The Euler–Bernoulli Limit of Thin Brittle Linearized Elastic Beams

Abstract

We show that the linear brittle Griffith energy on a thin rectangle $\varGamma$-converges after rescaling to the linear one-dimensional brittle Euler–Bernoulli beam energy.

In contrast to the existing literature, we prove a corresponding sharp compactness result, namely a suitable weak convergence after subtraction of piecewise rigid motions with the number of jumps bounded by the energy.

1 Introduction

We consider a model of brittle linearly elastic Euler–Bernoulli beams of fixed length $L>0$ and variable thickness $h \searrow 0$. We define the undeformed beam as

$$ \varOmega _{h} := (0,L) \times (-h/2,h/2) \subset \mathbb{R}^{2}. $$

We fix throughout the paper a positive definite stiffness tensor $\mathbb{C}\in \mathbb{R}^{2\times 2 \times 2 \times 2}$ such that

$$ \mathbb{C}_{ijkl} = \mathbb{C}_{jikl} = \mathbb{C}_{klij}\quad \text{ and }F:\mathbb{C}F \geq c|F+F^{T}|^{2} $$

(1)

for some $c>0$.

This allows us to define the energy of a displacement field $w\in \mathrm {GSBD}^{2}(\varOmega _{h})$ as

$$ F_{h}(w) := \frac{1}{h^{3}}\int _{\varOmega _{h}} \frac{1}{2} e w : \mathbb{C}e w \,dx + \frac{\beta}{h} \mathcal {H}^{1} (J_{w}), $$

(2)

where $\mathcal {H}^{1}$ denotes the 1-dimensional Hausdorff measure and $J_{w}$ is the jump set of the function $w$, see Sect. 5 (also for the definition of the space $\mathrm {GSBD}^{2}(\varOmega _{h})$ and its properties).

The elastic prefactor $1/h^{2}$ is chosen so that no stretching occurs and we recover the bending regime in the limit. The fracture prefactor $\beta/h$ denotes the renormalized material toughness (the actual material toughness is $\beta h^{2}$), which has to scale as $1/h$ to recover the number of fracture points in the limit. This energy is not necessarily understood as a model for a single material as the thickness tends to 0. Instead, it is valid for materials and scales $h$ such that $\mathbb{C} \sim \beta h^{2}$.

We note that by the symmetry of ℂ, for $w\in \mathrm {GSBV}(\varOmega _{1})$ we have $ew : \mathbb{C}ew = \nabla w : \mathbb{C}\nabla w$ almost everywhere, where $ew = (\nabla w + \nabla w ^{T})/2$. For the more general definition of $ew$ and the space $\mathrm {GSBD}^{2}(\varOmega _{h})$, see Sect. 5.

Because we let $h\to 0$, we perform the usual change of variables: Let $y \in L^{2}(\varOmega ;\mathbb{R}^{2})$ and $h>0$. Define $w \in L^{2}(\varOmega _{h})$ by $w(x_{1},x_{2}) = y(x_{1},x_{2} / h)$ and correspondingly define the energy $E_{h}: L^{0}(\varOmega _{1}) \to [0,\infty ]$ by

$$ E_{h}(y) = \textstyle\begin{cases} F_{h}(w) &\text{ if } w \in \mathrm {GSBD}^{2}(\varOmega _{h}), \\ + \infty &\text{ else,} \end{cases} $$

(3)

where $F_{h}$ is defined in (2). Formally, for regular functions $y$ (e.g. $y \in \mathrm {SBV}^{2}(\varOmega _{1};\mathbb{R}^{2})$) by a change of variables it holds

$$ E_{h}(y) = \frac{1}{h^{2}} \int _{\varOmega _{1}} \frac{1}{2} (\partial _{1} y, \frac{1}{h} \partial _{2} y) : \mathbb{C}(\partial _{1} y, \frac{1}{h} \partial _{2} y) \,dx + \beta \int _{Jy} |(\nu _{1},\frac{1}{h}\nu _{2})| \,d\mathcal {H}^{1}, $$

(4)

where $\nu \in S^{1}$ is the measure-theoretic normal to the jump set $J_{y}$. In general by the definition it is not true that $F_{h}(y) < \infty$ implies that $\nabla y$ exists as a function in $L^{2}(\varOmega _{1};\mathbb{R}^{2\times 2})$. We will use Korn’s inequality for functions with a small jump set (see [12]) to establish the relation (4) on a large set.

We show that the sequence of energies $E_{h}$ $\varGamma$-converges to a limit energy $E_{0}$ which is only finite on the following set of admissible limit configurations

$$ \begin{aligned} \mathcal {A}:= \{(y,J)\,:\,&y\in \mathrm {SBV}^{2}(\varOmega _{1};\mathbb{R}^{2}), D_{2} y = 0, \partial _{1} y_{1} = 0, \partial _{1} y_{2}\in \mathrm {SBV}(\varOmega _{1};\mathbb{R}), \\ &J_{y} \cup J_{\partial _{1} y} \subseteq J\times (-1/2,1/2), J \subset (0,L)\text{ finite}\}.\end{aligned} $$

In other words, $y$ is a function only of $x_{1}$, $y_{1}$ is piecewise constant on $(0,L)$, and $y_{2}$ is piecewise $W^{2,2}$ on $(0,L)$ with both $y_{2}$ and $\partial _{1} y_{2}$ jumping only on the finite fracture set $J \subset (0,L)$. The limit energy $E_{0}:L^{1}(\varOmega _{1};\mathbb{R}^{2})\to [0,\infty ]$ is given by (see Theorem 2)

$$ E_{0}(y,J) := \textstyle\begin{cases} \int _{\varOmega _{1}} \frac{a}{24}|\partial _{1}\partial _{1} y_{2}|^{2} \,dx + \beta \#J&,\text{ if }(y,J)\in \mathcal {A}, \\ \infty &,\text{ otherwise}. \end{cases} $$

We note that $E_{0}$ is in fact a one-dimensional energy

$$ E_{0}(y,J) = \int _{0}^{L} \frac{a}{24} |y_{2}''|^{2} \,dx_{1} + \beta \#J $$

for $y(x_{1},x_{2}) = (y_{1}(x_{1}),y_{2}(x_{1}))$, with $y_{1}:(0,L)\to \mathbb{R}$ piecewise constant, which for $y_{1}=0$ coincides with the one-dimensional version of the Blake–Zisserman model for image denoising, [37] (see also [13–15] for its analysis).

If $y_{1}\in \mathrm {SBV}((0,L))$ is piecewise constant and $y_{2},y_{2}' \in \mathrm {SBV}((0,L))$, then a natural choice for the fracture set is $J = J_{y} \cup J_{y'}$. For this choice of $J$ we have in fact $(y,J)\in \mathcal {A}$ and $E_{0}(y):= E_{0}(y,J)$ is finite. We also use the notation $y\in \mathcal {A}$. We allow for larger fracture sets in order to accomodate the compactness statement in Theorem 1. In fact, by the right choice of piecewise rigid motions, $y$ and $\partial _{1} y$ need not jump at all, even though every $y_{h}$ has large jumps. The larger fracture set $J$ allows us to keep track of where the rigid motions jump as well.

The bending constant $a>0$ is defined as usual in Euler–Bernoulli beam theory as

$$ a := \inf _{b,c\in \mathbb{R}} \begin{pmatrix} 1 & b \\ 0&c \end{pmatrix} :\mathbb{C}\begin{pmatrix} 1 & b \\ 0&c \end{pmatrix} . $$

(5)

The vector $(b,c)^{T}$ can be seen as an optimal shear response to a unit curvature. We note that unlike in Euler–Bernoulli beam theory, more complex models such as Ehrenfest–Timoshenko beam theory keep track of the additional shear variable in addition to the displacement $y$, leading to generally higher energy.

As is usual in the considered scaling regime in dimension reduction, the limit energy penalizes bending moments, which are not penalized in $E_{h}$. The emergence of a bending energy can be seen heuristically by taking

$$ y_{h}(x_{1},x_{2}) := y(x_{1},0) - x_{2} h \nabla y_{2}(x_{1},0) - \frac{1}{2} x_{2}^{2} h^{2} \partial _{1}\partial _{1} y(x_{1},0) \begin{pmatrix} b \\ c \end{pmatrix} , $$

where we need to subtract $x_{2} h \nabla y_{2}(x_{1},0)$ from $y$ so that the symmetric part of the matrix $(\partial _{1} y_{h}, \frac{1}{h} \partial _{2} y_{h})$ converges to 0. The precise calculation is found in Sect. 8.

Similar $\varGamma$-convergence results have already been proven in the $n$-dimensional setting, see [1, 8]. However, here we show a stronger complementing compactness theorem, see also the discussion in Sect. 9. The complementing compactness result can be illustrated as follows. Already without the possibility of fracture it is clear that sequences of functions with a bounded elastic energy are not precompact in a reasonable way as the elastic energy is invariant under the addition of rigid motions which form a non-compact set. Using Korn’s inequality, in this setting it can be expected that one can identify a sequence of rigid motions $A_{h} x + b_{h}$, $A_{h} \in \mathrm {Skew}(2)$, $b_{h} \in \mathbb{R}^{2}$ such that the difference of $w_{h}$ and the rigid motions is precompact after being rescaled to $\varOmega _{1}$. Additionally, fracture can occur and different rigid motions might be present on different parts of $\varOmega _{h}$ which have been broken apart from one another. However, the form of the energy $E_{h}$ suggests that the only way to break apart larger parts of $\varOmega _{h}$ is along essentially vertical cracks. Hence, a reasonable compactness result needs to identify the different parts of $\varOmega _{h}$ which have been broken apart from one another along vertical lines together with the corresponding dominant rigid motions, and additionally detect the asymptotically vanishing part of $\varOmega _{h}$ that is disconnected from the rest of $\varOmega _{h}$ along non-vertical lines. In fact, we show that for an energy-bounded sequence $y_{h}$ there are $x_{2}$-independent, piecewise-constant functions $A_{h}$ and $b_{h}$ and asymptotically vanishing sets $\omega _{h}$ such that the sequence is precompact in $L^{2}(\varOmega _{1};\mathbb{R}^{2})$. Moreover, the functions $A_{h}$ and $b_{h}$ are constructed carefully enough so that the modified sequence does not have asymptotically more jump than $y_{h}$ which is important for meaningful asymptotic lower bounds, see Theorem 1. A key tool in this analysis will be a Korn’s inequality for $\mathrm {GSBD}^{2}(\varOmega )$, see [12].

Next, we present a brief overview over existing results in the literature.

2 Elasticity, Beams, and Fracture

2.1 Geometric and Linearized Elasticity

We provide in this section a brief overview over the relevant theories. First we start with unfractured homogeneous hyperelastic materials. Here a stress-free reference configuration $\varOmega \subset \mathbb{R}^{d}$ undergoes a deformation $u:\varOmega \to \mathbb{R}^{d}$. The geometric hyperelastic energy is then given by

$$ \int _{\varOmega }W(\nabla u)\,dx, $$

where $W:\mathbb{R}^{d\times d} \to [0,\infty )$ denotes the $C^{2}(\mathbb{R}^{d\times d})$ hyperelastic energy density. We make the physical assumptions that $W(\mathrm {id}) = 0$, i.e. $u(x) = x$ has the lowest possible energy, and $W(RA) = W(A)$ for $R\in SO(d)$, i.e. rotations have no effect on the energy. The most-studied energy densities are those with quadratic growth at $SO(d)$ and at $\infty$, where $\operatorname{dist}^{2}(A,SO(d)) \lesssim W(A) \lesssim \operatorname{dist}^{2}(A,SO(d))$. A central result in the theory of hyperelastic materials is the geometric rigidity result by Friesecke, James, Müller [25], which states that for open connected Lipschitz domains $\varOmega \subset \mathbb{R}^{d}$, there exists a constant $C(\varOmega )>0$ such that

$$ \min _{R\in SO(d)} \int _{\varOmega } |\nabla u - R|^{2}\,dx \leq C( \varOmega ) \int _{\varOmega }\operatorname{dist}^{2}(\nabla u, SO(d))\,dx. $$

(6)

In particular, we have that whenever $\int _{\varOmega }W(\nabla u_{k})\,dx\to 0$, up to subsequences and fixed rotations $R_{k}\in SO(d)$ and shifts $b_{k}\in \mathbb{R}^{d}$, we have $R_{k}^{T} (u_{k}(x) - b_{k}) \rightharpoonup x$ weakly in $H^{1}(\varOmega ;\mathbb{R}^{d})$. For deformations with small hyperelastic energy, we may thus write $R_{k}^{T}(u_{k}(x) - b_{k}) = x + w_{k}(x)$, with $w_{k}(x)$ converging weakly to zero in $H^{1}(\varOmega ;\mathbb{R}^{d})$. A Taylor expansion of the energy yields

$$ \int _{\varOmega } W(\nabla u_{k})\,dx \approx \frac{1}{2} \int _{\varOmega } \nabla w_{k}(x):\mathbb{C}\nabla w_{k}(x)\,dx, $$

where $\mathbb{C}= D^{2}W(\mathrm {id})\in \mathbb{R}^{d\times d\times d \times d}$. The quadratic growth conditions on $W$ and Schwarz’s theorem then guarantee (1). For a rigorous derivation via $\varGamma$-convergence, see [22].

The resulting quadratic form dealing with infinitesimal displacements $|w| \ll 1$ is commonly referred to as linearized elasticity, and forms an important part of the physics and engineering literature, see e.g. [30]. In particular, it is often simpler to deal with than the geometrically nonlinear version.

For example, applying (6) to small deformations yields Korn’s inequality

$$ \min _{A\in \mathrm {Skew}(d)}\int _{\varOmega }|\nabla u - A|^{2} \,dx \leq C( \varOmega ) \int _{\varOmega }|\nabla u + \nabla u^{T}|^{2} \,dx, $$

which can be proved using elementary methods and was in fact proved by Korn in [31].

