# Coarse-graining of a discrete model for edge dislocations in the regular
triangular lattice

R. Alicandro  DIEI, Università di Cassino e del Lazio meridionale, via Di
Biasio 43, 03043 Cassino (FR), Italy<EMAIL_ADDRESS>,  L. De Luca
Istituto per le Applicazioni del Calcolo “M. Picone”, IAC-CNR, via dei Taurini
19, 00185 Roma, Italy<EMAIL_ADDRESS>,  G. Lazzaroni  DiMaI, Università
di Firenze, Viale Morgagni 67/a, 50134 Firenze, Italy
<EMAIL_ADDRESS>,  M. Palombaro  DISIM, Università dell’Aquila,
Via Vetoio, 67100 L’Aquila, Italy<EMAIL_ADDRESS>and  M.
Ponsiglione  Dip. di Matematica, Univ. Roma-I “La Sapienza”, Piazzale Aldo
Moro 5, 00185 Roma, Italy<EMAIL_ADDRESS>
###### Abstract.

We consider a discrete model of planar elasticity where the particles, in the
reference configuration, sit on a regular triangular lattice and interact
through nearest neighbor pairwise potentials, with bonds modeled as linearized
elastic springs. Within this framework we introduce plastic slip fields, whose
discrete circulation around each triangle detects the possible presence of an
edge dislocation.

We provide a $\Gamma$-convergence analysis, as the lattice spacing tends to
zero, of the elastic energy induced by edge dislocations in the energy regime
corresponding to a finite number of geometrically necessary dislocations.

Keywords: Dislocations, Topological singularities, Plasticity, Discrete to
continuum limits, Gamma-convergence.

MSC2020: 74C05, 58K45, 70G75, 74G65, 49J45.

###### Contents

  1. 1 The model and the main result
  2. 2 The continuum case
  3. 3 Proof of the main result
    1. 3.1 Preliminary results
    2. 3.2 Proof of Theorem 1.1
  4. 4 Degeneracies of the model and possible kinematic constraints

## Introduction

Dislocations are line defects in the periodic structure of crystals and are
considered the main microscopic mechanism of plastic flow. Idealized straight
dislocations are classified into two types, edge and screw, while in real
crystals dislocations are actually curved lines of mixed type [12, 11]. From a
mathematical viewpoint, dislocations can be seen as topological line
singularities around which the elastic strain has non trivial circulation
given by a vector of the underlying lattice and referred to as Burgers vector.

Here we focus on planar elasticity, where the relevant dislocations are of
edge nature and can be seen as topological point singularities of the elastic
strain. Specifically, we study the discrete elastic energy induced by a finite
system of edge dislocations in a finite portion
$\Omega\cap\varepsilon\mathbb{T}$ of the regular triangular lattice
$\varepsilon\mathbb{T}$ , where $\varepsilon$ is the lattice spacing.

As customary in the linearized framework, we adopt the additive decomposition
of the discrete deformation gradient $\mathrm{d}u$ , defined on pairs of
nearest neighbors, into an elastic and a plastic part. The latter is
represented by an additional variable $\sigma$ , referred to as slip, defined
on pairs of nearest neighbors and taking values in the set of lattice vectors.
In this way we identify dislocations as points around which the discrete
circulation of the plastic slip $\sigma$, or, equivalently, the discrete
circulation of $\mathrm{d}u-\sigma$ , is non trivial. Such a procedure agrees
with the formalism of the eigenstrains considered in [2].

We focus on the simple case where nearest neighbors interact through
linearized pairwise potentials; specifically, in absence of singularities, the
energy functional is defined by

$G_{\varepsilon}(u):=\frac{1}{2\varepsilon^{2}}\sum_{\genfrac{}{}{0.0pt}{1}{i,j\in\Omega\cap\varepsilon\mathbb{T}}{|i-j|=\varepsilon}}\big{(}\mathrm{d}u(i,j)\cdot(j-i)\big{)}^{2}\,.$

The above energy can be formally derived by linearization of suitable
nonlinear frame invariant functionals (see Remark 1.2); further rigorous
linearization results in terms of $\Gamma$-convergence are provided in [1, 4,
16]. In the limit as $\varepsilon\to 0$ the energy
$\frac{1}{\varepsilon^{2}}G_{\varepsilon}$ approximates, up to a pre-factor,
the continuum isotropic elastic energy with Lamé parameters both equal to $1$
(see Remark 3.1).

In presence of a plastic slip field $\sigma$, the energy functional should
depend only on the elastic strain $\mathrm{d}u-\sigma$ , and hence the
resulting elastic energy reads as

$F_{\varepsilon}(u,\sigma):=\frac{1}{2\varepsilon^{2}}\sum_{\genfrac{}{}{0.0pt}{1}{i,j\in\Omega\cap\varepsilon\mathbb{T}}{|i-j|=\varepsilon}}\big{(}(\mathrm{d}u(i,j)-\sigma(i,j))\cdot(j-i)\big{)}^{2}\,.$

It is well known that the energy induced by an isolated edge dislocation, both
in the discrete and in the continuum setting, is of order
$\varepsilon^{2}|\log\varepsilon|$, while short dipoles of opposite
dislocations induce a much smaller energy of order $\varepsilon^{2}$. The aim
of the present paper is to determine the asymptotic behavior (as
$\varepsilon\to 0$) of $F_{\varepsilon}$ in the energy regime
$\varepsilon^{2}|\log\varepsilon|$ . This corresponds to a finite distribution
of geometrically necessary edge dislocations, i.e., to the superposition of a
finite number of isolated dislocations plus clusters of singularities with
total Burgers vector equal to zero (usually referred to as statistically
stored dislocations).

Our results are obtained within the rigorous formalism of $\Gamma$-convergence
and consist in a compactness property for the dislocation measures and in the
derivation of the effective limit energy induced by dislocations.
Specifically, our analysis shows that the discrete dislocation density
generated by the plastic slip $\sigma$ converges, in the sense of flat
convergence (1.7), to a finite sum of Dirac deltas
$\sum_{k=1}^{K}b^{k}\delta_{x^{k}}$ with $b^{k}\in\mathbb{T}$ . The effective
energy (namely the $\Gamma$-limit) “counts” the limiting singularities with a
coefficient $\varphi(b^{k})$ given by the explicit formula (2.13). These
results rely on a similar analysis performed in [8, 6] in a semi-discrete
setting within the so-called core radius approach.

We restrict our analysis to configurations of plastic slips inducing
dislocations with minimal mutual distance larger than or equal to
$\sqrt{3}\varepsilon$ . Roughly speaking, this means that two neighboring
triangles cannot both contain a dislocation. Such a mild separation assumption
(MS) guarantees that each dislocation induces a core energy of order
$\varepsilon^{2}$; the latter estimate is essential also in the core radius
approach, and represents the starting point in the so-called ball construction
technique [15], which is devised to provide sharp lower bounds. This is not a
mere technical assumption; indeed, removing it, one can exhibit unphysical
configurations with zero stored elastic energy where all triangles contain a
dislocation (see Remark 3.5). In this respect, it seems that our linearized
model (without assuming (MS)) fails to describe the core energy stored in a
single triangle and induced by the presence of a dislocation. On the other
hand, under the mild separation assumption (MS), each dislocation is
surrounded by an annulus of “elastically deformed” triangles where a finite
amount of energy is stored, as it follows from an application of Korn’s
inequality.

Although our model provides a good description of “non-pathological”
dislocation configurations, it exhibits some degeneracies, due to the discrete
linearized framework relying on a reference configuration, as well as to the
presence of the slip variable (see Section 4). A natural way to rule out such
degeneracies could be to include kinematic constraints on the slip fields
mimicking (in a discrete framework) pure shear/deviatoric stress conditions
that are typically assumed in (continuum) elasto-plasticity. In this respect,
in Section 4, we discuss a possible constraint on the slip fields, based on
the (formal) linearization of the volume preserving condition on dislocation-
free $\varepsilon$-triangles; we remark that our $\Gamma$-convergence analysis
still holds true under such a constraint.

Finally, some comments are in order. First, we do believe that a similar
discrete-to-continuum analysis could be developed in the
$\varepsilon^{2}|\log\varepsilon|^{2}$ regime, which corresponds to a limit
diffused distribution of dislocations and to an effective energy accounting
for both elastic and plastic effects (see [9] for an analogous result within
the core-radius approach). Second, it would be interesting to look at the
coarse-graining for the energy $F_{\varepsilon}$ in even higher regimes, such
as that of the grain boundaries (see the recent papers [10, 13, 7]). Third, a
challenging issue would be to deal directly with nonlinear models built on the
deformed configurations. An intriguing intermediate attempt in this direction
could be to consider hybrid models combining the mathematical efficiency of
linearized theories together with the mechanical understanding of dislocations
in terms of interpenetrating pentagon-heptagon pairs usually observed in the
deformed configurations.

#### Notation.

For every $m,n\in\mathbb{N}$ and for every matrix $M\in\mathbb{R}^{m\times n}$
, we denote by $M^{*}\in\mathbb{R}^{n\times m}$ the transpose matrix of $M$ ,
i.e., such that $M^{*}_{ij}=M_{ji}$ for every $i=1,\ldots,n$ and
$j=1,\ldots,m$ . In what follows the elements $(x_{1},x_{2})$ of
$\mathbb{R}^{2}$ will be also identified (whenever it is convenient) with
column vectors $\left(\begin{array}[]{l}x_{1}\\\
x_{2}\end{array}\right)=(x_{1}\quad x_{2})^{*}$. We denote by
$\\{e_{1},e_{2}\\}$ the canonical basis of $\mathbb{R}^{2}$ , where
$e_{1}=(1,0)$ and $e_{2}=(0,1)$ . Given two vectors $a=(a_{1},a_{2})\,,\
b=(b_{1},b_{2})\in\mathbb{R}^{2}$ , we set $a\wedge b:=a_{1}b_{2}-a_{2}b_{1}$
; moreover, we set $a^{\perp}:=(-a_{2},a_{1})$ .

For every open set $U$ with $\partial U$ smooth and for every $s\in\partial U$
, we denote by $\tau(s)$ the tangent vector to $\partial U$ at $s$ defined as
$\tau(s)=n^{\perp}(s)$, where $n(s)$ denotes the outer normal unitary vector
to $\partial U$ at $s$.

For every $p\in\mathbb{R}^{2}$ and for every $0<r<R$ we define the annulus
$A_{r,R}(p):=B_{R}(p)\setminus\overline{B}_{r}(p)$ , where, for every $\rho>0$
, $B_{\rho}(p)$ and $\overline{B}_{\rho}(p)$ denote the open and the closed
ball centered at $p$ with radius $\rho$ , respectively.

Moreover, for every bounded open set $A\subset\mathbb{R}^{2}$ and for every
$\beta\in L^{2}(A;\mathbb{R}^{2\times 2})$ the symbol $\mathrm{Curl}\,\beta$
denotes the row-by-row distributional curl of $\beta$ ; formally,

$\mathrm{Curl}\,\beta=(\partial_{x_{1}}\beta_{12}-\partial_{x_{2}}\beta_{11},\partial_{x_{1}}\beta_{22}-\partial_{x_{2}}\beta_{21})\,.$

Analogously, the symbol $\mathrm{Div}\,\beta$ will denote the row-by-row
distributional divergence of $\beta$ , formally given by

$(\mathrm{Div}\,\beta)_{i}=\partial_{x_{1}}\beta_{i1}+\partial_{x_{2}}\beta_{i2}\,,\qquad
i\in\\{1,2\\}\,.$

Finally, $\mathcal{M}(\mathbb{R}^{2};\mathbb{R}^{2})$ denotes the class of
$\mathbb{R}^{2}$ valued Radon measures on $\mathbb{R}^{2}$ .

##  1\. The model and the main result

In this section we introduce our model and state the main result.

#### Reference lattice.

We set $\nu:=\frac{1}{2}e_{1}+\frac{\sqrt{3}}{2}e_{2}$ and
$\eta:=-\frac{1}{2}e_{1}+\frac{\sqrt{3}}{2}e_{2}$ . Let
$\mathbb{T}:=\mathrm{span}\,_{\mathbb{Z}}\\{e_{1},\nu\\}$ and set

$T^{+}:=\mathrm{conv}\\{0,e_{1},\nu\\}\quad\textrm{ and }\quad
T^{-}:=\mathrm{conv}\\{0,e_{1},-\eta\\}\,,$

where, for every $a,b,c\in\mathbb{R}^{2}$, the set $\mathrm{conv}\\{a,b,c\\}$
denotes the convex envelope of the points $a$, $b$, $c$, i.e., the (closed)
triangle with vertices at $a$, $b$, $c$ . For every $\varepsilon>0$ we denote
by $\mathcal{T}_{\varepsilon}$ the family of the triangles $T_{\varepsilon}$
of the form $i+\varepsilon T^{\pm}$, with $i\in\varepsilon\mathbb{T}$ .

