A homogenization bending shell theory for multiscale materials from $3D$ nonlinear elasticity

TL;DR

This paper develops homogenized bending shell theories from 3D nonlinear elasticity, accounting for multiple small parameters related to material scales and shell thickness, resulting in different asymptotic models.

Contribution

It introduces a multiscale homogenization approach for nonlinear shell models considering three small parameters, expanding the theoretical framework for complex materials.

Findings

Derivation of various asymptotic shell theories based on parameter ratios

Inclusion of multiple homogenization scales in shell modeling

Extension of nonlinear elasticity to multiscale shell structures

Abstract

We derive homogenized bending shell theories starting from three dimensional nonlinear elasticity. The original three dimensional model contains three small parameters: the two homogenization scales $\varepsilon$ and $\varepsilon^2$ of the material properties and the thickness $h$ of the shell. Depending on the asymptotic ratio of these three parameters, we obtain different asymptotic theories.

Equations

$$\gamma :=\underset{h\to 0}{\lim }\frac{h}{\epsilon (h)}\in [0,\infty ]$$

$${\gamma }_1:=\underset{h\to 0}{\lim }\frac{h}{\epsilon (h)}\text{and}{\gamma }_2:=\underset{h\to 0}{\lim }\frac{h}{{\epsilon }^2(h)}.$$

$$J^h(v)=\frac{1}{h}{\int }_{S^h}W\left({\Theta }^h(x),r(x)/\epsilon ,r(x)/{\epsilon }^2,\nabla v(x)\right)dx,$$

$$\underset{h\to 0}{\lim }\sup h^{-2}{J}^h(u^h)<+\infty .$$

$$\Phi (S)=\omega \times \{0\}.$$

$${\tau }_{\beta }(x)=({\partial }_{\beta }\xi )(\Phi (x))\text{for all}x\in S,$$

$${\tau }^{\alpha }\cdot {\tau }_{\beta }={\delta }_{\alpha \beta }\text{on}S,$$

$$n=\frac{{\tau }_1\times {\tau }_2}{|{\tau }_1\times {\tau }_2|}.$$

$$T_S(x)=I-n(x)\otimes n(x).$$

$$a\odot b:=\frac{1}{2}(a\otimes b+b\otimes a),$$

$$L^2(S,$$

$$\{Z\in L^2(S,W^{1,2}(\mathcal{Y},{\mathbb{R}}^3)):Z(x)\in W^{1,2}(\mathcal{Y},T_{x}S)\text{for a.e.}x\in S\}.$$

$$\text{S}(x)\tau =({\nabla }_{\tau }n)(x)\text{for all}x\in S,\tau \in T_{x}S.$$

$$(u^{\ast }{\text{S}}_u)\tau =u^{\ast }({\text{S}}_u{D}_{\tau }u)$$

$${\text{S}}_u^r=u^{\ast }{\text{S}}_u-\text{S}.$$

$$S^h=\left\{x+tn(x):x\in S\text{and}t\in (-\frac{h}{2},\frac{h}{2})\right\}.$$

$$d\pi =T_S(\pi )(I+t\text{S}(\pi ))(I+t\text{S}(\pi )T_S(\pi ){)}^{-1}.$$

$$|d\pi -(I-t\text{S}(\pi )T_S(\pi ))|\le C{t}^2\text{on}{S}^1.$$

$$({\text{Hess}}_{\mathcal{Y}}f)(x,y)={\partial }_{y_{\alpha }}{\partial }_{y_{\beta }}f(x,y){\tau }^{\alpha }(x)\odot {\tau }^{\beta }(x),$$

$$({\text{Hess}}_{\mathcal{Z}}f)(x,y,z)={\partial }_{z_{\alpha }}{\partial }_{z_{\beta }}f(x,y,z){\tau }^{\alpha }(x)\odot {\tau }^{\beta }(x),$$

$$({\text{Def}}_{\mathcal{Y}}v)(x,y)={\left(\text{sym}{\nabla }_{y}v(x,y)\right)}_{\alpha \beta }{\tau }^{\alpha }(x)\odot {\tau }^{\beta }(x).$$

$$({\text{Def}}_{\mathcal{Z}}v)(x,y,z)={\left(\text{sym}{\nabla }_{z}v(x,y,z)\right)}_{\alpha \beta }{\tau }^{\alpha }(x)\odot {\tau }^{\beta }(x).$$

$$\Xi (z^{\prime },z_3)=\xi (z^{\prime })+z_{3}n\left(\xi (z^{\prime })\right)\text{for all}{z}^{\prime }\in \omega \text{and}{z}_3\in \mathbb{R}.$$

$${\Theta }^h(x)=\pi (x)+\frac{t(x)}{h}n(x).$$

$${\Theta }^h\circ \Xi =\Xi \circ {\tilde{\Theta }}^h\text{on}{\Omega }^h.$$

$$({\tilde{\nabla }}_h\tilde{y})\circ {\tilde{\Theta }}^h=\nabla \tilde{u}\text{on}\Omega .$$

$$({\nabla }_{h}y)\circ {\Theta }^h=\nabla u\text{on}{S}^h.$$

$${\tilde{\nabla }}_h\tilde{y}=\nabla u(\Xi )\left((\nabla \Xi )\circ ({\tilde{\Theta }}^h{)}^{-1}\right)$$

$$\nabla {\Theta }^h=\left(T_S+\frac{1}{h}(n\otimes n+t\text{S})\right)(I+t\text{S}{)}^{-1}\text{on}{S}^h.$$

$${\nabla }_{h}y=\nabla y\left(T_S+\frac{1}{h}(n\otimes n+ht\text{S})\right)(I+ht\text{S}{)}^{-1}\text{on}{S}^1.$$

$$\underset{h\to 0}{\lim }{\int }_{S^1}{f}^h(x)\phi \left(x,\frac{r(x)}{\epsilon },\frac{r(x)}{{\epsilon }^2}\right)dx={\int }_{S^1}{\int }_{\mathcal{Y}}{\int }_{\mathcal{Y}}f(x,y,z)\phi \left(x,y,z\right)dzdydx,$$

Full text

Key words Homogenization, dimension reduction, nonlinear elasticity, bending shell theory, multiscale convergence.

2010 AMS Subject Classification: 35B27; 74B20; 74K25; 74Q05; 74E30.

1 Introduction

Since the early ’90s the search for lower dimensional models describing thin three-dimensional structures has been of great interest for its implications in the mechanics of materials. The nonlinear models for plates and shells for homogeneous material have been derived rigorously by means of $\Gamma$-convergence [3], starting from three dimensional nonlinear elasticity. Hierarchies of limit models have been deduced by $\Gamma-$convergence, depending on the scaling of the elastic energy $J^{h}$ with respect to the thickness parameter $h\in(0,1]$. More precisely, we note that $J^{h}\sim 1$ for membrane, $J^{h}\sim h^{2}$ for bending and $J^{h}\sim h^{4}$ for von-Kármán regime.

The first results in that direction for membrane regime for plates and shells can be found in [13] and [14], respectively. The nonlinear bending theory for plates was derived in [8] and the corresponding theory for shells in [7].

Recently, models of homogenized bending plates and shells for heterogeneous material were derived by simultaneous use of homogenization and dimensional reduction techniques in the special case when relation between thickness of the body $h$ and oscillations of the material $\varepsilon(h)$ is given by the existence of limit

[TABLE]

with some assumptions on stored energy density. Different limit models can be obtained depending on values of $\gamma$ ($\gamma=0$, $\gamma\in(0,+\infty)$ and $\gamma=+\infty$) see e.g. [10, 11, 12] and [16].

In [4], authors consider scaling of energy corresponding to bending plates theory by introducing two different scales for the oscillation of material, namely, $\varepsilon(h),\varepsilon^{2}(h)$. More precisely, they assume existence of limits

[TABLE]

In fact, they consider various cases of i.e. $\gamma_{1}=0$ and $\gamma_{2}=+\infty$; $\gamma_{1}\in(0,+\infty)$ and $\gamma_{2}=+\infty$; $\gamma_{1}=+\infty$ and $\gamma_{2}=+\infty$.

