Upper and lower bounds on the set of recoverable strains and on effective energies in cubic-to-monoclinic martensitic phase transformations

A major open problem in the mathematical analysis of martensitic phase transformations is the derivation of explicit formulae for the set of recoverable strains and for the relaxed energy of the system. These are governed by the mathematical notion of quasiconvexity. Here we focus on bounds on these quasiconvex hulls and envelopes in the setting of geometrically-linear elasticity. Firstly, we will present mathematical results on triples of transformation strains. This yields further insight into the quasiconvex hull of the twelve transformation strains in cubic-to-monoclinic phase transformations. Secondly, we consider bounds on the energy of such materials based on the so-called energy of mixing thus obtaining a lamination upper bound on the quasiconvex envelope of the energy. Here we present a new algorithm that yields improved upper bounds and allows us to relate numerical results for the lamination upper bound on the energy with theoretical inner bounds on the quasiconvex hull of triples of transformation strains.


Introduction
In this article we study martensitic phase transformations in the context of shape memory alloys (SMA), which have a large variety of technological applications, e.g., in medical or satellite technology. SMAs are a class of materials which show very flexible behaviour known as pseudoelasticity (stress based) and shape-memory effect (temperature based). This relies on a spontaneous diffusion-less solid-solid phase transformation from austenite to martensite and vice-versa. A key feature is the formation of finescale microstructures in the martensitic phase composed of different martensitic variants.
Though these materials are very flexible, there is a limit after which the material deforms plastically. This article is concerned with the question of characterizing this limit. Phrased differently, we aim to characterize all strains that allow recovery of the original shape of the material upon heating above a critical temperature. We follow the variational approach proposed by Ball and James [1], see [2] for an introduction. The mathematical notion related to the addressed question is that of quasiconvexity (in the sense of Morrey). The objects we seek are the so-called quasiconvex hull of the set of (phase) transformation strains and the quasiconvex envelope of the related free energy density. Unfortunately these are extremely difficult to evaluate. Quasiconvex hulls are known only in some special cases. One therefore considers inner and outer bounds on the quasiconvex hull such as the polya e-mail: anja.schloemerkemper@mathematik.uni-wuerzburg.de convex hull, the rank-one convex hull and the lamination convex hull, respectively, see, e.g., [3,4] for details.
We work in the setting of geometrically-linear elasticity and focus on mono-crystalline materials. We assume that each variant obeys a linear-elastic material law from which we obtain the energy where e ( j) is the transformation strain and C is the (variantindependent) elastic modulus. This yields the microscopic energy density where n is the number of variants. The overall effective macroscopic energy is then given by the relaxation of the microscopic energy, which corresponds to calculating the quasiconvex envelope, QW(e, c), corresponding to the vector of variant fractions If the individual transformation strains are pairwise compatible (see Def. 1 below), it is known [5] that their quasiconvex hull/envelope and convex hull/envelope coincide. This applies to the cases of cubic-to-tetragonal, cubic-to-trigonal and cubic-to-orthorhombic phase transformations but it does not apply to cubic-to-monoclinic phase transformations, which however are of great interest, e.g., in NiTi, which has a wide industrial application.
We study the corresponding compatible hulls theoretically in §2 and investigate the envelope of the corresponding energy numerically in §4, this is motived by [7] and [8], respectively. A comparison of the results yields a striking correspondence.

Stress-free situation
In the stress-free situation we are interested in the zero, i.e., minimal, energy states of the system which are given by the quasiconvex hull of E. In order to obtain some further insight into this task, we follow ideas of [7,8] and consider inner/upper and outer/lower bounds on the quasiconvex hull/envelope of triples of these 12 strains. It turns out that these depend on compatibility properties of the strains: c . Then e 1 and e 2 are compatible (or e 1 and e 2 are symmetrized rank-one connected), This allows to easily determine specific examples for the five cases above, see [7] for details and further examples. The exact definition of the quasiconvex hull Q(E) of E is not relevant here. We therefore refer to [7] and references therein for the definition and for further definitions and properties stated below. Crucial for the following considerations are the bounds where C(E) denotes the convex hull and L(E) the lamination convex hull of E. However we need the definition of the symmetrized rank-one convex hull, R(E), which is based on the notion of symmetrized rank-one convex functions: and for all e 1 , e 2 ∈ S 3×3 with e 1 e 2 , Definition 3. Let E ⊂ S 3×3 be compact. Then the symmetrized rank-one convex hull of E is We begin our analysis with a well-known result: if E consists of three pairwise compatible strains as in Case 5, then R(E) = C(E) is the triangle with these strains as extreme points. Next we will recall some results related to Case 1:  Table 1]. In the computations we choose the material parameters for NiTi: α = 0.0243, β = −0.0437, δ = 0.0580 and ε = 0.0427, cf. [6].

