Mircea Sofonea
email: sofonea@univ-perp.fr

Flavius Pȃtrulescu
email: fpatrulescu@ictp.acad.ro

A Viscoelastic Contact Problem with Adhesion and Surface Memory Effects *

Keywords: existence, fixed point, mathematical model AMS Subject Classification: 74M15, 74G25, 74G30, 74D05, 49J40

We consider a mathematical model which describes the quasistatic contact between a viscoelastic body and an obstacle, the so-called foundation. The material's behavior is modelled with a constitutive law with long memory. The contact is with normal compliance, unilateral constraint, memory effects and adhesion. We present the classical formulation of the problem, then we derive its variational formulation and prove an existence and uniqueness result. The proof is based on arguments of variational inequalities and fixed point.

Introduction

Processes of contact with adhesion are important in many industrial settings where parts, usually nonmetallic, are glued together. For this reason, a considerable effort has been made in their modeling, analysis, numerical analysis, and numerical simulations and, as a result, the engineering and computational literature on this related topics is extensive. Moreover, the mathematical literature devoted to the analysis of adhesive contact process is rapidly growing.

General models with adhesion can be found in [START_REF] Frémond | Equilibre des structures qui adhèrent à leur support[END_REF][START_REF] Frémond | Adhérence des solides[END_REF][START_REF] Frémond | Non-Smooth Thermomechanics[END_REF][START_REF] Raous | A consistent model coupling adhesion, friction and unilateral contact[END_REF][START_REF] Shillor | Models and Analysis of Quasistatic Contact[END_REF] and the references therein. In particular, a description of the derivation of various equations and conditions related to the adhesive contact can be found in [START_REF] Frémond | Non-Smooth Thermomechanics[END_REF]. The analysis of various contact models with adhesion, including existence and uniqueness results for weak solutions, can be found in [START_REF] Bonetti | Analysis of a unilateral contact problem taking into account adhesion and friction[END_REF][START_REF] Chau | Variational and numerical analysis of a quasistatic viscoelastic contact problem with adhesion[END_REF][START_REF] Chau | Dynamic frictionless contact with adhesion[END_REF][START_REF] Cocu | Existence results for unilateral quasistatic contact problems with friction and adhesion[END_REF][START_REF] Sofonea | Analysis and Approximation of Contact Problems with Adhesion or Damage[END_REF][START_REF] Touzaline | Study of a viscoelastic frictional contact problem with adhesion[END_REF][START_REF] Touzaline | Analysis of a contact adhesive problem with normal compliance and nonlocal friction[END_REF], for instance. In carrying out this analysis, a systematic use of results on elliptic and evolutionary variational inequalities, convex analysis, nonlinear equations with monotone operators, and fixed points of operators was made. The numerical analysis of quasistatic and dynamic models of adhesive contact can be found in [START_REF] Sofonea | Analysis and Approximation of Contact Problems with Adhesion or Damage[END_REF]. There, fully discrete schemes were considered and error estimates were derived. Moreover, an application of the theory in the medical field of prosthetic limbs was described in [START_REF] Rojek | Contact problems with friction, adhesion and wear in orthopaedic biomechanics. I: General developments[END_REF][START_REF] Rojek | Contact problems with friction, adhesion and wear in orthopaedic biomechanics. II: Numerical implementation and application to implanted knee joints[END_REF]. There, the bonding arises between the artificial limb and the tissue and is of considerable importance, since debonding may lead to decrease in the person's ability to use the limb. The main ingredient in the models presented in all the above mentioned papers is the introduction of a surface internal variable, the bonding field, which describes the fractional density of active bonds on the contact surface. As a fraction its values are restricted to 0 ≤ β ≤ 1. When β = 1 at a point of the contact surface, the adhesion is complete and all the bonds are active; when β = 0 all the bonds are inactive, severed, and there is no adhesion; finally, when 0 < β < 1 the adhesion is partial and only a fraction β of the bonds is active.

In this paper, we cover the modelling and variational analysis of a new contact problem with adhesion within the infinitesimal strain theory. The evolution of the bonding field is described by a general first order ordinary differential equation, already used in the previously cited papers. Nevertheless, we introduce three novelties in the contact model, which make the difference with our previous papers. First, we describe material's behavior by a viscoelastic constitutive law with long memory. Second, we model the adhesive contact with a normal compliance condition with unilateral condition which takes into account the memory effect of the surfaces. A similar condition was introduced in [START_REF] Sofonea | Analysis of a history-dependent frictionless contact problem[END_REF][START_REF] Sofonea | A viscoplastic contact problem with normal compliance, unilateral constraint and memory term[END_REF] in the study of frictionless contact process without adhesion. Also, a contact condition with normal compliance, unilateral constraint and adhesion was used in [START_REF] Jarušek | On the solvability of dynamic elastic-visco-plastic contact problems with adhesion[END_REF]. There, in contrast with this paper, the memory effects of the foundation were neglected, the process was assumed to be dynamic and the material's behavior was described with an elastic-visco-plastic constitutive law. The third novelty arises in the fact that, unlike a large number of references, the adhesive contact problem considered in this paper are formulated on the unbounded interval of time R + = [0, ∞). This implies the use of the framework of Fréchet spaces of continuous functions, instead of that of the classical Banach spaces of continuous functions defined on a bounded interval of time, used in our previous papers.

The rest of the paper is organized as follows. In Section 2 we introduce some notations and preliminary material. In Section 3 we provide a detailed description of the model of adhesive contact. Then, in Section 4, we list the assumptions on the data, derive the variational formulation of the problem and state our main existence and uniqueness result, Theorem 2. The proof is provided in Section 5. It is based on arguments of variational inequalities and fixed point.

Notations and Preliminaries

In this short section we present the notations we shall use and some preliminary material. For further details we refer the reader to [START_REF] Han | Quasistatic Contact Problems in Viscoelasticity and Viscoplasticity[END_REF][START_REF] Shillor | Models and Analysis of Quasistatic Contact[END_REF][START_REF] Sofonea | Mathematical Models in Contact Mechanics[END_REF]. Everywhere in this paper we use the notation N for the set of positive integers and R + will represent the set of nonnegative real numbers, i.e. R + = [0, +∞). For a given r ∈ R we denote by r + its positive part, i.e. r + = max{r, 0}. Let d ∈ N. Then, we denote by S d the space of second order symmetric tensors on R d . The inner product and norm on R d and S d are defined by

u • v = u i v i , v = (v • v) 1 2 ∀u, v ∈ R d , σ • τ = σ ij τ ij , τ = (τ • τ ) 1 2 ∀σ, τ ∈ S d .
Here and below the indices i, j, k, l run between 1 and d and, unless stated otherwise, the summation convention over repeated indices is used.

Let Ω be a bounded domain Ω ⊂ R d (d = 1, 2, 3) with a Lipschitz continuous boundary Γ and let Γ 1 be a measurable part of Γ such that meas(Γ 1 ) > 0. We use the notation x = (x i ) for a typical point in Ω ∪ Γ and we denote by ν = (ν i ) the outward unit normal at Γ . Also, an index that follows a comma represents the partial derivative with respect to the corresponding component of the spatial variable, e.g. u i,j = ∂u i /∂x j . We use standard notations for the Lebesgue and Sobolev spaces associated to Ω and Γ and, moreover, we consider the spaces

V = v = (v i ) ∈ H 1 (Ω) d : v = 0 on Γ 1 , Q = τ = (τ ij ) ∈ L 2 (Ω) d×d : τ ij = τ ji .
These are real Hilbert spaces endowed with the inner products

(u, v) V = Ω ε(u) • ε(v) dx, (σ, τ ) Q = Ω σ • τ dx
and the associated norms • V and • Q , respectively. Here ε represents the deformation operator given by

ε(v) = ε ij (v) , ε ij (v) = 1 2 (v i,j + v j,i ) ∀v ∈ H 1 (Ω) d .
Completeness of the space (V, • V ) follows from the assumption meas(Γ 1 ) > 0, which allows the use of Korn's inequality.

For an element v ∈ V we still write v for the trace of v on the boundary Γ . We denote by v ν and v τ the normal and the tangential component of v on Γ , respectively, defined by v ν = v • ν, v τ = v -v ν ν. Let Γ 3 be a measurable part of Γ . Then, by the Sobolev trace theorem, there exists a positive constant c 0 which depends on Ω, Γ 1 and Γ 3 such that

v L 2 (Γ3) d ≤ c 0 v V ∀v ∈ V. (2.1)
For a regular function σ : Ω ∪ Γ → S d we denote by σ ν and σ τ the normal and the tangential components of the vector σν on Γ , respectively, and we recall that σ ν = σν • ν and σ τ = σν -σ ν ν.

We also introduce the space of fourth order tensor fields given by

Q ∞ = E = (E ijkl ) E ijkl = E jikl = E klij ∈ L ∞ (Ω), 1 ≤ i, j, k, l ≤ d
and we recall that Q ∞ is a real Banach space with the norm

E Q∞ = 1≤i,j,k,l≤d E ijkl L ∞ (Ω) .
Moreover, a simple calculation shows that

Eτ Q ≤ E Q∞ τ Q ∀E ∈ Q ∞ , τ ∈ Q. (2.2)
Given a normed space (X, • X ) we use the notation C(R + ; X) for the space of continuous functions defined on R + with values in X. It is well known that, if X is a Banach space, then C(R + ; X) can be organized in a canonical way as a Fréchet space, i.e., as a complete metric space in which the corresponding topology is induced by a countable family of seminorms. Details can be found in [START_REF] Corduneanu | Problèmes globaux dans la théorie des équations intégrales de volterra[END_REF] and [START_REF] Massera | Linear Differential Equations and Function Spaces[END_REF], for instance. Also, for a subset K ⊂ X we still use the symbol C(R + ; K) for the set of continuous functions defined on R + with values in K. Finally, for n ∈ N and K ⊂ X we denote by C([0, n]; K) the set of continuous functions defined on [0, n] with values in K.

