Jtam.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

41, 3, pp. 675-691, Warsaw 2003

NON-ASSOCIATED SUPERELASTICITY IN ROTATING

FRAME FORMULATION

Benoit Vieille

Mohammed Lamine Boubakar

Christian Lexcellent

Laboratory of Applied Mechanics LMARC, Besancon, France

e-mail: benoit.vieille01@edu.univ-fcomte.fr

In order to describe and predict the superelastic finite strains behaviour
of Shape Memory Alloys, a kinematic description with directors is propo-
sed. Compatible with either isotropic or anisotropic behaviours, it allows a
direct extension from the small pertubation formalism.Particularly, a gene-
ral framework is proposed for the description of the Shape Memory Alloys
superelastic behaviour under 3D proportional loadings.

Key words: Shape Memory Alloys, superelasticity, finite strains, normality
rule, non-associated constitutive framework

Notations

Tensors will be denoted by bold leters. Their juxtaposition implies the
usual double contraction operation. Superposed S and A indicate respectively
a symmetric tensor (•)S and an antisymmetric tensor (•)A. A superposeddot
(̇•) indicates the usual time derivative. ∂X(•) denotes the partial derivative of
(•) with respect to X. I is the second order identity tensor. Finally, ‖•‖ is
the Euclidean norm.

1. Introduction

Among the differentmaterials considered to be smart, i.e. adapting them-
selves to their environment, the ShapeMemory Alloys do present interesting
properties, namely the shapememory effect and the superelasticity.Thosepro-
perties rest on a phase transformation: themartensitic transformation, from a



676 B.Vieille et al.

parent phase called austenite to a product phase called martensite, occurring
in the material subjected to a stress and/or temperature. Superelasticity is
certainly one of the most studied properties of that kind of material. Up to
now, many SMA constitutive models in 1D have been developed in the past
fewyears for structural elements such aswires, rods andbeams.However, very
few 3D models have been studied to represent the behaviour of SMA in 3D
structures.

The present work aims atmodelling the SMAfinite strains superelasticity
behaviour thanks to a rotating frame formulation. This one is based on the
director vectors concept presented in the following part. Within this appro-
ach, the transformation gradient decomposition allows to get the partition of
a cumulated tensorial deformation in a formalism close to the small perturba-
tions one. Using the notion of ”constrained equilibrium”, the thermodynamic
description is based, like in the small strains hypothesis, on an assumed non-
convex specific free energy function and two independent normal dissipative
processes, i.e. one for the forward transformation, the other for the reverse
transformation. The evolution laws of both transformations are derived from
a suitable maximum dissipation principle. The consequent constitutive equ-
ations can be time-discretized thanks to amethod based on the forwardEuler
time discretization (explicit integration) and then implemented in a finite ele-
ment code.

2. Finite transformations kinematics

The approach used to build a finite transformations kinematics for the
SMA thermomechanical behaviour study is based on the concept of the defor-
med intermediate configuration introduced for the first time by Eckart (1948)
and the notion of director vectors due to Cosserat and Cosserat (1909), resu-
med byMandel (1971).

In the case of the SMA, the director vectors notion allows to orient and
then to fix a non-relaxed intermediate configuration associated to the phase
transition in the material. Assuming an elastic behaviour independent of the
state phase, the following decomposition of the total transformation gradient
is introduced

F=(I+εe)qFtr (2.1)

Ftr is the transformation gradient due to the phase transition allowing to
connect a reference configuration to an intermediate configuration in which a



Non-associated superelasticity in rotating... 677

director frame linked somehow to thematerial internal structure preserves its
initial orientation. The evolution of this frameup to the current configuration,
in which the elastic deformations εe (‖εe‖ ≪ 1) are measured, is defined by
rotation q (q⊤q= I; detq=1).
The SMA are characterized by a possible reversible phase transition, au-

stenite < − > martensite. Their behaviour being associated with a solid-solid
phase transition according to specific planes called habit planes, an orthonor-
med direct frame defined by these planes as an average of their orientation
can be considered as the director (Fig.1).

