A variational derivation of Stoney-like formulas for self-stressed bilayered plates

Since the beginning of the 20th century, it is known that the sponta-neous bending of heterogeneous bilayered plates correlates with the self-stress due to the contrast in the material properties of the two layers, and that this correlation can be exploited to gauge the internal stress state. Over the last decades, ever-growing device miniaturization has made stress assessment and even stress engineering an area of major technological interest. In this paper, we obtain two eﬀective 2D models accounting for the spontaneous bending of devices comprised of a thin substrate and a much thinner coating by applying a Γ -convergence technique to the standard 3D linear hyperelastic model of a bilayered plate. Our procedure is characterized by the introduction of two distinct smallness parameters plus three independent energy scaling parameters.


Introduction
We wish to investigate the interplay between disparate length (and material) scales in the mechanics of the spontaneous bending of self-stressed bilayered plates.To do so, we apply a (somewhat nonstandard) Γ -convergence technique to deduce specific 2D models from the standard 3D linear hyperelastic theory.After giving a quick account of the academic and technological background of our work in Section 1.1, in Section 1.2 we provide the reader with a guide to the technicalities to be found in the rest of the paper and a summary of the results we achieve.Section 2 contains a precise formulation of the 3D linear hyperelastic theory underlying our asymptotic analysis.On this basis, in Section 3 we construct an ε-sequence of artificial 3D problems and the corresponding ε-sequence of energy functionals, parameterized by four independent scaling exponents.Their compactness is studied in Section 4, where the (small) subset of the parameter space relevant to our purpose is progressively selected.In Section 5, after studying the Γ -limit of the sequence of energy functionals constructed in Section 3, we finally identify the distinguished limit(s) appropriate to our modeling problem.The ensuing 2D theories are presented in Section 6 and illustrated with a simple plane-strain application in Section 7. Some remarks and a few hints for future work close the paper (Section 8).

Background
In the 1909's paper [18] where, via an elementary strength-of-materials argument, Stoney obtained his later renowned formula relating coating prestress with substrate bending, he reported the rationale of his own experiments and the aim of his simple model as follows: It seemed, therefore, that metals are deposited under tension, and if so that they should strain the material on which they were deposited so as to bend it; and that by the amount of this bending the tension under which they were deposited could be determined; and it was found that when nickel was deposited on one side of a steel sheet of metal very considerable bending took place. . . .The amount of bending, combined with the thickness of the deposit, enabled the tension under which it is deposited to be calculated.
Translated into the notations we use in this paper, Stoney's formula reads where κ St is the (average) plate curvature and σ ˝the (hypothesized) film tension.Since the substrate thickness q h could be directly measured, the (much smaller) thickness p h of the deposited film could be determined by differential weighing, and the relevant elastic modulus M (equal for film and substrate) reasonably estimated, through (1) Stoney was able to infer the film tensionunaccessible to measurement-from a rough measure of the plate curvature.
Despite the enormous progress made during the intervening century in fabrication and measurement technologies, the ideas embodied in (1) still form the basis for the estimation of self-stress in modern devices that Stoney could not even imagine.In the last several decades, stimulated by the escalating developments in applications ranging from surface coating of mechanical parts to microelectronics, growing efforts have been devoted to assess the validity of (1) and to advance proposals for improving it.In 1997, with a shell finite element analysis combined with an experimental study of circular wafer specimens with in-plane isotropic properties subjected to a uniform biaxial prestress, Finot et al. [3] showed that the curvature is still spherical but non-uniform already for moderately large deflections.A couple of years later, Freund, Floro & Chason [4] extended (1) to not-so-thin films, accounting also for differing moduli between film and substrate.Apparently independently of [3], they also pointed out that moderately large deflections suffice to falsify the assumption of a uniform curvature.In 2005-7, Huang and coworkers [9][10][11] extended (1) to cover non-uniform prestress.A comprehensive, compact introduction to the original Stoney formula and its offshoots is given by Freund & Suresh in the second chapter of their extensive book [5].
Nowadays, the capability to fabricate smaller and smaller components is broadening the application of thin coatings in micro-and nano-electromechanical devices.Even though high-resolution techniques-such as X-ray and focused ion beam (see the 2016 tutorial [?] by Chason & Guduru and the 2018 review article [1] by Abadias, Chason et al.)-are now available for in-situ characterization of residual stress, the conventional wafer curvature method based on Stoney's formula is still largely used, due to its simplicity.Residual stress needs to be checked since, if too large, it can have detrimental effects damaging the film and/or causing delamination.However, it can also be used for good.To an increasing extent, finely engineered residual stress states are put to use for developing well-controlled, small-scale architectures.More than a decade ago, Guidi & coworkers [7,6] fine-tuned the shape of a crystalline undulator by a patterned deposition of silicon nitride onto a silicon wafer.The electrons or positrons channeled into the periodically curved crystal follow its bent crystalline planes, thus emitting electromagnetic radiation.More recently, Backe & coworkers [2,20] obtained the same effect by varying periodically in space the concentration of the different components of a Si-Ge crystalline alloy epitaxially growing on a silicon substrate.In a very recent overview [19], Truong et al. present an innovative pathway allowing for unconventional 3D micro-and nano-systems to be produced out of thin planar films that popup and self-assemble when activated by suitably prestressed substrates or by external stimuli, either photonic or thermal.
In the rest of this paper, we shall use the terms 'film' and 'coating' as synonyms.

