Modelling of Phase Transformations in magnetostrictive materials - - PowerPoint PPT Presentation

▶
modelling of phase transformations in magnetostrictive
SMART_READER_LITE
LIVE PREVIEW

Modelling of Phase Transformations in magnetostrictive materials - - PowerPoint PPT Presentation

Nov 28, 2023 610 likes •1.07k views

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 a s Roub cek Charles University, Prague &

slide-1
SLIDE 1

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

slide-2
SLIDE 2

Phase transformations in NiMnGa The model and its analysis Some other phenomena to be involved

Content of the talk:

1 Phase transformations in NiMnGa

Martensitic/austenitic transformation Ferro/para-magnetic transformation Coupling of transformations: magnetostriction

2 The model and its analysis

Partly linearized ansatz Analysis: semi-implicit discretisation, a-priori estimates Analysis: convergence

3 Some other phenomena to be involved

General nonlinear ansatz Pinning effects

Tom´ aˇ s Roub´ ıˇ cek (Workshop, MFF, Prague, March 31, 2012) Phase transformations in NiMnGa

slide-3
SLIDE 3

Phase transformations in NiMnGa The model and its analysis Some other phenomena to be involved Martensitic/austenitic transformation Ferro/para-magnetic transformation Coupling 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

Tom´ aˇ s Roub´ ıˇ cek (Workshop, MFF, Prague, March 31, 2012) Phase transformations in NiMnGa

slide-4
SLIDE 4

Phase transformations in NiMnGa The model and its analysis Some other phenomena to be involved Martensitic/austenitic transformation Ferro/para-magnetic transformation Coupling 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 a lower-symmetrical latice: martensite phase, lower heat capacity.

Tom´ aˇ s Roub´ ıˇ cek (Workshop, MFF, Prague, March 31, 2012) Phase transformations in NiMnGa

slide-5
SLIDE 5

Phase transformations in NiMnGa The model and its analysis Some other phenomena to be involved Martensitic/austenitic transformation Ferro/para-magnetic transformation Coupling 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 a lower-symmetrical latice: martensite phase, lower heat capacity. the lower-symmetrical latice occurs in several variants;

Tom´ aˇ s Roub´ ıˇ cek (Workshop, MFF, Prague, March 31, 2012) Phase transformations in NiMnGa

slide-6
SLIDE 6

Phase transformations in NiMnGa The model and its analysis Some other phenomena to be involved Martensitic/austenitic transformation Ferro/para-magnetic transformation Coupling 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 a lower-symmetrical latice: martensite phase, lower heat capacity. the lower-symmetrical latice occurs in several variants;

Tom´ aˇ s Roub´ ıˇ cek (Workshop, MFF, Prague, March 31, 2012) Phase transformations in NiMnGa

slide-7
SLIDE 7

Phase transformations in NiMnGa The model and its analysis Some other phenomena to be involved Martensitic/austenitic transformation Ferro/para-magnetic transformation Coupling 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 a lower-symmetrical latice: martensite phase, lower heat capacity. the lower-symmetrical latice occurs in several variants; each of them can be rotated:

Tom´ aˇ s Roub´ ıˇ cek (Workshop, MFF, Prague, March 31, 2012) Phase transformations in NiMnGa

slide-8
SLIDE 8

Phase transformations in NiMnGa The model and its analysis Some other phenomena to be involved Martensitic/austenitic transformation Ferro/para-magnetic transformation Coupling 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 a lower-symmetrical latice: martensite phase, lower heat capacity. the lower-symmetrical latice occurs in several variants; each of them can be rotated:

Tom´ aˇ s Roub´ ıˇ cek (Workshop, MFF, Prague, March 31, 2012) Phase transformations in NiMnGa

slide-9
SLIDE 9

Phase transformations in NiMnGa The model and its analysis Some other phenomena to be involved Martensitic/austenitic transformation Ferro/para-magnetic transformation Coupling 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 a lower-symmetrical latice: martensite phase, lower heat capacity. the lower-symmetrical latice occurs in several variants; each of them can be rotated:

Tom´ aˇ s Roub´ ıˇ cek (Workshop, MFF, Prague, March 31, 2012) Phase transformations in NiMnGa

slide-10
SLIDE 10

Phase transformations in NiMnGa The model and its analysis Some other phenomena to be involved Martensitic/austenitic transformation Ferro/para-magnetic transformation Coupling 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 a lower-symmetrical latice: martensite phase, lower heat capacity. the lower-symmetrical latice occurs in several variants; each of them can be rotated:

Tom´ aˇ s Roub´ ıˇ cek (Workshop, MFF, Prague, March 31, 2012) Phase transformations in NiMnGa

slide-11
SLIDE 11

Phase transformations in NiMnGa The model and its analysis Some other phenomena to be involved Martensitic/austenitic transformation Ferro/para-magnetic transformation Coupling of transformations: magnetostriction

Crystalographical options of lower-symmetrical martensite: Self-accomodation of a microstructure in martensite

Tom´ aˇ s Roub´ ıˇ cek (Workshop, MFF, Prague, March 31, 2012) Phase transformations in NiMnGa

slide-12
SLIDE 12

Phase transformations in NiMnGa The model and its analysis Some other phenomena to be involved Martensitic/austenitic transformation Ferro/para-magnetic transformation Coupling of transformations: magnetostriction

Crystalographical options of lower-symmetrical martensite: Self-accomodation of a microstructure in martensite

Tom´ aˇ s Roub´ ıˇ cek (Workshop, MFF, Prague, March 31, 2012) Phase transformations in NiMnGa

slide-13
SLIDE 13

Phase transformations in NiMnGa The model and its analysis Some other phenomena to be involved Martensitic/austenitic transformation Ferro/para-magnetic transformation Coupling of transformations: magnetostriction

Crystalographical options of lower-symmetrical martensite: Self-accomodation of a microstructure in austenite and martensite

Tom´ aˇ s Roub´ ıˇ cek (Workshop, MFF, Prague, March 31, 2012) Phase transformations in NiMnGa

slide-14
SLIDE 14

Phase transformations in NiMnGa The model and its analysis Some other phenomena to be involved Martensitic/austenitic transformation Ferro/para-magnetic transformation Coupling of transformations: magnetostriction

Crystalographical options of lower-symmetrical martensite: Self-accomodation of a microstructure (example of CuAlNi)

Courtesy of . V´ aclav Nov´ ak and Petr ˇ Sittner, Institute of Physics, Academy of Sciences, Czech Rep.

Tom´ aˇ s Roub´ ıˇ cek (Workshop, MFF, Prague, March 31, 2012) Phase transformations in NiMnGa

slide-15
SLIDE 15

Phase transformations in NiMnGa The model and its analysis Some other phenomena to be involved Martensitic/austenitic transformation Ferro/para-magnetic transformation Coupling of transformations: magnetostriction

Schematic stress/strain response of SMM: low temperature vs high temperature quasiplasticity pseudoelasticity

Tom´ aˇ s Roub´ ıˇ cek (Workshop, MFF, Prague, March 31, 2012) Phase transformations in NiMnGa

slide-16
SLIDE 16

Phase transformations in NiMnGa The model and its analysis Some other phenomena to be involved Martensitic/austenitic transformation Ferro/para-magnetic transformation Coupling of transformations: magnetostriction

Experiments by L.Straka, V.Nov´ ak, 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 293 K (left) and 323 K (right):

1 2 3 4 5 6 7 8 10 20 30 40 50 compression strain [%] pressure [MPa] 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 20 40 60 80 100 120 compression strain [%] pressure [MPa]

Tom´ aˇ s Roub´ ıˇ cek (Workshop, MFF, Prague, March 31, 2012) Phase transformations in NiMnGa

slide-17
SLIDE 17

Phase transformations in NiMnGa The model and its analysis Some other phenomena to be involved Martensitic/austenitic transformation Ferro/para-magnetic transformation Coupling of transformations: magnetostriction

Computational simulations: Compression experiment with NiMnGa (001)-oriented singlecrystal Reorientation of martensite during a compression experiment at 293 K.

1 2 3 4 5 6 7 8 −40 −20 20 40 60 80 100 120 140 compression strain [%] pressure [MPa] 1 2 3 4 5 6 20 40 60 80 100 120 compression strain [%] pressure [MPa]

Stress/strain response during a compression experiment at 293 K and at 323 K.

