A hierarchy of multilayered plate models

TL;DR

This paper derives a hierarchy of multilayered plate models from 3D nonlinear elasticity using $ extGamma$-convergence, capturing effects of heterogeneity, pre-stress, and different elastic regimes in thin films.

Contribution

It introduces a systematic derivation of layered plate theories from 3D elasticity, including effects of heterogeneity, pre-stress, and multiple elastic regimes, using $ extGamma$-convergence.

Findings

Derivation of linearised Kirchhoff, von Kármán, and linear plate theories with spontaneous curvature.

Effective elastic constants expressed via moments of material properties.

Inclusion of pre-stress effects in the derived plate models.

Abstract

We derive a hierarchy of plate theories for heterogeneous multilayers from three dimensional nonlinear elasticity by means of $\Gamma$-convergence. We allow for layers composed of different materials whose constitutive assumptions may vary significantly in the small film direction and which also may have a (small) pre-stress. By computing the $\Gamma$-limits in the energy regimes in which the scaling of the pre-stress is non-trivial, we arrive at linearised Kirchhoff, von K{\'a}rm{\'a}n, and fully linear plate theories, respectively, which contain an additional spontaneous curvature tensor. The effective (homogenised) elastic constants of the plates will turn out to be given in terms of the moments of the pointwise elastic constants of the materials.

Equations

$$E_h(\tilde{y})={\int }_{{\Omega }_h}W(z,\nabla \tilde{y}(z))\mathrm{d}z.$$

$$J_h(y)={\int }_{{\Omega }_1}W(x,{\nabla }_{h}y)\mathrm{d}x,$$

$$J_h^{\beta }(y)=\frac{1}{h^{\beta }}{\int }_{{\Omega }_1}W({\nabla }_{h}y).$$

$$I_{\mathrm{Ki}}(y)=\frac{1}{24}{\int }_{\omega }Q(\mathrm{II}-a_1\mathrm{Id})-a_2\mathrm{d}x,$$

$${\mathcal{I}}_{\mathrm{vK}}^{\theta }(u,v)=\frac{\theta }{2}{\int }_{\omega }{Q}_2({\nabla }_{s}u+\frac{1}{2}\nabla v\otimes \nabla v)\mathrm{d}x+\frac{1}{24}{\int }_{\omega }{Q}_2({\nabla }^{2}v-I)\mathrm{d}x.$$

$${\Omega }_1:=\omega \times (-1/2,1/2)\subset {\mathbb{R}}^3$$

$$y:{\Omega }_1\to {\mathbb{R}}^3,x\mapsto y(x)=\tilde{y}(x_1,x_2,h{x}_3),$$

$$Y:=W^{1,2}({\Omega }_1;{\mathbb{R}}^3).$$

$$\alpha \in (2,\infty ),$$

$${\mathcal{I}}_{\alpha }^h(y)=\frac{1}{h^{2\alpha -2}}{\int }_{{\Omega }_1}{W}_{\alpha }^h(x_3,{\nabla }_{h}y(x))\mathrm{d}x,$$

$$W_{\alpha }^h(x_3,F)=W_0(x_3,F(I+h^{\alpha -1}{B}^h(x_3))),F\in {\mathbb{R}}^{3\times 3}.$$

$$W_{\alpha =3}^h(x_3,F)=W_0\left(x_3,F\left(I+h^2\sqrt{\theta }{B}^h(x_3)\right)\right),F\in {\mathbb{R}}^{3\times 3},$$

$$\omega (s):=\underset{-1/2<t<1/2}{\mathrm{esssup}}\underset{|F|⩽s}{\sup }|W_0(t,I+F)-\frac{1}{2}{Q}_3(t,F)|$$

$$W_0(t,F)=W_0(t,RF).$$

$$\underset{-1/2<t<1/2}{\mathrm{essinf}}{W}_0(t,F)⩾C{\mathrm{dist}}^2(F,\mathrm{SO}(3)),$$

$$Q_3(t,F):=D^2{W}_0(t,I)[F,F]=\frac{{\partial }^2{W}_0(t,I)}{\partial F_{ij}\partial F_{ij}}{F}_{ij}{F}_{ij},$$

$$D^2{W}_0(t,R)[FR,FR]=D^2{W}_0(t,I)[F,F]=Q_3(t,F).$$

$$W_0(t,I+F)⩽C{\mathrm{dist}}^2(I+F,\mathrm{SO}(3)).$$

$$Q_2(t,G):=\underset{c\in {\mathbb{R}}^3}{\min }{Q}_3(t,\hat{G}+c\otimes e_3)\text{, for }t\in \left(-1/2,1/2\right),G\in {\mathbb{R}}^{2\times 2},$$

$$Q_2(t,G)=Q_3(t,\hat{G}+\mathcal{L}(t,G)\otimes e_3).$$

$$Q_2(t,F)\gtrsim |F{|}^2\text{leavevmode}\text{leavevmode}\forall F\in {\mathbb{R}}_{\mathrm{sym}}^{2\times 2}\text{ and }|\mathcal{L}(t,F)|\lesssim |F|\text{leavevmode}\text{leavevmode}\forall F\in {\mathbb{R}}^{2\times 2}$$

$${\overline{Q}}_2(E,F):={\int }_{-1/2}^{1/2}{Q}_2(t,E+tF+\check{B}(t))\mathrm{d}t,$$

$${\overline{Q}}_2^{\star }(F):=\underset{E\in {\mathbb{R}}^{2\times 2}}{\min }{\overline{Q}}_2(E,F)=\underset{E\in {\mathbb{R}}_{\mathrm{sym}}^{2\times 2}}{\min }{\int }_{-1/2}^{1/2}{Q}_2(t,E+tF+\check{B}(t))\mathrm{d}t.$$

$$\underset{h\to 0}{\mathrm{lsup}}{\mathcal{I}}_{\alpha }^h(y^h)⩽C.$$

$$X_{\alpha}=\left\{\begin{array}[]{ll}W^{1,2}(\omega;\mathbb{R})&\text{if }\alpha\in(2,3),\\ W^{1,2}(\omega;\mathbb{R}^{2})\times W^{1,2}(\omega;\mathbb{R})&\text{if }\alpha\geqslant 3,\end{array}\right.$$

$$X_{\alpha}\coloneqq\left\{\begin{array}[]{lll}W^{1,2}(\omega;\mathbb{R})&\text{if}&\alpha\in(2,3),\\ W^{1,2}(\omega;\mathbb{R}^{2})\times W^{1,2}(\omega;\mathbb{R})&\text{if}&\alpha\geqslant 3.\end{array}\right.$$

$$\rho :Y\to Y,y^h\mapsto \rho (y^h):=(R^h{)}^{\top }{y}^h-c^h$$

$$P_{\alpha }^h(y^h)\to w\text{weakly in }{X}_{\alpha },$$

$$P^{h}_{\alpha}:Y\rightarrow X_{\alpha},y^{h}\mapsto\left\{\begin{array}[]{ll}v_{\alpha}^{h},&\text{\quad if }\alpha\in(2,3),\\ (u_{\theta}^{h},v_{\theta}^{h})&\text{\quad if }\alpha=3,\\ (u_{\alpha}^{h},v_{\alpha}^{h}),&\text{\quad if }\alpha>3,\end{array}\right.$$

$$\left\{\begin{array}[]{lll}u_{\alpha}^{h}(x^{\prime})&\coloneqq&\frac{1}{h^{\gamma}}\int_{-1/2}^{1/2}(\rho(y^{h})^{\prime}(x^{\prime},x_{3})-x^{\prime})\,\mathrm{d}x_{3},\\ v_{\alpha}^{h}(x^{\prime})&\coloneqq&\frac{1}{h^{\alpha-2}}\int_{-1/2}^{1/2}\rho(y^{h})_{3}(x^{\prime},x_{3})\,\mathrm{d}x_{3},\end{array}\right.$$

Full text

1. 1 Introduction
2. 2 The setting
3. 3 Main results
4. 4 Compactness and identification of limit strain
5. 5 $\mathbf{\Gamma}$-convergence of the hierarchy
6. 6 $\Gamma$-convergence of the interpolating theory
7. 7 Approximation and representation theorems

1 Introduction

The derivation of effective theories for thin structures such as beams, rods, plates and shells is a classical problem in continuum mechanics. Fundamental results in formulating adequate dimensionally reduced theories for three-dimensional elastic objects have already been obtained by Euler [17], Kirchhoff [27] and von Kármán [44], cf. also [30, 9, 10].

A physical plate, given by a domain $\Omega_{h}=\omega\times(-h/2,h/2)\subset\mathbb{R}^{3}$, is identified with a hyperelastic body of height $h$ “much smaller” than the lengths of the sides of $\omega$. The plane domain $\omega\subset\mathbb{R}^{2}$ constitutes the mid-layer of the plate. We assume that the body has a (possibly non-homogeneous) stored energy density $W$ (precise conditions on $W$ will be specified later) and, after deformation by $\tilde{y}:\Omega_{h}\to\mathbb{R}^{3}$, the total elastic energy

[TABLE]

The problem amounts to identifying effective functionals in the limit $h\to 0$ operating on dimensionally reduced deformations of the mid-plane. In spite of its long history, rigorous results in this direction relating classical models for plates to the parent three-dimensional elasticity theory have only been obtained comparably recently.

In order to avoid working on a changing domain, a rescaling $x_{3}=z_{3}/h$ is performed to obtain a fixed $\Omega_{1}$. We set $z_{h}(x_{1},x_{2},x_{3})=(x_{1},x_{2},hx_{3})$ and we consider instead of a deformation $\tilde{y}:\Omega_{h}\rightarrow\mathbb{R}^{3}$, the rescaled one $y_{h}:\Omega_{1}\rightarrow\mathbb{R}^{3},y_{h}(x)=\tilde{y}(z_{h}(x))$. We define the energy per unit volume as $J_{h}=\frac{1}{h}E_{h}$, which after a change of variables can be seen to be

[TABLE]

where $\nabla_{h}=(\partial_{1},\partial_{2},\partial_{3}/h)$.

After first results in linear elasticity had been established, see [2, 4], perhaps the first work to derive a non-linearly elastic, lower dimensional theory with a rigorous analysis using variational convergence was [3] for the case of strings. In the context of nonlinear plates we consider the rescaled functionals

[TABLE]

For $\beta=0$, inspired by the work in [3], in [28] a non-linear membrane theory is derived. The range $\beta\in(0,1)$ is the so-called constrained membrane regime, analysed in detail in [11]. To the best of our knowledge, the regime $\beta\in[1,2)$ remains not very well explored, except under certain kinds of boundary conditions or assumed admissible deformations, see, e.g., [5] and the work in [13].

Most significant in view of our setup are the contributions to the cases $\beta\geq 2$. In [21] Friesecke, James and Müller prove the fundamental geometric rigidity estimate which carries Korn’s inequality to the nonlinear setting and utilise it to obtain the non-linear Kirchhoff theory of pure bending under an isometry constraint in case $\beta=2$. This estimate is at the core of most of the later developments in this area. In their seminal paper [22], the same authors exploit the quantitative geometric rigidity estimate of [21] in a systematic investigation of limits for the whole range of scalings $\beta\in[2,\infty)$, deriving the first hierarchy of limit models. They also provide a thorough (albeit succinct) overview of the state of the art around 2006. The lecture [34, Chapter 2] provides a nice waltkthrough of this paper, as well as abundant references and open problems as of 2017.

