Riccardo Scala
email: riccardo.scala@wias-berlin.de

Nicolas Van Goethem
email: vangoeth@fc.ul.pt

Dislocations

Keywords: finite elasticity, variational problem, Peach-Koehler force, constraint reaction, integer-multiplicity currents

In the presence of dislocations, the elastic deformation tensor F is not a gradient, but satisfies the condition Curl F = Λ T L (with the dislocation density Λ L , a tensor-valued measure concentrated in the dislocation L and F ). Then F ∈ L p with 1 ≤ p < 2. This peculiarity is at the origin of the mathematical difficulties encountered with dislocations at the mesoscopic scale, which are here modeled by integral 1-currents which are free to form complex geometries in the bulk. In this paper, we first consider an energy minimization problem among the couples (F, L) of strains and dislocations, and then we exhibit a constraint reaction field arising at minimality, due to the satisfaction of the condition on the deformation curl, hence providing explicit expressions of the Piola-Kirchhoff stress and Peach-Koehler force. Moreover, it is shown that the Peach-Koehler force is balanced by a defect-induced configurational force, a sort of line tension. The functional spaces needed to mathematically represent dislocations and strains are also analyzed and described in a preliminary part of the paper.

Introduction

Dislocations in elastic bodies are at the origin of dissipative phenomena, and in particular their motion is responsible for the plastic behavior of single crystals. A dislocation loop L is a closed curve in Ω. Outside the dislocation, i.e. in Ω \ L, the body is considered as perfectly elastic. This scale of matter description is called the mesoscopic, or the continuum scale. Nonetheless, it is not easy to understand the physical nature of a mesoscopic dislocation. In fact, it is not a material line, since it can be equivalently generated by an excess or a lack of lattice atoms. Moreover, contrarily to fracture, it cannot even be defined as a mere singularity in the reference configuration where deformation fields would be unbounded. In fact, a dislocation must be viewed as a singularity of the deformation field whose support lies in the current configuration (see, e.g., [START_REF] Acharya | Driving forces and boundary conditions in continuum dislocation mechanics[END_REF][START_REF] Scala | Currents and dislocations at the continuum scale[END_REF]). Therefore, dislocation location and field singularity are bound notions. Specifically, the support of the curl of the deformation field (which in the presence of dislocations is not a gradient) is identified with the dislocation density field. This definition is at the basis of the present work, since a constraint reaction will be generated by the satisfaction of the latter relation between model variables.

1.1. Mathematical and physical properties of dislocations. The intrinsic mathematical difficulties generated by dislocations are fundamentally different from those encountered in the mathematical modeling of fracture mechanics. In particular, the displacement is not an appropriate model variable, as opposed to most of Solid Mechanics problems. Furthermore, the stress and strain fields are not square-integrable and so the less tractable L p spaces with 1 ≤ p < 2 must be considered, and bounds on the model fields are given in terms of the curl and the divergence (in place of the full gradient), given in measure spaces (instead of Sobolev spaces). Moreover, we believe that in order to model single crystals with dislocations, where complex geometries such as dislocation networks (cf. Fig. 1) are observed [START_REF] Zaiser | Dislocation patterns in crystalline solids-phenomenology and modelling[END_REF], one can hardly rely on the assumption of a periodic array of straight dislocations. Therefore, one is forced to build specific mathematical tools step by step, which should provide

• An appropriate functional framework.

• A geometric description of the lines. To achieve the latter, the mathematical formalism of currents as briefly described in Section 1.2 has been proposed. In this framework, a cluster as depicted in Fig. 1 is modeled as a continuum dislocation [START_REF] Scala | Currents and dislocations at the continuum scale[END_REF]. The formalism of currents to study and model dislocations clusters has been introduced in the pioneering works [START_REF] Hochrainer | Moving dislocations in finite plasticity: a topological approach[END_REF][START_REF] Hochrainer | Fundamentals of a continuum theory of dislocations[END_REF] and then adopted in more recent contributions in the theory of continuum dislocations, as in [START_REF] Conti | Modeling of dislocations and relaxation of functionals on 1-currents with discrete multiplicity[END_REF][START_REF] Conti | The line-tension approximation as the dilute limit of linear-elastic dislocations[END_REF]. The notion of integral current with coefficient in a group, also adopted by the authors in the companion paper [START_REF] Scala | Currents and dislocations at the continuum scale[END_REF] and [START_REF] Scala | Graphs of torus-valued harmonic maps with application to a variational model for dislocations[END_REF], is the main tool to treat dislocations networks. This is due, principally, to the ability of dislocation lines to annihilate, sum, and form complex structures with specific rules for summing the Burgers vectors, which belong to a specific group. Moreover the formalism of currents in general has showed to be much useful in order to treat variational problems in the theory of continuum dislocations (see [START_REF] Scala | Graphs of torus-valued harmonic maps with application to a variational model for dislocations[END_REF][START_REF] Scala | Currents and dislocations at the continuum scale[END_REF]). Restricting ourselves to a quasi-static regime, we assume that the optimal networks result from minimization laws (note that such minimization states are reached very fast in actual crystals such as pure copper, where resistance to dislocation motion is negligible [START_REF] Berdichevsky | Continuum theory of dislocations revisited[END_REF]).

Therefore, the first purpose of this paper is to establish the functional setting appropriate to describe mesoscopic dislocations. The main features are that (i) when Sobolev spaces W 1,p are considered, exponent p is in the "bad range" 1 ≤ p < 3/2, and (ii) the second grade variable is the curl instead of the gradient, and the curl must be a concentrated Radon measure. Minimization problems in this range are considered in [START_REF] Scala | Currents and dislocations at the continuum scale[END_REF] where, aware of [START_REF] Müller | Existence of minimizers for a polyconvex energy in a crystal with dislocations[END_REF], the main tools used are integral currents and Cartesian maps.

We shall provide elements for an analysis of the space of L p -tensors whose curl is bounded in a measure space, and in particular put in light and study the homeomorphism between this space and the space of solenoidal Radon measures, which in the model application will be the space of dislocation densities. The second purpose is to compute the first variation of the energy with respect to the strain and the dislocation associated density. These will allow us to determine a configurational force, capable of driving the dislocations outside equilibrium, which as far as the deformation part of the energy is concerned, is the well-known Peach-Koehler force.

1.2.

A quick survey on currents and dislocations at the continuum scale. In [START_REF] Scala | Currents and dislocations at the continuum scale[END_REF] we proposed a mathematical model for a countable family of dislocations in an elastic body Ω, here considered as the current (as opposed to "reference") configuration. Motivated by physical reasons [START_REF] Hirth | Theory of dislocations[END_REF][START_REF]Crystal Growth-From fundamentals to technology[END_REF][START_REF] Zaiser | Dislocation patterns in crystalline solids-phenomenology and modelling[END_REF][START_REF] Zubov | Nonlinear Theory of Dislocations and Disclinations in Elastic Bodies[END_REF], we consider finite elasticity near the line with a less-thanquadratic strain energy, while linear elasticity is a valid assumption away from the dislocations. Since the dislocation loop is the singularity set for stress and strain, the deformation gradient field F is incompatible, meaning that 1 -Curl F = Λ T = 0 in Ω, (1.1) with F the (inverse) deformation tensor, and where the dislocation density Λ is a Radon measure in M( Ω, M 3 ) concentrated on the dislocation set L. Here, L is a dislocation network in the current (i.e., deformed) configuration. Clearly if Λ = 0 then F is a gradient and there are no dislocations in the bulk. Moreover, conservation properties for dislocations imply that their density is solenoidal, Div Λ T = 0.

(1.

2) The explicit formula for Λ shows a linear dependence on the line orientation τ and on the Burgers vector b (i.e., Λ := τ ⊗ bδ L ), where for crystallographic reasons, the value of the Burgers vector is constrained to belong to a countable lattice in R 3 .

In the proposed formalism, currents (for which the main reference is [START_REF] Federer | Geometric measure theory[END_REF]) are used to describe dislocations at the mesoscopic scale. Specifically, dislocations are described by integermultiplicity 1-currents, which are mathematical objects generalizing the concept of curves, and are assumed closed to account for the property (1.2), implying that every dislocation is a loop or ends at the crystal boundary. A brief survey of the mathematical formalism can be found in Section 3.1, while for details we refer to [START_REF] Scala | Currents and dislocations at the continuum scale[END_REF]. For a so-called dislocation current L we will denote the associated density by Λ = Λ L . Whatever the model be, in this paper we are merely concerned with variations at optimality, thus modeling and existence issues are not discussed.

The starting point of the present work is the minimum problem min

(F,L)∈A W(F, Λ L ), (1.3)
where the energy

W(F, Λ L ) = W e (F ) + W dislo (Λ L ), (1.4)
satisfies some appropriate convexity and coerciveness conditions, while A is the space of admissible couples of deformation and dislocation currents. Among the properties of admissibility, we require that F and L be related by condition (1.1), and that F be the gradient of a Cartesian map away from L. Therefore, both F and L are represented by particular types of integral currents.