Calculations, visualizations: courtesy of Marcel Arndt, Universit¨ at Bonn.

Tom´ aˇ s Roub´ ıˇ cek (Workshop, MFF, Prague, March 31, 2012) Phase transformations in NiMnGa

slide-18
SLIDE 18

Phase transformations in NiMnGa The model and its analysis Some other phenomena to be involved Martensitic/austenitic transformation Ferro/para-magnetic transformation Coupling of transformations: magnetostriction

Transformation in magnetic materials: low temperature (below Currie point): highly-ordered, ferromagnetic state very low temperature: the Heissenberg constraint |m| = Ms is well satisfied but in higher temperatures the deviation from it can be large in outer field high temperature (above Currie point Tc): dis-ordered, paramagnetic state

G.Bertotti: Hysteresis in Magnetism. Academic Press, San Diego, 1998.

Tom´ aˇ s Roub´ ıˇ cek (Workshop, MFF, Prague, March 31, 2012) Phase transformations in NiMnGa

slide-19
SLIDE 19

Phase transformations in NiMnGa The model and its analysis Some other phenomena to be involved Martensitic/austenitic transformation Ferro/para-magnetic transformation Coupling of transformations: magnetostriction

Both martensite/austenite and ferro/para-magnetic transformations are coupled: Strong dependence of thermo-mechanical response on magnetic field in Ni2MnGa single crystals – for example:

K.Ullakko, J.K.Huang, C.Kantner, R.C.O’Handley, V.V.Kokorin in Appl. Phys. Lett. 69 (1996), 1966–1968.

Tom´ aˇ s Roub´ ıˇ cek (Workshop, MFF, Prague, March 31, 2012) Phase transformations in NiMnGa

slide-20
SLIDE 20

Phase transformations in NiMnGa The model and its analysis Some other phenomena to be involved Martensitic/austenitic transformation Ferro/para-magnetic transformation Coupling of transformations: magnetostriction

Other phenomena to be captured: electric resistivity depending on temperature and phase (an example in NiTi):

V.Nov´ ak, P.ˇ Sittner, G.N.Dayananda, F.M.Braz-Fernandes, K.K.Mahesh, Materials Science and Engineering A 481-482 (2008) 127-133.

Tom´ aˇ s Roub´ ıˇ cek (Workshop, MFF, Prague, March 31, 2012) Phase transformations in NiMnGa

slide-21
SLIDE 21

Phase transformations in NiMnGa The model and its analysis Some other phenomena to be involved Partly linearized ansatz Analysis: semi-implicit discretisation, a-priori estimates Analysis: convergence

Variables (minimal scenario): u displacement, E(u) = 1

2(∇u)⊤+ 1 2∇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.

Tom´ aˇ s Roub´ ıˇ cek (Workshop, MFF, Prague, March 31, 2012) Phase transformations in NiMnGa

slide-22
SLIDE 22

Phase transformations in NiMnGa The model and its analysis Some other phenomena to be involved Partly linearized ansatz Analysis: semi-implicit discretisation, a-priori estimates Analysis: 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.

Tom´ aˇ s Roub´ ıˇ cek (Workshop, MFF, Prague, March 31, 2012) Phase transformations in NiMnGa

slide-23
SLIDE 23

Phase transformations in NiMnGa The model and its analysis Some other phenomena to be involved Partly linearized ansatz Analysis: semi-implicit discretisation, a-priori estimates Analysis: 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.

Tom´ aˇ s Roub´ ıˇ cek (Workshop, MFF, Prague, March 31, 2012) Phase transformations in NiMnGa

slide-24
SLIDE 24

Phase transformations in NiMnGa The model and its analysis Some other phenomena to be involved Partly linearized ansatz Analysis: semi-implicit discretisation, a-priori estimates Analysis: 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.

Tom´ aˇ s Roub´ ıˇ cek (Workshop, MFF, Prague, March 31, 2012) Phase transformations in NiMnGa

slide-25
SLIDE 25

Phase transformations in NiMnGa The model and its analysis Some other phenomena to be involved Partly linearized ansatz Analysis: semi-implicit discretisation, a-priori estimates Analysis: 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.

