Phase transformations in NiMnGa The model and its analysis Some other phenomena to be involved Modelling of Phase Transformations in magnetostrictive materials like NiMnGa Tom´ aˇ s Roub´ ıˇ cek Charles University, Prague & Academy of Sciences of the Czech Rep. & University of West Bohemia reflecting collaboration with Giuseppe Tomassetti and M.Arndt, M.Griebel, V.Nov´ ak, P.Plech´ aˇ c, P.Podio-Guidugli, K.R.Rajagopal, P. ˇ Sittner, C.Zanini and others. Tom´ aˇ s Roub´ ıˇ cek (Workshop, MFF, Prague, March 31, 2012) Phase transformations in NiMnGa
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@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
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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
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
Phase transformations in NiMnGaThe model and its analysis
Some other phenomena to be involved
Martensitic/austenitic transformationFerro/para-magnetic transformationCoupling of transformations: magnetostriction
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
Phase transformations in NiMnGaThe model and its analysis
Some other phenomena to be involved
Martensitic/austenitic transformationFerro/para-magnetic transformationCoupling of transformations: magnetostriction
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
Phase transformations in NiMnGaThe model and its analysis
Some other phenomena to be involved
Martensitic/austenitic transformationFerro/para-magnetic transformationCoupling of transformations: magnetostriction
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
Phase transformations in NiMnGaThe model and its analysis
Some other phenomena to be involved
Martensitic/austenitic transformationFerro/para-magnetic transformationCoupling of transformations: magnetostriction
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
Phase transformations in NiMnGaThe model and its analysis
Some other phenomena to be involved
Martensitic/austenitic transformationFerro/para-magnetic transformationCoupling of transformations: magnetostriction
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
Phase transformations in NiMnGaThe model and its analysis
Some other phenomena to be involved
Martensitic/austenitic transformationFerro/para-magnetic transformationCoupling of transformations: magnetostriction
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
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
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
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
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
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
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
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
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
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:
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
Phase transformations in NiMnGaThe model and its analysis
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
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.
Tomas Roubıcek (Workshop, MFF, Prague, March 31, 2012) Phase transformations in NiMnGa
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.
Tomas Roubıcek (Workshop, MFF, Prague, March 31, 2012) Phase transformations in NiMnGa
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:
Tomas Roubıcek (Workshop, MFF, Prague, March 31, 2012) Phase transformations in NiMnGa
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
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