C Stolz

General Relationships between Micro and Macro Scales for the Non-linear Behaviour of Heterogeneous Media

This chapter is mainly concerned with the determinat ion of general relationships bet ween microscopic and macroscopic mechani cal properties for elastoplastic material with or without damage. The overall properties ar e determined in terms of the unkno wn properties of each constituent phase of the heterogeneous body. At first we must define a representative volume element (RVE) of the heterogeneous material for which the macroscopic mechani cal fields ar e some spatial av erage of the microscopic one. The determinat ion of local quantities is ac hieved by solving some particular boundary value problem on the R VE,Jrom which macroscopic quantities ar e derived. The essential structure of micro-macro relationships is presented in the case of elastoplastic material. A generalisation is given to take the temperature field into ac count. We present al so a model of damage and its av eraging process. Final ly we discuss the determinati on of macroscopic temperature through continuum thermodynam ics.

INTRODUCTION

There are two main ste p s in the determination of the overall p ro p erties of a material in terms of the known p ro p erties of the constituent p hases.

First, the com p lex geometry of the n-phase body and the micro-mechanical behaviour of each of the phases being given, one has to solve a complicated boundary value problem. The determination of the local mechanical fields (stress a, strain e, plastic strain eP) is achieved by a localization process that requires some particular boundary conditions, from which macroscopic fields (stress :I:, strain E, ... ) are derived. A comprehensive review of methods to solve some particular localization process is given by [START_REF] Willis | Overall properties of composites[END_REF], [START_REF] Walpole | On the overall elastic moduli of composite materials[END_REF] and [START_REF] Hashin | Analysis of composite materials[END_REF].

The second step is concerned with the derivation of relationships between microscopic tensors and macroscopic tensors, which differ only by a length scale. Generally the macroscopic tensor is some spatial average of the microscopic one. Such a relationship is presented in continuum thermodynamics by [START_REF] Francfort | Thermodynamique et lois de comportement thermo-mecanique homogeneisees[END_REF] and [START_REF] Germain | Continuum thermodynamics[END_REF]. This theoretical approach is summarized in the last section of this chapter.

This chapter is mainly concerned with the second basic problem, i.e. the determination of general relationships between microscopic and macroscopic fields for elastoplastic materials and for some model of damage. The essential structure of micro-macro relationships in elastoplastic materials is given in the previous works of [START_REF] Hill | The elastic behaviour of a crystalline aggregate[END_REF][START_REF] Hill | Continuum micromechanism of elastoplastic polycrystals[END_REF] and [START_REF] Mandel | Proc. 11th Int. Congr. Appl. Mech[END_REF][START_REF] Mandel | Plasticite Classique et Viscoplasticite[END_REF]. After a general review and a second part concerned with the determination of the macroscopic elastic modulus of a heterogeneous material, we generalize the above-mentioned works and also our own previous results in plasticity to take the temperature field into account in Section 4.

In Section 5 we shall discuss the micromechanics of damage and its global average. Damage phenomena are due to microcracking in the body, for instance debonding between dissimilar materials, or at grain boundaries in polycrystals, and to the growth of holes or microcavities. Even if we assume small strain, the evolution of damage implies change of geometry, which is a non-linear and irreversible process.

In the last section we present a discussion about the determination of macroscopic temperature.

PRELIMINARY CONCEPTS

We consider a volume V, in which we distinguish two different scales, the micro one where the material properties vary from point to point like a highly heterogeneous medium, and the macro one where the properties are those of a homogeneous continuum.

To describe a heterogeneous medium it is essential to define a volume element which is small enough to allow us to distinguish the microscopic heterogeneities and sufficiently large to represent with accuracy the overall behaviour. This volume element is called a representative volume element (RVE); for example, the RVE of a periodic composite is simply the elementary cell [START_REF] Suquet | Plasticite et homogeneisation, These Universite Pierre et Marie Curie[END_REF]. For a randomly distributed composite of a polycrystalline aggregate it is necessary to have a minimum of information about the geometry of the constituent phases and to make assumptions like statistical homogeneity or ergodicity in order to define the RVE in a statistical sense [START_REF] Beran | Statistical Continuum Theories[END_REF][START_REF] Fokin | Statistical properties of inhomogeneous solid media: central moment fu nctions of material characteristics[END_REF][START_REF] Kroner | Linear properties of random media: the systematic theory[END_REF][START_REF] Willis | Overall properties of composites[END_REF][START_REF] Hashin | Analysis of composite materials[END_REF].

The problem is to characterize the overall behaviour only from a knowledge of the local constitutive equation of each constituent phase of the heterogeneous medium. On the microscopic level the mechanical properties vary from point to point with the local scale y while the average quantities vary smoothly with the macroscopic scale x.

With every microscopic quantity f (stress a, mass per unit volume p, deformation e, heat flux q, ... ) we can associate its macroscopic value

J by an averaging process on the RVE, extended eventually in the cases for which RVE contains holes or rigid inclusions:

-11.

f = <f) = V / (y) dv Y (2.1)
It is clear that eqn (2.1) defines a unique macro state quantity for each micro state.

Conversely, starting with a macro state, the definition of a suitable corresponding micro state requires complementary information or a localization process to determine the nearest possible micro state of the real one. For the choice of a representative volume element suitable boundary conditions must be prescribed and must satisfy some requirements [START_REF] Francfort | Thermodynamique et lois de comportement thermo-mecanique homogeneisees[END_REF][START_REF] Germain | Continuum thermodynamics[END_REF].

In particular, for study of the macroscopic behaviour these boundary conditions can be homogeneous; in this case one prescribes homogeneous stress conditions (T d = �.n on a V) or homogeneous strain conditions (U = E.y for yE aV) (where� and E are two symmetric tensors).

For these boundary conditions, a local stress distribution a and a local deformation e are developed over the volume V.

In the case of homogeneous loading the stress a verifies the equation of equilibrium and the boundary conditions Dually, if u = u over av it is clear that

1 i 1 E = V ovl(u ® n + n ® u) da = (e) = e
(2.7) and the microstress is such that

L (Cfn -an)® y da = 0 av (2.8)
and the macrostress I: = Cf = (a).

Some other boundary conditions can be used; in particular, for periodic structures [START_REF] Sanchez-Palencia | Comportement local et macroscopique d'un type de milieux physiques heterogenes[END_REF][START_REF] Suquet | Plasticite et homogeneisation, These Universite Pierre et Marie Curie[END_REF], the elementary cell V generates the geometry of the whole by periodicity, all the local fields are periodic functions and the Hill-Mandel macro homogeneity condition is a necessary condition resulting from periodicity.

The construction of the macroconstitutive law from the microscopic behaviour can proceed as follows: starting from some macroscopic quantities, using a localization process to determine local fields, then the averaging process determines the missing macro quantities.

OVERALL PROPERTIES OF AN ELASTIC COMPOSITE MATERIAL

The main difficult problem is to determine effectively the overall property of a composite material. We shall restrict our discussion to some methods for the determination of the overall moduli for an elastic material.

