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 1

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

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] 3

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 4

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, 5 upper bounds by construction

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 6

2.4 Derivation of a reduced model for the twinned-Martensite to Austenite interface Phase indicator function: χ ∈ {− 1 , 0 , 1 } , χ = χ ( x 1 , x 2 ) Displacement field: u = ( u 1 , u 2 ), u = u ( x 1 , x 2 ) 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 } 2 � � �   �  0 χ � � 7 1 2 ( ∇ + ∇ t ) u − Elastic energy : dx 1 dx 2 � � �  � χ 0 � � � �

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 8

Simplification 1): Impose position of twinned-Martensite to Austenite interface Position of interface { x 2 = 0 } :   ∈ {− 1 , 1 } for x 1 > 0   χ = 0 for x 1 < 0   Nondimensionalize length by restriction to x 1 ∈ ( − 1 , 1), regime of interest η ≪ 1 Impose (artificial) L -periodicity in x 2 � � 9 � 1 Interfacial energy η (0 , 1) × [0 ,L ) |∇ χ | + L 2

Simplification 2): Impose shear direction � 0 � Favor twin normal n = 1 � 2 � by imposing shear direction a = . i. e. 0 u 2 ≡ 0 but u 1 = u 1 ( x 1 , x 2 ) 1    ∂ 1 u 1 2 ∂ 2 u 1 Strain 1 2 ( ∇ + ∇ t ) u = 1  2 ∂ 2 u 1 0 � 1 � L 0 ( ∂ 1 u 1 ) 2 + 2( 1 2 ∂ 2 u 1 − χ ) 2 dx 2 dx 1 Elastic energy − 1 10

Simplification 3): Anisotropic rescaling and limit 1 � η � (0 , 1) × [0 ,L ) |∇ χ | 2 L ( − 1 , 1) × (0 ,L ) ( ∂ 1 u 1 ) 2 + 2( 1 � � 2 ∂ 2 u 1 − χ ) 2 dx + Ansatz for rescaling x 2 = η α ˆ ⇒ ∂ 2 = η − α ˆ L = η α ˆ x 2 = ∂ 2 . L, u 1 = 2 η α ˆ ∂ 1 u 1 = 2 η α ∂ 1 ˆ ⇒ ∂ 2 u 1 = 2ˆ u 1 = ∂ 2 ˆ u 1 , u 1 . 1 ∂ 1 χ � η � � � (0 , 1) × [0 ,L ) | | 2 η − α ˆ ˆ L ∂ 2 χ 11 u 1 ) 2 + 2(ˆ � � ( − 1 , 1) × (0 ,L ) 4 η 2 α ( ∂ 1 ˆ u 1 − χ ) 2 d ˆ + ∂ 2 ˆ x

Seek nontrivial limit: elastic part u 1 ) 2 + 2(ˆ 4 η 2 α ( ∂ 1 ˆ u 1 − χ ) 2 Elastic energy density: ∂ 2 ˆ Penalization of ˆ ∂ 2 ˆ u 1 − χ ≫ penalization of ∂ 1 ˆ u 1 Neclegting ∂ 1 ˆ u 1 no option — otherwise no elastic effect Hence constraint ˆ ∂ 2 ˆ u 1 − χ = 0 in limit. 12

Seek nontrivial limit: interfacial part ∂ 1 χ η � � Interfacial energy density: 2 | | η − α ˆ ∂ 2 χ Penalization of ˆ penalization of ∂ 1 χ ∂ 2 χ ≫ Constraint ˆ ∂ 2 χ = 0 no option — otherwise no twin Hence have to neglect penalization of ∂ 1 χ 13 η 1 − α 2 | ˆ Interfacial energy density ∂ 2 χ | in limit.

Seek nontrivial limit: choice of α u 1 ) 2 + η 1 − α Total energy density 4 η 2 α ( ∂ 1 ˆ 2 | ˆ ∂ 2 χ | For balance need η 2 α ∼ η 1 − α ⇒ α = 1 3 2 1 3 1 L ˆ Rescaling of energy density: L E = η E ˆ 2 2 Prediction from 1 3 1 L ˆ L E = η E : energy density ∼ η 3 ˆ 1 1 Prediction from x 2 = η 3 ˆ x 2 : twin width ∼ η 3 14 ... provided limit model makes sense for ˆ L ≫ 1

Limit model is singular � 1 � ˆ � 1 L � x 2 dx 1 + 1 u 1 ) 2 d ˆ L ) | ˆ Minimize 4 0 ( ∂ 1 ˆ ∂ 2 χ | dx 1 subject to 2 [0 , ˆ − 1 0 � � ∈ {− 1 , 1 } for x 1 > 0 ˆ ∂ 2 ˆ u 1 = χ . = 0 for x 1 < 0 � 1 | ˆ ∂ 2 χ | just counts transitions between 1 and -1 2 [0 , ˆ L ) Infinite twin refinement: Elastic energy = ⇒ ˆ u 1 = const = 0 for x 1 < 0 = u 1 ( x 1 , · ) → 0 ˆ as x 1 ↓ 0 ⇒ = ⇒ χ ( x 1 , · ) ⇀ 0 as x 1 ↓ 0 � L ) | ˆ = ∂ 2 χ ( x 1 , · ) | ↑ ∞ as x 1 ↓ 0 ⇒ [0 , ˆ � 1 � L ) | ˆ Interfacial energy = ∂ 2 χ ( x 1 , · ) | dx 1 < ∞ ⇒ 15 [0 , ˆ 0 ... does limit model have finite energy?

