Derivation of von K\'arm\'an plate theory in the framework of three-dimensional viscoelasticity

TL;DR

This paper derives a viscoelastic von Kármán plate model from three-dimensional nonlinear viscoelasticity using a rigorous dimension reduction approach, combining nonlinear plate theory and gradient flow methods.

Contribution

It introduces a novel derivation of viscoelastic von Kármán plates from 3D models within a finite-strain, Kelvin-Voigt framework, integrating gradient flow theory.

Findings

Derived a 2D viscoelastic plate model from 3D viscoelasticity.

Established weak solutions for the viscoelastic von Kármán plate equations.

Unified nonlinear plate theory with gradient flow approach.

Abstract

We apply a quasistatic nonlinear model for nonsimple viscoelastic materials at a finite-strain setting in the Kelvin's-Voigt's rheology to derive a viscoelastic plate model of von K\'arm\'an type. We start from time-discrete solutions to a model of three-dimensional viscoelasticity where the viscosity stress tensor complies with the principle of time-continuous frame-indifference. Combining the derivation of nonlinear plate theory by Friesecke, James and M\"{u}ller, and the abstract theory of gradient flows in metric spaces by Sandier and Serfaty we perform a dimension-reduction from 3D to 2D and identify weak solutions of viscoelastic form of von K\'arm\'an plates.

Equations

$$\displaystyle-{\rm div}\Big{(}\partial_{F}W(\nabla w)+\partial_{\dot{F}}R(\nabla w,\partial_{t}\nabla w)\Big{)}=f\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}e_{3}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\ \ \ \text{ in $[0,T]\times\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\Omega_{h}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}$.}$$

$$\displaystyle-{\rm div}\Big{(}\partial_{F}W(\nabla w)+\varepsilon\mathcal{L}_{P}(\nabla^{2}w)+\partial_{\dot{F}}R(\nabla w,\partial_{t}\nabla w)\Big{)}=f\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}e_{3}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\ \ \ \text{ in $[0,T]\times\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\Omega_{h}$,}$$

$$\displaystyle{\rm div}\big{(}\mathbb{C}_{W}\big{(}e(u)+\tfrac{1}{2}\nabla v\otimes\nabla v\big{)}\big{)}$$

$$\displaystyle-{\rm div}\Big{(}\mathbb{C}_{W}\Big{(}e(u)+\frac{1}{2}\nabla v\otimes\nabla v\Big{)}\nabla v\Big{)}+\frac{1}{12}{\rm div}\,{\rm div}\big{(}\mathbb{C}_{W}\nabla^{2}v\big{)}$$

$$\displaystyle\begin{cases}0={\rm div}\Big{(}\mathbb{C}_{W}\big{(}e(u)+\frac{1}{2}\nabla v\otimes\nabla v\big{)}+\mathbb{C}_{R}\big{(}e(\partial_{t}u)+\nabla\partial_{t}v\otimes\nabla v\big{)}\Big{)},&\vspace{0.1cm}\textstyle\\ f=-{\rm div}\Big{(}\Big{(}\mathbb{C}_{W}\big{(}e(u)+\frac{1}{2}\nabla v\otimes\nabla v\big{)}+\mathbb{C}_{R}\big{(}e(\partial_{t}u)+\nabla\partial_{t}v\otimes\nabla v\big{)}\Big{)}\nabla v\Big{)}&\vspace{0.1cm}\textstyle\\ \quad\quad+\tfrac{1}{12}{\rm div}\,{\rm div}\Big{(}\mathbb{C}_{W}\nabla^{2}v+\mathbb{C}_{R}\nabla^{2}\partial_{t}v\Big{)}&\ \ \ \ \text{in }[0,\infty)\times S,\end{cases}$$

$$|\partial \varphi {|}_{\mathcal{D}}(y):=\underset{z\to y}{\mathrm{limsup}}\frac{(\varphi (y)-\varphi (z){)}^{+}}{\mathcal{D}(y,z)}.$$

$$\displaystyle\nabla_{h}y:=\Big{(}\nabla^{\prime}y,\frac{1}{h}y_{,3}\Big{)}=\nabla w.$$

$$({\nabla }_h^{2}y{)}_{ijk}:=h^{-{\delta }_{3j}-{\delta }_{3k}}({\nabla }^{2}y{)}_{ijk}=({\nabla }^{2}w{)}_{ijk}={\partial }_{jk}^2{w}_i\text{for i,j,k∈{1,2,3}},$$

$$\displaystyle\begin{split}(i)&\ \ W\text{ continuous and \color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}$C^{3}$ \color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}in a neighborhood of $SO(3)$},\\ (ii)&\ \ \text{frame indifference: }W(QF)=W(F)\text{ for all }F\in\mathbb{R}^{3\times 3},Q\in SO(3),\\ (iii)&\ \ W(F)\geq c\operatorname{dist}^{2}(F,SO(3)),\ W(F)=0\text{ iff }F\in SO(3),\end{split}$$

$$\begin{array}{rl}(i)& \text{frame indifference: }P(QZ)=P(Z)\text{ for all }Z\in {\mathbb{R}}^{3\times 3\times 3},Q\in SO(3),\\ (ii)& \text{P is convex and C1},\\ (iii)& \text{growth condition: For all Z∈R3×3×3 we have }\\ & c_1|Z{|}^p\le P(Z)\le c_2|Z{|}^p,|{\partial }_{Z}P(Z)|\le c_2|Z{|}^{p-1}\end{array}$$

$$R(F,\dot{F})=R(QF,Q(\dot{F}+AF))\forall Q\in SO(3),A\in {\mathbb{R}}_{\mathrm{skew}}^{3\times 3}$$

$$R(F,\dot{F}):=\underset{\epsilon \to 0}{\lim }\frac{1}{2{\epsilon }^2}{D}^2(F+\epsilon \dot{F},F)=\frac{1}{4}{\partial }_{F_1^2}^2{D}^2(F,F)[\dot{F},\dot{F}]$$

$$(i)$$

$$(ii)$$

$$(iii)$$

$$(iv)$$

$$(v)$$

$$\forall Q_1,Q_2\in SO(3),\forall F_1,F_2\in G{L}_{+}(3),$$

$$(vi)$$

$$\displaystyle\mathscr{S}_{h}=\Big{\{}y\in W^{2,p}(\Omega;\mathbb{R}^{3}):\ y(x^{\prime},x_{3})=\begin{pmatrix}x^{\prime}\\ hx_{3}\end{pmatrix}+\begin{pmatrix}h^{2}\hat{u}(x^{\prime})\\ h\hat{v}(x^{\prime})\end{pmatrix}-$$

$$\displaystyle\text{for }x^{\prime}\in\partial S,\ x_{3}\in I\Big{\}},$$

$$\displaystyle\big{(}\mathcal{L}^{h}_{P}(\nabla^{2}_{h}y)\big{)}_{ij}=-{\rm div}_{h}(\partial_{Z}P(\nabla^{2}_{h}y))_{ij*},\ \ \ \ i,j\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\in\{1,\ldots,3\}$$

$$\displaystyle\begin{cases}-{\rm div}_{h}\Big{(}\partial_{F}W(\nabla_{h}y)+\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}h^{\beta_{1}}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\mathcal{L}^{h}_{P}(\nabla^{2}_{h}y)+\partial_{\dot{F}}R(\nabla_{h}y,\partial_{t}\nabla_{h}y)\Big{)}=\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}h^{\beta_{2}}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}fe_{3}&\text{in }[0,\infty)\times\Omega\\ y(0,\cdot)=y^{h}_{0}&\text{in }\Omega\\ y(t,\cdot)\in\mathscr{S}_{h}&\text{for }t\in[0,\infty)\end{cases}$$

$$\displaystyle\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}I^{\beta_{1},\beta_{2}}_{h}(y)\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}=\int_{\Omega}W(\nabla_{h}y(x))\,dx+h^{\beta_{1}}\int_{\Omega}P(\nabla^{2}_{h}y(x))\,dx-h^{\beta_{2}}\int_{\Omega}f(x)y_{3}(x)\,dx$$

$$\displaystyle Y^{n}_{h,\tau}={\rm argmin}_{y\in\mathscr{S}_{h}}\Phi_{h}(\tau,Y^{n-1}_{h,\tau};y),\ \ \ \Phi_{h}(\tau,y_{0};y_{1}):=\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}I^{\beta_{1},\beta_{2}}_{h}(y_{1})\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}+\frac{1}{2\tau}\mathcal{D}^{2}(y_{0},y_{1}),$$

$$\displaystyle\mathcal{D}(y_{0},y_{1}):=\Big{(}\int_{\Omega}D^{2}(\nabla_{h}y_{0},\nabla_{h}y_{1})\Big{)}^{1/2}.$$

$$\displaystyle\tilde{Y}_{h,\tau}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}(0,\cdot)\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}=Y^{0}_{h,\tau},\ \ \ \tilde{Y}_{h,\tau}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}(t,\cdot)\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}=Y^{n}_{h,\tau}\ \text{for}\ t\in((n-1)\tau,n\tau],\ n\geq 1.\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}$$

$${\varphi }_h(y)=h^{-4}{I}_h^{4-p\alpha ,3}(y)=\frac{1}{h^4}{\int }_{\Omega }W({\nabla }_{h}y(x))dx+\frac{1}{h^{\alpha p}}{\int }_{\Omega }P({\nabla }_h^{2}y(x))dx-\frac{1}{h}{\int }_{\Omega }f(x)y_3(x)dx$$

$$\displaystyle\mathcal{D}_{h}(y_{0},y_{1})=h^{-2}\mathcal{D}(y_{0},y_{1})=h^{-2}\Big{(}\int_{\Omega}D^{2}(\nabla_{h}y_{0},\nabla_{h}y_{1})\Big{)}^{1/2}.$$

$$\displaystyle u(x^{\prime}):=\frac{1}{h^{2}}\int_{I}\Big{(}\begin{pmatrix}y^{h}_{1}\\ y^{h}_{2}\end{pmatrix}(x^{\prime},x_{3})-\begin{pmatrix}x_{1}\\ x_{2}\end{pmatrix}\Big{)}\,dx_{3},\ \ \ \ \ v(x^{\prime}):=\frac{1}{h}\int_{I}y^{h}_{3}(x^{\prime},x_{3})\,dx_{3},$$

$$Q_W^2(G)=\underset{a\in {\mathbb{R}}^3}{\min }{Q}_W^3(G^{\ast }+a\otimes e_3+e_3\otimes a)$$

$$Q_D^3(F)=\frac{1}{2}{\partial }_{F_1^2}^2{D}^2(\mathbf{Id},\mathbf{Id})[F,F],Q_D^2(G)=\underset{a\in {\mathbb{R}}^3}{\min }{Q}_D^3(G^{\ast }+a\otimes e_3+e_3\otimes a).$$

$$Q_W^3(F)={\mathbb{C}}_W^3[F,F]\forall F\in {\mathbb{R}}^{3\times 3},Q_W^2(G)={\mathbb{C}}_W^2[G,G]\forall G\in {\mathbb{R}}^{2\times 2}$$

$$\displaystyle{\mathscr{S}}_{0}=\{(u,v)\in W^{1,2}(S;\mathbb{R}^{2})\times W^{2,2}(S):\ u=\hat{u},\ v=\hat{v},\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\ \nabla^{\prime}v=\nabla^{\prime}\hat{v}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\ \text{ on }\partial S\}.$$

Full text

1. Introduction

Dimension-reduction problems play a significant role in nonlinear analysis and numerics because they allow for simpler computational approaches, still preserving main features of the bulk system. In this context, it is important that a clear relationship between the full three-dimensional problem and its lower-dimensional counterpart is made rigorous. The last decades have witnessed remarkable progress in that direction through the use of variational methods, particularly by $\Gamma$-convergence [18] together with quantitative rigidity estimates [25]. Among the large body of results, we mention here only the rigorous justification of membrane theory [31, 32], bending theory [16, 25, 39], and von Kármán theory [26, 30] for plates as variational limits of nonlinear three-dimensional elasticity for vanishing thickness. In particular, we refer to [26] for the derivation of a hierarchy of different plate models and for a thorough literature review.

In the present work, we apply a similar scenario of deriving plate theories to problems in nonlinear viscoelasticity: starting from a three-dimensional model of a nonsimple viscoelastic material at a finite strain setting in Kelvin’s-Voigt’s rheology (i.e., a spring and a damper coupled in parallel), recently treated by the authors in [23], we derive a model of von Kármán (vK) viscoelastic plates.

In [23], the existence of weak solutions for such a three-dimensional model of nonsimple viscoelastic materials has been established with the help of gradient flows in metric spaces developed in [3, 42]. The notion of a nonsimple (or second-grade) material refers to the fact that the elastic energy depends also on the second gradient of the deformation. This concept, first suggested by Toupin [44, 45], has proved to be useful in modern mathematical elasticity, see e.g. [10, 7, 8, 36, 37, 41]. We also refer to [15, 21] where thermodynamical consistency of such models has been shown. We point out that this approach seems to be currently unavoidable in order to obtain the existence of solutions in the nonlinear viscoelastic setting, see [23, 37] and [35] for a general discussion about the interplay between the elastic energy and viscous dissipation. Nevertheless, a main justification is the observation in [23] that, in the small strain limit, the problem leads to the standard system for linearized viscoelasticity without second gradient.

In the present work, we consider a thin plate of thickness $h$ and pass to the dimension-reduction limit $h\to 0$ in the vK energy regime. We show that this gives rise to effective equations in terms of suitably rescaled in-plane and out-of-plane displacements which feature membrane and bending terms both in the elastic and the viscous stress. This represents a dissipative counterpart of the purely elastic vK theory which was first formulated more than hundred years ago [46]. Besides identifying the correct 2D limiting equations of viscous vK plates, which to the best of our knowledge has not been done in previous literature, the main goals of this contribution are twofold: (1) we show existence of solutions to the effective 2D system, and (2) we prove rigorously that these solutions are in a certain sense the limits of solutions to the 3D equations as $h\to 0$, see Theorem 2.2 and Theorem 2.3 for details.

Let us mention that there are previous works on viscoelastic plates [11, 40], some even including inertial effects [12, 13]. Their starting point, however, is already a plate model. Our model, derived rigorously from three-dimensional viscoelasticity, is new and, if viscosity is dropped, it reduces to the well-known model of elastic plates [26]. Let us emphasize that for purely elastic models neglecting viscosity various existence results were obtained for Föppl-von Kármán plates without resorting to a second-grade material, see e.g. [9, 26, 30, 33, 34] . We refer to [24] for a numerical study of vK viscoelastic plates, and mention that in that paper we have also proved the existence of solutions to the viscoelastic vK plate equations by means of converging numerical discretizations. This work, however, already relies on the plate model derived in the present paper. For the derivation of a plate model without viscosity but with inertia we refer to [1].

We now describe our setting in more detail. Without inertia, a nonlinear viscoelastic material in Kelvin’s-Voigt’s rheology satisfies the system of equations

[TABLE]

Here, $[0,T]$ is a process time interval with $T>0$, $\Omega_{h}:=S\times(-\frac{h}{2},\frac{h}{2})\subset\mathbb{R}^{3}$ is a smooth, thin, bounded domain representing the reference configuration, and $w:[0,T]\times\Omega_{h}\to\mathbb{R}^{3}$ is a deformation mapping with deformation gradient $\nabla w$. Moreover, $W:\mathbb{R}^{3\times 3}\to[0,\infty]$ is a stored energy density representing a potential of the first Piola-Kirchhoff stress tensor $\partial_{F}W:=\partial W/\partial F$ and $F\in\mathbb{R}^{3\times 3}$ is the placeholder of $\nabla w$. The function R:\mathbb{R}^{3\times 3}\times\mathbb{R}^{3\times 3}\to[0,\infty)\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0} denotes a (pseudo)potential of dissipative forces, where $\dot{F}\in\mathbb{R}^{3\times 3}$ is the placeholder of $\partial_{t}\nabla w$. Finally, $f:\Omega_{h}\to\mathbb{R}$ is a volume density of external forces, which is considered independent of time and the deformation $y$, and for simplicity acting on $\Omega_{h}$ only in normal direction $e_{3}$.

We assume that $W$ is a frame-indifferent function, i.e., $W(F)=W(QF)$ for $Q\in{\rm SO}(3)$ and $F\in\mathbb{R}^{3\times 3}$. This implies that $W$ depends on the right Cauchy-Green strain tensor $C:=F^{\top}F$, see e.g. [17]. The second term on the left-hand side of (1.1) is the stress tensor ${S}(F,\dot{F}):=\partial_{\dot{F}}R(F,\dot{F})$ which has its origin in viscous dissipative mechanisms of the material. We point out that its potential $R$ plays an analogous role as $W$ in the case of purely elastic, i.e., non-dissipative processes. Naturally, we require that $R(F,\dot{F})\geq R(F,0)=0$. The viscous stress tensor must comply with the time-continuous frame-indifference principle meaning that ${S}(F,\dot{F})=F\tilde{S}(C,\dot{C})$, where $\tilde{S}$ is a symmetric matrix-valued function and $\dot{C}$ denotes the time derivative of the right Cauchy-Green strain tensor $C$. This condition constraints $R$ so that $R(F,\dot{F})=\tilde{R}(C,\dot{C})$ for some nonnegative function $\tilde{R}$, see [4, 5, 35]. In the following, we suppose that the material is homogeneous, i.e., neither the elastic stored energy density nor the dissipation depend on material points. Moreover, for technical reasons, we will restrict our analysis to the case of zero Poisson’s ratio in the out-of-plane direction, see (2.17)-(2.18) and Remark 5.11 for some details in that direction. Such an assumption, also present in other works (see e.g. [9]), simplifies the analysis.