We end this section with the following result which will be used in Section 5 of the paper. Theorem 1. Let (X, • X ) be a Banach space, K a nonempty closed subset of X and let Λ : C(R + ; K) → C(R + ; K) be a nonlinear operator. Assume that there exists k ∈ N with the following property: for all n ∈ N there exist two constants c n ≥ 0 and d n ∈ [0, 1) such that

Λu(t) -Λv(t) k X ≤ c n t 0 u(s) -v(s) k X ds + d n u(t) -v(t) k X for all u, v ∈ C(R + ; K) and for all t ∈ [0, n]. Then the operator Λ has a unique fixed point η * ∈ C(R + ; K).
Theorem 1 was proved in [START_REF] Sofonea | A fixed point result with applications in the study of viscoeplastic frictionless contact problems[END_REF] in the case when K = X. Nevertheless, a careful examination of the proof shows that the theorem is still valid for operators Λ :

C(R + ; K) → C(R + ; K), provided that K is a nonempty closed part of X.
The modification in proof are straightforward and, therefore, we do not provide the details. We only mention that the difference consists in the use of the Banach fixed point argument for contractive maps defined on the set C([0, n]; K) with values on C([0, n]; K), for all n ∈ N, instead of contractive maps defined on the space C([0, T ]; X) with values in C([0, T ]; X).

Problem Statement

The physical setting is as follows. A viscoelastic body occupies a bounded domain Ω ⊂ R d (d = 1, 2, 3) with a Lipschitz continuous boundary Γ , divided into three measurable parts Γ 1 , Γ 2 and Γ 3 , such that meas(Γ 1 ) > 0. The body is subject to the action of body forces of density f 0 , is fixed on Γ 1 and is submitted to the action of surface tractions of density f 2 on Γ 2 . Moreover, the body is in adhesive contact on Γ 3 with an obstacle, the so-called foundation. We adopt the framework of the small strain theory, we assume that the contact process is quasistatic and we study it in the interval of time R + = [0, ∞). To derive a mathematical model which corresponds to this physical setting we need to precise the constitutive law of the material, the balance equations and the boundary conditions, as well.

In this paper we assume that the material's behavior follows a viscoelastic constitutive law with long memory of the form

σ(t) = Aε u(t) + t 0 B(t -s)ε u(s) ds in Ω, (3.1)
where, here and below, u denotes the displacement field, σ represents the stress field, ε(u) is the linearized strain tensor and t ∈ R + represents the time variable. Also, A and B represent the elasticity operator and the relaxation tensor, respectively, and are assumed to verify the following conditions:

                                     (a) A : Ω × S d → S d . (b) There exists L A > 0 such that A(x, ε 1 ) -A(x, ε 2 ) ≤ L A ε 1 -ε 2 ∀ε 1 , ε 2 ∈ S d , a.e. x ∈ Ω. (c) There exists m A > 0 such that A(x, ε 1 ) -A(x, ε 2 ) • (ε 1 -ε 2 ) ≥ m A ε 1 -ε 2 2 ∀ε 1 , ε 2 ∈ S d , a.e. x ∈ Ω. (d) The mapping x → A(x, ε) is measurable on Ω, for any ε ∈ S d . (e) The mapping x → A(x, 0) belongs to Q, (3.2) B ∈ C(R + ; Q ∞ ). (3.3)
Note that in (3.1) and below, in order to simplify the notation, we do not indicate explicitly the dependence of various functions on the spatial variable x. Various examples and mechanical interpretation concerning viscoelastic constitutive laws of the from (3.1) can be found in [START_REF] Drozdov | Finite Elasticity and Viscoelasticity-A Course in the Nonlinear Mechanics of Solids[END_REF][START_REF] Sofonea | Variational Inequalities with Applications. A Study of Antiplane Frictional Contact Problems[END_REF][START_REF] Sofonea | Mathematical Models in Contact Mechanics[END_REF].

Next, since process is quasistatic, we shall use the equilibrium equation

Div σ(t) + f 0 (t) = 0 in Ω, (3.4)
where Div denotes the divergence operator for tensor valued functions, i.e. Div σ = (σ ij,j ). This equation shows that at each time moment the external forces are balanced by the internal stresses. Moreover, since the body is fixed on Γ 1 and given tractions are acting on Γ 2 we impose the following displacement-traction conditions:

u(t) = 0 on Γ 1 , σ(t)ν = f 2 (t) on Γ 2 .
Math. Model. Anal., [START_REF] Sofonea | A fixed point result with applications in the study of viscoeplastic frictionless contact problems[END_REF](5):607-626, 2014.

In addition, we assume that the densities of body forces and surface tractions have regularity

f 0 ∈ C R + ; L 2 (Ω) d , f 2 ∈ C R + ; L 2 (Γ 2 ) d . (3.5)
We now turn to the description on the adhesive contact conditions on the surface Γ 3 in which our main interest is. First, we assume that the penetration is limited by a bound g > 0 and, therefore, at each time moment t ∈ R + , the normal displacement satisfies the inequality

u ν (t) ≤ g on Γ 3 . (3.6)
Next, we assume that the normal stress has an additive decomposition of the form

σ ν (t) = σ D ν (t) + σ R ν (t) + σ M ν (t) + σ A ν (t) on Γ 3 , (3.7)
where the functions σ D ν (t), σ R ν (t), σ A ν (t) and σ M ν (t) describe the deformability, the rigidity, the adhesive and the surface memory properties of the foundation. We assume that σ D ν (t) satisfies the normal compliance contact condition:

-σ D ν (t) = p ν u ν (t) on Γ 3 . (3.8)
Here p ν is a given function which satisfies

                               (a) p ν : Γ 3 × R → R + . (b) There exists L ν > 0 such that p ν (x, r 1 ) -p ν (x, r 2 ) ≤ L ν |r 1 -r 2 | ∀r 1 , r 2 ∈ R, a.e. x ∈ Γ 3 . (c) p ν (x, r 1 ) -p ν (x, r 2 ) (r 1 -r 2 ) ≥ 0 ∀r 1 , r 2 ∈ R, a.e. x ∈ Γ 3 . (d) The mapping x → p ν (x, r) is measurable on Γ 3 ,
for any r ∈ R. (e) p ν (x, r) = 0 for all r ≤ 0, a.e. x ∈ Γ 3 .

(3.9)

The part σ R ν (t) of the normal stress satisfies the Signorini condition in the form with a gap function, i.e.

σ R ν (t) ≤ 0, σ R ν (t) u ν (t) -g = 0 on Γ 3 . (3.10)
Details on the Signorini condition and normal compliance function can be found in [START_REF] Han | Quasistatic Contact Problems in Viscoelasticity and Viscoplasticity[END_REF][START_REF] Klarbring | Frictional contact problems with normal compliance[END_REF][START_REF] Klarbring | On friction problems with normal compliance[END_REF][START_REF] Shillor | Models and Analysis of Quasistatic Contact[END_REF], for instance. Here we restrict ourselves to recall that the normal compliance condition describes the contact with a deformable foundation and the Signorini contact condition describes the contact with a perfectly rigid foundation.

The function σ M ν (t) satisfies the condition

             σ M ν (t) ≤ t 0 b(t -s) u + ν (s) ds, σ M ν (t) = 0 if u ν (t) < 0, on Γ 3 , σ M ν (t) = - t 0 b(t -s) u + ν (s) ds if u ν (t) > 0, (3.11)
where b is a surface memory function which verifies

b ∈ C R + ; L ∞ (Γ 3 ) , b(t, x) ≥ 0 for all t ∈ R + , a.e. x ∈ Γ 3 . (3.12)
Details on this condition can be found in [START_REF] Sofonea | A viscoplastic contact problem with normal compliance, unilateral constraint and memory term[END_REF]. Finally, the contribution of the bonding to the normal traction, σ A ν (t), satisfies

σ A ν (t) = γ ν β 2 (t) R u ν (t) on Γ 3 , (3.13)
where R is the truncation function given by

R(s) =      L if s < -L, -s if -L ≤ s ≤ 0, 0 if s > 0. (3.14)
Here and below L > 0 is the characteristic length of the bond, beyond which it stretches without offering any additional resistance (see, e.g., [START_REF] Raous | A consistent model coupling adhesion, friction and unilateral contact[END_REF]) and γ ν represents an adhesion coefficient. More details on this condition can be found in [START_REF] Sofonea | Analysis and Approximation of Contact Problems with Adhesion or Damage[END_REF].

We combine (3.7), (3.8), (3.13) and denote

-σ M ν (t) = ξ(t) to see that σ R ν (t) = σ ν (t) + p ν u ν (t) -γ ν β 2 (t) R u ν (t) + ξ(t) on Γ 3 . (3.15)
Then we substitute equality (3.15) in (3.10) and use (3.6), (3.11) and (3.12) to obtain the following contact condition

                           u ν (t) ≤ g, σ ν (t) + p ν u ν (t) + ξ(t) -γ ν β 2 (t) R u ν (t) ≤ 0, u ν (t) -g σ ν (t) + p ν u ν (t) + ξ(t) -γ ν β 2 (t) R u ν (t) = 0, 0 ≤ ξ(t) ≤ t 0 b(t -s) u + ν (s) ds, on Γ 3 . ξ(t) = 0 if u ν (t) < 0, ξ(t) = t 0 b(t -s) u + ν (s) ds if u ν (t) > 0, (3.16)
To complete our model we assume that the resistance to tangential motion is generated mainly by the glue, and the frictional traction can be neglected. In particular, when all the adhesive bonds are inactive, or broken, the motion is frictionless. Thus, the tangential traction depends on the intensity of adhesion and on the tangential displacement, but only up to the bond length L, that is

-σ τ (t) = p τ β(t) R * u τ (t) on Γ 3 , (3.17)
where the truncation operator R * is defined by

R * (v) =    v if v ≤ L, L v v if v ≥ L. (3.18)
Then, p τ (β) acts as the stiffness or spring constant, and the traction is in direction opposite to the displacement. The tangential function p τ satisfies

                               (a) p τ : Γ 3 × R → R + . (b) There exists L τ > 0 such that p τ (x, β 1 ) -p τ (x, β 2 ) ≤ L τ |β 1 -β 2 | ∀β 1 , β 2 ∈ R, a.e. x ∈ Γ 3 . (c) There exists M τ > 0 such that p τ (x, β) ≤ M τ ∀β ∈ R, a.e. x ∈ Γ 3 . (d) The mapping x → p τ (x, β) is measurable on Γ 3 ,
for any β ∈ R. (e) p τ (x, 0) = 0 a.e. x ∈ Γ 3 .

