Michaël Peigney

On the energy-minimizing strains in martensitic microstructures -Part 2: Geometrically linear theory

Keywords: Energy minimization, Lamination, Microstructures, Phase transformation, Shape-memory alloys

This paper addresses the theoretical prediction of the set of energyminimizing (or stress-free) strains that can be realized by martensitic microstructure. Polyconvexification and related notions are used to derive some upper bounds (in the sense of inclusion of sets). Lower bounds are obtained from lamination techniques. The geometrically linear setting (infinitesimal strains) is considered in the present Part 2. Three-, four-, and twelve-well problems are considered. In particular, the structure of the set of energyminimizing strains in cubic to monoclinic transformations is investigated in detail. That investigation is notably supported by three-dimensional vizualisations obtained by considering four-well restrictions.

Introduction

The peculiar properties of shape memory alloys are the macroscopic result of a diffusion-less solid/solid phase transformation that occurs at a microscopic level. Adopting the framework of non-linear elasticity, the microscopic behavior is classically described by a multi-well free energy Ψ. The macroscopic (or effective) free energy is then obtained as the relaxation (or quasiconvexification) of Ψ. Because of the non-convex structure of Ψ, the relaxation procedure is notoriously difficult to perform. Exact solutions are only known in few special cases. This paper focuses on the set of strains that minimize the macroscopic energy. Those so-called recoverable strains can be interpreted as stress-free macroscopic strains. As explained by [START_REF] Bhattacharya | Energy minimization and the recoverable strains in polycrystalline shape memory alloys[END_REF], they play an important role in the properties of shape memory alloys.

The problem can be formulated either in the geometrically non-linear setting (finite strains) or in the geometrically linear setting (infinitesimal strains). The present Part 2 is devoted to the geometrically linear theory, whereas the geometrically non-linear theory has been considered in Part 1 [START_REF] Peigney | On the energy-minimizing strains in martensitic microstructures -Part 1: geometrically non-linear theory[END_REF]. Although the geometrically non-linear theory is more accurate, the problem is significantly more tractable in the geometrically linear theory. Consider for instance the case of n geometrically compatible phases. In the geometrically linear setting, it can be proved that the set of energyminimizing macroscopic strains is equal to the convex hull of the strains that minimize Ψ [START_REF] Bhattacharya | Comparison of the geometrically nonlinear and linear theories of martensitic transformations[END_REF]. In contrast, in the geometrically non-linear setting, the problem is much more complex and has been solved exactly only in the case n = 2 [START_REF] Ball | Proposed experimental tests of a theory of fine microstructure and the two-well problem[END_REF]. It has to be mentioned that, even in the geometrically linear setting, substantial difficulties remain where the phases are not all pairwise compatible: analytical expressions have been only obtained in the cases of two and three phases [START_REF] Kohn | Relaxation of a double-well energy[END_REF][START_REF] Smyshlyaev | On the relaxation of a three-well energy[END_REF].

The outline of the present Part 2 is as follows. Using distinctive properties of the relaxation, we first derive a general upper bound (denoted by P K) on the set of strains that minimize the macroscopic energy (Section 2). This is accomplished by adapting a method used in Part 1 to the geometrically linear theory. That upper bound is compared with existing bounds from the literature. In particular, for the three-well problem (Section 3), the bound obtained is shown to coincide with the results of [START_REF] Smyshlyaev | On the relaxation of a three-well energy[END_REF]. Four-well problems are considered in Section 4. The upper bound P K is compared with lower bounds constructed by a sequential lamination algorithm. Three-dimensional visualizations of the various bounding sets are given for some examples related to the cubic to monoclinic transformations. Those examples serve two purposes. First, they illustrate the gap between the lower and upper bounds considered, giving an appreciation of those bounds. Second, they provide a first insight in the full study of the twelve-well problems corresponding to cubic to monoclinic transformations, which is the focus of Section 5. Taking the 12 variants into account adds some complications in the derivation of meaningful bounds. In particular, the sequential lamination algorithm introduced in Section 4 cannot be used directly as it involves prohibitive calculation costs. An alternative strategy is detailed for constructing relevant lower bounds in that case. The structure of the bounds obtained is investigated in detail, distinguishing the cases of monoclinic-I and monoclinic-II martensite.

2. Upper bound on QK in the geometrically linear theory Let Ψ denote the free energy density of the material at the microscopic level. To account for the phase transformation between austenite and martensite, Ψ is generally modeled as a function with multiple wells. We denote by K the set of strains that minimize Ψ. In the geometrically linear setting, K is a discrete set, i.e.

K = {e 1 , • • • , e n }.
(2.1)

At a temperature below the transformation temperature, the strains e 1 , • • • , e n in (2.1) are the transformation strains of the martensitic variants. Consider a crystal occupying a domain Ω. The effective free energy of the material is the relaxation (or quasiconvexification) of Ψ, defined as

QΨ(ē) = inf e∈A(ē) 1 |Ω| Ω Ψ(e) dx (2.2)
where

A(ē) = {e| ∃u(x) ∈ W 1,∞ (Ω, R 3 ) such that e = (∇u + ∇ T u)/2 in Ω; u(x) = ē.x on ∂Ω}. (2.
3)

The multiple-well structure of Ψ entails that the infimum in (2.2) is generally not attained, which makes the calculation of QΨ a far from trivial matter. Also note that QΨ is independent on the domain Ω considered (see e.g. [START_REF] Dacorogna | Direct methods in the calculus of variations[END_REF] and references therein).

In this paper we are interested in estimating the set of effective strains that minimize QΨ. That set, denoted by QK, is referred to as the quasiconvex hull of K. The first step in our study is to derive a general upper bound on QK. This is accomplished by using a distinctive property of QK, namely that any strain ē in QK can be written as

ē = R 3×3 s edν(e) (2.4)
for some Young measure ν supported on K [START_REF] Ball | Proposed experimental tests of a theory of fine microstructure and the two-well problem[END_REF][START_REF] Müller | Variational models for microstructure and phase transitions[END_REF]. A notable consequence of that property is that QK only depends on Ψ through its set K of minimizers. Young measures notably have the following properties [START_REF] Kinderlehrer | Characterizations of young measures generated by gradients[END_REF]:

ν ≥ 0; R 3×3 s dν(e) = 1; (2.5) h( R 3×3 s edν(e)) ≤ R 3×3
s h(e)dν(e) for any quasiconvex function h. (2.6)

In the present case, K has the discrete structure (2.1). The measure ν in (2.4) is supported on K and therefore can be written as

ν = n r=1 θ r δ er (2.7)
where δ er is the Dirac mass at e r . The property (2.5) implies that θ = (θ 1 , • • • , θ n ) belongs to the set T n defined as The conclusion is that any ē in QK can be written as ē = n r=1 θ r e r for some θ ∈ T n verifyingr,s θ r θ s (e r -e s ) * ≥ 0. The set QK is thus included in the set P K defined as

T n = {θ = (θ 1 , • • • , θ n )|θ r ≥ 0;
P K = { n r=1 θ r e r |θ ∈ T n ; - n r,s=1 θ r θ s (e r -e s ) * ≥ 0}.
(2.10)

The notation P K is motivated by the fact that the set in (2.10) is related to the notion of polyconvexity used in the geometrically non-linear theory (see Part 1). From (2.10) it can be seen that P K is included in the convex hull CK of {e 1 , • • • , e n }, given by CK = { n r=1 θ r e r |θ ∈ T n }. Moreover, P K is equal to CK if -(e r -e s ) * is positive for all {r, s}. Let us interpret this last condition: for a fixed pair {r, s}, the symmetric tensor e r -e s can be decomposed as e r -e s = 3 i=1 i u i ⊗ u i where ( 1 , 2 , 3 ) are the eigenvalues of e r -e s and (u 1 , u 2 , u 3 ) is an orthonormal basis. The adjugate tensor (e r -e s ) * is then equal to

2 3 u 1 ⊗ u 1 + 1 3 u 2 ⊗ u 2 + 1 2 u 3 ⊗ u 3 .
Consequently, -(e r -e s ) * is positive if and only if i j ≤ 0 for all i = j, i.e. if and only if one eigenvalue is equal to 0 and the two others are of opposite sign. This last condition can be shown [START_REF] Bhattacharya | Comparison of the geometrically nonlinear and linear theories of martensitic transformations[END_REF] to be equivalent to the fact that the strains e r and e s are compatible, i.e. there exists some vectors (u, v) such that e r -e s = u ⊗ v + v ⊗ u.

(2.11)

Hence P K is equal to CK if the strains {e 1 , • • • , e n } are pairwise compatible.
In that case, it is actually known that QK = CK [START_REF] Bhattacharya | Comparison of the geometrically nonlinear and linear theories of martensitic transformations[END_REF].

