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 geometrically linear three-well problem, CPAM 62, 2009 2 Approximate rigidity of twins, local case Capella, O.: A quantitative rigidity result for the cubic-to-tetragonal
phase transition in the geometrically linear theory with interfacial energy, Proc. Roy. Soc. Edinburgh A, to appear Optimal microstructure of Martensitic inclusions Knüpfer, 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 3 Chap 3. 3-d models cubic-to-tetragonal, [cubic-to-orthorombic]
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 5 2.5 Self-consistency of reduced model, lower bounds by interpolation, 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} ) 7 Elastic energy: 1 2 ( + t )u 0 χ χ 0 2 dx 1 dx 2
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 8 Simplification 3 Anisotropic rescaling and limit
Simplification 1): Impose position of twinned-martensite to Austenite interface Position of interface {x 2 = 0}: χ { 1,1} for x 1 > 0 = 0 for x 1 < 0 Nondimensionalize length by restriction to x 1 ( 1,1), regime of interest η 1 9 Impose (artificial) L-periodicity in x 2 Interfacial energy η (12 (0,1) [0,L) χ +L )
Simplification 2): Impose shear direction Favor twin normal n = ( ) 0 1 by imposing shear direction a = ( ) 2 0. i. e. u 2 0 but u 1 = u 1 (x 1,x 2 ) Strain 1 2 ( + t )u = 1u 1 1 2 2 u 1 1 2 2 u 1 0 10 Elastic energy 1 1 L 0 ( 1u 1 ) 2 +2( 1 2 2u 1 χ) 2 dx 2 dx 1
Simplification 3): Anisotropic rescaling and limit 1 L( η 2 + (0,1) [0,L) χ ( 1,1) (0,L) ( 1u 1 ) 2 +2( 1 2 2u 1 χ) 2 dx Ansatz for rescaling x 2 = η αˆx 2 = 2 = η αˆ 2. L = η αˆl, u 1 = 2η α û 1 = 2 u 1 = 2ˆ 2 û 1, 1 u 1 = 2η α 1 û 1. ) 11 1 ˆL( η 2 + 1 χ (0,1) [0,L) ( η αˆ 2 χ ) ( 1,1) (0,L) 4η2α ( 1 û 1 ) 2 +2(ˆ 2 û 1 χ) 2 dˆx )
Seek nontrivial limit: elastic part Elastic energy density: 4η 2α ( 1 û 1 ) 2 +2(ˆ 2 û 1 χ) 2 Penalization of ˆ 2 û 1 χ penalization of 1 û 1 Neclegting 1 û 1 no option otherwise no elastic effect 12 Hence constraint ˆ 2 û 1 χ = 0 in limit.
Seek nontrivial limit: interfacial part Interfacial energy density: η 2 ( 1 χ ) η αˆ 2 χ Penalization of ˆ 2 χ penalization of 1 χ Constraint ˆ 2 χ = 0 no option otherwise no twin Hence have to neglect penalization of 1 χ 13 Interfacial energy density η1 α 2 ˆ 2 χ in limit.
Seek nontrivial limit: choice of α Total energy density 4η 2α ( 1 û 1 ) 2 + η1 α 2 ˆ 2 χ For balance need η 2α η 1 α α = 1 3 Rescaling of energy density: 1 L E = η2 3 1ˆLÊ Prediction from 1 L E = η2 3 1ˆLÊ: energy density η2 3 Prediction from x 2 = η 1 3ˆx 2 : twin width η 1 3 14... provided limit model makes sense for ˆL 1
Limit model is singular Minimize 4 1 1 ˆL 0 ( 1û 1 ) 2 dˆx 2 dx 1 + 1 2 1 { { 1,1} for x1 > 0 ˆ 2 û 1 = χ = 0 for x 1 < 0 1 ˆ 2 2 χ just counts transitions between 1 and -1 [0,ˆL) Infinite twin refinement: 0 [0,ˆL) ˆ 2 χ dx 1 subject to }. 15 Elastic energy = û 1 = const = 0 for x 1 < 0 = û 1 (x 1, ) 0 as x 1 0 = Interfacial energy = = χ(x 1, ) 0 as x 1 0 [0,ˆL) ˆ 2 χ(x 1, ) as x 1 0 1... does limit model have finite energy? 0 [0,ˆL) ˆ 2 χ(x 1, ) dx 1 <
2.5 Self consistency of reduced model, upper bounds by construction, lower bounds by interpolation Proposition 2 [Kohn, Müller] Functional: E = 4 1 1 L 0 ( 1u 1 ) 2 dx 2 dx 1 + 1 2 1 0 [0,L) 2χ dx 1. Admissible configurations: u 1,χ L-periodic in x 2 with { } { 1,1} for x1 > 0 2 u 1 = χ. = 0 for x 1 < 0 Then universal C < such that 16 i) upper bound L (u 1,χ) E CL, ii) lower bound L,(u 1,χ) E 1 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 L 0 χ2 dx 1 C ( L 0 u2 1 dx 2 )1 3 ( L 0 2χ dx 2 sup x 2 χ )2 3. Holds in any d as χ L 2 C(d) 1 1 1 χ 3 L 2 χ 3 χ 1 L 1 3 L 18 Simpler version of χ L 4 3 C(d) 1 χ 2 3 L 2 χ 1 3 L 1 (Cohen-Dahmen-Daubechies-Devore)
2.8 Loss of rigidity by convex integration Proposition 3 [Müller, Sverák] M s. t. 1 2 (M +Mt ) intconv{e 0,E 1,E 2 } Ω R 2 open, bdd. u: R 2 R 2 with u L 2 loc 19 u = M in R 2 Ω, 1 2 ( + t )u {E 0,E 1,E 2 } a. e. on Ω.