Fig. 1. Director frame (RD) for the Representative Volume Element (a) and the
director frames linked to the habit planes (b)

The expression of the total transformation gradient (2.1) leads to the fol-
lowing decomposition of the material strain rate tensor D = [ḞF−1]S in the
intermediate configuration

ḋq = ε̇
e
q+ ḋ

tr
q ḋ

tr
q =
[
Ḟ
tr(Ftr)−1

]S
(2.2)

where
(•)q =q

⊤(•)q



678 B.Vieille et al.

and d representing a cumulated tensorial deformation in the sense introduced
in Gillormini et al. (1993).
The intermediate configuration can be defined by decomposition of the

material spin W= [ḞF−1]A in the current configuration, by solving

q̇=(W−Wtr)q q
∣∣∣
t=0
= I Wtr =q

[
Ḟtr(Ftr)−1

]A
q⊤ (2.3)

That is why the knowledge of the spin Wtr of the material milieu, as
regards its internal structure, is needed. Beyond the micro-macro approach
seeming to be more natural to get this knowledge, the use of anti-symmetric
isotropic tensorial functions representation theorems or the choice of a kine-
matic rotation consistent toward aphenomenological approach, are both other
possibilities to assess the value of Wtr. For random or pseudo-random distri-
butions of the habit planes, a family of objective kinematics rotations q can
be defined by solving the following differential problem (Boubakar, 1994)

q̇=
[
(1−α)W+αṘR⊤

]
q q

∣∣∣
t=0
= I α ∈]0,1] (2.4)

where R is the proper rotation associated with F.

3. Constitutive frame in the intermediate configuration

The specific Helmholtz free energy for a three-dimensional formulation
of the SMA behaviour is given by the following relation in the intermediate
configuration, where εeq = dq −d

tr
q (2.2) is the rotated elastic strain tensor

from the current configuration, z is the martensite volume fraction and T is
the temperature (Boubakar and Lexcellent, 2002)

ψ = ψ(εeq,z,T) = ψ
el(εeq)+ψ

ch−th(z,T)+ψdi(z,T) (3.1)

ψdi(z,T) represents a specific coherency energy (Müller, 1989), and

ψel(εeq)=
1

2ρ
(εeq)

⊤
C
e
ε
e
q

(3.2)

ψch−th(z,T)= ua0R−Ts
a
0R+Cv

[
(T −TR)−T ln

T

TR

]
−zπ0(T)

π0(T)= (u
a
0R−u

m
0R)−T(s

a
0R−s

m
0R) is thedriving force for temperature-induced

phase transition at stress-free state (a representing the austenitic phase and



Non-associated superelasticity in rotating... 679

m the martensitic phase), ua0R, u
m
0R, s

a
0R and s

m
0R being the values of internal

energies and entropies at the chosen reference temperature TR.
The fourth order elastic stiffness tensor Ce, the specific heat Cv and the

mass density ρ are assumed to be the same for both phases.
From the Clausius-Duhem inequality, the mechanical dissipation is then

here

Dmech =(ḋtrq )
⊤
σq −Rż ­ 0 R =−ρπ0(T)+ρ∂zψ

di(z,T) (3.3)

where σq = q
⊤
σq is the rotated Cauchy stress tensor σ of the current con-

figuration. The forward transformation (a → m) is characterized by ż > 0
and the reverse one (a ← m) by ż < 0. Assuming a normal dissipative process
for each transformation, Dmech can be definedby two independentdissipation
functions of ḋtrq and ż

Dmech =






Dam
[
(ḋtrq , ż); (z,T)

]
if ż > 0

Dma
[
(ḋtrq , ż); (z,T)

]
if ż < 0

(3.4)