Relations with existing bounds on the energy

Any lower bound Q -Ψ on the relaxation QΨ automatically generates an upper bound on QK (in the sense of inclusion of sets). Indeed, if

Q -Ψ is such that Q -Ψ ≤ QΨ, then QK ⊂ {ē|Q -Ψ(ē) ≤ 0}.
It is interesting to compare such an upper bound with the bound P K in (2.10). In the geometrically linear setting, lower bounds on QΨ have been proposed in the case where

Ψ(e) = min 1≤r≤n+1 Ψ r
(2.12) with Ψ r (e) = 1 2 (e -e r ) : L : (e -e r ) for r ≤ n,

Ψ n+1 (e) = 1 2 e : L : e + m.
(2.13)

In those expressions, Ψ r and Ψ n+1 are the free energy of martensite variant r and of the austenite, respectively. At a temperature below the transformation temperature, the minimum energy m of the austenite is strictly positive. The tensor L is a symmetric positive definite elasticity tensor, assumed to take the same value for all the phases. In that case, as detailed by [START_REF] Govindjee | The free energy of mixing for n-variant martensitic phase transformations using quasi-convex analysis[END_REF], the relaxation QΨ has the structure

QΨ(ē) = inf θ∈T n+1 n+1 r=1 θ r Ψ r (ē) + h(θ) (2.14)
where h(θ) can be interpreted as a mixing energy between the phases. Solving the relaxation problem is equivalent to determining the function h. [START_REF] Govindjee | The free energy of mixing for n-variant martensitic phase transformations using quasi-convex analysis[END_REF] considered the lower bound h 0 on h provided by the following formula:

h 0 (θ) = 1 2 n r,s=1 θ r θ s e r : L : e s - 1 2 n r=1
θ r e r : L : e r .

(2.15)

The resulting lower bound on QΨ is equal to the convexification of Ψ, as defined by

CΨ(ē) = inf θ∈T n+1 CΨ(ē, θ) (2.16) with CΨ(ē, θ) = 1 2 (ē - n r=1 θ r e r ) : L : (ē - n r=1
θ r e r ) + mθ n+1 .

The upper bound on QK that is deduced from (2.16) is the convex hull CK.

If the strains (e 1 , • • • , e n ) are not all pairwise compatible, the sets QK and P K may be strictly smaller than CK (see Section 4 for some examples). A lower bound h 1 that improves on (2.15) has been proposed by [START_REF] Peigney | A non-convex lower bound on the effective free energy of polycrystalline shape memory alloys[END_REF].

The corresponding lower bound on QΨ is expressed as

QΨ(ē) ≥ inf θ∈T n+1 CΨ(ē, θ) + h 1 (θ)
where (2.17) In that equation, K(a) is the symmetric fourth-order tensor such that (1/2)ē : K(a) : ē = -a : ē * for all ē, and M (a) is defined by

h 1 (θ) = sup a≥0|L-K(a)≥0
M (a) = -K(a) -K(a) : (L -K(a)) -1 : K(a).
(2.18)

The corresponding bound on QK is given by

{ n r=1 θ r e r |θ ∈ T n+1 ; θ n+1 = 0; h 1 (θ) ≤ 0}. (2.19)
That bound is now compared with the bound P K in (2.10). Consider a given θ ∈ T n+1 such that h 1 (θ) ≤ 0 and θ n+1 = 0. We have

0 ≥ n r,s=1 θ r θ s (e r -e s ) : M (a) : (e r -e s ) (2.20)
for any a ≥ 0 verifying L -K(a) ≥ 0. Let a 0 ≥ 0 be fixed. For t positive sufficiently small, the tensor L -K(ta 0 ) is positive. Moreover, at the first order in t, the tensor M (ta 0 ) is equal to -tK(a 0 ). We thus obtain from (2.20) that

0 ≥ n r,s=1
-θ r θ s (e r -e s ) : K(a 0 ) : (e r -e s ),

i.e. that 0 ≤ -a 0 : ( r,s θ r θ s (e r -e s ) * ). This proves that the tensor n r,s=1 θ r θ s (e r -e s ) * is positive. Consequently, any strain in the set (2.19) is in the P K defined by (2.10). Conversely, consider a given θ ∈ T n such that 0 ≤ -n r,s=1 θ r θ s (e r -e s ) * . Setting θ n+1 = 0, we have from the definition (2.18) of M (a):

4h(θ) = n+1 r,s=1 θ r θ s (e r -e s ) : M (a) : (e r -e s ) = a : ( n r,s=1 θ r θ s (e r -e s ) * ) - n r,s=1 τ rs : (L -K(a)) -1 : τ rs (2.21)
where τ rs = K(a) : (e r -e s ). For any a ≥ 0 such that L -K(a) ≥ 0, the two terms on the right-hand side of (2.21) are negative. Therefore, any strain in P K is in the set (2.19). The conclusion is that the bound P K in (2.10) coincides with the set deduced from the energy bound (2.17). Note however that the latter only applies when the microscopic energy is piecewise quadratic of the form (2.12), whereas the bound P K does not rely on that assumption.

The tree-well problem

The quasiconvexification of the free energy Ψ in (2.12) has been thoroughly studied by [START_REF] Smyshlyaev | On the relaxation of a three-well energy[END_REF] in the case n = 3. Using a Hashin-Shtrickman type variational formulation and considering known restrictions on H-measures, these authors derived a lower bound Q SW Ψ that improves on the convexification of Ψ, and obtained a sufficient condition for that lower bound Q SW Ψ to coincide with the exact value of QΨ. Here we consider an example given by [START_REF] Smyshlyaev | On the relaxation of a three-well energy[END_REF], for which the calculations can be done in closed form: the elasticity tensor L in (2.12) is taken as isotropic (i.e. L ijpq = λδ ij δ pq + µ(δ ip δ jq + δ iq δ jp )), and the strains (e 1 , e 2 , e 3 ) in (2.12) are taken as

e 1 = diag(λ 1 , λ 2 , λ 3 ) , e 2 = diag(µ 1 , µ 2 , µ 3 ) , e 3 = 0. (3.1)
The eigenvalues λ i and µ i can always be represented in the form

λ j = R j cos 1 2 χ j , µ j = -R j sin 1 2 χ j , (3.2)
for some scalars R j and χ j . To alleviate the notations, we set:

s i = sin 1 2 χ i , c i = cos 1 2 χ i , s(i+j) = sin 1 2 (χ i +χ j ) , s(i-j) = sin 1 2 (χ i -χ j ).
(3.3)

We assume that the following condition, introduced by Smyshlyaev and Willis (1998), is satisfied for all permutation {i, j, k} of {1, 2, 3}:

R j R k s(j -i)s(k -i) < 0. (3.4)
The bound Q SW Ψ obtained by [START_REF] Smyshlyaev | On the relaxation of a three-well energy[END_REF] takes the following expression:

Q SW Ψ(ē) = inf (θ 1 ,θ 2 )∈T 2 { 1 2 (ē -θ 1 e 1 -θ 2 e 2 ) : L : (ē -θ 1 e 1 -θ 2 e 2 ) + Î(θ 1 , θ 2 )} (3.5) with T 2 = {(θ 1 , θ 2 )|0 ≤ θ 1 ; 0 ≤ θ 2 ; θ 1 + θ 2 ≤ 1}. (3.6) The function Î in (3.5) is of the form Î(θ 1 , θ 2 ) = inf (kr,φr) 3 r=1 F (φ r ) (3.7)
where the infimum is taken over values (k r , φ r ) such that k r ≥ 0 and

θ 1 (1 -θ 1 ) -θ 1 θ 2 -θ 1 θ 2 θ 2 (1 -θ 2 ) = 3 r=1 k r sin 2 1 2 φ r sin 1 2 φ r cos 1 2 φ r sin 1 2 φ r cos 1 2 φ r cos 2 1 2 φ r .
(3.8) The exact expression of the function F in (3.7) can be found in Smyshlyaev and Willis (1998)(eqn(7.16)). For our purpose, it is sufficient to mention that F is a positive function. Let Q SW K = {ē|Q SW Ψ(ē) ≤ 0} be the upper bound on QK deduced from Q SW Ψ. Since L and F are positive, Q SW Ψ(ē) is negative if and only if ē = θ 1 e 1 + θ 2 e 2 for some (θ 1 , θ 2 ) ∈ T 2 verifying Î(θ 1 , θ 2 ) = 0. When the condition (3.4) is satisfied, it has been shown by [START_REF] Smyshlyaev | On the relaxation of a three-well energy[END_REF] that Î(θ 1 , θ 2 ) = 0 if and only if the infimum over φ r in (3.7) is attained for φ r = χ r . Consequently, the upper bound Q SW K consists of tensors θ 1 e 1 + θ 2 e 2 for which (θ 1 , θ 2 ) ∈ T 2 and there exists k r ≥ 0 verifying