In this paper, we deduce a multiscale version of the results in [12] and it is a natural follow up of [4]. We consider the scaling of energy corresponding to bending shell theory and assume that the shell $S^{h}\subset\mathbb{R}^{3}$ of thickness $h$ undergoes the action of two homogeneity scales $\varepsilon(h)$ and $\varepsilon^{2}(h)$ according to the assumption (1.1). The rescaled nonlinear elastic energy is given by

[TABLE]

for every deformation $v\in W^{1,2}(S^{h};\mathbb{R}^{3})$, where the stored energy density $W$ is periodic in its second and third arguments and satisfies the usual assumptions in nonlinear elasticity as well as the nondegeneracy condition (see Section 4) adopted from [12] and [11]. More precisely, we consider a sequences of deformations $(u^{h})\subset W^{1,2}(S^{h};\mathbb{R}^{3})$ satisfying

[TABLE]

We seek to identify the effective energy associated to the rescaled elastic energies $\left\{\displaystyle\frac{J^{h}(u^{h})}{h^{2}}\right\}$ for different values of $\gamma_{1}$ and $\gamma_{2}$. We restrict ourselves to convex shells and our analysis requires the following approaches:

• (1)

Dimension reduction techniques, in particular, the quantitative rigidity estimate and approximation schemes developed by Friesecke, James and Müller in their famous work on the derivation of nonlinear plate theories [8].

• (2)

Homogenization methods, in particular, three-scale convergence see [2, 4, 5] and [6].

We derive different asymptotic theories depending on the parameters $\gamma_{1}$ and $\gamma_{2}$ (see Section 4) .

The main result of our article is Theorem 4.1. Starting from a limiting deformation with belongs to the space $W^{2,2}(S^{h};\mathbb{R}^{3})$ we use the density result from [12] to construct the required recovery sequence. The paper is organized as follows: in Section 2, we introduce the definition and some properties of three-scale convergence for shells. Section 3 describes the setting of our problem and introduce the assumptions on energy density. In Section 4, we mention our main result and prove it in Section 5 deriving bounds for rescaled elastic energy depending on different values of $\gamma_{1}$ and $\gamma_{2}$. Finally, in Section 6, we analize the homogenized model for convex shell case.

Contents

1. 1 Introduction

2. 2 Preliminaries

3. 2.1 Homogenization for shells

4. 2.1.1 Surfaces and shells in $\mathbb{R}^{3}$

5. 2.1.2 Three-scale convergence on shells

6. 3 Elasticity framework and intermediate results

7. 3.1 Asymptotic energy functionals

8. 4 Main result

9. 5 Proof of main result

10. 5.1 Proof of lower bound

11. 5.2 Proof of upper bound

12. 6 Convex shells

2 Preliminaries

2.1 Homogenization for shells

We begin by introducing some further notation. Set $Y=[-\frac{1}{2},\frac{1}{2})^{2}$ and we denote by $\mathcal{Y}=\mathbb{R}^{2}/\mathbb{Z}^{2}$ the set $Y$ endowed with the torus topology. For all $k\in\mathbb{N}\cup\{0\}$ the set of all $f\in C^{k}(\mathbb{R}^{2})$ with $D^{\alpha}f(\cdot+z)=D^{\alpha}f$ for all $z\in\mathbb{Z}^{2}$ and all multiindices $\alpha$ of order up to $k$ is denoted by $C^{k}(\mathcal{Y})$.

We denote by $C^{k}_{0}$ functions with compact support. For any open set $A$, we denote by $L^{2}(\mathcal{Y})$, $W^{1,2}(\mathcal{Y})$ and $W^{1,2}(A\times\mathcal{Y})$ the Banach spaces obtained as closures of $C^{\infty}(\mathcal{Y})$ and $C^{\infty}(\overline{A},C^{\infty}(\mathcal{Y}))$ with respect to the norm in $L^{2}(Y)$, $W^{1,2}(Y)$ and $W^{1,2}(A\times Y)$, respectively. An additional dot (e.g. in $\dot{L}^{2}(\mathcal{Y})$) denotes functions with average zero over $\mathcal{Y}$.

2.1.1 Surfaces and shells in $\mathbb{R}^{3}$

Let $h\in(0,1]$, $\kappa\in(0,1)$ and let $\omega\subset\mathbb{R}^{2}$ be a bounded domain with $C^{3,\kappa}$ boundary. Set $I=(-\frac{1}{2},\frac{1}{2})$, $\Omega^{h}=\omega\times(hI)$, and $\Omega=\omega\times I$. From now on, $S\subset\mathbb{R}^{3}$ denotes (the relative interior of) an embedded compact connected oriented surface with boundary. For convenience we assume that $S$ is parametrized by a single chart. More precisely, we assume that there exist an open set $V\subset\mathbb{R}^{3}$ containing the closure of $S$ and an open set $U\subset\mathbb{R}^{3}$ containing $\overline{\omega}\times\{0\}$ and $C^{3,\kappa}$ diffeomorphism $\Phi:V\to U$ such that

[TABLE]

Then $\xi:\omega\to S$, defined by $\xi(z)=\Phi^{-1}(z,0)$, is a global $C^{3,\kappa}$ chart for $S$.

By $W^{2,2}_{\text{iso}}(S)$ we denote the $W^{2,2}(S)$ isometries of the surface $S$ into $\mathbb{R}^{3}$. The space $W^{2,\infty}_{\text{iso}}(S)$ is defined similarly.

By Theorem 2.1 in [12] we have that $u\in W^{2,2}_{\text{iso}}(S)$ is equivalent to $u\circ\xi\in W^{2,2}_{g}(\omega)$, for $g=(\nabla\xi)^{T}(\nabla\xi)$ the Riemannian metric on $\omega$ induced by $\xi$.

As usual, $T\,S$ denotes the tangent bundle over $S$ and $N\,S$ the normal bundle. A basis of the tangent space $T_{x}S$ is given by

[TABLE]

where $\beta=1,2$. We view $T_{x}S$ as a subspace of $\mathbb{R}^{3}$ and write $\sigma\cdot\tau$ the scalar product of both spaces.

The dual basis of the tangent space $T_{x}S$ is denoted by $(\tau^{1}(x),\tau^{2}(x))$. So by definition

[TABLE]

where $\delta_{\alpha\beta}$ is the Kronecker symbol. We frenquently indentify $T^{*}_{x}(S)$ with $T_{x}(S)$ via the scalar product. Define the normal $n:S\to\mathbb{S}^{2}$ by

[TABLE]

The orthogonal projection onto $T_{x}S$ is

[TABLE]

The tensor products $T\,S\otimes T\,S$ etc. are defined fiberwise. $T^{*}_{x}S\otimes T^{*}_{x}S$ will be regarded as a subspace of $\mathbb{R}^{3\times 3}$.

If $E$ and $F$ are vector spaces (or bundles) then the space of all symmetric products

[TABLE]

with $a\in E$ and $b\in F$ is denoted by $E\odot F$.

Section $B$ of $T^{*}S\otimes T^{*}S$ will frequently regarded as maps from $S$ into $\mathbb{R}^{3\times 3}$ via the embedding $\iota$ defined by $\iota(B)=B(T_{S},T_{S})$. By definition, $B(T_{S},T_{S}):S\to\mathbb{R}^{3}$ takes the vector fields $v,w:S\to\mathbb{R}^{3}$ into the function $x\mapsto B(x)(T_{S}(x)v(x),T_{S}(x)w(x))$.

