An analysis of nonclassical austenitemartensite interfaces in CuAlNi
J. M. Ball, K. Koumatos, H. Seiner
Mathematical Institute, University of Oxford, 2429 St. Giles’, Oxford OX1 3LB
Institute of Thermomechanics ASCR, Dolejškova 5, 182 00 Prague 8, Czech Republic
Faculty of Nuclear Sciences and Physical Engineering, CTU in Prague, Trojanova 13, 120 00 Prague 2, Czech Republic
Keywords: martensitic transformations, nonclassical austenitemartensite interfaces, cubictoorthorhombic
Abstract
Ball & Carstensen [2, 3], theoretically investigated the possibility of the occurrence of nonclassical austenitemartensite interfaces and studied the cubictotetragonal case extensively. Here, we aim to present an analysis of such interfaces recently observed by Seiner et al. [12] in CuAlNi single crystals, undergoing a cubictoorthorhombic transition. We show that they can be described by the nonlinear elasticity model for martensitic transformations and we make some predictions regarding the volume fractions of the martensitic variants involved, as well as the habit plane normals.
Introduction
A classical austenitemartensite interface is one in which the undistorted austenite meets a simple laminate of martensite and these have been broadly studied. In recent years, a theory of martensitic transformations has been developed which allows austenitemartensite interfaces to occur in which the microstructure of the martensite is more complicated; these are referred to as nonclassical interfaces and have hitherto not been systematically observed. In this nonlinear elasticity model, in which interfacial energy is neglected, microstructures are identified as limits of minimizing sequences for the total free energy
(1) 
Here, represents the reference configuration of undistorted austenite at the transformation temperature and denotes the deformed position of particle . The freeenergy function depends on the deformation gradient and the temperature . By frame indifference, for all , and for all rotations ; that is for all matrices in . By adding an appropriate function of , we may assume that . At , the energy wells of the freeenergy function are given by for the undistorted austenite and , , for the distinct variants of martensite, where each is a positive definite, symmetric matrix. Hence, with equality if and only if . For , the martensite wells, , minimize , whereas for the minimum is given by the austenite well, , where is the thermal expansion coefficient of the austenite and .
A nonclassical planar austenitemartensite interface corresponds to a choice of habitplane normal such that there exists a sequence of deformations for which as . We require that, for , the values of the deformation gradient tend to , i.e. corresponds to the austenite phase; without loss of generality, this is equivalent to except possibly for a set of zero volume. For , we require that, as , tends in a suitable way to the set of martensite energy wells. In fact, we require that the Young measure of is supported in the set (for details see e.g. [1]).
From now on, we shall make the assumption that the martensitic microstructure is homogeneous; that is, for , the macroscopic deformation gradient is independent of . The set of all such matrices is called the quasiconvexification of and is denoted by . It can be shown that is also the set of such that there exists a sequence of deformations with on the boundary of , , and in the above sense.
If we know for a given set of martensite wells, to ensure geometric compatibility, we need to examine whether it is possible to establish a rankone connection between and ; that is, we need to find vectors and such that
(2) 
where, by frame indifference, we have chosen the identity matrix to represent the austenite energy well. Unfortunately, we only have a characterization of for two martensitic wells (), i.e. when . In this case, any can be obtained as the macroscopic deformation gradient of a double laminate (see [5]). Even though is unknown in the case of three tetragonal wells, Ball & Carstensen [3] were able, using the twowell calculation, to characterize exactly the values of the deformation parameters for a cubictotetragonal transformation which permit nonclassical interfaces. They also presented results for a cubictoorthorhombic transformation, though not of the type occurring in CuAlNi.
Given a matrix , the question as to whether we can solve the equation
(3) 
for vectors and is answered by the following lemma in [4, 8].
Lemma 1. Let be a nonsingular matrix that is not a rotation. Then the wells and are rankone connected if and only if the middle eigenvalue of is 1. Then for some if and only if m is a nonvanishing multiple of one of the two vectors, , where are the three eigenvalues of with corresponding eigenvectors , , .
Having outlined a brief description of the model, we proceed to the experimental observations on a CuAlNi single crystal.
Experimental Observations
The first micrographs of interfaces between austenite and crossing twins were obtained by Seiner et al. [13], who documented that such interfaces can form during the shape recovery process of CuAlNi single crystals. However, these interfaces were only weakly nonclassical, i.e. they were classical interfaces weakly disturbed by a negligible volume fraction of compound twins intersecting the firstorder laminate of the TypeII twins. These observations motivated the authors towards more intensive research in this field – the experiment was improved in order to increase the volume fraction of the compound twins in the martensitic microstructure. The resulting experimental procedure is briefly outlined below, and will appear in more details in [12].
The specimen examined in this case was a 3.93.84.2mm rectangular parallelepiped of the austenitic phase of CuAlNi, cut from a single crystal of this alloy such that the normals to the specimen faces had approximately the principal crystallographic directions 100. The original single crystal was grown by a Bridgman method at the Institute of Physics ASCR in Prague. The specimen was subjected to the following sequence of mechanical and thermal loadings (see Fig. 1):

At room temperature, the specimen was transformed into a single variant of 2H martensite by applying uniaxial compression (in a bench vice). Due to the effect called mechanical stabilization of martensite, the specimen did not return to austenite after unloading, but remained as a single variant of martensite.