Tom´ aˇ s Roub´ ıˇ cek (Workshop, MFF, Prague, March 31, 2012) Phase transformations in NiMnGa

slide-26
SLIDE 26

Phase transformations in NiMnGa The model and its analysis Some other phenomena to be involved Partly linearized ansatz Analysis: semi-implicit discretisation, a-priori estimates Analysis: 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× . m γ(|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

  • internal

energy

+ µ0 2 |h|2

magnetic energy

+ ̺ 2|. u|2

kinetic energy

dx =

  • Ω

f0·. u

  • power of

external load

dx + boundary power. Gibbs’ relation: ψ = ε − sθ with entropy s = −φ′

θ

internal energy: ε = ϑ+ψ(E, m, 0)+ 1

2CH∇E.

. .∇E+ 1

2λ|∇m|2, enthapy ϑ =

θ

0 c(·).

Tom´ aˇ s Roub´ ıˇ cek (Workshop, MFF, Prague, March 31, 2012) Phase transformations in NiMnGa

slide-27
SLIDE 27

Phase transformations in NiMnGa The model and its analysis Some other phenomena to be involved Partly linearized ansatz Analysis: semi-implicit discretisation, a-priori estimates Analysis: 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

Tom´ aˇ s Roub´ ıˇ cek (Workshop, MFF, Prague, March 31, 2012) Phase transformations in NiMnGa

slide-28
SLIDE 28

Phase transformations in NiMnGa The model and its analysis Some other phenomena to be involved Partly linearized ansatz Analysis: semi-implicit discretisation, a-priori estimates Analysis: convergence

Fully implicit time-discretisation + regularization: Recursive formula for the 5-tuple (uk

τ, mk τ, ϑk τ, ek τ , hk τ) solving the system

Momentum-equilibrium equation: ̺uk

τ−2uk−1 τ

+uk−2

τ

τ 2 − div

  • Sk

τ − div Hk τ

  • = f k

τ − µ0(∇hk τ)⊤mk τ

with Sk

τ := σE(E(uk τ), mk τ, ϑk τ) + DE

uk

τ−uk−1 τ

τ

  • +τ|E(uk

τ)|η−2E(uk τ), and

Hk

τ := DH∇E

uk

τ−uk−1 τ

τ

  • +CH∇E(uk

τ)+τ|∇E(uk τ)|η−2∇E(uk τ),

Landau-Lifshitz-Gilbert equation: αmk

τ−mk−1 τ

τ − mk

τ

γ(|mk

τ|)×mk τ−mk−1 τ

τ − λ∆mk

τ + σm(E(uk τ), mk τ, ϑk τ)

− µ0hk

τ = τdiv

  • |∇mk

τ|η−2

∇mk

τ

  • − τ|mk

τ|η−2mk τ,

Tom´ aˇ s Roub´ ıˇ cek (Workshop, MFF, Prague, March 31, 2012) Phase transformations in NiMnGa

slide-29
SLIDE 29

Phase transformations in NiMnGa The model and its analysis Some other phenomena to be involved Partly linearized ansatz Analysis: semi-implicit discretisation, a-priori estimates Analysis: convergence

Heat equation: ϑk

τ − ϑk−1 τ

τ − div

  • K0(Ek

τ, mk τ, ϑk τ)∇ϑk τ

  • = S(Ek

τ, mk τ, ϑk τ)ek τ :ek τ

+

  • 1−

√τ 2

  • DE

uk

τ−uk−1 τ

τ

  • :E

uk

τ−uk−1 τ

τ

  • + DH∇E

uk

τ−uk−1 τ

τ . . .∇E uk

τ−uk−1 τ

τ

  • +
  • 1−

√τ 2

  • α
  • mk

τ−mk−1 τ

τ

  • 2

+ A

  • Ek

τ, mk τ, ϑk τ; Ek τ−Ek−1 τ

τ , mk

τ−mk−1 τ

τ

  • ,

Maxwell system: hk

τ−hk−1 τ

τ + curl ek

τ

µ0 = ∇mk

τ

uk

τ−uk−1 τ

τ − mk

τ−mk−1 τ

τ + divuk

τ−uk−1 τ

τ mk

τ,

curl hk

τ − S(Ek τ, mk τ, ϑk τ) ek τ = τ|ek τ |η−2ek τ ,

for k = 1, ..., Kτ := T/τ, where we abbreviated Ek

τ = E(uk τ) 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.