For any vector bundle $E$ over $S$ we denote by $L^{2}(S,E)$ the space of all $L^{2}-$sections of $E$. The spaces $W^{1,2}(S,E)$ etc. are defined similarly. For any vector bundle $E$ over $S$ with fibers $E_{x}$, we denote by $L^{2}(\mathcal{Y},E)$ the vector bundle over $S$ with fibers $L^{2}(\mathcal{Y},E_{x})$. The bundles $W^{1,2}(\mathcal{Y},E)$ etc. are defined similarly. For example, $L^{2}-$sections of the bundle $W^{1,2}(\mathcal{Y},T\,S)$ are given by

[TABLE]

For a function $f:S\to\mathbb{R}$ its differential $df$ is given by $df(x)\tau=\nabla_{\tau}f(x)$ for all $\tau\in T_{x}S$. Here $\nabla_{\tau}f$ denotes the directional derivative of $f$ in direction of the tangent vector $\tau$. We extend these definitions componentwise to maps into $\mathbb{R}^{3}$. By $\nabla$ we denote the usual gradient on $\mathbb{R}^{3}$ (or on $\mathbb{R}^{2}$).

As usual, the Weingarten map S of $S$ is the differential of the normal, i.e.,

[TABLE]

We extend $\textbf{S}(x)$ trivially to $\mathbb{R}^{3}$ by setting $\textbf{S}(x)=\textbf{S}(x)T_{S}(x)$.

For an immersion $u:S\to\mathbb{R}^{3}$ denote by $\textbf{S}_{u}$ the Weingarten map for the surface $u(S)$. We define its pullback to $S$ by setting

[TABLE]

for all smooth tangent vector fields $\tau$ to $S$. Here by definition, $u^{*}(D_{\sigma}u)=\sigma$ for all smooth tangent vector fields $\sigma$ to $S$. As in we will encounter the relative Weingarten map

[TABLE]

The nearest point of retraction $\pi$ of a tubular neightborhood of $S$ onto $S$ satisfies $\pi(x+tn(x))=x$ for small $|t|$ and all $x\in S$. After rescaling the ambient space, we may assume that the curvature of $S$ is as small as we please. Therefore, we may assume without loss of generality that $\pi$ is well-defined on a domain containing the closure of the set $\{x+tn(x):x\in S,-1/2<t<1/2\}$, and that $|Id+t\textbf{S}(x)|\in(1/2,3/2)$ for all $t\in(-\frac{1}{2},\frac{1}{2})$ and all $x\in S$.

For a subset $\overline{S}\subset S$ and $h\in(0,1]$ we define $\overline{S}^{h}=\{x+tn(x):x\in\overline{S},\,-h/2<t<h/2\}$. In particular, the whole shell is, by definition,

[TABLE]

We introduce the map $r=\Phi\circ\pi$. Moreover, we introduce the function $t:S^{1}\to\mathbb{R}$ by setting $t(x)=(x-\pi(x))\cdot n(x)$ for all $x\in S^{1}$. By virtue of Lemma 2.21 in [11] we have the following identity on $S^{1}$:

[TABLE]

(Here and elsewhere we write $T_{S}(\pi)$ instead of $T_{S}\circ\pi$ etc.). Hence there exists a constant $C$ depending only on $S$ such that

[TABLE]

Abusing notations, maps $f:S\to\mathbb{R}^{k}$ will often be extended to $S^{1}$ by setting $f=f\circ\pi$. We extend $r,T_{S}$ and S in this way, too.

For functions $f\in L^{2}(S,W^{2,2}(\mathcal{Y}))$ the expression $\text{Hess}_{\mathcal{Y}}$ is the section of the bundle $L^{2}(\mathcal{Y},T\,S\odot T\,S)$ over $S$ given by

[TABLE]

where $(\nabla^{2}_{y}f)_{\alpha\beta}=\partial_{y_{\alpha}}\partial_{y_{\beta}}f$. Analogously, for functions $f\in L^{2}(S\times\mathcal{Y},W^{2,2}(\mathcal{Y}))$ the expression $\text{Hess}_{\mathcal{Z}}$ is the section of the bundle $L^{2}(\mathcal{Y}\times\mathcal{Y},T\,S\odot T\,S)$ over $S$ given by

[TABLE]

where $(\nabla^{2}_{z}f)_{\alpha\beta}=\partial_{z_{\alpha}}\partial_{z_{\beta}}f$.

For $v\in L^{2}(S,W^{1,2}(\mathcal{Y};\mathbb{R}^{2}))$ we define the section $\text{Def}_{\mathcal{Y}}v$ of the bundle $L^{2}(\mathcal{Y},T^{*}S\odot T^{*}S)$ by

[TABLE]

For $v\in L^{2}(S\times\mathcal{Y},W^{1,2}(\mathcal{Y};\mathbb{R}^{2}))$ we define the section $\text{Def}_{\mathcal{Z}}v$ of the bundle $L^{2}(\mathcal{Y}\times\mathcal{Y},T^{*}S\odot T^{*}S)$ by

[TABLE]

Here and elsewhere $\nabla_{y}$ is the gradient in $\mathcal{Y}$ with respect to the variable $y$ and $\nabla_{z}$ is the grandient in $\mathcal{Z}$ with respect to the variable $z$ (and not some directional derivative).

We define the map $\Xi:\omega\times\mathbb{R}\to\mathbb{R}^{3}$ by

[TABLE]

As in [10] we will use the diffeomorphism $\widetilde{\Theta^{h}}:\Omega^{h}\to\Omega$ given by $\widetilde{\Theta^{h}}(z_{1},z_{2},z_{3})=(z_{1},z_{2},z_{3}/h)$, and for a map $\widetilde{y}:\Omega\to\mathbb{R}^{3}$ we introduce the scaled gradient $\widetilde{\nabla}_{h}y=(\partial_{1}y,\partial_{2}y,\frac{1}{h}\partial_{3}y)$. The counterpart of $\widetilde{\Theta^{h}}$ on the shell is the diffeomorphism $\Theta^{h}:S^{h}\to S^{1}$ given by

[TABLE]

It is easy to see that

[TABLE]

For given $u:S^{h}\to\mathbb{R}^{3}$ we define its pulled back version $\widetilde{u}:\Omega^{h}\to\mathbb{R}^{3}$ by $\widetilde{u}=u\circ\Xi$. We also define its rescaled version $y:S^{1}\to\mathbb{R}^{3}$ by $y(\Theta^{h})=u$ on $S^{h}$ and we define the pulled back version $\widetilde{y}$ of this map by $\widetilde{y}=y\circ\Xi$. Then it is easy to see that

[TABLE]

We define the rescaled gradient $\nabla_{h}y$ of $y$ by the condition

[TABLE]

Using (2.3) and (2.4) it is clear to see that

[TABLE]

and using (2.1) we can see that

[TABLE]

Finally, to express $\nabla_{h}y$ in terms of $\nabla y$, we insert the definition of $y$ into (2.4) and use (2.1) to find

[TABLE]

2.1.2 Three-scale convergence on shells

Recall that we extend the chart $r$ trivially from $S$ to $S^{1}$. We make the following definitions:

• (i)

A sequence $(f^{h})\subset L^{2}(S^{1})$ is said to converge weakly three-scale on $S^{1}$ to the function $f\in L^{2}(S^{1},L^{2}(\mathcal{Y}\times\mathcal{Y}))$ as $h\to 0$, provided that the sequence $(f^{h})$ is bounded in $L^{2}(S^{1})$ and

[TABLE]

for all $\varphi\in C^{0}_{c}\left(S^{1},C^{0}(\mathcal{Y}\times\mathcal{Y})\right)$.

• (ii)

We say that $f^{h}$ strongly three-scale converges to $f$ if, in addition,

[TABLE]

• (iii)

For a sequence $(f^{h})\subset L^{2}(S^{1})$ and for $f_{1}\in L^{2}(S^{1}\times\mathcal{Y}\times\mathcal{Y})$ with $\int_{\mathcal{Y}}f_{1}(\cdot,\cdot,z)dz=0$ for almost every $x\in S^{1}\times\mathcal{Y}$, we write $f^{h}\overset{osc,Z}{\rightharpoonup}f_{1}$ provided that

[TABLE]

for all $\varphi\in C^{\infty}_{0}(S^{1};C^{\infty}(\mathcal{Y}))$, all $\rho\in C^{\infty}(\mathcal{Y})$ with $\int_{\mathcal{Y}}\rho(z)dz=0$.

