenol 4 Results 5 Conclusions Elastic part ( J = det F = √ I 3 ) Ω ∗ e = 1 � 1 � + 1 2( J 2 − 1) − ln J 2 G (tr ¯ C = J − 2 / 3 C ¯ 2 K C − 3) , coupled magneto-elastic part Ω ∗ m =Ω ∗ m ( I 1 , I 3 , I 4 , I 5 , I 6 ) = − 1 2 µ 0 I 4 − g (4) ( I 1 , I 3 , h 4 ) − g (5) ( I 1 , I 3 , h 5 ) − g (6) ( I 1 , I 3 , h 6 ) h i = √ I i , i = 4 , 5 , 6 . Magnetoelastic interaction – RK ECCM 2018 8/14

1 Introduction 1 Introduction 2 Constitutive eq 3 Galfenol 4 Results 5 Conclusions 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 Application to galfenol 3 Galfenol 4 Results 5 Conclusions Functional form of the g ( i ) -functions µ 0 α 4 g (4) ( I 1 , I 3 , I 4 ) = f ′ ( h 4 ; Γ) ln [cosh ( f ( h 4 ; Γ))] , where Γ = Γ( I 1 , I 3 ) = I 1 I − 1 / 3 − 3 3 n f ( h 4 ; Γ) = ξ (4) � ξ (4) (Γ) h 4 /h 0 − η (4) � 0 (Γ) h 4 /h 0 + (Γ) � , i i i =1 and �•� denotes the Macaulay brackets. Magnetoelastic interaction – RK ECCM 2018 10/14

1 Introduction 1 Introduction 2 Constitutive eq 3 Galfenol 4 Results 5 Conclusions 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 Some results 3 Galfenol Some parameters: 4 Results 5 Conclusions α 4 = M sat = 1370 kA/m , χ m = 400 , h 0 = M sat /χ m ⇒ It also appears that when n = 1 : ξ (4) η (4) ξ (4) ∼ exp( − Γ) , ∼ C Γ , ≈ 0 . 65 ( constant ) 0 1 1 Model with solid lines and experimental data with dashed lines Magnetoelastic interaction – RK ECCM 2018 12/14

1 Introduction 1 Introduction 2 Constitutive eq 3 Galfenol 4 Results 5 Conclusions 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 Concluding remarks and future work 3 Galfenol 4 Results 5 Conclusions 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