Samir Adly
email: samir.adly@unilim.fr

Oanh Chau
email: oanh.chau@univ-reunion.fr

Mohamed Rochdi
email: mohamed.rochdi@univ-reunion.fr

SOLVABILITY OF A CLASS OF THERMAL DYNAMICAL CONTACT PROBLEMS WITH SUBDIFFERENTIAL CONDITIONS

Keywords: 2000 Mathematics Subject Classification. Primary: (2000) 74M10, 74M15, 74F25, 74H20, 74H25, 34G20 long memory thermo-visco-elasticity, sub-differential contact condition, dynamical process, fixed point, evolution inequality

We study a class of dynamic thermal sub-differential contact problems with friction, for long memory visco-elastic materials, which can be put into a general model of system defined by a second order evolution inequality, coupled with a first order evolution equation. We present and establish an existence and uniqueness result, by using general results on first order evolution inequality, with monotone operators and fixed point methods.

1. Introduction. Despite of numerous recent progress, contact mechanics still remains a rich domain of various new problems, and the literature devoted to the subject is more and more extensive. An early attempt at the study of contact problems for elastic viscoelastic materials within the mathematical analysis framework was introduced in the pioneering reference works [START_REF] Ciarlet | I : Three-Dimensional Elasticity[END_REF][START_REF] Duvaut | Les Inéquations en Mécanique et en Physique[END_REF][START_REF] Necas | On the solution of variational inequality to the Signorini problem with small friction[END_REF][START_REF] Kikuchi | Contact problems in elasticity[END_REF][START_REF] Panagiotopoulos | Inequality Problems in Meechanics and Applications[END_REF]. More recently, the dynamic Signorini's problem for a cracked viscoelastic body is studied in [START_REF] Cocou | Existence of a solution to a dynamic unilateral contact problem for a cracked viscoelastic body[END_REF]. Mathematical analysis of unilateral contact problems involving static, quasi-static and dynamic process may be found in the recent self-contained book [START_REF] Ch | Unilateral Contact Problems, Variational Methods and Existence Theorems[END_REF]. Further extensions to non convex contact conditions with non-monotone and possible multi-valued constitutive laws led to the active domain of non-smooth mechanic within the framework of the so-called hemivariational inequalities, for a mathematical as well as mechanical treatment we refer to [START_REF] Goeleven | Variational and Hemivariational Inequalities, Theory, Methods and Applications, Volume I: Unilateral Analysis and Unilateral Mechanics[END_REF][START_REF] Panagiotopoulos | Hemivariational Inequalities, applications in Mechanics and Engineering[END_REF]. Quasi-static contact problems, by taking into account the parameter of the temperature field, were analyzed in [START_REF] Awbi | Quasistatic Thermovisoelastic Frictional Contact Problem with Damped Response[END_REF], where the friction is described by a general normal damped response condition, there the existence and uniqueness of weak solution has been established. The existence of solutions for thermal viscoelastic dynamic contact problems with Coulomb friction law were treated in [START_REF] Ch | Unilateral Contact Problems, Variational Methods and Existence Theorems[END_REF]. This work is a companion paper of the results obtained in [START_REF] Chau | On a dynamic bilateral contact problem with friction for viscoelastic materials[END_REF]. In [START_REF] Chau | On a dynamic bilateral contact problem with friction for viscoelastic materials[END_REF] we studied a class of dynamical long memory viscoelastic problems, where the contact is bilateral and frictional obeying the Tresca's law. The variational formulation of the problem leads to a second order evolution inequality on the displacement field. Then an existence and uniqueness result of "weak" solution on displacement has been proved. Here we consider a class of dynamical long memory viscoelastic problems, the new feature in this paper is that the contact is defined by a general sub-differential condition on the velocity, and with heat exchange, putting then the problem into a coupled system, defined by a second order evolution inequality on the displacement field, and a differential equation which governs the evolution of the temperature field. On the other hand we investigate an existence and uniqueness result of "strong" solution on displacement and temperature fields, i.e. solution with more regularity. For this proposal we specify stronger assumptions on the data and operators, and use a new method based on a version of first order evolution inequality with monotonicity, convexity and fixed point theory.

The paper is organized as follows. In Section 2 we describe the mechanical problem, specify the assumptions on the data to derive the variational formulation, and then we state our main existence and uniqueness result. In Section 3, we give the proof of the claimed result.

2. Statement of the problem. In this section we study a class of thermal contact problems with sub-differential conditions, for long memory visco-elastic materials. We describe the mechanical problems, list the assumptions on the data and derive the corresponding variational formulations. Then we state an existence and uniqueness result on displacement and temperature fields, which we will prove in the next section.

The physical setting is as follows. A visco-elastic body occupies the domain Ω with surface Γ that is partionned into three disjoint measurable parts Γ 1 , Γ 2 and Γ 3 , such that meas(Γ 1 ) > 0. Let [0, T ] be the time interval of interest, where T > 0. The body is clamped on Γ 1 × (0, T ) and therefore the displacement field vanishes there. We also assume that a volume force of density f 0 acts in Ω × (0, T ) and that surface tractions of density f 2 act on Γ 2 × (0, T ). The body may come in contact with an obstacle, the foundation, over the potential contact surface Γ 3 . The model of the contact is specified by a general sub-differential boundary condition, where thermal effects may occur in the frictional contact with the basis. We are interested in the dynamic evolution of the body.

Let us recall now some classical notations, see e.g. [START_REF] Duvaut | Les Inéquations en Mécanique et en Physique[END_REF][START_REF] Nečas | Mathematical Theory of Elastic and Elastoplastic Bodies: an introduction[END_REF] for further details. We denote by S d the space of second order symmetric tensors on R d (d = 2, 3), while " • " and | • | will represent the inner product and the Euclidean norm on S d and R d . Let Ω ⊂ R d be a bounded domain with a Lipschitz boundary Γ and let ν denote the unit outer normal on Γ. Everywhere in the sequel the indexes i and j run from 1 to d, summation over repeated indices is implied and the index that follows a comma represents the partial derivative with respect to the corresponding component of the independent variable. We also use the following notation:

H = L 2 (Ω) d , H = { σ = (σ ij ) | σ ij = σ ji ∈ L 2 (Ω), 1 ≤ i, j ≤ d}, H 1 = { u ∈ H | ε(u) ∈ H }, H 1 = { σ ∈ H | Div σ ∈ H }.
Here ε : H 1 -→ H and Div : H 1 -→ H are the deformation and the divergence operators, respectively, defined by :

ε(u) = (ε ij (u)), ε ij (u) = 1 2 (u i,j + u j,i ), Div σ = (σ ij,j ).
The spaces H, H, H 1 and H 1 are real Hilbert spaces endowed with the canonical inner products given by :

(u, v) H = Ω u i v i dx, (σ, τ ) H = Ω σ ij τ ij dx, (u, v) H1 = (u, v) H + (ε(u), ε(v)) H , (σ, τ ) H1 = (σ, τ ) H + (Div σ, Div τ ) H .
We recall that C denotes the class of continuous functions; and C m , m ∈ N * the set of m times differentiable functions. Finally D(Ω) denotes the set of infinitely differentiable real functions with compact support in Ω; and W m,p , m ∈ N, 1 ≤ p ≤ +∞ for the classical Sobolev spaces; and

H m 0 (Ω) := {w ∈ W m,2 (Ω), w = 0 on Γ}, m ≥ 1.
To continue, the mechanical problem is then formulated as follows.

Problem Q : Find a displacement field u : Ω × [0, T ] -→ R d and a stress field σ : Ω × [0, T ] -→ S d and a temperature field θ : Ω × [0, T ] -→ R + such that for a.e. t ∈ (0, T ):

σ(t) = Aε( u(t)) + Gε(u(t)) + t 0 B(t -s) ε(u(s)) ds -θ(t) C e in Ω (2.1) ü(t) = Div σ(t) + f 0 (t) in Ω (2.2) u(t) = 0 on Γ 1 (2.3) σ(t)ν = f 2 (t) on Γ 2 (2.4) u(t) ∈ U, ϕ(w) -ϕ( u(t)) ≥ -σ(t)ν • (w -u(t)) ∀w ∈ U on Γ 3 (2.5) θ(t) -div(K c ∇θ(t)) = -c ij ∂ ui ∂ x j (t) + q(t) on Ω (2.6) -k ij ∂ θ ∂ x j (t) n i = k e (θ(t) -θ R ) on Γ 3 (2.7) θ(t) = 0 on Γ 1 ∪ Γ 2 (2.8) θ(0) = θ 0 in Ω (2.9) u(0) = u 0 , u(0) = v 0 in Ω (2.10)
Here, (2.1) is the Kelving Voigt's long memory thermo-visco-elastic constitutive law of the body, σ the stress tensor, A is the viscosity operator, G for the elastic operator, C e := (c ij ) represents the thermal expansion tensor, and B is the so called tensor of relaxation which defines the long memory of the material, as an important particular case, when B ≡ 0, we find again the usual visco-elasticity of short memory. In (2.2) is the dynamic equation of motion where the mass density ≡ 1. On the contact surface, the general relation (2.5) is a sub-differential boundary condition, where

D(Ω) d ⊂ U ⊂ H 1
represents the set of contact admissible test functions, σν denotes the Cauchy stress vector on the contact boundary and ϕ : Γ 3 × R d -→ R is a given function. Various situations may be modelled by such a condition, and some concrete examples will be recalled below. The differential equation (2.6) describes the evolution of the temperature field, where K c := (k ij ) represents the thermal conductivity tensor, q(t) the density of volume heat sources. The associated temperature boundary condition is given by (2.7), where θ R is the temperature of the foundation, and k e is the heat exchange coefficient between the body and the obstacle, and p τ : Γ 3 × R + -→ R + is a given tangential function. Finally, u 0 , v 0 , θ 0 represents the initial displacement, velocity and temperature, respectively.