We write $f^{h}\overset{3}{\rightharpoonup}f$ to denote weak three-scale convergence and $f^{h}\overset{3}{\rightarrow}f$ to denote strong three-scale convergence. If $f^{h}\overset{3}{\rightharpoonup}f$ then $f^{h}\rightharpoonup\int_{\mathcal{Y}}\int_{\mathcal{Y}}f(\cdot,y,z)dzdy$ weakly in $L^{2}$. If $f^{h}$ is bounded in $L^{2}(S^{1})$ then it has a subsequence which converges weakly three-scale to some $f\in L^{2}(S^{1};L^{2}(\mathcal{Y}\times\mathcal{Y}))$. These and other facts can be deduced from the corresponding results on planar domains (cf. [1] and [17]) by means of the following simple observations.

Defining $\tilde{f}=f^{h}\circ\Xi$ and $\tilde{f}(\Xi(z),y,w)$, and taking

[TABLE]

a change of variable shows that (2.8) is equivalent to

[TABLE]

where $z^{\prime}$ is the projection of $z$ onto $\mathbb{R}^{2}$.

The following Lemma can be also adapted to the curved setting following the same pattern as above (writting just $S^{1}$ instead of $\Omega$).

Lemma 2.1**.**

• (i)

Let $f_{0}$ and $f^{1}\in H^{1}(\Omega)$ such that $f^{h}\rightharpoonup f_{0}$ weakly in $H^{1}(\Omega)$ then there exist $\phi\in L^{2}(\Omega;\dot{H}^{1}(\mathcal{Y};\mathbb{R}^{3}))$ and $\psi\in L^{2}(\Omega\times\mathcal{Y};\dot{H}^{1}(\mathcal{Y};\mathbb{R}^{3\times 3}))$ such that

[TABLE]

• (ii)

Let $f_{0}$ and $f^{h}\in H^{1}(\Omega)$ be such that $f^{h}\rightharpoonup f_{0}$ weakly in $H^{1}(\Omega)$ and assume that $\nabla f^{h}\overset{3}{\rightharpoonup}\nabla f_{0}+\nabla_{y}\phi+\nabla_{z}\psi$ for some $\phi\in L^{2}(\Omega;\dot{H}^{1}(\mathcal{Y};\mathbb{R}^{3}))$ and $\psi\in L^{2}(\Omega\times\mathcal{Y};\dot{H}^{1}(\mathcal{Y};\mathbb{R}^{3\times 3}))$. Then $\frac{f^{h}}{\varepsilon^{2}}\overset{osc,Z}{\rightharpoonup}\psi$.

• (iii)

Let $f_{0}$ and $f^{h}\in H^{1}(\Omega)$ be such that $f^{h}\rightharpoonup f_{0}$ weakly in $H^{2}(\Omega)$ and assume that $\nabla f^{h}\overset{3}{\rightharpoonup}\nabla f_{0}+\nabla_{y}\phi+\nabla_{z}\psi$ for some $\phi\in L^{2}(\Omega;\dot{H}^{1}(\mathcal{Y};\mathbb{R}^{3}))$ and $\psi\in L^{2}(\Omega\times\mathcal{Y};\dot{H}^{1}(\mathcal{Y};\mathbb{R}^{3\times 3}))$. Moreover, if $\nabla^{2}f^{h}\overset{3}{\rightharpoonup}\nabla^{2}f_{0}+\nabla^{2}_{y}\phi+\nabla^{2}_{z}\psi$ for some $\phi\in L^{2}(\Omega;\dot{H}^{2}(\mathcal{Y};\mathbb{R}^{3}))$ and $\psi\in L^{2}(\Omega\times\mathcal{Y};\dot{H}^{2}(\mathcal{Y};\mathbb{R}^{3\times 3}))$. Then

$\displaystyle\frac{f^{h}}{\varepsilon^{4}}\overset{osc,Z}{\rightharpoonup}\psi$.

Proof.

The proof of (i) can be found in Theorem 1.2 in [2]. The proof of (ii) is an extension of Lemma 3.7 in [10], and that of (iii) is similar. ∎

3 Elasticity framework and intermediate results

Throught this paper we assume that $\varepsilon:(0,1)\to(0,1)$ denotes a function such that the limits

[TABLE]

and

[TABLE]

exist in $[0,\infty]$. We will frequently write $\varepsilon$ instead of $\varepsilon(h)$, but always with the understanding that $\varepsilon$ depends on $h$. There are five possible regimes: $\gamma_{1},\gamma_{2}=+\infty$; $0<\gamma_{1}<+\infty$ and $\gamma_{2}=+\infty$; $\gamma_{1}=0$ and $\gamma_{2}=+\infty$; $\gamma_{1}=0$ and $0<\gamma_{2}<+\infty$; $\gamma_{1}=0$ and $\gamma_{2}=0$. We focus initially on the first three regimes, that is on the cases in which $\gamma_{2}=+\infty$, after the case $\gamma_{1}=0;\gamma_{2}\in(0,+\infty)$ and then the last one.

From now on, we consider $\mathbb{R}^{3\times 3}$: set of all real square matrices of order $3$, the set of all rotations in $\mathbb{R}^{3}$

[TABLE]

and we fix a Borel measurable energy density

[TABLE]

with the following properties:

• $\bullet$

$W(\cdot,y,z,\textbf{F})$ is continuous for almost every $y,z\in\mathbb{R}^{2}$ and $\textbf{F}\in\mathbb{R}^{3\times 3}$.

• $\bullet$

$W(x,\cdot,\cdot,\textbf{F})$ is $\mathcal{Y}\times\mathcal{Y}-$periodic for all $x\in S^{1}$ and almost every $\textbf{F}\in\mathbb{R}^{3\times 3}$.

• $\bullet$

For all $(x,y,z)\in S^{1}\times\mathcal{Y}\times\mathcal{Y}$ we have $W(x,y,z,I)=0$ and $W(x,y,z,\textbf{R}\textbf{F})=W(x,y,z,\textbf{F})$ for all $\textbf{F}\in\mathbb{R}^{3\times 3}$, $\textbf{R}\in SO(3)$.

• $\bullet$

There exist constants $0<\alpha\leq\beta$ and $\rho>0$ such that for all $(x,y,z)\in S^{1}\times\mathcal{Y}\times\mathcal{Y}$ we have

[TABLE]

• $\bullet$

For each $(x,y,z)\in S^{1}\times\mathcal{Y}\times\mathcal{Y}$ there exists a quadratic form $\mathscr{Q}(x,y,z,\cdot):\mathbb{R}^{3\times 3}\to\mathbb{R}$ such that

[TABLE]

Clearly $\mathscr{Q}(\cdot,y,z,\cdot)$ is continuous for almost every $(y,z)\in\mathbb{R}^{2}\times\mathbb{R}^{2}$ and $\mathscr{Q}(x,\cdot,\cdot,\textbf{G})$ is $\mathcal{Y}\times\mathcal{Y}-$periodic for all $x\in S^{1}$ and all $\textbf{G}\in\mathbb{R}^{3\times 3}$.

The elastic energy per unit thickness of a deformation $u^{h}\in H^{1}(S^{h};\mathbb{R}^{3})$ of the shell $S^{h}$ is given by

[TABLE]

In order to express the elastic energy in terms of the new variables, we associate with $y:S^{1}\to\mathbb{R}^{3}$ the energy

[TABLE]

By a change of variables we have

[TABLE]

where again $y^{h}(\Theta^{h})=u^{h}$. Using (2.6) and (2.7) we see that there exists a constant $C$ such that

[TABLE]

We will consider the following result that appears in Proposition 4.2, [11].

