NSF-PIRE Summer School Geometrically linear theory for shape memory - - PowerPoint PPT Presentation

NSF-PIRE Summer School Geometrically linear theory for shape memory alloys: the effect of interfacial energy

Felix Otto Max Planck Institute for Mathematics in the Sciences Leipzig, Germany

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Goal of mini-course Introduction to 3 recent works on microstructure or absence thereof in cubic-to-tetragonal phase transformation Approximate rigidity of twins, periodic case

Capella, O.: A rigidity result for a perturbation of the geomet- rically linear three-well problem, CPAM 62, 2009

Approximate rigidity of twins, local case

Capella, O.: A quantitative rigidity result for the cubic-to-tetragonal

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phase transition in the geometrically linear theory with interfa- cial energy, Proc. Roy. Soc. Edinburgh A, to appear

Optimal microstructure of Martensitic inclusions

Kn¨ upfer, Kohn, O.: Nucleation barriers for the cubic-to-tetragonal phase transformation, CPAM, to appear

See www.mis.mpg.de for copies (Otto, Publications, Shape-Memory Alloys)

Structure of mini-course

• Chap 1. Kinematics
• Chap 2. 2-d models

square-to-rectangular, hexagonal-to-rhombic

• Chap 3. 3-d models

cubic-to-tetragonal, [cubic-to-orthorombic]

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Structure of Chapter 1 on kinematics 1.1 Strain a geometrically linear description 1.2 Rigidity of skew symmetric gradients 1.3 Twins and rank-one connections 1.4 Triple junctions are rare 1.5 Quadruple junctions are more generic

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Structure of Chapter 2 on 2-d models Square-to-rectangular phase transformation 2.1 Derivation of the linearized two-well problem 2.2 Rigidity of twins 2.3 Elastic and interfacial energies 2.4 Derivation of a reduced model for twinned-Martensite to Austenite interface 2.5 Self-consistency of reduced model, lower bounds by interpolation, upper bounds by construction

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Structure of Chapter 2 on 2-d models, cont Hexagonal-to-rhombic phase transformation 2.6 Derivation of the linearized three-well problem 2.7 Twins and sextuple junctions 2.8 Loss of rigidity by convex integration

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2.4 Derivation of a reduced model for the twinned-Martensite to Austenite interface Phase indicator function: χ ∈ {−1, 0, 1}, χ = χ(x1, x2) Displacement field: u = (u1, u2), u = u(x1, x2) Interfacial energy: η

• length of interface between {χ = 1} and {χ = −1}

+ length of interface between {χ = 1} and {χ = 0} + length of interface between {χ = −1} and {χ = 0}

• Elastic energy:
• 1

2(∇ + ∇t)u −

  0 χ

χ 0

 

• 2

dx1dx2

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2.4 Derivation of a reduced model for the twinned-Martensite to Austenite interface Simplification 1 Impose position of twinned-Martensite to Austenite interface Simplification 2 Impose shear direction Simplification 3 Anisotropic rescaling and limit

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Simplification 1): Impose position of twinned-Martensite to Austenite interface Position of interface {x2 = 0}: χ

  

∈ {−1, 1} for x1 > 0 = 0 for x1 < 0

  

Nondimensionalize length by restriction to x1 ∈ (−1, 1), regime of interest η ≪ 1 Impose (artificial) L-periodicity in x2 Interfacial energy η

• 1

2

• (0,1)×[0,L)|∇χ| + L
• 9

Simplification 2): Impose shear direction Favor twin normal n =

1

• by imposing shear direction a =

2

• . i. e.

u2 ≡ 0 but u1 = u1(x1, x2) Strain 1

2(∇ + ∇t)u =

  ∂1u1

1 2∂2u1 1 2∂2u1

 

Elastic energy

1

−1

L

0 (∂1u1)2 + 2(1 2∂2u1 − χ)2dx2dx1

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Simplification 3): Anisotropic rescaling and limit 1 L