2.2 Thin Elastic Structures

In contrast to full bodies, lower dimensional structures have potentially lots of isometric embeddings into $\mathbb{R}^{d}$. A famous example is the Nash–Kuiper theorem [35], which states that for every Riemannian $m$-manifold $M$ and every smooth 1-Lipschitz map $f:M\to \mathbb{R}^{d}$ with $d> m$, there is an isometric $C^{1}$ immersion of $M$ into $\mathbb{R}^{d}$ that is arbitrarily close in $L^{\infty}(M)$ to $f$.

Compare that to open sets $\varOmega \subset \mathbb{R}^{d}$, where every isometric $C^{1}$ deformation $w:\varOmega \to \mathbb{R}^{d}$ must be a rigid motion by (6).

Thin structures are slightly thickened versions of submanifolds. The simplest nontrivial example is the Euler–Bernoulli beam $\varOmega _{h} := (0,L) \times (-h/2,h/2) \subset \mathbb{R}^{2}$. In the geometrically nonlinear setting, a deformation carries low hyperelastic energy if the midsection $(0,L) \times \{0\}$ is isometrically embedded, i.e. $|\partial _{1} u(x_{1},0)| = 1$. In that case,

$$ \int _{\varOmega _{h}} W(\nabla u)\,dx \approx h^{3}\int _{0}^{L} \frac{a}{24}|\partial _{1}\partial _{1} u(x_{1},0)|^{2}\,dx_{1}, \text{ if }|\partial _{1} u(x_{1},0)| = 1, $$

(7)

where $a>0$ is defined by (5), and $\mathbb{C}= D^{2}W(\mathrm {id})$.

In contrast, starting with the linearized elastic energy, we find that

$$ \int _{\varOmega _{h}} \frac{1}{2} ew:\mathbb{C}ew \,dx \approx h^{3} \int _{0}^{L} \frac{a}{24}|\partial _{1} \partial _{1} w_{2}(x_{1},0)|^{2}\,dx_{1}, \text{ if } \partial _{1} w_{1}(x_{1},0) = 0, $$

which is the linearized version of the bending energy (7), c.f. [19]. The resulting one-dimensional energy is named the Euler–Bernoulli energy after its originators. See e.g. [4, 30] for further reading.

We note that in the scaling regime $\int _{\varOmega _{h}} W(\nabla u)\,dx \approx h$, stretching is possible and dominates the energy over bending. The theory is generally called string theory, see e.g. [4]. For its two-dimensional analogue, the so-called membrane theory, see, for example, [5, 6, 32, 33].

Generalizing from thin structures in the plane to thin structures in three-dimensional space, we differentiate between beams or rods of the type $R_{h} := (0,L) \times hS \subset \mathbb{R}^{3}$, with $S\subset \mathbb{R}^{2}$ open, bounded, connected, and plates $P_{h}:= (0,L)\times (0,L) \times (-h/2,h/2)$. Both, linear and nonlinear variational models exist for both, see e.g. [34] for beams and [25, 27] for plates. We note also that shells, which are curved analogues of plates, have been similarly studied, see e.g. [26].

2.3 Griffith’s Model of Fracture

Fracture is one of multiple failure modes in elastic structures. Fracture occurs along codimension-one hypersurfaces called cracks, where the deformation is discontinuous. We differentiate between cohesive fracture, where the energy depends on the magnitude of the discontinuity, and brittle fracture, which we discuss in this article, where the total energy depends only on the surface measure of the crack.

For an open reference configuration $\varOmega \subset \mathbb{R}^{d}$ and a displacement field $w:\varOmega \to \mathbb{R}^{d}$ which is $C^{1}$ outside a closed rectifiable hypersurface $\varGamma \subset \varOmega$, we define the Griffith brittle fracture energy (see [23, 29]) as

$$ E(w) := \inf \left \{\int _{\varOmega \setminus \varGamma } \frac{1}{2} ew: \mathbb{C}ew\,dx + \beta \mathcal {H}^{d-1}(\varGamma )\,:\,\varGamma \subset \varOmega \mbox{ closed} \text{ s.t. } w\in C^{1}(\varOmega \setminus \varGamma ) \right \}. $$

Here $\beta >0$ is the material toughness, i.e. the surface tension of the crack surface. Expectedly, the space of piecewise $C^{1}$ deformations generally does not contain the minimizers of $E$, which led to the characterization of the energy space for $E$ in [21], the space of generalized functions of bounded deformation $\mathrm {GSBD}^{2}(\varOmega )$, whose definition and key properties we recount in Sect. 5.

The study of fracture in thin materials has seen advancement in recent years. In the nonlinear setting in [11] the authors study the derivation of a membrane theory in which stretching is dominant, see also [10]. Recently, Schmidt showed in [36] that the nonlinear version of $E_{h}$ $\varGamma$-converges to

$$ \int _{0}^{L} \frac{a}{24} |y''(x_{1})|^{2}\, dx_{1} + \beta \#(J_{y} \cup J_{\partial _{1} y}),\text{ if }y\in \mathrm {SBV}((0,L);\mathbb{R}^{2}), |y'(x_{1})| = 1 \text{ a.e.,} $$

which is the nonlinear analogue to the limit energy $E_{0}$. In [8] and [1] the authors study the asymptotics of an $n$-dimensional analogue of the energy $E_{h}$, see also [2, 7, 9] for the antiplane setting. Using a slightly different rescaling of the function $y_{h}$, c.f. [19], the authors obtain the limiting energy

$$ \int _{\varOmega _{1}} Q(\nabla y) \, dx + \beta \mathcal {H}^{d-1}(J_{y}), $$

where $Q$ is a quadratic form and $y$ is of the form $y_{i}(x_{1},\dots ,x_{d}) = \bar{y}_{i}(x_{1},\dots ,x_{d}) - x_{d} \partial _{i} y_{d}(x_{1},\dots ,x_{d})$ for $i=1,\dots , d-1$ and $y_{d}$ does not depend on $x_{d}$. Although very similar to the result presented here, we note that the used techniques are rather different. In order to identify the specific form of the limiting $y$ in [8] the authors study the distributional symmetric gradient of $y$ together with convolution techniques, in [1] the authors use a delicate approximation argument in $\mathrm {GSBD}^{2}$. In contrast our proofs are based on rigidity arguments which are much closer to the techniques used in [36], see also [25, 34]. This allows to obtain more control on the rescaled gradient $\nabla _{h} y_{h} = (\partial _{1} y_{h}, \frac{1}{h} \partial _{2} y_{h}, \frac{1}{h} \partial _{3} y_{h})$. In the presented setting this enables us to obtain an improved compactness statement and a short proof for the identification of the limiting configurations. Moreover, in other problems the additional control of $\nabla _{h} y_{h}$ is crucial. For example in the derivation of a rod theory the information about torsion is stored in the limit of $\nabla _{h} y_{h}$ and cannot be seen in the limiting $y$, see [28].

We note that our result deals with the slightly simpler linear energy but uses different methods, which can be generalized to the linear theory in higher dimensions.

3 Notation

Throughout the paper $C>0$ is a generic constant that may change from line to line. Moreover, we use standard notation $x = (x_{1},\dots ,x_{d})$ for vectors in $\mathbb{R}^{d}$. In particular, we will identify vectors with its transpose wherever it simplifies notation. At several points, the components of vector-valued functions with a subscript, e.g. $w_{h}:\mathbb{R}^{2}\to \mathbb{R}^{2}$, are denoted $(w_{h})_{1}$, $(w_{h})_{2}$. We say that two vectors $v,w\in \mathbb{R}^{d}$ are parallel, $v\|w$, if they are linearly dependent. The space of symmetric and skew-symmetric $\mathbb{R}^{d\times d}$ matrices will be denoted by $\mathrm {Symm}(d)$ and $\mathrm {Skew}(d)$, respectively. We use standard notation for the Lebesgue measure $\mathcal{L}^{d}$ and the $s$-dimensional Hausdorff measure $\mathcal{H}^{s}$. Moreover, for a Lebesgue-measurable set $B \subseteq \mathbb{R}^{d}$ we write $|B|$ for $\mathcal{L}^{d}(B)$. Moreover, we use standard notation for Lebesgue and Sobolev spaces $L^{p}$ and $W^{1,p}$. Lastly, for a set of finite perimeter $E \subseteq \mathbb{R}^{d}$ we write $\partial ^{*} E$ for its reduced boundary, cf. [3].

4 Main Results

We now state our main results. We start with the compactness result.

Theorem 1

Let $E_{h}$ be defined as in (3), (4) and $(y_{h})_{h} \subseteq L^{2}(\varOmega _{1};\mathbb{R}^{2})$ such that $\sup _{h > 0} E_{h}(y_{h}) < \infty$. Then there is a subsequence (not relabeled), a pair $(y,J)\in \mathcal {A}$, a sequence of sets $\sigma _{h} \subset \varOmega _{1}$ of finite perimeter with measure theoretic normal $\nu \in S^{1}$, and sequences of piecewise constant functions $\overline{A}_{h}:(0,L) \to \mathrm {Skew}(2)$, $\overline{b}_{h}:(0,L) \to \mathbb{R}^{2}$ jumping only on the finite set $J = \{j_{1},\ldots ,j_{N}\} \subset (0,L)$ such that

• (i)
  $$ \lim _{h\to 0} \int _{\varOmega _{1}\setminus \sigma _{h}} |y_{h}(x) - \overline{A}_{h}(x_{1})(x_{1},hx_{2})^{T} - \overline{b}_{h}(x_{1}) - y(x)|^{2} \,dx = 0 . $$
• (ii)
  $$ |\sigma _{h}| \to 0 \quad \textit{ and }\quad \sup _{h>0} \int _{ \partial ^{*}\sigma _{h}} |(\nu _{1},\frac{1}{h} \nu _{2})|\,d\mathcal {H}^{1} < \infty . $$

• (iii) We have $J_{y} \cup J_{\partial _{1} y} \cup J_{\overline{A}_{h}}\cup J_{ \overline{b}_{h}} \subseteq J$ and

  $$ \#J \leq \left \lfloor \liminf _{h\to 0} \int _{J_{y_{h}}} |(\nu _{1}, \frac{1}{h} \nu _{2})| \,d\mathcal {H}^{1} \right \rfloor . $$

  Here $\lfloor \cdot \rfloor :[0,\infty )\to \mathbb{N}$ is the integer part function $x\mapsto \max \{n\in \mathbb{N}\,:\,n\leq x\}$.

Remark 1

In other words, $\overline{A}_{h}$ and $\overline{b}_{h}$ only jump where the limit jump density of $y_{h}$ is at least one. We emphasize that one may not replace (iii) by the better estimate

$$ \# J \leq \left \lfloor \int _{J_{y_{h}}} |(\nu _{1},\frac{1}{h} \nu _{2})| \,d\mathcal {H}^{1} \right \rfloor . $$

To see this, consider the sequence of triangles in $\overline{\varOmega _{h}}$ with vertices $t_{h} := (L/2, h/2 - h^{4})$, $l_{h} := (L/2 - h^{4}, h/2)$, $r_{h} := (L/2 + h^{4}, h/2)$, whose sidelengths are $O(h^{4})$, and define the piecewise affine displacement

$$ w_{h}(x) := \textstyle\begin{cases} 0 &x_{1} < L/2, x \notin \operatorname{conv}(t_{h},r_{h},l_{h}) \\ \frac{1}{h}(x-t_{h})^{\perp}&x_{1} > L/2, x\notin \operatorname{conv}(t_{h},r_{h},l_{h}) \\ \frac{1}{h}(v^{\perp}\otimes v)(x-t_{h}) &x\in \operatorname{conv}(t_{h},r_{h},l_{h}), \end{cases} $$

where $v = (e_{1}+e_{2})/\sqrt{2} = (r_{h} - t_{h})/|r_{h} - t_{h}|$. See Fig. 1 for a sketch of the corresponding deformation.

Fig. 1

Sketch of the deformation $x+w_{h}(x)$ in Remark 1. The length of the jump is slightly less than the height. Elastic stress, while high, is contained to the small triangle $\operatorname{conv}(t_{h},l_{h},r_{h})$

This displacement field jumps on the line segment $\gamma _{h} := \{L/2\}\times (-h/2, h/2 - h^{4})$ and has elastic energy

$$ \int _{\varOmega _{h} \setminus \gamma _{h}} |ew_{h}|^{2}\,dx = \int _{ \operatorname{conv}(t_{h},r_{h},l_{h})} |\nabla w_{h}|^{2}\,dx = h^{8}/h^{2} = h^{6} \ll h^{3}. $$

So even though $\mathcal {H}^{1}(J_{w_{h}}) = h-h^{4}$, i.e.

$$ \left \lfloor \int _{J_{y_{h}}} |(\nu _{1},\frac{1}{h} \nu _{2})| \,d \mathcal {H}^{1} \right \rfloor = \lfloor 1 - h^{3} \rfloor = 0, $$

there are no constants $\overline{A}_{h}$, $\overline{b}_{h}$ such that $w_{h} - \overline{A}_{h} x - \overline{b}_{h}$ is bounded on most of $\varOmega _{h}$. To achieve convergence in measure, we have to allow $\overline{A}_{h}$, $\overline{b}_{h}$ to jump once at $L/2$.

Remark 2

Even for $\bar{A}_{h}=0$, $\bar{b}_{h}=0$ the compactness result only guarantees convergence in measure, but not in $L^{1}(\varOmega _{1};\mathbb{R}^{2})$. As a result, we have that the minimizers of $E_{h} + F$ converge in measure to minimizers of $E_{0} + F$ whenever $F$ is continuous under convergence in measure. Nontrivial linear functionals $F$ are of course not continuous under convergence in measure. Consider for example $F(y) := \int _{\varOmega _{1}} y_{2} \,dx$, and

$$ y_{h}(x) := \textstyle\begin{cases} -\frac{1}{h^{5}} e_{2} ,& \text{ if }x\in B((L/2,0),h^{2}) \\ 0,& \text{ otherwise.} \end{cases} $$

Then $E_{h}(y_{h}) \to 0$, $y_{h} \to 0$ in measure but not in $L^{1}$, and $F(y_{h}) \to -\infty$.