To derive the variational formulation of the mechanical problems (2.1)-(2.10) we need additional notations. Thus, let V denote the closed subspace of H 1 defined by

D(Ω) d ⊂ V = { v ∈ H 1 | v = 0 on Γ 1 } ∩ U ; E = {η ∈ H 1 (Ω), η = 0 on Γ 1 ∪ Γ 2 }. Since meas Γ 1 > 0, Korn's inequality holds : there exists C K > 0 which depends only on Ω and Γ 1 such that ε(v) H ≥ C K v H1 ∀v ∈ V A proof
of Korn's inequality may be found in [START_REF] Nečas | Mathematical Theory of Elastic and Elastoplastic Bodies: an introduction[END_REF], p.79. On V we consider the inner product given by

(u, v) V = (ε(u), ε(v)) H ∀ u, v ∈ V,
and let • V be the associated norm, i.e.

v V = ε(v) H ∀ v ∈ V.
It follows that • H1 and • V are equivalent norms on V and therefore (V, • V ) is a real Hilbert space. Moreover, by the Sobolev's trace theorem, we have a constant C 0 > 0 depending only on Ω, Γ 1 , and

Γ 3 such that v L 2 (Γ3) ≤ C 0 v V ∀ v ∈ V.
In the study of the mechanical problem (2.1)-(2.10), we assume that the viscosity operator A : Ω × S d -→ S d , (x, τ ) -→ (a ijkh (x) τ kh ) is linear on the second variable and satisfies the usual properties of ellipticity and symmetry, i.e.

       (i) a ijkh ∈ W 1,∞ (Ω); (ii) Aσ • τ = σ • Aτ ∀σ, τ ∈ S d , a.e. in Ω; (iii) there exists m A > 0 such that Aτ • τ ≥ m A |τ | 2 ∀τ ∈ S d , a.e. in Ω (2.11)
The elasticity operator G : Ω × S d -→ S d satisfies :

             (i) there exists L G > 0 such that |G(x, ε 1 ) -G(x, ε 2 )| ≤ L G |ε 1 -ε 2 | ∀ε 1 , ε 2 ∈ S d , a.e. x ∈ Ω ; (ii) x -→ G(x, ε) is Lebesgue measurable on Ω, ∀ε ∈ S d ; (iii) the mapping x -→ G(x, 0) ∈ H (2.12)
The relaxation tensor

B : [0, T ] × Ω × S d -→ S d , (t, x, τ ) -→ (B ijkh (t, x) τ kh ) satisfies      (i) B ijkh ∈ W 1,∞ (0, T ; L ∞ (Ω)); (ii) B(t)σ • τ = σ • B(t)τ ∀σ, τ ∈ S d , a.e. t ∈ (0, T ), a.e. in Ω (2.13)
We suppose the body forces and surface tractions satisfy

f 0 ∈ W 1,2 (0, T ; H), f 2 ∈ W 1,2 (0, T ; L 2 (Γ 2 ) d ) (2.14)
For the thermal tensors and the heat sources density, we suppose that

C e = (c ij ), c ij = c ji ∈ L ∞ (Ω), q ∈ W 1,2 (0, T ; L 2 (Ω)) (2.15)
The boundary thermic data satisfy

k e ∈ L ∞ (Ω; R + ), θ R ∈ W 1,2 (0, T ; L 2 (Γ 3 )) (2.16)
The thermal conductivity tensor verifies the usual symmetry and ellipticity : for some c k > 0 and for all (ξ i ) ∈ R d ,

K c = (k ij ), k ij = k ji ∈ L ∞ (Ω), k ij ξ i ξ j ≥ c k ξ i ξ i .
(2.17)

Finally we have to put technical assumptions on the initial data and the sub-differential condition on the contact surface, in order to use classical results on first order set valued evolution equations. Many various possibilities could be considered (see e.g. ([2, 3, 11]). Here we use a general theorem taken in ( [START_REF] Lions | Quelques méthodes de résolution des problèmes aux limites non linéaires[END_REF]) p. 46, in a simplified case, which is enough for our proposal and applications.

We assume that the initial data satisfy the conditions

u 0 ∈ V, v 0 ∈ V ∩ H 2 0 (Ω) d , θ 0 ∈ E ∩ H 2 0 (Ω) (2.18)
On the contact surface, the following frictional contact function

ψ(w) := Γ3 ϕ(w) da verifies                                          (i) ψ : V -→ R
is well defined, continuous and convex;

(ii) there exists a sequence of differentiable convex functions

(ψ n ) : V -→ R such that ∀w ∈ L 2 (0, T ; V ), T 0 ψ n (w(t)) dt -→ T 0 ψ(w(t)) dt, n -→ +∞;
(iii) for all sequence (w n ) and w in W 1,2 (0, T ; V ) such that

w n w, w n w weakly in L 2 (0, T ; V ), then lim inf n-→+∞ T 0 ψ n (w n (t)) dt ≥ T 0 ψ(w(t)) dt (iv) ∀w ∈ V, (w = 0 on Γ 3 =⇒ ∀n ∈ N, ψ n (w) = 0 V ) (2.19)
Here ψ n (v) denotes the Fréchet derivative of ψ n at v.

To continue, using Green's formula, we obtain the variational formulation of the mechanical problem Q in abstract form as follows.

Problem QV :

Find u : [0, T ] → V , θ : [0, T ] → E satisfying a.e. t ∈ (0, T ):                    ü(t) + A u(t) + B u(t) + C θ(t), w -u(t) V ×V + ( t 0 B(t -s) ε(u(s)) ds, ε(w) -ε( u(t))) H + ψ(w) -ψ( u(t)) ≥ f (t), w -u(t) V ×V ∀w ∈ V ; θ(t) + K θ(t) = R u(t) + Q(t) in E ; u(0) = u 0 , u(0) = v 0 , θ(0) = θ 0 .
Here, the operators and functions A, B :

V -→ V , C : E -→ V , ψ : V -→ R, K : E -→ E , R : V -→ E , f : [0, T ] -→ V , and Q : [0, T ] -→ E are defined by ∀v ∈ V , ∀w ∈ V , ∀τ ∈ E, ∀η ∈ E: A v, w V ×V = (A(εv), εw) H ; B v, w V ×V = (G(εv), εw) H ; Cτ, w V ×V = -(τ C e , εw) H ; f (t), w V ×V = (f 0 (t), w) H + (f 2 (t), w) (L 2 (Γ2)) d ; Q(t), η E ×E = Γ3 k e θ R (t) η dx + Ω q(t) η dx; K τ, η E ×E = d i,j=1 Ω k ij ∂τ ∂x j ∂η ∂x i dx + Γ3 k e τ • η da; R v, η E ×E = - Ω c ij ∂v i ∂x j η dx.
Theorem 1. Assume that (2.11)-(2.19) hold, then there exists an unique solution {u, θ} to problem QV with the regularity :

u ∈ W 2,2 (0, T ; V ) ∩ W 2,∞ (0, T ; H) θ ∈ W 1,2 (0, T ; E) ∩ W 1,∞ (0, T ; F ). ( 2

.20)

Before giving the proof, we show in the following two typical examples of sub-differential conditions.

Example 1. Tresca's friction law.

The contact condition on Γ 3 is bilateral, and satisfies (see e.g. [START_REF] Duvaut | Les Inéquations en Mécanique et en Physique[END_REF][START_REF] Panagiotopoulos | Inequality Problems in Meechanics and Applications[END_REF]):

   u ν = 0, |σ τ | ≤ g, |σ τ | < g =⇒ uτ = 0, |σ τ | = g =⇒ uτ = -λσ τ , for some λ ≥ 0, on Γ 3 × (0, T ).
Here g represents the friction bound, i.e., the magnitude of the limiting friction traction at which slip begins, with g ∈ L ∞ (Γ 3 ), g ≥ 0 a.e. on Γ 3 . We deduce the admissible displacement space:

V := {w ∈ H 1 ; with w = 0 on Γ 1 ; w ν = 0 on Γ 3 }, and the sub-differential contact function:

ϕ(x, y) = g(x)|y τ (x) | ∀x ∈ Γ 3 , y ∈ R d ,
where y τ (x) := y -y ν(x) ν(x), y ν(x) := y • ν(x), with ν(x) the unit normal at x ∈ Γ 3 . We have then

ψ(v) := Γ3 g |v τ | da, ∀v ∈ V
is well defined on V with the property: for some c > 0,

|ψ(w) -ψ(v)| ≤ c v -w L 2 (Γ3) d , ∀v, w ∈ V.
Let us show that assumptions in (2.19) are verified. Firstly it is clear that ψ : V -→ R is convex. By using the continuous embedding from V into L 2 (Γ 3 ) d and the last inequality, we find that

ψ is Lipschitz continuous on V.
This gives (2.19)(i). To approximate the function ψ, we use the sequence

ψ n (v) := Γ3 g |v τ | 2 + 1 n da, ∀v ∈ V, ∀n ∈ N * .
Now we claim that ψ n ( n ∈ N * ) is Frechet differentiable and the Frechet derivative of ψ n is given by

