Relaxation and creep phenomena in shape memory alloys. Part II: Stress relaxation and strain creep...

Z. angew. Math. Phys. 51 (2000) 419–4480044-2275/00/030419-30 $ 1.50+0.20/0c© 2000 Birkhauser Verlag, Basel

Zeitschrift fur angewandteMathematik und Physik ZAMP

Relaxation and creep phenomena in shape memory alloys.Part II: Stress relaxation and strain creep during phasetransformation

X. Balandraud, E. Ernst and E. Soos

Abstract. In Part I of the paper we use a simplified 1D thermomechanical model to describethe behavior of SMAs. We prove that this model can describe the existence of the hysteresis loopand the pseudoelastic behavior of SMAs. In Part II of the paper we show that the same modelpredicts the stress relaxation and the strain creep during martensitic transformation. We presentexperimental results concerning stress relaxation and strain creep. The theoretically calculatedand the experimentally deduced stress relaxation and strain creep curves are in satisfactoryagreement.

Mathematics Subject Classification (1991). 80A22.

Keywords. Shape memory alloys, stress relaxation, strain creep, thermomechanical models,experimental results.

1. Introduction

In Part I (see [1]) of our paper we use a simplified 1D version of Pham’s 3Dthermomechanical model (see Pham [2]) to describe the behavior of SMAs. Weshow that our model can describe the existence of the hysteresis loop and thepseudoelastic behavior of SMAs during traction - compression tests. We alsopresent the experimental results allowing us to determine the material parametersfor the studied shape memory Cu - Zn - Al alloy.

Vacher [3] and Muller [4] have reported the existence of relaxation and creepphenomena in SMA during deformation induced phase transformation.

In Sections 2 and 3 of Part II of our paper we show that our simplified 1Dmodel can predict stress relaxation and strain creep during direct austenitic -martensitic transformation, as well as stress creep and strain relaxation duringinverse martensitic - austenitic transformation.

In Section 4 of Part II we present our experimental results concerning stressrelaxation and strain creep during direct austenitic - martensitic transformation.Our results show the strong influence of temperature variations on stress relaxation

420 X. Balandraud et al. ZAMP

and on strain creep.In Section 5 of Part II we compare the predictions of the model with the

obtained experimental data. There exists a good agreement between the theoreti-cally calculated and experimentally determined stress relaxation and strain creepcurves.

The last section contains some concluding remarks concerning both parts ofthe paper, as well as our opinion concerning the way in which the model can beimproved, by including a non - vanishing intrinsic dissipation.

The significance of all symbols is given in Part I of our paper.

2. Stress relaxation and creep

In this section we study the predictions of the homogeneous 1D model for thestress relaxation and stress creep problem. This phenomenon occurs when, duringa hard tensile test at a certain moment during the phase transformation, the strainis held constant.

Our aim is to determine the relaxation or creep values for the stress, tempera-ture and volume fraction, that is the asymptotic values of those functions whenthe time goes to infinity.

The main result of this section shows that the asymptotic value of the stressdepends piecewise linearly on the value of the strain at which the loading or un-loading was stopped (see (2.34) and (2.49)). This result will be compared in thelast section with the experimental data.

For simplicity, the superscript E used in Part I will be omitted. We first recallthe relations governing the thermomechanic behavior of a cylindrical specimenduring the direct and inverse phase transformation.

According to the equations (5.1)–(5.5) from Part I we have:

σ(t) = E(ε(t) − gβ(t)), (2.1)

σ(t) = p(T0 + θ(t)− Ta + β(t)4T ), (2.2)

θ(t) +1τθ(t) =

L

Cβ(t). (2.3)

From Eqs. (2.1) and (2.2) we get the relations expressing the stress and themartensitic volume fraction as functions of the strain and of the temperature:

σ(t) =pE

Eg + p4T [4Tε(t) + g (T0 + θ(t) − Ta)] , (2.4)

β(t) =1

Eg + p4T [Eε(t)− p (T0 + θ(t)− Ta)] . (2.5)

Vol. 51 (2000) Stress relaxation and strain creep in SMA 421

2.1. The case of the direct transformation

In the first part of this section we consider the case when the loading ceases duringthe direct martensitic transformation.

Let εMS , tMS and, respectively εMF , tMF , be the values of the strain and timefor which the martensitic transformation starts and, respectively, ends, and let usdenote with θMF the temperature at the moment when the martensitic transfor-mation is complete. We remark that tMS , εMF , tMF , and θMF actually dependon the value ε(t) = a of the strain rate.

We denote by t the moment at which the loading is stopped during a directmartensitic transformation. From this moment, the strain is held constant untilthe end of the experiment. Obviously, the moment t satisfies the inequalities

tMS < t < tMF . (2.6)

Let us denote by ε, σ, β and θ the values of ε, σ, β and θ at the time t. Sincethe axial deformation is fixed, for t > t we have

ε(t) = ε, (2.7)

that isε(t) = 0, (2.8)

while the martensitic volume fraction at t satisfies the phase transformation con-dition

0 < β < 1. (2.9)

As β varies continuously, there is a value t∗ > t such as (2.9) is fulfilled forevery intermediate value of the time variable t. Accordingly, the time evolution ofthe temperature is described by the equation

θ +1τE

θ =EL

C(Eg + p4T ) + pLε, (2.10)

where the relaxation time τE is given by relation (5.23) from Part I, i.e.