Next, we state the main $\varGamma$-convergence result.

Theorem 2

The $\varGamma$-limit of $E_{h}$ as defined in (3), (4) with respect to the convergence in Theorem 1is $E_{0}$. More precisely, we have

• (i) For any pair $(y,J) \in \mathcal {A}$ there is a sequence $y_{h}\in \mathrm {SBV}^{2}(\varOmega _{1};\mathbb{R}^{2})$ such that $y_{h} \to y$ in $L^{\infty}(\varOmega _{1};\mathbb{R}^{2})$, $\int _{J_{y_{h}}} |(\nu _{1},\nu _{2}/h)|\,d\mathcal {H}^{1} = \#J$ for every $h>0$, and $\lim _{h\to 0} E_{h}(y_{h}) = E_{0}(y,J)$.

• (ii) Let $(y,J)\in \mathcal {A}$ and $y_{h}: \varOmega _{1} \to \mathbb{R}^{2}$, $\omega _{h}\subset \varOmega _{1}$, and $\bar{A}_{h}:(0,L)\to \mathrm {Skew}(2)$, $\bar{b}_{h}:(0,L)\to \mathbb{R}^{2}$ two piecewise constant functions, such that the conditions of Theorem 1hold. Then $E_{0}(y,J) \leq \liminf _{h\to 0} E_{h}(y_{h})$.

Remark 3

Note that the conditions for the lower bound (ii) in Theorem 2 are satisfied if $y_{h} \to y$ in $L^{2}$, c.f. Theorem 1 above.

Remark 4

For an isotropic material it holds $\mathbb{C} F : F = 2\mu |F_{sym}|^{2} + \lambda \operatorname{tr}(F)^{2}$ where the Lamé coefficients satisfy $\mu >0$ and $2\mu + \lambda > 0$. A straightforward computation then shows for the bending constant from (5) that $a = 2 \mu + 2 \mu h/(2\mu+\lambda)$.

5 The Space $\mathrm {GSBD}^{2}$ and Korn’s Inequality

We use the space of generalized functions of special bounded deformation with integrability 2, written $\mathrm {GSBD}^{2}(\varOmega )$, as the effective domain of the energies $F_{h}$, c.f. [21]. This space is the natural topological function space for the brittle Griffith fracture model. It is analogous to the $\mathrm {GSBV}^{p}$ spaces that are widely used in image segmentation, see [3] for the definition and the properties of the spaces $\mathrm {GSBV}^{p}$. In order to recall the definition of the space $\mathrm {GSBD}^{p}$, we first recall the definition of the jump set and the approximate symmetric gradient.

Definition 1

• (i) Let $\varOmega \subset \mathbb{R}^{d}$ be open, $x\in \varOmega$, and $v:\varOmega \to \mathbb{R}^{N}$ be measurable. The blow-up limit of $v$ at $x$, if it exists, is defined as the measurable function $\mathrm {blow\,up}_{x} v :\mathbb{R}^{d} \to \mathbb{R}^{N}$ such that

  $$ v\left (\frac{\cdot - x}{r} \right ) \stackrel{r \to 0}{\longrightarrow} \mathrm {blow\,up}_{x} v \text{ in measure on }B(0,1). $$

  Note that the function $\mathrm {blow\,up}_{x} v$ is positively 0-homogeneous, i.e. $\mathrm {blow\,up}_{x} v(ty) = \mathrm {blow\,up}_{x} v(y)$ for all $y\in \mathbb{R}^{d}$, $t>0$. If a constant blow-up limit exists, it is also called the approximate limit of v at x. By standard measure theoretic arguments, the approximate limit of a measurable function exists at almost every point.

• (ii) The jump set $J_{v}$ of a measurable function $v:\varOmega \to \mathbb{R}^{N}$ is the set of all points $x\in \varOmega$ where $\mathrm {blow\,up}_{x} v$ exists and is of the form

  $$ \mathrm {blow\,up}_{x} v (y) = \textstyle\begin{cases} a,&\text{ if }y\cdot \nu > 0 \\ b,&\text{ otherwise}, \end{cases} $$

  for some $\nu \in S^{d-1}$, $a,b\in \mathbb{R}^{N}$, $a \neq b$. For $x\in J_{v}$, we denote by $[v](x) := b-a\in \mathbb{R}^{N}$ the jump of $v$ at $x$ and $\nu \in S^{d-1}$ the measure-theoretic normal of $J_{v}$ at $x$.

• (iii) A function $v: \varOmega \to \mathbb{R}^{d}$ is said to have an approximate symmetric gradient $ev(x) \in \mathrm {Symm}(d)$ at $x \in \varOmega$ if it holds

  $$ \mathrm {blow\,up}_{x} \frac{ \left ( u(y) - u(x) - ev(x) (y-x) \right ) \cdot (y-x) }{|x-y|^{2}} = 0. $$

Next we recall the definitions of the spaces $\mathrm {BD}(\varOmega )$, $\mathrm {SBD}^{p}(\varOmega )$ and $\mathrm {GSBD}(\varOmega )$.

Definition 2

• (i) Let $\varOmega \subset \mathbb{R}^{d}$ be open. A vector field $v\in L^{1}(\varOmega ;\mathbb{R}^{d})$ is said to be of bounded deformation, $\mathrm {BD}(\varOmega )$, if

  $$ \sup \left \{\int _{\varOmega }v \cdot (\nabla \cdot \phi ) \,dx\,:\, \phi \in C_{c}^{1}(\varOmega ; \mathrm {Symm}(d)) , \|\phi \|_{\infty }\leq 1 \right \} < \infty . $$

• (ii) The vector field is said to be of special bounded deformation with integrability $p\in (1,\infty )$, $\mathrm {SBD}^{p}(\varOmega )$, if there is a tensor field $ev\in L^{p}(\varOmega ;\mathrm {Symm}(d))$ such that

  $$ \int _{\varOmega }-v \cdot (\nabla \cdot \phi )\,dx = \int _{\varOmega }ev: \varPhi \,dx + \int _{J_{v}} [v] \cdot \phi \nu \,d\mathcal {H}^{d-1} $$

  for all $\phi \in C_{c}^{1}(\varOmega ; \mathrm {Symm}(d))$ and $\mathcal {H}^{d-1}(J_{v}) < \infty$, where $J_{v}$ is the jump set of $v$.

• (iii) A measurable vector field $v:\varOmega \to \mathbb{R}^{d}$ is said to be of generalized special bounded deformation, $\mathrm {GSBD}(\varOmega )$, if there exists a bounded positive Radon measure $\lambda \in \mathcal{M}_{+}(\varOmega )$ such that for every $\xi \in \mathbb{R}^{d}$ and for $\mathcal {H}^{d-1}$-almost every $\theta \in \xi ^{\perp }\subset \mathbb{R}^{d}$ the function $f_{\xi ,\theta}: \varOmega _{\xi ,\theta} \to \mathbb{R}$, where $\varOmega _{\xi ,\theta} := \{t\in \mathbb{R}\,;\,\theta +t\xi \in \varOmega \}$, defined as

  $$ f_{\xi ,\theta}(t) := u(\theta +t\xi ) \cdot \xi , $$

  is in $\mathrm {SBV}_{loc}(\varOmega _{\xi ,\theta})$ and it holds for every Borel set $B \subset \varOmega$

  $$ \int _{\xi ^{\perp}} |D f_{\xi ,\theta}| (B_{\xi ,\theta }\setminus J^{1}_{\xi ,\theta }) + \mathcal{H}^{0} (B_{\xi ,\theta } \cap J^{1}_{\xi ,\theta }) \, d \mathcal{H}^{n-1}(\theta ) \leq \lambda (B), $$

  where as before $B_{\xi ,\theta} = \{ t \in \mathbb{R}: \theta +t\xi \in B\}$, and $J^{1}_{\xi ,\theta }:= \{t\in J_{f_{\xi ,\theta }}\,:\,|[f_{\xi , \theta }(t)]|\geq 1\}$ denotes the points on the real line where $f_{\xi ,\theta }$ has a large jump.

We recall from [21] that for $v \in \mathrm {GSBD}(\varOmega )$ it can be shown that the approximate symmetric gradient $ev \in L^{1}(\varOmega ;\mathbb{R}^{d \times d})$ exists $\mathcal{L}^{d}$-a.e. and the jump set $J_{v}$ is a countably $\mathcal{H}^{d-1}$-rectifiable set with measure-theoretic normal $\nu$. Moreover, it holds for $\xi \in S^{d-1}$ and $\mathcal{H}^{d-1}$-a.e. $\theta \in \xi ^{\perp}$

$$ Df_{\xi ,\theta} = ev(\theta +t \xi ) \xi \cdot \xi \, dt + \sum _{t \,:\,\theta +t\xi \in J_{v}} [v]\cdot \xi \delta _{t}. $$

Finally, we define the space $\mathrm {GSBD}^{p}(\varOmega )$.

Definition 3

Let $p \in (1,\infty )$. We say that $v \in \mathrm {GSBD}^{p}(\varOmega )$ if $v \in \mathrm {GSBD}(\varOmega )$, $ev \in L^{p}(\varOmega )$ and $\mathcal{H}^{d-1}(J_{v}) < \infty$.

For fine properties of the functions in $\mathrm {GSBD}^{p}$ we refer to [21].

We now state a strong version of Korn’s inequality for functions in $\mathrm {GSBD}^{2}(\varOmega )$ in any dimension, which is found in [12], for an earlier two-dimensional version see also [20], [24], or [18].

Proposition 1

Let $\varOmega \subset \mathbb{R}^{d}$ be open, bounded, connected with Lipschitz boundary. Then there is a constant $C(\varOmega )>0$ such that for all $w\in \mathrm {GSBD}^{2}(\varOmega )$ there is a function $\bar{w} \in W^{1,2}(\varOmega ;\mathbb{R}^{d})$ and a set of finite perimeter $\omega \subseteq \varOmega$ such that $w = \bar{w}$ on $\varOmega \setminus \omega$.

$$ |\omega | + \mathcal {H}^{d-1}(\partial \omega ) \leq C(\varOmega ) \mathcal {H}^{d-1}(J_{w}) $$

and

$$ \int _{\varOmega } |e \bar{w}|^{2} \, dx \leq C(\varOmega ) \int _{\varOmega } |ew|^{2} \, dx. $$

Moreover, there exists a matrix $A\in \mathrm {Skew}(d)$ and a vector $b\in \mathbb{R}^{d}$ such that

$$ \int _{\varOmega \setminus \omega } |\nabla w - A|^{2} + |w - Ax - b|^{2} \,dx \leq C(\varOmega ) \int _{\varOmega }|ew|^{2}\,dx. $$

Remark 5

• (i) It is clear by a change of variables that the constant $C(\varOmega )$ is scaling invariant, i.e. that $C(\lambda \varOmega ) =C(\varOmega )$ for all $\lambda >0$.

• (ii) We note that the estimate is useless if $\mathcal {H}^{d-1}(J_{w})$ is too large with respect to $\varOmega$, as then we can simply take $\omega = \varOmega$.

We shall use this result to define good rectangles in $\varOmega _{h}$, noting that we never use the extension $\bar{w}$, only the bounds on the bad set $\omega$. For the rest of the article (excluding the Appendix) we work only in $d=2$.

Definition 4

Let $h,\delta > 0$, and $w \in \mathrm {GSBD}^{2}(\varOmega _{h})$. We consider all the size-$h$ rectangles $Q_{z} := (z- h, z + h) \times (-h/2,h/2)$ with $z=h,2h,\ldots , (\lfloor L/h \rfloor - 1) h$. We call a size-$h$ rectangle $Q_{z}$ $\delta$-good with respect to $w$ if

$$ \mathcal {H}^{1}(J_{w} \cap Q_{z}) \leq \delta h, $$

and bad otherwise. For a $\delta$-good size-$h$ rectangle we denote by $\omega _{z} \subseteq Q_{z}$ the exceptional set from Proposition 1.

Remark 6

Again, let $h,\delta > 0$ and $w \in \mathrm {GSBD}^{2}(\varOmega _{h})$. We note that if $\delta \leq \delta _{0}$ for some universal constant $\delta _{0}$, then we have on a $\delta$-good rectangle $Q_{z}$ by Proposition 1 (and the scaling invariance from Remark 5) that $|Q_{z}\setminus \omega _{z}| > 7|Q_{z}|/8$ and there is a unique pair $A_{z}\in \mathrm {Skew}(2)$, $b_{z}\in \mathbb{R}^{2}$ defined as

$$ (A_{z},b_{z}) := \arg \min _{A,b} \int _{Q_{z} \setminus \omega _{z}} \frac{1}{h^{2}}|w(x) - Ax -b|^{2} + |\nabla w(x) - A|^{2}\,dx, $$

where minimization runs over $\mathrm {Skew}(2) \times \mathbb{R}^{2}$. Proposition 1 yields that

$$ \int _{Q_{z} \setminus \omega _{z}} \frac{1}{h^{2}}|w(x) - A_{z} x -b_{z}|^{2} + |\nabla w(x) - A_{z}|^{2}\,dx \leq C \int _{Q_{z}} |ew|^{2}\,dx. $$

Then we see that for two neighboring $\delta$-good rectangles $Q_{z}$, $Q_{z+h}$ we have

$$ |A_{z} - A_{z+h}|^{2} + |b_{z} - b_{z+h}|^{2} \leq C h^{-2} \int _{Q_{z} \cup Q_{z+h}} |e w|^{2}, $$

(8)

for some universal constant $C$.