Following the study in [23], we consider a version of (1.1) for second-grade materials where the elastic stored energy density (and the first Piola-Kirchhoff stress tensor, too) depends also on the second gradient of $w$. In this case, we get

[TABLE]

where $\varepsilon>0$ is small and $\mathcal{L}_{P}$ is a first order differential operator which corresponds to an additional term $\int_{\Omega_{h}}P(\nabla^{2}w)$ in the stored elastic energy, associated to a convex and frame-indifferent density $P$. (We refer to (2.9) for more details.) As already mentioned before, this idea by Toupin [44, 45] has proved to be useful in mathematical elasticity because it brings additional compactness to the problem. For example, concerning existence theory for second-grade materials, no convexity properties of $W$ are needed, in particular, we do not have to assume that $W$ is polyconvex [6, 17]. Moreover, it is shown in [27] that, if $W$ satisfies suitable and physically relevant growth conditions (as $W(F)\to\infty$ if ${\rm det}\,F\to 0$), then every minimizer of the elastic energy is a weak solution to the corresponding Euler-Lagrange equations.

In [26], it has been shown that for forces scaling like $\sim h^{3}$, which corresponds to an energy per thickness of $\sim h^{4}$, the nonlinear elastic energy can be related rigorously by $\Gamma$-convergence as $h\to 0$ to the so-called von Kármán functional. This functional is given in terms of rescaled in-plane displacements $u$ and out-of-plane displacements $v$. The corresponding Euler-Lagrange equations take the form

[TABLE]

where $e(u):=(\nabla u+(\nabla u)^{\top})/2$ denotes the linear strain tensor, and $\mathbb{C}_{W}$ is the tensor of elastic constants, derived suitably from $W$ (see (2.17)–(2.19) below for details). The first equation corresponds to the membrane strain, which was used already earlier in Föppl’s work [22] and leads to a nonlinearity in the vK equations. The second equation includes also a bending contribution. In the present context, we will see that the passage from the nonlinear elastic energy to the vK functional by $\Gamma$-convergence remains true if the nonlinear energy is enhanced by a second gradient term $\varepsilon\int_{\Omega_{h}}P(\nabla^{2}w)$ for certain scalings of $\varepsilon$, see Theorem 5.6.

In the frame of viscoelastic materials, we address the relation between the nonlinear equations (1.2) to the following equations for viscoelastic vK plates:

[TABLE]

where $\mathbb{C}_{R}$ is the tensor of viscosity coefficients which is derived from the dissipation potential $R$. More precisely, we prove existence of solutions to (1.3) and make the dimension reduction rigorous, i.e., we show that solutions to (1.2) converge to solutions of (1.3) in a specific sense. The solutions have to be understood in a weak sense, see (2.22) for the exact definition. We point out that the same relation is expected to hold also for the original problem of simple materials (1.1) (i.e., only the first gradient of $w$ is considered), but a proof seems unreachable (or at least rather difficult) at the moment. We emphasize that a nonsimple material model is used here because viscous phenomena are considered, too. In fact, dissipation due to viscosity leads to the loss of weak lower semicontinuity in semidiscretized variational problems, see [35] for a detailed discussion.

Our general strategy is to treat the system of quasistatic viscoelasticity in the abstract setting of metric gradient flows [3], where the underlying metric is given by a dissipation distance suitably related to the potential $R$ (see (2.6) below). To our best knowledge, this was formulated for the first time in [35] for simple materials. Similar to [23], our starting point is the existence of time-discrete solutions to the three-dimensional equations (1.2). Here, the second gradient term allows to obtain an existence result without polyconvexity conditions [6] for the dissipation which seems to be incompatible with frame indifference. Existence of solutions to the equations (1.3) of viscous vK plates is guaranteed by identifying them as limits of solutions to the nonlinear three-dimensional equations (1.2). In this context, we follow the abstract framework of sequences of metric gradient flows, developed in [38, 42, 43]. In using this theory, the challenge lies in proving that the additional conditions needed to ensure convergence of gradient flows are satisfied.

More specifically, to use the abstract convergence result, lower semicontinuity of (i) the energies, (ii) the metrics, and (iii) the local slopes is needed. (i) The estimate for the energies essentially follows from [26] where we show that it still holds for nonsimple materials if the contribution of the second gradient in terms of $\varepsilon$ (see (1.2)) is chosen sufficiently small, cf.  Theorem 5.6. (ii) The lower semicontinuity of the metrics can be established in a very similar fashion. (iii) The lower semicontinuity of the local slopes, however, is very technical and the core of our argument. We briefly explain the main idea. The local slope of an energy $\phi$ in a metric space with metric $\mathcal{D}$ is defined by

[TABLE]

Consider a sequence of deformations $(w^{h})_{h}$ of thin plates converging to a limit $(u,v)$. The first step in the proof is to show that the local slope in the limiting setting at some configuration $y=(u,v)$ can be determined by considering only variations of the form $z=(u_{s},v_{s})=(u,v)+s(\tilde{u},\tilde{v})$ for $s>0$ small. This follows from a specific representation, see Lemma 4.9, which is based on some generalized convexity properties. (Recall, however, that the vK model is actually nonconvex.) Then the crucial step is to choose sequences $(w^{h}_{s})_{h,s}$ where $w^{h}_{s}\to w^{h}$ represents a competitor sequence for the local slope in the 3D setting. For each $s>0$, $(w^{h}_{s})_{h}$ has to be constructed as a mutual recovery sequence of $(u_{s},v_{s})$, i.e., for both the elastic energies and the dissipations. In this context, the rate of convergence needs to be linear in $s$ as $h\to 0$. The realization is in fact quite technical and the details are contained in Theorem 5.10. An important step in this analysis is to understand the strain difference of the configurations $w^{h}$ and $w^{h}_{s}$, see Lemma 5.5.

Let us mention that a derivation of a nonlinear bending theory (see [25]) in the setting of viscoelasticity seems to be even more involved and remains an open problem: the fact that deformations are generically not near a single rigid motion does not comply with the model investigated in [23]. Even more severely, the characterization of the local slope in the limiting two-dimensional setting appears to be very difficult due to the nonlinear isometry constraint.

The plan of the paper is as follows. In Section 2, we introduce the nonlinear three-dimensional viscoelastic system and its two-dimensional vK limiting version in more detail. Then we state our main results. In particular, Proposition 2.1 shows existence of time-discrete solutions to the three-dimensional problem (1.2). In Theorem 2.2 we present an existence result for solutions to the vK plate equations (1.3) in the “time-continuous setting”, which is based on identifying these solutions with so-called curves of maximal slope [20]. Finally, Theorem 2.3 shows the relationship between the two systems which relies on an abstract convergence result for curves of maximal slope and their approximation via the minimizing movement scheme.

Section 3 is devoted to definitions of generalized minimizing movements and curves of maximal slope. Here, we also recall results about sequences of curves of maximal slope and their approximation by minimizing movements. Section 4 is devoted to properties of the elastic energies and the dissipation distances. In particular, we show that the dissipation distances give rise to complete metric spaces, and derive some generalized convexity properties in the limiting 2D setting, as well as a representation of the local slope. Section 5 discusses the relation between the three- and two-dimensional setting, compactness properties, and $\Gamma$-convergence results. Moreover, here we prove the fundamental lower semicontinuity properties for local slopes. Finally, proofs of our main results can be found in Section 6.

2. The model and main results

2.1. Notation

In what follows, we use standard notation for Lebesgue spaces, $L^{p}(\Omega)$, which are measurable maps on $\Omega\subset\mathbb{R}^{d}$, $d=2,3$, integrable with the $p$-th power (if $1\leq p<+\infty$) or essentially bounded (if $p=+\infty$). Sobolev spaces, i.e., $W^{k,p}(\Omega)$ denote the linear spaces of maps which, together with their weak derivatives up to the order $k\in\mathbb{N}$, belong to $L^{p}(\Omega)$. Further, $W^{k,p}_{0}(\Omega)$ contains maps from $W^{k,p}(\Omega)$ having zero boundary conditions (in the sense of traces). To emphasize the target space $\mathbb{R}^{k}$, $k=1,2,3$, we write $L^{p}(\Omega;\mathbb{R}^{k})$. If $k=1$, we write $L^{p}(\Omega)$ as usual. We refer to [2] for more details on Sobolev spaces and their duals. We also denote the components of vector functions $y$ by $y_{1}$, $y_{2}$, and $y_{3}$, and so on.

If $A\in\mathbb{R}^{d\times d\times d\times d}$ and $e\in\mathbb{R}^{d\times d}$ then $Ae\in\mathbb{R}^{d\times d}$ is such that for $i,j\in\{1,\ldots,d\}$ we define $(Ae)_{ij}:=A_{ijkl}e_{kl}$ where we use Einstein’s summation convention. An analogous convention is used in similar occasions, in the sequel. By $\mathbf{Id}\subset\mathbb{R}^{3\times 3}$ we denote the identity matrix. We often drop $dx$ at the end of integrals if the integration variable is clear from the context. Finally, at many spots, we closely follow notation introduced in [3] and [26] to ease readability of our work.

2.2. The setting

We first introduce a 3D setting following the setup in [26]. We consider a right-handed orthonormal system $\{e_{1},e_{2},e_{3}\}$ and $S\subset\mathbb{R}^{2}$ open, bounded with Lipschitz boundary, in the span of $e_{1}$ and $e_{2}$. Let $h>0$ small. We consider deformations $w:S\times(-\frac{h}{2},\frac{h}{2})\to\mathbb{R}^{3}$. It is convenient to work in a fixed domain $\Omega=S\times I$ with $I:=(-\frac{1}{2},\frac{1}{2})$ and to rescale deformations according to $y(x)=w(x^{\prime},hx_{3})$ such that $y:\Omega\to\mathbb{R}^{3}$, where we use the abbreviation $x^{\prime}=(x_{1},x_{2})$. We also introduce the notation $\nabla^{\prime}y=y_{,1}\otimes e_{1}+y_{,2}\otimes e_{2}$ for the in-plane gradient, and the scaled gradient

[TABLE]

Moreover, we define the scaled second gradient by

[TABLE]

where $\delta_{3j},\delta_{3k}$ denotes the Kronecker delta.

Stored elastic energy density and body forces: We assume that $W:\mathbb{R}^{3\times 3}\to[0,\infty]$ is a single well, frame-indifferent stored energy density with the usual assumptions in nonlinear elasticity. We suppose that there exists $c>0$ such that

[TABLE]

where $SO(3)=\{Q\in\mathbb{R}^{3\times 3}:Q^{\top}Q=\mathbf{Id},\,\det Q=1\}$. Moreover, for $p>3$, let $P:\mathbb{R}^{3\times 3\times 3}\to[0,\infty]$ be a higher order perturbation satisfying

[TABLE]

for $0<c_{1}<c_{2}$. Finally, $f\in L^{\infty}(\Omega)$ denotes a volume normal force, i.e., a force oriented in $e_{3}$ direction. Note that more general forces could in principle be included, e.g., boundary forces (see [30]). This is neglected here for the sake of simplicity rather than generality.

Dissipation potential and viscous stress: We now introduce a dissipation potential. We follow here the discussion in [35, Section 2.2] and [23, Section 2]. Consider a time dependent deformation $y:[0,T]\times\Omega\to\mathbb{R}^{3}$. Viscosity is not only related to the strain rate $\partial_{t}\nabla_{h}y(t,x)$ but also to the strain $\nabla_{h}y(t,x)$. It can be expressed in terms of a dissipation potential $R(\nabla_{h}y,\partial_{t}\nabla_{h}y)$, where $R:\mathbb{R}^{3\times 3}\times\mathbb{R}^{3\times 3}\to[0,\infty)$. An admissible potential has to satisfy frame indifference in the sense (see [4, 35])

[TABLE]

for all $F\in GL_{+}(3)$ and $\dot{F}\in\mathbb{R}^{3\times 3}$, where $GL_{+}(3)=\{F\in\mathbb{R}^{3\times 3}:\det F>0\}$ and $\mathbb{R}^{3\times 3}_{\rm skew}=\{A\in\mathbb{R}^{3\times 3}:A=-A^{\top}\}$.

From the point of modeling, it is more convenient to assume the existence of a (smooth) global distance $D:GL_{+}(3)\times GL_{+}(3)\to[0,\infty)$ satisfying $D(F,F)=0$ for all $F\in GL_{+}(3)$. From this, an associated dissipation potential $R$ can be calculated by

[TABLE]

for $F\in GL_{+}(3)$, $\dot{F}\in\mathbb{R}^{3\times 3}$. Here, $\partial^{2}_{F_{1}^{2}}D^{2}(F_{1},F_{2})$ denotes the Hessian of $D^{2}$ in direction of $F_{1}$ at $(F_{1},F_{2})$, which is a fourth order tensor. For some $c>0$ we suppose that $D$ satisfies

[TABLE]

Note that conditions (i),(iii) state that $D$ is a true distance when restricted to symmetric matrices with nonnegative determinants. We cannot expect more due to the separate frame indifference (v). We also point out that (v) implies (2.5) as shown in [35, Lemma 2.1]. Note that in our model we do not require any conditions of polyconvexity [6] neither for $W$ nor for $D$. One possible example of $D$ satisfying (2.2) might be D(F_{1},F_{2})=\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}|F_{1}^{\top}F_{1}-F_{2}^{\top}F_{2}|\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}. This leads to R(F,\dot{F})=|\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}F^{\top}\dot{F}+\dot{F}^{\top}F\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}|^{2}/2 which is a standard choice. For further examples we refer to [35, Section 2.3].

Equations of viscoelasticity in 3D: Following the study in [30], we define clamped boundary conditions as follows. We consider functions $\hat{u}\in W^{2,\infty}(S;\mathbb{R}^{2})$ and $\hat{v}\in W^{3,\infty}(S)$ which represent in-plane and out-of-plane boundary conditions, respectively. We introduce the set of admissible configurations by

[TABLE]

where $I=(-\frac{1}{2},\frac{1}{2})$. Following [23] we formulate the equations of viscoelasticity for a nonsimple material involving the perturbation $P$ (cf. (2.4)). We introduce a differential operator associated to $P$. To this end, we recall the notation of the scaled gradients in (2.1)-(2.2). For i,j\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\in\{1,\ldots,3\}, we denote by $(\partial_{Z}P(\nabla^{2}_{h}y))_{ij*}$ the vector-valued function $((\partial_{Z}P(\nabla^{2}_{h}y))_{ijk})_{k=1,2,3}$. We also introduce the scaled (distributional) divergence ${\rm div}_{h}g$ for a function $g\in L^{1}(\Omega;\mathbb{R}^{3})$ by ${\rm div}_{h}g=\partial_{1}g_{1}+\partial_{2}g_{2}+\frac{1}{h}\partial_{3}g_{3}$. We define

[TABLE]

for $y\in\mathscr{S}_{h}$. Let $\beta_{1},\beta_{2}>0$. The equations of nonlinear viscoelasticity can be written as

[TABLE]

for some $y^{h}_{0}\in\mathscr{S}_{h}$, where $\partial_{F}W(\nabla_{h}y)$ $+h^{\beta_{1}}\mathcal{L}^{h}_{P}(\nabla^{2}_{h}y)$ denotes the first Piola-Kirchhoff stress tensor and $\partial_{\dot{F}}R(\nabla_{h}y,\partial_{t}\nabla_{h}y)$ the viscous stress with $R$ as introduced in (2.6). As no surface forces are applied, we implicitly assume zero Neumann boundary conditions for the stress and the hyperstress on $S\times\{-1/2,1/2\}$, i.e., on the top and the bottom of the cylinder $\Omega$, see e.g. [28] for a specific form of such conditions. As (2.2) prescribes only the values of the function but not of the derivative, on the lateral boundary there arise additional Neumann conditions from the second deformation gradient. (We again refer to [28] for details.) As we will see below, however, they do not affect the effective 2D plate model. Suitable scalings $h^{\beta_{1}}$ and $h^{\beta_{2}}$ related to $\mathcal{L}^{h}_{P}$ and the normal force $f$, respectively, will be discussed below. Note that $\varepsilon=h^{\beta_{1}}$ in (1.2).

Time-discrete solutions to (2.10): The first auxiliary goal of this paper is to show existence of time-discrete solutions to (2.10) for small $h>0$. For this, we introduce a functional $I^{\beta_{1},\beta_{2}}_{h}:W^{2,p}(\Omega;\mathbb{R}^{3})\to\mathbb{R}$ describing the elastic energy of the body by

[TABLE]

for a deformation $y:W^{2,p}(\Omega;\mathbb{R}^{3})\to\mathbb{R}^{3}$. We note that the energy takes into account the different scalings of the terms in (2.10).

We use an approximation scheme solving suitable time-incremental minimization problems: consider a fixed time step $\tau>0$ and suppose that an initial datum $y_{0}^{h}\in\mathscr{S}_{h}$ is given. Set $Y^{0}_{h,\tau}=y_{0}^{h}$. Whenever $Y^{0}_{h,\tau},\ldots,Y^{n-1}_{h,\tau}$ are known, $Y^{n}_{h,\tau}$ is defined as (if existent)

[TABLE]

where $\mathcal{D}$ denotes the global dissipation distance between two deformations, defined by

[TABLE]

Suppose that, for a choice of $\tau$, a sequence $(Y^{n}_{h,\tau})_{n\in\mathbb{N}}$ solving (2.12) exists. We define the piecewise constant interpolation by

[TABLE]

In the following, $\tilde{Y}_{h,\tau}$ will be called a time-discrete solution. We often drop the $x$-dependence and write $\tilde{Y}_{h,\tau}(t)$ for a time-discrete solution at time $t$. Note that the existence of such solutions is usually guaranteed by the direct method of the calculus of variations under suitable compactness, coercivity, and lower semicontinuity assumptions, see Proposition 2.1 and its proof. We point out that, in the setting of general (but not thin) bodies with suitable boundary conditions imposed on the entire $\partial\Omega$, it has been shown in [23, Theorem 2.1] that time-discrete solutions indeed converge to weak solutions to the system (2.10) as $\tau\to 0$. In the present context, time-discrete solutions will be the starting point to pass to a two-dimensional, time-continuous framework.

