Heterogeneous elastic plates with in-plane modulation of the target curvature and applications to thin gel sheets

TL;DR

This paper derives a rigorous 2D Kirchhoff plate theory for heterogeneous elastic sheets with in-plane modulated target curvature, and applies it to model shape changes in swelling gel sheets, advancing understanding of thin elastic structures.

Contribution

It introduces a novel derivation of a Kirchhoff plate model from 3D elasticity for heterogeneous materials with in-plane modulated spontaneous strains, using $ ext{Gamma}$-convergence.

Findings

Derived a 2D Kirchhoff plate model from 3D elasticity for heterogeneous sheets.

Characterized the energy penalizing deviations from target curvature.

Applied the model to swelling-induced shape changes in gel sheets.

Abstract

We rigorously derive a Kirchhoff plate theory, via $\Gamma$-convergence, from a three-di\-men\-sio\-nal model that describes the finite elasticity of an elastically heterogeneous, thin sheet. The heterogeneity in the elastic properties of the material results in a spontaneous strain that depends on both the thickness and the plane variables $x'$. At the same time, the spontaneous strain is $h$-close to the identity, where $h$ is the small parameter quantifying the thickness. The 2D Kirchhoff limiting model is constrained to the set of isometric immersions of the mid-plane of the plate into $\mathbb{R}^3$, with a corresponding energy that penalizes deviations of the curvature tensor associated with a deformation from a $x'$-dependent target curvature tensor. A discussion on the 2D minimizers is provided in the case where the target curvature tensor is piecewise constant. Finally, we apply the derived plate theory to the modeling of swelling-induced shape changes in heterogeneous thin gel sheets.

Equations

$$N={\overline{N}}^h(z)=\overline{N}+hg\left(z^{\prime },\frac{z_3}{h}\right),z=(z^{\prime },z_3)\in {\Omega }_h,$$

$$\big{(}\alpha+hb(z^{\prime},z_{3}/h)\big{)}{\rm SO}(3),\qquad b=\Theta g,$$

$$\overline{W}^{h}(z,F)\,=\,W\big{(}FV^{h}(z^{\prime},z_{3}/h)\big{)},$$

$${\overline{U}}^h(z)={\mathbb{I}}_3+hB\left(z^{\prime },\frac{z_3}{h}\right),$$

$$\overline{\mathcal{E}}^{h}(v)\,=\,\int\limits_{\Omega_{h}}\overline{W}^{h}\big{(}z,\nabla v(z)\big{)}\,{\rm d}z.$$

$${\rm curl}\big{(}{\rm curl}\,\check{D}\big{)}\,=\,0,\qquad\quad\mbox{with}\qquad\check{D}(z^{\prime}):=\!\!\int\limits_{\nicefrac{{-1}}{{2}}}^{\nicefrac{{1}}{{2}}}\!\!\check{B}(z^{\prime},t)\,{\rm d}t\qquad\mbox{for a.e. }z^{\prime}\in\omega,$$

$${\mathcal{E}}^0(y)=\frac{1}{24}{\int }_{\omega }{Q}_2\left({\mathrm{A}}_y(z^{\prime })-\overline{A}(z^{\prime })\right)\mathrm{d}{z}^{\prime }+ad.t.,$$

$$\overline{A}(z^{\prime })=12{\int }_{\text{nicefrac}-12}^{\text{nicefrac}12}t\check{B}(z^{\prime },t)\mathrm{d}t,\text{mbox}fora.e.z^{\prime }\in \omega .$$

$$\hat{G}=\left(\begin{array}[]{c|c}G&\begin{array}[]{c}0\\ 0\end{array}\\ \hline\cr 0\quad 0&0\end{array}\right).$$

$$\text{mbox}thereexistsaclosedsubset\Sigma \subset \partial \omega \text{mbox}with{\mathcal{H}}^1(\Sigma )=0\text{mbox}suchthat$$

$$\text{mbox}theouterunitnormalexistsandiscontinuouson\partial \omega \setminus \Sigma .$$

$$B^h\to B^0=:B\text{mbox}in{\mathrm{L}}^{\infty }(\Omega ,\mathrm{Sym}(3)),\text{mbox}ash\to 0,$$

$$U^h(x):={\mathbb{I}}_3+h{B}^h(x)\text{mbox}fora.e.x\in \Omega \text{mbox}andforeveryh\ge 0,$$

$$W^h(x,F)=W^h(x,RF)\text{mbox}foreveryF\in {\mathbb{R}}^{3\times 3}\text{mbox}andeveryR\in \mathrm{SO}(3);$$

$$\operatorname*{ess\,sup}_{x\in\Omega}\big{\|}W^{h}(x,\cdot)-W\big{\|}_{C^{2}(\mathcal{\overline{U}})}\to 0,\qquad\mbox{ as }h\to 0;$$

$$W^{h}(x,F)\geq C{\rm dist}^{2}\big{(}F,{\rm SO}(3)U^{h}(x)\big{)},\qquad\mbox{for every }F\in\mathbb{R}^{3\times 3}.$$

$$Q_3(F):=D^{2}W({\mathbb{I}}_3)[F,F],\text{mbox}foreveryF\in {\mathbb{R}}^{3\times 3}.$$

$$Q_{2}(G):=\min_{d\in\mathbb{R}^{3}}Q_{3}\big{(}\hat{G}+d\otimes\mathsf{f}_{3}\big{)},$$

$$\overline{Q}_{2}(x^{\prime},G):=\min_{D\in\mathbb{R}^{2\times 2}}\int_{\nicefrac{{-1}}{{2}}}^{\nicefrac{{1}}{{2}}}Q_{2}\big{(}D+tG-\check{B}(x^{\prime},t)\big{)}\,{\rm d}t,$$

$$\overline{Q}_{2}(x^{\prime},G)=\min_{D\in{\rm Sym}(2)}\int_{\nicefrac{{-1}}{{2}}}^{\nicefrac{{1}}{{2}}}Q_{2}\big{(}D+tG-\check{B}(x^{\prime},t)\big{)}\,{\rm d}t.$$

$$D_{\min }(x^{\prime }):={\int }_{\text{nicefrac}-12}^{\text{nicefrac}12}\check{B}(x^{\prime },t)dt.$$

$${\overline{Q}}_2(x^{\prime },G)={\int }_{\text{nicefrac}-12}^{\text{nicefrac}12}{Q}_2\left({\int }_{\text{nicefrac}-12}^{\text{nicefrac}12}\check{B}(x^{\prime },s)\mathrm{d}s+tG-\check{B}(x^{\prime },t)\right)\mathrm{d}t$$

$$\underset{D\in \mathrm{Sym}(2)}{\min }{\int }_{\text{nicefrac}-12}^{\text{nicefrac}12}{Q}_2(D+tG-\check{B}(x^{\prime },t))\mathrm{d}t$$

$$=$$

$$\overline{Q}_{2}(x^{\prime},G)\,=\,\frac{1}{12}Q_{2}\left(G-12\int_{\nicefrac{{-1}}{{2}}}^{\nicefrac{{1}}{{2}}}t\check{B}(x^{\prime},t)\,{\rm d}t\right)+\int_{\nicefrac{{-1}}{{2}}}^{\nicefrac{{1}}{{2}}}Q_{2}\big{(}\check{B}(x^{\prime},t)\big{)}\,{\rm d}t\\ -Q_{2}\left(\int_{\nicefrac{{-1}}{{2}}}^{\nicefrac{{1}}{{2}}}\check{B}(x^{\prime},t)\,{\rm d}t\right)-12Q_{2}\left(\int_{\nicefrac{{-1}}{{2}}}^{\nicefrac{{1}}{{2}}}t\check{B}(x^{\prime},t)\,{\rm d}t\right),$$

$$\bigg{|}W^{h}\Big{(}x,U^{h}(x)+F\Big{)}-W\Big{(}\mathbb{I}_{3}+F\Big{)}\bigg{|}\leq\varepsilon|F|^{2},$$

$$\Big{|}W^{h}\Big{(}x,U^{h}(x)+F\Big{)}\Big{|}\leq C|F|^{2}.$$

$${\rho }^0(F):=W({\mathbb{I}}_3+F)-\frac{1}{2}{D}^{2}W({\mathbb{I}}_3)[F{]}^2\text{mbox}and\rho (s):=\underset{|F|\le s}{\sup }|{\rho }^0(F)|$$

$$\rho (s)/s^2\to 0\text{mbox}ass\to 0.$$

$$\nabla^{\prime}y:=\left(\partial_{1}y\bigg{|}\,\partial_{2}y\right)\qquad\mbox{and}\qquad\nabla_{h}y:=\left(\nabla^{\prime}y\,\bigg{\lvert}\,\frac{1}{h}\partial_{3}y\right).$$

$$\mathcal{E}^{h}(y):=\int_{\Omega}W^{h}\big{(}x,\nabla_{h}y(x)\big{)}\,{\rm d}x,\quad\quad\mbox{for every }\,y\in{\rm W}^{1,2}(\Omega,\mathbb{R}^{3}).$$

Full text

1. 1 Introduction

2. 1.1 Notation

3. 2 Three-dimensional model and derivation of the corresponding Kirchhoff plate model

4. 3 2D energy minimizers

5. 3.1 $x^{\prime}$-dependent target curvature tensor $\overline{A}$ and pointwise minimizers

6. 3.2 The case of piecewise constant $\overline{A}$.

7. 4 Application to thin gel sheets

8. 5 Conclusions

1. Introduction

Plants [6, 15] and other natural systems [8, 7] are able to perform complex shape changes that produce curved configurations, often starting from flat initial states. These shape changes usually involve thin structures, such as membranes, plates or shells, and exploit some internal activation or the responsiveness of the material to non-mechanical external triggers, such as changes in humidity. By mimicking natural behaviors and architectures, synthetic, polymer-based thin sheets have been fabricated that can spontaneously deform in response to non-mechanical stimuli. In particular, in these systems curvature arises from heterogeneous in-plane [16, 26, 25, 45, 37] or through-the-thickness strains [1, 2, 38, 39, 40, 44], which are induced by heterogeneous material properties, including variable anisotropy. Thus, to study and control the emerging shapes, the derivation of plate theories for materials with heterogeneous response to external stimuli has become as a topic of interest in both the mathematical and the physical literature [9, 5, 10, 43, 33, 34].

In this framework, we wish to contribute by drawing attention to some new plate theories and corresponding mathematical problems in dimension reduction inspired by shape morphing applications involving polymer gels. Specifically, in these applications one wants to program the material properties of a thin gel sheet $\Omega_{h}=\omega\times(-h/2,h/2)$, where $\omega\subseteq\mathbb{R}^{2}$ is the mid-plane and $0<h\ll 1$ is the thickness, in order to endow it with a controlled curvature that is realizable, upon swelling, at the minimum energy cost. Practically, curvature of the sheet can be obtained by imprinting a heterogeneous density $N$ of polymer chains, which corresponds to a heterogeneous shear modulus of the polymer network. For concreteness, we consider the case where $N$ is a (small) perturbation of order $h$ of the average value $\overline{N}$, that is,

[TABLE]

for some bounded function $g:\omega\times(-1/2,1/2)\to\mathbb{R}$. Referring to the classical Flory-Rehner model [17] for isotropic polymer gels, we obtain as a consequence of the above assumption on $N$ that the free energy density $\overline{W}^{h}$ associated with the system is minimized at

[TABLE]

where the (dimensionless) constants $\alpha$ and $\Theta$ are functions of the material parameters appearing in the expression of $\overline{W}^{h}$, including $\overline{N}$. We refer the reader to (4.4) for the explicit expression of $\overline{W}^{h}$ and to the whole Section 4 for more details on this 3D model. We recall that $\alpha$ corresponds to the free-swelling stretch of a homogeneous gel (i.e. $g=0$ in (1.1)) with respect to its dry state, and is hence greater than one.

Intuitively, the connection between the density of polymer chains in (1.1) and the energy minimizers in (1.2) offers a way to program minimum-energy strain fields that, as we will see in the following, induce a target curvature for the system. This mechanism of generation of curvature through heterogeneous elastic properties is poorly explored in the mathematical literature of active or “pre-strained” materials, and thus constitutes a novel ingredient of our theory. Tipically, these materials are modeled by 3D energy densities of the form (see, for example, [42])

[TABLE]

for a certain (frame indifferent) homogeneous energy density $W$ minimized at ${\rm SO}(3)$, where the “pre-stretch” $V^{h}$ is generally a smooth, invertible tensor field that represents (the inverse of) an active stretch, growth, plasticity or other inelastic phenomena. In these models, $V^{h}$ plays the role of a parameter that is externally controlled, without any dependence on the elastic properties of the system (or on other parameters of the energy). Instead, in models based on the Flory-Rehner energy and on the relations (1.1)-(1.2), there is an intimate connection between material parameters and minimum energy deformations.