We now show a stronger version of (8) for two $\delta$-good rectangles that are separated by a sequence of bad rectangles – as long as there is not enough jump to separate the two rectangles completely:

Proposition 2

Let $\eta \in (0,1)$. Then there is a constant $\delta (\eta )>0$ such that for all $N\in \mathbb{N}$ there is a constant $C(\eta ,N)>0$ such that the following holds:

Let $h>0$, $w_{h} \in \mathrm {GSBD}^{p}(\varOmega _{h})$, $Q_{z}$ and $Q_{z'}$ be two $\delta (\eta )$-good size-$h$ rectangles with $|z-z'|\leq Nh$, and

$$ \mathcal {H}^{1}(J_{w_{h}} \cap \operatorname{conv}(Q_{z} \cup Q_{z'})) \leq (1-\eta ) h. $$

Then

$$ |A_{z} - A_{z'}|^{2} + |b_{z} - b_{z'}|^{2} \leq C(\eta , N) h^{-2} \int _{\operatorname{conv}(Q_{z} \cup Q_{z'})} |e w_{h}|^{2}\,dx. $$

(9)

Here, the matrices $A_{z},A_{z'} \in \mathrm {Skew}(2)$ and vectors $b_{z},b_{z'} \in \mathbb{R}^{2}$ are the matrices and vectors given by Proposition 1on the squares $Q_{z}$ and $Q_{z'}$, respectively.

Proof

We assume without loss of generality that $h=1$ (by rescaling) and that $A_{z'},b_{z'} = 0$. This is achieved by replacing $w_{h}$ with $w_{h}(x) - A_{z'}x - b_{z'}$, which has the same jump set and elastic strain. We will write $w=w_{h}$ since $h=1$. We also assume that $z'> z$.

We show that there are three pairs $(p_{i},q_{i})\in Q_{z} \times Q_{z'}$, $i=1,2,3$, such that

$$ \left | \left (A_{z} p_{i} + b_{z} \right )\cdot (p_{i}- q_{i}) \right |^{2} \leq C(\eta , N) \int _{\operatorname{conv}(Q_{z} \cup Q_{z'})} |ew|^{2} \,dx. $$

(10)

If we can make sure that the three pairs are in general position, Lemmas 1 and 2 in the Appendix yield the upper bounds on $|A_{z}|$, $|b_{z}|$.

The reason why we need three pairs in general position, i.e. specifically not three parallel line segments, is geometric. Given two line segments connecting $Q_{z}$ and $Q_{z'}$, there is a rigid motion of only $Q_{z'}$ that leaves the lengths of both identical. The same holds for three parallel line segments. Lemmas 1 and 2 in the Appendix make this geometric fact quantitative in the geometrically linear setting and show that in two dimensions, three line segments in general position prevent a rigid motion of only one rectangle. See Fig. 2 for a visual sketch. The rest of the proof is devoted to finding three line segments that are in general position, do not carry too much elastic energy, do not intersect the jump set, and are representative.

Fig. 2

Two good rectangles $Q_{z}$, $Q_{z'}$ are separated by a series of bad rectangles, but without enough jump set to completely separate the two. Then we can find three line segments connecting the good rectangles that prevent the two from being infinitesimally rotated or shifted against one another

We first show how to construct the first two pairs, where $(p_{i} - q_{i}) \| e_{1}$ (the notation $v \| w$ means that the vectors $v,w\in \mathbb{R}^{2}$ are linearly dependent):

Start with the set of horizontal lines that do not cross $J_{w}$ in the sense of slicing for $\mathrm {GSBD}$ functions, or more precisely

$$ H_{1} := \{x_{2} \in (-1/2,1/2)\,:\, x_{1} \mapsto e_{1}\cdot w(x_{1},x_{2}) \textrm{ is absolutely continuous on }(z - 1, z' + 1)\}. $$

We have by the segment regularity in $\mathrm {GSBD}^{2}$ (see [21, Theorem 8.1]) that $\mathcal {H}^{1}(H_{1}) \geq \eta$, since

$$ 1-\mathcal {H}^{1}(H_{1}) \leq \frac{1}{h} \mathcal {H}^{1}(J_{w_{h}} \cap \operatorname{conv}(Q_{z} \cup Q_{z'})) \leq \frac{(1-\eta ) h}{h}. $$

We intersect $H_{1}$ with three large subsets of $(-1/2,1/2)$, namely

By Fubini’s theorem and Markov’s inequality we have $\mathcal {H}^{1}(H_{2}),\mathcal {H}^{1}(H_{3}),\mathcal {H}^{1}(H_{4}) \geq 1- \frac{\eta}{8}$, and thus

$$ \mathcal {H}^{1}(H_{1} \cap H_{2} \cap H_{3} \cap H_{4}) \geq \eta - \frac{3\eta}{8} > \frac{\eta}{2}. $$

By Proposition 1 we have

$$ \frac{8}{\eta}(|\omega _{z}| + |\omega _{z'}|) \leq \frac{16}{\eta} C \delta . $$

If $\delta < \frac{\eta}{16C}$, this ensures that whenever $t\in H_{3} \cap H_{4}$, there are points $x\in Q_{z}\setminus \omega _{z}$, $y\in Q_{z'} \setminus \omega _{z'}$ with $x_{2} = y_{2} = t$, $|x - y| \geq 1$, and

$$ |w(x) - A_{z} x -b_{z}| + |w(y)| \leq \frac{8}{\eta}\left ( \int _{Q_{z} \setminus \omega _{z}} |w(x) - A_{z}x - b|\,dx + \int _{Q_{z'} \setminus \omega _{z'}} |w(x)|\,dx \right ). $$

(11)

This allows us to choose the first two pairs $(p_{i},q_{i})\in (Q_{z}\setminus \omega _{z}) \times (Q_{z'} \setminus \omega _{z'})$, $i =1,2$ such that $(p_{i})_{2} = (q_{i})_{2}\in H_{1}\cap H_{2}\cap H_{3} \cap H_{4}$,

$$ |p_{i} - q_{i}| \geq 1, \quad |(p_{1})_{2} - (p_{2})_{2}| \geq \frac{\eta}{2}. $$

By the definitions of $H_{1}$, $H_{2}$, $H_{3}$, $H_{4}$ we then have

$$ \begin{aligned} &|(p_{i}-q_{i}) \cdot (A_{z} p_{i} +b_{z})| \\ \leq & |(p_{i}-q_{i}) \cdot (w(p_{i}) - w(q_{i}))| + |(p_{i}-q_{i}) \cdot (w(p_{i}) - A_{z} p_{i} - b_{z})| + |(p_{i}-q_{i}) \cdot w(q_{i})| \\ \leq & C(\eta ,N) \sqrt{\int _{\operatorname{conv}(Q_{z} \cup Q_{z'})} |ew(x)|^{2} \,dx}, \end{aligned} $$

where, to estimate the first term, we used the fundamental theorem of calculus

$$ \begin{aligned} (p_{i}-q_{i}) \cdot (w(p_{i}) - w(q_{i})) &= \int _{[p_{i},q_{i}]} (p_{i} - q_{i}) \cdot \nabla w\frac{p_{i} -q_{i}}{|p_{i} - q_{i}|}\,d\mathcal {H}^{1} \\ &= \int _{[p_{i},q_{i}]} (p_{i} - q_{i}) \cdot ew \frac{p_{i} -q_{i}}{|p_{i} - q_{i}|}\,d\mathcal {H}^{1}. \end{aligned} $$

That we may do so for almost every segment not intersecting the jump set is proved for $\mathrm {GSBD}$ functions in e.g. [21].

We now repeat the above argument to obtain one more pair $(p_{3},q_{3})$ with $p_{3}\in Q_{z}\setminus \omega _{z}$, $q_{3}\in Q_{z'}\setminus \omega _{z'}$. Instead of a horizontal line segment, we choose $(p_{3} - q_{3} ) \| e_{\theta }:= \frac{1}{\sqrt{1+\theta ^{2}}}(1, \theta )$ with $\theta := \frac{\eta ^{2}}{N+2}>0$. We define analogously to before

As before, we have $\mathcal {H}^{1}(D_{1}) \geq \eta$, $\mathcal {H}^{1}(D_{2}), \mathcal {H}^{1}(D_{3}), \mathcal {H}^{1}(D_{4}) \geq 1- \frac{\eta}{8}$, so that

$$ \mathcal {H}^{1}(D_{1} \cap D_{2} \cap D_{3} \cap D_{4}) \geq \frac{\eta}{2}. $$

This allows us to pick a diagonal line $z+t+\mathbb{R}e_{\theta}$ with $t\in D_{1} \cap D_{2} \cap D_{3} \cap D_{4}$ and if $\delta < \frac{\eta}{32C}$, we find a pair $p_{3}\in Q_{z}\setminus \omega _{z}$, $q_{3}\in Q_{z'}\setminus \omega _{z'}$ on the diagonal line with $|p_{3} - q_{3}| \geq 1$ and such that (11) holds. Note that as long as $|t \pm 1/2| > \frac{\eta}{4}$, the diagonal line intersects both $Q_{z}$ and $Q_{z'}$. By the definitions of $D_{1}$, $D_{2}$, $D_{3}$, $D_{4}$, (10) holds also for $(p_{3},q_{3})$.

Define the linear map $F\in \mathrm{Lin}(\mathrm {Skew}(2)\times \mathbb{R}^{2}; \mathbb{R}^{3})$ by

$$ F(A,b) := ((Ap_{i} + b) \cdot (p_{i}-q_{i}))_{i=1,2,3}. $$

By (10) we have

$$ |F(A_{z},b_{z})| \leq C(\eta ,N) \sqrt{\int _{\operatorname{conv}(Q_{z} \cup Q_{z'})} |ew|^{2}\,dx }. $$

Using the identity $F^{-1} = (\det F)^{-1} (\operatorname{cof}F)^{T}$, we have

$$ |A_{z}| + |b_{z}| \leq |\det F|^{-1} |F|^{2} |F(A_{z},b_{z})|. $$

We clearly have $|F| \leq (N+3)^{3}$. By the direct calculations in Lemmas 1 and 2, found in the Appendix, we may estimate

$$ |\det F| \geq |p_{1} - q_{1}||p_{2} - q_{2}||p_{3} - q_{3}|\, \frac{\eta}{2}|\sin \theta |^{2} \geq c(\eta ,N), $$

since all three pairs have distance at least 1, the parallel lines have distance at least $\eta /2$, and the angle of the diagonal line is $\theta$. This shows that

$$ |A_{z}| + |b_{z}| \leq C(\eta ,N) \sqrt{\int _{\operatorname{conv}(Q_{z} \cup Q_{z'})} |ew|^{2}\,dx } $$

whenever $\delta < \eta/32C$, completing the proof. □

Remark 7

We note here that the above procedure may be generalized to $d=3$ using Lemma 3 and employing 6 segments instead of 3. We also note that we only used the bound on $|\omega _{z}|$, not the one on its perimeter. In theory, it is also possible to employ a similar construction in nonlinear elasticity, using a geometric rigidity estimate as in [25] for $\mathrm {GSBV}^{2}$ functions with a small jump set.

Remark 8

The choice of line segments connecting two elastic bodies in order to prevent independent rigid motions of either body is encountered in civil engineering in the context of trusses. The proof above, and its three-dimensional version, show that many such trusses exist. Lemmas 2 and 3, found in the Appendix, are potentially useful to the engineering community in the construction of optimal trusses in bridges, scaffolding, towers etc.

6 Proof of Compactness

Here we combine estimates (8) and (9) to show that $\nabla w_{h}$ is very close in $L^{2}(\varOmega _{h};\mathbb{R}^{2\times 2})$ to some $A_{h}\in \mathrm {SBV}^{2}((0,L);\mathrm {Skew}(2))$ and $\#J_{A_{h}} \leq \left \lfloor \liminf _{h\to 0} \frac{1}{h} \mathcal {H}^{1}(J_{w_{h}}) \right \rfloor$. An additional Poincaré inequality then yields estimates for $w_{h}$.

This will imply the compactness result and be useful for the proof of the lower bound.

Proposition 3

Let $w_{h}\in \mathrm {GSBD}^{2}(\varOmega _{h})$ with $\sup _{h>0} F_{h}(w_{h}) < \infty$. Let

$$ M:= \left \lfloor \liminf _{h\to 0} \frac{1}{h} \mathcal {H}^{1}(J_{w_{h}}) \right \rfloor . $$

Then there is a subsequence (not relabeled), a sequence of sets $\omega _{h} \subset \varOmega _{h}$ and sequences of piecewise affine functions $A_{h} \in \mathrm {SBV}^{2}((0,L);\mathrm {Skew}(2))$, $b_{h}\in \mathrm {SBV}^{2}((0,L);\mathbb{R}^{2})$ such that

• (i) $\frac{1}{h} |\omega _{h}| \to 0$ and $\frac{1}{h} \mathcal {H}^{1}(\partial ^{*}\omega _{h}) \leq C$.

• (ii) $\#(J_{A_{h}} \cup J_{b_{h}}) \leq M$ and

  $$ \sup _{h > 0}\int _{0}^{L} |A_{h}'(x_{1})|^{2} + |b_{h}'(x_{1})|^{2} \,dx_{1} < \infty . $$
• (iii)
  $$ \sup _{h > 0} \frac{1}{h^{3}}\int _{\varOmega _{h} \setminus \omega _{h}}| \nabla w_{h}(x) - A_{h}(x_{1})|^{2} \,dx < \infty . $$
• (iv)
  $$ \sup _{h > 0} \frac{1}{h^{3}} \int _{\varOmega _{h} \setminus \omega _{h}} |w_{h}(x) - x_{2}A_{h}(x_{1})e_{2} - b_{h}(x_{1})|^{2}\,dx < \infty . $$

Proof

First, we may assume that $\liminf _{h \to 0} \frac{1}{h} \mathcal {H}^{1}(J_{w_{h}}) = \lim _{h\to 0} \frac{1}{h} \mathcal {H}^{1}(J_{w_{h}}) = M+\alpha$ with $\alpha \in [0,1)$.

Define $\eta := (1-\alpha)/2(M+1) > 0$. Then for $h>0$ small enough we have

$$ \frac{1}{(1-\eta )h} \mathcal {H}^{1}(J_{w_{h}}) < M+1. $$

In addition, let $\delta = \delta (\eta ) > 0$ be as in Proposition 2.