η

2

• (0,1)×[0,L)|∇χ|

+

• (−1,1)×(0,L)(∂1u1)2 + 2(1

2∂2u1 − χ)2dx

• Ansatz for rescaling

x2 = ηαˆ x2 = ⇒ ∂2 = η−αˆ ∂2. L = ηαˆ L, u1 = 2ηαˆ u1 = ⇒ ∂2u1 = 2ˆ ∂2ˆ u1, ∂1u1 = 2ηα∂1ˆ u1. 1 ˆ L

η

2

• (0,1)×[0,L)|
• ∂1χ

η−αˆ ∂2χ

• |

+

• (−1,1)×(0,L) 4η2α(∂1ˆ

u1)2 + 2(ˆ ∂2ˆ u1 − χ)2dˆ x

• 11

Seek nontrivial limit: elastic part Elastic energy density: 4η2α(∂1ˆ u1)2 + 2(ˆ ∂2ˆ u1 − χ)2 Penalization of ˆ ∂2ˆ u1 − χ ≫ penalization of ∂1ˆ u1 Neclegting ∂1ˆ u1 no option — otherwise no elastic effect Hence constraint ˆ ∂2ˆ u1 − χ = 0 in limit.

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Seek nontrivial limit: interfacial part Interfacial energy density:

η 2|

• ∂1χ

η−αˆ ∂2χ

• |

Penalization of ˆ ∂2χ ≫ penalization of ∂1χ Constraint ˆ ∂2χ = 0 no option — otherwise no twin Hence have to neglect penalization of ∂1χ Interfacial energy density

η1−α 2 |ˆ

∂2χ| in limit.

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Seek nontrivial limit: choice of α Total energy density 4η2α(∂1ˆ u1)2 + η1−α

2 |ˆ

∂2χ| For balance need η2α ∼ η1−α ⇒ α = 1

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Rescaling of energy density:

1 LE = η

2 3 1

ˆ L ˆ

E Prediction from 1

LE = η

2 3 1

ˆ L ˆ

E: energy density ∼ η

2 3

Prediction from x2 = η

1 3ˆ

x2 : twin width ∼ η

1 3

... provided limit model makes sense for ˆ L ≫ 1

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Limit model is singular Minimize 4

1

−1

ˆ

L 0 (∂1ˆ

u1)2dˆ x2dx1 + 1

2

1

• [0,ˆ

L)|ˆ

∂2χ|dx1 subject to ˆ ∂2ˆ u1 = χ

• ∈ {−1, 1}

for x1 > 0 = 0 for x1 < 0

• .

1 2

• [0,ˆ

L)

|ˆ ∂2χ| just counts transitions between 1 and -1

Infinite twin refinement: Elastic energy = ⇒ ˆ u1 = const = 0 for x1 < 0 = ⇒ ˆ u1(x1, ·) → 0 as x1 ↓ 0 = ⇒ χ(x1, ·) ⇀ 0 as x1 ↓ 0 = ⇒

• [0,ˆ

L) |ˆ

∂2χ(x1, ·)| ↑ ∞ as x1 ↓ 0 Interfacial energy = ⇒

1

• [0,ˆ

L) |ˆ

∂2χ(x1, ·)|dx1 < ∞ ... does limit model have finite energy?

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2.5 Self consistency of reduced model, upper bounds by construction, lower bounds by interpolation Proposition 2 [Kohn, M¨ uller] Functional: E = 4

1

−1

L

0 (∂1u1)2dx2dx1 + 1 2

1

• [0,L) |∂2χ|dx1.

Admissible configurations: u1, χ L-periodic in x2 with ∂2u1 = χ

• ∈ {−1, 1}

for x1 > 0 = 0 for x1 < 0

• .

Then ∃ universal C < ∞ such that i) upper bound ∀ L ∃ (u1, χ) E ≤ CL, ii) lower bound ∀ L, (u1, χ) E ≥ 1

CL.

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Proof of Proposition 2 i) (Construction)

• W. l. o. g. L = 1.

