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What’s Shape Memory Alloy ?
PART ONEIntroduction 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 TemperatureMf : Austensite -> Martensite Finish Temperature
As : Martensite -> Austensite Start TemperatureAf : Martensite -> Austensite Finish Temperature
A
A
M
M
Hysteresis size = ½ (As – Af + Ms -
Mf)
How SMA works ? One path-loading
Fig 3. Shape Memory Effect of an SMA.
M D-M A
Example about # of Variants of Martensite ［ KB03］
Fig 4. Example of many “cubic-tetragonal” martensite variants.
How SMA works ? One path-loading
M D-M A
T-MFig 5. Fig 6. Loading path.
Austenite directly to detwinned martensite
Fig 7. Temperature-induced phase transformation with applied load.
D-M
A
Austenite directly to detwinned martensite
M
D-M
A
Fig 8. Fig 9. Thermomechanical loading
Pseudoelastic Behavior
Fig 10. Pseudoelastic loading path
D-M
Fig 11. Pseudoelastic stress-strain diagram.
Summary: Shape memory alloy (SMA) phases and crystal structures
Fig 12. How SMA works.
① Maximum recoverable strain② Thermal/Stress Hysteresis size③ 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.
PART TwoCofactor
Conditions
• 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
New findings: extremely small hysteresis width when λ2 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 λ2 1
Fig 15.
Conditions of compatibility for twinned martensiteDefinition. (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)
Conditions of compatibility for twinned martensiteDefinition. (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 (λ1-1)(1-λ3) ≥ 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 (λ1-1)(1-λ3) ≥ 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 2566-2587(2013))
(★)(★★)
-1/2 β
PART ThreeEnergy barriers of
Aust.-Mart. Interface transition
layers
Conditions to minimize hysteresis
• Conditions:
• Geometrical explanations of these conditions:1) det U = 1 means no volume change2) 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 Ti50Ni50-
xPdx, x~11:From 2000 ~ 160000
Introduce facial energy per unit area κ: Fig 17.
A simple transition layer
Do Tayor expansion for φ near θc:
Let’s identify hysteresis size
Fig 17.
General CaseSome Gamma-Convergence Problem Fig 18.
PART FourNew Fancy SMA
• 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)
Theory driven to find –or- create new materials
Functional stability of AuxCu55-xZn45 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.
References1. [KB] Bhattacharya K. Microstructure of martensite: why it forms and
how it gives rise to the shape-memory effect[M]. Oxford University Press, 2003.
2. 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.
3. 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): 2566-2587.
4. Zhang Z, James R D, Müller S. Energy barriers and hysteresis in martensitic phase transformations[J]. Acta Materialia, 2009, 57(15): 4332-4352.
5. 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.
6. 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.
7. 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!