Proposition 3.1**.**

Let $(w^{h})$ be a bounded sequence in $H^{2}(S;\mathbb{R}^{3})$ such that $\frac{1}{h}q_{w^{h}}$ is bounded in $L^{2}(S;T^{*}S\otimes T^{*}S)$. Then there exist $w_{0}\in H^{2}(S)$, $w_{1}\in L^{2}(S;\dot{H}^{2}(\mathcal{Y};\mathbb{R}^{3}))$ and $B\in L^{2}(S,\dot{L}^{2}(\mathcal{Y};T^{*}S\otimes T^{*}S))$ such that, after passing to a subsequence, $q_{w^{h}}/h\overset{2}{\rightharpoonup}B$ and $\text{Hess}\,w^{h}\overset{2}{\rightharpoonup}\text{Hess}\,w_{0}+\text{Hess}_{\mathcal{Y}}\,w_{1}$. Set $B_{w}=\int_{Y}B(\cdot,y)\,dy$. Then the following condition are true:

• (i)

If $h\gg\varepsilon^{2}$ then there exists a unique $v\in L^{2}(S;\dot{H}^{1}(\mathcal{Y};\mathbb{R}^{2}))$ such that

[TABLE]

• (ii)

If $h\sim\varepsilon^{2}$ and if we set $\frac{1}{\gamma_{2}}=\lim_{h\to 0}\frac{\varepsilon^{2}}{h}$, then there exists a unique $v\in L^{2}(S;\dot{H}^{1}(\mathcal{Y};\mathbb{R}^{2}))$ such that

[TABLE]

• (iii)

If $h\ll\varepsilon^{2}$, then there exists a unique $v\in L^{2}(S;\dot{H}^{1}(\mathcal{Y};\mathbb{R}^{2}))$ such that

[TABLE]

We adapt Lemma 4.3 in [11] to our case for $\gamma_{1},\gamma_{2}\in[0,+\infty]$ in order to obtain the following result:

Lemma 3.1**.**

Let $(w^{h})\in H^{1}(S^{1};\mathbb{R}^{3})$ be such that

[TABLE]

Then there exist a map $w_{0}\in H^{1}(S;\mathbb{R}^{3})$ and a field $H_{\gamma_{1},\gamma_{2}}\in L^{2}(S\times I\times\mathcal{Y}\times\mathcal{Y};\mathbb{R}^{3\times 3})$ of the form

[TABLE]

for some $w_{1}\in L^{2}(S\times I;W^{1,2}(\mathcal{Y}))$, $w_{2}\in L^{2}(S\times I\times\mathcal{Y};W^{1,2}(\mathcal{Y}))$ and $w_{3}\in L^{2}(S\times\mathcal{Y}\times\mathcal{Y};W^{1,2}(I))$, such that, up to a subsequence, $w^{h}\to w_{0}$ in $L^{2}$ and

[TABLE]

Here, $\tau^{3}=n$, $w_{0}$ is the weak limit in $H^{1}(S)$ of $\int_{I}w^{h}(x+tn(x))dt$ and $\hat{H}_{\gamma_{1},\gamma_{2}}\in L^{2}(S^{1}\times\mathcal{Y}\times\mathcal{Y};\mathbb{R}^{3\times 3})$ is defined by $\hat{H}_{\gamma_{1},\gamma_{2}}(x,y,z)=H_{\gamma_{1},\gamma_{2}}(\pi(x),t(x),y,z)$.

Proof. The hypotheses imply, e.g. by (2.7) , that the $w^{h}$ are uniformly bounded in $H^{1}(S^{1})$, so up to a subsequence $w^{h}\rightharpoonup:w_{0}$ in $H^{1}(S^{1})$. Set $\tilde{w}^{h}=w^{h}\circ\Xi$, so clearly $\tilde{w}^{h}$ is uniformly bounded in $L^{2}(\Omega)$. From the uniform $L^{2}-$bound on $\nabla_{h}w^{h}$ and from (2.5) we deduce that $\tilde{\nabla}_{h}\tilde{w}^{h}$ is uniformly bounded in $L^{2}(\Omega)$. Hence there is $\tilde{w}_{0}\in H^{1}(\Omega;\mathbb{R}^{3})$ with $\partial_{3}\tilde{w}_{0}=0$ such that $\tilde{w}^{h}\rightharpoonup\tilde{w}_{0}$ weakly in $H^{1}(\Omega;\mathbb{R}^{3})$; clearly $\tilde{w}_{0}=w_{0}\circ\Xi$, so (since $\partial_{3}\tilde{w}_{0}$=0) in particular $w_{0}$ is the trivial extension of a map defined on $S$. In the case $\gamma_{1}=0$ and $\gamma_{2}=+\infty$ by uniform boundedness in $L^{2}(\Omega)$, there exist (see Theorem 3.2 in [4]) $\tilde{w}_{1}\in L^{2}(\Omega;W^{1,2}(\mathcal{Y})))$, $\tilde{w}_{2}\in L^{2}(\Omega\times\mathcal{Y};W^{1,2}(\mathcal{Y}))$ and $\overline{w}_{3}\in L^{2}(\omega\times\mathcal{Y}\times\mathcal{Y};W^{1,2}(I))$ with $\partial_{y_{i}}\overline{w_{3}}=\partial_{z_{i}}\overline{w_{3}}=0$ for $i=1,2$ such that, up to the extraction of a (not relabeled) subsequence,

[TABLE]

By (2.5) the left-hand side equals $(\nabla_{h}w)(\Xi)\nabla\Xi(\tilde{\Theta}^{-1}_{h})$. As $\nabla\Xi(\tilde{\Theta}^{-1}_{h})$ converges uniformly on $S^{1}$ to $(\partial_{1}\xi,\partial_{2}\xi,n(\xi))$ (extended trivially in the $x_{3}-$direction), we conclude:

[TABLE]

On the right-hand side we use

[TABLE]

and $(\partial_{\alpha}\tilde{w}_{0})\circ\Xi^{-1}=dw_{0}(\tau_{\alpha})$ to obtain the claim when $\gamma_{1}=\gamma_{2}=+\infty$, after defining $(w_{1})_{i}=(\tilde{w}_{1}\circ r)\cdot\tau_{i}$, $(w_{2})_{i}=(\tilde{w}_{2}\circ r)\cdot\tau_{i}$ and $(w_{3})_{i}=(\overline{w}_{3}\circ r)\cdot\tau_{i}$ for $i=1,2,3$. The other four cases are proven similarly using Theorem 3.2 in [4] in where we are able to cover the two last cases($\gamma_{1}=\gamma_{2}=0$; $\gamma_{1}=0\,\text{and}\,\,0<\gamma_{2}<+\infty$) which are valid for plates and are extended easily to shells too. $\square$

3.1 Asymptotic energy functionals

Next we will introduce the asymptotic energy functionals. In order to do so, we need the definition of the relaxation fields and the cell formulae. Recall that $a\odot b=\frac{1}{2}(a\otimes b+b\otimes a)$. We make the following definitions:

[TABLE]

and for $(\zeta,\eta,\varphi,\mu)\in L^{2}(S,D(\mathcal{U}_{0,+\infty}))$ define

[TABLE]

[TABLE]

and for $(\zeta,\eta,\rho,c)\in L^{2}(S,D(\mathcal{U}_{+\infty,+\infty})$ define

[TABLE]

For $\gamma_{1}\in(0,\infty)$

[TABLE]

and for $(\zeta,\eta,\rho)\in L^{2}(S,D(\mathcal{U}_{\gamma_{1},+\infty}))$ define

[TABLE]

By embedding $D(\mathcal{U}_{0,+\infty})$ trivially into $L^{2}(S,D(\mathcal{U}_{0,\infty}))$, we can regard $\mathcal{U}_{0,+\infty}$ as a map from $D(\mathcal{U}_{0,+\infty})$ into $L^{2}(S,L^{2}(I\times\mathcal{Y}\times\mathcal{Y},\mathbb{R}^{3\times 3}_{\text{sym}}))$.

For $\gamma_{2}\in(0,+\infty)$ set

[TABLE]

and for $(\zeta,\eta,\varphi,\mu)\in L^{2}(S,D(\mathcal{U}_{0,\gamma_{2}}))$ define

[TABLE]

For each $x\in S$ the fiberwise action $\mathcal{U}^{(x)}_{0,+\infty}$ of $\mathcal{U}_{0,+\infty}$ is

[TABLE]

for all $(\zeta,\eta,\varphi,\mu)\in D(\mathcal{U}_{0,+\infty})$.