Preamble and outline of the paper
The bulk of this paper (Sections 2-5) is quite technical.For the benefit of the reader who might wish for a gentle introduction to the subject, we outline here our procedure and give a quick overview of our main results.This introductory information should be sufficient for the reader to grasp the mechanical content of Sections 6 and 7, before delving deeper into the convergence technicalities.
We consider a bilayered plate-shaped 3D elastic body, comprised of a thin substrate and a much thinner film.Our aim is to determine, via Γ -convergence, which scalings produce effective 2D pictures of the phenomenon of interest, namely, the self-bending triggered by the film-substrate mismatch.For convenience, we parameterize space with a system of Cartesian coordinates adapted to the reference shape of our system, and often identify points with their coordinates.In the reference configuration, the film and the substrate occupy, respectively, the right cylindrical regions where ω is a region in the x 3 " 0 coordinate plane, p h R is the film thickness, and q h R is the substrate thickness.Film and substrate adhere at their interface.The reference configuration, unstressed for the substrate, is stressed for the film, where the uniform membrane prestress T R ˝is tensile.Within the frame of linear 3D elasticity, we assume the elastic energy to be given by the following functional of the displacement field u : where C R is the (uniform) elasticity tensor of the substrate, L R is the (uniform) incremental elasticity tensor of the film, ∇u is the displacement gradient, and the strain Eu is the symmetric part of ∇u.All integrals on 3D domains, such as p Ω R and q Ω R in (3), are meant with respect to volume (omitted in notation).The devices motivating our modeling effort are typically characterized by the following dimensions: p h R « 0.1´0.3µm, q h R « 0.2´0.3mm, while the representative linear size L ω of the film-substrate interface ω is « 0.1 m.The two ratios prompt the idea of looking for an effective 2D model of the 3D functional (3) and bear evidence that in doing so one should account for (at least) two distinct scales of smallness.In Section 3, we shall pursue this idea by constructing a sequence of energy functionals tF ε u, parameterized by a small parameter ε, To stress the fact that this sequence is an instrumental mathematical artifact associated with the 'real' problem, the original functional in (3) and the thicknesses of the two layers in (4) are labeled with a superscript R, where R stands for 'real'.We start defining a sequence of domains parameterized by ε (Section 3.1).The reference shapes of the ε-film and the ε-substrate are, respectively, p Ω ε :" ωˆs 0, p h ε r , q Ω ε :" ωˆs´q h ε , 0 r , where, as suggested by (4), and the exponent s is to be tuned in such a way that p h ε R " p h R .In addition to s, we introduce three more parameters (Section 3.2), namely, the exponent t to scale the prestress T R ˝, the exponent ℓ to scale the film elasticity tensor L R (independently of the substrate elasticity tensor C, which we do not rescale): and the weighting factor η :" to be applied to the second term in (3), the only one affected by the skewsymmetric part of ∇u.In terms of the rescaled quantities, the energy functional (3) rewrites as The ε-sequence we construct is essentially obtained by replacing ε R with ε in (8) and applying the piecewise linear change of variables that maps p Ω ε , q Ω ε into the ε-independent cylinders p Ω :" ωˆs 0, L ω r , q Ω :" ωˆs´L ω , 0 r , respectively.The typical element of the sequence reads where p u, p H ε p u, and p E ε p u denote, respectively, the displacement, its gradient, and the strain in the film, as rescaled by the change of variables (9), while q u and q E ε q u denote, respectively, displacement and strain in the rescaled substrate.
The sequence tF ε u depends on several parameters.Before studying the variational limit of the sequence tF ε u as ε goes to zero, we preliminary sift the parameter space, excluding progressively portions of that space that cannot possibly yield limits relevant to our purpose, i. e., resulting in effective 2D models that exhibit no bending due to film prestress.For brevity, we shall say that we discard the parameter values that do not satisfy the self-bending requirement.On the contrary, the selected ones, and the reduced models they produce, will be said to possess the self-bending property.Since each of the three quadratic integrands in (10) is nonnegative, it is apparent that the energy can be minimized by a nontrivial displacement only if the linear integrand is nonzero.Hence, the preliminary selection criterion we apply is that this term should be preserved in the limit.We first prove (Lemma 1) that if t `ℓ `s ě 0, then (under the hypothesis that the elasticity tensors L, C are positive definite) sequences of displacements with bounded energy have L 2 -bounded scaled strains.Next, we argue (Section 4.2) that enforcing the self-bending requirement restricts the above inequality to an equality: t `ℓ `s " 0. The structure of the limit strains (characterized in Lemma 3), in combination with the selfbending requirement, forces q ě s ´ℓ .Then, we establish (Lemma 4) that the limit displacement fields p u, q u are of Kirchhoff-Love type, and determine the limit glueing conditions at the film-substrate interface.Jointly with the self-bending requirement, these conditions enforce s ď ℓ ď 3s .Combining the previous results, we prove (Lemma 5) that either ℓ " s or ℓ " 3s should hold.Finally, an analysis of the geometric stiffness due to the prestress (Lemmas 6, 7) leaves us with only two viable alternatives: either ℓ " s & q ě 2 , or ℓ " 3s & q ě 2`2s .
In Section 5, we compute the Γ -limit of the sequence tF ε u.The study of the limit behavior of the sequence of energies (Lemma 10) identifies the prospective Γ -limits, for which we prove the so-called liminf inequality.Enforcing the self-bending requirement on the limit energy rules out the second alternative (ℓ " 3s).So, we are left only with the set of parameters ℓ " s pñ t " ´2 ℓq & q ě 2 , for which we prove (Lemma 11) the so-called recovery sequence condition.In conclusion, our main result is encapsulated in Theorem 2, formulated under the assumption that L and C are monoclinic with respect to the x 3 " 0 coordinate plane, i.e., L αβγ3 " L α333 " C αβγ3 " C α333 " 0 pα, β, γ " 1, 2q.
Before presenting the limit energy we obtain in two variants-one for q " 2, the other for q ą 2 , qualitatively different between them-a comment on the two equalities ℓ " s , t " ´2 ℓ is in order.The discriminating parameter q, on the contrary, is best discussed afterwards.On one hand, the larger s, the thinner the film relative to the substrate (cf.( 6)), and hence the weaker the self-bending effect of the film-substrate mismatch.On the other hand, the larger ℓ, the stiffer the film relative to the substrate (cf.(7a)).It comes to no surprise that the distinguished limit(s) are obtained for ℓ " s .The other equality (t " ´2 ℓ ) strikes the balance between the first and the third term in (8) (cf.(7a)).We find it appropriate to transcribe here the limit energy (61) rescaled as in (5), i. e., multiplied by ε R : where all integrals are meant with respect to area (omitted in notation), the multiplier χ q differentiates whether q " 2 or q ą 2 χ q " 1 if q " 2 , χ q " 0 if q ą 2 , (11b) and the functions ξ R " pξ R α | α " 1, 2q and ξ R 3 , defined on ω, parameterize à la Kirchhoff-Love the displacement field defined on p ∇ and E are, respectively, the in-plane gradient and in-plane strain; L R and C R denote the planar elasticity tensors, as defined in Section 5.The fact that the contribution on the last line of (11a) is nonzero only for q " 2, as pointed out in (11b), makes clear the role played by q in the ε-sequence of energies (10).If this exponent is too small (q ă 2), the term ε t`q T ˝p H ε p u ¨p H ε p u asymptotically obliterates the linear term ε t T ˝¨p E ε p u , thus breaking the self-bending property.If it is too large (q ą 2), the self-bending property is guaranteed, but the energy contribution from the skew-symmetric part of the displacement gradient is lost in the limit.The critical value separating the two regimes (q " 2) establishes a delicate balance between the two terms, resulting in a Föppl-von Kármánlike effective theory.This theory and the standard linear theory stemming for larger values of q are presented in Section 6.Both of them produce (different) extensions of Stoney's formula, that we obtain and illustrate in Section 7.