Scaling and displacement fields: Due to the scaling of the boundary conditions in (2.2), the energy of time-discrete solutions $\tilde{Y}_{h,\tau}$ is expected to be small in terms of $h$. More specifically, in our setting it will turn out that the energy is of order $h^{4}$ which corresponds to the so-called von Kármán regime. (For an exhaustive treatment of different scaling regimes we refer the reader to [26].) As $y_{3}$ scales like $h$, see (2.2), a suitable choice for the scaling of the forces in (2.11) is therefore $h^{3}$, i.e., we set $\beta_{2}=3$. Moreover, we choose $\beta_{1}=4-p\alpha$ for some $0<\alpha<1$. On the one hand, this will imply that $\|\nabla^{2}_{h}\tilde{Y}_{h,\tau}\|_{L^{p}(\Omega)}$ is small, more precisely, of order $h^{\alpha}$, cf. (2.4)(iii). On the other hand, $\alpha<1$ will ensure that the higher order perturbation will vanish in the effective 2D limiting model.

Based on this discussion, we introduce the rescaled nonlinear energy $\phi_{h}:W^{2,p}(\Omega;\mathbb{R}^{3})\to[0,\infty]$ by

[TABLE]

for $y\in\mathscr{S}_{h}$. Similarly, for $y_{0},y_{1}\in\mathscr{S}_{h}$, the rescaled global dissipation distance is given by

[TABLE]

Following the discussion in [26], for $y\in\mathscr{S}_{h}$ we introduce the corresponding averaged, scaled in-plane and out-of-plane displacements which measure the deviation from the mapping $(x^{\prime},0)$:

[TABLE]

where again $I=(-\frac{1}{2},\frac{1}{2})$. In a similar fashion, given a time-discrete solution $\tilde{Y}_{h,\tau}$, we introduce averaged functions $\tilde{U}_{h,\tau}$ and $\tilde{V}_{h,\tau}$ dependent on $(t,x^{\prime})$. Via the minimizing movement scheme, we will later see that along a sequence $(\tilde{Y}_{h,\tau}(t))_{h,\tau}$ we get $\tilde{U}_{h,\tau}(t)\rightharpoonup u(t)$ weakly in $W^{1,2}(S;\mathbb{R}^{2})$ and $\tilde{V}_{h,\tau}(t)\to v(t)$ strongly in $W^{1,2}(S)$ with $v(t)\in W^{2,2}(S)$ for each $t\geq 0$, when $h,\tau\to 0$. The main goal of our work is to understand which equations are solved by $(u(t),v(t))$.

Quadratic forms: To formulate the effective 2D problem for the scaled in-plane and out-of-plane displacements, we need to consider various quadratic forms. First, we define $Q_{W}^{3}:\mathbb{R}^{3\times 3}\to\mathbb{R}$ by $Q_{W}^{3}(F)=\partial^{2}_{F^{2}}W(\mathbf{Id})[F,F]$. One can show that it depends only on the symmetric part $\frac{1}{2}(F^{\top}+F)$ and that it is positive definite on $\mathbb{R}^{3\times 3}_{\rm sym}=\{A\in\mathbb{R}^{3\times 3}:A=A^{\top}\}$, cf. Lemma 4.1. We also introduce $Q_{W}^{2}:\mathbb{R}^{2\times 2}\to\mathbb{R}$ by

[TABLE]

for $G\in\mathbb{R}^{2\times 2}$, where the entries of $G^{*}\in\mathbb{R}^{3\times 3}$ are given by $G^{*}_{ij}=G_{ij}$ for $i,j\in\{1,2\}$ and zero otherwise. Note that (2.17) corresponds to a minimization over stretches in the $e_{3}$ direction. We will assume that the minimum in (2.17) is attained for $a=0$. Similarly, we define

[TABLE]

(Notice that $Q_{D}^{3}(F)=2R(\mathbf{Id},F)$ with $R$ from (2.6).) We again assume that the minimum is attained for $a=0$.

The assumption that $a=0$ is a minimum in (2.17)-(2.18) corresponds to a model with zero Poisson’s ratio in the $e_{3}$ direction. This assumption is not needed in the purely static analysis [26, 30]. In our setting, it is only needed in the proof of lower semicontinuity of slopes, see Theorem 5.10. Dropping this assumption would lead to a considerably more involved limiting description which we do not want to pursue here. We refer to Remark 5.11 for some details in that direction.

We also introduce corresponding symmetric fourth order tensors $\mathbb{C}^{d}_{W}$ and $\mathbb{C}^{d}_{D}$, $d=2,3$, satisfying

[TABLE]

and likewise for $\mathbb{C}_{D}^{d}$, $d=2,3$.

Equations of viscoelasticity in 2D: We now present the effective 2D equations. By recalling definition (2.16) and the boundary conditions in the 3D setting (2.2), we first introduce the relevant space by

[TABLE]

The clamped boundary conditions correspond to the ones considered in [30]. We emphasize that Neumann conditions in (2.10) on the lateral boundary arising from the second deformation gradient do not affect the plate equations (2.21) below since our scaling makes the second gradient vanish as the thickness of the domain tends to zero.

Given $(u_{0},v_{0})\in\mathscr{S}_{0}$, we consider the equations

[TABLE]

where $\mathbb{C}^{2}_{W}$ and $\mathbb{C}^{2}_{D}$ are defined in (2.19), and $\odot$ denotes the symmetrized tensor product. Note that the frame indifference of the energy and the dissipation (see (2.3)(ii) and (2.2)(v), respectively) imply that the contributions only depend on the symmetric part of the strain $e(u):=\frac{1}{2}(\nabla^{\prime}u+(\nabla^{\prime}u)^{\top})$ and the strain rate $e(\partial_{t}u):=\frac{1}{2}(\partial_{t}\nabla^{\prime}u+\partial_{t}(\nabla^{\prime}u)^{\top})$. Here, ${\rm div}_{2}$ denotes the distributional divergence in dimension two.

We also say that $(u,v)\in W^{1,2}([0,\infty);{\mathscr{S}}_{0})$ is a weak solution of (2.21) if $u(0,\cdot)=u_{0}$, $v(0,\cdot)=v_{0}$ and for a.e. $t\geq 0$ we have

[TABLE]

for all $\varphi_{u}\in W^{1,2}_{0}(S;\mathbb{R}^{2})$ and $\varphi_{v}\in W^{2,2}_{0}(S)$. Note that (2.22a) corresponds to two scalar equations and (2.22b) corresponds to one scalar equation, respectively. Our goal will be to show that time-discrete solutions to (2.10), as introduced in (2.13), converge to weak solutions to (2.21) in a suitable sense.

We also mention that the equations are related to the von Kármán functional

[TABLE]

for $(u,v)\in{\mathscr{S}}_{0}$: actually, we will also see that ${\phi}_{0}$ can be related to $\phi_{h}$ (see (2.14)) in the sense of $\Gamma$-convergence, cf. Section 5.2 below. A similar relation holds for $\mathcal{D}_{h}$, introduced in (2.15), and the global dissipation distance in the 2D setting, defined by

[TABLE]

for $(u_{0},v_{0}),(u_{1},v_{1})\in{\mathscr{S}}_{0}$.

2.3. Main results

Define the sublevel sets $\mathscr{S}_{h}^{M}:=\{y\in\mathscr{S}_{h}:\phi_{h}(y)\leq M\}$. Our general strategy will be to show that the spaces $(\mathscr{S}_{h}^{M},\mathcal{D}_{h})$ and $({\mathscr{S}}_{0},{\mathcal{D}}_{0})$ are complete metric spaces (see Lemma 4.5 and Lemma 4.6 below) and to follow the methodology developed in [3].

To show existence of solutions to the equations, we will apply the theory of [3] about curves of maximal slope. By using the property that in Hilbert spaces curves of maximal slope can be related to gradient flows, we then find weak solutions to (2.21). To understand the relation of solutions in the 3D and 2D setting, we will employ an abstract convergence result for curves of maximal slope and their approximation via the minimizing movement scheme, see [38, 43]. The relevant results about curves of maximal slope are recalled in Section 3.

Our first main result addresses the existence of time-discrete solutions to the 3D problem.

Proposition 2.1** (Time-discrete solutions in the 3D setting).**

Let $M>0$ and $\mathscr{S}_{h}^{M}=\{y\in\mathscr{S}_{h}:\phi_{h}(y)\leq M\}$. Let $\beta_{1}=4-\alpha p$ and $\beta_{2}=3$. Let $y_{0}^{h}\in\mathscr{S}_{h}^{M}$. Then, for $h>0$ sufficiently small only depending on $M$, the sequence of minimization problems in (2.12) has a solution, and gives rise to a time-discrete solution $\tilde{Y}_{h,\tau}$ with $\tilde{Y}_{h,\tau}(0)=y_{0}^{h}$.

Note that the existence of weak (time-continuous) solutions to (2.10) has been addressed in [23] for bulk materials. However, due to the thinness of the domain representing the body and the imposed boundary conditions, the results obtained there are not applicable. This is due to the fact that specific constants depend on the domain and blow up for vanishing thickness. Therefore, here we only prove existence of time-discrete solutions.

For the main definitions and notation for curves of maximal slope and strong upper gradients we refer to Section 3.1. In particular, we write $|\partial{\phi}_{0}|_{{\mathcal{D}}_{0}}$ for the (local) slopes, see Definition 3.1. For the 2D problem we obtain the following results.

Theorem 2.2** (Solutions in the 2D setting).**

The limiting 2D problem has the following properties:

(i) (Curves of maximal slope) For all $(u_{0},v_{0})\in{\mathscr{S}}_{0}$ there exists a curve of maximal slope $(u,v):[0,\infty)\to{\mathscr{S}}_{0}$ for ${\phi}_{0}$ with respect to the strong upper gradient $|\partial{\phi}_{0}|_{{\mathcal{D}}_{0}}$ satisfying $(u,v)(0)=(u_{0},v_{0})$.

(ii) (Relation to PDE) For all $(u_{0},v_{0})\in{\mathscr{S}}_{0}$, each curve of maximal slope $(u,v):[0,\infty)\to{\mathscr{S}}_{0}$ with $(u,v)(0)=(u_{0},v_{0})$ is a weak solution to the partial differential equations (2.21) in the sense of (2.22).

We mention that in [24, Theorem 4.1] we provide a slightly different proof of Theorem 2.2(i), without using the fact that the 2D model is the limit of the time-discrete 3D model. However, the proof still heavily relies on the properties of $\phi_{0}$, ${\mathcal{D}}_{0}$, and $|\partial{\phi}_{0}|_{{\mathcal{D}}_{0}}$ derived in the present paper.

Finally, we study the relation of time-discrete solutions (2.13) and weak solutions to the equations (2.21). To this end, we need to specify the topology of the convergence. Given $y^{h}\in\mathscr{S}_{h}$ and $(u,v)\in{\mathscr{S}}_{0}$, we say $y^{h}\stackrel{{\scriptstyle\pi\sigma}}{{\to}}(u,v)$ as $h\to 0$ if the corresponding averaged and scaled displacements fields defined in (2.16), denoted by $u^{h}$ and $v^{h}$, satisfy $u^{h}\rightharpoonup u$ weakly in $W^{1,2}(S;\mathbb{R}^{2})$ and $v^{h}\to v$ strongly in $W^{1,2}(S)$. (The symbol $\pi\sigma$ is used because of the abstract convergence result, see Section 3.2.)

Theorem 2.3** (Relation between 3D and 2D systems).**

*Let $\beta_{1}=4-p\alpha$ and $\beta_{2}=3$. Let $(u_{0},v_{0})\in{\mathscr{S}}_{0}$ be an initial datum.

(i) Then, for the family of sequences of initial data we have*

[TABLE]

*(ii) We consider a sequence $(y^{h}_{0})_{h}\in\mathcal{B}(u_{0},v_{0})$, a null sequence $(\tau_{h})_{h}$, and a sequence of time-discrete solutions $\tilde{Y}_{h,\tau_{h}}$ as in (2.13) with $\tilde{Y}_{h,\tau_{h}}(0)=y^{h}_{0}$. *

Then, there exists a curve of maximal slope $(u,v):[0,\infty)\to{\mathscr{S}}_{0}$ for ${\phi}_{0}$ with respect to $|\partial{\phi}_{0}|_{{\mathcal{D}}_{0}}$ satisfying $u(0)=u_{0}$ and $v(0)=v_{0}$ such that up to a subsequence (not relabeled) there holds

[TABLE]

We point out that (2.25) corresponds to the existence of recovery sequences for the static problem. We note that the existence of the time-discrete solutions $\tilde{Y}_{h,\tau_{h}}$ in (ii) is guaranteed by Proposition 2.1. Item (ii) shows the convergence of time-discrete solutions to (2.10) to (time-continuous) weak solutions to (2.21). Moreover, we also have convergence of the energies. From now on we set $f\equiv 0$ for convenience. The general case indeed follows with minor modifications, which are standard.

3. Preliminaries: Curves of maximal slope

In this section we recall the relevant definitions about curves of maximal slope and present a convergence result of time-discrete solutions to curves of maximal slope.

3.1. Definitions

We consider a complete metric space $(\mathscr{S},\mathcal{D})$. We say a curve $y:(a,b)\to\mathscr{S}$ is absolutely continuous with respect to $\mathcal{D}$ if there exists $m\in L^{1}(a,b)$ such that

[TABLE]

The smallest function $m$ with this property, denoted by $|y^{\prime}|_{\mathcal{D}}$, is called metric derivative of $y$ and satisfies for a.e. $t\in(a,b)$ (see [3, Theorem 1.1.2] for the existence proof)

[TABLE]

We define the notion of a curve of maximal slope. We only give the basic definition here and refer to [3, Section 1.2, 1.3] for motivations and more details. By $h^{+}:=\max(h,0)$ we denote the positive part of a function $h$.

Definition 3.1** (Upper gradients, slopes, curves of maximal slope).**

We consider a complete metric space $(\mathscr{S},\mathcal{D})$ with a functional $\phi:\mathscr{S}\to(-\infty,+\infty]$.

(i) A function $g:\mathscr{S}\to[0,\infty]$ is called a strong upper gradient for $\phi$ if for every absolutely continuous curve \color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}y:\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}(a,b)\to\mathscr{S} the function $g\circ y$ is Borel and

[TABLE]

(ii) For each $y\in\mathscr{S}$ the local slope of $\phi$ at $y$ is defined by

[TABLE]

(iii) An absolutely continuous curve $y:(a,b)\to\mathscr{S}$ is called a curve of maximal slope for $\phi$ with respect to the strong upper gradient $g$ if for a.e. $t\in(a,b)$

[TABLE]

3.2. Curves of maximal slope as limits of time-discrete solutions

We consider a sequence of complete metric spaces $(\mathscr{S}_{k},\mathcal{D}_{k})_{k}$, as well as a limiting complete metric space $(\mathscr{S},\mathcal{D})$. Moreover, let $(\phi_{k})_{k}$ be a sequence of functionals with $\phi_{k}:\mathscr{S}_{k}\to[0,\infty]$ and $\phi:\mathscr{S}\to[0,\infty]$.

We introduce time-discrete solutions for the energy $\phi_{k}$ and the metric $\mathcal{D}_{k}$ by solving suitable time-incremental minimization problems: consider a fixed time step $\tau>0$ and suppose that an initial datum $Y^{0}_{k,\tau}$ is given. Whenever $Y_{k,\tau}^{0},\ldots,Y^{n-1}_{k,\tau}$ are known, $Y^{n}_{k,\tau}$ is defined as (if existent)

[TABLE]

We suppose that for a choice of $\tau$ a sequence $(Y_{k,\tau}^{n})_{n\in\mathbb{N}}$ solving (3.1) exists. Then we define the piecewise constant interpolation by

[TABLE]

We call $\tilde{Y}_{k,\tau}$ a time-discrete solution. Note that the existence of such solutions is usually guaranteed by the direct method of the calculus of variations under suitable compactness, coercivity, and lower semicontinuity assumptions.

Our goal is to study the limit of time-discrete solutions as $k\to\infty$. To this end, we need to introduce a suitable topology for the convergence. First, although $\mathcal{D}$ naturally induces a topology on the limiting space $\mathscr{S}$, it is often convenient to consider a weaker Hausdorff topology $\sigma$ on $\mathscr{S}$ to have more flexibility in the derivation of compactness properties (see [3, Remark 2.0.5]). We assume that for each $k\in\mathbb{N}$ there exists a map $\pi_{k}:\mathscr{S}_{k}\to\mathscr{S}$. Given a sequence $(z_{k})_{k}$, $z_{k}\in\mathscr{S}_{k}$, and $z\in\mathscr{S}$, we say

[TABLE]

We suppose that the topology $\sigma$ satisfies

[TABLE]

Moreover, assume that there exists a $\sigma$-sequentially compact set $K_{N}\subset\mathscr{S}$ such that for all $k\in\mathbb{N}$

[TABLE]

Specifically, for a sequence $(z_{k})_{k}$ with $\phi_{k}(z_{k})\leq N$, we find a subsequence (not relabeled) and $z\in\mathscr{S}$ such that $\pi_{k}(z_{k})\stackrel{{\scriptstyle\sigma}}{{\to}}z$. We suppose lower semicontinuity of the energies and the slopes in the following sense: for all $z\in\mathscr{S}$ and $(z_{k})_{k}$, $z_{k}\in\mathscr{S}_{k}$, we have

[TABLE]

We now formulate the main convergence result of time-discrete solutions to curves of maximal slope, proved in [38, Section 2].