ψ n (v).h = Γ3 g (v τ , h τ ) R d |v τ | 2 + 1 n da, ∀h ∈ V.
(2.21) Indeed. Let fix n ∈ N * and v ∈ V . For any h ∈ V :

ψ n (v + h) -ψ n (v) = Γ3 g |v τ + h τ | 2 + 1 n -|v τ | 2 + 1 n da, thus ψ n (v + h) -ψ n (v) = Γ3 g |v τ + h τ | 2 -|v τ | 2 |v τ + h τ | 2 + 1 n + |v τ | 2 + 1 n da, and
ψ n (v + h) -ψ n (v) = Γ3 g 2 (v τ , h τ ) R d + |h τ | 2 |v τ + h τ | 2 + 1 n + |v τ | 2 + 1 n da.
As

|v τ + h τ | 2 + 1 n + |v τ | 2 + 1 n ≥ 2 √ n , we have Γ3 g |h τ | 2 |v τ + h τ | 2 + 1 n + |v τ | 2 + 1 n da ∈ O( h 2 L 2 (Γ3) d ) as h -→ 0 V . Hence Γ3 g |h τ | 2 |v τ + h τ | 2 + 1 n + |v τ | 2 + 1 n da ∈ O( h 2 V ) as h -→ 0 V .
We obtain as h -→ 0 V :

2 (v τ , h τ ) R d |v τ + h τ | 2 + 1 n + |v τ | 2 + 1 n - (v τ , h τ ) R d |v τ | 2 + 1 n = (v τ , h τ ) R d |v τ | 2 + 1 n -|v τ + h τ | 2 + 1 n O(1),
where O(1) denotes some scalar function of h bounded in a neighborhood of 0 V .

Previous computations show that

|v τ | 2 + 1 n -|v τ + h τ | 2 + 1 n = (v τ , h τ ) R d O(1) as h -→ 0 V . Thus 2 (v τ , h τ ) R d |v τ + h τ | 2 + 1 n + |v τ | 2 + 1 n - (v τ , h τ ) R d |v τ | 2 + 1 n = O(|h τ | 2 R d ) as h -→ 0 V . Consequently Γ3 g 2 (v τ , h τ ) R d |v τ + h τ | 2 + 1 n + |v τ | 2 + 1 n da = Γ3 g (v τ , h τ ) R d |v τ | 2 + 1 n da + O( h 2 V ) as h -→ 0 V .
This ends the proof of the claim as

h ∈ V -→ Γ3 g (v τ , h τ ) R d |v τ | 2 + 1 n da
is a linear continuous functional.

To continue, we deduce from the Frechet derivative's formula that ψ n is of class C 1 . Direct algebraic computations show that for all α ≥ 0, β ≥ 0 such that α + β = 1, and for all reals x and y, n ≥ 1:

(α x + β y) 2 + 1 n ≤ α x 2 + 1 n + β y 2 + 1 n . (2.22)
Then ψ n is convex for all n ≥ 1. The convergence property in (2.19)(ii) follows from Lebesgue's dominated convergence theorem.

To justify (2.19)(iii), consider a sequence (w n ) and w in W 1,2 (0, T ; V ) satisfying

w n w, w n w weakly in L 2 (0, T ; V ). As ψ n (w n ) ≥ ψ(w n ), ∀n ∈ N * , then lim inf n-→+∞ T 0 ψ n (w n (t)) dt ≥ lim inf n-→+∞ T 0 ψ(w n (t)) dt.

Now using the compact embedding theorem, we obtain

w n w, w n w weakly in L 2 (0, T ; V ) =⇒ w n -→ w stronly in L 2 (0, T ; L 2 (Γ 3 ) d ).
Because of the continuous embedding from V into L 2 (Γ 3 ) d , we deduce that

w n -→ w stronly in L 2 (0, T ; V ).
As ψ is Lipschitz continuous on V , we have that the functional

w ∈ L 2 (0, T ; V ) -→ T 0 ψ(w(t)) dt
is well defined and is continuous. Consequently The normal damped response contact condition with Tresca's friction law (see e.g. [START_REF] Jarušek | Dynamic contact problems with small Coulomb friction for viscoelastic bodies[END_REF]) is defined by:

lim inf n-→+∞ T 0 ψ(w n (t)) dt = lim n-→+∞ T 0 ψ(w n (t)) dt = T 0 ψ(w(t)) dt
   -σ ν = k 0 | uν | r-1 uν , |σ τ | ≤ g, |σ τ | < g =⇒ uτ = 0, |σ τ | = g =⇒ uτ = -λσ τ , for some λ ≥ 0, on Γ 3 × (0, T ).
Here 0 < r < 1 and g, k 0 ∈ L ∞ (Γ 3 ), g ≥ 0, k 0 ≥ 0. The coefficient k 0 represents the hardness of the foundation, and g the friction threshold. The admissible displacement space is given by:

V := {w ∈ H 1 ; with w = 0 on Γ 1 }
and the sub-differential contact function

ϕ(x, y) = 1 r + 1 k 0 (x) |y ν(x) | r+1 + g(x)|y τ (x)| ∀x ∈ Γ 3 , y ∈ R d .
Then denoting by p := r + 1, we have the contact function well defined on V by

ψ(v) := Γ3 k 0 p |v ν | p da, + Γ3 g |v τ | da, ∀v ∈ V.
Here 1 < p < 2. Then the mapping x ≥ 0 -→ x p is convex, which implies that ψ is convex on V . Using the continuous embeddings from V into L 2 (Γ 3 ) d , and from L 2 (Γ 3 ) into L p (Γ 3 ), we verify also that ψ is Lipschitz continuous on V.

To approximate the function ψ we use the sequence

ψ n (v) := Γ3 k 0 p |v ν | 2 + 1 n p 2 da + Γ3 g |v τ | 2 + 1 n da, ∀v ∈ V, ∀n ∈ N * .
We verify that the Frechet derivative of ψ n is given by

ψ n (v).h := Γ3 k 0 v ν h ν (|v ν | 2 + 1 n ) 1-p 2 da + Γ3 g (v τ , h τ ) R d |v τ | 2 + 1 n da, ∀h ∈ V.
From the fact that the mapping x ≥ 0 -→ x p is convex increasing, and using (2.22), we verify that ψ n is convex for all n ≥ 1. Similarly we have the conditions (ii)-(iv) in (2.19) by the same arguments in the previous example.

Then we conclude that the assumptions in (2.19) are also satisfied in this example.

We begin by

Lemma 1. For all η ∈ W, there exists an unique

v η ∈ W 1,2 (0, T ; V ) ∩ W 1,∞ (0, T ; H) satisfying          vη (t) + A v η (t), w -v η (t) V ×V + (η(t), ε(w) -ε(v η (t))) H + ψ(w) -ψ(v η (t)) ≥ f (t), w -v η (t) V ×V , ∀ w ∈ V, a.e. t ∈ (0, T ); v η (0) = v 0 . (3.1)
Moreover, ∃c > 0 such that ∀η 1 , η 2 ∈ W:

v η2 (t) -v η1 (t) 2 H + t 0 v η1 -v η2 2 V ≤ c t 0 η 1 -η 2 2 H , ∀t ∈ [0, T ]. (3.2)
Proof. Let η ∈ W. The existence and uniqueness of v η follows straightly from Theorem 2, where we apply F defined by for all t ∈ [0, T ],

F (t), w V ×V := f (t), w V ×V -(η(t), ε(w)) H , ∀w ∈ V.
The assumptions in (2.14)

imply that F ∈ W 1,2 (0, T ; V ). Now let η 1 , η 2 ∈ L 2 (0, T ; V ). In (3.1) we take (η = η 1 , w = v η2 (t)), then (η = η 2 , w = v η1 (t)).
Adding the two inequalities, we deduce that for a.e. t ∈ (0; T ):

vη2 (t) -vη1 (t), v η2 (t) -v η1 (t) V ×V + A v η2 (t) -A v η1 (t), v η2 (t) -v η1 (t) V ×V ≤ -(η 2 (t) -η 1 (t), ε(v η2 (t)) -ε(v η1 (t))) H .
Then integrating over (0, t), from (2.11)(iii) and from the initial condition on the velocity, we obtain:

∀t ∈ [0, T ], v η2 (t) -v η1 (t) 2 H + m A t 0 v η2 (s) -v η1 (s) 2 V ds ≤ - t 0 (η 2 (s) -η 1 (s), ε(v η2 (s)) -ε(v η1 (s))) H ds.
Then (3.2) follows.

Here and below, we denote by c > 0 a generic constant, whose value may change from lines to lines.

Lemma 2. For all η ∈ W, there exists an unique

θ η ∈ W 1,2 (0, T ; E) ∩ W 1,∞ (0, T ; F ) satisfying θη (t) + K θ η (t) = R v η (t) + Q(t), in E , a.e. t ∈ (0, T ), θ η (0) = θ 0 . (3.3)
Moreover, ∃c > 0 such that ∀η 1 , η 2 ∈ W:

θ η1 (t) -θ η2 (t) 2 F ≤ c t 0 v η1 -v η2 2 V , ∀t ∈ [0, T ]. (3.4) and θη1 (t) -θη2 (t) 2 F ≤ c t 0 v η1 -v η2 2
V , a.e. t ∈ (0, T ).

