F Dell'isola

C Z Wo Źniak
email: cwozniak@ippt.gov.pl

On phase transition layers in certain micro-damaged two-phase solids

Keywords: microcracks, micro-damaged materials, two-phase solids, interfacial layers

A continuum model of a moving transition layer separating two different solid phases of a certain micro-damaged material is proposed. An application of this model to the analysis of the propagation of this layer in a micro-cracked solid is shown.

Introduction

In the recent literature some interest is paid to the study of propagation of discontinuity surfaces in continuous media. These surfaces are introduced as models of a large variety of phenomena (see for instance [START_REF] Kosinski | Field Singularities and Wave Analysis in Continuum Mechanics[END_REF]) also including various kinds of phase transitions. When these phenomena need more detailed description the modelization is improved by endowing discontinuity surfaces with more sophisticated structure. This is done, for instance, by introducing certain surface fields, [START_REF] Dell'isola | On the derivation of thermomechanical balance equations for continuous systems with nonmaterial interface[END_REF][START_REF] Dell'isola | A phenomenological approach to phase transition in classical field theory[END_REF][START_REF] Dell'isola | Deduction of thermodynamic balance laws for bidimensional nonmaterial directed continua modelling interphase layers[END_REF], describing the passage from one phase to another one. However, this passage may not always be abrupt and hence the interface often has to be regarded as a certain three-dimensional layer (see [START_REF] Dell'isola | Deduction of thermodynamic balance laws for bidimensional nonmaterial directed continua modelling interphase layers[END_REF] and the review paper [START_REF] Rubinstein | On mathematical models for solid-liquid zones in a two-phase monocomponent system and in binary alloys 275-283[END_REF] on mushy regions arising in phase transitions) in which we can assume to deal with a certain mixture of both phases. This also occurs in some particular cases of solid-solid phase transitions, [START_REF] Miller | Interfacial Phenomena: Equilibrium and Dynamic Effects[END_REF]. Thus, the question arises how to describe of the interfacial layer properties and the conditions under which this layer can exist and propagate.

In this paper we formulate some necessary existence and propagation conditions for a moving transition layer separating two different solid phases of a micro-damaged medium. These two solid phases correspond to different macroregions in which microcracks are all open or all closed respectively, [START_REF] Woźniak | Laminated media with interface defects[END_REF][START_REF] Woźniak | On the dynamic behaviour of micro-damaged stratified media[END_REF].

Generally speaking, the term phase means here a state of a solid specified by a known constitutive relation. Hence the differences in material behaviour for different states of a solid are assumed to arise not only because of the structural phase transition (e.g. martensitic transformations described at the microlevel in [START_REF] Ball | Fine phase mixtures as minimizers of energy[END_REF], [START_REF] Collins | The computation of the austenitic-martensitic phase transition[END_REF] and [START_REF] James | Theory of diffusionless phase transition in PDEs and Continuum Models of Phase Transition Lecture Notes[END_REF]), but because of the existence of different strain energy functions related to different classes of local deformations. Such a situation was considered in [START_REF] Woźniak | On the dynamic behaviour of micro-damaged stratified media[END_REF] where the micro-damaged body behaviour, at the macrolevel, was described in terms of two different strain energy functions related to two different (and varying in time) macroregions (phases) with either all open or all closed microcracks. A phase transition takes place where microcracks are partly closed and partly open and hence in the part of the space in which a strain energy function (at a macrolevel) is not well defined. The considerations carried out in [START_REF] Woźniak | On the dynamic behaviour of micro-damaged stratified media[END_REF] were related to situations in which the aforementioned interfacial state can occur exclusively on certain discontinuity surfaces (which can arise and disappear in the deformation process) separating both phases. In this contribution we investigate the more general situation in which the interfacial state can be present in a certain spatial layer separating the two phases or situated inside one of them and moving across the medium.

The scope of the paper can be outlined as follows:

In Section 2 we introduce certain auxiliary concepts related to the kinematic description of the interfacial layer moving across a solid and separating two different phases. The main hypothesis, formulated in Section 3, is that this interfacial layer may be represented as a certain arrangement of both phases forming a definite banded periodic structure. After scaling down this structure we arrive at the mathematical model of the interfacial layer representing a particular fine mixture of both phases. We adapt here the modelling procedure proposed in [START_REF] Woźniak | A nonstandard method of modelling of thermoelastic periodic composites[END_REF] and [START_REF] Cz | Refined macrodynamics of periodic structures[END_REF] for composite materials. A similarity between a composite material structure and a fine phase mixture was already noted in [START_REF] Ball | Fine phase mixtures as minimizers of energy[END_REF]. The considerations carried out in Sections 2 and 3 are rather general, being related to an arbitrary solid. This solid can show itself in two different phases, each characterized by a separate analytical expression for the strain energy function, and in which the interface is modelled as a shell-like layer moving across the region occupied by the body in its motion. The obtained model is applied in Section 4 to the study of interfacial layers moving across the micro-damaged medium described in [START_REF] Woźniak | On the dynamic behaviour of micro-damaged stratified media[END_REF]. The solution to a special problem, illustrating the general considerations, is given in Section 5.

The conclusions formulated in Section 6 end the paper.

The main results of this contribution are related to a motion of interfacial layers in a micro-cracked solid and can be summarized as follows:

(1) the material behaviour of the interfacial layer depends not only on the material properties of both phases, but also on the propagation speed of the layer, (2) not every propagation speed of the layer is possible, (3) if the interfacial layer does not propagate across the medium, then its material properties are similar to those of a micro-layered two-component laminate [START_REF] Bakhvalov | Processes Averaging in Periodic Media[END_REF], [START_REF] Bensoussan | Asymptotic Analysis for Periodic Structures[END_REF], [START_REF] Woźniak | A nonstandard method of modelling of thermoelastic periodic composites[END_REF] and [START_REF] Cz | Refined macrodynamics of periodic structures[END_REF].

To simplify the treatment only diffusionless phase transitions and exclusively purely mechanical motions are considered throughout the paper; the modelling of more complex phenomena will be presented separately.

Preliminaries

Let B R be a known reference placement of a solid in the euclidean three-space E and p(; t ) : B R ! E be its configuration at an arbitrary time instant t.

Let us denote by the set of all non-singular 3 3 real matrices interpreted as deformation gradients F t (X) := rp(X; t ) 2 in an arbitrary configuration of a body. Let W + R ; W R be two different convex differentiable real-valued functions, both defined in , satisfying for every F 2 and for an arbitrary rotation matrix R the conditions

W + R (RF) = W + R ( F ) ; W R ( R F ) = W R ( F ) ;
i.e. being invariant under arbitrary rotations. Let there also be known two disjoined non-empty subsets + and of such that 0 := n ( + [ ) 6 = ;:

In the subsequent considerations we shall deal with an elastic homogeneous solid. The main assumption is that to every substantial point of this solid a strain energy function W R is assigned

W R (F) := W + R (F) if F 2 + ; W R (F) if F 2 ;
while for F 2 0 the strain energy function is not well defined. This means that for some values of the deformation gradient (i.e. for F 2 0 ) the material response of the solid under consideration is not determined.

Under the aforementioned conditions we deal with what will be called a two-phase solid, each phase being denoted by superscript + or .

For an overview of the problems related to the physical meaning and the interpretation of strain energy functions when dealing with micro-cracked materials see [START_REF] Woźniak | On the dynamic behaviour of micro-damaged stratified media[END_REF] and the references therein.

An example of the above situation was described in [START_REF] Woźniak | On the dynamic behaviour of micro-damaged stratified media[END_REF] where a certain macro-model of a micro-damaged body (with a continuous distribution of micro-cracks) was investigated.

In the aforementioned paper, as well as in the present one, + and are disjoined sets of deformation gradients such that if F 2 + then micro-cracks are all open and if F 2 then they are all closed. To these two phases of considered micro-cracked solid two different forms of strain energy function are assigned W + R ; W R respectively, since in both cases the responses of the material are different. At the same time in [START_REF] Woźniak | On the dynamic behaviour of micro-damaged stratified media[END_REF] the strain energy function was not well defined on certain surfaces separating both phases.

In this paper we study those micro-cracked materials for which it is physically meaningful to assume that certain regions exist in which not all micro-cracks are all open or all closed and hence a strain energy in these regions is not well defined.