τE =C(Eg + p4T ) + pL

C(Eg + p4T )τ > τ. (2.11)

By using (2.8), the differential equation (2.10) becomes for every t < t < t∗

θ +1τE

θ = 0. (2.12)

In accord with the results obtained in the first part of the Section 5 from Part I,the initial value θ satisfies

θ(t) = θ > 0. (2.13)


The evolution of the temperature θ(t) is then given for t < t < t∗ by therelation

θ(t) = θe−t−tτE . (2.14)

Replacing the values (2.7) and (2.14) in (2.4) and (2.5) we get the evolution ofthe stress and of the volume fraction for t < t < t∗

σ(t) =pE

Eg + p4T

[ε4T + g

(T0 + θe−

t−tτE − Ta

)], (2.15)

and

β(t) =1

Eg + p4T

[Eε− p

(T0 + θe−

t−tτE − Ta

)]. (2.16)

These relations remain valid as long as the phase transformation condition issatisfied.

As θ > 0, relation (2.15) implies that the axial stress σ(t) is a decreasingfunction of time. Hence the model predicts a stress relaxation. This behavior isin agreement with existing experimental data (see Vacher [3] and Muller [4]).

The relation (2.16) implies that the volume fraction of martensite β(t) is an in-creasing function of the time, therefore our model indicates that the direct marten-sitic transformation continues even after the elongation was stopped.

Since the volume fraction of martensite β(t) is an increasing function of time,there are two possibilities: either

β(t) < 1, for every t > t, (2.17)

or there exists a moment tM such as

β(t) < 1 for t < t < tM and β(tM ) = 1. (2.18)

Let us consider these two cases one by one.We first study the case when the relation (2.17) is fulfilled. Since β(t) is

increasing, we have0 < β∞(ε) ≤ 1. (2.19)

We can now compute the asymptotic values for the stress and for the volumefraction, since relations (2.15) and (2.16) are valid for every t > t. We get

σ∞(ε) =pE

Eg + p4T [ε4T + g (T0 − Ta)] , (2.20)

andβ∞(ε) =

1Eg + p4T [Eε− p (T0 − Ta)] . (2.21)


From (2.19) and (2.21) we have

ε < ε = g +p

E(Ta − Tm). (2.22)

That is the necessary and sufficient condition for (2.17) to take place.Consequently, if (2.22) is satisfied, the direct martensitic transformation takes

place for every t > t, and both the austenitic and the martensitic phases aresimultaneously present in the relaxed (final) state of the specimen. Moreover, thetemperature θ(t) is strictly decreasing, and its asymptotic value is zero, i.e.

limt→∞

θ(t) = θ∞(ε) = 0. (2.23)

In the second case, that is when (2.22) is not fulfilled, the martensitic trans-formation ends at the time tM . According to the constitutive equation (2.1) andto the evolution law (3.12) from Part I, the time behavior of the axial stress σ(t)and of the volume fraction β(t), for t > tM , is described by the equations

σ(t) = E(ε− g) and β(t) = 1. (2.24)

For t > tM , the temperature satisfies the differential equation

θ +1τθ = 0, (2.25)

with the initial conditionθ(tM ) = θe−

tM−tτE . (2.26)

Hence the temperature is given, for t > tM , by the following expression:

θ(t) = θe−tM−tτE e−

t−tMτ . (2.27)

Accordingly, the asymptotic values for σ(t), β(t) and θ(t) are in the second case

σ∞(ε) = E(ε− g), β∞(ε) = 1 and θ∞(ε) = 0. (2.28)

In order to summarize the obtained results, we recall that

εMS =p

E(T0 − Ta) and εMF = g +

p

E(T0 + θMF − Ta), (2.29)

andθMF > 0. (2.30)

By using (2.22), (2.29) and (2.30), we get

εMS < ε < εMF . (2.31)


We can summarize (2.20), (2.21), (2.23) and (2.28) in

θ∞(ε) = 0, for εMS < ε < εMF , (2.32)

β∞(ε) ={ 1

Eg+p4T [Eε− p (T0 − Ta)] for εMS < ε ≤ ε,1 for ε ≤ ε < εMF,

(2.33)

We observe that the relation

σ∞(ε) ={ Ep

Eg+p4T [ε4T + g (T0 − Ta)] for εMS < ε < ε,

E(ε− g) for ε < ε < εMF.

(2.34)

We observe that the relaxation values of the temperature, volume fraction andstress do not depend directly on the strain rate ε(t) = a, imposed to the specimenup to the time t. The equation (2.34) implies that the asymptotic value of thestress depends piecewise linearly on the value of the strain at which the loadinghas been stopped.

In the same time, from (2.22) and (2.34) we get

σ = σ∞(ε) = p(T0 − Tm). (2.35)

2.2. The case of the inverse transformation

In the second part of this section, we shall study the case when the loading isstopped during the inverse martensitic - austenitic transformation.

We denote by tAS , εAS and θAS the time, strain and temperature at whichthe inverse transformation starts, and by tAF , εAF ,and θAF , the similar quantitiescorresponding to the end of the inverse transformation. We remark that all thesevariables actually depend on the strain rate ε(t) = a of the experiment.

Let t be the moment at which the unloading is stopped during the inverseaustenitic transformation. After this moment the strain is held constant. Obviouslywe have

tAS < t < tAF . (2.36)

Let us denote by ε, σ, β and θ the values of ε, σ, β and θ at the time t. Since,for t > t, the strain is held constant,

ε(t) = ε. (2.37)

As we already know, θAS is positive, i.e.

θAS > 0, (2.38)

and the evolution of the temperature between tAS and t is described by the relation(see Eq. (5.73) from Part I)

θ(t) = θAS e− t−tAS

τE − aτEΓE(

1− e−t−tASτE

). (2.39)


Hence we get

θ = θ(t) = θAS e− t−tAS

τE − aτEΓE(

1− e−t−tASτE

). (2.40)

Consequently, if t is “near” tAS , then θ is positive, but if the difference betweent and tAS becomes important, θ might become negative. The same conclusionfollows from the inequalities (5.127) from Part I.