This variational approach has been extended and revisited in a variety of different contexts, among them more complex shell geometries [20], more basic atomistic models [40, 8], or more complicated material properties as incompressibility [12], brittleness [43] or oscillatory dependence on the space variable [37, 24, 25]. Moreover, the convergence of equilibria and even dynamic solutions have been established [36, 33, 1].

The focus in this contribution is on materials whose reference configuration is subjected to stresses (one speaks of pre-strained or pre-stressed bodies) and whose energy density exhibits a dependence on the out-of-plane direction (modelling multilayered plates). Examples of these situations are heated materials, crystallisations on top of a substrate and multilayered plates.

For $\beta=2$ the second author derived in [41, 42] an effective Kirchhoff theory for stored energy densities of the form $W(x_{3},F)=W_{0}(x_{3},F(I+hB^{h}(x_{3})))$, depending explicitly on the out-of-plane coordinate $x_{3}$ and a “mismatch tensor” $B^{h}(x_{3})$ which measures the deviation of the energy well $\operatorname{argmin}W(x_{3},\cdot)$ from the rigid motions $\operatorname{argmin}W_{0}(x_{3},\cdot)=\mathrm{SO}(3)$. We remark that the regime $\beta=2$ is precisely adapted to capture the effects of a misfit $hB^{h}$ scaling linearly $h$. In the simplest case with linearly changing $B^{h}(x_{3})=ax_{3}\operatorname{Id}$ one obtains a $\Gamma$-limit $I_{\rm Ki}$ with

[TABLE]

if $y\in\mathcal{A}$ (and $I_{\rm Ki}(y)=+\infty$ if not), where $\mathcal{A}$ is a suitable class of admissible deformations (isometric immersions). $Q$ is a quadratic form acting on the shape tensor $\operatorname{II}$ (the second fundamental form of $y$). The coefficients of $Q$ and the numbers $a_{1},a_{2}$ can be explicitly computed. In [41, 42] also a thorough investigation of the shape of energy minimisers (for free boundary conditions) is provided which shows that the optimal configurations are rolled-up portions of cylinders whose winding direction is determined by the material parameters and the misfit tensor.

The main goal of our work is to extend such an analysis to the energy regimes $\beta>2$ in order to allow for more general pre-strain scalings of the form $h^{\alpha-1}B^{h}(x_{3})$, $\alpha>2$. A main source of motivation are physical experiments which show that there are situations in which optimal configurations are spherical caps (paraboloids with positive Gauß curvature) rather than cylinders, [32, 39, 18, 19, 26, 16]. We will see that indeed this discrepancy can be explained in terms of different energy scaling regimes, where the von Kármán scaling $\beta=4$ is critical. In the present paper we lay the foundation for this by deriving effective plate theories for pre-strained multilayers. We analyse the functionals obtained here in depth in our companion paper [15].

Indeed there are previous results for $\beta=4$ in particular. With the aim to model, e.g., growth processes in plants, in [29] the authors derive the von Kármán functional with a spontaneous curvature term for pre-stressed plates. However, their setup is not comparable to our situation. On the one hand, it is even more general as an explicit $(x_{1},x_{2})$ dependence of the misfit is allowed. On the other hand, there is no explicit $x_{3}$ dependence as would be necessary to model multilayers. Very recently, these results have been extended to other scalings and significantly $x_{3}$-dependent misfits, see [31]. However, the treatment of energy densities which may vary considerably in the thin film direction is, as we will see, subtle. A main source of technical difficulties is the fact that in our situation we can no longer expect the mid plane to follow the limiting plate deformation exactly. This phenomenon can be observed already in the simplest situation of a bilayer with one layer being much softer than the other. If rolled up, the unstretched plane will move into the stiffer layer, to an extent which depends on the local curvature. Moreover, we introduce an additional fine scale $\theta$ at a critical exponent, cf. below.

Yet it turns out that in our setup the von Kármán case $\beta=4$ is in fact a rather straightforward extension of [22, 42]. The regime $\beta>4$ is however a bit more involved. In contrast to the homogeneous case in [22], the dependence on the in-plane variable may be non-trivial so it cannot be discarded by setting it to [math] without loss of generality. The scaling in the linearised Kirchhoff case $\beta\in(2,4)$ turns out to be the most difficult. In order to construct recovery sequences we need to provide a representation result for symmetric tensor fields on $\omega$ in terms of symmetrised gradients and solutions to the non-elliptic Monge-Ampere equation $\det\nabla^{2}v=0$, cf. Theorem 13. In all cases the resulting effective functionals are explicitly computed with homogenised material constants that can be calculated from the first moments in $x_{3}$ of the individual elasticity constants of the various layers.

From a modelling point of view, a main novelty is our introducing a new interpolating regime in between the linearised Kirchhoff case $\beta\char 60\relax 4$ and the fully linear case $\beta>4$. This is motivated by our findings in [15] which show that minimisers (after rescaling) coincide for all $\beta\in(2,4)$ (parts of a cylinder) and for all $\beta\in(4,\infty)$ (parts of a parabolic cap). We introduce a new scaling regime $\theta h^{4}$ with $\theta\in(0,\infty)$ and obtain von Kármán functionals that upon varying $\theta$ continuously connect the extreme cases $\theta\to 0$ and $\theta\to\infty$, which turn out to reduce to the functionals obtained for $\beta>4$ and $\beta\char 60\relax 4$, respectively. In the simplest non-trivial example, the prototypical limit functional is of von Kármán type:

[TABLE]

In contrast to the cases $\beta\neq 4$ minimisers of this functional are not explicit. We discuss their behaviour in detail in [15], in particular, how they interpolate in between $\beta\char 60\relax 4$ and $\beta>4$.

Outline

Having fixed the precise setup in Section 2, in Section 3 we present our main results: Theorem 1 on $\Gamma$-convergence in a hierarchy of energy scalings and Theorem 2 on the asymptotic behaviour of the interpolating von Kármán functional for $\theta\rightarrow 0$ or $\theta\rightarrow\infty$. We then recall some basic results on compactness and explicit representations for the limit strains from [22] in Section 4. Proofs of lower and upper bounds in Theorem 1 are collected in Section 5, where we obtain (1) and more general functionals. In Section 6 we show how the von Kármán functional interpolates between different theories. Finally, in Section 7 we prove some density and matrix representation theorems essential for the construction of recovery sequences and identification of minimisers in the linearised Kirchhoff regime.

Notation

We denote by $e_{1},e_{2},e_{3}$ the standard basis vectors in $\mathbb{R}^{3}$ and write $x=(x^{\prime},x_{3})\in\mathbb{R}^{3},x^{\prime}\in\mathbb{R}^{2}$. The spaces of symmetric and antisymmetric $n\times n$ matrices are $\mathbb{R}^{n\times n}_{\operatorname{sym}}$ and $\mathbb{R}^{n\times n}_{\operatorname{ant}}$, respectively. $A_{\operatorname{sym}}=\operatorname{sym}A=\tfrac{1}{2}(A+A^{\top})$ is the symmetric part and $A_{\operatorname{ant}}=\operatorname{ant}A=\tfrac{1}{2}(A-A^{\top})$ the antisymmetric part of a square matrix $A$.

Attaching a row and a column of zeros to a matrix $G\in\mathbb{R}^{2\times 2}$ leads to $\hat{G}\coloneqq\sum_{\alpha,\beta=1}^{2}G_{\alpha\beta}e_{\alpha}\otimes e_{\beta}\in\mathbb{R}^{3\times 3}$, conversely, $\check{B}\in\mathbb{R}^{2\times 2}$ is the matrix resulting from the deletion of the third row and column of any $B\in\mathbb{R}^{3\times 3}$. If $Q(\cdot)$ is a quadratic form, we denote the associated bilinear form by $Q[\cdot,\cdot]$.

For $f:\mathbb{R}^{3}\rightarrow\mathbb{R}$ a scalar function $\nabla f=(\partial_{1}f,\partial_{2}f,\partial_{3}f)^{\top}$ is a column vector, whereas for $y:\mathbb{R}^{3}\rightarrow\mathbb{R}^{3}$ we have $\nabla y\in\mathbb{R}^{3\times 3}$ with rows $\nabla^{\top}y_{i}$, i.e., $(\nabla y)_{ij}=y_{i,j}=\partial_{j}y_{i},i,j\in\{1,2,3\}$. Its left $3\times 2$ submatrix is $\nabla^{\prime}y$, its rescaled gradient $\nabla_{h}y=(\partial_{1}y,\partial_{2}y,1/h\,\partial_{3}y)$. Moreover, $\nabla_{s}u=\frac{1}{2}(\nabla u+\nabla^{\top}u)$, is the symmetrised gradient of $u:\mathbb{R}^{2}\rightarrow\mathbb{R}^{2}$, $\nabla^{2}v$ the Hessian matrix of $v:\mathbb{R}^{n}\rightarrow\mathbb{R}$.

Let $\omega\subset\mathbb{R}^{2}$. We set $\hat{\nabla}v\coloneqq(\partial_{1}v,\partial_{2}v,0)^{\top}\in\mathbb{R}^{3}$ for $v:\omega\rightarrow\mathbb{R}$, $\hat{\nabla}u\coloneqq\sum_{\alpha,\beta=1}^{2}(\nabla^{\prime}u)_{\alpha\beta}e_{\alpha}\otimes e_{\beta}\in\mathbb{R}^{3\times 3}$ for $u:\omega\rightarrow\mathbb{R}^{2}$ and $\hat{\nabla}b\coloneqq\sum_{\alpha=1}^{3}\sum_{\beta=1}^{2}(\nabla^{\prime}b)_{\alpha\beta}e_{\alpha}\otimes e_{\beta}\in\mathbb{R}^{3\times 3}$ for $b:\omega\rightarrow\mathbb{R}^{3}$.

The norm on Sobolev spaces is $\|v\|_{k,p,\Omega}=\|v\|_{W^{k,p}(\Omega)}$. We will omit the domain when it is clear from the context.

We abbreviate $A_{\theta}\coloneqq\nabla_{s}u_{\theta}+\tfrac{1}{2}\nabla v_{\theta}\otimes\nabla v_{\theta}$, mostly in Section 6 and set $(f)_{\omega}\coloneqq\frac{1}{|\omega|}\int_{\omega}f(x^{\prime})\,\mathrm{d}x^{\prime}$ is the average of $f$ over $\omega$.

2 The setting

As described in Section 1, we consider a sequence of increasingly thin domains $\Omega_{h}\coloneqq\omega\times(-h/2,h/2)\in\mathbb{R}^{3}$ and rescale them to

[TABLE]

where $\omega\subset\mathbb{R}^{2}$ is bounded with Lipschitz boundary. As a consequence of the rescaling, instead of maps $\tilde{y}:\Omega_{h}\rightarrow\mathbb{R}^{3}$, we consider the rescaled deformations

[TABLE]

belonging to the space

[TABLE]

For each scaling333In the notation of Section 1 we have $\beta=2\alpha-2$ .