Three-dimensional hyperelastic formulation
Let ω Ă R 2 be a bounded open set with smooth boundary Bω, and let B D ω be a subset, with positive length, of Bω.Let p h R and q h R be two positive real numbers, and We shall refer to p Ω R , q Ω R , and Ω R as the regions occupied by, respectively, the film, the substrate, and the bilayered plate in their reference configuration.We impose Dirichlet boundary conditions on The substrate is assumed to be linearly hyperelastic, homogeneous, and stress-free in the reference configuration, so that the Piola stress is with C R the elasticity tensor, u the displacement field on Ω R , and Eupxq :" sym∇upxq " 1 2 `∇upxq `∇upxq J ˘(12) the strain, i.e., the symmetric part of the displacement gradient ∇u.The elasticity tensor C R has the major and minor symmetries, i. e., C R A¨B " C R B ¨A for all symmetric double tensors A, B, and C R W " 0 for all skew tensor W .The film is assumed to be linearly hyperelastic and homogeneous, but stressed in the reference configuration.Let T R ˝be the film prestress and L R be its incremental elasticity tensor, so that the Piola stress is (see [8,13,17]) The incremental elasticity tensor L R has the major and minor symmetries.If body and surface forces are zero, the total energy of the system is (we recall from Section 1.2 that all integrals on 3D domains are meant with respect to volume, omitted in notation).We shall assume that there exist The prestress T R ˝is assumed to be planar, i.e., and tensile in its plane: i. e., there exists C T ą 0 such that Equations ( 15) and ( 16) imply For simplicity, we impose u " 0 on B D Ω R and assume that B D Ω R has positive two-dimensional Hausdorff measure.But in actual fact, the analysis that follows carries through also if B D Ω R is empty, provided that the Neumann boundary data on B N Ω R " BΩ R are normal to all infinitesimal rigid displacements and we quotient out by this finite-dimensional subspace of H 1 pΩ R ; R 3 q.This point being clarified, in the sequel we shall feel free to minimize the total energy F R on so that the 3D problem to solve is: Thanks to assumptions ( 14), ( 16), it can be shown (see, for instance, [12,15]) that problem (P R ) has a unique solution.Notably, this solution is not the null displacement, due to the prestress T R ˝, characterized by properties ( 15), ( 16).