The following T 3 -lemma [7, Lemma 4.3] yields an easy characterization of T 3 s:
It turns out that all triples of pairwise incompatible strains in E are T 3 s, see [7, §3 and §7]. For T 3 s it is known (cf., e.g., [10, §3] where e 1,2 , e 2,3 , e 3,1 are defined by Hence, in this case, an inner bound on the quasiconvex hull is given by the inner triangle and the "arms" of the T 3 , cf. Figure 2. An outer bound is the convex hull of the three transformation strains.
Next we consider Case 3: (These exist since det(e 0 − e 1 ) = 0 and det(e 0 − e 2 ) = 0.) Then λ 1 λ 2 = 0 and (note that we allow for λ 1 = λ 2 ). Then We omit the proof since it is similar to those for the preceding cases. In the case λ 1 = λ 2 there can be another compatible line.

Lamination upper bound and corresponding numerical approach
The lamination upper bound relies on earlier works [9,11,12] and [13,14]. It involves the so-called free energy of mixing w mix : Δ n → R, which allows to write the quasiconvex envelope of the energy as QW (e, c) = n j=1 c j W j (e) + w mix (c) .
The upper bound of our numerical approach is obtained by an upper bound on the energy of mixing, which satisfies for all c 1 , c 2 ∈ Δ n , where ψ is given in the references above. In [13,14], an upper bound of w mix based on ( w ≥ 0 on the vertices of Δ n } . Since the condition (3) has to be checked for all c 1 , c 2 ∈ Δ n with c 1 c 2 , it is in general not possible to compute w upp . Our approach will be different. Starting from an upper bound of w mix , we will use (3) to obtain a possibly improved upper bound. Note that the construction ofw is independent of whether c 1 and c 2 are compatible or not.
By construction, we have w mix ≤w ≤ w, which implies that the idea of Lemma 8 above can be used to obtain iteratively improved upper bounds. To start the iteration, an initial upper bound is needed. Here, it turns out that the trivial Taylor upper bound w 0 (c) = 0 is enough to get sensible iterates and is exact for the pure variants found at the vertices of the triangle configurations considered in the cases mentioned above.
We present the idea of the method for a mixture of three pure phases as a special case of the twelve possible martensitic variants in NiTi. For convenience we now relabel the strains 1, 2, 3. The method can be extended to more phases, but it is expected that the computational cost grows exponentially with the number of phases.
In the simplex Δ 3 we generate points with uniform dis- In the next step, we generate all possible triplets of points (c 1 , c 2 , c 3 ) ∈ C 3 with c 2 = 1 2 (c 1 + c 3 ). Even though alternative construction methods of the triplets, such as the more general c 2 = (ac 1 + bc 3 )/(a + b), could be used to potentially improve the quality of the bound, they would also increase the computational effort significantly and are therefore not evaluated here. All triplets considered are collected in the set G.
Then for any triplet (c 1 , c 2 , c 3 ) ∈ G we apply (4) to obtain an improved upper bound of w mix . In order to do so, the value of ψ at c 1 − c 3 has to be evaluated. Here, we can employ the fact that ψ is positively homogeneous of degree 2. Moreover, the distribution of points of C in the simplex is highly regular and symmetric, and hence the number of evaluations of ψ is much lesser than the number of triplets in G. This means that ψ can be evaluated offline, i.e., before the actual iteration starts.
The iteration procedure can be described as follows.
1. Generate C, G. Due to Lemma 8, all iterates w k generated by the above algorithm are upper bounds of w mix . In our numerical tests, the iteration was stopped if the maximum change between two successive iterates was smaller than 10 −12 .

