Decrease hysteresis for Shape Memory Alloys

1. Decrease hysteresis for Shape Memory Alloys Jin Yang; Caltech MCE Grad Email: yangjin@caltech.edu

2. What’sShapeMemoryAlloy?

3. PART ONE Introduction of Shape Memory Effects

4. Two Stable phases at different temperature Fig 1. Different phases of an SMA

5. SMA’sPhaseTransition As:Martensite->AustensiteStartTemperature Af:Martensite->AustensiteFinishTemperature Hysteresis size = ½ (As – Af + Ms - Mf) M A M A Mf:Austensite->MartensiteFinishTemperature Ms:Austensite->MartensiteStartTemperature Fig 2. MartensiteFractionv.s.Temperature

6. How SMA works ? One path-loading D-M A M Fig 3. Shape Memory Effect of an SMA.

7. Example about # of Variants of Martensite［KB03］ Fig 4. Example of many “cubic-tetragonal” martensite variants.

8. How SMA works ? One path-loading D-M A M T-M Fig 5. Fig6. Loading path.

9. Austenitedirectly to detwinnedmartensite D-M A Fig 7. Temperature-induced phase transformation with applied load.

10. Austenitedirectly to detwinnedmartensite D-M A M Fig 8. Fig9. Thermomechanical loading

11. Pseudoelastic Behavior D-M Fig 10. Pseudoelastic loading path Fig11. Pseudoelastic stress-strain diagram.

12. Summary: Shape memory alloy (SMA) phases and crystal structures Fig12. How SMA works.

13. What SMA’s pratical properties we care about ? • Maximum recoverable strain • Thermal/Stress Hysteresis size • Shift of transition temperatures • Other fatigue and plasticity problems and other factors, e.g. expenses… Fig13. SMA hysteresis & shift temp.

14. PART Two Cofactor Conditions

15. New findings: extremely small hysteresis width when λ2 1 • Nature Materials, (April 2006; Vol 5; Page 286-290) • Combinatorial search of thermoelastic shape-memory alloys with extremely small hysteresis width • Ni-Ti-Cu & Ni-Ti-Pb Fig14.

16. New findings: extremely small hysteresis width when λ2 1 • Adv. Funct. Mater. (2010), 20, 1917–1923 • Identification of Quaternary Shape Memory Alloys with Near-Zero Thermal Hysteresis and Unprecedented Functional Stability Fig15.

17. Conditions of compatibility for twinned martensite Definition. (Compatibility condition) Two positive-definite symmetric stretch tensors Ui and Uj are compatible if: , where Q is a rotation, n is the normal direction of interface, a and Q are to be decided. Result 1 [KB Result 5.1] Given F and G as positive definite tensors, rotation Q, vector a ≠0, |n|=1, s.t. iff: (1) C = G-TFTFG-1≠Identity (2) eigenvalues of C satisfy: λ1≤ λ2 =1 ≤ λ3 And there are exactly two solutions given as follow: (k=±1, ρ is chosen to let |n|=1)

18. Conditions of compatibility for twinned martensite Definition. (Compatibility condition) Two positive-definite symmetric stretch tensors Ui and Uj are compatible if: , where Q is a rotation, n is the normal direction of interface, a and Q are to be decided. Result 2 (Mallard’s Law)[KB Result 5.2] Given F and G as positive definite tensors, (i) F=Q’FR for some rotation Q’ and some 180° rotation R with axis ê; (ii)FTF≠GTG, then one rotation Q, vector a ≠ 0, |n|=1, s.t. And there are exactly two solutions given as follow: (ρ is chosen to let |n|=1) Need to satisfy some conditions; Usually there are TWO solutions for each pair of {F,G} ;

19. Austenite-Martensite Interface (★) (★★) Fig16.

20. Austenite-Martensite Interface (★) (★★) Need to check middle eigenvalue of is 1. Which is equivalent to check: Order of g(λ) ≤6, actually it’s at most quadratic in λ and it’s symmetric with 1/2. so it has form: And g(λ) has a root in (0,1)  g(0)g(1/2) ≤ 0. and use this get one condition; Another condition is that from 1 is the middle eigenvalue  (λ1-1)(1-λ3) ≥ 0