The heterogeneous body is characterized by n phases with different elastic properties. The polycrystal aggregate is a particular case of a heterogeneous body, where the phases differ in the orientation of each crystal. Self-consistent theories which apply to polycrystal aggregates need knowledge of the properties of each grain [START_REF] Hill | Continuum micromechanism of elastoplastic polycrystals[END_REF]Berveiller and Zaoui, 1979). [START_REF] Kroner | Bounds of effective elastic moduli of disordered materials[END_REF][START_REF] Kroner | Linear properties of random media: the systematic theory[END_REF], by a statistical approach, has compared the homogenization method of periodic media with the self consistent scheme. The periodic model is a representative model of perfect order; one needs to solve a boundary value problem in a single cell, and the self-consistent method is more appropriate to model perfect disorder,

Bounds of the Elastic Modulus for a Heterogeneous Body

The medium is assumed to be elastic; each phase r has an elastic constitutive law with modulus 1\ (e = A,. a, nr = A; 1 ). We suppose that the characteristic function </>, of each phase is known:

</>,(y) = 0 </>,(y) = 1
We have the equality y does not belong to phase r l y belongs to phase r

1 n V L L </>, (y) dv Y = 1 r = I (3.1) (3.2)
and the concentration of the r phase is c, = bL ct>,(y) dv y (3.3) Several works [START_REF] Beran | Use of classical variational principles to determine bounds for the effective bulk modulus in heterogeneous media[END_REF]Hashin and Shtrikman, 1962;[START_REF] Hill | Elastic properties of reinforced solids: some theoretical principles[END_REF] are related to the bounds of the moduli using minimum principles. For all displacements u* kinematically admissible with given boundary conditions on a v the real field minimizes the energy:

l fl U(u*) = V 2 e* rr s* dv y (3.4)
then for a localization process, which describes a homogeneous deformation on iJV (u* = e.y,y E iJV), the following inequality holds:

U < } v e•I J. rr,<Pr(Y) dv y . 8 r V
From the macrohomogeneity hypothesis we have

U - 1 - 1- l __ =-erre = -ae = -ae 2 2 2 (3.5) (3.6)
and, noting ff the overall modulus (Cf = ff e), one can show that 1fv -1f is positive semi-definite, where the estimate 1fv for the overall modulus is the Voigt average [START_REF] Voigt | Ueber die Bezichung zwischen den beiden Elastizitats constanten isotrope Korper[END_REF]:

n 1fv = I c,n, (3.7) r = I
Dually, we use the minimum of complementary energy for a localization process which prescribes homogeneous tensions on a V:

U < 2�0'! f A,cp,(y) dvff r = I
Then, with the macrohomogeneity hypothesis, noting A the overall modulus (Cf = Aa), the inequality shows that Av-A is positive semi-definite and A;1 is an estimate of ff in the Reuss sense:

nR = A;l = (I c,A. r l r = I (3.9)
The two estimates rrv and rrR are lower and upper bounds. The best estimates are such that the localization process approaches the real solution. In the case of periodic media the answer is clear; for such a material n., with many elementary cells, if homogeneous tension or homogeneous deformation are applied on an., far from the boundary, the field solution of the boundary value problem is a periodic field; the averages of stress or strain on V are equal to the prescribed value. The real localization process is the periodic localization process, so the overall moduli obtained with homogeneous prescribed macrostress or macrostrain are equal.

For other composites, it is necessary to have complementary information. The calculus of overall properties is related to the determination of fields which satisfy equilibrium equations; perfect bonding across phase interfaces and some simplifications must be made to derive explicit results.

In the self-consistent scheme, for example, we consider that the inclusions are spherical or ellipsoidal, and when the medium is statistically uniform the equivalent body is assumed to be isotropic. To estimate the overall moduli, we can consider that each inclusion is embedded in a homogeneous matrix with the overall moduli A as elastic properties, and subject to a homogeneous field at infinity. The interaction problem between a matrix and a spherical or ellipsoidal inclusion is well known [START_REF] Eshelby | The determination of the elastic field of an ellipsoidal inclusion and related problem[END_REF][START_REF] Eshelby | Elastic inclusion and inhomogeneities[END_REF]. The microstress in the inclusion is homogeneous and an explicit result can be given in the formal equation a (r) = I: + J, : (E -e(r))

(3.10)

The interaction is described by J, which depends on the overall moduli A, on the inclusion geometry and on orientation (Berveiller and Zaoui, 1979).

If we take the statistical average and the local behaviour into account, we have the equation

I: = (er, ) I: AE and£ = (e) ] A = (�(� + J,r 1 (A + J,) ) (3.11) (3.12)
with unknown A Usually this equation is complex to solve; the determination ofJ, is not the easiest thing to do. Each phase seems to see the other phases through the homogeneous material; we have not given a good description of the interactions between all the inclusions. Some developments due to Berveiller and Zaoui (1979), [START_REF] Fokin | Statistical properties of inhomogeneous solid media: central moment fu nctions of material characteristics[END_REF] and Andrieux (1983) can be used to solve this problem with greater accuracy.

The first estimates (Voigt and Reuss estimates) are established using the minimull_l energy principle, and the bounds are obtained for extremal values associated with homogeneous boundary conditions. A successful method is to determine a suitable displacement field to have a better estimate. For this reason it is convenient to introduce a homogeneous comparison material with modulus Ao and to set the local modulus A equal to A0 with some perturbation oA: A= Ao +oA If e is the real deformation over V, the stress polarization p = cr-Aoe = 8A.e verifies (3.13)

(3.14) div(p + Aoe) = 0 over V (3.15)
and if we know the Green's function for the comparison material we have u(x) = u0(x) + L G(x, y) div p(y) dv y

(3.16)

The deformation e is the sum of the homogeneous deformation E = e0 and the last term of eqn (3.16), due to heterogeneity. In this integral formulation we can substitute the variational expression. Such methods are presented by Willis (1981) and Hashin and Shtrikman (1962ab, 1963). These authors have obtained bounds for overall moduli by different approximations of p and different averaging processes.

The Case of a Macrohomogeneous Body

In the last section we saw that the estimate of the elastic modulus is not the same when one prescribes macrohomogeneous strain or stress over V. But when the body can be considered as macro homogeneous in the sense of [START_REF] Hill | The elastic behaviour of a crystalline aggregate[END_REF][START_REF] Hill | Continuum micromechanism of elastoplastic polycrystals[END_REF] and [START_REF] Mandel | Proc. 11th Int. Congr. Appl. Mech[END_REF] (3.17)

and we can prove the existence of a localization tensor Aijpq such that

(3.18)

The tensor A is the elastic localization tensor introduced by Mandel (1964) in the case of a localization process in homogeneous macro stresses.

As (er) = l: for all prescribed l:, we have

(Aijpq) = �(S;p o1q + O;q o1p) (3.19) At fixed subscripts (p, q) Aijpq
Apqrs = (Aijpq"AgkiAklrs) = t.A N A (3.24)
In the following we consider only the case of a macrohomogeneous body.

STUDY OF A THERMOELASTOPLASTIC MATERIAL

The Macroscopic Constitutive Law

In this example we consider a thermoelastoplastic material and we assume that there is no effect of viscosity. The absolute temperature T is given and the total deformation e is the sum of an elastic part ee and an irreversible part eP:

E = Se+ eP (4.1)
The irreversible part has a standard evolution; the rate tP is normal to a yield surface delimited by a convex function (j> of the stress a: e P = a(j>

(4.2)
a a

The reversible part ee is related to a and to the increase of the temperature from its initial valu e T0:

ee = A 0 a+ a(T-T0) (4.3) 4.2

. Global Domain of Elasticity

At the local level there exists an elastic domain 't' defined by ' G' = la/ (j>(a) < Oj. At the macro level there exists an elastic domain rff defined as follows.

Let two macrostresses I: and I:* belong to rt, at each point of V; the corresponding microstresses a and a* are elements of 'G' and the difference a -a* is deduced from I: -I:* by means of a localization process with purely elastic behaviour.

So it is obvious that (4.4)

where A is the localization tensor introduced in the last section. I: is the deformation deduced from a purely elastic unloading with return of the temperature to its initial value [START_REF] Mandel | Plasticite Classique et Viscoplasticite[END_REF][START_REF] Halphen | Sur la thermodynamique du polycristal metallique[END_REF].