It is easy to see that for an arbitrary configuration p(X; t ) ; X 2 B R ; of the solid under consideration the set B R can be partitioned into three mutually disjoined subsets

B + R (t) := fX 2 B R : F t (X) 2 + g ; B R (t) := fX 2 B R : F t (X) 2 g ; B 0 R (t) := X 2 B R : F t (X) 2 0 :
The subsets B + R (t) and B R (t) are respectively occupied by the phases with strain energies

W + R (F t (X)) and W + R (F t (X)). When X 2 B 0
R the strain energy is not well defined. A decomposition similar to that given above can also be introduced in the spatial region B t := p(B R ; t ) occupied by the solid in the configuration p(; t ) . Those points which belong to B 0 R (t) for every t constitute in B R (and hence in B t := p(B R ; t ) ) the interfacial part of the considered solid. This interfacial part can either separate the parts B + R (t) and B R (t) of B R occupied by the different phases in which the solid can present itself, or can have its boundary in common with only one of the parts B + R (t) and B R (t).

It was shown in [START_REF] Woźniak | On the dynamic behaviour of micro-damaged stratified media[END_REF] that these interfaces (occurring there as smooth surface elements) can appear and disappear in short time intervals and hence their motion may be not continuous. This is a main difference between an interfacial surface separating two phases of a solid and a moving discontinuity surface representing a wave in a material continuum.

The aim of this paper is to introduce and apply a new physical description for the moving interfacial region B 0 R (t) under the assumption that it is constituted by a system of thin layers.

Some of these layers can separate different solid phases or be situated inside one of them. We restrict ourselves to the study of the local properties of the interface so that we consider only one thin shell-like interfacial layer element J R (t); J R ( t ) B 0 R ( t ) with t 2 T := ( t o ; t f ) . Let M R (t) stand for the midsurface of this element and let it be parametrized by X = r(; t ) ; = ( 1 ; 2 ) 2 ;

being a regular plane region. Let the surface M R (t) be oriented by the field of unit normals n(; t ) . We will also assume that all considered interfacial elements have, for every t 2 T, a constant thickness 2h(t) which is small compared to the curvature radius of the midsurface M R (t): Under these assumptions the layer J R (t) can be parametrized by means of the well-known shell-like parametrization X = X(; ; t ) = r ( ; t ) + n ( ; t ) ;

(1) where = ( 1 ; 2 ) 2 parametrize the midsurface M R (t) and 2 ( h(t); h ( t )). It can be seen that the motion in B R t 2 T ! J R (t) of the interfacial layer element is uniquely determined by the motion of its midsurface t 2 T ! M R (t) and by the mapping t 2 T ! h(t):

(

Let us take into acount the inverse of the relation 1 setting = (X; t ) ; X 2 J R ( t ) ; t 2 T :

(

We remark that once the motion of the interface layer is known, the function () is also known. Treating now X and t as independent variables we can define c(X; t ) : = @( X ; t ) @t ; X2J R ; t 2 T

as the local propagation speed of the interfacial layer at X 2 J R ; t 2 T .

In the sequel we will tacitly assume that at least one interfacial layer element (either disjoining two phases or included into one of them) exists and its motion in B R t 2 T ! J R (t) B R is described by ( 1) and ( 2) which are representing a non-material shell-like element with time-varying thickness moving across B R .

On phase transition layers in certain micro-damaged two-phase solids 179

Modelling procedure

The aim of the subsequent analysis is to propose a physical description for the moving interfacial layer whose kinematics was described at the end of the previous section. The idea of the approach proposed is based on the heuristic hypothesis that the interfacial layer can be modelled, at a macrolevel, as a certain fine mixture of both phases. This fine mixture will be obtained performing a limit passage under the modelling assumption that the interface is constituted by a banded heterogeneous structure in which both phases occupy non-intersecting systems of very thin sublayers. For an example of the limit passage leading from a microdescription to a macrodescription for certain micro-cracked solids the reader is referred to [START_REF] Woźniak | On the dynamic behaviour of micro-damaged stratified media[END_REF].

Let us separate a layer J R (t) into a large number n of sublayers with equal thickness " = "(t) := h(t)=n separated by the surfaces = 0; = "; = 2" ; : : : ; = n" = h(t):

Define by h " (); 2 ( h ( t ) ; h ( t )) a continuous function such that h " () = + 1 2 " for = "; 3" ; : : : h " ( ) = 1 2 " for = 0; 2"; 4" ; : : : we obtain @h " @ () =

+ 1 if 2I 1 if 2I + :
The above saw-like function was used in modelling periodic-laminates, [START_REF] Woźniak | A nonstandard method of modelling of thermoelastic periodic composites[END_REF], [START_REF] Cz | Refined macrodynamics of periodic structures[END_REF], [START_REF] Woźniak | Laminated media with interface defects[END_REF],

and will be applied here under the assumption that sublayers for which the gradient of h " is equal to +1 are occupied by -phase and those for which it is equal to 1 by + phase.

Hence in the first step of modelling it is assumed that the interfacial element is an aggregate of both phases which occupy thin sublayers of thickness ". Thus, we deal with a certain banded structure inside every J R (t) whose motion (in the three-space E) can be assumed to have a similar form to that used for laminated materials [START_REF] Woźniak | Laminated media with interface defects[END_REF] and given by p(X; t ) = P ( X ; t ) + h " ( ) Q ( X ; t ) ;X 2 J R ( t ) ; t 2 T ;

(

) 5
where P(X; t ) ; Q ( X ; t ) are functions which in every ball of radius " can be treated as constants [START_REF] Woźniak | Laminated media with interface defects[END_REF] and is given by (3). The second term in (5) describes the disturbances in the motion t ! J R (t) caused by the heterogeneity of the material filling the adjacent sublayers [START_REF] Cz | Refined macrodynamics of periodic structures[END_REF].

180 F. dell'Isola and Cz. Woźniak

In the second step of modelling we take into account (3) and ( 4), treating X and t as independent variables (i.e. using the Lagrangean description of motion). After passing to the limit for " ! 0 we obtain F(X; t ) : = r p ( X ; t ) = r P ( X ; t ) +

+ 1 1 r ( X ; t ) Q ( X ; t ) _ p ( X ; t ) = _ P ( X ; t ) + + 1 1 c ( X ; t ) Q ( X ; t ) (6)
for X 2 J R (t); t 2 T , where the signs + and are related to the situation in which X belongs to or + phase respectively.

Thus, we have arrived at the description of a certain fine mixture: the deformation gradients in X at the instant t of the phase + and have to satisfy respectively the conditions rP(X; t ) + r ( X ; t ) Q ( X ; t ) 2 ; r P ( X ; t ) r ( X ; t ) Q ( X ; t ) 2 + :

The last conditions have to be verified for every X 2 J R (t) at every instant t and constitute existence conditions for the proposed model of interfacial layer. Equation ( 7) imposes restrictions both on the motion of the interfacial layer t 2 T ! J R (t) across B R (because r(X; t ) depends on this motion) as well as on the motion of this layer in E, described by the fields P and Q. The aforementioned vector fields, following [START_REF] Woźniak | A nonstandard method of modelling of thermoelastic periodic composites[END_REF], will be called macrodeformation and corrector fields respectively, while rP will be referred to as macrodeformation gradient field.

In order to obtain field equations for P() and Q() we apply a procedure similar to that used in micromodelling some composite materials, [START_REF] Woźniak | A nonstandard method of modelling of thermoelastic periodic composites[END_REF], [START_REF] Cz | Refined macrodynamics of periodic structures[END_REF], [START_REF] Woźniak | Laminated media with interface defects[END_REF]. Let us define the mean value hW R i of the strain energy hW R i(rP;

Q) := 1 2 [W + R (rp) + W R ( r p )]; (8)
where

W + R (rp) = W + R ( r P r Q); W R (rp) = W R ( r P + r Q ) (9)
and the mean value of the kinetic energy

1 2 h R _ p 2 i( _ P; Q) := 1 2 R ( _ P 2 + c 2 Q 2 ) ( 10
)
where R is the mass density in the reference configuration (constant in both phases).