For each $x\in S$ we define $L^{(x)}_{0,+\infty}(I\times\mathcal{Y}\times\mathcal{Y})=\mathcal{U}^{(x)}_{0,+\infty}(D(\mathcal{U}_{0,+\infty}))$, i.e.,

[TABLE]

This is a subspace of $L^{2}(I\times\mathcal{Y}\times\mathcal{Y},\mathbb{R}^{3\times 3}_{\text{sym}})$. We denote by $L_{0,+\infty}(I\times\mathcal{Y}\times\mathcal{Y})$ the vector bundle over $S$ with fibers $L^{(x)}_{0,+\infty}(I\times\mathcal{Y}\times\mathcal{Y})$; in what follows we will frequently omit the index $(x)$ for the fibers. The bundle $L_{\gamma_{1},+\infty}(I\times\mathcal{Y}\times\mathcal{Y})$, for $\gamma_{1}\in(0,\infty]$ and $L_{0,\gamma_{2}}(I\times\mathcal{Y}\times\mathcal{Y})$, for $\gamma_{2}\in(0,\infty)$ are defined analogously. The elements of these spaces are the relaxation fields.

For $\gamma_{1}\in[0,+\infty]$, $\gamma_{2}=+\infty$ and $x\in S$, we define $\mathscr{Q}_{\gamma_{1},+\infty}(x,\cdot);T^{*}_{x}S\otimes T^{*}_{x}S\to\mathbb{R}$ by setting

[TABLE]

Here the infimum is taken over all $U\in L^{(x)}_{\gamma_{1},+\infty}(I\times\mathcal{Y}\times\mathcal{Y})$ and all $p\in T^{*}_{x}S\otimes T^{*}_{x}S$.

For $\gamma_{1}=0$, $\gamma_{2}\in(0,+\infty)$ and $x\in S$, we define $\mathscr{Q}_{0,\gamma_{2}}(x,\cdot);T^{*}_{x}S\otimes T^{*}_{x}S\to\mathbb{R}$ by setting

[TABLE]

Here the infimum is taken over all $U\in L^{(x)}_{0,\gamma_{2}}(I\times\mathcal{Y}\times\mathcal{Y})$ and all $p\in T^{*}_{x}S\otimes T^{*}_{x}S$.

Note that $\mathscr{Q}_{\gamma_{1},+\infty}(x,q)=\mathscr{Q}_{\gamma_{1},+\infty}(x,\text{sym}\,q)$ for all $x\in S$ and all $q\in T^{*}_{x}S\otimes T^{*}_{x}S$. For $x\in S$ and $q\in T^{*}_{x}S\odot T^{*}_{x}S$ define the homogeneous relaxation (cf. [15]):

[TABLE]

Then it is easy to see that

[TABLE]

where the infimum is taken over all $\zeta\in W^{1,2}(\mathcal{Y},\mathbb{R}^{2})$, all $\varphi\in W^{2,2}(\mathcal{Y})$, all $\eta\in L^{2}(I\times\mathcal{Y};W^{1,2}(\mathcal{Y};\mathbb{R}^{3}))$ and all $p\in T^{*}_{x}S\odot T^{*}_{x}S$. In the case when the material is homogeneous in the thickness direction, we have

[TABLE]

The analogous formula holds for $\mathscr{Q}_{0,\gamma_{2}}$.

As in [12], for all $x\in S$ and all $q\in T^{*}_{x}S\odot T^{*}_{x}S$ we have

[TABLE]

It is not difficult to show that for all $\gamma_{1}\in[0,+\infty]$, $\gamma_{2}=+\infty$ and $x\in S$ the map $\mathscr{Q}_{\gamma_{1},+\infty}(x,\cdot)$ is quadratic and that there exist $c_{1},c_{2}>0$ such that for all $x\in S$ we have

[TABLE]

For $\gamma_{1}\in[0,+\infty]$ and $\gamma_{2}=+\infty$ we define $I_{\gamma_{1},+\infty}:W^{1,2}(S;\mathbb{R}^{3})\to\mathbb{R}$ by setting

[TABLE]

For $\gamma_{2}\in(0,+\infty)$ the functional $I_{0,\gamma_{2}}:W^{1,2}(S;\mathbb{R}^{3})\to\mathbb{R}$ is defined analogously, by replacing $\mathscr{Q}_{\gamma_{1},+\infty}$ by $\mathscr{Q}_{0,\gamma_{2}}$.

4 Main result

For a given sequence $(u^{h})\subset W^{1,2}(S^{h};\mathbb{R}^{3})$ we continue to define the sequence $(y^{h})\subset W^{1,2}(S^{1},\mathbb{R}^{3})$ of rescaled deformations by $y^{h}(\Theta^{h})=u^{h}$. We recall the compactness result for sequences with finite bending energy, cf. Theorem 1 in [9] for a proof.

Let $(u^{h})\subset W^{1,2}(S^{h};\mathbb{R}^{3})$ satisfies

[TABLE]

Proposition 4.1**.**

Let $(u^{h})\subset W^{1,2}(S^{h},\mathbb{R}^{3})$ be such that (4.1) holds. Then there exist $u\in W^{2,2}_{\text{iso}}(S)$ such that (after passing to subsequences and extending $u$ and $n$ trivially to $S^{1}$), as $h\to 0$ we have

[TABLE]

Here $Q\in W^{1,2}(S,\text{SO}(3))$ is determined by the condition $Q\tau=\nabla_{\tau}u$ for all smooth tangent vector fields $\tau$ along $S$.

We denote by $\widetilde{W}^{2,2}_{\text{iso}}(S)$ the set of those maps $u\in W^{2,2}_{\text{iso}}(S)$ for which there exists $(u^{h})\subset W^{2,\infty}_{\text{iso}}(S)$ converging strongly to $u$ in $W^{2,2}(S)$. The reason to introduce this space is that we are able to construct the recovery sequence only for limiting deformations $u$ belonging to this space. Theorem 2.1 in [12] plays an essential role in this construction. The following $\Gamma-$convergence result is our main result:

Theorem 4.1**.**

Let $\gamma_{1}\in[0,+\infty]$ and $\gamma_{2}=+\infty$. Then the following are true:

• (i)

Let $(u^{h})\subset W^{1,2}(S^{h},\mathbb{R}^{3})$ be such that (4.1) holds and such that $y^{h}-\frac{1}{|S^{1}|}\int_{S^{1}}y^{h}\to u$ strongly in $L^{2}(S^{1})$ for some $u\in L^{2}(S^{1},\mathbb{R}^{3})$. Then

[TABLE]

• (ii)

If, in addition, $S$ is simply connected, then for every $u\in\widetilde{W}^{2,2}_{\text{iso}}(S)$ there exists $(u^{h})\subset W^{1,2}(S^{h};\mathbb{R}^{3})$ satisfying (4.1), and such that $y^{h}\to u$ strongly in $W^{1,2}(S^{1})$. Moreover,

[TABLE]

Furthermore, for $\gamma_{1}=0$ and $\gamma_{2}\in(0,+\infty)$ the items $(i)$ and $(ii)$ hold replacing $I_{\gamma_{1},+\infty}$ by $I_{0,\gamma_{2}}$.

5 Proof of main result

5.1 Proof of lower bound

We consider a sequence $(u^{h})\subset W^{1,2}(S^{h},\mathbb{R}^{3})$ satisfying (4.1) and we set $y^{h}(\Theta^{h})=u^{h}$. The following lemma is analogous to Lemma 3.3 in [12] and is essentially contained in [7]. It is a consequence of Theorem 3.1 in [8] and of the arguments in [4] and [9].