Theorem 3.2**.**

Suppose that (3.4)-(3.6) hold. Moreover, assume that $|\partial\phi|_{\mathcal{D}}$ is a strong upper gradient for $\phi$. Consider a null sequence $(\tau_{k})_{k}$. Let $(Y^{0}_{k,\tau_{k}})_{k}$ with $Y^{0}_{k,\tau_{k}}\in\mathscr{S}_{k}$ and $\bar{z}_{0}\in\mathscr{S}$ be initial data satisfying

[TABLE]

Then for each sequence of discrete solutions $(\tilde{Y}_{k,\tau_{k}})_{k}$ starting from $(Y^{0}_{k,\tau_{k}})_{k}$ there exists a limiting function $z:[0,+\infty)\to\mathscr{S}$ such that up to a subsequence (not relabeled)

[TABLE]

as $k\to\infty$, and $z$ is a curve of maximal slope for $\phi$ with respect to $|\partial\phi|_{\mathcal{D}}$. In particular, $z$ satisfies the energy identity

[TABLE]

The statement is a combination of convergence results for curves of maximal slope [23, 43] with their approximation by time-discrete solutions via the minimizing movement scheme. We refer to [23, Theorem 3.6] for an abstract convergence result for curves of maximal slope in a setting where conditions (3.4)-(3.6) hold. We also mention the version in the seminal work [43] where condition (3.4) is replaced by a lower bound condition on the metric derivatives along the sequence.

For the proof of Theorem 3.2 we refer to [38, Section 2]. We also mention [23, Theorem 3.7] for a formulation of the result which is a bit closer to the statement given here. Strictly speaking, in [38], only the case of a single metric space is considered. The generalization to a sequence of spaces, however, is straightforward, cf. [43, equation (2.7)]. Note that the nonnegativity of the energies $\phi_{k}$ can be generalized to a suitable coerciveness condition, see [3, (2.1.2b)] or [38, (2.5)]. This is not included here for the sake of simplicity.

Let us also mention the recently obtained variant [14] where it is not necessary to require that $|\partial\phi|_{\mathcal{D}}$ is a strong upper gradient, cf. [3, Definition 1.2.1 and Definition 1.2.2] for the definition of strong and weak upper gradients. This comes at the expense of the fact that the lower semicontinuity along the sequence $(\phi_{k})_{k}$ (see (3.6)) has to be replaced by a continuity condition along $(\phi_{k})_{k}$ for sequences $(\pi_{k}(z_{k}))_{k}$ converging with respect to the metric $\mathcal{D}$.

4. Properties of energies and dissipation distances

In this section we prove basic properties of the energies and dissipation distances, and we establish properties for the local slope in the 2D setting. Let $h>0$ and $0<\alpha<1$. We recall the definition of the nonlinear energy and the dissipation distance in (2.14) and (2.15), respectively. We also recall (2.2) and the notation for the sublevel sets $\mathscr{S}_{h}^{M}=\{y\in\mathscr{S}_{h}:\phi_{h}(y)\leq M\}$. In the whole section, $C\geq 1$ and $0<c\leq 1$ indicate generic constants, which may vary from line to line and depend on $M$, $S$, the exponent $p>3$ (see (2.4)), $\alpha$, on the constants in (2.3), (2.4), (2.2), and on the boundary data $\hat{u}$ and $\hat{v}$. However, all constants are always independent of the small parameter $h$ and the deformations $y$.

4.1. Basic properties

We start with some properties about the Hessian of $W$ and $D^{2}$. By $\partial^{2}D^{2}$ we denote the Hessian and by $\partial^{2}_{F_{1}^{2}}D^{2},\partial^{2}_{F_{2}^{2}}D^{2}$ the Hessian in direction of the first or second entry of $D^{2}$, respectively. Moreover, we define ${\rm sym}(F)=\frac{1}{2}(F+F^{\top})$ for $F\in\mathbb{R}^{d\times d}$, $d=2,3$. Recall the definitions of the quadratic forms in (2.17)-(2.18). By $\mathbf{Id}\subset\mathbb{R}^{3\times 3}$ we again denote the identity matrix.

Lemma 4.1** (Properties of Hessian).**

(i) $\partial^{2}_{F_{1}^{2}}D^{2}(Y,Y)=\partial^{2}_{F_{2}^{2}}D^{2}(Y,Y)$ for all $Y\in\mathbb{R}^{3\times 3}$ in a neighborhood of $SO(3)$ such that $\partial^{2}D^{2}(Y,Y)$ exists.

(ii) There exists a constant $c>0$ such that $Q^{d}_{W}(F)=Q^{d}_{W}({\rm sym}(F))\geq c|{\rm sym}(F)|^{2}$ and $Q^{d}_{D}(F)=Q^{d}_{D}({\rm sym}(F))\geq c|{\rm sym}(F)|^{2}$ for all $F\in\mathbb{R}^{d\times d}$, $d=2,3$.

(iii) There exists $C>0$ such that for all $F_{0},F_{1}\in\mathbb{R}^{3\times 3}$ in a neighborhood of $SO(3)$ there holds $|W(F_{1})-W(F_{0})-\frac{1}{2}(Q_{W}^{3}(F_{1}-\mathbf{Id})-Q_{W}^{3}(F_{0}-\mathbf{Id}))|\leq\sum\nolimits_{k=1}^{3}C|F_{0}-\mathbf{Id}|^{3-k}|F_{1}-F_{0}|^{k}$.

Proof.

For the proof of (i) and (ii) we refer to [23, Lemma 4.1]. To see (iii), we perform a Taylor expansion. First, we find

[TABLE]

We observe that $DW(F_{0})=DW(\mathbf{Id})+D^{2}W(\mathbf{Id})(F_{0}-\mathbf{Id})+{\rm O}(|F_{0}-\mathbf{Id}|^{2})$ and $DW(\mathbf{Id})=0$ by (2.3)(iii). Moreover, $|D^{2}W(F_{0})-D^{2}W(\mathbf{Id})|\leq C|F_{0}-\mathbf{Id}|$ by the regularity of $W$. Thus, we get

[TABLE]

By recalling that $Q_{W}^{3}(F)=D^{2}W(\mathbf{Id})[F,F]$ for $F\in\mathbb{R}^{3\times 3}$ and the fact that $D^{2}W(\mathbf{Id})[\cdot,\cdot]$ is symmetric in the two entries, an elementary computation yields

[TABLE]

The result follows by combination of (4.1) and (4.2). ∎

The following geometric rigidity result will be a key ingredient for our analysis.

Lemma 4.2** (Rigidity in thin domains).**

For $h$ sufficiently small, for all $y\in\mathscr{S}_{h}^{M}$ there exists a mapping $R(y)\in W^{1,2}(S;SO(3))$ satisfying

[TABLE]

where $C_{q}$ depends also on q\in\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}[1,\infty)\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}. (In (i), $R(y)$ is extended to $\Omega$ by $R(y)(x^{\prime},x_{3})=R(y)(x^{\prime})$.)

Proof.

Property (i) is based on geometric rigidity [25] and is proved in [26, Theorem 6, Remark 5], where we use that $\|\operatorname{dist}(\nabla_{h}y,SO(3))\|^{2}_{L^{2}(\Omega)}\leq CMh^{4}$ by (2.3)(iii), (2.14), and the fact that $y\in\mathscr{S}_{h}^{M}$. Also (ii)-(iv) are proved there with a rotation $\bar{Q}$ in place of $\mathbf{Id}$. The fact that we may choose $\bar{Q}=\mathbf{Id}$ is due to the boundary conditions, which break the rotational invariance, see [30, Lemma 13].

We now show (v). By the definition of $\phi_{h}$ and (2.4)(iii) we get $\|\nabla^{2}_{h}y\|_{L^{p}(\Omega)}\leq Ch^{\alpha}$ for all $y\in\mathscr{S}^{M}_{h}$, where the constant depends on $M$. In particular, by (2.1)-(2.2) this implies

[TABLE]

As $p>3$, Poincaré’s inequality yields some $F\in\mathbb{R}^{3\times 3}$ such that

[TABLE]

for a constant additionally depending on $\Omega$ and $p$. This implies $\|\nabla_{h}y-F\|_{L^{\infty}(\Omega)}\leq Ch^{\alpha}$. Along with (ii), the triangle inequality, and $\alpha<1$ we obtain $|F-\mathbf{Id}|\leq Ch^{\alpha}$. This concludes the proof of (v).

Finally, we show (vi). A careful inspection of the proof of [26, Theorem 6] shows that $R(y)(x^{\prime})$ may be defined as the nearest-point projection onto $SO(3)$ of

[TABLE]

where $I=(-1/2,1/2)$ and $\psi\in C_{c}^{\infty}((-1,1)^{2})$ denotes a standard mollifier, i.e., $\psi\geq 0$ and $\int_{(-1,1)^{2}}\psi=1$. Then (vi) follows from (v). ∎

By Lemma 4.2 we get that $\nabla_{h}y$ is approximated by the $SO(3)$-valued function $R(y)$. As the energy is invariant under rotations, the energy of $y$ is essentially controlled by the distance of $R(y)^{\top}\nabla_{h}y$ from $\mathbf{Id}$. To this end, we introduce the quantity

[TABLE]

In the following we set for shorthand $H_{Y}:=\frac{1}{2}\partial^{2}_{F_{1}^{2}}D^{2}(Y,Y)=\frac{1}{2}\partial^{2}_{F_{2}^{2}}D^{2}(Y,Y)$ for $Y\in\mathbb{R}^{3\times 3}$ in a neighborhood of $SO(3)$. Given a deformation $y\in\mathscr{S}^{M}_{h}$, we also introduce the mapping $H_{\nabla_{h}y}:\Omega\to\mathbb{R}^{3\times 3\times 3\times 3}$ by $H_{\nabla_{h}y}(x)=H_{\nabla_{h}y(x)}$ for $x\in\Omega$. Note that this is well defined for $h$ sufficiently small by (4.2)(v). Recall the definition of $\mathcal{D}_{h}$ and ${\mathcal{D}}_{0}$ in (2.15) and (2.2), respectively, and the definition of $Q_{W}^{3}$, $Q_{D}^{3}$ in (2.17)-(2.18).

Lemma 4.3** (Dissipation and energy).**

Let $h$ sufficiently small. Then, for all $y,y_{0},y_{1}\in\mathscr{S}_{h}^{M}$ and all open subsets $U\subset\Omega$ we have

[TABLE]

Proof.

As a preparation, we observe that by the uniform bound on $\nabla_{h}y_{0}$, $\nabla_{h}y_{1}$ (see (4.2)(v)) and a Taylor expansion at $(\nabla_{h}y_{0},\nabla_{h}y_{0})$ we obtain for all open subsets $U\subset\Omega$

[TABLE]

We recall (4.4) and define G(y_{i})\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}:=\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}h^{2}G^{h}(y_{i})\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}=R(y_{i})^{\top}\nabla_{h}y_{i}-\mathbf{Id}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}, $i=0,1$, for convenience. Using the separate frame indifference (2.2)(v) we have

[TABLE]

Thus, by $h^{4}\mathcal{D}_{h}(y_{0},y_{1})^{2}=\int_{\Omega}D^{2}(\nabla_{h}y_{0},\nabla_{h}y_{1})$ and again by Taylor expansion we also get

[TABLE]

We now show (i). By the regularity of $D$ and (4.2)(v) we get $\|H_{\nabla_{h}y_{0}}-\mathbb{C}^{3}_{D}\|_{\infty}\leq Ch^{\alpha}$, where the fourth order tensor $\mathbb{C}^{3}_{D}$ associated to $Q^{3}_{D}$ is defined in (2.19). Therefore, we obtain

[TABLE]

for all open $U\subset\Omega$. By using (4.5) and again (4.2)(v) we get (i).

To see (ii), we observe $\|H_{R(y_{0})^{\top}\nabla_{h}y_{0}}-\mathbb{C}^{3}_{D}\|_{\infty}\leq C\|\nabla_{h}y_{0}-R(y_{0})\|_{\infty}\leq Ch^{\alpha}$ by the regularity of $D$ and (4.2)(v),(vi). Thus, we get

[TABLE]

In a similar fashion, (4.2)(v),(vi) also imply $\|G(y_{i})\|_{\infty}\leq Ch^{\alpha}$ for $i=0,1$ and thus

[TABLE]

This together with (4.6)-(4.1) (divided by $h^{4}$), and $G^{h}(y_{i})=h^{-2}G(y_{i})$ for $i=0,1$ yields

[TABLE]

This shows the first inequality of (ii). To see the second inequality, we use (4.2)(i) and (4.4).

We now show (iii) and (iv). We use the frame indifference of $W$ and Lemma 4.1(iii) (with $F_{i}=R(y_{i})^{\top}\nabla_{h}y_{i}=\mathbf{Id}+G(y_{i})$ for $i=0,1$) to obtain

[TABLE]

where $\Delta(y_{0})$ and $\Delta(y_{1})$ are defined in the statement of the lemma. The fact that $\|G(y_{i})\|_{\infty}\leq Ch^{\alpha}$ for $i=0,1$ (see (4.2)(v),(vi)) and Hölder’s inequality yield

[TABLE]

Using $G^{h}(y_{i})=h^{-2}G(y_{i})$ for $i=0,1$ and (4.2)(i) we obtain the first inequality of (iv). The second inequality follows again by (4.2)(i). Finally, to see (iii), we apply (iv) for $y_{0}=y$ and $y_{1}=\mathbf{id}$, where we use $\Delta(y_{1})=0$. ∎

4.2. Metric spaces and their properties

In this section we prove that $(\mathscr{S}^{M}_{h},\mathcal{D}_{h})$ and $({\mathscr{S}}_{0},{\mathcal{D}}_{0})$ are complete metric spaces. We start with the 3D setting. Recall the definition of $\phi_{h}$ and $\mathcal{D}_{h}$ in (2.14)-(2.15). As a preparation, we address the positivity of $\mathcal{D}_{h}$.

Lemma 4.4** (Positivity of $\mathcal{D}_{h}$).**

Let $M>0$ and let $h$ sufficiently small. Let $y_{0},y_{1}\in\mathscr{S}^{M}_{h}$ with $\mathcal{D}_{h}(y_{0},y_{1})=0$. Then $y_{0}=y_{1}$.

Proof.

It is convenient to formulate the problem for the original (not rescaled) functions $w_{0}$ and $w_{1}$ defined on $\Omega_{h}=S\times(-\frac{h}{2},\frac{h}{2})$. To explain the main idea, we first assume that $\Omega_{h}$ is the union of pairwise disjoint cubes of sidelength $h$ up to a set of negligible measure. Denote the family of cubes by $\mathcal{Q}$. By $\mathcal{Q}_{1}\subset\mathcal{Q}$ we denote the cubes whose boundaries share at least one face with $\partial S\times(-\frac{h}{2},\frac{h}{2})$. Let $\mathcal{Q}_{2}\subset\mathcal{Q}\setminus\mathcal{Q}_{1}$ be the cubes whose boundaries share at least one face with a cube in $\mathcal{Q}_{1}$. In a similar fashion, we define $\mathcal{Q}_{i}$, $i\geq 2$, and find $\mathcal{Q}=\bigcup_{i=1}^{I}\mathcal{Q}_{i}$ for some $I\in\mathbb{N}$.

We now first show that $w_{0}=w_{1}$ on each $Q\in\mathcal{Q}_{1}$. To this end, fix $Q\in\mathcal{Q}_{1}$. From Lemma 4.3(i) (in terms of $w_{0},w_{1}$ instead of $y_{0},y_{1}$) we get

[TABLE]

Since $w_{1}=w_{0}$ on $\partial S\times(-\frac{h}{2},\frac{h}{2})$, we get that $w_{1}=w_{0}$ on at least one face of $\partial Q$. Then Korn’s inequality (see, e.g., [26, Proposition 1]) implies

[TABLE]

Together with Lemma 4.1(ii) this shows

[TABLE]

For $h$ sufficiently small, (4.8)-(4.9) along with $\int_{Q}D^{2}(\nabla w_{0},\nabla w_{1})=0$ show $\nabla w_{1}=\nabla w_{0}$ a.e. on $Q$. Since $w_{1}=w_{0}$ on at least one face of $\partial Q$, this also gives $w_{1}=w_{0}$ a.e. on $Q$, as desired.

We now proceed iteratively to show that $w_{1}=w_{0}$ on each $Q\in\mathcal{Q}$: suppose that the property has already been shown for all $Q\in\bigcup_{i=1}^{j}\mathcal{Q}_{i}$. Then $w_{1}=w_{0}$ on each $Q\in\mathcal{Q}_{j+1}$ follows from the above arguments noting that $w_{1}=w_{0}$ on at least one face of $\partial Q$ since $w_{1}=w_{0}$ on all squares $Q\in\bigcup_{i=1}^{j}\mathcal{Q}_{i}$.

This shows $w_{1}=w_{0}$ on $\Omega_{h}$ in the case that $\Omega_{h}$ is the union of pairwise disjoint cubes of sidelength $h$ up to a set of negligible measure. In the general case, we may cover $\Omega_{h}$ by cubes of sidelength $h$, again denoted by $\mathcal{Q}$, such that for each $Q\in\mathcal{Q}$ the set $Q\cap\Omega_{h}$ is bilipschitzly equivalent to a cube of sidelength $h$ with a controlled Lipschitz constant. We note that the constant in Korn’s inequality can be chosen uniformly for these sets. We may again decompose $\mathcal{Q}$ into pairwise disjoint families $(\mathcal{Q}_{i})_{i}$ and show iteratively that $w_{1}$ coincides with $w_{0}$ on all cubes. ∎

Lemma 4.5** (Properties of $(\mathscr{S}^{M}_{h},\mathcal{D}_{h})$ and $\phi_{h}$).**

Let $M>0$. For $h>0$ sufficiently small we have

• (i)

$(\mathscr{S}^{M}_{h},\mathcal{D}_{h})$* is a complete metric space.*

• (ii)

Compactness: If $(y_{n})_{n}\subset\mathscr{S}^{M}_{h}$, then $(y_{n})_{n}$ admits a subsequence converging weakly in $W^{2,p}(\Omega;\mathbb{R}^{3})$ and strongly in $W^{1,\infty}(\Omega;\mathbb{R}^{3})$.

• (iii)

Topologies: The topology induced by $\mathcal{D}_{h}$ coincides with the weak $W^{2,p}(\Omega;\mathbb{R}^{3})$ topology.