If b is external body force, T R is the first Piola-Kirchhof stress tensor in the vicinity of the layer and n R is the unit outward normal to @J R (t), we obtain the action functional for the considered system as follows:

A(rP; _ P ; Q ) = Z t f t 0 ( A 1 ( t ) + A 2 ( t ))dt;
On phase transition layers in certain micro-damaged two-phase solids 181

A 1 (t) := Z J R t [ 1 2 R ( _ P 2 + c 2 Q 2 ) h W R i ( r P ; Q ) R b P ] d v ( X ) ; A 2 ( t ) : = Z @ J R t (T R n R )Pda(X): (11)
The principle of stationary action leads to the following system of equations

Div S R R P + b = 0; H R + R c 2 Q = 0; (12)
where S R := @hW R i @rP ; H R := @hW R i @Q :

(13)
Equations ( 12) and ( 13) have to be satisfied in J R (t) for every t 2 T. The above equations, together with the boundary condition

(S R T R ) n R = 0;
describe the behaviour of the interfacial layer element when it is possible to model it as a continuous fine mixture of the phases + and , i.e. when the existence conditions ( 7) hold and hence hW R i is well defined by means of ( 8) and ( 9). Hence ( 12) and ( 13) represent the mathematical model for the interfacial layers under consideration. Equation (13) defines a system of internal forces S R and H R in the interfacial layer; S R represents the first Piola-Kirchhoff total stress tensor and H R can be called propagational forces, as they are equal to zero if the local propagation speed c = c(X; t ) is zero.

The interfacial layer was modelled above as a two-phase fine mixture with equal phase concentrations. However, physically motivated situations may exist in which such concentrations are different and which can also depend on the variable . In order to describe these situations denote by + and (both belonging to [0;1] and such that + + = 1) the concentrations respectively of + and phases inside the interfacial layer J R (t). We will assume that their distribution

+ = + (); = (); 2 ( h(t); h ( t ))
is known. For the sake of simplicity we have assumed that both fields + and are independent of 2 and t 2 T.

Following the approach used above we define i := 0; 2"; 4" ; : : : ; h ( t ) 6 i 6 h ( t ) and partition every part of the layer J R (t); bounded by the coordinate surfaces = i "; = i + " (" being previously determinated), into the two sublayers ( i "; i " + 2" ( i )); ( i " + 2" ( i ); i + " ) closed cracks respectively) and the sets of admissible local deformations are assumed to be given, in terms of the infinitesimal strain tensor e, by + := fe : n en > + g ;W + R ( e ) = 1 2 E + ( n en) 2 ; := fe : n en < g ;W R ( e ) = 1 2 E ( n en) 2 ;

where the constants + ; ; + > ; which we can call strain thresholds, play the role of the phase-defining thresholds. Moreover, the constants E + and E are longitudinal elastic moduli relative to the uniaxial strain and to the phase with all open and all closed microcracks respectively. They are such that E > E + : when the cracks are open the stiffness of the material is lower than in the case of closed cracks. In the model discussed in [START_REF] Woźniak | Laminated media with interface defects[END_REF] and [START_REF] Woźniak | On the dynamic behaviour of micro-damaged stratified media[END_REF] it is assumed that + = = 0: In the sequel we will study the propagation of a longitudinal plane displacement wave, opening (or closing) microcracks, in which the phase change occurs inside a layer. We will use the following denotations:

(1) U is the component of displacement vector field along n and U ;n its derivative in the direction of n, (2) (t) is the position of the midsurface of the interfacial layer and 2h(t) its thickness at any instant t,

(3) S is the normal stress component, S = n S R n, (4)
we define the longitudinal wave propagation speeds in both phases by

c + := s E + R ; c := s E R :
The average strain energy for the considered continuum and displacement fields is easily evaluated once (8) is recalled

hW R i = 1 2 Ẽ(U ;n ) 2 + E Q U ;n + 1 2 Ẽ(Q) 2
where

Ẽ := 1 2 (E + + E ); E := E E + :
Therefore, we obtain the following expression for the normal stress S in terms of Q and U ;n S = ẼU ;n + E Q while the second from [START_REF] Sun | Time-harmonic waves in a stratified medium propagating in the direction of the layering[END_REF] becomes

ẼQ + EU ;n R c 2 Q = 0:
This last equation allows us to express Q in terms of U ;n and c 2

Q = E Ẽ R c 2 U ;n
On phase transition layers in certain micro-damaged two-phase solids 185 so that S can be expressed in terms of U ;n and c 2 as follows

S = Ẽ E 2 Ẽ R c 2 ! U ;n : (19)
We will call interfacial longitudinal modulus the quantity

E(c 2 ) := Ẽ E 2 Ẽ R c 2 :
The interfacial longitudinal modulus must be positive everywhere inside the interfacial layer. This physical condition implies the following restrictions on the propagation speed c c < c + or c > c :

(20)

The field U has to satisfy the hyperbolic equation implied by the first from [START_REF] Sun | Time-harmonic waves in a stratified medium propagating in the direction of the layering[END_REF],

S ;n R d 2 U dt 2 + b = 0 in B 0 R (t); (21)
in which the expression (19) for S has to be substituted. Equation (21) has to be considered together with the continuity conditions for S and U on both surfaces separating the interfacial layer from the remaining part of the solid. If this layer is included between + and phase, then the position of these surfaces is given by U ;n ((t) + h ( t ) ; t ) = + ;U ;n ((t) h(t); t ) = :

It has to be emphasized that the above analysis has a physical sense only if condition (7) holds inside the interfacial layer. In the framework of the linearized theory the aforementioned condition yields

U ;n + Q = 1 E Ẽ R c 2 ! U ;n < ; U ;n Q = 1 + E Ẽ R c 2 ! U ;n > + : (22)
These inequalities must be satisfied at every point of the interfacial layer if it is to be modelled as an ideal mixture of phases + and , as done in the present paper.

Outside the interfacial layer the standard equations of linear elasticity with strain energy, equal respectively to W + R and W R , are assumed to hold.

We remark that:

(1) the interfacial longitudinal modulus reduces to the effective modulus found in [START_REF] Woźniak | A nonstandard method of modelling of thermoelastic periodic composites[END_REF] when c vanishes;

(2) when c tends to c + or to c respectively from lower or higher values then at (t) + h ( t )

or (t) h(t) respectively the interfacial longitudinal modulus vanishes.

Illustrative example

The analysis carried out in Section 2-4 was quite formal and one can ask whether the model of the interfacial layer presented here satisfies the existence conditions [START_REF] Dell'isola | A phenomenological approach to phase transition in classical field theory[END_REF] or ( 22) and hence has physical sense. In order to give an answer to the above question, using the results found in Section 4, we will consider below a simple problem illustrating the behaviour of microdamaged solids under consideration.

Let the micro-cracked solid occupy a thick layer bounded by coordinate planes x = 0; x = L and the vector n, determining the orientation of microcracks, coincide with the versor of x-axis. The solid is subject to a body force acting along the x-axis and having constant value b and to the external pressure p = p(t) applied at the plane x = 0. Moreover, let the displacement of the solid be equal to zero at x = L:

Hence, we deal with an uniaxial strain state corresponding to a displacement field with only one non-vanishing component U(x; t) in which the material properties of the solid are determined by the elasticity moduli E + and E . We must also recall that we assume as known the two strain thresholds + and with + > > 0. It means that if U ;x > + or U ;x < then the microcracks are all open or all closed and the corresponding stress strain relations are given respectively by S = E + U ;x or S = E U ;x :

Our aim is to find the displacement field U(x; t); x 2 [ 0 ; L ] ; t 2 T :For the sake of simplicity its evolution will be treated only in the quasi-stationary case.

It can be easily shown that the evolution of the midsurface and of the thickness of the interfacial layer, provided that it exists, is given by

(t) = 1 R b p ( t ) + E + + E 2 ! ; h ( t ) = + E + + E 2 R b = : h = const (
On phase transition layers in certain micro-damaged two-phase solids 187

In the region occupied by the interfacial layer the stress-strain relation cannot be specified in the framework of standard elasticity theory. Thus, we model it using the results found in Sections 3 where the conditions (20) are assumed to be verified. Since (t); c ( t ) and h are given by ( 23), ( 25) and (26) together with the continuity conditions for the displacement U and the stress S for x = (t) h and (t) + h as well as with the boundary conditions U(0; t ) = 0 ; E + U ; 1 ( 0 ; t ) = p ( t ) 8 t 2 T allow, with simple algebra, the analytical expression for the unique solution of the considered evolution problem, provided that for every t 2 T the conditions (24) are verified. which represent the conditions imposed on the material parameters E; Ẽ; R ; and + under which the interfacial layer, propagating at the instant t 2 T with the speed c(t) 2 ( 1; c + ) [ ( c ; 1 ) , can be modelled as an ideal mixture of the two considered solid phases.

