Convergence of Equilibria for Incompressible Elastic Plates in the von K\'arm\'an Regime

We prove convergence of critical points $u^h$ of the nonlinear elastic energies $E^h$ of thin incompressible plates $\Omega^h=\Omega \times (-h/2, h/2)$, which satisfy the von K\'arm\'an scaling: $E^h(u^h)\leq Ch^4$, to critical points of the appropriate limiting (incompressible von K\'arm\'an) functional.


Introduction and the main result
In this paper we prove convergence of critical points of the nonlinear elastic energies on thin incompressible plates in the von Kármán scaling regime, to critical points of the appropriate limiting (incompressible von Kármán) functional.

Elastic energy of thin incompressible plates.
Let Ω ⊂ R 2 be an open, bounded, simply connected domain. For h > 0, define Ω h to be the 3d plate with the midplate Ω and thickness h: The elastic energy of a deformation u h ∈ W 1.2 (Ω h , R 3 ) of the homogeneous plate Ω h , scaled by its unit thickness, is given by: while the total energy, relative to the external force with the density f h ∈ L 2 (Ω h , R 3 ), is: The elastic energy density W in : R 3×3 → [0, ∞] in (1.1) is assumed to be infinite at compressible deformations: The effective density W : R 3×3 → [0, ∞) above, which acts when det F = 1, is required to satisfy the following conditions: (vi) (regularity) W is of class C 1 on R 3×3 + . (vii) (local regularity) W is of class C 2 in a small neighborhood of SO(3).
The growth conditions in (iv) and (v) will be crucial in the present analysis. Condition (iv) has been introduced in the context of [6] and it allows to use the nonlinear version of Korn's inequality [5], ultimately serving to control the local deviations of the deformation u h from rigid motions, by the elastic energy I h (u h ). Condition (v) has been introduced in [1] (see also [2]) in the context of inner variations, in order to control the related strain in terms of the energy. Both conditions are compatible with other requirements above.
1.2. Notation. Given a matrix F ∈ R n×n , we denote its trace by Tr F and its transpose by F T . The symmetric part of F is given by sym F = 1 2 (F + F T ). The cofactor of F is the matrix: cof F , where [cof F ] ij = (−1) i+j detF ij and eachF ij ∈ R (n−1)×(n−1) is obtained from F by deleting its ith row and jth column. The identity matrix is denoted by Id n .
In what follows, we shall use the matrix norm |F | = (Tr(F T F )) 1/2 , which is induced by the inner product: F 1 : F 2 = Tr(F T 1 F 2 ). To avoid notational confusion, we will often write F 1 : F 2 instead of F 1 : F 2 . In general, 3 × 3 matrices will be denoted by F and 2 × 2 matrices will be denoted by F ′′ . Unless noted otherwise, F ′′ is the principal 2 × 2 minor of F .
Finally, by C k b (R n , R s ) we denote the space of continuous functions whose derivatives up to the order k are continuous and bounded in R n .
1.3. The limiting energy. The following 2d energy functional has been rigorously derived in [10] as the Γ-limit of the scaled incompressible energies h −4 I h in (1.1), when h → 0: acting on couples w ∈ W 1,2 (Ω, R 2 ), v ∈ W 2,2 (Ω, R). The fields (w, v) may be identified as the in-plane and the out-of-plane displacements, respectively. Roughly speaking, any minimizing sequence of e 3 and´Ω f = 0, will have the structure: among all rotations R, while c h ∈ R 3 are constant translation vectors. Moreover, (w, v,R) minimize the following total limiting energy: A precise formulation of the statements above can be found in [9].
The energy in (1.4) is the incompressible version of the von Kármán functional, which has been derived (for compressible case, i.e. without the assumption that det ∇u h = 1) by means of Γ-convergence in [6]. The quadratic forms Q in 2 differ from the standard Q 2 in [6] in as much as minimization in (1.5) below is taken over the out-of-plane stretches which preserve the incompressibility constraint. Namely, Q in 2 in (1.4) are given as: (1.5) Both forms Q above are positive semidefinite, and strictly positive definite on symmetric matrices. We also introduce the linear operators L in 2 : R 2×2 → R 2×2 and L 3 : R 3×3 → R 3×3 such that: Note that symmetric operators L are uniquely given by: L(F 1 ) : 1.4. Critical points and the incompressible inner variations. Following [2], we now define the critical points u h of the functionals J h in (1.2) with respect to inner variations, that is requesting that the derivative of J h at an incompressible equilibrium u h be zero: This requirement is translated into the following condition: We refer to section 2 for the derivation and discussion of (1.7). Let us only note now that the incompressible inner variations: used in definition of minimizers of J h , and also they replace the inner variations u h ǫ (x) = u h (x) + ǫφ(u h (x)) considered in [2] and [11] in the compressible case. 1.5. The main result. The following is our main result: Assume that: for a constant C > 0 independent of h. Then there exists a sequence of proper rotationsR h ∈ SO(3), and translations c h ∈ R 3 , such that for the renormalized deformations: the following convergences hold, up to a subsequence in h, as h → 0: (iii) For the scaled out-of-plane displacements: For the scaled in-plane displacements: The limiting displacements (w, v) solve the following Euler-Lagrange equations of the functional (1.4), expressed in the variational form:
We note that (1.8) are automatically satisfied by any minimizing sequence of u h of the total energy J h , under the assumption that f h (x) = h 3 f (x ′ )e 3 [6]. Also, (1.7) holds for every minimum of J h (see Theorem 2.3), and the assertions (i) -(v) are then a direct consequence [10] of the fact that 1 h 4 J h Γ-converges to J . In general, Γ-convergence does not assure that a limit of a sequence of equlibria is an equilibrium of the Γ-limit. In the present situation, this turns out to be the case.
1.6. Relation to other works. Our work is largely inspired by [11] and [10]. To put it in a larger perspective, recall that one of the fundamental questions in the mathematical theory of elasticity has been to rigorously justify various 2d plate models present in the engineering literature, in relation to the three-dimensional theory. This goal has been largely accomplished in [6], where a hierarchy of limiting 2d energies has been derived; the distinct theories are differentiated by their validity in the corresponding scaling regimes h β , β ≥ 2, i.e. in presence of assumption (1.8) where h 4 is replaced by h β .
Under the additional incompressibility constraint, the works [3,4] proved compactness properties and the Γ-convergence of the functionals 1 h β I h as in (1.1), for the so-called Kirchhoff scaling β = 2, while [10] treated the case β = 4 including as well a more complex case of shells when the midsurface Ω is a generic 2d hypersurface in R 3 . In view of the fundamental property of Γ-convergence, it follows that the global almost-minimizers of the energies (1.2) converge to the minimizers of the limiting energy (given by (1.4) in the von Kármán regime).
Regarding convergence of stationary points for thin plates, the first result has been obtained in [12] under the von Kármán scaling β = 4 (see also [7] for an extension to thin shells). These results relied on the crucial assumption that the elastic energy density W is differentiable everywhere and its derivative satisfies a linear growth condition: |DW (F )| ≤ C(|F | + 1). This assumption is contradictory with the physically expected non-interpenetration condition, and subsequently it has been removed in [11] and exchanged with Ball's condition (1.3), while the equilibrium equations have been rephrased in terms of the inner variations. In the present paper we follow the same approach; indeed the concept of inner variations comes up naturally in the context of incompressible elasticity.
To conclude, we now comment on the isotropic case. For an isotropic energy density W with the Lamé constants λ and µ, the Euler-Lagrange equations (1.12) -(1.13) of (1.4) are: where v is the out-of-plane displacement, while the in-plane displacement w can be recovered through the Airy stress potential Φ, by means of: As expected, the system (1.14) can be now obtained as the incompressible limit, i.e. when passing with the Poisson ratio ν → 1 2 , of the classical (compressible) von Kármán system: is the Poisson ratio, and B = S 12(1−ν 2 ) is bending stiffness. By the change of variable Φ = 2µΦ 1 one can eliminate the parameter µ entirely and write (1.14) in its equivalent form:

Incompressible inner variations and critical points
Following [2], we want to define the critical points u h of the functionals J h in (1.2) by taking inner variations. That is, we request that the derivative of J h at an incompressible equilibrium u h be zero along all curves ǫ → u h ǫ of incompressible deformations of Ω h having the form: Assuming sufficient smoothness of Φ, the above immediately implies: On the other hand, any divergence-free vector field φ generates a path of incompressible deformations. We recall this standard fact below, for the sake of completeness.
We are now ready to derive the equilibrium equations (1.7). The result is essentially similar to Theorem 2.4 [2], which dealt with the compressible inner variations u h ǫ = u h (x) + ǫφ • u h of a deformation u h with clamped boundary conditions. The growth condition (1.3) will be crucial in passing to the limit in the nonlinear term in J h , to which end we are going to use the following Lemma from [2]: and |A − Id| < γ, then: Proof. For the notational convenience, in what follows we drop the index h and write U instead of Ω h , which stands now for a fixed open bounded domain in R 3 . It is easy to notice that: It directly implies that: To treat the nonlinear term, consider: Since the integrand below converges to 0 pointwise by (2.4), and it is bounded by the function 2 ∇φ L ∞ |DW (∇u)(∇u) T | which is integrable in view of (1.3), we obtain: Therefore, the left hand side in (2.5) converges to 0 as well. This completes the proof.

