Comptes Rendus Mécanique

. Westudythetransientresponseofathermoelasticstructuremadeoftwothree-dimensionalbodies connected by a thin adhesive layer. Once more we highlight the powerful ﬂexibility of Trotter’s theory of approximation of semi-groups of operators acting on variable spaces: considering the geometrical and physicalcharacteristicsofthethinlayeras parameters ,weareabletoshowinaunitarywaythatthissituation leads to a huge variety of limit models the properties of which are detailed. In particular, according to the relative behaviors of the di ﬀ erent parameters involved, new features are evidenced such as the apparition of an added speciﬁc heat coe ﬃ cient for the interface or of additional thermomechanical state variables deﬁned not only on the limit geometric interface but on its cartesian product by any interval of real numbers.


Setting the problem
We pursue our investigations on thin junctions initiated in [1,2], then further developed in [3][4][5][6][7][8][9][10][11], and hereafter consider the situation of a transient multi-physical coupling within the scope of linear thermoelasticity. As in our previous works, we have chosen to use notations that may seem daunting but have the advantage of conveying all the information necessary to express the complexity of the studied problem.
Because interphases play a crucial role in the analysis of structure assemblies, the studies devoted to bonding problems cover a huge landscape. Within the scope of Mechanics, the reader may refer to [12] and the references quoted therein for a good introduction. Rigorous mathematical approach to dynamical situations, however, are scarce. And because steady state cases should in fact be considered as particular cases of transient phenomena, the theory of approximation of semi-groups offers, from our point of view, an almost perfect tool dedicated to the mathematical modeling in Physics of continuous media. The interested reader will find good presentations of this theory in classical textbooks such as [13,14], while Trotter's fundamental contribution [15] is presented and harnessed in various physical applications in [16]. More recently this theory has been the subject of a revival (see [17]) particularly because of the large number of problems it can address. Here, the power of this method will appear in three ways: first, despite the large number of parameters involved, we are able to carry out a rigorous mathematical study of this transient problem in a unitary manner (Section 2); second, this unitary study reveals a very wide variety of limit models (Section 3); third, we are able to extract new thermomechanical features from our models, such as the appearance of an additional specific heat coefficient for the interface or additional state variables (Section 4).
(1.1) Thus the problem (P s ) of determining the evolution in the framework of small perturbations of the assembly, whose state is denoted by U s = (u s , v s , θ s ), u s , v s , θ s being the fields of displacement, velocity and temperature increment with respect to T 0 , involves a sextuplet s := (ε, ρ L , µ L , β L , κ L , α L ) of data so that all the fields will be thereafter indexed by s. If U 0 s = (u 0 s , v 0 s , θ 0 s ) is the given initial state, a formulation of (P s ) could be: where t denotes the time, e(u) is the linearized strain associated with the field of displacement u, and I is the identity matrix of S 3 .

Existence and uniqueness of a solution to
we seek z s = (u s , θ s ) in the form where z e s is the unique solution to where for all open set G of R N , 1 ≤ N ≤ 3, H 1 γ (G, R N ) denotes the subset of the Sobolev space
The remaining part z r s of z s will be involved in an evolution equation governed by a mdissipative operator A s in a Hilbert space H s of possible states with finite thermomechanical (strain, kinetic, thermal) energy defined by (2.8) and endowed with the following inner product and norm: It is straightforward to check the following.

Proposition 2.1. Operator A s is m-dissipative and, for all
14) with f s equal to f ε /ρ ε in Ω ε and 0 in B ε . So the Hille-Yosida theorem (see [13]) leads to: and U 0 s belongs to (u e s (0), 0, θ e s (0)) + D(A s ), then (2.14) has a unique solution such that U r s belongs to C 1 ([0, T ], H s ). Hence there exists a unique (u s , θ s ) in which does satisfy

A mathematical analysis of the asymptotic behavior
Now we regard the sextuplet s of geometrical and thermomechanical data as a sextuplet of parameters taking values in a countable subset of (0, +∞) 6 with a single cluster point s in {0}×[0, +∞] 5 and study the asymptotic behavior of U s in order to suggest a simplified but accurate enough model for the genuine physical situation. We will show that, depending on the relative behavior of (ρ L , µ L , β L , κ L , α L ) with respect to ε, numerous (100!) limit models appear  The physical properties of the adhesive layer corresponding to the various values of I will be conveyed in Section 4 through brief comments.