Step 1 Building block for branched structure on (0, 1) × (0, 1) Step 2 Rescaling construction on (0, H) × (0, 1) Step 3 Concatenation construction on (0, 1) × (0, 1)

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Proof of Proposition 2 i) (lower bound) Lemma 7 ∃ universal C < ∞ ∀ L-periodic u1(x2), χ(x2) related by ∂2u1 = χ with

L

0 χ2dx1 ≤ C

L

0 u2 1dx2

1

3 L

0 |∂2χ|dx2 sup x2

|χ|

2

3

. Holds in any d as χL2 ≤ C(d)|∇|−1χ

1 3

L2 ∇χ

1 3

L1 χ

1 3

L∞

Simpler version of χ

L

4 3 ≤ C(d)|∇|−1χ 2 3

L2 ∇χ

1 3

L1

(Cohen-Dahmen-Daubechies-Devore)

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2.8 Loss of rigidity by convex integration Proposition 3 [M¨ uller, Sver´ ak] ∀ M s. t. 1

2(M + Mt) ∈ int conv{E0, E1, E2}

∀ Ω ⊂ R2 open, bdd. ∃ u: R2 → R2 with ∇u ∈ L2

loc

∇u = M in R2 − ¯ Ω, 1 2(∇ + ∇t)u ∈ {E0, E1, E2}

• a. e. on Ω.

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Step 1: Conti’s construction = Lemma 5 Consider for λ = 1

4:

M0 = 1

λ

• 1
• , M1 =

1 1−λ

• −1

1

• , M2 =

1 1−λ2

• 1

−λ λ −1

• ,

M3 =

1 1−λ2

• −1 −λ

λ 1

• , M4 = 1

λ

• −1
• , Ω = (−1, 1)2.

Then ∃ Ω0, · · · Ω4 ⊂ Ω finite of convex, open sets ∃ u: R2 → R2 Lipschitz s. t. ∇u = in R2 − ¯ Ω, ∇u = Mi in Ωi, |Ω0| =

1 2λ|Ω|.

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Step 2: Deformation and rotation of Conti’s construction ∀ M, M0, M1 s. t. M = 1

4M0 + 3 4M1 with

M1 − M0 = a ⊗ n for some a ∈ R2, n ∈ S1, a · n = 0 ∀ ǫ > 0 ∃ ˜ M1, · · · , ˜ M4 s. t. | ˜ M1−M1|, | ˜ M2/3−M2|, | ˜ M4−M| < ǫ, where M2 := 1

5M0 + 4 5M1.

∃ Ω ⊂ R2 open, bdd., ˜ Ω1, · · · , ˜ Ω4 ⊂ Ω finite of convex, open sets ∃ u: R2 → R2 Lipschitz with ∇u = M in R2 − ¯ Ω, ∇u = ˜ Mi in ˜ Ωi, ∇u = M0 in Ω − (˜ Ω1 ∪ · · · ∪ ˜ Ω4), |˜ Ω1 ∪ · · · ∪ ˜ Ω4| ≤

7 8|Ω|.

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Step 3: Application to hexagonal-to-rhombic ∀ M s. t. 1

2(M + Mt) ∈ int conv{E0, E1, E2}

∃ ˜ M1, · · · , ˜ M4 s. t. 1

2( ˜

Mi + ˜ Mt

i ) ∈ int conv{E0, E1, E2}

∃ Ω ⊂ R2 open, bdd., ˜ Ω1, · · · , ˜ Ω4 ⊂ Ω finite of convex, open sets ∃ u: R2 → R2 Lipschitz with ∇u = M in R2 − ¯ Ω, ∇u = ˜ Mi in ˜ Ωi,

1 2(∇ + ∇t)u

∈ {E0, E1, E2} in Ω − (˜ Ω1 ∪ · · · ∪ ˜ Ω4), |˜ Ω1 ∪ · · · ∪ ˜ Ω4| ≤

7 8|Ω|.

