►plateau

SMA orthodontic wires

B

A

C

D

σ

ε

steel SMA

SMAdamper

)(0 tsinFxkxcxm ω=++ &&&

Timpano

26/9/97

7/10/97

In order to reduce the value of the seismic forces applied to the “timpano”, the junctions between the “timpano” and the roofare made of SMA (constant force and energy dissipation).

SMA modeling

• Microscopic Thermodynamics Models

• Macroscopic Phenomenological Models

• Micromechanics Based Macroscopic Models

Microscopic Thermodynamics Modelsphenomomenological thermodynamics to describe an infinitesimal volume in an infinite domain

explanation of the micro-scale behavior, such as nucleation, interface motion and growth of a martensite plate.

extremely helpful for understanding the phenomenon but difficult to apply for engineering applications.

Abeyaratne and Knowles. Journal of the Mechanics and Physics of Solids 1990Ball and James. Archives of Rational Mechanics and Analysis 1987Barschand Krumhansl Martensite ASM International 1992Falk Z. Physik B-Condensed Matter 1983

Micromechanics Based Macroscopic Modelsthermodynamics to describe the transformation and micromechanics to estimate the interaction energy due to the transformation in the material

interaction energy: key factor in the transformation mechanism

knowledge of the microstructural evolution is required

assumptions at the microstructure level have been made to approximate the interaction energy

Fischer and Tanaka. International Journal of Solids and Structures 1992Raniecki et al. Arch. Mech. 1992Sun and Hwang. Journal of the Mechanics and Physics of Solids 1993Patoor et al. Mechanics of Phase Transformations and Shape Memory Alloys 1994

Macroscopic Phenomenological Modelsphenomenological thermodynamics curve fitting from experimental data

based on the phase diagram of SMA transformation

martensite volume fraction used as an internal variable

suitable for engineering applications due to their simplicity

accurate models (built on experimental data)

lack of multi-dimensional experimental tests

difficult to extend phenomenological models to 2-D and 3-D cases

three-dimensional macroscopic phenomenological models proposed with some successAuricchio and Taylor. Comp Methods Appl. Mech.Engrg. 1997Brinson. J. of Intell. Mat. Syst. and Struct. 1993Bekker and Brinson. Journal of the Mechanics and Physics of Solids 1997Ivshin and Pence. Int. J. Engng. Sci. 1994Liang and Rogers. Journal of Engineering Mathematics 1992Sato and Tanaka. Res. Mech. 1988Boyd and Lagoudas. Int. J. Plasticity 1995Liang and Rogers. Journal of Engineering Mathematics 1992

SuperelasticSuperelastic 1D SMA model1D SMA model

ξξAA austeniteaustenite volume volume fractionfraction

ξξSS single single variantvariant martensitemartensite volume volume fractionfraction

1A Sξ ξ+ =

1S Aξ ξ= −

Considered phase trasformations

( )( )S LEσ ε ξ ε ±= −

Stress-strain relationship

maximum residual strain( )Lε±

Evolution of the Young modulus

( )( )

A S

S A S

E EEE E E

ξξ

=+ −

ΕΕAA austeniteaustenite YoungYoung modulusmodulus

ΕΕSS martensitemartensite YoungYoung modulusmodulus

Reuss homogenization

σsAS~σfAS~

austenite - martensite evolution( ) ( )

( ) ( )

when

1= 1

AS ASs f

S S ASf

σ σ σ

ξ ξ σσ σ

± ±

±

≤ ≤

− −−

& &

σsSA~

σfSA~

εL

martensite - austenite evolution( ) ( )

( )

when

1=

SA SAs f

S S SAf

σ σ σ

ξ ξ σσ σ

± ±

±

≤ ≤

−& &

ε

σEvolution of the internal parameter

Considered phase trasformations

Note( ) ( )

( ) ( ) ( ) ( )

when

1 1= 1 =1

AS ASs f

SS S AS AS

Sf f

σ σ σ

ξξ ξ σ σξσ σ σ σ

± ±

± ±

≤ ≤

− − −−− −

&& & &

( ) ( )( )

,

0

1=1

S t t

ASs

SAS

S f

ξ σ

σ

ξ σξ σ σ± ±

−− −∫ ∫&

&

( )