The local deformation defined by

eE = NaE + e1 (4.8)
is related to the macro one E E by the lemma of macrohomogeneity and

(4.9)

and for all stresses a' statically admissible with I:' = (a') we have (4.10)

For the particular choice of a' =A I:', and taking into account that r0

is self-equilibrated, we can split EE into two terms:

(4.11)

We recognize the macroscopic modulus A= IAA0A and the result obtained is in agreement with the definition of the macroscopic elastic deformation.

In the same way, the global deformation E = (e) must satisfy eqn ( 4.10), and for the difference E P = E -E Ew e have the definition of the irreversible macroscopic deformation: E = 1Ae p p (4.12)

The irreversible part of the macrodeformation is not the mean of the micro one because the field of plastic deformation is not kinematically admissible [START_REF] Mandel | Proc. 11th Int. Congr. Appl. Mech[END_REF][START_REF] Mandel | Plasticite Classique et Viscoplasticite[END_REF][START_REF] Hill | Continuum micromechanism of elastoplastic polycrystals[END_REF].

The relations (4.11) and (4.12) describe the form of the stress-strain relation, and moreover these equations are true for quasi-static evolutions. If there exists a viscoplastic potential </J for the rate of the plastic deformation, such that (4.13) it is obvious that at the macro level the mean potential [START_REF] Mandel | Plasticite Classique et Viscoplasticite[END_REF][START_REF] Rice | Inelastic constitutive relations for solids: an internal variable theory and its application to metal plasticity[END_REF] <I>(:E, r) = (/> ( a) At the micro level, the second law of thermodynamics gives the rate of dissipation:

d = pTs + div q -q • � > o
Using the conservation of energy, pe = asdivq (4.17 We have assumed that q is defined by q = -awl a� where w is a convex function of � . therefore the term -q • �is positive, and when we admit the decomposition of the dissipation rate in two parts, the term due to the irreversible mechanism is also positive, a t P > 0. By integration over V the total plastic dissipation rate g p =as P can be given in terms of macroscopic variables:

(4.20)

where Wb = !r A0r is the energy of the self-equilibrated stresses. The last result is obtained by application of eqn (4.10); for the plastic deformation rate and the real stresses a= A:E + r.

The physical meaning of this result is that perfect plastic behaviour at the micro level presents hardening behaviour on the macro scale.

THE CASE OF A SOLID HAVING LOCALLY DAMAGED ZONES

Description of the Model of Damage

It is observed that the reduction of material rigidity is generally due to the evolution of defects such as cavities, cracks, etc. Under external loading, these zones cannot support tensile stresses. It is proposed to characterize the damaged material only by the property that the stress tensor vanishes in the damaged zone. Therefore the stress vector must vanish on the boundary iJz; of any damaged zone z; (e1 •-; = 0 on iJz;).

If the volume V contains many damaged parts with total volume z(t), a function of the time t, the volume of the sound material is !l(t) = V-z(t); we note iJ!l;, the common boundary between 0 and z.

The mechanical properties of the sound material are defined by the thermodynamical potential.

The definition of the average must be changed because f is only defined in n:

J = (j)0 = � .{(dv (5.1)
We suppose that the mass per unit volume is constant:

vx En p(x) = Po (5.2)
Our study is concerned with the behaviour of material introduced by To determine the evolution of damaged zones, it is necessary to introduce a rupture criterion. Rupture occurs when a threshold value (in stress, deformation or energy) is reached; when the strain is increasing the stress vanishes. However, the tensor A depends on the actual volume O(t) of the sound material and its rate A is related to the propagation of the damaged zones [START_REF] Bui | Relations entre grandeurs microscopiques et macroscopiques pour un solide anelastique ayant des zones endommagees[END_REF].

Determination of the Evolution of the Damaged Zone for a Thermoelastic Material

Consider a thermoelastic material with free energy w(e, T) which depends only on the deformation e and the absolute temperature T. At each point of V we define the energy density e and the entropy s per unit mass. For simplicity we assume that damaged zones have no heat conductivity and that they are inert (e = 0, s = 0, v = 0).

The first law of thermodynamics is the conservation of energy, at the macro level:

� + K = P cal + Pext (5.5)
The variation of internal energy and kinetic energy K = !pu2 is equal to the heat flux and the power of external loading. We consider only quasi-static evolution so that the contribution of kinetic energy is negligible. The heat flux determines P eal:

P eal = -( q • -; d a Jav (5.6)
and by means of the equilibrium equation we have, for the power of external loading, P ext = at -J anv d a an;

(5.7)

The localization process depends on the evolution of damaged zones and all the quantities are changed with the variation of n. Ifjis a regular mechanical quantity and defined on n, we have the equation

f = f--f cda . -;-1 1 V an• I (5.8)
where c is the normal velocity of the boundary an;. So we have the local equation corresponding to (5.5):

qn at-divq over n l on ani with et> > 0 .

The production of entropy is necessarily positive:

;f + ( 9__ • n d a > 0
Jan T I By eqn (5.8) we can write the inequality (5.10) in another form:

(5.9)

(5.10) p0s+�divq -q • ;; -tL n (�n+p0sc)da > 0 (5.11)

I
For an elastic material we have (5.12)

With the conductivity law q = -aw I a� where � = V TIT and w convex, function of �. we have

1 - --q • Vq > 0 T (5.13)
Therefore the fracture dissipation rate is equal to the last term of (5.11):

Using eqns (5.9),

[hr = � { (qn + P o se ) da > 0

Jan;

1 { w [hr =
VJ P o T e da > 0 an;

(5.14) (5.15) Per unit of area the thermodynamical force associated with e has the dimensions of entropy. We assume a brittle criterion [START_REF] Nguyen | Normal dissipativity and energy criteria in fr acture[END_REF][START_REF] Suquet | Plasticite et homogeneisation, These Universite Pierre et Marie Curie[END_REF][START_REF] Stolz | Analyse du comportement et fo rmulation du probleme en vitesse pour un solide thermoelastique fr agile avec zone endommagee[END_REF]:

y = W T < .r;: e = 0 l y = .r;: e > 0
(5.16)

The two criteria Y = .r;: and a"�: = 0 must be satisfied along the moving part of the boundary an;, and we must introduce the transport condition at a geometrical point of an; by the convected derivative 01r:

x E ani {j,

1

. f(x + elit, t + lit) -f(x, t) �f = lm --'----'---,---�--"-'---_:_

ar-o lit

On an; the two conditions are written as follows:

� . � .... .. � !')(an ) = (a+ en •V a) • n + a!')n = . t - T!')(Y) = W-WT-Ten VY
It is easy to show that, in the plane case (with normal k), !') -; = de . 7 ds (5.17)

(5.18)

(5.19) 7 is the tangent vector to ani with s the curvilinear coordinate.

Taking into account the equilibrium equation

- -- - t' •Va •t' +n • Va• n = 0 (5.20)
.

-and a particular form of a = at" ® t" + azk ® k , the equations (oV = oVq U oVr = oVu U oV,., oVq n oVr = c/J, oV,. n oVu = c/J) and on o!l; we have c = 0; ii: = 0; q • n = 0 l (5.24) c -=1= 0; �(an) = 0; �(!/) = 0; q • n =po ec

The solution of this problem is a stationary point of the functional [START_REF] Stolz | Analyse du comportement et fo rmulation du probleme en vitesse pour un solide thermoelastique fr agile avec zone endommagee[END_REF]: Moreover, by subtraction in eqns (5.31) (with the same choice a' =A�') one obtains

i < 8 T > J c 2 - - p a ---e cda + -pTn • V(f/
(5.33)

The complementary term is due to the redistribution of residual stresses inside n, when the geometry changes with the propagation of the damaged zones.

