@let@token modelling of phase transformations in ...prusv/ncmm/... · shape-memory materials(smm):...
TRANSCRIPT
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Phase transformations in NiMnGaThe model and its analysis
Some other phenomena to be involved
Modelling of Phase Transformations inmagnetostrictive materials like NiMnGa
Tomas Roubıcek
Charles University, Prague&
Academy of Sciences of the Czech Rep.&
University of West Bohemia
reflecting collaboration with
Giuseppe Tomassettiand
M.Arndt, M.Griebel, V.Novak, P.Plechac, P.Podio-Guidugli,
K.R.Rajagopal, P.Sittner, C.Zanini and others.
Tomas Roubıcek (Workshop, MFF, Prague, March 31, 2012) Phase transformations in NiMnGa
![Page 2: @let@token Modelling of Phase Transformations in ...prusv/ncmm/... · Shape-memory materials(SMM): alloys (=SMAs) or intermetalics. The mechanism behindshape-memory e ect(=SME): higher](https://reader035.vdocument.in/reader035/viewer/2022081617/602db37bf938dd296d06a0b4/html5/thumbnails/2.jpg)
Phase transformations in NiMnGaThe model and its analysis
Some other phenomena to be involved
Content of the talk:
1 Phase transformations in NiMnGaMartensitic/austenitic transformationFerro/para-magnetic transformationCoupling of transformations: magnetostriction
2 The model and its analysisPartly linearized ansatzAnalysis: semi-implicit discretisation, a-priori estimatesAnalysis: convergence
3 Some other phenomena to be involvedGeneral nonlinear ansatzPinning effects
Tomas Roubıcek (Workshop, MFF, Prague, March 31, 2012) Phase transformations in NiMnGa
![Page 3: @let@token Modelling of Phase Transformations in ...prusv/ncmm/... · Shape-memory materials(SMM): alloys (=SMAs) or intermetalics. The mechanism behindshape-memory e ect(=SME): higher](https://reader035.vdocument.in/reader035/viewer/2022081617/602db37bf938dd296d06a0b4/html5/thumbnails/3.jpg)
Phase transformations in NiMnGaThe model and its analysis
Some other phenomena to be involved
Martensitic/austenitic transformationFerro/para-magnetic transformationCoupling of transformations: magnetostriction
Shape-memory materials (SMM): alloys (=SMAs) or intermetalics.The mechanism behind shape-memory effect (=SME): higher temperatures:
atoms tend to form a latice with high symmetry (mostly cubic):austenite phase, higher heat capacity
Tomas Roubıcek (Workshop, MFF, Prague, March 31, 2012) Phase transformations in NiMnGa
![Page 4: @let@token Modelling of Phase Transformations in ...prusv/ncmm/... · Shape-memory materials(SMM): alloys (=SMAs) or intermetalics. The mechanism behindshape-memory e ect(=SME): higher](https://reader035.vdocument.in/reader035/viewer/2022081617/602db37bf938dd296d06a0b4/html5/thumbnails/4.jpg)
Phase transformations in NiMnGaThe model and its analysis
Some other phenomena to be involved
Martensitic/austenitic transformationFerro/para-magnetic transformationCoupling of transformations: magnetostriction
Shape-memory materials (SMM): alloys (=SMAs) or intermetalics.The mechanism behind shape-memory effect (=SME): higher temperatures:
atoms tend to form a latice with high symmetry (mostly cubic):austenite phase, higher heat capacitya lower-symmetrical latice: martensite phase, lower heat capacity.
Tomas Roubıcek (Workshop, MFF, Prague, March 31, 2012) Phase transformations in NiMnGa
![Page 5: @let@token Modelling of Phase Transformations in ...prusv/ncmm/... · Shape-memory materials(SMM): alloys (=SMAs) or intermetalics. The mechanism behindshape-memory e ect(=SME): higher](https://reader035.vdocument.in/reader035/viewer/2022081617/602db37bf938dd296d06a0b4/html5/thumbnails/5.jpg)
Phase transformations in NiMnGaThe model and its analysis
Some other phenomena to be involved
Martensitic/austenitic transformationFerro/para-magnetic transformationCoupling of transformations: magnetostriction
Shape-memory materials (SMM): alloys (=SMAs) or intermetalics.The mechanism behind shape-memory effect (=SME): higher temperatures:
atoms tend to form a latice with high symmetry (mostly cubic):austenite phase, higher heat capacitya lower-symmetrical latice: martensite phase, lower heat capacity.the lower-symmetrical latice occurs in several variants;
Tomas Roubıcek (Workshop, MFF, Prague, March 31, 2012) Phase transformations in NiMnGa
![Page 6: @let@token Modelling of Phase Transformations in ...prusv/ncmm/... · Shape-memory materials(SMM): alloys (=SMAs) or intermetalics. The mechanism behindshape-memory e ect(=SME): higher](https://reader035.vdocument.in/reader035/viewer/2022081617/602db37bf938dd296d06a0b4/html5/thumbnails/6.jpg)
Phase transformations in NiMnGaThe model and its analysis
Some other phenomena to be involved
Martensitic/austenitic transformationFerro/para-magnetic transformationCoupling of transformations: magnetostriction
Shape-memory materials (SMM): alloys (=SMAs) or intermetalics.The mechanism behind shape-memory effect (=SME): higher temperatures:
atoms tend to form a latice with high symmetry (mostly cubic):austenite phase, higher heat capacitya lower-symmetrical latice: martensite phase, lower heat capacity.the lower-symmetrical latice occurs in several variants;
Tomas Roubıcek (Workshop, MFF, Prague, March 31, 2012) Phase transformations in NiMnGa
![Page 7: @let@token Modelling of Phase Transformations in ...prusv/ncmm/... · Shape-memory materials(SMM): alloys (=SMAs) or intermetalics. The mechanism behindshape-memory e ect(=SME): higher](https://reader035.vdocument.in/reader035/viewer/2022081617/602db37bf938dd296d06a0b4/html5/thumbnails/7.jpg)
Phase transformations in NiMnGaThe model and its analysis
Some other phenomena to be involved
Martensitic/austenitic transformationFerro/para-magnetic transformationCoupling of transformations: magnetostriction
Shape-memory materials (SMM): alloys (=SMAs) or intermetalics.The mechanism behind shape-memory effect (=SME): higher temperatures:
atoms tend to form a latice with high symmetry (mostly cubic):austenite phase, higher heat capacitya lower-symmetrical latice: martensite phase, lower heat capacity.the lower-symmetrical latice occurs in several variants;each of them can be rotated:
Tomas Roubıcek (Workshop, MFF, Prague, March 31, 2012) Phase transformations in NiMnGa
![Page 8: @let@token Modelling of Phase Transformations in ...prusv/ncmm/... · Shape-memory materials(SMM): alloys (=SMAs) or intermetalics. The mechanism behindshape-memory e ect(=SME): higher](https://reader035.vdocument.in/reader035/viewer/2022081617/602db37bf938dd296d06a0b4/html5/thumbnails/8.jpg)
Phase transformations in NiMnGaThe model and its analysis
Some other phenomena to be involved
Martensitic/austenitic transformationFerro/para-magnetic transformationCoupling of transformations: magnetostriction
Shape-memory materials (SMM): alloys (=SMAs) or intermetalics.The mechanism behind shape-memory effect (=SME): higher temperatures:
atoms tend to form a latice with high symmetry (mostly cubic):austenite phase, higher heat capacitya lower-symmetrical latice: martensite phase, lower heat capacity.the lower-symmetrical latice occurs in several variants;each of them can be rotated:
Tomas Roubıcek (Workshop, MFF, Prague, March 31, 2012) Phase transformations in NiMnGa
![Page 9: @let@token Modelling of Phase Transformations in ...prusv/ncmm/... · Shape-memory materials(SMM): alloys (=SMAs) or intermetalics. The mechanism behindshape-memory e ect(=SME): higher](https://reader035.vdocument.in/reader035/viewer/2022081617/602db37bf938dd296d06a0b4/html5/thumbnails/9.jpg)
Phase transformations in NiMnGaThe model and its analysis
Some other phenomena to be involved
Martensitic/austenitic transformationFerro/para-magnetic transformationCoupling of transformations: magnetostriction
Shape-memory materials (SMM): alloys (=SMAs) or intermetalics.The mechanism behind shape-memory effect (=SME): higher temperatures:
atoms tend to form a latice with high symmetry (mostly cubic):austenite phase, higher heat capacitya lower-symmetrical latice: martensite phase, lower heat capacity.the lower-symmetrical latice occurs in several variants;each of them can be rotated:
Tomas Roubıcek (Workshop, MFF, Prague, March 31, 2012) Phase transformations in NiMnGa
![Page 10: @let@token Modelling of Phase Transformations in ...prusv/ncmm/... · Shape-memory materials(SMM): alloys (=SMAs) or intermetalics. The mechanism behindshape-memory e ect(=SME): higher](https://reader035.vdocument.in/reader035/viewer/2022081617/602db37bf938dd296d06a0b4/html5/thumbnails/10.jpg)
Phase transformations in NiMnGaThe model and its analysis
Some other phenomena to be involved
Martensitic/austenitic transformationFerro/para-magnetic transformationCoupling of transformations: magnetostriction
Shape-memory materials (SMM): alloys (=SMAs) or intermetalics.The mechanism behind shape-memory effect (=SME): higher temperatures:
atoms tend to form a latice with high symmetry (mostly cubic):austenite phase, higher heat capacitya lower-symmetrical latice: martensite phase, lower heat capacity.the lower-symmetrical latice occurs in several variants;each of them can be rotated:
Tomas Roubıcek (Workshop, MFF, Prague, March 31, 2012) Phase transformations in NiMnGa
![Page 11: @let@token Modelling of Phase Transformations in ...prusv/ncmm/... · Shape-memory materials(SMM): alloys (=SMAs) or intermetalics. The mechanism behindshape-memory e ect(=SME): higher](https://reader035.vdocument.in/reader035/viewer/2022081617/602db37bf938dd296d06a0b4/html5/thumbnails/11.jpg)
Phase transformations in NiMnGaThe model and its analysis
Some other phenomena to be involved
Martensitic/austenitic transformationFerro/para-magnetic transformationCoupling of transformations: magnetostriction
Crystalographical options of lower-symmetrical martensite:
Self-accomodation of a microstructure in martensite
Tomas Roubıcek (Workshop, MFF, Prague, March 31, 2012) Phase transformations in NiMnGa
![Page 12: @let@token Modelling of Phase Transformations in ...prusv/ncmm/... · Shape-memory materials(SMM): alloys (=SMAs) or intermetalics. The mechanism behindshape-memory e ect(=SME): higher](https://reader035.vdocument.in/reader035/viewer/2022081617/602db37bf938dd296d06a0b4/html5/thumbnails/12.jpg)
Phase transformations in NiMnGaThe model and its analysis
Some other phenomena to be involved
Martensitic/austenitic transformationFerro/para-magnetic transformationCoupling of transformations: magnetostriction
Crystalographical options of lower-symmetrical martensite:
Self-accomodation of a microstructure in martensite
Tomas Roubıcek (Workshop, MFF, Prague, March 31, 2012) Phase transformations in NiMnGa
![Page 13: @let@token Modelling of Phase Transformations in ...prusv/ncmm/... · Shape-memory materials(SMM): alloys (=SMAs) or intermetalics. The mechanism behindshape-memory e ect(=SME): higher](https://reader035.vdocument.in/reader035/viewer/2022081617/602db37bf938dd296d06a0b4/html5/thumbnails/13.jpg)
Phase transformations in NiMnGaThe model and its analysis
Some other phenomena to be involved
Martensitic/austenitic transformationFerro/para-magnetic transformationCoupling of transformations: magnetostriction
Crystalographical options of lower-symmetrical martensite:
Self-accomodation of a microstructure in austenite and martensite
Tomas Roubıcek (Workshop, MFF, Prague, March 31, 2012) Phase transformations in NiMnGa
![Page 14: @let@token Modelling of Phase Transformations in ...prusv/ncmm/... · Shape-memory materials(SMM): alloys (=SMAs) or intermetalics. The mechanism behindshape-memory e ect(=SME): higher](https://reader035.vdocument.in/reader035/viewer/2022081617/602db37bf938dd296d06a0b4/html5/thumbnails/14.jpg)
Phase transformations in NiMnGaThe model and its analysis
Some other phenomena to be involved
Martensitic/austenitic transformationFerro/para-magnetic transformationCoupling of transformations: magnetostriction
Crystalographical options of lower-symmetrical martensite:
Self-accomodation of a microstructure (example of CuAlNi)
Courtesy of.
Vaclav Novak and Petr Sittner,Institute of Physics,
Academy of Sciences, Czech Rep.