( ) ( ),

ASt s

S t AS ASf s

σ σξσ σ

±

± ±

−=

−

( ) ( )( ) ( ) ( )( ),ln 1 ln + lnAS AS ASS t f t f sξ σ σ σ σ± ± ±− − = − − −

( ) ( )

( )

( ) ( )

( ), ,

1 1ln ln1 1

AS AS AS ASf s f s

AS ASS t S tf t f t

σ σ σ σξ ξσ σ σ σ

± ± ± ±

± ±

⎛ ⎞⎛ ⎞ − −= =⎜ ⎟⎜ ⎟⎜ ⎟ ⎜ ⎟− −− −⎝ ⎠ ⎝ ⎠

ResultsNDC Ni-Ti alloy provided by Nitinol Device & Components

GAC Ni-Ti alloy provided by GAC International Inc.

Beam problem

1. Kinematics

2. Constitutive model

3. Equilibrium equations

4. Numerical procedures (fiber model)

σsAS~σfAS

~

σsSA~ σfSA

~

εL

ε

σAppendix: An analytical solution

equal material response in tension and compression,

equal elastic properties betweenaustenite and martensite,

equal initial and final values forthe phase transformations,

pure bending loading state.

rectangular cross-section b×h

1 1 1

2 2 2

1 1 1 1 1 2

2 2 22

3 3 1 3

( ) with 02

( ) with 02

( ) ( ) with

( ) with2

( ) with2

AS

ASL

SA ASp p p

SAL p

SAp

hy E y y

hy E y y

E y y

hE y

hE y

ε σ

ε ε σ

η ε η ε η σ σ η

η ε η ε σ η

η ε η σ η

⎡ ⎤⎢ ⎥⎣ ⎦

, ,

⎡ ⎤⎢ ⎥ ,⎣ ⎦

,

: = ≤ ≤

: − = ≤ ≤

⎡ ⎤: − = − ≤ ≤⎣ ⎦

: − = ≤ ≤

: = ≤ ≤

σsAS~σfAS

~

σsSA~ σfSA

~

εL

ε

σ

unloadingloading

1

y

1

2

3

y

6

y

y2

ζ

4

5 ηη

η3

1

2

ζ

unloadingsection

11’22’

33’44’

σsAS~

σfAS~

σsSA~

σfSA~

εL

ε

σ

1 2 3 4

1’2’

3’ 4’

1’2’

3’4’

strain

stress

Comparison between analytical and numerical solutions(GAC Ni-Ti alloy)

AnalyticalScheme 1 Scheme 2

0 0.1 0.2 0.3 0.4 0.50

10

20

30

40

50

60

Curvature [1/mm]

Mom

ent

[N*m

m]

bending moment versus the curvature

Scheme 1Scheme 2

0 0.1 0.2 0.3 0.4 0.5 0.60

5

10

15

20

25

30

35

40

Curvature [1/mm]

Mom

ent

[N*m

m]

Scheme 1Scheme 2

-0.5 0 0.5 1 1.5 2 2.5 3 3.50

5

10

15

20

25

30

35

40

Axial strain [%]

Mom

ent

[N*m

m]

A SE E≠

curvature and axial strain versus the bending moment

Stress distribution in the cross-section evolution during the loading process

1. no phase transformation, linear elastic material response, neutral axis at zero.

2. phase transformation on the part of the cross-section in traction (stress transformation higher in compression than in tension), then transformation in compression; neutral axis starts to move downward and axial deformation shows up.

3. material restiffening when the phase transformation is completed in traction; progressive upward movement of the neutral axis, reduction of the axial deformation..

Unloading process stress pattern more and more complex: combination of the neutral axis movement and the different response in tension and in compression.