In dislocation theory an energy like (1.4) was used by Lazar and co-workers in [START_REF] Agiasofitou | On the nonlinear continuum theory of dislocations: a gauge field theoretical approach[END_REF][START_REF] Anastassiadis | The gauge theory of dislocations: conservation and balance laws[END_REF], where the decomposition in an elastic and a dislocation part in this form was first proposed by Kröner [START_REF] Kröner | Dislocations in crystals and in continua: A confrontation[END_REF]. From a mathematical viewpoint, that is, with variational techniques in appropriate functional spaces, Problems (1.3) has been discussed and first solved in [START_REF] Müller | Existence of minimizers for a polyconvex energy in a crystal with dislocations[END_REF] with a single fixed dislocation loop in the crystal bulk (thus implying a minimization in F only), and later extended in [START_REF] Scala | Currents and dislocations at the continuum scale[END_REF] for an unfixed countable family of dislocation currents satisfying certain boundary conditions. Existence of minimizers is based on the assumption (classical in fracture mechanics) that the number of clusters is bounded. 1.3. Formal derivation of the Peach-Koehler force. From the standpoint of configurational force theory (as in [START_REF] Gurtin | Configurational Forces as Basic Concepts of Continuum Physics[END_REF]), or as a result of invariance properties and Noether Theorem (as proposed by [START_REF] Agiasofitou | On the nonlinear continuum theory of dislocations: a gauge field theoretical approach[END_REF]), the Eshelby stress E = W I -F T P appears as a crucial quantity, with W the energy density, and P the first Piola-Kirchhoff stress. Assuming that Div P = 0, i.e. that static Equilibrium holds, one immediately finds that Peach -Koehler force = Div E = Λ T L × P ,

where the brackets stand to emphasize that the vectorial product takes place in a certain function space, as a duality product. Eq. (1.5) is known as the Peach-Koehler force (see, e.g. [START_REF] Hirth | Theory of dislocations[END_REF] for a straight dislocation): it is a force due to the equilibrium between the dislocation and the adjacent elastic medium. In particular the functional choice is provided by the Physics considered, that is, in the case of dislocations, whether one considers the macroscale (with Sobolev fields and no line singularity), or on the contrary, the mesoscale, as in this work, where Geometric measure theory and related functional spaces must be considered. Let us remark that the strong form Div P = 0 is classically obtained by the Euler-Lagrange equation Ω P • ∇udx = 0 for all test function u, provided the integration by parts is valid. The point is that at our scale of matter description, P turns out to belong to a Lebesgue space and thus PN is not defined at the boundary, precluding the use of the Divergence theorem. For this reason, Div P = 0 must follow from another procedure, namely P will be defined as the curl of a constraint reaction L (in the sense of [START_REF] Fosdick | Stress as a constraint reaction in rigid bodies[END_REF]), in appropriate function space, and due to the satisfaction of (1.1).

On the other hand, observe that the rightmost member of (1.5) has no rigorous meaning at the mesoscale, since Λ L is a measure and P a Lebesgue-integrable field.

1.4. Scope of the work. It is the scope of the present work to elucidate the functional setting allowing one to mathematically establish (1.5). To the knowledge of the authors, such a proof was inexistent in the literature, since the variational problem was unsolved until [START_REF] Scala | Currents and dislocations at the continuum scale[END_REF], at the mesoscale, in finite-strain elasticity, and for curved dislocations (i.e., loops) and dislocation networks. To this respect it has to be emphasized that our point of view is completely different from that of [START_REF] Agiasofitou | On the nonlinear continuum theory of dislocations: a gauge field theoretical approach[END_REF], where the relation (1.5) is derived by postulating existence of a minimum (and hence the validity of Euler-Lagrange equations), and without any concern for the functional space setting by which the Physics of dislocations is modeled.

Considering the existence of minimizers of Problem (1.3), in the present paper we analyze the variation of W at the minimum points with respect to L, which by a formal chain rule writes as

δ L W(F, Λ L ) = δ F W(F, Λ L )δ L F + δ Λ W(F, Λ L )δ L Λ L .
Note first that W writes as the sum of a deformation and a defect part, the first depending on F , the second on Λ L = -( Curl F ) T . However both variables are related to L in an intrinsic manner, and hence a precise meaning must be given to the above chain rule expression.

The first aim of this paper is of theoretical nature: basically, it consists in giving a precise meaning to δ L F and δ L Λ L , that will be achieved by proving a series of preliminary results. As far as the second term is concerned, the geometric analysis made in [START_REF] Scala | Currents and dislocations at the continuum scale[END_REF] and synthesized in Section 3.1 is used as basis, but here completed by putting the concentrated measure Λ L in duality with a certain continuous tensor, called the constraint reaction. One difficulty is related to the identification of the dual space of Radon measures which are concentrated in closed lines, since in general it is not true that this set is a subspace of continuous functions. This first result will in particular require to invert the curl operator. As far as the deformation part of the energy is concerned, we have already mentioned that it is not a gradient, since to satisfy constraint (1.1), it must read as F = ∇u + Curl V (expression recognized as a tensor Helmholtz-Weyl type decomposition). As a matter of fact, F will depend on L through the solution of -Curl Curl V = Λ T L , which is an equation to consider with care, since it is not an elliptic PDE. In this paper, use will also be made of Helmholtz and Friedrich/Maxwell type decompositions in L p (see, e.g., [START_REF] Galdi | An introduction to the mathematical theory of the Navier-Stokes equations[END_REF][START_REF] Kozono | L r -variational inequality for vector fields and the Helmholtz-Weyl decomposition in bounded domains[END_REF]), where by Maxwell it is intended estimates of a vector/tensor with respect to their curl and divergence [START_REF] Neff | Maxwell meets Korn: a new coercive inequality for tensor fields in R N ×N with square-integrable exterior derivative[END_REF][START_REF] Neff | On a canonical extension of Korn's first and Poincaré's inequalities to H(Curl)[END_REF][START_REF] Neff | Poincaré meets Korn via Maxwell: extending Korn's first inequality to incompatible tensor fields[END_REF][START_REF] Yanagisawa | Hodge decomposition of L r -vector fields on a bounded domain and its application to the Navier-Stokes equations[END_REF]. The crucial fact being that by equation (1.1), the L p -norm of the deformation gradient is estimated by the dislocation density norm, here intended as total variation of the Radon measure.

A direct consequence of the results in this work, discussed in Section 4.5 is to set the basis of a model of evolution in time of dislocations, in the sense that computing δ L W amounts to consider that a certain (configurational) force exerted on the dislocations is vanishing. Therefore, a moving dislocation will evolve with a velocity proportional to this force, as documented in dislocation theories [START_REF] Acharya | Driving forces and boundary conditions in continuum dislocation mechanics[END_REF][START_REF] Hirth | Theory of dislocations[END_REF], and originating from the variation of the deformation part of the energy. In the final Theorem 6 we show that at optimality, there is a balance of forces, one of which being the well-known Peach-Koehler force F , while the other is a line-tension term, G , provided by the variation of the defect part of the energy (see also [START_REF] Conti | Singular kernels, multiscale decomposition of microstructure, and dislocation models[END_REF]). In fact the identity F = -G holding at minimality, might be considered as a constitutive law for F , since G is given explicitly in terms of the dislocation energy density, and the line curvature, normal and tangent vectors. Let us emphasize that time evolution per se is not considered in the present work. 1.5. Structure of the paper. In Section 2, the theoretical results required as preliminaries are stated and proved, unless their proof is found elsewhere in the literature. An important result is the existence of a constraint reaction, given in Section 2.4, relying on the important result on the invertibility of the curl operator as found in Section 2.3. Section 3 contains three subsections where the mathematical properties of a dislocation model in this setting are given and discussed. In particular, the functional relations between the deformation and the defect variables are given (important relations are here (3.13) and (3.14)), their admissibility is studied, and minimization results in appropriate spaces are recalled. In Section 4, the generic results of previous sections are applied to a more specific dislocation model. The scope here is to compute the first variation of the energy at the minimum points, eventually yielding the Peach-Koehler force expression in Section 4.3. In Section 4.5, a shape optimization view of minimality provides a balance of configurational forces, which is applied to an example. All preliminary results of this paper are required to derive this force expression, collected in Theorem 6. 1.6. A remark. This paper has been written in two parts, the first, i.e., Section 2, where all theoretical results are stated and proved without even referring to dislocations. Indeed, the functional spaces described in this section are broader than those needed for dislocations, and hence the results more general. Instead, Sections 3 and 4 are specifically devoted to the study of dislocations, and hence the previously statements are particularized. Moreover, in order to be self-contained, the essence of [START_REF] Scala | Currents and dislocations at the continuum scale[END_REF] is recalled in simple terms in Section 3.1.

Theoretical setting and preliminary results

2.1. Notations and conventions. The class of 3 × 3 matrices is denoted by M 3 := R 3×3 . In the following definitions the codomain space R is either tensor-valued, R = M 3 , or vector valued, R = R 3 . Then R stands for R 3 or R, respectively. Symbol M stands for finite Radon measures, while D denotes the topological vector space of smooth functions with compact support. The subset of R-valued solenoidal finite Radon measures on an open set X ⊂ R 3 reads

M div (X, R) := {µ ∈ M(X, R) s.t. µ, Dϕ = 0 ∀ϕ ∈ C 1 0 (X, R )}, (2.1)
where the product (here intended in the sense of finite Radon measures) yields, in the case R = M 3 , a real tensor whose components read ( µ ij , D j ϕ k ) ik . Recall that ϕ ∈ C 1 0 (X, R ) if it is of class C 1 and if for every > 0 there exists a compact set K ⊂ X such that |ϕ(x)| and |Dϕ(x)| are smaller than for any x ∈ X \ K.

Observe that M div (X, R) is a closed subspace of M(X, R) and hence is a Banach space, endowed with the total variation norm |µ|(X) = sup{ µ, ϕ : ϕ ∈ C (X, R), ϕ ∞ ≤ 1} (see [START_REF] Ambrosio | Functions of bounded variation and free discontinuity problems[END_REF] for details on vector-and tensor-valued Radon measures on metric spaces). A particular subclass of M div (X, R) will be the family of (transpose of) the dislocation densities 2 (Λ )

T ∈ M div ( Ω, M 3 ), where Ω ⊃ Ω is an open set containing only dislocations loops. For a tensor A and a vector N we use the convention that

(N × A) ij = -(A × N ) ij = -jkl A ik N l . Moreover the curl of a tensor A is defined componentwise as ( Curl A) ij = jkl D k A il . As a consequence one has Curl A, ψ = -A il , jkl D k ψ ij = A il , lkj D k ψ ij = A, Curl ψ , (2.2)
for every ψ ∈ D(Ω, M 3 ). In general, if ψ has not compact support, it holds

Curl A, ψ = A, Curl ψ + ∂Ω (N × A) • ψdS. (2.3)
Note that with this convention one has Div Curl A = 0 in the sense of distributions, since componentwise the divergence is classically defined as ( Div A) i = D j A ij . 3 The following lemma characterizes the dislocation measures as a particular subclass of the solenoidal measures.

Lemma 1. Let µ ∈ M div (Ω, R 3×3 ) be a measure that is absolutely continuous with respect to the H 1 -measure restricted on a simple Lipschitz curve L with tangent vector τ and such that L is either closed or ends at the boundary. Then µ is a dislocation measure, that is, there exists a constant vector b such that

µ = b ⊗ τ H 1 L . (2.4)
We drop the proof which is quite simple, and refer to [START_REF] Scala | Graphs of torus-valued harmonic maps with application to a variational model for dislocations[END_REF]. Let us denote by M dislo (Ω, M 3 ) the class of the transpose of such measure as (2.4).

