Michaël Peigney
email: michael.peigney@polytechnique.org

Improved bounds on the energy-minimizing strains in martensitic polycrystals

Keywords: Shape memory alloys, Polycrystal, Energy minimization, Finite strains PACS 46.25.g, 62.20.fg, 81.30.fk

This paper is concerned with the theoretical prediction of the energy-minimizing (or recoverable) strains in martensitic polycrystals, considering a nonlinear elasticity model of phase transformation at finite strains. The main results are some rigorous upper bounds on the set of energy-minimizing strains. Those bounds depend on the polycrystalline texture through the volume fractions of the different orientations. The simplest form of the bounds presented is obtained by combining recent results for single crystals with a homogenization approach proposed previously for martensitic polycrystals. However, the polycrystalline bound delivered by that procedure may fail to recover the monocrystalline bound in the homogeneous limit, as is demonstrated in this paper by considering an example related to tetragonal martensite. This motivates the development of a more detailed analysis, leading to improved polycrystalline bounds that are notably consistent with results for single crystals in the homogeneous limit. 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.

Introduction

The peculiar properties of shape memory alloys (SMAs) -such as the shape memory effect or the superelastic behavior -are the macroscopic result of a diffusionless solid/solid phase transformation between different crystallographic structures, known as austenite and martensite [START_REF] Otsuka | Shape memory materials[END_REF][START_REF] Bhattacharya | Microstructure of Martensite[END_REF]. Since the austenite has a crystallographic structure with higher symmetry than the martensite, several martensitic variants need to be distinguished depending on the orientation of the martensitic lattice with respect to the austenitic lattice. Each martensitic variant is characterized by a transformation strain that is defined as the deformation gradient between the austenitic and martensitic lattices. The number of martensitic variants and the corresponding transformation strains depend on the alloy considered. MnCu and MnNi are examples of alloys undergoing a cubic to tetragonal transformation (i.e. the austenite has a cubic lattice and the martensite has a tetragonal lattice). In that case, there are 3 martensitic variants. Widely used alloys such as NiTi or γCuAlNi obey a cubic-to-monoclinic transformation, in which case there are 12 martensitic variants to be considered.

The phase transformation between austenite and martensite can be triggered both by mechanical and thermal loading. The most striking illustration of that phenomenon is the shape memory effect displayed by alloys such as NiTi or CuAlNi: cooling down a stress-free sample below a critical temperature T 0 transforms the homogeneous austenite (stable at high temperature) to a martensitic microstructure, in which the martensitic variants arrange themselves so as to produce a stress-free state with no macroscopic deformation. Once in the cooled state, deforming the sample mechanically entails a phase transformation of some martensitic variants to others. After unloading, a macroscopic residual stress-free strain is observed as a result from the cooperative effect of the microscopic transformation strains in each variant. Heating the sample transforms the martensite back in austenite, thus restoring the initial configuration. We note that, in a polycrystalline sample (which is the common form of commercially available SMAs), the formation of austenite/martensite microstructures is typically observed in each grain. This leads one to distinguish between three length scales: the microscopic scale of the austenite/martensite microstructures, the mesoscopic scale of the constitutive grains, and the macroscopic scale of the polycrystal consisting in numerous grains. The macroscopic scale can be interpreted as the length scale of a representative volume element (RVE) of the polycrystal. Expressing the macroscopic free energy in terms of the mesoscopic free energies in the constitutive grains is a long standing topic in the study of polycrystals and granular materials (see e.g. [START_REF] Misra | Thermomechanics-based nonlinear rate-dependent coupled damage-plasticity granular micromechanics model[END_REF] for a recent work in the field of granular materials).

A possible route to study the formation of microstructures in SMAs is to adopt a nonlinear elasticity model of phase transformation [START_REF] Ball | Proposed experimental tests of a theory of fine microstructure and the two-well problem[END_REF]. The general principle is that, under a prescribed loading, the system tends to minimize its free energy. This approach amounts to consider that the evolutions are thermodynamically reversible and to look for stable equilibria. 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 energy Ψ is obtained as the relaxation (or quasiconvexification) of Ψ , which essentially amounts to solve an optimal design problem with respect to the martensite/austenite geometric arrangement (see Section 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 energy Ψ obtained by homogenization of the constitutive free energy functions.

Assuming the microscopic free energy Ψ to be known, determining its relaxation Ψ largely remains an open problem. Estimating the macroscopic free energy Ψ 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 energy-minimizing 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 [START_REF] Bhattacharya | Energy minimization and the recoverable strains in polycrystalline shape memory alloys[END_REF]. 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 along prescribed directions (see e.g. [START_REF] Zhao | Texture development and transformation strain of a cold-rolled Ti50-Ni45-Cu5 alloy[END_REF]). 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).

The problem can be formulated either in the geometrically nonlinear setting or in the geometrically linear setting. The latter assumes small deformations with respect to a reference configuration. Although that assumption can be regarded as questionable for shape memory alloys (notably because of possibly large rotation effect), it allows the analysis to be simplified significantly. In the geometrically linear setting, the exact expression of the relaxation of a (piecewise quadratic) double-well energy can been obtained [START_REF] Kohn | Relaxation of a double-well energy[END_REF][START_REF] Pipkin | Elastic materials with two preferred states[END_REF]. Lower and upper bounds have been proposed for the relaxation of a three-or more-well energy [START_REF] Smyshlyaev | On the relaxation of a three-well energy[END_REF][START_REF] Govindjee | The free energy of mixing for n-variant martensitic phase transformations using quasi-convex analysis-variant martensitic phase transformations using quasi-convex analysis[END_REF][START_REF] Govindjee | An upper bound to the free energy of mixing by twin-compatible lamination for n-variant martensitic phase transformations[END_REF][START_REF] Peigney | A non-convex lower bound on the effective free energy of polycrystalline shape memory alloys[END_REF]. The structure of the set of mesoscopic energy-minimizing strains has been studied in detail, notably for the most challenging case of monoclinic martensite [START_REF] Bhattacharya | Energy minimization and the recoverable strains in polycrystalline shape memory alloys[END_REF][START_REF] Chenchiah | Non-laminate microstructures in monoclinic-i martensite[END_REF][START_REF] Peigney | On the energy-minimizing strains in martensitic microstructures-part 2: Geometrically linear theory[END_REF]. Concerning polycrystals, bounds are available for the macroscopic energy-minimizing strains as well as for the macroscopic free energy [START_REF] Hackl | An upper bound to the free energy of n-variant polycrystalline shape memory alloys[END_REF][START_REF] Peigney | A non-convex lower bound on the effective free energy of polycrystalline shape memory alloys[END_REF][START_REF] Bhattacharya | Energy minimization and the recoverable strains in polycrystalline shape memory alloys[END_REF][START_REF] Shu | The influence of texture on the shape-memory effects in polycrystals[END_REF].

In this paper, we focus on upper bounds of the macroscopic energy-minimizing strains of martensitic polycrystals, in the geometrically non-linear setting. We begin in Section 2 by describing the nonlinear elasticity model of phase transformation and presenting the upper bound of [START_REF] Peigney | On the energy-minimizing strains in martensitic microstructures-part 1: Geometrically nonlinear theory[END_REF] for single crystals. In Section 3 we combine the monocrystalline bound with the methodology of [START_REF] Peigney | Recoverable strains in composite shape-memory alloys[END_REF] to derive explicit upper bounds for polycrystals. It turns out, however, that the upper bounds that are obtained may fail to recover the single crystal bound in the homogeneous limit, as discussed in Section 4. Motivated by that observation, we modify the polycrystalline approach so as to take the special structure of the single crystal bound into account (Section 5). This results in new upper bounds for polycrystals, which improve on the bounds of Section 3 and are consistent with the single crystal bound in the homogeneous limit. In Section 6, we study a two-orientation/three-well polycrystal and compare the proposed upper bounds with lamination lower bounds.

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. 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

K = {F|Ψ (F) = 0}.
The principle of frame indifference implies that Ψ (R.F) = Ψ (F) for any rotation R and deformation gradient F. Moreover, in the case of shape memory alloys, Ψ has a multi well structure. Accordingly, the set K takes the form

K = m i=1 SO(3)U i where U 1 , • • • , U m are
given symmetric positive definite tensors that depend both on the temperature and on the alloy considered. At a temperature T below the transformation temperature T 0 , the number m of wells is equal to the number nv of martensitic variants. The corresponding transformation strains 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 pre-superscript t denotes the transpose operator). At T = T 0 , both the austenite and the martensite variants minimize the microscopic energy, so that m = nv + 1. In general, the deformation gradient U nv+1 corresponding to the austenite is not symmetry related to any of the martensitic transformation strain. At T > T 0 , there is only one well which corresponds to the austenite phase.

Consider a reference configuration where a domain Ω is occupied by a single crystal of shape memory alloy. Each material particle is identified by its location x in the reference configuration. The deformation of the crystal is described by a mapping u : Ω → R 3 which gives the location (in the deformed configuration) of the material particle x. We study the equilibrium of the crystal under boundary conditions of the form u(x) = F.x where F is a given mesoscopic deformation gradient (here and in the following, the superscript . denotes quantities defined at the mesoscopic scale). The effective free energy at equilibrium is given by

Ψ ( F) = inf F∈A ( F) Ψ (F) (1)
where . denotes volume average over the domain Ω and the set A ( F) of admissible deformation gradient fields is defined by

A ( F) = {F|det F > 0, ∃u(x) ∈ W 1,∞ (Ω , R 3 ) such that F = ∇u in Ω ; u(x) = F.x on ∂ Ω }. (2)
The infimum problem in (1) is easily solved if F ∈ K: in that case, the homogeneous field F(x) = F realizes the infimum and we have Ψ ( F) = Ψ ( F) = 0. The situation gets more complicated if F / ∈ K: in that case, a minimizer of (1) does not necessarily exist. This is a consequence of the multi well structure of Ψ .The non existence of a minimizer physically corresponds to the formation of microstructures at a very fine scale (twinned laminates are examples of microstructures which are frequently observed in shape memory alloys, see e.g. [START_REF] Otsuka | Shape memory materials[END_REF]). In such a situation, the evaluation of Ψ ( F) requires to study the weak convergence of minimizing sequences, which involves mathematical concepts such as Young measures [START_REF] Kinderlehrer | Characterizations of young measures generated by gradients[END_REF] and quasiconvexity [START_REF] Dacorogna | Direct methods in the calculus of variations[END_REF]. The function F → Ψ ( F) is mathematically referred to as the quasiconvexification (or relaxation) of Ψ and can be interpreted as the mesoscopic energy of the material. Let K be the set of deformation gradients that minimize Ψ . Since Ψ is positive and vanishes on K, the minimum value of Ψ is equal to 0 and we have

K = { F| Ψ ( F) = 0}.
The set K is also known as the quasiconvex hull of K [START_REF] Dacorogna | Direct methods in the calculus of variations[END_REF][START_REF] Müller | Variational models for microstructure and phase transitions[END_REF]. The exact calculation of K still remains a largely open question. In the framework of finite strains as considered in this paper, the problem has been solved exactly [START_REF] Ball | Proposed experimental tests of a theory of fine microstructure and the two-well problem[END_REF][START_REF] Bhattacharya | Relaxation on some multi-well problems[END_REF] only in the case where

(i) the strains U 1 , • • • , U m have same determinant, (ii) there exists µ > 0 and v ∈ R 3 such that U i v = µv for all 1 ≤ i ≤ m. (3)
The assumption (3ii) show that problems of the form (3) are fundamentally two-dimensional. Commonly used shape-memory alloys, such as TiNi or CuAlNi, do not fulfill the assumption (3ii) and the exact expression of K for such materials is unknown. In such a situation, one may look for bounds (in the sense of inclusion of sets) to obtain some rigorous information on K. In this paper we primarily focus on upper bounds (the question of constructing lower bounds is addressed in Section 6).