A candidate for the limit behavior
From now on, C denotes various constants which may vary from line to line and we use the convention 0 × ∞ = ∞ × 0 = 0.
3.1.1. The limit space H I This candidate could be determined by studying the asymptotic behavior of sequences with bounded total thermomechanical energy. For the sake of notation simplicity, and when no confusion ensues, we will use the same symbols U s , u s , v s and θ s to denote both the elements of general sequences and the solution to (P s ). It will appear that in some cases the thermomechanical state of the "limit structure", where the three-dimensional adhesive layer is geometrically replaced by the surface S it shrinks to, does not involve the sole state variables of the adhering bodies but additional thermomechanical state variables not necessarily defined on S but in B := S × (−1, 1) which accounts for the limit behavior of the adhesive layer.
It is convenient to introduce the following "scaling operators" which transform a field y s defined on B ε into a field y sB defined on B in such a way that a bounded energy for y s is equivalent to a bounded "scaled" energy for y sB : (3.1) Clearly: In view of following Proposition 3.1, it is natural to recall some classical notions. Let For an element y of H 1 (Ω \ S, R N ), 1 ≤ N ≤ 3, we will denote its restrictions to the open sets Ω ± by y ± which is an element of H 1 (Ω ± , R N ). The symbols γ S (y + ) and γ S (y − ) will denote the trace of y + and y − , respectively, on S. Of course, for y in H 1 (Ω, R N ), γ S (y) will denote the trace of y on S. We also use y : it is well known that a continuous mapping γ S ± is defined on H ∂ 3 (B, R N ) for the traces on S ± := S ± e 3 with values in L 2 (S ± , R N ), and, from now on, γ S ± (y) is treated as an element of We will use the following Hilbert spaces and norms: spaces of displacement fields H I u : , and they clearly are complete relatively to the inner product (u, u ) I 1 : = Note that for a field y in H 1 (S, R 2 ) we also denote the symmetrized gradient of y by e(y).
spaces of velocity fields H I v : spaces of temperature fields H I θ : spaces of limit states H I : we have in H I and a not relabeled subsequence such that Anyway the boundedness of (u s , u s ) 1,s implies that there exists u + Ω in H 1 Γ M D+ (Ω + , R 3 ) and a not relabeled subsequence such that ((T ε u s ) + , γ S ((T ε u s ) + )) converges weakly in H 1 (Ω + , R 3 ) and strongly in L 2 (S, R 3 ) toward (u + Ω , γ S (u + Ω )), respectively. By using Korn inequality and a cutoff function η such that η( (3.14) Hence assumption (H2) implies that there exist . Moreover, when I M2 = 2, one has e u B = 0 and u Ω belongs to H 1 (Ω, R 3 ).
, so that, by using the space R of infinitesimal rigid displacements and the weak convergence of (S u,4 ε u s ) 3 in H ∂ 3 (B ) toward γ S (u + Ω3 ), it is routine to establish that there exist a not relabeled subsequence and some (u Moreover as there exists an infinitesimal rigid displacement ρ s such that γ S + (T ε u s /ε) + γ S + (ρ s ) strongly converges in L 2 (S, R 2 ) toward γ S ± (u B ) , one deduces that e( γ S (u Ω )) = 0.

