On the physically motivated choice of internal variables describingmartensitic phase transformations

Claus Oberste-Brandenburg∗

Institute of Mechanics, Ruhr-University Bochum, 44780 Bochum, Germany

Phenomenological models which describe phase transformation in solids often employ a single scalar variable, the mass (orvolume) fraction of one of the phases involved. However, the orientation of the applied stress has a major influence on themicrostructure developing during the phase transition and thus the overall mechanical behavior. Therefore, additional internalvariables of higher order have to be introduced in order to capture this aspect.

The choice of these internal variables is often based on purely phenomenological considerations. In contrast to this, basedon local observations at the phase boundary, a thermodynamically dual pair of second order tensors is introduced here.

Constitutive relations for shape menory alloys in the pseudoelastic and pseudoplastic range are proposed.

1 Introduction

During the pseudoelastic transformation in shape memory alloys (SMA), not only the absolute value of the applied stress, butalso its direction has a significant influence on the onset and progress of the transformation. The aim of the contribution is toprovide a simple macroscopic approach to cover this aspect.

As the driving force for the movement of the phase transition front, the projection of the Eshelby Tensor, or asymmetricchemical potential tensor, in direction normal to the front can be identified (cf. e.g. [1]). Following the reasoning by Grinfeld[2], the whole tensor µ, the difference of asymmetric chemical potential tensor of the phases, is used to “characterize thestate of the substance in the entire vicinity of a material particle, independently of the choice of any elementary area”.Thus, introducing the tensorial measure µ as the thermodynamic force raises the question of the choice of the associatedthermodynamic flux. The thermodynamic flux used within this context is formally defined by the integration over the interfaceΛ0 as

ξ =1

Mtot

∫Λ0

ρ UN N ⊗ N dA0 (1)

where UN denotes the velocity of the phase interface, ρ the mass density, both measures in the referential configuration, andMtot the total mass of the body under consideration. Considering homogeneous phases, the total dissipation due to the phasetransition can be described as

T sΛ = µ : ξ ≥ 0. (2)

where µ denotes now the difference of the phases considered (cf. [3], [4], and [5]). The aim of this contribution is to developa set of constitutive equations which describe the relation between these measures. As a first approach, a function ξ = ξ(µ, ξ)is proposed. The use of the tensor ξ as an argument for ξ is important as only situations where the probability for eachorientation is the same, i.e. only spheroidal inclusions exist, could be considered if omitted.

2 Kinetic relation

A threshold function F ξ = F ξ(µ, ξ) is introduced for which F ξ = 0 is required for an ongoing phase transformation.Furthermore,

ξ =⟨λξ

⟩ξ

∂P ξ

∂µ(3)

where P ξ is a scalar potential and λξ a scalar variable, both to be specified later.⟨λξ

⟩ξ

= λξ if F ξ = 0 and zero otherwise.

Use of the consistency condition F ξ = 0 leads to

λξ = −(

∂F ξ

∂µ: µ

) (∂F ξ

∂ξ:

∂P ξ

∂µ

)−1

. (4)

∗ Corresponding author: e-mail: cob@tm.bi.ruhr-uni-bochum.de, Phone: +49 234 32 22114, Fax: +49 234 32 02114

PAMM · Proc. Appl. Math. Mech. 5, 313–314 (2005) / DOI 10.1002/pamm.200510133

© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

The use of the decomposition ξ = 13 1 ξ + ξ′ with ξ = tr

(ξ)

might prove useful in the following. It can be shown, that ξ

describes the change of the mass fraction of the phase under consideration. Thus, the deviatoric part bears information aboutthe current orientation of the transformation front. The transition to the application for shape memory alloys is connected tothe fact that the observed microstructural orientation is strongly related to the orientation of the interface. This leads to theidea that one may associate growth with only hydrostatic contribution to ξ as growth of twinned martensite whereas a changeof ξ where the deviatoric part also changes is associated with a growth where oriented martensite forms.