Tom´ aˇ s Roub´ ıˇ cek (Workshop, MFF, Prague, March 31, 2012) Phase transformations in NiMnGa

slide-30
SLIDE 30

Phase transformations in NiMnGa The model and its analysis Some other phenomena to be involved Partly linearized ansatz Analysis: semi-implicit discretisation, a-priori estimates Analysis: convergence

We use the discrete scheme recursively, starting from k = 1 by using u0

τ = u0,τ,

u−1

τ

= u0,τ−τv0, m0

τ = m0,τ,

ϑ0

τ = ˆ

c(θ0), h0

τ = h0,

Existence of (uk

τ, mk τ, ϑk τ, ek τ , hk τ):

η large enough (namely η > 8) ⇒ pseudomonotone coercive operator ⇒ Br´ ezis’ theorem ⇒ uk

τ∈W 2,η(Ω;I

R3), mk

τ∈W 1,η(Ω;I

R3), ϑk

τ∈W 1,2(Ω),

hk

τ∈L2,η′ curl (Ω;I

R3), ek

τ ∈Lη,2 curl (Ω;I

R3) where Lp,q

curl (Ω; I

R3) :=

  • v ∈Lp(Ω; I

R3); curl v ∈Lq(Ω; I R3)

  • .

Non-negativity: ϑk

τ ≥ 0.

Tom´ aˇ s Roub´ ıˇ cek (Workshop, MFF, Prague, March 31, 2012) Phase transformations in NiMnGa

slide-31
SLIDE 31

Phase transformations in NiMnGa The model and its analysis Some other phenomena to be involved Partly linearized ansatz Analysis: semi-implicit discretisation, a-priori estimates Analysis: 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;I

R3))∩W 1,2(I;W 2,2(Ω;I R3)) ≤ C,

  • mτ
  • L∞(I;W 1,2(Ω;I

R3))∩W 1,2(I;L2(Ω;I R3)) ≤ C,

  • ¯

ϑτ

  • L∞(I;L1(Ω)) ≤ C,
  • hτ
  • L∞(I;L2(Ω;I

R3)) ≤ C,

  • ¯

eτ

  • L2(Q;I

R3) ≤ C,

  • E(¯

uτ)

  • L∞(I;W 1,η(Ω;)) ≤ Cτ −1/η,
  • mτ
  • L∞(I;W 1,η(Ω;I

R3)) ≤ Cτ −1/η,

  • eτ
  • Lη(Q;I

R3) ≤ Cτ −1/η

based on the semi-convexity of the functional (for enough small τ > 0) (E, m) → ϕ(E, m) + τ η |E|η + τ η |m|η + DE:E + α|m|2 2√τ

Tom´ aˇ s Roub´ ıˇ cek (Workshop, MFF, Prague, March 31, 2012) Phase transformations in NiMnGa

slide-32
SLIDE 32

Phase transformations in NiMnGa The model and its analysis Some other phenomena to be involved Partly linearized ansatz Analysis: semi-implicit discretisation, a-priori estimates Analysis: 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;I

R3))∩W 1,2(I;W 2,2(Ω;I R3)) ≤ C,

  • mτ
  • L∞(I;W 1,2(Ω;I

R3))∩W 1,2(I;L2(Ω;I R3)) ≤ C,

  • ¯

ϑτ

  • L∞(I;L1(Ω)) ≤ C,
  • hτ
  • L∞(I;L2(Ω;I

R3)) ≤ C,

  • ¯

eτ

  • L2(Q;I

R3) ≤ C,

  • E(¯

uτ)

  • L∞(I;W 1,η(Ω;)) ≤ Cτ −1/η,
  • mτ
  • L∞(I;W 1,η(Ω;I

R3)) ≤ Cτ −1/η,

  • eτ
  • Lη(Q;I

R3) ≤ Cτ −1/η

based on the semi-convexity of the functional (for enough small τ > 0) (E, m) → ϕ(E, m) + τ η |E|η + τ η |m|η + DE:E + α|m|2 2√τ

