decrease hysteresis for shape memory alloys

Decrease hysteresis for Shape Memory Alloys

Jin Yang; Caltech MCE Grad Email: yangjin@caltech.edu

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!

Thank you ! Jin Yang

yangjin@caltech.edu