The Euler-Bernoulli limit of thin brittle linearized elastic beams

We show that the linear brittle Griffith energy on a thin rectangle $\Gamma$-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


Introduction
We consider a model of brittle linearly elastic Euler-Bernoulli beams of fixed length L > 0 and variable thickness h 0. We define the undeformed beam as We fix throughout the paper a positive definite Cauchy stress tensor C ∈ R 2×2×2×2 such that (1) C ijkl = C jikl = C klij and F : for some c > 0. This allows us to define the energy of a displacement field w ∈ GSBD 2 (Ω h ) as (2) where H 1 denotes the 1-dimensional Hausdorff measure and J w is the jump set of the function w, see Section 5 (also for the definition of the space GSBD 2 (Ω h ) and its properties).
The elastic prefactor 1 h 3 is chosen so that no stretching occurs and we recover the bending regime in the limit.The fracture prefactor β h denotes the material toughness, which has to scale as 1  h to recover the number of fracture points in the limit.We note that by the symmetry of C, for w ∈ GSBV(Ω 1 ) we have ew : Cew = ∇w : C∇w almost everywhere, where ew = (∇w + ∇w T )/2.For the more general definition of ew and the space GSBD 2 (Ω h ), see Section 5.
Because we let h → 0, we perform the usual change of variables: Let y ∈ L 2 (Ω; R 2 ) and h > 0. Define w ∈ L 2 (Ω h ) by w(x 1 , x 2 ) = y(x 1 , x 2 /h) and correspondingly define the energy where F h is defined in (2).Formally, for regular functions y (e.g.y ∈ SBV 2 (Ω 1 ; R 2 )) by a change of variables it holds (4) where ν ∈ 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) < ∞ implies that ∇y exists as a function in L 2 (Ω 1 ; R 2×2 ).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 Γ-converges to a limit energy E 0 which is only finite on the following set of admissible limit configurations A := {y ∈ SBV 2 (Ω 1 ; R 2 ) : D 2 y = 0, ∂ 1 y 1 = 0, ∂ 1 y 2 ∈ SBV(Ω 1 ; R)}.
The bending constant a > 0 is defined as usual in Euler-Bernoulli beam theory as (5) a := inf b,c∈R 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 where we need to subtract x 2 h∇y 2 (x 1 , 0) from y so that the symmetric part of the matrix (∂ 1 y h , 1 h ∂ 2 y h ) converges to 0. The precise calculation is found in Section 8. Similar Γ-convergence results have already been proven in the n-dimensional setting, see [2,8].However, here we show a stronger complementing compactness theorem, see also the discussion in Section 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 ∈ Skew(2), b h ∈ R 2 such that the difference of w h and the rigid motions is precompact after being rescaled to Ω 1 .Additionally, fracture can occur and different rigid motions might be present on different parts of Ω 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 Ω h is along essentially vertical cracks.Hence, a reasonable compactness result needs to identify the different parts of Ω 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 Ω h that is disconnected from the rest of Ω h along non-vertical lines.In fact, we show that for an energy-bounded sequence y h there are x 2 -independent, piecewiseconstant functions A h and b h and asymptotically vanishing sets ω h such that the sequence . 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 4.1.A key tool in this analysis will be a Korn's inequality for GSBD 2 (Ω), 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 Ω ⊂ R d undergoes a deformation u : Ω → R d .The geometric hyperelastic energy is then given by ˆΩ W (∇u) dx, where W : R d×d → [0, ∞) denotes the C 2 (R d×d ) hyperelastic energy density.We make the physical assumptions that W (id) = 0, i.e. u(x) = x has the lowest possible energy, and W (RA) = W (A) for R ∈ 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 ∞, where dist 2 (A, SO(d)) W (A) dist 2 (A, SO(d)).A central result in the theory of hyperelastic materials is the geometric rigidity result by Friesecke, James, Müller [24], which states that for open connected Lipschitz domains Ω ⊂ R d , there exists a constant C(Ω) > 0 such that (6) min In particular, we have that whenever ´Ω W (∇u k ) dx → 0, up to subsequences and fixed rotations x weakly in H 1 (Ω; R d ).For deformations with small hyperelastic energy, we may thus write where The quadratic growth conditions on W and Schwarz's theorem then guarantee (1).For a rigorous derivation via Γ-convergence, see [20].
The dynamics of the resulting quadratic form dealing with infinitesimal displacements |w| 1 are commonly referred to as linearized elasticity, and form an important part of the physics and engineering literature, see e.g.[28].In particular, they are often times simpler to deal with than the geometrically nonlinear version.
For example, applying (6) to small deformations yields Korn's inequality min which can be proved using elementary methods and was in fact proved by Korn in [29].
2.2.Thin elastic structures.In contrast to full bodies, lower dimensional structures have potentially lots of isometric embeddings into R d .A famous example is the Nash-Kuiper theorem [33], which states that for every Riemannian m-manifold M and every smooth 1-Lipschitz map f : Compare that to open sets Ω ⊂ R d , where every isometric C 1 deformation w : Ω → R d must be a rigid motion by (6).
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) × hS ⊂ R 3 , with S ⊂ R 2 open, bounded, connected, and plates P h := (0, L) × (0, L) × (−h/2, h/2).Both, linear and nonlinear variational models exist for both, see e.g.[32] for beams and [24,25] for plates.We note also that shells, which are curved analogues of plates, have been similarly studied, see e.g.[23].
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 Ω ⊂ R d and a displacement field w : Ω → R d which is C 1 outside a closed rectifiable hypersurface Γ ⊂ Ω, we define the Griffith brittle fracture energy (see [21,27]) as Here β > 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 [19], the space of generalized functions of bounded deformation GSBD 2 (Ω), whose definition and key properties we recount in Section 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 [34] that the nonlinear version of E h Γ-converges to ˆL 0 a 24 |y (x 1 )| 2 dx 1 + β#(J y ∪ J ∂ 1 y ), if y ∈ SBV((0, L); R 2 ), |y (x 1 )| = 1 a.e., which is the nonlinear analogue to the limit energy E 0 .In [8] and [2] the authors study the asymptotics of an n-dimensional analogue of the energy E h , see also [7,1,9] for the antiplane setting.Using a slightly different rescaling of the function y h , c.f. [18], the authors obtain the limiting energy where Q is a quadratic form and y is of the form 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 [2] the authors use a delicate approximation argument in GSBD 2 .In contrast our proofs are based on rigidity arguments which are much closer to the techniques used in [34], see also [24,32].This allows to obtain more control on the rescaled gradient 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 ∇ h y h is crucial.For example in the derivation of a rod theory the information about torsion is stored in the limit of ∇ h y h and cannot be seen in the limiting y, see [26].
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.