Let 1 ≤ p < ∞ and let Ω ⊂ R 3 be an arbitrary open set. We introduce the vector space of tensor-valued fields

BC p (Ω, R 3×3 ) := {F ∈ L p (Ω, R 3×3 ) s.t. Curl F ∈ M div (Ω, R 3×3 )}, (2.5)
which, as endowed with the norm

F BC p := F p + | Curl F |(Ω), (2.6)
turns out to be a Banach space. 2 The transpose is taken to be consistent to the second author's references on dislocations [START_REF] Van Goethem | A distributional approach to the geometry of 2D dislocations at the continuum scale[END_REF]. This convention was originally taken from Kröner [START_REF] Kröner | Continuum theory of defects[END_REF]. 3 In this paper we therefore follow the transpose of Gurtin's notation convention [START_REF] Cermelli | On the characterization of geometrically necessary dislocations in finite plasticity[END_REF] but care must be payed since the curl and divergence of tensor fields are given alternative definitions in the literature (including the second author references [START_REF] Van Goethem | Thermodynamic forces in single crystals with dislocations[END_REF][START_REF] Van Goethem | A distributional approach to the geometry of 2D dislocations at the continuum scale[END_REF] where the current curl would write Curl A = -A × ∇).

Remark 1. One might define BC p (Ω, M 3 ) by specifying only that Curl F ∈ M(Ω, M 3 ) and considering the solenoidal property of µ as a direct consequence of the distributional identity Div Curl F = 0 in Ω.

Helmholtz decomposition for tensor fields.

Lemma 2. Let G ∈ L p (Ω, M 3 ) with 1 < p < ∞ and Ω be a bounded open and simply-connected set with C 1 boundary. There exists a unique solution (up to a constant) φ ∈ W 1,p (Ω, R 3 ) of

-∆φ = Div G in Ω ∂ N φ = -GN on ∂Ω . (2.7) Moreover such solution satisfies Dφ p ≤ C G p .
Proof. This Lemma is a direct tensor extension of the theorems of existence and uniqueness of Neumann problem as shown in [START_REF] Simader | The Dirichlet problem for the Laplacian in bounded and unbounded domains[END_REF] (see also [20, Lemma III.1.2 and Theorem III.1.2]).

Remark that Eq. (2.7) is a formal strong form meaning that the following weak form is solved [START_REF] Yanagisawa | Hodge decomposition of L r -vector fields on a bounded domain and its application to the Navier-Stokes equations[END_REF]:

-∇φ, ∇ϕ = G, ∇ϕ ∀ϕ ∈ W 1,p (Ω, M 3 ).
(2.8)

In particular, observe that the trace GN is not well-defined on the domain boundary. This issue will be addressed by Lemma 3. Let us define

L p div (Ω, M 3 ) := {F ∈ L p (Ω, M 3 ) s.t. Div F = 0} (2.9) L p curl (Ω, M 3 ) := {F ∈ L p (Ω, M 3 ) s.t. Curl F = 0} (2.10)
The following result can be proven (see [START_REF] Van Goethem | Cauchy elasticity with dislocations in the small strain assumption[END_REF]):

Lemma 3. Let Ω ⊂ R 3 be a bounded open set with boundary of class C 1 and let F ∈ L p (Ω, R 3×3 ) be such that Div F ∈ L p (Ω, R 3 ). Let us define the distribution F N as F N, γ(ϕ) := Div F, ϕ + F, Dϕ (2.11)
for all ϕ ∈ W 1,p (Ω, R 3 ), with γ(ϕ) ∈ W 1/p,p (∂Ω, R 3 ) the boundary trace of ϕ, and where •, • always means the duality product in appropriate spaces. Then F N ∈ W -1/p,p (∂Ω, R 3 ) := W 1/p,p (∂Ω, R 3 ) .

Similarly it holds true:

Lemma 4. Let Ω ⊂ R 3 be a bounded open set with boundary of class C 1 and let F ∈ L p (Ω, R 3×3 ) be such that Curl F ∈ L p (Ω, R 3×3 ). Then F × N ∈ W -1/p,p (∂Ω, R 3 ) := W 1/p,p (∂Ω, R 3 ) is defined as F × N, γ(ϕ) := Curl F, ϕ -F, Curl ϕ (2.12) for all ϕ ∈ W 1,p (Ω, R 3 ), with γ(ϕ) ∈ W 1/p,p (∂Ω, R 3 ) the boundary trace of ϕ. Let 1 < p < ∞. In virtue of the previous two lemmas, if V ∈ L p (Ω, R 3 ) is such that Div V ∈ L p (Ω, R), then it is well-defined its normal trace V N ∈ W -1/p,p (∂Ω) := W 1/p,p (∂Ω) on ∂Ω. Similarly, if V ∈ L p (Ω, R 3 ) with Curl V ∈ L p (Ω, R 3 ), then its antinormal trace on ∂Ω is V ×N ∈ W -1/p,p (∂Ω, R 3
) and is defined as in (2.12). These properties can be straightforwardly applied to tensor-valued maps

V ∈ L p (Ω, M 3 ), so that if Div V ∈ L p (Ω, R 3 ) it is well-defined its normal trace V N ∈ W -1/p,p (∂Ω, R 3 ) on ∂Ω (componentwise, (V N ) i = V ij N j ).
Similarly for the antinormal trace V × N (componentwise, jlp V il N p ), it belongs to W -1/p,p (∂Ω, M 3 ) as soon as Curl V ∈ L p (Ω, M 3 ) respectively (see also [START_REF] Kozono | L r -variational inequality for vector fields and the Helmholtz-Weyl decomposition in bounded domains[END_REF] and references therein).

Let us introduce the spaces

V p (Ω) := V ∈ L p div (Ω, M 3 ) s.t. Curl V ∈ L p (Ω, M 3 ), V × N = 0 on ∂Ω , (2.13) Ṽp (Ω) := V ∈ L p div (Ω, M 3 ) s.t. Curl V ∈ L p (Ω, M 3 ), V N = 0 on ∂Ω . (2.14)
The following estimate can be found in [START_REF] Kozono | L r -variational inequality for vector fields and the Helmholtz-Weyl decomposition in bounded domains[END_REF].

Lemma 5 (Kozono-Yanagisawa). Let Ω ⊂ R 3 be a bounded open set with boundary of class

C 1 and assume F ∈ V p (Ω) or F ∈ Ṽp (Ω). Then F ∈ W 1,p (Ω, R 3×3
) and there exists a positive constant C = C(Ω) such that

∇F p ≤ C ( Curl F p + F p ) . (2.15)
This shows that V p (Ω) and Ṽp (Ω) are closed subspaces in W 1,p (Ω, M 3 ). By virtue of Lemma 5 and for simply connected and bounded domains, a better estimate can be found in [START_REF] Wahl | Estimating ∇u by div u and curl u[END_REF]. Note that the following is a classical result for smooth functions with compact support [START_REF] Wahl | Estimating ∇u by div u and curl u[END_REF].

Lemma 6 (von Wahl). Let Ω be a simply-connected and bounded domain and let

F ∈ V p (Ω) or F ∈ Ṽp (Ω). Then it holds ∇F p ≤ C Curl F p . (2.16)
As a direct consequence the following result holds.

Lemma 7.

Let Ω be a simply-connected and bounded domain and let

F ∈ V p (Ω) or F ∈ Ṽp (Ω). Then Curl F = 0 ⇐⇒ F = 0.
We remark that, when F ∈ Ṽp (Ω), Lemma 7 amounts to proving the uniqueness property of Lemma 2. Moreover, in [START_REF] Kozono | L r -variational inequality for vector fields and the Helmholtz-Weyl decomposition in bounded domains[END_REF], a more general statement is established without the simplyconnectedness assumption. In general, for Ω a smooth and bounded subset of R 3 , Curl F = Div F = 0 has a non-trivial solution. In particular Kozono and Yanagisawa [START_REF] Yanagisawa | Hodge decomposition of L r -vector fields on a bounded domain and its application to the Navier-Stokes equations[END_REF] show that the solutions belong to a subspace of C ∞ ( Ω, M 3 ) with positive finite dimension, depending on the Betti numbers of Ω.

The following result is well known in the Hilbertian case L 2 but is not classical for the general Banach space L p . It is basically proven with help of Lemma 2 (for a complete proof see [START_REF] Kozono | L r -variational inequality for vector fields and the Helmholtz-Weyl decomposition in bounded domains[END_REF][START_REF] Yanagisawa | Hodge decomposition of L r -vector fields on a bounded domain and its application to the Navier-Stokes equations[END_REF], cf. also [START_REF] Galdi | An introduction to the mathematical theory of the Navier-Stokes equations[END_REF][START_REF] Neff | Maxwell meets Korn: a new coercive inequality for tensor fields in R N ×N with square-integrable exterior derivative[END_REF]).

Theorem 1 (Helmholtz-Weyl-Hodge-Yanagisawa). Let 1 < p < ∞ and let Ω be a bounded, simply-connected and smooth open set in R 3 . For every F ∈ L p (Ω, M 3 ), there exist u 0 ∈ W 1,p 0 (Ω, R 3 ) and a solenoidal V ∈ Ṽp (Ω), such that

F = Du 0 + Curl V, L p (Ω, M 3 ) = ∇W 1,p 0 (Ω, R 3 ) ⊕ Curl Ṽp (Ω) , (2.17)
Alternatively, there exist u ∈ W 1,p (Ω, R 3 ), and a solenoidal V 0 ∈ V p (Ω), such that

F = Du + Curl V 0 , L p (Ω, M 3 ) = ∇W 1,p (Ω, R 3 ) ⊕ Curl V p (Ω) , (2.18)
Moreover the decompositions are unique, in the sense that u 0 , V , V 0 are uniquely determined, while u is unique up to a constant, and it holds Du 0 p , Du p ≤ C F p , respectively.