Upper bounds on K can be obtained using known restrictions on Young measures [START_REF] Peigney | On the energy-minimizing strains in martensitic microstructures-part 1: Geometrically nonlinear theory[END_REF]. Some notations are in order. We introduce the frame-indifferent function Φ : R 3×3 → R defined by

Φ(M) = max R∈SO(3) tr(R.M). (4)
The supremum problem defining Φ(M) can be solved in closed form : we have indeed

Φ(M) = λ 3 + λ 2 + λ 1 sgn (det M) (5)
where 0 ≤ λ 1 ≤ λ 2 ≤ λ 3 are the eigenvalues of t M.M. Let F * denote the adjugate of a matrix F ∈ R 3×3 , i.e.

F * = det F. t F -1 . For any a ∈ R 3×3 , b ∈ R 3×3 and c ∈ R, the function h defined by h(F) = tr(F.a + F * .b) + c det F (6)
is known to be quasiconvex [START_REF] Dacorogna | Direct methods in the calculus of variations[END_REF], i.e. satisfies h( F) ≤ h(F) for all F and F ∈ A ( F).

The functions Φ in (4) and h in (6) are related by the following identity, which will be frequently used in this paper: sup R∈SO(3)

h(R.F) = Φ(F.a + F * .b) + c det F. (8)
Finally, we set

T = {θ = (θ 1 , • • • , θ m ) ∈ R m : θ i ≥ 0; m ∑ i=1 θ i = 1}. (9)
Using those notations, it can be proved [START_REF] Peigney | On the energy-minimizing strains in martensitic microstructures-part 1: Geometrically nonlinear theory[END_REF] that the set K+ defined by

K+ = { F ∈ R 3×3 : ∃θ ∈ T such that det F = m ∑ i=1 θ i det U i and 0 ≥ sup (a,b)∈C {Φ( F.a + F * .b) - m ∑ i=1 θ i Φ(U i .a + U * i .b)} } (10)
is an upper bound on K, i.e. satisfies K ⊂ K+ . For a given F in K+ , the vector θ in (10) can be interpreted as the volume fractions of the different wells in a microstructure realizing F. In general there is no uniqueness for θ . This is connected to the fact that, except in special cases, energy-minimizing deformation gradients F can be realized by several microstructures.

In [START_REF] Govindjee | An upper bound to the free energy of mixing by twin-compatible lamination for n-variant martensitic phase transformations[END_REF], C is a given arbitrary subset of R 3×3 ×R 3×3 . Note that each choice of C generates a corresponding bound on K. In principle, the best bound is obtained by taking C as large as possible, i.e. C = R 3×3 × R 3×3 .

In practice, however, one may restrict C to a smaller family to ease the calculations and notably obtain closed form expressions.

For later reference, we close this section with a brief derivation of the bound K+ . A fundamental property of K is that any given F ∈ K can be written as

F = R 3×3 A dν(A) (11)
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]. For our purpose, we only need to record the following properties of Young measures [START_REF] Kinderlehrer | Characterizations of young measures generated by gradients[END_REF]:

(i) ν ≥ 0 , R 3×3 dν(A) = 1, (ii) f R 3×3 A dν(A) ≤ R 3×3 f (A) dν(A) for any f quasiconvex. ( 12
)
Moreover, since K = ∪ m i=1 K i where the wells K i = SO(3)U i are closed and disjoint, the Young measure ν in (11) can be written as

ν = m ∑ i=1 ν i
where ν i is a positive measure supported on K i [START_REF] Ball | Proposed experimental tests of a theory of fine microstructure and the two-well problem[END_REF]. Using the properties [START_REF] Govindjee | The free energy of mixing for n-variant martensitic phase transformations using quasi-convex analysis-variant martensitic phase transformations using quasi-convex analysis[END_REF][START_REF] Hackl | An upper bound to the free energy of n-variant polycrystalline shape memory alloys[END_REF] with quasiconvex functions h of the form (6), we obtain

h( F) ≤ m ∑ i=1 SO(3)U i h(A) dν i (A).

The identity (8) gives

h(A) ≤ Φ(U i .a + U * i .b) + c det U i for any A ∈ SO(3)U i . Consequently we find h( F) ≤ m ∑ i=1 θ i Φ(U i .a + U * i .b) + c det U i (13)
where

θ i = R 3×3 dν i (A).
As a consequence of (12i), the vector θ = (θ 1 , • • • , θ n ) belongs to the set T in [START_REF] Eucken | The effect of textures on shape memory behaviour[END_REF]. Let now R be an arbitrary rotation. Observing that F → h( R.F) is quasiconvex and proceeding as in ( 13), we obtain

h( R. F) ≤ m ∑ i=1 θ i Φ(U i .a + U * i .b) + c det U i .
Taking the supremum over R in this last equation and using the identity (8), we get

Φ( F.a + F * .b) + c det F ≤ m ∑ i=1 θ i {Φ(U i .a + U * i .b) + c det U i }. ( 14
)
The conclusion is that for any F ∈ K, there exists θ ∈ T such that ( 14) holds for any (a, b, c). In other words, the set K+ in ( 10) is an upper bound on K. As detailed in [START_REF] Peigney | On the energy-minimizing strains in martensitic microstructures-part 1: Geometrically nonlinear theory[END_REF], the bound given by ( 10) coincides with K for the reference cases where the exact expression of K is available, such as problems of the form (3).

Polycrystal

Now consider a polycrystal occupying a domain Ω . We can decompose Ω as Ω = ∪ n r=1 Ω r where each subdomain Ω r is formed by grains with the same orientation. The microscopic free energy in Ω r can be written as

Ψ r (F) = Ψ ( t R r .F.R r ) (15)
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 microscopic free energy of the polycrystal is a heterogeneous function of the form

Ψ (F, x) = n ∑ r=1 χ r (x)Ψ r (F).
The set K(x) of deformation gradients minimizing F → Ψ (F, x) is thus

K(x) = m i=1 SO(3)U i (x) (16)
where

U i (x) = ∑ n r=1 χ r (x)U r i and U r i = R r .U i . t R r . The functions χ 1 , • • • , χ n together with the rotations R 1 , • • • , R n fully
describe the texture of the polycrystal, i.e. the shapes, distributions and orientations of the grains.

Consider boundary conditions of the form u(x) = F.x where F is a given macroscopic deformation gradient (here and in the following, the superscript . denotes macroscopic quantities). At equilibrium, one expects the formation of microstructures in each grain. Assuming the scale of such microstructures to be much smaller than the scale of the grains, the macroscopic free energy Ψ ( F) of the polycrystal is given by

Ψ ( F) = min F∈A ( F) n ∑ r=1 χ r Ψ r (F) (17)
where Ψ r is the relaxation of Ψ r , as defined in [START_REF] Ball | Fine phase mixtures as minimizers of energy[END_REF] (see e.g. [START_REF] Bhattacharya | Microstructure of Martensite[END_REF] for a detailed justification). In the following, we primarily focus on the set K of deformation gradients that minimize the macroscopic energy, i.e.

K = { F| Ψ ( F) = 0}.
In view of (17), we have the following characterization of K:

K = { F|∃F ∈ A ( F); F(x) ∈ K(x) for all x ∈ Ω }. (18)
Hence the distinctive property of strains F in K 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. Let Kr be the quasi convex hull of ∪ m i=1 SO(3)U r i , i.e. the set of strains minimizing Ψ r . For any F in

∩ n r=1
Kr , the homogeneous field

F(x) = F is in A ( F) and satisfies the constraint F(x) ∈ K(x) in Ω . The set K-= ∩ n r=1
Kr is thus a lower bound on K, in the sense that K-⊂ K. That lower bound (referred to as the Taylor bound) has been studied in detail in the geometrically linear setting [START_REF] Bhattacharya | Energy minimization and the recoverable strains in polycrystalline shape memory alloys[END_REF]. Note that the bound Konly depends on the rotations R 1 , • • • , R n , and not on the functions χ 1 , • • • , χ n , therefore ignoring a lot of information about the texture.

Using the quasiconvexity of the function h in ( 6), upper bounds on K that take one-point statistics of the functions χ r can be derived [START_REF] Peigney | Recoverable strains in composite shape-memory alloys[END_REF]. Let indeed F be in K and consider a deformation gradient field F in A ( F) such that F(x) ∈ K(x) for all x (recall that such F exists by [START_REF] Misra | Thermomechanics-based nonlinear rate-dependent coupled damage-plasticity granular micromechanics model[END_REF]). Using the property [START_REF] Chenchiah | Non-laminate microstructures in monoclinic-i martensite[END_REF] and noting that ∑ r χ r = 1, we have

h( F) ≤ h(F) = n ∑ r=1 χ r h(F) . ( 19
)
Observe from (8

) that h(F(x)) ≤ Φ(F(x).a + F * (x).b) + c det F(x). For x ∈ Ω r , we have F(x) ∈ Kr and consequently h(F(x)) ≤ Φ(F(x).a + F * (x).b) + c det F(x) ≤ sup F∈ Kr {Φ(F.a + F * .b) + c det F}.
Substituting in [START_REF] Müller | Variational models for microstructure and phase transitions[END_REF] gives

h( F) ≤ n ∑ r=1 χ r sup F∈ Kr {Φ(F.a + F * .b) + c det F} . (20)
The frame-indifference of K implies that R. F also satisfies [START_REF] Otsuka | Shape memory materials[END_REF] for any rotation R, which using (8) leads to

Φ( F.a + F * .b) + c det F ≤ n ∑ r=1 χ r sup F∈ Kr {Φ(F.a + F * .b) + c det F} . (21)
Let C denote a given subset of R 3×3 × R 3×3 . From ( 21) we obtain that any F in K necessarily satisfies

sup c sup (a,b)∈C Φ( F.a + F * .b) + c det F - n ∑ r=1 χ r sup F∈ Kr {Φ(F.a + F * .b) + c det F} ≤ 0. ( 22
)
The set of deformation gradient F verifying ( 22) is thus an upper bound of K. That upper bound has been used in [START_REF] Peigney | Recoverable strains in composite shape-memory alloys[END_REF] on some examples where the constitutive single crystals fulfill the conditions (3), in which case the sets Kr are known. For polycrystals that do not satisfy (3), the direct application of the bound ( 22) is hampered by the fact that Kr is unknown. In such case, calculating the supremum over Kr that appears in [START_REF] Peigney | Recoverable strains in composite shape-memory alloys[END_REF] remains out of reach. Such a difficulty can be overcome by using the results from Section 2. Let indeed Kr + be the upper bound of Kr defined in Eq. [START_REF] Govindjee | An upper bound to the free energy of mixing by twin-compatible lamination for n-variant martensitic phase transformations[END_REF]. Since Kr ⊂ Kr + , we have sup Kr Φ(F.a