In the particular case of a time-independent behaviour (plasticity-like beha-
viour), these functions are convex, positively homogeneous of order one. Since
the forward transformation (a → m) can be initiated in any loading direction,
Dam is a quasi-positively homogeneous function defining a full convex cone
in an eight-dimensional space such that Dam[(0,0);(z,T)] = 0. Following the
generalized standard material theory (Halphen and Nguyen, 1975), the ther-
modynamic forces σq and R belong then to the subdifferential ∂Dam of Dam
and the dual variables ḋtrq and ż belong to the subdifferential ∂D

∗
am of the

indicator function of the convex domain Ωam = {(σq,R)/ϕam(σq,R) ¬ 0},
i.e. the elasto-dissipative domain, ϕam defining a yield function. Thus

(ḋtrq )
⊤(σq − σ̂q)− (R− R̂)ż ­ 0 ∀

[
(σq,σ̂q),(R,R̂)

]
∈ Ωam (3.5)

what means that among all the admissible thermodynamic forces, the ones
associated with a given set (ḋtrq , ż) maximize the mechanical dissipation, i.e.

Dmech = max
(̂σq,R̂)∈Ωam

{
(ḋtrq )

⊤
σ̂q−R̂ż

}
= min
(̂σq,R̂)∈Ωam

{
−
(
(ḋtrq )

⊤
σ̂q−R̂ż

)}
(3.6)

Considering the associated unconstrainedminimization problemusing the fol-
lowing Lagrange function

Lam(σ̂q,R̂, λ̇am)=−(ḋ
tr
q )
⊤
σ̂q+ R̂ż + λ̇amϕam(σ̂q, R̂) (3.7)



680 B.Vieille et al.

the evolution laws giving ḋtrq and ż for a forward transformation (a → m)
are obtained classically from the Lam optimality conditions. In the standard
Kuhn-Tucker form

˙dtrq = λ̇am∂
σ̂q

ϕam ż =−λ̇am∂R̂ϕam ϕam =0

λ̇am ­ 0 λ̇amϕam =0
(3.8)

It can be stated from the yield condition ϕam = 0 that σq = σ̂q and
R = R̂. TheLagrangemultiplier λ̇am is derived fromthe consistency condition
ϕ̇am =0.
The normality assumption contained in these relations has been verified

experimentally in Fig.2 for CuAlBe alloys (Bouvet, 2001) where the arrows
represent the phase transformation direction.

Fig. 2. Transformation outset surface in the case of bicompression and
tension-internal pressure tests (Bouvet, 2001)

The following formalyield functioncanbe introduced for the forward trans-
formation

ϕam(σq,R)= Σ(σq)−Σam(R)¬ 0 (3.9)

what leads to (Boubakar and Lexcellent, 2002)

ḋ
tr
q = λ̇am∂σqΣ ż = λ̇am∂RΣam =

λ̇am
γΣ

(3.10)

Hence
ḋtrq = żγΣ∂σqΣ (3.11)



Non-associated superelasticity in rotating... 681

γΣ is the maximum phase transition strain in the direction of the threshold
stress Σam. Following the definition of the elasto-dissipative domain Ωam in
the thermodynamic forces space, the threshold stress Σam is defined as a
function of R. However, from the expression of R in Eq. (3.3), it can be al-
so considered as a constitutive function of z and T describing the effect of
interaction between differently oriented crystals in the shapememory polycry-
stalline alloys. The scalar-valued function Σ of the rotated Cauchy stresses
tensor or effective stress, is positively homogeneous of the first order andmust
verify Σ(Qσ⊤q Q) = Σ(σQ) for any rotation Q of the intermediate configu-
ration in the case of an isotropic behaviour. If the volumetric changes are
negligible, Σ depends on the second and the third invariant of σq (Boubakar
et al., 20020 in order to account for the superelasticity asymmetric character
(Fig.3). The invariants of σq are also those of σ since σq is obtained from σ
thanks to an endomorphic transformation.