For $z \in \{h,2h, \dots , (\lfloor L/h \rfloor -1) h\}$, we write $Q_{z} = (z-h,z+h) \times (-h/2,h/2)$. We recall from Definition 4 that the size-$h$ rectangle $Q_{z}$ is called $\delta$-good with respect to $w_{h}$ if $\mathcal {H}^{1}(J_{w_{h}} \cap Q_{z}) \leq \delta h$. We write

$$ \mathcal{G}_{h} = \{Q_{z}: z\in \{h,2h,\dots , (\lfloor L/h \rfloor -1) h\}, Q_{z} \text{ is a $\delta $-good rectangle} \}. $$

Then we apply Proposition 1 to each $Q_{z} \in \mathcal{G}_{h}$ and obtain matrices $A_{z} \in \mathrm {Skew}(2)$, vectors $b_{z} \in \mathbb{R}^{2}$ and sets of finite perimeter $\omega _{z} \subseteq Q_{z}$ such that

$$ \int _{Q_{z} \setminus \omega _{z}} |\nabla w_{h} - A_{z}|^{2} + h^{-2} |w_{h} - A_{z} (x - (z,0)) - b_{z}|^{2} \,dx \leq C \int _{\varOmega }|e w_{h}|^{2} \,dx, $$

(12)

and

$$ |\omega _{z}| + \mathcal {H}^{1}(\partial ^{*} \omega _{z}) \leq C \mathcal {H}^{1}(J_{w_{h}} \cap Q_{z}). $$

Let us recall from Proposition 2, c.f. also Remark 6, that for the choice of $\delta$ and two neighboring rectangles $Q_{z},Q_{z+h} \in \mathcal{G}_{h}$ it holds

$$ h^{2} |A_{z} - A_{z+h}|^{2} + h^{2}|b_{z} - b_{z+h}|^{2} \leq C \int _{Q_{z} \cup Q_{z+h}} |e w_{h}|^{2}. $$

(13)

Moreover, we write

$$ \mathcal{B}_{h} = \{Q_{z}: z\in \{h,2h,\dots , (\lfloor L/h \rfloor -1) h\}, Q_{z} \text{ is a $\delta $-bad rectangle} \}. $$

As $E_{h}(w_{h}) \leq C$, we obtain that $\#\mathcal{B}_{h} \leq C/\delta$. Let us now denote by $(C_{h}^{k})_{k=1}^{K_{h}}$ the connected components of $\bigcup _{Q_{z} \in \mathcal{B}_{h}} Q_{z}$, where by construction $K_{h} \leq C/\delta$. We define the connected components with a large jump set as

$$ \mathcal{J}_{h} = \left \{ C_{h}^{k}: \frac{1}{h} \mathcal {H}^{1}(J_{w_{h}} \cap C_{h}^{k}) \geq 1 - \eta \right \}. $$

By the choice of $\eta$ we find that $\# \mathcal{J}_{h} < M+1$ which implies that $\# \mathcal{J}_{h} \leq M$. Moreover, by Proposition 2 we find for our choice of $\delta$ and $C_{h}^{k} \notin \mathcal{J}_{h}$ that

$$ h^{2} |A_{z} - A_{z'}|^{2} + h^{2} |b_{z} - b_{z'}|^{2} \leq C(\eta , N) \int _{\operatorname{conv}(Q_{z} \cup Q_{z'})} |e w_{h}|^{2}\,dx, $$

(14)

where $Q_{z}, Q_{z'} \in \mathcal{G}_{h}$ are the good rectangles neighboring $C_{h}^{k}$ and $N = |z-z'| / h \leq \#\mathcal{B}_{h} \leq \frac{C}{\delta}$. We note that the leftmost and rightmost connected component of bad rectangles may only have one neighboring good rectangle, in which case no estimate is necesssary.

Now we can construct the functions $A_{h} \in \mathrm {SBV}((0,L);\mathrm {Skew}(2))$ by linearly interpolating the values $A_{z}$ between neighboring good rectangles and over connected components of bad rectangles which do not carry a lot of jump set, see Fig. 3. Precisely, we define

If not already defined, we extend this definition constantly on the left and right onto the entire interval $(0,L)$, creating no additional derivative or jump. By our construction, we have $A_{h}(z)= A_{z}$ whenever $Q_{z}$ is $\delta$-good.

Fig. 3

Interpolation procedure to construct $A_{h}$

We define the function $b_{h}$ by interpolating in a similar fashion between the values $b_{z}$.

First we notice that the functions $A_{h}$ and $b_{h}$ can only jump at the center of connected components in $\mathcal{J}_{h}$. Consequently,

$$ \#(J_{A_{h}} \cup J_{b_{h}}) \leq \#\mathcal{J}_{h} \leq M. $$

Moreover, using (13) and (14) we obtain (note that $A_{h}'$ and $b_{h}'$ denote the absolutely continuous part of $D A_{h}$ and $D b_{h}$, respectively)

$$\begin{aligned} \int _{0}^{L} |A_{h}'|^{2} + |b_{h}'|^{2} \,dx_{1} \leq C(\eta ) h^{-3} \int _{\varOmega _{h}} |e w_{h}|^{2} \, dx \lesssim C(\eta ) \end{aligned}$$

This shows (ii).

Next, we define the exceptional set $\omega _{h}$ as the union of the exceptional sets on the good rectangles, all bad rectangles and a boundary layer, i.e.

$$ \omega _{h} = \bigcup _{Q_{z} \in \mathcal{G}_{h}} \omega _{z} \cup \bigcup _{Q_{z} \in \mathcal{B}_{h}} Q_{z} \cup ((\lfloor L/h \rfloor -1 )h, L) \times (-h/2,h/2). $$

It follows from the properties of $\omega _{z} \subseteq Q_{z}$ that

$$ \mathcal {H}^{1}(\partial ^{*} \bigcup _{Q_{z} \in \mathcal{G}_{h}} \omega _{z}) \leq \sum _{Q_{z} \in \mathcal{G}_{h}} \mathcal {H}^{1}(\partial ^{*} \omega _{z}) \leq C\sum _{Q_{z} \in \mathcal{G}_{h}} \mathcal {H}^{1}(Q_{z} \cap J_{w_{h}}) \leq 2C \mathcal {H}^{1}(J_{w_{h}}). $$

By the subadditivity of the squareroot and the isoperimetric inequality it follows that

$$\begin{aligned} \mathcal{L}^{2}\left ( \bigcup _{Q_{z} \in \mathcal{G}_{h}} \omega _{z} \right )^{\frac{1}{2}} \leq \sum _{Q_{z} \in \mathcal{G}_{h}} \mathcal{L}^{2}(\omega _{z})^{\frac{1}{2}} \leq C \sum _{Q_{z} \in \mathcal{G}_{h}} \mathcal {H}^{1}(\partial ^{*} \omega _{z}) \leq 2 C \mathcal {H}^{1}(J_{w_{h}}). \end{aligned}$$

Hence, it follows (recall $\# \mathcal{B}_{h} \leq \frac{C}{\delta }$)

$$\begin{aligned} &\mathcal {H}^{1}(\partial ^{*} \omega _{h}) \leq C \mathcal {H}^{1}(J_{w_{h}}) + 2 \#\mathcal{B}_{h} h + h \leq C(1+1/\delta ) h, \\ &\mathcal{L}^{2}(\omega _{h}) \leq C \mathcal {H}^{1}(J_{w_{h}})^{2} + \# \mathcal{B}_{h} h^{2} + h^{2} \leq C(1+1/\delta ) h^{2}, \end{aligned}$$

which shows (i).

To prove (iii) we estimate using (12) and Hölder’s inequality

$$\begin{aligned} &\int _{\varOmega _{h} \setminus \omega _{h}} |\nabla w_{h}(x) - A_{h}(x_{1})|^{2} \, dx \\ \leq &2 \sum _{Q_{z} \in \mathcal{G}_{h}} \int _{Q_{z} \setminus \omega _{z}} |\nabla w_{h}(x) - A_{z}|^{2} + |A_{z} - A_{h}(x_{1})|^{2} \, dx \\ \leq &C \sum _{Q_{z} \in \mathcal{G}_{h}} \int _{Q_{z}} |ew_{h}|^{2} \, dx + \sum _{Q_{z} \in \mathcal{G}_{h}} 2h^{3} \int _{z-h}^{z+h} |A_{h}'(x_{1})|^{2} \, dx_{1} \\ \leq &2C \int _{\varOmega _{h}} |ew_{h}|^{2} \, dx + 4 h^{3} \int _{0}^{L} |A_{h}'(x_{1})|^{2} \, dx_{1} \\ \leq &C h^{3}. \end{aligned}$$

(15)

Note that for the last inequality we used (ii).

It remains to show (iv). We recall that by Proposition 1 we have for all $Q_{z} \in \mathcal{G}_{h}$ that

$$ \int _{Q_{z} \setminus \omega _{z}} |w_{h}(x) - A_{z}x - b_{z}|^{2} \, dx \leq C h^{2} \int _{Q_{z}} |ew|^{2}\,dx. $$

Hence, we obtain from the definition of $\omega _{h}$ and similar estimates as in (15) that

$$\begin{aligned} &\int _{\varOmega _{h} \setminus \omega _{h}} |w_{h}(x) - A_{h}(x_{1}) x - b_{h}(x_{1})|^{2} \, dx \\ \leq &\sum _{Q_{z} \in \mathcal{G}_{h}} \int _{Q_{z} \setminus \omega _{z}} |w_{h}(x) - A_{h}(x_{1}) x - b_{h}(x_{1})|^{2} \, dx \\ \leq &C \sum _{Q_{z} \in \mathcal{G}_{h}} \int _{Q_{z} \setminus \omega _{z}} |w_{h}(x) - A_{z} x - b_{z}|^{2} + |A_{z} - A_{h}(x_{1})|^{2} + |b_{z} - b_{h}(x_{1})|^{2} \, dx \\ \leq &C \sum _{Q_{z} \in \mathcal{G}_{h}} h^{2} \int _{Q_{z}} |e w_{h}|^{2} \, dx + \sum _{Q_{z} \in \mathcal{G}_{h}} 2h^{3} \int _{z-h}^{z+h} |A_{h}'(x_{1})|^{2} + |b_{h}'(x_{1})|^{2} \, dx_{1} \\ \leq & C h^{2} F_{h}(w_{h}) + C h^{3} \int _{0}^{L} |A_{h}'(x_{1})|^{2} + |b_{h}'(x_{1})|^{2} \, dx_{1} \\ \leq &C h^{3}. \end{aligned}$$

 □

We can now prove Theorem 1:

Proof of Theorem 1

Let us define $T_{h}: \varOmega _{h} \to \varOmega _{1}$ by $T_{h}(x_{1},x_{2}) = (x_{1},x_{2}/h)$. Then we write $\sigma _{h} = T_{h}(\omega _{h})$, where $\omega _{h}$ is the set constructed in Proposition 3. From Proposition 3 (i) we immediately obtain (ii).

Next, we write

$$ M = \left \lfloor \liminf _{h\to 0}\int _{J_{y_{h}}} (\nu _{1}, \frac{1}{h} \nu _{2}) \, d\mathcal {H}^{1} \right \rfloor . $$

Let $A_{h}$ and $b_{h}$ be the functions from Proposition 3. Recall from (ii) in Proposition 3 that $A_{h} \in \mathrm {SBV}((0,L);\mathrm {Skew}(2))$ and $b_{h} \in \mathrm {SBV}((0,L);\mathbb{R}^{2})$ with

$$ \#(J_{A_{h}} \cup J_{b_{h}}) \leq M \text{ and } \sup _{h} \int _{0}^{L} |A_{h}'(x_{1})|^{2} + |b_{h}'(x_{1})|^{2}\,dx_{1} < \infty . $$

(16)

We write $J_{A_{h}} \cup J_{b_{h}}\cup \{0,L\} := \{t_{0}^{h}, \dots , t_{N_{h}+1}^{h} \}$, where $N_{h}\leq M$ is the actual number of jumps of $(A_{h},b_{h})$, and define the function $\bar{A}_{h}: (0,L) \to \mathrm {Skew}(2)$ by (see also Fig. 4)

$${\overline{A}}_h(t)={⨏}_{t_i}^{t_{i+1}}{A}_h(s)ds\text{ if }t\in (t_i,t_{i+1}).$$

It follows immediately that $\bar{A}_{h}$ is piecewise constant and $J_{\bar{A}_{h}} \subseteq J_{A_{h}}$. By a minor modification of $\bar{A}_{h}$, if necessary, we may assume that $J_{\bar{A}_{h}} = J_{A_{h}}$.

Fig. 4

Sketch of the construction of $\bar{A}_{h}$. The function $A_{h}$ evolves from piecewise interpolation. Its graph (more precisely, the graph of the upper right entry of the matrix field $A_{h}$) is sketched in blue. The corresponding graph of $\bar{A}_{h}$ (i.e. the upper right entry of the matrix field $\bar{A}_{h}$) is sketched in green. The field $\bar{A}_{h}$ is constant between jump points of $A_{h}$

We deduce from (16) that $\| A_{h} - \bar{A}_{h} \|_{L^{\infty}(\varOmega _{1};\mathbb{R}^{2\times 2})} \leq C$ and that $A_{h} - \bar{A}_{h}$ is a bounded sequence in $\mathrm {SBV}^{2}((0,L);\mathrm {Skew}(2))$. By the compactness properties of the space $\mathrm {SBV}^{2}$, there exists a (not relabeled) subsequence and $A \in \mathrm {SBV}^{2}((0,L);\mathrm {Skew}(2))$ such that $A_{h} - \bar{A}_{h} \to A$ in $L^{1}((0,L);\mathbb{R}^{2\times 2})$, $A_{h}' \rightharpoonup A'$ in $L^{2}((0,L);\mathbb{R}^{2\times 2})$ and $\#J_{A} \leq \liminf _{h} \# J_{A_{h}}$.