The specimen was rotated by 90 and uniaxial compression was applied again. In this case, the loading induced the reorientation of martensite into another variant via compound twinning (for an analysis of the relation between the direction of applied compression and the activated twinning systems in CuAlNi, see [9]). The reorientation was not fully completed. Instead, the loading was interrupted at the moment when the specimen contained comparable volume fractions of both variants. By such a procedure, we obtained a finely compound twinned specimen.

The finely compound twinned specimen was heated from one side using a gas lighter, which induced the shape recovery process, i.e. the thermally driven return of the specimen into austenite. As the compound twins cannot form any compatible interface with austenite, the transition was achieved by formation of an interfacial microstructure, which ensured a compatible connection between the mechanically stabilized martensite (the compound twins) and austenite. This interfacial microstructure was formed by TypeII twins crossing the original compound microstructure and getting arranged into a so called Xinterface (for more details of formation of Xinterfaces in CuAlNi see [13], for the theoretical analysis of this microstructure, see [10]). By removing the heating in the middle of the course of the transition, the interfacial microstructure was stopped, and the nonclassical interfaces between austenite and the two mutually crossing systems of twins (compound and TypeII) were observable by optical microscopy. An example is given in Fig. 2.
This procedure was repeated several times, which enabled the capturing of several optical micrographs of the nonclassical interfaces. An interesting observation was that, as shown in Fig. 3, the interface between the austenite and the system of crossing twins was never exactly planar, but rather slightly curved. This results from the fact that the pattern of compound twins induced in the specimen by the uniaxial compression in stage (b) of the experimental procedure is never exactly homogeneous. With varying volume fraction of the compound twins, the admissible orientation of the habit plane varies as well, as will be shown in the theoretical analysis given in the following section. However, a more complete analysis of the curved interface remains to be done.
Analysis of NonClassical Interfaces
In this section, we present an analysis for the above nonclassical austenitemartensite interface via the nonlinear elasticity model for martensitic transformations. The martensitic region on one side of the observed interface consists of twin crossings involving four martensitic variants. Since the quasiconvexification of more than two wells is unknown, we are not able to analyze all the possibilities for nonclassical interfaces in CuAlNi. However, we can do this in the twin crossing case. Firstly, we shall concentrate on the martensite phase and try to construct the microstructure consisting of compound and TypeII twin crossings, as shown in Fig. 4.
Let and be two martensitic variants able to form a TypeII twin and , the respective compound counterparts. Clearly and can also form a TypeII twin and we proceed by writing the compatibility equations for the parallelogram microstructure. These are:
(4) 
(5) 
(6) 
(7) 
where and are, respectively, the shearing vector and the normal to the twinning plane for the system of variants and . Also, denotes the mutual rotation between variants and .
If the above compatibility equations hold, a necessary and sufficient condition for the entire parallelogram microstructure to be compatible is that
(8) 
Equation (8) is necessary, since both sides describe the mutual rotation between and , and sufficiency follows by showing that (4)(7) imply that the normals , , and are coplanar.
Let
(9) 
(10) 
which represent the macroscopic deformation gradients corresponding to the TypeII structures. From [7], there are solutions of the twinning relations (6) and (7) with
(11) 
and we make this choice in accordance with the experimental observations. Then the geometry of the parallelogram structure requires to be the same in (9) and (10).
At this stage we note that . To see this, consider equation (4). We get that
Therefore, by taking determinants, we obtain
Clearly and thus
(12) 
Hence, for , we have
Similarly, and we deduce that . This result has a simple physical interpretation. Since the determinants have the meaning of volume change and the microstructures and are just mixtures of variously rotated single variants, the volume must remain the same.
Now, assume that the volume fraction of variant in , as well as that of in , is . Then, the macroscopic deformation gradient of the entire microstructure is
(13) 
Whether the interface between the parallelogram microstructure and the austenite can be formed is a question of finding and such that the middle eigenvalue of is equal to 1, which is equivalent to finding a rankone connection between the austenite and . In particular, by Lemma 1,
(14) 
Forming the Interface
By manipulating the compatibility equations and making use of equations (8) and (11) we can see that
All normals cannot be parallel to each other and hence, using a result in [10], we deduce that
(15) 
Hence, there exists some such that
(16) 
By identifying the rotation and using the formulae for the twinning solutions in Result 5.2 of [6], as well as the relations between them [7], we obtain that
Our goal is to deduce an expression for which will enable us to find solutions of (14) for . Towards this end, we shall seek a rankone connection between and . Indeed, using (8) and equations (4)(7),
Having that and using (16) we get
(17) 
where
Combining equation (17) with (13) we get that
and using the expression for we obtain
(18) 
Then,
(19) 
We shall first calculate . Hence we have,
However, from (17)
and taking determinants, we see that
Since we get that
(20) 
From (19), we have
and it remains to calculate .