A necessary condition to (27), as + > ; is c < c + ; 8 t 2 T (28) which excludes the higher propagation speeds.

In concluding this Section we consider two special cases of the conditions (27):

(1) Let us assume that = + . In this case (as E > 0) the conditions (27) are equivalent to condition (28), if one recalls (20). If we assume that = + = 0, (as done in [START_REF] Woźniak | On the dynamic behaviour of micro-damaged stratified media[END_REF]) then (23) implies that the interfacial layer becomes an interfacial surface.

(2) If + > 0 > then the conditions (27) are never satisfied. In this case the proposed model of the interfacial layer as fine mixture of both phases does not make sense from the physical viewpoint. Indeed our reasoning proves that two phases with the phase-defining thresholds verifying + > 0 > cannot coexist as constituents of the fine mixture introduced in this paper.

  and linear in every interval between aforementioned values of . Setting I := ( 0 ; " ) [ ( 2 "; 3") [ [ ( 2 "; ") [ ( 4"; 3") [ I + : = ( "; 2") [ (3";4") [ [ ( "; 0) [ ( 3"; 2") [

  and 4. Combining (19) and (21) we obtain

	Ẽ	E 2 Ẽ R c 2 (t)	!	U ;xx (x;t) + R b = 0
	if x 2 ((t) h; (t) + h ) ;	(26)

  However, the model applied may have no physical sense if this solution does not verify the existence condition (22) which now reads U ;x (x;t) > + ; 8 x 2 ( ( t ) h; (t) + h ) :

	1	E Ẽ R c 2 (t)	!	U ;x (x;t) < ;
	1	E Ẽ R c 2 ( t )
	The last inequalities imply that
	1	E Ẽ R c 2 (t)	!	<	+	;	1	E Ẽ R c 2 (t)	!	> +	(27)

!

occupied respectively by the + and phases. Let h " (); 2 ( h(t) + 2 "; h(t)) be a continuous function, linear inside each of the aforementioned sublayers, which on the surfaces separating sublayers attains the values h " ( i + ") = h " ( i " ) = " minf + ( i ); ( i ) g ; h " ( i " + 2 " ( i )) = " minf + ( i ); ( i ) g : 2 ( h ( t ) ; h ( t ))

we see that @h " @ () = ' + ( ) if 2 ( i " + 2" ( i ); i + " ) ; ' ( )if 2 ( i "; i " + 2" ( i )):

Passing to the limit " ! 0; instead of ( 6) we now obtain

where = (X; t ) .

The interfacial layer existence conditions become rP(X; t ) + ' + ( ( X ; t )) r(X; t ) Q ( X ; t ) 2 ; r P ( X ; t ) + ' ( (

and must hold for every X 2 B R ; t 2 T . Under [START_REF] Woźniak | Laminated media with interface defects[END_REF] we obtain

and the mean strain energy will be given by hW R i(rP; Q) := + ((X; t ))W + R (rp) + ( ( X ; t ))W R (rp):

For the mean value of the kinetic energy we obtain

On phase transition layers in certain micro-damaged two-phase solids 183

From the principle of stationary action we obtain again (12 1 ) in which the mean strain energy hW R i is now defined by ( 16) and ( 17). The layer existence conditions are now given by ( 15) while ( 122 ) is replaced by

where H R is given by ( 132 ) in which the means strain energy is given by ( 16) and ( 17).

In order to explain the above generalization of the proposed modelling procedure we shall consider two special cases. First, assume that it is physically motivated the supposition that the passage across the interface from one phase to another is continuous. In this case we can set

;

provided that the passage from phase to + phase takes place in the -axis direction. Second, let the interfacial layer be situated inside, for instance, the + phase. In some cases it may be reasonable to require that this interface is constituted by a mixture of + and phases in which concentration varies smoothly. Such a situation will be described by setting:

This case constitutes a continuum model of the physical situation investigated in [START_REF] Woźniak | On the dynamic behaviour of micro-damaged stratified media[END_REF], where the existence of such interfacial layers, but with a discrete structure, was shown.

It should be emphasized that if there is no physical information about the internal structure of the interfacial layer, then the assumption + = = 0:5 and hence the model introduced before seem to be the most reasonable.

Applications

In this section we want to apply the model developed up to now to the study of the propagation of a particular interfacial layer. More precisely, we consider a certain variant of the microcracked material of which the macromodel was introduced and investigated in [START_REF] Woźniak | Laminated media with interface defects[END_REF] and [START_REF] Woźniak | On the dynamic behaviour of micro-damaged stratified media[END_REF]: the considered solid body is assumed damaged because of the presence of planar stochastically distributed microcracks (parallel to the laminae interfaces) whose planes are oriented in B R by a regular field n of unit vectors. It will be assumed that the region B R can be partitioned into three regions: i.e. the region in which the microcracks are open (phase +), the region in which the microcracks are closed (phase ) and the interfacial layer in which the microcracks are partly closed and partly open. In the last case we assume that the strain energy is not defined.

We will limit our study to the case (1) of linear elastic micro-cracked materials in which the vector field n is constant, (2) of interfacial layer propagation in the direction of n.

Therefore, we will restrict our consideration to uniaxial strain states. In this case the strain energy functions W + R and W R (the strain energy in the medium with all open and with all

Conclusions

Summarizing the general results obtained in Section 3 we can conclude that:

(1) in the proposed model the properties of the medium constituting the interfacial layer can be easily determined once the constitutive properties of phases + and and the motion of the layer are known,

(2) since c is the relative velocity in the interfacial layer with respect to the referencial space B R then the equations ( 12) and ( 13) are Galilean invariant,

(3) the second from ( 13) is an algebraic relation Q and rP. The definition (8) of average strain energy implies that using it one can express Q in terms of rP and c 2 and therefore the strain energy, for the introduced interfacial fine mixture can be regarded as a function of the macrodeformation gradient rP and the speed c, (4) if the interfacial layer coincides with a material one (i.e. if it is not moving across B R ) then the speed c vanishes and the equations ( 12) and ( 13) reduce to those found in [START_REF] Cz | Refined macrodynamics of periodic structures[END_REF] where some macromodels for micro-laminated materials are considered.

Moreover, in Section 4 of this paper we have proven that:

(1) the interfacial layers can be modelled as fine mixtures of both phases only if the propagation speed satisfies the inequalities (20), (2) the material behaviour of the medium filling the interfacial layer depends not only on the material properties of both phases but also on the propagation speed of the layer, (3) since the interfacial longitudinal elastic modulus has to be always positive not every propagation speed of the interfacial layer is possible.

Finally, in Section 5 we have shown that the class of systems described by the model proposed in this paper is not empty.

The aforementioned results lead to the conclusion that the proposed approach to the modelling of interfacial layers in the micro-damaged two-phase solids seems to deserve both mathematical and physical interest.


Jean-Francois Ponge
email: jean-francois.ponge@wanadoo.fr

Fungal communities: relation to resource succession

An example of substrate succession: the colonization of pine needles by fungus flora

.

 Mitchell and Millar 1978b), iii) climate [START_REF] Van Maanen | Distribution of three fungi colonising fallen Pinus sylvestris needles along altitudinal transects[END_REF][START_REF] Gourbière | Associations between three fungi on pine needles and their variation along a climatic gradient[END_REF]), but did not seem to be affected by surface grazing of the needles [START_REF] Mclean | Does selective grazing by mites and collembola affect litter fungal community structures?[END_REF].

A more complete pattern, including penetration of pine needles by mycorrhizal fungi and soil micro-and mesofauna, was observed by [START_REF] Ponge | Étude écologique d'un humus forestier par l'observation d'un petit volume, premiers résultats. I. La couche L 1 d'un moder sous pin sylvestre[END_REF][START_REF] Ponge | Étude écologique d'un humus forestier par l'observation d'un petit volume. II. La couche L 2 d'un moder sous Pinus sylvestris[END_REF][START_REF] Ponge | Étude écologique d'un humus forestier par l'observation d'un petit volume. III. La couche F 1 d'un moder sous Pinus sylvestris[END_REF][START_REF] Ponge | Ecological study of a forest humus by observing a small volume. I. Penetration of pine litter by mycorrhizal fungi[END_REF]Ponge ( , 1991a) by scrutinizing successive layers within a small surface of Scots pine litter (5x5 cm). The succession of organisms, both microbial and animal, which took place in pine needles, from death of foliage to disappearance in humus, was summarized in Figure 1, taking into account numerous possible short-cuts which were found to occur currently. It should be highlighted that all organisms involved in this successional course mineralized organic matter, through excretory as well as respiratory pathways. In the course of this successional process, which can be considered at first sight as a processing chain sensu [START_REF] Heard | Processing chain ecology: resource condition and interspecific interactions[END_REF], the substrate was observed to change, as long as resources were exploited by successive inhabitants of pine needles, and part of these resources was lost [START_REF] Berg | Nutrient dynamics in some decomposing leaf and needle litter types in a Pinus sylvestris forest[END_REF] or used for the build-up of microbial and animal biomass [START_REF] Stark | Nutrient cycling pathways and litter fungi[END_REF][START_REF] Hasegawa | Carbon and nutrient dynamics in decomposing pine needle litter in relation to fungal and faunal abundances[END_REF].