[TABLE]

and for all deformations $y\in Y$, define the scaled elastic energy per unit volume:

[TABLE]

where $\nabla_{h}=(\partial_{1},\partial_{2},\partial_{3}/h)^{\top}$ is the gradient operator resulting after the change of coordinates described in Section 1. For the sake of conciseness, we will present most results below for all scalings simultaneously, adding the parameter $\alpha$ to much of the notation. The energy density for $\alpha\neq 3$ is given by

[TABLE]

where $B^{h}:\left(-1/2,1/2\right)\rightarrow\mathbb{R}^{3\times 3}$ describes the internal misfit and $W_{0}$ the stored energy density of the reference configuration. In the regime $\alpha=3$ we include an additional parameter $\theta>0$ controlling further the amount of misfit in the model:

[TABLE]

and we later write $\tilde{B}^{h}=\sqrt{\theta}B^{h}$. Note that given the choice $h^{\alpha-1}$ for the scaling of the misfit, the fact that in the limit it will be again scaled quadratically forces the choice of a scaling of $h^{-2(\alpha-1)}$ for the energy, since otherwise one would compute trivial (vanishing or infinite) energies in the limits. This will become apparent in the computation of the lower bounds in Theorem 3. Our assumptions for $B^{h}$ and $W_{0}$ are those of [42, Assumption 1.1]:

Assumption 1

1. a)

For a.e. $t\in\left(-1/2,1/2\right)$, $W_{0}(t,\cdot)$ is continuous on $\mathbb{R}^{3\times 3}$ and $C^{2}$ in a neighbourhood of $\operatorname{SO}(3)$ which does not depend on $t$. 2. b)

The quadratic form $Q_{3}(t,\cdot)=D^{2}W_{0}(t,I)[\cdot,\cdot]$ is in $L^{\infty}\left(\left(-1/2,1/2\right);\mathbb{R}^{9\times 9}\right)$. 3. c)

The map

[TABLE]

shall satisfy $\omega(s)=o(s^{2})$ as $s\rightarrow 0$. 4. d)

For all $F\in\mathbb{R}^{3\times 3}$ and all $R\in\operatorname{SO}(3)$

[TABLE] 5. e)

For a.e. $t\in\left(-1/2,1/2\right),W_{0}(t,F)=0$ if $F\in\operatorname{SO}(3)$ and

[TABLE]

for all $F\in\mathbb{R}^{3\times 3}$ and some $C>0$. 6. f)

$B^{h}\rightarrow B$* in $L^{\infty}\left(\left(-1/2,1/2\right);\mathbb{R}^{3\times 3}\right)$.*

The Hessian

[TABLE]

for $t\in\left(-1/2,1/2\right),F\in\mathbb{R}^{3\times 3}$ is twice the quadratic form of linear elasticity theory, which results after a linearisation of $W_{0}$ around the identity. By Assumption 1.e it is positive definite on symmetric matrices and vanishing on antisymmetric matrices. We note in passing two consequences of the above conditions. First, frame invariance (Assumption 1.d) extends to the second derivative where defined, i.e.

[TABLE]

Second, the energy $W_{0}$ grows at most quadratically in a neighbourhood of $\operatorname{SO}(3)$, i.e. for small $|F|$ it holds that:

[TABLE]

Define $Q_{2}$ to be the quadratic form on $\mathbb{R}^{2\times 2}$ obtained by relaxation of $Q_{3}$ among stretches in the $x_{3}$ direction:

[TABLE]

where $e_{3}=(0,0,1)\in\mathbb{R}^{3}$. (See the last paragraph of Section 1 for the definition of $\hat{G}$.) This process effectively minimises away the effect of transversal strain. Solving the minimisation problem yields a map $\mathcal{L}:I\times\mathbb{R}^{2\times 2}\rightarrow\mathbb{R}^{3}$, linear in its second argument, which attains the minimum:

[TABLE]

In particular, also the $Q_{2}(t,\cdot)$ are positive definite on symmetric matrices and vanishing on antisymmetric matrices. In fact, by Assumption 1.b and 1.e we have the bounds

[TABLE]

uniformly in $t\in(-1/2,1/2)$.

For the regimes $\alpha\geqslant 3$, we define the effective form

[TABLE]

with $E,F\in\mathbb{R}^{2\times 2}$ (see the last paragraph of Section 1 for the definition of $\check{B}$). For $\alpha\in(2,3)$ we consider its relaxation

[TABLE]

For the case $\alpha=3$, we include an additional parameter $\theta>0$ as discussed in page 2 and later write $\tilde{B}=\sqrt{\theta}B$. Both $\overline{Q}_{2}$ and $\overline{Q}_{2}^{\star}$ are non-negative quadratic forms (see [15] for formulae explicitly relating these to $Q_{3}(t,\cdot)$, $t\in(-1/2,1/2)$).

For fixed $\alpha\in(2,\infty)$ we say that a sequence $(y^{h})_{h>0}\subset Y$ has finite scaled energy if there exists some constant $C>0$ such that

[TABLE]

This definition will be central for many of the arguments below. After some corrections we will have precompactness of such sequences, thus essentially proving that the family $\mathcal{I}^{h}_{\alpha}$ is equicoercive, the essential condition for the fundamental theorem of $\Gamma$-convergence showing convergence of minimisers and energies. This compactness takes place in adequate target ambient spaces

[TABLE]

equipped with the weak topology.444Because the weak topology is not 1st countable, for $\Gamma$-convergence one argues that one may consider bounded sets, where it is metrisable.

An essential ingredient in arguments with $\Gamma$-convergence is the choice of sequential convergence to obtain (pre-)compactness. For the lower bounds we may suppose that a sequence $(y^{h})_{h>0}$ has finite scaled energy, which enables Lemma 1 for the identification of the limits. This requires us to work with the corrected deformations $\rho(y^{h})\coloneqq(\overline{R}^{h})^{\top}y^{h}-\overline{c}^{h}$, for some constants $\overline{R}^{h}\in\operatorname{SO}(3)$ and $\overline{c}^{h}\in\mathbb{R}^{3}$ depending on $y^{h}$, see (12).555These maps “remove” rigid movements from the $y^{h}$ bringing them close to the identity. Note that the energy is not affected by this change because of frame invariance (Assumption 1.d). We choose to encode this transformation into the definition of $\Gamma$-convergence via maps $P^{h}_{\alpha}$ (Definition 2) for general transformations $\rho$ with arbitrary $R^{h}\in\operatorname{SO}(3)$ and $c^{h}\in\mathbb{R}^{3}$. Despite adding clutter to the notation, this helps to highlight and isolate the technical requirement of the sequences involved with special rigid transformations.666We only require that there be some constants $R^{h},c^{h}$ for $P^{h}$-convergence. In order to obtain compactness and in the lower bounds we will take the specific ones given in Lemma 1 whereas for the recovery sequences we will use $R^{h}=I,c^{h}=0$.

Definition 1

Let $Y\coloneqq W^{1,2}(\Omega_{1};\mathbb{R}^{3})$ and

[TABLE]

We say that a sequence $(y^{h})_{h>0}\subset Y$ $P^{h}$-converges to some $w\in X_{\alpha}$ if and only if there exist constants $R^{h}\in\operatorname{SO}(3),c^{h}\in\mathbb{R}^{3}$ which define maps

[TABLE]

such that

[TABLE]

where

[TABLE]

and we defined:

For $\alpha\neq 3$ and $x^{\prime}\in\omega$, the scaled, averaged and corrected in-plane and out-of-plane displacements:

[TABLE]

where

[TABLE]

For $\alpha=3$ and $x^{\prime}\in\omega$, introducing the additional parameter $\theta>0$:

[TABLE]

For $\alpha=3$, we overload the notation with the parameter $\theta$ writing $(u_{\theta}^{h},v_{\theta}^{h})$ and $P_{\theta}^{h}$ instead of $(u_{\alpha}^{h},v_{\alpha}^{h})$ or $P^{h}_{\alpha}$, letting the letter used in the subindex resolve ambiguity.

With Definition 1 we can specify precisely what we mean by $\Gamma$-convergence of the energies (2): 777We refer to the notes [6] for a quick introduction to $\Gamma$-convergence.

Definition 2

Let $\alpha>2$. We say that the family of scaled elastic energies $\{\mathcal{I}_{\alpha}^{h}:Y\rightarrow\mathbb{R}\}_{h>0}$, $h>0$, $\Gamma$-converges via maps $P^{h}$ to $\mathcal{I}_{\alpha}:X_{\alpha}\rightarrow\mathbb{R}$ iff:

1. a)

Lower bound:* For every $w\in X_{\alpha}$ and every sequence $(y^{h})_{h>0}\subset Y$ which $P^{h}$-converges to $w$ as $h\rightarrow 0$ it holds that*

[TABLE] 2. b)

Upper bound:* For every $w\in X$ there exists a recovery sequence $(y^{h})_{h>0}\subset Y$ which $P^{h}$-converges to $w$ as $h\rightarrow 0$ and*

[TABLE]

Finally, we identify what the space of admissible displacements for the limit theories will be:

[TABLE]

where the space of out-of-plane displacements with singular Hessian

[TABLE]

will be central in the linearised Kirchhoff theory. We will define the functionals to be $+\infty$ for inadmissible displacements in $X_{\alpha}\backslash X_{\alpha}^{0}$.

3 Main results

Our first goal is to prove that in the pre-strained setting described above one has a hierarchy of plate models à la [22]. The proof is split into several theorems in Section 5. For notation we refer to the end of Section 1, for details on our particular use of $\Gamma$-convergence, see Definition 2.

Theorem 1** **(Hierarchy of effective theories)

Let

[TABLE]

If $\alpha\in(2,3)$ and $\omega$ is convex, then the elastic energies $\mathcal{I}_{\alpha}^{h}$ $\Gamma$-converge to the linearised Kirchhoff energy888Convexity of the domain is required for the representation theorems in Section 7 which are used in the construction of the recovery sequence for $\alpha\in(2,3)$.

[TABLE]

where $\overline{Q}^{\star}_{2}$ is defined in (6). See Theorems 3 and 4.

If $\alpha=3$ and $\theta>0$ then the energies $\mathcal{I}_{\theta}^{h}\coloneqq\frac{1}{\theta}\mathcal{I}_{\alpha=3}^{h}$ $\Gamma$-converge to the von Kármán type energy999Again, we slightly overload the notation in what would be a double definition of $\mathcal{I}_{3}^{h}$, trusting the letter used in the subindex to dispel the ambiguity.

[TABLE]

where $\overline{Q}_{2}$ is defined in (5). See Theorems 3 and 5.

Finally, if $\alpha>3$ then $\mathcal{I}_{\alpha}^{h}$ $\Gamma$-converges to the linearised von Kármán energy

[TABLE]

See Theorems 3 and 6.

Moreover, in all cases $\alpha>2$ there exists a subsequence (not relabelled) such hat $(y^{h})_{h>0}$ $P^{h}$-converges to $v\in X_{\alpha}$ (if $\alpha\in(2,3)$), respectively $(u,v)\in X_{\alpha}$ (if $\alpha\geq 3$), see Lemma 1.

Remark 1

We will not be considering body forces for simplicity, but including them in the analysis as in [22] is straightforward. 2. 2.

A standard argument shows that almost minimisers of $\mathcal{I}_{\alpha}^{h}$ $P^{h}$-converge (up to subsequences) to minimisers of the limiting functional $\mathcal{I}_{\rm lKi}$, respectively $\mathcal{I}_{\rm vK}$, respectively $\mathcal{I}_{\rm lvK}$. 3. 3.