Let $\Omega\subset\mathbb{R}^{2}$ be a bounded open set with Lipschitz
continuous boundary, representing the domain of definition of the relevant
fields in the model. For every $\varepsilon>0$, we set

$\mathcal{T}_{\varepsilon}(\Omega):=\\{T_{\varepsilon}\in\mathcal{T}_{\varepsilon}\,:\,T_{\varepsilon}\subset\Omega\\}$

and we define
$\Omega_{\varepsilon}:=\bigcup_{T_{\varepsilon}\in\mathcal{T}_{\varepsilon}(\Omega)}T_{\varepsilon}$.
Moreover, we set
$\Omega_{\varepsilon}^{0}:=\Omega_{\varepsilon}\cap\varepsilon\mathbb{T}$ and
we denote by $\Omega_{\varepsilon}^{1}$ the family of nearest neighbor bonds
in $\Omega_{\varepsilon}$, i.e.,
$\Omega_{\varepsilon}^{1}:=\\{(i,j)\in\Omega_{\varepsilon}^{0}\times\Omega_{\varepsilon}^{0}\,:\,|i-j|=\varepsilon\\}$.
Trivially, $(i,j)\in\Omega_{\varepsilon}^{1}$ if and only if
$(j,i)\in\Omega_{\varepsilon}^{1}$ .

In the following we will generalize the notation introduced above to general
subsets of $\mathbb{R}^{2}$ (not necessarily open). In particular, for every
triangle $T_{\varepsilon}\in\mathcal{T}_{\varepsilon}$ , we have

$(T_{\varepsilon})_{\varepsilon}^{1}=\\{(i,j)\in(T_{\varepsilon}\cap\varepsilon\mathbb{T})\times(T_{\varepsilon}\cap\varepsilon\mathbb{T})\,:\,i\neq
j\\}\,.$

For every map $V:(T_{\varepsilon})_{\varepsilon}^{1}\to\mathbb{R}^{2}$ , we
define the discrete circulation of $V$ on the “boundary of $T_{\varepsilon}$”
as

(1.1) $\mathrm{d}V(T_{\varepsilon}):=V(i,j)+V(j,k)+V(k,i)\,,$

where $(i,j,k)$ is a triple of counter-clockwise oriented vertices of
$T_{\varepsilon}$ .

#### Displacement, plastic slip, dislocation measure.

We denote by $\mathcal{D}_{\varepsilon}(\Omega)$ the class of displacements
$u:\Omega_{\varepsilon}^{0}\to\mathbb{R}^{2}$; moreover, for every
$u\in\mathcal{D}_{\varepsilon}(\Omega)$ we define the discrete gradient
$\mathrm{d}u:\Omega_{\varepsilon}^{1}\to\mathbb{R}^{2}$ of $u$ as
$\mathrm{d}u(i,j):=u(j)-u(i)$ for every $(i,j)\in\Omega_{\varepsilon}^{1}$.

We define the class of plastic slips $\mathcal{S}_{\varepsilon}(\Omega)$ as

$\mathcal{S}_{\varepsilon}(\Omega):=\Big{\\{}\sigma:\Omega_{\varepsilon}^{1}\to\varepsilon\mathbb{T}\,:\,\sigma(i,j)=-\sigma(j,i)\textrm{
for every }(i,j)\in\Omega_{\varepsilon}^{1}\Big{\\}}\,.$

For every $\sigma\in\mathcal{S}_{\varepsilon}(\Omega)$ we define the discrete
dislocation measure associated to $\sigma$ as

$\mu[\sigma]:=-\sum_{T_{\varepsilon}\in\mathcal{T}_{\varepsilon}(\Omega)}\mathrm{d}\sigma(T_{\varepsilon})\delta_{x_{T_{\varepsilon}}}\,,$

where $\mathrm{d}\sigma$ is defined in (1.1) and $x_{T_{\varepsilon}}$ denotes
the barycenter of the triangle $T_{\varepsilon}$ . Notice that for every
$u\in\mathcal{D}_{\varepsilon}(\Omega)$ and for every
$\sigma\in\mathcal{S}_{\varepsilon}(\Omega)$ it holds

$\mu[\sigma]=\sum_{T_{\varepsilon}\in\mathcal{T}_{\varepsilon}(\Omega)}\mathrm{d}(\mathrm{d}u-\sigma)(T_{\varepsilon})\delta_{x_{T_{\varepsilon}}}\,.$

The class of admissible dislocation measures, denoted by
$X_{\varepsilon}(\Omega)$, is the family of measures of the form
$\mu=\sum_{T_{\varepsilon}\in\mathcal{T}_{\varepsilon}(\Omega)}b(T_{\varepsilon})\delta_{x_{T_{\varepsilon}}}$
with $b(T_{\varepsilon})\in\varepsilon\mathbb{T}$ and satisfying the following
mild separation property:

(MS) for every $T_{\varepsilon}\in\mathcal{T}_{\varepsilon}(\Omega)$ with
$\mu(T_{\varepsilon})\neq 0$, we have $\partial
T_{\varepsilon}\cap\partial\Omega_{\varepsilon}=\emptyset$ and
$\mu(T^{\prime}_{\varepsilon})=0$ for every
$T^{\prime}_{\varepsilon}\in\mathcal{T}_{\varepsilon}(\Omega)$ with $\partial
T^{\prime}_{\varepsilon}\cap\partial T_{\varepsilon}\neq\emptyset$.

Finally, we set

$X(\Omega):=\Big{\\{}\mu=\sum_{k=1}^{K}b^{k}\delta_{x^{k}}\,:\,K\in\mathbb{N}\,,\,b^{k}\in\mathbb{T}\,,\,x^{k}\in\Omega\Big{\\}}\,.$

#### The energy functional and the main result.

We are now in a position to define the energy functionals
$F_{\varepsilon}:\mathcal{D}_{\varepsilon}(\Omega)\times\mathcal{S}_{\varepsilon}(\Omega)\to[0,+\infty)$
as

(1.2)
$F_{\varepsilon}(u,\sigma):=\frac{1}{2\varepsilon^{2}}\sum_{(i,j)\in\Omega_{\varepsilon}^{1}}[(\mathrm{d}u(i,j)-\sigma(i,j))\cdot(j-i)]^{2}\,,$

where $\cdot$ denotes the standard scalar product in $\mathbb{R}^{2}$ .

We will consider also localized versions of the functional
$F_{\varepsilon}(u,\sigma)$ in (1.2). More specifically, for every set
$A\subset\mathbb{R}^{2}$ , we define
$F_{\varepsilon}(\cdot,\cdot;A):\mathcal{D}_{\varepsilon}(A)\times\mathcal{S}_{\varepsilon}(A)\to[0,+\infty)$
as

(1.3) $F_{\varepsilon}(u,\sigma;A):=\frac{1}{2\varepsilon^{2}}\sum_{(i,j)\in
A_{\varepsilon}^{1}}[(\mathrm{d}u(i,j)-\sigma(i,j))\cdot(j-i)]^{2}\,,$

so that $F_{\varepsilon}(u,\sigma;\Omega)=F_{\varepsilon}(u,\sigma)$ .

Since in our analysis the relevant parameter is the dislocation measure $\mu$
associated to $\sigma$, we let the energy functionals depend only on (the
admissible measures) $\mu$, by defining
$\mathcal{F}_{\varepsilon}:X_{\varepsilon}(\Omega)\to[0,+\infty)$ as

$\mathcal{F}_{\varepsilon}(\mu):=\inf_{\genfrac{}{}{0.0pt}{1}{(u,\sigma)\in\mathcal{D}_{\varepsilon}(\Omega)\times\mathcal{S}_{\varepsilon}(\Omega)}{\mu[\sigma]=\mu}}F_{\varepsilon}(u,\sigma)\,.$

For every $b\in\mathbb{T}$, we set

(1.4)
$\varphi(b):=\frac{1}{3\pi}\min\bigg{\\{}\sum_{i=1}^{3}|z_{i}|\,:\,z_{1},z_{2},z_{3}\in\mathbb{Z}\,,\quad
b=z_{1}e_{1}+z_{2}\nu+z_{3}\eta\bigg{\\}}\,.$

Our main result is the following theorem.

######  Theorem 1.1.

The following $\Gamma$-convergence result holds true.

  * (i)

(Compactness) Let
$\\{\mu_{\varepsilon}\\}_{\varepsilon}\subset\mathcal{M}(\mathbb{R}^{2};\mathbb{R}^{2})$
be such that $\mu_{\varepsilon}\in X_{\varepsilon}(\Omega)$ for every
$\varepsilon>0$ . If $\mathcal{F}_{\varepsilon}(\mu_{\varepsilon})\leq
C\varepsilon^{2}|\log\varepsilon|$ , then, up to a subsequence,
$\frac{\mu_{\varepsilon}}{\varepsilon}\stackrel{{\scriptstyle\mathrm{flat}}}{{\rightarrow}}\mu$
(as $\varepsilon\to 0$) for some $\mu\in X(\Omega)$ .

  * (ii)

($\Gamma$-liminf inequality) For every
$\mu=\sum_{k=1}^{K}b^{k}\delta_{x^{k}}\in X(\Omega)$ and for every
$\\{\mu_{\varepsilon}\\}_{\varepsilon}\subset\mathcal{M}(\mathbb{R}^{2};\mathbb{R}^{2})$
with $\mu_{\varepsilon}\in X_{\varepsilon}(\Omega)$ for every $\varepsilon>0$
and such that
$\frac{\mu_{\varepsilon}}{\varepsilon}\stackrel{{\scriptstyle\mathrm{flat}}}{{\rightarrow}}\mu$
(as $\varepsilon\to 0$) it holds

(1.5) $\liminf_{\varepsilon\to
0}\frac{\mathcal{F}_{\varepsilon}(\mu_{\varepsilon})}{\varepsilon^{2}|\log\varepsilon|}\geq\frac{\sqrt{3}}{2}\sum_{k=1}^{K}\varphi(b^{k})\,.$

  * (iii)

($\Gamma$-limsup inequality) For every
$\mu=\sum_{k=1}^{K}b^{k}\delta_{x^{k}}\in X(\Omega)$ there exists
$\\{\mu_{\varepsilon}\\}_{\varepsilon}\subset\mathcal{M}(\mathbb{R}^{2};\mathbb{R}^{2})$
such that $\mu_{\varepsilon}\in X_{\varepsilon}(\Omega)$ for every
$\varepsilon>0$ ,
$\frac{\mu_{\varepsilon}}{\varepsilon}\stackrel{{\scriptstyle\mathrm{flat}}}{{\rightarrow}}\mu$
(as $\varepsilon\to 0$) and

(1.6) $\limsup_{\varepsilon\to
0}\frac{\mathcal{F}_{\varepsilon}(\mu_{\varepsilon})}{\varepsilon^{2}|\log\varepsilon|}\leq\frac{\sqrt{3}}{2}\sum_{k=1}^{K}\varphi(b^{k})\,.$

The convergence appearing in Theorem 1.1 is the flat convergence , that is the
convergence with respect to the flat norm, defined by

(1.7) $\|\mu\|_{\mathrm{flat}}:=\sup_{\genfrac{}{}{0.0pt}{1}{\phi\in
C^{0,1}_{\mathrm{c}}(\Omega)}{\|\phi\|_{C^{0,1}}\leq
1}}\Big{|}\int_{\Omega}\phi\,\mathrm{d}\mu\Big{|}\,,\qquad\textrm{for every
}\mu\in\mathcal{M}(\mathbb{R}^{2};\mathbb{R}^{2})\,,$

where $C^{0,1}(\Omega)$ is the space of Lipschitz continuous functions endowed
with the norm

$\|\phi\|_{C^{0,1}}:=\sup_{x\in\Omega}|\phi(x)|+\sup_{\genfrac{}{}{0.0pt}{1}{x,y\in\Omega}{x\neq
y}}\frac{|\phi(x)-\phi(y)|}{|x-y|}\,,$

and $C^{0,1}_{\mathrm{c}}(\Omega)$ is the subspace of $C^{0,1}$ functions
compactly supported in $\Omega$ .