The equilibrium equation (1.7)
In this section, we review several facts from [6] and [11], to set the stage for a proof of Theorem 1.1 and to rewrite the equation (1.7) using the change of variables (1.9).
The first crucial step in the dimension reduction argument of [6] is finding the appropriate approximations of the deformations gradients u h . Under the sole assumption: an application of a nonlinear verion of Korn's inequality [5], yields existence of rotation fields in Ω, so that: Recall that Ω 1 = Ω×(− 1 2 , 1 2 ) is the common domain of the rescaled deformations y h (x ′ , x 3 ) = (R h ) T u h (x ′ , hx 3 )− c h , and the typical point in Ω 1 is denoted by x = (x ′ , x 3 ). Then, the detailed analysis in [6] shows that convergences in (i) -(iv) of Theorem 1.1 hold, as a consequence of (1.8) implying (3.1). The constant rotationsR h ∈ SO(3) are given by:R where the orthogonal projection P SO(3) onto SO(3) above is well defined; see also [8] for detailed calculations. Further, there holds: and upon defining the matrix fields A h ∈ W 1,2 (Ω, R 3×3 ): it also follows that: The same convergence holds strongly in L q (Ω, R 3×3 ) for each q ≥ 1.

Proof. By (3.2), (3.3), and applying the Poincaré-Wirtinger inequality on segments {x
Together with (1.10), the above inequality implies the second assertion in (3.6). The first assertion follows then directly in view of (1.11).
Define the strain G h ∈ L 2 (Ω 1 , R 3×3 ) and the scaled stress E h ∈ L 1 (Ω 1 , R 3×3 ) as: We now gather the fundamental properties of E h and G h from [11], that will be used in the sequel.
where G is the limiting strain whose principal 2 × 2 minor G ′′ satisfies: with: is symmetric, and there holds: Moreover, calling χ h the characteristic function of B h , we have: The below more convenient form of the equilibrium condition will be repeatedly used in the proof of Theorem 1.1.

Lemma 3.3. Condition (1.7) is equivalent to:
which satisfies ψ ∈ C 1 b and divψ = 0, and moreover: Use now (1.7) with the divergence-free test function ψ: The formula (3.13) now follows directly, in view of:

Identification of the operators in (1.12) -(1.13)
Lemma 4.1. Let G ∈ R 3×3 and a symmetric matrix E ∈ R 3×3 satisfy: Then: Proof. Since L and Q depend only on the symmetric parts of their arguments, we may without loss of generality assume that G is symmetric.
Firstly, by definitions in (1.5), (1.6), it follows that for every F ′′ ∈ R 2×2 there is a unique tangential minimizer d = d(F ′′ ) ∈ R 2 , in the sense that: The second identity above is just the Euler-Lagrange equation for the minimization in (1.5). By convexity of this minimization problem, it also follows that d is linear: Observe now that: where we repeatedly used the assumptions on G and E, and (4.2). Consequently, by uniqueness of the minimizer d, it follows that: Take any F ′′ ∈ R 2×2 . By (4.2) and (4.3), we see that: Expanding the above and removing Q 2 (G ′′ ) and Q 2 (F ′′ ) from both sides, we obtain: by (4.4) and assumptions on E and G. The expression (4.1) follows now directly.
Therefore, recalling Lemma 3.2 (iii), we observe that the limiting stress and strain satisfy the assumptions of Lemma 4.1 pointwise almost everywhere. We now record the following simple conclusion which will be used in deriving the Euler-Lagrange equations (1.12), (1.13).