The exploitation of internal pine needle tissues was found to begin by the use of cell contents, with weak signs of cell wall destruction. Sections done in needle parts colonized by Lophodermium pinastri (Schrad.) Chev., an ascomyete infecting senescent needles, showed that the fungus was present as hyphae living in the mesophyll tissue, without any penetration of plant cell walls. In the mesophyll tissue, files of cells appeared collapsed, without any starch grains, but cells were still entire, although with a distinct browning of their walls. No profound change occurred in the stele, except a distinct browning of phloem cell walls [START_REF] Ponge | Étude écologique d'un humus forestier par l'observation d'un petit volume, premiers résultats. I. La couche L 1 d'un moder sous pin sylvestre[END_REF](Ponge , 1991a)). Clearly, the action of the fungus was external and limited to full use of cell wall contents, but browning of cell walls was indicative of its cellulolytic power [START_REF] Kirk | Degradation and conversion of lignocelluloses[END_REF]. All needles colonized by this fungus showed typical black diaphragms delineating territories, each occupied by a clone, and black fruit bodies between the epidermis and the hypodermis. At this stage, entire needles or, more often, needle parts, can be occupied by another senescence stage fungus, which fructifies once the needle is on the ground: the coelomycete Ceuthospora pinastri (Fr.) Höhn., pycnidial imperfect state of the ascomycete Phacidium lacerum Fr., improperly identified as Fusicoccum bacillare Saccardo & Penzig by [START_REF] Kendrick | Biological aspects of the decay of Pinus silvestris leaf litter[END_REF]. Some needles, detached from the tree before reaching maturity, were infected by Lophodermella spp., a stele-invading pathogenic ascomycete [START_REF] Williamson | Histochemistry of Corsican pine needles infected by Lophodermella sulcigena (Rostr.) v.Höhn[END_REF], Mitchell et al. 1978).

The second main colonizer, Verticicladium trifidum Preuss, conidial state of the ascomycte Desmazierella acicola Lib., was resting as small melanized stroma in ostiola of needles colonized by L. pinastri. D. acicola was also observed to behave as a first colonizer when needles were still not infected at the time they fell on the ground. When needle parts were colonized by L. pinastri or C. pinastri, while other parts were still not colonized, then V. trifidum was observed to occur first in fungus-free needle sections, later on extending its colonies to the whole needle. In no case V. trifidum and L. pinastri were found living together in the same section. Whatever happened before, all needles became progressiveley colonized by V. trifidum and the lower layer of entire needles was entirely made of black, softened needles resulting from the activity of this fungus. It has been shown by [START_REF] Kendrick | Biological aspects of the decay of Pinus silvestris leaf litter[END_REF] to live several years within the same needle. Several stages were observed during the time this dematiaceous fungus occupied a needle. First, it appeared as thickwalled hyphae growing longitudinally at the inside of resin canals and protoxylem tracheids, but cells from phloem, mesophyll and transfusion tissues were also penetrated altogether [START_REF] Ponge | Étude écologique d'un humus forestier par l'observation d'un petit volume, premiers résultats. I. La couche L 1 d'un moder sous pin sylvestre[END_REF](Ponge , 1991a)). At this stage, the only tissue which remained free of fungus was the metaxylem, but all other lignified tissues (transfusion tissue, protoxylem) remained intact, with transparent and refringent cell walls (except after previous occupation by L. pinastri). In some needles, starch grains were still present in mesophyll cells, testifying for V. trifidum as a first colonizer. In other needles, previous occupancy by L. pinastri or C. pinastri was attested by the presence of hard, recalcitrant tissues, such as diaphragms or pycnidial walls, respectively. At this stage, blackening of the needles was restricted to the vicinity of stomata, where V. trifidum filled sub-stomatic chambers with its black stromata. Melanization of pine cells appeared to occur only in stomatal guard cells and nearby hypodermal cells. The next step was the further development of V. trifidum, which formed black stromata in all internal tissues, particularly in the transfusion tissue [START_REF] Ponge | Étude écologique d'un humus forestier par l'observation d'un petit volume. II. La couche L 2 d'un moder sous Pinus sylvestris[END_REF]. Tracheids of the transfusion tissue disappeared progressively by lysis, leaving only areolae visible under the phase contrast light microscope. Melanization of pine needles affected the entire hypodermis, the cell walls of which appeared covered internally by thick black deposits, despite the absence of fungal penetration. The late development of V. trifidum was thus responsible for blackening and softening of pine needles, which made them palatable to soil saprophagous fauna [START_REF] Hayes | Studies on the feeding preferences of some phthiracarid mites (Acari: Oribatidae)[END_REF]. At this stage, having gained enough energy from the near entire consumption of needle internal tissues, this fungus fructified abundantly, in the form of dense bushes of black conidiophores protruding from stomatal apertures.

At the stage of the late development of V. trifidum, needles were actively penetrated by members of soil mesofauna, particularly oribatid mites and enchytraeids. A succession was observed from oribatids to enchytraeids, the latter group invading preferably needles which had been previously excavated by oribatids, which filled them with their excrements [START_REF] Ponge | Étude écologique d'un humus forestier par l'observation d'un petit volume. III. La couche F 1 d'un moder sous Pinus sylvestris[END_REF](Ponge , 1991a)). However, several instances were found of enchytraeids penetrating directly needles previously invaded by V. trifidum or even only L. pinastri [START_REF] Ponge | Étude écologique d'un humus forestier par l'observation d'un petit volume, premiers résultats. I. La couche L 1 d'un moder sous pin sylvestre[END_REF]. Defecation by enchytraeids occurred mainly outside pine needles, contrary to oribatid mites, except in most superficial needles where environmental conditions were probably too dry outside pine needles. Within oribatid faeces, pine material appeared to be finely ground by mouth parts of mites and became humified during the intestinal transit, as assessed by optical properties of gut contents. Pine cell walls took a brown and amorphous aspect, with fuzzy contour, indicating strong transformation of both cellulose and lignin [START_REF] Kilbertus | Étude ultrastructurale de quelques étapes de la biodégradation dans la nature du tissu lignifié de Pinus nigra Arn. subsp. laricio (Poir.) Maire[END_REF][START_REF] Saur | Alimentary studies on the collembolan Paratullbergia callipygos using transmission electron microscopy[END_REF]). Pine material seemed much less transformed in enchytraeid faeces, at least when these animals did not reingest oribatid faeces. Despite abundance and intense activity, enchytraeid worms contributed poorly to humification, contrary to oribatids, as this was observed also by [START_REF] Toutain | Étude ultrastructurale des processus de biodégradation. II. Modèle enchytréides-litière de feuillus[END_REF] in beech litter. Penetration by microfauna (nematodes, amoebae) was observed, using holes done by bigger animals. At this stage a bacterial development was prominent within and around collapsed pine needles, following inoculation of microbial germs by soil fauna [START_REF] Macfadyen | The animal habitat of soil bacteria[END_REF][START_REF] Kilbertus | Étude ultrastructurale de quelques étapes de la biodégradation dans la nature du tissu lignifié de Pinus nigra Arn. subsp. laricio (Poir.) Maire[END_REF][START_REF] Touchot | Rôle d'un collembole (Folsomia candida) au cours de la dégradation des litières de charme et de chêne, en présence ou en absence d'argile[END_REF].

Given size and shape of the cells, these bacterial colonies were supposed to include nitrogen-fixing strains [START_REF] Ponge | Étude écologique d'un humus forestier par l'observation d'un petit volume. III. La couche F 1 d'un moder sous Pinus sylvestris[END_REF].