(3.5)

Proof. The existence and uniqueness result verifying (3.3) follows from classical result on first order evolution equation, which can be seen as a particular case of Theorem 2 applied to the Gelfand evolution triple (see e.g. [START_REF] Zeidler | Nonlinear Functional Analysis and its Applications, II/A, Linear monotone operators[END_REF] p . 416).

E ⊂ F ≡ F ⊂ E .
We verify that the operator K : E -→ E is linear continuous and strongly monotone, and from the expression of the operator R,

v η ∈ W 1,2 (0, T ; V ) =⇒ R v η ∈ W 1,2 (0, T ; F ), as Q ∈ W 1,2 (0, T ; E ) then R v η + Q ∈ W 1,2 (0, T ; E ).
Now for η 1 , η 2 ∈ W, we have for a.e. t ∈ (0; T ):

θη1 (t) -θη2 (t), θ η1 (t) -θ η2 (t) E ×E + K θ η1 (t) -K θ η2 (t), θ η1 (t) -θ η2 (t) E ×E = R v η1 (t) -R v η2 (t), θ η1 (t) -θ η2 (t) E ×E .
Then integrating the last property over (0, t), using the strong monotonicity of K and the Lipschitz continuity of R : V -→ E , we deduce (3.4).

To continue, from θη ∈ L 2 (0, T ; E)

and from (3.3) which implies

θη (t) + K θη (t) = R u η (t) + Q(t) a.e. t ∈ (0, T ) =⇒ θη ∈ L 2 (0, T ; E ),
we deduce that θη ∈ C([0, T ]; F ).

Then that for η 1 , η 2 ∈ W, we have for a.e. t ∈ (0; T ): θη1 (t) -θη2 (t), θη1 (t) -θη2 (t) E ×E + K θη1 (t) -K θη2 (t), θη1 (t) -θη2 (t) E ×E = R u η1 (t) -R u η2 (t), θη1 (t) -θη2 (t) E ×E .

Integrating the last property over (0, t), and with similar arguments we deduce θη1 (t) -θη2 (t) 2 F ≤ c Then from (2.12), (2.13), and Lemma 2, we deduce that for all η 1 , η 2 ∈ W, for all t ∈ [0, T ]: We conclude from the last inequality by contracting principle that the operator Λ has a unique fixed point η * ∈ W. We verify then that the functions

Λ η 1 (t) -Λ η 2 (t)

t 0 u η1 -u η2 2 V 0 B

 020 , a.e. t ∈ (0, T ). And(3.5) follows.Proof of Theorem 1. We have now all the ingredients to prove the Theorem 1. Consider the operator Λ : W → W defined by for all η ∈ W:Λ η (t) = G(ε(u η (t))) + t (t -s) ε(u η (s)) ds -θ η (t) C e , ∀t ∈ [0, T ],whereu η (t) = u 0 + t 0 v η (s) ds, ∀t ∈ [0, T ]; u η ∈ W 2,2 (0, T ; V ).

u(t) := u 0 + t 0 v

 0 η * , ∀t ∈ [0, T ], θ := θ η * are solutions to problem QV with the regularity (2.20), the uniqueness follows from the uniqueness in Lemma 1 and Lemma 2.

  ., which gives the property (2.19)(iii). Finally (2.19)(iv) is immediate from the formula stated for the Frechet derivative of ψ n (see (2.21)).

	Example 2. Thermal contact problem with normal damped response and Tresca's
	friction law

  Now using (3.6) and (3.7), after some algebraic manipulations, we have for any β > 0:

	Again from (2.13) and (2.12), we have
		d dt	0	t	B(t -s) ε(u η1 (s)) ds -	0	t	B(t -s) ε(u η2 (s)) ds	2 H
							t
		≤ c u η1 (t) -u η2 (t) 2 V + c	u η1 (s) -u η2 (s) 2 V ds
							0
			t
		≤ c			v η1 (s) -v η2 (s) 2 V ds.
			0
	and
		d dt	G(ε(u η1 (t))) -G(ε(u η2 (t)))	2 H	≤ c v η1 (t) -v η2 (t) 2 V .
	Then
	d dt	Λ η 1 (t) -Λ η 2 (t)	2 H	≤ c v η1 (t) -v η2 (t) 2 V + c	0	t	v η1 (s) -v η2 (s) 2 V ds.	(3.7)
									(3.6)

2 H ≤ c θ η1 (t) -θ η2 (t) 2 F + c t 0 v η1 (s) -v η2 (s) 2 V ds ≤ c t 0 v η1 (s) -v η2 (s) 2 V ds. t 0 e -βτ Λ η 1 (τ ) -Λ η 2 (τ ) 2 H + Λ η 1 (τ ) -Λ η 2 (τ ) 2 H dτ ≤ c β t 0 e -βτ η 1 (τ ) -η 2 (τ ) 2 H + η1 (τ ) -η2 (τ ) 2 H dτ.

3.

Proof of Theorem 1. The idea is to bring the second order inequality to a first order inequality, using monotone operator, convexity and fixed point arguments, and will be carried out in several steps.

Let us introduce the velocity variable

The system in Problem QV is then written for a.e. t ∈ (0, T ):

Various abstract formulations concerning the existence and uniqueness result on parabolic variational inequalities of the second kind, could be found in the literature, depending on the assumptions on the operators and data (see e.g. [START_REF] Brézis | Problèmes unilatéraux[END_REF][START_REF] Duvaut | Les Inéquations en Mécanique et en Physique[END_REF][START_REF] Lions | Quelques méthodes de résolution des problèmes aux limites non linéaires[END_REF]). We will use a version taken in ( [START_REF] Duvaut | Les Inéquations en Mécanique et en Physique[END_REF]) p. 46, which is sufficient for our proposal, and which we recall as follows.

Theorem 2. Let A : V -→ V be linear continuous coercive, {ψ; (ψ n )} verify the hypotheses (2.19), F ∈ W 1,2 (0, T ; V ), and v 0 ∈ V satisfy: there exists a sequence (v n 0 ) in V , there exists a bounded sequence (h n ) in H such that v n 0 -→ v 0 in V and for all n ∈ N,

We can apply the Theorem 2 by using (2.11); the conditions on v 0 in (2.18) imply that there exists a sequence

To continue, we assume in the sequel that the conditions (2.11)-(2.19) of the Theorem 1 are satisfied. Let define W := {η ∈ W 1,2 (0, T ; H), η(0) = G(εu 0 ) -θ 0 C e }.


Andrea Ceglia
email: aceglia@b-phot.org

Peter Cosyns

Nadine Schibille

Wendy Meulebroeck

Unravelling provenance and recycling of Late Antique glass with trace elements

Keywords: Archaeological glass, Cyrpus, Late-Antiquity, LA-ICP-MS

come

Introduction

The last decades have seen an evolution of the analytical tools used to characterize and provenance glass, highlighting the every growing need of trace element analyses to identify meaningful groups [START_REF] Dussubieux | Recent Advances in Laser Ablation ICP-MS for Archaeology[END_REF]. With the contribution of archaeometry, it is now widely accepted that in the first millennium AD, the glass industry was mostly organized into large primary factories located in Egypt and the Levant and secondary workshops where raw glass chunks were remelted and shaped into objects [START_REF] Freestone | The origins of Byzantine glass from Maroni Petrera, Cyprus[END_REF]. Nevertheless, we are still far from having a clear understanding of the first millennium glass industry. There are many geographical gaps that would give us useful insights into the connections between areas and even if the number of publications on the topic keeps increasing (Bugoi et al., 2016;[START_REF] Cholakova | Compositional identification of 6th c. AD glass from the Lower Danube[END_REF][START_REF] Gliozzo | Late Antique and Early Medieval Glass Vessels from Faragola (Italy)[END_REF]Maltoni et al., 2016;Maltoni and Silvestri, 2016;[START_REF] Phelps | Natron glass production and supply in the late antique and early medieval Near East: The effect of the Byzantine-Islamic transition[END_REF]Schibille et al., 2016b,a;[START_REF] Silvestri | Chemical and Oxygen Isotopic Composition of Roman and Late Antique Glass from Northern Greece[END_REF] much work still needs to be carried out. We therefore started working on Cypriot glass material from the early Christian period, because Cyprus occupies a central hub of the commercial routes between the Near East and the rest of the Roman Empire. The study of the archaeological glass on the island can thus offer a way to reach a wider understanding of the glass industry as a whole and add another piece of information to the puzzle of ancient glass distribution [START_REF] Freestone | The origins of Byzantine glass from Maroni Petrera, Cyprus[END_REF][START_REF] Ceglia | Shedding light on the glass industry of ancient Cyprus: archaeological questions, methodology and preliminary results[END_REF]Ceglia et al., 2015a[START_REF] Ceglia | Light through glass: The spectrum of Late Antique glass from Cyprus[END_REF][START_REF] Bonnerot | Technology and materials of Early Christian Cypriot wall mosaics[END_REF].

Previous research concentrated on the glass finds from three ecclesiastical sites: Agioi Pente at Yeroskipou, Maroni-Petrera and Kalavasos-Kopetra all located on the south-west to central-south coastline of Cyprus (Ceglia et al., 2015a[START_REF] Ceglia | Light through glass: The spectrum of Late Antique glass from Cyprus[END_REF]. (Ceglia et al., 2015a[START_REF] Ceglia | Light through glass: The spectrum of Late Antique glass from Cyprus[END_REF] have carried out chemical and spectroscopic analysis by means of EPMA and optical absorption spectroscopy in order to characterise the material and to study the redox state of iron in naturally coloured glasses.