Tom´ aˇ s Roub´ ıˇ cek (Workshop, MFF, Prague, March 31, 2012) Phase transformations in NiMnGa

slide-33
SLIDE 33

Phase transformations in NiMnGa The model and its analysis Some other phenomena to be involved Partly linearized ansatz Analysis: semi-implicit discretisation, a-priori estimates Analysis: 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;I

R3))∩W 1,2(I;W 2,2(Ω;I R3)) ≤ C,

  • mτ
  • L∞(I;W 1,2(Ω;I

R3))∩W 1,2(I;L2(Ω;I R3)) ≤ C,

  • ¯

ϑτ

  • L∞(I;L1(Ω)) ≤ C,
  • hτ
  • L∞(I;L2(Ω;I

R3)) ≤ C,

  • ¯

eτ

  • L2(Q;I

R3) ≤ C,

  • E(¯

uτ)

  • L∞(I;W 1,η(Ω;)) ≤ Cτ −1/η,
  • mτ
  • L∞(I;W 1,η(Ω;I

R3)) ≤ Cτ −1/η,

  • eτ
  • Lη(Q;I

R3) ≤ Cτ −1/η,

+ a special nonlinear test of the heat equation + Gagliardo-Nirenberg interpolation

  • ∇¯

ϑτ

  • Lr (Q;I

Rd) ≤ Cr

with r < 5/4.

Tom´ aˇ s Roub´ ıˇ cek (Workshop, MFF, Prague, March 31, 2012) Phase transformations in NiMnGa

slide-34
SLIDE 34

Phase transformations in NiMnGa The model and its analysis Some other phenomena to be involved Partly linearized ansatz Analysis: semi-implicit discretisation, a-priori estimates Analysis: convergence

Further a-priori estimates:

  • ̺..

u i

τ

  • L2(I;W 2,2(Ω;I

R3)∗)+Lη′(I;W 2,η(Ω;I R3)∗) ≤ C,

  • ̺..

u i

τ − τdiv

  • |E(¯

uτ)|η−2E(¯ uτ)

  • + τdiv2

|∇E(¯ uτ)|η−2∇E(¯ uτ)

  • L2(I;W 2,2(Ω;I

R3)∗) ≤ C,

  • .

ϑτ

  • L1(I;W 3,2(Ω)∗) ≤ C,
  • curl ¯

hτ + τ|¯ eτ|η−2¯ eτ

  • L2(Q;I

R3) ≤ C,

  • .

hτ

  • L2(I;L2

curl ,0(Ω;I

R3)∗) ≤ C.

Tom´ aˇ s Roub´ ıˇ cek (Workshop, MFF, Prague, March 31, 2012) Phase transformations in NiMnGa

slide-35
SLIDE 35

Phase transformations in NiMnGa The model and its analysis Some other phenomena to be involved Partly linearized ansatz Analysis: semi-implicit discretisation, a-priori estimates Analysis: convergence

Convergence for τ → 0: Step 0: Banach’ selection principle: uτ → u strongly in W 1,2(I; W 2,2(Ω; I R3)), mτ → m strongly in W 1,2(I; W 1,2(Ω; I R3)), ¯ ϑτ → ϑ strongly in Ls(Q) with any s < 5/3, ¯ eτ → e strongly in L2(Q; I R3), ¯ hτ → h weakly* in L∞(I; L2(Ω; I R3)), and, moreover (with hb from not-mentioned boundary conditions) ¯ hτ−¯ hb,τ → h−hb weakly in Lη′(I; L2,η′

curl ,0(Ω; I

R3)), and curl ¯ hτ + τ|¯ eτ|γ−2¯ eτ → curl h weakly in L2(Q; I R3×3). for a subsequence. Then we want to prove that any (u, m, ϑ, h, e) obtained in this way is a weak solution to the considered IBVP (after the transformation θ → ϑ) which also preserves the total energy.

Tom´ aˇ s Roub´ ıˇ cek (Workshop, MFF, Prague, March 31, 2012) Phase transformations in NiMnGa

slide-36
SLIDE 36

Phase transformations in NiMnGa The model and its analysis Some other phenomena to be involved Partly linearized ansatz Analysis: semi-implicit discretisation, a-priori estimates Analysis: 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, while the quasilinear regularizing terms vanish, e.g.