θ 1 (1 -θ 1 ) -θ 1 θ 2 -θ 1 θ 2 θ 2 (1 -θ 2 ) = 3 r=1 k r s 2 r s r c r s r c r c 2 r .
(3.9)

Let us now compare this result with the bound P K in (2.10). Define the tensor E(θ 1 , θ 2 ) by

E(θ 1 , θ 2 ) = θ 1 (1 -θ 1 -θ 2 )e * 1 + θ 2 (1 -θ 1 -θ 2 )e * 2 + θ 1 θ 2 (e 1 -e 2 ) * . (3.10)
The set P K in (2.10) consists of tensors θ 1 e 1 + θ 2 e 2 such that (θ 1 , θ 2 ) ∈ T 2 and -E(θ 1 , θ 2 ) ≥ 0. Some simple calculations give

e * 1 = diag(R 2 R 3 c 2 c 3 , R 1 R 3 c 1 c 3 , R 1 R 2 c 1 c 2 ), e * 2 = diag(R 2 R 3 s 2 s 3 , R 1 R 3 s 1 s 3 , R 1 R 2 s 1 s 2 ), (e 1 -e 2 ) * = diag(R 2 R 3 (c(2 -3) + s(2 + 3)), R 1 R 3 (c(1 -3) + s(1 + 3)), R 1 R 2 (c(1 -2) + s(1 + 2))).
(3.11)

The tensor -E(θ 1 , θ 2 ) is diagonal, and positive if and only if -E ii (θ 1 , θ 2 ) ≥ 0 for i = 1, 2, 3. Using (3.11), we obtain three inequalities of the form

R j R k [θ 1 θ 2 s(j + k) + θ 1 (1 -θ 1 )c j c k + θ 2 (1 -θ 2 )s j s k ] ≤ 0 (3.12)
where j = k. We now show that the conditions (3.12) are actually equivalent to the requirements (3.9). Let us introduce the following matrix:

∆ =   s 2 1 s 2 2 s 2 3 c 2 1 c 2 2 c 2 3 -s 1 c 1 -s 2 c 2 -s 3 c 3   . (3.13)
The condition (3.9) can be rewritten as

∆   k 1 k 2 k 3   =   θ 1 (1 -θ 1 ) θ 2 (1 -θ 2 ) -θ 1 θ 2   with k r ≥ 0. (3.14)
Calculating the determinant of ∆ gives det ∆ = s(3 -2)s(1 -2)s(1 -3). By (3.4), the matrix ∆ is thus invertible and the condition (3.9) is equivalent to

∆ -1   θ 1 (1 -θ 1 ) θ 2 (1 -θ 2 ) -θ 1 θ 2   ≥ 0. (3.15)
The inversion of ∆ yields

∆ -1 = 1 det ∆   c 2 c 3 s(3 -2) s 2 s 3 s(3 -2) -s(3 + 2)s(3 -2) c 1 c 3 s(1 -3) s 1 s 3 s(1 -3) -s(1 + 3)s(1 -3) c 1 c 2 s(2 -1) s 1 s 2 s(2 -1) -s(1 + 2)s(2 -1)   . (3.16)
Substituting (3.16) in (3.15), we obtain the three following inequalities

1 s(i -j)s(i -k) [θ 1 θ 2 s(j + k) + θ 1 (1 -θ 1 )c j c k + θ 2 (1 -θ 2 )s j s k ] ≥ 0 (3.17)
where {i, j, k} is a permutation of {1, 2, 3}. Because of the condition (3.4), this inequality is equivalent to (3.12). Therefore, the upper bound P K coincides with Q SW K. It has to be emphasized that the mathematical arguments used by Smyshlyaev and Willis (namely a variational formulation of a Hashin-Shtrickman type) are of a different nature than those used for deriving the bound P K.

Four-well problems

In this section, K is assumed to be of the form K = {e 1 , e 2 , e 3 , e 4 }. The upper bound P K in (2.10) is compared with lower bounds resulting from sequential lamination techniques [START_REF] Kohn | Relaxation of a double-well energy[END_REF]. The construction of such lower bounds relies on the fact that if two given strains e and e are compatible in the sense of (2.11), then any strain in the line segment [e, e ] is realized by a simple laminate. As mentioned earlier, the compatibility condition can be reformulated as λ 2 = 0, where λ 1 ≤ λ 2 ≤ λ 3 are the eigenvalues of ee . That condition can easily be proved to be equivalent to det(e -e ) = 0, ( Rank-r laminates can be constructed by using that argument in an iterative fashion. The set of strains realized by such laminates forms a lower bound on QK, denoted by R r K. We now detail a formal algorithm to construct R r K. That algorithm consists in determining a representation of R r K in the form

R r K = i∈Ir [a i r , b i r ] (4.3)
where a i r and b i r are strains in CK. Setting R 0 K equal to K, the set

R r+1 K is constructed from R r K by going through each pair of line segments {[a i r , b i r ], [a j r , b j r ]} in (4.
-a j r ) -x(b i r -a i r ) + b i r -b j r ). (4.4)
That equation is a third-degree polynomial in (x, x ). For any root (x, x ) in [0, 1] × [0, 1] verifying (4.2), the strains in the line segment [e, e ] are realized by rank-(r + 1) laminates of strains in K. Taking the union of all such line segments gives the representation of R r+1 K in the form (4.3). That algorithm makes it clear that the dimension of the manifold

R r+1 K (denoted by dim R r+1 K) is bounded from above by 1 + dim R r K. It follows that dim R r K ≤ r. (4.5)
For r > 1, that algorithm may be difficult to execute by hand, but it can easily be implemented numerically. In that regard, a dramatic escalation in needed computational resources is observed as r increases. Computational time typically varies as M r where M is a constant (depending on the size of K). As a notable consequence, it proved difficult to get beyond r = 3 in the calculations. Also note that the above construction of lamination bounds is not restricted to four-well problems.

The different sets introduced so far satisfy the chain of inclusion

K = R 0 K ⊂ R 1 K ⊂ • • • ⊂ R r K ⊂ QK ⊂ P K ⊂ CK ⊂ vect(K)
where vect(K) is the vectorial space spanned by K. Since all those sets are included in vect(K), the gap between two of them can be quantified by comparing their measures in vect(K), as defined by

|S| = e∈vect(K) χ S (e)de (4.6)
where χ S is the characteristic function of the set S considered (i.e. χ S (e) is equal to 1 if e ∈ S, and null otherwise). Consider the linear mapping f defined as

f : R 3 → R 3×3 s (θ 1 , θ 2 , θ 3 ) → 3 r=1 θ r e r + (1 - 3 r=1 θ r )e 4 . (4.7)
That mapping is injective if e 1 -e 4 , e 2 -e 4 , e 3 -e 4 are linearly independent, which is assumed from now on. The mapping f in (4.7) serves two purposes. First, since it defines a one-to-one mapping between R 3 and R 3×3 s , the mapping f provides 3-dimensional representations of the convex hull K and other related bounds on QK. In particular, the convex hull K is represented by the tetrahedron

T 3 = {(θ 1 , θ 2 , θ 3 ) ∈ R 3 + | 3 r=1 θ r = 1}
and the upper bound P K is represented by the three-dimensional set f -1 (P K) given by (4.8) Second, the mapping f being affine, it allows for a simple calculation of measures in (4.6). We have indeed

f -1 (P K) = {θ ∈ T 3 |0 ≤ - 3 r,s=1
|S| = J θ∈f -1 (S ) dθ (4.9)
where J is the Jacobian of f and is equal to the mixed product [e 1 -e 4 , e 2e 4 , e 3 -e 4 ]. Since f -1 (CK) = T 3 and θ∈T 3 dθ = 1/6, we obtain

|S| |CK| = 6 θ∈f -1 (S )
dθ.

(4.10)

The ratio |S|/|CK| is thus directly obtained from the volume of the threedimensional set f -1 (S). Note that |CK| and J are strictly positive if e 1e 4 ,e 2 -e 4 ,e 3 -e 4 are linearly independent. However, even in that case, it is generally not ensured that |QK| > 0.