The definition of $\bar{b}_{h}$ is completely analogous. Again, we may assume without loss of generality that $J_{\bar{b}_{h}} = J_{b_{h}}$ and obtain that $\| b_{h} - \bar{b}_{h}\|_{L^{\infty}((0,L);\mathbb{R}^{2})} \leq C$. Then there exists a (not relabeled) subsequence and $b \in \mathrm {SBV}^{2}((0,L);\mathbb{R}^{2})$ such that $b_{h} - \bar{b}_{h} \to b$ in $L^{1}((0,L);\mathbb{R}^{2})$, $b_{h}' \rightharpoonup b'$ in $L^{2}((0,L);\mathbb{R}^{2})$, and $\# J_{b} \leq \liminf _{h \to 0} \# J_{b_{h}}$.

Let us now define the functions $u_{h}: \varOmega _{h} \to \mathbb{R}^{2}$ by

$$ u_{h}(x) = (A_{h}(x_{1}) - \bar{A}_{h}(x_{1}) )x + (b_{h}(x_{1}) - \bar{b}_{h}(x_{1})). $$

It follows that $u_{h} \in \mathrm {SBV}^{2}(\varOmega _{h};\mathbb{R}^{2})$. Moreover, we obtain from the bounds and properties of $A_{h} - \bar{A}_{h}$ and $b_{h} - \bar{b}_{h}$ that

$$ \| u_{h} \|_{L^{\infty}(\varOmega _{h};\mathbb{R}^{2})} \leq C, \int _{\varOmega _{h}} |\nabla u_{h}|^{2} \, dx \leq C h \text{ and } J_{u_{h}} \subseteq (J_{A_{h}} \cup J_{b_{h}}) \times (-h/2,h/2). $$

(17)

Accordingly, we define the rescaled functions $\bar{u}_{h} \in \mathrm {SBV}^{2}(\varOmega _{1}; \mathbb{R}^{2})$ by

$$ \bar{u}_{h}(x_{1},x_{2}) = u_{h}(x_{1},hx_{2}) = (A_{h}(x_{1}) - \bar{A}_{h}(x_{1}) ) (x_{1},hx_{2})^{T} + (b_{h}(x_{1}) - \bar{b}_{h}(x_{1})), $$

(18)

which satisfies by (17)

$$ \| \bar{u}_{h} \|_{L^{\infty}(\varOmega _{1};\mathbb{R}^{2})} \leq C, \int _{ \varOmega _{1}} |\nabla \bar{u}_{h}|^{2} \, dx \leq C \text{ and } J_{ \bar{u}_{h}} \subseteq (J_{A_{h}} \cup J_{b_{h}}) \times (-1/2,1/2). $$

In particular, $\bar{u}_{h}$ is a bounded sequence in $\mathrm {SBV}^{2}(\varOmega _{1};\mathbb{R}^{2})$. Moreover, by the definition, (18), and the compactness properties of $A_{h} - \bar{A}_{h}$ and $b_{h} - \bar{b}_{h}$ it follows $\bar{u}_{h} \to y$ in $L^{1}(\varOmega _{1};\mathbb{R}^{2})$, where

$$ y(x_{1},x_{2}) = A(x_{1})(x_{1},0)^{T} + b(x_{1}). $$

In particular, $y$ does not depend on $x_{2}$. In order to show $y \in \mathcal{A}$ we still need to prove that $\partial _{1} y_{1} = 0$ and $\partial _{1} y_{2} \in \mathrm {SBV}(\varOmega _{1})$.

Next, let us define $\tilde{w}_{h}: \varOmega _{h} \to \mathbb{R}^{2}$ by . From Proposition 3 (i) and (iii) we deduce that $\tilde{w}_{h} \in \mathrm {GSBV}^{2}(\varOmega _{h};\mathbb{R}^{2})$ and

$$\begin{aligned} \int _{\varOmega _{h}} |\nabla \tilde{w}_{h}|^{2} \, dx &= \int _{ \varOmega _{h} \setminus \omega _{h}} |\nabla w_{h} - \bar{A}_{h}|^{2} \, dx \\ &\leq 2 \int _{\varOmega _{h} \setminus \omega _{h}} |\nabla w_{h} - A_{h}|^{2} + |A_{h} - \bar{A}_{h}|^{2} \, dx \\ &\leq C(h^{3} + h). \end{aligned}$$

Moreover, $\mathcal {H}^{1}(J_{\tilde{w}_{h}}) \leq \mathcal {H}^{1}(\partial ^{*} \omega _{h}) + \mathcal {H}^{1}(J_{w_{h}}) + h \#(J_{A_{h}} \cup J_{b_{h}}) \leq Ch$. It follows from Proposition 3 (iv) that

$$ \int _{\varOmega _{h}} |\tilde{w}_{h}|^{2} \, dx \leq C h. $$

The above shows that the functions $\tilde{y}_{h}: \varOmega _{1} \to \mathbb{R}^{2}$ defined by

form a bounded sequence in $L^{2}(\varOmega _{1};\mathbb{R}^{2})$, are in $\mathrm {GSBV}^{2}(\varOmega _{1};\mathbb{R}^{2})$ and satisfy

$$ \int _{\varOmega _{1}} | \nabla _{h} \tilde{y}_{h}|^{2} \, dx \leq C \text{ and } \int _{J_{\tilde{y}_{h}}} |(\nu _{1}, \frac{1}{h} \nu _{2})| \, d\mathcal {H}^{1} \leq C, $$

where $\nu \in S^{1}$ is the measure-theoretic normal to $J_{\tilde{y}_{h}}$. It follows immediately that $\tilde{y}_{h}$ is a bounded sequence in $\mathrm {GSBV}^{2}(\varOmega _{1};\mathbb{R}^{2})$.

Moreover, it follows from Proposition 3 (iv) that

As $\mathcal{L}^{2}(\sigma _{h}) \to 0$ (c.f. Proposition 3 (i)), it follows that $\tilde{y}_{h} \to y$ in $L^{2}(\varOmega _{1};\mathbb{R}^{2})$, which is (i). By the compactness properties of $\mathrm {GSBV}^{2}(\varOmega _{1};\mathbb{R}^{2})$ it even follows that

$$ \nabla \tilde{y}_{h} \rightharpoonup \nabla y \text{ in } L^{2}( \varOmega _{1};\mathbb{R}^{2\times 2}). $$

On the other hand, we find that

However, this implies that $A = (\partial _{1} y, c)$ for a function $c \in \mathrm {SBV}^{2}(\varOmega _{1};\mathbb{R}^{2})$. As $A$ is skew-symmetric, it follows immediately that $\partial _{1} y_{1} = 0$. Moreover, we deduce from the fact that $A \in \mathrm {SBV}^{2}((0,L);\mathrm {Skew}(2))$ and that $\partial _{1} y \in \mathrm {SBV}^{2}(\varOmega _{1})$. This concludes the proof of $y,\partial _{1} y \in \mathrm {SBV}((0,L);\mathbb{R}^{2})$.

It remains to define the set $J$ and to show the inequalities in (iii).

First, by extracting a subsequence, we can ensure that the number of jumps $N_{h} = \#(\bar{A}_{h} \cup \bar{b}_{h}) \leq M$ is constant. Then by extracting a further subsequence, we can ensure that each succesive jump point converges to some $t_{i} := \lim _{h\to 0}t^{h}_{i}\in [0,L]$. By dropping those $t_{i}$ that are either 0 or $L$, we arrive at the set $J := \{t_{1},\ldots ,t_{N}\}$ with $N\leq M$. By our construction, $J_{y} \cup J_{\partial _{1} y} \subset J$.

At this stage, $J_{\bar{A}_{h}} \cup J_{\bar{b}_{h}}$ is not contained in $J$ but very close to it or to $\{0,L\}$. We define slightly different piecewise constant functions by moving the jumps, calling the new functions $\bar{\bar{A}}_{h}:(0,L)\to \mathrm {Skew}(2)$ and $\bar{\bar{b}}_{h}:(0,L)\to \mathbb{R}^{2}$, with $J_{\bar{\bar{A}}_{h}} \cup J_{\bar{\bar{b}}_{h}} \subseteq J$. We also enlarge $\sigma _{h}$ to contain the union of at most $M$ short rectangles between $J_{\bar{A}_{h}} \cup J_{\bar{b}_{h}}$ and $J$. This modification leaves the estimate $|\sigma _{h}|\to 0$ intact, while increasing the surface term $\int _{\partial ^{*} \sigma _{h}} | (\nu _{1}, \frac{1}{h} \nu _{2})| \,d\mathcal {H}^{1}$ by at most $2M$. This proves (iii). □

7 The Lower Bound

In this section, we will prove the lower bound claimed in Theorem 2. As usual we may consider sequences $(y_{h})_{h}$ which are equibounded in energy. In particular, due to the compactness theorem, up to a subsequence this sequence has a limit $y$ in the sense of Theorem 1. First we prove the following result to identify the limit of the rescaled gradient of $y_{h}$.

Proposition 4

Let $y_{h}$ be a sequence as in Theorem 1with limit $(y,J)\in \mathcal{A}$. Define the function $W_{h}: \varOmega _{1} \to \mathbb{R}$ as

where $\sigma _{h} = \{(x_{1},x_{2}) \in \varOmega _{1}: (x_{1},hx_{2}) \in \omega _{h} \}$ for $\omega _{h} \subseteq \varOmega _{h}$ from Proposition 3. Then up to subsequences $W_{h}$ converges weakly in $L^{2}(\varOmega _{1})$ to a function $W \in L^{2}(\varOmega _{1})$ of the form

$$ W(x_{1},x_{2})= -x_{2} \, \partial _{1} \partial _{1} y_{2}(x_{1}) + T(x_{1}), $$

where $T \in L^{2}((0,L))$.

Remark 9

The proof of Proposition 4 is based on the following observation: $\nabla y_{h}$ is very close to $A_{h}\in \mathrm {SBV}^{2}((0,L);\mathrm {Skew}(2))$, whereas $y_{h} - \bar{A}_{h} x - \bar{b}_{h} \to y\in \mathcal{A}$. Assuming exact equality $\nabla y_{h}(x) = A_{h}(x_{1})$, it follows from Schwarz’s Theorem and skew-symmetry of $A_{h}$ that

$$ \partial _{2} \partial _{1} (y_{h})_{1} = \partial _{1} \partial _{2} (y_{h})_{1} = -\partial _{1} \partial _{1} (y_{h})_{2} = -\partial _{1} \partial _{1} (y_{h} - \bar{A}_{h} x - \bar{b}_{h})_{2} \rightharpoonup -\partial _{1} \partial _{1} y_{2}. $$

Since $y$ only depends on $x_{1}$, the structure $\partial _{1} (y_{h})_{1} \stackrel {*}{\rightharpoonup }-x_{2} \partial _{1}\partial _{1} y_{2}(x) + T(x_{1})$ follows immediately. In general, we do not have exact equality $\nabla y_{h} = A_{h}$. In fact, $y_{h}$ is not twice differentiable, and of course may have jumps. In the proof, we use second differences of $y_{h}$ instead of second derivatives.

Remark 10

Simlarly, one could identify the limit of as a function of the form $-x_{2} \, A'(x_{1}) e_{2} + T(x_{1})$, where $A_{h}$ are the skew-symmetric functions constructed in Proposition 3 and $A$ is the corresponding limit constructed in the proof of Theorem 1. As $A$ is skew-symmetric it holds $A'(x_{1}) e_{2} \cdot e_{2} = 0$. Hence, it is sufficient to determine the first component in this setting. In the derivation of rod theories, this first entry only carries the information on bending but not on torsion, see [28].

Proof

Boundedness of $E_{h}$ implies immediately that $W_{h}$ is bounded in $L^{2}(\varOmega _{1};\mathbb{R}^{2})$, a weakly convergent subsequence exists.

Let us fix such a limit $W \in L^{2}(\varOmega _{1};\mathbb{R}^{2})$ and a corresponding (not relabeled) subsequence.

To show the special form of $W$, we consider for fixed $z\in (0,1)$ the second difference $V_{h}:(0,L-s_{h}) \times (-1/2,1/2 - z) \to \mathbb{R}^{2}$, depending on $s_{h}>0$, defined by

Here, whenever all of $y_{h}$, , $A_{h}$ (from Proposition 3), $y$ and $\nabla y$ are absolutely continuous along the boundary of the rectangle spanned by the four points $x$, $x+ze_{2}$, $x+ze_{2}+s_{h} e_{1}$, $x+ s_{h} e_{1}$, and 0 otherwise.

As the jump sets of $y$, $\nabla y$, and $A_{h}$ are purely vertical, it holds

Here, $\nu$ denotes the measure theoretic normal to $J_{y_{h}}$ and $\partial ^{*} \sigma _{h}$, respectively. By Theorem 1 (ii) and the bounds on the energy we have

$$ \mathcal {H}^{1}(J_{y_{h}}) + \mathcal {H}^{1}(\partial ^{*} \sigma _{h}) + \#(J_{A_{h}} \cup J_{y} \cup J_{\nabla y}) \leq C $$

for all $h>0$ and

$$ \lim _{h\to 0} \left ( \int _{J_{y_{h}}} |\nu \cdot e_{2}|\,d\mathcal {H}^{1} + \int _{\partial ^{*} \sigma _{h}} |\nu \cdot e_{2}|\,d\mathcal {H}^{1}\right ) = 0, $$

so that

as long as $s_{h} \to 0$.

Now let $f\in C_{c}^{\infty}((0,L) \times (-1/2,1/2-z))$ be a test function. Then we have for $s_{h} \to 0$, with $\psi _{h}(x,t) = \partial _{1} (y_{h})_{1}(x+ze_{2} + ts_{h}e_{1})/h - \partial _{1} (y_{h})_{1}(x+ts_{h}e_{1})/h$,

Conversely, we have for e.g. $s_{h} = h^{\frac{1}{2}}$ with $\varphi _{h}(x,t) = \partial _{2} (y_{h})_{1}(x+tze_{2} + s_{h} e_{1})/h s_{h} - \partial _{2} (y_{h})_{1}(x+tze_{2})/h s_{h}$

For the second equality, we used the independence of $A_{h}$, $y$ from $x_{2}$ and the bound , c.f. (iii) in Proposition 3.