When no damage occurs eqn (5.30) shows that r is self-equilibrated and the classical relation ( 4.20) is recovered. However, the existence of a viscoplastic potential at the microlevel does not ensure the existence of a macro-viscoplastic potential E1R. Only the first term can be deduced from the average potential cl>(�, p) = </>(a).

Study of the Dissipation Rate

The global energy W is the sum of a reversible part and an embedded energy Wb in the residual stresses r. For a quadratic form w(e -eP), the dissipation 1J can be decomposed into three terms, the first due to heat conduction, the second due to plasticity atP and the third due to fracture 1Jr.

So the mechanical part of £2 is given by £2 = af:P + -WC ds ;> 0 -

1 1 V iJO• 1
(5.34)

With the hypothesis of macrohomogeneity applied to actual stresses, we can deduce :Yr = � J wc da = ai:-ai:P -: t G I;EE + wb ) (5.36) Using the macrohomogeneity hypothesis ai: = I;E where wb = !rA 0 r. If the damaged zones do not propagate, 9r = 0 and the macro compliance A is constant, we recover dwb/dt = rA 0 r, i.e. the variation of the stored energy is only due to residual stresses, as in classical plasticity [START_REF] Mandel | Plasticite Classique et Viscoplasticite[END_REF][START_REF] Bui | Relations entre grandeurs microscopiques et macroscopiques pour un solide anelastique ayant des zones endommagees[END_REF].

If there is propagation of damaged zones, the first term !(I;EE -±EE)

does not vanish. It is analogous to Irwin's energy release rate for crack propagation [START_REF] Bui | Mecanique de la Rupture Fragile[END_REF] and it contributes to the dissipation rate.

THERMODYNAMICAL CONSIDERATIONS

Relationships between Micro and Macro Scales

In the preceding sections we were concerned by means of averaging processes and applications to define what are macroscopic mechanical quantities. We have said nothing about the macroscopic temperature or we have implicitly admitted that the temperature is given or corresponds to the temperature of the equilibrium state.

We have restricted our exposition to macro homogeneous bodies; can we then give general relations for other materials?

To be specific, one will be concerned with materials that admit thermodynamical potentials, e(e, a, s) internal energy or w(e, a,(}) free energy, where e is the deformation, a the internal variables, (} the absolute temperature and s the entropy. Then the local equation of state may be written e.g. ( 6.1)

where a is the elastic stress (this supposes no viscosity), and A are the thermodynamical forces associated with the variation of the internal variables a.

To determine the field of temperature, we must add to the constitutive equations (6.1) the heat conduction law. Noting � = V (log 0), we assume the following form for the generalized Fourier relation between heat flux and temperature: q = (6.2) when w(�) is a convex no-negative function off

As in the first section, it is clear that with every microscopic quantity f we can associate its macroscopic value! by an averaging process on the RVE V extended eventually in the cases for which V contains holes, damaged zones, or rigid inclusions:

J = (f) = .!_ ( f dv V )v (6.3)
and we have remarked that some constraints are necessary to determine a micro state corresponding to a given macro one. For the choice of a representative volume element suitable boundary conditions must be prescribed and must satisfy special requirements [START_REF] Francfort | Thermodynamique et lois de comportement thermo-mecanique homogeneisees[END_REF][START_REF] Germain | Continuum thermodynamics[END_REF]:

R1 -Assume that the macro state (0', e, �, w) is given; the microscopic quantities f must then satisfy:

-the local constitutive law;

-the boundary conditions on av;

-the relation of averaging;

-the equation of equilibrium in stress and flux diva = 0 and div q = 0 over V.

R2 -The Hill-Mandel macrohomogetl � ity condition must be fulfilled by all fields a*, q* with 0'* =(a*) and q* = (q*) and by all fields s* and �* kinematically and thermally admissible:

-(6.4) 0'*6* = (a* s*) l q*� * = (q* �*)

We have given particular localization processes for periodic structures and for homogeneous boundary conditions.

R3 -The process of localization must ensure the existence and uniqueness of the microscopic response. The construction of the macro constitutive law from the microscopic behaviour can proceed as follows:

-starting from some macro quantities; -using a localization process to determine local fields then the averaging process to determine the missing macro quantities.

Study of Systems in Equilibrium

In this case the homogeneous deformation E = 6, the set of internal variables a and the temperature T of th e macroscopic description are prescribed. Near the equilibrium we assume that the temperature is quasi-uniform 0 = T (but the presence of heterogeneity implies that the gradient � is not uniform, � * � ). We wish to find the local quantities a, s, q, �as functions of the macro state E, T, a, E =�;they are solutions of the boundary value problem (9):

E = (s) E = (�) ow a = p 08 (s, a, T) q = ow a� � = VlogO = 0
) (}' } (6.5) div y q = 0 Boundary conditions of localization For a given localization process this solution is unique (R3). Let The global description has the same form as the local one, and the equation (6.7) gives the constitutive stress-strain relation. Similarly it is possible to give the generalized Fourier relation between q and ""(.

If we consider that near a state of equilibrium the two principles of thermodynamics are valid at the microlevel pe = e1 tdiv q l (6.9) pOs + div q-q • � > o we can write the same relations at the macrolevel, if all the equations (6.6)--(6.9) are valid at each time, which means that the thermostatic description and the Gibbs equation are true for the equivalent homogeneous medium if they are valid for each of the constituent phases [START_REF] Francfort | Thermodynamique et lois de comportement thermo-mecanique homogeneisees[END_REF][START_REF] Germain | Continuum thermodynamics[END_REF].

Non-equilibrium Evolution

When the system is far from an equilibrium state, in particular when the local absolute temperature () is not uniform on V, the equilibrium equation and the property of macrohomogeneity are not valid. Some complementary terms due to inertia phenomena must be taken into account. However, we can try to define the macroscopic temperature by the concept of a local accompanying equilibrium state.

At each time, the volume element V must be isolated and one has to study the dynamical evolution toward an equilibrium state and find the micro quantities which characterize this equilibrium. When the volume element V is isolated, quantities such as the strain E = 6 and energy e, together with some internal parameter a, must be kept constant. Inside the RVE various physical dissipative mechanisms have taken place and each of them is described by the internal variable a. The dimensions of the RVE and the inertia effect define a characteristic timet", and we must distinguish two kinds of internal parameters a and a'. The first family a can be considered as constant; the second a', having a relaxation time shorter than t", must be considered a function of time.

Under the assumption that in this transition towards equilibrium the two principles of thermodynamics are valid, the equilibrium state is the solution of the problem (6.5) where E = (e,), a = a, for a fl la'l and e = (e1) Examples are given by [START_REF] Francfort | Homogeneisation et oscillations rapides en thermo elasticite lineaire[END_REF].

The major difficulty in this study is to define correctly the absolute temperature for a macroscopic body which has microheterogeneity. In the first part of this chapter we have studied only the quasi-static evolution of the RVE, for which the absolute temperature is defined as in Section 6.2.

CONCLUSION

We have presented some aspects of the general relations between micro and macro scales in heterogeneous media, and we have tried to exhibit some of the most significant lines of research in the mechanical behaviour of composite materials. This domain is still in a state of expansion, not only in solid mechanics but also in fluid mechanics and in mixtures theory. In the next few years progress may be expected in many directions: better mathematical consistency, deepening of the physical interpretation and significance.

  case the local deformation e derives from a local displacement u which verifies eij = i (�:� + :::) over V ] the Hill-Mandel macrohomogeneity condition is fulfilled by all field a* with a* = (a*) and by all e* kinematically admissible so Cf*e* = (a* e*) (2.4) With these conditions it is obvious that, in the case of homogeneous loading, (2.5) and the microdisplacement u fluctuates around the homogeneous displacement U such that J (U-u) ® n da 0

  the difference between the two estimates is negligible. So when one prescribes homogeneous boundary conditions in stress (Td = I: •-; on oV, with I: a symmetric tensor) the local stress is obtained as the solution of an elastic linear problem: Equation of equilibrium: div er = 0 over V er . -: = r on av Elastic constitutive law: e = N(y) :er (Vy E V) Existence of a local displacement u such that eij = !(��� + ��)

  satisfies the equations of equilibrium Aijpqj = 0 (3.20) with the boundary conditions AijJK!IJ = !(o;p o1q + O;q o1p) • n1 and the deformation satisfies the condition of compatibility. (3.21) (3.22) With the help of a macro homogeneity condition, for all prescribed I: we have (3.23) The macroscopic modulus A (E = A l:) is determined as a function of the local one (e = A0 er) by replacing er in eqn (3.23) by A I::

4. 3 .