These studies provided new evidences on the glass compositions found in three Late Antique sites in Cyprus. By comparing them to the known primary glass groups produced in Syria, Palestine and Egypt we were able to identify glasses of the following chemical compositions: Levantine 1, Egypt 1, HIMTa and HIMTb, HLIMT1 , HIT, two Roman fragments and a plant ash object (Ceglia et al., 2015a, 2016, andreferences therein).

In this work we re-analysed the material presented in Ceglia et al. (2015a) using LA-ICP-MS in order to implement the complementary trace element data of these objects to further elucidate earlier research results. We discuss trace element patterns to refine the chemical characterisation of the primary production groups represented among the three 5th to 7th century AD Cypriot assemblages. Furthermore, the trace elements enable us to ascertain the evidence of recycling within the dataset. We propose here new trace element thresholds to detect recycling tailored on the different glass groups.

Experimental

The objects studied total 179 glass fragments, mostly naturally coloured, from the three Cypriot sites of Yeroskipou-Agioi Pente, Maroni-Petrera and Kalavasos-Kopetra. In Appendix A a more detailed description can be found. In this paper 58 elements were determined by Laser Ablation Inductively Coupled Plasma Mass Spectrometry (ICP-MS) at the Centre Ernest-Babelon of the IRAMAT (Orléans) [START_REF] Gratuze | Glass characterization using laser ablation-inductively coupled plasma-mass spectrometry methods[END_REF]Schibille et al., 2016a). The operating conditions of the 193 nm laser were set at an energy of 4 to 6 mJ, with a repetition rate of 10 Hz and a spot size diameter of 100 µm allowing a micro sampling invisible to the naked eye. Take-up time was 50 seconds and the measurements were carried out on a list of pre-selected isotopes. For silicon, the 28 Si isotope was employed as internal standard.

The analyses were carried out on a single spot on the polished sections previously used for EPMA avoiding any surface contamination or corrosion interference. The so-obtained signal intensities were converted by means of an average response factor KY, determined using a combination of five different standard reference materials (SRM). Detection limits vary according to the ablation parameters (spot size diameter and laser repetition rate) and to the optimisation parameter of the mass spectrometer. Typical detection limits for soda-lime glasses are given in (Gratuze, 2014, Table 13.4). Corning A and NIST SRM612 were regularly analysed as unknown samples to determine the accuracy and precision of the data, which were determined by comparing the obtained results against reported values for Corning A [START_REF] Brill | Chemical composition characterization of ancient glass finds from TroesmisTurcoaia, Romania[END_REF][START_REF] Wagner | Critical assessment of the elemental composition of Corning archeological reference glasses by LA-ICP-MS[END_REF][START_REF] Vicenzi | Archeological Reference Glasses: New Additions to the Smithsonian[END_REF] and NIST612 [START_REF] Jochum | Determination of reference values for NIST SRM 610-617 glasses following ISO guidelines[END_REF]. The data are presented in the supplementary material (Appendix B). Accuracy is within 4% relative for all major elements in NIST612 and Corning A, with the exception of alumina (8%) and lime (15%) in the latter reference glass. The accuracy for most trace elements in NIST 612 and Corning A are within 10% with the majority within 5%.

Results

Although EPMA is generally considered more accurate for major elements over LA-ICP-MS, the quantification presented in this work, obtained with the latter technique, is in very good agreement with the previous EPMA data (Ceglia et al., 2015a). In Figure 1 we show the relation for major and minor elements analysed by the two methods. Pearson's linear correlation coefficient varies from 0.97 to 1 indicating a strong positive correlation. The angular coefficient are very close to one -ranging between 0.96 for Fe 2 O 3 and 1.07 for TiO 2 -suggesting a very good agreement between the two techniques. Two samples SF34, KK293 have contrasting values of MnO and CaO between the two techniques. We believe that this is due to inhomogeneity of these samples rather than a problem with the measurements.

As already seen in (Ceglia et al., 2015a) all glasses, with the exception of a plant ash fragment, are low magnesia soda-lime-silica glass typical of the Roman and Early Byzantine period. In (Ceglia et al., 2015a) 6 groups were defined: Levantine 1, Foy 2, HIMTa, HIMTb, Egypt 1 and HIT. The major, minor and trace elements composition obtained by LA-ICP-MS analyses for all samples is reported in the supplementary material (Appendix C). Please note that in this work HLIMT is referred to as Foy 2. In this paper a further refinement is presented as two more groups have been recognized: High Mn Levantine 1 and High Fe Foy 2. Furthermore, thanks to LA-ICP-MS, some glasses with doubtful assignments have been confirmed or reassigned to another group (for more detail please see section 4.2). The reassignment of samples to specific glass types was achieved by a reiterative analysis of the full data -major, minor and trace elements -together with a multidimensional principal component analysis (PCA) carried out using Matlab. To perform the statistical analysis we selected 14 elements that are believed to be representative of the two main components of glass sand and natron (Si, Na, Mg, K, Ca, Al, Fe, Ti, Li, B, Zr, Sr, La and Ce) deliberately not including colouring / decolouring elements. This allows us to discriminate between glass types using different sand sources and/or recipes without interferences due to recycling or additives. The principal component scores are calculated on the mean-scaled LA-ICP-MS data. Principal components 1 and 2 account for more than 70% of the variability and separate the data into distinct primary production groups In Figure 3 we show the trace elements patterns of the main compositional groups normalized to the upper continental crust composition. This type of representation has been largely used since its first use in [START_REF] Freestone | Primary Glass from Israel and the Production of Glass in Late Antiquity and the Early Islamic Period[END_REF] as it allows a quick overview of several elements. The distributions are all very similar with exception of Sr, Zr, Ce and La. Barium is also showing some differences but it should be noted that this element is highly variable in group HIMTa. In general, it is possible to see an increase of the trace levels from Levantine to HIMTb glass. Fig. 3: Concentration of trace elements normalized to the upper continental crust [START_REF] Kamber | A new estimate for the composition of weathered young upper continental crust from alluvial sediments, Queensland, Australia[END_REF] for a selection of elements.

Table 1 summarizes the compositions of what we considered "pristine" or not recycled glasses presented in this work. It reports also the average composition of the raw chunks analysed in [START_REF] Foy | Caracterisation des verres de la fin de l'antiquite en mediterranee occidentale: l'emergence de nouveaux courants commerciaux[END_REF] as a source of comparison.

How we distinguished between recycled and unrecycled glass is detailed in section 4.3. In the discussion on recycling we have not considered high Fe Foy 2 and high Mn Levantine 1 because the former consists of too few samples and the latter is not an homogeneous group.

Table 1: Average concentrations (µ) and standard deviation (σ) of the pristine glass for each group. The last three groups are taken from [START_REF] Foy | Caracterisation des verres de la fin de l'antiquite en mediterranee occidentale: l'emergence de nouveaux courants commerciaux[END_REF]. In parenthesis there is the number of samples selected. The criteria to choose non-recycled glass are explained in the text. In Appendix D the average data are reported for all the 58 elements analysed. In addition, in Appendix E all samples belonging to each group are reported along with their chemical composition.

Discussion

In order to ensure a smooth flow this section has been divided into three parts, each tackling one of the three main points of this paper. First, we will use the trace elements to discuss the differences among the glass types identified and to compare them with data available in the literature. Then, we will examine in detail which glasses were reassigned with respect to (Ceglia et al., 2015b). Finally we present a thorough analysis of recycling tailored to the different compositional groups.

Provenancing with trace elements

LA-ICP-MS gave us the possibility to further characterise the glasses previously analysed by EPMA (Ceglia et al., 2015a). The trace elements can be related to accessory minerals present in the sand such as feldspar, pyroxene, amphibole, zircon, monazite and more. Many of the trace and Rare Earth Elements (REE)

are not influenced by the use of decolourants or by recycling. They can thus serve as reliable indicators to distinguish sand sources. Other researchers have already discussed interesting relations based on elements such as Zr and Ti [START_REF] Aerts | Change in silica sources in Roman and post-Roman glass[END_REF], Zr, Ti, Cr and La [START_REF] Shortland | Trace element discriminants between Egyptian and Mesopotamian Late Bronze Age glasses[END_REF][START_REF] Walton | Evidence for the trade of Mesopotamian and Egyptian glass to Mycenaean Greece[END_REF] and Zr, Sr and Ba [START_REF] Freestone | Primary Glass from Israel and the Production of Glass in Late Antiquity and the Early Islamic Period[END_REF][START_REF] Paynter | Analyses of colourless Roman glass from Binchester, County Durham[END_REF][START_REF] Silvestri | The coloured glass of Iulia Felix[END_REF][START_REF] Silvestri | The colourless glass of Iulia Felix[END_REF]. According to [START_REF] Brems | Trace Element Analysis in Provenancing Roman Glass-Making[END_REF] the most diagnostic trace elements are Ti, Cr, Sr, Zr, and Ba.

The concentration of most trace elements increases from Levantine 1 to HIMTb (Figure 3). This is a well-known phenomenon, associated with the maturer sand source used for Levantine 1 compared to Egyptian productions such as HIMT glass.