• (iv)

Lower semicontinuity: $\mathcal{D}_{h}(y_{n},y)\to 0$    $\Rightarrow$   $\liminf_{n\to\infty}\phi_{h}(y_{n})\geq\phi_{h}(y)$.

Proof.

We start with (ii). We recall (2.4)(iii) and (2.14), and find \|\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\nabla^{2}_{h}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}y\|^{p}_{L^{p}(\Omega)}\leq CMh^{p\alpha} for all $y\in\mathscr{S}^{M}_{h}$. This together with (4.2)(v) and the boundary conditions (2.2) shows $\sup_{y\in\mathscr{S}^{M}_{h}}\|y\|_{W^{2,p}(\Omega)}<\infty$. Since $p>3$, (ii) follows from a standard compactness argument.

We now show (iii). (a) We first suppose that $y_{n}\rightharpoonup y$ weakly in $W^{2,p}(\Omega;\mathbb{R}^{3})$. As $p>3$, this implies $y_{n}\to y$ strongly in $W^{1,\infty}(\Omega;\mathbb{R}^{3})$ and thus $\mathcal{D}_{h}(y_{n},y)\to 0$ by dominated convergence. (b) On the other hand, assume that $\mathcal{D}_{h}(y_{n},y)\to 0$. Item (ii) yields that a subsequence (not relabeled) of $(y_{n})_{n}$ converges weakly in $W^{2,p}(\Omega;\mathbb{R}^{3})$ to some $\tilde{y}$. By (a) we find $\mathcal{D}_{h}(y_{n},\tilde{y})\to 0$. The triangle inequality, see (2.2)(iii), then shows $\mathcal{D}_{h}(y,\tilde{y})\leq\lim_{n\to\infty}(\mathcal{D}_{h}(y,y_{n})+\mathcal{D}_{h}(y_{n},\tilde{y}))=0$. This yields $y=\tilde{y}$ by Lemma 4.4, and therefore $y_{n}\rightharpoonup y$ weakly in $W^{2,p}(\Omega;\mathbb{R}^{3})$. As the limit is independent of the subsequence, the convergence actually holds for the whole sequence.

The proof of (iv) follows again by the dominated convergence theorem and the convexity of $P$, see (2.4)(ii).

Finally, we show (i). Apart from the positivity, all properties of a metric follow directly from (2.2) and (2.15). The positivity has has been addressed in Lemma 4.4. It therefore remains to show that $(\mathscr{S}^{M}_{h},\mathcal{D}_{h})$ is complete. Let (y_{k})_{k}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\subset\mathscr{S}_{h}^{M}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0} be a Cauchy sequence with respect to $\mathcal{D}_{h}$. By (ii) and (iii) we find $y\in W^{2,p}(\Omega;\mathbb{R}^{3})$ and a subsequence (not relabeled) such that $\lim_{k\to\infty}\mathcal{D}_{h}(y_{k},y)=0$. By (iv) and the trace theorem we get $y\in\mathscr{S}_{h}^{M}$. The fact that $(y_{k})_{k}$ is a Cauchy sequence now implies that the whole sequence $y_{k}$ converges to $y$ with respect to $\mathcal{D}_{h}$. This concludes the proof. ∎

Similar properties can be derived in the 2D setting. Recall the definition of $\phi_{0}$ and $\mathcal{D}_{0}$ in (2.23) and (2.2), respectively. For convenience, we will use the notations

[TABLE]

Lemma 4.6** (Properties of $({\mathscr{S}}_{0},{\mathcal{D}}_{0})$ and ${\phi}_{0}$).**

We have:

• (i)

$({\mathscr{S}}_{0},{\mathcal{D}}_{0})$* is a complete metric space.*

• (ii)

Compactness: If $(u_{n},v_{n})_{n}\subset{\mathscr{S}}_{0}$ is a sequence with $\sup_{n}{\phi}_{0}(u_{n},v_{n})<+\infty$, then $(u_{n},v_{n})_{n}$ is bounded in $W^{1,2}(S;\mathbb{R}^{2})\times W^{2,2}(S)$.

• (iii)

Topologies: The metrics ${\mathcal{D}}_{0}$ and $d((u,v),(u^{\prime},v^{\prime})):=\|u-u^{\prime}\|_{W^{1,2}(S)}+\|v-v^{\prime}\|_{W^{2,2}(S)}$ are equivalent. In particular, $\|v-v^{\prime}\|_{W^{2,2}(S)}\leq C{\mathcal{D}}_{0}((u,v),(u^{\prime},v^{\prime}))$ for $C=C(S)>0$.

• (iv)

Continuity: ${\mathcal{D}}_{0}((u_{n},v_{n}),(u,v))\to 0$    $\Rightarrow$    $\lim_{n\to\infty}{\phi}_{0}(u_{n},v_{n})={\phi}_{0}(u,v)$.

Proof.

We first prove (ii). By Lemma 4.1(ii) and the triangle inequality we find

[TABLE]

for a universal constant $C>0$. This together with Korn’s and Poincaré’s inequality, and the fact that $u=\hat{u}$ on $\partial S$ gives

[TABLE]

Now consider $(u,v)\in{\mathscr{S}}_{0}$ with ${\phi}_{0}(u,v)\leq M$ for $M>0$. Then by Lemma 4.1(ii) and (2.23) we find $\|(\nabla^{\prime})^{2}v\|_{L^{2}(S)}\leq C$, where the constant depends on $M$. Since $v=\hat{v}$ and $\nabla^{\prime}v=\nabla^{\prime}\hat{v}$ on $\partial S$, by an argumentation similar to (4.2) we also get

[TABLE]

where the constant additionally depends on $\|\hat{v}\|_{W^{2,2}(S)}$. Then ${\phi}_{0}(u,v)\leq M$ together with (4.2) and the embedding $W^{2,2}\hookrightarrow W^{1,4}$ (in dimension two) yields $\|u\|_{W^{1,2}(S)}\leq C$, where the constant also depends on $\|\hat{u}\|_{W^{1,2}(S)}$. This shows property (ii).

We now address (iii). As a preparation, consider $(u_{0},v_{0}),(u_{1},v_{1})\in{\mathscr{S}}_{0}$. Arguing similarly as before, using Lemma 4.1(ii) and Poincaré’s inequality, we find

[TABLE]

Here, we used that $v_{0}$ and $v_{1}$ as well as their first derivatives coincide on $\partial S$. Repeating the argumentation leading to (4.2) we find

[TABLE]

Here, in the first step, we used Korn’s and Poincaré’s inequality. In the second step, we applied Lemma 4.1(ii) and the triangle inequality.

We now suppose that $\mathcal{D}_{0}((u_{n},v_{n}),(u,v))\to 0$ (see (2.2)). Then $v_{n}\to v$ strongly in $W^{2,2}(S)$ by (4.13). This also implies $v_{n}\to v$ in $W^{1,4}(S)$ and thus, in view of (4.14), we find that $u_{n}\to u$ strongly in $W^{1,2}(S;\mathbb{R}^{2})$. Therefore, the sequence converges with respect to the metric $d$ defined in the statement of (iii). Conversely, given a sequence $(u_{n},v_{n})_{n}$ with $u_{n}\to u$ strongly in $W^{1,2}(S;\mathbb{R}^{2})$ and $v_{n}\to v$ in $W^{2,2}(S)$, we also get $v_{n}\to v$ in $W^{1,4}(S)$. We observe that $\mathcal{D}_{0}$ is continuous with respect to this convergence, i.e., ${\mathcal{D}}_{0}((u_{n},v_{n}),(u,v))\to 0$. This yields the equivalence of the metrics, and (4.13) shows the second part of (iii).

The proof of (iv) is similar, noting that also ${\phi}_{0}$ is continuous with respect to the topology induced by $\mathcal{D}_{0}$, see (2.23).

We finally prove (i). The positivity and the completeness follow from (iii). Thus, the only thing left to show is the triangle inequality. It can be derived by an elementary computation, using that ${\mathcal{D}}^{2}_{0}$ is the sum of two quadratic forms. We omit the details. ∎

Remark 4.7**.**

For later purposes, we remark that the proof of (ii) can be generalized: a similar argument shows that for given $(\tilde{u},\tilde{v})\in{\mathscr{S}}_{0}$ and $v\in W^{2,2}(S)$ we get

[TABLE]

where $C$ depends on $M$, $\hat{u}$ and $\hat{v}$. This follows by repeating the argument in (4.2)-(4.12) and using \|B(\tilde{v},{v})\|_{L^{2}(S)}\leq C\|\tilde{v}\|_{W^{1,4}(S)}\|v\|_{W^{1,4}(S)}\leq C\|\tilde{v}\|_{W^{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}2,2}(S)}\|v\|_{W^{2,2}(S)}.

4.3. Generalized geodesics and properties of slopes in the 2D setting

In this section, we first derive convexity properties for the energy and the dissipation distance in the 2D setting along generalized geodesics. (We refer to [35, Section 3.2, Section 3.4] for a discussion about generalized geodesics in a related problem.) Afterwards, we derive fundamental properties for the local slope which will be instrumental to use the theory in [3]. For $M>0$, we define the sublevel sets ${\mathscr{S}}_{0}^{M}=\{(u,v)\in{\mathscr{S}}_{0}:\ {\phi}_{0}(u,v)\leq M\}$.

Lemma 4.8** (Convexity and generalized geodesics in the 2D setting).**

Let $M>0$. Then there exist smooth functions $\Phi^{1},\Phi^{2}_{M}:[0,\infty)\to[0,\infty)$ satisfying $\lim_{t\to 0}\Phi^{1}(t)/t=1$ and $\lim_{t\to 0}\Phi^{2}_{M}(t)/t=0$ such that for all $(u_{0},v_{0})\in{\mathscr{S}}_{0}^{M}$ and all $(u_{1},v_{1})\in{\mathscr{S}}_{0}$ we have

[TABLE]

where $u_{s}:=(1-s)u_{0}+su_{1}$ and $v_{s}:=(1-s)v_{0}+sv_{1}$, $s\in[0,1]$.

Proof.

For convenience, we first introduce some abbreviations and provide some preliminary estimates. We let $D=\mathcal{D}_{0}((u_{0},v_{0}),(u_{1},v_{1}))$. We use the notation defined in (4.10) and also set $B_{\rm diff}=B(v_{0}-v_{1},v_{0}-v_{1})$. Particularly, by Lemma 4.6(iii) and a Sobolev embedding we observe that

[TABLE]

For brevity, we also introduce

[TABLE]

for $s\in[0,1]$. (The notation $G_{0}$ is borrowed from [26], see also Lemma 5.3 below.) With the definition of $\mathcal{D}_{0}$ in (2.2) and Lemma 4.1(ii) we get

[TABLE]

Similarly, we observe by (2.23) (with $f\equiv 0$), Lemma 4.1(ii), and the fact that $(u_{0},v_{0})\in\mathscr{S}^{M}_{0}$

[TABLE]

We now start with the proof of (i). First, we observe

[TABLE]

We will show that there exists $C>0$ independent of $s$ such that

[TABLE]

for $s\in[0,1]$. Then recalling the definition of $\mathcal{D}_{0}$ in (2.2), (i) follows for the function $\Phi^{1}(t)=\sqrt{t^{2}+Ct^{3}+Ct^{4}}$. To show (4.19), recalling (4.10), we obtain by an elementary expansion

[TABLE]

where the last equality follows from the definition of $B_{\rm diff}$. By recalling (4.16) this implies

[TABLE]

Recalling also $\mathbb{C}^{2}_{D}$ in (2.19), an expansion and the Cauchy-Schwartz inequality then yield

[TABLE]

By using (4.15) and (4.17) we now see that (4.19) holds. This concludes the proof of (i).

Recall the definition of $\phi_{0}$ in (2.23). We show (ii) for the function \Phi^{2}_{M}(t)=C\sqrt{M}t^{2}+\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}Ct^{3}+\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}Ct^{4} for some $C>0$. Due to convexity of s\mapsto\int_{S}\frac{1}{24}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}Q^{3}_{W}(\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}(\nabla^{\prime})^{2}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}v_{s}), it suffices to show

[TABLE]

In view of (2.19) and (4.20), an elementary expansion yields

[TABLE]

Then taking the integral we obtain

[TABLE]

By using $\|G^{1}_{0}\|_{L^{2}(S)}\leq\|G^{0}_{0}\|_{L^{2}(S)}+\|G^{1}_{0}-G^{0}_{0}\|_{L^{2}(S)}$, (4.15), and (4.17)-(4.18) we get (4.21). This concludes the proof of (ii). ∎

Now we derive representations and properties of the local slope corresponding to ${\phi}_{0}$. Recall the notation ${\mathscr{S}}_{0}^{M}=\{(u,v)\in{\mathscr{S}}_{0}:\ {\phi}_{0}(u,v)\leq M\}$.

Lemma 4.9** (Local slope in the 2D setting).**

Let $M>0$. The local slope for the energy ${\phi}_{0}$ admits the representation

[TABLE]

for all $(u,v)\in{\mathscr{S}}^{M}_{0}$, where $\Phi^{1}$ and $\Phi^{2}_{M}$ are the functions from Lemma 4.8. The slope is a strong upper gradient for ${\phi}_{0}$.

*Proof. *We follow the lines of the proofs of Theorem 2.4.9 and Corollary 2.4.10 in [3]. Let $M>0$ and $(u,v)\in\mathscr{S}_{0}^{M}$. Let $\Phi^{1},\Phi^{2}_{M}$ be the functions from Lemma 4.8, and recall that $\lim_{t\to 0}\Phi^{1}(t)/t=1$ and $\lim_{t\to 0}\Phi^{2}_{M}(t)/t=0$. We recall the definition of the local slope in Definition 3.1 and obtain

[TABLE]

where the supremum is taken over functions in $\mathscr{S}_{0}$. In the second equality we used that $(u^{\prime},v^{\prime})\to(u,v)$ means ${\mathcal{D}}_{0}((u,v),(u^{\prime},v^{\prime}))\to 0$, and the fact that $\lim_{t\to 0}\Phi^{1}(t)/t=1$ and $\lim_{t\to 0}\Phi^{2}_{M}(t)/t=0$.

To see the other inequality, we fix $(u^{\prime},v^{\prime})\neq(u,v)$. It is not restrictive to suppose that

[TABLE]

Define $u_{s}=(1-s)u+su^{\prime}$ and $v_{s}=(1-s)v+sv^{\prime}$ for $s\in[0,1]$. By Lemma 4.8 with $(u_{0},v_{0})=(u,v)$ and $(u_{1},v_{1})=(u^{\prime},v^{\prime})$ we get

[TABLE]

Note $(u_{s},v_{s})\to(u,s)$ as $s\to 0$ with respect to the topology induced by ${\mathcal{D}}_{0}$, see Lemma 4.6(iii). Therefore,

[TABLE]

The claim now follows by taking the supremum with respect to $(u^{\prime},v^{\prime})$.

It remains to show that $|\partial{\phi}_{0}|_{{\mathcal{D}}_{0}}$ is a strong upper gradient. Let us first recall from [3, Lemma 1.2.5] that in the complete metric space $({\mathscr{S}}_{0},{\mathcal{D}}_{0})$, the global slope

[TABLE]

is a strong upper gradient for ${\phi}_{0}$ since ${\phi}_{0}$ is ${\mathcal{D}}_{0}$-lower semicontinuous (see Lemma 4.6(iv)). Moreover, [3, Lemma 1.2.5] also states that $|\partial{\phi}_{0}|_{{\mathcal{D}}_{0}}$ is a weak upper gradient for ${\phi}_{0}$ in the sense of [3, Definition 1.2.2]. We do not repeat the definition of weak upper gradients, but only mention that weak upper gradients are also strong upper gradients if for each absolutely continuous curve $z:(a,b)\to{\mathscr{S}}_{0}$ with respect to $\mathcal{D}_{0}$ satisfying $|\partial{\phi}_{0}|_{{\mathcal{D}}_{0}}(z)|z^{\prime}|_{{\mathcal{D}}_{0}}\in L^{1}(a,b)$, the function ${\phi}_{0}\circ z$ is absolutely continuous.

To check that ${\phi}_{0}\circ z$ is absolutely continuous, we first extend the curve $z$ continuously to $[a,b]$ and introduce the compact metric space $\mathscr{S}^{\prime}=z([a,b])$ with the metric induced by $\mathcal{D}_{0}$. Choose $M>0$ such that $\mathscr{S}^{\prime}\subset\mathscr{S}_{0}^{M}$, which is possible due to (2.23), (2.2), and the fact that ${\rm diam}(\mathscr{S}^{\prime}):=\sup_{s,t\in[a,b]}\mathcal{D}_{0}(z(s),z(t))<+\infty$. Let $\mathcal{G}^{\prime}_{{\phi}_{0}}$ be the global slope as introduced in (4.22) with respect to $\mathscr{S}^{\prime}$ (instead of ${\mathscr{S}}_{0}$). The representation of the local slope implies

[TABLE]

for all (u,v)\in\mathscr{S}^{\prime}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\subset\mathscr{S}_{0}^{M}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}, where

[TABLE]

In particular, since $|\partial{\phi}_{0}|_{{\mathcal{D}}_{0}}(z)|z^{\prime}|_{{\mathcal{D}}_{0}}\in L^{1}(a,b)$, it follows \color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\mathcal{G}^{\prime}_{{\phi}_{0}}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}(z)|z^{\prime}|_{{\mathcal{D}}_{0}}\in L^{1}(a,b). As discussed above, $\mathcal{G}^{\prime}_{{\phi}_{0}}$ is a strong upper gradient. Thus, we indeed get that ${\phi}_{0}\circ z$ is absolutely continuous, see Definition 3.1. $\Box$

5. Relation between 3D and 2D setting

In this section we consider sequences $(y^{h})_{h}$ with $y^{h}\in\mathscr{S}^{M}_{h}=\{y\in\mathscr{S}_{h}:\ \phi_{h}(y)\leq M\}$. In Section 5.1 we derive compactness properties and identify suitable limiting objects in terms of scaled in-plane and out-of-plane displacement fields. Afterwards, in Section 5.2 we derive a $\Gamma$-convergence result and prove lower semicontinuity for the local slopes along the passage from the 3D to the 2D setting. As in Section 4, $C>0$ indicates a generic constant which is independent of $h$, but may depend on $M$, $S$, $p>3$, $\alpha\in(0,1)$, on the constants in (2.3), (2.4), (2.2), and the boundary data $\hat{u}$ and $\hat{v}$.