Remark 2. When F is smooth with compact support, decompositions such as (2.17) and (2.18) are classically given by Stokes theorem and explicit formulae involving the divergence and the curl of F (see [START_REF] Wahl | Estimating ∇u by div u and curl u[END_REF], [START_REF] Bolik | Estimating ∇u in terms of div u, curl u, either (ν, u) or ν × u and the topology[END_REF]).

Remark 3. Let F be of class C 1 . In the particular case Curl F = 0 the Helmholtz decomposition is trivial when Ω is a simply-connected domain. Indeed it is well-known that in such a case there exists u ∈ C 2 (Ω, R 3 ) satisfying F = Du. This result extends for F ∈ L p with 1 < p < +∞ as shown in [START_REF] Galdi | An introduction to the mathematical theory of the Navier-Stokes equations[END_REF]. See [START_REF] Kozono | L r -variational inequality for vector fields and the Helmholtz-Weyl decomposition in bounded domains[END_REF] for a complete treatment of Helmholtz decomposition in L p , relying on the pioneer paper [START_REF] Fujiwara | An Lr-theorem of the Helmholtz decomposition of vector fields[END_REF]. Moreover, if Div F = 0 then, by Theorem 1, F = Curl V with V ∈ Ṽp (Ω). Remark that for smooth functions F , this result holds for any simply-connected domain with Lipschitz boundary.

Remark 4. Smoothness of the boundary is a strong requirement which is needed for the following reason: (2.17) and (2.18) require in principle to solve a Poisson equation ∆u = Div F with the right-hand side in some distributional (viz., Sobolev-Besov) space for which smoothness of the boundary is needed. It is known [START_REF] Fabes | Boundary layers on Sobolev-Besov spaces and Poisson's equation for the Laplacian in Lipschitz domains[END_REF] that for a Lipschitz boundary the solution holds for restricted p (namely 3/2 -≤ p ≤ 3 + ) for some = (Ω) > 0. Note that for p = 2 a Lipschitz boundary would be sufficient. Proof. Take any ϕ ∈ W 1,p (Ω, R 3 ). By parts integration (equations (2.11) and (2.12)), it holds

( Curl V )N, ϕ ∂Ω = Curl V, Dϕ = V × N, Dϕ ∂Ω = 0.
Since ϕ is arbitrary, the proof is achieved.

By Lemma 8, the function u of (2.18) is found by solving (2.7) with φ = u and G = -F . This also gives a meaning to the condition ∂ N u = F N . A key equation behind the results of this work is the following system:

   -Curl F = µ T in Ω Div F = 0 in Ω F N = 0 on ∂ Ω, , (2.19)
with µ T a Radon measure in M div ( Ω, M 3 ). Note that the transpose is here put for convenience.

In fact, the right-hand side is a general tensor-valued solenoidal bounded Radon measure.

Existence and uniqueness of a solution is given by Theorem 2 below, for the proof of which Lemma 2 (or Lemma 7) will be required.

The following result is first given for general solenoidal measures then slightly improved for dislocation measures. The existence part is a straightforward consequence of the main result of [START_REF] Bourgain | New estimates for the Laplacian, the divcurl, and related Hodge systems[END_REF], whereas some further details can be found in [START_REF] Scala | Graphs of torus-valued harmonic maps with application to a variational model for dislocations[END_REF]Appendix].

Theorem 2 (Biot-Savart). Let µ be a tensor-valued Radon measure such that µ T ∈ M div ( Ω, M 3 ). Then there exists a unique F in BC 1 div ( Ω, M 3 ) solution of (2.19). Moreover F belongs to BC p div ( Ω, M 3 ) for all p with 1 ≤ p < 3/2 and for all such p there exists a constant C > 0 satisfying F p ≤ C|µ|( Ω).

(2.20)

Moreover, if µ = τ ⊗ b H 1 L ,
for some b ∈ R 3 and a C 2 -closed curve L in Ω with unit tangent vector τ , then the solution F belongs to BC p div ( Ω, M 3 ) for all p < 2. Let us remark that the regularity assumption on the curve L is here necessary, since there exist examples of a measure concentrated on a rectifiable curve such that the associated deformation F is not in L p with 3/2 < p < 2, as shown in [START_REF] Conti | The line-tension approximation as the dilute limit of linear-elastic dislocations[END_REF].

By uniqueness, there exists a linear one-to-one and onto correspondence between the spaces M div ( Ω, M 3 ) and BC p div ( Ω, M 3 ). Thus the map Curl

-1 : M div ( Ω, M 3 ) → BC p div ( Ω, M 3 ), ν → F = -Curl -1 (ν), (2.21)
is well-defined and linear. Therefore, we may write

BC p div ( Ω, M 3 ) := Curl -1 M div ( Ω, M 3 ) . (2.22)
Moreover, for any F ∈ BC p div ( Ω, M 3 ) we recover by Eq. (2.20) the L p -counterpart of Maxwell relation in L 2 [START_REF] Neff | Maxwell meets Korn: a new coercive inequality for tensor fields in R N ×N with square-integrable exterior derivative[END_REF], that is,

F p ≤ C| Curl F |( Ω).
(2.23)

Remark 5. In case Ω is not simply-connected the uniqueness of solution of problem (2.19) does not hold. In such a case, Lemma 7 would also not hold, since the problem might exhibit non-trivial solutions, as shown in [START_REF] Yanagisawa | Hodge decomposition of L r -vector fields on a bounded domain and its application to the Navier-Stokes equations[END_REF].

2.4.

Existence of a constraint reaction. In the next sections we will deal with a linear and continuous map,

Φ : BC p ( Ω, M 3 ) → R, (2.24)
that is such that |Φ(F )| ≤ C F p for some C > 0, and satisfying

L p curl ( Ω, M 3 ) ⊂ ker Φ. (2.25)
An important result for maps of this kind is now stated and proved.

Theorem 3. Let 1 < p < 3/2 and let Φ be a linear and continuous map on L p ( Ω, M 3 ) satisfying Φ(Du) = 0 for every u ∈ W 1,p ( Ω, R 3 ). Then there exist two maps L and L belonging to

C( Ω, M 3 )∩W 1,p ( Ω, M 3 ), with 3 < p < ∞, 1/p+1/p = 1, such that, for every F ∈ BC p ( Ω, M 3 ), Φ(F ) = Curl L, F = Curl L, F = L, Curl F , (2.26)
and satisfying Div L = Div L = 0 in Ω, N × L = 0 and LN = 0 on ∂ Ω.

Proof. Since Φ is linear and continuous it holds

Φ(F ) = T, F , (2.27)
for some T ∈ L p ( Ω, M 3 ). Now, for every ϕ ∈ C ∞ ( Ω, R 3 ) we have T, Dϕ = Φ(Dϕ) = 0, proving that (i) Div T = 0 in Ω and, integrating by parts, that (ii) TN = 0 on ∂ Ω. By Theorem 1 (Eq. (2.17) or (2.18)), there exist a unique

L ∈ L p div ( Ω, M 3 ) satisfying N × L = 0 on ∂ Ω and a unique L ∈ L p div ( Ω, M 3 ) with LN = 0 on ∂ Ω, such that Curl L + Du = Curl L + Du 0 = T, (2.28)
for some u and u 0 as in Theorem 1. Since Div T = 0 in Ω, one has u 0 = 0 and from Curl LN = TN = 0 on ∂ Ω, Du = 0. By Maxwell-Friedrich-type inequality (i.e., the generalization of (2.15), see [START_REF] Yanagisawa | Hodge decomposition of L r -vector fields on a bounded domain and its application to the Navier-Stokes equations[END_REF]), i.e.,

∇L p ≤ C ( Curl L p + Div L p + L p ) , (2.29)
the fact that L ∈ L p ( Ω, M 3 ) with Curl L ∈ L p ( Ω, M 3 ) and Div L = 0, imply that L ∈ W 1,p ( Ω, M 3 ), which since 3 < p ≤ ∞ entails by Sobolev embedding that

L ∈ C ( Ω, M 3 ). (2.30)
The same is true for L. Integrating by parts the identities (2.26) we get, since

N × L = 0 on ∂ Ω, Φ(F ) = Curl L, F = L, Curl F ,
achieving the proof.

In the applications, Φ will be the first variation of the deformation part of the energy. In the sequel we will restrict to those variations whose deformation curl is concentrated in a closed curve, and, specifically, is associated to some dislocation density measure. This latter notion will be made clear in Section 3.1.

Energy minimization of dislocation networks

The keypoint of this work is to perform variations around the minima of Problem (1.3) in the largest possible functional spaces. As far as the deformation part of the energy is concerned, this amounts to proving the existence of an appropriate Lagrange multiplier to account for the constraint (1.1). This will be achieved thanks to Theorem 3. In principle, variations can be made with respect to (i) F , (ii) the dislocation density Λ and (iii) the dislocation set L. In the first case one recovers the equilibrium equations, where the Piola-Kirchhoff stress is written as the curl of the constraint reaction. The second case is more delicate since the space of variations is not a linear space (due to the so-called crystallographic assumption), thus creating a series of difficulties which we do not address further. Most interesting is the variation with respect to the line, that is, with respect to infinitesimal Lipschitz variations of the optimal dislocation cluster L . The difficulty here is that both F and Λ depend on L. In the case of Λ, the dependence is explicit since L is in some sense the support of Λ = Λ L (see (3.4)). In the case of F , the dependence is implicit since it holds

F = ∇u + F • , (3.1)
where F depends on L through the relation Curl

F • = -(Λ L )
T . Therefore, since the energy consists of one term in F and another in Λ, variations of the energy with respect to L (that is, with respect to its support L) will require an appropriate version of the chain rule. This computation is the main objective of Section 4, which to be carried out carefully requires a series of preliminary steps, collected in the present section. In order to be self-contained, results from [START_REF] Scala | Currents and dislocations at the continuum scale[END_REF] are first recalled, while rewritten in a concise form. We refer to [START_REF] Scala | Currents and dislocations at the continuum scale[END_REF] and [START_REF] Scala | Graphs of torus-valued harmonic maps with application to a variational model for dislocations[END_REF] for a full discussion of the results and of the models. In the next two sections, the results from Section 2 are applied to continuum dislocations. The main results are relations (3.13) and (3.14).