Tomas Roubıcek (Workshop, MFF, Prague, March 31, 2012) Phase transformations in NiMnGa
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Phase transformations in NiMnGaThe model and its analysis
Some other phenomena to be involved
Martensitic/austenitic transformationFerro/para-magnetic transformationCoupling of transformations: magnetostriction
Schematic stress/strain response of SMM:
low temperature vs high temperature
quasiplasticity pseudoelasticity
Tomas Roubıcek (Workshop, MFF, Prague, March 31, 2012) Phase transformations in NiMnGa
![Page 16: @let@token Modelling of Phase Transformations in ...prusv/ncmm/... · Shape-memory materials(SMM): alloys (=SMAs) or intermetalics. The mechanism behindshape-memory e ect(=SME): higher](https://reader035.vdocument.in/reader035/viewer/2022081617/602db37bf938dd296d06a0b4/html5/thumbnails/16.jpg)
Phase transformations in NiMnGaThe model and its analysis
Some other phenomena to be involved
Martensitic/austenitic transformationFerro/para-magnetic transformationCoupling of transformations: magnetostriction
Experiments by L.Straka, V.Novak, M.Landa, O.Heczko, 2004:Compression experiment: reorientation of tetragonal martensite in a(001)-oriented singlecrystal NiMnGa under temperature 293 K:
Stress-strain diagram at temperature 293K (left) and 323K (right):
0 1 2 3 4 5 6 7 80
10
20
30
40
50
compression strain [%]
pres
sure
[MP
a]
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50
20
40
60
80
100
120
compression strain [%]
pres
sure
[MP
a]
Tomas Roubıcek (Workshop, MFF, Prague, March 31, 2012) Phase transformations in NiMnGa
![Page 17: @let@token Modelling of Phase Transformations in ...prusv/ncmm/... · Shape-memory materials(SMM): alloys (=SMAs) or intermetalics. The mechanism behindshape-memory e ect(=SME): higher](https://reader035.vdocument.in/reader035/viewer/2022081617/602db37bf938dd296d06a0b4/html5/thumbnails/17.jpg)
Phase transformations in NiMnGaThe model and its analysis
Some other phenomena to be involved
Martensitic/austenitic transformationFerro/para-magnetic transformationCoupling of transformations: magnetostriction
Computational simulations:Compression experiment with NiMnGa (001)-oriented singlecrystal
Reorientation of martensite during a compression experiment at 293K .
0 1 2 3 4 5 6 7 8−40
−20
0
20
40
60
80
100
120
140
compression strain [%]
pres
sure
[MP
a]
0 1 2 3 4 5 60
20
40
60
80
100
120
compression strain [%]
pres
sure
[MP
a]
Stress/strain response during a compression experiment at 293K and at 323K .Calculations, visualizations: courtesy of Marcel Arndt, Universitat Bonn.
Tomas Roubıcek (Workshop, MFF, Prague, March 31, 2012) Phase transformations in NiMnGa
![Page 18: @let@token Modelling of Phase Transformations in ...prusv/ncmm/... · Shape-memory materials(SMM): alloys (=SMAs) or intermetalics. The mechanism behindshape-memory e ect(=SME): higher](https://reader035.vdocument.in/reader035/viewer/2022081617/602db37bf938dd296d06a0b4/html5/thumbnails/18.jpg)
Phase transformations in NiMnGaThe model and its analysis
Some other phenomena to be involved
Martensitic/austenitic transformationFerro/para-magnetic transformationCoupling of transformations: magnetostriction
Transformation in magnetic materials:
low temperature (below Currie point): highly-ordered, ferromagnetic statevery low temperature: the Heissenberg constraint |m| = Ms is well satisfiedbut in higher temperatures the deviation from it can be large in outer fieldhigh temperature (above Currie point Tc): dis-ordered, paramagnetic state
G.Bertotti: Hysteresis in Magnetism.Academic Press, San Diego, 1998.
Tomas Roubıcek (Workshop, MFF, Prague, March 31, 2012) Phase transformations in NiMnGa
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Phase transformations in NiMnGaThe model and its analysis
Some other phenomena to be involved
Martensitic/austenitic transformationFerro/para-magnetic transformationCoupling of transformations: magnetostriction
Both martensite/austenite and ferro/para-magnetic transformations are coupled:
Strong dependence of thermo-mechanical response on magnetic fieldin Ni2MnGa single crystals – for example:
K.Ullakko, J.K.Huang, C.Kantner, R.C.O’Handley, V.V.Kokorinin Appl. Phys. Lett. 69 (1996), 1966–1968.
Tomas Roubıcek (Workshop, MFF, Prague, March 31, 2012) Phase transformations in NiMnGa
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Phase transformations in NiMnGaThe model and its analysis
Some other phenomena to be involved
Martensitic/austenitic transformationFerro/para-magnetic transformationCoupling of transformations: magnetostriction
Other phenomena to be captured:
electric resistivity depending on temperature and phase (an example in NiTi):
V.Novak, P.Sittner, G.N.Dayananda, F.M.Braz-Fernandes, K.K.Mahesh,Materials Science and Engineering A 481-482 (2008) 127-133.
Tomas Roubıcek (Workshop, MFF, Prague, March 31, 2012) Phase transformations in NiMnGa
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Phase transformations in NiMnGaThe model and its analysis
Some other phenomena to be involved
Partly linearized ansatzAnalysis: semi-implicit discretisation, a-priori estimatesAnalysis: convergence
Variables (minimal scenario):u displacement, E(u) = 1
2 (∇u)>+ 12∇u = small-strain tensor,
m magnetisation,θ temperature,h magnetic field,e electric field.
Basic concepts: small strains, Kelvin-Voigt rheology, 2nd-grade materials,electric displacement current (∼ electric-field energy) neglected,
⇒ eddy-current approximation of the Maxwell equations,partly linear free energy ϕ(E,m, θ) = ϕ0(E,m) + θϕ1(E,m):
⇒ heat capacity c = −ϕ′′θθ = −ϕ′′θθ(θ),cross-effects neglected (no Peltier/Seeback effects).
Main parameters of the model:K = K(E,m, θ) thermal conductivity,S = S(E,m, θ) electrical conductivity, c = c(θ) heat capacity,γ = γ(|m|) effective gyromagnetic ratio, µ0 vacuum permeability,α magnetic-dissipation constant, λ magnetic exchange-energy constant,% mass density, f0 bulk force (inertial and load),D viscosity tensor,CH hyperelasticity tensor, DH hyperviscosity tensor.
