TP3“ModellierungundHomogenisierungmagneto ... · 2018. 6. 4. · −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
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FOR 1509 Ferroische FunktionsmaterialienMehrskalige 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.
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m3
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hard
hard easyeasy
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
[ ddt
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
ηm×
(m× [δmΨmat − κ0ms(h−∇φ)]
)in B .
Preserving the constraint on the magnetization director in an
m
a1 a2∆ϑ1
∆ϑ2
∆m
∆w = ∆ϑ1a1 + ∆ϑ2a2
S2
TmS2
n
x∈B
φ
ΩB
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]∆ϑ)]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 × 10−5s
dt = 2 × 10−5s
dt = 10−5s
dt = 10−5s
dt = 5 × 10−6s
dt = 5 × 10−6s
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=argstatϕ,φ,I
∫
B
[ ddt
Ψ(F ,H,I) + Φv(I)]dV
with Ψ = U(J) + Ψe(Fnet
) + Ψv(Fnet
,I) + Ψmag(F ,H), and
Φv given by the specific viscoelastic model under consideration.
F F
E(H) FnetFnet E(h)
E1 E1
E2 E2
E3 E3
E′1
E′2
E′3
e′1
e′2
e′3
e1 e1
e2 e2
e3 e3
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.
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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
xx
x x
y
yy
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. Zah, C. Miehe, A Finite Deformation Microsphere
Model for Magneto-Visco-Elastic Response in Magnetorheologi-
cal Elastomers PAMM, 2013.
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