Examples from the monoclinic-I transformation

The bounds detailed previously are now illustrated on some four-well problems related to the cubic to monoclinic-I transformation. There are 12 martensitic variants in that transformation, each variant being compatible with seven of the others (see Tables 1 and2). In order to have a first insight in the structure of QK I , we consider only four of the twelve transformations strains in Table 1. There are obviously a large number of four-well problems that can be constructed that way. The structure of the corresponding bounds is strongly dependent on the number of pairwise transformation strains in the four-well restriction that is considered. Rather than carrying out an exhaustive study of all the possibilities, we present a selection of examples which proves to be illustrative for our purpose, notably for the subsequent study of the twelve-well problem.

The first case we consider is K I 4 = {e I 1 , e I 2 , e I 6 , e I 11 }. The three-dimensional set f -1 (P K I 4 ) is represented in Figure 1 for Ti-49.75Ni. The values of the

    α δ - δ α - - - β     α -δ - -δ α - β     α -δ -δ α - - β   e I 5 e I 6 e I 7 e I 8   α δ β δ α     α - δ - β - δ - α     α - -δ - β -δ α     α -δ β - -δ - α   e I 9 e I 10 e I 11 e I 12   β α δ δ α     β - - - α δ - δ α     β - - α -δ -δ α     β - α -δ - -δ α  
Table 1: Transformation strains in the cubic to monoclinic-I transformation.

Variant 1 2 3 4 5 6 7 8 9 10 11 12 . 1 1 1 1 . 1 . 1 .

1 . 1 . 1 1 . 1 . 1 . 1 . 1 1 1 . 1 1 . 1 . . 1 . 1 1 1 1 . . 1 . 1 1 . 1 . 1 . 1 . . 1 1 1 1 . . 1 . 1 . 1 1 . 1 1 . 1 1 . 1 . 1 . 1 1 . 1 . 1 1 . . 1 . 1 1 1 1 . 1 . . 1 1 . . 1 1 . . 1 . 1 1 1 10 . 1 1 . . 1 1 . 1 . 1 1 11 1 . . 1 . 1 1 . 1 1 . 1 12
. 1 1 . 1 . . 1 1 1 1 . two types of cubic to monoclinic transformations. [START_REF] Bhattacharya | Energy minimization and the recoverable strains in polycrystalline shape memory alloys[END_REF] proved that the set QK I is not convex, and conjectured that a similar property holds for monoclinic-II martensite. Such a property does not follow directly from the fact that the set P K II of a given tensor ē is generally not unique. However, consider the intersection of the convex hull CK II with the subspace H defined by the equations ē12 = ē13 = δ/2, assuming that δ > 0 (the variants in Table 3 can always be numbered in such a way that this condition is satisfied). For any ē in CK II ∩ studied using convex bounds. Since QK is not convex, such bounds can only be strict. [START_REF] Shu | The influence of texture on the shapememory effects in polycrystals[END_REF] notably used the convex hull of CK to bound QK from above. A convex lower bound has been proposed in closed form by [START_REF] Bhattacharya | Energy minimization and the recoverable strains in polycrystalline shape memory alloys[END_REF]. Noting that e 2i-1 and e 2i are compatible for each i = 1, • • • , 6 (see Tables 234), [START_REF] Bhattacharya | Energy minimization and the recoverable strains in polycrystalline shape memory alloys[END_REF] observed that QK contains the strains e i = (e 2i-1 +e 2i )/2. The strains e 1 , • • • , e 6 can be verified to be pairwise compatible (they actually correspond to the six transformation strains of the cubic to orthorombic transformation). As a consequence, QK contains the convex hull S of e 1 , • • • , e 6 . As proved by [START_REF] Bhattacharya | Energy minimization and the recoverable strains in polycrystalline shape memory alloys[END_REF], the set S is formed by the tensors ē satisfying the conditions:

  α + δ 0 δ α - 0 0 0 β     α - δ 0 δ α + 0 0 0 β     α + -δ 0 -δ α - 0 0 0 β     α - -δ 0 -δ α + 0 0 0 β   e II 5 e II 6 e II 7 e II 8   α + 0 δ 0 β 0 δ 0 α -     α - 0 δ 0 β 0 δ 0 α +     α + 0 -δ 0 β 0 -δ 0 α -     α - 0 -δ 0 β 0 -δ 0 α +   e II 9 e II 10 e II 11 e II 12   β 0 0 0 α - δ 0 δ α +     β 0 0 0 α + δ 0 δ α -     β 0 0 0 α - -δ 0 -δ α +     β 0 0 0 α + -δ 0 -δ α -  
tr ē = 2α + β; min(α, β) ≤ ēii ≤ max(α, β) for i=1, 2, 3; |ē jk | ≤ ēii -α β -α
δ for all {i, j, k} permutation of {1, 2, 3}.

(5.

3)

It can be proved that S has a non-empty interior in the 5-dimensional space {ē|trē = 2α + β} [START_REF] Bhattacharya | Energy minimization and the recoverable strains in polycrystalline shape memory alloys[END_REF]. Consequently, QK is of dimension 5, contrary to the four variant cases considered in Section 4 for which the sets QK I 4 and QK II 4 are manifolds of dimension 3. A schematic representation of the convex bounds S and CK is given in Figure 6. In that Figure, the 12 transformation strains are represented as cocyclic points. Compatible transformation strains are connected by a line segment (for simplicity, each transformation strain is represented as being compatible with only two of the others). The lower bound S (shown in green) is constructed as the convex hull of six particular points (which are middle points of line segments connecting compatible transformation strains). The upper bound CK is the union of the red and green domains.

The gap between the convex bounds S and CK can be quantified by comparing the measures of those two sets. Those measures are defined as in (4.6), except that vect K is now the 5-dimensional space {ē| tr ē = 2α + β}. As mentioned above, the set S has a non-empty interior in vect K and therefore has a non-zero measure. The exact value of |S| can be calculated from the analytical expression (5.3) of S. In lack of a closed-form expression of CK, its measure is obtained numerically using the algorithm of [START_REF] Barber | The quickhull algorithm for convex hulls[END_REF] for calculating convex hulls in vectorial spaces of arbitrary dimension. The calculations of |S| and |CK| have been performed for several materials, using measurements of lattice parameters from the literature [START_REF] Knowles | Crystallography of the martensitic transformation in equiatomic nickel-titanium[END_REF][START_REF] Seo | Tem investigation of the microstructure and defects of CuZr martensite[END_REF][START_REF] Saburi | Configurations of martensite variants in Cu-Zn-Ga[END_REF][START_REF] Tadaki | Thermoelastic nature and crystal structure of the Cu-Zn martensite related to the shape memory alloy[END_REF][START_REF] Chakravorty | Electron microscopy of internally faulted CuZnAl martensite[END_REF]. Corresponding values of the ratio |S|/|CK| are reported in the first column of Table 6. As can be observed in that table, the ratio |S|/|CK| is relatively low (< 0.18) in all the examples considered, especially in the case of monoclinic-I martensite for which it does not exceed 0.07. This is an indication that the gap between the two bounds S and CK is relatively large. In the following, we aim at refining our knowledge of QK by deriving tighter bounds.

Non-convex bounds

In this section, we investigate the structure of QK by deriving non-convex upper bounds and lower bounds similar in nature to those considered previously for the four-well problem. The consideration of the 12 variants, however, add some complication in the derivation of those bounds. Concerning the upper bound P K, the main difficulty is that the decomposition of a strain ē in the form ē = 12 r=1 θ r e r (5.4) is not unique. More precisely, consider the mapping g defined as

g : R 12 → R 3×3 s (θ 1 , • • • , θ 12 ) → 12 r=1
θ r e r .

(5.5)

Except for special values of the lattice parameters, Ker g is a vectorial space of dimension 7. Denoting by g the (injective) restriction of g on (Ker g) ⊥ , the set of solutions to (5.4) is the affine space defined as g -1 (ē) = g-1 (ē) + Ker g.

(5.6)

Let A be the bounded subset of R 12 defined as

A = {θ ∈ T 12 | - r,s θ r θ s (e r -e s ) * ≥ 0}.
From (2.10), the distinctive property of strains ē in P K is

g -1 (ē) ∩ A = ∅. (5.7)
Assuming that ē is given and using a numerical approach, it is relatively easy to determine if (5.7) is satisfied or not. Difficulties in deriving meaningful non-convex lower bounds are more substantial. It has already be mentioned that QK has a non-empty interior, i.e. that |QK| > 0. The relation (4.5) shows that it is necessary to take at least r = 5 for the corresponding bound R r K to have a non-zero measure. However, for twelve-well problems, it proves difficult to get beyond r = 2 in the numerical calculations. Note that rank-2 laminates have been considered by [START_REF] Govindjee | An upper bound to the free energy of mixing by twin-compatible lamination for n-variant martensitic phase transformations[END_REF] for bounding the effective energy in cubic to monoclinic transformations.