When θ > 0, the analysis made in the previous subsection applies withoutmodifications. Thus the asymptotic values of the temperature, volume fractionand stress are those given by (2.32)–(2.34).

Let us consider the case θ < 0. Since β < 1, there is a value t∗ > t such as (2.9)is satisfied for every intermediate value of the time variable t, which implies thatrelations (2.14), (2.15) and (2.16) are true.

Relation (2.15) implies that the axial stress is an increasing function of time.It follows that our model predicts a creep of the stress, in good agreement withlaboratory experiments (see Muller [4]). Similarly, the relation (2.16) shows thatthe volume fraction is a decreasing function of the time, hence the model impliesa continuation of the inverse austenitic transformation even after the unloadingwas stopped.

If the conditionεMS =

p

E(T0 − Ta) < ε, (2.41)

is satisfied, then, by using (2.21) for t > t, we obtain

β(t) > β∞(ε) > 0. (2.42)

Consequently, if condition (2.41) is satisfied, then the asymptotic values of θ(t),σ(t) and β(t) are those given by Eqs. (2.23), (2.20) and (2.21). Therefore, in thefinal state of the specimen both phases are simultaneously present.

If condition (2.41) is not satisfied, then (2.16) implies that there is tM > t, s.a.

β(t) > 0 for t < t < tM , and β(tM ) = 0. (2.43)

Hence the austenitic transformation ends at the time tM .According to the constitutive equation (2.1) and the evolution law (3.10) from

Part I, the time behavior of the stress and of the volume fraction is described fort > tM by the formula

σ(t) = Eε, β(t) = 0. (2.44)

Therefore, the asymptotic values for the stress and volume fraction are

σ∞(ε) = Eε, β∞(ε) = 0. (2.45)

Consequently, if the restriction (2.41) is not satisfied, the final state of the spe-cimen is a pure austenitic one, and the asymptotic value of the stress lies on the“first elastic line”.


In order to summarize the results, let us notice that, according to the equation(5.127) of Part I

θEAF < 0 = θEMS < θEAS < θEMF ,

and, as we already know from (5.121) from Part I and from (2.31)

0 < εAF < εMS < ε < εAS < εMF . (2.46)

The obtained results can be expressed by the following relations

θ∞(ε) = 0, for εAF < ε < εAS , (2.47)

β∞(ε) =

1 for ε < ε < εAS ,

1Eg+p4T [Eε− p (T0 − Ta)] for εMS < ε < ε,

0 for εAF < ε < εMS ,

(2.48)

and

σ∞(ε) =

E(ε− g) for ε < ε < εAS,

EpEg+p4T [ε4T + g (T0 − Ta)] for εMS < ε < ε

Eε for εAF < ε < εMS .

(2.49)

The Fig. 2.1 represents the asymptotic value of the volume fraction with respectto the value of the strain at which the loading, respectively the unloading processis stopped.

ε ε

(ε)β (ε)β

εεAF εMS εAS εMFε ^MS εMF

0

1

0

1

ε

Figure 2.1. The direct and the inverse transformation: volume fraction versus strain

The dependence of the asymptotic value of the stress on the value of the im-posed constant strain ε, both in direct and in inverse transformation is representedin Fig. 2.2.


^

ε

ASσ

σMS

σAF

AFε MSε εAS εMF

σMF

σ

εε

0

σ

σ

( )T

( )T

Figure 2.2. Relaxation and creep of the axial stress

3. Strain creep and relaxation

In this section we shall study the predictions of our homogeneous 1D model for thestrain relaxation and strain creep problem. This phenomenon occurs when duringa soft tensile test at a certain moment during the phase transformation the stressis held constant.

We determine the relaxation or creep values for the strain, temperature andthe volume fraction, that is their asymptotic values when the time goes to infinity.

The main result of this section shows that the asymptotic value of the straindepends piecewise linearly on the value of the stress at which the loading or un-loading was stopped (see (3.34) and (3.50)).

Let us recall the relations governing the thermomechanical behavior of a cylin-drical specimen during the direct or inverse phase transformation in a soft test.

For simplicity, we shall omit the superscript S, used in Part I. According to theequations (5.1) - (5.5) from Part I we have:

σ(t) = E(ε(t) − gβ(t)), (3.1)

σ(t) = p(T0 + θ(t)− Ta + β(t)4T ), (3.2)

θ(t) +1τθ(t) =

L

Cβ(t). (3.3)

From Eqs. (3.1) and (3.2) we obtain the equations expressing the strain andthe martensitic volume fraction as functions of the stress and of the temperature:

ε(t) =1Eσ(t) +

g

p4T [σ(t) − p (T0 + θ(t) − Ta)] , (3.4)


andβ(t) =

1p4T [σ(t) − p (T0 + θ(t)− Ta)] . (3.5)

3.1. The case of the direct transformation

In the first part of this section we consider the case when the loading ceases duringthe direct martensitic transformation.

Let σMS , tMS and, σMF , tMF , be the values of the stress and time for which themartensitic transformation starts and ends, respectively. Let us denote with θMF

the temperature at the moment when the martensitic transformation is complete.We remark that tMS , σMF , tMF , and θMF actually depends on the value σ(t) = bof the stress rate.