######  Remark 1.2.

We illustrate how to formally derive (1.2) from a nonlinear frame invariant
model with nearest neighbor pairwise interaction potentials, in the case where
no dislocation is present, i.e., $\sigma\equiv 0$. We assume that, given a
deformation $v\colon\Omega_{\varepsilon}^{0}\to\mathbb{R}^{2}$, the total
interaction energy is

$E_{\varepsilon}(v)=\varepsilon^{2}\sum_{(i,j)\in\Omega_{\varepsilon}^{1}}\psi\Big{(}\Big{|}\frac{\mathrm{d}v(i,j)}{\varepsilon}\Big{|}\Big{)}\,,$

where $\psi$ is a $C^{2}$ function such that $\psi(1)=\psi^{\prime}(1)=0$ and
$\psi^{\prime\prime}(1)>0$ , so that the identity is an equilibrium
configuration. Expressing the energy in terms of the displacement
$u(x)=\frac{v(x)-x}{\delta}$, with $\delta>0$, and rescaling by $\delta^{2}$,
the total interaction energy reads as

$E^{\delta}_{\varepsilon}(x+\delta
u)=\frac{\varepsilon^{2}}{\delta^{2}}\sum_{(i,j)\in\Omega_{\varepsilon}^{1}}\psi\Big{(}\Big{|}\frac{j-i}{\varepsilon}+\delta\,\frac{\mathrm{d}u(i,j)}{\varepsilon}\Big{|}\Big{)}\,.$

A second order Taylor expansion of $E^{\delta}_{\varepsilon}(x+\delta u)$ with
respect to $\delta$ about the point $\delta=0$ gives

$E_{\varepsilon}^{\delta}(x+\delta
u)=F_{\varepsilon}(u)+\mathrm{o}_{\delta}(1)\,,$

where

$F_{\varepsilon}(u)=\frac{1}{2\varepsilon^{2}}\psi^{\prime\prime}(1)\sum_{(i,j)\in\Omega_{\varepsilon}^{1}}[\mathrm{d}u(i,j)\cdot(j-i)]^{2}\,,$

which coincides with (1.2) up to a multiplicative constant when $\sigma\equiv
0$. We refer to [10] for a formal derivation of (1.2) starting from a
nonlinear model with discrete plastic slips.

##  2\. The continuum case

In order to prove Theorem 1.1 we will use the corresponding result in the
continuum setting [6], that we briefly recall here. Let $A$ be an open bounded
subset of $\mathbb{R}^{2}$ with Lipschitz continuous boundary. For every
$\mu\in X(A)$ we set

(2.1)
$A_{\varepsilon}(\mu):=A\setminus\bigcup_{x\in\mathrm{supp}\,(\mu)}\overline{B}_{\varepsilon}(x)$

and we define the set $\mathscr{A}\\!\mathscr{S}_{\varepsilon}(\mu)$ of
admissible strains associated to $\mu$ as

(2.2) $\displaystyle\mathscr{A}\\!\mathscr{S}_{\varepsilon}(\mu):=$
$\displaystyle\Big{\\{}\beta\in L^{2}(A_{\varepsilon}(\mu);\mathbb{R}^{2\times
2})\,:\,\mathrm{Curl}\,\beta=0\,\textrm{ in }A_{\varepsilon}(\mu)\,,$
$\displaystyle\int_{\partial
U}\beta\,\tau\,\mathrm{d}\mathcal{H}^{1}=\mu(U)\textrm{ for every open set
}U\subset A\textrm{ with }\partial U\subset A_{\varepsilon}(\mu)\textrm{
smooth}\Big{\\}}\,.$

Let $\mathbb{C}$ be an elasticity tensor, i.e., a linear operator from
$\mathbb{R}^{2\times 2}$ into itself satisfying the following property: There
exist two constants $0<c_{1}<c_{2}$ such that

$c_{1}|\beta^{\mathrm{sym}}|^{2}\leq\frac{1}{2}\mathbb{C}\beta:\beta\leq
c_{2}|\beta^{\mathrm{sym}}|^{2}\qquad\textrm{for every
}\beta\in\mathbb{R}^{2\times 2}\,,$

where $\beta^{\mathrm{sym}}:=\frac{1}{2}(\beta+\beta^{*})$ .

The elastic energy of a field
$\beta\in\mathscr{A}\\!\mathscr{S}_{\varepsilon}(\mu)$ in the body $A$ is
given by

$E_{\varepsilon}(\beta;A_{\varepsilon}(\mu)):=\frac{1}{2}\int_{A_{\varepsilon}(\mu)}\mathbb{C}\beta:\beta\,\mathrm{d}x\,,$

and the energy induced by the dislocation distribution $\mu$ in the body $A$
is defined by

(2.3)
$\mathcal{E}_{\varepsilon}(\mu;A):=\inf_{\beta\in\mathscr{A}\\!\mathscr{S}_{\varepsilon}(\mu)}E_{\varepsilon}(\beta;A_{\varepsilon}(\mu))+|\mu|(A)\,,\qquad\textrm{for
every }\mu\in X(A)\,.$

In (2.3) , the first addendum on the right hand side is the elastic energy
induced by the dislocation measure $\mu$ , whereas the second addendum plays
the role of a plastic core energy.

In order to introduce the self-energy of an edge dislocation, for every
$b\in\mathbb{T}$ we first define the strain field
$\beta_{\mathbb{R}^{2}}^{b,\mathbb{C}}$ satisfying the circulation condition

$\mathrm{Curl}\,\beta=b\delta_{0}\textrm{ in }\mathbb{R}^{2}$

and the equilibrium equation

$\mathrm{Div}\,\mathbb{C}\beta=0\textrm{ in }\mathbb{R}^{2}\,.$

As shown in [6], the strain field $\beta_{\mathbb{R}^{2}}^{b,\mathbb{C}}$ is
given, in polar coordinates, by

(2.4)
$\beta^{b,\mathbb{C}}_{\mathbb{R}^{2}}(\rho,\theta):=\frac{1}{\rho}\big{(}f^{b,\mathbb{C}}(\theta)\otimes(-\sin\theta,\cos\theta)+g^{b,\mathbb{C}}\otimes(\cos\theta,\sin\theta)\big{)}\,,$

where the constant $g^{b,\mathbb{C}}\in\mathbb{R}^{2}$ and the function
$f^{b,\mathbb{C}}\in C^{0}([0,2\pi];\mathbb{R}^{2})$ , with $f(0)=f(2\pi)$ and
$\int_{0}^{2\pi}f^{b,\mathbb{C}}(\omega)\,\mathrm{d}\omega=b$ , are uniquely
determined by the vector $b$ and the tensor $\mathbb{C}$ .

The corresponding displacement $u^{b,\mathbb{C}}_{\mathbb{R}^{2}}$ (i.e., such
that $\nabla
u^{b,\mathbb{C}}_{\mathbb{R}^{2}}=\beta^{b,\mathbb{C}}_{\mathbb{R}^{2}}$) is
computed explicitly in the literature (see, for instance, [3, formula 4.1.25])
and, in polar coordinates, is given by

(2.5)
$u^{b,\mathbb{C}}_{\mathbb{R}^{2}}(\rho,\theta)=F^{b,\mathbb{C}}(\theta)+g^{b,\mathbb{C}}\log\rho\,,$

where
$F^{b,\mathbb{C}}(\theta)=\int_{0}^{\theta}f^{b,\mathbb{C}}(\omega)\,\mathrm{d}\omega$
, for $\theta\in[0,2\pi)$. Note that the displacement above is uniquely
determined up to a constant.

For every $b\in\mathbb{T}$ we set

(2.6)
$\psi^{\mathbb{C}}(b):=\int_{0}^{2\pi}\frac{1}{2}\mathbb{C}\,\Gamma^{b,\mathbb{C}}(\theta):\Gamma^{b,\mathbb{C}}(\theta)\,\mathrm{d}\theta=\frac{1}{|\log
r|}\int_{B_{1}\setminus\overline{B}_{r}}\frac{1}{2}\mathbb{C}\beta^{b,\mathbb{C}}_{\mathbb{R}^{2}}:\beta^{b,\mathbb{C}}_{\mathbb{R}^{2}}\,\mathrm{d}x\,,\qquad
0<r<1\,,$

where we have set
$\Gamma^{b,\mathbb{C}}(\theta):=\big{(}f^{b,\mathbb{C}}(\theta)\otimes(-\sin\theta,\cos\theta)+g^{b,\mathbb{C}}\otimes(\cos\theta,\sin\theta)\big{)}$
for every $\theta\in[0,2\pi]$ . Finally, for every $b\in\mathbb{T}$ we define

(2.7)
$\varphi^{\mathbb{C}}(b):=\min\bigg{\\{}\sum_{i=1}^{N}|z_{i}|\psi^{\mathbb{C}}(b_{i})\,:\,z_{i}\in\mathbb{Z}\,,\,b_{i}\in\mathbb{T}\,,\,N\in\mathbb{N}\,,\,\sum_{i=1}^{N}z_{i}b_{i}=b\bigg{\\}}\,.$

The following result is a slight variant of [6, Theorem 4] .

######  Theorem 2.1.

The following $\Gamma$-convergence result holds true.

  * (i)

(Compactness) Let $\\{\mu_{\varepsilon}\\}_{\varepsilon}\subset X(A)$ and let
$\\{\beta_{\varepsilon}\\}_{\varepsilon}$ be a sequence of fields with
$\beta_{\varepsilon}\in\mathscr{A}\\!\mathscr{S}_{\varepsilon}(\mu_{\varepsilon})$
(for every $\varepsilon>0$) such that

(2.8)
$E_{\varepsilon}(\beta_{\varepsilon};A_{\varepsilon}(\mu_{\varepsilon}))+|\mu_{\varepsilon}|(A)\leq
M|\log\varepsilon|\,;$

then, up to a subsequence,
$\mu_{\varepsilon}\stackrel{{\scriptstyle\mathrm{flat}}}{{\rightarrow}}\mu$
(as $\varepsilon\to 0$) for some $\mu\in X(A)$ .

  * (ii)

($\Gamma$-liminf inequality) For every
$\mu=\sum_{k=1}^{K}b^{k}\delta_{x^{k}}\in X(A)$ , for every
$\\{\mu_{\varepsilon}\\}_{\varepsilon}\subset X(A)$ with
$\mu_{\varepsilon}\stackrel{{\scriptstyle\mathrm{flat}}}{{\rightarrow}}\mu$
(as $\varepsilon\to 0$) and $|\mu_{\varepsilon}|(A)\leq C|\log\varepsilon|$
and for every sequence of fields $\\{\beta_{\varepsilon}\\}_{\varepsilon}$
with
$\beta_{\varepsilon}\in\mathscr{A}\\!\mathscr{S}_{\varepsilon}(\mu_{\varepsilon})$
(for every $\varepsilon>0$) it holds

(2.9) $\liminf_{\varepsilon\to
0}\frac{E_{\varepsilon}(\beta_{\varepsilon};A_{\varepsilon}(\mu_{\varepsilon}))}{|\log\varepsilon|}\geq\sum_{k=1}^{K}\varphi^{\mathbb{C}}(b^{k})\,;$

  * (iii)

($\Gamma$-limsup inequality) For every
$\mu=\sum_{k=1}^{K}b^{k}\delta_{x^{k}}\in X(A)$ there exists a sequence of
measures $\\{\mu_{\varepsilon}\\}_{\varepsilon}\subset X(A)$ such that
$\mu_{\varepsilon}\stackrel{{\scriptstyle\mathrm{flat}}}{{\rightarrow}}\mu$
(as $\varepsilon\to 0$) and

(2.10) $\limsup_{\varepsilon\to
0}\frac{\mathcal{E}_{\varepsilon}(\mu_{\varepsilon};A)}{|\log\varepsilon|}\leq\sum_{k=1}^{K}\varphi^{\mathbb{C}}(b^{k})\,.$

The proof of Theorem 2.1 can be obtained by arguing verbatim as in the proof
of [6, Theorem 4]. In fact, the minor differences between Theorem 2.1 and [6,
Theorem 4] are the following. First, in [6] the admissible strains $\beta$
should satisfy the condition

$\int_{A_{\varepsilon}(\mu)}(\beta-\beta^{*})\,\mathrm{d}x=0\,.$

Such a condition can be always enforced in view of the invariance of the
elastic energy with respect to translations. Second, in [6, Theorem 4] the
compactness property is stated enforcing (2.8) only for the optimal
$\beta_{\varepsilon}$. Third, in [6, Theorem 4], the lower bound (2.9) is
provided for the functional $\mathcal{E}_{\varepsilon}$; there, the assumption
$|\mu_{\varepsilon}|(A)\leq C|\log\varepsilon|$ is automatically satisfied by
sequences with equibounded energy. However, assuming
$|\mu_{\varepsilon}|(A)\leq C|\log\varepsilon|$ , the same proof provides the
same lower bound for
$\frac{1}{|\log\varepsilon|}E_{\varepsilon}(\beta_{\varepsilon};A_{\varepsilon}(\mu_{\varepsilon}))$
.