An alternative approach which proves to be fruitful is to consider simple laminates involving compatible strains in S and R r K, i.e. to consider the set S r K defined as

S r K = e∈R r K,e ∈S
det(e-e )=0

[e, e ].

(5.8)

That set is a lower bound on QK and contains S for r ≥ 1. The motivation for introducing the set S r K in (5.8) is that, contrary to R r K, it has a nonzero measure for any r ≥ 1. In particular, we can use values of r that are low enough for the computations to remain tractable (typically r ≤ 2). The expectation -which remains to be confirmed at that point -is that S r K is strictly larger than R r K ∪ S.

The following property, illustrated in Figure 6 (middle), is easily proved from (5.8):

A given strain ē is in S r K if there exists a strain e ∈ R r K such that : (i) e is compatible with ē, (ii)the half line {e + y(ē -e)|y ≥ 0} intersects S.

(5.9) That characterization of S r K is well suited to numerical implementation. Assume indeed that the decomposition (4.3) of R r K is known (using for instance the algorithm detailed in Section 4). To test if a given strain ē is in S r K, we can go through each line segment [a r i , b r i ] in (4.3) and look for strains e = xa r i +(1-x)b r i that are compatible with ē. Corresponding values of x ∈ [0, 1] are roots of the 3rd-degree polynom det(xa r i + (1 -x)b r i -ē). For any such e, we can use the relations (5.3) to determine if S has a non-empty intersection with the half-line {e + y(ē -e)|y ≥ 0}. If this is the case, then the strain ē considered is in S r K.

Examples

In order to quantify the improvement brought upon by the bounds S r K and P K, we first calculate their measures and compare the results to the measures of S and CK. A Monte-Carlo method is used to evaluate the integrals (4.6) that define |S r K| and |P K|. In those integrals, the functions χ P K (ē) and χ SrK (ē) are calculated using the characterizations (5.7) and (5.9). Table 6 shows the results obtained for several materials, both for monoclinic-I and monoclinic-II martensite.

A first observation is that in all cases, the lower bounds S r K dramatically improve on the convex lower bound S. The upper bound P K significantly improves on CK for monoclinic-II martensite, but remains close to CK for monoclinic-I martensite (in a similar way to the four-well example K I 4 of Section 4.1). As a corollary, the gap between the best available bounds (namely S 2 K and P K) is greatly reduced compared to the gap between the convex bounds (S and CK), yielding a much improved estimate of QK.

Comparing the results for monoclinic-I martensite with those for monoclinic- II martensite, we observe that

|P K I | |CK I | < |P K II | |CK II | ,
i.e. the set P K I is thus closer to the convex hull CK I than the set P K II is to the set CK II . The best available lower bound S 2 K satisfies the same property, that is to say

|S 2 K I | |CK I | < |S 2 K II | |CK II | .
That situation is reminiscent of the four-well examples presented in Section 4, for which similar relations were observed. This might be an indication that the set QK I is closer to its corresponding convex hull than the set QK II .

Although calculating measures is instructive in getting a global picture, it does not fully characterize the various bounding sets introduced, nor the estimate of QK that results from those bounds: the gap between S 2 K and P K is typically expected to be larger in certain directions than in others. To better illustrate that point, we consider effective strains ē(ω, τ ) of the form

ē(ω, τ ) = 1 3 (2α + β)I + τ (u(ω) ⊗ v(ω) + v(ω) ⊗ u(ω)), (5.10) with u(ω) = cos(ω)u 1 + sin(ω)u 2 , v(ω) = -sin(ω)u 1 + cos(ω)u 2 .
In those expressions, (u 1 , u 2 , u 3 ) is an orthonormal basis of the cubic austenitic lattice. For latter reference, note that the matrix representation of ē(ω, τ )

in the basis (u 1 , u 2 , u 3 ) is ē(ω, τ ) = 2α + β 3 I + τ  
-sin 2ω cos 2ω 0 cos 2ω sin 2ω 0 0 0 0   .

(5.11)

The strain ē(ω, τ ) can be achieved by cooling down a stress-free sample below the transformation temperature and subsequently applying a simple shear of amplitude 2τ (between the directions u(ω) and v(ω)) in the cooled state. Cooling down a stree-free sample indeed produces the so-called selfaccommodated state in which there is no austenite and all the martensitic variants appear in equal volume fraction, resulting in a macroscopic strain (2α + β)/3I. We are interested in bounding the values of (ω, τ ) for which ē(ω, τ ) ∈ QK. As monoclinic martensite is invariant in the u 1 → -u 1 symmetry, the conditions ē(ω, τ ) ∈ QK and ē(ω, -τ ) ∈ QK are equivalent.

It is therefore sufficient to consider only positive values of τ . Let

Sτ (ω) = sup{τ |ē(ω, τ ) ∈ S}.
Since S is convex and closed, we have ē(ω, τ ) ∈ S for any positive τ such that τ ≤ Sτ (ω). It can be calculated from (5.3) that

Sτ (ω) = min( A | sin 2ω| , δ 3| cos 2ω| ) (5.12) where A = min( 2α + β 3 -min(α, β), - 2α + β 3 + max(α, β)).
In a similar fashion, we define

Cτ (ω) = sup{τ |ē(ω, τ ) ∈ CK} , Qτ (ω) = sup{τ |ē(ω, τ ) ∈ QK}, P τ (ω) = sup{τ |ē(ω, τ ) ∈ P K} , S r τ (ω) = sup{τ |ē(ω, τ ) ∈ S r K}. The relations S ⊂ S 1 K ⊂ S 2 K ⊂ QK ⊂ P K ⊂ CK imply that Sτ ≤ S 1 τ ≤ S 2 τ ≤ Qτ ≤ P τ ≤ Cτ.
(5.13)

Since CK is convex and closed, we have ē(ω, τ ) ∈ CK for any 0 ≤ τ ≤ Cτ (ω). Even though P K and S r K are not convex, numerical computations show that they satisfy a similar property, i.e. ē(ω, τ ) ∈ P K for any 0 ≤ τ ≤ P τ (ω), and ē(ω, τ ) ∈ S r K for any 0 ≤ τ ≤ S r τ (ω). The functions Sτ , S r τ , P τ and Cτ thus completely characterize the intersection of the various bounding sets with strains of the form (5.10). Those functions are represented in Figure 7 for Ni-49.75Ti (monoclinic-I martensite) and in Figure 8 for β 1 Cu-14Al-4Ni (monoclinic-II martensite). It is obvious from the definition (5.10) that all those functions are of period π. As can be observed in Figures 78, they are also invariant in the ω → π/2 -ω and ω → π -ω symmetries. This can be verified to stem from the invariance of monoclinic martensite in the (u 1 , u 2 ) → (u 2 , u 1 ) and u 2 → -u 2 transformations, respectively. Let us now study the results of Figures 7-8 in more detail. Because of the mentioned symmetries, we limit our attention to ω ∈ [0, π/4]. Concerning the monoclinic-I example (Figure 7), it can be proved that Sτ coincides with Cτ on an interval of the form [ω 1 , π/4]. The value of ω 1 is given by tan 2ω 1 = α -β δ .

Indeed, for ω 1 ≤ ω ≤ π/4, it can easily be calculated that A/| sin 2ω| reaches the minimum in (5.12). Use of (5.11) gives ē22 (ω, Sτ (ω)) = max(α, β).

Note that any ē ∈ CK I satisfies max i=1,2 ēii ≤ max r e I r,12 = max(α, β).

Consider now a given τ such that τ > Sτ (ω). The expression (5.11) shows that ē22 (ω, τ ) > ē22 (ω, Sτ (ω)) = max(α, β). Therefore we have ē(ω, τ ) / ∈ CK I and consequently τ > Cτ (ω). By letting τ tend towards Sτ (ω) from above, we obtain that Sτ (ω) ≥ Cτ (ω). Comparing with (5.13) we can conclude that Sτ (ω) = Cτ (ω). For ω 1 ≤ ω ≤ π/4, the bounds Sτ and Cτ are thus optimal and give the exact value of Qτ (ω).