(3.19)

We follow [START_REF] Frémond | Equilibre des structures qui adhèrent à leur support[END_REF][START_REF] Frémond | Adhérence des solides[END_REF][START_REF] Sofonea | Analysis and Approximation of Contact Problems with Adhesion or Damage[END_REF] and assume that the bonding field satisfies the unilateral constraint 0 ≤ β(t) ≤ 1 on Γ 3 .

(3.20)

Moreover, its evolution is governed by the differential equation

β(t) = -γ ν β(t) R u ν (t) 2 -ε a + on Γ 3 , (3.21)
in which ε a represents the Dupré energy and R is the truncation operator given by

R(s) =      -L if s < -L, s if -L ≤ s ≤ L, L if s > L. (3.22)
We complete this differential equation with the initial condition

β(0) = β 0 on Γ 3 (3.23)
and we assume that the adhesion coefficient, γ ν , the Dupré energy ε a , and initial bonding field, β 0 , satisfy the conditions

γ ν ∈ L ∞ (Γ 3 ), γ ν ≥ 0, ε a ∈ L ∞ (Γ 3 ), ε a ≥ 0, (3.24)
β 0 ∈ L 2 (Γ 3 ), 0 ≤ β 0 ≤ 1 a.e. on Γ 3 . (3.25)
We gather the above equations and conditions to obtain the following formulation of the mechanical problem of quasistatic adhesive contact with normal compliance, unilateral constraint and surface memory term.

Problem P. Find a displacement field u : Ω × R + → R d , a stress field σ : Ω × R + → S d and a adhesion field β : Γ 3 × R + → [0, 1] such that σ(t) = Aε u(t) + t 0 B(t -s)ε u(s) ds in Ω, (3.26) Div σ(t) + f 0 (t) = 0 in Ω, (3.27) u(t) = 0 on Γ 1 , (3.28) σ(t)ν = f 2 (t) on Γ 2 , (3.29) -σ τ (t) = p τ β(t) R * u τ (t) on Γ 3 , (3.30)
β(t) = -γ ν β(t) R u ν (t) 2 -ε a + on Γ 3 (3.31)
for all t ∈ R + , there exists ξ :

Γ 3 × R + → R which satisfies u ν (t) ≤ g, σ ν (t) + p ν u ν (t) + ξ(t) -γ ν β 2 (t) R u ν (t) ≤ 0, u ν (t) -g σ ν (t) + p ν (u ν (t)) + ξ(t) -γ ν β 2 (t) R u ν (t) = 0, 0 ≤ ξ(t) ≤ t 0 b(t -s) u + ν (s) ds, ξ(t) = 0 if u ν (t) < 0, ξ(t) = t 0 b(t -s) u + ν (s) ds if u ν (t) > 0                            on Γ 3 , (3.32)
for all t ∈ R + and, moreover,

β(0) = β 0 on Γ 3 . (3.33)
The unique weak solvability of the contact problem P will be stated in Section 4 and proved in Section 5. We end this current section with some additional comments of the contact condition (3.32), which represents one of the novelties of this paper.

First, we recall that (3.32) describes a condition with unilateral constraint, since inequality (3.6) holds at each time moment. The rest of the comments in this paragraph, together with the corresponding equalities and inequalities, are valid for a given point x on the contact surface Γ 3 . Nevertheless, we recall that, for simplicity, we skip the dependence of various functions on x. Assume that at a given moment t there is penetration which did not reach the bound g, i.e. 0 < u ν (t) < g. Then, (3.32) yields

-σ ν (t) = p ν u ν (t) + t 0 b(t -s) u + ν (s) ds. (3.34)
This equality shows that at the moment t, the reaction of the foundation depends both on the current value of the penetration (represented by the term p ν (u ν (t))) and on the history of the penetration (represented by the integral term in (3.34)). Assume now that at a given moment t there is separation between the body and the foundation, i.e. u ν (t) < 0. Then, (3.32) shows that

σ ν (t) = γ ν β 2 (t) R u ν (t) ,
i.e. the reaction of the foundation is nonnegative and depends on adhesion coefficient, on the square of intensity of adhesion and on the normal displacement, but as it does not exceed the bound length L. Once it exceeds it the normal traction remains constant and |σ ν (t)| ≤ γ ν L.

In conclusion, condition (3.32) shows that delimitation takes place when there is separation. When there is penetration the contact stress is given by a normal compliance condition with memory term of the form (3.34) but up to the limit g. When the limit g is reached, the stress is given by a Signorini-type unilateral condition. This condition can be interpreted physically as follows. The foundation is assumed to be made of a hard material covered with a thin layer made of a soft adhesive material with thickness g. The layer has adhesive viscoelastic behavior, i.e. is deformable, allows penetration and develops memory effects. The hard material is perfectly rigid and, therefore, it does not allow penetration. To summarize, the foundation has a rigid-adhesiveviscoelastic behavior; its adhesive-viscoelastic behavior is caused by the layer of the soft material while its rigid behavior is caused by the hard material.

Variational Formulation and Main Result

We now turn to the variational formulation of Problem P and, to this end, we assume in what follows that (u, σ, β) represents a triple of regular functions which satisfy (3.26)-(3.33). We introduce the set of admissible displacements and the set of admissible bonding fields, respectively, defined by

U = v ∈ V : v ν ≤ g a.e. on Γ 3 }, (4.1)
Z = ω ∈ L 2 (Γ 3 ) : 0 ≤ ω ≤ 1 a.e. on Γ 3 . (4.2)
Let v ∈ U and t ∈ R + be given. We use the Green's formula to see that

Ω σ(t) • ε(v) dx + Ω Div σ(t) • v dx = Γ σ(t)ν • v da ∀v ∈ V
and, combining this equality with the equilibrium equation (3.27), we find that

Ω σ(t)• ε(v)-ε u(t) dx = Ω f 0 (t)• v -u(t) dx+ Γ σ(t)ν • v -u(t) da.
Then, we split the surface integral over Γ 1 , Γ 2 and Γ 3 and, since vu(t) = 0 a.e. on Γ 1 , σ(t)ν = f 2 (t) on Γ 2 , we deduce that

Ω σ(t) • ε(v) -ε u(t) dx = Ω f 0 (t) • v -u(t) dx + Γ2 f 2 (t) • v -u(t) da + Γ3 σ(t)ν • v -u(t) da.
Moreover, since

σ(t)ν • v -u(t) = σ ν (t) v ν -u ν (t) + σ τ (t) • v τ -u τ (t) on Γ 3 ,
taking into account condition (3.30) we obtain

Ω σ(t) • ε(v) -ε u(t) dx = Ω f 0 (t) • v -u(t) dx + Γ2 f 2 (t) • v -u(t) da (4.3)
+ Γ3 σ ν (t) v ν -u ν (t) da - Γ3 p τ β(t) R * u τ (t) • v τ -u τ (t) da.
We write now

σ ν (t) v ν -u ν (t) = σ ν (t) + p ν u ν (t) + ξ(t) -γ ν β 2 (t) R u ν (t) (v ν -g) + σ ν (t) + p ν u ν (t) + ξ(t) -γ ν β 2 (t) R u ν (t) g -u ν (t) -p ν u ν (t) + ξ(t) -γ ν β 2 (t) R u ν (t) v ν -u ν (t) on Γ 3 ,
then we use the contact condition (3.32) and definition (4.1) of the set U to see that

σ ν (t) v ν -u ν (t) ≥ -p ν u ν (t) + ξ(t) -γ ν β 2 (t) R u ν (t) v ν -u ν (t) on Γ 3 . (4.4)
We use (3.32), again, and the hypothesis (3.12) on function b to deduce that

t 0 b(t -s) u + ν (s) ds v + ν -u + ν (t) ≥ ξ(t) v ν -u ν (t) on Γ 3 . (4.5)
Then we add the inequalities (4.4) and (4.5) and integrate the result on Γ 3 to find that

Γ3 σ ν (t) v ν -u ν (t) da + Γ3 t 0 b(t -s) u + ν (s) ds v + ν -u + ν (t) da ≥ - Γ3 p ν u ν (t) v ν -u ν (t) da + Γ3 γ ν β 2 (t) R u ν (t) v ν -u ν (t) da. (4.6)
Finally, we combine (4.3) and (4.6) to obtain that

Ω σ(t) • (ε(v) -ε u(t) dx + Γ3 p ν u ν (t) v ν -u ν (t) da + Γ3 p τ β(t) R * u τ (t) • v τ -u τ (t) da + Γ3 t 0 b(t -s) u + ν (s) ds v + ν -u + ν (t) da - Γ3 γ ν β 2 (t) R u ν (t) v ν -u ν (t) da ≥ Ω f 0 (t) • v -u(t) dx + Γ2 f 2 (t) • v -u(t) dx ∀v ∈ U. (4.7)
Next, we use the Riesz representation Theorem to define the operator P : V → V and the function f : R + → V by equalities

(P u, v) V = Γ3 p ν (u ν )v ν da ∀u, v ∈ V, (4.8)
(f (t), v) V = Ω f 0 (t) • v dx + Γ2 f 2 (t) • v da ∀v ∈ V. (4.9)
It follows from assumptions (3.9) and (2.1) that

(P u -P v, u -v) V ≥ 0, P u -P v V ≤ c 2 0 L ν u -v V ∀u, v ∈ V, (4.10)
which shows that P : V → V is a monotone Lipschitz continuous operator. Moreover, the regularity (3.5) implies that

f ∈ C(R + ; V ). (4.11)
We also consider the functional j :

Z × V × V → R defined by j(β, u, v) = Γ3 p τ β(t) R * u τ (t) • v τ -γ ν β 2 (t) R u ν (t) v ν da. (4.12)
Then, we use (4.7) and notations (4.8)-(4.12) to deduce that

σ(t), ε(v) -ε u(t) Q + P u(t), v -u(t) V + j β(t), u(t), v -j β(t), u(t), u(t) + t 0 b(t -s) u + ν (s) ds, v + ν -u + ν (t) L 2 (Γ3) ≥ f (t), v -u(t) V . (4.13)
Also we integrate the differential equation (3.31) with the initial condition (3.33) to obtain that

β(t) = β 0 - t 0 γ ν β(s) R u ν (s) 2 -ε a + ds on Γ 3 . (4.14)
Finally, we recall that the unilateral constraints imposed to the displacement and bonding field, combined with the definitions (4.1) and (4.2), yield

u(t) ∈ U, β(t) ∈ Z. (4.15)
We now gather the constitutive law (3.26), the variational inequality (4.13), the integral equation (4.14) and the unilateral constraints (4.15) to obtain the following variational formulation of Problem P.