The sequence of 3D energy functionals
In this section, we define a sequence of problems intimately related to problem (P R ).We construct this sequence by rescaling not only its domain, as is commonly done, but all the quantities involved, as proposed in [16].The sequence is then used to derive, via Γ -convergence, a 2D problem that approximates (P R ).In Section 3.1, after defining a sequence of ε-shrinking domains, we map them onto an ε-independent domain and compute the corresponding change of variables.Then, in Section 3.2, we reformulate the original problem (P R ) in terms of the transformed variables and define the sequence of ε-energy functionals that will be analyzed in Sections 4 and 5.

A sequence of domains
Let ε P s 0, 1s and Comparing ( 18) with ( 6), one has that q h ε R " q h R and p h ε R " p h R .We now let and make a change of variables so as to map the domains p Ω, q Ω, Ω ε onto the ε-independent domains p Ω :" p Ω 1 , q Ω :" q Ω 1 , and Ω :" Ω , and B D Ω are defined accordingly.Let be defined, respectively, by To each function u : Ω ε Ñ R 3 we associate the two functions defined, respectively, by The condition that u be continuous at the interface between p Ω ε and q Ω ε translates into the interface condition p up¨, ¨, 0q " q up¨, ¨, 0q.
Differentiating (20) yields We let p H ε p u :" ∇p u p p P ε q ´1 " p∇uq˝p p ε , q H ε q u :" ∇q u p q P ε q ´1 " p∇uq˝q p ε , (23a) Definitions (23a) are perhaps more transparent when written in components: 3.2 Rewriting the three-dimensional problem P R We shall now use the change of variables introduced in Section 3.1 to rewrite (P R ) on the domains p Ω and q Ω.In doing so, we will introduce four more scaling parameters that will play a major role in the subsequent analysis.Using the fact that p Ω R " p Ω ε R and q Ω R " q Ω ε R , we can use the mappings p p ε R and q p ε R to rewrite the energy F R on the domains p Ω and q Ω.After letting the energy F R puq defined in (13) rewrites as Next, we introduce three scaling parameters, i. e., the real numbers t, q, and ℓ (whose role and significance is discussed in Section 1.2) and, based on the energy scaling (5) rewrite the energy in the form where Let us remark that, once parameters t, q and ℓ are set, quantities T ˝, η, L, and C are problem data.With the energy representation (25) in mind, for every ε P s 0, 1s we introduce the energy functional its domain A being defined as In Sections 4 and 5, we study the behavior of the minimizers and the Γ -limit of the sequence tF ε u, as ε goes to zero.In doing so, we shall focus attention on the values of the four scaling parameters s, t, q, and ℓ that deliver Γ -limits accounting for the spontaneous bending of self-stressed bilayered plates.In the terms introduced in Section 1.2, we will discard the parameter values that do not satisfy the self-bending requirement.The selected ones, and the reduced models they produce, will be said to possess the self-bending property.

Compactness
In this section, we study the compactness of the energy functional (27).The compactness of sequences with finite energy will be proved by exploiting the elastic stiffness of both coating and substrate (cf.( 14)) on one hand and, on the other, the prestress stiffness due to the coating-substrate mismatch (cf.( 17)).

Elastic stiffness
Our first lemma, in which only the elastic stiffness plays a role, investigates under which condition sequences of displacements with finite energy have bounded rescaled strains.

The importance of the linear term
Here, by combining a mechanical and a mathematical argument, we obtain restrictions on the parameters t and ℓ introduced in Section 3.2.As we shall see, a key role is played by the linear term in the energy functional (27), i. e., The energy functional (27) is comprised of four terms, the one above linear and the other three quadratic and positive definite: cf. ( 14) and (17).Hence, the linear term is the only one that can make the total energy associated with a nonzero displacement from the reference configuration smaller than that associated with the null displacement.As a consequence, if this term vanishes as ε goes to zero, the limit functional will fail to satisfy the selfbending requirement.Rewriting the linear term in the form and using Lemma 1 we see that, if t `ℓ `s ą 0 , this term converges to zero as ε goes to zero.In conclusion, the self-bending requirement and Lemma 1 imply that t `ℓ `s " 0.