Tomas Roubıcek (Workshop, MFF, Prague, March 31, 2012) Phase transformations in NiMnGa
![Page 22: @let@token Modelling of Phase Transformations in ...prusv/ncmm/... · Shape-memory materials(SMM): alloys (=SMAs) or intermetalics. The mechanism behindshape-memory e ect(=SME): higher](https://reader035.vdocument.in/reader035/viewer/2022081617/602db37bf938dd296d06a0b4/html5/thumbnails/22.jpg)
Phase transformations in NiMnGaThe model and its analysis
Some other phenomena to be involved
Partly linearized ansatzAnalysis: semi-implicit discretisation, a-priori estimatesAnalysis: convergence
The equations:
Momentum equilibrium:
%..u − div
(ϕ′E(E(u),m, θ) + DE(
.u)− div
(CH∇E(u) + DH∇E(
.u)))
= f0 − µ0∇h>m,
Landau-Lifshitz-Gilbert equation:
α.m − m× .m
γ(|m|)− λ∆m + ϕ′m(E(u),m, θ) = µ0h,
heat equation
c(θ).θ−div
(K(E(u),m, θ)∇θ
)= S(E(u),m, θ)e:e + DE(
.u):E(
.u) + DH∇E(
.u)
...∇E(.u)
+ α| .m|2 + θϕ′′Eθ(E(u),m, θ):E(.u)+θϕ′′mθ(E(u),m, θ)· .m,
Maxwell system (in eddy-current approximation):
µ0(.h +
.m) + curl e = −µ0(div
.u)m − µ0∇m
.u,
ε0.e − curl h + S(E(u),m, θ)e = 0.
Tomas Roubıcek (Workshop, MFF, Prague, March 31, 2012) Phase transformations in NiMnGa
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Phase transformations in NiMnGaThe model and its analysis
Some other phenomena to be involved
Partly linearized ansatzAnalysis: semi-implicit discretisation, a-priori estimatesAnalysis: convergence
The equations:
Momentum equilibrium:
%..u − div
(ϕ′E(E(u),m, θ) + DE(
.u)− div
(CH∇E(u) + DH∇E(
.u)))
= f0 − µ0∇h>m,
Landau-Lifshitz-Gilbert equation:
α.m − m× .m
γ(|m|)− λ∆m + ϕ′m(E(u),m, θ) = µ0h,
heat equation
c(θ).θ−div
(K(E(u),m, θ)∇θ
)= S(E(u),m, θ)e:e + DE(
.u):E(
.u) + DH∇E(
.u)
...∇E(.u)
+ α| .m|2 + θϕ′′Eθ(E(u),m, θ):E(.u)+θϕ′′mθ(E(u),m, θ)· .m,
Maxwell system (in eddy-current approximation):
µ0(.h +
.m) + curl e = −µ0(div
.u)m − µ0∇m
.u,
ε0.e − curl h + S(E(u),m, θ)e = 0.
Tomas Roubıcek (Workshop, MFF, Prague, March 31, 2012) Phase transformations in NiMnGa
![Page 24: @let@token Modelling of Phase Transformations in ...prusv/ncmm/... · Shape-memory materials(SMM): alloys (=SMAs) or intermetalics. The mechanism behindshape-memory e ect(=SME): higher](https://reader035.vdocument.in/reader035/viewer/2022081617/602db37bf938dd296d06a0b4/html5/thumbnails/24.jpg)
Phase transformations in NiMnGaThe model and its analysis
Some other phenomena to be involved
Partly linearized ansatzAnalysis: semi-implicit discretisation, a-priori estimatesAnalysis: convergence
The equations: ...analysed for slow loading ⇒ pinning terms still needed
Momentum equilibrium: K.R.Rajagopal + T.R., 2003
%..u − div
(ϕ′E(E(u),m, θ) + DE(
.u)− div
(CH∇E(u) + DH∇E(
.u)))
= f0 − µ0∇h>m,
Landau-Lifshitz-Gilbert equation:
α.m − m× .m
γ(|m|)− λ∆m + ϕ′m(E(u),m, θ) = µ0h,
heat equation
c(θ).θ−div
(K(E(u),m, θ)∇θ
)= S(E(u),m, θ)e:e + DE(
.u):E(
.u) + DH∇E(
.u)
...∇E(.u)
+ α| .m|2 + θϕ′′Eθ(E(u),m, θ):E(.u)+θϕ′′mθ(E(u),m, θ)· .m,
Maxwell system (in eddy-current approximation):
µ0(.h +
.m) + curl e = −µ0(div
.u)m − µ0∇m
.u,
ε0.e − curl h + S(E(u),m, θ)e = 0.
Tomas Roubıcek (Workshop, MFF, Prague, March 31, 2012) Phase transformations in NiMnGa
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Phase transformations in NiMnGaThe model and its analysis
Some other phenomena to be involved
Partly linearized ansatzAnalysis: semi-implicit discretisation, a-priori estimatesAnalysis: convergence
Derivation of the Maxwell system:ms = magnetisation in the physical space,“magnetic” part of the Maxwell system:
µ0(.h +
.ms) + curl e = 0
m = magnetisation in the reference configuration related with ms by
det(I+∇u(t, x))ms(t, x+u(t, x)) = m(t, x).
Differentiation in time:
det(I+∇u)( .ms + (I+∇u)−>(div
.u)ms +∇ms
.u)
=.m.
Small-displacement approximation x + u ≈ x , which entails:I+∇u ≈ I, ms ≈ m, and ∇ms ≈ ∇m, so that
.ms ≈
.m − (div
.u)m −∇m .u ←− occuring as r.h.s. of the Maxwell system
.u is not considered small⇒ small but very fast mechanical vibrations
in some experiments on frequencies about 1 MHz or more.
Tomas Roubıcek (Workshop, MFF, Prague, March 31, 2012) Phase transformations in NiMnGa
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Phase transformations in NiMnGaThe model and its analysis
Some other phenomena to be involved
Partly linearized ansatzAnalysis: semi-implicit discretisation, a-priori estimatesAnalysis: convergence
Energetics:Test: momentum eq. by
.u, LLG by
.m, heat eq. by 1, Maxwell by (h, e):
Use: cancelation of the gyromagnetic term: m× .mγ(|m|) ·
.m = 0,
+ cancelation of curl-terms + the identity:∫Ω
µ0
((div
.u)m +∇m .u)︸ ︷︷ ︸
r.h.s. of Maxwell eq.
·h dx =
∫Ω
µ0div(m ⊗ .u)·h dx=
∫Ω
µ0
(div((
.u ⊗m)h)− (m ⊗ .u):∇h
)dx
=
∫Γ
µ0(m·h)(.u·n)dS −
∫Ω
µ0∇h>m︸ ︷︷ ︸r.h.s. of
momentum equation
· .u dx
d
dt
∫Ω
ε︸︷︷︸internalenergy
+µ0
2|h|2︸ ︷︷ ︸
magneticenergy
+%
2|.u|2︸ ︷︷ ︸
kineticenergy
dx =
∫Ω
f0·.u︸︷︷︸
power ofexternal load
dx + boundary power.
Gibbs’ relation: ψ = ε− sθ with entropy s = −φ′θinternal energy: ε = ϑ+ψ(E,m, 0) + 1
2CH∇E...∇E + 1
2λ|∇m|2, enthapy ϑ =
∫ θ0c(·).