We denote by t the moment at which the loading is stopped. From this instantthe stress is held constant until the end of the experiment. We have

tMS < t < tMF , (3.6)

Let us denote by ε, σ, β and θ the values of ε, σ, β and θ at the time t. Sincefor t > t the axial stress is fixed, we have

σ(t) = σ, (3.7)

that isσ(t) = 0, (3.8)

while the volume fraction of martensite at t satisfies the phase transformationcondition

0 < β < 1. (3.9)

As β varies continuously, there is a value t∗ > t such as (3.9) is fulfilled forevery intermediate value of the time variable t. Accordingly, the time evolution ofthe temperature is described by the equation

θ +1τSθ =

L

p(L+ C4T )σ, (3.10)

where the relaxation time τS is given by the relation (5.24) from Part I, i.e.

τS =C4T + L

C4T τ > τ. (3.11)

By using (3.8), the differential equation (3.10) becomes for t < t < t∗

θ +1τSθ = 0. (3.12)


As in the first part of the Section 5 of Part I, the initial value fulfills

θ(t) = θ > 0. (3.13)

The evolution of the temperature θ(t) in the interval (t, t∗) is then given by therelation

θ(t) = θe−t−tτS . (3.14)

Replacing the values (3.7) and (3.14) of the imposed stress and temperature in(3.4) and (3.5) we get

ε(t) =σ

E+

g

p4T

[σ − p

(T0 + θe−

t−tτS − Ta

)], (3/15)

and

β(t) =1

p4T

[σ − p

(T0 + θe−

t−tτS − Ta

)], (3.16)

for every t < t < t∗.These relations remain valid as long as the phase transformation condition (3.9)

is fulfilled.As θ > 0, relation (3.15) implies that the axial strain ε(t) is an increasing

function of the time. Therefore, the thermomechanical model studied in this paperpredicts a strain creep, fact in good accord with existing experimental data (seeVacher [3]).

The relation (3.16) implies that the volume fraction of martensite β(t) is anincreasing function of time. Thus our model indicates that the direct martensitictransformation continues even after the traction was stopped.

Since the martensite volume fraction β(t) is an increasing function of time,there are two possibilities, either

β(t) < 1, for every t > t, (3.17)

or there is a time moment tM such as

β(t) < 1 for t < t < tM , and β(tM ) = 1. (3.18)

Let us consider these two cases one by one.We first study the case when relation (3.17) is fulfilled. Since β(t) is increasing,

we have0 < β∞(σ) ≤ 1. (3.19)

We can now compute the asymptotic values for the stress and for the volumefraction, since relations (3.15) and (3.16) are valid for every t > t. We get

ε∞(σ) =σ

E+

g

p4T [σ − p (T0 − Ta)] , (3.20)


andβ∞(σ) =

1p4T [σ − p (T0 − Ta)] . (3.21)

From (3.19) and (3.21) we have

σ < σ = p(T0 − Tm). (3.22)

That is the necessary and sufficient condition for (3.17) to take place.Consequently, if (3.22) is satisfied, the direct martensitic transformation takes

place for every t > t, and both the austenitic and the martensitic phases aresimultaneously present in the relaxed (final) state of the specimen. Moreover,the temperature θ(t) is strictly decreasing, and its asymptotic value is obviouslyvanishing, so

θ∞(σ) = 0. (3.23)

In the second case, that is when (3.22) is not fulfilled, the martensitic transfor-mations ends at the time tM . According to the constitutive equation (2.1) and tothe evolution law (3.12) of Part I, for t > tM , the time behavior of the axial strainσ(t) and of the volume fraction β(t) is described by the equations

ε(t) =1Eσ + g and β(t) = 1, (3.24)

while the temperature satisfies for t > tM the differential equation

θ +1τθ = 0, (3.25)

with the initial conditionθ(tM ) = θe−

tM−tτS . (3.26)

Consequently, for t > tM , the temperature is given by the following expression:

θ(t) = θe−tM−tτS e−

t−tMτ . (3.27)

Hence, the asymptotic values for ε(t), β(t) and θ(t) in the second case are

ε∞(σ) =1Eσ + g, β∞(σ) = 1 and θ∞(σ) = 0. (3.28)

In order to summarize the obtained results, let us recall that

σMS = p(T0 − Ta) and σMF = p(T0 + θMF − Ta), (3.29)

and thatθMF > 0. (3.30)


By using (3.22), (3.29) and (3.30), we get

σMS < σ < σMF . (3.31)

We can summarize (3.20), (3.21), (3.23) and (3.28) in

θ∞(σ) = 0, for σMS < σ < σMF , (3.32)

β∞(σ) ={ 1

p4T [σ − p (T0 − Ta)] for σMS < σ < σ,

1 for σ < σ < σMF,

(3.33)

and

ε∞(σ) =

{σE + g


σE + g for σ < σ < σMF.

(3.34)

The equation (3.34) implies that the asymptotic value of the strain dependspiecewise linearly on the value of the stress at which the loading have been stopped.

In the same time, from (3.7) and (3.34) it follows

ε ≡ ε∞(σ) =p

E(T0 − Tm) + g. (3.35)

3.2. The case of the inverse transformation

The second part of this section is dedicated to the study of the case when thecompression is stopped during the inverse austenitic transformation.

We denote by tAS , σAS and θAS the time, stress and temperature at which theinverse transformation starts, and by tAF , σAF , and θAF , the similar quantitiescorresponding to the end of the inverse transformation. We remark that all thesevariables actually depends on the stress rate σ(t) = b of the experiment.

Let t be the moment at which the unloading is stopped during the inverseaustenitic transformation. After this moment the stress is held constant. Obviouslywe have

tAS < t < tAF . (3.36)

Let us denote by ε, σ, β and θ the values of ε, σ, β and θ at the time t. Sincefor t > tM the stress is held constant, we have

σ(t) = σ. (3.37)

As we already know, θAS is positive, i.e.