With the help of elementary computations the effective quadratic forms $\overline{Q}^{\star}_{2},\overline{Q}_{2}$ can be rewritten in terms of the moments in $t$ of the individual $Q_{3}(t,\cdot)$. This is made explicit in [15].

The functional $\mathcal{I}_{\rm lKi}$ is said to model a linearised Kirchhoff regime because the isometry condition $\nabla^{\top}y\nabla y=I$ of the Kirchhoff model is replaced by $\det\nabla^{2}v=0$, a necessary and sufficient condition for the existence of an in-plane displacement $u$ such that $\nabla u+\nabla^{\top}u+\nabla v\otimes\nabla v=0$. This condition is to leading order equivalent to $\nabla^{\top}y\nabla y=I$ for deformations $y=(h^{2\alpha-4}u,h^{\alpha-2}v)$.101010In the numerical analysis literature, the denomination linear Kirchhoff is sometimes used for a pure bending regime without constraints. The functional $\mathcal{I}^{\theta}_{\rm vK}$ is of von Kármán type with in-plane and out-of-plane strains interacting in a membrane energy term, and a bending energy term. For simple choices of $Q_{2}$ and $B^{h}$, one recovers the classical functional (1). Finally, we say that the third limit $\mathcal{I}_{\rm lvK}$, models a linearised von Kármán (or fully linear) regime by analogy with the classical equivalent, but it is of a different kind than the one expected from the hierarchy derived in [22], since it again features an interplay between in-plane and out-of-plane components.111111This is in contrast to [22]. In our setting with the additional dependence on the $x_{3}$ coordinate, it is not possible to simply drop terms while bounding below the energy in the proof of the lower bound as is done in [22, p. 211] because of the difficulty in building recovery sequences later. For $\alpha\in(2,3)$ we introduce an additional relaxation and make use of representation Theorem 13 to construct them, but for $\alpha>3$, no such result is available. One could think that minimising globally, $\inf_{\nabla_{s}u}\int Q_{2}\left(t,\nabla_{s}u+\ldots\right)$, might be a way of discarding in-plane displacements to recover the standard theory, but this yields a functional which is not local and therefore lacks an integral representation (see e.g. [7, Chapter 9]). Note that even if we pick $Q_{2}$ independent of $t$ and $B=0$, we do not recover the functional of [22] because ours keeps track of both in-plane and out-of-plane displacements which is essential to capture the effect of pre-stressing with the internal misfit $B^{h}$.

Our second goal is to show that the limit energy $\mathcal{I}^{\theta}_{\rm vK}$ interpolates between $\mathcal{I}_{\rm lKi}$ and $\mathcal{I}_{\rm lvK}$ as the parameter $\theta$ moves from $\infty$ to [math], so that one can say that the theory of von Kármán type bridges the other two. More precisely, in Section 6 we prove:

Theorem 2** **(Interpolating regime)

The following two $\Gamma$-limits hold:

[TABLE]

if $\omega$ is convex (Theorems 8 and 9) and:

[TABLE]

(Theorems 10 and 11). Furthermore, sequences $(u_{\theta},v_{\theta})_{\theta>0}$ of bounded energy $\mathcal{I}^{\theta}_{\rm vK}$ are precompact in suitable spaces as $\theta\uparrow\infty$ or $\theta\downarrow 0$ (Theorem 7).

Example. The easiest non-trivial situation is given by a linear internal misfit in a homogeneous material with

[TABLE]

Then

[TABLE]

for $(u,v)\in W^{1,2}(\omega;\mathbb{R}^{2})\times W^{2,2}(\omega;\mathbb{R})$. We refer to [14] for more worked out examples.

4 Compactness and identification of limit

strain

We collect here some basic results proving compactness of sequences of scaled energy and providing explicit representations for the limit strains, as required for the proofs of $\Gamma$-convergence in Section 3. These results are direct consequences of the homogeneous case treated in [22, Lemma 1]. We recall the definition of the scaled elastic energies (2):

[TABLE]

Lemma 1

Let $\alpha\in(2,\infty)$ and let $(y^{h})_{h>0}\subset Y$ have finite scaled $\mathcal{I}^{h}_{\alpha}$ energy. For every $h>0$ there exist constants $\overline{R}^{h}\in\operatorname{SO}(3)$ and $c^{h}\in\mathbb{R}^{3}$ such for the corrected deformations

[TABLE]

there exist rotations $R^{h}:\omega\rightarrow\operatorname{SO}(3)$ (extended constantly along $x_{3}$ to all of $\Omega_{1}$ outside $\{0\}\times\omega$) approximating $\nabla_{h}\tilde{y}^{h}$ in $L^{2}(\Omega_{1})$. Quantitatively:

[TABLE]

Furthermore,

[TABLE]

Finally there exists a subsequence (not relabelled) such that for the scaled and averaged in-plane and out-of-plane displacements from (7) there exist $(u,v)\in W^{1,2}(\omega;\mathbb{R}^{2})\times W^{2,2}(\omega)$ such that, if $\alpha\neq 3$:

[TABLE]

If $\alpha=3$ an analogous result holds with $u_{\theta}^{h}$ and $v_{\theta}^{h}$ from (8).

In particular, in the sense of Definition 1 we have that $(y^{h})_{h>0}$ $P^{h}$-converges to $v\in X_{\alpha}$ (if $\alpha\in(2,3)$), respectively $(u,v)\in X_{\alpha}$ (if $\alpha\geq 3$).

Proof.

This is exactly a particular case of [22, Lemma 1], estimates (84) and (85) and estimates (86) and (87), once we prove that if $(y^{h})_{h>0}$ have finite scaled $\mathcal{I}^{h}_{\alpha}$ energy, then they have finite scaled energy in the sense of [22].

Note first that among all choices we can make for the energy density $W$ which fulfil the assumptions in [22], we can pick $\operatorname{dist}^{2}(\cdot,\operatorname{SO}(3))$. Therefore we will bound this quantity. Write $d(F)\coloneqq\operatorname{dist}(F,\operatorname{SO}(3))$. We begin by using Assumption 1.e:

[TABLE]

Consider now the following:

[TABLE]

But then we are done since:

[TABLE]

∎

Lemma 2

121212This is almost word for word [22, Lemma 2] with the very minor addition of the factors $\theta,\sqrt{\theta}$. For other scaling choices see [22, p.208]. Note that this is inspired by [9, Theorem 5.4.2] (itself based in [9, Theorem 1.4.1.c]).

Let $\alpha\in(2,\infty)$ and let $(y^{h})_{h>0}$ be a sequence in $Y$ which $P^{h}$-converges to $(u,v)\in X_{\alpha}$ in the sense of Theorem 1 and $R^{h}:\omega\rightarrow\operatorname{SO}(3)$ (extended constantly along $x_{3}$ to all of $\Omega_{1}$ outside $\{0\}\times\omega$) such that

[TABLE]

Then:

[TABLE]

where

[TABLE]

and

[TABLE]

where the submatrix $\check{G}\in\mathbb{R}^{2\times 2}$ is affine in $x_{3}$:

[TABLE]

and

[TABLE]

[TABLE]

and

[TABLE]

Proof.

See [22, p. 208–209]. ∎

5 $\mathbf{\Gamma}$-convergence of the

hierarchy

This section proves the lower (Theorem 3) and upper bounds (Theorems 4, 5 and 6) required for deriving the hierarchy of models in Theorem 1.

An important result of [22] is that for small $h>0$ deformations $y^{h}$ of finite scaled energy are, up to rigid motions, roughly the trivial map $(x^{\prime},x_{3})\mapsto(x^{\prime},hx_{3})$. The factor by which they fail to (almost) be the identity is essential for the linearisation step in the proof below as well as for the identification of the limit strains of weakly convergent sequences of scaled displacements. We must account for these rigid motions if compactness is to be achieved, in particular because deformations might “wander to infinity” without altering the elastic energy. Lemmas 1 and 2 gather these ideas more precisely. In particular, the last statement of Lemma 1 provides the required compactness.

Recall that we are always using weak convergence in the spaces $X_{\alpha}$.

Theorem 3** **(Lower bounds)

Let $\alpha\in(2,3)$. If $(y^{h})_{h>0}\subset Y$ is a sequence $P^{h}_{\alpha}$-converging to $v\in X_{\alpha}$, then

[TABLE]

Now let $\alpha=3$. If $(y^{h})_{h>0}\subset Y$ is a sequence $P^{h}_{\theta}$-converging to $(u,v)\in X_{\alpha}$, then for all $\theta>0$

[TABLE]

Finally, let $\alpha>3$. If $(y^{h})_{h>0}\subset Y$ is a sequence $P^{h}_{\alpha}$-converging to $(u,v)\in X_{\alpha}$, then

[TABLE]

Proof.

If $\alpha=3$, we define $\tilde{B}^{h}\coloneqq\sqrt{\theta}B^{h}$ and $\tilde{B}=\sqrt{\theta}B$, otherwise $\tilde{B}\coloneqq B$ and $\tilde{B}^{h}\coloneqq B^{h}$. Following closely the techniques in [21, 22, 41, 42] we use a Taylor expansion of the energy around the identity which allows us to cancel or identify its lower order terms. For this we must correct the deformations with an approximation by rotations and work in adequate sets where there is control over higher order terms.

Upon passing to a subsequence (not relabelled) which realises ${\operatorname{linf}}_{h\rightarrow 0}\mathcal{I}_{\alpha}^{h}(y^{h})$ as its limit, we may w.l.o.g. assume that $(y^{h})_{h>0}$ has finite scaled $\mathcal{I}^{h}_{\alpha}$ energy and pass to further subsequences in the following.

Step 1: Approximation by rotations

We will be working with the corrected deformations

[TABLE]

as given in Lemma 1. For simplicity we use the same notation $y^{h}$ for these functions. Also by Lemma 1 there exist rotations $R^{h}:\omega\rightarrow\operatorname{SO}(3)$ (extended constantly along $x_{3}$ to all of $\Omega_{1}$ outside $\omega\times\{0\}$) which approximate $\nabla_{h}y^{h}$ in $L^{2}(\Omega_{1})$ and are close to the identity, as required for the identification of the limit strain in Lemma 2.

