Bounds on the recoverable deformations of polycrystalline SMAs at finite strain

. This communication is concerned with the theoretical prediction of the recoverable strains (i.e. the strains that can be recovered by the shape memory e ﬀ ect) in polycrystalline SMAs. The analysis is carried out in the ﬁnite strain setting, considering a nonlinear elasticity model of phase transformation. The main results are some rigorous upper bounds on the set of recoverable strains. Those bounds depend on the polycrystalline texture through the volume fractions of the di ﬀ erent orientations. A two-orientation polycrystal of tetragonal martensite is studied as an illustration. In that case, analytical expressions of the upper bounds are derived and the results are compared with lower bounds obtained by considering laminate textures. The issue of applying the proposed method to complex polycrystalline textures is commented on.


Introduction
A possible route to study the formation of microstructures in SMAs is to adopt a nonlinear elasticity model of phase transformation [1]. The general principle is that, under a prescribed loading, the system tends to minimize its free energy. Assuming the microscopic, mesoscopic and macroscopic scales to be well separated, the energy minimization principle leads to different expressions of the free energy at each scale. Denoting the microscopic free energy by Ψ, the mesoscopic energyΨ is obtained as the relaxation (or quasiconvex envelope) of Ψ, which essentially amounts to solve an optimal design problem with respect to the martensite/austenite geometric arrangement (see Sect. 2 for a precise definition). Viewing group of grains with the same orientation as individual homogeneous materials (governed by mesoscopic free energies), the polycrystal can be regarded as a composite material with a macroscopic energyΨ obtained by homogenization of the constitutive free energy functions.
Assuming the microscopic free energy Ψ to be known, determining its relaxationΨ largely remains an open problem. Estimating the macroscopic free energyΨ is even more challenging as stress and strain compatibility conditions between the grains need to be taken into account. Of special interest are the strains that minimize the mesoscopic (resp. macroscopic) free-energy. Those energyminimizing strains can indeed be interpreted as the recoverable strains of a monocrystalline (resp. polycrystalline) shape memory alloy, i.e. the strains that can be recovered by the shape memory effect [2,3]. Knowing the set of recoverable strains is crucial for designing SMA systems. Experiments only give partial insight in the structure of that set, as they usually only give measurements a e-mail: michael.peigney@polytechnique.org along prescribed directions (see e.g. [4]). In this paper, we propose theoretical bounds on the whole set of recoverable strains, i.e. in the space of three-dimensional deformation gradients. Those bounds are expressed in terms of the lattice parameters and of statistical information on the polycrystalline texture (namely the orientation distribution function). Such data can be obtained experimentally using X-ray diffraction or EBSD (Electron Back Scattering Diffraction).
Most work related to that topic has been carried out in the geometrically linear setting, i.e. assuming small deformations with respect to a reference configuration [2,[5][6][7][8][9][10][11][12][13]. In this paper, we focus on upper bounds of the recoverable strains of martensitic polycrystals, in the geometrically non-linear setting. The set of mesoscopic energyminimizing strains has been obtained in closed-form for a double-well energy [1]. Using known restrictions on Young measures [1,14], an upper bound on the mesoscopic energy-minimizing strains has been proposed in the case of three or more wells [15]. Regarding polycrystals, a general method has been introduced in [16] for generating upper bounds on the set of macroscopic recoverable strains, assuming that the set of recoverable strains of the constitutive single crystals (or at least an upper bound on it) is known. The approach used in [16] is based on the translation method [17][18][19], which has proved to be a powerful tool in various problems related to homogenization [20][21][22]. In this communication, we first present in Sect. 2 the monocrystalline bound [15] and subsequently combine it with the methodology of [16] to derive explicit upper bounds for polycrystals (Sect. 3). It turns out, however, that the obtained bounds may fail to recover the single crystal bound in the homogeneous limit. Motivated by that observation, we modify the polycrystalline approach so as to take the special structure of the single crystal bound into account (Sect. 4). This results in new upper bounds for polycrystals, which improve on the bounds of Sect. 3 and are consistent with the single crystal bound in the homogeneous limit. A two-orientation/three-well polycrystal is studied as an illustrative example in Sect. 5.