Fig. 3. Yield functions and transformation directions for each phase transition.
Proportional loading case

If the forward transformation is interrupted, when unloading, the reverse
transformation (a ← m) can not occur in any direction, i.e. Dma (3.4) is



682 B.Vieille et al.

not a full cone, since the phase transition strainsmust be recovered during the
reverse transformation.FromEq. (3.11), Dma[(ḋ

tr
q , ż);(z,T)]≡ Dma[ż;(z,T)].

The simplest positively homogenous dissipation function of order one for the
reverse transformation can then be chosen such as:
— for the forward transformation, Πam(z,T) > 0

Dam[ż;(z,T)] = Πam(z,T)ż (3.12)

— for the reverse transformation, Πma(z,T) < 0

Dma[ż;(z,T)] = Πma(z,T)ż (3.13)

In this case, the normality rule ż =−λ̇ma∂Πφma with (Boubakar and Lexcel-
lent, 2002)

φma = Π −Πma(z,T)­ 0
(3.14)

Π = γΣ(∂σqΣ)
⊤
σq+ρπ0−ρ∂zψ

di

allows to build a unique yield function for the reverse transformations in the
stress space, considering proportional loading-unloading paths (Fig.3). Thus

ϕma =(∂σqΣ)
⊤
σq −Σma(z,T)­ 0

(3.15)

γΣΣma =−ρπ0+ρ∂zψ
di(z,T)+Πma(z,T)

Within the context of a phenomenological approach, the threshold stress
Σma is defined from a uniaxial test (Boubakar and Lexcellent, 2002).
Following the generalized standard theory of materials, the direction of

the reverse transformation for a given stress state must be normal at the
corresponding point in the stress space to a convex domain Ωκ containing
Ωma = {Π/Π −Πma ­ 0} and turned towards the outside of this domain, i.e.
a convex constraint domain is necessary to assure a global minimum for the
instrinsic dissipation. The use of the maximum dissipation principle in order
to derive the evolution laws giving ḋtrq and ż needs thedefinition of apotential
function κ(σq,R) such that

κ(σq,R) > 0 when Π −Πma > 0

κ(σq,R)= 0 when Π −Πma =0
(3.16)

Since for proportional loading

∂σqΣ =
ḋtrq

‖ḋtrq ‖
=
dtrq

‖dtrq ‖
(3.17)



Non-associated superelasticity in rotating... 683

the following function can be introduced

κ(σq,R)=
(dtrq )

⊤

‖dtrq ‖
σq −Σma(R)­ 0 Σma(R)≡ Σma(z,T) (3.18)

and then finally

ḋ
tr
q =−λ̇ma∂σqκ =−λ̇ma

dtrq

‖dtrq ‖
ż = λ̇ma∂Rκ (3.19)

λ̇ma > 0 is given by the consistency condition ϕ̇ma =0.

4. Application to the RLT models family

Theprevious sections presented a plasticity-type phenomenologicalmodel-
ling of the SMA superelasticity. Generalizing the main proposed approaches
up to date, it enables to take into account different response configurations
(Fig.4) by defining suitable hardening laws, in absence of a good knowledge
about the nucleation and the growth of martensite into austenite (Boubakar
et al., 2002). As shown in Boubakar et al. (2002), in the case of small strain
hypothesis, the definition of a history-variable set allows to take into account
a strong history-depedence of the behaviour.
Defining a relatively simple interaction energy ψdi, theRLTmodels family

allows to account quitewell for the transformation configurations presented in
the Fig.4a where the place of the internal loops transition is located along a
diagonal line. Concerning the choice of ψdi, the simplest function z(1−z) such
as ψdi =0 for z =0 (pure austenite) and z =1 (pure martensite), has been
introduced byMüller (1989, 1991), Huo andMüller (1993). Different features
of these models can be specified within the proposed approach in Table 1.