Prestress stiffness
Lemma 1 is altogether independent of the prestress stiffness, which we study in the next Lemma 2 Let F ε be the functional defined in (32) and let pp u ε , q u ε q P A be such that sup ε F ε pp u ε , q u ε q ă `8.

Limit of deformation measures
Lemma 1 and Lemma 2 give independent bounds on the rescaled strain and the rescaled displacement gradient.To compare these two bounds we shall use the following Korn-like inequality, whose proof slightly improves the one given in [14].
Theorem 1 There exists C K ą 0 such that, for every w P H 1 p p Ω; R 3 q with w " 0 on B D p Ω, one has and Proof.Given w P H 1 p p Ω; R 3 q, let r w P H 1 p p Ω; R 3 q be defined by r w :" p P ε w.By (23), it follows that p H ε w " ∇w p p P ε q ´1 " p p P ε q ´1∇ r w p p and p E ε w " p p P ε q ´1E r w p p P ε q ´1.
Thus, for ε small enough, it follows that By Korn's inequality, on a fixed domain there exists C K ą 1, independent of r w, such that Hence, Since C K ą 1, from (37)-(39) we deduce inequality (34).Inequality (35) follows at once from (34) and (37).In fact, since p P ε w " pw 1 , w 2 , ε 2p2s`1q w 3 q, we have Ωq .l Similarly one can prove that Ωq ě } q P ε w} 2 for every w P H 1 p q Ω; R 3 q with w " 0 on B D q Ω.From Lemmas 1 and 2 we deduce, by means of Theorem 1, the convergence results in the following Lemma 3 Let pp u ε , q u ε q P A be such that sup ε F ε pp u ε , q u ε q ă `8.Then, up to subsequences, for some p E P L 2 p p Ω; R 3ˆ3 q, q E P L 2 p q Ω; R 3ˆ3 q, and p H iα P L 2 p p Ωq for i " 1, 2, 3, and α " 1, 2.Moreover, p H βα " 0 for ℓ ą s ´q, p E βα " 0 for ℓ ă s ´q, and p H 3α " 0 for ℓ ą 5s `2 ´q, where α, β " 1, 2 .
Proof.The weak convergence of the subsequences follows at once from Lemmas 1 and 2. We now prove the moreover part.By Lemma 1 and Theorem 1, we have that p p H ε p u ε q βα ε ℓ´s and p p H ε p u ε q 3α ε ℓ´s ε 1`2s   are bounded in L 2 p p Ωq. Thus, from the equality we deduce that p H βα " 0 for ℓ ą s ´q.With a similar argument we prove that p H 3α " 0 for ℓ ą 5s `2 ´q.Since we prove also that p E αβ " 0 for ℓ ă s ´q .l From Lemma 3 we can find further restrictions on the exponent ℓ.In fact, by (31) and Lemma 3 we deduce that the linear term (30) converges to By means of (15) and of (26) we deduce that implying that L 0 p p Eq " 0 if ℓ ă s ´q.Then, motivated by the argument given in Section 4.2, we require that ℓ ě s ´q . (42)