Another interesting feature of the Flory-Rehner energy and, correspondingly, of the family of energy densities we consider, is that they are not representable in the pre-stretch form (1.3), with V^{h}=\big{(}1+hb/\alpha\big{)}^{-1}\mathbb{I}_{3} and with energy density $W$ minimized at $\alpha{\rm SO}(3)$ (see Remark 4.1). This feature depends on the different structure of such an energy with respect to the models based on the representation (1.3), which originates from physical considerations. More precisely, in the latter models $\overline{W}^{h}$ has the physical meaning of purely elastic energy, while in models for polymer gels $\overline{W}^{h}$ is the sum of two energy contributions (elastic and mixing energies) that concurrently define the energy minimum, but none of them is separately minimized at ${\rm SO}(3)(V^{h})^{-1}$. However, the corresponding rescaled densities converge uniformly to some homogeneous density $W$.

Motivated by the above observations and discarding for the moment the scalar parameter $\alpha$, which can be accounted for by a simple change of variable, we thus consider the slightly more general setting of a material characterized by a spontaneous stretch distribution $\overline{U}^{h}$ of the form

[TABLE]

where $B:\omega\times(-1/2,1/2)\to{\rm Sym}(3)$ is a given (bounded) strain distribution.

The term “spontaneous” for the distribution $\overline{U}^{h}$ refers to the tendency of the system to deform, at each point $z$, according to a deformation whose gradient coincides with $\overline{U}^{h}(z)$, in order to attain the energy minimum pointwise. However, generally there is no (orientation-preserving) deformation defined globally in $\Omega_{h}$, whose gradient coincides with $\overline{U}^{h}$ in the whole of $\Omega_{h}$. Equivalently, in the words of Mechanics, $\overline{U}^{h}$ is not kinematically compatible, or, in the words of Riemannian geometry, the Riemann curvature tensor associated with $\overline{U}^{h}$ does not vanish identically throughout $\Omega_{h}$.

It is now appropriate to notice that the 3D setting just described can be seen as a generalization of the setting considered in [42] (see also [41]), where the pre-stretch is of the same form as in (1.4), except that the $z^{\prime}$-dependence is not considered. At the same time, the relevant case where the pre-stretch in (1.3) is only $z^{\prime}$-dependent has been addressed in [31] and [10] and has given rise to the fortunate route of the mathematical treatment of the “non-Euclidean plate theories” (see also [28]), introduced from a physical and mechanical view point by the pioneering work of Sharon and coauthors in [18] and [26]. All in all, our theory stands between those of [42], on one hand, and of [31] and [10], on the other hand, and, to the best of our knowledge, represents the first attempt to considering Kirchhoff plate theories originated by 3D energies characterized by pre-stretches or spontaneous stretches which are heterogeneous in plane as well as along the thickness. Pre-stretches of the form (1.4) have been very recently treated in [13] and [27] to derive corresponding rod models with misfit. Moreover, similar prestretches have been considered in [30] to obtain 2D models in the case of scaling orders higher than the Kirchhoff one.

The central result of this paper is the derivation of a Kirchhoff plate theory from the 3D model outlined above. With abuse of notation, we again denote by $\overline{W}^{h}$ the energy density associated with this system, which is minimized, for every $z\in\Omega_{h}$, at ${\rm SO}(3)\overline{U}^{h}(z)$. Hence, the total free energy associated with a deformation $v:\Omega_{h}\to\mathbb{R}^{3}$ is

[TABLE]

Then, the same arguments as in [42] (which are in turn a slight variant of those employed in the seminal work [20]) can be used to find the corresponding limiting Kirchhoff plate model, under the assumption that

[TABLE]

where $\check{B}:\omega\to{\rm Sym}(2)$ is obtained from the spontaneous strain distribution $B$ appearing in (1.4) by omitting the third row and the third column. Condition (1.5) deserves some comments. It guarantees that $\check{D}$ is a symmetrized gradient, and in turn allows for the construction of a standard ansatz for the recovery sequence. When instead condition (1.5) is violated, usual arguments such as local modifications or perturbation arguments seem insufficient to prove the same $\Gamma$-limit. In fact, we believe that the general $\Gamma$-limit has to include a nonlocal term, which can be interpreted as a “first order stretching term”. To conclude the comments on condition (1.5), let us add a trivial but important observation: the difficulties one encounter in removing the compatibility assumption on the matrix field $\check{D}$ do not originate from the dependence of the spontaneous strain on the thickness variable, since they persist even in the case where such a dependence is absent.

The Kirchhoff model resulting from the dimension reduction is governed by the energy functional

[TABLE]

on each ${\rm W}^{2,2}$-isometry $y$, where ad.t. stays for “additional terms” not depending on $y$.

In the above expression, the quadratic form $Q_{2}$ is defined via a standard relaxation of the second differential of the limiting density $W$ at $\mathbb{I}_{3}$ (see formulas (2.6) and (2.7)), the symbol ${\rm A}_{y}$ stands for the pull-back of the second fundamental form associated with $y(\omega)$ (see (2.21)), and the target curvature tensor $\overline{A}$ is defined as

[TABLE]

It is readily seen that in the case where the prestretch depends only on the thickness variable, a constant target cuvature $\overline{A}$ is produced. For polymer gels, this expression makes the anticipated connection between heterogeneous density $N$ of polymer chains (encoded by $\check{B}$) and curvature more evident, even if not fully explicit. Further, under some approximations or using numerical methods, such a relation can be made explicit and thus can be actually employed in the design of shape morphing gel plates. In general, our derivation, which relies on an accurate description of the 3D swelling energy, offers an advantage over theories based on purely elastic energies with “pre-stretch”, where such a connection must be plugged in artificially.

It is worth mentioning that beam theories derived from 2D energies of the form (1.6), in the limit as $\varepsilon\to 0$ when $\omega=(-\ell/2,\ell/2)\times(\varepsilon/2\times\varepsilon/2)$, can be found in [3] for the case $\overline{A}$ constant and in [19] in the case $\overline{A}=\overline{A}(x_{1})$. To use a common terminology, these 1D theories may describe narrow ribbons of soft active materials.

To give some insight on the minimizers of the derived 2D model (1.6), we focus on the case where the spontaneous strain $B$ is an odd function of the thickness variable (which trivially fulfills condition (1.5)), being at the same time a piecewise constant function of the planar variable. This case leads in turn to a piecewise constant target curvature tensor. In Section 3, we recall that in the case where $\overline{A}$ is constant, then a minimizer of the 2D energy $\mathcal{E}^{0}$ actually minimizes the integrand function pointwise, and the corresponding deformed configuration is a piece of cylindrical surface (see Lemma 3.3 and the discussion preceding it). In the case of a piecewise constant $\overline{A}$, some conditions (specified in Theorem 3.9) under which cylindrical surfaces can be patched together resulting into an isometry must be fulfilled for the pointwise minimizer to exist. When these conditions hold, an example of minimum energy configuration, a patchwork of cylindrical surfaces, is sketched in Figure 1.

The paper is structured as follows: we deal with the theoretical results concerning dimension reduction in Sections 2 and 3, and then we apply them to the case of thin gel sheets, in Section 4. In the final section, we draw some conclusions.

We end this section by introducing some general notation which will be used throughout the paper.

1.1. Notation

For fixed $n\in\mathbb{N}$ we will denote by

• •

$\mathbb{R}^{n\times n}$ the vector space of real $n\times n$ matrices and by $\mathbb{I}_{n}\in\mathbb{R}^{n\times n}$ the identity matrix,

• •

${\rm Sym}(n):=\{M\in\mathbb{R}^{n\times n}:M^{\scriptscriptstyle{\mathsf{T}}}=M\}$ the vector space of symmetric matrices, where by $M^{\scriptscriptstyle{\mathsf{T}}}\in\mathbb{R}^{n\times n}$ we denote the transpose of the matrix $M\in\mathbb{R}^{n\times n}$,

• •

${\rm Skew}(n):=\{M\in\mathbb{R}^{n\times n}:M^{\scriptscriptstyle{\mathsf{T}}}=-M\}$ the set of skew-symmetric matrices,

• •

${\rm SO}(n):=\{M\in\mathbb{R}^{n\times n}:M^{\scriptscriptstyle{\mathsf{T}}}M=\mathbb{I}_{n},{\rm det}(M)=1\}$ the set of all rotations of $\mathbb{R}^{n}$,

• •

${\rm Orth}(n):=\{M\in\mathbb{R}^{n\times n}:M^{\scriptscriptstyle{\mathsf{T}}}M=\mathbb{I}_{n}\}$ the set of all orthogonal transformations of $\mathbb{R}^{n}$,

• •

${\rm Trs}(n):=\{T_{v}:=\cdot+v:v\in\mathbb{R}^{n}\}$ the set of all translations in $\mathbb{R}^{n}$. Sometimes, to distinguish between translations in $\mathbb{R}^{2}$ and $\mathbb{R}^{3}$, we will denote by $\tau_{v}$ the elements of ${\rm Trs}(2)$,

• •

$M_{{\rm sym}}:=\frac{M+M^{\scriptscriptstyle{\mathsf{T}}}}{2}$ the symmetric part of the matrix $M\in\mathbb{R}^{n\times n}$,

• •

${\rm tr}\,M$ the trace of the matrix $M$ and ${\rm tr}^{2}M:=({\rm tr}M)^{2}$,

• •

$|M|:=\sqrt{\sum_{i,j=1}^{n}|m_{ij}|^{2}}=\sqrt{{\rm tr}(M^{\scriptscriptstyle{\mathsf{T}}}M)}$, Frobenius norm of a matrix $M=[m_{ij}]_{i,j=1}^{n}\in\mathbb{R}^{n\times n}$,

• •

$\mathcal{L}^{n}$ the $n$-dimensional Lebesgue measure,

• •

$\mathcal{H}^{n}$ the $n$-dimensional Hausdorff measure.

Furthermore, we give the following definitions:

• •

$\check{F}\in\mathbb{R}^{2\times 2}$ is the $2\times 2$ submatrix of $F\in\mathbb{R}^{3\times 3}$ obtained by omitting the last row and the last column of $F$,

• •

given $G\in\mathbb{R}^{2\times 2}$, the matrix $\hat{G}\in\mathbb{R}^{3\times 3}$ associated to $G$ is defined as

[TABLE]

We denote by $\{\mathsf{e}_{1},\mathsf{e}_{2}\}$ the standard basis of $\mathbb{R}^{2}$ and by $\{\mathsf{f}_{1},\mathsf{f}_{2},\mathsf{f}_{3}\}$ the standard basis of $\mathbb{R}^{3}$. An open connected subset of $\mathbb{R}^{2}$ will be called domain. Sometimes, for the sake of brevity, an open subset of $\mathbb{R}^{2}$ with Lischitz boundary will be called a Lipschitz subset of $\mathbb{R}^{2}$. The closure of a set $S\subseteq\mathbb{R}^{2}$ is denoted by $\overline{S}$ or by ${\rm cl}(S)$.

2. Three-dimensional model and derivation of the

corresponding Kirchhoff plate model

Throughout the paper $\omega\subseteq\mathbb{R}^{2}$ will be a simply-connected, bounded domain with Lipschitz boundary satisfying the following condition:

[TABLE]

The requirement that $\omega$ is a simply-connected domain has to do with the “compatibility” condition of Theorem 2.8 below, which is imposed on the tensor-valued map $D_{\rm min}$ defined by (2.2) and (2.9). The condition (2.1) is a standard requirement on the domain in order to have some density results for the space of ${\rm W}^{2,2}$-isometric immersions of $\omega$ into $\mathbb{R}^{3}$ (see, e.g., [23] and [24]).