  • Q

τ|E(¯ uτ)|η−2E(¯ uτ):E(v) dxdt

  • ≤ τE(¯

uτ)η−1

Lη(Q;I R3×3)E(v)Lη(Q;I R3×3)

≤ Cτ 1/ηE(v)Lη(Q;I

R3×3) → 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 i

τ−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 = lim

τ→0

  • Q

̺. ui

τ · .

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; I R3)).

Tom´ aˇ s Roub´ ıˇ cek (Workshop, MFF, Prague, March 31, 2012) Phase transformations in NiMnGa

slide-37
SLIDE 37

Phase transformations in NiMnGa The model and its analysis Some other phenomena to be involved Partly linearized ansatz Analysis: semi-implicit discretisation, a-priori estimates Analysis: 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 inf

τ→0

  • Q

DE(. uτ):E(. uτ) + DH∇E(. uτ):∇E(. uτ) + α|. mτ|2 + S(¯ Eτ, ¯ mτ, ¯ ϑτ)¯ eτ·¯ eτdx ≤ lim sup

τ→0

  • Q

DE(. uτ):E(. uτ) + DH∇E(. uτ):∇E(. uτ) + α|. mτ|2 + S(¯ Eτ, ¯ mτ, ¯ ϑτ)¯ eτ·¯ eτdx ≤ lim sup

τ→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.

Tom´ aˇ s Roub´ ıˇ cek (Workshop, MFF, Prague, March 31, 2012) Phase transformations in NiMnGa

slide-38
SLIDE 38

Phase transformations in NiMnGa The model and its analysis Some other phenomena to be involved Partly linearized ansatz Analysis: semi-implicit discretisation, a-priori estimates Analysis: convergence

Step 4: Limit passage in the heat equation: Having proved the strong convergence in Step 2, the right-hand side of the heat equation converges strongly in L1(Q) and this limit passage is then 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 perform rigorously this test and sum it with mechanical/magnetic energy balance

  • btained already in Step 2.

Tom´ aˇ s Roub´ ıˇ cek (Workshop, MFF, Prague, March 31, 2012) Phase transformations in NiMnGa

slide-39
SLIDE 39

Phase transformations in NiMnGa The model and its analysis Some other phenomena to be involved General nonlinear ansatz Pinning effects

Fully nonlinear coupling: more symmetry ∼ higher heat capacity heat 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 = ˆ c(E, m, θ) := θ c(E, m, Θ)dΘ c(E, m, θ). θ = ∂ˆ c(E, m, θ) ∂t − c1(E, m, θ):. E − c2(E, m, θ)·. m, c1(E, m, θ) = θ c′

E(E, m, Θ)dΘ

and c2(E, m, θ) = θ c′

m(E, m, Θ)dΘ.

Tom´ aˇ s Roub´ ıˇ cek (Workshop, MFF, Prague, March 31, 2012) Phase transformations in NiMnGa

slide-40
SLIDE 40

Phase transformations in NiMnGa The model and its analysis Some other phenomena to be involved General nonlinear ansatz Pinning effects

Define: T (E, m, ·) := [ˆ 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.

Tom´ aˇ s Roub´ ıˇ cek (Workshop, MFF, Prague, March 31, 2012) Phase transformations in NiMnGa

slide-41
SLIDE 41

Phase transformations in NiMnGa The model and its analysis Some other phenomena to be involved General nonlinear ansatz Pinning effects

Thus, in terms of the 5-tuple (u, m, ϑ, e, h), the original system transforms 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,

Tom´ aˇ s Roub´ ıˇ cek (Workshop, MFF, Prague, March 31, 2012) Phase transformations in NiMnGa

slide-42
SLIDE 42

Phase transformations in NiMnGa The model and its analysis Some other phenomena to be involved General nonlinear ansatz Pinning 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 system expands 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

  • .