Limit displacements
We shall now characterize the limit of the rescaled displacements.After letting KLp p Ωq :" tw P H 1 D p p Ω; R 3 q : pEwq iα " 0u,
Lemma 4 Let pp u ε , q u ε q P A be a sequence of displacements satisfying (29), repeated here for the reader's convenience: Then, up to subsequences for some p u P KLp p Ωq and q u P KLp q Ωq such that Moreover, with the notations introduced in Lemma 3, we have that p E αβ " pEp uq αβ , q E αβ " pEq uq αβ .
Proof.From (29), (35), and (40) we deduce that and, possibly by passing to a subsequence, we have that for some p u P H 1 D p p Ω; R 3 q and q u P H 1 D p q Ω; R 3 q.Since, due to (23), we obtain Ωq, implying that p u P KLp p Ωq.We show similarly that q u P KLp q Ωq and p q E ε q u ε q αβ á pEq uq αβ in L 2 p q Ωq.
Since pp u ε , q u ε q P A and we have that p q P ε q u ε q α p¨, ¨, 0q " ε ℓ´s p p P ε p u ε q α ε ℓ´s p¨, ¨, 0q, and p q P ε q u ε q 3 p¨, ¨, 0q " ε ℓ´3s p p P ε p u ε q 3 ε ℓ´s p¨, ¨, 0q.By (21) and the continuity of the trace with respect to weak convergence in H 1 , we deduce all the properties of the traces of p u and q u in the lemma statement.l Since s ě 0, ℓ ą 3s implies ℓ ą 0, so that, according to Lemma 4, q u 3 p¨, ¨, 0q " 0, q u α p¨, ¨, 0q " 0.
Since q u P KLp q Ωq, these equalities imply that q u " 0 if ℓ ą 3s, from which we conclude that, if ℓ ą 3s, the elastic stiffness of the substrate becomes asymptotically negligible with respect to that of the coating, so that their mismatch is accommodated by stretching the relatively soft substrate, without any bending.Consequently, from now on we consider that On the other hand, for ℓ ă s we have that p u α p¨, ¨, 0q " 0, p u 3 p¨, ¨, 0q " 0.
Since p u P KLp p Ωq, it follows that p u " 0 and hence p E αβ " 0, i. e., L 0 p p Eq " 0 (see (41)).Then, in order to satisfy the self-bending requirement, from now on we require also that ℓ ě s .
‚ For ℓ " 3s, we have p u α p¨, ¨, 0q " 0 and q u 3 p¨, ¨, 0q " p u 3 p¨, ¨, 0q, implying q ξ α " 1 2 L ω B α q ξ 3 , q ξ 3 " p ξ 3 and hence (48).l We observe that, if s ă ℓ ă 3s, substrate and coating become asymptotically disconnected.Such a limit obviously violates the self-bending requirement, so we discard it by requiring that either ℓ " s or ℓ " 3s . (50) The consequences of Lemma 2 will now be studied in Lemma 6 Let pp u ε , q u ε q P A be a sequence of displacements satisfying (33), i. e., Then, up to subsequences, for some p w α P L 2 p0, 1; H 1 D pωqq and p w 3 P H 1 D pωq, such that B 3 p w 3 " 0 and Moreover, with the notations introduced in Lemma 3, we have that p H iα " B α p w i .
Proof.From (33) we have that Hence, by Poincaré's inequality, we have that up to a subsequence and for some p w P L 2 p0, 1; and ℓ ě s ´q, combined with s ě 0, implies that 3s `ℓ `q `2 ą 0, we have It follows that B 3 p w 3 " 0 and Then, from Lemma 4 and the equality we deduce (51).l The sequences of Lemma 4 are compared with those of Lemma 6 in the next Lemma 7 With the notation of Lemma 4 and Lemma 6, we have Proof.From the equality (52) follows, while the equality Recalling the dichotomy (50), we summarize the results obtained for ℓ " s and for ℓ " 3 s, respectively, in Section 4.5.1 and in Section 4.5.2.

The limit energy functional
In this section, we study the Γ -limit of the functional F ε defined in (32).
After introducing the relevant definitions and notations, we first prove the socalled liminf inequality for ℓ " s and ℓ " 3s (Lemma 10).Analyzing the results (Sections 5.1 and 5.2) leads us to discard the case ℓ " 3s, since it fails to satisfy the self-bending requirement.In Section 5.3, we prove the recovery sequence condition for the remaining case ℓ " s (Lemma 11).Lastly, the results obtained in the liminf and recovery sequence lemmas are recapitulated in Theorem 2. Let Let F : A lim Ñ R be defined as where for all A P R 2ˆ2 sym , with adb :" 1 2 pabb`bbaq for all a, b P R 3 and e 3 " p0, 0, 1q.
Lemma 10 [ Liminf inequality ] Let F ε and F be defined as in (32) and (58), respectively.For every pp u, q uq P A lim and every sequence pp u ε , q u ε q P A such that we have that lim inf εÑ 0 Proof.Without loss of generality, we may suppose that lim inf εÑ 0 since otherwise there is nothing to prove.Then, from Lemma 3 it follows that for some p E P L 2 p p Ω; R 3ˆ3 q, q E P L 2 p q Ω; R 3ˆ3 q, and p H iα P L 2 p p Ωq, with i " 1, 2, 3 ; α " 1, 2 .By convexity (cf.( 14), ( 17) and ( 26) where the second inequality uses (59), while the final equality follows from Lemmas 8 and 9. l

The lower bound for ℓ " s
Let ∇ and E be, respectively, the in-plane gradient and the in-plane strain.From (55) we obtain and p H 3α " χ q `∇ξ 3 ˘α with χ q " 1 if q " 2 , χ q " 0 if q ą 2 .
Substituting ( 55) and ( 60) in (58), we obtain the limit energy in terms of the Kirchhoff-Love displacement fields ξ, ξ 3 , defined on ω: after integration over the thickness, reads where the membrane energy is accounted for in the first line and the bending energy in the last line, whereas the middle line couples in-plane stretching and flexure.

The lower bound for ℓ " 3s
Recall from 5.1 that ∇ and E are, respectively, the in-plane gradient and the in-plane strain.From (57) we obtain and p H 3α " r χ q `∇ξ 3 ˘α with r χ q " 1 if q " 2s `2 , r χ q " 0 if q ą 2s `2 .(62b) Substituting ( 57) and ( 62) in (58), we obtain the limit energy in terms of the Kirchhoff-Love displacement fields ξ, ξ 3 , defined on ω: which, after integration over the thickness, reads The transverse displacement ξ 3 enters this energy only in two non-negative terms, depending on its first and second gradients.Hence, its minimizer has constant transverse displacement, patently breaking the self-bending property.
In conclusion, we discard the case ℓ " 3s and concentrate on the only case left, namely,