It is possible to identify certain elements and ratios that can be very effective to differentiate the different glass types. We focussed on three components: the REE-bearing mineral fraction, the amount of zircon and the amount of Sr in the carbonaceous fraction. The most abundant REEs are Ce, La and Nd. In our dataset

Nd is strongly correlated with La. Therefore to characterize the sand the ratio Ce/La seems to be very promising. The content of Zr appears to be a reliable marker for differences in the silica source, usually introduced in the form of zircon, a zirconium silicate (ZrSiO 4 ). To compare the zircon content in different glass types, we normalized the Zr content to the silica concentration. The third parameter is the ratio of Sr to CaO which is indicative of the carbonaceous fraction of the sand.

However, it must be noted that strontium can also be incorporated into the glass batch as constituent of clay minerals, feldspars or the manganese bearing minerals [START_REF] Ganio | Trade routes across the Mediterranean: a Sr/Nd isotopic investigation on Roman colourless glass[END_REF][START_REF] Cholakova | Compositional identification of 6th c. AD glass from the Lower Danube[END_REF]. In Figures 4a and4b we report the relations between these three markers that can separate different types of glass very well.

The sand of the Syro-Palestinian coast has a ratio of Ce to La of 1.8-1.9, only 40-55 ppm of Sr per wt% of CaO and the lowest zircon-to-silica ratios. Group Foy 2 has lower Ce/La (roughly 1.6-1.7), a higher zircon fraction and the highest Sr/Ca ratio. However, this does not apply to the high-Fe Foy 2 group that should be considered a contemporary yet different production to Foy 2 as already pointed out by Schibille et al. (2016a). Egypt 1, HIMTa and HIMTb have a high Zr/Si ratio with HIMTa having a relatively large spread. Conversely the ratio between strontium and calcium is constant across these groups (about 70 ppm per wt% of CaO). The pattern of REEs minerals, however, differ notably between the three groups with HIMTb having the highest contribution of La and Egypt 1 the lowest.

One of the two HIT fragments (ID821) plots together with the blurred group of HIMTa, while the other HIT sample (NSF1065) has a much lower Sr/Ca ratio compared to any of the other glass types, which prompts questions about its origin which are at the moment unresolved. In Figure 4c Occasionally the Sr to CaO ratio increases in groups with high Mn due to the addition of strontiomelane as decolouring agent [START_REF] Ganio | Trade routes across the Mediterranean: a Sr/Nd isotopic investigation on Roman colourless glass[END_REF][START_REF] Cholakova | Compositional identification of 6th c. AD glass from the Lower Danube[END_REF]. Indeed, the biplot of Sr/CaO and MnO shows that there is a gradual increase from Levantine 1 over High Mn Levantine 1 to HIMT glass (Figure 5). Given that Levantine 1 represents the natural levels of Sr/CaO due to inclusions of seashells, the higher content of Sr/CaO in HIMT glass could be explained by the addition of the decolouring minerals. Therefore the Sr/CaO ratio in HIMT prior to the addition of MnO would be similar to Levantine 1 and due to the inclusion of seashells, which would support Nenna's proposition of the Mediterranean coastline of the Sinai as region of provenance of HIMT glass [START_REF] Nenna | Egyptian glass abroad: HIMT glass and its markets[END_REF]. Nevertheless, it cannot be stated conclusively whether the source of Mn in HIMT is strontiomelane or not for several reasons. First of all one of the HIT glasses (ID821) has similar Sr/CaO and Zr/SiO 2 to HIMTa although it has no manganese. Similarly, Egypt 1 glass has higher Sr/CaO than Levantine 1 glass even if it contains no manganese.The higher amount of Sr/CaO in the Egyptian glass could be explained by diagenetic modification of calcite into aragonite and/or by the presence of other Sr-bearing minerals. The use of strontiomelane was suggested to explain the high content of Sr in Foy 2 glass based on the correlation between this element and MnO [START_REF] Cholakova | Compositional identification of 6th c. AD glass from the Lower Danube[END_REF]. In our case, there is no evidence of a correlation with Mn. The high Sr/CaO can be explained either by the use of sand/raw materials richer in Sr and/or by the use of a different source of strontiomelane since this mineral is highly variable in Sr content [START_REF] Meisser | Strontiomelane, SrMn4+ 6Mn3+ 2O16, a new mineral species of the cryptomelane group from St. Marcel -Praborna, Aosta Valley, Italy[END_REF].

Another interesting pattern that can be noted is the relationship between titanium and niobium. This is partially related to the Zr-Ti relation often used in glass studies (see for examples [START_REF] Brems | Trace Element Analysis in Provenancing Roman Glass-Making[END_REF]), but it provides a different view because Nb seems to be added to the batch almost exclusively with Ti. The geochemical association of Nb with Ti-bearing minerals as rutile and ilmenite has long been recognised. It is known that its content varies with the type of rock in which the titanium minerals occur [START_REF] Fleischer | Geochemical association of niobium (columbium) and titanium and its geological and economic significance Foster HE, Jackson CM (2010) The composition of late Romano-British colourless vessel glass: glass production and consumption[END_REF].

Therefore, Nb could be an interesting element for provenancing ancient glass since its concentration depends on the quantity, type and origin of Ti minerals that are added to the glass batch as sand impurities. Variations in the Nb/Ti ratio imply different glass types (Figure 6a). Levantine, HIMT and Foy 2 compositions, we notice that their data confirms the existence of two linear correlations between Nb and Ti (Figure 6b). Having the same Nb/Ti ratio, of course, does not mean that the glass origin was the same, but it implies that the minerals introducing these elements to the glass batch are the same, probably derived from the same type of rock. Therefore it is interesting to note that Foy 2, high-Fe Foy 2 and Levantine 1 cluster on the same correlation lines. This suggests that these three types of glass productions exploited sand sources that have the same Nb-Ti bearing minerals, which differ from the minerals occurring in the sands used for Egypt 1 and HIMT.

Reassignment of glass

One of the aims of this paper was to refine the classification of the glasses analysed by Ceglia et al. (2015a). In that paper the material was clustered on the basis of the major and minor element composition but in some cases it was difficult to determine the group because of a mixed glass chemistry. Eight glasses were assigned to Foy 2 with a question mark -two from Yeroskipou-Agioi Pente (ID817e and ID828), four from Maroni-Petrera (NSF1007a, SF42, SF109a and SF113) and two from Kalavasos-Kopetra (KK293 and KK303) -based on the iron, titanium, manganese and calcium contents. With the complete chemical data we can now reconsider the assignments of these glasses.

The major and minor oxides composition of samples ID817e and ID828 from Yeroskipou-Agioi Pente are in good agreement with the profile of Foy 2 glass except for the high Fe 2 O 3 (2.35 and 3.47 wt% respectively). The trace elements confirmed their similarity to this group having an elevated content of Sr (613 and 560 ppm), Zr (84 and 83 ppm), Li (higher than 5 ppm) and being impure with Sb (77 and 62 ppm). In view of their iron concentration they have to be attributed to the High-Fe Foy 2 group as already reported in [START_REF] Cholakova | Compositional identification of 6th c. AD glass from the Lower Danube[END_REF] and (Schibille et al., 2016b). These glasses have a more intense colour than Foy 2 glasses due to the higher concentration of Fe. In [START_REF] Ceglia | Light through glass: The spectrum of Late Antique glass from Cyprus[END_REF] we have reported the optical and colour analysis of these glasses. Sample ID817e has a colour similar to HIMTa 6: Niobium and titanium are strongly correlated in Ti-bearing minerals as rutile and ilmenite. Our dataset shows the existence of two different Nb/Ti ratios which is confirmed in the glass analysed by [START_REF] Gliozzo | Colourless glass from the Palatine and Esquiline hills in Rome (Italy). New data on antimony-and manganese-decoloured glass in the Roman period[END_REF], (Schibille et al., 2016a) and [START_REF] Conte | Butrint (Albania) between Eastern and Western Mediterranean glass production: EMPA and LA-ICP-MS of Late Antique and Early Medieval finds[END_REF]. The dashed lines are calculated with the least square method on the data presented in this paper.

glasses while ID828 had all optical parameters in the range of the HIMTb glasses.

Similarly (Schibille et al., 2016a) reports that the high Fe Foy 2 glass has olive to yellow-green colours, typical of HIMT glass.

The association of the three Maroni-Petrera samples NSF1007a, SF42 and SF109a to Foy 2 had been doubtful due to their lower K 2 O and CaO contents, while SF113 shows higher TiO 2 and Al 2 O 3 . The trace element patterns substantiate the attribution of samples NSF1007a, SF42, SF109a to Foy 2, even if the content of Sr (about 430 ppm) and Li (less than 5 ppm) are lower than for most Foy 2 glasses. However, the ratios of Sr/Ca and Ce/La fit well with the rest of the group. Sample SF113 is assigned to HIMTa because of its Sr/Ca and Nb/Ti ratios, despite its low Zr/Si ratio.

One of the two glasses from Kalavasos-Kopetra originally categorised as Foy 2 (sample KK293) shows a discrepancy in the CaO content between EPMA (5 wt%) and LA-ICP-MS (7.45 wt%). Nevertheless both glasses have Sr/Ca, Zr/Si and Ce/La ratios in line with the values of the Foy 2 group and can therefore be assigned to this group.

One of the glasses originally attributed to Levantine 1 (sample SF34) has higher amounts of Ti (0.12 wt%) with respect to the typical values for this glass type.

Trace elements confirm that this glass falls into the Foy 2 group instead as the ratios of Ce/La and Zr/Si are similar to this group. On the other hand the amount of strontium normalized to calcium is low, leaving a certain degree of uncertainty to this assignment. Six Levantine 1 samples (ID108, NSF1020a, NSF1020b, SF1, SF17, SF103) have MnO higher than 0.5 wt% and they are therefore here sub-grouped in the high Mn Levantine 1 glass type.