Now, we specialize the functions $\psi^{\mathbb{C}}$ and
$\varphi^{\mathbb{C}}$ to the particular isotropic case we deal with in the
discrete-to-continuum limit. More specifically, let $\mathbb{C}$ be the
isotropic elasticity tensor with Lamé parameters both equal to $1$ , i.e.,

(2.11)
$\mathbb{C}\beta:\beta:=|\mathrm{tr}\,\beta|^{2}+2|\beta^{\mathrm{sym}}|^{2}\,,$

By [5, formulas (3.3) and (3.4)], for the specific choice of $\mathbb{C}$ in
(2.11) we have that the constant $g^{b,\mathbb{C}}$ and the function
$f^{b,\mathbb{C}}$ in (2.4) are given by

(2.12) $\displaystyle g^{b,\mathbb{C}}=$
$\displaystyle-\frac{1}{6\pi}b^{\perp}\,,$ $\displaystyle
f^{b,\mathbb{C}}(\theta)=$
$\displaystyle\frac{1}{2\pi}b-\frac{1}{3\pi}(-b_{1}\cos(2\theta)-b_{2}\sin(2\theta),b_{2}\cos(2\theta)-b_{1}\sin(2\theta))\,;$

therefore, by straightforward computations, in this case

$\psi(b):=\psi^{\mathbb{C}}(b)=\frac{1}{3\pi}|b|^{2}\,,\qquad
b\in\mathbb{T}\,,$

and hence, by (2.7),

(2.13) $\displaystyle\varphi(b)=\varphi^{\mathbb{C}}(b)=$
$\displaystyle\frac{1}{3\pi}\min\bigg{\\{}\sum_{i=1}^{N}|z_{i}|\,|b_{i}|^{2}\,:\,z_{i}\in\mathbb{Z}\,,\,b_{i}\in\mathbb{T}\,,\,N\in\mathbb{N}\,,\,\sum_{i=1}^{N}z_{i}b_{i}=b\bigg{\\}}$
$\displaystyle=$
$\displaystyle\frac{1}{3\pi}\min\bigg{\\{}\sum_{i=1}^{3}|z_{i}|\,:\,z_{1},z_{2},z_{3}\in\mathbb{Z}\,,\quad
b=z_{1}e_{1}+z_{2}\nu+z_{3}\eta\bigg{\\}}\,,$

which is exactly (1.4).

##  3\. Proof of the main result

This section is devoted to the proof of Theorem 1.1. In Subsection 3.1 below
we collect some auxiliary results that will be instrumental to prove Theorem
1.1.

###  3.1. Preliminary results

We start by deriving the continuum isotropic elasticity tensor associated to
our discrete functional.

######  Remark 3.1.

Let $\varepsilon>0$ and let $T_{\varepsilon}\in\mathcal{T}_{\varepsilon}$ .
Let moreover $u:T_{\varepsilon}\cap\varepsilon\mathbb{T}\to\mathbb{R}^{2}$ and
$\sigma:(T_{\varepsilon})_{\varepsilon}^{1}\to\varepsilon\mathbb{T}$ be such
that $\sigma(i,j)=-\sigma(j,i)$ for every
$(i,j)\in(T_{\varepsilon})_{\varepsilon}^{1}$ . If
$\beta\in\mathbb{R}^{2\times 2}$ satisfies

$\beta\,(j-i)=\mathrm{d}u(i,j)-\sigma(i,j)\qquad\textrm{for every
}(i,j)\in(T_{\varepsilon})_{\varepsilon}^{1},$

then, by straightforward computations, we have that

(3.1) $\displaystyle F_{\varepsilon}(u,\sigma;T_{\varepsilon})=$
$\displaystyle\varepsilon^{2}\Big{(}|e_{1}^{*}\,\beta\,e_{1}|^{2}+|\nu^{*}\,\beta\,\nu|^{2}+|\eta^{*}\,\beta\,\eta|^{2}\Big{)}=\varepsilon^{2}\frac{3}{8}\mathbb{C}\beta:\beta=\frac{\sqrt{3}}{2}\int_{T_{\varepsilon}}\mathbb{C}\beta:\beta\,\mathrm{d}x\,,$

where $F_{\varepsilon}(\cdot,\cdot;T_{\varepsilon})$ is defined in (1.3) and
$\mathbb{C}$ is given in (2.11).

In the next lemma we construct, far from the singularities, a (continuous)
strain field $\beta$ that is compatible with a given distribution of
dislocations and whose (continuous) energy behaves like the discrete energy
$F_{\varepsilon}$ . Such an estimate, together with a bound on the total
variation, will allow us to deduce the $\Gamma$-liminf inequality in Theorem
1.1(ii) directly from the analogous statement in the continuous setting
(Theorem 2.1(ii)).

######  Lemma 3.2.

Let $\varepsilon>0$ and let
$(u,\sigma)\in\mathcal{D}_{\varepsilon}(\Omega)\times\mathcal{S}_{\varepsilon}(\Omega)$
. Let moreover

(3.2)
$\mathcal{K}_{\varepsilon}:=\\{T_{\varepsilon}\in\mathcal{T}_{\varepsilon}(\Omega)\,:\,\mu[\sigma](T_{\varepsilon})=0\\}\,,\qquad
K_{\varepsilon}:=\bigcup_{T_{\varepsilon}\in\mathcal{K}_{\varepsilon}}T_{\varepsilon}\,.$

Then, there exists a piecewise constant (namely, constant on each triangle
$T_{\varepsilon}\in\mathcal{K}_{\varepsilon}$) field
$\beta^{u,\sigma,\mathcal{K}_{\varepsilon}}\in
L^{2}(K_{\varepsilon};\mathbb{R}^{2\times 2})$ such that:

  * (a)

$\mathrm{d}u(i,j)-\sigma(i,j)=\beta^{u,\sigma,\mathcal{K}_{\varepsilon}}_{|_{T_{\varepsilon}}}\,(j-i)$
for every $i,j\in T_{\varepsilon}$ with
$T_{\varepsilon}\in\mathcal{K}_{\varepsilon}$ ;

  * (b)

$F_{\varepsilon}(u,\sigma;K_{\varepsilon})=\frac{\sqrt{3}}{2}\int_{K_{\varepsilon}}\frac{1}{2}\mathbb{C}\beta^{u,\sigma,\mathcal{K}_{\varepsilon}}:\beta^{u,\sigma,\mathcal{K}_{\varepsilon}}\,\mathrm{d}x+\frac{1}{4\varepsilon^{2}}\sum_{\genfrac{}{}{0.0pt}{1}{(i,j)\in
K^{1}_{\varepsilon}}{i,j\in\partial
K_{\varepsilon}}}|(\mathrm{d}u(i,j)-\sigma(i,j))\cdot(j-i)|^{2}$ , where
$F_{\varepsilon}$ is defined in (1.3) and $\mathbb{C}$ is defined in (2.11) ;

  * (c)

$\mathrm{Curl}\,\beta^{u,\sigma,\mathcal{K}_{\varepsilon}}=0$ in
$K_{\varepsilon}$ and $\int_{\partial
U}\beta^{u,\sigma,\mathcal{K}_{\varepsilon}}\,\tau\,\mathrm{d}\mathcal{H}^{1}=\mu[\sigma](U)$
for every smooth open set $U\subset\Omega$ such that $\partial U\subset
K_{\varepsilon}$ .

###### Proof.

Let $T_{\varepsilon}=\mathrm{conv}\\{i,j,k\\}\in\mathcal{K}_{\varepsilon}$ and
assume that the triple $(i,j,k)$ is counterclockwise oriented. Let
$v_{T_{\varepsilon}}\in\mathcal{D}_{\varepsilon}(T_{\varepsilon})$ be defined
by

$v_{T_{\varepsilon}}(i)=0\,,\quad
v_{T_{\varepsilon}}(j)=\mathrm{d}u(i,j)-\sigma(i,j)\,,\quad
v_{T_{\varepsilon}}(k)=v_{T_{\varepsilon}}(j)+\mathrm{d}u(j,k)-\sigma(j,k)\,,$

and notice that the discrete gradient $\mathrm{d}v_{T_{\varepsilon}}$ of
$v_{T_{\varepsilon}}$ agrees with $\mathrm{d}u-\sigma$ . For every
$T_{\varepsilon}\in\mathcal{K}_{\varepsilon}$ let
$\tilde{v}_{T_{\varepsilon}}:T_{\varepsilon}\to\mathbb{R}^{2}$ be the
piecewise affine interpolation of $v_{T_{\varepsilon}}$ and let
$\beta^{u,\sigma,\mathcal{K}_{\varepsilon}}:K_{\varepsilon}\to\mathbb{R}^{2\times
2}$ be the map defined by

$\beta^{u,\sigma,\mathcal{K}_{\varepsilon}}:=\sum_{T_{\varepsilon}\in\mathcal{K}_{\varepsilon}}\mathrm{D}\tilde{v}_{T_{\varepsilon}}\,\chi_{T_{\varepsilon}}\,,$

where $\mathrm{D}\tilde{v}_{T_{\varepsilon}}$ denotes the (continuous)
gradient of the function $\tilde{v}_{T_{\varepsilon}}$ . Clearly
$\beta^{u,\sigma,\mathcal{K}_{\varepsilon}}\in
L^{2}(K_{\varepsilon};\mathbb{R}^{2\times 2})$ . Moreover, by construction, we
have that property (a) is satisfied; furthermore, since for every
$T_{\varepsilon}\in\mathcal{K}_{\varepsilon}$ and for every $(i,j)\in
T_{\varepsilon}^{1}$ it holds

$\beta^{u,\sigma,\mathcal{K}_{\varepsilon}}_{|_{T_{\varepsilon}}}\,(j-i)=\mathrm{d}u(i,j)-\sigma(i,j)\,,$

by Remark 3.1 we get

$\displaystyle F_{\varepsilon}(u,\sigma;K_{\varepsilon})=$
$\displaystyle\frac{1}{2}\sum_{T_{\varepsilon}\in\mathcal{K}_{\varepsilon}}F_{\varepsilon}(u,\sigma;T_{\varepsilon})+\frac{1}{4\varepsilon^{2}}\sum_{\genfrac{}{}{0.0pt}{1}{(i,j)\in
K^{1}_{\varepsilon}}{i,j\in\partial
K_{\varepsilon}}}|(\mathrm{d}u(i,j)-\sigma(i,j))\cdot(j-i)|^{2}$
$\displaystyle=$
$\displaystyle\frac{\sqrt{3}}{2}\int_{K_{\varepsilon}}\frac{1}{2}\mathbb{C}\beta^{u,\sigma,\mathcal{K}_{\varepsilon}}:\beta^{u,\sigma,\mathcal{K}_{\varepsilon}}\,\mathrm{d}x+\frac{1}{4\varepsilon^{2}}\sum_{\genfrac{}{}{0.0pt}{1}{(i,j)\in
K^{1}_{\varepsilon}}{i,j\in\partial
K_{\varepsilon}}}|(\mathrm{d}u(i,j)-\sigma(i,j))\cdot(j-i)|^{2}\,,$

i.e., property (b).

Finally, we prove that also (c) is satisfied. To this end, we notice that if
$T_{\varepsilon}^{1}$ and $T_{\varepsilon}^{2}$ are two triangles sharing one
edge $(i,j)$ , then, by construction,
$\beta^{u,\sigma,\mathcal{K}_{\varepsilon}}_{|_{T_{\varepsilon}^{1}}}\,(j-i)=\beta^{u,\sigma,\mathcal{K}_{\varepsilon}}_{|_{T_{\varepsilon}^{2}}}\,(j-i)$
so that $\mathrm{Curl}\,\beta^{u,\sigma,\mathcal{K}_{\varepsilon}}=0$ on
$K_{\varepsilon}$ . Now, since for every
$T_{\varepsilon}\in\mathcal{K}_{\varepsilon}$ ,

$\int_{\partial
T_{\varepsilon}}\beta^{u,\sigma,\mathcal{K}_{\varepsilon}}\,\tau\,\mathrm{d}\mathcal{H}^{1}=0\,,$

by using Stokes’ Theorem, for every smooth open set $U\subset\Omega$ such that
$\partial U\subset K_{\varepsilon}$, we have

$\int_{\partial
U}\beta^{u,\sigma,\mathcal{K}_{\varepsilon}}\,\tau\,\mathrm{d}\mathcal{H}^{1}=\int_{U\cap\partial
K_{\varepsilon}}\beta^{u,\sigma,\mathcal{K}_{\varepsilon}}\,\tau\,\mathrm{d}\mathcal{H}^{1}=-\sum_{T_{\varepsilon}\subset
U}\mathrm{d}\sigma(T_{\varepsilon})=\mu[\sigma](U)\,,$

which concludes the proof of property (c) and of the whole lemma. ∎

The following result allows to extend the field $\beta$ constructed in Lemma
3.2 above up to the boundary of $\Omega$ .