2.5 Self consistency of reduced model, upper bounds by construction, lower bounds by interpolation Proposition 2 [Kohn, M¨ uller] Functional: � 1 � L � 1 � 0 ( ∂ 1 u 1 ) 2 dx 2 dx 1 + 1 E = 4 [0 ,L ) | ∂ 2 χ | dx 1 . 2 − 1 0 Admissible configurations: u 1 , χ L -periodic in x 2 with � � ∈ {− 1 , 1 } for x 1 > 0 ∂ 2 u 1 = χ . = 0 for x 1 < 0 Then ∃ universal C < ∞ such that i) upper bound ∀ L ∃ ( u 1 , χ ) E ≤ CL, 16 E ≥ 1 ii) lower bound ∀ L, ( u 1 , χ ) C L.

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) 17

Proof of Proposition 2 i) (lower bound) Lemma 7 ∃ universal C < ∞ ∀ L -periodic u 1 ( x 2 ) , χ ( x 2 ) related by ∂ 2 u 1 = χ with � 1 � 2 � L �� L 3 �� L 3 0 χ 2 dx 1 ≤ C 0 u 2 0 | ∂ 2 χ | dx 2 sup 1 dx 2 | χ | . x 2 1 1 1 � χ � L 2 ≤ C ( d ) �|∇| − 1 χ � 3 3 3 Holds in any d as L 2 �∇ χ � L 1 � χ � L ∞ 2 1 3 ≤ C ( d ) �|∇| − 1 χ � 3 3 Simpler version of � χ � L 2 �∇ χ � 4 L 1 L 18 (Cohen-Dahmen-Daubechies-Devore)

2.8 Loss of rigidity by convex integration Proposition 3 [M¨ uller, Sver´ ak] ∀ M s. t. 1 2 ( M + M t ) ∈ int conv { E 0 , E 1 , E 2 } ∀ Ω ⊂ R 2 open, bdd. ∃ u : R 2 → R 2 with ∇ u ∈ L 2 loc in R 2 − ¯ ∇ u = M Ω , 1 2( ∇ + ∇ t ) u ∈ { E 0 , E 1 , E 2 } a. e. on Ω . 19

Step 1: Conti’s construction = Lemma 5 Consider for λ = 1 4 : � � � � � � 0 1 0 − 1 1 − λ M 0 = 1 1 1 , M 1 = , M 2 = , 1 − λ 2 λ 1 − λ 0 0 1 0 − 1 λ � � � � − 1 − λ 0 0 1 , M 4 = 1 , Ω = ( − 1 , 1) 2 . M 3 = 1 − λ 2 λ λ 1 − 1 0 Then ∃ Ω 0 , · · · Ω 4 ⊂ Ω finite � of convex, open sets ∃ u : R 2 → R 2 Lipschitz s. t. in R 2 − ¯ = 0 Ω , ∇ u = in Ω i , ∇ u M i 1 20 | Ω 0 | = 2 λ | Ω | .

Step 2: Deformation and rotation of Conti’s construction ∀ M, M 0 , M 1 s. t. M = 1 4 M 0 + 3 4 M 1 with for some a ∈ R 2 , n ∈ S 1 , a · n = 0 M 1 − M 0 = a ⊗ n ∃ ˜ M 1 , · · · , ˜ ∀ ǫ > 0 M 4 s. t. where M 2 := 1 5 M 0 + 4 | ˜ M 1 − M 1 | , | ˜ M 2 / 3 − M 2 | , | ˜ M 4 − M | < ǫ, 5 M 1 . Ω 4 ⊂ Ω finite � of convex, open sets ∃ Ω ⊂ R 2 open, bdd. , Ω 1 , · · · , ˜ ˜ ∃ u : R 2 → R 2 Lipschitz with in R 2 − ¯ ∇ u = M Ω , ˜ in ˜ ∇ u = M i Ω i , in Ω − (˜ Ω 1 ∪ · · · ∪ ˜ ∇ u = M 0 Ω 4 ) , 21 7 | ˜ Ω 1 ∪ · · · ∪ ˜ Ω 4 | 8 | Ω | . ≤

Step 3: Application to hexagonal-to-rhombic ∀ M s. t. 1 2 ( M + M t ) ∈ int conv { E 0 , E 1 , E 2 } M 4 s. t. 1 M t ∃ ˜ M 1 , · · · , ˜ 2 ( ˜ M i + ˜ i ) ∈ int conv { E 0 , E 1 , E 2 } Ω 4 ⊂ Ω finite � of convex, open sets ∃ Ω ⊂ R 2 open, bdd. , Ω 1 , · · · , ˜ ˜ ∃ u : R 2 → R 2 Lipschitz with in R 2 − ¯ = Ω , ∇ u M ˜ in ˜ = Ω i , ∇ u M i 1 2 ( ∇ + ∇ t ) u in Ω − (˜ Ω 1 ∪ · · · ∪ ˜ ∈ { E 0 , E 1 , E 2 } Ω 4 ) , 7 | ˜ Ω 1 ∪ · · · ∪ ˜ Ω 4 | ≤ 8 | Ω | . 22