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 , . . ., x d ) for vectors in 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 : R 2 → R 2 , are denoted (w h ) 1 , (w h ) 2 .We say that two vectors v, w ∈ R d are parallel, v w, if they are linearly dependent.The space of symmetric and skew-symmetric R d×d matrices will be denoted by Symm(d) and Skew(d), respectively.We use standard notation for the Lebesgue measure L d and the s-dimensional Hausdorff measure H s .Moreover, for a Lebesgue-measurable set B ⊆ R d we write |B| for 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 ⊆ R d we write ∂ * E for its reduced boundary, cf.[3].

Main results
We now state our main results.We start with the compactness result.
Theorem 4.1.Let E h be defined as in (3) and Then there is a subsequence (not relabeled), a function y ∈ A, a sequence of sets σ h ⊂ Ω 1 of finite perimeter with measure theoretic normal ν ∈ S 1 , and sequences of piecewise constant functions (iii) We have and, for every fixed h > 0, Here Remark 4.2.In other words, A h and 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 To see this, consider the sequence of triangles in Ω 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), and define The length of the jump is slightly less than the height.Elastic stress, while high, is contained to the small triangle conv(a h , b h , c h ).
See Figure 1 for a sketch of the corresponding deformation.
This displacement field jumps on the line segment ) and has elastic energy Remark 4.3.Even for Āh = 0, bh = 0 the compactness result only guarantees convergence in measure, but not in L 1 (Ω 1 ; 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) := ´Ω1 y 2 dx, and Next, we state the main Γ-convergence result.
Theorem 4.4.The Γ-limit of E h as defined in (3) with respect to the convergence in Theorem 4.1 is E 0 .More precisely, we have (i) For any y ∈ A there is a sequence Remark 4.5.Note that the conditions for the lower bound (ii) in Theorem 4.4 are satisfied if y h → y in L 2 , c.f. Theorem 4.1 below.