+ F * .b) + c det F ≤ sup Kr + Φ(F.a + F * .b) + c det F. Therefore, we obtain from (22) that any F in K necessarily satisfies sup c sup (a,b)∈C {Φ( F.a + F * .b) + c det Ū - n ∑ r=1 χ r sup F∈ Kr + {Φ(F.a + F * .b) + c det F}} ≤ 0. ( 23
)
The calculation of the right-hand side in [START_REF] Peigney | A non-convex lower bound on the effective free energy of polycrystalline shape memory alloys[END_REF] can be further simplified if C = C , i.e. if the bound [START_REF] Peigney | A non-convex lower bound on the effective free energy of polycrystalline shape memory alloys[END_REF] and the bound Kr + given 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] and are calculated using the same set of tensors (a, b). In that case, we have indeed

sup F∈ Kr + {Φ(F.a + F * .b) + c det F} = max 1≤i≤m {Φ(U r i .a + U r, * i .b) + c det U r i } (24)
for any (a, b) ∈ C . That last property is easily proved: for a given F in Kr + , Eq. ( 10) yields

Φ(F.a + F * .b) + c det F ≤ m ∑ i=1 θ i {Φ(U r i .a + U r, * i .b) + c det U r i } (25)
for some θ ∈ T. Since any θ in T verifies θ i ≥ 0 and ∑ i θ i = 1, the right hand side of ( 25) is bounded from above by max 1≤i≤m {Φ(U r i .a

+ U r, * i .b) + c det U r i }. Therefore we have sup F∈ Kr + {Φ(F.a + F * .b) + c det F} ≤ max 1≤i≤m {Φ(U r i .a + U r, * i .b) + c det U r i }.
The converse inequality follows directly from the fact that U r i ∈ Kr 23) and [START_REF] Peigney | On the energy-minimizing strains in martensitic microstructures-part 1: Geometrically nonlinear theory[END_REF] gives

+ for i = 1, • • • , m. Combining (
K ⊂ K0 + (26)
where

K0 + = F ∈ R 3×3 : 0 ≥ sup c sup (a,b)∈C Φ( F.a + F * .b)+ c det F - n ∑ r=1 χ r max 1≤i≤m {Φ(U r i .a + U r, * i .b)+ c det U r i } . (27)
In the most usual situation when all the transformation strains U r i have the same determinant, the set K0

+ can be rewritten more simply as

K0 + = F ∈ R 3×3 : det F = det U 1 1 , 0 ≥ sup (a,b)∈C {Φ( F.a + F * .b) - n ∑ r=1 χ r max 1≤i≤m Φ(U r i .a + U r, * i .b)} . ( 28
)
The set K0 + 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.

For a discrete C , checking if a given deformation gradient F is in K0 + amounts to check if F satisfies a finite number of constraints. A similar remark holds if C is not discrete but with the form

N j=1 w∈R 3 {(u j ⊗ w, u j ⊗ w)}.
In that case, Eq. ( 5) indeed shows that Φ( F.

u j ⊗ w + F * .u j ⊗ w) = F.u j + F * .u j . w so that the condition 0 ≥ sup (a,b)∈C {Φ( F.a + F * .b) - n ∑ r=1 χ r max 1≤i≤m Φ(U r i .a + U r, * i .b)} is equivalent to 0 ≥ max 1≤ j≤N { F.u j + F * .u j - n ∑ r=1 χ r max 1≤i≤m U r i .u j + U r, * i .u j },
i.e. to a set of N constraints on F.

Homogeneous limit

In this Section we study the behavior of the polycrystalline bound (27) in the limit χ 1 → 1. In that case, the set K0 + in Eq. ( 27) is characterized by

0 ≥ sup c sup (a,b)∈C Φ( F.a + F * .b) + c det F -max 1≤i≤m {Φ(U 1 i .a + U 1, * i .b) + c det U 1 i } . (29)
In the homogeneous limit χ 1 → 1, there is no distinction between the mesoscopic scale and the macroscopic scale. Since the bound K0 + is the result of substituting the monocrystalline upper bound [START_REF] Govindjee | An upper bound to the free energy of mixing by twin-compatible lamination for n-variant martensitic phase transformations[END_REF] in the general prescription [START_REF] Peigney | Recoverable strains in composite shape-memory alloys[END_REF], one would expect the set defined by [START_REF] Smyshlyaev | On the relaxation of a three-well energy[END_REF] to coincide with the monocrystalline bound K+ generated from the same family C . This can be proved to be indeed the case for problems of the form (3), but it is not true in general. To illustrate that point, consider the cubic to tetragonal transformation at T < T 0 : we have

K 1 = 3 i=1 SO(3)U 1 i where U 1 1 =   η 2 0 0 0 η 1 0 0 0 η 1   , U 1 2 =   η 1 0 0 0 η 2 0 0 0 η 1   , U 1 3 =   η 1 0 0 0 η 1 0 0 0 η 2   . (30)
These matrix representations are relative to the reference orthonormal basis (v 1 , v 2 , v 3 ) of the cubic austenitic lattice in orientation 1. The parameters (η 1 , η 2 ) are non-negative and distinct. We assume in the following that η 2 < η 1 .

We determine the bounds K+ in [START_REF] Govindjee | An upper bound to the free energy of mixing by twin-compatible lamination for n-variant martensitic phase transformations[END_REF] and K0 + in (29) corresponding to the family C of tensors (a, b) defined by

C = j∈{1,2,3},w∈R 3 {(v j ⊗ w, 0), (0, v j ⊗ w)}. (31)
Let us first determine the set K+ in [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 a fixed F in K+ , we can see from [START_REF] Govindjee | An upper bound to the free energy of mixing by twin-compatible lamination for n-variant martensitic phase transformations[END_REF] that det F = η 2 1 η 2 . Moreover, there exists θ ∈ T such that

0 ≥ Φ( F.a + F * .b) - 3 ∑ i=1 θ i Φ(U 1 i .a + U 1, * i .b) (32)
for all (a, b) in the family C defined by [START_REF] Ye | Interior point algorithms: theory and analysis[END_REF]. Any such (a, b) can either be written as (v j ⊗ w, 0) or (0, v j ⊗ w).

In the first case, the definition (4) gives Φ(F.a + F * .b) = F.v j . w so that (32) becomes

0 ≥ F.v j - 3 ∑ i=1 θ i U 1 i .v j .
Observing that

U 1 i .v j = η 1 + (η 2 -η 1 )δ i j and recalling that ∑ 3 i=1 θ i = 1, we obtain 0 ≥ F.v j -η 1 -θ j (η 2 -η 1 ). (33)
In the case (a, b) = (0, v j ⊗ w), a similar reasoning leads to

0 ≥ F * .v j -η 1 η 2 -θ j η 1 (η 1 -η 2 ). (34)
This inequalities ( 33) and (34) can be combined as as

F * .v j -η 1 η 2 η 1 (η 1 -η 2 ) ≤ θ j ≤ η 1 -F.v j η 1 -η 2 . ( 35
)
Any θ in T is such that θ j ≥ 0 and ∑ 3 j=1 θ j = 1. It follows that any F in K+ satisfies

0 ≤ η 1 -F.v j η 1 -η 2 ; F * .v j -η 1 η 2 η 1 (η 1 -η 2 ) ≤ 1; F * .v j -η 1 η 2 η 1 (η 1 -η 2 ) ≤ η 1 -F.v j η 1 -η 2 for j = 1, 2 , 3;
-3η 2 + 1 η 1 3 ∑ j=1 F * .v i ≤ η 1 -η 2 ≤ 3η 1 - 3 ∑ j=1 F.v j ; (36)
which can be rewritten more simply as

F.v j ≤ η 1 , F * .v j ≤ η 2 1 , η 1 F.v j + F * .v j ≤ η 1 (η 1 + η 2 ) for j = 1, 2, 3; 3 ∑ j=1 F.v j ≤ 2η 1 + η 2 , 3 ∑ j=1 F * .v j ≤ η 1 (2η 2 + η 1 ). (37)
Conversely, for any F satisfying (37), it can be verified that there exists θ ∈ T satisfying [START_REF] Ye | On homogeneous and self-dual algorithm for linear complementarity problems[END_REF] for all (a, b) in C . Therefore, the inequalities (37) (complemented by the restriction det F = η 2 1 η 2 ) characterize the upper bound K+ that is generated by the family C defined in Eq. [START_REF] Ye | Interior point algorithms: theory and analysis[END_REF].

Let us now calculate the bound K0 + in [START_REF] Smyshlyaev | On the relaxation of a three-well energy[END_REF]. Since the transformation strains U 1 1 , U 1 2 , U 1 3 in (30) have same determinant, we obtain from (29) that any F in K0

+ satisfies det F = η 2 1 η 2 and 0 ≥ Φ( F.a + F * .b) -max 1≤i≤3 Φ(U 1 i .a + U 1, * i .b) (38)
for all (a, b) in C . Using the special form of the tensors (a, b) defined in [START_REF] Ye | Interior point algorithms: theory and analysis[END_REF] and the corresponding values of

Φ(U 1 i .a + U 1, * i .b), the inequality (38) is found to be equivalent to F.v j ≤ η 1 , F * .v j ≤ η 2 1 for j = 1, 2, 3. (39)
Comparing (39) with (37) shows that K+ ⊂ K0 + . That inclusion is actually strict: consider indeed the deformation gradient F0 defined by

F0 = (η 2 1 η 2 ) 1/3 I + ( η 1 η 2 2/3 -1 v 1 ⊗ v 2 .
We have det F0 = η 2 1 η 2 and

F0 .v 1 = F0 .v 3 = (η 2 1 η 2 ) 1/3 , F0 .v 2 = η 1 , F * 0 .v 2 = F * 0 .v 3 = (η 2 1 η 2 ) 2/3 , F * 0 .v 1 = η 5/3 1 η 1/3 2 .
Since η 2 < η 1 , it can easily be verified that F0 satisfies (39) and therefore is in K0 + . However, we have

η 1 F0 .v 2 + F * 0 .v 2 = η 2 1 + (η 2 1 η 2 ) 2/3 > η 1 (η 1 + η 2 )
. Therefore, F0 does not satisfy (37) and is not in K+ . This example shows that the bound in [START_REF] Pipkin | Elastic materials with two preferred states[END_REF] may fail to recover the single crystal bound in the homogeneous limit. This is an indication that some information is lost when directly plugging the monocrystalline bound [START_REF] Govindjee | An upper bound to the free energy of mixing by twin-compatible lamination for n-variant martensitic phase transformations[END_REF] in the general prescription [START_REF] Peigney | Recoverable strains in composite shape-memory alloys[END_REF]. In the following, we derive an improved upper bound on K that notably coincides with the single crystal bound (10) in the homogeneous limit.