######  Lemma 3.3.

Let $\varepsilon>0$ and let
$(u,\sigma)\in\mathcal{D}_{\varepsilon}(\Omega)\times\mathcal{S}_{\varepsilon}(\Omega)$
be such that $\mu[\sigma]\in X_{\varepsilon}(\Omega)$ ; let moreover
$\mathcal{K}_{\varepsilon}$ and $K_{\varepsilon}$ be defined as in (3.2) and
let $\beta=\beta^{u,\sigma,\mathcal{K}_{\varepsilon}}$ be the field provided
by Lemma 3.2. Then, there exists a field
$\hat{\beta}=\hat{\beta}^{u,\sigma,\mathcal{K}_{\varepsilon}}\in
L^{2}(K_{\varepsilon}\cup(\Omega\setminus\Omega_{\varepsilon});\mathbb{R}^{2\times
2})$ such that

  * (i)

$\hat{\beta}=\beta$ in $K_{\varepsilon}$ ;

  * (ii)

$\mathrm{Curl}\,\hat{\beta}=0$ (in the sense of distributions);

  * (iii)

$\int_{K_{\varepsilon}\cup(\Omega\setminus\Omega_{\varepsilon})}\mathbb{C}\hat{\beta}:\hat{\beta}\,\mathrm{d}x\leq
C\int_{K_{\varepsilon}}\mathbb{C}\beta:\beta\,\mathrm{d}x$ , for some constant
$C$ independent of $\varepsilon$ .

###### Proof.

It is enough to notice that, in view of (MS), each of the triangles
$T_{\varepsilon}\in\mathcal{T}_{\varepsilon}(\Omega)$ touching
$\partial\Omega_{\varepsilon}$ satisfies $\mu[\sigma](T_{\varepsilon})=0$ .
Therefore, in order to construct a field $\hat{\beta}$ satisfying properties
(i), (ii), and (iii), it is enough to extend locally by reflection the field
$\beta$ provided by Lemma 3.2. ∎

We conclude this subsection showing how assumption (MS) allows to estimate the
total variation of the dislocation measure $\mu$ with the elastic energy
$\mathcal{F}_{\varepsilon}(\mu)$ .

######  Lemma 3.4.

There exists $\bar{C}>0$ such that
$|\frac{\mu}{\varepsilon}|(\Omega)\leq\bar{C}\frac{1}{\varepsilon^{2}}\mathcal{F}_{\varepsilon}(\mu)$
for every $\varepsilon>0$ and for every $\mu\in X_{\varepsilon}(\Omega)$ .

###### Proof.

Let $\varepsilon>0$ and $\mu\in X_{\varepsilon}(\Omega)$ be fixed. Let
moreover
$(u,\sigma)\in\mathcal{D}_{\varepsilon}(\Omega)\times\mathcal{S}_{\varepsilon}(\Omega)$
with $\mu[\sigma]=\mu$ . For every $p\in\mathrm{supp}\,\mu$ let
$\mathcal{K}_{\varepsilon}^{p}$ be the set of the triangles
$T_{\varepsilon}\neq T_{\varepsilon}^{p}$ sharing at least one vertex with the
triangle $T_{\varepsilon}^{p}$ centered at $p$ and let
$K_{\varepsilon}^{p}:=\bigcup_{T_{\varepsilon}\in\mathcal{K}_{\varepsilon}^{p}}T_{\varepsilon}$
. By the very definition of $X_{\varepsilon}(\Omega)$, we have that
$\mu(T_{\varepsilon})=0$ for every
$T_{\varepsilon}\in\bigcup_{p\in\mathrm{supp}\,\mu}\mathcal{K}_{\varepsilon}^{p}$.
Let $\beta^{u,\sigma,\mathcal{K}_{\varepsilon}^{p}}$ be the map defined in
Lemma 3.2 above. Recalling the notation for the annulus $A_{r,R}(p)$ , for
every $p\in\mathrm{supp}\,\mu$ we set

$S_{\varepsilon}^{p}:=\frac{1}{|A_{\frac{\sqrt{3}}{3}\varepsilon,\frac{5\sqrt{3}}{12}\varepsilon}(p)|}\int_{A_{\frac{\sqrt{3}}{3}\varepsilon,\frac{5\sqrt{3}}{12}\varepsilon}(p)}(\beta^{u,\sigma,\mathcal{K}^{p}_{\varepsilon}})^{{\mathrm{skew}}}\mathrm{d}y\,,$

where, for every $\beta\in\mathbb{R}^{2\times 2}$ , we have set
$\beta^{\mathrm{skew}}:=\frac{1}{2}(\beta-\beta^{*})$ . By Korn’s inequality,
there exists a constant $\kappa$ (independent of $\varepsilon$ and $p$) such
that

$\displaystyle\int_{A_{\frac{\sqrt{3}}{3}\varepsilon,\frac{5\sqrt{3}}{12}\varepsilon}(p)}|(\beta^{u,\sigma,\mathcal{K}^{p}_{\varepsilon}})^{\mathrm{sym}}|^{2}\mathrm{d}x\geq$
$\displaystyle\kappa\int_{A_{\frac{\sqrt{3}}{3}\varepsilon,\frac{5\sqrt{3}}{12}\varepsilon}(p)}\big{|}\beta^{u,\sigma,\mathcal{K}^{p}_{\varepsilon}}-S_{\varepsilon}^{p}\big{|}^{2}\mathrm{d}x$
$\displaystyle\geq$
$\displaystyle\kappa\int_{\frac{\sqrt{3}}{3}\varepsilon}^{\frac{5\sqrt{3}}{12}\varepsilon}\int_{\partial
B_{\rho}(p)}\big{|}(\beta^{u,\sigma,\mathcal{K}^{p}_{\varepsilon}}-S_{\varepsilon}^{p})\,\tau\big{|}^{2}\,\mathrm{d}\mathcal{H}^{1}\,\mathrm{d}\rho$
$\displaystyle\geq$ $\displaystyle\frac{\kappa}{2\pi}\log
2|\mu(p)|^{2}=\frac{\kappa}{2\pi}\log
2\,\varepsilon^{2}\Big{|}\frac{\mu}{\varepsilon}(p)\Big{|}^{2}\geq\frac{\kappa}{2\pi}\log
2\,\varepsilon^{2}\Big{|}\frac{\mu}{\varepsilon}(p)\Big{|}\,,$

where the second inequality follows by Fubini theorem, estimating
$|\beta^{u,\sigma,\mathcal{K}^{p}_{\varepsilon}}-S_{\varepsilon}^{p}|$ from
below by its tangential part, and the third one is a consequence of Jensen’s
inequality using Lemma 3.2(c) together with

$\int_{\partial
B_{\rho}(p)}S^{p}_{\varepsilon}\,\tau\,\mathrm{d}\mathcal{H}^{1}=0\,.$

By assumption (MS), Lemma 3.2(b) and the very definition of $\mathbb{C}$ in
(2.11), using that the annuli
$A_{\frac{\sqrt{3}}{3}\varepsilon,\frac{5\sqrt{3}}{12}\varepsilon}(p)$ are
pairwise disjoint and contained in $K_{\varepsilon}^{p}$ , we thus deduce that

$\displaystyle F_{\varepsilon}(u,\sigma)\geq$
$\displaystyle\frac{\sqrt{3}}{2}\sum_{p\in\mathrm{supp}\,\mu}\frac{1}{2}\int_{K_{\varepsilon}^{p}}\mathbb{C}\beta^{u,\sigma,\mathcal{K}_{\varepsilon}^{p}}:\beta^{u,\sigma,\mathcal{K}_{\varepsilon}^{p}}\,\mathrm{d}x\geq
C\sum_{p\in\mathrm{supp}\,\mu}\int_{A_{\frac{\sqrt{3}}{3}\varepsilon,\frac{5\sqrt{3}}{12}\varepsilon}(p)}|(\beta^{u,\sigma,\mathcal{K}^{p}_{\varepsilon}})^{\mathrm{sym}}|^{2}\mathrm{d}x$
$\displaystyle\geq$ $\displaystyle
C\varepsilon^{2}\Big{|}\frac{\mu}{\varepsilon}\Big{|}(\Omega)\,,$

which, taking the infimum over the pairs
$(u,\sigma)\in\mathcal{D}_{\varepsilon}(\Omega)\times\mathcal{S}_{\varepsilon}(\Omega)$
with $\mu[\sigma]=\mu$ , provides the claim for $\bar{C}=\frac{1}{C}$ . ∎

######  Remark 3.5.

Without the assumption (MS), Lemma 3.4 does not hold. As an example, for every
$\varepsilon>0$ , let $u_{\varepsilon}\equiv 0$ and

$\sigma_{\varepsilon}(i,j):=\left\\{\begin{array}[]{ll}+\sqrt{3}\varepsilon\,e_{2}&\textrm{if
}j=i+\varepsilon e_{1}\,,\\\ -\sqrt{3}\varepsilon\,e_{2}&\textrm{if
}j=i-\varepsilon e_{1}\,,\\\ 0&\textrm{elsewhere in
}\Omega_{\varepsilon}^{1}\,.\end{array}\right.$

Trivially,
$|\frac{\mu[\sigma_{\varepsilon}]}{\varepsilon}|(\Omega)\sim\frac{1}{\varepsilon^{2}}$,
$\big{\|}\frac{\mu[\sigma_{\varepsilon}]}{\varepsilon}\big{\|}_{\mathrm{flat}}\sim\frac{1}{\varepsilon}$,
and
$\mathcal{F}_{\varepsilon}(\mu[\sigma_{\varepsilon}])=F_{\varepsilon}(u_{\varepsilon},\sigma_{\varepsilon})\equiv
0$ .

###  3.2. Proof of Theorem 1.1

We are now in a position to prove Theorem 1.1.

###### Proof of Theorem 1.1(i).

For every $\varepsilon>0$ , let
$(u_{\varepsilon},\sigma_{\varepsilon})\in\mathcal{D}_{\varepsilon}(\Omega)\times\mathcal{S}_{\varepsilon}(\Omega)$
with $\mu[\sigma_{\varepsilon}]=\mu_{\varepsilon}$ be such that

(3.3) $F_{\varepsilon}(u_{\varepsilon},\sigma_{\varepsilon})\leq
2\mathcal{F}_{\varepsilon}(\mu_{\varepsilon})\,.$

We let

(3.4)
$\mathcal{K}_{\varepsilon}:=\\{T_{\varepsilon}\in\mathcal{T}_{\varepsilon}(\Omega)\,:\,\mu_{\varepsilon}(T_{\varepsilon})=0\\}\,\qquad
K_{\varepsilon}:=\bigcup_{T_{\varepsilon}\in\mathcal{K}_{\varepsilon}}T_{\varepsilon}\,,$

and we set
$\beta_{\varepsilon}:=\beta^{u_{\varepsilon},\sigma_{\varepsilon},\mathcal{K}_{\varepsilon}}$
and
$\hat{\beta}_{\varepsilon}:=\hat{\beta}^{u_{\varepsilon},\sigma_{\varepsilon},\mathcal{K}_{\varepsilon}}$,
where $\beta^{u_{\varepsilon},\sigma_{\varepsilon},\mathcal{K}_{\varepsilon}}$
and
$\hat{\beta}^{u_{\varepsilon},\sigma_{\varepsilon},\mathcal{K}_{\varepsilon}}$
are the fields provided by Lemmas 3.2 and 3.3, respectively. Furthermore, we
set $\tilde{\mu}_{\varepsilon}:=\frac{\mu_{\varepsilon}}{\varepsilon}$ and
$\tilde{\beta}_{\varepsilon}:=\frac{\hat{\beta}_{\varepsilon}}{\varepsilon}$ .
Recalling the definition of $\Omega_{\varepsilon}(\tilde{\mu}_{\varepsilon})$
in (2.1) and using, in order of appearance, Lemma 3.3(iii), Lemma 3.2(b),
(3.3) and the energy bound, we have

(3.5)
$\displaystyle\frac{1}{2}\int_{\Omega_{\varepsilon}(\tilde{\mu}_{\varepsilon})}\mathbb{C}\tilde{\beta}_{\varepsilon}:\tilde{\beta}_{\varepsilon}\,\mathrm{d}x=$
$\displaystyle\frac{1}{2\varepsilon^{2}}\int_{\Omega_{\varepsilon}(\tilde{\mu}_{\varepsilon})}\mathbb{C}\hat{\beta}_{\varepsilon}:\hat{\beta}_{\varepsilon}\,\mathrm{d}x\leq\frac{C}{2\varepsilon^{2}}\int_{K_{\varepsilon}}\mathbb{C}\beta_{\varepsilon}:\beta_{\varepsilon}\,\mathrm{d}x$
$\displaystyle\leq$
$\displaystyle\frac{C}{\varepsilon^{2}}F_{\varepsilon}(u_{\varepsilon},\sigma_{\varepsilon})\leq\frac{C}{\varepsilon^{2}}\mathcal{F}_{\varepsilon}(\mu_{\varepsilon})\leq
C|\log\varepsilon|\,;$

moreover, by Lemma 3.4 and by the energy bound, we have

(3.6) $|\tilde{\mu}_{\varepsilon}|(\Omega)\leq C|\log\varepsilon|\,.$

In view of Lemma 3.3(ii) and of Lemma 3.2(c), we have
$\tilde{\beta}_{\varepsilon}\in\mathscr{A}\\!\mathscr{S}_{\varepsilon}(\tilde{\mu}_{\varepsilon})$
for every $\varepsilon>0$ (with $A=\Omega$ in (2.2)) , so that, by (3.5) and
(3.6), applying Theorem 2.1(i) with
$\mu_{\varepsilon}=\tilde{\mu}_{\varepsilon}$ and
$\beta_{\varepsilon}=\tilde{\beta}_{\varepsilon}$, we deduce the claim. ∎

###### Proof of Theorem 1.1(ii).

We can assume without loss of generality that
$\mathcal{F}_{\varepsilon}(\mu_{\varepsilon})\leq
C\varepsilon^{2}|\log\varepsilon|$ , which in view of Lemma 3.4, implies that
$\big{|}\frac{\mu_{\varepsilon}}{\varepsilon}\big{|}(\Omega)\leq
C|\log\varepsilon|$ .