We are interested in a thin sheet $\Omega_{h}:=\omega\times(-h/2,h/2)$, with $0<h\ll 1$, of a material characterized by a spontaneous stretch given at each point of $\Omega_{h}$ in the form \overline{U}^{h}(z)=\mathbb{I}_{3}+hB\big{(}z^{\prime},\frac{z_{3}}{h}\big{)}, for a suitable spontaneous strain $B\in{\rm L}^{\infty}(\Omega,{\rm Sym}(3))$. The stretch $\overline{U}^{h}$ being spontaneous for the material is modeled by introducing a energy density whose minimum state is precisely $\overline{U}^{h}(z)$ at each point $z$, modulo superposed rigid body rotations. We denote by $U^{h}$ the spontaneous stretch given in terms of the rescaled variable $x\in\Omega:=\Omega_{1}$. Namely, $U^{h}(x)=\overline{U}^{h}(x^{\prime},hx_{3})$ so that $U^{h}=\mathbb{I}_{3}+hB$.

More in general, we consider a family $\mathcal{B}=\{B^{h}\}_{h\geq 0}$ of spontaneous strains such that

[TABLE]

the corresponding family $\{U^{h}\}_{h\geq 0}$ of spontaneous stretches defined as

[TABLE]

and the associated family $\{W^{h}\}_{h>0}$ of (rescaled) energy density functions $W^{h}:\Omega\times\mathbb{R}^{3\times 3}\to[0,+\infty]$, which we suppose to be Borel functions satisfying the following properties:

• (i)

for a.e. $x\in\Omega$, the map $W^{h}(x,\cdot)$ is frame indifferent, i.e.

[TABLE]

• (ii)

for a.e. $x\in\Omega$, $W^{h}(x,\cdot)$ is minimized precisely at ${\rm SO}(3)U^{h}(x)$;

• (iii)

there exists an open neighbourhood $\mathcal{U}$ of ${\rm SO}(3)$ and $W\in C^{2}(\overline{\mathcal{U}})$ such that

[TABLE]

• (iv)

there exists a constant $C>0$, independent of $h$, such that for a.e. $x\in\Omega$ it holds that

[TABLE]

The most interesting scenarios occur when the Cauchy-Green distribution $C^{h}$ associated with the spontaneous stretch distribution $U^{h}$ $-$ namely, $C^{h}(x):=(U^{h}(x))^{2}$ $-$ is not kinematically compatible, i.e. there is no orientation-preserving deformation $v^{h}:\Omega\to\mathbb{R}^{3}$ such that $(\nabla v^{h})^{\scriptscriptstyle{\mathsf{T}}}\nabla v^{h}=C^{h}$ in $\Omega$. We also recall that, since $C^{h}(x)$ is a positive definite symmetric matrix, the distribution $C^{h}$ can be interpreted as a metric on $\Omega$ and that, in this framework, the kinematic compatibility of $C^{h}$ is equivalent to the condition that the Riemann curvature tensor associated with $C^{h}$ vanishes identically in $\Omega$ (see [11] and [31]).

Definition 2.1** (Admissible family of free-energy densities).**

Given $\mathcal{B}=\{B^{h}\}_{h\geq 0}$ satisfying (2.2) and the associated family $\{U^{h}\}_{h\geq 0}$ defined in (2.3), we call $\mathcal{B}\text{-}$admissible a family $\{W^{h}\}_{h>0}$ of Borel functions from $\Omega\times\mathbb{R}^{3\times 3}$ to $[0,+\infty]$ fulfilling ${\rm(i)\text{-}(iv)}$.

Given a $\mathcal{B}\text{-}$admissible family $\{W^{h}\}_{h>0}$ of free-energy densities, with associated limiting density function $W$, using a standard notation we define the following quadratic form:

[TABLE]

Moreover, for every $G\in\mathbb{R}^{2\times 2}$, we set

[TABLE]

referring to Subsection 1.1 for the notation $\hat{G}$. Observe that the limiting density $W$ inherits properties (i), (ii) and (iv) from convergence (2.4). From this fact one can deduce that $Q_{2}$ is indeed a quadratic form and that $Q_{k}$, for $k=2,3$, has the following properties:

• •

$Q_{k}$ is positive semi-definite on $\mathbb{R}^{k\times k}$ and positive definite when restricted to ${\rm Sym}(k)$,

• •

$Q_{k}(F)=0$ for every $F\in{\rm Skew}(k)$,

• •

$Q_{k}$ is strictly convex on ${\rm Sym}(k)$.

The proof of some of the listed properties can be found for instance in [10, 20]. We also refer to [14, Proposition 11.9] for a useful characterization of quadratic forms.

Our limiting 2D model will be related to the 2D density function $\overline{Q}_{2}:\omega\times\mathbb{R}^{2\times 2}\to[0,+\infty)$ defined as

[TABLE]

for a.e. $x^{\prime}\in\omega$ and every $G\in\mathbb{R}^{2\times 2}$, where $\check{B}$ is related to the 3D model through (2.2), using the notation introduced in Subsection 1.1. Since $Q_{2}$ does not depend on the skew-symmetric part of its argument,

we can think of $\overline{Q}_{2}$ to be defined only on $\omega\times{\rm Sym}(2)$ as

[TABLE]

This minimum problem can be solved explicitly, as stated by the following lemma.

Lemma 2.2**.**

For a.e. $x^{\prime}\in\omega$ and every $G\in{\rm Sym}(2)$, the minimizer in (2.8) is unique and coincides with

[TABLE]

In other words, we have that

[TABLE]

for a.e. $x^{\prime}\in\omega$ and every $G\in{\rm Sym}(2)$.

Proof.

By using the bilinear form associated with $Q_{2}$ it is easy to see that for a.e. $x^{\prime}\in\omega$ and every $G\in{\rm Sym}(2)$ it holds

[TABLE]

From this equality, the thesis trivially follows.

∎

Note that $D_{\rm min}$, which is in principle dependent on $G$ from its definition, turns out to be independent of $G$ in the end. This is not the case when, e.g., the limiting density function $W$ depends explicitly on $x_{3}$, not just through its spontaneous stretch, see [42]. Note also from hypothesis (2.2) that $D_{\rm min}\in{\rm L}^{\infty}(\omega,{\rm Sym}(2))$. Finally, observe for future reference that from (2.10) one can rewrite $\overline{Q}_{2}$ in the more explicit form

[TABLE]

for a.e. $x^{\prime}\in\omega$ and every $G\in{\rm Sym}(2)$.

Before passing to the rigorous derivation of the 2D model, we provide a technical lemma consisting in two estimates for the family $\{W^{h}\}$ of energy densities and for its uniform limit $W$ defined in a neighbourhood $\mathcal{U}$ of ${\rm SO}(3)$. They are elementary consequences of properties (ii) and (iii) of Definition 2.1 (hence we omit their proof). These estimates will be used in the proof of the $\Gamma\text{-}\liminf$ and the $\Gamma\text{-}\limsup$.

Lemma 2.3**.**

Let $\bar{r}>0$ be such that $B_{2\bar{r}}(\mathbb{I}_{3})$ is contained in $\mathcal{U}$. Then for every $\varepsilon>0$ there exists $h_{\varepsilon}>0$ and $C>0$ such that for a.e. $x\in\Omega$, every $F\in B_{\bar{r}}(0)$ and every $h\in(0,h_{\varepsilon}]$ it holds that

[TABLE]

[TABLE]

We also introduce two auxiliary functions that will be used in the proof of $\Gamma$-convergence result. Letting $\bar{r}>0$ be as in Lemma 2.3 above, we define

[TABLE]

for every $F\in B_{\bar{r}}(0)$ and every $s>0$ .

As a direct consequence of the regularity of $W$, we have that

[TABLE]

In order to state and prove the following compactness and $\Gamma$-convergence results we will use the standard notation

[TABLE]

Moreover, given a $\mathcal{B}\text{-}$admissible family \big{\{}W^{h}\big{\}}_{h>0} of energy densities in the sense of Definition 2.1, for every $h>0$ we define the rescaled free energy functional $\mathcal{E}^{h}:{\rm W}^{1,2}(\Omega,\mathbb{R}^{3})\to[0,+\infty]$ as

[TABLE]

Theorem 2.4** (Compactness).**

Let \big{\{}y^{h}\big{\}}_{h>0}\subseteq{\rm W}^{1,2}\big{(}\Omega,\mathbb{R}^{3}\big{)} be a sequence which satisfies

[TABLE]

Then \big{\{}\nabla_{h}y^{h}\big{\}}_{h>0} is precompact in {\rm L}^{2}\big{(}\Omega,\mathbb{R}^{3\times 3}\big{)}, that is: there exists a (not relabeled) subsequence such that \nabla_{h}y^{h}\to\left(\nabla^{\prime}y\big{|}\nu\right) in {\rm L}^{2}\big{(}\Omega,\mathbb{R}^{3\times 3}\big{)}, where $\nu(x):=\partial_{1}y(x)\wedge\partial_{2}y(x)$. Moreover, the limit \left(\nabla^{\prime}y\big{|}\nu\right) has the following properties:

• $(i)$

\left(\nabla^{\prime}y\big{|}\nu\right)(x)\in{\rm SO}(3)* for a.e. $x\in\Omega$,*

• $(ii)$

\left(\nabla^{\prime}y\big{|}\nu\right)\in{\rm W}^{1,2}\big{(}\Omega,\mathbb{R}^{3\times 3}\big{)}* and*

• $(iii)$

\left(\nabla^{\prime}y\big{|}\nu\right)* is independent of $x_{3}$.*

In other words, the limiting deformation $y$ belongs to the class ${\rm W}^{2,2}_{{\rm iso}}(\omega)$ defined as in (2.20).

To prove this compactness result, we can use the same argument as in the proof of the corresponding result in [20] where the spontaneous stretch is $\mathbb{I}_{3}$ in place of our $U^{h}=\mathbb{I}_{3}+hB$. Note that the same argument holds in the case of spontaneous stretch of the form $\mathbb{I}_{3}+h^{\alpha}B$ with $\alpha\geq 1$.

Proof of Theorem 2.4.

We will show that the sequence \big{\{}\nabla_{h}y^{h}\big{\}}_{h>0}\subseteq{\rm L}^{2}\big{(}\Omega,\mathbb{R}^{3\times 3}\big{)} satisfies

[TABLE]

The thesis then directly follows by applying Theorem 4.1 from [20]. Fix $h>0$ and $F\in\mathbb{R}^{3\times 3}$. For a.e. $x\in\Omega$ there exists $R_{h,F}(x)\in{\rm SO}(3)$ such that

[TABLE]

We have the following estimate:

[TABLE]

for a.e. $x\in\Omega$. By (2.17) and (2.19) we have that (2.18) holds true. ∎

Given a bounded Lipschitz domain $S\subset\mathbb{R}^{2}$, the class of the isometries of $S$ into $\mathbb{R}^{3}$ is denoted by

[TABLE]

For the sake of brevity, we equivalently use the symbol ${\rm W_{\rm iso}^{2,2}}(S)$. We recall that for a given y\in{\rm W}^{2,2}\big{(}\omega,\mathbb{R}^{3}\big{)} the pull-back of the second fundamental form of $y(\omega)$ at the point $y(x^{\prime})$ is given by

[TABLE]

As we are going to see, the 2D limiting model will depend on deformations $y\in{\rm W_{\rm iso}^{2,2}}(\omega,\mathbb{R}^{3})$ through ${\rm A}_{y}$.

More precisely, the limiting model

will be described by the energy functional $\mathcal{E}^{0}:{\rm W}^{1,2}(\Omega,\mathbb{R}^{3})\to[0,+\infty]$ defined as

[TABLE]

where $\overline{Q}_{2}$ is defined through (2.2) and (2.10).

We also recall that smooth functions are dense in the class of ${\rm W}^{2,2}$-isometric immersions, as stated in the following theorem proved in [23].

Theorem 2.5**.**

Assume that $S\subseteq\mathbb{R}^{2}$ is a bounded Lipschitz domain which satisfies (2.1). Then ${\rm W}^{2,2}_{{\rm iso}}(S)\cap C^{\infty}(\overline{S},\mathbb{R}^{3})$ is ${\rm W}^{2,2}$-strongly dense in ${\rm W}^{2,2}_{{\rm iso}}(S)$.

This density result will be used for the construction of the recovery sequence in the proof of the $\Gamma\text{-}\limsup$ convergence result below.