Step 2: Rewriting of the deformation gradient

The functions

[TABLE]

are uniformly bounded in $L^{2}$ by invariance of the norm by rotations:

[TABLE]

Now, by the frame invariance of $W^{h}(x_{3},\cdot)$

[TABLE]

where we have set

[TABLE]

Step 3: Cutoff function

We will be expanding $W_{0}(x_{3},I+h^{\alpha-1}A^{h})$ around $I$, but in order to apply the Taylor expansion successfully we need to stay where $W_{0}$ is twice differentiable, that is we must control $\operatorname{dist}(I+h^{\alpha-1}A^{h},\operatorname{SO}(3))$. We achieve this by multiplying with a cutoff function ${\large\chi}^{h}$, defined as the characteristic function of the “good set” $\{x\in\Omega_{1}:|G^{h}|\leqslant h^{-1/2}\}$. Here we have:

[TABLE]

which, because $|R^{h}|\equiv\sqrt{3}$, implies that ${\large\chi}^{h}|\nabla_{h}y^{h}|\leqslant C.$ Consequently, since the $\tilde{B}^{h}$ are uniformly bounded as well:

[TABLE]

and then

[TABLE]

so in the good sets we may indeed expand around $I$ for small values of $h$. Now, the sequence $(G^{h})_{h>0}$ is bounded in $L^{2}$ by (15) so we may extract a subsequence converging weakly in $L^{2}$ to some $G\in L^{2}(\Omega_{1})$, which we consider from now on without relabelling. Furthermore the sequence $({\large\chi}^{h})_{h>0}$ is essentially bounded and ${\large\chi}^{h}\rightarrow 1$ in measure in $\Omega_{1}$. Indeed $|\{|{\large\chi}^{h}-1|>\varepsilon\}|=|\{|G^{h}|>h^{-1/2}\}|\rightarrow 0\text{ as }h\rightarrow 0$ because $\|G^{h}\|_{0,2,\Omega_{1}}\leqslant C$ uniformly. Consequently we have

[TABLE]

Analogously, the sequence $({\large\chi}^{h}\tilde{B}^{h})_{h>0}$ is essentially bounded and converges in measure to $\tilde{B}$ because $|\{|{\large\chi}^{h}\tilde{B}^{h}-\tilde{B}|>\varepsilon\}|\leqslant|\{|\tilde{B}^{h}-\tilde{B}|>\varepsilon\}|+|\{{\large\chi}^{h}=0\}\cap\{|\tilde{B}|>\varepsilon\}|\rightarrow 0$. Hence, using again the strong convergence $(R^{h})^{\top}\nabla_{h}y^{h}\rightarrow I$ in $L^{2}(\Omega_{1})$ (Lemma 1):

[TABLE]

So we conclude

[TABLE]

Step 4: Taylor expansion

Because $W_{0}(x_{3},\cdot)_{|\operatorname{SO}(3)}\equiv 0$, for any fixed $x_{3}$ the lower order terms of its Taylor expansion

[TABLE]

vanish and we have (for small enough $h$, as explained above)

[TABLE]

where $\eta^{h}(x_{3},h^{\alpha-1}{\large\chi}^{h}A^{h})=o(h^{2\alpha-2}|{\large\chi}^{h}A^{h}|^{2})$ represents the higher order terms. Defining the uniform bound

[TABLE]

we have $\omega(s)=o(s^{2})$ by Assumption 1.c, and integrating over the rescaled domain $\Omega_{1}$ we obtain the estimate:

[TABLE]

Step 5: The limit inferior

In order to pass to the limit, for the first integral on the right hand side of (5) we use that $Q_{3}$ is positive semidefinite, therefore convex and continuous, and the integral is sequentially weakly lower semicontinuous. For the second integral we use again Assumption 1.c and the fact that $|h^{\alpha-1}{\large\chi}^{h}A^{h}|\leqslant h^{1/2}$ to obtain the bound (uniform over $\Omega_{1}$):

[TABLE]

But then, because ${\large\chi}^{h}A^{h}$ converges weakly in $L^{2}$, we have $\|{\large\chi}^{h}A^{h}\|_{0,2,\Omega_{1}}^{2}\leqslant C$ and

[TABLE]

as $h\rightarrow 0$. Taking the $\lim\inf$ at both sides of (5) we have:

[TABLE]

where the last estimate follows trivially from the definition of $Q_{2}$.

If $\alpha\geqslant 3$, by Lemma 2 the limit strain $\check{G}$ has the representation

[TABLE]

with $G_{1}$ and $\operatorname{sym}G_{0}$ given respectively by (13) and (14) as:

[TABLE]

and

[TABLE]

We plug both into the last integral and use the fact that $Q_{2}(x_{3},\cdot)$ vanishes on antisymmetric matrices to obtain

[TABLE]

In particular, if $\alpha=3$, we have again:

[TABLE]

If $\alpha\in(2,3)$, then $\operatorname{sym}G_{0}$ is unknown, so we must further relax the integrand. With the definition of $\overline{Q}_{2}^{\star}$ we see that the final integral above is

[TABLE]

∎

We proceed now with the computation of the recovery sequences for each of the three regimes discussed. We assume convexity of the domain in order to apply the representation theorems in Section 7.

Theorem 4** **(Upper bound, linearised Kirchhoff regime)

Assume $\omega$ is convex, let $\alpha\in(2,3)$ and $v\in X_{\alpha}\coloneqq W^{1,2}(\omega)$. There exists a sequence $(y^{h})_{h>0}\subset Y$ which $P^{h}$-converges to $v$ such that

[TABLE]

with $\mathcal{I}_{\rm lKi}$ defined as in (9) by

[TABLE]

Proof.

We set $\varepsilon=h^{\alpha-2}$, so that $h\ll\varepsilon\ll 1$ and $h^{2}\ll\varepsilon h\ll 1$.

Step 1: Setup and recovery sequence

The functional $\mathcal{I}_{\rm lKi}$ is strongly continuous on $W^{2,2}_{sh}(\omega)$ by the continuity and 2-growth of $\overline{Q}^{\star}_{2}$. By Theorem 12 we have a set $\mathcal{V}_{0}$ of smooth maps with singular Hessian which is $W^{2,2}$-dense in $W^{2,2}_{sh}$, see (33). Therefore, by a standard argument (see, e.g., [6]) it is enough to construct here the recovery sequence. Take then a smooth function $v\in\mathcal{V}_{0}$. Because $\|\nabla v\|_{\infty}\char 60\relax C$, for $\varepsilon$ small enough there exist by [22, Theorem 7] in-plane displacements $u_{\varepsilon}\in W^{2,2}(\omega;\mathbb{R}^{2})\cap W^{2,\infty}(\omega;\mathbb{R}^{2})$ with uniform bounds in $\varepsilon$ such that the deformations

[TABLE]

are isometries.131313The uniform bounds for $\|u_{\varepsilon}\|_{2,2}$ follow from [22, Theorem 7], equation (181), and those for $\|u_{\varepsilon}\|_{2,\infty}$ from the explicit construction done in the proof, in particular equations (183), (186) and (190). That is: $\nabla^{\top}\overline{y}_{\varepsilon}\nabla\overline{y}_{\varepsilon}=I_{2}$, where

[TABLE]

Additionally the following normal vectors are unitary in $\mathbb{R}^{3}$:

[TABLE]

where the rest $r_{\varepsilon}$ satisfies

[TABLE]

by virtue of $\|u_{\varepsilon}\|_{2,\infty}\leqslant C$ and $\|\nabla v\|_{\infty}\leqslant C$. Consequently the matrices

[TABLE]

are in $\operatorname{SO}(3)$ for every $x^{\prime}\in\omega$, with the remaining matrix $\tilde{r}_{\varepsilon}$ satisfying

[TABLE]

by the same arguments as before. Now, for some smooth functions $\alpha,g_{1},g_{2}\in C^{\infty}(\overline{\omega};\mathbb{R})$, $g\coloneqq(g_{1},g_{2})$ and $d\in L^{\infty}(\Omega_{1};\mathbb{R}^{3})$ with $\nabla^{\prime}d\in L^{\infty}(\Omega_{1};\mathbb{R}^{3\times 2})$ and $D^{h}\in C^{\infty}(\overline{\Omega}_{1};\mathbb{R}^{3})$ to be determined later, set

[TABLE]

We will prove

[TABLE]

as well as $P^{h}_{\alpha}(y^{h})\rightarrow v$ in $W^{1,2}$ for some constants $R^{h}\in\operatorname{SO}(3),c^{h}\in\mathbb{R}^{3}$.

Step 2: Preliminary computations

In order to compute the limit of $\frac{1}{h^{2\alpha-2}}\int_{\Omega_{1}}W_{0}(x_{3},\nabla_{h}y^{h}(I+\varepsilon hB^{h}))$ we start with the gradient of the recovery sequence:

[TABLE]

For the term in $h$ and any $i\in\{1,2,3\}$ and $j\in\{1,2\}$ we have

[TABLE]

Also: $\frac{1}{h}\partial_{3}[(x_{3}-\alpha)b_{\varepsilon}]=\frac{1}{h}b_{\varepsilon}$, so that

[TABLE]

Substituting back into the gradient yields:

[TABLE]

Because we intend to use the frame invariance of the energy, we will need the product of $\nabla_{h}y^{h}$ with $R_{\varepsilon}^{\top}=I+\mathcal{O}(\varepsilon)$. First we have:

[TABLE]

where we have subsumed terms $o(\varepsilon h)$ into the $o(1)$ inside $A^{h}$. Using $|b_{\varepsilon}|\equiv 1$ and $\overline{y}_{\varepsilon,i}\perp b_{\varepsilon}$ we also have $R_{\varepsilon}^{\top}b_{\varepsilon}=e_{3}$. Therefore

[TABLE]

Step 3: Convergence of the energies

The next step is a Taylor expansion around the identity. Given that the energy is scaled by $(\varepsilon h)^{-2}$, only those terms scaling as $\varepsilon h$ in (19) will remain: anything beyond that will not be seen and anything below will make the energy blow up. This means that we must choose $D^{h}$ so that $F^{h}=o(\varepsilon h)$. In [22], [42] it was possible for the authors to obtain exactly $F^{h}=0$ by choosing $D^{h}$ adequately, but in our case this will not be possible.141414Technically, this is due to the fact that the term $he_{3}\otimes\hat{\nabla}\alpha$ is a row in a matrix instead of a column, which makes it impossible to exactly compensate because $R_{\varepsilon}^{\top}\nabla_{h}D^{h}$ effectively only provides a column vector to work with. Indeed,

R_{\varepsilon}^{\top}\nabla_{h}D^{h}=\nabla_{h}D^{h}+\varepsilon\left(\begin{array}[]{cc}0&\nabla v\\ -\nabla^{\top}v&0\end{array}\right)\nabla_{h}D^{h}+\tilde{r}^{\top}_{\varepsilon}\nabla_{h}D^{h},

so in order to cancel $he_{3}\otimes\hat{\nabla}\alpha$ we must have that the leading term $\nabla_{h}D^{h}$ be of order $h$. But then $\nabla_{h}D^{h}=\left(\nabla^{\prime}D^{h},\frac{1}{h}D_{,3}^{h}\right)$ requires that $D^{h}$ scale at least as $h^{2}\ll\varepsilon h\ll 1$ so we “lose” the first two columns of $\nabla_{h}D^{h}$. If we set $D^{h}\coloneqq h^{2}D$ for some smooth $D$, we have

[TABLE]

This means that we must solve the equations $\tilde{F}^{h}=o(\varepsilon)$. Although these have no solution the symmetrised version does,151515Dividing by $h$ we arrive at:

\left\{\begin{array}[]{rll}D_{1,3}+\varepsilon v_{,1}D_{3,3}&=&\alpha_{,1}+o(\varepsilon),\\ D_{2,3}+\varepsilon v_{,2}D_{3,3}&=&\alpha_{,2}+o(\varepsilon),\\ D_{3,3}-\varepsilon v_{,1}D_{1,3}-\varepsilon v_{,2}D_{2,3}&=&o(\varepsilon),\end{array}\right.

with solution:

$D(x^{\prime},x_{3})=x_{3}\hat{\nabla}\alpha+x_{3}\varepsilon\nabla v\cdot\nabla\alpha e_{3}.$

so that for every smooth choice of $\alpha$ we can pick a bounded $D^{h}$ such that

[TABLE]

a fact that we will exploit next. By frame invariance, (19) and $F^{h}=h\tilde{F}^{h}$, we can write

[TABLE]

Because of (20) by our choice of $D$ we need to subtract the antisymmetric part of $\tilde{F}^{h}$, which we do by means of another rotation and frame invariance:

[TABLE]

Now whenever $h$ is small enough that $I+hC^{h}$ belongs to the neighbourhood of $\operatorname{SO}(3)$ where $W_{0}$ is twice differentiable, we can apply Taylor’s theorem and the fact that $Q_{3}$ vanishes on antisymmetric matrices to see that, as $h\rightarrow 0$:

[TABLE]

where

[TABLE]

We choose

[TABLE]

with $\mathcal{L}$ the map from (3), which by (3) and (4) is linear in the second component and satisfies $|\mathcal{L}(t,A)|\lesssim|A|$ uniformly in $t$, and $B_{\cdot 3}$ the third column of $B$. Because the matrix $(\alpha-x_{3})\nabla^{2}v+\nabla_{s}g+\check{B}_{s}$ is bounded uniformly in $x^{\prime}$, by the bound (4) the map

[TABLE]

is in $W^{1,\infty}(\Omega_{1};\mathbb{R}^{3})$ and $y^{h}\in W^{1,2}$ as required (for the derivatives with respect to $x^{\prime}$ note that $v,g$ are smooth and $B$ independent of $x^{\prime}$).