The space GSBD 2 and Korn's inequality
We use the space of Generalized functions of Special Bounded Deformation with integrability 2, written GSBD 2 (Ω), as the effective domain of the energies F h , c.f. [19].This space is the natural topological function space for the brittle Griffith fracture model.It is analogous to the GSBV p spaces that are widely used in image segmentation, see [3] for the definition and the properties of the spaces GSBV p .In order to recall the definition of the space GSBD p , we first recall the definition of the jump set and the approximate symmetric gradient.Definition 5.1.
(i) Let Ω ⊂ R d be open, x ∈ Ω, and v : Ω → R N be measurable.The approximate limit of v at x, if it exists, is defined as the measurable function → ap limv x in measure on B(0, 1).
Note that the function ap lim x v is positively 0-homogeneous, i.e. ap lim x v(ty) = ap lim x v(y) for all y ∈ R d , t > 0.
(ii) The jump set J v of a measurable function v : Ω → R N is the set of all points x ∈ Ω where ap lim x u exists and is of the form Next we recall the definitions of the spaces BD(Ω), SBD p (Ω) and GSBD(Ω).
(ii) The vector field is said to be of special bounded deformation with integrability , where J v is the jump set of v. (iii) A measurable vector field v : Ω → R d is said to be a function of generalized special bounded deformation, GSBD(Ω), if there exists a bounded Radon measure measure λ ∈ M + (Ω) such that for every ξ ∈ R d and for where B ξ,y = {t ∈ R : y + tξ ∈ B}.
We recall from [19] that for v ∈ GSBD(Ω) it can be shown that the approximate symmetric gradient ev ∈ L 1 (Ω; R d×d ) exists L d -a.e. and the jump set J v is a countably H d−1 -rectifiable set with measure theoretic normal ν v .Moreover, it holds for ξ ∈ S d−1 and Eventually, we define the set GSBD p (Ω).
For fine properties of the functions ins GSBD p we refer to [19].We now state a strong version of Korn's inequality for functions in GSBD 2 (Ω) in any dimension, which is found in [12], for an earlier two-dimensional version see also [22] or [16].Then there is a constant C(Ω) > 0 such that for all w ∈ GSBD 2 (Ω) there is a function w ∈ W 1,2 (Ω; R d ) and a set set of finite perimeter ω ⊆ Ω such that w = w on Ω \ ω.
Moreover, there exists a matrix A ∈ Skew(d) and a vector b ∈ R d such that We note that the estimate is useless if H d−1 (J w ) is too large with respect to Ω, as then we can simply take ω = Ω.
We shall use this result to define good rectangles in Ω h , noting that we never use the extension w, only the bounds on the bad set ω.For the rest of the article (excluding the appendix) we work only in d = 2. Definition 5.5.Let h, δ > 0, and w ∈ GSBD 2 (Ω h ).We consider all the rectangles and bad otherwise.For a good rectangle we denote by ω z ⊆ Q z the exceptional set from Proposition 5.4.
Remark 5.6.Again, let h, δ > 0 and w ∈ GSBD 2 (Ω h ).We note that if δ ≤ δ 0 for some universal constant δ 0 , then we have on a good rectangle Q z by Proposition 5.4 that where minimization runs over Skew(2) × R 2 .Proposition 5.4 yields that ˆQz\ωz 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.
Then we see that for two neighboring δ-good rectangles Q z , Q z+h we have for some universal constant C.
We now show a stronger version of ( 8) for two 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 5.7.Let η ∈ (0, 1).Then there is a constant δ(η) > 0 such that for all N ∈ N there is a constant C(η, N ) > 0 such that the following holds: Here, the matrices A z , A z ∈ Skew(2) and vectors b z , b z ∈ R 2 are the matrices and vectors given by Proposition 5.4 on 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 If we can make sure that the three pairs are in general position, Lemmas A.3 and A.5 yield the upper bounds on |A z |, |b z |.See Figure 2 for a visual sketch.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 ∈ R 2 are linearly dependent): Start with the set of horizontal lines that do not cross J w in the sense of slicing for GSBD functions, or more precisely We have by the segment regularity in GSBD 2 (see [19]) that H 1 (H 1 ) ≥ η.
We intersect H 1 with three large subsets of (−1/2, 1/2), namely By Fubini's theorem and Markov's inequality we have , and thus By Proposition 5.4 we have This allows us to choose the first two pairs By the definitions of H 1 , H 2 , H 3 , H 4 we then have where in order to estimate the first term we used the fundamental theorem of calculus The fact that we may do so for almost every segment not intersecting the jump set is proved for GSBD functions in e.g.[19].
We now repeat the above argument to obtain one more pair (p 3 , q 3 ) with N +2 > 0. We define analogously to before As before, we have This allows us to pick a diagonal line z , we find a pair p 3 ∈ Q z \ ω z , q 3 ∈ Q z \ ω z on the diagonal line with |p 3 − q 3 | ≥ 1 and such that (11) holds.Note that as long as |t ± 1/2| > η 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 ∈ Lin(Skew(2) × R 2 ; R 3 ) by By (10) we have Using the identity We clearly have |F | ≤ (N + 3) 3 .By the direct calulations in Lemmas A.3 and A.5, we may estimate since all three pairs have distance at least 1, the parallel lines have distance at least η/2, and the angle of the diagonal line is θ.This shows that whenever δ < η 32C , completing the proof.
Remark 5.8.We note here that the above procedure may be generalized to d = 3 using Lemma A.6 and employing 6 segments instead of 3. We also note that we only used the bound on |ω 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 [24] for GSBV 2 functions with a small jump set.
Remark 5.9.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 A.5 and A.6 are potentially useful to the engineering community in the construction of optimal trusses in bridges, scaffolding, towers etc.