θAS > 0, (3.38)


and the evolution of the temperature between tAS and t is described by the relation(see Eq. (5.73) of Part I)

θ(t) = θAS e− t−tAS

τS − aτSΓS(

1− e−t−tASτS

). (3.39)

Hence we get

θ = θ(t) = θAS e− t−tAS

τS − bτSΓS(

1− e−t−tASτS

). (3.40)

Consequently, if t is ”near” tAS , then θ is positive, but if the difference betweent and tAS becomes important, θ might become negative. The same conclusionfollows from the inequalities (5.128) of Part I.

When θ > 0, the analysis made in the previous subsection is valid withoutmodifications. Thus the asymptotic values of the temperature, volume fractionand strain are those given by (3.32)–(3.34).

Let us consider the case θ < 0. Since β < 1 , there is a value t∗ > t such as(3.9) is satisfied for every intermediate value of the time variable t, which impliesthat the relations (3.14), (3.15) and (3.16) are true.

Since θ < 0, relation (3.15) implies that the axial strain is a decreasing functionof time. It follows that the thermomechanical model predicts a relaxation of strain.

Similarly, relation (3.16) shows that the volume fraction is also a decreasingfunction of time. Hence the model implies a continuation of the inverse austenitictransformation even after the unloading (compression) was stopped.

If the conditionσMS = p(T0 − Ta) > σ, (3.41)

is satisfied, then by using (3.21) we obtain

β(t) > β∞(σ) > 0. (3.42)

for every t > t.Consequently, condition (3.41) is satisfied, then the asymptotic values of θ(t),

σ(t) and β(t) are those given by the formulas (3.23), (3.20) and (3.21). Therefore,in the final state of the specimen both phases are simultaneously presented.

If condition (3.41) is not fulfilled, then (3.16) implies that there is tM > t, suchas

β(t) > 0, for t < t < tM , and β(tM ) = 0. (3.43)

Hence the austenitic transformation ends at the time tM .According to the general constitutive equation (2.1) and the evolution law

(3.10) of Part I, the time behavior of the stress and of the volume fraction isdescribed for t > tM by the formula

ε(t) =1Eσ, β(t) = 0. (3.44)


Therefore, the asymptotic values for the stress and for the volume fraction are

ε∞(σ) =1Eσ, β∞(σ) = 0. (3.45)

Consequently, if the restriction (3.41) is not satisfied, the final state of the spe-cimen is a pure austenitic one, and the asymptotic value of the stress lies on the“first elastic line”.

In order to summarize the results, let us remark that

σAS = p(T0 + θAS − Tm), σAF = p(T0 + θAF − Ta), (3.46)

and that using the relations (5.20) of Part I, (3.22), (3.29) and (3.46) we mayconclude that

0 < σAF < σMS < σ < σAS < σMF . (3.47)

The predictions of the model for this case might be expressed by the relations

θ∞(σ) = 0, for σAF < σ < σAS , (3.48)

β∞(σ) =

1 for σ < σ < σAS ,

1p4T [σ − p (T0 − Ta)] for σMS < σ < σ,

0 for σAF < σ < σMS ,

(3.49)

and

ε∞(σ) =

1Eσ + g for σ < σ < σAS ,σE + g


1Eσ for σAF < σ < σMS ,

(3.50)

Fig. 3.1 represents the asymptotic value of the volume fraction with respect tothe value of the stress at which the loading, respectively the unloading, process isstopped.

σ σ

(σ)β (σ)β

σσAF σMS σAS σMFσ ^MS σMF

0

1

0

1

σ

Figure 3.1. The direct and the inverse transformation: volume fraction versus stress


MF

AFσ

σMS

σAF

AFε εMS εAS εMF

σ

σ

εε

σ

ε

σ

0( )T

( )T

Figure 3.2. Relaxation and creep of the axial strain

The dependence of the asymptotic value of the strain on the value of the con-stant stress, both in direct and in inverse transformation (relations (3.34) and(3.50)) is represented in Fig 3.2.

Using the relations (2.22) and (3.35) we get

ε = ε∞(σ) = ε; (3.51)

analogously, from (2.35) and (3.22) it follows that

σ = σ∞(ε) = σ. (3.52)

The above relations do not imply that the curve of Fig. 2.2, which characterizesthe relaxation and the creep of the axial stress in a hard test coincides with thecurve from the Fig. 3.2, where are recollected the results of the relaxation andthe creep of the axial strain in a soft test. Indeed, even if ε = ε and if σ = σ, thecritical temperatures θMS , θMF , θAS , θAF , and hence the critical stresses σMS ,σMF , σAS , σAF , and the corresponding critical strains εMS , εMF , εAS , εAF , aredifferent, since their values depend on the strain or on the stress rate.

4. Experimental results

In this section we present our experimental results obtained in the performedtensile tests. The used thermomechanical device and SMA sample are presentedin Section 4 of Part I. Our relaxation, creep and ”isothermal” tests are analyzed


taking into account the observed time evolution of the axial strain, axial stressand surface temperature. We particularly focus our attention on the evolution ofthe temperature during the performed tests. One of our aims was to study theconnection between the variation of the temperature and the stress relaxation andstrain creep during the phase transformation.

4.1. Stress relaxation during martensitic transformation

All the tests were performed at a room temperature T0 = 313 K. The chronologyof every test is as follows (see Fig. 4.1a and 4.1b):1. The first part of the test is strain controlled, with a strain rate ε = 10−3 s−1

(a b).2. The strain ε, reached at the end of the first stage, is held constant during 120

seconds (b c). Tests differ from each other in the strain level ε reached at theend of the first period (b).