21. Austenite-MartensiteInterface (★) Result 3 [KB Result 7.1] Given Ui and vector a, n that satisfy the twinning equation (★), we can obtain a solution to the aust.-martensite interface equation (★★), using following procedure: (Step 1) Calculate: The austenite-martensite interface eq has a solution iff: δ≤ -2 and η ≥ 0; (Step 2) Calculate λ (VOlUME fraction for martensites) (Step 3) Calculate C and find C’s three eigenvalues and corresponding eigenvectors. And ρ is chosen to make |m|=1 and k = ±1. (★★) Need to satisfy some conditions; Usually there are Four solutions for each pair of {Ui, Uj} ;

22. Austenite-Martensite Interface (★) (★★) What if Order of g(λ) < 2,β=0; g(λ) has a root in (0,1), Now, λ is free only if belongs to (0,1). Another condition is that from 1 is the middle eigenvalue  (λ1-1)(1-λ3) ≥ 0

23. Cofactor conditions (★) • Under certain denegeracy conditions on the input data U, a, n, there can be additional solutions of (★★), and these conditions called cofactor conditions: • Simplified equivalent form: (Study of the cofactor condition. JMPS 2566-2587(2013)) (★★) -1/2 β 

24. PART Three Energy barriers of Aust.-Mart. Interface transition layers

25. Conditions to minimize hysteresis Objective in this group meeting talk: --- Minimization of hysteresis of transformation • Conditions: • Geometrical explanations of these conditions: • det U = 1 means no volume change • middle eigenvalue is 1 means there is an invariant plane btw Aust. and Mart. • cofactor conditions imply infinite # of compatible interfaces btw Aust. and Mart. or

26. A simple transition layer Using linear elasticity theory, we can see the C region’s energy: Area of C region: Energy: Fig17. We can check there is solution for C:

27. A simple transition layer Introduce facial energy per unit area κ: Fig17. Where ξ is geometric factor related with m, n, A, a; And it’s can be changed largely as for various twin systems for Ti50Ni50-xPdx, x~11: From 2000 ~ 160000

28. A simple transition layer Do Tayor expansion for φ near θc: Let’s identify hysteresis size Fig17.

29. General Case Some Gamma-Convergence Problem Fig18.

30. PART Four New Fancy SMA

31. Theory driven to find –or- create new materials • Nature, (Oct 3, 2013; Vol 502; Page 85-88) • Enhanced reversibility and unusual microstructure of a phase-transforming material • Zn45AuxCu(55-x) (20 ≤ x ≤30)(Cofactor conditions satisfied)

32. Functional stability of AuxCu55-xZn45 alloys during thermal cycling Fig19.

33. Unusual microstructure Various hierarchical microstructures in Au30 Fig20.

34. Why Riverine microstructure is possible? Planar phase boundary (transition layer); Planar phase boundary without Trans-L; A triple junction formed by Aust. & type I Mart. twin pair; (c)‘s 2D projection; A quad junction formed by four variants; (e)’s 2D projection; Curved phase boundary and riverine microstructure. Fig21.

35. Details of riverine microstructure Fig22. Fined twinned & zig-zag boundaries

36. References [KB] Bhattacharya K. Microstructure of martensite: why it forms and how it gives rise to the shape-memory effect[M]. Oxford University Press, 2003. Song Y, Chen X, Dabade V, et al. Enhanced reversibility and unusual microstructure of a phase-transforming material[J]. Nature, 2013, 502(7469): 85-88. Chen X, Srivastava V, Dabade V, et al. Study of thecofactor conditions: Conditions of supercompatibility between phases[J]. Journal of the Mechanics and Physics of Solids, 2013, 61(12): 2566-2587. Zhang Z, James R D, Müller S. Energy barriers and hysteresis in martensitic phase transformations[J]. ActaMaterialia, 2009, 57(15): 4332-4352. James R D, Zhang Z. A way to search for multiferroic materials with “unlikely” combinations of physical properties[M]//Magnetism and structure in functional materials. Springer Berlin Heidelberg, 2005: 159-175. Cui J, Chu Y S, Famodu O O, et al. Combinatorial search of thermoelastic shape-memory alloys with extremely small hysteresis width[J]. Nature materials, 2006, 5(4): 286-290. Zarnetta R, Takahashi R, Young M L, et al. Identification of Quaternary Shape Memory Alloys with Near‐Zero Thermal Hysteresis and Unprecedented Functional Stability[J]. Advanced Functional Materials, 2010, 20(12): 1917-1923. Thanks Gal for help me understand one Shu’s paper!

37. Thank you ! Jin Yang yangjin@caltech.edu