The upper bound for ℓ " s
Up to this point, our results are independent of the symmetry group of the elasticity tensors L and C. For simplicity, we shall now assume that they both belong to the monoclinic group with x 3 " 0 symmetry plane, i. e., Making use of this hypothesis in (59) gives for all A P R 2ˆ2 sym , with e 3 " p0, 0, 1q and together with analogous results for C .
Lemma 11 [ Recovery sequence ] Let ℓ " s, q ě 2, and F ε , F be defined by (32) and (58), respectively.For every pp u, q uq P A lim there exists a sequence pp u ε , q u ε q P A such that and lim εÑ0 F ε pp u ε , q u ε q ď Fpp u, q uq.
Proof.Let pp u, q uq P A lim .Then, by Lemma 8 there exist ξ P H 1 D pω; R 2 q and we can find sequences p ψ ε P H 1 D pωq and q Then, pp u ε , q u ε q P A and Moreover, From ( 68) and (35) we have that p P ε p u ε is bounded in H 1 p p Ω; R 3 q, so that from (66) we deduce that p P ε p u ε á p u in H 1 p p Ω; R 3 q, as stated in (65) 1 .Analogous calculations done for the substrate lead to from which it follows that The weak convergence in (65) 2 follows from (40).
We are now ready to take the limit of F ε pp u ε , q u ε q.Using (32) and ( 67)-( 69), we obtain Fpp u, q uq :" lim where only the second equality relies on (64).l Lemmas 10 and 11 are summarized in the following Theorem 2 Let ℓ " s, t " ´2s, and q ě 2.Then, the sequence {F ε } defined by as in (27)-(28) Γ -converges to F as given in (70) in the following sense: 1. [ Liminf inequality ] For every pp u, q uq P A lim and every sequence pp u ε , q u ε q P A such that we have lim inf εÑ0 F ε pp u ε , q u ε q ě Fpp u, q uq.

[ Recovery sequence ]
For every pp u, q uq P A lim there is a sequence pp u ε , q u ε q P A such that and lim εÑ0 F ε pp u ε , q u ε q ď Fpp u, q uq.
6 Effective two-dimensional theories The limit energy, first given in (58) as a function of the displacement fields p u, q u (defined, respectively, on p Ω and q Ω ), is rewritten in (61) in terms of the Kirchhoff-Love displacement fields ξ, ξ 3 defined on ω.Both representations make use of the 'fictitious' parameters introduced in (26) for constructing the sequence of ε-energy functionals (27).Here, keeping in mind that ℓ " s (cf.(63)) and hence t " ´2s, by way of definitions ( 18) and (26) we recover the corresponding 'real' quantities and we express the limit energy (61) in terms of them.
To achieve this goal, we recall from ( 5) and ( 25) that the energy is scaled in such a way that ε R F ε R " F R , take into account the different scalings of the gradient in the film and in the substrate (22), and define the 'real' Kirchhoff-Love displacement fields as guided by Lemmas 4 and 5, where we learn from (46) 2 that ξ α and ξ 3 parameterize the Kirchhoff-Love displacement field associated with q u, while (43) 2 tells us that q u α and q u 3 are, respectively, the weak limits in H 1 of q u ε α and ε ´1 q u ε 3 .Our final result is then with χ q " 1 for q " 2 and χ q " 0 for q ą 2 .Functional(s) (72) encapsulate(s) the linear mechanics of an elastic bilayered plate comprised of a thin film, modeled as a tensionally prestressed extensible membrane, and a comparatively much thicker, prestress-free substrate.The distinguished limit we obtain for q " 2 captures à la Föppl-von Kármán the effect of the tensile prestress on "moderate" plate rotations.In Section 7, both theories are applied to a simple problem, whose solution(s) deliver(s) Stoney-like formulas, consistently approaching the original Stoney formula for extremely thin coatings of stiffness comparable to that of the substrate.