Numerical results
Next we report about the results of our numerical experiments. The algorithm above was implemented in Matlab and run on a standard desktop computer. The tests were performed with grid size 1/N = 1/256 resulting in ≈ 33, 000 points in C. The set G contained ≈ 17.6 × 10 6 triplets. The minimization contained in the function ψ was evaluated for 240 search directions. We used the cubic-tomonoclinic-I transformation of NiTi, where we took the material parameters summarized in Table 1; these parameters were also used for the upper bounds computed in [8], cf. § 5. The C tensor is parametrized by three values C 11 = 140, C 12 = 110, C 44 = 32. In the figures, the vertices of the triangles are chosen such that the first mentioned strain is in the bottom left corner; the others follow anti-clockwise.
Firstly, we investigated the upper bound for the triple {e (1) , e (6) , e (12) }. These transformation strains are pairwise incompatible (Case 1), hence from the above considerations we expect to see T 3 structures in the upper bound. The total energy surface obtained from the upper bound can be seen in Figure 1, where the vertices (0, 0), (1, 0), and ( 1 2 , √ 3 2 ) correspond to the phases e (1) , e (6) , and e (12) . Figure 2 shows the logarithmic scaling of the total energy. Furthermore, the corresponding T 3 -configuration is superimposed; it shows a very good correspondence.
As can be seen from these figures, the upper bound of the total energy is close to zero in the T 3 region: the value of the total energy at the barycenter of {e (1) , e (6) , e (12) } is about 1.84 × 10 −6 , while the maximum total energy is of order 3 · 10 −3 . This indicates that our approach is successful at computing a sensible upper bound. Moreover, these results prompt the question as to whether the iteratively improved upper bounds would converge to the quasiconvex envelope of the energy if the grid size 1/N goes to zero or if more complex patterns of G are considered.
Secondly, we study Case 2, namely the triple e (1) , e (7) , and e (12) , cf. Figure 3, as well as the triple e (1) , e (6) , and e (8) , cf. Figure 4. This also reflects the theoretical behaviour. The first example is related to the last part of Lemma 6: the upper bound is zero at the compatible line [e (1) , e (7) ]; otherwise it is clearly not zero, which corresponds to the fact that there are no compatible lines within the triangle.  (1) , e (6) , e (12) }.  (1) , e (6) , e (12) }, logarithmic scale. The corresponding T 3configuration is superimposed.  (1) , e (7) , e (12) }, logarithmic scale. Note that there is a blue line on the bottom of the triangle that reflects the compatibility between the strains e (1) and e (7) .
The second example shows exactly the bound asserted in Lemma 7.
Next we present our result for Case 3, in which the strains e (1) and e (7) are compatible as are e (7) and e (10) , but e (1) and e (10) are not. Further, there is no other compatible direction in the triangle, i.e., λ 1 = λ 2 = 0 in Lemma 5. This computation, see Figure 5, also reflects the corresponding theoretical result, see Lemma 3, which says that an inner bound on the compatible hull is given by the compatible lines [e (1) , e (7) ] and [e (7) , e (10) ].
Both triples have two compatible edges and one incompatible edge as well as an additional compatible line.
First, we present upper bounds of the total energy for the three phases {e (1) , e (2) , e (7) }. Here, the phase e (1) is compatible to e (2) and to e (7) , while the latter two are incompatible, i.e. we are in Case 4. The upper bound on the total energy by our method can be found in Figure 6 in logarithmic scaling. The value of the total energy at the barycenter of the phases is 1.69×10 −7 , while the maximum of the total energy is of the order of 3·10 −3 . This maximal value yields an estimate of the difference between the upper bound and the convex envelope of the energy. Note that the convex envelope is zero on the whole triangle spanned by the three strains considered. Hence, as in [8], we obtain that the difference between these two hulls is small.
As can be observed, the upper bound closely approximates the expected set of zeros of the total energy, which is sketched by a solid line in the figure. Thus it confirms the non-symmetric shape of the inner bound on the quasiconvex hull. Moreover, the logarithmic figure appears to indicate that there is a sharp transition between the zero set of the energy and energy states outside of the inner triangle.
Second, we studied {e (1) , e (3) , e (12) }. This again shows a non-symmetric triangle, which however is much smaller, cf. Figure 7. Note the correspondence with the theoreti-   (1) , e (2) , e (7) }, logarithmic scale. Note that the compatible lines are superimposed. Finally, our computations show that the lamination upper bound in Case 5 of the three pairwise compatible strains {e (1) , e (2) , e (3) } is almost zero on the whole triangle (no figure shown). Since in this case, the lamination convex hull equals the convex hull, this is exactly as expected.

Conclusion
In this article we have demonstrated a close relationship between theoretical predictions on inner bounds of compatible hulls of transformation strains and numerical upper bounds on the quasiconvex envelopes of the corresponding energy. This work was motived by observations in [8] about the lamination upper bound which yields energy values showing some relation to the corresponding T 3 -configurations. With the new algorithm presented in this article we obtain much better correspondence. Indeed, the algorithm shows that the lamination upper bound is (almost) zero at the T 3 -configurations.
Similarly, for the other cases of triples of strains we obtain a strikingly correspondence between the inner bound on the quasiconvex hull of the strains and the (almost) zero set of the upper bound of the quasiconvex envelope of the energy.
Another observation made in [8] is related to the convex envelope. In the case of pairwise compatible triples of strains, the convex envelope happens to be identically zero on the triangle spanned by the three strains. It turns out that the relative difference between the energy values of the lamination upper bound and the convex envelope is small, even in the setting of T 3 s, which involves incompatibilities, the difference is 3 · 10 −3 . On the other hand it shows well that there is an interesting structure that can be seen in the lamination upper bound on the energy as well as in the inner bound on the set of recoverable strains. Whether these structures are also present in the quasiconvex envelopes and hulls remains an open problem.