Theorem 2.6** ($\Gamma$-limit).**

The following convergence results hold true:

(i)* $\Gamma\text{-}\liminf$: for every sequence $\{y^{h}\}_{h>0}$ and every $y$ such that $y^{h}\rightharpoonup y$ weakly in ${\rm W}^{1,2}(\Omega,\mathbb{R}^{3})$, it holds*

[TABLE]

(ii)* $\Gamma\text{-}\limsup$: under the hypothesis*

[TABLE]

with $D_{\rm min}$ defined by (2.2) and (2.9), we have that for every $y\in{\rm W}^{1,2}(\Omega,\mathbb{R}^{3})$ there exists a sequence $\{y^{h}\}_{h>0}$ such that $y^{h}\to y$ in ${\rm W}^{1,2}(\Omega,\mathbb{R}^{3})$, fulfilling

[TABLE]

The convergence results of the previous theorem amount to saying that the sequence of energy functionals $\frac{1}{h^{2}}\mathcal{E}^{h}$ $\Gamma$-converge to $\mathcal{E}^{0}$, as $h\to 0$, in the strong and weak topology of ${\rm W}^{1,2}(\Omega,\mathbb{R}^{3})$. The operator ${\rm curl}$ inside the parenthesis in condition (2.23) acts on a $2\!\times\!2$ matrix by taking the ${\rm curl}$ of each row, giving as a result a two-dimensional vector. We postpone the proof of the theorem after the following example.

Example 2.7**.**

Note that when $D_{\rm min}$ is constant, condition (2.23) is trivially satisfied. In particular, recalling definition (2.9), condition (2.23) is trivially satisfied whenever the map $x\mapsto\check{B}$ is constant in $x^{\prime}$. At the same time, the same condition is satisfied with $D_{\rm min}\equiv 0$ by every map $x\mapsto\check{B}(x)$ which is nothing but odd in $x_{3}$. We also note that it is possible to realize $D_{\rm min}\equiv C\neq 0$ through a map $x\mapsto\check{B}(x)$ which is not constant in $x^{\prime}$. To construct such an example, one can fix $B_{m}\in{\rm Sym}(2)\setminus\{0\}$ and define

[TABLE]

where $\{\omega_{i}\}_{i=1}^{N}$ is a partition of $\omega$ and \{\check{B}_{i}\}_{i=1}^{N}\subseteq{\rm L}^{\infty}\big{(}(-1/2,1/2),\mathbb{R}^{3\times 3}\big{)} is a family of functions satisfying

[TABLE]

and such that $\check{B}_{i}(x_{3})\neq\check{B}_{j}(x_{3})$ for every $i\neq j$ and every $x_{3}$. This gives rise to $\check{B}$ which is piecewise constant in $x^{\prime}$ (but not constant in the same variable), and in turn to

[TABLE]

Note also that the above defined map $\check{B}$ can give rise to a non-constant tensor valued map $x^{\prime}\mapsto\int_{\nicefrac{{-1}}{{2}}}^{\nicefrac{{1}}{{2}}}t\check{B}(x^{\prime},t)\,{\rm d}t$, which is interpreted in Section 3 (in each point $x^{\prime}$) as the target curvature tensor which appears in the 2D limiting model. Indeed, in the case of $N=2$, by choosing $\check{B}_{1}(x_{3}):=(x_{3}+1)\mathbb{I}_{2}$ and $\check{B}_{2}(x_{3}):=(x_{3}^{3}+1)\mathbb{I}_{2}$ for all $x_{3}\in(-1/2,1/2)$, we obtain a simple example of $\check{B}$ for which $D_{\rm min}$ is constant, while the tensor-valued map $x^{\prime}\mapsto\int_{\nicefrac{{-1}}{{2}}}^{\nicefrac{{1}}{{2}}}t\check{B}(x^{\prime},t)\,{\rm d}t$ is piecewise constant.

∎

The proof of the $\Gamma\text{-}\liminf$ is a straightforward adaptation to the case of a family of energy densities $\{W^{h}\}$ with wells {\rm SO}(3)\big{(}\mathbb{I}_{3}+hB^{h}\big{)}, of the corresponding result in [20] pertaining the case of a homogeneous $W$ (minimized at ${\rm SO}(3)$). For the construction of the recovery sequence in the proof of the $\Gamma\text{-}\limsup$ one has instead to add an additional term with respect to the classical construction (see the third summand on the right-hand side of (2.26)). Such additional term gives rise, in the limit as $h\to 0$, to a symmetrized gradient (see formula (2.29)), in a position where the map $D_{\rm min}$ should appear in order to match the $\Gamma$-limit (cfr. (2.8) and (2.22)). For this purpose, condition (2.23) guarantees that the map $D_{\rm min}$ is a symmetrized gradient, thanks to Theorem 2.8. Throughout the following proof $\overline{C}$ is a generic positive constant, varying form line to line and independent of all other quantities.

Proof of Theorem 2.6.

(i) $\Gamma\text{-}\liminf$: Let $y\in{\rm W}^{1,2}(\Omega,\mathbb{R}^{3})$ and $\{y^{h}\}$ be such that $y^{h}\rightharpoonup y$ weakly in ${\rm W}^{1,2}(\Omega,\mathbb{R}^{3})$. Assume that \liminf_{h\to 0}\mathcal{E}^{h}\big{(}y^{h}\big{)}/h^{2}<+\infty, otherwise the proof is trivial. Then, as shown in [20] and up to a (not relabeled) subsequence, there exists a family of piecewise constant maps $R^{h}:Q_{h}\to{\rm SO}(3)$ such that

[TABLE]

and $R^{h}\to(\nabla^{\prime}y|\nu)$ in ${\rm L}^{2}(\Omega,\mathbb{R}^{3})$ as $h\to 0$, where $Q_{h}:=\bigcup_{Q_{a,3h}\subseteq\omega}Q_{a,h}$ and $Q_{a,h}:=a+(-h/2,h/2)^{2}$ for every $h>0$ and $a\in h\mathbb{Z}^{2}$. Moreover, the sequence $G^{h}:\Omega\to\mathbb{R}^{3\times 3}$ defined by

[TABLE]

converges weakly in ${\rm L}^{2}(\Omega,\mathbb{R}^{3\times 3})$, as $h\to 0$, to some $G\in{\rm L}^{2}(\Omega,\mathbb{R}^{3\times 3})$ such that

[TABLE]

Letting $\chi_{h}$ be the characteristic function of the set Q_{h}\cap\big{\{}|G^{h}(x)|\leq 1/\sqrt{h}\big{\}} we also have that $\chi_{h}G^{h}\rightharpoonup G$ in ${\rm L}^{2}(\Omega,\mathbb{R}^{3\times 3})$ as $h\to 0$. Now, by denoting $A^{h}:=G^{h}-B^{h}$ and by using also the convergence in (2.2) we have

[TABLE]

By using frame indifference of $W^{h}$, (2.15) and the estimate (2.12) from Lemma 2.3, we have that for a fixed $\varepsilon>0$, there exists $\bar{h}>0$ such that the following estimates hold for every $h\in(0,\bar{h})$:

[TABLE]

where $\rho^{0}$ and $\rho$ are defined in (2.14) . Since $Q_{3}$ is lower semicontinuous in the weak topology of ${\rm L}^{2}(\Omega,\mathbb{R}^{3\times 3})$ and since (2.15) holds , passing to $\liminf$ as $h\to 0$ in the above inequality we obtain

[TABLE]

where $\overline{C}>0$ is such that $||A^{h}||_{{\rm L}^{2}(\Omega,\mathbb{R}^{3\times 3})}\leq\overline{C}$. Finally, by letting $\varepsilon\to 0$ and by using the fact that $Q_{3}(F)\geq Q_{2}(\check{F})$ for every $F\in\mathbb{R}^{3\times 3}$ we get that

[TABLE]

which proves $\Gamma\text{-}\liminf$ inequality.

(ii) $\Gamma\text{-}\limsup$: Let us prove $\Gamma\text{-}\limsup$ inequality for a given $y\in{\rm W}^{2,2}_{{\rm iso},0}(\omega):={\rm W}^{2,2}_{{\rm iso}}(\omega)\cap C^{\infty}(\overline{\omega},\mathbb{R}^{3})$. Once we have proved it, $\Gamma$-$\limsup$ inequality will follow for any $y\in{\rm W}^{2,2}_{{\rm iso}}(\omega)$ by the density result of Theorem 2.5 and the continuity of the limiting functional $\mathcal{E}^{0}$ with respect to ${\rm W}^{2,2}$ convergence. Suppose that $\mathcal{E}^{0}(y)<+\infty$ (otherwise the proof is trivial). Let $d\in C^{\infty}_{c}(\Omega,\mathbb{R}^{3})$ and define $D:\Omega\to\mathbb{R}^{3}$ by

[TABLE]

Let $\tilde{g}:=(\tilde{g}_{1},\tilde{g}_{2})\in C^{\infty}_{c}(\mathbb{R}^{2},\mathbb{R}^{2})$. We consider the family of functions $y^{h}$ of the form

[TABLE]

for every $x\in\Omega$ and every $h>0$, whose ($h$-rescaled) gradient $\nabla_{h}y^{h}$ reads as

[TABLE]

for every $x\in\Omega$ and every $h>0$, One can easily verify that, in particular, $\{y^{h}\}_{h>0}\subseteq{\rm W}^{2,\infty}(\Omega,\mathbb{R}^{3})$ and that it converges in ${\rm W}^{1,2}$ to $y$, as $h\to 0$. Denote by R(x^{\prime}):=\big{(}\nabla^{\prime}y(x^{\prime})\big{|}\nu(x^{\prime})\big{)} for every $x^{\prime}\in\omega$. Set

[TABLE]

and note that $C^{h}$ converges in ${\rm L}^{\infty}(\Omega,\mathbb{R}^{3\times 3})$ to the function

[TABLE]

With this notation, we have that $R^{\scriptscriptstyle{\mathsf{T}}}(x^{\prime})\nabla_{h}y^{h}(x)=U^{h}(x)+hC^{h}(x)$ for a.e. $x\in\Omega$ with $U^{h}$ given by (2.3). By the frame indifference of $W^{h}(x,\cdot)$, boundedness of $C^{h}$ and $B^{h}$ in ${\rm L}^{\infty}$-norm and the estimates (2.12) and (2.13) from Lemma 2.3, there exists $\overline{C},\bar{h}>0$ such that

[TABLE]

for a.e. $x\in\Omega$ and every $0<h\leq\overline{h}$. Moreover,

[TABLE]

pointwise almost everywhere in $\Omega$, as $h\to 0$. Then, by applying dominated convergence theorem we get that

[TABLE]

as $h\to 0$. To proceed, for a.e. $x\in\Omega$ we denote by $\overline{F}(x)$ the $2\times 2$ part of the argument of $Q_{3}$ in the above integral. Now let $\ell:{\rm Sym}(2)\to\mathbb{R}^{3}$ be the map that associates to every $F\in{\rm Sym}(2)$ the unique element of {\rm argmin}_{c\in\mathbb{R}^{3}}\,Q_{3}\big{(}\hat{F}+(c\otimes{\sf f}_{3})_{{\rm sym}\,}\big{)}. By writing down the first order necessary condition for the minimum problem defining $\ell(F)$, one can easily deduce that the map $\ell$ is linear. Define $\overline{d}:\Omega\to\mathbb{R}^{3}$ as

[TABLE]

for a.e. $x\in\Omega$. Since $\overline{F}_{{\rm sym}\,}\in{\rm L}^{\infty}(\Omega,{\rm Sym}(2))$, $B\in{\rm L}^{\infty}(\Omega,{\rm Sym}(3))$ and $y$ and $\tilde{g}$ are smooth vector fields, it follows that $\overline{d}$ belongs to ${\rm L}^{2}(\Omega,\mathbb{R}^{3})$. By choosing $d$ to be equal to $\bar{d}$, one can readily check that

[TABLE]

Observe further that

[TABLE]

Since $y\in{\rm W}^{2,2}_{{\rm iso},0}(\omega)$, it holds that $\partial_{i}y\cdot\partial_{j}y=\delta_{ij}$ and $\partial_{i}\partial_{j}y\cdot\partial_{k}y=0$ for every $i,j,k=1,2$. In turn, we have that

[TABLE]

Now, by direct computation we obtain

[TABLE]

Finally, by definition of $Q_{2}$, (2.27) and (2.28) it holds that

[TABLE]

Therefore, the density of $C_{c}^{\infty}(\Omega,\mathbb{R}^{3})$ in ${\rm L}^{2}(\Omega,\mathbb{R}^{3})$ and a diagonal argument give us that

[TABLE]

The compatibility assumption (2.23) on $D_{{\min}}$ and Theorem 2.8 guarantee the existence of the map $w\in{\rm W}^{1,2}(\omega,\mathbb{R}^{2})$ such that $D_{{\min}}(x^{\prime})=\nabla_{{\rm sym}}w(x^{\prime})$ for a.e. $x^{\prime}\in\omega$. Thus, by using the density of $C_{c}^{\infty}(\mathbb{R}^{2},\mathbb{R}^{2})$ (when restricted to $\omega$) in ${\rm W}^{1,2}(\omega,\mathbb{R}^{2})$ and a diagonal argument one more time, we prove $\Gamma$-$\limsup$ inequality for a given $y\in{\rm W}^{2,2}_{{\rm iso},0}(\omega)$. ∎

The following result is used in the proof of the $\Gamma\text{-}\limsup$. It can be found in [12, Theorem 3.2] and has to do with the so-called Saint-Venant compatibility condition in ${\rm L}^{2}$ .