For the final equality, we used the weak convergence $A_{h}'\rightharpoonup \begin{pmatrix} 0 & - \partial _{1} y_{2} \\ \partial _{1} y_{2} & 0\end{pmatrix}$ in $L^{2}((0,L);\mathrm {Skew}(2))$, c.f. the proof of Theorem 1.

It follows that $W(x+ze_{2}) - W(x) = -z \partial _{1}\partial _{1} y_{2}(x)$ for almost every $x\in \varOmega _{1}$.

We define $T(x_{1}) := \int _{-1/2}^{1/2} W(x_{1},z)\,dz$. It follows for almost every $x_{1} \in (0,L)$

$$\begin{aligned} W(x_{1},x_{2}) - T(x_{1}) & = \int _{-1/2}^{1/2} W(x_{1},x_{2}) - W(x_{1},z) \, dz \\ & = \int _{-1/2}^{1/2} -(x_{2}-z) \, (\partial _{1} \partial _{1} y_{2}(x_{1}) ) \, dz \\ &= -x_{2} \, (\partial _{1} \partial _{1} y_{2}(x_{1})). \end{aligned}$$

 □

Now we can prove the lower bound in Theorem 2.

Proof of Theorem 2 (i)

By standard arguments we may always assume that it holds $\liminf _{h\to 0} E_{h}(y_{h}) = \lim _{h \to 0} E_{h}(y_{h})$ and $\sup _{h} E_{h}(y_{h}) < \infty$.

Then we notice that by the assumed convergence it holds that, c.f. Theorem 1 (iii),

$$ \#J \leq \liminf _{h\to 0} \int _{J_{y_{h}}} |(\nu _{1},\frac{1}{h} \nu _{2})| \, d \mathcal {H}^{1}. $$

(19)

Next, we show that

$$ \liminf _{h \to 0} \frac{1}{2} \int _{\varOmega _{1}} \mathbb{C} \nabla _{h} y_{h} : \nabla _{h} y_{h} \, dx \geq \frac{1}{24} \int _{\varOmega _{1}} a |\partial _{1} \partial _{1} y_{2}|^{2} \, dx, $$

where $a>0$ is defined in (5). First, we apply Proposition 3 to the function $w_{h}(x_{1},x_{2}) = y_{h}(x_{1},x_{2}/h)$ to obtain sets $\omega _{h} \subseteq \varOmega _{h}$ and $A_{h} \in \mathrm {SBV}^{2}((0,L);\mathrm {Skew}(2))$ for which the properties (i), (ii) and (iii) of Proposition 3 hold. Again we denote by $\sigma _{h}$ the analogue of $\omega _{h} \subseteq \varOmega _{h}$ in $\varOmega _{1}$. Then (up to a subsequence) it holds by Proposition 4 that the sequence converges weakly in $L^{2}(\varOmega _{1})$ to a function $W \in L^{2}(\varOmega _{1})$ of the form

$$ W(x_{1},x_{2}) = -x_{2} \, (\partial _{1} \partial _{1} y_{2}(x_{1}) ) + T(x_{1}), $$

where $T \in L^{2}((0,L))$. It follows by the definition of the bending constant $a$, c.f. (5), that

$$\begin{aligned} \liminf _{h\to 0} \frac{1}{h^{2}} \frac{1}{2} \int _{\varOmega _{1}} \mathbb{C} \nabla _{h} y_{h} : \nabla _{h} y_{h} \, dx \geq &\liminf _{h \to 0} \frac{1}{2} \int _{\varOmega _{1}} a \,|W_{h}(x)|^{2} \, dx \\ \geq & \frac{1}{2} \int _{\varOmega _{1}} a \, |W(x)|^{2} \, dx. \end{aligned}$$

Using the specific form of $W$ we compute

$$\begin{aligned} \int _{\varOmega _{1}} a\, |W |^{2} \,dx = &\int _{\varOmega _{1}} a \, |-x_{2} \partial _{1} \partial _{1} y(x_{1}) + T(x_{1})|^{2} dx \\ \geq &\int _{0}^{L} \int _{-1/2}^{1/2} \left (a \, x_{2}^{2} \left ( \partial _{1} \partial _{1} y_{2}(x_{1}) \right )^{2} - 2a x_{2} \,( \partial _{1} \partial _{1} y_{2}(x_{1})) \, T(x_{1}) \right )\, dx_{2} dx_{1} \\ =& \frac{1}{12} \int _{0}^{L} a \, |\partial _{1} \partial _{1} y_{2}(x_{1}) |^{2} \, dx_{1}. \end{aligned}$$

(20)

Combining (19) and (20) yields

$$ \liminf _{h \to 0} E_{h}(y_{h}) \geq \frac{1}{24}\int _{0}^{L} a | \partial _{1} \partial _{1} y_{2}(x_{1}) |^{2} \, dx_{1} + \beta \# J. $$

 □

8 The Upper Bound

In this section we prove the existence of a recovery sequence for the energy $E_{0}$, i.e. we show Theorem 2 (ii).

Proof of Theorem 2 (ii)

We may assume that $(y,J) \in \mathcal{A}$ (otherwise there is nothing to show). Hence, $y \in \mathrm {SBV}^{2}(\varOmega _{1}; \mathbb{R}^{2})$ does not depend on $x_{2}$, $\partial _{1} y_{2} \in \mathrm {SBV}^{2}(\varOmega _{1})$ and $\partial _{1} y_{1} = 0$. In addition, $J\subset (0,L)$ is finite and $y$, $\partial _{1} y$ jump only on $J\times (-1/2,1/2)$.

In particular, we remark that $\partial _{1} \partial _{1} y_{2} \in L^{2}(\varOmega _{1})$ does not depend on $x_{2}$. Therefore we can find for every $\eta >0$ a function $g_{\eta} \in H^{1}(\varOmega _{1})$ which does not depend on $x_{2}$ such that $\| -\partial _{1} \partial _{1} y_{2} - g_{\eta} \|_{L^{2}(\varOmega _{1})} \leq \eta$. By the embedding $H^{1}((0,L)) \hookrightarrow L^{\infty}((0,L))$ in one dimension it follows that $\| g_{\eta} \|_{L^{\infty}(\varOmega _{1})} \leq C(\eta )$. In addition, we note that since $\partial _{1} y_{2} \in \mathrm {SBV}^{2}(\varOmega _{1})$ does not depend on $x_{2}$ we obtain that $\partial _{1} y_{2} \in L^{\infty}(\varOmega _{1})$. Since $\partial _{2} y_{2} = 0$, we obtain $\nabla y_{2} \in L^{\infty}(\varOmega _{1};\mathbb{R}^{2})$.

Moreover, we recall from the definition of the bending coefficient $a$, c.f. (5), that there exist $b,c \in \mathbb{R}$ such that

$$ a = \mathbb{C} \begin{pmatrix} 1 && b \\ 0 && c \end{pmatrix} : \begin{pmatrix} 1 && b \\ 0 && c \end{pmatrix} . $$

Now we define $y_{h}: \varOmega _{1} \to \mathbb{R}^{2}$ by

$$ y_{h}(x_{1},x_{2}) = y(x_{1},0) - x_{2} h \nabla y_{2}(x_{1},0) + \frac{1}{2} x_{2}^{2} h^{2} g_{\eta}(x_{1},0) \begin{pmatrix} b \\ c \end{pmatrix} . $$

One sees immediately that $y_{h} \to y$ in $L^{\infty}(\varOmega _{1};\mathbb{R}^{2})$.

Moreover, the jumps of $y_{h}$ only occur in $x_{1}$-direction and $J_{y_{h}} = J_{y} \cup J_{\partial _{1} y} \subseteq J$. If $J \setminus (J_{y} \cup J_{\partial _{1} y})$ is nonempty, we add a small piecewise constant function to $y_{h}$ to ensure that $J_{y_{h}} = J$.

In addition, $\nabla _{h} y_{h} = (\partial _{1} y_{h},\frac{1}{h} \partial _{2} y_{h})$ is given by

$$\begin{aligned} &\begin{pmatrix} - hx_{2} \, \partial _{1} \partial _{1} y_{2} + \frac{1}{2} x_{2}^{2} h^{2} \, b \,\partial _{1} g_{\eta} & -\partial _{1} y_{2} + x_{2} h \, b g_{ \eta} \\ \partial _{1} y_{2} + \frac{1}{2} x_{2}^{2} h^{2} \, c \, \partial _{1} g_{\eta} & x_{2} h \,c g_{\eta} \end{pmatrix} \\ &= \underbrace{\begin{pmatrix} 0 & -\partial _{1} y_{2} \\ \partial _{1} y_{2} & 0 \end{pmatrix}}_{ \text{skew-symmetric}} + \begin{pmatrix} - hx_{2} \, \partial _{1} \partial _{1} y_{2} + \frac{1}{2} x_{2}^{2} h^{2} \, b \, \partial _{1} g_{\eta} & x_{2} h \, b g_{\eta} \\ \frac{1}{2} x_{2}^{2} h^{2} \, c \,\partial _{1} g_{\eta} & x_{2} h \, c g_{\eta} \end{pmatrix}. \end{aligned}$$

Plugging into the elastic energy and using the $x_{2}$-independence of the occuring functions we find

$$\begin{aligned} &\frac{1}{2} \int _{\varOmega _{1}} \mathbb{C} (\partial _{1} y_{h}, \frac{1}{h} \partial _{2} y_{h}) : (\partial _{1} y_{h},\frac{1}{h} \partial _{2} y_{h}) \, dx \\ =& \frac{1}{2} \int _{\varOmega _{1}} h^{2} x_{2}^{2} \mathbb{C} \begin{pmatrix} -\partial _{1} \partial _{1} y_{2} & b g_{\eta} \\ 0 & b g_{\eta} \end{pmatrix} : \begin{pmatrix} -\partial _{1} \partial _{1} y_{2} & b g_{\eta} \\ 0 & b g_{\eta} \end{pmatrix} \\ &\quad + x_{2}^{3}h^{3} \, \mathbb{C} \begin{pmatrix} - \,\partial _{1} \partial _{1} y_{2} & b g_{\eta} \\ 0 & c g_{\eta} \end{pmatrix} : \begin{pmatrix} b (\partial _{1} g_{\eta}) & 0 \\ c(\partial _{1} g_{\eta})& 0 \end{pmatrix} \\ &\quad + \frac{1}{4} x_{2}^{4} h^{4} (\partial _{1} g_{\eta})^{2} \, \mathbb{C} \begin{pmatrix} b & 0 \\ c& 0 \end{pmatrix} : \begin{pmatrix} b & 0 \\ c& 0 \end{pmatrix} \, dx \\ = & \frac{1}{2} \int _{\varOmega _{1}} h^{2} x_{2}^{2} \mathbb{C} \begin{pmatrix} -\partial _{1} \partial _{1} y_{2} & -(\partial _{1} \partial _{1} y_{2}) \, b \\ 0 & -(\partial _{1} \partial _{1} y_{2}) \, c \end{pmatrix} : \begin{pmatrix} -\partial _{1} \partial _{1} y_{2} & -(\partial _{1} \partial _{1} y_{2}) \, b \\ 0 & -(\partial _{1} \partial _{1} y_{2}) \, c \end{pmatrix} \\ &\quad - h^{2} x_{2}^{2} \mathbb{C} \begin{pmatrix} 0 & \partial _{1} \partial _{1} y_{2} + g_{\eta} \, b \\ 0 & \partial _{1} \partial _{1} y_{2} + g_{\eta} \, c \end{pmatrix} : \begin{pmatrix} - 2\partial _{1} \partial _{1} y_{2} & -(\partial _{1} \partial _{1} y_{2} - g_{\eta}) \, b \\ 0 & -(\partial _{1} \partial _{1} y_{2} - g_{\eta}) \, c \end{pmatrix} \\ &\quad + \frac{1}{4} x_{2}^{4} h^{4} (\partial _{1} g_{\eta})^{2} \, \mathbb{C} \begin{pmatrix} b & 0 \\ c& 0 \end{pmatrix} : \begin{pmatrix} b & 0 \\ c& 0 \end{pmatrix} \, dx \\ \leq & \frac{1}{2} \int _{\varOmega _{1}} h^{2} x_{2}^{2} a \, | \partial _{1} \partial _{1} y_{2}|^{2} \, dx + C h^{2} (\| \partial _{1} \partial _{1} y_{2} \|_{L^{2}(\varOmega _{1})} + \eta ) \| \partial _{1} \partial _{1} y_{2} + g_{\eta} \|_{L^{2}(\varOmega _{1})} + C h^{4} C( \eta ) \\ \leq & \frac{1}{2} \int _{\varOmega _{1}} h^{2} x_{2}^{2} a \, | \partial _{1} \partial _{1} y_{2}|^{2} \, dx + C \eta h^{2} (\| \partial _{1} \partial _{1} y_{2} \|_{L^{2}(\varOmega _{1})}+\eta ) + C h^{4} C(\eta ). \end{aligned}$$

As $J_{y_{h}} = J \times (-1/2,1/2)$ is purely vertical, it follows that

$$\begin{aligned} &\limsup _{h\to 0} E_{h}(y_{h}) \\ = &\limsup _{h\to 0} h^{-2}\int _{\varOmega _{1}} \frac{1}{2} \mathbb{C}( \nabla _{h} y_{h}): (\nabla _{h} y_{h}) \, dx + \beta \int _{J_{y_{h}}} |(\nu _{1},\nu _{2}/h)| \, d\mathcal {H}^{1} \\ \leq &\frac{1}{2} \int _{\varOmega _{1}} x_{2}^{2} a \, |\partial _{1} \partial _{1} y_{2}|^{2} \, dx + C\eta + \beta \# J \\ = &\frac{1}{24} \int _{\varOmega _{1}} a \,| \partial _{1}\partial _{1} y_{2}|^{2} \, dx + \beta \# J. \end{aligned}$$

A diagonal argument in $\eta$ and $h$ finishes the proof. □

9 Discussion

In this section we want to briefly comment on the presented results and discuss future directions. First, we acknowledge that the presented $\varGamma$-convergence result is very similar to the $n$-dimensional results already obtained in [8] and [1]. However, we note that the compactness result presented in this paper and also the topology used for the $\varGamma$-convergence result is more general. In [8] uniform $L^{\infty}$-bounds are assumed a priori, in [1] the authors invoke a compactness result in $\mathrm {GSBD}$ due to Chambolle and Crismale, [17]. The latter result essentially yields for a sequence $(y_{h})_{h} \subseteq \mathrm {GSBD}(\varOmega _{1})$ such that $\sup _{h} \int _{\varOmega _{1}} |e(w_{h})|^{2} \, dx < \infty$ and $\sup _{h} \mathcal{H}^{d-1}(J_{y_{h}}) < \infty$ that there is a (not relabeled) subsequence for which the set $A = \{ x \in \varOmega _{1}: y_{h}(x) \to \infty \}$ has finite perimeter and outside $A$ one has $y_{h} \to y$ pointwise a.e. for some $y \in \mathrm {GSBD}(\varOmega _{1})$. Moreover, $\mathcal{H}^{d-1}(\partial ^{*} A \cup J_{y}) \leq \liminf _{h} \mathcal{H}^{d-1}(J_{y_{h}})$. The compactness statement in this work also characterizes the limiting behavior of the sequence $y_{h}$ on the set $A$. Namely, we identify suitable rigid motion which we locally subtract from $y_{h}$ such that the remaining sequence is essentially compact in $L^{2}(\varOmega _{1};\mathbb{R}^{2})$. In addition, these rigid motions do not create additional jump in the limit, c.f. Theorem 1.