Proof of compactness
Here we combine estimates ( 8) and ( 9) to show that ∇w h is very close in L 2 (Ω h ; R 2×2 ) to some A h ∈ SBV 2 ((0, L); Skew(2)) and #J A h ≤ lim inf h→0 1 h H 1 (J w h ) .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.
Then there is a subsequence (not relabeled), a sequence of sets ω h ⊂ Ω h and sequences (iv) Proof.First, we may assume that lim inf h→0 This implies for all η > 0 small enough that it holds 1 In addition, let δ = δ(η) > 0 be as in Proposition 5.7.For z ∈ {h, 2h, . . ., ( L/h − 1)h}, we write Then we apply Proposition 5.4 to each Q z ∈ G h and obtain matrices A z ∈ Skew(2), vectors b z ∈ R 2 and sets of finite perimeter ω z ⊆ Q z such that (12) Let us recall from Proposition 5.7, c.f. also Remark 5.6, that for the choice of δ and two neighboring rectangles Moreover, we write We define the connected components with a large jump set as By the choice of η we find that #J h < M + 1 which implies that #J h ≤ M .Moreover, by Proposition 5.7 we find for our choice of δ and Now we can construct the functions A h ∈ SBV((0, L); 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 Figure 3. Precisely, we define If not already defined, we extend this definition constantly onto the intervals (0, h) and (( L/h − 1)h, L).
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 J h .Consequently, Moreover, using ( 13) and ( 14) we obtain (note that A h and b h denote the absolutely continuous part of DA h and Db h , respectively) This shows (ii).Next, we define the exceptional set ω h as the union of the exceptional sets on the good rectangles, all bad rectangles and a boundary layer, i.e.(b) Sketch of the interpolation procedure to construct A h .Between neighboring good cubes with center kh and (k + 1) we interpolate linearly.In connected components of the union of the bad rectangles in which the jump set of w h is larger than 1 − η the function A h jumps (left).
In connected components in which the jump set of w h is less than 1 − η we interpolate using Proposition 5.7.It follows from the properties of By the the subadditivity of the squareroot and the isoperimetric inequality it follows that which shows (i).
In order to prove (iii) we estimate using ( 12) and Hölder's inequality Note that for the last inequality we used (ii).