At this stage, needles became highly friable and most of them were left as small fragments embedded in animal faecal deposits, mostly of enchytraeid origin, which were permeated by dense mycelial webs of mycorrhizal fungi. Dematiaceous (melanized) hyphae of the ascomycete Cenococcum geophilum Fr. and hyaline hyphae of the basidiomycete Hyphodontia sp. were found to arise from monopodial jet-black and coral-like orange-brown mycorrhizae, respectively. Penetration of remaining needles by C. geophilum was prominent [START_REF] Ponge | Étude écologique d'un humus forestier par l'observation d'un petit volume. III. La couche F 1 d'un moder sous Pinus sylvestris[END_REF][START_REF] Ponge | Ecological study of a forest humus by observing a small volume. I. Penetration of pine litter by mycorrhizal fungi[END_REF](Ponge , 1991a)), the fungus passing from its aerial to its submerged form, but resources used by this fungus at the inside of pine needles were not identified, although observations on other humus components concluded to its chitinolytic and cellulolytic activity.

Anyhow, the profuse development of mycorrhizal fungi around and within animal faeces and pine needle remains let us to suppose that they used and translocated nutrients released by microbial and animal activity at the inside of pine needles [START_REF] Bending | The structure and function of the vegetative mycelium of ectomycorrhizal plants. V. Foraging behaviour and translocation of nutrients from exploited litter[END_REF]. It should be highlighted that the bacterial development registered before this stage seemed to be arrested by mycorrhizal fungi, maybe under the influence of their antibiotic activity [START_REF] Krywolap | An antibiotic produced by the mycorrhizal fungus Cenococcum graniforme[END_REF][START_REF] Marx | The influence of ectotrophic mycorrhizal fungi on the resistance of pine roots to pathogenic infections. I. Antagonism of mycorrhizal fungi to root pathogenic fungi and soil bacteria[END_REF][START_REF] Suay | Screening of basidiomycetes for antimicrobial activities[END_REF].

Studies on other coniferous species

Numerous parralels can be found with studies on other conifers. In particular, we must highlight the paramount work done on fir needles (Abies alba Mill.) by [START_REF] Gourbière | Structure spatio-temporelle de la mycoflore des premiers stades de décomposition des aiguilles d'Abies alba[END_REF][START_REF] Gourbière | Structure spatio-temporelle de la mycoflore décomposant les aiguilles d'Abies alba: place de Marasmius androsaceus[END_REF], [START_REF] Gourbière | Microscopie de la mycoflore des aiguilles de sapin (Abies alba). I. Thysanophora penicilloides[END_REF], [START_REF] Gourbière | Activité et développement d'Adoristes ovatus (C.L. Koch, 1839) (Acarien, Oribate) dans les aiguilles d'Abies alba Mill. Relations avec la décomposition et les microflores fongiques[END_REF][START_REF] Gourbière | Microscopie de la mycoflore des aiguilles de sapin (Abies alba). II. Lophodermium piceae[END_REF]Gourbière et al. ( , 1987[START_REF] Gourbière | Microscopie de la mycoflore des aiguilles de sapin blanc (Abies alba). IV. Décomposition de la cuticule, de l'hypoderme et de l'épiderme[END_REF], [START_REF] Gourbière | Décomposition des aiguilles d'Abies alba: hétérogénéité du substrat et de la mycoflore, rôle de Marasmius androsaceus[END_REF], [START_REF] Savoie | The role of polysaccharidase enzymes in the organisation of the fungal communities of a forest leaf litter[END_REF], 1988[START_REF] Betina | Mycotoxins: chemical, biological and environmental aspects[END_REF] and [START_REF] Savoie | Cellulase activities in senescent coniferous needles and in needle litter naturally colonized by various fungi[END_REF]. They described a fungal succession quite similar to that observed on pine needles. Fir needles were first colonized by Lophodermium piceae (Fckl.) Höhn., a vicariant of L. pinastri, then by Thysanophora penicilloides (Roum.) Kendrick, in place of V. trifidum in pines. The segregation between T. penicilloides and L. piceae was similar to that observed between V. trifidum and L. pinastri. However, a prominent difference was that V. trifidum was observed to remain in pine needles for several years [START_REF] Kendrick | Biological aspects of the decay of Pinus silvestris leaf litter[END_REF], which allowed it to exploit most internal resources of decaying needles, while T. penicilloides (or L. piceae in the absence of further replacement by T. penicilloides) was succeeded within a few months by the white rot basidiomycete Marasmius androsaceus (L.: Fr.) Fr. (Gourbière et al. 1987[START_REF] Gourbière | Structure spatio-temporelle de la mycoflore décomposant les aiguilles d'Abies alba: place de Marasmius androsaceus[END_REF]), thus it did not participate to a great extent to the degradation of cell wall material [START_REF] Gourbière | Microscopie de la mycoflore des aiguilles de sapin (Abies alba). I. Thysanophora penicilloides[END_REF][START_REF] Gourbière | Microscopie de la mycoflore des aiguilles de sapin (Abies alba). II. Lophodermium piceae[END_REF]). The penetration of fir needles by rhizomorphs of M. androsaceus, which could occur soon after needle fall, was retarded when needles or parts of needles had been previously colonized by L. piceae. This phenomenon was possibly due to the existence of diaphragms, which may act as physical barriers [START_REF] Ponge | Étude écologique d'un humus forestier par l'observation d'un petit volume, premiers résultats. I. La couche L 1 d'un moder sous pin sylvestre[END_REF]. The presence of M. androsaceus has been often recorded in pines, too [START_REF] Lehmann | The fungal succession on normal and urea-treated pine needles[END_REF], Mitchell & Millar 1978a[START_REF] Soma | Ecological studies of soil organisms with references to the decomposition of pine needles. I. Soil macrofaunal and mycofloral surveys in coastal pine plantations[END_REF][START_REF] Ponge | Étude écologique d'un humus forestier par l'observation d'un petit volume. II. La couche L 2 d'un moder sous Pinus sylvestris[END_REF], 1991a[START_REF] Cox | Effects of fungal inocula on the decomposition of lignin and structural polysaccharides in Pinus sylvestris litter[END_REF]), but its presence in coniferous litter seems to be erratic, probably due to the needle-byneedle colonisation ability of its rhizomorph system [START_REF] Macdonald | The rhizomorphs of Marasmius androsaceus Fries[END_REF][START_REF] Gourbière | Décomposition des aiguilles d'Abies alba: hétérogénéité du substrat et de la mycoflore, rôle de Marasmius androsaceus[END_REF]). The importance of the time of fall for the colonisation of coniferous needles by M. androsaceus or other internal fungi (T. penicilloides on fir or V. trifidum on pine) was suggested by [START_REF] Ponge | Étude écologique d'un humus forestier par l'observation d'un petit volume. II. La couche L 2 d'un moder sous Pinus sylvestris[END_REF] and demonstrated experimentally by [START_REF] Gourbière | Structure spatio-temporelle de la mycoflore décomposant les aiguilles d'Abies alba: place de Marasmius androsaceus[END_REF].

How to explain the observed successions?