 3 On the Decomposition of the Macroscopic Deformation Let I: be the real m acro stress and a the corresponding micro one. At the initial temperature T0, the local response in purely elastic behaviour is (4.5)The stress field r = a -aE is a self-equilibrated stress and is the sum of two self-equilibrated stresses: macroscopic elastic deformation due to the difference in absolute temperature T -T 0 and to the given macrostress

  potential associated with EP: ( a = A:E + r)

  relation between the thermodynamical potentials e and w(e•, I), e = w -Ts, we can deduce d = asP -q� > 0 (4.19)

  Bui and eo-workers[START_REF] Bui | Propagation dynamique d'une zone endommagee dans un solide elastique-fragile en mode Ill et en regime permanent[END_REF][START_REF] Bui | Relations entre grandeurs microscopiques et macroscopiques pour un solide elastique-fragile ayant des zones endommagees[END_REF] to study fracture, pictured in Fig. I.

Fig. 1 .

 1 Fig. 1. (a) Elastic-brittle or (b) elastoplastic-brittle behaviour.

T

  Knowing at each time t the thermomechanical fields and the geometry of !l(t), we search for the rate quantities and the normal velocity c on an;. The rate equations are diva= o pTs + divq = o 8 = !(�v + �Tv) � = �logT x E !l a= A0e +AT aw q = --a� with the boundary conditions q . n = Qd on avq on av;. T = r on avT ) (m = ffd on av;,.

  field r is self-equilibrated on O(t) we have, from eqn (5.8), r is generally not self-equilibrated; we have inside n a redistribution of residual stresses. The rate e and A 0 aE are two kinematically admissible fields of deformation, then the macroscopic deformations E and EE are given by the relations (5.3). For all a' statically admissible with �• = (a') in the localization process we have For the particular choice of a' =A�' we have EE = 1AAoaE = A±+ A� (5.31) (5.32) which defines exactly the variations of the reversible part EE of the macrodeformation (EE =A�).

  p w (E; a, T) = (pw(s, a, T))(6.6) where s is the solution of the problem (9). Using the hypothesis of macrohomogeneity of Hill (R2), we obtain aw < aw oe ) < ae )and noting that e = E + 11 with (q) = 0 for all E, so (oe/oE) same way it can be proved that s = -aw;ar T = a-e;a-s (6.8)

  are prescribed, and we obtain � • -ae (E ,. r•) and a'• are the values of the macroscopic temperature and the values of internal parameters at the equilibrium state, respectively. If the actual state is far from the equilibrium, it is obvious that � • *(a,).

ACKNOWLEDGEMENTS

The author wishes to thank H. D. Bui, K Dang Van and Q. S. Nguyen for discussion and for guidance.

On the set V= {v, T/v = vd on avw t = Td on av rl• Once the velocity c is determined, we are able to know the variation of all macro quantities by application of eqn (5.8). In the following part we suppose that c is determined, and our point of interest is the macro scopic behaviour of a RVE V, and we reduce our consideration on elastoplastic material, with free energy of the form w(e-eP). The influence of temperature could be treated as in the last example (Section 4).

Evolution of Macroscopic Quantities

Here we determine the evolution of the macroscopic modulus and we decompose the rate of deformation. By comparison of the two expressions forE and E in eqns (5.3), the application of eqn (5.8) shows that

(5.26) for all kinematically admissible e. One particular field e is AgpqA pqm so it is obvious to obtain (Bui et al., 1982)

A = 'AAoA = � ( 'AAoAc da Jan;

(5.27)

The elastic modulus varies with the propagation of the damaged zones; this variation must appear in the reversible part of the rate of the elastic macroscopic deformation.

When the damaged zone does not propagate inn the local stress field is a =AI:' + r where r is a self-equilibrated stress on the actual geometry. For an increasing ± of the macrostress from the value I: such that the damaged zone has a normal velocity c of propagation, the local stresses are (;with a corresponding rate of deformation

(5.28) in the case of an elastoplastic material. Let 6-E be the fictitious stresses obtained with purely elastic behaviour with the same velocity of propagation c:

(5.29)

where A is related to the variation of the localization tensor A due to the change in the geometry of n induced by the damaged zone propagation.


V Jakšić

B Landon

A Panati

A note on reflectionless Jacobi matrices

The property that a Jacobi matrix is reflectionless is usually characterized either in terms of Weyl mfunctions or the vanishing of the real part of the boundary values of the diagonal matrix elements of the resolvent. We introduce a characterization in terms of stationary scattering theory (the vanishing of the reflection coefficients) and prove that this characterization is equivalent to the usual ones. We also show that the new characterization is equivalent to the notion of being dynamically reflectionless, thus providing a short proof of an important result of [BRS]. The motivation for the new characterization comes from recent studies of the non-equilibrium statistical mechanics of the electronic black box model and we elaborate on this connection.

Introduction

The purpose of this paper is two fold. First, we wish to advertise a certain point of view regarding full-line Jacobi matrices that appears absent from the literature. Secondly, we shall use this point of view to give a new characterization of reflectionless Jacobi matrices.

The point of view we wish to describe has its origin in recent studies of the non-equilibrium statistical mechanics of the electronic black box model. We will need here only the simplest variant of this model. The simple electronic black box (SEBB) model describes a quantum dot S coupled to two electronic reservoirs R l/r that we will colloquially call 'left' and 'right.' The system S is described by the Hilbert space C and energy ω ∈ R. The reservoir R l/r is described by a pair (h l/r , h l/r ), where h l/r is a single fermion Hilbert space and h l/r is a single fermion Hamiltonian. We set

h = h l ⊕ C ⊕ h r , h 0 = h l ⊕ ω ⊕ h r .
The fermionic second quantization of the pair (h, h 0 ) describes the uncoupled SEBB model R l + S + R r .

The junctions coupling S to the reservoirs are specified by a choice of unit vectors χ l/r ∈ h l/r and real numbers λ l/r describing the strength of the coupling. The corresponding tunnelling Hamiltonian is

h T = h T,l + h T,r , h T,l/r = λ l/r |1 χ l/r | + |χ l/r 1| .
Finally, the coupled EBB model is obtained by the fermionic second quantization of the pair (h, h) where

h = h 0 + h T .
In the literature, the Hamiltionian h is also known as the Friedrichs model or Wigner-Weisskopf atom.

In the study of the SEBB model it is no loss of generality to assume that χ l/r is a cyclic vector for h l/r . We shall further assume that the operators h l/r are bounded. The SEBB model is called non-trivial if the spectral measure ν l/r for h l/r and χ l/r is not supported on a finite set.

