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FOR 1509 Ferroische Funktionsmaterialien Mehrskalige Modellierung und experimentelle Charakterisierung TP3 “Modellierung und Homogenisierung magneto-mechanischen Materialverhaltens auf verschiedenen Skalen” C. Miehe G. Ethiraj University of Stuttgart University of Stuttgart Institute of Appled Mechanics (CE) Institute of Appled Mechanics (CE) Micromechanically Motivated Computational Modeling of Magneto-Mechanically Coupled Materials Phase Field Model for Micro-Magneto-Elasticity Micromagnetics allows for a truly multiscale viewpoint in the modeling of magneto-mechanically coupled materials since macroscopic behavior is determined by the domain structure and evolution of magnetic microstructure on the microscale. -1 -1 -1 m1 m2 m3 0 0 0 1 1 1 -1 -1 -1 m2 m1 m3 0 0 0 1 1 1 -1 -1 m2 m3 -1 m1 0 0 0 1 1 1 hard hard easy easy easy 6 directions 2 directions 8 directions Geometrically Exact Variational Principle The coupled boundary value problem of micro-magneto- elasticity is governed by the rate-type variational principle { ˙ u, ˙ φ, ˙ m} = arg stat ˙ u, ˙ φ, ˙ m B d dt Ψ ′ (u, m, φ) + Φ( ˙ m) dV The evolution of the magnetization director resulting from the variational principle is the Landau-Lifschitz Gilbert equation ˙ m = 1 η m × ( m × [δmΨmat − κ0ms( h −∇ φ)] ) in B . Preserving the constraint on the magnetization director in an m a 1 a 2 Δϑ 1 Δϑ 2 Δm Δw =Δϑ 1 a 1 +Δϑ 2 a 2 S 2 TmS 2 n x∈B φ Ω B u algorithmic setting is a challenge. In order to overcome this, we use the exponential map to update the magnetization {a1, a2, m}⇐ exp[1 × ([a1, a2]Δϑ)]{a1, a2, m} Alternatively, we may include a normalization step as a post- processing operation in order to satisfy the constraint. Numerical Results Starting from a random distribution, we compare the final con- figuration resulting from each method for different time-steps. The results of the projection method approaches those of the geometrically exact method with decreasing time-steps. Geometrically exact method Projection method dt =2 × 10 −5 s dt =2 × 10 −5 s dt = 10 −5 s dt = 10 −5 s dt =5 × 10 −6 s dt =5 × 10 −6 s Micromechanics of Magnetorheological Elastomers The coupled boundary value problem of magneto-visco-elasticity is compactly represented in the rate-type variational principle { ˙ ϕ, ˙ φ, ˙ I} = arg stat ˙ ϕ, ˙ φ, ˙ I B d dt Ψ(F , H, I )+Φ v ( ˙ I) dV with Ψ = U (J )+ ¯ Ψ e ( ¯ F net )+ ¯ Ψ v ( ¯ F net , I )+Ψmag (F , H), and Φ v given by the specific viscoelastic model under consideration. ¯ F ¯ F ¯ E(H) ¯ F net ¯ F net ¯ E(h) ˜ E 1 ˜ E 1 ˜ E 2 ˜ E 2 ˜ E 3 ˜ E 3 ˜ E ′ 1 ˜ E ′ 2 ˜ E ′ 3 ˜ e ′ 1 ˜ e ′ 2 ˜ e ′ 3 ˜ e 1 ˜ e 1 ˜ e 2 ˜ e 2 ˜ e 3 ˜ e 3 Due to the micromechanics of such materials, we are motivated to consider a multiplicative split of the deformation gradient. ¯ F net := ¯ F - case 1: total isochoric deformation, ¯ F ¯ E(H) - case 2: right multiplicative decomposition, ¯ E(h) ¯ F - case 3: left multiplicative decomposition. Numerical Results In dynamic compression tests, the model fits very well with ex- periments. This is shown in force vs. displacement plots below. -0.8 -0.6 -0.4 -0.2 0 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 -0.8 -0.6 -0.4 -0.2 0 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 Force(N) Force(N) Disp.(mm) Disp.(mm) Sim - Sim - Exp + Exp + H=0 H= 0.4 Results of a FEM simulation displays the modeling capability. λ λ λ λ h 0.62 1.52 0.62 1.52 0.27 2.00 0.61 2.10 x x x x x y y y y y z References [1] C. Miehe and G. Ethiraj, A Geometrically Consistent Incremen- tal Variational Formulation for Phase Field Models in Micromag- netics, CMAME, 2012. [2] C. Miehe, B. Kiefer, D. Rosato, An incremental variational for- mulation of dissipative magnetostriction at the macroscopic con- tinuum level IJSS, 2011. [3] G. Ethiraj, D. Z¨ah, C. Miehe, A Finite Deformation Microsphere Model for Magneto-Visco-Elastic Response in Magnetorheologi- cal Elastomers PAMM, 2013.