Single crystal
In the framework of nonlinear elasticity at finite strains, the microscopic behavior of a shape-memory alloy is described by its free energy density Ψ, which is a function of the deformation gradient F. The principle of frame indifference implies that Ψ(R.F) = Ψ(F) for any rotation R and deformation gradient F. We denote by K the set of deformation gradients that minimize Ψ. Without loss of generality, we can assume that the minimum value of Ψ is equal to 0, so that Ψ ≥ 0 and If the temperature T is below the transformation temperature T 0 (which is assumed throughout this paper), the set K takes the form where m is the number of martensitic variant and U 1 , · · · , U m are the transformations strains. The symmetric positive definite U 1 , · · · , U m are all symmetry related, i.e. for any (i, j) there exists a rotation R i j such that U j = t R i j .U i .R i j (here and in the following, the presuperscript t denotes the transpose operator). This is implies that U 1 , · · · , U m all have the same determinant η. It is convenient to introduce the set E defined as Consider a reference configuration where a domain Ω is occupied by a single crystal of shape memory alloy. The mesoscopic free energy of the single crystal is given bỹ where . denotes volume average over the domain Ω and the set A(F) of admissible deformation gradient fields is defined by The functionF →Ψ(F) is mathematically referred to as the quasiconvex envelope (or relaxation) of Ψ. This last denomination is justified by the fact thatΨ is the largest function such that: (i)Ψ ≤ Ψ, (ii)Ψ and quasiconvex, i.e. satisfiesΨ LetK be the set of deformation gradients that minimizẽ Ψ. SinceΨ is positive and vanishes on K, the minimum value ofΨ is equal to 0 and we havẽ The setK is also known as the quasiconvex hull of K [14,23]. The exact expressionK remains generally out or reach. Therefore, bounds onK (in the sense of inclusion of sets) are of interest. In that regard, it can be proved [15] that the setK + defined bỹ is an upper bound onK, i.e. satisfiesK ⊂K + . In (4), the set T is defined by and Φ is the frame indifferent function Φ : Note that The bound (4) is obtained using known restrictions on Young measures [1,14]. The crucial point is that the func- is quasiconvex for all a and b [24]. In (4), C is a given arbitrary subset of R 3×3 × R 3×3 : Each choice of C generates a corresponding bound onK. For a well chosen C, the bound given by (4) coincides with K for the reference cases where the exact expression ofK is available (see [15]).
Finally, for a givenF inK + , we note that the vector θ in (4) can be interpreted as the volume fractions of the different wells in a microstructure realizingF.

Polycrystal
Now consider a polycrystal occupying a domain Ω. We can decompose Ω as Ω = ∪ n r=1 Ω r where each sub-domain Ω r is formed by grains with the same orientation. The microscopic free energy in Ω r can be written as where R r is a rotation describing the orientation in Ω r relative to a reference single crystal. Defining χ r the characteristic function of Ω r (i.e. χ r (x) = 1 if x ∈ Ω r , and χ r (x) = 0 otherwise), the macroscopic free energyΨ(F) of the polycrystal is given bȳ whereΨ r is the relaxation of Ψ r , as defined in (1) (see e.g. [3] for a detailed justification). In the following, we primarily focus on the setK of deformation gradients that minimize the macroscopic energy, i.e.

02004-p.2
In view of (9), we havē whereK(x) is the quasiconvex hull of In (11), Eq. (10) shows that the distinctive properties of strainsF inK is that they can be realized by a deformation u(x) whose gradient F = ∇u satisfies the local constraint F(x) ∈K(x) at each point.
An upper bound onK that take one-point statistics of the functions χ r has been derived [16]. With the present notations, that bound is characterized by The upper bound in (12) has been used in [16] on some simple examples where the setsK r are known. In more general situations, the direct application of the bound (12) is hampered by the fact thatK r is unknown. Such a difficulty can be overcome by using the results from Sect. 2. Let indeedK r + be the upper bound ofK r defined in Eq.
(4). SinceK r ⊂K r + , we have Therefore, we obtain from (12)  (13) The calculation of the right-hand side in (13) can be further simplified if C = C, i.e. if the bound (13) and the boundK r + given by (4) are calculated using the same set of tensors (a, b). In such case it can be verified that We thus arrive at wherē The setK 0 + is an explicit upper bound that depends on one-point statistics of the texture, i.e. on the volume fractions χ r of the different orientations. The setK 0 + is defined by a set of nonlinear constraints onF.