The relevant transport phenomena in the SEBB model are linked to the choice of the initial state. Suppose that initially the reservoir R l/r is in equilibrium at inverse temperature β l/r and chemical potential µ l/r . If either β l = β r or µ l = µ r , then, in the large time limit, the resulting temperature/chemical potential differential induces a non-trivial steady state energy/charge flux between the left and the right reservoirs. The Landauer-Büttiker formalism relates the transport theory of the SEBB model (the values of the steady state energy and charge fluxes, full counting statistics of the energy and charge fluxes, etc.) to the scattering theory of the pair (h, h 0 ). For example, the steady state energy/charge flux is given by the Landauer-Büttiker formula that involves only the initial energy densities of the reservoirs and the scattering matrix of the pair (h, h 0 ). The same is true for the Levitov-Lesovik formulas for the large deviation functionals associated to the full counting statistics of energy/charge flux. We refer the reader to the lecture notes [JKP, JOPP] for additional information and references to the vast literature on these subjects.

Consider now a full-line Jacobi matrix J acting on H = ℓ 2 (Z) by

(Ju) k = a k u k+1 + a k-1 u k-1 + b k u k , (1.1)
where {a k } k∈Z and {b k } k∈Z are bounded sequences of real numbers and a k = 0 for every k. To such a matrix one associates a SEBB model as follows. For fixed n ∈ Z, let

H (l) n = ℓ 2 ((-∞, n -1]), H (r) n = ℓ 2 ([n + 1, ∞)). Let J (l) n and J (r)
n be the half-line Jacobi matrices with Dirichlet boundary conditions obtained by restricting

J to H (l) n and H (r) n , respectively. Setting h l/r = H (l/r) n , h l/r = J (l/r) n , ω = b n , χ l = δ n-1 , χ r = δ n+1 , λ l = a n-1 , λ r = a n , and J 0 = J (l) n + J (r) n + b n |δ n δ n |, (1.2)
one arrives at the SEBB model with

h 0 = J 0 , h = J.
Thus, a Jacobi matrix naturally induces a non-trivial SEBB model. In turn, a well-known orthogonal polynomial construction (see Theorem I.2.4 in [Si]) implies that every non-trivial SEBB model is unitarily equivalent to a Jacobi matrix SEBB model. This leads to the identification of the class of SEBB models with the class of Jacobi matrices.

This note is a first step in the exploration of this connection. We shall focus on the following point.

Although the study of the scattering theory of the pair (h, h 0 ) is completely natural from the point of view of non-equilibrium statistical mechanics, the study of the scattering theory of the corresponding pair of Jacobi matrices (J, J 0 ) is, to the best of our knowledge, virtually absent in the literature on Jacobi matrices. We shall exploit this connection to determine when a Jacobi matrix is reflectionless, a property which has attracted considerable attention in the recent literature on Jacobi matrices (see Chapter 8 in [Te] and Chapter 7 in [Si]). Motivated by recent studies of the SEBB model (and in particular by [JLP]) we shall propose a definition of reflectionless based on the scattering matrix of the pair (J, J 0 ) that is, in our opinion, physically and mathematically natural. We show that this definition is equivalent to the standard definitions appearing in the literature and also to the dynamical definition introduced in [BRS]. A consequence is a short, transparent proof of the main result of [BRS] which has settled a 25 year old conjecture raised in [DeSi].

The paper is organized as follows. In Section 2 we recall the standard definitions of reflectionless Jacobi matrices and the result of [BRS]. In Section 3 we introduce stationary reflectionless Jacobi matrices and state our main result. Its proof is given in Section 4. The concepts introduced in this paper go beyond reflectionless and shed light on the notions of the dynamical reflection probability of [DaSi] and the spectral reflection probability of [BRS, GNP, GS] by identifying them with the scattering matrix reflection probability. This point is discussed in Section 5.

Finally, we remark that, as in [BRS], our results can be extended to Schrödinger operators on the real line and CMV matrices. The extension to Schrödinger operators is somewhat technical due to the lack of convenient references to the stationary scattering theory of singular perturbations (induced by imposing boundary conditions). The details can be found in [Z]. The extension to CMV matrices can be found in [CLP].

Acknowledgment. The research of V.J. and B.L. was partly supported by NSERC. The research of A.P. was partly supported by ANR (grant 12-JS01-0008-01). We wish to thank J. Breuer, R. Frank and B.

Simon for useful discussions. A part of this work was done during the visit of the authors to Institute Henri Poincaré in Paris. We wish to thank M. Esteban and M. Lewin for their hospitality.

Three notions of reflectionless

The Green's function of the Jacobi matrix (1.1) is defined for z ∈ C\R by

G nm (z) = δ n , (J -z) -1 δ m .
By general principles (see [J] or any book on harmonic analysis) the boundary values

G nm (λ ± i0) = lim ε↓0 δ n , (J -λ ∓ iε) -1 δ m (2.3)
exist, are finite and non-vanishing for Lebesgue a.e. λ ∈ R.

The Weyl m-functions are defined for z ∈ C\R by

m (l) n (z) = δ n-1 , (J (l) n -z) -1 δ n-1 , m (r) n (z) = δ n+1 , (J (r) n -z) -1 δ n+1 .
The boundary values m (l/r) n (λ ± i0) are defined analogously to (2.3), and are also finite and non-vanishing for Lebesgue a.e. λ ∈ R.

We recall the terminology of [BRS]. Let e ⊂ R be a Borel set. A Jacobi matrix is called measure theoretically reflectionless on e if

Re [G nn (λ + i0)] = 0 (2.4)

for Lebesgue a.e. λ ∈ e and every n. A Jacobi matrix is called spectrally reflectionless on e if

a 2 n m (r) n (λ + i0)m (l) n+1 (λ -i0) = 1 (2.5)
for Lebesgue a.e. λ ∈ e and every n.

From the first of the two formulas

G nn (z) = - 1 a 2 n m (r) n (z) -m (l) n+1 (z) -1 = - 1 a 2 n-1 m (l) n (z) -m (r) n-1 (z) -1 , (2.6)
one derives that if (2.5) holds for λ and n, then (2.4) holds for the same λ and n. The following is the well-known converse (see [GKT, SY, Te] and Theorem 7.4.1 in [Si]).

Theorem 2.1 For Lebesgue a.e. λ ∈ R, the following are equivalent.

(1) Re [G nn (λ + i0)] = 0 for all n.

(2) Re [G nn (λ + i0)] = 0 for three consecutive n's.

(3) (2.5) holds for one n.

(4) (2.5) holds for every n.

An immediate and well-known consequence is

Corollary 2.2 J is measure theoretically reflectionless on e iff it is spectrally reflectionless on e.

Based on the ideas of Davies and Simon [DaSi], Breuer, Ryckman and Simon introduced in [BRS] the notion of being dynamically reflectionless. We first recall the setup of [DaSi] adapted to the Jacobi matrix case. For each n ∈ Z, let χ

(l)
n be the characteristic function of (-∞, n -1] and χ (r)

n of [n + 1, ∞). Set H ± l = ϕ ∈ H ac (J) for all n, lim t→±∞ χ (r) n e -itJ ϕ = 0 .
The elements of H ± l are the states that are concentrated on the left in the distant future/past t → ±∞. The sets H ± r are defined with χ

(l) n replacing χ (r)
n . The sets H + l/r are related to H - l/r by time-reversal, i.e.,

H + l/r = ϕ | ϕ ∈ H - l/r . (2.7)
In what follows H ac (A) denotes the absolutely continuous subspace for self-adjoint A and P ac (A) the projection onto this subspace. The result of Davies and Simon [DaSi] is:

Theorem 2.3 H ac (J) = H + l ⊕ H + r = H - l ⊕ H - r .
(2.8)

Remark. In [DaSi] this theorem was proven in the context of Schrödinger operators on the real line. In the discrete setting considered here the argument is considerably simpler. For completeness and later reference, we sketch the proof.