3. A return to zero stress is then performed with the stress rate σ = −10 MPa s−1

(c d). This zero stress is then held during 30 seconds (d e).To correctly see the stress relaxation amplitude (see Fig. 4.2b), an intermediatestage is added between the stages 2 and 3.

2’. A slight increase in stress is performed (c c’).

Let us remark that the condition (5.14) from Part I is fulfilled in all the followingtests.

Figures 4.1 show the evolution in time of parameters measured by sensors fora stress relaxation at ε = 1.65 %.

Figure 4.1. Stress relaxation test at ε = 1.65 % and T0 = 313 K. a) stress versus time b) strainversus time c) temperature versus time

During the complete test, performed at constant room temperature T0, the am-plitude of the temperature, measured by the thermocouple on the sample, reachesup to 3 K. At the end of the first period (b), the temperature variation is 1.8 K


over the initial temperature T0. When the strain is then held constant (b c),the sample returns at the room temperature T0, while the stress decreases until itreaches an asymptotic value : we measure a stress relaxation amplitude of -9 MPa.

The temperature increase, during the intermediate stage (c c’), is associatedwith the slight increase in stress. During the last part of the test (c’ e), we noticea temperature drop, and finally a return to the initial temperature T0; this returnto T0 begins before the stress reaches the zero value.

Figure 4.2a shows the connection between temperature and stress during thestress relaxation : we are able to know if one of these two variables reaches itsasymptotic value before the other one. Observing the slope near the point c, wecan conclude that the stress relaxation is directly linked to the return to the roomtemperature T0.

On the figure 4.2b, the point (σ∞(ε),ε) is marked with a star corresponding tothe asymptotic value σ∞(ε) of the stress at the fixed strain ε. The intermediatestage 2’ allows to correctly detect the position of this point.

Figure 4.2. Stress relaxation test at ε = 1.65 % and T0 = 313 K. a) temperature versus stressb) stress versus strain

Figure 4.3 presents the results of two stress relaxation tests, for ε = 0.89 %and ε = 1.65 %. Let us remark the superimposition of responses during the firstperiod (a b). We see that the stress relaxation is connected with the temperaturevariations near the initial temperature.

For strains below 0.3 %, the stress relaxation phenomenon was not detectedwith our load cell. Figure 4.4 exhibits that, for strain levels superior to 0.3 %, thepoints (σ∞(ε), ε) are located on a straight line.

4.2. Strain creep during martensitic transformation

We also performed several tests on the previous sample, at the same room tem-perature T0 = 313 K, for the following loading chronology (see Fig. 4.5a) :


Figure 4.3. Stress relaxation test at ε = 0.89 % and T0 = 313 K (- - -). Stress relaxation test atε = 1.65 % and T0 = 313 K (—).

Figure 4.4. Stress asymptotic values at fixed strains at T0 = 313 K.

1. During the loading period (a b), we have a stress rate σ = 10 MPa s−1.2. The stress σ, reached at the end of the first stage, is held constant during 120

seconds (b c). Tests differ from each other in the stress level σ reached at theend of the loading period (b).

3. A return to zero stress is then performed with the stress rate σ = −10 MPa s−1

(c d). This zero stress is then held during 30 seconds (d e).Let us notice that the condition (5.15) from Part I is fulfilled for all the tests thatfollows.

Figure 4.5 shows the time evolution of parameters measured by sensors for astrain creep test at σ = 111 MPa.

The temperature evolution is qualitatively similar to the one measured during astress relaxation test (see Fig. 4.5c). At the end of the loading period (b), we have a1.55 K temperature variation over the initial temperature. When the stress is heldconstant (b c), the sample returns at the room temperature T0, while the strain


Figure 4.5. Strain creep test at σ = 111 MPa and T0 = 313 K. a) stress versus time b) strainversus time c) temperature versus time

is still increasing to an asymptotic value. We measure a strain creep amplitude of+0.15 %. During the last part of the test (c e), we notice a temperature drop, andfinally a return to the initial temperature.

Observing the slope near the point c from Fig. 4.6a, we see the connectionbetween temperature and strain during the strain creep: the two variables reachtheir asymptotic values at the same time.

Figure 4.6. Strain creep test at σ = 111 MPa and T0 = 313 K. a) temperature versus strainb) stress versus strain

Figure 4.7 presents the results of two strain creep tests, for σ = 111 MPaand σ = 152 MPa. For all strain creep tests, we verified the superimposition ofresponses during the loading period (a b).

For stresses below 50 MPa, the strain creep phenomenon was not detected withour extension sensor. Figure 4.8 exhibits that, for stress levels superior to 50 MPa,the points (σ, ε∞(σ)) are located on a straight line.


Figure 4.7. Strain creep test at σ = 111 MPa and T0 = 313 K (—). Strain creep test atσ = 152 MPa and T0 = 313 K (- - -)

Figure 4.8. Strain asymptotic values at fixed stresses at T0 = 313 K.

4.3. The “isothermal” test

The test was performed with the same sample as before, with a room temperatureT0 = 313 K. It is strain controlled (see Fig. 4.9b) with the lowest speed available(ε = ± 6.6 × 10−6 s−1). In this case the sample is continuously as close aspossible to the thermal equilibrium with the exterior. Figure 4.9 shows the timedependent evolution of the stress, strain and temperature read by sensors. Duringthe ”isothermal” test, the temperature variation has not exceeded 0.2 K (Fig. 4.9c).

Figure 4.10 allows to compare the “isothermal” strain-stress curve with pre-vious results concerning stress relaxation and strain creep. We notice that theasymptotic values of strain and stress are on the loading part of the “isothermal”stress - strain curve. In this way we can again infer that the stress relaxation andthe stress creep are indeed related to the time evolution of the same parameter:the temperature of the sample.