3.1. Dislocation density measures. In the sequel, we will adopt Notations 1 and 2. In order to perform variations in F and Λ, we introduce an appropriate subspace of M div ( Ω, M 3 ) called the set of dislocation density measures and based upon the notion of integer-multiplicity (or integral) 1-currents.

In many applications, the Burgers vector is constrained by crystollagraphic properties to belong to a lattice. For simplicity this lattice will be assumed isomorphic to Z 3 . Let the lattice basis { b1 , b2 , b3 } be fixed, and define the set of admissible Burgers vectors as

B := {b ∈ R 3 : ∃β ∈ Z 3 such that b = β i bi }.
In the sequel, we will adopt the non-restrictive and simple choice B = Z 3 , i.e., bi = e i , the ith Euclidean base vector. Moreover, we write b ∈ Z 3 to mean b ∈ B.

Let L be an H 1 -rectifiable subset of Ω, τ the unit oriented tangent vector defined H 1 -a.e. on L, and θ : L → Z a H 1 -integrable integer-valued function. Then the integer-multiplicity 1-current L denoted by L := {L, τ, θ} is defined as

L(ω) := L ω, τ θ(x)dH 1 (x),
for every compactly supported and smooth 1-form ω defined in Ω. The (topological vector) space of such forms is denoted by D 1 ( Ω, R 3 ).

A dislocation can be described using the notion of integer-multiplicity 1-current. For every Burgers vector b ∈ Z 3 , we introduce the regular b-dislocation in Ω as the closed integral 1-current Lb := { Lb , τ b , θ b }, where Lb represents the union of a finite family of Lipschitz and closed curves in Ω, τ b represents its oriented unit tangent vector, and θ b an integer valued function on Lb called multiplicity. We define the regular b-dislocation L b := {L b , τ b , θ b } in Ω as the restriction of Lb to Ω, i.e., L b (ω) := Lb ∩ Ω ω, τ b θ b (x)dH 1 (x) for every compactly supported and smooth 1-form ω defined in Ω. Associated to any b-dislocation in Ω is its density, that is the measure ΛL b ∈ M( Ω, M 3 ), defined by

Λ Lb , w := Lb ((wb) * ), (3.2)
for every w ∈ D( Ω, M 3 ), where in the right-hand side ω := (wb) * is the covector (wb) * := w kj b j dx k . If we identify test functions w ∈ D( Ω, M 3 ) with 1-forms in D 1 ( Ω, R 3 ) 3 , then we can also identify the density Λ Lb with an integral 1-current with coefficients in the group Z 3 , as in (3.2). We will use the notation

Λ Lb = Lb ⊗ b.
Its counterpart in Ω is the restriction of Λ Lb to Ω, denoted by Λ L b , and characterized by

Λ L b = L b ⊗ b = τ b ⊗ bθ b H 1 L b .
A general dislocation L is a sequence of b-dislocations { Lb } b∈Z 3 . The associated dislocations densities in Ω and Ω are given by

Λ L = b∈Z 3 Λ Lb and Λ L = b∈Z 3 Λ L b , (3.3)
respectively. These definitions allow us to describe any dislocation showing a finite or countable family of Burgers vectors. However it can be shown that actually any dislocation current L can be split in the basis of R 3 , as the sum of three integral 1-currents (called canonical dislocation currents) L = L 1 + L 2 + L 3 , in such a way that Λ Li = Λ i = L i ⊗ e i for i = 1, 2, 3, and that Λ L = Λ 1 + Λ 2 + Λ 3 . With the notation L i = {L i , τ i , θ i }, we call L := ∪ i L i the dislocation set, which corresponds to the support of L as shown in [START_REF] Scala | Currents and dislocations at the continuum scale[END_REF].

A dislocation current α in V := Ω \ Ω is a boundary condition if it is the restriction to V of a closed dislocation current α in Ω. We finally define the class of admissible dislocations in Ω with respect to a given boundary condition α as the set of all dislocation currents L which are the restrictions to Ω of some closed dislocation current L in Ω such that L V = α. In the sequel we will always suppose that dislocation currents are admissible for a fixed boundary datum.

3.2.

Functional space representation of dislocation networks. We will restrict our attention to the class of continuum dislocations (c.d.), defined as follows: L is a continuum dislocation if for i = 1, 2, 3, there exists a 1-Lipschitz map

λ i : [0, M i ] → Ω such that Li = λ i [0, M i ],
the push-forward by λ i of the standard current given by integration on the interval [0, M i ], see [START_REF] Scala | Currents and dislocations at the continuum scale[END_REF]Section 2] for details (note that the latter definition is equivalent to the original one given in [START_REF] Scala | Currents and dislocations at the continuum scale[END_REF], thanks to [37, Theorem 4.5]). Moreover, since all such currents are boundaryless by definition, we can rescale the functions λ i and suppose they are defined on S 1 . These dislocations might be called clusters because their Lipschitz description allow for the formation of complex curves. Their counterparts in Ω are defined as above. In such a case, the density of a continuum dislocation in Ω can be written as the sum of the three measures

Λ L = 3 i=1 Λ i = 3 i=1 λ i [S 1 ] Ω ⊗ e i , (3.4)
that can be equivalently written as Λ i = ( λi ⊗ e i )λ i H 1 , where λ i H 1 is the push-forward of the 1-dimensional Hausdorff measure on S 1 through λ i (see, e.g. [START_REF] Krantz | Geometric Integration Theory[END_REF] for this notion).

If L is a continuum dislocation, then there exists a set C L ⊂ Ω containing the support of the density Λ L which is a continuum, i.e., a finite union of connected compact sets with finite 1-dimensional Hausdorff measure. Note that such a set is not unique, and that we can always take, for example, C L = ∪ 3 i=1 λ i (S 1 ).

Let us introduce the class of dislocation density measures with compact support in Ω as

M Λ ( Ω, M 3 ) := {μ ∈ M( Ω, M 3 ) : ∃ L, c.d., with density -(Λ L) T = μ}. ( 3.5)
Let λ ∈ W 1,1 (S 1 , M 3 ), with L := ∪ 3 i=1 λ i (S 1 ). We introduce θ i (P ) := {s ∈ (λ i ) -1 (P ) :

λi

|λ i | (s) = τ (P )} -{s ∈ (λ i ) -1 (P ) : λi |λ i | (s) = -τ (P )},
for every P ∈ L, which stands for the multiplicity of the dislocation with Burgers vector e i , where symbol denotes the cardinality of a set (the subtraction is due to overlapping loops with reverse orientations).

For every ϕ ∈ C c ( Ω, M 3 ), the density μλ := -(Λ L) T which is associated to λ reads

-μλ , ϕ = 3 k=1 S 1 ϕ(λ k (s)) • (e k ⊗ λk (s))dH 1 (s) = 3 k=1 S 1 (ϕ • λ k ) kj (s) λk j (s)ds. (3.6)
The latter can be also seen as the integration on the imagine L i of the curve λ i counted with its multiplicity θ i . It turns out that -μλ , ϕ = L ϕ ij (P )τ i j (P )θ i (P )dH 1 (P ).

(3.7) Here τ i j θ i dH 1 = ( λi ) j ds.

(3.8)

The counterpart of μλ in Ω is µ λ = μλ Ω. The correspondence between the arcs λ and the Burgers vectors of the dislocation will appear clearer in the following Remark.

Remark 6. When we deal with a dislocation L generated by a single loop with Burgers vector

b = (β 1 , β 2 , β 3 ) = β i e i , β i ∈ Z (b = 0), then we have a Lipschitz function γ b ∈ W 1,1 (S 1 , R 3 ) such that L = γ b [S 1 ] Ω and -µ T γ b = Λ L = L ⊗ b, that is the measure such that -µ γ b , ϕ = S 1 ϕ(γ b (s)) • (b ⊗ γb (s))ds = S 1 ϕ ij (γ b (s))b i γb j (s)ds = L ϕ ij τ i j b i θdH 1 , (3.9)
where θ(P ) represents the multiplicity of the dislocation and is defined for every P ∈ L as

θ(P ) := {s ∈ (γ b ) -1 (P ) : γb |γ b | (s) = τ (P )} -{s ∈ (γ b ) -1 (P ) : γb |γ b | (s) = -τ (P )}. (3.10)
For every µ ∈ M Λ ( Ω, M 3 ) it is easy to check that Div µ = 0 in Ω, since L i are closed integral currents. In fact for all ψ ∈ D( Ω, R 3 ) one has -Dψ, µ = Dψ,

3 k=1 e k ⊗ λk (λ k H 1 ) = 3 i=1 S 1 D j (ψ i • λ i ) λi j ds = S 1 D t (ψ k • λ k )dt = 0. We then get M Λ ( Ω, M 3 ) ⊂ M div ( Ω, M 3
). We can now identify the space M Λ ( Ω, M 3 ) with W 1,1 (S 1 , Ω3 ), through the map

T : W 1,1 (S 1 , Ω3 ) → M Λ ( Ω, M 3 ) s.t. T (λ) = -μ λ defined in (3.6). (3.11)
The map T is by definition onto, while for every λ ∈ W 1,1 (S 1 , Ω3 ) it holds

T (λ) M ≤ λ L 1 , (3.12)
implying the continuity of T . In general T is not an injective map, but it is injective up to a equivalence relation ∼ in W 1,1 (S 1 , Ω3 ) (viz, λ ∼ λ if and only if T (λ) = T (λ ) as measures).

As a consequence, it holds

T W 1,1 (S 1 , Ω3 ) = M Λ ( Ω, M 3 ), (3.13) T -1 M Λ ( Ω, M 3 ) = W 1,1 (S 1 , Ω3 ). (3.14)
3.3. Class of admissible deformations and existence of minimizers. In this section, we exhibit an existence result for minimizers of energies W satisfying some particular assumptions.