Proof. By (2.7) it suffices to prove that H ac (J) = H + l ⊕ H + r . Since s -lim t→∞ e itJ Ce -itJ P ac (J) = 0 for any compact C, we have that for any n

H + l/r = ϕ ∈ H ac (J) lim t→∞ χ (r/l) n e -itJ ϕ = 0 = ϕ ∈ H ac (J) lim t→∞ e itJ χ (l/r) n e -itJ ϕ = ϕ .
Since the commutators [χ (l/r) n

, J] are finite rank, trace-class scattering theory implies that the limits

P + l/r = s -lim t→∞ e itJ χ (l/r) n e -itJ P ac (J)
exist, and are furthermore independent of n. From the fact that e itJ commutes with P + l/r , we obtain that P ac (J) commutes with P + l/r , and so e -itJ P ac (J) = P + l/r .

[P + l/r ] * =
Hence, P + l/r are the orthogonal projections onto H + l/r . The statement now follows from the identity P ac (J) = e itJ χ (l) n e -itJ P ac (J) + e itJ χ (r) n e -itJ P ac (J) + e itJ |δ n δ n |e -itJ P ac (J).

✷

In what follows P e (A) denotes the spectral projection of a self-adjoint operator A onto a Borel set e and |S| the Lebesgue measure of a set S. In [BRS] the following notion was introduced.

Definition 2.4 A Jacobi matrix is called dynamically reflectionless on e if for any Borel set e 1 ⊂ e, P e1 (J)P ac (J) = 0 =⇒ |e 1 | = 0, (2.9)

and P e (J)[

H + l ] = P e (J)[H - r ].
The result of [BRS] is Theorem 2.5 J is dynamically reflectionless on e iff it is spectrally reflectionless on e.

Remark. Theorem 2.5 verifies the conjecture of Deift and Simon [DeSi] that the a.c. spectrum of ergodic Jacobi matrices is dynamically reflectionless.

The notion of stationary reflectionless

Recall that J 0 is given by (1.2). It follows from trace-class scattering theory that the wave operators

w ± = s -lim t→±∞ e itJ e -itJ0 P ac (J 0 )
exist and are complete (here, completeness means that Ran w ± = H ac (J)). Note that w ± depends on n (whenever this dependence is clear within the context, we shall not indicate it explicitly). By the spectral theorem we may identify H ac (J 0 ) with

H ac (J 0 ) = H ac (J (l) n ) ⊕ H ac (J (r) n ) = L 2 (R, dν l,ac ) ⊕ L 2 (R, dν r,ac )
where ν l/r,ac is the a.c. part of the spectral measure for J

(l/r) 0 and δ n-1 /δ n+1 . We recall the well-known formula dν l/r,ac dλ

(λ) = 1 π Im [m (l/r) n (λ + i0)]
(a pedagogical proof can be found in [J]). It follows from stationary scattering theory [Y] that the scattering matrix s = w * + w - acts as multiplication by a unitary 2 × 2 matrix

s(λ) = s ll (λ) s lr (λ) s rl (λ) s rr (λ)
on H ac (J 0 ) where

s jk (λ) = δ j,k + 2ia j a k G nn (λ + i0) Im [m (j) n (λ + i0)]Im [m (k) n (λ + i0)],
(3.10) j, k ∈ {l, r}, a l = a n-1 and a r = a n . In the current setting, the above formula can be also easily verified by a direct computation which we sketch in the appendix (see [La] for more details). Note that s lr (λ) = s rl (λ).

Motivated by the non-equilibrium statistical mechanics of the SEBB model (and in particular by the work [JLP]) we introduce Remark. In particular, this proposition implies that if the scattering matrix s(λ) is off-diagonal for Lebesgue a.e. λ ∈ e and some n, then it is so for all n.

We shall prove Theorem 3.3 J is dynamically reflectionless on e iff it is stationary reflectionless on e.

This result combined with Proposition 3.2 implies Theorem 2.5.

The proof of Proposition 3.2 is very simple. As we shall see, the proof of Theorem 3.3 is also simple due to the direct connection with scattering theory. The notion of stationary reflectionless naturally links the notions of spectral and dynamical reflectionless and is likely to play a role in future developments.

Proof of Theorem 3.3

Let

Σ l/r,ac = λ dν l/r,ac dλ (λ) > 0 , E = Σ l,ac ∪ Σ r,ac .
The set E is an essential support of the a.c. spectrum of J 0 . From unitarity and symmetry of the scattering matrix, we see that for Lebesgue a.e. λ,

s ll (λ) = 0 ⇐⇒ s rr (λ) = 0.
Note also that if J is stationary reflectionless on e, then |e\(Σ l,ac ∩ Σ r,ac )| = 0, and in particular |e\E| = 0. These two observations yield an equivalent formulation of stationary reflectionless that is more suitable for comparison with being dynamically reflectionless:

Lemma 4.1 J is stationary reflectionless on e iff |e\E| = 0 and, for some n, s ll (λ) = 0 for Lebesgue a.e. λ ∈ e ∩ Σ l,ac , s rr (λ) = 0 for Lebesgue a.e. λ ∈ e ∩ Σ r,ac .

Since J -J 0 is finite rank, E is an essential support of the a.c. spectrum of J, and so the condition (2.9) in the definition of dynamical reflectionless can be replaced by the equivalent condition

|e\E| = 0. (4.11)
The key observation is that the projections P ± l/r on the subspaces H ± l/r (recall the proof of Theorem 2.3) satisfy

P ± l/r = s -lim t→±∞ e itJ e -itJ0 χ (l/r) n e itJ0 e -itJ P ac (J) = w ± χ (l/r) n w * ± .
(4.12)

Above, we have used the fact that J 0 commutes with χ (l/r) n

and that Ran w * ± = H ac (J 0 ). Note that the P ± l/r commute with J and its spectral projections. For ϕ, ψ ∈ H, ϕ, P e (J)P + l P - l ψ = ϕ, P e (J)P + l P e (J)P - l ψ = w * + P e (J)ϕ,

χ (l) n w * + w -χ (l) n w * -P e (J)ψ = w * + ϕ, P e (J 0 )χ (l) n sχ (l) n P e (J 0 )w * -ψ , (4.13)
where the last line follows by the intertwining property of the wave operators. We write

w * + ϕ = ϕ l (λ) ⊕ ϕ r (λ) ∈ L 2 (R, dν l,ac ) ⊕ L 2 (R, dν r,ac ),
and the same for w * -ψ. With this notation, the last line of (4.13) becomes e ϕ l (λ)s ll (λ) ψ l (λ)dν l,ac (λ).

Since Ran w * ± = H ac (J 0 ), we conclude P e (J)P + l P - l = 0 ⇐⇒ s ll (λ) = 0 for Lebesgue a.e. λ ∈ e ∩ Σ l,ac .

A similar computation yields P e (J)P - l P + l = 0 ⇐⇒ s ll (λ) = 0 for Lebesgue a.e. λ ∈ e ∩ Σ l,ac , and two similar statements where P ± l is replaced by P ± r and s ll by s rr . This, together with Lemma 4.1 and (4.11), yields the theorem.

Remarks

The arguments used in the proof of Theorem 3.3 go beyond reflectionless and shed light on the notions of dynamical and spectral reflection probability.

Dynamical reflection probability.