Figure 4.9. “Isothermal” test at T0 = 313 K. a) stress versus time b) strain versus time c) tem-perature versus time.

Figure 4.10. “Isothermal” test at T0 = 313 K (—). ◦ : strain asymptotic values at fixed stresses(strain creep tests); * : stress asymptotic values at fixed strains (stress relaxation tests)

At the same time we can see that the asymptotic values of the stress and of thestrain are situated on the same straight line. Thus we can observe the existenceof a satisfactory agreement between the theoretical predictions of the thermome-chanical model (see Eqs. (2.49) and (3.50)), and the recorded experimental results(see Fig. 4.10).

In our opinion the existence of the hysteresis loop in an “isothermal” test isdue to the intrinsic dissipation which becomes important in situations in whichvery low stress or strain rates are involved, and the temperature variation be-comes negligible. Taking into account this fact, the model can be improved, re-placing the thermoelastic model by a thermoviscoelastic one, as this is done, forinstance, by Lobel [5] and by Chrysochoos [6]. In the future, we shall investigatethe consequences of such a replacement in problems concerning relaxation andcreep phenomena in SMAs.


5. Theory versus experiment

In this section we present qualitative, as well as quantitative comparisons betweenour experimental and theoretical results. The theoretical results taking into thevalues of the material parameters given in Table 1 of Section 4 from Part I wereobtained using the simplified 1D model presented in section 3 of Part I. In the lastpart of this section we also compare the theoretical and experimental results corre-sponding to the room temperature T0 = 322.5 K, using the values of the materialparameters determined from tests performed at room temperature T0 = 313 K.

5.1. Stress relaxation during martensitic transformation

The calculation was performed with material parameters presented in Table 1, andfor loading conditions corresponding to the Subsection 4.1. Figure 5.1 allows us tocompare the results of the model to those of the experiment for a stress relaxationtest for the fixed strain ε = 1.15 % at T0 = 313 K.

Figure 5.1. Experimental and theoretical results for a stress relaxation test for ε = 1.65 % atT0 = 313 K.


Figure 5.1b shows that the experimentally determined and the theoreticallycalculated stress relaxation curves are in a very good agreement, and that themodel can describe qualitatively, as well as quantitatively the stress relaxationphenomenon. Also from the Figure 5.1c, we can see that the prediction of themodel and the recorded experimental data concerning the variation of the tem-perature are in a satisfactory agreement. Figure 5.1d presents the time evolutionof the martensite volume fraction given by the model. The theoretical and experi-mental result presented in Figure 5.1b, c, d reveals the strong connection existingbetween the time evolution of the temperature and the time evolution of the stressand volume fraction during a stress relaxation test.

Figure 5.2 shows the results obtained in a stress relaxation test for the fixedstrain level ε = 1.15 % at T0 = 313 K . Figure 5.2b reveals again a satisfactoryagreement between the experimentally obtained and theoretically calculated stressrelaxation curves.

Figure 5.2. Experimental results for a stress relaxation test for ε = 1.15 % at T0 = 313 K

In Figure 5.3 we have compared the theoretical “isothermal” stress - strainrelation (see Eqs. (2.4) with θ = 0) corresponding to T0 = 313 K and the ex-perimentally obtained asymptotic values of the stress corresponding to variousfixed strain levels. We can conclude that the theoretical results agree with theexperimental data.

5.2. Strain creep during martensitic transformation

The calculation was performed with material parameters presented in Table 1 andfor loading conditions corresponding to the Subsection 4.2. Figure 5.4 allows usto compare the results of the model to those of the experiment for a strain creeptest for the fixed stress σ = 111 MPa at T0 = 313 K. Figure 5.4b shows that thetheoretically calculated and the experimentally determined strain creep curves arein a satisfactory agreement, and that the model is able to describe qualitatively, as


Figure 5.3. Theoretical “isothermal” stress - strain curve at T0 = 313 K (the continuous line); ?: experimental asymptotic values of the stress for fixed strains (stress relaxation tests).

Figure 5.4. Experimental and theoretical results for a strain creep test for σ = 111 MPa atT0 = 313 K

well as quantitatively, the strain creep phenomenon. In the same time, the Figure5.4c shows that the prediction of the model and the recorded experimental data


concerning the variation of the temperature are in a satisfactory agreement. InFigure 5.4d we present the time evolution of the martensite volume fraction givenby the model. The experimental and theoretical results presented in Figure 5.4b,c, d prove the existence of a strong connection between the time evolution of thetemperature and the time evolution of the strain and volume fraction during astrain creep test.

Figure 5.5 shows the results obtained in a strain creep test for the fixed stresslevel σ = 152 MPa at T0 = 313 K .

Figure 5.5. Experimental and theoretical results for a strain creep test for σ = 152 MPa atT0 = 313 K

Figure 5.5 b reveals a satisfactory agreement between the experimentally ob-tained and theoretically calculated strain creep curve.

In Figure 5.6 we have compared the theoretical ”isothermal” stress - straincurve (see Eqs. (3.4) with θ = 0) corresponding to T0 = 313 K, and the experi-mentally recorded asymptotic values of the strain corresponding to various fixedstress levels. We can observe a good agreement between the experimental and thetheoretical results.

5.3. Theory versus experiment at room temperature T0 = 322.5 K

In Fig. 5.7 we give the obtained theoretical and experimental results correspondingto a strain creep test at the stress level σ = 152 MPa.

Figure 5.8 presents the theoretical “isothermal” stress-strain curve correspond-ing to T0 = 322.5 K and on the same figure are given the asymptotic values ofthe stress (star) and of the strain (circle) recorded in stress relaxation and straincreep tests for T0 = 322.5 K.