Problem P V . Find a displacement field u : R + → U , a stress field σ : R + → Q and a bonding field β : R + → Z such that for all t ∈ R + we have

σ(t) = Aε u(t) + t 0 B(t -s)ε u(s) ds, (4.16
)

σ(t), ε(v) -ε u(t) Q + P u(t), v -u(t) V + t 0 b(t -s) u + ν (s) ds, v + ν -u + ν (t) L 2 (Γ3) + j β(t), u(t), v -u(t) ≥ (f (t), v -u(t)) V ∀v ∈ U, (4.17)
β(t) = β 0 - t 0 γ ν β(s) R u ν (s) 2 -ε a + ds. (4.18)
The existence of the unique solution of Problem P V is stated below and proved in the next section.

Theorem 2. Assume that (3.2), (3.3), (3.5), (3.9), (3.12), (3.19), (3.24), and (3.25) hold. Then, there exists a unique solution (u, σ, β) of Problem P V . Moreover, the solution satisfies,

u ∈ C(R + ; U ), σ ∈ C(R + ; Q), β ∈ C(R + ; Z). (4.19)
We conclude that, under assumptions of Theorem 2, Problem P has a unique weak solution with regularity (4. [START_REF] Sofonea | A fixed point result with applications in the study of viscoeplastic frictionless contact problems[END_REF]).

We end this section with some inequalities involving the functional j which will be used in the proof of Theorem 2. Below in this section β, β 1 , and β 2 denote elements of Z while u 1 , u 2 , and v represent elements of V ; recall also that u iν and u iτ denote the normal component and the tangential part of u i , for i = 1, 2; and, finally, c denotes a generic positive constant which may depend on Ω, Γ 1 , Γ 2 , Γ 3 , A, B, p ν , p τ , γ ν and L, but does not depend on t nor on the rest of the input data, and whose value may change from place to place.

First, we note that j is linear with respect to the last argument and, therefore,

j(β, u, -v) = -j(β, u, v). (4.20)
Next, using (4.12) we find that

j(β 1 , u 1 , u 2 -u 1 ) + j(β 2 , u 2 , u 1 -u 2 ) = Γ3 γ ν β 2 1 R(u 1ν ) -R(u 2ν ) (u 1ν -u 2ν ) da + Γ3 γ ν β 2 1 -β 2 2 R(u 2ν )(u 1ν -u 2ν ) da + Γ3 p τ (β 1 ) R * (u 1τ ) -R * (u 2τ ) • (u 2τ -u 1τ ) da + Γ3 p τ (β 1 ) -p τ (β 2 ) R * (u 2τ ) • (u 2τ -u 1τ ) da and since R(u 1ν ) -R(u 2ν ) (u 1ν -u 2ν ) ≤ 0 a.e. on Γ 3 , R * (u 1τ ) -R * (u 2τ ) • (u 2τ -u 1τ ) ≤ 0 a.e. on Γ 3 , we obtain j(β 1 , u 1 , u 2 -u 1 ) + j(β 2 , u 2 , u 1 -u 2 ) ≤ Γ3 γ ν β 2 1 -β 2 2 R(u 2ν )(u 1ν -u 2ν ) da + Γ3 p τ (β 1 ) -p τ (β 2 ) R * (u 2τ ) • (u 2τ -u 1τ ) da.
Math. Model. Anal., 19(5):607-626, 2014.

Using now the inequalities

| R(u 2ν )| ≤ L, R * (u 2τ ) ≤ L, |β 1 | ≤ 1, |β 2 | ≤
j(β 1 , u 1 , u 2 -u 1 ) + j(β 2 , u 2 , u 1 -u 2 ) ≤ c Γ3 |β 1 -β 2 | u 1 -u 2 da.
Next, we combine the previous inequality with (2.1) to obtain

j(β 1 , u 1 , u 2 -u 1 ) + j(β 2 , u 2 , u 1 -u 2 ) ≤ c β 1 -β 2 L 2 (Γ3) u 1 -u 2 V .
We now choose

β 1 = β 2 = β in (4.21) to find that j(β, u 1 , u 2 -u 1 ) + j(β, u 2 , u 1 -u 2 ) ≤ 0. (4.21)
Similar manipulations, based on the Lipschitz continuity of the truncation operators R and R * and on the boundedness of the function p τ , show that

j(β, u 1 , v) -j(β, u 2 , v) ≤ c u 1 -u 2 V v V . ( 4

Proof of Theorem 2

The proof of Theorem 2 will be carried out in several steps. To present it everywhere below we assume that the hypothesis (3.2), (3.3), (3.5), (3.9), (3.12), (3.19), (3.24), and (3.25) hold. Also, we use the product space X = Q×L 2 (Γ 3 )× L 2 (Γ 3 ), endowed with the norm

η X = θ Q + ζ L 2 (Γ3) + β L 2 (Γ3) ∀η = (θ, ζ, β) ∈ X. Let η = (θ, ζ, β) ∈ C(R + ; Q × L 2 (Γ 3 ) × Z).
In the first step we consider the following variational problem.

Problem P V η . Find a displacement field u η : R + → U such that, for all t ∈ R + , u η (t) ∈ U and Aε u η (t) , ε(v) -ε u η (t) Q + θ(t), ε(v) -ε u η (t) Q + P u η (t), v -u η (t) V + ζ + (t), v + ν -u + ην (t) L 2 (Γ3) + j β(t), u η (t), v -u η (t) ≥ f (t), v -u η (t) V ∀v ∈ U.
(5.1)

We have the following result concerning this problem.

Lemma 1. There exists a unique solution to Problem P V η which satisfies u η ∈ C(R + ; U ). Moreover, if u i represents the solution of Problem P V η for η = η i ∈ C(R + ; Q × L 2 (Γ 3 ) × Z), i = 1, 2, then there exists c > 0 such that u 1 (t) -u 2 (t) V ≤ c η 1 (t) -η 2 (t) X ∀t ∈ R + .

(5.2) (5.8)

Then, using (3.24) and (3.25) it follows that ω(t) ∈ L 2 (Γ 3 ) and, moreover, 0 ≤ ω(t) ≤ β 0 ≤ 1 a.e. on Γ 3 , for all t ∈ R + . We conclude from here that ω(t) ∈ Z. On the other hand, it is easy to see that t → Λ(t) is a continuous function from R + to the product space X = Q × L 2 (Γ 3 ) × L 2 (Γ 3 ), which concludes the first part of the lemma. where, here and below, c represent various positive constants which do not depend on n. We conclude from here that

Λη 1 (t) -Λη 2 (t) X ≤ c n t 0 u 1 (s) -u 2 (s) V ds + c t 0 β 1 (s) -β 2 (s) L 2 (Γ3) ds, (5.10)
where now c n represent various positive constants which depend on n. We now combine the inequalities (5.10) and (5.2) to deduce that Λη 1 (t) -Λη 2 (t) X ≤ c n t 0 η 1 (s) -η 2 (s) X ds and we note that, obviously, Q × L 2 (Γ 3 ) × Z is a closed subset of the space X. These ingredients allow us to apply Theorem 1 to conclude the proof. Now, we have all the ingredients needed to prove Theorem 2.

Lemma 2 .

 2 The operator Λ takes values in the setC(R + ; Q × L 2 (Γ 3 ) × Z). Moreover, it has a unique fixed point η * ∈ C(R + ; Q × L 2 (Γ 3 ) × Z).Proof. Let η = (θ, ζ, β) ∈ C(R + ; Q×L 2 (Γ 3 )×Z) and denote by ω the function defined byω(t) = β 0 -t 0 γ ν β(s) R u ην (s) 2 -ε a + ds + ∀t ∈ R + .

0 u 1 0 u 1 c t 0 u 1 0 β 1

 01010101 For the second part, we considerη 1 = (θ 1 , ζ 1 , β 1 ), η 2 = (θ 2 , ζ 2 , β 2 ) ∈ C(R + ; Q × L 2 (Γ 3 ) × Z)and, for the sake of simplicity, we use the notation u ηi = u i , for i = 1, 2. Let n ∈ N and let t ∈ [0, n]. We use assumptions (3.3) and (3.12) on B and b, respectively, inequalities (2.2) and (2.1) as well as the bounds|R(u 1ν (s))| ≤ L, |R(u 2ν (s))| ≤ L, |β 1 (s)| ≤ 1, |β 2 (s)| ≤ 1, valid a.e. on Γ 3 , for all s ∈ [0, t].After some elementary calculation we deduce thatΛη 1 (t) -Λη 2 (t) X ≤ c max s∈[0,n] B(s) Q∞ t (s) -u 2 (s) V ds + c max s∈[0,n] b(s) L ∞ (Γ3) t (s) -u 2 (s) V ds + (s) -u 2 (s) V ds + t (s) -β 2 (s) L 2 (Γ3) ds ,(5.9)

  1, valid a.e. on Γ 3 , and the property (3.19)(c) of the function p τ , we deduce that

Math. Model. Anal.,[START_REF] Sofonea | A fixed point result with applications in the study of viscoeplastic frictionless contact problems[END_REF](5):607-626, 2014.

The work of the second author has been partially supported by project No. POS-DRU/159/1.5/S/132400 at Babeş-Bolyai University.

Proof. Let t ∈ R + and consider the operator A ηt : V → V and the functional ϕ ηt : V → R defined by (A ηt u, v) V = Aε(u), ε(v) Q + (P u, v) V + j β(t), u, v ∀u, v ∈ V, (5.3)

∀v ∈ V.