For ω < ω 1 , the numerical results displayed in Figure 7 show that S 1 τ and S 2 τ significantly improve on Sτ . In particular, there is an interval [ω * , ω 1 ] on which the bounds S 2 τ and Cτ are found to coincide (up to the accuracy of the numerical calculations). Let us prove in particular that

Cτ (ω * ) = S 2 τ (ω * ) = τ * (5.14)
where tan 2ω * = (α -β)/(δ + 2 )

and τ * = 1 3 (α -β) 2 + (δ + 2 ) 2 .
That value τ * actually corresponds to the maximum of S 2 τ and Cτ (see Figure 7). To prove (5.14) we first note that

ē(ω * , τ * ) =   α+2β 3 δ+2 3 0 δ+2 3 α 0 0 0 2α+β 3   .
Observing that ē22 (ω * , τ * ) = α and that α > β (in the case of Ni-49.75Ti), we can show that τ * ≥ Cτ (ω * ) by a similar reasoning as used previously. We now prove that ē(ω * , τ * ) is realized by a rank-2 laminate. To that purpose, observe that ē(ω * , τ * ) can be written as In the case of β 1 Cu-14Al-4Ni, the functions P τ and Cτ are very close and therefore give a good approximation of the function Qτ characterizing the quasiconvex hull. The results for Ni-49.75Ti do not give the same level of closeness between P τ and Cτ , but a significant improvement is still observed compared to the convex lower bound Sτ .

In Part 1 of this paper [START_REF] Peigney | On the energy-minimizing strains in martensitic microstructures -Part 1: geometrically non-linear theory[END_REF], bounds on the energy-minimizing strains for the cubic to tetragonal transformation have been derived in the geometrically non-linear setting. Those bounds have been illustrated by considering effective strains which are the analogue of (5.10) in the geometrically non-linear theory, i.e. a simple shear (of amplitude τ and orientation ω) applied in the self-accommodated state. The results obtained are qualitatively similar to those displayed in Figure 8: the transformation considered is recoverable up to a maximum shear amplitude τ (ω) that depends on ω, the function ω → τ (ω) exhibiting localized peaks at specific values of ω. Such similarities should not obscure the fact that the arguments at play are of a different nature. In the geometrically linear example considered in Figure 8, the results that are obtained partly come from the fact that, because we are considering 12 transformation strains, the material is given a large number of degrees of freedom to accommodate a given strain ē. In contrast, if the cubic to tetragonal transformation is studied in the geometrically linear setting, it can easily be verified that the strain ē(ω, τ ) is recoverable only for τ = 0. In the geometrically non-linear theory, local rotations act as an additional degree of freedom for the material to accommodate strains. This is the main reason why, even for the cubic to tetragonal transformation, non-zero shear τ (ω) is predicted to be recoverable in that theory.

Concluding remarks

In difference with the geometrically non-linear theory, the minimizers of the microscopic energy in the geometrically linear theory form a discrete set. This contributes to major simplifications in the analysis. For instance, the upper bound (3.12) for the three-well problem is obtained with relatively simple calculations, whereas considerable effort is needed for the analog bound in the geometrically non-linear theory [START_REF] Peigney | On the energy-minimizing strains in martensitic microstructures -Part 1: geometrically non-linear theory[END_REF].

Four-and twelve-well problems have been studied extensively in this paper. There is a connection between the two: for cubic to monoclinic transformations, sufficient conditions on the effective strain ē can be found for the microstructures to be restricted to only four variants (see Appendix B). In particular, the four-well example {e 1 , e 2 , e 5 , e 6 } considered in Section 4 can be interpreted as the trace of QK on a particular hyperplane of {ē| tr ē = 2α + β}. As notably illustrated on the four-well examples of Section 4.2, the convex upper bound may significantly overestimate the set QK. It is interesting to observe, however, that it gives a relatively good approximation of QK in cubic to monoclinic transformations (especially for monoclinic-I martensite), performing better than the convex lower bound S in that regard.

The bounds presented could be improved in several ways. For instance, the lower bound S 2 K considered in Section 5 is not stable by lamination. Therefore, the set formed by simple laminates of compatible strains in S 2 K is expected to give a tighter lower bound. Note that the characterization (5.9) of S 2 K is simple enough for such a calculation to remain tractable. It would also be interesting to compare the results presented with the lower bound recently proposed by [START_REF] Chenchiah | Non-laminate microstructures in monoclinic-I martensite[END_REF] for monoclinic-

θ

  r h(e r ) for any quasiconvex function h. (2.8) Recall that a function h is quasiconvex in the linearized strain e if |Ω|h(ē) ≤ Ω h(e) dx for all ē and e ∈ A(ē). (2.9) Let e * denote the adjugate of e, defined by the relations e * ii = e jj e kk -e 2 jk , e * jk = e ji e ki -e jk e ii , for any {i, j, k} permutation of {1, 2, 3}. We use the notation M ≥ 0 to indicate that a second-order symmetric tensor M is positive, i.e. satisfies u.M .u ≥ 0 for all vectors u. It can be verified that the function e → -a : e * is quasiconvex provided that a ≥ 0 (see e.g. Peigney (2008)). Using such functions in (2.8) gives 0 ≤ a : (ē *be rewritten in a more convenient form. Using the equality r θ r = 1 and the fact that the function e → e * is quadratic, we have ē *r θ r e * r = ( r θ r e r ) * -θ r θ s (e r -e s ) * .

  r : M (a) : e r -θ r θ s e r : M (a) : e s .

(

  tr(e -e )) 2 -(e -e ) : (e -e ) ≤ 0. (4.2)

  3) and looking for strains e ∈ [a i r , b i r ] and e ∈ [a j r , b j r ] that are compatible. Writing e and e as e = xa i r + (1 -x)b i r , e = x a j r + (1 -x )b j r , we need to solve the equation 0 = det(e -e ) = det(x (b j r

θ

  r θ s (e r -e s ) * -2e 4 -e s ) * }.

4

  represented in Figure 5 is non-convex : the family (e II 1 , • • • , e II 12

Figure 6 :

 6 Figure 6: Schematic representations of the convex bounds (left) and of the lower bound S 1 K (right) for monoclinic martensite. The line segments form the set R 1 K. The lower bound S 1 K is constructed by looking for strains ē that are realized by simple lamination of a strain e in S and a strain e on a line segment in R 1 K, such as [e 2 , e 3 ] (middle).

Figure 7 :

 7 Figure 7: Bounds on (ω, τ ) such that ē(ω, τ ) ∈ QK I .

Figure 8 :

 8 Figure 8: Bounds on (ω, τ ) such that ē(ω, τ ) ∈ QK II .

Table 2 :

 2 Compatible variants (indicated by '1') in monoclinic-I martensite.

	e II 1	e II 2	e II 3	e II 4

Table 3 :

 3 Transformation strains in the cubic to monoclinic-II transformation.

	Variant 1 2 3 4 5 6 7 8 9 10 11 12
	1	. 1 1 1 1 . 1 . 1 .	1	.
	2	1 . 1 1 . 1 . 1 . 1	.	1
	3	1 1 . 1 1 . 1 . 1 .	1	.
	4	1 1 1 . . 1 . 1 . 1	.	1
	5	1 . 1 . . 1 1 1 . 1	.	1
	6	. 1 . 1 1 . 1 1 1 .	1	.
	7	1 . 1 . 1 1 . 1 . 1	.	1
	8	. 1 . 1 1 1 1 . 1 .	1	.
	9	1 . 1 . . 1 . 1 . 1 1 1
	10	. 1 . 1 1 . 1 . 1 .	1 1
	11	1 . 1 . . 1 . 1 1 1	.	1
	12	. 1 . 1 1 . 1 . 1 1 1	.

Table 4 :

 4 Compatible variants (indicated by the index '1') in monoclinic-II martensite.

	Monoclinic-I
	Variants 1,2,6,11 0.60	0.79
	Variants 1,3,6,8	0.82	0.94
	Monoclinic-II
	Variants 1,2,6,11 0.57	0.65
	Variants 1,3,6,8	0	0.006

Figure 4: 3D representation of the upper bound P K II 4 with K II 4 = {e II 1 , e II 2 , e II 6 , e II 11 }. K |R 3 K|/|CK| |P K|/|CK|

Table 5 :

 5 Measures of the bounding sets (expressed as fractions of |CK|) for some 4-well problems related to monoclinic martensite.

  |S|/|CK| |S 1 K|/|CK| |S 2 K|/|CK| |P K|/|CK|

	Monoclinic-I
	Ni-49.75Ti	0.0645	0.543	0.824	0.994
	CuZr	0.0543	0.547	0.865	0.998
	Monoclinic-II
	Cu-20Zn-12Ga	0.154	0.576	0.787	0.922
	Cu-39.3Zn	0.131	0.539	0.761	0.924
	Cu-15Zn-17Al	0.176	0.592	0.756	0.917
	β 1 Cu-14Al-4Ni	0.0915	0.470	0.706	0.918

Table 6 :