Recycling

Glass recycling appears to be a common activity during the Roman and Late Antique period. Early Christian Cyprus is no exception, even if the island is located very close to the primary production regions. Nevertheless, no clear recycling indicators have to date been identified (Freestone, 2015, and references therein). Usually, concentrations of elements related to (de)colouring activities above background levels are considered markers of recycling. These background levels have yet to be defined. It is generally accepted that the deliberate use of (de)colourants would induce an increase of these elements, above 1000 ppm. Unintentional additions of coloured fragments in a colourless (or naturally coloured) batch provokes a rise of the concentrations of elements associated with colours with respect to natural impurities in the raw materials. Therefore, when the concentrations of certain elements (normally Co, Zn, Sn, Cu and Pb) ranges between 100 ppm and 1000 ppm, it is generally interpreted as an indication of glass recycling [START_REF] Freestone | The origins of Byzantine glass from Maroni Petrera, Cyprus[END_REF][START_REF] Wedepohl | The use of marine molluskan shells for Roman glass and local raw glass production in the Eifel area (western Germany)[END_REF][START_REF] Degryse | Isotopic Discriminants Between Late Bronze Age Glasses From Egypt and the Near East[END_REF]Foster and Jackson, 2010). If glass objects present lower concentrations of these elements, i.e. between 1 ppm and 100 ppm, either glassmakers used "fresh" glass from primary workshops, or limited recycling occurred, or much attention was paid to the cullet selection in order to avoid contamination [START_REF] Silvestri | The coloured glass of Iulia Felix[END_REF][START_REF] Silvestri | The colourless glass of Iulia Felix[END_REF]. These guidelines are very general and do not address the intrinsic variability of glass groups, or, more appropriately, of sand sources. Hence, we have examined in detail the trace elements in the glasses of the Cypriot assemblage to propose a definition of recycling based on our large dataset. We defined thresholds for transition metals according to the glass type. When all the trace elements are below these thresholds, the sample can be considered to represent a pristine raw glass.

Levels of trace elements in excess of these thresholds are indicative of contamination through recycled cullet. Besides the elements normally considered (Co, Zn, Sn, Cu and Pb), Sb is another good marker for detecting recycling as its relative abundance in the upper continental crust is 0.2-0.45 ppm [START_REF] Jackson | From Roman to early medieval glasses. Many happy returns or a new birth?[END_REF][START_REF] Rudnick | Composition of the Continental Crust[END_REF]. Additionally, we take into account also K 2 O and P 2 O 5 . The relation between melting cycles and ashes was demonstrated by the experiment undertaken by [START_REF] Paynter | Experiments in the reconstruction of Roman woodfired glassworking furnaces: waste products and their formation processes[END_REF]. The furnace atmosphere is rich in potash vapour and particulate phosphate, which can partly be absorbed by the glass melt. As a result the content of both oxides increases accordingly to the exposure time of the melt to the fumes. The higher the number of re-melting events, the higher the concentration of the ash impurities in the glass.

For Levantine 1 initially we have set the thresholds between recycled and unrecycled glass based on the data from Apollonia published in [START_REF] Freestone | The origins of Byzantine glass from Maroni Petrera, Cyprus[END_REF]. The authors report the average trace elements contents of tank furnace glass and vessels from Apollonia. Using this as reference material, we defined as recycled the glass with more than 5 ppm of Co, 15 ppm of Cu and Zn, 35 ppm of Pb and 3 ppm of Ag. These would leave us with 67 glass samples. The average concentrations of recycling markers is very low, nevertheless several of these glasses have still 0.1 or more P 2 O 5 and few hundreds of ppm of Mn. Therefore some recycling is still present within this subgroup of fragments. Recently [START_REF] Phelps | Natron glass production and supply in the late antique and early medieval Near East: The effect of the Byzantine-Islamic transition[END_REF] have published LA-ICP-MS data of 5 samples tank furnace from Apollonia with extremely low values of these additives (1.5 ppm of Co and 1.3 of Cu, 6.4 of Zn, 3 of Pb and 0.03 of Ag). In view of this information we reduced the thresholds to 3 ppm of Co and Cu, 10 ppm of Zn and Pb and 1 ppm of Ag obtaining a group of 17 glasses. Even though these values could seem too stringent, the average composition of this small group resembles very well the primary furnace from Apollonia. Cypriot glass has 139 ppm of Mn 1.3 ppm of Co, 2.3 ppm of Cu, 6.1 ppm of Zn, 3.6 ppm of Pb, 0.04 ppm of Ag, 0.55 ppm of Sn, 0.02 ppm of Sb and 0.04 P 2 O 5 (see Table 1). These are probably the only glasses that did not undergo more than one or very few remelting cycles. The other Levantine objects appear to have been remelted several times with the addition of coloured cullet to the batch more or less evident in the trace element make-up.

It is much more complicated to evaluate the degree of remelting/recycling in the other groups, as no raw glass from primary factories has been recovered so far. In addition, Foy 2, HIMTa, HIMTb and Egypt 1 are made with raw materials containing more impurities increasing the natural levels of transition metal ions.

However, we can tentatively use the data of some raw glass chunks found in secondary glass workshops analysed by [START_REF] Foy | Caracterisation des verres de la fin de l'antiquite en mediterranee occidentale: l'emergence de nouveaux courants commerciaux[END_REF].

Within their data set, [START_REF] Foy | Caracterisation des verres de la fin de l'antiquite en mediterranee occidentale: l'emergence de nouveaux courants commerciaux[END_REF] have published the chemical composition of 23 glass chunks belonging to Série 2.1 and report the trace elements of 15 of those samples. The majority of these chunks have Pb and Cu lower than 100 ppm.

By removing the samples with more than 100 ppm of Cu and Pb from our Foy 2 group, we are left with a quite homogeneous group that could define threshold values for recycling: 7 ppm of Co, 43 ppm of Cu, 23 ppm of Zn, 59 ppm of Pb, 0.11 ppm of Ag, 6.73 ppm of Sn, 151 ppm of Sb (Table 1). The average composition of the Cypriot Foy 2 is very similar to that one published by [START_REF] Foy | Caracterisation des verres de la fin de l'antiquite en mediterranee occidentale: l'emergence de nouveaux courants commerciaux[END_REF] as well as to recently published Byzantine glass weights (Schibille et al., 2016a).

Nevertheless, in both cases there are still signs of recycling. The average content of Sb is higher than 100 ppm, well above the natural levels, and is roughly linearly correlated to lead, supporting the hypothesis that Pb is added to the batch together with Sb. Moreover the average composition has more than 0.1 wt% of P 2 O 5 which is proportional to K 2 O indicating that their content increased during the melting process (Figure 7). Since this glass type has been found at several places over the Mediterranean world as well as in continental Europe, the large amount of Sb in these objects suggests that Foy 2 was produced in Egyptian primary workshops where abundant amount of Sb-decoloured cullet was available to ease the melting process and reduce costs.

HIMTa and HIMTb correspond to Série 1 in [START_REF] Foy | Caracterisation des verres de la fin de l'antiquite en mediterranee occidentale: l'emergence de nouveaux courants commerciaux[END_REF], where they reported 12 glass chunks, three with more than 3 wt% of Fe 2 O 3 .The content of Sb in these glasses is very low, therefore, we initially considered all glass with Sb higher than 2 ppm as being recycled glass. Nine HIMTa glasses from Cyprus could be considered unrecycled. On average they have 13 ppm of Co, 58 ppm of Cu, 30 ppm of Zn, 13 ppm of Pb, 0.17 ppm of Ag and 1.74 ppm of Sn (Table 1). These objects have very low P 2 O 5 and K 2 O (0.046 and 0.39 wt% respectively), confirming that the glass did not undergo multiple or long remelting procedures.

Compared to the raw chunks (Table 1) the average of several markers is lower, but the difference is biased by the presence of one glass chunk with high recycling indicators (VRR173).

All the 5 samples of HIMTb have Sb lower than 1 ppm. However they clearly belong to three different melting events: 1) ID94 and ID589, 2)ID476 and ID572a and 3) ID496. Batch 1 has more than 100 ppm of Zn while the other two have about 60 ppm. Copper ranges between 64-89 ppm, lead between 7-35 ppm, cobalt between 11-13 ppm. Moreover all glasses have more than 0.1 P 2 O 5 . It is complicated to establish whether these glasses are recycled and what the levels of recycling markers in raw glass are. However, [START_REF] Foy | Caracterisation des verres de la fin de l'antiquite en mediterranee occidentale: l'emergence de nouveaux courants commerciaux[END_REF] report three raw chunks of this glass type (VRR49, VRR50 and VRR52 from Série 1), which have even higher levels of copper, lead and P 2 O 5 , paving the way to the hypothesis that the enrichment in these elements is happening at the primary level. The variability might be due either to the intrinsic inhomogeneity of the raw materials or, similarly to Foy 2, to the addition of extra ingredients to the batch other than sand and natron, i.e. recycled cullet. The addition of recycled material is a substantial hypothesis since high levels of Sb, Pb and Cu were found in the raw chunks of HIMTb composition [START_REF] Foy | Caracterisation des verres de la fin de l'antiquite en mediterranee occidentale: l'emergence de nouveaux courants commerciaux[END_REF].