(5.4)

We use (3.2), (4.10), (4.20), (4.21) and (4.22) to see that the operator A ηt is strongly monotone and Lipschitz continuous; moreover, it is easy to see that the functional ϕ ηt is convex and lower semicontinuous and, in addition, the set U is a closed convex nonempty subset of V . Using these ingredients, it follows from standard arguments on variational inequalities (see, for instance Theorem 2.8 in [START_REF] Sofonea | Mathematical Models in Contact Mechanics[END_REF]) that there exists a unique element u ηt ∈ U such that

Denote u η (t)=u ηt . Then, it follows from (5.3)-(5.5) that the element u η (t)∈U is the unique element which solves the variational inequality (5.1). We now prove the continuity of the function t → u η (t) : R + → V . To this end, let t 1 , t 2 ∈ R + and denote u η

We use standard arguments in (5.1) to find that

Therefore, (3.2), (4.10), (4.21) and (2.1) yield

where c is a positive constant. This inequality combined with (4.11) and the regularity of the functions θ, ζ, β show that u η ∈ C(R + ; V ). Thus, we conclude the existence part in Lemma 1. The uniqueness part follows from of the unique solvability of (5.1) for each t ∈ R + . Finally, the estimate (5.2) follows by using simuilar arguments as those used in the proof of the inequality (5.6).

We now consider the operator Λ which maps every element

for all t ∈ R + . Here u η ∈ C(R + ; U ) is the solution of Problem P V η provided in Lemma 1. We have the following result.

be the fixed point of Λ and let u * , σ * be the functions defined by u * (t) = u η * (t), (5.11)

for all t ∈ R + . We recall that η * = Λη * and, using the equalities (5.7), (5.11) and assumption (3.12) we deduce that

for all t ∈ R + . We show that (u * , σ * , β * ) satisfies the system (4.16)-(4.18). First, we note that (4.16) is a direct consequence of (5.12). Next, we write the variational inequality (5.1) for η = η * and use the equalities (5.11)-(5.14) to see that (4.17) holds. And, finally, we claim that (4.18) also holds. Indeed, let ω * be the function defined by

(5.16)

A careful examination of equalities (5.16) and (5.15) shows that the following properties hold, a.e. on Γ 3 : t → ω * (t) is a non increasing function, ω * (0) = β 0 ≥ 0 and, if there exists t 0 ∈ R + such that ω * (t 0 ) = 0, then ω * (t) = 0 for all t ≥ t 0 . We deduce from here that ω * (t) ≥ 0 for any t ∈ R + . Therefore, since (5.15) and (5.16) imply that β * (t) = ω * (t) + we find that

(5.17) Equality (4.18) is now a direct consequence of the equalities (5.16) and (5.17). This proves that the triple (u * , σ * , β * ) represents a solution of Problem P V . The regularity expressed in (4. [START_REF] Sofonea | A fixed point result with applications in the study of viscoeplastic frictionless contact problems[END_REF]) is a direct consequence of the Lemmas 1 and 2, combined with assumptions (3.2) and (3.3).

Uniqueness. The uniqueness of the solution follows from the uniqueness of the fixed point of operator Λ defined by (5.7) combined with the unique solvability of Problem P V η . Indeed, let (u, σ, β) be a solution of Problem P V which satisfies (4.19) for all t ∈ R + . We substitute equality (4.16) in (4.17) and, using (5.18), we deduce that u satisfies the inequality (5.1), at each time moment t ∈ R + .

On the other hand, it follows from Lemma 1 that Problem P V η has a unique solution, denoted u η , with regularity u η ∈ C(R + ; U ). Therefore, we conclude that u = u η .

(5.19)

We use (5.19) to see that


domain: sdv phys

text:
Violaine Ossola
email: violaine.ossola@gmail.com

Fabien Pottier

Charline Pinna

Katia Bougiouri

Aurélie Tournié

Anne Michelin

Christine Andraud

Doris Gomez

Marianne Elias

Clearwing butterflies challenge the thermal melanism hypothesis

Keywords: Altitude, Optics, Thermal melanism, Thermal properties, transparency

In contrast to most butterflies harboring opaque wing colorations, some species display large transparent patches on their wings. Wing transparency, which entails a dramatic reduction of pigmentation, raises the question of potential costs for vital functions, such as thermoregulation, especially along climatic gradients. The thermal melanism hypothesis posits that darker colorations should be favored in colder environments, which enables them to absorb more radiation and maintain a body temperature compatible with activity. This prediction extends to the near infrared (NIR) range, which represents a large proportion of solar radiation. Here we assess the implications of wing transparency for light absorption and thermal properties in 42 butterfly species from the neotropical tribe Ithomiini that range the extent of transparency, from fully opaque to highly transparent, and we test whether those species conform to the prediction of the thermal melanism hypothesis. We find that transparent wings are less efficient than opaque wings to absorb light across UV, Visible and NIR wavelength ranges, and are also less efficient to collect heat. Moreover, dark coloration occupies a lower proportion of wing area as altitude increases, and ithomiine species harbor more transparency at higher altitudes, where climatic conditions are colder, going strongly against the prediction of the thermal melanism hypothesis. We discuss these surprising results in light of recent studies suggesting that factors other than adaptation to cold, such as predation pressure, physiology or behavior, may have driven the evolution of wing patterns in Ithomiini.

Main Text

Introduction

Diurnal butterflies are known for the great variety of colors they display. The scales covering their wings produce colorations through pigments [START_REF] Stavenga | Coloration principles of nymphaline butterflies -thin films, melanin, ommochromes and wing scale stacking[END_REF] or structures [START_REF] Berthier | Iridescences: the physical colors of insects[END_REF][START_REF] Tada | Microscale Radiative Effects in Complex Microstructures of Iridescent Butterfly Wing Scales[END_REF]. Coloration is involved in multiple functions, including intraspecific signaling [START_REF] Kemp | Female butterflies prefer males bearing bright iridescent ornamentation[END_REF]; antipredator defenses such as crypsis and camouflage [START_REF] Endler | Progressive background in moths, and a quantitative measure of crypsis[END_REF], masquerade [START_REF] Skelhorn | Masquerade: Camouflage Without Crypsis[END_REF], aposematism and mimicry [START_REF] Poulton | The Experimental Proof of the Protective Value of Colour and Markings in Insects in reference to their Vertebrate Encmies[END_REF]; as well as thermal adaptation [START_REF]The rôle of butterfly wings in regulation of body temperature[END_REF].

The balance between these selective pressures drives the diversity of color patterns in butterfly wings [START_REF] Postema | Color under pressure: how multiple factors shape defensive coloration[END_REF].

In ectotherms, coloration has been shown to play an important role in determining body temperature and activity, since the degree of darkness is positively correlated with internal body temperature [START_REF] Heinrich | Mechanisms of Insect Thermoregulation[END_REF]. Darker butterfly wings absorb more light, and are associated with longer flight activity and larger flight distances in cold conditions than lighter wings [START_REF] Ellers | Functional ecological implications of intraspecific differences in wing melanization in Colias butterflies: WING DARKNESS, FLIGHT AND EGG MATURATION[END_REF][START_REF] Roland | Melanism and diel activity of alpine Colias (Lepidoptera: Pieridae)[END_REF].

The thermal melanism hypothesis (TMH) posits that darker colorations provide thermal advantages in cold environments [START_REF] Clusella Trullas | Thermal melanism in ectotherms[END_REF], especially for ectotherms, as dark surfaces are more efficient to absorb heat from the environment than light surfaces. Darker species or individuals are expected to live at high altitudes or latitudes, whereas lighter ones are more likely to be present at lower altitudes and latitudes, which offer warmer conditions. The existence of such melanism gradients has been shown in many case studies, across geographical regions and wide ranges of taxa, from reptiles to insects [START_REF] Clusella-Trullas | Testing the thermal melanism hypothesis: a macrophysiological approach[END_REF][START_REF] Guppy | The adaptive significance of alpine melanism in the butterfly Parnassius phoebus F. (Lepidoptera: Papilionidae)[END_REF][START_REF] Zeuss | Global warming favours lightcoloured insects in Europe[END_REF]. However, visible colors represent only a small part of the incident solar radiation reaching the earth's surface. Over 50% of the solar energy lies in the nearinfrared wavelengths (NIR, 700 to 2500 nm). NIR should then be of particular importance in ectotherm thermal capacities, more than ultraviolet to red (i. e. 300 to 700 nm, hereafter UV-Visible range) [START_REF] Stuart-Fox | Thermal consequences of colour and nearinfrared reflectance[END_REF], since they are not involved in other aspects of their ecology because they cannot be seen by most organisms. Recent studies on communities of European [START_REF] Kang | Climate predicts both visible and near-infrared reflectance in butterflies[END_REF] and Australian [START_REF] Munro | Climate is a strong predictor of near-infrared reflectance but a poor predictor of colour in butterflies[END_REF] butterflies confirmed that reflectance in the NIR range is strongly correlated with climatic factors, especially temperature, and more strongly compared to UV and Visible wavelengths. In colder environments, butterflies tend to reflect less and therefore absorb more in the NIR. These findings highlight that optical properties can be disconnected to a certain extent between NIR and UV-Visible ranges. By contrast, no clear trends exist regarding humidity or precipitation and wing darkness. Gloger's rule, which predicts that darker individual should be found in more humid climates, has been evidenced mainly for endotherms with little clear support in ectotherms [START_REF] Kang | Climate predicts both visible and near-infrared reflectance in butterflies[END_REF][START_REF] Delhey | Reconciling ecogeographical rules: rainfall and temperature predict global colour variation in the largest bird radiation[END_REF].