Improved bound for polycrystals
The boundK 0 + in (15) can be improved upon by taking the special structure of the monocrystalline bound (4) into account, as is now explained. Consider a givenF inK. By (10), there exists a field F ∈ A(F) such that F(x) ∈K(x) for all x ∈ Ω. Using the bound (4) onK(x), we know there exists θ(x) ∈ T such that for all a and b. Since χ r (x) ∈ {0, 1} and r χ r (x) = 1, Eq. (16) can be rewritten as For any r = 1, · · · , n and i = 1, · · · , m, define Taking volume averages in (17) yields for all a and b. The crucial point is that the function F → Φ(F.a + F * .b) is quasiconvex, which in view of (19) implies that for all a and b. The scalar θ r i can be interpreted as the volume fraction of martensitic variant i with orientation r. Note from (18) that { θ r i } 1≤r≤n 1≤i≤m belongs to the setT defined byT (21) The developments so far show that for anyF inK, there exists Θ ∈T verifying the inequality (20). This last statement can be rewritten asK wherē ESOMAT 2015 In a way similar to the boundK 0 + considered in Sect. 3, K + depends on the texture through the volume fractions χ r of the different orientations (the later indeed appear in the definition (21) of the setT).
For a givenF in E, let Q(F) be the subset of R n m defined by Observe that Q(F) is a convex set defined by a family of linear constraints. The set Q(F) can be interpreted as the set of volume fractions in the microstructures realizingF. The distinctive propery of strainsF inK + is that Q(F) is non empty. In the language of linear programming, checking whether the convex set Q(F) is non empty amounts to check feasibility of the linear constraints in (24) [25], which is not a direct calculation -even for a discrete C.
In that regard, it can be noted that interior-point methods [26] offer some efficient algorithms for detecting feasibility in large-scale linear programming problems. Interestingly, such algorithms, as the self-dual algorithm of Ye [27], have been used in other problems related to shapememory alloys [28] and could possibly be useful for calculating the boundK + in the case of a complex polycrystalline texture.

Upper bounds
We consider a polycrystal with two orientations, assuming without loss of generality that orientation 1 is the reference orientation. The constitutive single crystals obey a cubic to tetragonal transformation: These matrix representations are relative to the reference orthonormal basis (v 1 , v 2 , v 3 ) of the cubic austenitic lattice in orientation 1. All the results presented next are obtained with the lattice parameters of MnCu, i.e. η 1 = 1.0099, η 2 = 0.9656 [29]. The set K 2 of strains that minimize the microscopic free energy in orientation 2 can be written as where R 2 is the rotation taken as Textures satisfying the assumptions made so far (i.e. n = 2 with R 1 = I and R 2 given by Eq. (25) ) are observed in some ribbons of shape memory alloys [30]. Consider deformation gradients F(ω, δ) of the form where The deformation gradient F(ω, δ) is a simple shear between the directions u(ω) and v(ω) (Figure 1), followed by a uniform dilatation (η 2 1 η 2 ) 1/3 I. The parameter ω is the angle made by the shear directions (u(ω), v(ω)) with the directions (v 1 , v 2 ) of the cubic austenitic lattice in orientation 1.
The results of Sect. 3-4 allow one to bound the values (ω, δ) for which F(ω, δ) is recoverable. The solid lines in Fig. 2 shows the boundary of the domain where the boundK + is calculated using (23) with a well chosen class C of tensors (a, b) for which closed-form expressions can be obtained (see [24] for details). The volume fraction χ 1 is set equal to 0.7. Any recoverable deformation gradient F(ω, δ) is necessarily within the bounded domain Δ + delimited by the solid lines in Figures  2.
Similarly, the dotted lines in Fig. 2 show the boundary of the domain whereK 0 + is calculated using the same tensors (a, b) as for the boundK + . This allows one to appreciate the improvement brought by the consideration of (23) over (15).

Lower bound
The relations definingK + in (23) are necessary-but not sufficient -conditions for a deformation gradient to be recoverable. The issue is to determine which deformation gradients inK + are indeed recoverable for some polycrystalline texture that is compatible with the prescribed statistics (i.e. with prescribed volume fractions of the different orientations). Considering the special class of laminated textures and adapting an argument introduced in [1], a set of values (ω * , δ * ) for which F(ω * , δ * ) is recoverable can be constructed. That set is denoted by Δ − and shown in green in Figure 2. The green domain Δ − is found to fill most of the domain Δ + , meaning that most of the values of (ω, δ) in Δ + can be realized by laminate textures. The gap between Δ − and Δ + could possibly be reduced by considering more complex polycrystalline textures.

Concluding remarks
In this paper, some rigorous upper bounds on the recoverable strains of martensitic polycrystals have been obtained in the geometrically nonlinear setting. The main results are the boundsK 0 + andK + (defined in (15) and (23) respectively) that depend on the texture through the volume fractions of the different orientations. Those bounds are expressed in terms of a given family C of tensors (a, b), which acts as a free parameter in (15)-(23): each choice of C generates corresponding boundsK 0 + andK + . For a given (say discrete) C, the boundK + is tighter thanK 0 + but more difficult to calculate: whereas checking if a given deformation gradientF is inK 0 + is a direct calculation, checking ifF ∈K + amounts to detecting feasibility of a linear programming problem in R n m . Those bounds could be evaluated in closed form for the 2-orientation/3-variant polycrystal presented as an illustrative example. For more complex textures, it is clear that numerical calculations of the bounds will be necessary at some point, which requires adequate algorithms, as discussed in Sect. 4. A more theoretical line of investigation consists in deriving upper bounds taking more information on the texture (such as 2-point statistics) into account.