Among the five Egypt 1 fragments one, sample ID792, is surely made of recycled glass having 2224 ppm of Pb. Sample ID464 has 0.34 wt% of P 2 O 5 which suggests that it could have been recycled. The other 3 samples have on average 5 ppm of Co, 3 ppm of Cu, 22 ppm of Zn, 3 ppm of Pb, 0.14 ppm of Ag, 0.46 ppm of Sn, no Sb and low P 2 O 5 (Table 1). These three glasses may represent the natural levels of these elements in the Egypt 1 glass type.

The values reported in Table 1 could be taken as new recycling thresholds tailored on each compositional group. Nevertheless, one should remain cautious as the only glass that can be surely recognized as clean raw glass is Levantine 1 since [START_REF] Phelps | Natron glass production and supply in the late antique and early medieval Near East: The effect of the Byzantine-Islamic transition[END_REF] provided up to date data of samples from primary furnaces.

As discussed above Foy 2 glass is almost surely made of recycled cullet already at the primary production center which makes it difficult to identify recycling at the secondary level. The thresholds for the other types could only be inferred from the presence of recycling markers or at best comparing raw chunks found in secondary workshops. The content of elements such as Co, Cr, Ni, Zn increases from HIMTa to HIMTb. The high amount of P 2 O 5 possibly suggests long and/or repeated melting cycles. However, the concentration of Pb is much lower than the 100 ppm usually taken as recycling threshold.

In view of our analysis it appears that even though Cyprus was close to primary centers a large amount of glass was recycled. A source of recycling material would be mosaic tesserae of which the island is very rich. Contemporary Cypriot tesserae were opacified using tin based opacifiers: lead stannate for yellow and green (with the addition of copper) and SnO for white and blue (with the addition of cobalt) tesserae [START_REF] Bonnerot | Technology and materials of Early Christian Cypriot wall mosaics[END_REF]. Therefore recycling can be demonstrated by plotting Pb against Sn and Cu (Figure 8). We should note that an increase of Pb, Cu and Sn has been related also to the use of arsenical bronze (see [START_REF] Francesco | Preliminary chemical characterization of Roman glass from Pompeii[END_REF] and [START_REF] Cagno | Evidence of early medieval soda ash glass in the archaeological site of San Genesio (Tuscany)[END_REF]), but in view of the abundance of glass tesserae on the island the latter material was much more likely used. The positive correlation between Pb and Sn is rather evident, while between Pb and Cu it is clear only for Levantine 1. Lead and copper are less correlated in the other groups, probably because of the natural variations of these elements in the heavy minerals present in the raw materials. Most of the Cypriot Levantine 1 glass has been diluted with recycled mosaic tesserae. This practice is less easy to unravel for Foy 2 glass as the raw product has already high content of recycling indicators, however the common increase of Sn and Pb hints at a certain amount of tesserae recycling. Five out of 14 HIMTa samples and two out of five Egypt 1 glasses are made with the addition of recycled material, while all HIMTb from Cyprus contain natural (or lower) levels of colourant if compared to the raw glass chunks found in France. All results point to local Cypriot vessel glass production and thus the existence of secondary glass workshops in Cyprus. Nevertheless, analytical results need corroboration with the results of an ongoing typological study.

Conclusions

This work provides complementary trace elements data to the ongoing research on the 5th-7th century AD glass assemblages from Cypriot early Christian ecclesiastic buildings. The analysis of the trace elements allowed us to underline relationships between three different components of the sand used in primary production such as REEs, zircon and the carbonaceous fraction. Using the Nb-Ti biplot and the Ce/La, Sr/Ca and Zr/Si ratios we were able to further discriminate the major groups.

In addition, we wanted to evaluate and eventually refine the group assignments made by EPMA. Very interestingly most was confirmed by LA-ICP-MS analysis, but the latter technique helped assigning some glasses for which the major and minor elements gave doubtful information. Trace elements provide so much more information that should be preferred wherever possible as it is equally good for major, minor and trace elements analysis.

Finally the trace element analysis allowed us to investigate glass recycling. By excluding the samples which showed higher contents of recycling markers (Pb, Zn, Cu, Sb and Co) and, when available, by comparing our data with published raw glass, we were able to determine the composition of pristine glass of the main groups reported in Table 1. We have shown that previous thresholds are very high, e.g. 30 ppm of Sb [START_REF] Degryse | Glass Making in the Greco-Roman World[END_REF] or 10-20 ppm of Sb [START_REF] Rehren | Changes in glass consumption in Pergamon (Turkey) from Hellenistic to late Byzantine and Islamic times[END_REF].

Pristine glass, with exception of Foy 2, has no Sb. Sb levels in excess of 1-2 ppm are indicative of glass made with the addition of Sb-bearing cullet. With these new thresholds we have shown that it is almost impossible to find unrecycled glass during the Late Antique period even in a region in close proximity to primary production centers. The practice of recycling would likely become more important the farther we move from these primary production locations. Furthermore, the common increase of Pb and Sn in the Cypriot glass suggests that glassmakers used locally available glass mosaic tesserae as glass cullets.

(Fig. 1 :

 1 Fig.1: Relation between the EPMA and LA-ICP-MS data. The Parson r coefficient are better than 0.97 and the angular coefficient ranges between 0.96 and 1.07.

  and 4d Foy 2, High Fe Foy 2 and Levantine 1 glass from Schibille et al. (2016a) are overlapped to our data. The same trends are clearly evident confirming the definition of the glass groups.

Fig. 4 :Fig. 5 :

 45 Fig.4: REEs minerals are characterised by the relative amounts of Ce, and La. Investigating these elements helps define the geochemistry of the sand source of the glass types. In c and d the data from(Schibille et al., 2016a) is overlapped to confirm the discrimination power of these ratios.

  Foy 2 and Levantine 1 have the same ratio between Nb and Ti although remaining very distinctive in the absolute amount of elements. Conversely, Egypt 1, HIMT and HIT show a lower Nb/Ti ratio. The calculated correlation coefficients of about 0.97 for both groups, indicating positive linear relationships. Unfortunately niobium is not often reported in the literature and, if so, it is generally rounded to the ppm. Exceptions are the data published by Conte et al. (2014) on the Late Antique and Early Medieval glass finds from Butrint (Albania), Gliozzo et al. (2015) on the colourless glass from the Palatine and Esquiline hills in Rome (Italy) and Schibille et al. (2016b) on Byzantine glass weights. By plotting the Nb and Ti values from these papers for the glasses with

  Fig.6: Niobium and titanium are strongly correlated in Ti-bearing minerals as rutile and ilmenite. Our dataset shows the existence of two different Nb/Ti ratios which is confirmed in the glass analysed by[START_REF] Gliozzo | Colourless glass from the Palatine and Esquiline hills in Rome (Italy). New data on antimony-and manganese-decoloured glass in the Roman period[END_REF],(Schibille et al., 2016a) and[START_REF] Conte | Butrint (Albania) between Eastern and Western Mediterranean glass production: EMPA and LA-ICP-MS of Late Antique and Early Medieval finds[END_REF]. The dashed lines are calculated with the least square method on the data presented in this paper.

Fig. 7 :

 7 Fig. 7: Foy 2 glass exhibits a linear correlation between P 2 O 5 and K 2 O which it underwent to several remelting cycles. The dataset from Schibille et al. (2016a) and Foy et al. (2003) have been removed of the samples with Pb and Cu >100 ppm.

Fig. 8 :

 8 Fig.8: There is a linear relation between lead and tin possibly because of the recycling of tesserae opacified with lead stannate. Lead is also weakly correlated with copper. Note the logarithmic scale.

  Principal component 1 and 2 explain most of the variance of the dataset. The groups are separated, although some overlap remains. On the top-right the vectors of the first two principal components are shown.

		6 8 4			High Mn Lev 1 Lev 1 Foy 2 High Fe Foy 2 HIMTa HIMTb Egypt 1 HIT						SiO 2 CaO K 2 O Al 2 O 3	Sr	TiO 2 Na 2 O Li La Ce	Fe 2 O 3 B MgO Zr
		2
	PC2	0
		-2
		-4
		-6
		-8
			-4		-2			0			2		4		6				8		10
												PC1
	8 4 Fig. 2: 6 0.1 2 4 6 8 1 2 Normalized concentrations													High Mn Lev 1 High Fe Foy 2 Egypt 1		Lev 1 HIMT a		Foy 2 HIMT b
		Li	Ga	Rb	Sr	Y	Zr	Nb	Ba	La	Ce	Pr	Nd Sm	Eu	Gd	Tb	Dy	Ho	Er	Tm	Yb	Lu	Hf

Hereafter we will refer to this group as Foy

in order to be consistent with the latest accepted nomenclature. For more information about HIMTa and HIMTb see(Ceglia et al., 2015a). HIT is described in[START_REF] Rehren | The Early Byzantine HIMT glass from Dichin, Northern Bulgaria[END_REF]

Acknowledgements The authors are thankful to the Department of Antiquities for providing access to the material. In particular we wish to express our gratitude to Prof. Demetrios Michealides, Prof. Marcus Rautman and Prof. Sturt Manning for having granted us permission to study the material they excavated. We would like to thank also Bernard Gratuze for helping with the LA-ICP-MS measurements. This project has received funding from the European Research Council (ERC) under the European Unions Horizon 2020 research and innovation programme (grant agreement No. 647315 to NS). The funding organisations had no influence in the study design, data collection and analysis, decision to publish, or preparation of the manuscript.