Improved bound for polycrystals

The bound K0 + in ( 27) can be improved upon by taking the special structure of the monocrystalline bound (10) into account, as is now explained. Consider a given F in K. By [START_REF] Misra | Thermomechanics-based nonlinear rate-dependent coupled damage-plasticity granular micromechanics model[END_REF], there exists a field

F ∈ A ( F) such that F(x) ∈ K(x) for all x ∈ Ω . Recall that K(x) = m i=1 SO(3)U i (x) where U i (x) = ∑ n r=1 χ r (x)U r i .
Using the bound (10) on K(x), we know there exists θ (x) ∈ T such that

0 ≥ sup c sup (a,b)∈C Φ(F(x).a + F * (x).b) + c det F(x) - m ∑ i=1 θ i (x){Φ(U i (x).a + U * i (x).b) + c det U i (x)} . ( 40
)
Since χ r (x) ∈ {0, 1} and ∑ r χ r (x) = 1, Eq. ( 40) can be rewritten as

0 ≥ sup c sup (a,b)∈C Φ(F(x).a + F * (x).b) + c det F(x) - n ∑ r=1 m ∑ i=1 χ r (x)θ i (x){Φ(U r i .a + U r, * i .b) + c det U r i } . (41) For any r = 1, • • • , n and i = 1, • • • , m, define θ r i (x) = χ r (x)θ i (x). (42)
Taking volume averages in (41) yields

0 ≥ sup c sup (a,b)∈C Φ(F.a + F * .b) + c det F - n ∑ r=1 m ∑ i=1 θ r i {Φ(U r i .a + U r, * i .b) + c det U r i } (43)
The crucial point is that the function F → Φ(F.a + F * .b) + c det F is quasiconvex. Using [START_REF] Chenchiah | Non-laminate microstructures in monoclinic-i martensite[END_REF] and observing from (8) that h(F) ≤ Φ(F.a + F * .b) + c det F, we have indeed

h( F) ≤ h(F) ≤ Φ(F.a + F * .b) + c det F . (44)
Consider a given rotation R. Since R.F ∈ A ( R. F), we obtain, in a similar way to (44),

h( R. F) ≤ Φ( R.F.a + R.F * .b) + c det( R.F) . (45)
The function Φ being frame-indifferent, the last term in (45) is equal to Φ(F.a + F * .b) + c det F . Therefore, taking the supremum over R in (45) and using the identity (8), we get

Φ( F.a + F * .b) + c det F ≤ Φ(F.a + F * .b) + c det F .
That last inequality, holding for any F and F ∈ A ( F), proves that F → Φ(F.a + F * .b) is quasiconvex. Although it has not been stated explicitely up to this point, the quasiconvexity of Φ can be regarding as the main argument behind the bounds (10) and ( 27) considered in Sections 2-3. Here, in view of (43), the quasiconvexity of Φ implies that

0 ≥ sup c sup (a,b)∈C Φ( F.a + F * .b) + c det F - n ∑ r=1 m ∑ i=1 θ r i {Φ(U r i .a + U r, * i .b) + c det U r i } . (46)
The scalar θ r i can be interpreted as the volume fraction of martensitic variant i with orientation r. Note from (42) that { θ r i } 1≤r≤n 1≤i≤m belongs to the set T defined by

T = {Θ ∈ R n m |Θ r i ≥ 0 ; m ∑ i=1 Θ r i = χ r ∀r = 1, • • • , n}. (47)
The developments so far show that for any F in K, there exists Θ ∈ T verifying the inequality (46). This last statement can be rewritten as K ⊂ K+ (48)

where

K+ = F : ∃Θ ∈ T such that 0 ≥ sup c sup (a,b)∈C Φ( F.a + F * .b) + c det F - n ∑ r=1 m ∑ i=1 Θ r i {Φ(U r i .a + U r, * i .b) + c det U r i } . (49)
In the case where all the transformation strains have the same determinant, the set K+ can be rewritten more simply as

K+ = F : det F = det U 1 1 ; ∃Θ ∈ T such that 0 ≥ sup (a,b)∈C {Φ( F.a + F * .b) - n ∑ r=1 m ∑ i=1 Θ r i Φ(U r i .a + U r, * i .b)} .
(50) Eq. ( 48) means that K+ is an upper bound on the set of macroscopic energy-minimizing strains for the polycrystal. In a way similar to the bound K0 + considered in Sec. 3, K+ depends on the texture through the volume fractions χ r of the different orientations (the later indeed appear in the definition (47) of the set T). Observe that the bound K+ is always tighter than the bound K0 + obtained from the same set C . By (47), any

Θ ∈ T indeed satisfies m ∑ i=1 Θ r i (Φ(U r i .a + U r, * i .b) + c det U r i ) ≤ χ r max 1≤i≤m {Φ(U r i .a + U r, * i .b) + c det U r i } (51) for r = 1, • • • , n. It follows that K+ ⊂ K0 + .
We also note that, contrary to the bound K0 + , the bound K+ in (49) coincides with the monocrystalline bound K+ in the homogeneous limit. This shows that the inclusion K+ ⊂ K0 + can be strict, i.e. that the bound K+ can bring a genuine improvement.

Regarding the practical calculation of the bound K+ , observe that the inequality 0 ≥ Φ( 49) can be viewed as a linear constraint on Θ , parametrized by F. Deformation gradients F in K+ are characterized by the fact that those linear constraints (supplemented by the conditions Θ ∈ T) define a non empty set of R n m . In the language of linear programming, this amounts to detect feasibility of the linear constraints [START_REF] Ye | Interior point algorithms: theory and analysis[END_REF], which is not a direct calculation -even for a discrete C . In the next section, we show how the problem can be conveniently solved in the case of a two-orientation / three-well polycrystal.

F.a + F * .b) - ∑ r,i Θ r i Φ(U r i .a + U r, * i .b) in (
6 Analytical example

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. We are interested in estimating the energy-minimizing strains at T < T 0 , i.e. in the case where

K 1 = 3 i=1 SO(3)U 1 i with U 1 1 , U 1 2 , U 1 3
given by [START_REF] Talbot | Bounds for the effective constitutive relation of a nonlinear composite[END_REF]. The set K 2 of strains that minimize the microscopic free energy in orientation 2 can be written as

K 2 = R 2 .K 1 . t R 2 where R 2 ∈ SO(3).
Table 1 Values of (a, b) used for the cubic to tetragonal transformation

a b Φ 1 k (a, b) Φ 1 i (a, b) (i = k) v k ⊗ v k 0 η 2 η 1 0 v k ⊗ v k η 2 1 η 1 η 2 (v i ± v j ) ⊗ v k 0 √ 2η 1 η 2 1 + η 2 2 0 (v i ± v j ) ⊗ v k √ 2η 1 η 2 η 1 η 2 1 + η 2 2 η 1 η 2 1 -η 2 2 v k ⊗ v k - η 2 η 1 (η 2 1 -η 2 2 ) v k ⊗ v k 0 1 η 2 η 2 1 -η 2 2 v k ⊗ v k - 1 η 2 1 -η 2 2 v k ⊗ v k 1 0
We first illustrate how the prescription (49) can be used to derive a closed-form bound on K. By (49) we know that for any given F in K, there exists Θ ∈ T verifying

0 ≥ Φ( F.a + F * .b) - 2 ∑ r=1 3 ∑ l=1 Θ r l Φ r l (a, b) (52)
with the notation Φ r l (a, b) = Φ(U r l .a + U r, * l .b). Assume we can pick out (a, b) such that

Φ 1 i (a, b) = Φ 1 j (a, b) = Φ 1 k (a, b) (53)
for some permutation (i, j, k) of (1, 2, 3). Since

∑ 3 l=1 Θ 1 l = χ 1 , the relation (52) gives 0 ≥ Φ( F.a + F * .b) +Θ 1 k (Φ 1 i (a, b) -Φ 1 k (a, b)) -χ 1 Φ 1 i (a, b) - 3 ∑ l=1 Θ 2 l Φ 2 l (a, b).
Using the fact that Θ 2 l ≥ 0 and ∑ 3 l=1 Θ 2 l = χ 2 , we get

0 ≥ Φ( F.a + F * .b) +Θ 1 k (Φ 1 i (a, b) -Φ 1 k (a, b)) -χ 1 Φ 1 i (a, b) -χ 2 max 1≤l≤3 Φ 2 l (a, b).
That last inequality can be rewritten as

Θ 1 k ≤ -Φ( F.a + F * .b) + χ 1 Φ 1 i (a, b) + χ 2 max l Φ 2 l (a, b) Φ 1 i (a, b) -Φ 1 k (a, b) if Φ 1 i (a, b) -Φ 1 k (a, b) > 0, Θ 1 k ≥ -Φ( F.a + F * .b) + χ 1 Φ 1 i (a, b) + χ 2 max l Φ 2 l (a, b) Φ 1 i (a, b) -Φ 1 k (a, b) if Φ 1 i (a, b) -Φ 1 k (a, b) < 0. ( 54
)
Observe that tensors (a, b) of the form (31) verify the condition (53). In a separate study of the monocrystalline cubic to tetragonal transformation [START_REF] Peigney | On the energy-minimizing strains in martensitic microstructures-part 1: Geometrically nonlinear theory[END_REF], other tensors (a, b) verifying (53) have been found. Those tensors (a, b) are listed in Table 1, along with the corresponding values of Φ 1 l (a, b). The tensors (a, b) in Table 1 have been found in the course of a full optimization of [START_REF] Govindjee | An upper bound to the free energy of mixing by twin-compatible lamination for n-variant martensitic phase transformations[END_REF] with respect to tensors (a, b) that are diagonal in the reference cubic lattice (see [START_REF] Peigney | On the energy-minimizing strains in martensitic microstructures-part 1: Geometrically nonlinear theory[END_REF] for more details).

Substituting the values of Table 1 in (54), we find that

A 1 k ( F) ≤ Θ 1 k ≤ B 1 k ( F) ( 55
)
where A 1 k ( F) and B 1 k ( F) are explicit functions whose expressions are reported in Appendix A. A similar analysis can be performed to the orientation r = 2. This is simply achieved by replacing v k by R 2 .v k and swapping the roles of orientations 1 and 2 in the previous developments. Such a procedure leads to where the expressions of A 2 k ( F) and B 2 k ( F) are detailed in Appendix A. In view of (55-56), some necessary conditions for F to be in K are

A 2 k ( F) ≤ Θ 2 k ≤ B 2 k ( F) (56)
A r k ( F) ≤ B r k ( F) for k = 1, 2, 3; 3 ∑ k=1 A r k ( F) ≤ χ r ≤ 3 ∑ k=1 B r k ( F); (57)
with r = 1, 2. The second inequality in (57) stems from the fact that ∑ 3 k=1 Θ r k = χ r . The relations (57) define the upper bound K+ corresponding to the set of tensors (a, b) listed in Table 1 (along with their rotated versions obtained by replacing v k with R 2 .v k in Table 1).

So as to illustrate the bound obtained, consider deformation gradients F(ω, δ ) of the form

F(ω, δ ) = (η 2 1 η 2 ) 1/3 (I + δ u(ω) ⊗ v(ω)) (58)
where

u(ω) = cos ω v 1 + sin ω v 2 and v(ω) = -sin ω v 1 + cos ω v 2 .
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.

In the following, we are interested in estimating the values (ω, δ ) for which F(ω, δ ) is energy-minimizing. Except stated otherwise, all the results presented next are obtained with the lattice parameters of MnCu, i.e. η 1 = 1.0099, η 2 = 0.9656 [START_REF] Basinski | The cubic tetragonal transformation in manganese copper alloys[END_REF]. The rotation R 2 defining the orientation 2 is taken as

R 2 =   √ 2 2 - √ 2 2 0 √ 2 2 √ 2 2 0 0 0 1   . ( 59
)
Textures satisfying the assumptions made so far (i.e. n = 2 with R 1 = I and R 2 given by Eq. ( 59) ) are observed in some ribbons of shape memory alloys [START_REF] Eucken | The effect of textures on shape memory behaviour[END_REF]. Let ∆ + = {(ω, δ ) : F(ω, δ ) ∈ K+ } be the trace of K+ on deformation gradients of the form (58). The boundary of ∆ + is represented in Figures 234(solid lines in blue) for several values of χ 1 : the deformation gradient F(ω, δ ) satisfies Eq. (57) for any (ω, δ ) within the bounded domain ∆ + delimited by the solid lines in Figures 234. The solid curves in Figures 234can thus be interpreted as lower and upper bounds on the shear δ for F(ω, δ ) to be energy-minimizing.