It remains to show (iv). We recall that by Proposition 5.4 we have for all
Hence, we obtain from the definition of ω h and similar estimates as in (15) that , where ω h is the set constructed in Proposition 6.1.From Proposition 6.1 (i) we immediately obtain (ii).Next, we write Let A h and b h be the functions from Proposition 6.1.Recall from (ii) in Proposition 6.1 that A h ∈ SBV((0, L); Skew(2)) and b h ∈ SBV((0, L); R 2 ) with ( 16) We write J A h ∪ {0, L} = {t 1 , . . ., t N h } and define the function Āh : (0, L) → Skew( 2) by (see also Figure 4) It follows immediately that Āh is piecewise constant and J Āh ⊆ J A h .By a minor modification of Āh , if necessary, we may assume that J Āh = J A h .We deduce from ( 16) that A h − Āh L ∞ (Ω 1 ;R 2×2 ) ≤ C and that A h − Āh is a bounded sequence in SBV 2 ((0, L); Skew(2)).By the compactness properties of the space SBV 2 , there exists a (not relabeled) subsequence and A ∈ SBV 2 ((0, L); Skew(2)) such that The definition of bh is completely analogous.Again, we may assume without loss of generality that Jb h = J b h and obtain that b h − bh L ∞ ((0,L);R 2 ) ≤ C. Then there exists a (not relabeled) subsequence and b ∈ SBV 2 ((0, Let us now define the functions It follows that u h ∈ SBV 2 (Ω h ; R 2 ).Moreover, we obtain from the bounds and properties of A h − Āh and b h − bh that ( 17) Accordingly, we define the rescaled functions ūh ∈ SBV 2 (Ω 1 ; R 2 ) by ( 18) which satisfies by ( 17) In particular, ūh is a bounded sequence in SBV 2 (Ω 1 ; R 2 ).Moreover, by the definition, (18), and the compactness properties of A h − Āh and b h − bh it follows ūh → y in In particular, y does not depend on x 2 .In order to show y ∈ A we still need to prove that ∂ 1 y 1 = 0 and ∂ 1 y 2 ∈ SBV(Ω 1 ).Next, let us define wh : Ω h → R 2 by wh (x) = w h (x) − Āh (x 1 )x − bh (x 1 ) 1 Ω h \ω h .From Proposition 6.1 (i) and (iii) we deduce that wh The above shows that the functions ỹh : where ν ∈ S 1 is the measure theoretic normal to J ỹh .It follows immediately that ỹh is a bounded sequence in GSBV 2 (Ω 1 ; R 2 ).Moreover, it follows from Proposition 6.1 (iv) that As L 2 (σ h ) → 0 (c.f.Proposition 6.1 (i)), it follows that ỹh → y in L 2 (Ω 1 ; R 2 ), which is (i).By the compactness properties of GSBV 2 (Ω 1 ; R 2 ) it even follows that On the other hand, we find that However, this implies that A = (∂ 1 y, c) for a function c ∈ SBV 2 (Ω 1 ; R 2 ).As A is skewsymmetric, it follows immediately that ∂ 1 y 1 = 0.Moreover, we deduce from the fact that A ∈ SBV 2 ((0, L); Skew(2)) and that ∂ 1 y ∈ SBV 2 (Ω 1 ).This concludes the proof of y ∈ A.
It remains to show the inequalities in (iii).First, let us recall that by construction we have Consequently, it follows from Proposition 6.1 (ii) and the usual change of coordinates that which is the second inequality in (iii).In addition, we note that J ∂ 1 y ⊆ J A × (−1/2, 1/2).Moreover, by the specific form of y, (19), we have J y ⊆ J A ∪ J b .However, lower semicontinuity yields #( which yields using (20) the remaining first inequality of (iii).

The lower bound
In this section, we will prove the lower bound claimed in Theorem 4.4.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 4.1.First we prove the following result to identify the limit of the rescaled gradient of y h .