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Step 4: Concatenation ∀ M s. t. 1

2(M + Mt) ∈ int conv{E0, E1, E2}

∀ Ω ⊂ R2 open, bbd ∃ ˜ M1, · · · , ˜ M4 s. t. 1

2( ˜

Mi + ˜ Mt

i ) ∈ int conv{E0, E1, E2}

∃ ˜ Ω1, · · · , ˜ Ω4 ⊂ Ω countable of convex, open sets ∃ u: R2 → R2 Lipschitz with ∇u = M in R2 − ¯ Ω, ∇u = ˜ Mi in ˜ Ωi,

1 2(∇ + ∇t)u

∈ {E0, E1, E2} in Ω − (˜ Ω1 ∪ · · · ∪ ˜ Ω4), |˜ Ω1 ∪ · · · ∪ ˜ Ω4| ≤

7 8|Ω|.

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Step 5: Iteration via replacement ∀ M s. t. 1

2(M + Mt) ∈ int conv{E0, E1, E2}

∀N ∈ N ∀ Ω ⊂ R2 open, bbd ∃ ˜ M1, · · · , ˜ M4N s. t. 1

2( ˜

Mi + ˜ Mt

i) ∈ int conv{E0, E1, E2}

∃ ˜ Ω1, · · · , ˜ Ω4N ⊂ Ω countable of convex, open sets ∃ u: R2 → R2 Lipschitz with ∇u = M in R2 − ¯ Ω, ∇u = ˜ Mi in ˜ Ωi,

1 2(∇ + ∇t)u

∈ {E0, E1, E2} in Ω − (˜ Ω1 ∪ · · · ∪ ˜ Ω4N), |˜ Ω1 ∪ · · · ∪ ˜ Ω4N| ≤ (7

8)N|Ω|.

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3.1 3-d models, cubic-to-tetragonal phase transformation 3 stress-free strains = Martensitic variants: E1 :=

  

−2 1 1

   , E2 :=   

1 −2 1

   , E3 :=   

1 1 −2

  

6 Martensitic twins with normals: n ∈ {(0, 1, 1), (0, 1, −1), (1, 0, 1), (−1, 0, 1), (1, 1, 0), (1, −1, 0)} No twin between Austenite

     

and three Martensitic variants E1, E2, E3

• cf. Lemma 3

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Rigidity of twins Dolzmann & M¨ uller, Meccanica ’95 Proposition 4 (Dolzmann & M¨ uller) Let u: R3 ⊃ B1 → R3 be Lipschitz with 1 2(∇ + ∇t)u ∈ {E1, E2, E3}

• a. e. in B1.

Then u = one of the six Martensitic twins on Bδ

(with δ > 0 universal).

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Approximate rigidity of twins Elastic + interfacial energy on B1: E :=

• B1 |1

2(∇ + ∇t)u − (χ1E1 + χ2E2 + χ3E3)|2dx

+ η

• B1 |∇χ1| + |∇χ2| + |∇χ3|

Admissible phase functions χi ∈ {0, 1}, χ1 + χ2 + χ3 ≤ 1 Proposition 5 (Capella & O.) Suppose E ≪ η2/3. Then (u, χ1, χ2, χ3) ≈ Austenite or one of the six Martensitic twins.

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Optimal Martensitic inclusions Energy in whole space E :=

• R3 |1

2(∇ + ∇t)u − (χ1E1 + χ2E2 + χ3E3)|2dx

+

• R3 |∇χ1| + |∇χ2| + |∇χ3|

Volume of Martensitic inclusion V :=

• R3χ1 + χ2 + χ3dx

Proposition 6 (Kn¨ upfer & Kohn & O.) min

(u,χ1,χ2,χ3)of volume V

E ∼ V 9/11. ... energy barriers to nucleation

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Future directions Cubic-to-tetragonal: Nucleation barriers at faces, edges, corners of sample Cubic-to-orthorhombic (similar to hexagonal-to-rhombic ?): crossing twins rigid (for finite interfacial energy)? [R¨ uland] Cubic-to-orthorhombic: Nucleation barrier for ma- terials with nearly compatible Austenite-Martensite [Zhang-James-M¨ uller, Zwicknagl]

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