The limit operator A I
According to Trotter's theory of approximation of semi-groups of linear operators acting on sequences of variable Hilbert spaces [15,16], we examine the asymptotic behavior of the resolvent (I − A s ) −1 of A s in order to guess the limit operator A I . Proposition 2.1 implies that a sequence 4,s ≤ C in addition to |U s | s ≤ C that we already considered. For this purpose we introduce the following spaces G I T2 θ of temperatures and operators g θ B : θ such that (i) T ε θ s weakly converges in H 1 Γ T D (Ω I T2 ) toward θ Ω ; (ii) g I T2 (ε, S θ ε θ s ) converges weakly in L 2 (B, R 3 ) toward g θ B , when I T2 ≥ 1. Note that it is only when I T2 ≤ 1, that the "additional" temperature θ B depends on x 3 . We therefore make a supplementary assumption: so that it is reasonable to suggest the following forms (·, ·) I 4 and (·, ·) I 5 as "potential limits" of (·, ·) 4,s and (·, ·) 5,s : (3.18) It will be convenient to introduce for all θ in G I The boundedness of both ρ L B ε |v s | 2 dx and µ L B ε ae(v s ) · e(v s ) dx leads us, when I M1 = 1 and I M2 = 3, 4, to introduce a special space SH I v for velocities and consequently a special space SH I of limit possible states with finite energy.
then where, for all elements of SH I u ,ǔ is defined by: we are in a position to define operator A I in SH I by: (3.28) Similar to the case of A s , it can be checked easily that A I is m-dissipative and, more specifically, that for all φ = (φ 1 , φ 2 , φ 3 ) in SH I :  Consequently, the same statement as that of Theorem 2.1 is valid for the following equation, which will be shown to describe the asymptotic behavior of the solution to (P s ): where

Convergence
To prove the convergence of the solution U s to (P s ) toward U I , we use the framework of Trotter's theory of approximation of semi-groups of linear operators acting on variable spaces (see [15,16]) because U s and U I do not inhabit the same space. First let the representation operator P I s defined by in Ω ± ε satisfies (3.37). When I M2 = 3, 4, we use a trick of the mathematical derivation of Kirchhoff-Love theory of plates [18,19]. Let ϕ s , ψ s defined as follows:   (3.2) and (3.4), is equivalent to: which expresses that the energetic gap between the state U s and the image on the initial physical configuration O ε of the limit state U tends to zero when s goes tos! Lastly we conclude by making an additional assumption (H4) about the initial state and establishing the