Lemma 5.1**.**

Define

[TABLE]

Then there exist constants $C,c>0$ such that the following is true: if $h\leq c$ and $u\in W^{1,2}(S^{h};\mathbb{R}^{3})$, then there exists a map $\tilde{R}:\omega\to SO(3)$ which is constant on each cube $x+\delta Y$ with $x\in\delta\mathbb{Z}$ and there exist $\tilde{R}_{s}\in W^{1,2}(\omega;\mathbb{R}^{3})$ such that for each $a\in\mathbb{R}^{2}$ with $|a_{1}|\leq\delta$ and $|a_{2}|\leq\delta$ and for each $\tilde{\omega}\subset\omega$ with $\text{dist}\,(\tilde{\omega,\partial\omega})>c\delta$ we have:

[TABLE]

Proposition 5.1**.**

Let $\gamma_{1}\in(0,+\infty)$ and $\gamma_{2}=+\infty$. Let $(u^{h})\subset W^{1,2}(S^{h};\mathbb{R}^{3})$ satisfying (4.1) and let $u\in W^{2,2}_{\text{iso}}(S)$ such that (4.2) and (4.3) hold. Let $\tilde{\omega}\subset\mathbb{R}^{2}$ be a domain with $C^{1,1}$ boundary whose closure is contained in $\omega$ and set $\widetilde{S}=\xi(\tilde{\omega})$. Denote by $\tilde{R}^{h}:\omega\to SO(3)$ the piecewise constant map obtained by applying Lemma 5.1 to $u^{h}$ and define $R^{h}:S^{1}\to SO(3)$ by $R^{h}=\tilde{R}^{h}\circ r$. Define $G^{h}\in L^{2}(S^{1};\mathbb{R}^{3\times 3})$ by

[TABLE]

where $y^{h}(\Theta^{h})=u^{h}$. Then there exist $B\in L^{2}(\tilde{S},T^{*}\tilde{S}\odot T^{*}\tilde{S})$ and $(\zeta,\eta,\rho)\in L^{2}(\tilde{S},D(\mathcal{U}_{\gamma_{1},+\infty}))$ such that (up to passing to subsequences)

[TABLE]

Remark 5.1**.**

A similar result is true if $\gamma_{1}=+\infty$ or $\gamma_{1}=0$. In the former case $\mathcal{U}_{\gamma_{1},+\infty}(\zeta,\eta,\rho)$ in (5.2) must be replaced by $\mathcal{U}_{+\infty,+\infty}$, where $(\zeta,\eta,\rho,c)\in L^{2}(\widetilde{S},D(\mathcal{U}_{+\infty,+\infty}))$. In the latter case, it must be replaced by $\mathcal{U}_{0,+\infty}(\zeta,\eta,\varphi,\mu)$, where $(\zeta,\eta,\varphi,\mu)\in L^{2}(\widetilde{S},D(\mathcal{U}_{0,+\infty}))$. Likewise (5.2) holds for $\gamma_{1}=0$ and $\gamma_{2}\in(0,+\infty)$.

Proof.

Define $\overline{u}^{h}:S\to\mathbb{R}^{3}$ by setting

[TABLE]

Let $\tilde{R}^{h}_{s}:\tilde{\omega}\to\mathbb{R}^{3\times 3}$ be the maps obtained by applying Lemma 5.1 to $u^{h}$ and set $R^{h}_{s}=\tilde{R}^{h}_{s}\circ r$. On $\tilde{S}^{h}$ define $z^{h}$ via

[TABLE]

Clearly

[TABLE]

Let $\tau$ be a smooth tangent vector field along $S$. Then we have

[TABLE]

Observe that (2.2) implies that $\nabla_{\tau}\pi$ equals $\tau-t\textbf{S}\tau$ up to a term of higher order. Using this and rewriting the problem in coordinates, one can now argue as in Theorem 4.1 in [4] to deduce the claim for $\gamma_{1}\in[0,+\infty]$. The fields $\mathcal{U}_{\gamma_{1},+\infty}$ arise, essentially, due the Lemma 3.1. ∎

The remaining proof of the lower bound follows standard arguments: truncation, Taylor expansion and lower semicontinuity of integral functional with respect the three-scale convergence. Thus one obtains a lower bound on every $C^{1,1}$ bounded compactly contained subdomain $\tilde{\omega}$ of $\omega$. Exhausting $\omega$ with a sequence of such subdomains, Theorem 4.1 (i) follows. Detail for this argument can be found in [4] and [11].

Lemma 5.2**.**

Let $(y^{h})\subset H^{1}(S^{1};\mathbb{R}^{3})$, define $G^{h}:S^{1}\to\mathbb{R}^{3\times 3}$ by (5.1) and $G$ be such that $G^{h}\overset{3}{\rightharpoonup}G$. Then we have

[TABLE]

We refer to [11] and [17] for a proof of Lemma 5.2 in the case in which $\mathscr{Q}$ is independent of $z$. The proof in our setting is a straightforward adaptation.

5.2 Proof of upper bound

Let us introducing the recovery sequence. Recall Lemma 3.5 in [12].

Lemma 5.3**.**

Let $u\in W^{2,\infty}_{\text{iso}}(S)$ and define $\nu:S\to\mathbb{S}^{2}$ by

[TABLE]

Let $w\in W^{2,\infty}(S,\mathbb{R}^{3})$ and define $\mu\in W^{1,\infty}(S,\mathbb{R}^{3})$ by

[TABLE]

and define the deformations $v^{h}:S^{h}\to\mathbb{R}^{3}$ by

[TABLE]

Define $R\in W^{1,\infty}(S,SO(3))$ by $R=\nabla uT_{S}+\nu\otimes n$. Then there exists $Y^{h}\in L^{\infty}(S^{h},\mathbb{R}^{3\times 3})$ with $||Y^{h}||_{L^{\infty}(S^{h})}\leq Ch^{2}$ such that

[TABLE]

Remark 5.2**.**

The choice of our recovery sequences depends on the following two factors:

• (i)

It considers the inhomogeneity of material.

• (ii)

The energy density contains spatial variable which makes it necessary to choose a nonzero displacement $w$ in Lemma 5.3 and whose existence is guaranteed by Proposition $2.15$ in **[12]**.

Moreover, multilayered materials can be deduces as particular cases of Theorem 4.1(cf. [4] for corresponding problem for plates).

Proof of Theorem 4.1(ii).

By approximation, it is enough to prove the claim for $u\in W^{2,\infty}_{\text{iso}}(S)$ and, thanks also to Proposition 2.15 in [12], for all $B$ of the form $B=du\odot dw$ with $w\in W^{2,\infty}(S,\mathbb{R}^{3})$.

We will use the same notation as in the statement of Lemma 5.3; in particular the definition of $v^{h}$ in terms of $w$ and $u$. Moreover, we set $\sigma^{\alpha}=\nabla_{\tau^{\alpha}}u$.

Case $\gamma_{1}\in(0,+\infty)$ and $\gamma_{2}=+\infty$. Let $\zeta\in C^{1}_{0}(S,\dot{C}^{1}(I\times\mathcal{Y},\mathbb{R}^{2}))$, $\rho\in C^{1}_{0}(S,\dot{C}^{1}(I;\dot{C}^{1}(\mathcal{Y})))$ and $\eta\in C^{1}_{0}(S\times\mathcal{Y},\dot{C}^{1}(I\times\mathcal{Y}))$ and define the rescaled deformations $y^{h}:S^{1}\to\mathbb{R}^{3}$ by the following equation on $S^{h}$:

[TABLE]

Lemma 5.3 implies that on $S^{1}$

[TABLE]

where $\lim_{h\to 0}||\frac{o(h)}{h}||=0$.