Let $A\subset\subset\Omega$ be an open bounded subset of $\mathbb{R}^{2}$ with
Lipschitz continuous boundary such that $\mathrm{supp}\,\mu\subset A$ . Let
$\varepsilon>0$ small enough such that $A\subset\Omega_{\varepsilon}$ . Let
moreover
$(u_{\varepsilon},\sigma_{\varepsilon})\in\mathcal{D}_{\varepsilon}(\Omega)\times\mathcal{S}_{\varepsilon}(\Omega)$
with $\mu[\sigma_{\varepsilon}]=\mu_{\varepsilon}$ be such that

(3.7) $\mathcal{F}_{\varepsilon}(\mu_{\varepsilon})\leq
F_{\varepsilon}(u_{\varepsilon},\sigma_{\varepsilon})+\varepsilon^{2}\,,$

and let $\mathcal{K}_{\varepsilon}$ be defined by (3.4); we set
$\beta_{\varepsilon}:=\beta^{u_{\varepsilon},\sigma_{\varepsilon},\mathcal{K}_{\varepsilon}}$
, where
$\beta^{u_{\varepsilon},\sigma_{\varepsilon},\mathcal{K}_{\varepsilon}}$ is
the field provided by Lemma 3.2 ,
$\tilde{\mu}_{\varepsilon}:=\frac{\mu_{\varepsilon}}{\varepsilon}$ and
$\tilde{\beta}_{\varepsilon}:=\frac{\beta_{\varepsilon}}{\varepsilon}$ . By
(3.7) and Lemma 3.2 we have that
$\tilde{\beta}_{\varepsilon}\in\mathscr{A}\\!\mathscr{S}_{\varepsilon}(\tilde{\mu}_{\varepsilon})$
(for every $\varepsilon>0$) and

$\mathcal{F}_{\varepsilon}(\mu_{\varepsilon})\geq
F_{\varepsilon}(u_{\varepsilon},\sigma_{\varepsilon})-\varepsilon^{2}\geq\frac{\sqrt{3}}{2}\varepsilon^{2}E_{\varepsilon}(\tilde{\beta}_{\varepsilon};A_{\varepsilon}(\tilde{\mu}_{\varepsilon}))-\varepsilon^{2}\,,$

whence the claim follows by applying Theorem 2.1(ii) with
$\mu_{\varepsilon}=\tilde{\mu}_{\varepsilon}$ and
$\beta_{\varepsilon}=\tilde{\beta}_{\varepsilon}$ . ∎

###### Proof of Theorem 1.1(iii).

$x^{k_{1}\\!,\varepsilon}$$\gamma^{k_{1}\\!,\varepsilon}$$x^{k_{2}\\!,\varepsilon}$$\gamma^{k_{2}\\!,\varepsilon}$$i{+}\varepsilon\eta$$i$$i{-}\varepsilon\eta$
Figure 1. Geometric construction used in the proof of Theorem 1.1(iii). The
position of each dislocation is approximated with a point $x^{k,\varepsilon}$
sitting on the barycenter of a triangle of $\mathcal{T}_{\varepsilon}(\Omega)$
. The approximate displacement is the restriction to the lattice points of a
continuum displacement having a jump on each half-line
$\gamma^{k,\varepsilon}$ . The sets $S^{1}_{k,\varepsilon}$ defined in the
proof contain bonds across the jumps: for example, the points $i$ ,
$i{\pm}\varepsilon\eta$ displayed in the picture are such that both bonds
$(i,i{-}\varepsilon\eta)$ , $(i,i{+}\varepsilon\eta)$ belong to the set
$S^{1}_{k_{1},\varepsilon}$ .

Let $\mu=\sum_{k=1}^{K}b^{k}\delta_{x^{k}}\in X(\Omega)$ . By standard density
arguments in $\Gamma$-convergence we can assume that
$\varphi(b^{k})=\psi(b^{k})$ for every $k=1,\ldots,K$ , i.e., that $|b^{k}|=1$
for every $k=1,\ldots,K$ . Let $\varepsilon\mathbb{T}^{\prime}_{+}(\Omega)$ be
the set of the barycenters of the triangles $T_{\varepsilon}^{+}=i+\varepsilon
T^{+}\in\mathcal{T}_{\varepsilon}(\Omega)$ .

For every $k=1,\ldots,K$ and for every $\varepsilon>0$ , let
$x^{k,\varepsilon}\in\varepsilon\mathbb{T}^{\prime}_{+}(\Omega)$ be such that

$|x^{k,\varepsilon}-x^{k}|=\min\\{|y-x^{k}|\,:\,y\in\varepsilon\mathbb{T}^{\prime}_{+}(\Omega)\\}\,.$

For every $k=1,\ldots,K$ set (see Figure 1)

$\gamma^{k,\varepsilon}:=\left\\{\begin{array}[]{ll}\\{x^{k,\varepsilon}+\lambda
e_{1}\,:\,\lambda\geq 0\\}&\textrm{if }b^{k}=\pm\,e_{1}\,,\\\
\\{x^{k,\varepsilon}+\lambda\nu\,:\,\lambda\geq 0\\}&\textrm{if
}b^{k}=\pm\,\nu\,,\\\ \\{x^{k,\varepsilon}+\lambda\eta\,:\,\lambda\geq
0\\}&\textrm{if }b^{k}=\pm\,\eta\,.\end{array}\right.$

Let $\bar{u}^{k,\varepsilon}\in
C^{2}(\mathbb{R}^{2}\setminus\gamma^{k,\varepsilon};\mathbb{R}^{2})$ be a
function satisfying the following properties,

(3.8)
$S_{\bar{u}^{k,\varepsilon}}=\gamma^{k,\varepsilon}\,,\qquad[\bar{u}^{k,\varepsilon}]=b^{k}\textrm{
on
}\gamma^{k,\varepsilon}\,,\qquad\nabla\bar{u}^{k,\varepsilon}(\cdot)=\beta^{b_{k},\mathbb{C}}_{\mathbb{R}^{2}}(\cdot-x^{k,\varepsilon})\,,$

where $\beta^{b,\mathbb{C}}_{\mathbb{R}^{2}}$ is defined in (2.4), for the
choice of $g^{b,\mathbb{C}}$ and $f^{b,\mathbb{C}}(\theta)$ in (2.12). Note
that the function $\bar{u}^{k,\varepsilon}$ is uniquely determined up to a
constant. We set $u^{k,\varepsilon}:=\varepsilon\bar{u}^{k,\varepsilon}$ and
we define $u_{\varepsilon}:\varepsilon\mathbb{T}\to\mathbb{R}^{2}$ as

$u_{\varepsilon}(i):=\displaystyle\sum_{k=1}^{K}u^{k,\varepsilon}(i)\,.$

Furthermore, for every $k=1,\ldots,K$ we set

$S^{1}_{k,\varepsilon}:=\\{(i,j)\in\varepsilon\mathbb{T}\times\varepsilon\mathbb{T}\,:\,|j-i|=\varepsilon\,,\,[i,j]\cap\gamma^{k,\varepsilon}\neq\emptyset\,,\,\textrm{dist}(i,\gamma^{k,\varepsilon})<\textrm{dist}(j,\gamma^{k,\varepsilon})\\}\,,$

where $[i,j]$ denotes the segment line with endpoints $i$ and $j$, and we
define
$\sigma_{\varepsilon}:\varepsilon\mathbb{T}\times\varepsilon\mathbb{T}\to\varepsilon\mathbb{T}$
as $\sigma_{\varepsilon}=\sum_{k=1}^{K}\sigma^{k,\varepsilon}$ , where

$\sigma^{k,\varepsilon}(i,j):=\begin{cases}-\varepsilon b^{k}&\textrm{ if
}(i,j)\in S_{k,\varepsilon}^{1}\,,\\\ +\varepsilon b^{k}&\textrm{ if }(j,i)\in
S_{k,\varepsilon}^{1}\,,\\\ 0&\textrm{ elsewhere}\,.\end{cases}$

Abusing notation, we still denote by $u_{\varepsilon}$ and
$\sigma_{\varepsilon}$ the restrictions of $u_{\varepsilon}$ and
$\sigma_{\varepsilon}$ to $\Omega_{\varepsilon}^{0}$ and
$\Omega_{\varepsilon}^{1}$, respectively. Then,
$u_{\varepsilon}\in\mathcal{D}_{\varepsilon}(\Omega)$ and
$\sigma_{\varepsilon}\in\mathcal{S}_{\varepsilon}(\Omega)$ ; moreover, for
$\varepsilon$ small enough,

$\mu_{\varepsilon}:=\mu[\sigma_{\varepsilon}]=\varepsilon\sum_{k=1}^{K}b^{k}\delta_{{x^{k,\varepsilon}}}\in
X_{\varepsilon}(\Omega),$

and, by construction,
$\frac{\mu_{\varepsilon}}{\varepsilon}\stackrel{{\scriptstyle*}}{{\rightharpoonup}}\mu$
as $\varepsilon\to 0$ , which clearly implies
$\frac{\mu_{\varepsilon}}{\varepsilon}\stackrel{{\scriptstyle\mathrm{flat}}}{{\rightarrow}}\mu$
.