A thermomechanical presentation of the results
Here we intend to make more explicit the formulation (3.43) of the limit behavior of the structure.
To lighten notations we skip the superscript I in (u I , θ I ) and write (u, θ) instead; an over dot ˙ denoting differentiation with respect to time, the motion equation reads as: while the "energy" equation reads as: Clearly the fields of stress σ ± Ω , thermal flux q ± Ω , displacement and temperature in the adhering bodies that occupy Ω + and Ω − satisfy the following relations written in strong form: where n denotes the outward normal to Ω, together with a thermomechanical contact condition along S, the common boundary of Ω + and Ω − . This corresponds to the transient response to the loading ( f , g M , g T ) of each adhering body clamped on Γ M D ± maintained at a uniform temperature T 0 on Γ T D ± and thermomechanically linked along S. These contact conditions, which stem from the limit behavior of the adhesive layer, can be deduced from the various expressions of the two integrals on B in (4.1)-(4.2). The motion and "energy" equations will be formulated in the form: T Ω (θ Ω ) + T B (θ B ) = Γ T N g τ · θ Ω dH 2 , ∀θ ∈ G I T2 . When I M2 = 1, 2, as e v B = ∂ 3 v B ⊗ S e 3 , one has: (4.8) is equal to 2 Sρ LüΩ · v Ω dH 2 and vanishes whenρ L = 0. One deduces: so that the two adhering bodies are stuck together (i.e. u Ω = 0) whenρ L > 0 and perfectly stuck together (i.e. u Ω = σ Ω e 3 = 0) whenρ L = 0. When I M2 = 1, one has: and the mechanical contact condition reads as: Therefore, ifρ L = 0 andᾱ I L = 0 or θ B is independent of x 3 (i.e. I T2 > 1, which means an adhesive with a not too weak conductance), one deduces: There is an elastic pull-back with a residual term between the two adhering bodies. The other cases for the limit mechanical behavior of the adhesive layer, which clearly appears as a continuous distribution of thermoelastic strings orthogonal to S, will be discussed further. As regards to the thermal behavior, one has: We may therefore distinguish two main cases: I T2 ≤ 2 (i.e. an adhesive with a very weak or weak heat conductance) when g θ B = 0 or (0, ∂ 3 θ B ) and I T2 = 3, 4 (adhesive with high or very high heat conductance) when g θ B = ( ∇θ B , 0) or 0. If I T2 ≤ 2, the limit thermal behavior of the adhesive could be considered as the one of a continuous distribution of strings orthogonal to S with specific heat coefficientβ L , thermal conductivityκ I L (thus insulating strings when I T2 = 0 whereas perfectly conducting ones when I T2 = 2) subjected to heat sources of lineic densityμ I Lᾱ I L a L (∂ 3uB ⊗ S e 3 ) · I (vanishing when I M2 = 0 or 2). Therefore the thermal contact condition between the adherent bodies reads as: • I T2 = 0, θ Ω = 0, q ± Ω · e 3 = 0 : perfectly insulating interface, • I T2 = 2, θ Ω = 0, − q Ω · e 3 = 2β L γ S (θ Ω ) +μ I Lᾱ I L ( u Ω ⊗ S e 3 ) · a L I : imperfect thermal contact, perfect when I T1 = 0 (adhesive with a low specific heat coefficient) and I M2 = 0 or 2.
When I T2 = 1 we have: thus, whenβ L = 0, there exists a contact conduction whose contact conductance isκ I L and a sourceμ I Lᾱ I L a L u Ω ⊗ S e 3 · I ; the other cases will be treated further. When I T2 = 3, 4, we are indeed dealing with a material surface with a specific heat coefficient β L , a thermal conductivityκ I L (thus perfectly conducting when I T2 = 4) subjected to a heat sourcē µ I Lᾱ I L a L u Ω ⊗ S e 3 ·I . So, the imperfect thermal contact condition reads as (the symbol ∆ denoting the Laplacian with respect to the sole coordinates x 1 and x 2 ): An interesting phenomena is highlighted: the appearance of an added specific heat coefficient is always positive unlessᾱ I L (a L I ) i 3 = 0, 1 ≤ i ≤ 3, in the limit behavior of the adhesive layer, while the stiffness involves a L in place of a L as in the Kirchhoff-Love anisotropic plates theory (see [10,19,20]).
So, when I M2 = 3, we have (4.25) and the mechanical contact condition between the adhering bodies reads as: It looks like a deformable material surface which is stuck between the adhering bodies and enjoys only in-plane strains. The nature of the thermal behavior of the adhesive was already examined but here we can make more explicit the thermal contact conditions: • I T2 = 0, θ Ω = 0, ∓(q ± Ω · e 3 ) = 0 : perfectly insulating interface, L a L e(γ S (u Ω ) )· I +κ I L θ Ω : contact conduction, L a L e(γ S (u Ω ) ) · I : imperfect thermal contact, L a L e(γ S (u Ω ))· I : imperfect thermal contact, L a L e(γ S (u Ω ) ) · I : imperfect thermal contact. When I M2 = 4 one has: Because g T B = ( ∇T B , 0) when I T2 = 3, the limit behavior of the adhesive layer is then similar to the one observed by [21] for thin linearly thermoelastic plates: a flexural problem for the component of the displacement field normal to S with a coupled membrane-thermal problem for the in-plane component of the displacement and the temperature.
The mechanical contact condition between the adhering bodies reads as: • u Ω = 0 • e( γ S (u Ω )) = 0 (4.28) The material surface inserted between the two adhering bodies may be considered as a secondgrade elastic one, enjoying only a motion orthogonal to S (a flexural problem . . . ). On the other hand, the thermal contact condition reads as: it involves the additional variable u M B defined on S with values in R 2 satisfying All this corresponds to a thermomechanical material surface occupying S whose material constants are given byβ +β add L ,κ I L ,ᾱ I L ,μ I L a L subjected to an inner heat source and free of mechanical loading. Of course u M B may be eliminated and consequently the thermal contact condition along S is a nonlocal relation (in time, only) between the normal flux (q ± Ω · e 3 )( x, t ) at the courant time t and the whole history of γ S (θ Ω )( x, τ), 0 ≤ τ ≤ t .
When I T1 = 1 (high specific heat coefficient), as the additional temperature variable θ B does depend on x 3 , the limit thermoelastic behavior of the adhesive layer cannot be interpreted in terms of a material surface.
Thus, in order to clarify the thermomechanical condition between the two adhering bodies, difficulties occur when I M2 = 1 or 4 and/or I T2 = 1, which correspond to the cases when the additional state variable u B depends on x 3 (I M2 = 1), does not explicitly depends on the traces on S of the displacements of the adhering bodies (I M2 = 4) and/or the additional state variable θ B depends on x 3 (I T2 = 1). In some of these cases, by adding a condition like I M1 = 0 (light adhesive layer) or I T1 = 0 (low specific heat coefficient) we again meet thermomechanical contact conditions involving the traces on S of the state variables of the adhering bodies only as in the cases we listed previously. When (4.31) so that, except when I M1 = 0 and I T2 = 0, 1 orᾱ I L = 0, u B differs from Aff(u Ω ) and the mechanical contact condition (4.11) does not involves the sole instantaneous values of the traces on S of u ± Ω and θ ± Ω . Of course, as the equations governing the evolutions of u B and θ B can be solved in terms of the whole history of the traces on S of u ± Ω and θ ± Ω , the contact condition at ( x, t ) is a rather complex function of the history of γ S (u ± Ω )( x, ·) and γ S (θ ± Ω )( x, ·) and not only of the history of the jumps u Ω ( x, ·), θ Ω ( x, ·).
When I T2 = 1, if Aff(θ Ω ) is defined similarly as Aff(u Ω ) (see (4.13)), θ 0 (1 + x 3 )(β L +β add L )θ B +μ I Lᾱ I L a L D 2 γ S (u Ω3 ) · I x 3 dx 3 +κ I L θ Ω , if I M2 = 4 (4.33) involves the whole history of γ S (u ± Ω ) and γ S (θ ± Ω ). So, in every cases, the limit thermomechanical behavior of the two adhering bodies and of the adhesive layer are of the same (thermoelastic) type as that of the original situation. But, of course, peculiarities of the limit behavior of the layer and the thermomechanical contact condition which replaces it strongly depend on the relative behaviors of the geometric and thermomechanical parameters. The thermomechanical coupling perpetuates whenμ I Lᾱ I L e u B does not vanish which is the case when I M2 differs from 0 or 2 withᾱ I L positive.

