Decrease hysteresis for Shape Memory Alloys Jin Yang; Caltech MCE Grad

What’s Shape Memory Alloy ?

PART ONE Introduction of Shape Memory Effects

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

SMA’s Phase Transition Fig 2. Martensite Fraction v.s. Temperature Ms : Austensite -> Martensite Start Temperature Mf : Austensite -> Martensite Finish Temperature As : Martensite -> Austensite Start Temperature Af : Martensite -> Austensite Finish Temperature A A A A M M M M Hysteresis size = ½ (As – Af + Ms - Mf)

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

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

How SMA works ? One path-loading M M D-MD-M D-MD-M A A T-M Fig 5. Fig 6. Loading path.

Austenite directly to detwinned martensite Fig 7. Temperature-induced phase transformation with applied load. D-MD-M D-MD-M A A

Austenite directly to detwinned martensite M M D-MD-M D-MD-M A A Fig 8. Fig 9. Thermomechanical loading

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

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

① Maximum recoverable strain ② Thermal/Stress Hysteresis size ③ Shift of transition temperatures ④ Other fatigue and plasticity problems and other factors, e.g. expenses… What SMA’s pratical properties we care about ? Fig 13. SMA hysteresis & shift temp.

SMA facing challenges! High expenses; Fatigue Problem; Large temperature/stress hysteresis Narrow temperature range of operation Reliability

Since the crystal lattice of the martensitic phase has lower symmetry than that of the parent austenitic phase, several variants of martensite can be formed from the same parent phase crystal. Parent and product phases coexist during the phase transformation, since it is a first order transition, and as a result there exists an invariant plane (relates to middle eigenvalue is 1), which separates the parent and product phases. Summary: Shape memory alloy (SMA) phases and crystal structures

PART Two Cofactor Conditions

Nature Materials, (April 2006; Vol 5; Page ) Combinatorial search of thermoelastic shape-memory alloys with extremely small hysteresis width Ni-Ti-Cu & Ni-Ti-Pb New findings: extremely small hysteresis width when λ 2  1 Fig 14.

Adv. Funct. Mater. (2010), 20, 1917–1923 Identification of Quaternary Shape Memory Alloys with Near-Zero Thermal Hysteresis and Unprecedented Functional Stability New findings: extremely small hysteresis width when λ 2  1 Fig 15.

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 -T F T FG -1 ≠Identity (2) eigenvalues of C satisfy: λ 1 ≤ λ 2 =1 ≤ λ 3 And there are exactly two solutions given as follow: (k=±1, ρ is chosen to let |n|=1)

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} ;

Austenite-Martensite Interface (★)(★) ( ★★ ) Fig 16.

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  (λ 1 -1)(1-λ 3 ) ≥ 0

Austenite-Martensite Interface 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} ; (★)(★) ( ★★ )

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  (λ 1 -1)(1-λ 3 ) ≥ 0

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 (2013)) (★)(★) ( ★★ ) -1/2 β 

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

Conditions to minimize hysteresis Conditions: Geometrical explanations of these conditions: 1)det U = 1 means no volume change 2)middle eigenvalue is 1 means there is an invariant plane btw Aust. and Mart. 3)cofactor conditions imply infinite # of compatible interfaces btw Aust. and Mart. Objective in this group meeting talk: --- Minimization of hysteresis of transformation or

A simple transition layer We can check there is solution for C: Using linear elasticity theory, we can see the C region’s energy: Area of C region: Energy: Fig 17.

A simple transition layer Where ξ is geometric factor related with m, n, A, a; And it’s can be changed largely as for various twin systems for Ti 50 Ni 50-x Pd x, x~11: From 2000 ~ Introduce facial energy per unit area κ: Fig 17.

A simple transition layer Do Tayor expansion for φ near θ c : Let’s identify hysteresis size Fig 17.

General Case Some Gamma-Convergence Problem Fig 18.

PART Four New Fancy SMA

Nature, (Oct 3, 2013; Vol 502; Page 85-88) Enhanced reversibility and unusual microstructure of a phase-transforming material Zn 45 Au x Cu (55-x) (20 ≤ x ≤30) (Cofactor conditions satisfied) Theory driven to find –or- create new materials

Functional stability of Au x Cu 55-x Zn 45 alloys during thermal cycling Fig 19.

Unusual microstructure Various hierarchical microstructures in Au30 Fig 20.

Why Riverine microstructure is possible? a.Planar phase boundary (transition layer); b.Planar phase boundary without Trans-L; c.A triple junction formed by Aust. & type I Mart. twin pair; d.(c)‘s 2D projection; e.A quad junction formed by four variants; f.(e)’s 2D projection; g.Curved phase boundary and riverine microstructure. Fig 21.

Details of riverine microstructure Fined twinned & zig-zag boundaries Fig 22.

References 1.[KB] Bhattacharya K. Microstructure of martensite: why it forms and how it gives rise to the shape-memory effect[M]. Oxford University Press, Song Y, Chen X, Dabade V, et al. Enhanced reversibility and unusual microstructure of a phase-transforming material[J]. Nature, 2013, 502(7469): Chen X, Srivastava V, Dabade V, et al. Study of the cofactor conditions: Conditions of supercompatibility between phases[J]. Journal of the Mechanics and Physics of Solids, 2013, 61(12): Zhang Z, James R D, Müller S. Energy barriers and hysteresis in martensitic phase transformations[J]. Acta Materialia, 2009, 57(15): 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: 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): 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): Thanks Gal for help me understand one Shu’s paper!

Thank you ! Jin Yang