Now, all quantities being bounded, by dominated convergence:

[TABLE]

Note that a final step is required to obtain convergence to $\mathcal{I}_{\rm lKi}(v)$.

Step 4: Convergence of the deformations: $P^{h}_{\alpha}(y^{h})\rightarrow v$ in $W^{1,2}$

Choose $R^{h}\equiv I\in\operatorname{SO}(3),c^{h}\equiv 0\in\mathbb{R}^{3}$ in the definition of $\rho$ for (7). We have

[TABLE]

where in (18) we defined $y^{h}_{3}(x^{\prime},x_{3})=\varepsilon v(x^{\prime})+h(x_{3}-\alpha(x^{\prime}))b_{\varepsilon 3}(x^{\prime})+\mathcal{O}(\varepsilon h)$. Then:

[TABLE]

and consequently $\|P^{h}_{\alpha}(y^{h})-v\|_{0,2}\rightarrow 0$. An analogous computation for the derivatives shows strong convergence in $W^{1,2}$.

Step 5: Simultaneous convergence

Finally, as in [42, Theorem 3.2], in order for the energy to converge to the true limit, we must pick $\alpha$ and $g$ in (18) so as to approximate the minimum $\overline{Q}_{2}$. This is done with Corollary 1, substituting sequences of smooth functions $(\alpha_{k})_{k\in\mathbb{N}},(g_{k})_{k\in\mathbb{N}}$ for the functions $\alpha,g$. Then, for each fixed $k$ we have:

[TABLE]

and

[TABLE]

And by a diagonal argument we can find $(y^{h})_{h>0}$ whose energy converges to $\mathcal{I}_{\rm lKi}(v)$ while maintaining the convergence of the deformations. ∎

Theorem 5** **(Upper bound, von Kármán regime)

Let $\alpha=3$ and consider displacements $(u,v)\in X_{\alpha=3}\coloneqq W^{1,2}(\omega;\mathbb{R}^{2})\times W^{1,2}(\omega;\mathbb{R})$. There exists a sequence $(y^{h})_{h>0}\subset Y$ which $P^{h}_{\theta}$-converges to $(u,v)$ such that

[TABLE]

with $\mathcal{I}^{\theta}_{\rm vK}$ defined as in (10) by

[TABLE]

over $X_{\alpha=3}^{0}=W^{1,2}(\omega;\mathbb{R}^{2})\times W^{2,2}(\omega;\mathbb{R})$ and as $\infty$ elsewhere.

Proof.

In order to build the recovery sequence $(y^{h})_{h>0}$ we will use the map $\mathcal{L}:\left(-1/2,1/2\right)\times\mathbb{R}^{2\times 2}\rightarrow\mathbb{R}^{3}$ given by (3), which for each $t$ realises the minimum of $Q_{3}(t,\hat{A}+c\otimes e_{3}),A\in\mathbb{R}^{2\times 2}$, i.e.

[TABLE]

where the last equality follows from the fact that $Q_{2}$ vanishes on antisymmetric matrices. Recall from (4) that $\mathcal{L}(t,\cdot)$ is linear for every $t$ and that $|\mathcal{L}(t,A)|\lesssim|A|$ uniformly in $t$.

The functional $\mathcal{I}^{\theta}_{\rm vK}$ is clearly continuous in $X^{0}_{\alpha}=W^{1,2}(\omega;\mathbb{R}^{2})\times W^{2,2}(\omega;\mathbb{R})$ with the strong topologies, so a standard argument [6] shows that it is enough to consider $(u,v)\in C^{\infty}(\overline{\omega};\mathbb{R}^{2})\times C^{\infty}(\overline{\omega};\mathbb{R})$, which is dense in $X^{0}_{\alpha}$. We define:

[TABLE]

where $d\in W^{1,\infty}(\Omega_{1};\mathbb{R}^{3})$ is a vector field to be determined along the proof.

Step 1: Approximation of the energy

A direct computation yields

[TABLE]

For later use we note here the product:

[TABLE]

where we used that $E$ is antisymmetric. For any matrix $M$ with positive determinant we have the polar decomposition $M=U\sqrt{M^{\top}M}=U\sqrt{I+P}$, with $U\in\operatorname{SO}(3)$ and $P=M^{\top}M-I$. By the frame invariance of the energy and a Taylor expansion around the identity of the square root

[TABLE]

and, assuming that a Taylor expansion of $W_{0}$ around the identity can be carried, i.e. that $M$ is close enough to SO(3), this is equal to:

[TABLE]

In view of the definition of $W_{0}$, we set

[TABLE]

where $\tilde{B}^{h}=\sqrt{\theta}B^{h}\rightarrow\tilde{B}=\sqrt{\theta}B$ in $L^{\infty}$. Then

[TABLE]

To compute the first term in $h^{2}$, $N=2F_{s}+E^{\top}E$, we have

[TABLE]

and:161616We use the identities $(c\otimes e_{3})(e_{3}\otimes c)=c\otimes c$, $(c\otimes e_{3})(c\otimes e_{3})=c_{3}c\otimes e_{3}$, $(e_{3}\otimes c)(e_{3}\otimes c)=c_{3}e_{3}\otimes c$ and $(e_{3}\otimes c)(c\otimes e_{3})=|c|^{2}e_{3}\otimes e_{3}$.

[TABLE]

Since these quantities are independent of $h$, for sufficiently small $h$ the product $(M^{h})^{\top}M^{h}$ does lie close enough to $\operatorname{SO}(3)$ and we can perform the desired Taylor expansion:

[TABLE]

Define now \hat{G}_{0}\coloneqq\theta\big{(}\hat{\nabla}_{s}u+1/2\hat{\nabla}v\otimes\hat{\nabla}v\big{)}, $\hat{G}_{1}\coloneqq-\theta^{1/2}\hat{\nabla}^{2}v$ as in Lemma 2. Bringing the previous computations together we obtain:

[TABLE]

hence

[TABLE]

We now choose the vector field $d$ to cancel one term and attain the minimum for the others by solving for $\partial_{3}d$ in:

[TABLE]

that is:

[TABLE]

Consequently, we set:

[TABLE]

and we obtain

[TABLE]

As in the proof of Theorem 4, (3) and (4) imply that $d\in W^{1,\infty}(\Omega_{1};\mathbb{R}^{3})$.

Step 2: Convergence

By the previous step we have $\frac{1}{\theta h^{4}}W_{0}(x_{3},\nabla_{h}y^{h})\rightarrow\frac{1}{2\theta}Q_{2}\left(x_{3},G_{0}+x_{3}G_{1}+\sqrt{\theta}\check{B}_{s}\right)$ a.e. as $h\rightarrow 0$, and the sequence is uniformly bounded so we can integrate over the domain and pass to the limit:

[TABLE]

Step 3: Convergence of the recovery sequence

Note that $P^{h}_{\theta}(y)\rightarrow(u,v)$ in $X_{\alpha}$ as $h\rightarrow 0$ with the choice $R^{h}=I\in\operatorname{SO}(3),c^{h}=0\in\mathbb{R}^{3}$ in Definition 1 since

[TABLE]

∎

In the next result, there is a departure from the analogous functional in [22] beyond the dependence on the out-of-plane component $x_{3}$. In the preceding cases, if one sets $Q_{2}(t,A)\equiv Q_{2}(A)$, and $B\equiv 0$ then the same functionals are obtained as in that work. However, in the regime $\alpha>3$ their limit has no membrane term, but we have $\overline{Q}_{2}\left(\nabla_{s}u,-\nabla^{2}v\right)=\frac{1}{2}\int Q_{2}\left(\nabla_{s}u\right)+\frac{1}{24}\int Q_{2}(\nabla^{2}v)$, with the membrane term. The reason is that [22] discard the in-plane displacements by minimising them away. In their proofs, they drop the first term in the lower bound and build the recovery sequence with no $u$ term in $h^{\alpha-1}$.

Note that it is by keeping the membrane term that our model is able to take into account and respond to the pre-stressing (internal misfit) $B^{h}$, e.g. compressive or tensile stresses in wafers.

Theorem 6** **(Upper bound, linearised von Kármán regime)

Let $\alpha>3$ and consider displacements $(u,v)\in X_{\alpha}\coloneqq W^{1,2}(\omega;\mathbb{R}^{2})\times W^{1,2}(\omega;\mathbb{R})$. There exists a sequence $(y^{h})_{h>0}\subset Y$ which $P^{h}_{\alpha}$-converges to $(u,v)$ such that

[TABLE]

with $\mathcal{I}_{\rm lvK}$ defined as in (11) by

[TABLE]

on $X_{\alpha}^{0}$ and by $+\infty$ elsewhere.

Proof.

We follow closely the notation and path of proof of Theorem 5. By a standard density argument it is enough to consider $(u,v)\in X_{\alpha}\cap C^{\infty}(\overline{\omega})$. Define

[TABLE]

with $d\in W^{1,\infty}(\Omega_{1};\mathbb{R}^{3})$. Then

[TABLE]

and, using that $E_{s}=0$:

[TABLE]

Define now $M^{h}\coloneqq\nabla_{h}y^{h}(I+h^{\alpha-1}B^{h})$. A few computations lead to

[TABLE]

from which follows, after a Taylor approximation (recall from the proof of Theorem 5, that this can be done for sufficiently small $h$):

[TABLE]

Picking $d$ such that:

[TABLE]

e.g.