For the proofs, we refer to [START_REF] Scala | Currents and dislocations at the continuum scale[END_REF]. Let us introduce

BC p,Λ ( Ω, M 3 ) := {F ∈ BC p ( Ω, M 3 ) s.t. Curl F ∈ M Λ ( Ω, M 3 )}, (3.15) BC p,Λ (Ω, M 3 ) := {F ∈ BC p (Ω, M 3 ) s.t. ∃ F ∈ BC p,Λ ( Ω, M 3 ) : F = F Ω }. (3.16)
and its proper subspace

BC p,Λ div ( Ω, M 3 ) := {F ∈ BC p div ( Ω, M 3 ) s.t. Curl F ∈ M Λ ( Ω, M 3 )} (3.17)
in such a way that by Theorem 2 and (3.13), it holds

BC p,Λ div ( Ω, M 3 ) := Curl -1 M Λ ( Ω, M 3 ) = Curl -1 T W 1,1 (S 1 , Ω3 ) . (3.18)
In [START_REF] Scala | Currents and dislocations at the continuum scale[END_REF], we consider deformations F ∈ BC p,Λ (Ω, M 3 ) which also satisfy some regularity conditions outside the continuum dislocation set C L of the dislocation Λ L ∈ M Λ ( Ω, M 3 ). If F is an admissible deformation, we assume that F satisfies the following property: Let us recall the meaning of Cart p (B, R 3 ). If U is an open set on R 3 , the space of Cartesian maps on U , denoted by Cart p (U, R 3 ), is defined as the space of maps u : U → R 3 belonging to W 1,p (U, R 3 ) and satisfying the following conditions: adj(Du), det(Du) belong to L 1 (U, M 3 ) and ∂G u = 0, where G u is the rectifiable 3-current in U × R 3 carried by the graph of u (see [START_REF] Giaquinta | Cartesian currents in the calculus of variations II. Variational integrals[END_REF]). We denote by AD p ( Ω) := {F ∈ BC p,Λ ( Ω, M 3 ) : F satisfies (P ) above} (3.19)

AD p (Ω) := {F ∈ BC p (Ω, M 3 ) s.t. ∃ F ∈ AD p ( Ω) : F = F Ω }. ( 3
F α := {F ∈ AD p (Ω), 1 ≤ p < 2 : F := F χ Ω + F χ V ∈ AD p ( Ω), -Curl F = Λ L T in Ω for some closed dislocation current L in Ω}. (3.21)
In particular, note that the dislocation current L in the above definition must coincide with α in V . We denote by L the restriction to Ω of L.

Assumptions on the energy. We make the following assumption on the elastic energy

W(F, Λ L ) := W e (F ) + W dislo (Λ L ), (3.22)
with

W e (F ) := Ω W e (F )dx. (3.23)
As for the dislocation part, we assume that

W dislo (Λ L ) = W 1 dislo (Λ L ) + W 2 dislo (Λ L ). (3.24)
where the precise continuity, growth properties on the bulk and defect energies are discussed and motivated in [START_REF] Scala | Currents and dislocations at the continuum scale[END_REF]. Let us stress that following [START_REF] Conti | Modeling of dislocations and relaxation of functionals on 1-currents with discrete multiplicity[END_REF] (where no variational problem is solved), an expression for the line tension W 1 dislo is here taken as

W 1 dislo (µ) = L ψ(θb, τ )dH 1 , (3.25) when µ = b ⊗ γ [S 1 ] = b ⊗ θτ H 1 L
is the dislocation density of a cluster generated by the loop γ ∈ W 1,1 (S 1 , R 3 ) and Burgers vector b = β i e i , β i ∈ Z (b = 0), and takes the value +∞ if µ is not of this type. Here ψ : Z 3 × R 3 → R is a non-negative function satisfying ψ(0, •) = 0 and ψ(b, t) ≥ c b for a constant c > 0.

As for the term W 2 dislo (Λ L ), it is remarkable that, under the hypotheses needed to get existence of minimizers, it does not depend on small perturbations of the dislocation line set L. This will be strongly used in the subsequent section. Now the existence theorem is the following: Theorem 4. Under Notation 3 and suitable hypotheses on the energy W in (3.22) (see [START_REF] Scala | Currents and dislocations at the continuum scale[END_REF] for details) there exists a minimizer F of the problem min

F ∈Fα W(F, Λ L ). (3.26)
We write Curl F = Λ T L with L being the optimal dislocation network, whose support is denoted by L . It should be remarked that, due to the Dirichlet condition F = F on Ω\Ω for the admissible deformations gradients, the minimizer is not trivial and must satisfy -Curl F = Λ L for some closed dislocation current L coinciding with α in Ω \ Ω. An explicit example showing the non triviality of the solution can be found in [START_REF] Scala | Currents and dislocations at the continuum scale[END_REF]Section 5.4]. Note that such energies at the macroscale are considered in [START_REF] Neff | The relaxed linear micromorphic continuum: well-posedness of the static problem and relations to the gauge theory of dislocations[END_REF], where a variational problem is solved.

Configurational forces at optimal dislocation networks

Certain forces apply on the dislocation clusters, solutions to the above minimization problem. They are due to the combined effect of the deformation and defect part of the energy. The line having no mass, these forces must be understood as being of configurational nature. They are related to the presence of micro-structure, here dislocations, in an otherwise static elastic medium in equilibrium. All the results of the previous sections will allow us to prove Theorem 6, which consists of a balance of forces at minimality. Furthermore, minimality will entail Euler-Lagrange equations which physically correspond to the balance of forces and to the vanishing of virtual work done by the configurational force, recognized as the Peach-Koehler force.

4.1. Shape variation at optimality. Let F be a minimizer of W(F ). By Theorem 1 and Eq. (3.13),

F = Du + Curl -1 • T (λ ).
where Curl -1 is the solution of (2.19), for some λ ∈ W 1,1 (S 1 , Ω3 ), and let -(Λ

) T := T (λ ) = -Curl F on Ω.
Define the linear map

S : W 1,1 (S 1 , Ω3 ) → BC p,Λ div ( Ω, M 3 ) : S = Curl -1 • T. (4.1)
We first prove the following preliminary result:

Lemma 9. The map S : W 1,1 (S 1 , Ω3 ) → BC p,Λ div ( Ω, M 3 ) is Gâteaux derivable at λ in all directions λ. In particular DS(λ )[λ] ∈ M( Ω, M 3 ), for every W 1,1 (S 1 , (R 3 ) 3 )-variation λ, and it holds DS(λ )[λ], ϕ = 3 i=1 S 1 jkm (ϕ • λ ) im (s)( λ ) i k (s)λ i j (s)ds., (4.2)
for every ϕ ∈ C c (Ω, M 3 ) such that Div ϕ = 0.

Proof. Let Ψ ∈ D(Ω, M 3 ). From (3.6) and (3.11), we infer by a Taylor expansion of Ψ that the directional derivative of T at λ along a variation λ ∈ W 1,1 (S 1 , (R 3 ) 3 ) reads

DT (λ )[λ], Ψ = 3 i=1 S 1 (Ψ • λ ) ij (s) λi j (s) + D k (Ψ • λ ) ij (s)( λ ) i j (s)λ i k (s)ds.
Integrating by parts the last expression we get

DT (λ )[λ], Ψ = 3 i=1 S 1 D k (Ψ • λ ) ij (s)( λ ) i j (s)λ i k (s) -D k (Ψ • λ ) ij (s)( λ ) i k (s)λ i j (s)ds = 3 i=1 S 1 D j (Ψ • λ ) ik (s) -D k (Ψ • λ ) ij (s) ( λ ) i k (s)λ i j (s)ds = 3 i=1 S 1 jkm mpq D p (Ψ • λ ) iq (s)( λ ) i k (s)λ i j (s)ds. (4.3)
Let us now compute DS. Let ϕ ∈ C c (Ω, M 3 ) such that Div ϕ = 0. Then ϕ = Curl Ψ for some Ψ ∈ C 1 (Ω, M 3 ) and hence, by Theorem 2

1 S(λ + λ) -S(λ ), Curl Ψ = 1 Curl -1 (T (λ + λ) -T (λ )) , Curl Ψ = 1 (T (λ + λ) -T (λ )) , Ψ .
Letting → 0 yields the result by (4.3), achieving the proof.

4.2.

First Euler-Lagrange equation and the static Equilibrium. In this section we make variations of the deformation at minimality, assuming the optimal line fixed, and derive the classical strong form of finite-strain elasticity.

Regularity assumption on the energy. We make the assumption that the energy W e : L p ( Ω) → R in (3.23) is Fréchet differentiable in L p ( Ω) with the Fréchet derivative of F → W(F, Λ ) denoted by W F ∈ L p ( Ω). As a consequence, for every F ∈ L p ( Ω), it holds

(A 1 ) δW(F )[F ] := d d W(F + F, Λ ) | =0 = Ω W F • F dx = δW e (F )[F ], W F := W F (F , Λ ) = δW e (F ) ∈ L p ( Ω).
Note that this assumption is rather general, and is about the least we can assume on W e .

Variations F of the deformation F still satisfying the constraint -Curl (F + F ) = (Λ ) T must belong to AD p curl ( Ω) := {F ∈ AD p ( Ω) s.t. Curl F = 0}. Moreover, such variations at the minimum points of the energy W must provide a vanishing variation of W. Thus, being F such a solution, for every curl-free

F = Du ∈ L p ( Ω), it must hold δW (Du) = δW e (F )[Du] = 0. ( 4.4)
From (A 1 ), Eq. (4.4) and Theorem 3 it results that there exists L continuous such that

P := W F = δW e (F ) = Curl L ∈ L p ( Ω) (4.5)
satisfying the following strong form:

-Div P = 0 in Ω P N = 0 on ∂ Ω . (4.6)
One could wonder why Eq. (4.6) is not immediate from the relation Ω W F • ∇udx = 0, which is Euler-Lagrange in weak form. In fact, the integration by parts which is classically used in this context is not legitimate in the present case, simply because the Divergence Theorem does not hold for L p -fields, since P N has no meaning at the boundary. This is the reason why Theorem 3 is called and P obtained as the curl of constraint reaction L , being therefore automatically divergence-free, while P N has a meaning by Lemma 8 and Theorem 3.

Remark 7. By (4.6), P := W F is identified with the first Piola-Kirchhoff stress. Being P in L p ( Ω), and recalling that F ∈ L p ( Ω), means that the Kirchhoff stress P F is in L 1 ( Ω).

4.3.

The Peach-Koehler force as a stationary condition. In this section, we derive the second Euler-Lagrange equation of the system in equilibrium.