Proposition 7.1. Define the function W
where Then, in the setting of Theorem 4.1, every subsequence of W h has a subsequence converging weakly in L 2 (Ω 1 ) to a function W ∈ L 2 (Ω 1 ) of the form where T ∈ L 2 ((0, L)) and y ∈ A is any function for which the conclusion of Theorem 4.1 hold.
Remark 7.2.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 6.1 and A is the corresponding limit constructed in the proof of Theorem 4.1.As A is skew-symmetric it holds A (x 1 )e 2 • 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 [26].2)) be the fields from Proposition 6.1.By point (iii) of Proposition 6.1, we have In particular, we have that ´Ωh \ω h 1 ), so that every subsequence has a weakly convergent subsequence.
Let us fix such a limit W ∈ L2 (Ω 1 ; R 2 ) and a corresponding (not relabeled) subsequence.
Before we can prove the specific form of the limit W , let us recall from the proof of Theorem 4.1 that we have by construction for the sequences A h from Proposition 6.1 and the sequences Āh from the proof of Theorem 4.
and that it holds since Āh is piecewise constant.We note that for any other limit ỹ, sequences of functions Āh and bh , and sets σ h for which the conclusions of Theorem 4.1 hold it can be shown that it still holds ∂ 1 ỹ = ∂ 1 y + d, where d ∈ SBV((0, L); R 2 ) is piecewise constant.In particular, ∂ 1 ∂ 1 ỹ = ∂ 1 ∂ 1 y.This shows that we can restrict ourselves to the specific y ∈ A that we construct in the proof of Theorem 4.1.
To show the form of W , we consider for fixed z ∈ (0, 1/2) the second difference V h : Ω ∩ (Ω − s h e 1 ) × (−1/2, 1/2 − z) → R 2 , depending on s h > 0, defined by Here, 1 h (x) = 1 whenever y h , A h , and A 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 the boundary of the rectangle does not intersect ∂σ h , and 0 otherwise.
As the jump sets of A and A h a purely horizontal, it holds Here, ν denotes the measure theoretic normal to J y h and ∂ * σ h , respectively.By Theorem 4.1 (ii) and the bounds on the energy the right hand side tends to 0 as h → 0 as long as Conversely, we have for For the second equality, we used the independence of A h , A from x 2 and the bound It follows that W (x + ze 2 ) − W (x) = z (A 1,2 ) (x 1 ) for almost every x ∈ Ω 1 .Now, recall from above that )) for almost every x 1 ∈ (0, L).We define T (x 1 ) := ´1/2 −1/2 W (x 1 , z) dz.It follows for almost every x 1 ∈ (0, L) Now we can prove the lower bound in Theorem 4.4.
Proof of Theorem 4.4 (i).By standard arguments we may always assume that it holds lim inf h→0 E h (y h ) = lim h→0 E h (y h ) and sup h E h (y h ) < ∞.
Then we notice that by the assumed convergence it holds that, c.f. Theorem 4.1 (iii), ( 21) Next, we show that lim inf where a > 0 is defined in (5).First, we apply Proposition 6.1 to the function w h (x 1 , x 2 ) = y h (x 1 , x 2 /h) to obtain sets ω h ⊆ Ω h and A h ∈ SBV((0, L); Skew(2)) for which the properties (i), (ii) and (iii) of Proposition 6.1 hold.Again we denote by σ h the analogue of ω h ⊆ Ω h in Ω 1 .Then (up to a subsequence) it holds by Proposition 7.1 that the sequence where T ∈ L 2 ((0, L)).It follows by the definition of the bending constant a, c.f. (5), that lim inf Using the specific form of W we compute Combining ( 21) and (22)

The upper bound
In this section we prove the existence of a recovery sequence for the energy E 0 , i.e. we show Theorem 4.4 (ii).

One sees immediately that y
Moreover, the jumps of y h only occur in x 1 -direction and J y h ⊆ J y ∪ J ∂ 1 y .In addition, Plugging into the elastic energy and using the x 2 -independence of the occuring functions we find A diagonal argument in η and h finishes the proof.