Tomas Roubıcek (Workshop, MFF, Prague, March 31, 2012) Phase transformations in NiMnGa
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Phase transformations in NiMnGaThe model and its analysis
Some other phenomena to be involved
Partly linearized ansatzAnalysis: semi-implicit discretisation, a-priori estimatesAnalysis: convergence
Example of a free energy considered in NiMnGa:
in partly linearized ansatz:
A.T.Zayak, V.D.Buchelnikov, P.Entel: A Ginzburg-Landau theory for Ni-Mn-Ga.Phase Trans. 75 (2002), 243-256
Tomas Roubıcek (Workshop, MFF, Prague, March 31, 2012) Phase transformations in NiMnGa
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Phase transformations in NiMnGaThe model and its analysis
Some other phenomena to be involved
Partly linearized ansatzAnalysis: semi-implicit discretisation, a-priori estimatesAnalysis: convergence
Fully implicit time-discretisation + regularization:
Recursive formula for the 5-tuple (ukτ ,mkτ , ϑ
kτ , e
kτ , h
kτ ) solving the system
Momentum-equilibrium equation:
%ukτ−2uk−1
τ +uk−2τ
τ 2− div
(Skτ − divHk
τ
)= f kτ − µ0(∇hkτ )>mk
τ with
Skτ := σE(E(ukτ ),mk
τ , ϑkτ ) + DE
(ukτ−uk−1τ
τ
)+τ |E(ukτ )|η−2E(ukτ ), and
Hkτ := DH∇E
(ukτ−uk−1τ
τ
)+CH∇E(ukτ )+τ |∇E(ukτ )|η−2∇E(ukτ ),
Landau-Lifshitz-Gilbert equation:
αmkτ−mk−1
τ
τ− mk
τ
γ(|mkτ |)×mk
τ−mk−1τ
τ− λ∆mk
τ + σm(E(ukτ ),mkτ , ϑ
kτ )
− µ0hkτ = τdiv
(|∇mk
τ |η−2)∇mk
τ
)− τ |mk
τ |η−2mkτ ,
Tomas Roubıcek (Workshop, MFF, Prague, March 31, 2012) Phase transformations in NiMnGa
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Phase transformations in NiMnGaThe model and its analysis
Some other phenomena to be involved
Partly linearized ansatzAnalysis: semi-implicit discretisation, a-priori estimatesAnalysis: convergence
Heat equation:
ϑkτ − ϑk−1τ
τ− div
(K0(Ek
τ ,mkτ , ϑ
kτ )∇ϑkτ
)= S(Ek
τ ,mkτ , ϑ
kτ )ekτ :ekτ
+(
1−√τ
2
)DE(ukτ−uk−1
τ
τ
):E(ukτ−uk−1
τ
τ
)+ DH∇E
(ukτ−uk−1τ
τ
)...∇E(ukτ−uk−1
τ
τ
)+(
1−√τ
2
)α∣∣∣mk
τ−mk−1τ
τ
∣∣∣2 + A(
Ekτ ,m
kτ , ϑ
kτ ;
Ekτ−Ek−1
τ
τ,mkτ−mk−1
τ
τ
),
Maxwell system:
hkτ−hk−1τ
τ+
curl ekτµ0
= ∇mkτ
ukτ−uk−1τ
τ− mk
τ−mk−1τ
τ+ div
ukτ−uk−1τ
τmkτ ,
curl hkτ − S(Ekτ ,m
kτ , ϑ
kτ ) ekτ = τ |ekτ |η−2ekτ ,
for k = 1, ...,Kτ := T/τ , where we abbreviated Ekτ = E(ukτ ) and
A(E,m, ϑ;.E,.m) = θϕ′′Eθ(E,m, θ):
.E+θϕ′′mθ(E,m, θ)· .m
with θ = c−1(ϑ) and c ′ = c .
Tomas Roubıcek (Workshop, MFF, Prague, March 31, 2012) Phase transformations in NiMnGa
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Phase transformations in NiMnGaThe model and its analysis
Some other phenomena to be involved
Partly linearized ansatzAnalysis: semi-implicit discretisation, a-priori estimatesAnalysis: convergence
We use the discrete scheme recursively, starting from k = 1 by using
u0τ = u0,τ , u−1
τ = u0,τ−τv0, m0τ = m0,τ , ϑ0
τ = c(θ0), h0τ = h0,
Existence of (ukτ ,mkτ , ϑ
kτ , e
kτ , h
kτ ):
η large enough (namely η > 8) ⇒ pseudomonotone coercive operator⇒ Brezis’ theorem ⇒
ukτ∈W 2,η(Ω;IR3),
mkτ∈W 1,η(Ω;IR3),
ϑkτ∈W 1,2(Ω),
hkτ∈L2,η′
curl (Ω;IR3),
ekτ∈Lη,2curl (Ω;IR3)
where Lp,qcurl (Ω; IR3) :=v ∈Lp(Ω; IR3); curl v ∈Lq(Ω; IR3)
.
Non-negativity: ϑkτ ≥ 0.