In the following, the approach

F ξ = F ξµ−g(ξ) with F ξ

µ = c1 Iµ1 +c2

√−Jµ

2 +c33

√Jµ

3 and g(ξ) = 1+g1 Iξ1 +g2

√−Jξ

2 +g33√

Jξ3 (5)

where IA1 = tr (A), JA

2 = −tr(A′ · A′) /2, JA

3 = tr(A′ · A′ · A′) /3, and A′ denotes the deviatoric part of A is considered.

c1, c2, c3, g1, g2, and g3 are material parameters. In order to simplify the equations, c3 = 0 is set which does not alter thegeneral points derived in the what follows. Different approaches for P ξ are considered in the following.

2.1 Case P ξ = F ξµ

Note that for this choice the process maximizes dissipation. It yields

ξ = 3 c1 λξ and ξ′ = c2µ′

2√−Jµ

2

λξ with λξ = c1 Iµ1 − c2

Jµ2

2√−Jµ

2

. (6)

When using (6a) in (6b) to substitute λξ, a situation occurs where a change of the deviatoric part is only possible if thespherical part changes as well. Bearing in mind that the deviatoric part is associated with the orientation of the martensiteand the spherical part with its mass fraction, processes as e.g. detwinning of thermally induced martensite with zero austenitefraction, a process responsible for pseudoplasticity in certain temperature ranges and alloys, or reorientation after completetransformation for nonproportional loading path, are not possible. This contradiction leads to the extension described in thefollowing case.

2.2 Case P ξ = F ξµ − c1 Iµ

1 Iξ1

P ξ is now chosen such that for ξ = 1, a change of the deviatoric part of ξ is possible. This approach yields

ξ = 3 c1 (1−ξ) λξ, ξ′ = c2µ′

2√−Jµ

2

λξ with λξ =

[c1 Iµ

1 − c2Jµ

2

2√−Jµ

2

] ⎡⎣3 c1 g1 (1 − ξ) + c2 g2

ξ′ : µ′

4√

Jξ2Jµ

2

⎤⎦−1

.

Two positive aspects can be noticed: The differential equation for ξ automatically satisfies the condition 0 ≤ ξ ≤ 1 andeven for ξ = 1, a reorientation of the martensite is possible. The second remark holds also true for ξ = 0, which is rathernon-physical. Thus, a possible extension to circumvent this contradiction is discussed in the following.

2.3 Case P ξ = F ξµ − c1 Iµ

1 Iξ1 − c2

√−Jµ2 (1 − Iξ

1 )

Now,

ξ = 3 c1 (1−ξ) λξ, ξ′ = c2µ′ ξ

2√−Jµ

2

λξ with λξ =

[c1 Iµ

1 − c2Jµ

2

2√−Jµ

2

] ⎡⎣3 c1 g1 (1 − ξ) + c2 g2

ξ′ : µ′

4√

Jξ2Jµ

2

⎤⎦−1

.

The contradictions mentioned above are resolved by this approach. However, further modifications might be necessary whenforthcoming multiaxial experimental investigations indicate that the current approach does not describe the situation correctly.

References

[1] Heidug, W., Lehner, F., Thermodynamics of coherent phase transformations in nonhydrostatically stressed solids. Pure Appl. Geophys.123, 91–98 (1985).

[2] M. Grinfeld, Thermodynamic methods in the theory of heterogenous systems. Longman Scientific & Technical, 1991.[3] Oberste-Brandenburg, C., A unified tensorial driving force for phase transitions – calculation of the onset of the transformation. PAMM

2(1), pp. 200–201, 2003.[4] Oberste-Brandenburg, C., Calculation of the onset and progress of the martensitic transformation using tensorial measures for the

transformation kinetics. In D. C. Lagoudas (Ed.), Proc. of SPIE Vol. 5053 Smart Structures and Materials 2003: Active Materials:Behavior and Mechanics, pp. 317–326, SPIE publications, 2003.

[5] Oberste-Brandenburg, C., Bruhns, O.T., A tensorial description of the transformation kinetics of the martensitic transformation. Int. J.Plasticity, i, (20), pp. 2083-2109, 2004.

© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

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