It has been brought to our attention by a referee that there is a follow-up article by Chambolle and Crismale, [16], where the compactness result is extended. Analogously to our compactness result Theorem 1, the authors subtract a piecewise rigid motion $a_{h}$ from different regions of $\varOmega _{1}$ to obtain convergence in measure of the difference $y_{h} - a_{h}$. We cannot directly apply the compactness result in [16], since our setting features varying domains $\varOmega _{h}$.

Moreover, we note that the presented analysis is closely related to the techniques presented in [36], where the author derives a beam theory from a rotationally invariant model. A key role in the analysis plays a geometric rigidity estimate for functions $y \in \mathrm {SBV}$, [24], which subdivides the domain into different regions in which the function $y$ is close to a rigid motion, see [36, Theorem 3.5]. The estimate is essentially sharp in the sense that the perimeter of the identified regions is up to a small error controlled by the size of the jump set of $y$.

The presented techniques can be extended to derive a three-dimensional rod theory for brittle materials in the linearized setting. Similarly to two-dimensional beams, three-dimensional rods can undergo stretching and bending. However, in addition one can observe torsion, i.e. the twisting of the rod around its axis, see, for example, [34] and the references therein. In contrast to bending or stretching, in order to capture torsion in the limit, one has to keep track of the limit of $\nabla w_{h} \approx A_{h}$. Note that a three-dimensional version of Proposition 2 can be proved using Lemma 3 instead of Lemma 2. A corresponding paper is in preparation, [28].

Appendix:  Rigidity for Trusses

Definition 5

Let $d, N\in \mathbb{N}\setminus \{0\}$. Given $N$ pairs $(p_{i},q_{i}) \in \mathbb{R}^{d} \times \mathbb{R}^{d}$, $i=1,\ldots ,N$, consider the linear map $F: \mathrm {Skew}(d) \times \mathbb{R}^{d} \to \mathbb{R}^{N}$ defined by

$$ F(A, b) = \left ( (Ap_{i} + b)\cdot (p_{i} - q_{i}) \right )_{i=1}^{N}. $$

We are interested in the special case $N = \dim (\mathrm {Skew}(d)) + d = d(d+1)/2$, where $F\in \mathbb{R}^{N\times N}$ is a square matrix. We then ask for which pairs $(p_{i},q_{i})$ the associated $F$ is invertible, and what is its determinant.

Remark 11

Assume that $q_{i} = 0$ for all $i=1,\ldots ,N$. Then $F(A, b) = F(0,b)$ for all $A\in \mathrm {Skew}(d)$, since $Ax \cdot x = 0$ for all $x\in \mathbb{R}^{d}$. In particular, $F$ is not invertible.

Lemma 1

Let $d\in \mathbb{N}\setminus \{0\}$, $N = d(d+1)/2$, $p_{i} \neq q_{i}$. Then there exists a function $f_{d} : (G^{d})^{N} \to \mathbb{R}$ such that

$$ \det F = d_{1} \ldots d_{N} f_{d}(L_{1},\ldots ,L_{N}), $$

(21)

where $d_{i} = |p_{i} - q_{i}|$, and

$$ L_{i}= \left [\left (p_{i}, \frac{p_{i} - q_{i}}{|p_{i} - q_{i}|} \right )\right ]\in (\mathbb{R}^{d} \times S^{d-1}) / \sim $$

is the oriented line obtained by extending the line segment $[p_{i},q_{i}]$, which is an element of the oriented affine line Grassmannian $G^{d} := \mathbb{R}^{d} \times S^{d-1} / \sim$, with $(x,v) \sim (z,w)$ whenever $v = w$ and $x-z \| \ v$.

In addition, the function $f_{d}$ has the property $f_{d}\circ \sigma = \det \sigma f_{d}$ for all permutations $\sigma$. Replacing any one line $[(p_{i},v_{i})]$ with its opposite $[(p_{i},-v_{i})]$ flips the sign of $f_{d}$ and $f_{d}$ is invariant under a single rotation or translation applied to all lines $L_{1}, \dots , L_{N}$.

Proof

Let $L_{i} = [(p_{i},v_{i})] \in G^{d}$. We define $f_{d}: (G^{d})^{N} \to \mathbb{R}$ by

$$ f_{d}(L_{1},\dots , L_{N}) = \det F, $$

where $F: \mathrm {Skew}(d) \times \mathbb{R}^{d} \to \mathbb{R}^{N}$ is the linear mapping such that

$$ F(A,b) = ((Ap_{i} + b) \cdot v_{i})_{i=1}^{N}. $$

This definition is independent from the representative. Indeed, let $[(q_{i},w_{i})] = L_{i}$. Then $p_{i}-q_{i}$ is parallel to $v_{i}=w_{i}$. Since $A$ is skew-symmetric, we find that

$$ (Aq_{i} + b)\cdot w_{i} = (Ap_{i} + b) \cdot v_{i} + (A(q_{i}-p_{i})) \cdot w_{i} = (Ap_{i} + b)\cdot v_{i}. $$

This shows that $f_{d}$ is well-defined. The formula (21) and the properties of $f_{d}$ follow immediately from the properties of the determinant. □

Remark 12

By Remark 11 and Lemma 1, we have $\det F = 0$ if all extended lines intersect in the same point or at least two lines are the same. We now show that these are the only cases where this happens.

Lemma 2

Let $L_{1},L_{2},L_{3}\in G^{2}$ be three oriented lines. Then

• (i) If $L_{1}$, $L_{2}$, $L_{3}$ are parallel, then $f_{2}(L_{1},L_{2},L_{3}) = 0$.

• (ii) If $L_{1}$ intersects $L_{2}$ in $p\in \mathbb{R}^{2}$ with angle $\alpha \in (0,\pi /2)$ and $L_{3}$ in $q\in \mathbb{R}^{2}$ with angle $\beta \in (0,\pi /2)$, then

  $$ |f_{2}(L_{1},L_{2},L_{3})| = |p-q| |\sin \alpha | |\sin \beta |. $$

  (22)

Note that this covers all cases. Note that if $L_{1}$, $L_{2}$, $L_{3}$ are pairwise transversal, then formula (22) is independent of the order of $L_{1}$, $L_{2}$, $L_{3}$. If two of the three lines are parallel, we have to fix the order so that $L_{1}$ is the transversal direction. In this case $\alpha = \beta$ in (22), and this is indeed the case in the proof of Proposition 2.

Proof

We work with variables $a$, $b_{1}$, $b_{2}$, where $A = a \begin{pmatrix} 0&1 \\ -1&0\end{pmatrix}$ and $b = (b_{1},b_{2})^{T}$ which defines the standard basis of $\mathrm {Skew}(2) \times \mathbb{R}^{2}$. In these coordinates, if $L_{i} = [(p_{i}, v_{i})]$, the mapping from Definition 5 is represented by a matrix $\tilde{F} \in \mathbb{R}^{3\times 3}$ whose rows are $(p_{i}\wedge v_{i}, (v_{i})_{1}, (v_{i})_{2})\in \mathbb{R}^{3}$, where as usual $x \wedge y = x_{1} y_{2} - x_{2} y_{1}$ for $x,y \in \mathbb{R}^{2}$.

In case (i), $v_{1} = v_{2} = v_{3}$, the rows of $\tilde{F}$ are linearly dependent, and $\det \tilde{F} = 0$. Hence, $f(L_{1},L_{2},L_{3}) = 0$.

In case (ii), we apply a rigid motion, so that $v_{2} = (1,0)^{T}$, $p_{1} = p_{2} = p = (0,0)^{T}$. Then $p_{3} = q = \pm |p-q|v_{1}$. We calculate explicitly

$$\begin{aligned} |f_{2}(L_{1},L_{2},L_{3})| &= \left | \det \begin{pmatrix} 0&(v_{1})_{1}&(v_{1})_{2} \\ 0&1&0 \\ (p-q)\wedge v_{3}&(v_{3})_{1}&(v_{3})_{2} \end{pmatrix} \right | \\ &= |(p-q)\wedge v_{3}||v_{1} \wedge v_{2}| = |p-q||\sin \beta | | \sin \alpha | . \end{aligned}$$

 □

We now turn to the three-dimensional case. The space $\mathrm {Skew}(3) \times \mathbb{R}^{3}$ has the canonical basis

$$ (\alpha , \beta , \gamma , b_{1},b_{2},b_{3}) \mapsto \left ( \begin{pmatrix} 0 & \alpha & \beta \\ -\alpha & 0 & \gamma \\ -\beta & -\gamma & 0\end{pmatrix} , \begin{pmatrix} b_{1} \\ b_{2} \\ b_{3} \end{pmatrix} \right ). $$

If $L_{i} = [(p_{i}, v_{i})]$, the $i$th row of $F$ is then given by

$$ (p_{i} \times v_{i}, v_{i})\in \mathbb{R}^{6}, $$

where $p_{i} \times v_{i}$ denotes the usual cross product in $\mathbb{R}^{3}$.

Lemma 3

Let $L_{i} = [(p_{i},v_{i})]\in G^{3}$ for $i=1,\ldots ,6$, and assume that $v_{1} = v_{2} = v_{3}$. Define $\overline{v} := v - (v_{1}\otimes v_{1}) v \in (v_{1})^{\perp}$ for $v\in \mathbb{R}^{3}$ and $\overline{L}_{i} = [(\overline{x}_{i}, \frac{\overline{v}_{i}}{|\overline{v}_{i}|})]\in G^{3}$ for $i=4,5,6$ which we interpret as an element of $G^{2}$. Then

$$ |f_{3}(L_{1},\ldots ,L_{6})| = |(\overline{x}_{2}-\overline{x}_{1}) \wedge (\overline{x}_{3} - \overline{x}_{1})| |\overline{v}_{4}|| \overline{v}_{5}||\overline{v}_{6}| |f_{2}(\overline{L}_{4}, \overline{L}_{5}, \overline{L}_{6})|. $$

Proof

We apply a rigid motion so that $v_{1} = (1,0,0)^{T}$, $p_{1} = (0,0,0)^{T}$. Then we may interpret the vectors $\bar{v}_{i}$ and $\bar{x}_{i}$ to be elements of $\mathbb{R}^{2}$. In this way we can compute

$$ \begin{aligned} &|f_{3}(L_{1},\ldots ,L_{6})| \\ = &\begin{vmatrix} (0,0,0)^{T} & (p_{2} \times v_{1})^{T} & (p_{3} \times v_{1})^{T} & (p_{4} \times v_{4})^{T} & (p_{5} \times v_{5})^{T} & (p_{6} \times v_{6})^{T} \\ v_{1}^{T} & v_{1}^{T} & v_{1}^{T} & v_{4}^{T} & v_{5}^{T} & v_{6}^{T} \end{vmatrix} \\ =& \begin{vmatrix} (p_{2} \times v_{1})^{T} & (p_{3} \times v_{1})^{T} & (p_{4} \times v_{4})^{T} & (p_{5} \times v_{5})^{T} & (p_{6} \times v_{6})^{T} \\ (0,0)^{T} & (0,0)^{T} & \overline{v}_{4}^{T} & \overline{v}_{5}^{T} & \overline{v}_{6}^{T} \end{vmatrix} \\ =& |\overline{x}_{2} \wedge \overline{x}_{3}| \begin{vmatrix} (\overline{x}_{4} \wedge \overline{v}_{4})^{T} & (\overline{x}_{5} \wedge \overline{v}_{5})^{T} & (\overline{x}_{6} \wedge \overline{v}_{6})^{T} \\ \overline{v}_{4}^{T} & \overline{v}_{5}^{T} & \overline{v}_{6}^{T} \end{vmatrix} \\ =& |\overline{x}_{2} \wedge \overline{x}_{3}| |\overline{v}_{4} | | \overline{v}_{5}||\overline{v}_{6}| |f_{2}(\overline{L}_{4}, \overline{L}_{5}, \overline{L}_{6})|. \end{aligned} $$

 □

We see that we can guarantee that $f_{3}(L_{1},\ldots ,L_{6})$ is bounded away from 0 if the three parallel lines are chosen with a large cross-area, the other three lines are transversal enough and the intersection points of the projections of $L_{4}$, $L_{5}$, $L_{6}$ do not all coincide.