Tomas Roubıcek (Workshop, MFF, Prague, March 31, 2012) Phase transformations in NiMnGa
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Phase transformations in NiMnGaThe model and its analysis
Some other phenomena to be involved
Partly linearized ansatzAnalysis: semi-implicit discretisation, a-priori estimatesAnalysis: convergence
A-priori estimates (uτ , uτ etc. are interpolants over [0,T ]):
Energy-type test (by.uτ ,
.mτ , 1, hτ , eτ ) ⇒∥∥uτ∥∥W 1,∞(I ;L2(Ω;IR3))∩W 1,2(I ;W 2,2(Ω;IR3))
≤ C ,∥∥mτ
∥∥L∞(I ;W 1,2(Ω;IR3))∩W 1,2(I ;L2(Ω;IR3))
≤ C ,∥∥ϑτ∥∥L∞(I ;L1(Ω))≤ C ,∥∥hτ∥∥L∞(I ;L2(Ω;IR3))≤ C ,∥∥eτ∥∥L2(Q;IR3)
≤ C ,∥∥E(uτ )∥∥L∞(I ;W 1,η(Ω;))
≤ Cτ−1/η,∥∥mτ
∥∥L∞(I ;W 1,η(Ω;IR3))
≤ Cτ−1/η,∥∥eτ∥∥Lη(Q;IR3)≤ Cτ−1/η
based on the semi-convexity of the functional (for enough small τ > 0)
(E,m) 7→ ϕ(E,m) +τ
η|E|η +
τ
η|m|η +
DE:E + α|m|2
2√τ
Tomas Roubıcek (Workshop, MFF, Prague, March 31, 2012) Phase transformations in NiMnGa
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Phase transformations in NiMnGaThe model and its analysis
Some other phenomena to be involved
Partly linearized ansatzAnalysis: semi-implicit discretisation, a-priori estimatesAnalysis: convergence
A-priori estimates (uτ , uτ etc. are interpolants over [0,T ]):
Energy-type test (by.uτ ,
.mτ , 1, hτ , eτ ) ⇒∥∥uτ∥∥W 1,∞(I ;L2(Ω;IR3))∩W 1,2(I ;W 2,2(Ω;IR3))
≤ C ,∥∥mτ
∥∥L∞(I ;W 1,2(Ω;IR3))∩W 1,2(I ;L2(Ω;IR3))
≤ C ,∥∥ϑτ∥∥L∞(I ;L1(Ω))≤ C ,∥∥hτ∥∥L∞(I ;L2(Ω;IR3))≤ C ,∥∥eτ∥∥L2(Q;IR3)
≤ C ,∥∥E(uτ )∥∥L∞(I ;W 1,η(Ω;))
≤ Cτ−1/η,∥∥mτ
∥∥L∞(I ;W 1,η(Ω;IR3))
≤ Cτ−1/η,∥∥eτ∥∥Lη(Q;IR3)≤ Cτ−1/η
based on the semi-convexity of the functional (for enough small τ > 0)
(E,m) 7→ ϕ(E,m) +τ
η|E|η +
τ
η|m|η +
DE:E + α|m|2
2√τ
Tomas Roubıcek (Workshop, MFF, Prague, March 31, 2012) Phase transformations in NiMnGa
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Phase transformations in NiMnGaThe model and its analysis
Some other phenomena to be involved
Partly linearized ansatzAnalysis: semi-implicit discretisation, a-priori estimatesAnalysis: convergence
A-priori estimates (uτ , uτ etc. are interpolants over [0,T ]):
Energy-type test (by.uτ ,
.mτ , 1, hτ , eτ ) ⇒∥∥uτ∥∥W 1,∞(I ;L2(Ω;IR3))∩W 1,2(I ;W 2,2(Ω;IR3))
≤ C ,∥∥mτ
∥∥L∞(I ;W 1,2(Ω;IR3))∩W 1,2(I ;L2(Ω;IR3))
≤ C ,∥∥ϑτ∥∥L∞(I ;L1(Ω))≤ C ,∥∥hτ∥∥L∞(I ;L2(Ω;IR3))≤ C ,∥∥eτ∥∥L2(Q;IR3)
≤ C ,∥∥E(uτ )∥∥L∞(I ;W 1,η(Ω;))
≤ Cτ−1/η,∥∥mτ
∥∥L∞(I ;W 1,η(Ω;IR3))
≤ Cτ−1/η,∥∥eτ∥∥Lη(Q;IR3)≤ Cτ−1/η,
+ a special nonlinear test of the heat equation + Gagliardo-Nirenberg interpolation∥∥∇ϑτ∥∥Lr (Q;IRd )≤ Cr with r < 5/4.
Tomas Roubıcek (Workshop, MFF, Prague, March 31, 2012) Phase transformations in NiMnGa
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Phase transformations in NiMnGaThe model and its analysis
Some other phenomena to be involved
Partly linearized ansatzAnalysis: semi-implicit discretisation, a-priori estimatesAnalysis: convergence
Further a-priori estimates:∥∥%..u iτ
∥∥∥L2(I ;W 2,2(Ω;IR3)∗)+Lη′ (I ;W 2,η(Ω;IR3)∗)
≤ C ,
∥∥∥%..u iτ − τdiv
(|E(uτ )|η−2E(uτ )
)+ τdiv2
(|∇E(uτ )|η−2∇E(uτ )
)∥∥∥L2(I ;W 2,2(Ω;IR3)∗)
≤ C ,
∥∥.ϑτ∥∥L1(I ;W 3,2(Ω)∗)≤ C ,
∥∥curl hτ + τ |eτ |η−2eτ∥∥L2(Q;IR3)
≤ C ,
∥∥.hτ∥∥L2(I ;L2curl ,0(Ω;IR3)∗)
≤ C .
Tomas Roubıcek (Workshop, MFF, Prague, March 31, 2012) Phase transformations in NiMnGa
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Phase transformations in NiMnGaThe model and its analysis
Some other phenomena to be involved
Partly linearized ansatzAnalysis: semi-implicit discretisation, a-priori estimatesAnalysis: convergence
Convergence for τ → 0: Step 0: Banach’ selection principle:
uτ → u strongly in W 1,2(I ;W 2,2(Ω; IR3)),
mτ → m strongly in W 1,2(I ;W 1,2(Ω; IR3)),
ϑτ → ϑ strongly in Ls(Q) with any s < 5/3,
eτ → e strongly in L2(Q; IR3),
hτ → h weakly* in L∞(I ; L2(Ω; IR3)),
and, moreover (with hb from not-mentioned boundary conditions)
hτ−hb,τ → h−hb weakly in Lη′(I ; L2,η′
curl ,0(Ω; IR3)), and
curl hτ + τ |eτ |γ−2eτ → curl h weakly in L2(Q; IR3×3).
for a subsequence.
Then we want to prove that any (u,m, ϑ, h, e) obtained in this way is aweak solution to the considered IBVP (after the transformation θ 7→ ϑ)which also preserves the total energy.
Tomas Roubıcek (Workshop, MFF, Prague, March 31, 2012) Phase transformations in NiMnGa
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Phase transformations in NiMnGaThe model and its analysis
Some other phenomena to be involved
Partly linearized ansatzAnalysis: semi-implicit discretisation, a-priori estimatesAnalysis: convergence
Step 1: Convergence in the semilinear mechanical/magnetic/electro part:Aubin-Lions’ theorem: strong convergence of E(uτ ), mτ , and ϑτ .Then weak convergence suffices in semilinear terms, whilethe quasilinear regularizing terms vanish, e.g.∣∣∣ ∫
Q
τ |E(uτ )|η−2E(uτ ):E(v)dxdt∣∣∣ ≤ τ‖E(uτ )‖η−1
Lη(Q;IR3×3)‖E(v)‖Lη(Q;IR3×3)
≤ Cτ 1/η‖E(v)‖Lη(Q;IR3×3) → 0
for any smooth v .
Step 2: Mechanical/magnetic energy preservation:test respectively by
.u,.m, h, and e and make the by-part integration
%..u iτ−τdiv
(|E(uτ )|η−2E(uτ )
)+τdiv2
(|∇E(uτ )|η−2∇E(uτ )
) ζ∈L2(I ;W 2,2(Ω;R3)∗).
and then
〈ζ,w〉 = limτ→0
∫Q
%.uiτ ·
.w − τ |E(uτ )|η−2E(uτ ) : E(w)
+ τ |∇E(uτ )|η−2∇E(uτ )...∇E(w)dxdt =
∫Q
%.u · .wdxdt.