In the course of the above mentioned successional processes of coniferous needle decomposition, food and habitat resources for fungi change to a great extent. The exhaustion of cell contents by early colonizers is followed by the differential attack of cellulose-rich then of lignin-rich cell wall material [START_REF] Savoie | Équipement enzymatique des champignons dégradant les aiguilles d'Abies alba Mill[END_REF][START_REF] Cox | Effects of fungal inocula on the decomposition of lignin and structural polysaccharides in Pinus sylvestris litter[END_REF]. In the mean time, fungal, then bacterial biomass, is built-up, which constitutes a new food resource for further colonizers [START_REF] Berg | Fungal biomass and nitrogen in decomposing Scots pine needle litter[END_REF]. These changes are accompanied by an increase in nitrogen [START_REF] Berg | Dynamics of nitrogen ( 15 N) in decomposing Scots pine (Pinus sylvestris) needle litter. Long-term decomposition in a Scots pine forest VI[END_REF][START_REF] Hasegawa | Carbon and nutrient dynamics in decomposing pine needle litter in relation to fungal and faunal abundances[END_REF], water [START_REF] Virzo De Santo | Factors regulating early-stage decomposition of needle litters in five different coniferous forests[END_REF]) and metal content [START_REF] Laskowski | Dynamics of some mineral nutrients and heavy metals in decomposing forest litter[END_REF], while fungal metabolism produces organic acids [START_REF] Takao | Organic acid production by basidiomycetes. I. Screening of acid-producing strains[END_REF][START_REF] Hintikka | Occurrence of calcium oxalate in relation to the activity of fungi in forest litter and humus[END_REF][START_REF] Lapeyrie | Oxalic acid synthesis by the mycorrhizal fungus Paxillus involutus (Batsch. Ex Fr.)[END_REF][START_REF] Devêvre | Release of complexing organic acids by rhizosphere fungi as a factor in Norway spruce yellowing in acidic soils[END_REF], melanins [START_REF] Kuo | Inhibition of the lysis of fungi by melanins[END_REF][START_REF] Butler | Pathogenic properties of fungal melanins[END_REF]) and other metabolites, among them toxins and antibiotics have been widely reported [START_REF] Wilkins | Investigation into the production of bacteriostatic substances by fungi. Preliminary examination of the ninth 100 species, all basidiomycetes[END_REF][START_REF] Krywolap | An antibiotic produced by the mycorrhizal fungus Cenococcum graniforme[END_REF][START_REF] Land | Mycotoxin production by some wood-associated Penicillium spp[END_REF][START_REF] Betina | Mycotoxins: chemical, biological and environmental aspects[END_REF]). Tannins, terpenes, and other secondary metabolites of coniferous litter exert a selective effect on fungal communities [START_REF] Black | Spore germination and germ hyphal growth of microfungi from litter and soil in the presence of ferulic acid[END_REF][START_REF] Berg | Chemical components of Scots pine needles and needle litter and inhibition of fungal species by extractives. In: Structure and function of northern coniferous forests. An ecosystem study[END_REF][START_REF] Lindeberg | Stimulation of litter-decomposing basidiomycetes by flavonoids[END_REF][START_REF] Lindeberg | Stimulation of some litter-decomposing basidiomycetes by shikimic acid[END_REF], but are progressively degraded by microbial activity [START_REF] Rai | Utilization of cellulose and gallic acid by litter inhabiting fungi and its possible implication in litter decomposition of a tropical deciduous forest[END_REF][START_REF] Lorenz | Litter decomposition and humus characteristics in Canadian and German spruce ecosystems: information from tannin analysis and 13 C CPMAS NMR[END_REF]. Thus the internal biochemical environment of coniferous needles varies to a great extent during decomposition, which may interfere with fungal requirements [START_REF] Savoie | Cellulase activities in senescent coniferous needles and in needle litter naturally colonized by various fungi[END_REF]. The role of fauna should not be neglected, too. Needle-consuming animals create cavities [START_REF] Gourbière | Activité et développement d'Adoristes ovatus (C.L. Koch, 1839) (Acarien, Oribate) dans les aiguilles d'Abies alba Mill. Relations avec la décomposition et les microflores fongiques[END_REF], Ponge 1991a), comminute and humify organic matter [START_REF] Ponge | Étude écologique d'un humus forestier par l'observation d'un petit volume. III. La couche F 1 d'un moder sous Pinus sylvestris[END_REF](Ponge , 1991a(Ponge , 1991b)), mobilize nitrogen [START_REF] Faber | The interaction of Collembola and mycorrhizal roots in nitrogen mobilization in a Scots pine forest soil[END_REF]) and inoculate microbes [START_REF] Pherson | The role of invertebrates in the fungal colonization of leaf litter[END_REF][START_REF] Ponge | Étude écologique d'un humus forestier par l'observation d'un petit volume, premiers résultats. I. La couche L 1 d'un moder sous pin sylvestre[END_REF], 1985), thus they condition the inside of pine needles in a different way than fungi themselves. Still controversial, while highly probable, is the selection role of differential grazing on fungal successions [START_REF] Newell | Interaction between two decomposer basidiomycetes and a collembolan under sitka spruce: grazing and its potential effects on fungal distribution and litter decomposition[END_REF][START_REF] Klironomos | Feeding preferences of the collembolan Folsomia candida in relation to microfungal successions on decaying litter[END_REF][START_REF] Bengtsson | Patchiness and compensatory growth in a fungus-Collembola system[END_REF][START_REF] Mclean | Does selective grazing by mites and collembola affect litter fungal community structures?[END_REF]. On other substrates than coniferous needles it has been demonstrated that, in the absence of soil fauna, the net result of competition between fungal species was a decrease in the weight loss of the decaying substrate (Rayner et al. 1984[START_REF] Lussenhop | Interaction of competing fungi with fly larvae[END_REF], while the contrary was observed in the presence of grazing fauna [START_REF] Lussenhop | Interaction of competing fungi with fly larvae[END_REF]. It should not be forgotten, too, that pine material is more or less rapidly, but ineluctably, transformed into animal faeces where other microbial successions can be observed [START_REF] Van Der Drift | The significance of the break-down of oak litter by Enoicyla pusilla[END_REF][START_REF] Nicholson | Studies on the decomposition of the faecal pellets of a millipede (Glomeris marginata (Villers))[END_REF][START_REF] Hanlon | Some factors influencing microbial growth on soil animal faeces. II. Bacterial and fungal growth on soil animal faeces[END_REF][START_REF] Tajovský | Decomposition of faecal pellets of the millipede Glomeris hexasticha (Diplopoda) in forest soil[END_REF]).

8

We may wonder whether the observed successions are governed by resources, biochemical interference or other interactions between organisms. More probably, a complex of biological and non-biological factors is involved in fungal successions on decaying substrates, as this has been demonstrated in wood [START_REF] Boddy | Fungal community ecology and wood decomposition processes in angiosperms: from standing tree to complete decay of coarse woody debris[END_REF]. Unfortunately, only partial answers can be found in the published literature, given the high degree of specialization now achieved by soil microbiology, and the need for sophisticated methods to address adequately mechanisms.

However, some experimental and descriptive studies can throw light on the way by which fungal strains are replaced or cohabit in decaying pine needles. Sometimes, it will be necessary to address other fungal successions, such as those prevailing during wood decay [START_REF] Levy | The place of basidiomycetes in the decay of wood in contact with the ground[END_REF][START_REF] Coates | Fungal population and community development in cut beech logs. III. Spatial dynamics, interactions and strategies[END_REF][START_REF] Renvall | Community structure and dynamics of wood-rotting basidiomycetes on decomposing conifer trunks in northern Finland[END_REF][START_REF] Boddy | Fungal community ecology and wood decomposition processes in angiosperms: from standing tree to complete decay of coarse woody debris[END_REF][START_REF] Hendry | Abiotic variables effect differential expression of latent infections in beech (Fagus sylvatica)[END_REF] if similar mechanisms can be suspected to occur in decaying needles.