Tom´ aˇ s Roub´ ıˇ cek (Workshop, MFF, Prague, March 31, 2012) Phase transformations in NiMnGa

slide-43
SLIDE 43

Phase transformations in NiMnGa The model and its analysis Some other phenomena to be involved General nonlinear ansatz Pinning effects

Some references:

M.Arndt, M.Griebel, V. Nov´ ak, T. Roub´ ıˇ cek, P.ˇ Sittner: Martensitic transformation in NiMnGa single crystals: numerical simulations and

  • experiments. Int. J. Plasticity 22 (2006), 1943-1961.
  • P. Plech´

aˇ c, T. Roub´ ıˇ cek: Visco-elasto-plastic model for martensitic phase transformation in shape-memory alloys. M2AS 25 (2002), 1281–1298.

  • P. Podio-Guidugli, T. Roub´

ıˇ cek, G. Tomassetti: A thermodynamically-consistent theory of the ferro/paramagnetic transition. Archive Rat. Mech. Anal. 198 (2010), 1057-1094. K.R. Rajagopal, T. Roub´ ıˇ cek: On the effect of dissipation in shape-memory

  • alloys. Nonlinear Anal., Real World Appl. 4 (2003), 581–597.
  • T. Roub´

ıˇ cek: Nonlinearly coupled thermo-visco-elasticity. NoDEA, submitted.

  • T. Roub´

ıˇ cek, G. Tomassetti: Thermodynamics of shape-memory alloys under electric current. Zeit. angew. Math. Phys. 61 (2010), 1-20.

  • T. Roub´

ıˇ cek, G. Tomassetti: Ferromagnets with eddy currents and pinning effects: their thermodynamics and analysis. M3AS 21 (2011), 29-55.

  • T. Roub´

ıˇ cek, G. Tomassetti: Phase transformations in electrically conductive ferromagnetic shape-memory alloys, their thermodynamics and analysis. Archive

  • Rat. Mech. Anal., submitted.
  • T. Roub´

ıˇ cek, G. Tomassetti, C. Zanini: The Gilbert equation with dry-friction-type damping. J. Math. Anal. Appl., 355 (2009), 453–468.

Some preprints available on: http://www.karlin.mff.cuni.cz/~roubicek/trpublic.htm

Tom´ aˇ s Roub´ ıˇ cek (Workshop, MFF, Prague, March 31, 2012) Phase transformations in NiMnGa

slide-44
SLIDE 44

Phase transformations in NiMnGa The model and its analysis Some other phenomena to be involved General nonlinear ansatz Pinning effects

Some references:

M.Arndt, M.Griebel, V. Nov´ ak, T. Roub´ ıˇ cek, P.ˇ Sittner: Martensitic transformation in NiMnGa single crystals: numerical simulations and experiments. Int. J. Plasticity 22 (2006), 1943-1961.

  • P. Plech´

aˇ c, T. Roub´ ıˇ cek: Visco-elasto-plastic model for martensitic phase transformation in shape-memory

  • alloys. M2AS 25 (2002), 1281–1298.
  • P. Podio-Guidugli, T. Roub´

ıˇ cek, G. Tomassetti: A thermodynamically-consistent theory of the ferro/paramagnetic transition. Archive Rat. Mech. Anal. 198 (2010), 1057-1094. K.R. Rajagopal, T. Roub´ ıˇ cek: On the effect of dissipation in shape-memory alloys. Nonlinear Anal., Real World Appl. 4 (2003), 581–597.

  • T. Roub´

ıˇ cek: Nonlinearly coupled thermo-visco-elasticity. NoDEA, submitted.

  • T. Roub´

ıˇ cek, G. Tomassetti: Thermodynamics of shape-memory alloys under electric current. Zeit. angew.

  • Math. Phys. 61 (2010), 1-20.
  • T. Roub´

ıˇ cek, G. Tomassetti: Ferromagnets with eddy currents and pinning effects: their thermodynamics and analysis. M3AS 21 (2011), 29-55.

  • T. Roub´

ıˇ cek, G. Tomassetti: Phase transformations in electrically conductive ferromagnetic shape-memory alloys, their thermodynamics and analysis. Archive Rat. Mech. Anal., submitted.

  • T. Roub´

ıˇ cek, 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

Tom´ aˇ s Roub´ ıˇ cek (Workshop, MFF, Prague, March 31, 2012) Phase transformations in NiMnGa