In Lepidoptera, few species exhibit transparency on their wings, a feature that has evolved multiple times independently [START_REF] Gomez | Wing transparency in butterflies and moths: structural diversity, optical properties, and ecological relevance[END_REF]. Transparency is achieved by a large diversity of wing microstructures, including a combination of scale shape or insertion modification, reduced scale size or density, as well as a reduction of scale pigments [START_REF] Gomez | Wing transparency in butterflies and moths: structural diversity, optical properties, and ecological relevance[END_REF]. In many cases, transparency is enhanced by the presence of nanostructures on the wing membrane, which decreases reflection [START_REF] Pinna | Mimicry can drive convergence in structural and light transmission features of transparent wings in Lepidoptera[END_REF][START_REF] Pomerantz | Developmental, cellular, and biochemical basis of transparency in clearwing butterflies[END_REF][START_REF] Siddique | The role of random nanostructures for the omnidirectional anti-reflection properties of the glasswing butterfly[END_REF]. Transparency has attracted recent research effort regarding its now-established role in crypsis in various species [START_REF] Arias | Partial wing transparency works better when disrupting wing edges: Evidence from a field experiment[END_REF][START_REF] Arias | Transparency improves concealment in cryptically coloured moths[END_REF][START_REF] Arias | Transparency reduces predator detection in mimetic clearwing butterflies[END_REF], and its involvement in predator-prey signaling in some clades of mimetic Lepidoptera [START_REF] Pinna | Mimicry can drive convergence in structural and light transmission features of transparent wings in Lepidoptera[END_REF]. Yet, transparency may come at the cost of other functions involving wing coloration, notably thermoregulation. While thermoregulatory processes are well documented in opaque species, they are largely unknown in clearwing butterflies. A study focusing on wing transparency across the entire Lepidoptera order found that clearwing species tend to transmit less light, and thus absorb more, in high latitudes than in tropical climates, suggesting a thermal cost for butterflies [START_REF] Gomez | Wing transparency in butterflies and moths: structural diversity, optical properties, and ecological relevance[END_REF]. Yet, the actual impact of transparent wing patches on wing thermal properties needs to be investigated. Among Lepidoptera, Ithomiini is a remarkable neotropical tribe of ca. 400 species [START_REF] Chazot | Renewed diversification following Miocene landscape turnover in a Neotropical butterfly radiation[END_REF][START_REF] Doré | Anthropogenic pressures coincide with Neotropical biodiversity hotspots in a flagship butterfly group[END_REF], 80% of which exhibit transparency on their wings to some extent. Species are highly segregated along the elevational gradient, from warm lowlands up to colder highlands, as high as 3,000 meters [START_REF] Doré | Anthropogenic pressures coincide with Neotropical biodiversity hotspots in a flagship butterfly group[END_REF][START_REF] Chazot | Mutualistic Mimicry and Filtering by Altitude Shape the Structure of Andean Butterfly Communities[END_REF]. Because wing transparency in Ithomiini is achieved by a reduction in membrane coverage by scales [START_REF] Gomez | Wing transparency in butterflies and moths: structural diversity, optical properties, and ecological relevance[END_REF][START_REF] Pinna | Mimicry can drive convergence in structural and light transmission features of transparent wings in Lepidoptera[END_REF], and is therefore associated with a lower quantity of pigments, one might expect those areas to poorly absorb solar light. Previous studies of geographical distribution of Ithomiini wing patterns revealed that the proportion of clearwing species increases with altitude, to the extent that all species above 2000 m have transparent wings [START_REF] Doré | Anthropogenic pressures coincide with Neotropical biodiversity hotspots in a flagship butterfly group[END_REF][START_REF] Chazot | Mutualistic Mimicry and Filtering by Altitude Shape the Structure of Andean Butterfly Communities[END_REF], contrary to the predictions of the TMH. Yet, this observation is based on the UV-Visible range [START_REF] Pinna | Mimicry can drive convergence in structural and light transmission features of transparent wings in Lepidoptera[END_REF][START_REF] Doré | Anthropogenic pressures coincide with Neotropical biodiversity hotspots in a flagship butterfly group[END_REF][START_REF] Chazot | Mutualistic Mimicry and Filtering by Altitude Shape the Structure of Andean Butterfly Communities[END_REF], and optical transparency shown in that range may be dissociated from the absorption properties in the NIR.

Here, we investigate for the first time wing traits relevant for thermal adaptation in 42 transparent and opaque Ithomiini species that represent the diversity of wing colorations, by using Infrared imaging and spectrophotometry. We assess the distribution of these wing traits in relation to the climatic conditions the species are exposed to in a phylogenetic framework. Specifically, we characterize and quantify the optical and thermal properties of transparent and opaque wing patches to investigate two mutually exclusive scenarios:

(i) Absorption compensation: in this scenario, the higher transmission and lower absorption of wavelengths in the UV-Visible range of transparent wing areas are compensated by a high absorption in the NIR, and the thermal capacities of transparent wings may be similar to or even better than those of opaque wings. Under this hypothesis, adaptation to climatic conditions may drive the altitudinal distribution of wing patterns in Ithomiini.

(ii) Thermal cost: in this scenario, the disconnection between wavelength ranges is not significant enough, with transparent wings transmitting more and absorbing less in the NIR range than opaque wings, therefore, reflecting similar absorption patterns at shorter wavelengths. This results in transparency being costly in terms of intrinsic heating capacities. Under the thermal cost scenario, thermal properties may not be the main target of selection with regard to the altitudinal distribution of wing patterns in Ithomiini.

In addition, assuming that opaque patches -even if not covering a high proportion of the wing surface in transparent speciesmay follow the thermal melanism hypothesis, we tested whether they represent a higher proportion of wing surface in species found in colder climates, a mechanism that could also compensate for poorer thermal capacities of transparency under the thermal cost scenario. The absence of evidence of compensation mechanisms (disconnection NIR from UV-Visible range, or increased proportion of opaque patches with altitude) would imply that factors other than wing color patterns, such as behavior or metabolism, explain adaptation to colder highland habitats in Ithomiini.

Results

To assess the thermal adaptation of Ithomiini butterflies, we examined the thermal and optical properties of the wings of 42 species in relation to climatic conditions. Unless specified otherwise, all statistical tests implemented Bayesian mixed models with Markov Chain Monte Carlo correcting for phylogenetic relatedness, with an associated Bayesian posterior P-value set to 0.05.

Wing optical and thermal properties

We first tested whether optical properties in the NIR wavelengths were correlated with those properties in the UV-Visible range, and whether those properties differed between transparent and opaque species, as defined hereafter. We measured transmission and reflection spectra on a 5mmdiameter spot on one hindwing and one forewing for each specimen, spanning the UV-Visible (300-700 nm) and NIR (700-2500 nm). We used the physical principle stating that Transmission (λ) + Reflection (λ) + Absorption (λ) = 1 to compute the total absorption spectra over 300-2500 nm (Fig. 4C). The transmission of the patch in the UV-Visible range was set as a measure for the degree of optical transparency. We split the species in two categories: transparent, for which wing transmission was higher than 35%, and opaque for the others (Table S1, Fig. S1).

Overall, UV-Visible transmission (300-700 nm), i.e., transparency as perceivable by butterflies and predators, was a good predictor of both overall absorption (300-2500 nm) and NIR absorption (700-2500 nm, Table S2, Fig. 1). NIR and UV-Visible absorption were highly correlated (Table S2, Fig. 1), be the species transparent or opaque. More transparent patches absorbed less light than opaque patches whatever the wavelength considered, supporting the thermal cost scenario for transparency (Fig. S2).

To investigate the thermal capacities of transparent wings and test whether they differ from those of opaque wings, we heated up wings with a flash illumination, and imaged them with an infrared camera to record temperature across the wing and its variation through time (Fig. 4A-B).

Image analysis showed that the average heating capacity of the wing spot, represented by excess temperature ΔT (maximum temperature reached after illumination -initial temperature), was closely related to its transparency, as estimated by the transmission in the UV-Visible range (Table S3, Fig. 2). Hence, the more transparent the spot, the less it heated up, in agreement with the thermal cost scenario. ΔT of wing spots was also positively predicted by the mean total absorption of the spot and by its NIR absorption (Table S3, Fig. 2). Therefore, wings that absorbed more in the UV-Visible-NIR range or in the NIR range heated up more. In an in-depth exploration of the intra-wing variation in heating capacity, we extracted ΔT on various points on different parts of the wings (veins and cells, i.e., the parts between veins), either in darkly colored patches DP (black to brown colors), lightly colored patches LP (for opaque species only, yellow to orange colors), or transparent patches TP, and on light LV or dark veins DV. A close look at the different areas and structures of the wings revealed that dark patches heated up significantly more than any other coloration on the wings, transparent or light (DP>(LP,TP) in Table S4, Fig. S3). Moreover, light patches heated up significantly more than transparent patches (LP>TP in Table S5, Fig. S3). Dark patches heated up more than dark veins (DP>DV in Table S6, Fig. S3), but on the other hand, light veins absorbed more heat than light patches (LV>LP in Table S7, Fig. S3). In all cases, the position of the patch with respect to the thorax, proximal or distal, played a role in predicting ΔT, but this varied according to the color of the patch considered, showing no clear-cut trend (Table S4-7).

Wing trait distribution along climatic gradients

We tested whether climatic variables predicted the proportion of dark patches on the wings of all the species, as posited by the TMH. We extracted 19 published average climatic variables for each species (29) that included temperature and precipitation descriptors and performed a PCA. PC1 and PC2 explained respectively 49 % and 27.1 % of the total variation. Higher PC1 values corresponded to warmer climates, while higher PC2 values corresponded mainly to wetter and more time-stable climates (tables S8, S9, Fig. S4). Pearson's product-moment correlation tests showed that PC1 was strongly negatively correlated with altitude (R=-0.95, N=42, p <.0001), meaning that species inhabiting higher altitude habitats had a lower PC1 value. PC2 did not correlate with altitude (R=0.24, N=42, p=0.12). To test if opaque and transparent species had similar climatic ranges we used the Levene's heteroscedasticity test on climatic variables. We found that opaque species were restricted to warmer environments, while transparent species were ubiquitous and not associated to particular altitudes (F =110.71, p <.005) (transparent species: mean=-0.27, var=10.56, opaque species: mean=1.08, var=1.09), in contradiction with the TMH.

Opaque and transparent species did not significantly differ in their variance for PC2 (F=1.76, p=0.19) (transparent species: mean=0.29, var= 3.88, opaque species: mean=-1.89, var=3.45), showing that they did not segregate according to humidity or environmental stability gradients.

Overall, climate did not predict wing optical properties. The degree of transmission of the wing spots over the range UV-Visible were not predicted by PC1, with a negligible positive effect for PC2 which goes against Gloger's rule (Table S10, Fig. S5). There was no significant association of spot absorption in the UV-Visible-NIR (table S11) or NIR ranges (table S12) with PC1 or PC2. By contrast, the average ΔT of the whole wing surfaces was positively correlated to PC1 (Table S13, Fig. 3). In other words, species living in warm environments heated up more than butterflies living in colder environments. Across all species, the proportion of dark surface on the wings increased with PC1 and was therefore higher in warmer environments (Table S14, Fig. S6). We found a similar trend when restricting the analysis to only transparent species (Table S15, Fig. S7). We examined whether this proportion varied with climate within the proximal zones alone, because wing conduction could be limited, and a patch was expected to have a greater thermal effect when closer to the body. Again, the proportion of dark surface in the proximal zones increased with PC1 (Table S16, Fig. S8), and was therefore higher in warmer environments, in contradiction with the TMH. The proportion of dark surface did not correlate with PC2, except when considering only proximal zones, indicating that proximally darker wings were associated with more humid habitats. Finally, we found no effect of climatic variables on total wing area, a proxy for butterfly size (Table S17).