Theorem 2.8**.**

Let $S\subseteq\mathbb{R}^{2}$ be a simply-connected bounded domain with Lipschitz boundary and let $A\in{\rm L}^{2}(S,{\rm Sym}(2))$ . Then

[TABLE]

Moreover $w$ is unique up to rigid displacements.

Remark 2.9**.**

By standard arguments of $\Gamma$-convergence it can be shown that the above analysis holds also in the case when the appropriate body forces are present. More precisely, the above results can be applied to the sequence of functionals $\{\mathcal{F}^{h}\}_{h>0}$ defined by

[TABLE]

where $\{f^{h}\}_{h\geq 0}\subseteq{\rm L}^{2}(\Omega,\mathbb{R}^{3})$ is the family of body forces such that

[TABLE]

The sequence $\{\mathcal{F}^{h}\}$ $\Gamma-$converges, as $h\to 0$, to

[TABLE]

where $f(x^{\prime}):=\int_{\nicefrac{{-1}}{{2}}}^{\nicefrac{{1}}{{2}}}f^{0}(x^{\prime},t)\,{\rm d}t$ for a.e. $x^{\prime}\in\omega$. * ∎*

3. 2D energy minimizers

3.1. $x^{\prime}$-dependent target curvature tensor $\overline{A}$ and pointwise minimizers

In this section, we discuss the minimizers of the derived $2D$ model in some special cases. Recall that the $2D$ limiting energy functional $\mathcal{E}^{0}$ is given by

[TABLE]

where ${\rm W}^{2,2}_{{\rm iso}}(\omega)$ is the set of ${\rm W}^{2,2}$-isometric immersions of $\omega$ into $\mathbb{R}^{3}$, defined by (2.20). From formula (2.11), we have that

[TABLE]

for every $y\in{\rm W}^{2,2}_{{\rm iso}}(\omega)$, where ${\rm ad.t.}$ stays for “additional terms” (not depending on $y$) and is given by

[TABLE]

Recall that ${\rm A}_{y}$ is the pull-back of the second fundamental form associated with $y(\omega)$ (see (2.21)), hence it gives information on the curvature realized by the deformation $y$. On the other hand, when reading the expression for $\mathcal{E}^{0}$, it is natural to define the target curvature tensor

[TABLE]

which encodes the spontaneous curvature of the system. While, for a.e. $x^{\prime}$, the tensor $\overline{A}(x^{\prime})$ (which depends on $\check{B}$ and in turn on the family of spontaneous strains $\{B^{h}\}$, see formula (2.2)) is a given $2{\times}2$ symmetric matrix with possibly nonzero determinat, it is a well known result of differential geometry that every smooth $y\in{\rm W}^{2,2}_{{\rm iso}}(\omega)$ satisfies $\det{\rm A}_{y}=0$ in $\omega$. From [36, Lemma 2.5], one can deduce that the same property holds for any arbitrary $y\in{\rm W}^{2,2}_{{\rm iso}}(\omega)$, a.e. in $\omega$. Our aim is to determine explicitly some classes of minimizers. More precisely, introducing the notation

[TABLE]

and having in mind the inequality

[TABLE]

we will focus our attention on pointwise minimizers of $\mathcal{E}^{0}$. Namely, on those $y\in{\rm W}^{2,2}_{{\rm iso}}(\omega)$ such that

[TABLE]

To go on, let us consider the set

[TABLE]

for a.e. $x^{\prime}\in\omega$. Note that $\mathcal{N}(x^{\prime})\neq\mathchar 31\relax$ for a.e. $x^{\prime}\in\omega$, because $Q_{2}$ is a positive definite quadratic form (when restricted to ${\rm Sym}(2)$) and $\mathcal{F}$ is a closed subset of ${\rm Sym}(2)$. To accomplish our program, we would like to have some explicit representation of the elements of $\mathcal{N}(x^{\prime})$, for a.e. $x^{\prime}\in\omega$, also in view of the application which motivates our analysis (see Section 4.). Therefore, we restrict our attention to case of $W$ isotropic, i.e. such that

[TABLE]

This implies the existence of constants $\lambda\in\mathbb{R}$ and $\mu>0$, called Lamé moduli, such that

[TABLE]

for every $F\in\mathbb{R}^{3\times 3}$ (see [22]). In turn, from this expression one can easily show that

[TABLE]

where $\beta$ has the expression

[TABLE]

Since $Q_{3}$ is positive definite by its very definition, then we have that $\mu>0$ and $2\mu+3\lambda>0$. In turn, it holds that $\beta>-1/2$ and hence that $Q_{2}$ is positive definite. This fact guarantees in particular that the quantities appearing in the statement of Lemma 3.1 below are well defined.

Note that in the case when $\overline{A}$ is constant in $\omega$, pointwise minimizers of $\mathcal{E}^{0}$ always exist. More precisely, as noticed in [41] and [42] (see Lemma 3.3 below), any minimizer $y$ of $\mathcal{E}^{0}$ with $\overline{A}$ constant is characterized by the property ${\rm A}_{y}(x^{\prime})\equiv{\rm const.}\in\mathcal{N}$ for a.e. $x^{\prime}\in\omega$, where

[TABLE]

Clearly, in the case of nonconstant $\overline{A}$, this is not always true. Now, while the analysis of the minimizers of $\mathcal{E}^{0}$, with an arbitrary nonconstant $\overline{A}$, is behind the scope of the present paper, it is natural in our context to try to understand under which conditions the existence of pointwise minimizers of $\mathcal{E}^{0}$ is guaranteed. In Subsection 3.2 we answer this question in the case when $\overline{A}$ is piecewise constant. To do this, we need a structure result for the set $\mathcal{N}$ in the case of constant $\overline{A}$. This is the content of the following lemma.

Lemma 3.1**.**

Let $a$ and $b$ be two real numbers and let $\beta$ be given by (3.8). The following implications hold:

• $(i)$

If $\,\overline{A}=\left(\begin{matrix}a&0\\ 0&a\end{matrix}\right)\quad\,\mbox{ then }\quad\mathcal{N}=\left\{\rho^{\scriptscriptstyle{\mathsf{T}}}\left(\begin{matrix}\mathfrak{r}&0\\ 0&0\end{matrix}\right)\rho\,:\,\rho\in{\rm SO}(2)\right\}\,\mbox{with}\ \,\mathfrak{r}=a\frac{1+2\beta}{1+\beta}.$

• $(ii)$

If $\,\overline{A}=\left(\begin{matrix}a&0\\ 0&-a\end{matrix}\right)\quad\,\mbox{ then }\quad\mathcal{N}=\left\{\left(\begin{matrix}\mathfrak{r}&0\\ 0&0\end{matrix}\right),\left(\begin{matrix}0&0\\ 0&-\mathfrak{r}\end{matrix}\right)\right\}\,\mbox{with}\ \,\mathfrak{r}=\frac{a}{1+\beta}.$

• $(iii)$

If $\,\overline{A}=\left(\begin{matrix}a&0\\ 0&b\end{matrix}\right),\,|a|>|b|\quad\mbox{ then }\quad\mathcal{N}=\left\{\left(\begin{matrix}\mathfrak{r}&0\\ 0&0\end{matrix}\right)\right\}\,\mbox{with}\ \,\mathfrak{r}=a+\frac{b\beta}{1+\beta}.$

• $(iv)$

If $\,\overline{A}=\left(\begin{matrix}a&0\\ 0&b\end{matrix}\right),\,|b|>|a|\quad\mbox{ then }\quad\mathcal{N}=\left\{\left(\begin{matrix}0&0\\ 0&\mathfrak{r}\end{matrix}\right)\right\}\,\mbox{with}\ \,\mathfrak{r}=b+\frac{a\beta}{1+\beta}.$

Before giving the proof of the above statement, let us make a couple of comments. First, note that the lemma, though restricted to the case of $\overline{A}$ diagonal, covers all the interesting cases, from the simple observation that, with abuse of notation, $\mathcal{N}_{\overline{A}}=\bar{\rho}\,\mathcal{N}_{\overline{D}}\,\bar{\rho}^{\scriptscriptstyle{\mathsf{T}}}$, where $\bar{\rho}\in{\rm Orth}(2)$ is such that $\bar{\rho}^{\scriptscriptstyle{\mathsf{T}}}\,\overline{A}\,\bar{\rho}$ coincides with the diagonal matrix $\overline{D}$. Moreover, interpreting the elements of $\mathcal{N}$ as second fundamental forms of cylinders (see the discussion below), the parameter $\mathfrak{r}$, when different from zero, corresponds to the nonzero principal curvature. In this case, observe also that, with abuse of notation, the set $\mathcal{N}_{(ii)}$ is never a subset of $\mathcal{N}_{(i)}$ and that, as for the (two) elements of $\mathcal{N}_{(ii)}$, the elements of $\mathcal{N}_{(i)}$ are pairwise linearly independent. This can be easily read off from the simple fact that

[TABLE]

Finally, the set of the directions corresponding to $\pm\mathfrak{r}$ in the cases $(i)$, $(ii)$, $(iii)$, and $(iv)$ is given by $\{\rho\,\mathsf{e}_{1}:\rho\!\in\!{\rm SO}(2)\},\{\mathsf{e}_{1},\mathsf{e}_{2}\},\{\mathsf{e}_{1}\},\{\mathsf{e}_{2}\},$ respectively. This fact can be interpreted saying that, in order to reduce the energy, while in case $(i)$ rolling up along all the possible directions is equally favorable, in the remaining cases the system rolls up along the direction corresponding to the greater (in modulus) eigenvalue of the target curvature tensor $\overline{A}$.

Proof of Lemma 3.1.

Let $a,b\in\mathbb{R}$ and let $\overline{A}={\rm diag}(a,b)$. By representing any $F\in{\rm Sym}(2)$ by $\left(\begin{smallmatrix}\xi&\zeta\\ \zeta&\upsilon\end{smallmatrix}\right)$, $\zeta,\xi,\upsilon\in\mathbb{R}$ and recalling that $Q_{2}$ is of the form (3.7), the minimization problem to be solved is:

[TABLE]

Denote P:=\big{\{}(\xi,\upsilon)\in\mathbb{R}^{2}\big{|}\,\xi\upsilon\geq 0\big{\}} and define for every $(\xi,\upsilon)\in P$ the function

[TABLE]

so that the minimization problem becomes $\min_{(\xi,\upsilon)\in P}f(\xi,\upsilon).$ In the case when $a\neq b$, $f$ attains its minimum on \partial P=\big{\{}(\xi,\upsilon)\in\mathbb{R}^{2}\,|\,\xi\upsilon=0\big{\}}. With this said, $(ii)$, $(iii)$ and $(iv)$ easily follow by straightforward computations. To prove $(i)$, we first note that the set of stationary points of $f$ in ${\rm int}(P)$ is given by

[TABLE]

where

[TABLE]

Moreover, the value of $f$ at these stationary points coincides with the value of $f$ at the boundary of $P$. In turn,

[TABLE]

concluding the result of point $(i)$, and thus proving the lemma.

∎

To conclude the section, we give some definitions which will be useful later on. They regard the sub-class of ${\rm W}^{2,2}_{{\rm iso}}(\omega)$ consisting of cylinders. Given $r\in(0,+\infty]$, we define the map $C_{r}:\mathbb{R}^{2}\to\mathbb{R}^{3}$ as

[TABLE]

for every $x^{\prime}=(x_{1},x_{2})\in\mathbb{R}^{2}$. Then we define the family of maps

[TABLE]

and we call its elements cylinders. Note that the above defined family of cylinders includes also planes - the elements of ${\rm Cyl}$ with $r=+\infty$.