[TABLE]

the term with $\mathcal{L}$ in $Q_{2}$ cancels out and we obtain

[TABLE]

Note that as proved in Theorem 5, the properties of $\mathcal{L}$ imply that the function $d\in W^{1,\infty}(\Omega_{1};\mathbb{R}^{3})$ so the previous computations are justified. We have therefore

[TABLE]

and also $Q_{2}(x_{3},A)\lesssim|A|^{2}$ because $Q_{3}$ is in $L^{\infty}$ (Assumption 1.b). Because $u_{i},v\in C^{\infty}(\overline{\omega})$ and $B_{s}\in L^{\infty}$, all arguments of $Q_{2}$ are uniformly bounded and we can apply dominated convergence to conclude:

[TABLE]

Set now  $R=I\in\operatorname{SO}(3),c=0\in\mathbb{R}^{3}$ for the rigid transformation $\rho$ in Definition 1. It remains to note that indeed $P^{h}_{\alpha}(y^{h})\rightarrow(u,v)$ in $X_{\alpha}$:

[TABLE]

and the proof is complete. ∎

6 $\Gamma$-convergence of the interpolating

theory

**Notation **

Throughout this section we write $A_{\theta}\coloneqq\nabla_{s}u_{\theta}+\tfrac{1}{2}\nabla v_{\theta}\otimes\nabla v_{\theta}$ for the strain induced by a pair of displacements $(u_{\theta},v_{\theta})$. As before, $\theta>0$.

We now set to prove Theorem 2, which states that the functional of generalised von Kármán type that we found in the preceding section,

[TABLE]

interpolates between the two adjacent regimes as $\theta\rightarrow\infty$ or $\theta\rightarrow 0$. As $\theta$ approaches infinity, we expect the optimal energy configurations to approach those of the linearised Kirchhoff model, whereas with $\theta$ tending to zero they should approach the linearised von Kármán model.

For this section we restrict ourselves to spaces where Korn-Poincaré type inequalities hold.

Definition 3

Let

[TABLE]

and

[TABLE]

We set $X_{w}\coloneqq X_{u}\times X_{v}$ with the weak topologies.

Additionally, from now on we assume without loss that the barycenter of $\omega$ be the origin so that $\int_{\omega}x^{\prime}\,\mathrm{d}x^{\prime}=0$. Finally, for the limit $\theta\rightarrow\infty$ we require that $\omega$* be convex* and recall the definition of the space of maps with singular Hessian

[TABLE]

Remark 2

There is no loss of generality in reducing to the space $X_{u}\times X_{v}$: First we can always add an infinitesimal rigid motion to $u$ and and any affine function to $v$ without changing $\nabla_{s}u$ or $\nabla^{2}v$. Second, although the nonlinear term $\nabla v\otimes\nabla v$ does change after adding an affine function, the extra terms appearing happen to be a symmetric gradient which can be absorbed into $\nabla_{s}u$ with a little help: For any $g(x)=a\cdot x+b$ for $a,b\in\mathbb{R}^{2}$, we have

[TABLE]

where we set $z(x)\coloneqq(2v(x)+a\cdot x)a\in W^{2,2}(\omega;\mathbb{R}^{2})$. Therefore, for any fixed $u\in W^{1,2}(\omega;\mathbb{R}^{2}),v\in W^{2,2}(\omega)$ one can choose $g(x)=-[(\nabla v)_{\omega}\cdot x+(v)_{\omega}]$ and define

[TABLE]

with $r(x)=Rx+c$, for constants $R\coloneqq\frac{-1}{|\omega|}\int_{\omega}\nabla_{a}u+\nabla_{a}z\,\mathrm{d}x\in\mathbb{R}^{2\times 2}_{\operatorname{ant}}$ and $c\coloneqq\frac{-1}{|\omega|}\int_{\omega}u(x)+z(x)+Rx\,\mathrm{d}x$. For $\tilde{u},\tilde{v}$ we then have on the one hand $\int\tilde{u}=0,\int\nabla_{a}\tilde{u}=0$ and $\int\tilde{v}=0,\int\nabla\tilde{v}=0$ and on the other (note that $\nabla_{s}r=0$):

[TABLE]

as desired.

Our first theorem identifies the types of convergence required in order to obtain precompactness of sequences of bounded energy. We use these definitions of convergence for the computation of the $\Gamma$-limits.

Theorem 7** **(Compactness)

Let $(u_{\theta},v_{\theta})_{\theta>0}$ be a sequence in $X_{w}$ with finite energy

[TABLE]

Then:

The sequence $(v_{\theta})_{\theta\uparrow\infty}$ is weakly precompact in $W^{2,2}(\omega)$ and the weak limit is in $X_{v}\cap W^{2,2}_{sh}(\omega)$. Additionally $(u_{\theta})_{\theta\uparrow\infty}$ is weakly precompact in $W^{1,2}(\omega;\mathbb{R}^{2})$. 2. 2.

The sequence $(\theta^{1/2}u_{\theta},v_{\theta})_{\theta\downarrow 0}$ is weakly precompact in $W^{1,2}(\omega;\mathbb{R}^{2})\times W^{2,2}(\omega)$ and the weak limit is in $X_{u}\times X_{v}$.

Proof.

By assumption:

[TABLE]

and the uniform lower bound on $Q_{2}$ in (4) yields

[TABLE]

so that $\int_{-1/2}^{1/2}Q_{2}(x_{3},F(x_{3}))\gtrsim\int_{-1/2}^{1/2}|F(x_{3})|^{2}$. Now split the inner integral in half, and normalise to use Jensen’s inequality. In the upper half:

[TABLE]

An analogous computation for the lower half of the interval results in

[TABLE]

and bringing both bounds together we obtain:

[TABLE]

Two applications of Poincaré’s inequality to the second bound yield:

[TABLE]

Therefore a subsequence (not relabelled) $v_{\theta}\rightharpoonup v$ for some $v\in X_{v}$. Now consider (30) again and observe that with the Sobolev embedding $W^{1,2}(\omega)\hookrightarrow L^{4}(\omega)$ we know that

[TABLE]

Together with (30) this implies

[TABLE]

so, by the Korn-Poincaré inequality, the sequence $(u_{\theta})_{\theta>0}$ is bounded in $W^{1,2}$ when $\theta\rightarrow\infty$ and there exists a subsequence (not relabelled) $u_{\theta}\rightharpoonup u$ for some $u\in X_{u}$.

Now if $z_{\varepsilon}\rightharpoonup z$ in $W^{1,2}(\omega;\mathbb{R}^{2})$, by the compact Sobolev embedding $W^{1,2}\hookrightarrow L^{4}$ we have $z_{\varepsilon}\rightarrow z$ in $L^{4}$ and

[TABLE]

So $\nabla v_{\theta}\otimes\nabla v_{\theta}\rightarrow\nabla v\otimes\nabla v$ in $L^{2}$ and from (30) and lower semicontinuity of the norm we deduce

[TABLE]

By [22, Proposition 9] $v\in W^{2,2}_{sh}(\omega)$ since $\omega$ is convex, and this concludes the proof of the first statement.

For the second statement we take $\theta\downarrow 0$. It only remains to prove precompactness for $u_{\theta}$ since the previous computation for $(v_{\theta})_{\theta>0}$ applies for all $\theta$. But it follows directly from (31) above: again with the Korn-Poincaré inequality, the sequence $(\theta^{1/2}u_{\theta})_{\theta>0}$ is bounded in $W^{1,2}$, so it contains a weakly convergent subsequence $\theta^{1/2}u_{\theta}\rightharpoonup u\in X_{u}$. ∎

We begin the proof of $\Gamma$-convergence in Theorem 2 with the lower and upper bound and a few technical lemmas for the passage from $\alpha=3$ to $\alpha\char 60\relax 3$.

Theorem 8** **(Lower bound, von Kármán to linearised Kirchhoff)

Assume $\omega$ is convex and let $(u_{\theta},v_{\theta})_{\theta>0}$ be a sequence in $X_{w}$ such that $v_{\theta}\rightharpoonup v$ in $X_{v}$ as $\theta\rightarrow\infty$. Then

[TABLE]

Proof.

By Theorem 7 we only need to consider $v\in X_{v}^{0}\coloneqq X_{v}\cap W^{2,2}_{sh}(\omega)$, hence $\mathcal{I}_{\rm lKi}(v)\char 60\relax\infty$. We can minimise the inner integral pointwise and obtain a lower bound:

[TABLE]

As $\overline{Q}_{2}^{\star}$ is a convex quadratic form, we have by the convergence $\nabla^{2}v_{\theta}\rightharpoonup\nabla^{2}v$ in $L^{2}$:

[TABLE]

∎

Theorem 9** **(Upper bound, von Kármán to linearised Kirchhoff)

Assume $\omega$ is convex. Set $X^{0}_{v}\coloneqq X_{v}\cap W^{2,2}_{sh}(\omega)$ and fix some displacement $v\in X_{v}$. There exists a sequence $(u_{\theta},v_{\theta})_{\theta\uparrow\infty}\subset X_{w}$ such that $v_{\theta}\rightharpoonup v$ in $W^{2,2}(\omega)$ and $\mathcal{I}_{\rm vK}^{\theta}(u_{\theta},v_{\theta})\rightarrow\mathcal{I}_{\rm lKi}(v)$ as $\theta\rightarrow\infty$.

Proof.

By Theorem 12 we can work with functions $v\in\mathcal{V}_{0}$, see (33), which are smooth with singular Hessian, since they are dense in the restriction to $X_{v}$. By [22, Proposition 9] there exists a displacement $u:\omega\rightarrow\mathbb{R}^{2}$ in $W^{2,2}(\omega;\mathbb{R}^{2})$ such that

[TABLE]

Fix $\delta>0$ and, using Corollary 1, choose smooth functions $\alpha\in C^{\infty}(\overline{\omega}),g\in C^{\infty}(\overline{\omega};\mathbb{R}^{2})$ such that

[TABLE]

where $A_{\min}\in L^{\infty}(\omega;\mathbb{R}^{2\times 2}_{\operatorname{sym}})$ is defined as

[TABLE]

Define now the recovery sequence $(u_{\theta},v_{\theta})_{\theta>0}$ with

[TABLE]

Clearly $v_{\theta}=v-\theta^{-1/2}\alpha\rightarrow v$ as $\theta\rightarrow\infty$ in $W^{2,2}(\omega)$. Furthermore

[TABLE]

[TABLE]

and

[TABLE]

so that, using (32) and the fact that the product $\|\nabla\alpha\otimes\nabla\alpha\|_{0,2}=\|\nabla\alpha\|_{0,4}^{2}$ is bounded we have

[TABLE]

Now subtract and add $A_{\min}$ inside $Q_{2}$ and use Cauchy’s inequality to get

[TABLE]

We plug this in and obtain:

[TABLE]

The proof is concluded by letting $\delta\rightarrow 0$ and passing to a diagonal sequence. ∎

We finish the proof of Theorem 2 with the lower and upper bounds for the transition from $\alpha=3$ to $\alpha>3$. The lack of constraints in the limit functional makes the proofs straightforward.

Theorem 10** **(Lower bound, von Kármán to linearised von Kármán)

Let $(u_{\theta},v_{\theta})_{\theta>0}$ be a sequence in $X_{w}$ such that $(\theta^{1/2}u_{\theta},v_{\theta})\rightarrow(u,v)$ in $X_{w}$ as $\theta\rightarrow 0$. Then

[TABLE]

Proof.

We may assume that $\sup_{\theta>0}\mathcal{I}^{\theta}_{\rm vK}(u_{\theta},v_{\theta})\leqslant C$. Then by Theorem 7 $(\nabla v_{\theta})_{\theta>0}$ is bounded in $W^{1,2}$ and by the Sobolev embedding $W^{1,2}\hookrightarrow L^{4}$ we have as before $\|\nabla v_{\theta}\otimes\nabla v_{\theta}\|_{0,2}=\|\nabla v_{\theta}\|_{0,4}^{2}\leqslant C$. Consequently

[TABLE]

By convexity of the quadratic form $Q_{2}$ we conclude

[TABLE]

∎

Theorem 11** **(Upper bound, von Kármán to linearised von Kármán)

Let $(u,v)\in X_{w}$. There exists a sequence $(u_{\theta},v_{\theta})_{\theta>0}\subset X_{w}$ converging to $(u,v)\in X_{w}$ such that $\mathcal{I}_{\rm vK}^{\theta}(u_{\theta},v_{\theta})\rightarrow\mathcal{I}_{\rm lvK}(u,v)$ as $\theta\rightarrow 0$.