Concluding remarks
This rather lengthy and complex thermomechanical presentation of the results of our mathematical analysis exemplifies the flexibility of use but also the power of Trotter's theory of approximation of semi-groups of operators acting on variable spaces: it permits a unitary treatment with very few technicalities. Our proposal of simplified but accurate enough models for the behavior of the structure made of the two adhering bodies and the thin adhesive layer, which has to be formulated on the genuine reference configurations Ω ± ε and B ε is of course obtained through the Trotter representant P I s U I of the solution U I of the limit problem (3.43), s taking the values of the original data. When I M2 differs from 0, a variant of P I s U I may be used through the construct detailed in the proof of Proposition 3.3. Thus, from a computational and practical point of view, a finite element approximation can be implemented without meshing the thin layer occupied by the adhesive! It should be noted that, contrary to the cumbersome method-frequent in the literatureconsisting of firstly switching to a fixed abstract domain through a "scaling" (change of coordinates and unknowns), abstract domain where the convergence is formally or rigorously studied, and secondly returning-but not always-to the initial physical domain, we have hereby treated directly through the representation operator P I s the convergence of the initial problem where, obviously, the limit can be, according to index I, expressed in a fixed abstract domain defined through the "scaling" outlined above but which is used only when it is necessary to refine the determination of the asymptotic behavior of sequences of thermomechanical states with bounded energies.
To reduce the weight of our already copious study we have not detailed the cases I M 2 = 3 + α and I M 2 = 4 + β corresponding to: ; e(γ S (u Ω )) = 0 × 0 , I M 2 = 4 + β (5.2) endowed by the norm defined in (3.9) together with the convention ∞ × 0 = 0. Eventually the present study which corrects and improves [6] may be considered as a framework to assess the formal and partial modelings proposed in [22], concerning poroelasticity as it is well known that the equations involved in linear poroelasticity are the same as those in linear thermoelasticity, and [23].