Step 1: Conti s construction = Lemma 5 Consider for λ = 1 4 : M 0 = 1 λ ( 0 1 0 0 ), M 1 = 1 1 λ ( 0 1 1 0 ), M 2 = 1 1 λ 2 ( 1 λ λ 1 ), M 3 = 1 1 λ 2 ( 1 λ λ 1 ), M 4 = 1 λ ( 0 0 1 0 ), Ω = ( 1,1) 2. Then Ω 0, Ω 4 Ω finite of convex, open sets u: R 2 R 2 Lipschitz s. t. u = 0 in R 2 Ω, u = M i in Ω i, 20 Ω 0 = 1 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 M 1 M 0 = a n for some a R 2,n S 1,a n = 0 ǫ > 0 M 1,, M 4 s. t. M 1 M 1, M 2/3 M 2, M 4 M < ǫ, where M 2 := 1 5 M 0 + 4 5 M 1. Ω R 2 open, bdd., Ω 1,, Ω 4 Ω finite of convex, open sets u: R 2 R 2 Lipschitz with 21 u = M in R 2 Ω, u = M i in Ω i, u = M 0 in Ω ( Ω 1 Ω 4 ), Ω 1 Ω 4 8 7 Ω.
Step 3: Application to hexagonal-to-rhombic M s. t. 1 2 (M +Mt ) int conv{e 0,E 1,E 2 } M 1,, M 4 s. t. 1 2 ( M i + M t i ) int conv{e 0,E 1,E 2 } Ω R 2 open, bdd., Ω 1,, Ω 4 Ω finite of convex, open sets u: R 2 R 2 Lipschitz with u = M in R 2 Ω, u = M i in Ω i, 1 2 ( + t )u {E 0,E 1,E 2 } in Ω ( Ω 1 Ω 4 ), Ω 1 Ω 4 8 7 Ω. 22
Step 4: Concatenation M s. t. 1 2 (M +Mt ) int conv{e 0,E 1,E 2 } Ω R 2 open, bbd M 1,, M 4 s. t. 1 2 ( M i + M i t) int conv{e 0,E 1,E 2 } Ω 1,, Ω 4 Ω countable of convex, open sets u: R 2 R 2 Lipschitz with 23 u = M in R 2 Ω, u = M i in Ω i, 1 2 ( + t )u {E 0,E 1,E 2 } in Ω ( Ω 1 Ω 4 ), Ω 1 Ω 4 7 8 Ω.
Step 5: Iteration via replacement M s. t. 1 2 (M +Mt ) int conv{e 0,E 1,E 2 } N N Ω R 2 open, bbd M 1,, M 4 N s. t. 1 2 ( M i + M i t) int conv{e 0,E 1,E 2 } Ω 1,, Ω 4 N Ω countable of convex, open sets u: R 2 R 2 Lipschitz with 24 u = M in R 2 Ω, u = M i in Ω i, 1 2 ( + t )u {E 0,E 1,E 2 } in Ω ( Ω 1 Ω 4 N), Ω 1 Ω 4 N ( 7 8 )N Ω.
3.1 3-d models, cubic-to-tetragonal phase transformation 3 stress-free strains = Martensitic variants: E 1 := 2 0 0 0 1 0 0 0 1, E 2 := 1 0 0 0 2 0 0 0 1, E 3 := 1 0 0 0 1 0 0 0 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)} 25 No twin between Austenite 0 0 0 0 0 0 0 0 0 and three Martensitic variants E 1,E 2,E 3 cf. Lemma 3
Rigidity of twins Dolzmann & Müller, Meccanica 95 Proposition 4 (Dolzmann & Müller) Let u: R 3 B 1 R 3 be Lipschitz with 1 2 ( + t )u {E 1,E 2,E 3 } a. e. in B 1. Then u = one of the six Martensitic twins on B δ 26 (with δ > 0 universal).
Approximate rigidity of twins Elastic + interfacial energy on B 1 : E := + η B 1 1 2 ( + t )u (χ 1 E 1 +χ 2 E 2 +χ 3 E 3 ) 2 dx B 1 χ 1 + χ 2 + χ 3 Admissible phase functions χ i {0,1}, χ 1 +χ 2 +χ 3 1 Proposition 5 (Capella & O.) Suppose E η 2/3. 27 Then (u,χ 1,χ 2,χ 3 ) Austenite or one of the six Martensitic twins.
Optimal Martensitic inclusions Energy in whole space E := + R 3 1 2 ( + t )u (χ 1 E 1 +χ 2 E 2 +χ 3 E 3 ) 2 dx R 3 χ 1 + χ 2 + χ 3 Volume of Martensitic inclusion V := Proposition 6 (Knüpfer & Kohn & O.) R 3χ 1 +χ 2 +χ 3 dx 28 min (u,χ 1,χ 2,χ 3 )of volume V E V 9/11.... energy barriers to nucleation
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)? [Rüland] 29 Cubic-to-orthorhombic: Nucleation barrier for materials with nearly compatible Austenite-Martensite [Zhang-James-Müller, Zwicknagl]