In our opinion there exists a good agreement between the theoretical predictionsof the model and the recorded experimental data.


Figure 5.6. Theoretical “isothermal” stress - strain curve at T0 = 313 K (the continuous line);◦: experimental asymptotic values of the strain for various fixed stresses (strain creep tests)

6. Final remarks

In order to describe the behavior of SMAs we have used a thermomechanicalmodel. The free energy involved in the model is a convex function with respect tothe strain and to the martensitic volume fraction, and a concave one with respectto the temperature. The model takes into account the non isothermal character ofthe phase transformation and assumes zero intrinsic dissipation. To describe thebehavior of SMA bars in traction - compression tests we have used a simplified,homogeneous 1D version of the model, taking into account the heat loss on thelateral surface of the involved bars.

We have shown that the 1D model can describe the existence of the hysteresisloop and the pseudoelastic behavior of SMAs, if the parameters characterizingthe material and the traction - compression tests satisfy two well - determinedconditions (see Eqs. (5.14) and (5.15) of Part I). These restrictions are satisfiedby the material constants determined by us and by the parameters involved in theexperiments.

We have shown that the 1D model predicts the stress relaxation and the straincreep during austenitic - martensitic phase transformation, as well as the stresscreep and the strain relaxation during martensitic - austenitic phase transforma-tion. According to the model, the asymptotic values of the stress depend piecewiselinearly on the value of the strain at which the traction or the compression wasstopped (see Eq. (2.34) and (2.49)). The involved asymptotic values are situa-ted on the “isothermal” stress - strain curve. Similarly, according to the model,the asymptotic value of the strain depends piecewise linearly on the value of thestress at which the loading or unloading was stopped (see (3.34) and (3.50)). Theinvolved asymptotic values are situated on the same “isothermal” stress - straincurve. The above predictions of the model are in very good agreement with theobtained experimental data, recorded in stress relaxation and strain creep experi-ments realized at two different room temperatures (see Figures 4.4, 4.8, 5.3, 5.6


Figure 5.7. Experimental and theoretical results of a strain creep test at σ = 152 MPa andT0 = 322.5 K.

Figure 5.8. Theoretical “isothermal” test at T0 = 322.5 K (—); ◦ : strain asymptotic valuesat fixed stresses (experimental strain creep tests); *: stress asymptotic values at fixed strains(experimental stress relaxation tests)


and 5.8).Our experimental results reveal the strong connection existing between the

variations of the temperature and of the stress during stress relaxation (see Figures4.2a and 4.3b), as well as between the variation of the temperature and of the strainduring strain creep (see Figures 4.6a and 4.7b).

There exists a satisfactory agreement between the experimentally obtained andtheoretically calculated stress relaxation and strain creep curves, as this can be seeninspecting Figures 5.1b, 5.4b and 5.7b. Also the time evolution of the temperatureis satisfactory predicted by the model, according to the results given in Figures5.1c, 5.4c and 5.7c.

In this way we can conclude that our non isothermal and intrinsic dissipationlessmodel can describe in a satisfactory way the thermomechanical behavior of SMAsin the analysed circumstances.

However, Figures 4.2 and 4.3 from Part I show that the theoretical mechanicalhysteresis loop is smaller than that given by the experiment, while Fig. 4.10 revealsthe existence of the hysteresis loop in an isothermal” test, when the variations ofthe temperature are negligible.

This fact shows that, in order to describe the size of the hysteresis loop, ourthermoelastic model must be improved. According to the results of the Section4 of Part I, we would approach this objective by replacing our model with athermoviscoelastic one having a non - vanishing intrinsic dissipation.

Consequently, the investigation of the relaxation and the creep phenomena forSMAs in a thermoviscoelastic frame-work remains an important research theme.

Acknowledgments

We wish to thank Professor Andre Chrysochoos, Dr. Christian Licht, Dr. MireilleLobel, Professor Olivier Maisonneuve and Dr. Robert Peyroux for many helpfuland interesting discussions which stimulated us to undertake this work.

References

[1] Balandraud, X., Ernst, E., Soos. E., Relaxation and creep phenomena in shape memoryalloys. Part I: Hysteresis loop and pseudoelastical behavior, ZAMP 51 (2000), 171–203.

[2] Pham, H., Analyse thermomecanique du comportement d’un alliage a memoire de formede type CuZnAl, THESE, Univ. Montpellier II, 1994.

[3] Vacher, P., Etude du comportement pseudoelastique d’alliages a memoire de formeCuZnAl polycristallins, THESE, U.F.R. des Sciences et Techniques de l’Universite deFranche-Comte, 1991.

[4] Muller, I., Six lectures on shape memory, 1996 (private communication).[5] Lobel, M., Caracterisation thermomecanique d’alliages a memoire de forme de type NiTi

et CuZnAl. Domaine de transition et cinetique de changement de phase, THESE, Univ.Montpellier II, 1995.


[6] Chrysochoos, A., Pseudo-elastic and pseudo-viscoelastic models for shape memory alloys,Rev. Roum. Sci. Techn. - Mec. Appl., 43(6) (1999), in press.

X. BalandraudLaboratoire de Mecanique et Genie CivilU.M.R. 5508, Universite Montpellier IIC.C. 081, Place E. Bataillon34095 Montpellier Cedex 5, France(e-mail: [email protected])

E. Ernst and E. SoosInstitute of Mathematicsof the Romanian AcademyP.o. Box 1-764Ro 70700 Bucharest, Romania(e-mail: [email protected])

(Received: November 3, 1998)

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