⇒ ζ = %..u ⇒ %
..u is in duality with
.u ∈ L2(I ;W 2,2(Ω; IR3)).
Tomas Roubıcek (Workshop, MFF, Prague, March 31, 2012) Phase transformations in NiMnGa
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Phase transformations in NiMnGaThe model and its analysis
Some other phenomena to be involved
Partly linearized ansatzAnalysis: semi-implicit discretisation, a-priori estimatesAnalysis: convergence
Step 3: Strong convergence of ∇E(.uτ ) and
.mτ and eτ :∫
Q
DE(.u):E(
.u) + DH∇E(
.u):∇E(
.u) + α| .m|2 + S(E,m, ϑ)e·e
≤ lim infτ→0
∫Q
DE(.uτ ):E(
.uτ ) + DH∇E(
.uτ ):∇E(
.uτ ) + α| .mτ |2 + S(Eτ , mτ , ϑτ )eτ ·eτdx
≤ lim supτ→0
∫Q
DE(.uτ ):E(
.uτ ) + DH∇E(
.uτ ):∇E(
.uτ ) + α| .mτ |2 + S(Eτ , mτ , ϑτ )eτ ·eτdx
≤ lim supτ→0
Φ(u0τ , v0,m0τ , h0
)− Φ
(uτ (T ),
.uτ (T ),mτ (T ), hτ (T )
)+
∫Ω
τ
η|E(u0τ )|η +
τ
η|∇E(u0τ )|η +
τ
η|m0τ |ηdx −
∫Σ
gτ ·.uτdt +
∫Q
fτ ·.uτ+curl hb,τ ·eτ
+ µ0
(.hτ +
.mτ −∇mτ
.uτ − (div
.uτ )τ
)·hb,τ − A(Eτ , mτ , ϑτ ;
.Eτ ,
.mτ )dxdt
≤ Φ(u0, v0,m0, h0
)− Φ
(u(T ),
.u(T ),m(T ), h(T )
)−∫
Σ
g ·.udt +
∫Q
f ·.u + curl hb·e
+ µ0
(.h +
.m −∇m.u − (div
.u)m
)·hb − A(E,m, ϑ;
.E,.m)dxdt
=
∫Q
DE(.u):E(
.u) + DH∇E(
.u):∇E(
.u) + α| .m|2 + S(E,m, ϑ)e·e.
Tomas Roubıcek (Workshop, MFF, Prague, March 31, 2012) Phase transformations in NiMnGa
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Phase transformations in NiMnGaThe model and its analysis
Some other phenomena to be involved
Partly linearized ansatzAnalysis: semi-implicit discretisation, a-priori estimatesAnalysis: convergence
Step 4: Limit passage in the heat equation:Having proved the strong convergence in Step 2, the right-hand side ofthe heat equation converges strongly in L1(Q) and this limit passage isthen easy.
Step 5: Total energy preservation:
We have.ϑ ∈ L1(I ;W 3,2(Ω)∗), and realize the already proved the heat
equation, which is in duality with the constant 1, we can performrigorously this test and sum it with mechanical/magnetic energy balanceobtained already in Step 2.
Tomas Roubıcek (Workshop, MFF, Prague, March 31, 2012) Phase transformations in NiMnGa
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Phase transformations in NiMnGaThe model and its analysis
Some other phenomena to be involved
General nonlinear ansatzPinning effects
Fully nonlinear coupling:
more symmetry ∼ higher heat capacityheat capacity is higher in austenite than in martensite
⇒ shape-memory effect
c should depend rather directly on E (and also m, not only on θ)
fully general nonlinear ansatz ϕ(E,m, θ) instead of ϕ0(E,m) + θϕ1(E,m)
then the heat capacity c = −ψ′′θθ depends, beside θ, also on E and m.
Generalized enthalpy transformation:
ϑ = c(E,m, θ) :=
∫ θ
0
c(E,m,Θ)dΘ
c(E,m, θ).θ =
∂c(E,m, θ)
∂t− c1(E,m, θ):
.E− c2(E,m, θ)· .m,
c1(E,m, θ) =
∫ θ
0
c ′E(E,m,Θ)dΘ and c2(E,m, θ) =
∫ θ
0
c ′m(E,m,Θ)dΘ.
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Phase transformations in NiMnGaThe model and its analysis
Some other phenomena to be involved
General nonlinear ansatzPinning effects
Define: T (E,m, ·) := [c(E,m, ·)]−1
K0(E,m, ϑ) := K(E,m, T (E,m, ϑ))T ′ϑ(E,m, ϑ),
K1(E,m, ϑ) := K(E,m, T (E,m, ϑ))T ′E(E,m, ϑ),
K2(E,m, ϑ) := K(E,m, T (E,m, ϑ))T ′m(E,m, ϑ),
S(E,m, ϑ) := S(E,m, T (E,m, ϑ)),
A1(E,m, ϑ) := T (E,m, ϑ)ϕ′′Eθ(E,m, T (E,m, ϑ)) + c1(E,m, T (E,m, ϑ))
A2(E,m, ϑ) := T (E,m, ϑ)ϕ′′mθ(E,m, T (E,m, ϑ)) + c2(E,m, T (E,m, ϑ)),
σE(E,m, ϑ) := ϕ′E(E,m, T (E,m, ϑ)),
σm(E,m, ϑ) := ϕ′m(E,m, T (E,m, ϑ)).
Then the heat flux transforms to:
K(E,m, θ)∇θ = K(E,m, T (e, ϑ))∇T (E,m, ϑ)
= K0(E,m, ϑ)∇ϑ+K1(E,m, ϑ)∇E +K2(E,m, ϑ)∇m.
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Phase transformations in NiMnGaThe model and its analysis
Some other phenomena to be involved
General nonlinear ansatzPinning effects
Thus, in terms of the 5-tuple (u,m, ϑ, e, h), the original systemtransforms to the following 5 equations:
%..u − div
(σE(E(u),m, ϑ) + DE(
.u)
− div(CH∇E(u)+DH∇E(
.u)))
= f0 − µ0∇h>m,
α.m − m× .m
γ(|m|)− λ∆m + σm(E(u),m, ϑ) = p0 + µ0h,
.ϑ−div
(K0(E(u),m, ϑ)∇ϑ+K1(E(u),m, ϑ)∇E(u) +K2(E(u),m, ϑ)∇m
)= S(E(u),m, ϑ)e·e + DE(
.u):E(
.u) + DH∇E(
.u)
...∇E(.u) + α| .m|2
+A1(E(u),m, ϑ):E(.u) +A2(E(u),m, ϑ)· .m,
µ0(.h +
.m) + curl e = −µ0∇m
.u − µ0(div
.u)m,
curl h − S(E(u),m, ϑ)e = 0,
Tomas Roubıcek (Workshop, MFF, Prague, March 31, 2012) Phase transformations in NiMnGa
![Page 42: @let@token Modelling of Phase Transformations in ...prusv/ncmm/... · Shape-memory materials(SMM): alloys (=SMAs) or intermetalics. The mechanism behindshape-memory e ect(=SME): higher](https://reader035.vdocument.in/reader035/viewer/2022081617/602db37bf938dd296d06a0b4/html5/thumbnails/42.jpg)
Phase transformations in NiMnGaThe model and its analysis
Some other phenomena to be involved
General nonlinear ansatzPinning effects
Pinning effects: phase field χ = χ(E(u),m) and additional dissipation ζ(.χ).