5. Parametric identification

Most of the models developed for the SMA superelasticity are based on a
small perturbation formalism and their parameters are identified within this
hypothesis. Thus, the RLT model parameters are identified directly from a
pure tension test (Bouvet et al., 2000). However, in the case of a finite strains
analysis, the use of such parameters introduces errors. Willing to exploit the



684 B.Vieille et al.

Fig. 4. Shape memory alloys superelastic responses: (a) CuZnAl (Huo andMüller,
1993), (b) NiTi (Lexcellent and Tobushi, 1995) and (c) CuAlBe (Bouvet, 2001)



Non-associated superelasticity in rotating... 685

Table 1.RLTmodel characteristics

• Specific coherency energy

ψdi(z,T) = z(1−z)φit(T)

with φit(T)= u0−Ts0, u0 internal energy and s0 entropy

• State law

σq = ρ
∂ψel

∂εeq
=Ceεeq σ̇q =C

e
ε̇
e
q

Ce elastic stiffness tensor

• Forward transformation

φam = γΣϕam = Π −Πam(z,T)¬ 0 where Πam(z,T)= k1(z)

• Reverse transformation

φma = γΣϕma = Π −Πma(z,T)­ 0 where Πma(z,T)= k2(z)

• Thermodynamic force

Π = γΣ(∂σqΣ)
⊤
σq +ρπ0(T)−ρ(1−2z)φit(T)

• Transformation kinetics

k1(z)= 2φit(M
0
S)z +

s0−∆s0−2s0z

a1
ln(1−z)

k2(z)=−2φit(A
0
S)(1−z)+

s0+∆s0−2s0(1−z)

a2
ln(z)

M0S is the forward transformation start temperature at stress free sta-
te; A0S is the reverse transformation start temperature at stress free
state

• Equivalent stress

(σq)eq =σqf(yσq) where f(yσq)= 1+ayσq

σq =

√
3

2
dev(σq) : dev(σq) yσq =

27

2

det(dev(σq))

σq

a is the material parameter characterizing the tension-compression
dissymmetry



686 B.Vieille et al.

identification method perfected within the small strain context, the triaxial
nature of the tension test involves a logarithmic cumulated tensorial defor-
mation measure. This measure can be used in order to avoid the finite strain
parameters reidentification. Consequently, a part of these parameters can be
deduced from the one obtained for the small strain hypothesis using the follo-
wing approach.

γ representing the maximum transformation strain in tension, the loga-
rithmic cumulated tensorial deformation measure leads directly to a relation
connecting γSP to γFS (FS and SP denoting respectively Finite Strains and
Small Perturbations)

γFS = ln(1+γSP) (5.1)

Besides, although the transformation temperaturesdonot changewhatever
the approach used is (what means that the transformation start stresses do
not change either), the other parameters introduced previously do depend on
the same approach (Bouvet et al., 2000)

(∆u0)FS =
γFS
γSP
(∆u0)SP and (u0)FS =

γFS
γSP
(u0)SP

(∆s0)FS =
γFS
γSP
(∆s0)SP and (s0)FS =

γFS
γSP
(s0)SP

(5.2)

∆u0 and ∆s0 representingthe internal energy and the entropy variation, re-
spectively.

6. Applications

TheRLT thermomechanical behaviour evoked mentioned has been imple-
mented in the finite element code CASTEM2000R© in a spatial shell context.
The model numerical integration has been validated by performing a set of
different tests (Bouvet et al., 2001; Boubakar and Vieille, 2002).

6.1. Tension-compression test

In order to bring out the tension-compression asymmetry behaviour, tests
have beenperformed considering the experimental results given byOrgéas and
Favier (1996). These tests have beenmadeonNiTi sampleswhosematerial pa-
rameters are given inTable 2. The numerical simulation and the experimental
points coincide quite well (Fig.5).