Discussion

Thermal properties correlate with optical properties in clearwing butterflies

The thermal imaging of Ithomiini wings reveals that opaque wings are on average more efficient to heat up than transparent wings. Hence, transparency entails thermal costs, a conclusion supported by the close-up study of the different wing patches, in agreement with the thermal cost scenario.

The darker the color, the more the patch heats up. Therefore, transparent patches contribute little to wing heat absorption, in line with the long studied link between melanization and heating capacities [START_REF] Schmitz | Thermal characterization of butterfly wings-1. Absorption in relation to different color, surface structure and basking type[END_REF]. Consistent with this finding, the optical examination of the transparent wing spots reveals that the low absorption in the UV-Visible wavelength range is not compensated by a higher absorption in the invisible NIR range, and therefore wing transparency comes at the cost of light absorption. This result departs from findings in other organisms where visible reflectance tends to be a poor predictor of NIR reflectance [START_REF] Nussear | Misconceptions about colour, infrared radiation, and energy exchange between animals and their environments[END_REF]. Moreover, transparent spots always absorb less light than opaque ones, regardless of the wavelength considered. This is in agreement with the correlation of the UV-Visible and NIR spectra that has been previously described for transmission [START_REF] Gomez | Wing transparency in butterflies and moths: structural diversity, optical properties, and ecological relevance[END_REF] or reflection [START_REF] Kang | Climate predicts both visible and near-infrared reflectance in butterflies[END_REF][START_REF] Munro | Climate is a strong predictor of near-infrared reflectance but a poor predictor of colour in butterflies[END_REF] in interspecific studies of butterflies taxa. Yet, veins in transparent Ithomiini species are always covered with melanized scales. Those veins heat up significantly more than transparent and even lightly colored patches, and likely represent a major part of the overall wing thermal capacities by providing heat to the dynamic hemolymph flow system to the thorax [START_REF] Tsai | Physical and behavioral adaptations to prevent overheating of the living wings of butterflies[END_REF].

In Ithomiini, absorption decreases in long wavelengths, and is quite low in the NIR for all the studied species. An interspecific comparison of the absorption of longer wavelengths (~8-12 µm), especially infrared radiated by the environment, remains to be carried out as this wavelength range is supposed to have greater thermal impact. However, the thermal imaging results clearly show that transparent patches are poorer at accumulating heat than opaque patches, suggesting an absence of compensation in the far infrared range.

The altitudinal distribution of transparent species contradicts the thermal melanism hypothesis

This study confirms the counterintuitive altitudinal distribution of Ithomiini wing patterns [START_REF] Chazot | Mutualistic Mimicry and Filtering by Altitude Shape the Structure of Andean Butterfly Communities[END_REF]. While the opaque species are only found in warm climates, clearwing species are ubiquitous and widely distributed, from hot lowlands to the cold Andean highlands. Moreover, among clearwing species, the proportion of opaque patches on the wings decreases in colder environments, i.e. at higher altitudes, challenging the long history of studies supporting the TMH in butterflies and insects e.g. [START_REF] Guppy | The adaptive significance of alpine melanism in the butterfly Parnassius phoebus F. (Lepidoptera: Papilionidae)[END_REF][START_REF] Zeuss | Global warming favours lightcoloured insects in Europe[END_REF][START_REF] Hegna | To quiver or to shiver: increased melanization benefits thermoregulation, but reduces warning signal efficacy in the wood tiger moth[END_REF][START_REF] Roland | Effect of melanism of alpine Colias nastes butterflies (Lepidoptera: Pieridae) on activity and predation[END_REF], but adding to the observation that TMH does not apply to all taxa or communities. For instance a recent study found support for TMH in geometrids but not in noctuids found in the same study sites [START_REF] Heidrich | Noctuid and geometrid moth assemblages show divergent elevational gradients in body size and color lightness[END_REF]. Yet, these studies often focus on temperate climates.

Neotropical case studies are scarce and show mixed support to the TMH. Two closely related neotropical butterfly genera, Catastica and Leptophobia have a divergent coloration distribution along altitudinal gradients [START_REF] Dufour | Divergent melanism strategies in Andean butterfly communities structure diversity patterns and climate responses[END_REF], possibly explained by different behaviors and elevation ranges.

While body and wing colorations become darker with increasing altitude in Catastica, Leptophobia shows a reversed pattern, similar to that observed in Ithomiini. In the case of Ithomiini, not only the non-conformity to TMH is outstanding, but there is no high NIR absorbance to compensate the poor UV-Visible absorption that would make highland butterflies more adapted to cooler conditions. The NIR range is expected to be only subjected to thermoregulatory pressures, unlike UV-Visible wavelengths involved in vision. Indeed, NIR reflection spectra of Australian and European butterflies communities are well predicted by climatic conditions [START_REF] Kang | Climate predicts both visible and near-infrared reflectance in butterflies[END_REF][START_REF] Munro | Climate is a strong predictor of near-infrared reflectance but a poor predictor of colour in butterflies[END_REF]. In contrast to their conclusions, our study of the absorption properties of wing spots showed that in general, climatic conditions do not predict wing absorption in any radiation range. Moreover, the exploration of wing heating capacities shows that Ithomiini species living in colder environments are less efficient to heat up than species living in warmer climates, adding even more to the thermal cost of transparency. Finally, there is hardly any link between thermal or optical properties and humidity, and as such this study does not support Gloger's rule in Ithomiini.

Adaptation to cooler conditions is not the only thermal constraint faced by organisms, as high temperature may also be challenging. Here we found that opaque species, which absorb more radiation and have higher thermal capacities, live in environments where they are most exposed to heat. For ectotherms, avoiding overheating (i.e. increasing body temperature above a critical temperature threshold) can be challenging. Butterflies may have physiological adaptations [START_REF] Montejo-Kovacevich | Microclimate buffering and thermal tolerance across elevations in a tropical butterfly[END_REF] or behavioral adjustments, such as changing the position of their wings according to environmental radiations [START_REF] Tsai | Physical and behavioral adaptations to prevent overheating of the living wings of butterflies[END_REF]. Information on the metabolism and behavior of Ithomiini in natura would shed light on their altitudinal adaptation, but are currently lacking. In addition, microhabitat variability can result in individuals sharing the same overall environment being exposed to contrasting light and temperature conditions. Microhabitat preferences thus likely play a role in adjusting body temperature [START_REF] Montejo-Kovacevich | Microclimate buffering and thermal tolerance across elevations in a tropical butterfly[END_REF][START_REF] Kleckova | Thermoregulation and microhabitat use in mountain butterflies of the genus Erebia: Importance of fine-scale habitat heterogeneity[END_REF]. However, in lowland ithomiine communities, butterflies with transparent wing patterns mostly fly in the understory, close to the vegetation and out of direct sunlight, while the proportion of species with opaque wings increases with height in the canopy [START_REF] Willmott | Maintaining mimicry diversity: optimal warning colour patterns differ among microhabitats in Amazonian clearwing butterflies[END_REF]. The latter are therefore more exposed to solar radiation, another counterintuitive distribution in relationship to thermoregulation.

Transparency under multiple selection pressures

Colorations in butterflies, and therefore wing optical properties, are under multiple selection pressures as they are involved in anti-predator defenses, signaling to conspecifics and thermal adaptation, with potential trade-offs [START_REF] Postema | Color under pressure: how multiple factors shape defensive coloration[END_REF]. Consequently, selection for crypsis and aposematism could be detrimental for thermal capacities [START_REF] Heidrich | The dark side of Lepidoptera: Colour lightness of geometrid moths decreases with increasing latitude[END_REF]. Wood tiger moths are a good example of the phenomenon: larvae with large black patches are at a thermal advantage compared to larvae with equally black and orange patches, but the latter display a more efficient aposematic signal [START_REF] Lindstedt | THERMOREGULATION CONSTRAINS EFFECTIVE WARNING SIGNAL EXPRESSION[END_REF].

Imagoes display latitudinal variation in wing pattern, consistent with the TMH, which incurs cost in the efficiency of warning signals as melanic individuals experience more predation [START_REF] Hegna | To quiver or to shiver: increased melanization benefits thermoregulation, but reduces warning signal efficacy in the wood tiger moth[END_REF]. In Ithomiini, the existence and pervasiveness of transparency is surprising given that those butterflies are toxic and harbor aposematic signals. Previous work has shown that transparency in Ithomiini reduces detection and attacks by naïve predators [START_REF] Arias | Transparency reduces predator detection in mimetic clearwing butterflies[END_REF]. Even though transparent patches are likely part of the aposematic signal [START_REF] Pinna | Mimicry can drive convergence in structural and light transmission features of transparent wings in Lepidoptera[END_REF], transparency may reduce predator learning speed and memorization of prey, as indirectly suggested by the observation that unpalatability tends to be stronger in clearwing than in opaque species [START_REF] Mcclure | Why has transparency evolved in aposematic butterflies? Insights from the largest radiation of aposematic butterflies, the Ithomiini[END_REF]. Thus, transparency may incur a cost on aposematism, with a trade-off between crypsis and aposematism. A possible explanation for the unexpected altitudinal distribution of transparency in Ithomiini species may be different predation pressures at different altitudes in tropical mountains [START_REF] Roslin | Higher predation risk for insect prey at low latitudes and elevations[END_REF], which may favor crypsis over aposematism at higher altitudes, as observed in Andean arctiine moths or which conspicuousness decreases with height [START_REF] Fiedler | Aposematic Coloration of Moths Decreases Strongly along an Elevational Gradient in the Andes[END_REF].

Therefore, the altitudinal distribution of color patterns in ithomiine may be primarily driven by predation rather than climatic conditions.

Conclusion

In Ithomiini, the altitudinal distribution of wing patterns is unlikely driven by thermal constraints, and goes against the predictions of the thermal melanism hypothesis. Transparent wings have a low absorption of wavelengths in the UV-Visible-NIR range and this comes at a cost regarding heating capacities. Conversely, transparency in these butterflies has probably evolved in response to selection incurred by predation. The results question the critical significance of wing color pattern for thermal adaptation in tropical ectotherms and lead the way to further investigations about metabolic and behavioral thermoregulation, and their relative contributions to climatic adaptation in these species.