Remark 3.2**.**

Observe that any cylinder $y=T_{v}\circ R\circ C_{r}\circ\rho$ maps lines parallel to $\rho^{\scriptscriptstyle{\mathsf{T}}}\mathsf{e}_{2}$ to the lines of zero curvature - rulings. More in general, direct computations give

[TABLE]

so that

[TABLE]

∎

By direct computations one can see that a map $y=T\circ R\,\circ\,C_{r}\circ\rho\in{\rm Cyl}$ is an isometry whose second fundamental form is given by

[TABLE]

Now, let us go back to the set $\mathcal{F}$ defined in (3.4). From (3.11) and from the simple observation that $\mathcal{F}$ can be equivalently represented as

[TABLE]

one can prove that the set $\mathcal{F}$ coincides with the set of (constant) second fundamental forms of cylinders. This fact can in turn be used to show, in the case where the target curvature tensor $\overline{A}$ is constant, that

[TABLE]

This is the first step of the proof of Lemma 3.3 below. The second part of the proof consists then in showing that

[TABLE]

This property is at the core of our investigations in the following subsection and can be proved using some fine properties of isometric immersions

([23], [24] and [36]). The proof of the following lemma can be found in [42, Proposition 4.2].

Lemma 3.3**.**

Let $\overline{A}$ be constant (cfr. (3.1)–(3.3)) and let $y\in{\rm W}^{2,2}_{{\rm iso}}(\omega)$ be a minimizer of $\mathcal{E}^{0}$. Then $y=v|_{\omega}$ for some $v\in{\rm Cyl}$. In particular, $y$ has constant second fundamental form.

3.2. The case of piecewise constant $\overline{A}$.

In this subsection, we consider the case where the target curvature is a piecewise constant tensor valued map $x^{\prime}\mapsto\overline{A}(x^{\prime})$. More precisely, given $n\in\mathbb{N}$, $n\geq 2$, we say that the map \overline{A}\in{\rm L}^{\infty}\big{(}\omega,\mathbb{R}^{2\times 2}\big{)} is piecewise constant if it is of the form

[TABLE]

where $\{\omega_{k}\}_{k=1}^{n}$ is a partition of $\omega$ made of Lipschitz subdomains $\omega_{k}$ according to Definition 3.4 . Clearly, it is convenient distinguishing between two different neighboring subdomain only when the corresponding spontaneous curvature are different from each other. Namely, we suppose that $\overline{A}_{k}\neq\overline{A}_{j}$ for every $k\neq j$ such that $\partial\omega_{j}\cap\partial\omega_{k}\neq\mathchar 31\relax$. With such target curvature, our 2D energy functional takes the form

[TABLE]

We want to determine the conditions the map $x^{\prime}\mapsto\overline{A}(x^{\prime})$ has to satisfy in order to guarantee the existence of pointwise minimizers of $\mathcal{E}^{0}$, i.e. to guarantee that there exists $y\in{\rm W}^{2,2}_{{\rm iso}}(\omega)$ such that ${\rm A}_{y}(x^{\prime})\in\mathcal{N}(x^{\prime})$ for a.e. $x^{\prime}\in\omega$, where $\mathcal{N}(x^{\prime})$ is defined by (3.6). In view of (3.14), we equivalently look for the necessary and sufficient conditions such that

[TABLE]

where

[TABLE]

Note from (3.13) that a deformation satisfying (3.15) is, roughly speaking, a “patchwork” of cylinders. Therefore, conditions on $\overline{A}$ guaranteeing (3.15) translates into conditions under which cylinders can be patched together resulting into an isometry. This is the content of the main result of the present section, namely of Theorem 3.9 below. In order to state and prove it, we need a definition and a preliminary lemma.

Definition 3.4** (Lipschitz $n\text{-}$subdivision).**

Fix $n\!\in\!\mathbb{N}$, $n\geq 2$. A family $\{\omega_{k}\}_{k=1}^{n}$ of open, bounded and connected subsets of $\mathbb{R}^{2}$ is said to be a Lipschitz $n\text{-}$subdivision of $\omega$ provided it can be obtained via the following procedure:

• •

Call $\omega_{1}^{\prime}:=\omega$.

• •

Suppose that for every $k=1,\ldots,n-1$ there exists a continuous injective curve $\gamma_{k}:[0,1]\to{\rm cl}(\omega_{k}^{\prime})$ such that \partial\omega_{k}^{\prime}\cap[\gamma_{k}]=\big{\{}\gamma_{k}(0),\gamma_{k}(1)\big{\}} (note that $\gamma_{k}(0)\neq\gamma_{k}(1)$) and the two connected components of $\omega_{k}^{\prime}\setminus[\gamma_{k}]$ are Lipschitz. Then call $\omega_{k+1}^{\prime}$ one of such connected components.

• •

Once the domains $\omega_{1}^{\prime},\ldots,\omega_{n}^{\prime}$ are defined, let $\omega_{k}:=\omega_{k}^{\prime}\setminus{\rm cl}(\omega_{k+1}^{\prime})$ for every $k=1,\ldots,n-1$ and let $\omega_{n}:=\omega_{n}^{\prime}$.

In particular, the subdomains $\omega_{1},\ldots,\omega_{n}$ of $\omega$ are Lipschitz domains such that

[TABLE]

Remark 3.5**.**

Since each $\omega_{k}$ is a Lipschitz domain, one has that its boundary $\partial\omega_{k}$ has null $\mathcal{L}^{2}$-measure. In particular, we deduce that \mathcal{L}^{2}\big{(}\omega\setminus\bigcup_{k=1}^{n}\omega_{k}\big{)}=0. * ∎*

The following Lemma 3.6 will be the main ingredient for the proof of Theorem 3.9. It gives a “recipe” on how two cylinders can be patched together. We refer to Remark 3.8 below for the notation and the properties of roto-translations used in this section. We point out that the proof of Theorem 3.9 will be achieved via an induction argument, which relies upon Lemma 3.6 and the definition of Lipschitz subdivision of $\omega$.

We remark that, more in general, the very fact that $y$ is a ${\rm W}^{2,2}$-isometry is sufficient to deduce that $\omega$ consists, up to a null set, of finitely many subdomains (touching each other on a finite union of line segments) on which $y$ is either a plane, or a cylinder, or a cone or “tangent developable”. This description can be obtained as a consequence of some fine properties of the class ${\rm W}^{2,2}_{{\rm iso}}(\omega)$ - see [36] for $\omega$ convex and [23], [24] for a more general $\omega$.

Lemma 3.6**.**

Let $\gamma:[0,1]\to{\rm cl}(\omega)$ be a continuous injective curve such that $[\gamma]\cap\partial\omega=\{\gamma(0),\gamma(1)\}$ and such that two connected components $\omega_{1}$ and $\omega_{2}$ of $\omega\setminus[\gamma]$ are Lipschitz. Let $y_{1},y_{2}\in{\rm Cyl}$, say $y_{1}=T_{v_{1}}\circ R_{1}\circ C_{r_{1}}\circ\rho_{1}$ and $y_{2}=T_{v_{2}}\circ R_{2}\circ C_{r_{2}}\circ\rho_{2}$, with $r_{1},r_{2}\in(0,+\infty)$ such that ${\rm det}\rho_{1}=-{\rm det}\rho_{2}$ whenever $r_{1}=r_{2}$. The map defined as

[TABLE]

belongs to ${\rm W}^{2,2}_{{\rm iso}}(\omega)$ if and only if the following conditions hold:

• $(i)$

$[\gamma]$* is a line segment spanned by some $\mathsf{e}\in\mathbb{R}^{2}\setminus\{0\}$ ;*

• $(ii)$

$\rho^{\scriptscriptstyle{\mathsf{T}}}_{1}\mathsf{e}_{2}$* and $\rho^{\scriptscriptstyle{\mathsf{T}}}_{2}\mathsf{e}_{2}$ are parallel to $\mathsf{e}$. This in particular implies that $\rho_{1}\rho_{2}^{\scriptscriptstyle{\mathsf{T}}}={\rm diag}(\sigma_{1},\sigma_{2})$, for some $\sigma_{1},\sigma_{2}\in\{\pm 1\}$;*

• $(iii)$

Setting w_{k}:=\rho_{k}\big{(}\gamma(0)-(0,0)\big{)} and $\theta_{k}:=(w_{k}\cdot\mathsf{e}_{1})/r_{k}$, for $k=1,2$, we have

[TABLE]

Proof.

(Necessity) Here, we show that if the deformation $y:=y_{1}\chi_{\omega_{1}}+y_{2}\chi_{\omega_{2}}$ is in ${\rm W}^{2,2}_{{\rm iso}}(\omega)$, then it complies with conditions $(i)$, $(ii)$ and $(iii)$. First of all, we recall from [35, Proposition 5] that the very condition ${\rm W}^{2,2}_{{\rm iso}}(\omega)$ implies $y\in C^{1}(\omega,\mathbb{R}^{3})$. At the same time, from the specific expression of $y$ we have that $\nabla y=\nabla y_{k}$ in $\omega_{k}$ for $k=1,2$, where

[TABLE]

This expression says in particular that $\nabla y$ is bounded and in turn that $y\in C^{1}(\overline{\omega},\mathbb{R}^{3})$. Let us first prove the necessity of the conditions $(i)$, $(ii)$ and $(iii)$ in the case when $\gamma(0)=(0,0)$. The continuity of $y$ and $\nabla y$ at the point $(0,0)$ gives, respectively, that $v_{1}=v_{2}$ (obtained by imposing $y_{1}(0,0)=y_{2}(0,0)$), and

[TABLE]

(obtained from $\nabla y_{1}(0,0)=\nabla y_{2}(0,0)$ and from expression (3.18)), which proves $(iii)$ . The continuity of $\nabla y$ gives also that $\nabla y_{1}(\gamma(t))=\nabla y_{2}(\gamma(t))$ for each $t\in[0,1]$, that is

[TABLE]

In turn, using the second condition in (3.19) and the notation $\rho_{1}\rho_{2}^{\scriptscriptstyle{\mathsf{T}}}=\left(\begin{matrix}m_{1}&m_{2}\\ m_{3}&m_{4}\end{matrix}\right)$, we have

[TABLE]

By the equality between the elements of the first row in the above expression one deduces that $\rho_{1}^{\scriptscriptstyle{\mathsf{T}}}\mathsf{e}_{2}$ and $\rho^{\scriptscriptstyle{\mathsf{T}}}_{2}\mathsf{e}_{2}$ must be parallel. This proves one part of the statement in $(ii)$ and implies, in particular, that $\rho_{1}\rho_{2}^{\scriptscriptstyle{\mathsf{T}}}={\rm diag}(m_{1},m_{4})$ with $m_{1},m_{4}\in\{\pm 1\}$. In order to conclude the proof of $(ii)$ and in the same time prove $(i)$ , we need to show that $[\gamma]$ is a line segment parallel to $\rho_{1}^{\scriptscriptstyle{\mathsf{T}}}\mathsf{e}_{2}$ (and to $\rho_{2}^{\scriptscriptstyle{\mathsf{T}}}\mathsf{e}_{2}$). Observe that $\rho_{1}\rho_{2}^{\scriptscriptstyle{\mathsf{T}}}={\rm diag}(m_{1},m_{4})$ implies $\rho_{2}^{\scriptscriptstyle{\mathsf{T}}}\mathsf{e}_{1}=m_{1}\rho_{1}^{\scriptscriptstyle{\mathsf{T}}}\mathsf{e}_{1}$ , so that the equation (3.20) simplifies to

[TABLE]

for every $t\in[0,1]$. By differentiating the above equality restricted to the first elements of the first and second rows one gets

[TABLE]

It turns out that (3.21) and (3.22) can be satisfied only if

[TABLE]

which implies that $[\gamma]$ is a line segment parallel to $\rho_{1}^{\scriptscriptstyle{\mathsf{T}}}\mathsf{e}_{2}$ (thus accordingly also to $\rho_{2}^{\scriptscriptstyle{\mathsf{T}}}\mathsf{e}_{2}$). To prove previous assertion, we distinguish two cases:

• •

if $r_{1}\neq r_{2}$, call $s:=m_{1}/m_{4}$ and fix $t\in[0,1]$. Condition (3.21) grants that $\gamma(t)\cdot\rho_{1}^{\scriptscriptstyle{\mathsf{T}}}{\sf e}_{1}/r_{1}$ and $s\gamma(t)\cdot\rho_{1}^{\scriptscriptstyle{\mathsf{T}}}{\sf e}_{1}/r_{2}$ have the same sine and cosine. Then, since sine and cosine cannot simultaneously vanish, (3.22) yields $\dot{\gamma}(t)\cdot\rho_{1}^{\scriptscriptstyle{\mathsf{T}}}{\sf e}_{1}/r_{1}=s\dot{\gamma}(t)\cdot\rho_{1}^{\scriptscriptstyle{\mathsf{T}}}{\sf e}_{1}/r_{2}$, whence necessarily $\dot{\gamma}(t)\cdot\rho_{1}^{\scriptscriptstyle{\mathsf{T}}}{\sf e}_{1}=0$.