By frame invariance of $W$ and using (3.1), we deduce from (5.3) that

[TABLE]

pointwise on $S^{1}$. From this we readily deduce

[TABLE]

**Case ** $\gamma_{1}=\gamma_{2}=+\infty$. This is similar to the previous case. So we only state the formula for the recovery sequence. For $\zeta\in C^{1}_{0}(S,\dot{C}^{1}(I\times\mathcal{Y},\mathbb{R}^{2}))$, $\rho\in C^{1}_{0}(S,\dot{C}^{1}(I;\dot{C}^{1}(\mathcal{Y})))$, $\eta\in C^{1}_{0}(S\times\mathcal{Y},\dot{C}^{1}(I\times\mathcal{Y}))$ and $c\in C^{1}_{0}(S,C^{1}_{0}(I,\mathbb{R}^{3}))$, we define $y^{h}:S^{1}\to\mathbb{R}^{3}$ by the following equation on $S^{h}$:

[TABLE]

**Case ** $\gamma_{1}=0$ and $\gamma_{2}=+\infty$. For $\zeta\in C^{1}_{0}(S,\dot{C}^{1}(\mathcal{Y},\mathbb{R}^{2}))$, $\varphi\in C^{2}_{0}(S,\dot{C}^{2}(\mathcal{Y}))$, $\eta\in C^{1}_{0}(S\times\mathcal{Y},\dot{C}^{1}(I\times\mathcal{Y}))$ and $\mu\in C^{1}_{0}(S,C^{1}_{0}(I\times\mathcal{Y},\mathbb{R}^{3}))$, we define $y^{h}:S^{1}\to\mathbb{R}^{3}$ by the following equation on $S^{h}$:

[TABLE]

Case $\gamma_{1}=0$ and $\gamma_{2}\in(0,+\infty)$. For $\zeta\in C^{1}_{0}(S,\dot{C}^{1}(\mathcal{Y},\mathbb{R}^{2}))$, $\varphi\in C^{2}_{0}(S,\dot{C}^{2}(\mathcal{Y}))$, $\eta\in C^{1}_{0}(S\times\mathcal{Y},\dot{C}^{1}(I\times\mathcal{Y}))$ and $\mu\in C^{1}_{0}(S,C^{1}_{0}(I\times\mathcal{Y},\mathbb{R}^{3}))$, we define $y^{h}:S^{1}\to\mathbb{R}^{3}$ by the following equation on $S^{h}$:

[TABLE]

We leave the details to the reader. $\square$

6 Convex shells

In this section, we shall identify the $\Gamma-$limit for convex shells in the remaining case $\gamma_{1}=\gamma_{2}=0$, i.e. $h\ll\varepsilon^{2}$. We wish to ilustrate the stronger influence of the geometry in this case. For obtaining the limit model we shall closely follow the arguments used in [11] as follows: We work under the assumption that $S$ is uniformly convex, $i.e.$, there exists $C>0$ such that

[TABLE]

For $x\in S$ we define a relaxation operator with the values in $L^{2}(I\times\mathcal{Y}\times\mathcal{Y};\mathbb{R}^{3\times 3}_{\text{sym}})$ as follows: Set $D(\mathcal{U}_{0,0})=\dot{L}^{2}(\mathcal{Y};\mathbb{R}^{2\times 2}_{\text{sym}})\times L^{2}(I\times\mathcal{Y};\dot{W}^{1,2}(\mathcal{Y};\mathbb{R}^{3}))\times L^{2}(I\times\mathcal{Y};\mathbb{R}^{3})$ and for all $(\dot{B},\eta,\mu)\in L^{2}(S,D(\mathcal{U}_{0,0}))$ define

[TABLE]

As usual, we introduce the vector bundle $L_{0,0}(I\times\mathcal{Y}\times\mathcal{Y})$ of relaxation fields to be the range of $\mathcal{U}_{0,0}$ similarly to the bundles $L_{0,+\infty}(I\times\mathcal{Y}\times\mathcal{Y})$ introduced earlier. As in the previous cases, each fiber of $L_{0,0}(I\times\mathcal{Y}\times\mathcal{Y})$ is a closed subspace of $L^{2}(I\times\mathcal{Y}\times\mathcal{Y};\mathbb{R}^{3\times 3}_{\text{sym}})$. We also define the functional $I_{0,0}:W^{1,2}(S;\mathbb{R}^{3})\to\mathbb{R}$ by setting

[TABLE]

with the quadratic form $\mathscr{Q}_{0,0}(x,\cdot):T^{*}S\odot T^{*}S\to\mathbb{R}$ given by

[TABLE]

Here the infimum is taken over all $U\in L^{(x)}_{0,0}(I\times\mathcal{Y}\times\mathcal{Y})$ and all $p\in T^{*}_{x}S\otimes T^{*}_{x}S$.

We introduce the space

[TABLE]

By Fourier transform it can be easily seen that $FL(S;\dot{C}^{\infty}(\mathcal{Y}))$ is dense in $L^{2}(S;\dot{H}^{m}(\mathcal{Y}))$, for any $m\in\mathbb{N}_{0}$.

Lemma 6.1**.**

(see Lemma 6.1 in [11]) Assume (6.1) is satisfied and let $\dot{B}\in L^{2}(S;\dot{L}^{2}(\mathcal{Y};T^{*}S\otimes T^{*}S))$. Then there exists unique $w\in L^{2}(S;\dot{H}^{1}(\mathcal{Y};\mathbb{R}^{2}))$ and $\varphi\in L^{2}(S;\dot{L}^{2}(\mathcal{Y}))$ such that

[TABLE]

Moreover, if $\dot{B}_{ij}\in FL(S;\dot{C}^{\infty}(\mathcal{Y}))$ for every $i,j=1,2$ then $w_{i}\in FL(S;\dot{H}^{1}(\mathcal{Y}))$, for $i=1,2$ and $\varphi\in FL(S;\dot{H}^{1}(\mathcal{Y}))$.

Theorem 6.1**.**

Under the hypotheses and with the notation of Theorem 4.1 and assuming, in addition, that $S$ is uniformly convex and that $h\ll\varepsilon^{2}$, moreover, the following are true:

• $\bullet$

We have

[TABLE]

• $\bullet$

If, in addition, $S$ is simply connected, then for every $u\in\widetilde{W}^{2,2}_{\text{iso}}(S)$ there exists $(u^{h})\subset W^{1,2}(S^{h};\mathbb{R}^{3})$ satisfying (4.1) and such that $y^{h}\to u$, strongly in $W^{1,2}(S^{1})$. Moreover,

[TABLE]

Proof.

We only sketch the proof. As in Proposition 5.1 there exist $B\in L^{2}(S,T^{*}S\odot T^{*}S)$ and $(\zeta,\eta,\varphi,\mu)\in L^{2}(S,D(\mathcal{U}_{0,+\infty}))$ such that (5.2) is satisfied. Using Proposition 3.1 (iii) as well Lemma 6.1, we conclude that $\varphi=0$. Thus by Proposition 5.1 there exists $(\dot{B},\eta,\mu)\in L^{2}(S,D(\mathcal{U}_{0,0}))$ , where $\dot{B}=\text{Def}_{\mathcal{Y}}\zeta$ and $B\in L^{2}(S,T^{*}S\odot T^{*}S)$ such that the maps $G^{h}$ defined as in (5.1) converge weakly three-scale to

[TABLE]

Hence the lower bound part follows readily from the Lemma 5.2 and definition of the functional $I_{0,0}$.

To prove the upper bound part we consider $\dot{B}$ with $(\dot{B})_{ij}\in FL(S;\dot{C}^{\infty}(\mathcal{Y}))$ for $i,j=1,2$. From Lemma 6.1 there exists $z\in(FL(S;\dot{C}(\mathcal{Y})))^{2}$ and $\varphi\in FL(S;\dot{C}^{\infty}(\mathcal{Y}))$ solving the system $\text{Def}_{\mathcal{Y}}z+\varphi\textbf{S}=\dot{B}$. We choose $\zeta\in C^{1}_{0}(S,\dot{C}^{1}(\mathcal{Y},\mathbb{R}^{2}))$, $\eta\in C^{1}_{0}(S\times\mathcal{Y},\dot{C}^{1}(I\times\mathcal{Y}))$ and $\mu\in C^{1}_{0}(S,C^{1}_{0}(I\times\mathcal{Y},\mathbb{R}^{3}))$ and we define $y^{h}:S^{1}\to\mathbb{R}^{3}$ by the following equation on $S^{h}$:

[TABLE]

∎

Acknowledgment

This work was inspired by the great contributions of Prof. Igor Velčić to the theory of thin structures. P. Hernández-Llanos was supported by Agencia Nacional de Investigación y Desarrollo, FONDECYT Postdoctorado 2023 grant no. 3230202. The warm hospitality at the Instituto de Ciencias de la Ingeniería of the Universidad de O’Higgins is gratefully acknowledged by P. Hernández-Llanos.