Similarly, we set

∆ 0 + = {(ω, δ ) : F(ω, δ ) ∈ K0 + }
where K0 + is calculated by applying [START_REF] Shu | The influence of texture on the shape-memory effects in polycrystals[END_REF] with the tensors (a, b) listed in Table 1 (along with their rotated versions obtained by replacing v k with R 2 .v k ). The boundary of ∆ 0 + is plotted in dotted lines in Figures 234. This allows one to appreciate the improvement brought by the consideration of (49) over [START_REF] Pipkin | Elastic materials with two preferred states[END_REF]. For instance, Fig. 2 Bounds on the shear δ for MnCu, χ 1 = 0.9.

in the case χ 1 = 0.7 (Figure 3), the relative improvement of the bound on the shear δ varies between 0 and 52% (depending on the angle ω considered), with an average of approximatively 35%. In the case χ 1 = 0.5 (Figure 4), the relative improvement is non-negative for all ω, with an average of approximatively 26% .

Lamination lower bound

Although the fact that K+ significantly improves on K0 + is promising, it does not give any clue regarding the sharpness of the bound. The relations defining K+ (49) are indeed necessary-but not sufficient -conditions for a deformation gradient to be energy-minimizing. The issue is to determine which deformation gradients in K+ are indeed energy-minimizing for some polycrystalline texture that is compatible with the prescribed statistics (i.e. with prescribed volume fractions of the different orientations). In order to address that question, we consider the special class of laminated textures and adapt an argument introduced in [START_REF] Ball | Proposed experimental tests of a theory of fine microstructure and the two-well problem[END_REF]: let F 1 ∈ K1 and

F 2 ∈ K2 be rank-1 connected, i.e. such that R.F 2 -F 1 = p ⊗ n (60)
for some vectors (p, n) and some rotation R. Following [START_REF] Ball | Proposed experimental tests of a theory of fine microstructure and the two-well problem[END_REF], we know that the effective deformation gradient

χ 1 F 1 + (1 -χ 1 )
R.F 2 is energy-minimizing for an (infinitely fine) laminate texture mixing orientations 1 and 2 in proportions χ 1 and 1χ 1 , respectively. The vector n in (60) corresponds to the normal to the interfaces in such a laminate texture. We wish to use that argument to construct values of (ω, δ ) such that F(ω, δ ) is energy-minimizing for some well-chosen laminate texture. This requires to find deformation gradients (F 1 , F 2 ) that fulfill the two following conditions:

(1)

F 1 ∈ K1 , F 2 ∈ K2 . (2) F 1 and F 2 are rank-1 connected.
The condition [START_REF] Ball | Fine phase mixtures as minimizers of energy[END_REF] means that F i is energy-minimizing for a single crystal with orientation i. The issue of finding deformation gradients that are energy-minimizing for a single crystal of tetragonal martensite has been addressed in [START_REF] Peigney | On the energy-minimizing strains in martensitic microstructures-part 1: Geometrically nonlinear theory[END_REF]: it can be shown that K1 contains all the deformation gradients of the form

  u 11 u 12 0 u 12 u 22 0 0 0 (η 2 1 η 2 ) 1/3   (61)
with (u 11 , u 22 , u 12 ) verifying

u 11 u 22 -u 2 12 = (η 2 1 η 2 ) 2/3 , u 2 11 + u 2 22 + 2u 2 12 + 2|u 12 (u 11 + u 22 )| ≤ η 2 1 + (η 1 η 2 2 ) 2/3 . ( 62
) Since K2 = R 2 . K1 . t R 2
, it follows that K2 contains all the deformation gradients of the form

R 2 .   u 11 u 12 0 u 12 u 22 0 0 0 (η 2 1 η 2 ) 1/3   . t R 2 (63)
where (u 11 , u 22 , u 12 ) are submitted to the same restrictions as (62). As explained in [START_REF] Peigney | On the energy-minimizing strains in martensitic microstructures-part 1: Geometrically nonlinear theory[END_REF], the deformation gradient in (61) can be realized by some fourth-rank laminated microstructure that involves the three martensitic variants.

Let now F 1 ∈ K1 and F 2 ∈ K2 be two deformation gradients of the form (61) and (63), respectively. We claim that F 1 and F 2 are rank-1 connected. This can be checked by applying a general procedure proposed in [START_REF] Ball | Fine phase mixtures as minimizers of energy[END_REF][START_REF] Khachaturyan | Theory of Structural Transformation in solids[END_REF]: setting

U i = t F i .F i and C = U -1 1 .U 2 2 .U -1
1 , given deformation gradients F 1 and F 2 are rank-1 connected if and only if the eigenvalues

λ 1 ≤ λ 2 ≤ λ 3 of C are such that λ 2 = 1. (64)
In that case, the vectors p and n solving the twinning equation ( 60) are given by

p = λ 3 (1 -λ 1 ) λ 3 -λ 1 u 1 + κ λ 1 (λ 3 -1) λ 3 -λ 1 u 3 , U -1 1 .n = ( √ λ 3 - √ λ 1 √ λ 3 -λ 1 )(-1 -λ 1 u 1 + κ λ 3 -1 u 3 ) (65)
where κ ∈ {-1, 1} and u i is a normalized eigenvector of C for the eigenvalue λ i .

In the present case, observe from Eqs. ( 59)-( 61)-(63) that det C = 1 and C.v 3 = v 3 , so that the condition (64) is necessarily satisfied. We can thus conclude that there exists a polycrystalline texture having a volume fraction χ 1 of orientation 1 and for which the deformation gradient G defined by

G = F 1 + (1 -χ 1 )p ⊗ n (66)
is energy-minimizing, i.e. is in the set K. The final step is to observe that the deformation gradient G in (66) can be written as G = R * .F(ω * , δ * ) for some rotation R * and some well-chosen values (ω * , δ * ). To that purpose, we first note that v 3 being an eigenvector of the symmetric tensor C for the eigenvalue λ 2 = 1, the two eigenvectors (u 1 , u 3 ) in (65) are orthogonal to v 3 . It follows from the expressions (65) that the vectors p and n are also orthogonal to v 3 . We thus have

G.v 3 = t G.v 3 = (η 2 1 η 2 ) 1/3 v 3 Using matrix representations in the basis (v 1 , v 2 , v 3 ), the tensor t G.G/(η 2
1 η 2 ) 2/3 can therefore be written as

S 0 0 1
where S is a symmetric positive definite matrix in R 2×2 with a determinant equal to 1. Let (µ 1 , µ 2 ) be the eigenvalues of S, ordered in such a way that µ 1 ≤ µ 2 . Since µ 1 µ 2 = det S = 1, we have necessarily µ 1 ≤ 1 ≤ µ 2 . Now the continuous function ω → u(ω).S.u(ω) takes values in [µ 1 , µ 2 ], so there exists ω * verifying u(ω * ).S.u(ω * ) = 1.

Setting

δ * = u(ω * ).S.v(ω * ), (68)
the relation det S = 1 implies that the matrix representation of S in (u(ω * ), v(ω * )) is

1 δ * δ * 1 + δ * ,2 .
It follows that t G.G and t F(ω * , δ * ).F(ω * , δ * ) have the same matrix representation in the basis (u(ω), v(ω), v 3 ), i.e. that t G.G = t F(ω * , δ * ).F(ω * , δ * ). That last equality implies that G = R * .F(ω * , δ * ) for some rotation R * . Since K is frame-indifferent and G is in K, we can conclude that F(ω * , δ * ) is in K for the values (ω * , δ * ) given by ( 67)-( 68).

As a first example of the above construction, set s = (η 1 /η 2 ) 1/3 and consider the deformation gradient F 0 1 ∈ K1 defined by

F 0 1 = (η 2 1 η 2 ) 1/3 diag(s, 1/s, 1) (69)
which satisfies (61-62). We take

F 1 = F 0 1 and F 2 = R 2 .F 0 1 . t R 2 .
Setting t = s 4 , the eigenvalues λ 1 and λ 3 of C are

λ 1 = 1 4t (1 + 2t + t 2 -(t -1) t 2 + 6t + 1), λ 3 = 1 4t (1 + 2t + t 2 + (t -1) t 2 + 6t + 1),
and the corresponding eigenvectors are

u 1 = v 2 - ( √ t 2 + 6t + 1 + t + 1) 2t v 1 , u 3 = v 2 + √ t 2 + 6t + 1 -t -1 2t v 1 .
Substituting in Eq. ( 65) gives two energy-minimizing strains for the polycrystal. The corresponding values of (ω * , δ * ) are shown as red dots in Figures 234. A corresponding laminate texture is shown in Figure 5(a-b) for χ 1 = 0.7, η 1 = 1.5, η 2 = 1. Layers of material with orientation 1 (shown in blue) are alternated with layers of material with orientation 2 (shown in red). In the reference configuration, Figure 5(a), the normal to the layers is taken as a vector n that is a solution of (65). In the deformed configuration, Figure 5(b), a homogeneous deformation gradient F 0 1 (resp. F 0 1 + p ⊗ n) is applied in the layers with orientation 1 (resp. 2). The macroscopic deformation gradient F(ω * , δ * ) that is realized that way is of the form (58). In Figure 5(a), the cubic domain is chosen in a such a way that the edges of the top section are oriented along the shear directions (u(ω * ), v(ω * )).

The deformation of each monocrystalline layer in Figure 5(b) is achieved by some geometric arrangement of the martensitic variants at the microscopic level. Such a geometric arrangement is represented in Figure 5(c) for a layer with orientation 1. As detailed in Appendix, the deformation in such a layer can be achieved by a second-rank laminate involving only variants 2 and 3 (in proportion approximatively equal to 0.3338 and 0.6662, respectively). The deformation of a layer with orientation 2 can be realized by a similar microstructure.

The procedure described so far can be repeated for any F 1 and F 2 of the form (61)-(63), thus generating a set of values (ω * , δ * ) for which F(ω * , δ * ) is energy-minimizing. That set is denoted by ∆ -in the following. Although difficult to carry out by hand, the calculation of ∆ -can conveniently be performed numerically. The results are represented as green dotted domains in Figures 234. The corners of the domain ∆ -correspond to all combinations of F 1 in {(η 2 1 η 2 ) 1/3 diag(s, 1/s, 1), (η 2 1 η 2 ) 1/3 diag(1/s, s, 1)} and F 2 in R 2 .{(η 2 1 η 2 ) 1/3 diag(s, 1/s, 1), (η 2 1 η 2 ) 1/3 diag(1/s, s, 1)}. t R 2 . As can be observed in Figures 234, the green domain ∆ -fills most of the domain ∆ + , which means that most of the values of (ω, δ ) in ∆ + can be realized by laminate textures. The gap between ∆ -and ∆ + could possibly be reduced by considering more complex polycrystalline textures.

Bounds on the volume fractions

The functions (A r i , B r i ) that define the bound K+ in (57) are of special interest as they give some information on the martensitic microstructures realizing any given energy-minimizing deformation gradient F. From (55), A r i ( F) and B r i ( F) are indeed lower and upper bounds on the volume fractions of martensitic variant i in orientation r.