Materials and Methods

Specimen sampling, climatic data and phylogeny

We studied wings of dried Ithomiini butterfly specimens, preserved in glassine envelops, collected between 2005 and 2018 in south America. We focused on 42 species represented by two or more females to reduce the potential variance due to sexual differences (Table S1). Measurements were taken on one hindwing and one forewing of each specimen. The selected species spanned a large range of wing coloration, from fully opaque to highly transparent, and are evenly distributed along the altitudinal gradient and across the phylogeny. GPS data on species distributions were obtained from a database that contains more than 3500 occurrences of specimens of the species studied here [START_REF] Doré | Anthropogenic pressures coincide with Neotropical biodiversity hotspots in a flagship butterfly group[END_REF]. We extracted the corresponding 19 climate variables (Table S8-9., Fig. S4) from WorldClim2 historical climatic data [START_REF] Fick | WorldClim 2: new 1-km spatial resolution climate surfaces for global land areas[END_REF] with a resolution of 2.5 arc-minutes. Variables were averaged for each species. For the phylogenetic comparative analyses, we used the most comprehensive phylogeny of the Ithomiini to date [START_REF] Chazot | Renewed diversification following Miocene landscape turnover in a Neotropical butterfly radiation[END_REF], which comprises 340 out of 393 extant taxa.

Spectrophotometry measurements

To quantify the extent to which wings transmit and absorb light, we performed spectrophotometry measurements on a spot of 5 mm of diameter on the hindwing and forewing of each specimen. For transparent species, the spot was in the proximal transparent patch. For opaque species, the spot was located on the center of the proximal zone of the wing. Since a wing can reflect and transmit a large proportion of the incident light, we measured both total (specular + diffuse) reflection and total (specular + diffuse) transmission spectra of each patch, with an Agilent CARY 5000 UV-Visible-NIR spectrophotometer, equipped with an integrating sphere, over the range 300-2500 nm, which includes ultraviolet (UV), Visible, and near-infrared (NIR) wavelengths. Wings were maintained in a black paper mask with an opening of 5 mm of diameter. Spectra were measured relative to a dark (paper mask without the wings, light off) and to a white reference (Spectralon white diffuse reflectance standard), with the paper mask for reflection, nothing for transmission, light on. The resulting total absorption spectra were calculated for each wavelength λ as Absorption (λ) = 1 -(Transmission (λ) + Reflection (λ)) according to the law of conservation of energy (Fig. 4).

The negative values due to the noise of the signal were fixed to zero with the function fixneg of the package Pavo in R [START_REF] Maia | pavo : an R package for the analysis, visualization and organization of spectral data[END_REF], and each spectrum was smoothed with the loess function in R. We quantified mean absorption at different wavelength ranges, in the UV-Visible between 300-700 nm, in the NIR between 700-2500 nm , and the total 300-2500 nm. Mean transmission in the UV-Visible range 300-700 nm was measured as a proxy for animal-visible transparency. We divided the species into two categories, transparent and opaque, based on a threshold of mean transmission at 35% in the UV-Visible, matching butterflies and predators' visual perception (Table S16).

Thermal measurements

To characterize and compare the wing thermal characteristics, we measured wing temperature kinetics following instant exposure to a flash light (Fig. 4B). We conducted stimulated heating measurements on one hindwing and one forewing for each specimen, with a Neewer professional photo system flash lamp N-300W set at maximum power, which emits over the UV-Visible-IR range.

Wings were recorded by an infrared camera T650sc 2019 FLIR Systems equipped with a FLIR lens T197-915. Emissivity was set at 0.95. Wings were kept at a constant ambient temperature of 23°C prior to the experiment. We placed each single wing horizontally on a custom grid made of 3 human hair in a frame to minimize contact with any surface, 6 cm below the camera and the flash and 30 cm above a table covered with white paper, to avoid background interference with the measure.

The flash lasted for 1/800 seconds. We started the recording a few seconds before activating the flash, and let the record run for 10 seconds. Each movie was recorded at 30 images.second -1 . From the movies, we extracted the thermal kinetics -temperature as a function of time -of different regions of interest on the wings with FLIR ResearchIR Max software, and more specifically the excess temperature ΔT as the difference between the maximal temperature attained by the region of interest after the flash hits minus the temperature prior to the flash. We extracted mean ΔT of the overall wing surface, mean ΔT of the spot measured in spectrophotometry and finally ΔT of precise points of interest on each wings as follows: 6 points in the dark colored patches DP (black to brown colors), 6 points in light colored patches LP (yellow to orange colors and for opaque species only), 6 points in the transparent patches TP (for transparent species only), and 8 points on veins, assigned to a category depending on their color: dark veins DV (black to brown) and light veins LV (yellow to orange).

Image analysis

We used digital photography to quantify the proportion of dark pigmented surface on the full wings of all the specimens. Melanization in proximal zones of the wing can be of primarily importance in thermoregulation as the zone is closer to the thoracic muscles [START_REF]The rôle of butterfly wings in regulation of body temperature[END_REF]. We also quantified the color proportion separately on the proximal and distal zone of each wing. We set the limit between the zones as the perpendicular line to the middle of the transect line from the wing insertion to its apex.

We cropped the photos for each zone considered and the background was filled with green, a color ignored when calculating the color proportions. In R, we used the getKMeanColors function from colordistance package [START_REF] Weller | Quantitative color profiling of digital images with earth mover's distance using the R package colordistance[END_REF] to sort the pixels in clusters according to their RGB values. We set the number of clusters to two, corresponding to mainly dark and brown, versus transparent, or any other light colors in opaque wings. With ImageJ 1.52 [START_REF] Schindelin | Fiji: an open-source platform for biological-image analysis[END_REF] we computed the total surface of each wing in mm².

Statistical analyses

We explored various scenarios linking absorption, transmission, thermal capacities and climatic variables across the 42 species of the dataset (see supplementary materials for model details and results). Because specimens are not independent and the species are related through the phylogeny, we implemented for each test Bayesian mixed models with Markov Chain Monte Carlo, correcting for phylogenetic relatedness, using the R package MCMCglmm (50) and the phylogeny from [START_REF] Chazot | Renewed diversification following Miocene landscape turnover in a Neotropical butterfly radiation[END_REF]. We controlled for chain convergence visually and with the Heidelberg stationarity test diagnostic function of convergence heidel.diag from the R package coda [START_REF] Plummer | CODA: Convergence Diagnosis and Output Analysis for MCMC[END_REF]. To get accurate estimations of posterior distributions and avoid chain autocorrelation, we adjusted the iterations, burn-in and thinning for each model to ensure an effective sample size across all parameters of at least 1000. Models were run with a weakly informative prior for random effect and residual variances (V = 1, nu = 0.002) not to constrain the exploration of parameter values. We selected the best model with a backward selection of fixed parameters based on Bayesian P-value of 0.05, within the 95% confidence interval excluding zero. In all the models, species identity was implemented as a random factor. We included all individuals in the models unless specified otherwise. Corresponding measured transmission (dashed lines) and reflection (dotted lines) spectra in percentage, for a spot of 5 mm of diameter on the proximal forewing of each butterfly, from 300 to 2500 nm, and the resulting computed absorption spectra (solid lines) for each wavelength λ as Absorption(λ)+ Transmission(λ) + Reflection(λ) = 1.

Figure 1 .

 1 Figure 1. Relationship between absorption and transmission spectra at different ranges: overall UV-Visible-NIR (300-2500 nm) absorption against UV-Visible (300-700 nm) transmission, NIR (700-2500 nm) absorption against UV-Visible transmission, NIR (700-2500 nm) absorption against UV-Visible (300-700 nm) absorption. All relationships are significant and the solid line is drawn from the prediction of the Bayesian phylogenetic mixed model (see Table S2) (N opaque = 72, N transparent = 235).

Figure 2 .

 2 Figure 2. The thermal properties of wing spots represented by the Excess Temperature ΔT (T°final -T°initial) under a flash illumination, extracted from the thermal imaging, against their optical properties at different ranges: UV-Visible (300-700 nm) transmission, UV-Visible-NIR (300-2500 nm) absorption, and NIR (700-2500 nm) absorption. All relationships are significant and the solid line is drawn from the prediction of the Bayesian phylogenetic mixed model (see TableS3) (N

  ) (N opaque = 72, N transparent = 235).

Figure 3 .

 3 Figure 3. Wings mean Excess Temperature ΔT (T°final -T°initial) under a flash illumination, extracted from the thermal imaging, as a function of climatic principal component PC1. The relationship is significant and the solid line is drawn from the prediction of the Bayesian phylogenetic mixed model (see TableS13) (N opaque = 72, N transparent = 235).

Figure 4 .

 4 Figure 4. Examples of optical and thermal imaging on the wings of four species of different opacity. A: Photos under standard illumination of the hindwings and forewings of four of the species studied, Heterosais nephele, Melinaea menophilus, Hypomenithis theudelinda, Thyridia psidii. B: Visualization of the thermal imaging in false colors of the same wings under a flash light. C:

Acknowledgments

This work was funded by Clearwing ANR project (ANR-16-CE02-0012), HFSP project on transparency (RGP0014/2016). VO is recipient of a doctoral fellowship from the doctoral school FIRE, Learning Planet Institute. We are strongly grateful to authorities in Peru and Ecuador for providing collecting permits (Ecuador: 005-IC-FAU-DNBAPVS/MA, 019-IC-FAU-DNBAPVS/MA021; Peru: 236-2012-AG-DGFFS-DGEFFS, 201-2013-MINAGRI-DGFFS/DGEFFS, 002-2015-SERFOR-DGGSPFFS, 373-2017-SERFOR-DGGSPFFS; Gobierno Regional San Martín PEHCBM: 124-2016-GRSM/PEHCBM-DMA/EII-ANP/JARR). We warmly thank Raúl Aldaz, Paula Santacruz and Alexandre Toporov for providing help in the field, and Santiago Villamarin, the Museo de Historia Natural and Professor Gerardo Lamas for their support. We also thank Jean-Luc Bodnar and Kamel Mouhoubi from Reims University for technical advice with IR imaging, Pierre de Villemereuil for his help with Bayesian analysis, Maël Doré for bioclimatic and distribution data sharing, Willy Daney de Marcillac