The first result we want to underline is that near all fungal strains involved in the degradation of forest litter are known to have cellulolytic activities [START_REF] Hudson | The development of the saprophytic fungal flora as leaves senesce and fall[END_REF][START_REF] Savoie | Decomposition of cellulose by the species of the fungal succession degrading Abies alba needles[END_REF]. In vitro, microfungi from the phylloplane, generally classified as "sugar" fungi [START_REF] Garrett | Ecological groups of soil fungi: a survey of substrate relationships[END_REF], prove also able to oxidatively cleave phenolic compounds [START_REF] Hogg | Micro-fungi on leaves of Fagus sylvatica. II. Duration of survival, spore viability and cellulolytic activity[END_REF][START_REF] Haider | Synthesis and transformation of phenolic compounds by Epicoccum nigrum in relation to humic acid formation[END_REF][START_REF] Rai | Utilization of cellulose and gallic acid by litter inhabiting fungi and its possible implication in litter decomposition of a tropical deciduous forest[END_REF]. However, we have shown that early colonizers of coniferous needles, such as Lophodermium spp., did not attack lignified cell walls [START_REF] Ponge | Étude écologique d'un humus forestier par l'observation d'un petit volume, premiers résultats. I. La couche L 1 d'un moder sous pin sylvestre[END_REF][START_REF] Gourbière | Microscopie de la mycoflore des aiguilles de sapin (Abies alba). II. Lophodermium piceae[END_REF]), such attack being rather performed slowly by secondary (or late primary) colonizers such as V. trifidum and T. penicilloides [START_REF] Gourbière | Microscopie de la mycoflore des aiguilles de sapin (Abies alba). I. Thysanophora penicilloides[END_REF][START_REF] Ponge | Étude écologique d'un humus forestier par l'observation d'un petit volume. III. La couche F 1 d'un moder sous Pinus sylvestris[END_REF]) and, much more rapidly, by non-specific white rots such as M. androsaceus and Mycena galopus (Pers.: Fr.) Kummer [START_REF] Frankland | Autecology and the mycelium of a woodland litter decomposer[END_REF][START_REF] Ponge | Étude écologique d'un humus forestier par l'observation d'un petit volume. II. La couche L 2 d'un moder sous Pinus sylvestris[END_REF][START_REF] Gourbière | Décomposition des aiguilles d'Abies alba: hétérogénéité du substrat et de la mycoflore, rôle de Marasmius androsaceus[END_REF], Gourbière et al. 1987[START_REF] Cox | Effects of fungal inocula on the decomposition of lignin and structural polysaccharides in Pinus sylvestris litter[END_REF]). Despite differences in fungal enzymic properties, in particular in the possession of phenoloxydases [START_REF] Kirk | Degradation and conversion of lignocelluloses[END_REF][START_REF] Hammel | Driven by nature: plant litter quality and decomposition[END_REF], the segregation of fungal colonies on the same needle [START_REF] Gourbière | Structure spatio-temporelle de la mycoflore des premiers stades de décomposition des aiguilles d'Abies alba[END_REF], Ponge 1991a) and switch-over effects of previous occupants on the fungal succession [START_REF] Gourbière | Structure spatio-temporelle de la mycoflore décomposant les aiguilles d'Abies alba: place de Marasmius androsaceus[END_REF]) point on the importance of biological interactions [START_REF] Rayner | Interspecific mycelial interactions. An overview[END_REF][START_REF] Wicklow | The role of competition in the structuring of fungal communities[END_REF][START_REF] Boddy | Interspecific combative interactions between wood-decaying basidiomycetes[END_REF]. Most of these interactions are based on the defence of the fungal individualistic territory by short-distance biochemical interference [START_REF] Rayner | Interspecific mycelial interactions. An overview[END_REF][START_REF] Wicklow | Interference competition. In: The fungal community, its organization and role in the ecosystem[END_REF] or, in the case of Lophodermium diaphragms, by physical barriers [START_REF] Ponge | Étude écologique d'un humus forestier par l'observation d'un petit volume, premiers résultats. I. La couche L 1 d'un moder sous pin sylvestre[END_REF].

The nutrient status of coniferous needles may have an impact on the fungal succession, as this was demonstrated by [START_REF] Lehmann | The fungal succession on normal and urea-treated pine needles[END_REF] and Mitchell & Millar (1978b): the application of lime or urea to decaying litter favoured the more nutrient-demanding ascomycetes (early colonizers) and disfavoured less-demanding white rot basidiomycetes (late colonizers), while the decomposition rate was increased [START_REF] Sanchez | Loblolly pine needle decomposition and nutrient dynamics as affected by irrigation, fertilization, and substrate quality[END_REF]. This could indicate that early colonizers are potentially able to fulfill the whole decomposition process but lack nutrients to i) exploit existing resources, ii) antagonize better-equipped fungi. These, and more especially cellulolytic basidiomycetes, are able to derive micro-and macronutrients from the degradation of recalcitrant compounds such as cell walls and tanninprotein complexes [START_REF] Saitô | Coactions between litter-decomposing hymenomycetes and their associated microorganisms during decomposition of beech litter[END_REF][START_REF] Entry | Litter decomposition and nutrient release in ectomycorrhizal mat soils of a Douglas fir ecosystem[END_REF], starting with the production of low energycost oxalic acid, non-enzymatically active during early stages of cellulose degradation [START_REF] Hintikka | Studies on white-rot humus formed by higher fungi in forest soils[END_REF][START_REF] Schmidt | A proposed role for oxalic acid in nonenzymatic wood decay by brown-rot fungi[END_REF], followed by high energy-cost enzymic production at later stages of degradation [START_REF] Kirk | Degradation and conversion of lignocelluloses[END_REF][START_REF] Hammel | Driven by nature: plant litter quality and decomposition[END_REF].

All these results point on biological processes as key factors that determine fungal successions at the inside of decaying coniferous needles. Colonization and dispersal are two fundamental steps of the development of fungal communities, at least from the point of view of the individualistic mycelium [START_REF] Ogawa | Ecology of higher fungi in Tsuga diversifolia and Betula ermani-Abies mariesii forests of subalpine zone[END_REF], Rayner et al. 1984[START_REF] Dowson | Outgrowth patterns of mycelial cordforming basidiomycetes from and between woody resource units in soil[END_REF][START_REF] Dahlberg | Size, distribution and biomass of genets in populations of Suillus bovinus (L.: Fr.) Roussel revealed by somatic incompatibility[END_REF][START_REF] Gourbière | Competition between unit-restricted fungi: a metapopulation model[END_REF]. Intra-and inter-specific competition contribute in turn to the shape of the community, by restricting each fungus both in space and time [START_REF] Rayner | Interspecific mycelial interactions. An overview[END_REF][START_REF] Boddy | Interspecific combative interactions between wood-decaying basidiomycetes[END_REF], 2001). Such interactive processes, including founder effects, i.e. the advantage given to the first invader, have been demonstrated to play an important role in plant successions [START_REF] Connell | Mechanisms of succession in natural communities and their role in community stability and organization[END_REF][START_REF] Finegan | Forest succession[END_REF][START_REF] Pickett | A hierarchical consideration of causes and mechanisms of succession[END_REF][START_REF] Grime | Dominant and subordinate components of plant communities: implications for succession, stability and diversity[END_REF][START_REF] Mccook | Understanding ecological community succession: causal models and theories, a review[END_REF] as well as in fungal successions [START_REF] Tribe | Interactions of soil fungi on cellulose film[END_REF][START_REF] Coates | Fungal population and community development in cut beech logs. III. Spatial dynamics, interactions and strategies[END_REF][START_REF] Frankland | Mechanisms in fungal succession[END_REF][START_REF] Renvall | Community structure and dynamics of wood-rotting basidiomycetes on decomposing conifer trunks in northern Finland[END_REF][START_REF] Niemelä | Interactions of fungi at late stages of wood decomposition[END_REF][START_REF] Hendry | Abiotic variables effect differential expression of latent infections in beech (Fagus sylvatica)[END_REF]. [START_REF] Gourbière | Proportion of needles colonized by one fungal species in coniferous litter: the dispersal hypothesis[END_REF] modelized the persistence and the extinction of a fungal species colonizing a number of discrete resource units, and applied this model to the experimental colonization of fir needles. Their results showed that the model, the parameters of which were determined by the experiment, accounted for the observed distribution of needles colonized in the field by the same fungus. Later on, they extended their model to two competing species, demonstrating that both species could coexist even in the absence of any trade-off between competitive and colonization abilities, but that the outcome of competition depended on a founder effect [START_REF] Gourbière | Competition between unit-restricted fungi: a metapopulation model[END_REF]. Recent discoveries did not prove unequivocally that biological traits of individuals/species and their interactions are the only reasons for fungal successions, but rather that biological patterns and processes play a decisive role in the way by which species are assembled both in space and time, as this has been recognized for a long time in plant communities [START_REF] Watt | Pattern and process in the plant community[END_REF].

Conclusion: a story of coniferous needles

A hypothetical scheme, which explains most of the variation observed in the fungal colonisation of coniferous needles can be drawn, on the basis of present knowledge.

Colonization of the inside of needles starts by the penetration of a restricted array of fungal strains which are able to withstand the biochemical environment of coniferous foliage (phenols, terpenes, carbon dioxide). This early colonisation occurs while needles are still attached on the tree, during the senescence stage. This step can be precociously achieved when fungal pathogens penetrate the needle, which causes its premature fall. Once fallen on the ground, the development of these early colonizers goes further, at the inside of territories delineated by barriers (biochemical, physical) created by each individualistic mycelium, until reproduction organs are produced. As far as original toxic compounds are degraded, and fungal defences are alleviated (for instance following full use of energy for fructification), colonisation may progress through the development of other, less specialized strains, already present as resting organs at the needle surface or able to transport energy from needle to needle through rhizomorphs. Litter-dwelling animals play an active role in the dissemination of fungal spores and, possibly if not clearly demonstrated, by stimulating or impeding the development of some fungal strains, according to their feeding preferences.

Legends of figures

Fig. 1 .

 1 Fig. 1. Succession of organisms observed during the decomposition of Scots pine needles

Mitchell & Millar 1978a), ii) the nutrient status of the foliage[START_REF] Lehmann | The fungal succession on normal and urea-treated pine needles[END_REF]