5.1. Compactness and identification of limiting strain

Given $y^{h}\in\mathscr{S}_{h}^{M}$, we define the averaged, scaled in-plane and out-of-plane displacement fields by

[TABLE]

where $I=(-\frac{1}{2},\frac{1}{2})$. By recalling the boundary conditions (2.2) and (2.20) we observe $(u^{h},v^{h})\in{\mathscr{S}}_{0}$.

In view of the rigidity estimate in Lemma 4.2, a deformation $y^{h}\in\mathscr{S}_{h}^{M}$ is close to the affine map $(x^{\prime},x_{3})\mapsto(x^{\prime},hx_{3})$ and thus the displacements defined in (5.1) are suitably controlled. More specifically, we have the following compactness result.

Lemma 5.1** (Compactness for displacements).**

Consider a sequence $(y^{h})_{h}$ with $y^{h}\in\mathscr{S}^{M}_{h}$ for all $h$. Then up to passing to a subsequence (not relabeled) we find $(u,v)\in{\mathscr{S}}_{0}$ such that

[TABLE]

For the proof we refer to [26, Lemma 1] and [30, Lemma 13].

Remark 5.2** (Ansatz for recovery sequences).**

Note that the convergence results in Lemma 5.1 are compatible with the ansatz

[TABLE]

for $(u,v)\in{\mathscr{S}}_{0}$ and $d^{h}\in W^{2,\infty}_{0}(S;\mathbb{R}^{3})$, which additionally satisfy the regularity

[TABLE]

We observe that the boundary conditions (2.2) are satisfied, i.e., $y^{h}\in\mathscr{S}_{h}$. For later purposes, we note that the scaled gradient is given by

[TABLE]

We also note that the (scaled) second gradient has the form

[TABLE]

and $\frac{1}{h^{2}}y_{,33}^{h}=0$. By (5.4) this implies \|(\nabla^{\prime})^{2}y^{h}\|_{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\infty}\leq Ch, $\|\nabla^{\prime}(\frac{1}{h}y^{h}_{,3})\|_{\infty}\leq Ch$ and $y_{,33}^{h}=0$ for a constant $C$ depending on $C^{\prime}$. Consequently, by using $\alpha<1$ and (2.4)(iii) we compute

[TABLE]

Later in Theorem 5.6 and Theorem 5.10 we will use this ansatz to construct recovery sequences for the energy and the dissipations. We remark that, on the one hand, our ansatz is slightly simpler than the one in [26, (119)] since we consider a model with zero Poisson’s ratio in $e_{3}$ direction, see Remark 5.7 and Remark 5.11 for more details. On the other hand, some additional effort arises from the fact that our 3D model contains second gradient terms. **

In (4.4) we have already introduced the quantity $G^{h}(y^{h})=h^{-2}(R(y^{h})^{\top}\nabla_{h}y^{h}-\mathbf{Id})$ which essentially controls the energy of $y^{h}$. By (4.2)(i) we get that $G^{h}(y^{h})$ converges weakly in $L^{2}$ (up to a subsequence) to some $G$. The next result shows that the symmetric part of the in-plane components of $G$ can be identified in terms of the in-plane displacement $u$ and the out-of-plane displacement $v$.

Lemma 5.3** (Identification of scaled limiting strain).**

Consider the setting of Lemma 5.1. Let $R(y^{h}):S\to SO(3)$ be the corresponding mappings from Lemma 4.2. Then

[TABLE]

Moreover, we find $G\in L^{2}(\Omega;\mathbb{R}^{3\times 3})$ such that

[TABLE]

The $2\times 2$ submatrix $G^{\prime\prime}$ given by $G^{\prime\prime}_{ij}=G_{ij}$ for $i,j\in\{1,2\}$ satisfies

[TABLE]

with

[TABLE]

Here, we again use the notation $e(u)={\rm sym}(\nabla^{\prime}u)$, see (4.10). For the proof we refer to [26, Lemma 1, Lemma 2] (see also [30, Lemma 16]) from which we also adopted the notation for ease of readability. Note that, in fact, an inspection of the proof shows that the mappings $R(y^{h})$ introduced there can be chosen as the ones from Lemma 4.2. In particular, the result shows that the relevant components of $G$ are affine in the thickness variable $x_{3}$. In the following, we will also use the notation

[TABLE]

Note that $G(u,v)\in L^{2}(\Omega;\mathbb{R}^{2\times 2}_{\rm sym})$.

Remark 5.4**.**

Using (5.8) and the fact that $Q_{W}^{2}$ only depends on the symmetric part of matrices (see Lemma 4.1(ii)), the energy ${\phi}_{0}(u,v)$ defined in (2.23) (with $f\equiv 0$) can be written as

[TABLE]

In view of the definition of ${\mathcal{D}}_{0}$ in (2.2), a similar computation yields

[TABLE]

In the following we will mainly use the representation in terms of an integral over $\Omega$ as it is convenient for many proofs. Only at the very end we will pass to an integral over $S$ as indicated in (5.4)-(5.10). **

The following lemma about strong convergence of strain differences will be instrumental in the proof of the lower semicontinuity of the local slopes. We introduce the notation $W^{2,p}_{0,\partial S}(\Omega;\mathbb{R}^{3})=\{y\in W^{2,p}(\Omega;\mathbb{R}^{3}):y(x^{\prime},x_{3})=0\text{ for }x^{\prime}\in\partial S,\ x_{3}\in I\}$, where $I=(-\frac{1}{2},\frac{1}{2})$. We also define ${\rm skew}(F)=\frac{1}{2}(F-F^{\top})$ for $F\in\mathbb{R}^{3\times 3}$.

Lemma 5.5** (Strong convergence of strain differences).**

Let $M>0$. Let $(y^{h})_{h}$ be a sequence with $y^{h}\in\mathscr{S}^{M}_{h}$ and let (z^{h}_{s})_{s,h}\subset W^{2,p}_{0,\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\partial S}(\Omega;\mathbb{R}^{3}), $h>0$, $s\in(0,1)$, be functions such that

[TABLE]

Then the following holds for a subsequence of $(y^{h})_{h}$ (not relabeled):

(a) For all $h$ sufficiently small, $w^{h}_{s}:=y^{h}+z^{h}_{s}$ lies in $\mathscr{S}^{M^{\prime}}_{h}$ for some $M^{\prime}=M^{\prime}(M)>0$.

(b) Let $(G^{h}(y^{h}))_{h}$, $(G^{h}(w^{h}_{s}))_{h}$ be the sequences in Lemma 5.3 and let $G_{y}$, $G^{s}_{w}$ be their limits. Then there exists $C=C(M)>0$ and $\rho(h)$ with $\rho(h)\to 0$ as $h\to 0$ such that

[TABLE]

(c) Let $(u,v)$ and $(\bar{u}_{s},\bar{v}_{s})$ be limits corresponding to $y^{h}$ and $w^{h}_{s}$, respectively, as given in Lemma 5.1. Then ${\rm sym}(G^{s}_{w}-G_{y})\,e_{3}=E^{s}\,e_{3}-\frac{1}{2}(|\nabla^{\prime}{v}|^{2}-|\nabla^{\prime}\bar{v}_{s}|^{2})e_{3}$ a.e. in $\Omega$.

We note that weak convergence of the strain differences is already guaranteed by Lemma 5.3. The important point here is that we actually obtain strong convergence with linear control in terms of $s$, see (b). Moreover, (c) provides a characterization of the out-of-plane components of the limiting strain difference. Later in the proof of Theorem 5.10 we will use this lemma to construct competitor sequences for the local slope in the 3D setting.

Proof.

Let $R(y^{h})$ be the $SO(3)$-valued mappings given by Lemma 4.2. For brevity, we introduce notations for the symmetric and skew-symmetric part of $\nabla_{h}z^{h}_{s}$ by

[TABLE]

The crucial point is to find a suitable $SO(3)$-valued mapping $R(w^{h}_{s})$ associated to $w^{h}_{s}=y^{h}+z^{h}_{s}$ satisfying the properties stated in Lemma 4.2 (Step 1). Once $R(w^{h}_{s})$ has been defined, we can prove properties (a)-(c) (Step 2).

Step 1: Definition of $R(w^{h}_{s})$. We first define

[TABLE]

on $S$. By (4.2)(iii) and (5.5)(i) we can check that $\tilde{R}$ is in a small tubular neighborhood of $SO(3)$ and satisfies $\|\nabla^{\prime}\tilde{R}\|^{2}_{L^{2}(S)}\leq Ch^{2}$. We let $R(w^{h}_{s})\in W^{1,2}(S;SO(3))$ be the map obtained from $\tilde{R}$ by nearest-point projection on $SO(3)$, see [26, Remark 5] for a similar argument. By (5.5)(i) and ${\accentset{\rule{4.20004pt}{0.8pt}}{F}}(z^{h}_{s})(x^{\prime})\in\mathbb{R}^{3\times 3}_{\rm skew}$ for all $x^{\prime}\in S$, it is elementary to check that

[TABLE]

Indeed, this follows from the fact that |((\mathbf{Id}+A-\frac{1}{2}A^{\top}A)^{\top}(\mathbf{Id}+A-\frac{1}{2}A^{\top}A))^{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}1/2\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}}-\mathbf{Id}|\leq C|A|^{3} for all $A\in\mathbb{R}^{3\times 3}_{\rm skew}$. Along with (5.5)(iii) we thus get

[TABLE]

We now check that $R(w^{h}_{s})$ satisfies the properties stated in Lemma 4.2, see (4.2). First, $\|\nabla^{\prime}\tilde{R}\|^{2}_{L^{2}(S)}\leq Ch^{2}$ implies $\|\nabla^{\prime}R(w^{h}_{s})\|^{2}_{L^{2}(S)}\leq Ch^{2}$. Moreover, (5.5)(i), (5.12), and the fact that $R(y^{h})$ satisfies (4.2)(iv) shows $\|R(w^{h}_{s})-\mathbf{Id}\|_{L^{q}(S)}\leq C_{q}h$ for $q\in[1,\infty)$. In a similar fashion, (4.2)(vi) and $\alpha<1$ yield $\|R(w^{h}_{s})-\mathbf{Id}\|_{L^{\infty}(S)}\leq Ch^{\alpha}$. It thus remains to check (4.2)(i), i.e., that

[TABLE]

holds. For notational convenience, we denote by $\omega^{h}_{i}\in L^{2}(\Omega;\mathbb{R}^{3\times 3})$, $i\in\mathbb{N}$, (generic) matrix valued functions whose $L^{2}$-norm is controlled in terms of a constant independent of $h$ and $s$. By (4.2)(v) (applied for $y^{h}$), (5.5)(i), and (5.12) we find

[TABLE]

We now consider the asymptotic expansion of $R(y^{h})^{\top}(\nabla_{h}y^{h}+\nabla_{h}z^{h}_{s})$ in terms of $h$: in view of (5.5)(ii), $\nabla_{h}z^{h}_{s}=E(z^{h}_{s})+F(z^{h}_{s})$, and $\|R(y^{h})-\mathbf{Id}\|_{L^{\infty}(\Omega)}\leq Ch^{\alpha}$ (see (4.2)(vi)) we find

[TABLE]

In a similar fashion, by using (4.2)(i),(iv) (for $y^{h}$) and (5.5)(i),(ii), we compute

[TABLE]

as well as

[TABLE]

By inserting these three estimates in (5.14) and using (5.5)(i) we find

[TABLE]

where in the last step we used $F(z^{h}_{s})^{\top}+F(z^{h}_{s})=0$. We now check that (5.13) holds. Indeed, it suffices to use (5.15), (5.5)(i),(ii), $\omega_{6}^{h}\in L^{2}(\Omega;\mathbb{R}^{3\times 3})$, and the fact that (4.2)(i),(iv) holds for $y^{h}$. In conclusion, this implies that the mapping $R(w^{h}_{s})$ satisfies the properties stated in Lemma 4.2.

Step 2: Proof of the statement. We are now in a position to prove the statement.

(a) By (4.2)(v) and (5.5)(i), $\nabla_{h}w^{h}_{s}$ is in a neighborhood of $\mathbf{Id}$. Thus, by (2.3)(iii) there holds $\int_{\Omega}W(\nabla_{h}w^{h}_{s})\leq C\int_{\Omega}\operatorname{dist}^{2}(\nabla_{h}w^{h}_{s},SO(3))$, and then by (5.13) we get $\int_{\Omega}W(\nabla_{h}w^{h}_{s})\leq M^{\prime}h^{4}$ for some $M^{\prime}$ sufficiently large (depending on $M$). Moreover, from (2.4)(iii) and the triangle inequality we get $P(\nabla^{2}_{h}w^{h}_{s})\leq CP(\nabla^{2}_{h}y^{h})+CP(\nabla^{2}_{h}z^{h}_{s})$. Therefore, by possibly passing to a larger $M^{\prime}$, we obtain $h^{-\alpha p}\int_{\Omega}P(\nabla^{2}_{h}w^{h}_{s})\leq M^{\prime}$ by (2.4)(iii), (5.5)(i), $\alpha<1$, and the fact that $y^{h}\in\mathscr{S}_{h}^{M}$. Summarizing, since $z^{h}_{s}\in W^{2,p}_{0,\partial S}(\Omega;\mathbb{R}^{3})$ and thus $w^{h}_{s}$ also satisfies the boundary conditions (2.2), we have shown that $w^{h}_{s}\in\mathscr{S}^{M^{\prime}}_{h}$ for some $M^{\prime}=M^{\prime}(M)>0$ independent of $h$. In particular, this implies that the statement of Lemma 4.2 holds for $w^{h}_{s}$ with $R(w^{h}_{s})$ as defined in Step 1.

(b) By $(u,v)$ we denote the limit corresponding to $y^{h}$ as given in Lemma 5.1. Recalling the definition $G^{h}(y^{h})=h^{-2}(R(y^{h})^{\top}\nabla_{h}y^{h}-\mathbf{Id})$ we find by (5.7), (5.5)(iv), and (5.15) that

[TABLE]

where ${\rho}(h)\to 0$ as $\rho\to 0$. By $G^{s}_{w}$ and $G_{y}$ we denote the weak $L^{2}$-limits of $G^{h}(w^{h}_{s})$ and $G^{h}(y^{h})$, respectively, which exist by Lemma 5.3. Then (5.16) implies

[TABLE]

and the first part of (b) holds. The second part of (b) is a consequence of (4.2)(iv) (for $y^{h}$), (5.5)(i),(ii), (5.15), and the fact that $\omega^{h}_{6}\in L^{2}(\Omega;\mathbb{R}^{3\times 3})$.

(c) By $(\bar{u}_{s},\bar{v}_{s})$ we denote the limit corresponding to $w^{h}_{s}$ as given in Lemma 5.1. By (5.7) (for $w^{h}_{s}$ and $y^{h}$, respectively), (5.5)(i),(iv), and (5.12) we observe that pointwise a.e. in $\Omega$ there holds

[TABLE]

Then by (5.17) and an expansion we get

[TABLE]

where in the last step we used that $A(v)\in\mathbb{R}^{3\times 3}_{\rm skew}$ pointwise a.e. in $\Omega$ and thus $A(v)^{\top}A(v)=-(A(v))^{2}$. Therefore, recalling the definition of $A(v)$ in (5.7) we obtain

[TABLE]

This concludes the proof. ∎

5.2. $\Gamma$-convergence and lower semicontinuity of slopes

In this section we establish a $\Gamma$-convergence result for the energies which is essentially proved in [26, 30]. However, some adaptions are necessary due to the second order perturbation $P$ in the energy. For an exhaustive treatment of $\Gamma$-convergence we refer the reader to [18]. Afterwards, we prove lower semicontinuity for the dissipation distances and the local slopes which is fundamental to use the theory in [38, 43] (see also Section 3.2, in particular (3.4) and (3.6)).

We first fix a topology for the convergence of the scaled in-plane and out-of-plane displacements induced by the compactness result in Section 5.1, see (5.1). We define mappings \pi_{h}:\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\mathscr{S}_{h}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\to{\mathscr{S}}_{0} by $\pi_{h}(y^{h})=(u^{h},v^{h})$ for each $y^{h}\in\mathscr{S}_{h}$, where $u^{h}$ and $v^{h}$ are the scaled in-plane and out-of-plane displacements corresponding to $y^{h}$ (see (5.1)). We say

[TABLE]

We also say $y^{h}\stackrel{{\scriptstyle\pi\sigma}}{{\to}}(u,v)$ if $\pi_{h}(y^{h})\stackrel{{\scriptstyle\sigma}}{{\to}}(u,v)$, cf. (3.3). Recall the definitions (2.14) and (2.23).

Theorem 5.6** ($\Gamma$-convergence).**

*Suppose that $W$ and $P$ satisfy the assumptions (2.3) and (2.4). Then $\phi_{h}$ converges to ${\phi}_{0}$ in the sense of $\Gamma$-convergence. More precisely,

(i) (Lower bound) For all $(u,v)\in{\mathscr{S}}_{0}$ and all sequences $(y^{h})_{h}$ such that $y^{h}\stackrel{{\scriptstyle\pi\sigma}}{{\to}}(u,v)$ we find

[TABLE]

(ii) (Optimality of lower bound) For all $(u,v)\in{\mathscr{S}}_{0}$ there exists a sequence $(y^{h})_{h}$, $y^{h}\in\mathscr{S}_{h}$ for all $h$, such that $y^{h}\stackrel{{\scriptstyle\pi\sigma}}{{\to}}(u,v)$ and

[TABLE]

Proof.