As a first example, calculating A r i ( F) and B r i ( F) for the deformation gradient F corresponding to Figure 5 gives the restrictions

0 ≤ Θ 1 1 ≤ 0.0844, 0 ≤ Θ 2 1 ≤ 0.1879, 0.3686 ≤ Θ 1 2 ≤ 0.5723, 0.0329 ≤ Θ 2 2 ≤ 0.3, 0.1838 ≤ Θ 1 3 ≤ 0.3433, 0.0782 ≤ Θ 2 3 ≤ 0.3. (70)
The polycrystalline texture and the martensitic microstructures displayed in Figure5 correspond to Θ

1 1 = 0, Θ 1 2 0.4662, Θ 1 3 0.2337, Θ 2 1 = 0, Θ 2 2 0.2, Θ 2 3 0.1.
Such values are compatible with the bounds (70), as expected.

In Figure 6, the bounds (A 1 i , B 1 i ) are displayed as functions of the shear δ , for several values of ω. The volume fraction χ 1 is set to 0.7. It is important to note that, because of the condition ∑ i Θ 1 i = χ 1 , the volume fractions Θ 1 i cannot reach any arbitrary value that is compatible with the bounds (A 1 i , B 1 i ). For instance in the case δ = 0, the upper bounds B 1 i on Θ 1 i are all equal to 1/3 but the volume fractions Θ 1 i cannot be simultaneously equal to the 1/3 (as otherwise the constraint ∑ i Θ 1 i = χ 1 = 0.7 would not be satisfied). There is no general trend concerning the variations of (A 1 i , B 1 i ) with δ . For instance, in the examples displayed in Figure 6, the function B 1 1 can either be decreasing, increasing or non monotonic, depending on the value of ω considered. Some intuitive insight in the behavior of the bounds (A 1 i , B 1 i ) can be obtained in the infinitesimal strain approximation, using the idea that the martensitic variants that are most likely to develop are those which are the most favorably oriented with respect to the applied strain, i.e. those that maximize the quantity z i (ω, δ ) = tr(ε 1 i .ε(ω, δ )). In that last equation, ε(ω, δ ) = (F(ω, δ ) + t F(ω, δ ))/2 -I is the infinitesimal strain associated to F(ω, δ ) and ε 1 i = (U 1 i + t U 1 i )/2 -I is the infinitesimal transformation strain for variant i. For the case at hand, we find

z 1 (ω, δ ) = δ 2 (η 1 -η 2 ) sin ω , z 2 (ω, δ ) = - δ 2 (η 1 -η 2 ) sin ω , z 3 (ω, δ ) = 0.
The cases (a),(b) and (c) in Figure 6 correspond to sin ω > 0. Therefore, for positive values of δ , variant 1 is favorably oriented and variant 2 is unfavorably oriented. Accordingly, when δ grows from 0, the bounds

(A 1 1 , B 1 1
) initially increase while the bounds (A 1 2 , B 1 2 ) decrease. This is in line with the intuition that more of variant 1 and less of variant 2 is expected to develop as δ increases from 0. Since z 3 (ω, δ ) = 0, variant 3 is neither favorably nor unfavorably oriented. Accordingly, the corresponding bounds (A 1 3 , B 1 3 ) remains stationary (for δ small enough). The case (d) in Figure 6 is slightly different as it corresponds to sin ω = 0. In such case we have z i (ω, δ ) = 0 for all i, which explains that all the bounds (A 1 i , B 1 i ) remains stationary. All of this reasoning is limited to the infinitesimal strain approximation, i.e. to small values for δ . As δ becomes large, rotation effects become significant and the qualitative arguments explained above do no longer apply. In particular, in the case (a), we can observe that B 1 1 decreases for large values of δ . The transformation F(ω, δ ) ceases to be energy-minimizing when one of the constraints

A 1 i < B 1 i or ∑ i A 1 i ≤ χ 1 < ∑ i B 1 i is violated.
The corresponding value of δ is displayed as a vertical dashed line in Figure 6. For instance, the constraint A

1 3 < B 1 3 (resp. A 1 2 < B 1 2
) is the limiting one in the case ω = 0.2 (resp. ω = 0.75), see Figure 6 (a) and (b). For ω = 1, Figure 6(c), the limiting constraint is ∑ i A 1 i < χ 1 . All the remarks made so far on the bounds (A 1 i , B 1 i ) for orientation 1 can be transposed to orientation 2. In particular, there are some values of ω for which the limiting constraints are associated with (A 2 i , B 2 i ).

Concluding remarks

In this paper, some rigorous upper bounds on the energy-minimizing strains of martensitic polycrystals have been obtained in the geometrically nonlinear setting. The main results are the bounds K0 + and K+ (defined in ( 27) and (49) 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 ( 27)-(49): each choice of C generates corresponding bounds K0 + and K+ . For a given (say discrete) C , the bound K+ is tighter than K0 + but more difficult to calculate: whereas checking if a given deformation gradient F is in K0 + is a direct calculation, checking if F ∈ K+ amounts to detecting feasibility of a linear programming problem in R n m . In this paper, the bounds K0 + and K+ have been used to study a 2-orientation/3-variant polycrystal, in which case the calculations could be performed in closed form (for a well chosen C ). For more complex textures, it is clear that numerical calculations of the bounds will be necessary at some point, which requires adequate algorithms. In that regard, it can be noted that interior-point methods offer some efficient algorithms for detecting feasibility in large-scale linear programming problems. Interestingly, such algorithms, as the self-dual algorithm of Ye [START_REF] Ye | On homogeneous and self-dual algorithm for linear complementarity problems[END_REF], has been used in other problems related to shape-memory alloys [START_REF] Peigney | Numerical simulation of shape memory alloys structures using interior-point methods[END_REF] and could possibly be useful for calculating the bound K+ in the case of a complex polycrystalline texture. A more theoretical line of investigation consists in deriving upper bounds taking more information on the texture (such as 2-point statistics) into account. Now consider a deformation gradient V + on the boundary of the domain in Figure 7, and let V -be its symmetric with respect to the u 23 = 0 axis. We can write

V + = η 1 0 0 0 v 22 v 23 0 v 23 v 33 , V -= η 1 0 0 0 v 22 -v 23 0 -v 23 v 33
and we assume that v 23 ≥ 0. Using the fact that V + satisfies (71) as an equality, we can relate (v 22 , v 33 ) to v 23 by the expressions

v 22 = 1 2 (S + ε S 2 -4P) , v 33 = 1 2 (S -ε S 2 -4P) (72)
with S = -v 23 + v 2 23 + (η 1 + η 2 ) 2 , P = η 1 η 2 + v 2 23 and ε = ±1. Following [START_REF] Ball | Proposed experimental tests of a theory of fine microstructure and the two-well problem[END_REF], deformation gradients on the boundary of the domain in Figure 7 can be realized by a simple laminate of variants 2 and 3, i.e. there exists c ± in [0, 1] and R ± ∈ SO(3) such that

V ± = R ± .(U 2 + (1 -c ± )b ⊗ n) (73)
where (b, n) are solutions of the twinning equation (U 2 + b ⊗ n).U -1 3 ∈ SO(3). There are actually two sets of vectors (b, n) solving the twinning equation. We denote them by (b + , n + ) and (b -, n -). For (η 1 , η 2 ) = (1.5, 1), we have n + = x(v 2 -v 3 ) and n -= -x(v 2 + v 3 ) with x 0.6934. It turns out that V + satisfies (73) with (b, n) = (b + , n + ), while V -satisfies (73) with (b, n) = (b -, n -).

Eq. (73) expresses the fact that V ± is realized by a laminate with a direction of lamination equal to n, mixing variants 2 and 3 in proportion c ± and 1c ± respectively. As V + and V -are symmetric with respect to the u 23 = 0 axis in Figure 7, it can be proved that c + = c -. The exact value of c + depends on the deformation gradient V + considered.

The same argument as used in Section 6.2 shows that V + and V -are rank-1 connected. Therefore K23 contains the deformation gradient

H(c) = V + + (1 -c)b ⊗ n
where (b , n ) solve the twinning equation (V + + b ⊗ n ).V -1 -∈ SO(3). The parameter c can take any value in [0, 1]. Let s be the operator that maps any given F in GL + (3) to its symmetric definite positive part U in the polar decomposition F = R.U. When c varies from 0 to 1, the tensor s(H(c)) remains in K23 and varies between V + and V -, as represented in Figure 7 (red lines). Since the twinning equation (V + + b ⊗ n ).V -1 -∈ SO(3) generally admits two sets of solutions (b , n ), there are two branches c → s(H(c)) to be considered, as can be seen in Figure 7. Independently on the branch considered, it can be verified that v 2 .s(H(1/2)).v 3 = 0, i.e. the curve v → s(H(c)) in Figure 7 crosses the u 23 = 0 axis for c = 1/2. It follows that s(H(1/2)) as a matrix representation of the form diag (η 1 , y, η 1 η 2 /y) with y = v 2 .s(H(1/2)).v 2 . Hence s(H(1/2)) is equal to F 0 1 if

v 2 .s(H(1/2)).v 2 = (η 1 η 2 2 ) 1/3 . ( 74
)
Recall that H(1/2) depends on V + , the latter being parametrized by v 23 . The function v 23 → v 2 .s(H(1/2)).v 2 is represented in Figure 8 for one of the branches c → H(c), with ε = 1 in (72). As can be observed on Figure 8, the condition (74) is satisfied for

Fig. 1

 1 Fig. 1 Representation of the deformation I + δ u(ω) ⊗ v(ω) : reference (left) and deformed (right) configuration.

Fig. 3

 3 Fig. 3 Bounds on the shear δ for MnCu, χ 1 = 0.7.

Fig. 4

 4 Fig. 4 Bounds on the shear δ for MnCu, χ 1 = 0.5.

Fig. 5

 5 Fig. 5 Example of energy-minimizing strain in a laminated texture, χ 1 = 0.7: (a) reference configuration; (b) deformed configuration; (c) details of the martensitic microstructure in a monocrystalline layer.

Fig. 6

 6 Fig. 6 Lower bound A 1 i and upper bound B 1 i on the volume fractions θ 1 i for MnCu, χ 1 = 0.7: ω = 0.2 (a), ω = 0.75 (b), ω = 1 (c), ω = π/2 (d).

Fig. 7

 7 Fig. 7 Representation of K23 .

Fig. 8

 8 Fig. 8 Determination of the microstructural parameter v 23 .

A Bounds on the volume fractions for a two-orientation polycrystal of tetragonal martensite

Substituting the values of Table 1 in (54) and recalling that 0 ≤ Θ 1 k ≤ χ 1 , we find that

where

,

In those expressions, the indexes (i, j) are such that (i, j, k) is a permutation of (1, 2, 3). Replacing v k by R 2 .v k and swapping the roles of orientations 1 and 2, we obtain in a similar fashion that

Although somewhat lengthy, the obtained expressions are fully explicit and easy to calculate.

B Construction of a microstructure realizing the deformation gradient F 0 1 in Eq. ( 69)

In this Appendix, we determine a martensitic microstructure realizing the deformation gradient F 0 1 in (69) for a reference single crystal with transformation strains given by [START_REF] Talbot | Bounds for the effective constitutive relation of a nonlinear composite[END_REF]. To that purpose, we first observe that F 0 1 is in the quasiconvex hull of SO(3)U 2 ∪ SO(3)U 3 (which is denoted by K23 from now on). Following [START_REF] Ball | Proposed experimental tests of a theory of fine microstructure and the two-well problem[END_REF], symmetric positive definite tensors U in K23 are indeed characterized by matrix representations (in the basis (v 1 , v 2 , v 3 ) of the cubic austenitic lattice) of the form

The fact that F 0 1 is in K23 means that F 0 1 can be realized by a microstructure involving variants 2 and 3 only. Also observe from the characterization given above that symmetric positive definite tensors U in K23 can be parametrized by their components (u 23 , u 22 ). In Figure 7 is represented the set of values (u 23 , u 22 ) corresponding to symmetric positive definite tensors U in K23 .