An illustrative plane-strain problem
In order to illustrate the 2D effective theory encapsulated in the energy functional (72), we assume that the region ω in the x 3 " 0 coordinate plane defining the reference shape of the bilayered plate is the rectangle and that the tensile pre-stress T R ˝is uniaxial: For simplicity, we also assume that both film and substrate are elastically isotropic, so that either L R or C R is fully characterized by two independent moduli, such as the Young modulus E and the Poisson ratio ν.We shall look for plane-strain minimizers of the energy (72), supposing that W is large enough for this to be sensible.Under this hypothesis, the Kirchhoff-Love displacement fields (71) can be parameterized with two scalar functions of the x 1 coordinate, v and w, as follows: Henceforth, we write x for x 1 and introduce a few notations adapted to the problem at hand.We define the thickness ratio (nonnegative and typically ! 1) and the membrane stiffness ratio (nonnegative and typically « ϑ ) where x M and | M denote the plane-strain moduli of the film and the substrate, respectively: Since no confusion may arise from now on, for the sake of clarity we introduce the simplified notations and write accordingly ϑh for p h R .Then, substituting (75) in the effective energy (72) and integrating over the width of ω, we obtain where a prime denotes differentiation with respect to the x coordinate and σ ˝is the prestress intensity introduced in (74).The energy (79) is the sum of a (positive definite symmetric) quadratic form and a linear form of the pair pv 1 , w 2 q, plus a (positive) term quadratic in w 1 (if χ q ‰ 0).The quadratic form encodes the constitutive equations where N is the membrane traction (force per unit length) and M is the bending moment (couple per unit length).Be it noted that stress moment M is taken with respect to the origin, located on the film-substrate interface, whence the significant elastic coupling between the membrane strain ϵ " v 1 and the bending curvature κ " w 2 .The stationarity condition of the energy functional (79) delivers the Euler-Lagrange equations v p2q ´α w p3q " 0, (81a) where equipped with the Neumann boundary conditions M 1 p¯Lq " χ q ϑ h σ ˝w1 p¯Lq.and to combine (81a) with (84), rewritten in the form v p2q ´α w p3q " 0, (88a) In what follows, we present separately the solution to (88) under boundary conditions (83) for χ q " 0 (Section 7.1) and χ q " 1 (Section 7.2).
is stiffer than that for q ą 2. In the sample problem we are discussing, (97) reduces to the integral Interestingly, the coupling between v and w embodied in (88) with χ q ‰ 0 is such that the L 2 norm of w 1 can be depleted only through a subtle redistribution of the rotation from the center towards the ends of the plate.Figure 1 illustrates the solutions we obtain for two sample bilayers, sharing the same half-length L " 50 mm, mismatch strain ϵ m " 5 ¨10 ´4, and membrane stiffness ratio ζ " 10 ´3, while having different substrate thickness: h " 0.1 mm and h " 0.2 mm, respectively.According to (92), the thicker bilayer has characteristic length ℓ ˝" 82 mm, the thinner one ℓ ˝" 41 mm.The solutions (89) for q ą 2 and (93) for q " 2 are represented by dashed and solid lines, respectively.
As expected, the transverse diplacement and the curvature are larger for q ą 2 (dashed lines) than for q " 2 (solid lines), the stiffness being increased by the addition of the positive term (98) to the energy.More interestingly, the average curvature is appreciably smaller than the Stoney curvature κ St , which is indistinguishable from the value given by (91) 2 and (96) on the scale of the figure.This difference is especially significant for the applications of Stoney-like formulas to the assessment of the residual stress in a wafer, where what is really measured-as done by Stoney himself-is the differential deflection between two measuring stations, from which an average curvature is obtained.With respect to the constant curvature obtained for q ą 2, the average curvature predicted by the model with q " 2 is reduced by 11% for the thicker bilayer (left) and as much as 31% for the thinner bilayer (right).In comparison, the correction to the Stoney formula embodied in (91) 2 is negligible, being of the order of a few per thousands.

Concluding remarks and future outlook
In this paper, we have addressed the problem of correlating the spontaneous bending of a bilayered plate with the self-stress due to the contrast between the material properties of the two layers.We have rationally deduced two Stoney-like formulas: one that includes only a minor correction to the original Stoney formula, similar to others in the literature (see, e. g., [4]), and another which is new and predicts a significantly non-constant curvature already in the linear range.
These formulas have been obtained from two reduced 2D models, deduced via Γ -convergence from a sequence of 3D models of self-stressed linearly elastic bilayers.This construction makes the main body of the paper and constitutes the most original part of it.The Γ -convergence technique we use is nonstandard, the sequence of variational problems being constructed by suitably rescaling not only their domain but also the material parameters involved.This allows us to fully exploit the contrast between the orders of magnitude of the different quantities involved, and to tune the scaling parameters in such a way so as to obtain reduced models appropriate to the specific application, i. e., effectively describing the self-bending phenomenon.
Finally, to illustrate and compare the two 2D models, we have applied them to the study of a rectangular isotropic bilayer, uniaxially prestressed under plain-strain conditions.It would be interesting, and relevant to practical applications, to apply them to more general geometries and material symmetries, as well as prestress conditions.
It has been observed (see, e. g., [19]) that prestressed narrow rectangular bilayers do not only self-bend but can also self-twist.Studying this behavior with a Γ -convergence technique is stimulating, since it would imply one more smallness parameter.The outcome could consist in effective beam-like 1D models, deduced in cascade from the 2D models discussed in the present paper.
and L A ¨A " L `A `fL pAq e 3 b e 3 ˘¨`A `fL pAq e 3 b e 3 ˘(64b) (79)), which clearly penalizes the rotation w 1 , all prefactors being positive.

Fig. 1 :
Fig.1: Transverse displacement (top) and bending curvature (bottom) of the thicker (left) and the thinner bilayer (right).The solutions (89) for q ą 2 and (93) for q " 2 are represented by dashed and solid lines, respectively. p ). Modulo an immaterial rigid displacement, the solution to (88) with boundary conditions (83) reads the mismatch strain ϵ m and the Stoney curvature κ St , defined as