Non-associated superelasticity in rotating... 687

Table 2.Tension/compression asymmetry – NiTi material parameters

E ρ ∆s0 s0 M
0
s A

0
sν γΣ a1 a2 a

[GPa] [kg/m3] [J/(kg·K)] [J/(kg·K)] [K] [K]

55 0.29 6500 0.055 0.5 0.2 0.189 75 0 280 300

Fig. 5. Asymmetry effect between tension-compression

6.2. Partial loading or unloading

Indeed, consideringmore complex tests, the stress distribution in the struc-
ture is not uniform, that is why the direct and reverse transformations outset
differs from one structure element to another. Whatever the case is, partial
loading or unloading, the unstable balance line is the line where direct and
inverse transformations start (Fig.6). Thematerial used for this test is aNiTi
whose properties are given in Table 3 (Bouvet et al., 2000). The numerical
simulation and the experimental points coincide quite well for the external
loops.

Table 3. Internal loops – NiTi material parameters

E ρ ∆s0 s0 M
0
s A

0
sν γΣ a1 a2 a

[GPa] [kg/m3] [J/(kg·K)] [J/(kg·K)] [K] [K]

52 0.3 6700 0.05 5 5 0.2 50 0 200 284



688 B.Vieille et al.

Fig. 6. (a) Partial loading; (b) partial unloading

6.3. Bulging test

A circular plate embedded around its circumference is submitted to a hy-
drostatic pressureuniformlydistributed on its surface.Meshing, geometry and
deformed shape are shown in Fig.7. Thematerial used for this test is CuZnAl
whose parameters are presented in Table 4 (Rogueda, 1993). In Fig.8 is pre-
sented the RLTmodel simulation for two thin films. Besides, the elastic finite
strains simulations are comparedwith the analytical results obtained by using
theBeams andLin equations established for this example (BBeams, 1959; Lin,
1990).

Table 4.Bulging test – CuZnAlmaterial parameters

E ρ ∆s0 s0 M
0
s A

0
s

ν γΣ a1 a2 a
[GPa] [kg/m3] [J/(kg·K)] [J/(kg·K)] [K] [K]

58.6 0.2 8500 0.0358 0.05 0.5 0.142 22.46 4.06 282 295



Non-associated superelasticity in rotating... 689

Fig. 7. Bulging test – deformed shape and dimensions

Fig. 8. Elastic and superelastic bulging test modelling

7. Conclusion

A non-material rotating frame formulation is proposed for the SMA su-
perelastic finite strains calculation. It assumes that the alloy elastic behaviour
is independent of the phase state. This approach takes place in the irrever-
sible processes thermodynamics framework and assumes the behaviour to be
independent from the deformation rate. It presents a formalism close to small
pertubations. A numerical scheme in two stages has been adopted, consisting
in an elastic prediction followed possibly by a superelastic correction (Bouvet



690 B.Vieille et al.

et al., 2001). The numerical results obtained for a set of different tests allow
to validate the SMA superelastic effect numerical integration in the particular
case of theRLTmodel andusing the shell finite elements. However, additional
tests are necessary to confirm the proposed approach. It is the purpose of a
runningwork involving a set of tests under non-proportional loadings. Indeed,
the proposed modelling can be easily extended to that kind of loading.

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Sformułowanie problemu niestowarzyszonej supersprężystości

w obrotowym układzie współrzędnych

Streszczenie

W pracy przedstawiono metodę przewidywania i opisu stanu skończonych su-
persprężystych odkształceń określających zachowanie się stopów z pamięcią kształtu.
Metodę opartona reprezentacji kinematycznej stanuotrzymanej zapomocąwektorów
kierunkowych.Metoda ta, jako spójnadlamateriałów izotropowych i anizotropowych,
umożliwia wnioskowanie wprost z formalizmu techniki perturbacyjnej bazującej na
małymparametrze.Wszczególności, zaproponowanoogólnepodejście do opisu super-
sprężystego zachowania się stopów z pamięcią kształtu poddanych proporcjonalnym
obciążeniom przestrzennym.

Manuscript received December 12, 2002; accepted for print March 5, 2003