Two further properties of G and E
In this section we derive the two fundamental properties of the incompressible stress and strain, allowing for pointwise application of Lemma 4.1, and ultimately leading to formulas in (4.6).
Lemma 5.1. The limiting strain G(x) is traceless, for almost every x ∈ Ω 1 .
We now prove the remaining property of the strain E in (4.5). The strategy of proof is the same as in the later proofs of the Euler-Lagrange equations; we will apply the equilibrium equation (3.13) to appropriate test functions φ h , such that after passing to the limit with h → 0 only some chosen terms will survive, yielding the week formulation of (4.5). One difficulty with (3.13) is that it only allows for globaly bounded φ h . For this reason, following [11], we introduce a family of truncation functions θ h which coincide with the identity on intervals (−ω h , ω h ) with a suitable rate of convergence of ω h → ∞.
Lemma 5.2. Let {ω h } be a sequence of positive numbers, increasing to +∞ as h → 0. There exists a sequence of nondecreasing functions θ h ∈ C 2 b (R, R) with the following properties: Proof. One may take: Proof. 1. Let η = (η 1 , η 2 ) ∈ C 2 b (R 3 , R 2 ) be a given test function, and define: Since ∂ 3 η 3 = −div η, the following test functions φ h ∈ C 1 b (R 3 , R 3 ) are divergence-free: and denoting ∇ tan the gradient in the tangential directions e 1 , e 2 , we have: The truncations θ h are chosen as in Lemma 5.2 and such that:

2.
Applying the equilibrium equation (3.13) with φ = φ h , we obtain: ). (5.5) Now, we will discuss the convergence as h → 0 of each term in (5.5). The first term converges to 0, 3. The second term in (5.5) when integrated over Ω 1 \B h , goes to 0 in view of (3.11) and of the pointwise boundedness of (θ h ′ y h 2). On the other hand, the limit of this integral over B h is the same as the limit of: because of (3.3). We now conclude that the integrals in (5.6) converge to: This follows by recalling (3.12) and observing that: The fourth term in (5.5) is bounded by: and it converges to 0 by (3.11), (3.12), (5.4) and the boundedness of . Finally, both terms in the right hand side of (5.5) are bounded by: which clearly converges to 0. Above, we used (5.2) and (5.4).

5.
In conclusion, passing to the limit with h → 0 in (5.5), results in: We now reproduce an argument from [11], in order to deduce that E 13,23 = 0. Take an arbitrary φ ∈ C 2 c (Ω, (Ω), and define: , and thus by (5.8) we obtain: Passing to the limit with k → ∞, it follows that: which concludes the proof.
6. Derivation of the first Euler-Lagrange equation (1.12) be a given test function, and let η 3 (x ′ ) = −div η(x ′ ). Given θ h as in Lemma 5.2, with: Denoting ∇ tan the gradient in the tangential directions e 1 , e 2 , we have:

2.
Applying the equilibrium equation (3.13) with φ = φ h , we obtain: Now, we will check convergence as h → 0 of each of the four terms in the identity (6.2). Regarding the first term, it converges to 0 when integrated over Ω 1 \ B h , by (3.11) and by the pointwise boundedness of (5.2). On the other hand, the limit of this integral over B h is the same as the limit of: because of the convergence in (3.3). Now, the limit of integrals in (6.3) equals: in view of (3.12) and: where we apply (3.7), and then (3.7) and (6.1) to conclude the convergence of both terms in the right hand side of the above displayed expression to 0.
Finally, the right hand side of (6.2) converges to 0 as well, as it is bounded by: In conclusion, passing to the limit with h → 0 in (6.2) we obtain: and thus the Euler-Lagrange equation (1.12) follows directly, in view of (4.6) and the density of test functions η as above in W 1,2 (Ω, R 2 ).

Derivation of the second Euler-Lagrange equation (1.13)
Lemma 7.1. For every η 3 ∈ C 3 b (R 2 , R), it follows that: Applying the equilibrium equation (3.13) with φ = φ h , we obtain: Recall that the tensor field A h in (3.4) is defined as: and therefore the left hand side of (7.2) can be written as: Let the sets B h be defined as in Lemma 3.2 (iv), for some exponent γ ∈ (0, 1). The first two terms in (7.4), when considered on Ω 1 \ B h , converge to 0 because they are bounded by: in view of (3.11) and |A h | ≤ C h . On the other hand, the same two terms while on B h , converge to: where we used the convergence (3.12) and the following strong convergences in L 3 (Ω 1 ): of A h to A by (3.5), of (R h ) TRh to Id by (3.3), and of ∇η 3 (y h ′ ) to ∇η 3 (x ′ ) in view of the Sobolev embedding and the strong convergence in W 1,2 (Ω 1 , R 2 ) in (3.6).