Regularity assumption on the stress. Regularity of the minimizers is a well-known open problem in mathematical elasticity. Indeed, almost no results exist, even with an energy growth with p ≥ 2 (i.e., without dislocations), as reported by J. Ball in [START_REF] Ball | Some open problems in elasticity[END_REF]. A related problem is the regularity of the Piola-Kirchhoff stress P . In order to derive the subsequent formulae, which are well-established by physicists, we will also appeal to an assumption, not on F , rather on some components of P . Let us consider the orthonormal curvilinear basis (τ , σ , ν ) on the optimal dislocation set L , with τ the unit tangent vector to L . Let us decompose P in this basis, i.e., P = P τ ⊗ τ + P . Physically, P τ represents the force dF exerted on a facet dS of normal τ , that is, on a section of the tubular neighborhood of the dislocation L , namely, dF = P τ dS. Since all such facets are crossed by the dislocation, they presumably correspond to singular forces, in such a way that no regularity assumption can be made on these components. We will therefore make a regularity assumption on the remaining components P . Let us emphasize that the optimal deformation tensor F is smooth in Ω \ L , and hence, by (4.6), P will also be smooth in Ω \ L . Therefore, it is assumed that (A 2 ) P is continuous in a neighborhood of L .

In fact, lack of continuity of these components would mean that the contact forces dF tend to infinity at L .

Validity of Assumption (A 2

). First, we remark that in linearized elasticity the stress behaves as ∼ 1/r and following [START_REF] Hirth | Theory of dislocations[END_REF][START_REF] Van Goethem | A distributional approach to 2D Volterra dislocations at the continuum scale[END_REF] one has P screw = 0 whereas P edge → ∞ as r → 0. Hence, as obvious, one must consider finite strain elasticity to discuss Assumption (A 2 ). Here again, the situation is not evident, since nonlinear stresses depend on the choice of the material (i.e., of the energy W e ) and on the physics which takes place at the singular line. We will thus follow L. Zubov who has reported the current state-of-the-art in [START_REF] Zubov | Nonlinear Theory of Dislocations and Disclinations in Elastic Bodies[END_REF]. About the screw dislocation, he first points out that for a Mooney (and neo-Hookean) incompressible material, one has divergence of P screw at the singularity, but he discards this case as being non-physical, i.e., not suitable for the creation of a screw dislocation [46, p.74]. Then he considers the Bartenev-Khazanovich among two others incompressible materials and finds P screw ∼ ln r, hence again lack of continuity, though better integrability properties. However, incompressibility is not assumed in general, and in particular not in the present paper (indeed, it would imply another constraint reaction [START_REF] Fosdick | A penalty interpretation for the Lagrange multiplier fields in incompressible multipolar elasticity theory[END_REF]). Therefore following Zubov again, one considers a Blatz-Ko material together with the physical observation that the creation of a screw dislocation takes place together with a cylindrical cavity, and this implies continuity of P screw at the singularity [46, Eqs. (3.2.12), (3.2.16), (3.2.24), (3.2.28), p.76], whereas P -P must not be continuous [START_REF] Zubov | Nonlinear Theory of Dislocations and Disclinations in Elastic Bodies[END_REF]Eqs. (3.2.12), p.76]. Because the technical difficulties are huge, Zubov does not compute in extenso the edge dislocation with a cavitation, but nonetheless we consider the following physical interpretation, as based on aforementioned Zubov's results and physical evidences of dislocation nucleation as reported by e.g. [START_REF] Berezhkova | Nitevidnye kristally (Thread-like crystals)[END_REF][START_REF] Cottrell | Theory of crystal dislocations[END_REF]: Assumption (A 2 ) holds true for a compressible material where a cavitation is found along any dislocation loop.

In practice Assumption (A 2 ) allows one to have a finite radius R in the reference configuration corresponding to r = 0 in the deformed configuration Ω. Furthermore, Zubov shows that R(0) is proportional, of the order of 10% the Burgers vector.

Note that the creation of such a cavity in single crystals is due to the nucleation process of dislocation loops resulting from the collapse of a void, i.e., a cluster of vacancies which has become unstable. 4.3.1. The Peach-Koehler force. For all F ∈ BC p ( Ω, R 3×3 ) we write F = Du+ Curl -1 (-Λ T ) = Du + S(λ) with u ∈ W 1,p ( Ω, R 3 ) (by Theorem 1 and Eq. (4.1)). Following this formalism, it is thus assumed that the energy W depends on the dislocation path λ ∈ W 1,1 (S 1 , Ω3 ) as defined in Eq. (3.11), viz.,

W • (Du, λ) := W(Du + S(λ), -T (λ)), (4.7)
and then

W • (Du, λ) = W(F, Λ) with Λ T = -Curl F if λ ∈ T -1 (-Λ T ).
Let us consider the energy at its minimum F :

W • (Du , λ ) := W(Du + S(λ ), -T (λ )) = W e (Du + S(λ )) + W defect (-T (λ )) =: W • e (λ ) + W • defect (λ ). (4.8)
Let us denote the variation of the energy W e by δ • W e (F ) := δW • e (λ ). The expression of the variation of the energy is then given by the following main result.

Theorem 5 (Work done by the Peach-Koehler force). Under the assumptions of Theorem 4, and hypotheses (A 1 ) and (A 2 ), one has

δ • W e (F ), λ = 3 i=1 L P × τ i T θ i • λ i • (λ ) -1 dH 1 , (4.9)
where we have employed notation (3.7).

Proof. We want to perform variations λ ∈ W 1,1 (S 1 , (R 3 ) 3 ) of W • e (λ ). Identifying W F with the Piola-Kirchhoff P as in (4.5), using Lemma 9 with ϕ = P, one has

δW • e (λ )[λ] = 3 i=1 S 1 λ i j jkm ( λ ) i k P im • λ ds = 3 i=1 S 1 λ i j jkm ( λ ) i k P im • λ ds, (4.10)
where Assumption (A 2 ) gives a meaning to P im on L (i.e., to P im • λ on S 1 ), hence to the duality pairing

δW • e (λ )[λ] = W F , DS(λ )[λ] . (4.11)
The proof is achieved.

The integrand in the last member of (4.9) is recognized as the Peach-Koehler force. Theorem 5 simply says that at minimality, the virtual work done by the Peach-Koehler force must vanish.

Remark 8. The duality pairing (4.11) holds as soon as one considers a mollification of W F , that is, if W F is assumed continuous. However, this assumption is stronger than (A2) which requires only the continuity of some physically relevant components (related to the formation of a cavitation at the line singularity). Furthermore, nothing guarantees that the variation with any mollification of W F would vanish, since it is strictly speaking not the minimum point. Thus, at the mesoscopic scale, the best assumption found is (A2) in order to be able to merely define the Peach-Koehler force as related to minimality.

According to Remark 8, the following section shows how the Peach-Koehler force would formally be recovered.

Formal derivation of the Peach-Koehler force from the Eshelby tensor. Recalling (3.23), we introduce the Eshelby tensor E writing componentwise as

E ij = δ ij W e -F ki P kj . (4.12)
Then, assuming that F and P are smooth enough,

∂ j E ij = ∂ i W e -∂ j F ki P kj -F ki ∂ j P kj . (4.13)
At minimality, one has ∂ j P kj = 0 and hence

∂ j E ij = ∂ i W e -∂ j F ki P kj = ∂ i W e -(∂ j F ki -∂ i F kj )P kj -∂ i F kj (W F ) kj , (4.14)
where the first and last terms of the right-hand side mutually cancel, whence

∂ j E ij = -(∂ j F ki -∂ i F kj )P kj = ijl lmn ∂ m F kn P kj = ijl ( Curl F ) kl P kj , (4.15)
that is

∂ j E ij = ilj Λ lk , P kj . (4.16)
Note that (4.16) has no rigorous meaning in our setting, i.e., at the mesoscale, since Λ being a Radon measure whereas P not being continuous, the duality pairing (4.16) is undefined. This is the reason why the Peach-Koehler force is established in our work by means of assumption (A2).

4.4. Configurational balance. Let L = γ (S 1 ) be a single smooth enough dislocation loop with tangent vector τ , normal vector ν, curvature κ, and total Burgers vector B. This is assumed for simplicity of exposition, but similar results can be stated for generals λ ∈ W 1,1 (S 1 , Ω3 ). We introduce

F := (P × τ ) T Bδ L , G := κ ψ(b, τ ) -∇ψ(b, τ ) • τ + ∇∇ψ(b, τ ) • ν ⊗ ν ν γ -1 δ L ,
the so-called Peach-Koehler force and line tension, respectively, where ψ is the energy density as introduced in (3.25).

Deriving strong forms of equilibrium from a variational problem is classically done provided some regularity of the minimizers is assumed, as summarized in the following theorem. Note that restricting to a single generating loop with Burgers vector b is chosen for the simplicity of the exposition. In order to well-define tangent and normal vectors, as well as line curvature, the following regularity assumption will be made on the optimal dislocation set L = γ (S 1 ):

(A 3 ) γ ∈ W 2,1 (S 1 , Ω).
Theorem 6. Under the assumptions of Theorem 4, assuming that ψ, ψ : Z 3 × R 3 → R + are of class C 2 , that the optimal dislocation network satisfies (A 3 ) and is associated to a single Burgers vector b, then minimality implies equilibrium of configurational forces, in the sense that the Peach-Koehler force F is balanced by the line tension G in L , i.e., F + G = 0.