Discussion
In this section we want to briefly comment on the presented results and discuss future directions.First, we acknowledge that the presented Γ-convergence result is very similar to the n-dimensional results already obtained in [8] and [2].However, we note that the compactness result presented in this paper and also the topology used for the Γ-convergence result is more general.In [8] uniform L ∞ -bounds are assumed a priori, in [2] the authors invoke a compactness result in GSBD due to Chambolle and Crismale, [17].The latter result essentially yields for a sequence (y h ) h ⊆ GSBD(Ω 1 ) such that sup h ´Ω1 |e(w h )| 2 dx < ∞ and sup h H d−1 (J y h ) < ∞ that there is a (not relabeled) subsequence for which the set A = {x ∈ Ω 1 : y h (x) → ∞} has finite perimeter and outside A one has y h → y pointwise a.e. for some y ∈ GSBD(Ω 1 ).Moreover, 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 (Ω 1 ; R 2 ).In addition, these rigid motions do not create additional jump in the limit, c.f. Theorem 4.1.Whether the used techniques can be generalized to higher dimensions to obtain an improved version of the result in [17], needs to be studied.
Moreover, we note that the presented analysis is closely related to the techniques presented in [34], 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 ∈ SBV, [22], which subdivides the domain into different regions in which the function y is close to a rigid motion, see [34,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, threedimensional 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, [32] 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 ∇w h ≈ A h .Note that a three-dimensional version of Proposition 5.7 can be proved using Lemma A.6 instead of Lemma A.5.A corresponding paper is in preparation, [26].
where d i = |p i − q i |, and is the oriented line obtained by extending the line segment [p i , q i ], which is an element of the oriented affine line Grassmannian i=1 .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 This shows that f d is well-defined.The formula (23) and the properties of f d follow immediately from the properties of the determinant.
Remark A.4.By Remark A.2 and Lemma A.3, 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.Note that this covers all cases.Note that if L 1 , L 2 , L 3 are pairwise transversal, then formula (24) 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 α = β in (24), and this is indeed the case in the proof of Proposition 5.7.In case (i), v 1 = v 2 = v 3 , the rows of F are linearly dependent, and det 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 = ±|p − q|v 1 .We calculate explicitly If L i = [(p i , v i )], the ith row of F is then given by (p i × v i , v i ) ∈ R 6 , where p i × v i denotes the usual cross product in R 3 .
Lemma A.6.Let L i = [(p i , v i )] ∈ G 3 for i = 1, . . ., 6, and assume that 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 vi and xi to be elements of R 2 .In this way we can compute We see that we can guarantee that f 3 (L 1 , . . ., 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.

( a )
Sketch of the situation in Ω h .The bad rectangles are sketched in red, the jump set of w h is sketched in blue.The size of the jump set of w h in the left connected component of the union of bad rectangles contains more than 1 − η, whereas in the right connected component it is less than 1 − η.

Figure 3 .
Figure 3. Interpolation procedure to construct A h .

Figure 4 .
Figure 4. Sketch of the construction of Āh .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 Āh (i.e. the upper right entry of the matrix field Āh ) is sketched in green.The field Āh is constant between jump points of A h .
G d := R d × S d−1 / ∼, with (x, v) ∼ (z, w) whenever v = w and x − z v.In addition, the function f d has the property f d • σ = det σf d for all permutations σ.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 , . . ., L N .Proof.Let L i = [(p i , v i )] ∈ G d .We define f d : (G d ) N → R by f d (L 1 , . . ., L N ) = det F,where F : Skew(d) × R d → R N is the linear mapping such that F (A, b) = ((Ap i + b)

Proof.
We work with variables a, b 1 , b 2 , where A = a 0 1 −1 0 and b = (b 1 , b 2 ) T which defines the standard basis of Skew(2) × R 2 .In these coordinates, ifL i = [(p i , v i )],the mapping from Definition A.1 is represented by a matrix F ∈ R 3×3 whose rows are(p i ∧ v i , (v i ) 1 , (v i ) 2 ) ∈ R 3, where as usual x ∧ y = x 1 y 2 − x 2 y 1 for x, y ∈ R 2 .
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.