• •

if $r_{1}=r_{2}$, by hypotheses we have that $\det\rho_{1}=-\det\rho_{2}$. Since $\rho_{1}\rho_{2}^{\scriptscriptstyle{\mathsf{T}}}={\rm diag}(m_{1},m_{4})$, we conclude that $m_{1}m_{4}=-1$, or equivalently $m_{1}=-m_{4}$. Now the first condition in (3.22) gives

[TABLE]

so that the map $t\mapsto\gamma(t)\cdot\rho_{1}^{\scriptscriptstyle{\mathsf{T}}}{\sf e}_{1}/r_{1}$ is constant and accordingly that $\dot{\gamma}(t)\cdot\rho_{1}^{\scriptscriptstyle{\mathsf{T}}}\mathsf{e}_{1}=0$ for every $t\in[0,1]$.

This concludes the proof of the necessary condition of the lemma in the case where $\gamma(0)=(0,0)$.

Considering now the case $v:=\gamma(0)-(0,0)\neq 0$, define $\hat{\omega}:=\omega-v$ and $\hat{y}_{k}:=y_{k}\circ\tau_{v}$, $k=1,2$ (recall form Section 1.1 that $\tau_{v}:=\cdot+v\in{\rm Trs}(2)$). By Remark 3.8, one can easily verify that

[TABLE]

where $\theta_{k}:=(w_{k}\cdot\mathsf{e}_{1})/r_{k}$ and $u_{k}:=v_{k}+R_{k}\circ C_{r_{k}}(w_{k})$, with w_{k}:=\rho_{k}\big{(}\gamma(0)-(0,0)\big{)}, for $k=1,2$. Observe that the domain $\hat{\omega}$ is partitioned into $\hat{\omega}_{1}$ and $\hat{\omega}_{2}$ by the subdivision curve $[\gamma]-v$ which satisfies the condition $\gamma(0)-v=(0,0)$. It is now clear that $y\in{\rm W}^{2,2}_{{\rm iso}}(\omega)$ implies $\hat{y}:=y\circ\tau_{v}=\hat{y}_{1}\chi_{\hat{\omega}_{1}}+\hat{y}_{2}\chi_{\hat{\omega}_{2}}\in{\rm W}^{2,2}_{{\rm iso}}(\hat{\omega})$, which further implies that $[\gamma]-v$ (and hence $[\gamma]$) is a line segment parallel to $\rho_{1}^{\scriptscriptstyle{\mathsf{T}}}\mathsf{e}_{2}$ and $\rho_{2}^{\scriptscriptstyle{\mathsf{T}}}\mathsf{e}_{2}$, implying $\rho_{1}\rho_{2}^{\scriptscriptstyle{\mathsf{T}}}={\rm diag}(\sigma_{1},\sigma_{2})$, for some $\sigma_{1},\sigma_{2}\in\{\pm 1\}$, and that

[TABLE]

which are precisely conditions $(i)$, $(ii)$ and $(iii)$.

(Sufficiency) Let $y_{1},y_{2}\in{\rm Cyl}$ satisfy conditions $(ii)$ and $(iii)$. Let $v:=\gamma(0)-(0,0)$ and let $\rho\in{\rm SO}(2)$ be a rotation which brings the line segment $[\gamma]-v$ to the vertical position. Let \underaccent{\bar}{y}_{k}:=y_{k}\circ\tau_{v}\circ\rho^{\scriptscriptstyle{\mathsf{T}}}. By denoting $u:=v_{1}+R_{1}\circ C_{r_{1}}(w_{1})$ and $R:=R_{1}\circ\hat{R}_{\theta_{1}}$ we have by $(iii)$ that \underaccent{\bar}{y}:=\underaccent{\bar}{y}_{1}\chi_{\omega_{1}}+\underaccent{\bar}{y}_{2}\chi_{\omega_{2}} is of the form

[TABLE]

where $\sigma_{k}^{1}\in\{\pm 1\}$ are such that $\rho_{1}\rho^{\scriptscriptstyle{\mathsf{T}}}={\rm diag}(\sigma_{1}^{1},\sigma_{2}^{1})$ (which follows form the fact that $\rho_{1}^{\scriptscriptstyle{\mathsf{T}}}\mathsf{e}_{2}\parallel[\gamma]$). By construction, \bar{y}\in C^{1}(\underaccent{\bar}{\omega},\mathbb{R}^{3}) with \underaccent{\bar}{\omega}=\rho(\omega-v). Simple computations give \partial_{1}\underaccent{\bar}{y},\partial_{2}\underaccent{\bar}{y}\in{\rm W}^{1,2}(\underaccent{\bar}{\omega},\mathbb{R}^{3}), which implies that \underaccent{\bar}{y}\in{\rm W}^{2,2}(\underaccent{\bar}{\omega},\mathbb{R}^{3}). Note also that \nabla\underaccent{\bar}{y}(x^{\prime})^{\scriptscriptstyle{\mathsf{T}}}\,\nabla\underaccent{\bar}{y}(x^{\prime})=\mathbb{I}_{3} for a.e. x^{\prime}\in\underaccent{\bar}{\omega}. Therefore \underaccent{\bar}{y}\in{\rm W}^{2,2}_{{\rm iso}}(\underaccent{\bar}{\omega}), thus accordingly y:=\underaccent{\bar}{y}\circ\rho\circ\tau_{-v}\in{\rm W}^{2,2}_{{\rm iso}}(\omega). ∎

Remark 3.7**.**

Observe that the condition “${\rm det}\rho_{1}=-{\rm det}\rho_{2}$ whenever $r_{1}=r_{2}$” permits to exclude the trivial case where we patch together pieces of cylinders $y_{1}$ and $y_{2}$ having the same curvatures (i.e. $\det\rho_{1}/r_{1}=\det\rho_{2}/r_{2}$, according to formula (3.11)). Clearly, this case does not force any condition on $[\gamma]$.

Moreover, an argument similar to that in the proof of Lemma 3.6 allows to prove necessary and sufficient conditions for having $y\in{\rm W}^{2,2}_{{\rm iso}}(\omega)$ of the form $y=y_{1}\chi_{\omega_{1}}+y_{2}\chi_{\omega_{2}}$ with, say, $y_{2}$ affine (using our terminology, a cylinder with $r_{2}=+\infty$). In this case, condition $(i)$ remains the same and condition $(ii)$ reduces to $\rho_{1}^{\scriptscriptstyle{\mathsf{T}}}\mathsf{e}_{2}\parallel[\gamma]$ (while $\rho_{2}\in{\rm Orth}(2)$ can be arbitrarily chosen). Moreover, for a chosen $\rho_{2}\in{\rm Orth}(2)$, condition $(iii)$ becomes

[TABLE]

with w_{k}:=\rho_{k}\big{(}\gamma(0)-(0,0)\big{)} and $\theta_{k}:=w_{k}\cdot\mathsf{e}_{1}/r_{k}$. * ∎*

Remark 3.8** (Properties of “roto-translations”).**

The following two properties, regarding the composition of cylinders, translations and rotations, can be easily proved.

• (i)

Fix $R\in{\rm SO}(3)$ and $T_{w}\in{\rm Trs}(3)$. Then $R\circ T_{w}=T_{Rw}\circ R$.

• (ii)

Let $\tau_{v}\in{\rm Trs}(2)$ and $\hat{R}_{\theta}\in{\rm SO}(3)$ be defined by

[TABLE]

Then $C_{r}\circ\tau_{v}=T_{C_{r}(v)}\circ\hat{R}_{(v\cdot\mathsf{e}_{1})/r}\circ C_{r}$, for every positive real number $r$.

In particular, property (ii) justifies the choice of the representation used for the elements in ${\rm Cyl}$ and it is useful for the proof of Lemma 3.6. * ∎*

Given a piecewise constant $\overline{A}$ and referring to Lemma 3.1 (see also the discussion after its statement), we set

[TABLE]

Recall that $\{0,\pm\mathfrak{r}_{k}\}$ are the eigenvalues (principal curvatures) of the (constant) curvature tensors ranging in $\mathcal{N}_{k}$.

Theorem 3.9**.**

Let $\overline{A}$ be of the form (3.14). Assume that $\mathfrak{r}_{k}\neq\mathfrak{r}_{j}$ for all $1\leq k<j\leq n$ such that \mathcal{H}^{1}\big{(}\partial\omega_{k}\cap\partial\omega_{j}\big{)}>0. Then there exists a pointwise minimizer $y\in{\rm W}^{2,2}_{{\rm iso}}(\omega)$ of $\mathcal{E}^{0}$ if and only if the following conditions are satisfied:

• $(a)$

$[\gamma_{k}]$* is a line segment with $\gamma_{k}(0),\gamma_{k}(1)\in\partial\omega$, for every $k=1,\ldots,n-1$;*

• $(b)$

\gamma_{k}\big{(}(0,1)\big{)}\cap\gamma_{j}\big{(}(0,1)\big{)}=\mathchar 31\relax* for all $k\neq j=1,\ldots,n-1$;*

• $(c)$

every non flat region $\omega_{k}$, i.e. $\omega_{k}$ with corresponding $\mathfrak{r}_{k}\neq 0$, satisfies: $\partial\omega_{k}\cap\omega$ consists of connected components which are orthogonal to some eigenvector (principal curvature direction) of the matrices of $\mathcal{N}_{k}$ corresponding to $\mathfrak{r}_{k}$.

Proof.

The sufficiency part of the statement follows by straightforward computations, as in the proof of Lemma 3.6. In order to prove necessity, we focus on the case $n=2$, when $\omega$ is subdivided into two Lipschitz subdomains $\omega_{1}$ and $\omega_{2}$ by a curve $\gamma:=\gamma_{1}$ as in Definition 3.4, since the general case can be achieved by an induction argument as a consequence of our definition of Lipschitz subdivision of the domain $\omega$.

Let $y\in{\rm W}^{2,2}_{{\rm iso}}(\omega)$ be a pointwise minimizer of $\mathcal{E}^{0}$. Note that on both subdomains $\omega_{1}$ and $\omega_{2}$ the target curvature tensor $\overline{A}$ is constant. Then by the definition of pointwise minimizers, by Lemma 3.3 and Lemma 3.1 we deduce that $y=y_{1}\chi_{\omega_{1}}+y_{2}\chi_{\omega_{2}}$, with $y_{k}=T_{v_{k}}\circ R_{k}\circ C_{1/|\mathfrak{r}_{k}|}\circ\rho_{k}\in{\rm Cyl}$, $k=1,2,$ where $\mathfrak{r}_{k}$ is given by (3.26) and $\rho_{k}$ is such that {\rm A}_{y_{k}}\equiv(\det\rho_{k})\,\rho_{k}^{\scriptscriptstyle{\mathsf{T}}}\,{\rm diag}\big{(}|\mathfrak{r}_{k}|,0\big{)}\rho_{k}\in\mathcal{N}_{k}. Since $\mathfrak{r}_{1}\neq\mathfrak{r}_{2}$, by Lemma 3.6 and Remark 3.7 we obtain that $[\gamma]$ must be a line segment and that $\rho_{k}^{\scriptscriptstyle{\mathsf{T}}}\mathsf{e}_{2}$ must be parallel to $[\gamma]$ (or equivalently that the eigenvector $\rho_{k}^{\scriptscriptstyle{\mathsf{T}}}{\sf e}_{1}$ of ${\rm A}_{y_{k}}$ is orthogonal to $[\gamma]$) whenever $\mathfrak{r}_{k}\neq 0$, $k=1,2$, which is precisely the statement of $(a)$ and $(c)$ in the case in which $n=2$.