 6 Measures of the bounding sets (expressed as fractions of |CK|).

lattice parameters are α = 0.0243, β = -0.0437, δ = 0.058, = 0.0427 [START_REF] Knowles | Crystallography of the martensitic transformation in equiatomic nickel-titanium[END_REF]. In Figure 1 (top), two snapshots from different viewpoints are put together for a better grasp of the three-dimensional representation of f -1 (P K I 4 ). Let us address a few comments on the shape of that set: since the pair {e I 1 , e I 6 } is not compatible (see Table 2), any θ in the line segment ]f -1 (e I 1 ), f -1 (e I 6 )[ is expected to be excluded from the set f -1 (P K I 4 ). For such a value of θ, the tensorr,s θ r θ s (e I r -e I s ) * indeed reduces to A(e I

1 -e I 6 ) * (with A = 0) and therefore is not positive. In a similar fashion, the pair {e I 2 , e I 11 } is also incompatible, and the corresponding edge of the tetrahedron T 3 is expected to be excluded from f -1 (P K I 4 ). As can be seen on Figure 1, there is actually a three-dimensional volume surrounding those two edges that is excluded from f -1 (P K I 4 ). As a result, the measure of P K I 4 is smaller than the measure of the convex hull CK

As as direct consequence of (4.5), it is necessary to take at least n = 3 for the corresponding set R n K I 4 to have a non-empty interior. The set R 3 K I 4 corresponding to rank-3 laminates is shown in Figure 1 (bottom). We find

6, which confirms that |QK I 4 | has a non-empty interior. Also observe that R 3 K I 4 is found to have a similar shape as P K I 4 . The gap between R 3 K I 4 and P K I 4 is illustrated in Figure 2, on which those two sets are superimposed. The gap between R 3 K I 4 and P K I 4 can be measured using relation (4.10), yielding

1 , e I 3 , e I 6 , e I 8 }. In that case, only two of the six pairs of strains in K I 4 are compatible (namely {e I 1 , e I 3 } and {e I 6 , e I 8 }). Nevertheless, the set P K I 4 is found to be quite close to the convex hull CK I 4 . As in the previous example, each edge connecting two incompatible strains is surrounded by a three-dimensional domain of strains that are not in P K I 4 . However, that domain is much smaller than in the previous example, resulting in a set P K I 4 that is much closer to the convex hull. More precisely, we find using (4.10) that |P K I 4 |/|CK I 4 | 0.94. Consideration of the lower bound R 3 K I 4 (not represented) leads to a similar conclusion (see Table 5).

Examples from the monoclinic-II transformation

We now examine some four-well problems related to monoclinic-II transformation. The 12 transformation strains of the cubic to monoclinic-II transformation are listed in Table 3. Each martensitic variant is compatible with seven of the others, just as for the cubic to monoclinic-I transformation (Table 4). We consider two examples of four-well problems that are the analogues of those studied previously for monoclinic-I martensite. Figure 4 4 can be interpreted in a similar way as P K I 4 , but the detailed shape of f -1 (P K II 4 ) is different from that of f -1 (P K I 4 ) and results in a smaller domain (see Table 5).

Let us now consider the quasiconvexification of 6 , e II 8 ] can be found. Comparing those results with those obtained previously for monoclinic-I martensite (see Table 5), we observe that similar relation is verified by the lower bounds, i.e

Analog inequalities are obtained between K I 4 and K II 4 :

That inequality shows that the quasiconvexhull

The relations (4.11)-(4.12) suggest that a property similar to (4.13) also holds between K I 4 and K II 4 , although this can not be proved rigorously from (4.11)-(4.12).

Twelve-well problems in monoclinic martensite

Convex bounds

In this Section we study the quasiconvexification of the sets K I = ∪ 12 r=1 {e I r } and K II = ∪ 12 r=1 {e II r }, thus considering the 12 transformation strains in the H, the decomposition (5.1) is unique and satisfies

Let us briefly justify this assertion. For a fixed ē in CK II ∩ H, there exists θ in T 12 such that ē = 12 r=1 θ r e II r . Using Table 3, we have ē12 = δ(θ 1 + θ 2θ 3 -θ 4 ) ≤ δ(θ 1 +θ 2 ). In a similar fashion, we have ē13 ≤ (θ 5 +θ 6 )δ. Therefore, the equations ē12 = ē13 = δ/2 imply that 1/2 ≤ θ 1 + θ 2 and 1/2 ≤ θ 5 + θ 6 . Since θ ∈ T 12 , these conditions can be realized only if 1/2 = θ 1 + θ 2 = θ 5 + θ 6 and θ r = 0 for r / ∈ {1, 2, 5, 6}. The decomposition (5.2) is thus obtained. The uniqueness of this decomposition comes from the fact that the tensors (e II 1 , e II 2 , e II 3 , e II 4 ) in 1 -e II 6 ) * is not positive. Since the decomposition (5.1) is unique, it can be concluded from (2.10) that (e II 1 + e II 6 )/2 is excluded from P K II , and a fortiori from QK II . This proves that QK II is not convex.

From here onwards we use the notation K to denote indifferently K I or K II . The corresponding transformations strains are denoted by e r (i.e. e r = e I r for K = K I , and e r = e II r for K = K II ). In the literature so far, the structure of QK in cubic to monoclinic transformations has been essentially Note from Table 1 that e I 1 and e I 9 (resp. e I 2 and e I 12 ) are compatible, so that the strain e (resp. e ) is in R 1 K I . It can be verified that det(e -e ) = 0, i.e that e and e are compatible. Therefore the strain ē(ω * , τ * ) is in R 2 K. As a consequence, we have τ * ≤ S 2 τ (ω * ). Combining the obtained relations Cτ (ω * ) ≤ τ * ≤ S 2 τ (ω * ) with (5.13), we can conclude that S 2 τ (ω * ) = Cτ (ω * ) = τ * .

The results for the monoclinic-II example (Figure 8) differ in several aspects. A first observation is that, contrary to the case of Ni-49.75Ti, the convex bounds Cτ and Sτ never coincide. As for monoclinic-I martensite, there exists values of ω such that max i ēii (ω, Sτ (ω)) = max(α, β). However, it can no longer be concluded that ē(ω, Sτ (ω)) / ∈ CK II for τ > Sτ (ω). The reason is that strains ē in CK II do not necessarily verify ēii ≤ max(α, β) (they only satisfy the less stringent restriction e ii ≤ max(α -, α + , β)).

A second observation is that the bound S 2 τ improves on Sτ for all value of ω. Hence, contrary to the case of monoclinic-I martensite, the convex lower bound Sτ is never optimal. Lastly, concerning upper bounds, we note that P τ (ω) is stricly lower than Cτ (ω) for some values of ω -that situation was not observed in the monoclinic-I example. The numerical calculations show that Cτ , P τ and S 2 τ all reach a maximum at the same value ω = π/4. It can be proved that

The proof is similar to the reasoning detailed previously for justifying (5.14).

The most delicate point consists in proving that ē(π/4, 2|β

That tensor can be written in the form

with x = (α -β)/6 . The lattice parameters in In the case K = K II 4 , the mapping f introduced in (4.7) specializes as

)), we have l(θ) ≥ 0 (resp. l(θ) = 0 ). It follows that f -1 l(C) is an interval of the form [0, η] with η ≥ 0. Assume that η > 0. Then for any η ≥ η > 0, there exists θ = {θ 1 , θ 3 , θ 6 , θ 8 } ∈ T 4 such that r,s=1,3,6,8 θ r θ s (e II r -e II s ) * ≤ 0 and θ 1 + θ 3 = η . Any such θ can be written in the form θ = (x, η -x, y, 1 -η -y), with 0 ≤ x ≤ η and 0 ≤ y ≤ 1 -η . Let η tend towards zero. At the first order in (x, η ), we have

For monoclinic-II martensite, it can easily be verified that u.(e II 6 -e II 8 ) * .u = 0 for any u ∈ vect(u 1 , u 3 ), and that u 3 .e.u 3 = 2 (α -β -) -δ 2 for e ∈ {e II 1 -e II 6 , e II 1 -e II 8 , e II 3 -e II 6 , e II 3 -e II 8 }. Therefore we get 1 2 r,s=1,3,6,8

The lattice parameters in β 1 Cu-14Al-4Ni are such that 2 (α -β -)-δ 2 > 0.

In that case, the relation r,s θ r θ s u 3 .(e II r -e II s ) * .u 3 ≤ 0 implies that η = 0, in contradiction with the starting assumption η > 0. If follows that η is necessarily equal to 0, i.e. that f -1 l(C) = {0}. Any e in C is such that Appendix B. Reduction to four-wells in cubic to monoclinic transformations

} be the set of transformation strains in a cubic to monoclinic transformation. We examine sufficient conditions on ē for the microstructures to be restricted on a given set of four variants, say {e 1 , e 2 , e 3 , e 4 }. Assume there exists a tensor N such that space V = {ē|trē = 2α + β}. The spaces V and W are respectively of dimension 5 and 3, so that W ⊥ is of dimension 2. Let (f 1 , f 2 ) denote an orthonormal basis of W ⊥ and define N (ω) = cos ω f 1 + sin ω f 2 for all ω.