(4.17)

Proof. Let us particularize (4.10) to the case where the density Λ is generated by one single loop γ ∈ W 1,1 (S 1 , Ω) with Burgers vector b = β i e i , β i ∈ Z (b = 0) (cf. Remark 6). For variations of the form γ + γ with γ ∈ W 1,∞ (S 1 , Ω), (4.10) becomes

δW • e (γ )[γ] = S 1 kjm (P • γ (s)) im τ k b i γ j (s) γ (s) ds = L kjm P im τ k b i γ j dH 1 = Ω kjm P im γ j dΛ ki . (4.18)
Using the notation introduced in (4.8) we write

W • (γ + γ) = W • e (γ + γ) + W • defect (γ + γ), (4.19)
We have δW

• (γ )[γ] = δW • e (γ )[γ] + δW • defect (γ )[γ]. ( 4
W 1 defect (µ) = S 1 ψ(b, γ γ (s)) γ(s) ds. (4.21)
Taking into account that the term W 2 defect (µ) does not change for small perturbations of the dislocation line, the first variation of (4.21) at the point γ ∈ W 1,1 (S 1 , Ω) can be explicitly computed and will coincide with δW

• defect (γ )[γ]. It holds δW • defect (γ )[γ] = = S 1 D k ψ(b, γ γ (s)) γk γ 2 -γ k γ j γj γ 2 (s) + ψ(b, γ γ (s)) γ j γj γ (s) ds, (4.22)
where D k ψ is the derivative of ψ with respect to the k-th component of its second variable. Denoting τ = γ γ , we integrate by parts to obtain δ Ŵdefect (γ )

[γ] = - S 1 ψ(b, τ ) τj -D k ψ(b, τ )τ k τj + D j D k ψ(b, τ ) τk -D p D k ψ(b, τ ) τk τ p τ j γ j ds,
where we dropped the variable s. Equivalently, recalling that τi = κν i and since

D j D k ψ(b, τ ) τk = τ i j τ i p D p D k ψ(b, τ ) τk + ν j ν p D p D k ψ(b, τ ) τk , it holds g j [b] := ψ(b, τ ) τ i j -D k ψ(b, τ )τ k τ i j + D j D k ψ(b, τ ) τk -D p D k ψ(b, τ ) τk τ i p τ i j = ψ(b, τ ) τ i j -D k ψ(b, τ )τ k τ i j + D p D k ψ(b, τ ) τk ν p ν j = κ (ψ(b, τ ) -D k ψ(b, τ )τ k + D p D k ψ(b, τ )ν p ν k ) ν j . (4.23)
Plugging the last expression in (4.20) and using (4.18), we obtain

δW • (γ )[γ] = S 1 kjm (P • γ ) im (s)b i γ k (s) -g j [b](s) γ j (s)ds. ( 4

.24)

From the condition δW • (γ )[γ] = 0 for all γ ∈ W 1,1 (S 1 , R 3 ), due to the minimality of γ , we then get from (4.24), F j + G j = 0, with

F j := kjm (P ) im B i τ k δ L and G j := ρ dislo (B)ν j δ L ,
where

ρ dislo (B) := -g [B]ε -1 , (4.25)
with g := g j ν j , ε = ε(P ) := γ • γ -1 (P ) , the local deformation of the curve at P ∈ L , B := θ(P )b, the total Burgers vector, and θ(P ) as defined by (3.10), the multiplicity of the dislocation (accounting for the loops of the cluster whose Burgers vector is a multiple of b).

The proof is achieved.

Remark 9. Actually, (4.17) holds at H 1 -a.e. P ∈ L, and not at all P . This is due to the fact that it might happen that a point P ∈ L is the overlapping of parts of γ which, although having the same tangent vector τ , do not have the same curvature κ nor the same orthogonal vector ν.

In the case where θ = 1 and the dislocation is parametrized by arc length (| γ | = 1), the balance of forces can be rewritten as

kjm P im b i τ k = g j [b] on L .
A modeling example. In [START_REF] Conti | Modeling of dislocations and relaxation of functionals on 1-currents with discrete multiplicity[END_REF] it is considered a potential W 1 dislo of the form (3.25) with ψ(b, τ

) := |b| 2 + η b, τ 2 , ( 4.26)
where η > 0 is a constant. In the particular case where b = βe 1 , β ≥ 1, it is shown that such energy is also lower semicontinuous.

In such a case, the above computations entail that

G j (P ) = |b| 2 -η b, τ 2 + 2η b, ν 2 κν j ,
so that at minimum of the energy, it holds where E and G stand for the configurational stress and the internal configurational force [22, p.34]. In our case G := ρ dislo [B]ν j δ L . Quoting Gurtin, such force is "related to the material structure of the body B; to each configuration of B there correspond a distribution of material and internal configurational forces that act to hold the material in place in that configuration. Such forces characterize the resistance of the material to structural changes and are basic when discussing temporal changes associated with phenomena such as the breaking of atomic bonds during fracture", to which we add and during dislocation motion.

Let us note that Agiasofitouand and Lazar [START_REF] Agiasofitou | On the nonlinear continuum theory of dislocations: a gauge field theoretical approach[END_REF] have also derived such a relation as (4.30) in the specific context of dislocations by means of invariance properties and the Noether Theorem (though, without solving any minimization problem). These authors showed that the translational balance laws of the elastic and dislocation parts give rise to the Peach-Koehler force and also give the interpretation that "the Peach-Koehler force is the interaction force between the elastic subsystem and the dislocation subsystem" (see also Eq. (5.39) in [START_REF] Anastassiadis | The gauge theory of dislocations: conservation and balance laws[END_REF]).

A brief glance at the dynamic problem. So far we have identified the stationarity condition as a balance of configurational work. This happens when minimality is reached.

Consider now a time-evolution problem involving dislocation lines. In principle, no variational problem drives its evolution instantaneously, but minimality might be reached as t → ∞ [START_REF] Berdichevsky | Continuum theory of dislocations revisited[END_REF]. So, a first remark is that before minimality is reached, one has Div E + g = 0, by definition, and hence there exists a nonzero momentum p such that, according to [22, p.46], -F T ṗ = Div E + G .

(4.29)

Hence, one might determine the motion of the line towards equilibrium, i.e., until ṗ = 0. Now, Gurtin further says [22, p.11] that as far as ṗ = 0 the internal configurational force remains "indeterminate when and only when the associated structures are fixed in the material". This is similar to the constraint forces in classical mechanics (as the line tension of the pendulum) which must not be determined to establish the motion equation. In particular no constitutive law for these forces is required in general.

However, we would like to emphasize that we have derived a constitutive law, since (4.30) rewrites as Peach -Koehler force = Div E = -ρ dislo [B]ν j δ L , (4.30)

where ρ dislo is given in extenso by (4.23) and (4.25) in terms of the dislocation energy.

Second Euler-Lagrange equation and the dislocation equation. In [START_REF] Agiasofitou | On the nonlinear continuum theory of dislocations: a gauge field theoretical approach[END_REF], the second Euler-Lagrange equations for our minimum problem is derived, that is, the differential of the total energy is computed with respect to divergence-free deformations G (recall that curl-free deformation were considered for the first EL equation in Section 4.2), and establish an equation relating dislocation density and stress. We would like to point out serious mathematical issues in order to give a meaning to a vanishing such variation, 0 = δW(F ) [G]. The principal reason (and the only which we discuss here) is that the differential d d W dislo (Λ + Λ) | =0 has no meaning in M Λ ( Ω, M 3 ), this space not being Banach, due to the the fact that tending to zero, the resulting Burgers vector might not be an integer, whereas the minimum is achieved in this class of measures, with a crystallographic Burgers vector.

Concluding remarks

On the way to mathematically understand time evolution of dislocations, the work achieved in [START_REF] Scala | Currents and dislocations at the continuum scale[END_REF] was the first step, allowing us to describe the geometry of dislocation clusters and to prove existence of solutions to a general variational problem. With the present contribution, our wish was to provide a further decisive step, since the result of Theorem 6 introduces two forces balancing each other at optimality, the first deriving from the elastic part of the energy and named after Peach and Koehler (well-known in dislocation models [START_REF] Hirth | Theory of dislocations[END_REF]), and the second deriving by shape variation of the defect part of the energy. Here crucial use has been made of the decomposition F = ∇u + F • where F • and Curl F • depend on the line. Such a force and such a balance of forces could be derived at the mesoscopic scale, without the required mathematical formalism, since there is subtle interplay between concentrated measures and Sobolev functions.

It turns out that the sum of these two forces naturally provides an expression of the velocity of the dislocation (for instance, a linear law is acceptable under certain working assumptions, see [START_REF] Acharya | Driving forces and boundary conditions in continuum dislocation mechanics[END_REF]). Of course, a non-vanishing velocity, i.e., a nonzero force, means that the solution does not coincide with energy minimization, as well known for evolution problems. In future work, it is our task to determine the dissipative effects, the balance equations, and analyze in detail the evolutionary scheme.

The force we here derived yields an important output in terms of modeling, but to achieve a proof of Theorem 6, a series of results have appeared about the mathematical nature of functional spaces for dislocation-induced deformations. These should also be considered as contributions to the general aim of understanding dislocation problems considered at the mesoscale in appropriate mathematical terms. Moreover, the paper has been written with a first part containing generic results, which are not related to dislocation models.

Figure 1 .

 1 Figure 1. Example of a continuum dislocation cluster.

Lemma 8 .

 8 Let Ω ⊂ R 3 be a bounded open set with boundary of class C 1 and let V ∈ V p (Ω). Then ( Curl V )N = 0 in the sense of Lemma 3.

2. 3 .

 3 Invertibility of the curl. Notations 1. Unless otherwise specified, the domains Ω we consider are bounded, smooth, and simply-connected subsets of R 3 , with outward unit normal N .Let us introduce the following notation.Notations 2.Given Ω, we denote by Ω another domain satisfying Notation 1 and such that Ω ⊂⊂ Ω.

  (P) For every ball B ⊂ Ω with B ∩ C L = ∅, there exists a Cartesian map u ∈ Cart p (B, R 3 ) such that F = Du in B.

. 20 ) 3 .

 203 NotationsLet Ω be the open set introduced in Notation 2 and let α be a boundary condition in V = Ω \ Ω (i.e. α = L V for a closed dislocation current L in Ω). We then fix F ∈ AD p ( Ω) such that -Curl F = Λ L T and define

. 20 )

 20 Let us now compute the variation of the defect part of the energy. For a dislocation density of the form µ = b ⊗ γ [S 1 ], (3.25) writes as

θ 2 ( 1 - 4 . 5 .

 2145 η) b, τ 2 + (1 + 2η) b, ν 2 κν j = jpk P ip θb i τ k . Some additional remarks.Formal balance of configurational forces. Eq. (4.18) yields by(4.16) and a slight abuse of notations,δW • e (γ )[γ] = Ω -∂ k E jk γ j dx.

(4. 27 )

 27 Therefore (4.24) and (4.27) can be rewritten as the virtual configurational work balance at minimality, i.e.,-Div E = G , (4.28)