S Pinson

Comparison of seaoor sound-speed prole measurement between the image source and reection seismology techniques

Keywords: numbers: 43.30 Ma, 43.30 Pc, 43.60 Fg, 43.60 Rw

In the context of sediment characterization, the image source method provides a fast and automated sound-speed prole measurement of the seaoor. In this letter, this method is compared to reection seismology methods. The comparison highlights similarities in the signal processing and a dierence in the basic equations of the wave reection travel time analysis. Understanding the links between the two methods helps sheds additional light on each method.

Introduction

A technique has recently been developed for rapidly estimating sub-seabed sound-speed proles using the concept of image sources.

The image source method (ISM) has been used for measurements from a vertical receive array 4,5 which yields a point estimate of sound speed vs depth in the seabed.

More recently it has been applied to horizontal array data 4,6 yielding the two-dimensional (depth vs range) sound-speed structure of the seabed to 50 m in sub-bottom and 14 km in range with resolutions roughly 1 m in depth and 5 m in range.

It turns out that the fundamental equations of the image source method can be understood and compared with those from classical seismic reection methods. This is the purpose of this letter, rst to explain the underlying connections between the two methods. The secondary purpose is to briey consider, in the light of the connections, relative strengths and possibilities of extension for the ISM.

The letter is organized as follows. First, the principles of both methods are presented. This is followed by a short discussion to explore some basic dierences and similarities between them.

Basic ideas of the methods

A typical measurement conguration for seaoor characterization by reection seismology is a source and a linear array of hydrophones (Fig. 1a). The source emits a low frequency, short, powerful pulse such that the wave goes deep into the seaoor. Then the wave is reected from geological interfaces and recorded by the hydrophone array.

For illustration purposes, a set of data has been simua) Electronic address: samuelpinson@yahoo.fr nmo equivalent sound-speed medium. In gray is the ray diagram for the image source l located in a water sound-speed c (0) medium.

interface l t (l) ⊥ × c (l) nmo θ (l) nmo θ (l) nmo image source located in a water sound-speed medium t ( l ) × c ( l ) n m o t ( l ) × c ( 0 ) θ (l) 0 z x
t (l) n ≈ t (l) ⊥ 2 + x n -x s c (l) nmo 2 , (1)
If one assumes that c

(l) nmo ≈ c (l) rms with: c (l) rms = l i=0 (c (i) ) 2 τ (i) ⊥ l i=1 τ (i) ⊥ , (2)
where c (l) rms is the root mean square sound speed, c (i) is the layer i sound speed and τ (i) ⊥ is the travel time at normal incidence in the layer i, then one can apply the Dix formula to obtain layer sound speeds 2 :

c (l) = (c (l) rms ) 2 t (l) ⊥ -(c (l-1) rms ) 2 t (l-1) ⊥ t (l) ⊥ -t (l-1) ⊥ . (3)
There are dierent ways to determine the normal move out sound speeds. One of them consists in calculating a velocity spectrum (velocity rather than sound speed is used in the seismic community) 10 . The velocity spectrum is calculated by summing the recorded signals along hyperbolas (colored curves on Fig. 1b) dened by Eq. ( 1):

V S(t ⊥ , c nmo ) = 1 N N n=1 s H n   t 2 ⊥ + x n -x s c nmo 2   2 , (4)
where s H n (t) is the analytic signal recorded at the hydrophone n (the superscript H stands for Hilbert transform). The velocity spectrum is not sensitive to low amplitude reected pulses thus making it dicult to detect low amplitude reections. The semblance velocity spectrum avoids that problem:

V S semb (t ⊥ , c nmo ) = 1 N N n=1 s H n t 2 ⊥ + xn-xs cnmo 2 2 1 N N n=1 s H n t 2 ⊥ + xn-xs cnmo 2 2 .
(

) 5
The semblance is a similarity measurement and is 1 when signals are identical along travel time hyperbolas and close to 0 when dierent. It is then possible to select local maxima of V S semb (t ⊥ , c nmo ) to calculate a soundspeed prole using Eq. (3).

Image source method

The idea of the ISM is to consider wave reections from geological interfaces as waves coming from image sources located symmetrically to the real source relative to the interfaces. The image sources are located in a water sound speed c (0) medium by migrating the recorded signals in the space domain:

I m (r) = 1 N N n=1 s H n (t n (r)) 2 , ( 6
)
where t n (r) = r n r /c (0) and r = (x, z) is the spatial coordinate.

This I m (r) function is displayed in Fig. 3a where the 3 images relative to the 3 interfaces are apparent at z ≈ -40, -50 and -62 m. But because their amplitudes are a function of impedance contrast, this means that low impedance contrast layers will be dicult to detect. So a semblance function of the migrated signals is used for an automatic detection:

I semb (r) = 1 N N n=1 s H n (t n (r)) 2 1 N N n=1 |s H n (t n (r))| 2 . (7)
The semblance function in Fig. 3d c and the angle of arrival θ (l) 0 , it is also possible to obtain the sound speed of the homogeneous media, equivalent in travel time, to the layered media. Eq. ( 1) of the echo travel time is rewritten using the angle of arrival on the sensor:

t (l) c ≈ 1 sin θ (l) nmo × |x c -x s | c (l) nmo , ( 8
)
where sin θ

(l)
nmo can be deduced from sin θ (l) 0 through the Snell-Descartes relation. Then the equivalent medium sound-speed is given by:

c (l) nmo = c (0) |x c -x s | t (l) c sin θ (l) 0 . ( 9
)
Finally, using the Eq. ( 3) one can obtain the layer sound speeds.

Link between the methods

One can notice the strong similarity between Eq. ( 4)

and Eq. ( 6), the velocity spectrum and the image source map respectively. Both consist of summing the signals along hyperbolic travel times which are driven by dierent parameters: c nmo and t ⊥ for the velocity spectrum and, x and z for the image source map.

The similarity is such that it is possible to change the image source map into a velocity spectrum through a change of variables from r = (x, z) to (c nmo , t ⊥ ) using Eq. ( 9) and simple trigonometry (Fig. 2):

c nmo = c (0) |x c -x s | t c sin θ 0 , (10)
and:

t ⊥ = t c cos θ nmo = t c 1 - c nmo c 0 sin θ 0 2 (11)
where

t c = r c -r /c 0 and θ 0 = tan -1 |xc-x| |zc-z| .
The result of this variable change is shown on Fig. 3b for the I m map and on Fig. 3e for the I semb map. Fig. 3c and 3f are a zoom on the NMO sound speeds of Fig. 3b and 3e. 6)) and I semb (Eq. ( 7)). The red point corresponds to the real source and the black points correspond to the hydrophones. (b) and (e): Im (Eq. ( 6)) and I semb maps with coordinates changed from x and z to cnmo and t ⊥ using Eq. ( 10) and Eq. ( 11) in order to obtain a velocity spectrum. (c) and (f): Zoom of (b) and (e). The colorbars on the right are related to the rows. The Im colormap is normalized by the maximum. I semb is intrinsically between 0 and 1.

shape for the migration function, the semblance shows a coherent value of 1 for the whole pulse duration (in the case of a good signal to noise ratio). This interpretation in terms of angle and range resolution is lost for the semblance velocity spectrum (Eq. ( 5)). It results in the impossibility of an accurate automatic detection of the NMO sound speeds using this semblance velocity spectrum. Because of that, using the ISM, the semblance function is only used to window small areas of the migration function (Eq. ( 6)) in which signals are summed coherently and in which accurate localization is performed using local maxima. The same process should be apply when using velocity spectrum.

Using Eq. ( 1) to dene the travel times implies the use of an horizontal array for seismic techniques. By contrast in ISM, since the sound-speed equations are a function of the array center coordinate r c , the array conguration is very exible as long as its aperture is not too large.

Indeed, the rst results of the ISM was obtained using a vertical array of hydrophones moored on the seaoor 5 .

Originally the ISM did not employ the NMO approximation to give sound-speed proles. Nevertheless, this formulation, borrowed from reection seismology, could be useful for eliminating the parallel layering restriction.

Conclusion

In its present form, the method requires the Born approximation (the multiple reections between the geologic interfaces are neglected), at and parallel geologic interfaces and, isotropic and homogeneous layers. The reection seismology methods have far fewer assumptions, given many decades of progress since their introduction.

For example, extension to include the anisotropy 8,9 or inhomogeneities/statistical parame- ters 3,7 . It is possible, now with a clearer understanding of the connections between the ISM and the classical methods, that ISM may be able to exploit some of these advances.

Figure 1 :

 1 Figure 1: (a): Sketch of the experiment. (b): Simulated seismogram of the seaoor reections.

10 .

 10 To determine c (l) nmo , the three sides of the right triangle are considered: the oset between the source and the receiver (x s -x n ), twice the depth of the interface t ⊥ is the null oset travel time (i.e. at normal incidence) of the interface l reection and t (l) n is the travel time of the interface l reection at the oset (x s -x n ). Using the Pythagorean theorem, the travel time t (l) n of the reection from interface l is written: source rs = (xs, 0) image source sensor rn = (xn, 0) rc = (xc, 0)

Figure 2 :

 2 Figure 2: Ray diagram for the reection on the interface l in a c (l)

6 .Fig. 2 )

 62 Fig.2). From the location of image sources, one can deduce the travel times t (l) c between the image source l and the array center r c and the angle of arrival θ (l) 0 (the subscript stands for the angle of incidence in water i.e. layer 0 and the upper index stands for the interface number on which the wave is reected). With these parameters it is then possible to deduce layer sound speeds and thicknesses using ray theory 1,46 . From the travel time t (l)

  Eq. (1) and Eq. (8) provides the basic dierence between classical reection seismology and the image source method. From the reection seismology point of view, the problem is written considering the right triangle dened by the source, the receiver and the image source in a c (l) nmo equivalent sound-speed medium. From the ISM point of view, the problem is written considering the source-receiver oset, the travel time and the angle of arrival on the receiver in a water sound speed c (0) medium. In reection seismology, one considers more realistic travel times by approximating the wave front shape as being ellipsoidal. From a physics point of view, reection seismology is better because ISM approximates the wave front as spherical. For this reason the ISM has to use a limited aperture array to enable a coherent echo summation in the image source localization process.As the ISM is an intuitive way of modeling the wave reections on interfaces, it becomes useful to understand some possibly unexpected phenomena associated with the velocity spectrum semblance function. Using the migration function (Eq. (6)) to nd the image sources, the resolution of the image location is driven by two dierent experimental parameters: the array aperture that drives the angle resolution of the images and the pulse shape that drives the range resolution of the images to the array. Using the semblance function for the ISM, the angle resolution remains the same but the range resolution is lost. Indeed, if the range resolution is driven by the pulse

Figure 3 :

 3 Figure 3: (a) and (d): Image source using respectively Im (Eq. (6)) and I semb (Eq. (7)). The red point corresponds to the real source and the black points correspond to the hydrophones. (b) and (e): Im (Eq. (6)) and I semb maps with coordinates changed from x and z to cnmo and t ⊥ using Eq. (10) and Eq. (11) in order to obtain a velocity spectrum. (c) and (f): Zoom of (b) and (e). The colorbars on the right are related to the rows. The Im colormap is normalized by the maximum. I semb is intrinsically between 0 and 1.