Making use of the expression for and (20)
Combining the last two equations and after some calculations we obtain that
(21) 
Here we have already used the fact that
(22) 
The above result requires some effort and is provided by investigating the axes of the rotations in the austenitic point group along with Result 5.2 in [6].
Using the expressions for and we get
(23) 
where
(24)  
(25)  
(26)  
It is trivial to observe that, for fixed , the expression multiplying is of rank 2 and similarly, for fixed , the expression multiplying is of the same rank. Hence, the determinant of will only contain terms with and in powers not greater than two. Letting we deduce that
(28) 
Using the relations between the twinning solutions for different martensitic variants [7], the above expression simplifies further by noting that for all
(29) 
and, respectively, for all
(30) 
Now, from equations (28)(30) we obtain the following form for ;
(31) 
From (23) and (31) it becomes easy to specify the coefficients in the expression for . Firstly,
(32) 
Noting that and , we deduce, after a simple calculation, that
(33) 
and
(34) 
As for the last coefficient, and the expression gets more complicated reducing to
(35) 
For , we only have the TypeII structure and we get
This becomes zero for ; call this , which agrees with the value obtained, for example, by Ball & James in [4].
Setting and solving for we deduce that, as long as
(36) 
then
(37) 
For and the above equation admits two solutions, namely and , and we see that branches starting from and are created consisting of values of that make the nonclassical interface possible. Provided that
(38) 
a necessary and sufficient condition for these to meet, i.e. for equation (37) to have solutions for all , is that
(39) 
If, in addition,
(40) 
then the solutions will be distinct.
Otherwise, if , simplifies even further and we obtain that, for all , and will suffice. Due to the symmetry of , branches are also created at the points and and the condition for these to meet is the same. A remaining question is whether it is the middle eigenvalue that is equal to . This is answered in a similar fashion to [4] and hence, we shall require that
(41) 
for all pairs that make the interface possible. This will also imply that the other eigenvalues of , and , are bounded away from , i.e. .
However, the formulae for the cubictoorthorhombic transformation get too involved and for this reason we will proceed numerically.
Numerical Results
In the remainder of this section we present a numerical calculation where, in accordance with [11], the lattice parameters for CuAlNi were chosen to be , and . The martensitic variants used are the ones obtained from the experimental observations; that is, , with , their compound counterparts. Following the above analysis, we calculated the zeros of , i.e. the values of that allow the interface to occur. Relations (36) and (38)(41) were satisfied and we plotted against as shown in Fig. 5.
It is easily seen that the value of does not change significantly from (corresponding to ), which would give the classical interface between the TypeII twinning system and the austenite.
Moreover, using the algebraic procedure given in [4], we calculated the different normals for on the curves of Fig. 5 and a plot of these is given in Fig. 6, where the normals are depicted as points on the unit sphere.
These normals lie on four segments of curves whose endpoints correspond to the normals of possible classical austenitemartensite interfaces. A similar calculation was performed in [2] for the cubictotetragonal transformation, where, in contrast to the predictions here, these segments were in fact arcs of circles on the unit sphere.
A more detailed comparison of these theoretical predictions with the observed nonclassical interfaces will appear in [12].
Conclusion
This paper brings a theoretical analysis of compatible interfaces between austenite and a crossing twins microstructure of 2H martensite of the CuAlNi shape memory alloy, where the crossing twins microstructure consists of TypeII and compound twinning systems. These interfaces were recently observed by optical microscopy during the shape recovery process of single crystals of this alloy [12]. The main aim of this paper is to show that these interfaces (although never observed in any shape memory alloy before) do not contradict the commonly accepted nonlinear elasticity model, but are, on the contrary, predictable by this model for arbitrary volume fraction of the compound laminate. Since the relation (37) between this volume fraction and a volume fraction of the TypeII laminate intersecting compatibly the compound twins was derived analytically for a general cubictoorthorhombic transition, the analysis brought by this paper can be easily applied to predict the existence of similar nonclassical interfaces in any other shape memory alloy with the same class of transition (e.g. CuAlMn) as well as for materials undergoing the cubictotetragonal transitions (since the tetragonal symmetry is a member of the orthorhombic symmetry class.)
The numerical simulations carried out in the last subsection of this paper reveal that there is a dependence between the compound volume fraction and the habit plane orientation for the CuAlNi alloy. This finding is consistent with the optical observations of slightly curved nonclassical interfaces between austenite and the crossingtwins microstructure with heterogeneous compound volume fraction (Fig. 3).
Acknowledgements
The experimental part of this paper was financially supported by the project No. 202/09/P164 of the Czech Science Foundation and by the institutional project of IT ASCR v.v.i. (CEZ:AV0Z20760514). Both supports are acknowledged by H.S.. The theoretical part of the paper (J.M.B. and K.K.) was supported by the EPSRC New Frontiers in the Mathematics of Solids (OxMOS) programme (EP/D048400/1) and the EPSRC Science and Innovation award to the Oxford Centre for Nonlinear PDE (EP/E035027/1). The project originated in the EU MULTIMAT network (MRTNCT2004505226).
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