Therefore, in order to prove (1.6) it is enough to show that

(3.9) $\limsup_{\varepsilon\to
0}\frac{F_{\varepsilon}(u_{\varepsilon},\sigma_{\varepsilon})}{\varepsilon^{2}|\log\varepsilon|}\leq\frac{\sqrt{3}}{2}\sum_{k=1}^{K}\psi(b^{k})\,.$

To this purpose, let $0<r<\frac{1}{4}\min\\{\textrm{dist}_{k_{1}\neq
k_{2}}(x^{k_{1}},x^{k_{2}})\,,\textrm{dist}(x^{k},\partial\Omega)\\}$ . In
analogy with the notation introduced in Section 1, for any open ball $B$ we
denote by $B_{\varepsilon}^{1}$ or $(B)_{\varepsilon}^{1}$ the family of
nearest neighbor bonds in $B$. In order to show (3.9), we preliminarily notice
that there exists a constant $C>0$ depending only on $K$ such that

$\displaystyle F_{\varepsilon}(u_{\varepsilon},\sigma_{\varepsilon})\leq$
$\displaystyle\frac{1}{2\varepsilon^{2}}\sum_{k=1}^{K}\sum_{(i,j)\in(B_{2r}(x^{k,\varepsilon}))_{\varepsilon}^{1}}[(\mathrm{d}u^{k,\varepsilon}(i,j)-\sigma^{k,\varepsilon}(i,j))\cdot(j-i)]^{2}$
$\displaystyle+\frac{1}{2\varepsilon^{2}}\sum_{k=1}^{K}\sum_{\genfrac{}{}{0.0pt}{1}{l=1}{l\neq
k}}^{K}\sum_{(i,j)\in(B_{2r}(x^{k,\varepsilon}))_{\varepsilon}^{1}}[(\mathrm{d}u^{l,\varepsilon}(i,j)-\sigma^{l,\varepsilon}(i,j))\cdot(j-i)]^{2}$
$\displaystyle+\frac{1}{\varepsilon^{2}}\sum_{k=1}^{K}\sum_{\genfrac{}{}{0.0pt}{1}{l=1}{l\neq
k}}^{K}\sum_{(i,j)\in(B_{2r}(x^{k,\varepsilon}))_{\varepsilon}^{1}}\big{[}(\mathrm{d}u^{k,\varepsilon}(i,j)-\sigma^{k,\varepsilon}(i,j))\cdot(j-i)\big{]}$
$\displaystyle\phantom{+\frac{C}{\varepsilon^{2}}\sum_{k=1}^{K}\sum_{\genfrac{}{}{0.0pt}{1}{l=1}{l\neq
k}}^{K}\sum_{(i,j)\in(B_{2r}(x^{k,\varepsilon}))_{\varepsilon}^{1}}}\qquad\big{[}(\mathrm{d}u^{l,\varepsilon}(i,j)-\sigma^{l,\varepsilon}(i,j))\cdot(j-i)\big{]}$
$\displaystyle+\frac{C}{\varepsilon^{2}}\sum_{k=1}^{K}\sum_{(i,j)\in\Omega_{\varepsilon}^{1}\setminus(\bigcup_{l=1}^{K}B_{r}(x^{l,\varepsilon}))_{\varepsilon}^{1}}[(\mathrm{d}u^{k,\varepsilon}(i,j)-\sigma^{k,\varepsilon}(i,j))\cdot(j-i)]^{2}\,.$

Therefore, (3.9) is proved once provided that for every fixed $k,l=1,\ldots,K$
with $l\neq k$ it holds

(3.10) $\displaystyle\limsup_{\varepsilon\to
0}\frac{1}{2\varepsilon^{4}|\log\varepsilon|}\sum_{(i,j)\in(B_{2r}(x^{k,\varepsilon}))_{\varepsilon}^{1}}[(\mathrm{d}u^{k,\varepsilon}(i,j)-\sigma^{k,\varepsilon}(i,j))\cdot(j-i)]^{2}=\frac{\sqrt{3}}{2}\psi(b^{k})\,,$
(3.11) $\displaystyle\limsup_{\varepsilon\to
0}\frac{1}{\varepsilon^{4}|\log\varepsilon|}\sum_{(i,j)\in(B_{2r}(x^{k,\varepsilon}))_{\varepsilon}^{1}}[(\mathrm{d}u^{l,\varepsilon}(i,j)-\sigma^{l,\varepsilon}(i,j))\cdot(j-i)]^{2}=0\,,$
(3.12) $\displaystyle\limsup_{\varepsilon\to
0}\frac{1}{\varepsilon^{4}|\log\varepsilon|}\sum_{(i,j)\in(B_{2r}(x^{k,\varepsilon}))_{\varepsilon}^{1}}\big{[}(\mathrm{d}u^{k,\varepsilon}(i,j)-\sigma^{k,\varepsilon}(i,j))\cdot(j-i)\big{]}$
$\displaystyle\phantom{+\frac{C}{\varepsilon^{2}}\sum_{k=1}^{K}\sum_{\genfrac{}{}{0.0pt}{1}{l=1}{l\neq
k}}^{K}\sum_{(i,j)\in(B_{2r}(x^{k,\varepsilon}))_{\varepsilon}^{1}}}\qquad\big{[}(\mathrm{d}u^{l,\varepsilon}(i,j)-\sigma^{l,\varepsilon}(i,j))\cdot(j-i)\big{]}=0\,,$
(3.13) $\displaystyle\limsup_{\varepsilon\to
0}\frac{1}{\varepsilon^{4}|\log\varepsilon|}\sum_{(i,j)\in\Omega_{\varepsilon}^{1}\setminus(\bigcup_{l=1}^{K}B_{r}(x^{l,\varepsilon}))_{\varepsilon}^{1}}[(\mathrm{d}u^{k,\varepsilon}(i,j)-\sigma^{k,\varepsilon}(i,j))\cdot(j-i)]^{2}=0\,.$

Now, by the very definition of $\beta^{b,\mathbb{C}}_{\mathbb{R}^{2}}$ in
(2.4), for every $b\in\mathbb{T}$ , we have that there exists a universal
constant $C>0$ such that

(3.14)
$|\beta^{b,\mathbb{C}}_{\mathbb{R}^{2}}(\rho,\theta)|\leq\frac{C}{\rho},\qquad\qquad|\nabla\beta_{\mathbb{R}^{2}}^{b,\mathbb{C}}(\rho,\theta)|\leq\frac{C}{\rho^{2}}\,;$

moreover, by the very definition of $u^{k,\varepsilon}$ and
$\sigma^{k,\varepsilon}$ ,

(3.15)
$\displaystyle\frac{\mathrm{d}u^{k,\varepsilon}(i,j)}{\varepsilon}-\frac{\sigma^{k,\varepsilon}(i,j)}{\varepsilon}=\int_{0}^{1}\beta^{b^{k},\mathbb{C}}_{\mathbb{R}^{2}}(i+t(j-i)-x^{k,\varepsilon})\,(j-i)\,\mathrm{d}t\,,$

for every $k=1,\ldots,K$ , for every
$T_{\varepsilon}\in\mathcal{T}_{\varepsilon}$ , and for every
$(i,j)\in(T_{\varepsilon})_{\varepsilon}^{1}$ . As a consequence, for every
$k=1,\ldots,K$ , for every $T_{\varepsilon}\in{\mathcal{T}_{\varepsilon}}$
with $\mu_{\varepsilon}(T_{\varepsilon})=0$ , for every
$(i,j)\in(T_{\varepsilon})_{\varepsilon}^{1}$, and for every $x\in
T_{\varepsilon}$ , we get

(3.16) $\displaystyle\
\frac{1}{\varepsilon^{2}}\Big{[}\Big{(}{\mathrm{d}u^{k,\varepsilon}(i,j)}-{\sigma^{k,\varepsilon}(i,j)}\Big{)}\cdot(j-i)\Big{]}^{2}=\
\Big{[}\Big{(}\frac{\mathrm{d}u^{k,\varepsilon}(i,j)}{\varepsilon}-\frac{\sigma^{k,\varepsilon}(i,j)}{\varepsilon}\Big{)}\cdot(j-i)\Big{]}^{2}$
$\displaystyle=$
$\displaystyle\varepsilon^{4}\Bigg{[}\int_{0}^{1}\Big{(}\frac{j-i}{\varepsilon}\Big{)}^{*}\beta^{b^{k},\mathbb{C}}_{\mathbb{R}^{2}}(i+t(j-i)-x^{k,\varepsilon})\Big{(}\frac{j-i}{\varepsilon}\Big{)}\,\mathrm{d}t\Bigg{]}^{2}$
$\displaystyle\leq$
$\displaystyle\varepsilon^{4}\int_{0}^{1}\Big{[}\Big{(}\frac{j-i}{\varepsilon}\Big{)}^{*}\beta^{b^{k},\mathbb{C}}_{\mathbb{R}^{2}}(i+t(j-i)-x^{k,\varepsilon})\Big{(}\frac{j-i}{\varepsilon}\Big{)}\Big{]}^{2}\,\mathrm{d}t$
$\displaystyle=$
$\displaystyle\varepsilon^{4}\Big{[}\Big{(}\frac{j-i}{\varepsilon}\Big{)}^{*}\beta^{b^{k},\mathbb{C}}_{\mathbb{R}^{2}}(x-x^{k,\varepsilon})\Big{(}\frac{j-i}{\varepsilon}\Big{)}\Big{]}^{2}$
$\displaystyle\
+\varepsilon^{4}\int_{0}^{1}\bigg{(}\Big{[}\Big{(}\frac{j-i}{\varepsilon}\Big{)}^{*}\beta^{b^{k},\mathbb{C}}_{\mathbb{R}^{2}}(i+t(j-i)-x^{k,\varepsilon})\Big{(}\frac{j-i}{\varepsilon}\Big{)}\Big{]}^{2}$
$\displaystyle\
\phantom{\int_{0}^{1}\Big{(}\Big{[}\Big{(}\frac{j-i}{\varepsilon}\Big{)}^{*}\beta^{b^{k},\mathbb{C}}_{\mathbb{R}^{2}}(i+t(j-i)-x^{k,\varepsilon})}-\Big{[}\Big{(}\frac{j-i}{\varepsilon}\Big{)}^{*}\beta^{b^{k},\mathbb{C}}_{\mathbb{R}^{2}}(x-x^{k,\varepsilon})\Big{(}\frac{j-i}{\varepsilon}\Big{)}\Big{]}^{2}\Big{)}\,\mathrm{d}t\bigg{)}\,,$

where the second equality follows from (3.15), and the inequality is a
consequence of Jensen’s inequality. Now, by (3.14) for every $k=1,\ldots,K$
and for every $\xi\in\\{\pm e_{1},\pm\nu,\pm\eta\\}$ it holds

$\big{|}\nabla\big{(}[\xi^{*}\beta^{b^{k},\mathbb{C}}_{\mathbb{R}^{2}}\xi]^{2}\big{)}(x-x^{k,\varepsilon})\big{|}\leq\frac{C}{|x-x^{k,\varepsilon}|^{3}}\,\qquad\textrm{
for every }x\neq x^{k,\varepsilon}\,.$

In particular, for every $k=1,\ldots,K$ , for every
$T_{\varepsilon}\in{\mathcal{T}_{\varepsilon}}$ with
$\mu_{\varepsilon}(T_{\varepsilon})=0$ , for every
$(i,j)\in(T_{\varepsilon})_{\varepsilon}^{1}$, and for every $x\in
T_{\varepsilon}$ ,

$\Big{[}\Big{(}\frac{j-i}{\varepsilon}\Big{)}^{*}\beta^{b^{k},\mathbb{C}}_{\mathbb{R}^{2}}(i+t(j-i)-x^{k,\varepsilon})\Big{(}\frac{j-i}{\varepsilon}\Big{)}\Big{]}^{2}-\Big{[}\Big{(}\frac{j-i}{\varepsilon}\Big{)}^{*}\beta^{b^{k},\mathbb{C}}_{\mathbb{R}^{2}}(x-x^{k,\varepsilon})\Big{(}\frac{j-i}{\varepsilon}\Big{)}\Big{]}^{2}\leq\frac{C\,\varepsilon}{|x-x^{k,\varepsilon}|^{3}}\,.$

Hence, for every $T_{\varepsilon}\in\mathcal{T}_{\varepsilon}$ with
$\mu_{\varepsilon}(T_{\varepsilon})=0$ , by (3.16) and integrating over
$T_{\varepsilon}$ , we get

(3.17) $\displaystyle
F_{\varepsilon}(u^{k,\varepsilon},\sigma^{k,\varepsilon};T_{\varepsilon})$
$\displaystyle\leq$
$\displaystyle\frac{4}{\sqrt{3}}\varepsilon^{2}\int_{T_{\varepsilon}}\Big{(}\big{|}e_{1}^{*}\beta^{b^{k},\mathbb{C}}_{\mathbb{R}^{2}}(x-x^{k,\varepsilon})e_{1}\big{|}^{2}+\big{|}\nu^{*}\beta^{b^{k},\mathbb{C}}_{\mathbb{R}^{2}}(x-x^{k,\varepsilon})\nu\big{|}^{2}+\big{|}\eta^{*}\beta^{b^{k},\mathbb{C}}_{\mathbb{R}^{2}}(x-x^{k,\varepsilon})\eta\big{|}^{2}\Big{)}\mathrm{d}x$
$\displaystyle+C\varepsilon^{3}\int_{T_{\varepsilon}}\frac{1}{|x-x^{k,\varepsilon}|^{3}}\,\mathrm{d}x$
$\displaystyle=$
$\displaystyle\sqrt{3}\varepsilon^{2}\int_{T_{\varepsilon}}\frac{1}{2}\mathbb{C}\beta_{\mathbb{R}^{2}}^{b^{k},\mathbb{C}}(x-x^{k,\varepsilon}):\beta_{\mathbb{R}^{2}}^{b^{k},\mathbb{C}}(x-x^{k,\varepsilon})\,\mathrm{d}x+C\varepsilon^{3}\int_{T_{\varepsilon}}\frac{1}{|x-x^{k,\varepsilon}|^{3}}\,\mathrm{d}x\,,$

where the equality follows by the second equality in (3.1) with $\beta$
replaced by $\beta_{\mathbb{R}^{2}}^{b^{k},\mathbb{C}}(x-x^{k,\varepsilon})$ .
Now, denoting by
$T_{\varepsilon}^{x^{k,\varepsilon}}\in\mathcal{T}_{\varepsilon}(\Omega)$ the
triangle centered at $x^{k,\varepsilon}$ , by (3.14) and (3.15) for every
$k=1,\ldots,K$ we get