In [DaSi] (see also [BRS]) Davies and Simon introduced the concept of dynamical reflection probability (also called reflection modulus) as follows. Recall that the projections P ± l/r satisfy (4.12). J commutes with P + l P - l P + l and, since J ↾ Ran P + l has simple spectrum, there exists a Borel function R + l on sp(J ↾ Ran P + l ) such that

P + l P - l P + l = R + l (J) ↾ Ran P + l . R +
l is unique (up to sets of Lebesgue measure zero) and satisfies 0 ≤ R + l (λ) ≤ 1. One extends R + l to R by setting R + l (λ) = 1 for λ ∈ sp(J ↾ Ran P + l ) and defines R - l , R ± r analogously. The functions R ± l/r are discused in Section 4 of [DaSi] in the Schrödinger case (their proofs easily extend to the Jacobi case). In this context, the main observation of this note is that the formula (4.12) implies the identity

R ± l/r (λ) = |s ll (λ)| 2 = |s rr (λ)| 2 .
Spectral reflection probability. To the best of our knowledge, the link between reflection probability and half-line m-functions was first observed in [GNP, GS] in the context of Schrödinger operators on the line. The definition of the spectral reflection probability of [GNP, GS] was based on suitable generalized eigenfunctions and was extended to Jacobi matrices in [BRS] as follows. Consider the case n = 0. For z ∈ C + , let ψ l/r (z) = {ψ (l/r) k (z)} k∈Z be the unique solution of the equation

a k ψ k+1 + a k-1 ψ k-1 + b k ψ k = zψ k (5.14)
that is square summable at ∓∞ and normalized by ψ (l/r) 0 = 1. These solutions are related to m-functions as

m (r) 0 (z) = - ψ (r) 1 (z) a 0 , m (l) 1 (z) = - 1 a 0 ψ (l) 1 (z)
.

(5.15)

For all k and Lebesgue a.e. λ the limit

lim ǫ↓0 ψ (l/r) k (λ + iǫ) = ψ (l/r) k (λ + i0)
exists and ψ (l/r) (λ + i0) solves (5.14) with z = λ. For Lebesgue a.e. λ ∈ Σ r,ac the solution ψ (r) (λ + i0) is not a multiple of a real solution and so ψ (r) (λ + i0) is also a solution linearly independent of ψ (r) (λ + i0).

Hence, for Lebesgue a.e. λ ∈ Σ r,ac we can expand

ψ (l) (λ + i0) = α(λ)ψ (r) (λ + i0) + β(λ)ψ (r) (λ + i0).
(5.16)

The spectral reflection probability of [GNP, GS, BRS] is

R r (λ) = β(λ) α(λ) 2 .
One extends R r (λ) to R by setting R r (λ) = 1 for λ ∈ Σ r,ac and defines R l (λ) analogously. Using (5.15) and (5.16) one computes

|R r (λ)| 2 = a 2 0 m (r) 0 (λ + i0)m (l) 1 (λ + i0) -1 a 2 0 m (r) 0 (λ + i0)m (l) 1 (λ + i0) -1 2 .
In this context the main observation of this note is that the formulas (2.6) and (3.10) yield the identity

R r (λ) = |s rr (λ)| 2 . Similarly, R l (λ) = |s ll (λ)| 2 .
These identities clarify the meaning of the spectral reflection probability and yield

R l/r = R ± l/r .
(5.17)

Much of the technical work in [BRS] was devoted to a direct proof of (5.17) via an implicit rederivation of the scattering matrix.

A Computation of the scattering matrix

We briefly sketch the derivation of the formula (3.10) for the scattering matrix of the pair (J, J 0 ). For details, we refer the reader to [La].

It suffices to consider the case n = 0 (recall 1.1). First, we shall show that for ϕ ∈ H,

w * ± ϕ = ϕ (±) l ⊕ ϕ (±) r ∈ H ac (J (l) 0 ) ⊕ H ac (J (r) 0 ),
where (recalling that a l = a -1 and a r = a 0 ),

ϕ (±) l/r (λ) = P ac (J (l/r) 0 )ϕ(λ) -a l/r δ 0 , (J -λ ∓ i0) -1 ϕ . For any ψ = ψ l ⊕ ψ r ∈ H ac (J (l) 0 ) ⊕ H ac (J (r) 0 ) we have, ψ, w * + ϕ = w + ψ, ϕ = lim t→∞ e itJ e -itJ0 ψ, ϕ = lim t→∞ ψ, e itJ0 e -itJ ϕ
By an abelian limit and the definition of J -J 0 we can rewrite the RHS as

lim t→∞ ψ, e itJ0 e -itJ ϕ = ψ, ϕ -i lim t→∞ t 0 ψ, e isJ0 (J -J 0 )e -isJ ϕ ds = ψ, ϕ -lim ε↓0 (L l (ε) + L r (ε)), where L l/r (ε) = i ∞ 0
e -εs a l/r ψ, e isJ0 δ -1/1 δ 0 , e -isJ ϕ ds.

Expanding the first inner product in the above integrand yields

L l/r (ε) = ia l/r R ψ l/r (λ) ∞ 0 δ 0 , e -is(J-λ-iε) ϕ ds dν l/r,ac (λ) = a l/r R ψ l/r (λ) δ 0 , (J -λ -iε) -1 ϕ dν l/r,ac (λ).
For ψ in a judiciously chosen dense set (see [La] or Proposition 7 in [JKP]) one can take the limit ε → 0 + inside the integral. This yields the formula for w * + . The computation for w * -is identical. To compute the scattering matrix, note that for ψ = ψ l ⊕ ψ r and ϕ = ϕ l ⊕ ϕ r in H ac The inner product in the above integrand equals a l e -isJ0 ψ, δ -1 w * -δ 0 , e -isJ0 ϕ + a r e -isJ0 ψ, δ 1 w * -δ 0 , e -isJ0 ϕ ,

where we have used the intertwining property of the wave operators. We use our formula for w * -to compute ψ, (s -½)ϕ = lim as ε → 0 + . For a suitable dense set of ψ and ϕ, this formal computation can be easily justified (see [La]).

Finally, we remark that the scattering matrix formula (3.10) is valid only after a 'unitarity' transformation which we describe now. For any ψ ∈ H ac (J 0 ), let ψ(λ) denote the vector (ψ l (λ), ψ r (λ)) ∈ C 2 . For ψ and ϕ in H ac (J 0 ), we have Multiplication by the matrix V (λ) is a unitary operator V : H ac (J 0 ) → L 2 (R, ρ l (λ)dλ)⊕L 2 (R, ρ r (λ)dλ), with ρ l/r the characteristic function of Σ l/r,ac . Our computation shows that the operator V sV -1 acts as multiplication by the 2 × 2 matrix s(λ) given by (3.10) on V H ac (J 0 ). In particular, this transformation ensures that s(λ) is unitary w.r.t. the standard inner product on C 2 .

ψ, ϕ = R V (λ)ψ(λ), V (

  s -lim t→∞ e itJ P ac (J)χ (l/r) n P ac (J)e -itJ * = s -lim t→∞ e itJ P ac (J)χ (l/r) n P ac (J)e -itJ = P + l

  (J 0 ), ψ, (s -½)ϕ = ψ, (w * + w --w * -w -)ϕ = (w + -w -)ψ, w -ϕ = limt→∞ (e itJ e -itJ0 -e -itJ e itJ0 )ψ, w -ϕ isJ (J -J 0 )e -isJ0 ψ, w -e isJ (J -J 0 )e -isJ0 ψ, w -ϕ ds.

  ll (ε) + H rl (ε) + H lr (ε) + H rr (ε)),whereH jk (ε) = a j a k R e -ε|s| R e isλ ψ j (λ)dν j,ac (λ) R e -isλ ′ G 00 (λ ′ + i0)ϕ k (λ ′ )dν k,ac (λ ′ ) ds,for j, k ∈ {l, r}. Formally, the computation is completed by noting thatH jk (ε) = a j a k R R ψ j (λ)ϕ k (λ ′ )G 00 (λ ′ + i0) R e is(λ-λ ′ )-ε|s| ds dν j,ac (λ)dν k,ac (λ ′ ),and that R e is(λ-λ ′ )-ε|s| ds → 2πδ(λ -λ ′ )

  λ)ϕ(λ) 2 dλ where •, • 2 denotes the standard inner product on C 2 and V (λ) is the 2 × 2 matrix