Three point bending

GAC wire

Numerical Experimental

0 0.5 1 1.5 2 2.5 3 3.5 4 4.50

1

2

3

4

5

6

7

8

9

Displacement [mm]

For

ce [

N]

Numerical Experimental

0 0.5 1 1.5 2 2.5 3 3.5 4 4.50

1

2

3

4

5

6

7

8

9

Displacement [mm]

For

ce [

N]

equal response in tension and compression

different material response in tension and in compression

Four point bending

NDC circular wire

Beam model3D model

0 0.5 1 1.5 2 2.5 3 3.5 40

10

20

30

40

50

60

Displacement [mm]

For

ce [

N]

Numerical Experimental

0 0.5 1 1.5 2 2.5 3 3.5 40

10

20

30

40

50

60

Displacement [mm]

For

ce [

N]

equal properties in traction and in compression and equal elastic modulibetween austenite and martensite.

applied force F versus midspan inflection

comparison between beam and three-dimensional analyses

comparison between numerical and experimental results

ThermomechanicalThermomechanical modelingmodeling of SEof SE--SMA SMA wireswires

−4 −3 −2 −1 0 1 2 3 410

15

20

25

30

35

Axial Strain [ % ]T

empe

ratu

re [

o C ]

typical superelastic response in tension and compression [Lim and McDowell (1999)]

Experimental data reveal: material response characterized by an hysteresis, indicating a mechanical dissipation;

superelastic response associated to significative temperature variations.

austenite-martensite phasetransformations occur in well definedstress-temperature ranges and theycan be induced through mechanicalloading-unloading patters, through thermal cooling-heating patterns or through combined mechanical-thermal patterns;

SMAs show strong thermo-mechanical constitutive coupling.

Single-phase material free-energy [Raniecki – Bruhns (1991)]

0 0 00

( ) ( ) ( ) ( ) ( ) ( ) logel el el el el TT T T T s T C T T TT

ψ ε ψ ψ ε ε⎡ ⎤

, = + − − , + − −⎢ ⎥⎣ ⎦

0 0 0u Tsψ = − 0( , 0)elT T ε= =

0u 0s

0T T=21

2el elEψ ε⎛ ⎞

⎜ ⎟⎝ ⎠

=

00

( ) logel el Ts s s CT Tψ ε∂

= − = + +∂

free-energy in the natural or reference state

internal energy and entropy in the reference state.

free-energy increase due to the elastic strain for

entropy increase due to the elastic strain for 0T T=( )el el els Eε ε α=

entropy

sC TT∂

=∂

heat capacity

2

0 0 0 00

1 ( ) ( ) log2

el el Tu Ts E T T E C T T TT

ψ ε ε α⎛ ⎞⎜ ⎟⎝ ⎠

⎡ ⎤= − + − − + − −⎢ ⎥

⎣ ⎦

final form of the free-energy

thermal expansion factorα

[ ] ( )0 1A A S S S int intu Ts u T s u Tsψ ξ ξ ξ⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦= − − ∆ − ∆ + − −

Material undergoing phase transformation

Au As

u∆ s∆ 00

A S

A S

u u us s s

∆ = − ≥∆ = − ≥

Su Ss

internal energy and the entropy of the austenite

internal energy difference and the entropy difference between the austenite and the martensiteinternal energy and the entropy of the martensite

int intu Ts− interaction term

( ) ( ) ( ) ( )20 0

0

1 log2 L S L S

TE s T T s E C T T TT

ψ ε ε ξ ε ε ξ α⎡ ⎤

= − − − − + − −⎢ ⎥⎣ ⎦

( ) ( )A A Su Ts u T sξ+ − − ∆ − ∆⎡ ⎤⎣ ⎦

free-energy form for a material undergoing a solid-solid PT

stress and entropy 0( ) ( )L Sel E s E T Tψ ψσ ε ε ξ αε ε∂ ∂

= = = − − −∂ ∂

0

log ( )A S L STs s s C s E

T Tψ ξ ε ε ξ α∂

= − = − ∆ + + −∂

heat equation divCT q b+ =&

tmc mecb H D= +

( )2 2

tmc LS SS

H T T T E E sT Tψ ψε αε ε αξ ξε ξ

∂ ∂⎡ ⎤= + = − + + ∆⎣ ⎦∂ ∂ ∂ ∂

& && &

( )mec LS SS

D T s uψ ψσε ε ε σξ ξε ξ

⎛ ⎞∂ ∂= − + = | | − ∆ + ∆⎜ ⎟∂ ∂⎝ ⎠

& && &

heat source

heat production associated to the thermomechanical coupling

heat production associated to dissipative mechanical processes