(i) The result is essentially proved in [30]. We give here the main steps for convenience of the reader. By the representation (5.4) and the fact that $P$ is nonnegative, for the lower bound it suffices to prove

[TABLE]

for all sequences $(y^{h})_{h}$ with $y^{h}\stackrel{{\scriptstyle\pi\sigma}}{{\to}}(u,v)$. We may suppose that $\liminf_{h\to 0}\phi_{h}(y^{h})$ is finite as otherwise there is nothing to prove. Thus, we can assume that $y^{h}\in\mathscr{S}^{M}_{h}$ for $M>0$ large enough. By Lemma 4.3(iii) we find

[TABLE]

Moreover, Lemma 5.1 and Lemma 5.3 imply that, possibly passing to a subsequence (not relabeled), $G^{h}(y^{h})\rightharpoonup G$ in $L^{2}(\Omega;\mathbb{R}^{3\times 3})$, where the $2\times 2$ submatrix $G^{\prime\prime}$ satisfies \color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{\rm sym}(G^{\prime\prime})\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}=G(u,v), see (5.8). This along with the lower semicontinuity in $L^{2}$ (note that $Q_{W}^{3}$ is a positive semidefinite quadratic form) yields $\liminf_{h\to 0}h^{-4}\int_{\Omega}W(\nabla_{h}y^{h})\geq\int_{\Omega}\frac{1}{2}Q^{3}_{W}(G)$. The fact that \color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{\rm sym}(G^{\prime\prime})\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}=G(u,v) and (2.17) give the desired lower bound (5.19).

(ii) By a general approximation argument in the theory of $\Gamma$-convergence it suffices to establish the optimality of the lower bound only for sufficiently smooth mappings, precisely for $u\in W^{2,\infty}(S;\mathbb{R}^{2})$ and $v\in W^{3,\infty}(S)$ with

[TABLE]

Indeed, a general $(u,v)$ can be approximated strongly in $W^{1,2}(S;\mathbb{R}^{2})\times W^{2,2}(S)$ by such functions, see Lemma 5.8 below. Note that the limiting energy ${\phi}_{0}$ is continuous with respect to this topology, see Lemma 4.6(iii),(iv).

Let us now construct recovery sequences for $u\in W^{2,\infty}(S;\mathbb{R}^{2})$ and $v\in W^{3,\infty}(S)$ satisfying (5.20). Define $d=-\frac{1}{2}|\nabla^{\prime}v|^{2}e_{3}\in W^{2,\infty}(S;\mathbb{R}^{3})$ and let $(d^{h})_{h}\subset W^{2,\infty}_{0}(S;\mathbb{R}^{3})$ with $d^{h}\to d=-\frac{1}{2}|\nabla^{\prime}v|^{2}e_{3}$ in $L^{2}(S;\mathbb{R}^{3})$ and \sup_{h}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\sqrt{h}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\|d^{h}\|_{W^{2,\infty}(S)}<\infty. We take the ansatz for $y^{h}$ as given in Remark 5.2. In Remark 5.2 we have already discussed that this ansatz is compatible with the convergence $y^{h}\stackrel{{\scriptstyle\pi\sigma}}{{\to}}(u,v)$. By the representation of the scaled gradient in (5.5), an elementary computation yields for the nonlinear strain

[TABLE]

where we used $\sup_{h}\sqrt{h}\|d^{h}\|_{W^{2,\infty}(S)}<\infty$. Since there holds $d^{h}\to d=-\frac{1}{2}|\nabla^{\prime}v|^{2}e_{3}$ in $L^{2}(S;\mathbb{R}^{3})$ and \sup_{h}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\sqrt{h}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\|d^{h}\|_{L^{\infty}(\Omega)}<\infty, we get

[TABLE]

for functions $\omega^{h}:\Omega\to\mathbb{R}^{3\times 3}$ with $\|\omega^{h}\|_{L^{2}(\Omega)}\to 0$ and \sup_{h}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\sqrt{h}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\|\omega^{h}\|_{L^{\infty}(\Omega)}<\infty. Therefore,

[TABLE]

where $G(u,v)^{*}\in L^{2}(\Omega;\mathbb{R}^{3\times 3})$ denotes the mapping with $(G(u,v)^{*})_{ij}=(G(u,v))_{ij}$ for $1\leq i,j\leq 2$ and zero otherwise, see (5.8). Taking the square root, using the frame indifference of $W$, (2.3)(i), and a Taylor expansion, we derive (cf. also [30, Proposition 19])

[TABLE]

as $h\to 0$. Definition (2.17) (and the assumption that the minimum is attained for $a=0$) yield $Q_{W}^{3}(G(u,v)^{*})=Q_{W}^{2}(G(u,v))$. Then (5.4) implies $\frac{1}{h^{4}}\int_{\Omega}W(\nabla_{h}y^{h})\to{\phi}_{0}(u,v)$. The proof is now concluded by observing $\lim_{h\to 0}h^{-\alpha p}\int_{\Omega}P(\nabla_{h}^{2}y^{h})=0$, see (5.2). ∎

Remark 5.7**.**

We remark that the assumption on $Q_{W}^{2}$ is actually not needed at the expense of a more involved recovery sequence, see [26, equation (119)]. However, the assumption will be instrumental for the lower semicontinuity of slopes, see Theorem 5.10 and Remark 5.11.

In the previous proof we have used the following density result.

Lemma 5.8** (Density of smooth functions with same boundary conditions).**

For each $(u,v)\in{\mathscr{S}}_{0}$ we find sequences $(u^{h})_{h}\subset W^{2,\infty}(S;\mathbb{R}^{2})$ and $(v^{h})_{h}\subset W^{3,\infty}(S)$ such that

[TABLE]

Proof.

The proof is standard: we approximate $u-\hat{u}$ and $v-\hat{v}$ by smooth functions with compact support in $S$ and add $\hat{u}\in W^{2,\infty}(S;\mathbb{R}^{2})$, $\hat{v}\in W^{3,\infty}(S)$, respectively. ∎

We now proceed with the lower semicontinuity of the dissipation distances. Recall the definitions in (2.15) and (2.2).

Theorem 5.9** (Lower semicontinuity of dissipation distances).**

Suppose that $D$ satisfies the assumptions (2.2). Let $M>0$. Then for sequences $(y_{1}^{h})_{h}$ and $(y^{h}_{2})_{h}$, $y_{1}^{h},y_{2}^{h}\in\mathscr{S}^{M}_{h}$, with $y^{h}_{1}\stackrel{{\scriptstyle\pi\sigma}}{{\to}}(u_{1},v_{1})$ and $y^{h}_{2}\stackrel{{\scriptstyle\pi\sigma}}{{\to}}(u_{2},v_{2})$ we have

[TABLE]

Proof.

The argument is similar to the one in (5.19), see the proof of Theorem 5.6(i), with the difference that we employ Lemma 4.3(ii) in place of Lemma 4.3(iii) and (5.10) in place of (5.4). ∎

We close this section with the fundamental property that the local slopes are lower semicontinuous along the passage from the 3D to the 2D setting. As emphasized before, this is crucial for the application of the theory in [38, 43], see (3.6). Recall the definition of $Q_{W}^{2}$, $Q_{D}^{2}$ in (2.17)-(2.18). The fact that the minimum is attained for $a=0$ implies

[TABLE]

for all $F\in\mathbb{R}^{3\times 3}$, where $F^{\prime\prime}$ denotes the $2\times 2$ matrix with entries $F^{\prime\prime}_{ij}=F_{ij}$ for $1\leq i,j\leq 2$, and $F^{*}$ denotes the $3\times 3$ matrix with entries $F^{*}_{ij}=F_{ij}$ for $1\leq i,j\leq 2$, and zero otherwise.

Theorem 5.10** (Lower semicontinuity of slopes).**

For each sequence $(y^{h})_{h}$ with $y^{h}\in{\mathscr{S}}_{h}^{M}$ such that $y^{h}\stackrel{{\scriptstyle\pi\sigma}}{{\to}}(u,v)$ we have

[TABLE]

Proof.

We divide the proof into several steps. We first define approximations of $(u,v)$ which allow us to work with more regular functions (Step 1). We then construct competitor sequences $(w^{h}_{s})_{h,s}$ for the local slope in the 3D setting satisfying $w^{h}_{s}\to y^{h}$ as $s\to 0$ (Step 2). Afterwards, we identify the limiting strain of the sequences $(w^{h}_{s})_{h}$ (Step 3), and we prove the lower semicontinuity (Step 4). Some technical estimates are contained in Steps 5–7.

Step 1: Approximation. By Lemma 5.8, for $\varepsilon>0$ we fix $u_{\varepsilon}\in W^{2,\infty}(S;\mathbb{R}^{2})$ and $v_{\varepsilon}\in W^{3,\infty}(S)$ with $(u_{\varepsilon},v_{\varepsilon})\in{\mathscr{S}}_{0}$ and

[TABLE]

This approximation will be necessary to construct sufficiently regular competitor sequences for the local slope of the 3D setting.

We further fix $\tilde{u}\in W^{2,\infty}(S;\mathbb{R}^{2})$ and $\tilde{v}\in W^{3,\infty}(S)$ with $(\tilde{u},\tilde{v})\in{\mathscr{S}}_{0}$, and satisfying $\tilde{u}\neq u,u_{\varepsilon}$, $\tilde{v}\neq v,v_{\varepsilon}$. The pair $(\tilde{u},\tilde{v})$ will represent the competitor in the local slope of the 2D setting, see Lemma 4.9. Below in (5.2), we will see that by approximation it is enough to work with functions of this regularity. The convex combinations

[TABLE]

will be the starting point for the construction of competitor sequences $(w^{h}_{s})_{h,s}$ for the 3D setting. In the following, $\tilde{C},C_{\varepsilon}$ denote generic constant which may vary from line to line, where $\tilde{C}$ may depend on $\tilde{u},u,\tilde{v},v$, and $C_{\varepsilon}$ additionally on $\varepsilon$.

Step 2: Construction of competitor sequences $(w^{h}_{s})_{h,s}$. We choose recovery sequences $y_{\varepsilon}^{h},\tilde{y}_{s}^{h}$ related to $(u_{\varepsilon},v_{\varepsilon})$ and $(\tilde{u}_{s},\tilde{v}_{s})$, exactly as in the proof of Theorem 5.6(ii): define $d_{\varepsilon}=-\frac{1}{2}|\nabla^{\prime}v_{\varepsilon}|^{2}e_{3}$, $\tilde{d}_{s}=-\frac{1}{2}|\nabla^{\prime}\tilde{v}_{s}|^{2}e_{3}$ and let $(d^{h}_{\varepsilon})_{h},(\tilde{d}^{h}_{s})_{h}\subset W^{2,\infty}_{0}(S;\mathbb{R}^{3})$ be sequences with $d^{h}_{\varepsilon}\to d_{\varepsilon}$ and $\tilde{d}^{h}_{s}\to\tilde{d}_{s}$ in $L^{2}(S;\mathbb{R}^{3})$. In view of (5.23), this can be done in such a way that there holds

[TABLE]

where $\rho(h)$ depends on $v_{\varepsilon}$, $\tilde{v}$, and satisfies $\rho(h)\to 0$ as $h\to 0$.

We take the ansatz for $y^{h}_{\varepsilon},\tilde{y}_{s}^{h}$ as given in Remark 5.2 and observe that $y^{h}_{\varepsilon},\tilde{y}^{h}_{s}$ satisfy the boundary conditions, i.e., ${z}^{h}_{s}:=\tilde{y}_{s}^{h}-y_{\varepsilon}^{h}\in W^{2,p}_{0,\partial S}(\Omega;\mathbb{R}^{3})$. For $h>0$ small and $s\in[0,1]$, we define

[TABLE]

By (5.24), (5.5)-(5.2) (with $\tilde{u}_{s},u_{\varepsilon}$ and $\tilde{v}_{s},v_{\varepsilon}$ in place of $u$ and $v$, respectively) and the fact that $(\tilde{u}_{s}-u_{\varepsilon},\tilde{v}_{s}-v_{\varepsilon})=s(\tilde{u}-u_{\varepsilon},\tilde{v}-v_{\varepsilon})$ we see

[TABLE]

This shows that the assumptions (5.5)(i)-(iii) are satisfied for $(z^{h}_{s})_{s,h}$ (for a constant $M=M(C_{\varepsilon})$). From (5.5) and (5.24) we also get that (5.5)(iv) holds for suitable $E^{s}$ and $F^{s}$. In particular, by the definition of $\tilde{d}^{h}_{s}$ and $d^{h}_{\varepsilon}$ we observe that

[TABLE]

Then Lemma 5.5(a) implies $w^{h}_{s}\in\mathscr{S}^{M^{\prime}}_{h}$ for a constant $M^{\prime}>0$ sufficiently large depending on $\varepsilon$, but independent of $s,h$. Thus, by (5.3), (5.25), $\tilde{d}^{h}_{s}\to{d}^{h}_{\varepsilon}$ in $L^{2}(S;\mathbb{R}^{3})$ as $s\to 0$ (see (5.24)), and a compactness argument (see Lemma 4.5(ii)), one can check that

[TABLE]

Step 3: Identification of limiting strains. Since the ansatz for $(y_{\varepsilon}^{h})_{h}$ and $(\tilde{y}_{s}^{h})_{h}$ is compatible with the convergence results in Lemma 5.1, the convergence in (5.18) holds, i.e., the scaled displacement fields corresponding to $(y_{\varepsilon}^{h})_{h}$ and $(\tilde{y}_{s}^{h})_{h}$ converge to $(u_{\varepsilon},v_{\varepsilon})$ and $(\tilde{u}_{s},\tilde{v}_{s})$, respectively. Thus, in view of (5.25), the scaled displacement fields corresponding to $w^{h}_{s}$ converge to $(u-u_{\varepsilon}+\tilde{u}_{s},v-v_{\varepsilon}+\tilde{v}_{s})$. In the following, it will be convenient to work with the convex combination defined by

[TABLE]

In fact, by (5.23) we see that the scaled displacement fields corresponding to $w^{h}_{s}$ converge to $(\hat{u}_{s}^{\varepsilon},\hat{v}_{s}^{\varepsilon})$.

The limits of the mappings $G^{h}(y^{h})$ and $G^{h}(w^{h}_{s})$ given by Lemma 5.3 are denoted by $G_{y}$ and $G^{s}_{w}$, i.e, we have (up to a subsequence)

[TABLE]

where the $2\times 2$ submatrices $G_{y}^{\prime\prime}$ and $(G^{s}_{w})^{\prime\prime}$ satisfy

[TABLE]

respectively. (Recall notation (5.8).) Above we have checked that the assumptions (5.5) hold for $(z^{h}_{s})_{s,h}$. We can therefore apply Lemma 5.5(b) and obtain

[TABLE]

where $\rho_{\varepsilon}(h)$ depends on $\varepsilon$ and satisfies $\rho_{\varepsilon}(h)\to 0$ as $h\to 0$. Moreover, in view of (5.23) and (5.29), an elementary but tedious computation leads to

[TABLE]

Then (5.27) and Lemma 5.5(c) yield ${\rm sym}(G_{y}-G^{s}_{w})\,e_{3}=s\langle\nabla^{\prime}v_{\varepsilon}-\nabla^{\prime}v,\nabla^{\prime}v_{\varepsilon}-\nabla^{\prime}\tilde{v}\rangle\,e_{3}$. Thus, by (5.22) and Hölder’s inequality we get

[TABLE]

Likewise, by (5.8), (5.29), and (5.31) one can check by an elementary expansion that

[TABLE]

where $g_{1},g_{2}$ depend on $u,\tilde{u},u_{\varepsilon},v,\tilde{v},v_{\varepsilon}$.

Step 4: Lower semicontinuity of slopes. We will show that there exist a continuous function $\eta_{\varepsilon}:[0,\infty)\to[0,\infty)$ with $\eta_{\varepsilon}(0)=0$ and a constant $\tilde{C}$ depending on $u,v,\tilde{u},\tilde{v}$ such that for all $s\in[0,1]$ there holds

[TABLE]

We defer the proof of (5.2) to Steps 5–7 below and now prove the lower semicontinuity. Recall the definition of $\phi_{h}$ in (2.14). By combining the three estimates in (5.2) we obtain for all $s\in[0,1]$

[TABLE]

Recall that $y^{h}\in\mathscr{S}^{M}_{h}$ and Theorem 5.6(i) imply $\phi_{0}(u,v)\leq M$. By applying Lemma 4.8 with $(u_{0},v_{0})=(u,v)$ and $(u_{1},v_{1})=(\hat{u}_{1}^{\varepsilon},\hat{v}_{1}^{\varepsilon})$ we get

[TABLE]

where $\Phi^{1}$ and $\Phi^{2}_{M}$ are the functions introduced in Lemma 4.8. Thus, in view of (5.28), Lemma 4.5(iii), and Definition 3.1, we find by letting $s\to 0$

[TABLE]

Letting $h\to 0$ we then derive

[TABLE]

We observe that $\hat{u}_{1}^{\varepsilon}\to\tilde{u}$ in $W^{1,2}(S;\mathbb{R}^{2})$ and $\hat{v}_{1}^{\varepsilon}\to\tilde{v}$ in $W^{2,2}(S;\mathbb{R}^{2})$ as $\varepsilon\to 0$, see (5.22) and (5.29). Thus, letting $\varepsilon\to 0$, using Lemma 4.6(iii),(iv), and then taking the supremum with respect to $(\tilde{u},\tilde{v})$ we get

[TABLE]

where $\bar{\mathscr{S}}^{\rm reg}_{0}\subset{\mathscr{S}}_{0}$ denotes the subset consisting of functions $u,v$ with regularity $W^{2,\infty}$ and $W^{3,\infty}$, respectively. Since each $(\tilde{u},\tilde{v})\in{\mathscr{S}}_{0}$ can be approximated in $W^{1,2}(S;\mathbb{R}^{2})\times W^{2,2}(S)$ by a sequence of functions in $\bar{\mathscr{S}}^{\rm reg}_{0}$ (see Lemma 5.8) and the right hand side is continuous with respect to that convergence (see Lemma 4.6), the previous inequality also holds for ${\mathscr{S}}_{0}$ instead of $\bar{\mathscr{S}}^{\rm reg}_{0}$. The representation given in Lemma 4.9 then implies

[TABLE]

To conclude the proof, it therefore remains to show (5.2).