Acknowledgements

The author would like to gratefully acknowledge Charles Holland for his help and stimulating discussions about that work. This work is funded by Direction Générale de l'Armement (DGA) through the Laboratoire Domaine Océaniques (LDO) of the Institut Universitaire Européen de la Mer (IUEM).


Velocity Measurement from the Spectral Phase of a Match-Filtered LFM Chirp.

Samuel Pinson

Abstract-Chirp signals are very convenient to perform acoustic measurement with high signal-to-noise ratios. But measurement using those signals may present some drawbacks if a transducer or a target is moving. Indeed match-filtered chirp signals are not Doppler tolerant. The Doppler effect affects their amplitude, delay and resolution. To perform a correct match filtering that includes the Doppler shift requires a prior knowledge of the Doppler velocity. In this paper, it is demonstrated that the Doppler velocity can be extracted directly from the Doppler cross-power spectrum. More precisely, the quadratic coefficient of the Doppler cross-power-spectrum phase is proportional to the relative velocity. Consequently, the method has a low computational cost.

Index Terms-Doppler effect, ambiguity function, linearfrequency-modulation (LFM) chirp, velocity measurement, pulse compression.

I. INTRODUCTION

W HEN USING A CHIRP for acoustic measurement, the match-filter output can be strongly affected by the Doppler effect. Pulse compression without a proper compensation of the Doppler effect affects amplitude, delay and resolution of the signal [START_REF] Lin | Wideband ambiguity function of broadband signals[END_REF]. To perform the correct correlation to compensate the relative motion between a source and a receiver or between an active system and a target, it is necessary to first measure the relative velocity or the target relative velocity. One way to measure the relative velocity is to calculate the maximum likelihood by cross-correlating the match-filter output with the ambiguity function calculated for a wide range of velocities. But that operation might be computationally expensive for embedded system or to deal with an important number of measurements.

For particular applications such as underwater acoustic communications [START_REF] Sharif | A computationally efficient doppler compensation system for underwater acoustic communications[END_REF] a rough velocity estimation can be made measuring the time interval between two pulses obtained after match filtering and use a phase locked loop [START_REF] Johnson | Improved Doppler tracking and correction for underwater acoustic communications[END_REF] to optimize the Doppler effect compensation. Under some conditions, it is possible to extract directly the information from the matchfiltered signal. In [START_REF] Altes | Sonar velocity resolution with a linearperiod-modulated pulse[END_REF], it is shown that the relative velocity can be measured from the instant frequency variation in a matchfiltered echo if the source signal is linear-period modulated (LPM). In this paper, it is shown that the relative velocity can be assessed directly from a selected echo in the signal when the chirp has a linear-frequency modulation (LFM). S. Pinson is with the Laboratório de Vibrac ¸ão e Acústica, Universidade Federal de Santa Catarina, SC 88040-370, Floriaónopolis, Brazil e-mail: samuelpinson@yahoo.fr.

II. THEORY

Considering a moving source transmitting a LFM chirp and a fixed receiver, the analytical emitted signal is given by:

s(t) = w(t) exp i2π f 0 t + α 2 t 2 , (1)
where w(t) is an envelope centered at the origin (typically a Gaussian or a rectangle function can be used). f 0 + αt is the instant frequency. The Doppler dependent received signal is:

r(t) = As(η(t -∆t)) , (2)
where A is an amplitude factor due to the wave propagation, η = 1 + v/c is the time-stretching factor, c is the sound-speed and v is the radial relative velocity and is defined positive when the source is moving towards the receiver. After match filtering, the received signal becomes:

χ r (t) = As(η(t -∆t)) * s(-t) , (3)
where * is the convolution operator and ∆t is the travel time between source and receiver.

A. Ambiguity function

To deduce the relative velocity v, one can match the received signal with numerous Doppler stretched emitted signals s(ηt) and find the maximum of correlation. Nevertheless, the emitted chirp has a long duration and it might be a better strategy to work on the match-filtered signal. Thus the compressed pulse can be windowed and the processing can be performed on a shorter signal. Then the match-filtered signal can be compared to the ambiguity function:

χ af (t, η) = s(η(t)) * s(-t) , (4)
by searching for the maximum of correlation between χ r (t) and χ af (t, η):

η = argmax η [χ r (t) * χ af (-t, η)] . (5)
This operation requires to compute convolutions for a large number of relative velocity and is consequently computationally expensive.

B. Doppler cross-power spectrum

Instead of correlating the match-filtered signal with the ambiguity function for various relative velocity, it is preferable to extract the velocity directly from the signal. More precisely, the velocity can be extracted from the Doppler cross-power spectrum:

FT{χ r (t)} = FT{r(t)} × FT{s(t)} * = R(f ) × S * (f ) , (6)
where F T {.} means the Fourier transform.

To derive the Doppler cross-power spectrum, the Fourier transform of the emitted LFM signal is first calculated:

S(f ) = FT{s(t)} = ∞ -∞ w(t) exp i2π f 0 t + α 2 t 2 -f t dt . (7)
Inside the integral, the exponential term is varying much faster than the smooth function w(t). This kind of integral can be evaluated by the asymptotic approximation of stationary phase [START_REF] Lin | Wideband ambiguity function of broadband signals[END_REF]:

∞ -∞ w(t) exp [iφ(t)] dt = w(t st ) i2π φ (t st ) exp [iφ(t st )] .
(8) where t st is the time of stationary phase where φ (t) = 0. The phase term in 7, its derivative and its second derivative are:

φ(t) = 2π f 0 t + α 2 t 2 -f t , φ (t) = 2π (αt + f 0 -f ) , φ (t) = 2πα , . (9)
One can see that the phase is stationary at t = (f -f 0 )/α. Using 8, the equation 7 results in:

S(f ) = w f -f 0 α i α exp -iπ (f -f 0 ) 2 α . (10)
To calculate the Fourier transform of the received signal, one can make the same calculation with the stationary phase t = (f /η -f 0 )/(ηα), or use the Fourier transform property FT{s(ηt)}(f ) = 1/ηS(f /η):

R(f ) = A η w f /η -f 0 α i α × exp -iπ (f /η -f 0 ) 2 α -i2πf ∆t . (11)
S(f ) has a wideband spectrum. So it is not possible to extract the velocity information by looking at the spectrum shift in frequency. But one can notice the Doppler phase dependence by looking at the argument of the Doppler crosspower spectrum:

arg [R(f ) × S * (f )] = π (f -f 0 ) 2 α -2πf ∆t-π (f /η -f 0 ) 2 α , (12
) which is a quadratic polynomial:

arg [R(f ) × S * (f )] = p 2 f 2 + p 1 f + p 0 , with: p 2 = π α 1 - 1 η 2 , p 1 = - 2π α f 0 1 - 1 η -2π∆t , p 0 = 0 . ( 13
)
The coefficient p 2 depends only on fixed parameters and the relative velocity. Note that the coefficient p 1 depends on the time delay ∆t plus an additional delay f 0 α -1 (1 -1/η) due to the Doppler effect. Making the approximations η 2 ≈ 1 + 2v/c and (1 + 2v/c)(1 -2v/c) ≈ 1, the relative velocity can be written as a simple function of the quadratic term coefficient:

v ≈ p 2 αc 2π . ( 14
)
For the active case (a fixed monostatic emitter/receiver and a moving target), we have:

η = 1 + v/c 1 -v/c ≈ 1 + 2v/c . ( 15
)
In that case, it is straightforward that equation 14 simply has to be divided by 2 to obtain the target relative speed (v ≈ p 2 αc/(4π) ).

C. Implementation

To extract the coefficient p 2 and deduce v, the following algorithm is proposed.

1) Perform the match filtering χ r (t) = r(t) * s(-t).

2) Make an automatic detection of the compressed pulse of interest. 3) Select a small portion of the signal that contains the pulse. 4) Compensate the pulse delay to center it around t = 0. 5) Compute the fast Fourier transform. 6) Unwrap the spectral phase. 7) Fit the unwrapped phase with a quadratic polynomial. 8) Calculate v using equation 14. Unwrapping the spectral phase is generally a delicate operation. This is why step 4 consists in delaying the selected pulse such that the spectral-phase variation is minimized.

III. EXAMPLE

For the example, a Gaussian function is chosen for the LFM chirp envelope w(t):

w(t) = 1 σ √ 2π exp - 1 2 t 2 σ 2 , (16)
where σ = 125 ms. The frequency is modulated around

f 0 = 1/2(f max + f min ) with α = (f max -f min )/(4σ).
Frequencies f min = 1 kHz and f max = 10 kHz are the instant frequencies at t = ∓2σ. The wave speed is set to the water sound speed c = 1500 m/s. A single pulse in the received signal is considered with a time delay ∆t = 200 ms. Match-filtered signals for v = -2, 0 and 2 m/s are plotted in Fig. 1. The effect of v = 0 on the simulated signals shows a lowered maximum amplitude, a time spreading with a varying instant frequency, and a shifted maximum peak time of the envelope. The additional delay corresponding to the Doppler effect f 0 α -1 (1 -1/η) = ±0.2 ms for v = ∓2 m/s is visible in Fig. 1a and1c.

In practice the match-filtered signal may be composed of multiple echoes. Taking the application example of an at sea seismic measurement with a hydrophone moored on the seafloor and a towed source, the pulse corresponding to the direct path between source and receiver must be distinct from the seafloor or sea-surface reflection such that they do not overlap. Thus the pulse can be detected by a peak detection and the signal portion containing that pulse can be extracted. The selected portion of the simulated pulse with v = 2 m/s is displayed in Fig. 2a. The pulse appears in the middle of the selected portion that is considered as a delay by the fast Fourier transform algorithm (FFT). The consequence is a rapid phase change as a function of frequency and possible difficulties to unwrap the phase. This is overcome by swapping left and right signal portion halves (Fig. 2b). Then the phase can be taken from the FFT for frequencies between f min and f max (Fig. 2c) and be unwrapped (Fig. 2d). The final step consists in fitting the unwrapped phase with a quadratic polynomial to obtain its coefficient p 2 and calculate v.

IV. CONCLUSION

A method has been described to measure radial relative velocity between a source and a receiver or between an active system and a target when using a LFM chirp source signal. It has a low computational cost and operates on matchfiltered signal if the echo of interest does not overlap another one. It has been demonstrated that the Doppler velocity is proportional to the quadratic coefficient of the Doppler crosspower-spectrum unwrapped phase. Thus there is no need to calculate various correlations to find the best match between the match-filtered signal and the ambiguity function.

Fig. 1 :

 1 Fig. 1: Simulated match-filtered signal for v =-2 m/s (a), 0 m/s (b) and 2 m/s (c)

Fig. 2 :

 2 Fig. 2: (a) Pulse extracted from the simulated signal. (b) Pulse with delay compensation. (c) Phase of the pulse Fourier transform. (d) Unwrapped phase.

ACKNOWLEDGMENT

The author wishes to gratefully thank Charles Holland who submitted this problem. This work is supported by the CAPES (Coordenac ¸ão de Aperfeic ¸oamento de Pessoal de Nível Superior from Brazil) through the young talent funding.