decrease hysteresis for shape memory alloys
DESCRIPTION
Jin group meeting_2014-may-21TRANSCRIPT
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
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-M
Fig 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 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; • Fa5gue Problem; • Large temperature/stress hysteresis • Narrow temperature range of opera5on • Reliability
• Since the crystal laCce of the martensi5c phase has lower symmetry than that of the parent austeni5c phase, several variants of martensite can be formed from the same parent phase crystal.
• Parent and product phases coexist during the phase transforma5on, since it is a first order transi5on, 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
QUj -Ui = a⊗n
• 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 è 1
Fig 14.
QUj -Ui = a⊗n
• 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 è 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-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)
∃ QF -G = a⊗n
QUj -Ui = a⊗n
a = ρλ3 1− λ1( )λ3 − λ1
e1 + kλ1 λ3 −1( )λ3 − λ1
e3⎛
⎝⎜⎜
⎞
⎠⎟⎟
; n =λ3 − λ1
ρ λ3 − λ1− 1− λ1G
Te1 + k λ3 −1GTe3( )
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)
QF -G = a⊗n
QUj -Ui = a⊗n
(Type Ι) a = 2 G−Te
| G−Te |−Ge
⎛
⎝⎜⎞
⎠⎟; n = e
∃
(Type ΙΙ) a = ρGe ; n = 2
ρe− GTGe
| Ge |2⎛
⎝⎜⎞
⎠⎟
Need to sa5sfy some condi5ons; Usually there are TWO solu5ons for each pair of {F,G} ;
Austenite-Martensite Interface
QUi -I = a⊗n
QUj -Ui = a⊗n;Q'(λQUj +(1-λ )Ui ) = I+b⊗m
(★) (★★)
Fig 16.
Austenite-Martensite Interface
QUi -I = a⊗n
QUj -Ui = a⊗n;Q'(λQUj +(1-λ )Ui ) = I+b⊗m
(★) (★★)
R'(Ui +λa⊗n) = I+b⊗m
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) ç g(0)g(1/2) ≤ 0. and use this get one condition; Another condition is that from 1 is the middle eigenvalue ç (λ1-1)(1-λ3) ≥ 0
Gλ = (Ui +λa⊗n)(Ui +λn⊗ a)
g λ( ) = det Ui + λn⊗a( ) Ui + λa⊗n( )− I⎡⎣ ⎤⎦ = 0
g λ( ) = β λ −1/ 2( )2 + η
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.
b = ρ
λ3(1− λ1)λ3 − λ1
e1 +kλ1(λ3 −1)λ3 − λ1
e3⎛
⎝⎜⎜
⎞
⎠⎟⎟
m=
λ3 − λ1ρ λ3 − λ1
− 1− λ1e1 +k λ3 −1e3( )Need to sa5sfy some condi5ons; Usually there are Four solu5ons for each pair of {Ui, Uj} ;
QUi -I = a⊗n
QUj -Ui = a⊗n;Q'(λQUj +(1-λ )Ui ) = I+b⊗m
(★) (★★)
δ = a⋅Ui Ui
2 − I( )−1n; η= tr Ui2( )−det Ui
2( )− 2+ | a |2
2δ
λ* = 1
21− 1+ 2
δ
⎛
⎝⎜
⎞
⎠⎟ λ = λ* or (1-λ*)
C = Ui + λn⊗a( ) Ui + λa⊗n( )
Austenite-Martensite Interface
QUi -I = a⊗n
QUj -Ui = a⊗n;Q'(λQUj +(1-λ )Ui ) = I+b⊗m
(★) (★★)
R'(Ui +λa⊗n) = I+b⊗m
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 ç (λ1-1)(1-λ3) ≥ 0
g λ( ) = det Ui + λn⊗a( ) Ui + λa⊗n( )− I⎡⎣ ⎤⎦ = β λ −1/ 2( )2 + η
g λ( ) = η= constant ≡ 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))
QUi -I = a⊗n
QUj -Ui = a⊗n;Q'(λQUj +(1-λ )Ui ) = I+b⊗m
(★) (★★)
λ2 =1
a⋅Ucof U2 − I( )n= 0trU2 −detU2 − | a |2 |n |2
4− 2 ≥ 0
λ2 =1XI :=|U-1e |=1 for Type I twin XII :=|Ue |= 1 for Type II twin
-‐1/2 β ß
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
det U( ) =1λ2 =1
a⋅Ucof U2 − I( )n= 0trU2 −detU2 − | a |2 |n |2
4− 2 ≥ 0
det U( ) =1
λ2 =1XI :=|U-1e |=1 for Type I twin XII :=|Ue |= 1 for Type II twin
or
A simple transition layer
C− I= f⊗mCv = AvCw =BwWe can check there is solution for C:
C = I + f⊗m ; f = b+ ε
αλ 1-λ( )a
Using linear elasticity theory, we can see the C region’s energy: Area of C region: Energy:
εα
2m ⋅n⊥
E = Area µ212
CA−1 − I( ) + CA−1 − I( )T( )2⎡⎣⎢
⎤⎦⎥
⎛⎝⎜
⎞⎠⎟
= εαw2m ⋅n⊥
µλ 2
4a ⋅cλ( )2 + | a |2 | cλ |
2( )⎛⎝⎜
⎞⎠⎟
where cλ = A−Tn + εα1− λ( )A−Tm
minαE⇒εwhµλ 2 1− λ( )ξ
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
E = 2κwhlε
+ εwhµλ 2 1− λ( )ξ +ϕ A,θ( )whl +ϕ I ,θ( )wh L − l( )
minαE⇒εwhµλ 2 1− λ( )ξ
Introduce facial energy per unit area κ:
minεE = 2whλ 2κµl 1− λ( )ξ
+ whl ϕ A,θ( )−ϕ I ,θ( )( ) + const
maxε , l
E =2λ 2whκµ 1− λ( )ξϕ I ,θ( )−ϕ A,θ( )( )
with lc =2λ 2κµ 1− λ( )ξϕ A,θ( )−ϕ I ,θ( )( )
Fig 17.
A simple transition layer
ϕ A,θ( )−ϕ I ,θ( ) = Lθc −θθc
L = θc
∂ϕ I θc( ),θc( )∂θ
−∂ϕ A θc( ),θc( )
∂θ⎛
⎝⎜
⎞
⎠⎟
Do Tayor expansion for φ near θc: Let’s identify hysteresis size H = 2 θc −θ( )
= 2λθc
L2κµ 1− λ( )ξ
lc
minεE = 2whλ 2κµl 1− λ( )ξ
+ whl ϕ A,θ( )−ϕ I ,θ( )( ) + const
minε , l
E =2λ 2whκµ 1− λ( )ξϕ I ,θ( )−ϕ A,θ( )( )
with lc =2λ 2κµ 1− λ( )ξϕ A,θ( )−ϕ I ,θ( )( )
Fig 17.
General Case
H = 2 θc −θ( )
= 2λθc
L2κµ 1− λ( )ξ
lc
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 • 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.
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, 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!