Step 5: Proof of (5.2)(i). By using Lemma 4.3(ii) and (5.2) we get

[TABLE]

Here, the last step follows from the fact that $Q_{D}^{3}(F)=Q_{D}^{3}({\rm sym}(F))$ for $F\in\mathbb{R}^{3\times 3}$, see Lemma 4.1(ii). Then, using (5.21) and (5.31) we find

[TABLE]

By (5.10) and (5.33) we conclude

[TABLE]

This yields (5.2)(i).

Step 6: Proof of (5.2)(ii). First, by Lemma 4.3(iv) and (5.2)(ii) we get

[TABLE]

Recall the definition of $\mathbb{C}_{W}$ in (2.19). An expansion and (5.2)(i) yield

[TABLE]

Inequalities (5.2)-(5.2), the weak convergence $G^{h}(y^{h})\rightharpoonup G_{y}$ in $L^{2}(\Omega;\mathbb{R}^{3\times 3})$ (see (5.30)) and (5.34) yield

[TABLE]

for some $\tilde{\rho}_{\varepsilon}(h)$, still satisfying $\tilde{\rho}_{\varepsilon}(h)\to 0$ as $h\to 0$. Using the fact that $Q_{W}^{3}(F)=Q_{W}^{3}({\rm sym}(F))$ (see Lemma 4.1(ii)), (5.21), (5.31), and (5.33) we conclude

[TABLE]

where the last step follows from (5.4).

Step 7: Proof of (5.2)(iii). By convexity of $P$ and the definition $w^{h}_{s}=y^{h}-y^{h}_{\varepsilon}+\tilde{y}^{h}_{s}$, see (5.25), we find

[TABLE]

By Hölder’s inequality and (2.4)(iii) we get

[TABLE]

Using $\phi_{h}(w^{h}_{s})\leq M^{\prime}$ since $w^{h}_{s}\in\mathscr{S}^{M^{\prime}}_{h}$ (see Lemma 5.5(a)) and (5.2)(i) we then derive

[TABLE]

where $C_{\varepsilon}$ depends also on $M^{\prime}$. By (5.39) and $1+\alpha(p-1)-\alpha p>0$, we finally get that (5.2)(iii) holds. ∎

Remark 5.11**.**

The previous proof is the only point where we need the assumption that the minimum in (2.17)-(2.18) is attained for $a=0$ which corresponds to a model with zero Poisson’s ratio in $e_{3}$ direction. Although this is a restrictive assumption, it is a good approximation for cellular materials such as cork. We also note that similar assumptions already appeared in the literature, see [9]. In fact, the sequence $(w^{h}_{s})_{h}$ has to be constructed in such a way that it is a recovery sequence (up to an $\varepsilon$-error) for both (5.2)(i) and (5.2)(ii). Without this assumption, a sound 2D model would necessarily have to depend on $Q_{W}^{3}$ and $Q_{D}^{3}$ (instead of $Q_{W}^{2}$ and $Q_{D}^{2}$) and extra variables in addition to $u$ and $v$ would be required to capture the extension or contraction of the vertical fibers along the evolution. (Still, we are not sure whether our analysis can be adapted to this case or not.)

6. Proof of the main results

In this section we give the proofs of Proposition 2.1-Theorem 2.3.

6.1. Existence of time-discrete solutions in 3D and passage from 3D to 2D

In this short subsection we prove Proposition 2.1 and Theorem 2.3.

Proof of Proposition 2.1.

Let $y^{h}_{0}\in\mathscr{S}_{h}^{M}=\{y\in\mathscr{S}_{h}:\phi_{h}(y)\leq M\}$ for some $M>0$. We recall that for the choices $\beta_{1}=4-\alpha p$ and $\beta_{2}=3$ we have $I_{h}^{\beta_{1},\beta_{2}}=h^{4}\phi_{h}$, see (2.14). Moreover, there holds $\mathcal{D}_{h}=h^{-2}\mathcal{D}$ by (2.15).

It is clear that the minimization problem (2.12) on $\mathscr{S}_{h}$ can be restricted to the set $\mathscr{S}_{h}^{M}$. Then the existence of solutions to the incremental problem (2.12) follows from the direct method of the calculus of variations: Lemma 4.5(ii),(iii) yield compactness with respect to the topology induced by $\mathcal{D}_{h}$ and Lemma 4.5(iv) implies lower semicontinuity. ∎

We now proceed with the proof of Theorem 2.3. We formulate our problem in the setting of Section 3.2. We consider the complete metric spaces $(\mathscr{S}_{h}^{M},\mathcal{D}_{h})$ and the limiting space $({\mathscr{S}}_{0},{\mathcal{D}}_{0})$ together with the functionals $\phi_{h}$ and ${\phi}_{0}$. Let $\sigma$ be the topology on ${\mathscr{S}}_{0}$ introduced in (5.18). Recall the definition of the mappings $\pi_{h}:\mathscr{S}^{M}_{h}\to{\mathscr{S}}_{0}$ defined by $\pi_{h}(y^{h})=(u^{h},v^{h})$ for each $y^{h}\in\mathscr{S}_{h}$ and the convergence $y^{h}\stackrel{{\scriptstyle\pi\sigma}}{{\to}}(u,v)$, see below (5.18) and see also (3.3).

Proof of Theorem 2.3.

We consider an initial datum $(u_{0},v_{0})\in{\mathscr{S}}_{0}$. We first see that the family of sequences of initial data $\mathcal{B}(u_{0},v_{0})$ defined in (2.25) is nonempty. This follows from Theorem 5.6(ii). We check that all assumptions of Theorem 3.2 are satisfied. First, (3.4) holds by Theorem 5.9 and (3.5) follows from Lemma 5.1. Also (3.6) is satisfied by the $\Gamma$-liminf inequality (Theorem 5.6(i)) and Theorem 5.10. Finally, the local slope $|\partial{\phi}_{0}|_{{\mathcal{D}}_{0}}$ is a strong upper gradient for $\phi_{0}$ by Lemma 4.9.

Now we consider a sequence $(y^{h}_{0})_{h}\in\mathcal{B}(u_{0},v_{0})$ and a null sequence $(\tau_{h})_{h}$. The definition of $\mathcal{B}(u_{0},v_{0})$ (see (2.25)) implies (3.2)(ii) with $\bar{z}_{0}=(u_{0},v_{0})$. In particular, as $\pi_{h}(y_{0}^{h})\stackrel{{\scriptstyle\pi\sigma}}{{\to}}(u_{0},v_{0})$, we get that the sequence $(\pi_{h}(y_{0}^{h}))_{h}$ is bounded in $W^{1,2}(S;\mathbb{R}^{2})\times W^{2,2}(S)$. In view of Lemma 4.6(iii), this yields (3.2)(i).

Let $\tilde{Y}_{h,\tau_{h}}$ be a sequence of time-discrete solutions as in (2.13) with $\tilde{Y}_{h,\tau_{h}}(0)=y^{h}_{0}$. Then the scalings $I_{h}^{4-\alpha p,3}=h^{4}\phi_{h}$ and $\mathcal{D}=h^{2}\mathcal{D}_{h}$ (see (2.14) and (2.15)) imply that $\tilde{Y}_{h,\tau_{h}}$ is also a time-discrete solution as in (3.1)-(3.2). The statement of Theorem 2.3 now follows from the abstract convergence result formulated in Theorem 3.2. ∎

6.2. Fine representation of the slope and solutions to the equations in 2D

This subsection is devoted to the proof of Theorem 2.2. We first note that Theorem 2.2(i) follows directly from Theorem 2.3. Therefore, we only need to show Theorem 2.2(ii). To this end, we derive a fine representation for the local slope in the 2D setting which will allow us to relate curves of maximal slope to solutions to the equations (2.21).

For the following proofs we introduce the abbreviation

[TABLE]

for $(u,v)\in{\mathscr{S}}_{0}$ and $\tilde{v}\in W^{2,2}(S)$. This definition captures the linear part of the difference of two strains. More precisely, with $G(u,v)$ and $G(\bar{u},\bar{v})$ as defined in (5.8), we have by an elementary computation

[TABLE]

Recall that $\mathbb{C}^{2}_{D}$ defined in (2.19) is a fourth order symmetric tensor inducing the quadratic form $G\mapsto Q_{D}^{2}(G)$ which is positive definite on $\mathbb{R}^{2\times 2}_{\rm sym}$ (cf. Lemma 4.1(ii)). Moreover, it maps $\mathbb{R}^{2\times 2}$ to $\mathbb{R}^{2\times 2}_{\rm sym}$, denoted by $G\mapsto\mathbb{C}_{D}^{2}G$ in the following. More precisely, the mapping $G\mapsto\mathbb{C}_{D}^{2}G$ from $\mathbb{R}^{2\times 2}_{\rm sym}$ to $\mathbb{R}^{2\times 2}_{\rm sym}$ is bijective. By $\sqrt{\mathbb{C}^{2}_{D}}$ we denote its (unique) root and by $\sqrt{\mathbb{C}^{2}_{D}}^{-1}$ the inverse of $\sqrt{\mathbb{C}^{2}_{D}}$, both mappings defined on $\mathbb{R}^{2\times 2}_{\rm sym}$. The same properties also hold with $\mathbb{C}_{W}^{2}$ in place of $\mathbb{C}_{D}^{2}$.

We now prove the following fine representation for the local slope.

Lemma 6.1** (Slope in the 2D setting).**

There exists a differential operator $\mathcal{L}:{\mathscr{S}}_{0}\to L^{2}(\Omega;\mathbb{R}^{2\times 2}_{\rm sym})$ satisfying

[TABLE]

such that the local slope at $(u,v)\in{\mathscr{S}}_{0}$ can be represented by

[TABLE]

*Proof. *To simplify the notation, we will write $(\bar{u},\bar{v})\to(u,v)$ instead of $\mathcal{D}_{0}((\bar{u},\bar{v}),(u,v))\to 0$. Recall the definition of the energy ${\phi}_{0}$ and the dissipation ${\mathcal{D}}_{0}$ in (2.23) and (2.2), as well as their representations in (5.4)-(5.10). By Definition 3.1(ii) we have

[TABLE]

This leads to

[TABLE]

Indeed, to see this, we use that $(\bar{u},\bar{v})\to(u,v)$ implies $G(\bar{u},\bar{v})\to G(u,v)$ strongly in $L^{2}(S;\mathbb{R}^{2\times 2})$ by Lemma 4.6(iii) and (5.8). Thus, we get

[TABLE]

Lemma 4.6(iii) and a Sobolev embedding also give

[TABLE]

This together with (5.10), (6.1)-(6.2), and the Cauchy-Schwartz inequality shows

[TABLE]

We introduce the space of test functions $\mathscr{T}=W^{1,2}_{0}(S;\mathbb{R}^{2})\times W^{2,2}_{0}(S)$. Due to the linearity of $H(\cdot,\cdot\,|v)$ we find

[TABLE]

where in the second step we used the properties of $\mathbb{C}^{2}_{D}$. We now consider the minimization problem

[TABLE]

where

[TABLE]

We note that the existence of a solution can be guaranteed by the direct method of the calculus of variations: to show coercivity, suppose $\mathcal{F}(u^{\prime},v^{\prime})\leq C$. We note that $\|H(u^{\prime},v^{\prime}|v)\|^{2}_{L^{2}(\Omega)}\leq C$ by Lemma 4.1(ii) with $C$ depending on $G(u,v)$. A standard argument involving Poincaré’s inequality and the boundary values yields $\|u^{\prime}\|_{W^{1,2}(S)}+\|v^{\prime}\|_{W^{2,2}(S)}\leq C$, where $C$ additionally depends on $\hat{u}$, $\hat{v}$, and $\|v\|_{W^{2,2}(S)}$. We refer to Remark 4.7 for details. Moreover, the functional is lower semicontinuous as it is convex in $H(u^{\prime},v^{\prime}|v)$ and $H(u^{\prime},v^{\prime}|v)$ is linear in $(u^{\prime},v^{\prime})$.

We denote a solution by $(u_{*},v_{*})\in\mathscr{T}$ and we observe that $(u_{*},v_{*})$ satisfies

[TABLE]

for all $(\varphi_{u},\varphi_{v})\in\mathscr{T}$. This equation can also be formulated as

[TABLE]

for all $(\varphi_{u},\varphi_{v})\in\mathscr{T}$, where we define the operator

[TABLE]

By the definition (6.1) and the regularity of the functions, we find $\mathcal{L}(u,v)\in L^{2}(\Omega;\mathbb{R}^{2\times 2}_{\rm sym})$. By (6.4), (6.5), and the definition of $\mathcal{L}$ we then get

[TABLE]

On the other hand, by a similar argument, in view of (6.4) and (6.5), we find

[TABLE]

where in the inequality we again distributed $\sqrt{\mathbb{C}^{2}_{D}}$ suitably to the two terms and used the Cauchy-Schwartz inequality. This concludes the proof. $\Box$

Following ideas in [3, Section 1.4], we now finally relate curves of maximal slope to solutions to the equations (2.21).

Proof of Theorem 2.2(ii).

Since $(u(t),v(t))$ is a curve of maximal slope, we get that ${\phi}_{0}(u(t),v(t))$ is decreasing in time, see (3.8). This together with Lemma 4.6(ii) gives

[TABLE]

Moreover, since $|({u},{v})^{\prime}|_{{\mathcal{D}}_{0}}\in L^{2}([0,\infty))$ by (3.8) and ${\mathcal{D}}_{0}$ is equivalent to the strong topology on $W^{1,2}(S;\mathbb{R}^{2})\times W^{2,2}(S)$ (see Lemma 4.6(iii)), we observe that $u$ and $v$ are absolutely continuous curves in the Hilbert spaces $W^{1,2}(S;\mathbb{R}^{2})$ and $W^{2,2}(S;\mathbb{R})$, respectively. By using [3, Remark 1.1.3] we observe that $u$ and $v$ are differentiable for a.e. $t$ with $\partial_{t}u(t)\in W^{1,2}(S;\mathbb{R}^{2})$ and $\partial_{t}v(t)\in W^{2,2}(S)$ for a.e. $t$. More precisely, we have

[TABLE]

and for all $0\leq s<t$, and a.e. in $S$ there holds

[TABLE]

As a preparation for the representation of the metric derivative, we now consider the difference $G(u(s),v(s))-G(u(t),v(t)$. For a.e. $t$ and a.e. $x\in\Omega$ we obtain by (6.2) and the linearity of $H(\cdot,\cdot\,|\,v(t))$

[TABLE]

Similarly, by (6.2), taking the integral over $\Omega$, using (6.2), Poincaré’s inequality, and Hölder’s inequality we get for all $0\leq s<t$

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We now estimate the metric derivative $|(u,v)^{\prime}|_{{\mathcal{D}}_{0}}$. By (5.10), (6.2), and Fatou’s lemma we get for a.e. $t$

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We now analyze the derivative $\frac{{\rm d}}{{\rm d}t}{\phi}_{0}(u(t),v(t))$ of the absolutely continuous curve ${\phi}_{0}\circ(u,v)$. Note that for a.e. $t$ we have $\lim_{s\to t}\int_{s}^{t}\|(\nabla^{\prime})^{2}\partial_{t}v(r)\|^{2}_{L^{2}(\Omega)}\,dr=0$ by (6.6) and, in a similar fashion, $\lim_{s\to t}(s-t)^{-1}\|\int_{s}^{t}H(\partial_{t}u(r),\partial_{t}v(r)\,|\,v(t))\|^{2}_{L^{2}(\Omega)}\,dr=0$ by (6.6) and Hölder’s inequality. Thus, using (5.4) and (6.2) we get for a.e. $t$ that

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By the property of $\mathcal{L}$ stated in (6.3) we get

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We find by Lemma 6.1, (6.2), and Young’s inequality

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where the last step is a consequence of the fact that $(u(t),v(t))$ is a curve of maximal slope with respect to ${\phi}_{0}$. Consequently, all inequalities employed in the proof are in fact equalities and we get

[TABLE]

pointwise a.e. in $\Omega$ for a.e. $t$. Multiplying the equation with $\sqrt{\mathbb{C}^{2}_{D}}$ from the left and testing with $H(\varphi_{u},\varphi_{v}\,|\,v(t))$ from the right for $(\varphi_{u},\varphi_{v})\in W^{1,2}_{0}(S;\mathbb{R}^{2})\times W^{2,2}_{0}(S)$, we obtain

[TABLE]

where we again used property (6.3). In the following we will again use the representation G(u(t),v(t))=\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{\rm sym}(G_{0})(t)\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}+x_{3}G_{1}(t) with the abbreviations

[TABLE]

introduced in (5.8). Consider also the corresponding time derivatives

[TABLE]

where for convenience we use the notation $\odot$ for the symmetrized vector product. Recall also (6.1) and observe that H(\partial_{t}u(t),\partial_{t}v(t)\,|\,v(t))=\partial_{t}G_{0}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}(t)\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}+x_{3}\partial_{t}G_{1}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}(t)\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}. Evaluating (6.11) for $\varphi_{v}=0$ leads to the equations

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We observe that (6.12) gives (2.22a). Evaluating (6.11) for $\varphi_{u}=0$ yields

[TABLE]

This gives the second equation (2.22b) and confirms that $(u,v)$ is a weak solution to (2.21). ∎

Acknowledgements This work was funded by the DFG project FR 4083/1-1 and by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy EXC 2044 -390685587, Mathematics Münster: Dynamics–Geometry–Structure. M.F. acknowledges support by the Alexander von Humboldt Stiftung and thanks for the warm hospitality at ÚTIA AVČR, where this project has been initiated. M.K. acknowledges support by the GAČR project 17-04301S and GAČR-FWF project 19-29646L.