Francesca Frontini
email: francesca.frontini@ilc.cnr.itmohamed.boukhaled@lip6.fr

Mohamed Amine Boukhaled

Mohamed Amine Boukhaled

Jean Gabriel Ganascia
email: jean-gabriel.ganascia@lip6.fr

Moliere's Raisonneurs: a quantitative study of distinctive linguistic patterns

Introduction and approach

Great authors of plays and novels are often renowned for the ability to create memorable characters that take on a life of their own and become almost as real as living persons to their readers/audience. The study of characterization, that is, of how it is that authors manage to achieve this, has become a well-researched topic in corpus stylistics: for instance [START_REF] Mahlberg | Corpus stylistics and Dickens's fiction[END_REF] attempts to identify typical lexical patterns for memorable characters in the work of Dickens by extracting those lexical bundles that stand out (namely are over-represented) in comparison to a general corpus. In other works, authorship attribution methods are applied to the different characters of a play to identify whether the author has been able to give each of them with a "distinct" voice. For instance [START_REF] Vogel | Computational Stylometry: Who's in a Play? In: Verbal and Nonverbal Features of Human-Human and Human-Machine Interaction[END_REF] compare the dialogues of individual characters in a Shakespeare play against the rest of the play or even against all plays in the Shakespearean corpus.

In [START_REF] Frontini | Linguistic Pattern Extraction and Analysis for Classic French Plays[END_REF], we propose a methodology for the extraction of significant patterns that enables literary critics to verify the degree of characterization of each character with respect to the others and to automatically induce a list of linguistic features that are significant and representative for that character. The proposed methodology relies on sequential data mining for the extraction of linguistic patterns and on correspondence analysis for the comparison of pattern frequencies in each character and the visual representation of such differences.

We chose to apply this analysis to Moliere's plays and the protagonists of those plays. In this work we focus on the figure of the raisonneurs, characters who take part in discussions with comical protagonists providing a counterpart to their follies. Such characters were interpreted at times as spokesmen for Moliere himself, and the voice of reason, at other times as comical characters themselves and no less foolish than their opponents.

Hawcroft's essay Reasoning with fools (2007) highlights the differences between five of these characters based on their role in the plot. Using this analysis as guidance, we compare significant linguistic patterns in order to see how these differences are marked by the author. We do this by adapting the discourse traits of each of them to the communicative function they needs to fulfill [START_REF] Biber | Register, genre, and style[END_REF].

Syntactic pattern extraction and ranking

In our study, we consider a syntagmatic approach based on a configuration similar to that proposed by [START_REF] Quiniou | What about sequential data mining techniques to identify linguistic patterns for stylistics?[END_REF]. The text is first segmented into a set of sentences, and then each sentence is mapped into a sequence of part of speech (PoS) tags. Tagging is automatically performed using TreeTagger [START_REF] Schmid | Treetagger| a language independent part-of-speech tagger[END_REF] 1 . For example the sentence J'aime ma maison où j'ai grandi.

is first mapped to a sequence of PoSTagged words; 2), (3)). A minimal filtering is applied, removing patterns with less than 5% of support; nevertheless sequential pattern mining is known to produce (depending on the window and gap size) a large quantity of patterns even on relatively small samples of texts.

In order to identify the most relevant patterns for each of the four characters we used correspondence analysis (CA), which is a multivariate statistical technique developed by [START_REF] Benzécri | Histoire et préhistoire de l'analyse des données. Partie V: l'analyse des correspondances[END_REF] and used for the analysis of all sorts of data, including textual data [START_REF] Lebart | Exploring textual data[END_REF]. CA allows us to represent both Moliere's characters and the (syntactic) patterns on a bi-dimensional space, thus making it visually clear not only which characters are more similar to each other but also which patterns are over/under-represented -that is, more distinctivefor each character or group of characters.

Moreover, patterns can be ranked according to their combined contribution on both axes, and those with the highest contribution can be retained, thus enabling the researcher to filter out less interesting 1 For a description of the French tagset see here:

Analysis and results

CA was performed with the R module FactoMiner [START_REF] Husson | FactoMineR: Multivariate Exploratory Data Analysis and Data Mining with R, R package version 1[END_REF] Table1: Characters and plays.

Figure 1 shows the result of the correspondence analysis, with the five raisonneurs printed as blue dots, the patterns printed as red triangles, and the 10 patterns with the highest contribution labeled with their identifiers. Filtering by contribution is crucial in our technique, which extracts over 9500 patterns, most of which are common to all characters (see central cloud in the plot) and thus not so interesting for our study.

The relative distances between the characters seem to match what is already known from literary criticism; first of all Béralde, who is the only character to express himself in prose, is isolated on the right of the x axis. In fact, it is not advisable to compare characters without distinguishing for prose and verse, but we have retained the example of Béralde to show how the proposed technique can easily identify differences in genre. As for the other characters, Hawcroft stresses the difference in the roles of Ariste, Philinte and Chysalde on the one hand and of Cléante on the other. The latter is a more pro-active character, more crucial to the plot; he is also less accommodating than the other three, who are depicted mostly as loyal friends and brothers, trying to help the hero to avoid the consequences of his foolish actions and beliefs. Instead, Cléante has also to worry about his sister's wellbeing: having to face not only the besotted brother in law, Orgon, but also the man who has duped him, Tartuffe.

In order to confirm this intuition, it is necessary to turn our attention to what it is that exactly causes the spatial distribution, namely the high contribution patterns, we find above. Our technique allows us not only to find the corresponding pattern for each identifier on the plot, but also to extract all underlying instances in the texts. Due to space constraints, only a brief demonstrative analysis will be performed.

Phylinte and Chrysalde are strongly associated with patterns containing prepositional phrases separated by commas. Such patterns are used in contexts where the characters give advice in a very cautious, indirect way. The overuse of punctuation itself, in these two characters, seems to be an indication that the character should be played as a Figure 1Correspondence analysis plot, with first 10 patterns for contribution. soft spoken person, who is fond of his friend and careful not to offend, e.g.:

(1) Pattern 1011 On the other hand, the patterns most associated with Cléante contain modal constructions, and are indicative of a more direct way of advising, and of stronger arguments, e.g.

(2) Pattern 1360 • Les bons et vrais dévots , qu' on doit suivre à la trace , Ne sont pas ceux aussi qui font tant de grimace .

• Et s' il vous faut tomber dans une extrémité , Péchez plutôt encore de cet autre côté .

Finally, the patterns extracted for Béralde are indicative of the greater simplicity and repetitiveness of his prose, and of the stereotypical role he has in the play, which is that of a man concerned with his brother, as in: • Oui , mon frère , puisqu' il faut parler à coeur ouvert ,

From this experiment it is therefore possible to conclude that the method described above is a promising one, as it not only verifies known facts about the characters in question, but also ground them on corpus based evidence.

Preliminary conclusions

Clustering techniques are commonly used in computer aided literary criticism. In order to prove that clusters are significant, statistical analysis can be later applied to verify that resulting clusters are significant. The strength of CA lies in the fact that it allows users to easily identify the reasons for certain texts to group together or to diverge. This helps to overcome the lack of transparency in the presentation of results which often disappoints experts when experimenting with similar techniques, thus making it a useful hermeneutical tool, in the sense of Ramsey (2011)'s algorithmic criticism.

  of 3 to 5 elements are extracted. Patterns can be made of PoS Tags only, or of a mix of PoS Tags and recurring lexical elements, with possible gaps (see examples (1), (

  http://www.cis.unimuenchen.de/~schmid/tools/TreeTagger/ patterns.

[

  PRO:PER ] [any word] [VER:infi] [PRP]

( 3 )

 3 Pattern 865 [,][DET:POS][any word][PUN]

  on five characters from five different plays (see Table1):

	Play	Raisonneur	Counterpart
	Ecole des femmes	Chrysalde	Arnolphe
	Ecole des maris	Ariste	Sganarelle
	Tartuffe	Cléante	Orgon
	Mysantrope	Phylinte	Alceste
	Malade imaginaire	Béralde	Argan

Acknowledgements

This work was supported by French state funds managed by the ANR within the Investissements d'Avenir programme under reference ANR-11-IDEX-0004-02, as well as by a scholarship from the Fondation Maison Sciences de l'Homme, Paris.