$F_{\varepsilon}(u^{k,\varepsilon},\sigma^{k,\varepsilon};T_{\varepsilon}^{x^{k,\varepsilon}})\leq
C\varepsilon^{2}\,,$

for some constant $C$ independent of $\varepsilon$ . Moreover, recalling the
notation previously introduced for the annuli, an integration in polar
coordinates shows that

$\int_{A_{\delta,2r}(x^{k,\varepsilon})}\frac{1}{|x-x^{k,\varepsilon}|^{3}}\,\mathrm{d}x\leq\frac{C}{\delta}\,,$

for every $0<\delta<2r$ . Therefore, by (3.17) and (2.6) , for every
$k=1,\ldots,K$ and for $c>0$ small enough (as for instance
$c<\frac{\sqrt{3}}{6}$) we obtain

(3.18)
$\displaystyle\frac{1}{2\varepsilon^{2}}\sum_{(i,j)\in(B_{2r}(x^{k,\varepsilon}))_{\varepsilon}^{1}}\Big{[}\Big{(}{\mathrm{d}u^{k,\varepsilon}(i,j)}-{\sigma^{k,\varepsilon}(i,j)}\Big{)}\cdot(j-i)\Big{]}^{2}$
$\displaystyle\leq$
$\displaystyle\frac{1}{2}\sum_{T_{\varepsilon}\in\mathcal{T}_{\varepsilon}(B_{2r}(x^{k,\varepsilon}))}F_{\varepsilon}(u^{k,\varepsilon},\sigma^{k,\varepsilon};T_{\varepsilon})$
$\displaystyle\leq$
$\displaystyle\frac{\sqrt{3}}{2}\varepsilon^{2}\int_{A_{c\varepsilon,2r}(x^{k,\varepsilon})}\frac{1}{2}\mathbb{C}\beta^{b^{k},\mathbb{C}}_{\mathbb{R}^{2}}(x-x^{k,\varepsilon}):\beta^{b^{k},\mathbb{C}}_{\mathbb{R}^{2}}(x-x^{k,\varepsilon})\,\mathrm{d}x+C\varepsilon^{2}$
$\displaystyle\leq$
$\displaystyle\varepsilon^{2}|\log\varepsilon|\frac{\sqrt{3}}{2}\psi(b^{k})+C\varepsilon^{2}\,,$

where the constant $C$ changes from line to line; this proves (3.10) .

Moreover, by (3.17) together with (3.14), for every $k,l=1,\ldots,K$ with
$l\neq k$ , we have

(3.19)
$\displaystyle\frac{1}{\varepsilon^{4}|\log\varepsilon|}\sum_{(i,j)\in(B_{2r}(x^{k,\varepsilon}))_{\varepsilon}^{1}}\big{[}(\mathrm{d}u^{l,\varepsilon}(i,j)-\sigma^{l,\varepsilon}(i,j))\cdot(j-i)\big{]}^{2}$
$\displaystyle\leq$
$\displaystyle\frac{1}{\varepsilon^{2}|\log\varepsilon|}\sum_{T_{\varepsilon}\in\mathcal{T}_{\varepsilon}(B_{2r}(x^{k,\varepsilon}))}F_{\varepsilon}(u^{l,\varepsilon},\sigma^{l,\varepsilon};T_{\varepsilon})$
$\displaystyle\leq$
$\displaystyle\frac{C}{|\log\varepsilon|}\Big{(}\int_{B_{2r}(x^{k,\varepsilon})}\mathbb{C}\beta^{b^{l},\mathbb{C}}_{\mathbb{R}^{2}}(x-x^{l,\varepsilon}):\beta^{b^{l},\mathbb{C}}_{\mathbb{R}^{2}}(x-x^{l,\varepsilon})\,\mathrm{d}x+\varepsilon\int_{B_{2r}(x^{k,\varepsilon})}\frac{1}{|x-x^{l,\varepsilon}|^{3}}\,\mathrm{d}x\Big{)}$
$\displaystyle\leq$ $\displaystyle\frac{C_{r}}{|\log\varepsilon|}\to
0\quad\textrm{ as }\varepsilon\to 0\,,$

where $C_{r}$ depends only on $r$ . Now, by (3.19) we obtain (3.11) .
Analogously, one can prove that also (3.13) holds true.

Finally, by the Hölder inequality, using (3.18) and (3.19), for every
$k,l=1,\ldots,K$ with $l\neq k$ we get

$\displaystyle\frac{1}{\varepsilon^{4}|\log\varepsilon|}\sum_{(i,j)\in(B_{2r}(x^{k,\varepsilon}))_{\varepsilon}^{1}}\big{[}(\mathrm{d}u^{k,\varepsilon}(i,j)-\sigma^{k,\varepsilon}(i,j))\cdot(j-i)\big{]}\big{[}(\mathrm{d}u^{l,\varepsilon}(i,j)-\sigma^{l,\varepsilon}(i,j))\cdot(j-i)\big{]}$
$\displaystyle\leq$
$\displaystyle\bigg{(}\frac{1}{\varepsilon^{4}|\log\varepsilon|}\sum_{(i,j)\in(B_{2r}(x^{k,\varepsilon}))_{\varepsilon}^{1}}\big{[}(\mathrm{d}u^{k,\varepsilon}(i,j)-\sigma^{k,\varepsilon}(i,j))\cdot(j-i)\big{]}^{2}\bigg{)}^{\frac{1}{2}}$
$\displaystyle\phantom{\frac{1}{\varepsilon^{4}|\log\varepsilon|}\bigg{(}\sum_{(i,j)\in(B_{2r}(x^{k,\varepsilon}))_{\varepsilon}^{1}}}\bigg{(}\frac{1}{\varepsilon^{4}|\log\varepsilon|}\sum_{(i,j)\in(B_{2r}(x^{k,\varepsilon}))_{\varepsilon}^{1}}\big{[}(\mathrm{d}u^{l,\varepsilon}(i,j)-\sigma^{l,\varepsilon}(i,j))\cdot(j-i)\big{]}^{2}\bigg{)}^{\frac{1}{2}}$
$\displaystyle\leq$ $\displaystyle\frac{C}{\sqrt{|\log\varepsilon|}}\to
0\qquad\textrm{as }\varepsilon\to 0\,,$

which proves (3.12) and concludes the proof of the $\Gamma$-limsup inequality.
∎

##  4\. Degeneracies of the model and possible kinematic constraints

We highlight that the functional (1.2) exhibits some degeneracies that are
typical of linearized energies as well as of discrete models built on a
reference configuration; the presence of the plastic slip field $\sigma$
actually makes the system even less rigid and produces even more unphysical
configurations.

As an example, let $\Omega:=[-1,1]^{2}$ and define
$u_{\varepsilon}^{\pm}\in\mathcal{D}_{\varepsilon}(\Omega)$ as

(4.1)
$u_{\varepsilon}^{\pm}(i):=\left\\{\begin{array}[]{ll}\pm\varepsilon\nu&\textrm{if
}i_{2}\leq 0\\\ 0&\textrm{elsewhere in
}\Omega_{\varepsilon}^{0}\,.\end{array}\right.$

Notice that $u_{\varepsilon}^{-}$ can be interpreted as a microscopic crack,
whereas $u_{\varepsilon}^{+}$ can be seen as an unphysical crack which
produces interpenetration of matter generating the superposition of two lines
of atoms. It is easy to check that there exist
$\sigma_{\varepsilon}^{\pm}\in\mathcal{S}_{\varepsilon}(\Omega)$ such that
$F_{\varepsilon}(u_{\varepsilon}^{\pm},\sigma_{\varepsilon}^{\pm})=0$ .
Moreover, up to slightly modifying $u_{\varepsilon}^{-}$, one can construct a
function $u_{\varepsilon}\in\mathcal{D}_{\varepsilon}(\Omega)$ such that the
crack does not disconnect $\Omega$ and has two ending points which in our
model are classified as dislocations with opposite Burgers vector. In
particular, what in our model is identified as a plastic slip, in some cases
should be rather understood as a microscopic crack opening.

Furthermore, we observe that the energy in (1.2) is invariant with respect to
integer dilations of the lattice. In order to see this, it is enough to fix
$\lambda\in\mathbb{Z}$ and to take the displacement
$u_{\varepsilon}^{\lambda}$ and the slip $\sigma_{\varepsilon}^{\lambda}$
defined by

(4.2) $u_{\varepsilon}^{\lambda}(i):=\lambda\varepsilon
i\,\qquad\qquad\sigma_{\varepsilon}^{\lambda}(i,j):=\varepsilon\lambda(j{-}i)\,.$

Trivially,
$\mathrm{d}u_{\varepsilon}^{\lambda}(i,j)\equiv\sigma_{\varepsilon}^{\lambda}(i,j)$
and hence
$F_{\varepsilon}(u_{\varepsilon}^{\lambda},\sigma_{\varepsilon}^{\lambda})\equiv
0$ .

In view of the degeneracies pointed out above, one may ask whether additional
suitable kinematic constraints on the plastic slips could prevent such
unphysical behaviors. In the present framework, it seems natural to
incorporate linearized constraints on the plastic slips, mimicking
deviatoric/pure shear volume preserving conditions on each
$\varepsilon$-triangle $T_{\varepsilon}\in\mathcal{T}_{\varepsilon}$ .

Let us discuss now a possible way to introduce such a kinematic constraint in
our model. In order to (formally) linearize the nonlinear volume preserving
constraint, consider deformation gradients $\mathrm{Id}+\delta z$ close to the
identity, where $z\in\mathbb{R}^{2\times 2}$ represents a plastic deformation
gradient, $\delta>0$ is a small parameter with respect to which the
linearization is performed and $\mathrm{Id}$ is the identity matrix in
$\mathbb{R}^{2\times 2}$ . Enforcing that the deformed configuration
$(\mathrm{Id}+\delta z)(T_{\varepsilon})$ has the same (oriented) area of
$T_{\varepsilon}$ we obtain

$[(k{-}i)+\delta\,z(k{-}i)]\wedge[(j{-}i)+\delta\,z(j{-}i)]=(k{-}i)\wedge(j{-}i)\,,$

for every triple $(i,j,k)$ of counterclockwise oriented vertices of
$T_{\varepsilon}$ . Neglecting lower order terms, this yields

(4.3) $z(k{-}i)\wedge(j{-}i)-z(j{-}i)\wedge(k{-}i)=0\,,$

for every triple $(i,j,k)$ as above. Summing (4.3) over two triples of the
type $(i,j,k)$, $(j,k,i)$, one can see that condition (4.3) is equivalent to
the well known trace free constraint on $z$ ; we refer the interested reader
to [14] for a rigorous derivation via $\Gamma$-convergence of such a kind of
trace free constraints starting from continuous nonlinear models in
plasticity, relying on multiplicative rather than additive decompositions of
the deformation gradient.

We now rewrite (4.3) in terms of the (linear) slip field $\sigma$ , which is,
by construction, given by

(4.4) $\sigma(i,j)=z(j{-}i)\qquad\qquad\textrm{for every bond }(i,j)\textrm{
of the triangle }T_{\varepsilon}\,.$

Condition (4.3) becomes

(4.5) $\sigma(i,k)\wedge(j{-}i)-\sigma(i,j)\wedge(k{-}i)=0\,,$

for every triple $(i,j,k)$ of counterclockwise oriented vertices of
$T_{\varepsilon}$ . As a consequence of (4.4), we have that
$\mu[\sigma](T_{\varepsilon})=0$ , and hence (4.5) is justified by the
reasoning above only on dislocation-free triangles. On the other hand, one can
easily check that the only condition (4.5) (assumed for every triple $(i,j,k)$
of counterclockwise oriented vertices of $T_{\varepsilon}$) implies that
$\mu[\sigma](T_{\varepsilon})=0$ . Summarizing, in our framework, a reasonable
linearization of the volume preserving constraint seems to be provided by
condition (4.5) assumed on all dislocation-free $\varepsilon$-triangles
$T_{\varepsilon}$ .

Notice that (4.5) rules out the degeneracies pointed out by the examples
above. More precisely, one can check that, given a displacement as in (4.1)
or(4.2), the minimum energy in a dislocation-free triangle among all slip
fields fulfilling (4.5) is strictly positive. On the other hand (MS)
guarantees that dislocation-free triangles are necessarily present. Finally,
our $\Gamma$-convergence result still holds true under the additional
constraint (4.5), which is satisfied by the recovery sequence constructed in
the proof of Theorem 1.1(iii).

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