Proof.

Define

[TABLE]

Clearly $(\theta^{1/2}u_{\theta},v_{\theta})\equiv(u,v)$ and using again $W^{1,2}\hookrightarrow L^{4}$ we have:

[TABLE]

Consequently:

[TABLE]

as stated. ∎

7 Approximation and representation

theorems

A key ingredient in the proofs of the upper bounds is the density of certain smooth functions in the space where the energy is minimised. In particular, for the case $\alpha\in(2,3)$ we obtain a result proving that $W^{2,2}$ maps with singular Hessian can be approximated by a specific set of smooth functions with the same property. In order to apply the results of [42] we may restrict ourselves to isometries which partition $\omega$ into finitely many so-called bodies and arms. More precisely, suppose $y:\omega\to\mathbb{R}^{3}$ is a $W^{2,2}$ isometric immersion and denote by $\operatorname{II}=\operatorname{II}_{(y)}$ its second fundamental form, i.e., $\operatorname{II}_{ij}=y_{,i}\cdot(y_{,1}\wedge y_{,2})_{,j}$. Then $\operatorname{II}$ is singular, and there exists $f_{y}\in W^{1,2}$ such that $\nabla f_{u}=\operatorname{II}$. We call $\gamma:[0,l]\to\omega$, parameterised by arclength, a leading curve if it is orthogonal to the inverse images of $f_{y}$ on regions where $f_{y}$ is not constant. We denote by $\kappa$ and $\nu$ the curvature and unit normal, respectively, i.e., $\gamma^{\prime\prime}=\kappa\nu$. In fact, $\kappa$ must be bounded, hence $\gamma\in W^{2,\infty}$. A subdomain $\omega^{\prime}\subset\omega$ is said to be covered by a curve $\gamma$ if

[TABLE]

As shown in [38], $\omega$ can be partitioned into so-called bodies and arms. Here a body is a connected maximal subdomain on which $y$ is affine and whose boundary contains more than two segments inside $\omega$. An arm is a maximal subdomain $\omega(\gamma)$ covered by some leading curve $\gamma$.

In [42] (built on [38]) it is shown that the set

[TABLE]

is dense in the $W^{2,2}$-isometries. Here we show that, additionally,

[TABLE]

is $W^{2,2}$-dense in $W^{2,2}_{sh}$.171717The density of $C^{2}(\omega)\cap W^{2,2}_{sh}(\omega)$ in $W^{2,2}_{sh}(\omega)$ was first announced in [38] to follow along the same lines as the density of smooth isometric immersions in the class of $W^{2,2}$ isometric immersions. As this seems not to be straightforward, we follow a different route reducing the density of $\mathcal{V}_{0}$ in $W^{2,2}_{sh}$ to the density of $\mathcal{A}_{0}$ in the set of $W^{2,2}$ isometric immersions. We are grateful to Peter Hornung for the help provided with this proof.

Theorem 12

Let $\omega\subset\mathbb{R}^{2}$ be a bounded, convex, Lipschitz domain. Then the set $\mathcal{V}_{0}$ is $W^{2,2}$-dense in $W^{2,2}_{sh}(\omega)$. In particular $\det\nabla^{2}v=0$ for all $v\in\mathcal{V}_{0}$.

Proof.

Step 1: Approximation

Let $v\in W^{2,2}_{sh}(\omega)$ and $\varepsilon>0$. By [22, Theorem 10], we can find some $\tilde{v}\in W^{2,2}_{sh}(\omega)\cap W^{1,\infty}(\omega)$ s.t. $\|v-\tilde{v}\|_{2,2}\char 60\relax\varepsilon/2$ and, for $\eta=\eta(\varepsilon)>0$ sufficiently small, $\|\nabla\eta\tilde{v}\|_{\infty}\char 60\relax 1/2$. One can now apply [22, Theorem 7] to construct an isometry $\tilde{y}\in W^{2,2}(\omega;\mathbb{R}^{3})$ whose out-of-plane component $\tilde{y}_{3}=\eta\tilde{v}$. By [42, Proposition 2.3] we find a smooth $y\in\mathcal{A}_{0}$ such that $\|y-\tilde{y}\|_{2,2}\char 60\relax\varepsilon\eta/2$ and in particular $\|y_{3}-\tilde{y}_{3}\|_{2,2}\char 60\relax\varepsilon\eta/2$. Setting $\psi\coloneqq y_{3}/\eta\in\mathcal{V}_{0}$ we conclude

[TABLE]

Step 2: Inclusion

Let $v\in\mathcal{V}_{0}$ with $\eta v=y_{3},\eta>0$ for some smooth isometry $y\in\mathcal{A}_{0}$. Recall that the second fundamental form $\mathrm{I}\mathrm{I}_{(y)}$ of any smooth isometric immersion $y$ is singular and the identity $\nabla^{2}y_{j}=-\mathrm{I}\mathrm{I}_{(y)}n_{j}$ holds for all $j\in\{1,2,3\}$, where $n=y_{,1}\wedge y_{,2}$.181818See [35, Proposition 3] for a proof for $W^{2,2}$ isometries on Lipschitz domains. Therefore $\det(\eta\nabla^{2}v)=\det(-\operatorname{II}_{(y)}n_{3})=0$ and the proof is complete. ∎

Remark 3

We note that the following similar statement can be proved using the same approximation arguments and [23, Theorem 1] (with the bonus of in addition holding for more general domains). Let $\omega\subset\mathbb{R}^{2}$ be a bounded, simply connected, Lipschitz domain whose boundary contains a set $\Sigma=\overline{\Sigma}\subset\partial\omega$ with $\mathcal{H}^{1}(\Sigma)=0$ such that on its complement $\partial\omega\backslash\Sigma$ the outer unit normal to $\omega$ exists and is continuous. Then the set $W^{2,2}_{sh}(\omega)\cap C^{\infty}(\overline{\omega})$ is $W^{2,2}$-dense in $W^{2,2}_{sh}(\omega)$.

Once one can work with smooth functions, the essential tool for the construction of the recovery sequences for $\alpha\in(2,3)$ is the following representation theorem for maps with singular Hessian and its corollary, both inspired by [42]. A crucial component in the proof of the result in that paper is the ability to use approximations partitioning the domain in regions over which they are affine. This is in close connection to the rigidity property for $W^{2,2}$-isometries proved in [38, Theorem II]: every point of their domain lies either on an open set or on a segment connecting the boundaries where the map is affine.

Theorem 13

Let $v\in\mathcal{V}_{0}$ and $A\in C^{\infty}(\overline{\omega};\mathbb{R}^{2\times 2}_{\operatorname{sym}})$ such that $A\equiv 0$ in a neighbourhood of $\{\nabla^{2}v=0\}$. There exist maps $\alpha,g_{1},g_{2}\in C^{\infty}(\overline{\omega})$ such that $\alpha=g_{i}=0$ on $\{\nabla^{2}v=0\}$ and

[TABLE]

Proof.

Let $\eta>0,y\in\mathcal{A}_{0}$ s.t. $\eta v=y_{3}$. Using that $\nabla^{2}y_{3}=-\mathrm{I}\mathrm{I}_{(y)}n_{3}$ holds by virtue of $y$ being an isometry, with $n=y_{,1}\wedge y_{,2}$ being the unit normal vector, we have that $A\equiv 0$ in a neighbourhood of $\{\operatorname{II}_{(y)}=0\}\cup\{n_{3}=0\}$, and

[TABLE]

We can apply [42, Lemma 3.3]191919Namely: If $y\in\mathcal{A}_{0}$ and $A\in C^{\infty}(\overline{\omega};\mathbb{R}^{2\times 2}_{\operatorname{sym}})$ vanishes over a neighbourhood of $N=\{\operatorname{II}_{(y)}=0\}$, then there exist $\tilde{\alpha},g_{1},g_{2}\in C^{\infty}(\overline{\omega})$ vanishing on $N$ such that $A=\nabla_{s}g+\tilde{\alpha}\operatorname{II}_{(y)}$. to $y$ in order to obtain functions $\tilde{\alpha},g_{1},g_{2}\in C^{\infty}(\overline{\omega})$ s.t. $\tilde{\alpha},g_{1},g_{2}=0$ on $\{\operatorname{II}_{(y)}=0\}$ and $A=\nabla_{s}g+\tilde{\alpha}\operatorname{II}_{(y)}$.

By examining the proof of this Lemma one can see that $\tilde{\alpha},g\equiv 0$ in a neighbourhood of $\left\{n_{3}=0\right\}$: since over bodies one has $\tilde{\alpha},g_{1},g_{2}=0$ by construction, we need only consider arms. On these sets, if $n_{3}$ vanishes at a point then it vanishes at a whole line perpendicular to the leading curve, because the latter is orthogonal to the level sets of the gradient. Now, because $A=0$ in a neighbourhood of this line, when solving the equations in the proof of the Lemma which determine $g$ then $\tilde{\alpha}$, one obtains $u_{2,s}=0$ and $u_{2,t}=0$, and with the boundary conditions $u_{2}=0$ then $u_{1}=0$ is a solution to the remaining equation. Hence $g=0$ and $\tilde{\alpha}=0$ on these lines. Since the functions so obtained are $C^{\infty}$, we can define $\alpha\coloneqq-\tilde{\alpha}\eta/n_{3}$ if $n_{3}\neq 0$ and $\alpha=0$ otherwise, and this is a smooth function such that

[TABLE]

∎

Corollary 1

Let $v\in\mathcal{V}_{0}$ and define for every $x^{\prime}\in\omega$

[TABLE]

Then $A_{\min}\in L^{2}(\omega;\mathbb{R}^{2\times 2}_{\operatorname{sym}})$ and there exist sequences of functions $\alpha_{k}\in C^{\infty}(\overline{\omega}),g_{k}\in C^{\infty}(\overline{\omega};\mathbb{R}^{2})$ such that

[TABLE]

Proof.

Let $k\in\mathbb{N}$ be arbitrary. First, on the set $\{\nabla^{2}v=0\}$ we trivially have $A_{\min}\equiv A_{0}$ a constant matrix. Now let $A_{k}\in C^{\infty}(\overline{\omega};\mathbb{R}^{2\times 2})$ with support in $\{\nabla^{2}v\neq 0\}$ such that

[TABLE]

and use Theorem 13 to pick smooth $\alpha_{k},\tilde{g}_{k}$ on $\overline{\omega}$ with

[TABLE]

Set $g_{k}(x^{\prime})=\tilde{g}_{k}(x^{\prime})+A_{0}x^{\prime}$. Then:

[TABLE]

∎

Acknowledgements

We are grateful to Peter Hornung for the help provided with the proof of Theorem 12. This work was financially supported by project 285722765 of the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation), “Effektive Theorien und Energie minimierende Konfigurationen für heterogene Schichten”.