Thus, in terms of the 6-tuple (u,m, ϑ, e, h, ω), the original systemexpands to the following six equations/inclusion:
%..u − div
(σE(E(u),m, ϑ) + DE(
.u) +χ′E(E(u),m)>ω
− div(CH∇E(u)+DH∇E(
.u)))
= f0 − µ0∇h>m,
α.m − m× .m
γ(|m|)− λ∆m + σm(E(u),m, ϑ) = p0 + µ0h−χ′m(E(u),m)>ω,
.ϑ−div
(K0(E(u),m, ϑ)∇ϑ+K1(E(u),m, ϑ)∇E(u) +K2(E(u),m, ϑ)∇m
)= S(E(u),m, ϑ)e·e + DE(
.u):E(
.u) + DH∇E(
.u)
...∇E(.u)+α| .m|2
+A1(E(u),m, ϑ):E(.u) +A2(E(u),m, ϑ)· .m
+ ζ(χ′E(E(u),m)E(
.u)+χ′m(E(u),m)
.m),
µ0(.h +
.m) + curl e = −µ0∇m
.u − µ0(div
.u)m,
curl h − S(E(u),m, ϑ)e = 0,
ω ∈ ∂ζ(χ′E(E(u),m)E(
.u)+χ′m(E(u),m)
.m).
Tomas Roubıcek (Workshop, MFF, Prague, March 31, 2012) Phase transformations in NiMnGa
![Page 43: @let@token Modelling of Phase Transformations in ...prusv/ncmm/... · Shape-memory materials(SMM): alloys (=SMAs) or intermetalics. The mechanism behindshape-memory e ect(=SME): higher](https://reader035.vdocument.in/reader035/viewer/2022081617/602db37bf938dd296d06a0b4/html5/thumbnails/43.jpg)
Phase transformations in NiMnGaThe model and its analysis
Some other phenomena to be involved
General nonlinear ansatzPinning effects
Some references:M.Arndt, M.Griebel, V. Novak, T. Roubıcek, P.Sittner: Martensitictransformation in NiMnGa single crystals: numerical simulations andexperiments. Int. J. Plasticity 22 (2006), 1943-1961.
P. Plechac, T. Roubıcek: Visco-elasto-plastic model for martensitic phasetransformation in shape-memory alloys. M2AS 25 (2002), 1281–1298.
P. Podio-Guidugli, T. Roubıcek, G. Tomassetti: A thermodynamically-consistenttheory of the ferro/paramagnetic transition. Archive Rat. Mech. Anal. 198(2010), 1057-1094.
K.R. Rajagopal, T. Roubıcek: On the effect of dissipation in shape-memoryalloys. Nonlinear Anal., Real World Appl. 4 (2003), 581–597.
T. Roubıcek: Nonlinearly coupled thermo-visco-elasticity. NoDEA, submitted.
T. Roubıcek, G. Tomassetti: Thermodynamics of shape-memory alloys underelectric current. Zeit. angew. Math. Phys. 61 (2010), 1-20.
T. Roubıcek, G. Tomassetti: Ferromagnets with eddy currents and pinningeffects: their thermodynamics and analysis. M3AS 21 (2011), 29-55.
T. Roubıcek, G. Tomassetti: Phase transformations in electrically conductiveferromagnetic shape-memory alloys, their thermodynamics and analysis. ArchiveRat. Mech. Anal., submitted.
T. Roubıcek, G. Tomassetti, C. Zanini: The Gilbert equation withdry-friction-type damping. J. Math. Anal. Appl., 355 (2009), 453–468.
Some preprints available on:http://www.karlin.mff.cuni.cz/~roubicek/trpublic.htm
Tomas Roubıcek (Workshop, MFF, Prague, March 31, 2012) Phase transformations in NiMnGa
![Page 44: @let@token Modelling of Phase Transformations in ...prusv/ncmm/... · Shape-memory materials(SMM): alloys (=SMAs) or intermetalics. The mechanism behindshape-memory e ect(=SME): higher](https://reader035.vdocument.in/reader035/viewer/2022081617/602db37bf938dd296d06a0b4/html5/thumbnails/44.jpg)
Phase transformations in NiMnGaThe model and its analysis
Some other phenomena to be involved
General nonlinear ansatzPinning effects
Some references:M.Arndt, M.Griebel, V. Novak, T. Roubıcek, P.Sittner: Martensitic transformation in NiMnGa singlecrystals: numerical simulations and experiments. Int. J. Plasticity 22 (2006), 1943-1961.
P. Plechac, T. Roubıcek: Visco-elasto-plastic model for martensitic phase transformation in shape-memoryalloys. M2AS 25 (2002), 1281–1298.
P. Podio-Guidugli, T. Roubıcek, G. Tomassetti: A thermodynamically-consistent theory of theferro/paramagnetic transition. Archive Rat. Mech. Anal. 198 (2010), 1057-1094.
K.R. Rajagopal, T. Roubıcek: On the effect of dissipation in shape-memory alloys. Nonlinear Anal., RealWorld Appl. 4 (2003), 581–597.
T. Roubıcek: Nonlinearly coupled thermo-visco-elasticity. NoDEA, submitted.
T. Roubıcek, G. Tomassetti: Thermodynamics of shape-memory alloys under electric current. Zeit. angew.Math. Phys. 61 (2010), 1-20.
T. Roubıcek, G. Tomassetti: Ferromagnets with eddy currents and pinning effects: their thermodynamicsand analysis. M3AS 21 (2011), 29-55.
T. Roubıcek, G. Tomassetti: Phase transformations in electrically conductive ferromagnetic shape-memoryalloys, their thermodynamics and analysis. Archive Rat. Mech. Anal., submitted.
T. Roubıcek, G. Tomassetti, C. Zanini: The Gilbert equation with dry-friction-type damping. J. Math.Anal. Appl., 355 (2009), 453–468.
Thanks a lot for your attention.
Some preprints available on:http://www.karlin.mff.cuni.cz/~roubicek/trpublic.htm
Tomas Roubıcek (Workshop, MFF, Prague, March 31, 2012) Phase transformations in NiMnGa