∎

Remark 3.10**.**

Let $k$ and $j$ be such that $\mathcal{H}(\partial\omega_{k}\cap\partial\omega_{j})>0$. Observe that when $\mathfrak{r}_{k}=\mathfrak{r}_{j}$ (this may happen, though $\overline{A}_{k}\neq\overline{A}_{j}$), this condition does not impose that $\partial\omega_{k}\cap\partial\omega_{j}$ is a line segment. Indeed, when $\mathfrak{r}_{k}=\mathfrak{r}_{j}$, a pointwise minimizer $y$, when restricted to $\omega_{k}$ and $\omega_{j}$, will be given by some cylinders $y_{k}$ and $y_{j}$ with $r_{k}=1/|\mathfrak{r}_{k}|$ and $r_{j}=1/|\mathfrak{r}_{j}|$, respectively, which have the same curvatures $\det\rho_{k}|\mathfrak{r}_{k}|=\mathfrak{r}_{k}=\mathfrak{r}_{j}=\det\rho_{j}|\mathfrak{r}_{j}|$. This fact, as observed in Remark 3.7, does not impose any further conditions on $\partial\omega_{k}\cap\partial\omega_{j}$. * ∎*

Note that, if the target curvature does not induce any flat region, the presence of a pointwise minimizer forces the subdivision lines $[\gamma_{k}]$ to be all parallel (see Figure 2, (A) and (B)). When instead a flat region is present in the subdivision, this can give rise to a pointwise minimizer, even if the $[\gamma_{k}]$ are not mutually parallel (see Figure 2, (C) and (D)). Finally, observe that in this case a subdomain of type $(iii)$ and $(iv)$ can coexist (tough they cannot be neighbors).

Point $(c)$ above implies that for every $k$ and $j$ such that $\omega_{k}$ and $\omega_{j}$ are neighbor (i.e. share a piece of boundary, in symbols $\mathcal{H}^{1}(\partial\omega_{k}\cap\partial\omega_{j})>0$) it cannot be that $A_{k}$ is of type $(iii)$ (see Lemma 3.1) and $A_{j}$ is of type $(iv)$ at the same time. This is because, if not so, from point $(c)$ above it would follow that the line segment $[\gamma]=\partial\omega_{k}\cap\partial\omega_{j}$ is simultaneously parallel to $\mathsf{e}_{2}$ and to $\mathsf{e}_{1}$, which is absurd. Hence, a reference domain endowed with target curvature as in Figure 3 does not admit a pointwise minimizer.

4. Application to thin gel sheets

In this section, we apply the reduced model derived in Section 2 to the study of thin sheets of polymer gel. In the present context, a polymer gel is a network of cross-linked polymer chains swollen with a liquid solvent. Denote by $v$ the volume per solvent molecule, by $\overline{N}$ the density of polymer chains in the reference volume and define $\mathbb{R}^{3\times 3}_{1}:=\{F\in\mathbb{R}^{3\times 3}:{\rm det}F\geq 1\}$. The dimensionless free-energy density for isotropic

and homogeneous

polymer gels is of Flory-Rehner type (see [17]) and is given by the function $W:\mathbb{R}^{3\times 3}_{1}\to\mathbb{R}$ defined as

[TABLE]

Here $\chi\in(0,1/2]$ and $\delta\geq 0$ are fixed dimensionless constants depending on the physical and chemical properties of the material and on environmental conditions, respectively. The function $W_{vol}^{\chi}:[1,+\infty)\to(-\infty,0]$ is of class $C^{\infty}$ on $(1,+\infty)$ and (right-) continuous at $1$, with

[TABLE]

Our attention is in particular focussed on a heterogeneous thin gel sheet occupying the reference configuration $\Omega_{h}$. More precisely, we suppose that the sheet is characterized by a $z$-dependent cross-linking density, which in turn determines a $z$-dependent density $\overline{N}^{h}$ of polymer chains. At the same time, we suppose that $\overline{N}^{h}$ is a perturbation of a constant value $\overline{N}$, namely

[TABLE]

where $g\in{\rm L}^{\infty}(\Omega)$ and

[TABLE]

Observe that the condition (4.3) is equivalent to $\int_{-\nicefrac{{1}}{{2}}}^{\nicefrac{{1}}{{2}}}g(x^{\prime},t)\,{\rm d}t=0$ for a.e. $x^{\prime}\in\omega$. Using the model energy density (4.1), we can describe this heterogeneous system via the family of densities

[TABLE]

for a.e. $z\in\Omega_{h}$, every $F\in\mathbb{R}^{3\times 3}_{1}$ and every $h>0$. Letting $\{W^{h}\}$ be the associated family of rescaled densities $W^{h}:\Omega\times\mathbb{R}^{3{\times}3}\to\mathbb{R}\cup\{+\infty\}$ defined by

[TABLE]

and declaring it to be equal $+\infty$ on $\mathbb{R}^{3{\times}3}\setminus\mathbb{R}^{3{\times}3}_{1}$, one can show (see the details in [4]) that there exist constants $\alpha>1$ and $\Theta\in\mathbb{R}\setminus\{0\}$, depending on $v$, $\overline{N}$, $\chi$ and $\delta$, such that for a.e. $x\in\Omega$ it holds

[TABLE]

Moreover, one can show that $\{W^{h}\}$ is a family of frame indifferent functions that uniformly converges (in the sense of $(iii)$ in Definition 2.1) to $W$ and have quadratic growth. The hypothesis (4.3) ensures that the spontaneous strain $B=b\mathbb{I}_{3}$ in this case satisfies the assumption (2.23).

By a suitable change of variable in order to switch from the energy wells that are $h$-close to $\alpha\mathbb{I}_{3}$ to those that are $h$-close to $\mathbb{I}_{3}$ and than using the theory developed in Section 2, we obtain the corresponding 2D Kirchhoff model, which is in this case given by the energy functional

[TABLE]

for every $y\in{\rm W}^{2,2}(\omega,\mathbb{R}^{3})$ satisfying $(\nabla y)^{\scriptscriptstyle{\mathsf{T}}}\nabla y=\alpha^{2}\mathbb{I}_{2}$ a.e. in $\omega$ (that we will briefly call an $\alpha$-isometry), and $+\infty$ otherwise in ${\rm W}^{1,2}(\Omega,\mathbb{R}^{3})$. The relation with the initial 3D model can be seen trough the target curvature tensor $\overline{A}$, given by

[TABLE]

and through the quadratic form $Q_{2}$ defined by

[TABLE]

and explicitly reads as

[TABLE]

where $G$ and $\Lambda(\alpha)$ are positive constants depending only on the (fixed) physical properties of the material.

Remark 4.1**.**

We remark that the (rescaled) energy densities $W^{h}(x,\cdot)$ defined by (4.4) are minimized on

[TABLE]

for some $\alpha>1$. Moreover, they uniformly converge to $W$ given by (4.1), which is minimized at $\alpha\,{\rm SO}(3)$. However, by directly confronting formulas (4.1) and (4.4), one can check that the densities $W^{h}$ cannot be rewritten in the “prestretch” form

[TABLE]

∎

Example 4.2**.**

Consider a thin film made of polymeric gel occupying the domain $\Omega_{h}$ where $\omega=(-d,d)\times(0,\ell)$ and with associated family of energy densities $\{\overline{W}^{h}\}$ given by (4.4). Suppose that the variation of the number of polymeric chains $\overline{N}^{h}$ given by (4.2) is such that the associated function $g$ is of the form

[TABLE]

with $g_{1},g_{2}\in{\rm L}^{\infty}(-1/2,1/2)$, satisfying $\int_{\nicefrac{{-1}}{{2}}}^{\nicefrac{{1}}{{2}}}g_{1}(t)\,{\rm d}t=\int_{\nicefrac{{-1}}{{2}}}^{\nicefrac{{1}}{{2}}}g_{2}(t)\,{\rm d}t=0$, and

[TABLE]

with $a_{1}$, $a_{2}$ non zero .

In turn, the limiting 2D model is characterized by the target curvature tensor $\overline{A}$ that equals $a(x^{\prime})\mathbb{I}_{2}$ at each $x^{\prime}\in\omega$, where $a(x^{\prime})=a_{1}$ if $x^{\prime}\in(-d,0]\times(0,\ell)$ and $a(x^{\prime})=a_{2}$ if $x^{\prime}\in(0,d)\times(0,\ell)$.

Finally, using the results of Lemma 3.1 and Theorem 3.9 , we can determine the minimizers of the limiting energy $\mathcal{E}^{0}$ . Note that we are in the case of Lipschitz $2$-subdivision of $\omega$ into subdomains $\omega_{1}:=(-d,0)\times(0,\ell)$ and $\omega_{2}:=(0,d)\times(0,\ell)$. Given that the subdivision curve is $[\gamma_{1}]=\partial\omega_{1}\cap\partial\omega_{2}=[0,\ell]$ (a line segment parallel to $\mathsf{e}_{2}$) and $\overline{A}$ is of type $(i)$ (see Lemma 3.1), a pointwise minimizer of $\mathcal{E}^{0}$ exists and is any (up to rotations and translations in $\mathbb{R}^{3}$) $\alpha$-isometry $y:=y_{1}\chi_{\omega_{1}}+y_{2}\chi_{\omega_{2}}$, with $y_{1},y_{2}$ given for every $(x_{1},x_{2})\in\omega$ by

[TABLE]

with

[TABLE]

(according to Lemma 3.1 and (4.8)) and appropriate choice (depending on the sign of $\mathfrak{r}_{k}$, $k=1,2$) of $\sigma_{i}\in\{-1,1\},i=0,1,2$.

Since the pull-back of the second fundamental form associated with $y_{1}(\omega_{1})$ and $y_{2}(\omega_{2})$ respectively, is given by

[TABLE]

it is clear that there exists two different (up to rotations and translation in $\mathbb{R}^{3}$) minimizing surfaces $y(\omega)$. The choice of $\sigma_{1}\in\{-1,1\}$ determines one of the two possible options for $y_{1}$, represented by a dashed or a full black line in Figure 4 below. For any chosen value of $\sigma_{1}$, the values of $\sigma_{2}$ and $\sigma_{0}$ are immediately determined by the sign of $\mathfrak{r}_{1}$ and $\mathfrak{r}_{2}$, respectively.

∎

5. Conclusions

In this paper, we have considered a family of 3D energy functionals that is relevant from the viewpoint of applications to shape morphing materials, especially in the context of swelling gels. As remarked in the Introduction, the starting 3D model (2.2)–(2.5) (which reduces to (4.4)–(4.6) in the specific gel case), which may be employed to accurately describe the swelling of polymer gels, is characterized by spontaneous stretches but not in general representable in the “pre-stretch form”. Another peculiarity of such a family of 3D energies is that the spontaneous stretches are naturally related to the elastic parameters of the material. Hence, heterogeneities in the stiffness can be exploited to program the target shape of the system.

Having in mind applications to free-swelling, thin gel sheets with heterogeneous stiffness, we have derived by dimension reduction from the aforementioned 3D model a Kirchhoff plate theory (Sections 2–3). This plate model, whose governing equations are (3.1)–(3.2), is then specialized to thin gel sheets in Section 4. A central result of the theory is the expression of the spontaneous curvature as a function of parameters that can be traced back to the three-dimensional stiffness field. The derivation of the limiting model is restricted to the case where the compatibility condition (1.5) is fulfilled by the spontaneous strain. As explained in the introduction, this fact allows us to perform a rigorous dimension reduction with standard arguments. Even though it is possible to realize experimentally simple systems that fulfill such condition, this paper raises and leaves open a mathematically relevant problem, that is, finding the general limiting Kirchhoff model without the restriction (1.5), whose complete solution would potentially give new insight into the dimension reduction from 3D elasticity to plate theory.

We have then investigated the pointwise minimizers of the 2D model, restricting the attention to the case where the target curvature is piecewise constant (see Figure 1, Figure 2 (B) and (D), for some sketches of the configurations which occur in this case). The interest in this special class of minimizers is twofold. On the one hand, such a class corresponds to some of the simplest structures that can be realized experimentally, which yet can find interesting engineering applications (i.e. foldable structures, see the forthcoming [4]). On the other hand, it opens the way to the study of a huge class of open minimum problems (that is the minimization of the functional (3.1)–(3.3) in ${\rm W}^{2,2}_{{\rm iso}}(\omega)$, given an arbitrary bounded $\overline{A}:\omega\to{\rm Sym}(2)$), for which ready-made analytical tools are not yet available.

Acknowledgements

This work has been funded by the European Research Council through the ERC Advanced Grant 340685-MicroMotility. We thank A. DeSimone for helpful discussions. We would also like to thank the anonymous referees for their careful reading of our paper, and for several constructive remarks.