LINE-TENSION MODEL FOR PLASTICITY AS THE Γ-LIMIT OF A NONLINEAR DISLOCATION ENERGY

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1 LINE-TENSION MODEL FOR PLASTICITY AS THE Γ-LIMIT OF A NONLINEAR DISLOCATION ENERGY LUCIA SCARDIA AND CATERINA IDA ZEPPIERI Abstract. In this paper we rigorously derive a line-tension model for plasticity as the Γ-limit of a nonlinear mesoscopic dislocation energy, without resorting to the introduction of an ad hoc cut-off radius. The Γ-limit we obtain as the length of the Burgers vector tends to zero has the same form as the Γ-limit obtained by starting from a linear, semi-discrete dislocation energy. The nonlinearity, however, creates several mathematical difficulties, which we tackled by proving suitable versions of the Rigidity Estimate in non-simply-connected domains and by performing a rigorous two-scale linearisation of the energy around an equilibrium configuration. Keywords: Nonlinear elasticity, dislocations, plasticity, rigidity estimate, Γ- convergence Mathematics Subect Classification: 49J45, 58K45, 74C05.. Introduction In this paper we rigorously derive a line-tension model for plasticity from a nonlinear mesoscopic dislocation model. Since the motion of dislocations is regarded as the main cause of plastic deformation, a large literature is focused on the problem of deriving plasticity theories from more fundamental dislocation models. There are a number of strain-gradient crystal plasticity models available in the engineering literature see, for example, [4, 4, 9, 20] and [26], all derived phenomenologically. A mathematically rigorous derivation of a plasticity model à la Fleck and Hutchinson [4] has been obtained in [6], starting from a semi-discrete dislocation model. A line-tension plasticity model has been obtained by Garroni and Müller see [7, 8] starting from a phase-field model for dislocations introduced in [2] see also more recent papers by Cacace and Garroni [6] and by Conti, Garroni and Müller [9]. Although a dislocation is a lattice defect, in most dislocation models it has been described in the framework of a continuum theory, in which the positions of the atoms are averaged out. Indeed this reduces enormously the total number of degrees of freedom: From all atom positions to a few geometric quantities displacement/deformation, dislocation line, slip planes, etc.. The starting point of our derivation is also a continuum dislocation model. The main novelty of our approach is that we consider a nonlinear dislocation energy, whereas most mathematical and engineering papers treat only a quadratic dislocation energy see, e.g.,[7, 6, 24], so that the constitutive relation between stress and strain is linear. These models are referred to as semidiscrete dislocation models. Clearly, the linear constitutive relation is not satisfactory close to the dislocations cores, where the strains are too large for the linear approximation to hold. Moreover, since the internal stress field caused by a dislocation decays as /r, with the distance r from the dislocation, the associated strain energy blows up at a dislocation. The conventional way of fixing this problem is to exclude in the computation of the energy the contribution in a

2 2 L. SCARDIA AND C.I. ZEPPIERI small tube or disc in the two-dimensional case around the dislocation. Therefore an internal scale, proportional to the interatomic distance, needs to be introduced. This is the so called core radius. The resulting strain-energy then diverges logarithmically in the core radius and has to be rescaled in order to obtain a finite energy in the limit of vanishing core radius. We remark that semi-discrete dislocation models are continuum models, but contrary to classical continuum models they are inherently size dependent, since they contain a small parameter, the core radius, which is reminiscent of the discrete structure of the crystal lattice. The core radius, though, is introduced only for mathematical reasons and it provides a poor representation of the structure of the lattice close to the dislocation. A possible approach to overcome this problem is to start from a purely discrete linear model for dislocations see [2, 24] and to compute the continuum limit as the lattice size tends to zero. Another possibility is to consider a more general, nonlinear constitutive relation between stress and strain. Indeed, the blow-up of the strain energy in the linear case is due to the fact that the energy density exhibits a non-integrable singularity at zero, being essentially /r 2 in the 2Dcase. Therefore, intuitively, it is possible to choose nonlinear stress-strain constitutive relations for which the strain energy in the core regions around the dislocations is finite. Here we follow this approach. Namely, we consider a nonlinear energy density W satisfying the following mixed growth conditions: C dist 2 F,SO2 F p + WF C 2 dist 2 F,SO2 F p +,. with C,C 2 > 0 and < p < 2 see [22], where similar energy densities are considered to study a dimensional reduction problem for bi-phase materials with dislocations at the interface. To our knowledge, the present paper is the first one in which a Γ-convergence analysis of a nonlinear dislocation energy is carried out. We treat idealised dislocations, pure edge dislocations, and we assume that the dislocation lines are straight and parallel. Therefore the problem is two-dimensional, living in the plane orthogonal to the direction of the dislocation lines, and involves only two components of the deformation. More precisely, given a displacement u of a domain Ω R 2 and denoting with β: Ω R 2 2 the elastic part of the strain u, the nonlinear elastic energy is given by Wβdx..2 Ω In this work we analyse the case of a finite number N of fixed dislocations, that we identify with N points in Ω representing the singularities of the strain field β. Notice that the mixed growth conditions. satisfied by W allow us to define the strain energy.2 in the whole domain Ω, hence also close to the dislocations. In fact, our choice of W entails that the nonlinear strain energy has a quadratic growth only for small strains; i.e., far from the dislocations, while it has a p-growth close to the dislocations. Therefore, since Wβ r is integrable at zero for < p < 2, the energy contribution in the core regions is p finite. At a first look, the nonlinear energy.2 does not contain explicitly the small parameter, say > 0, describing the underlying lattice in the original discrete dislocation problem. This lattice parameter is indeed recovered via the incompatibility condition that the strain β has to satisfy. The above condition asserts that the circulation of β around each dislocation is proportional to

3 LINE-TENSION MODEL FOR PLASTICITY AS THE Γ-LIMIT OF A NONLINEAR DISLOCATION ENERGY 3 the Burgers vector which is of the order of the lattice spacing ; i.e., Curlβ = ˆbi δ xi,.3 where ˆb,...,ˆb N S are the directions of the Burgers vectors corresponding to a system of fixed dislocations located at x,...,x N Ω. With this choice the dislocation density Curl β is fixed and therefore the energy.2 depends only on the strain β. It turns out that the dislocation energy.2 associated to a strain β satisfying the admissibility condition.3 is of order 2 in the lattice parameter. We notice that this scaling is the same as that of the linear dislocation energy see, e.g. [24]. In the linear case, however, one can equivalently assume that the Burgers vectors are fixed, of length one, and scale the dislocation energy by log. The scaled elastic energy corresponding to a strain β satisfying.3 is then defined by 2 Wβdx..4 Ω The logarithmic scaling of the energy and the topological singularities of the strain resemble some features of the Ginzburg-Landau model for vortices [5, 25], but in the considered nonlinear setting the connection with this model is quite formal. Some of the technical difficulties arising in the study of the asymptotic behaviour of.4 originate from the nonlinear nature of the model, and some from the specific growth assumptions. on the energy density W. In order not to deal with all of them at once, we start our analysis focusing on a model case in which the energy density W is nonlinear, but exhibits a quadratic nonlinearity. Hence, this model case needs to be regularised removing from the domain N core regions of radius > 0 around the N point defects x,...,x N. If the strains β satisfy a suitable variant of.3 see 2.4, the corresponding scaled-energy is then defined by 2 i= Ω Wβdx,.5 where W now behaves essentially as dist 2 F,SO2 see Section 2 for more details and Ω := Ω \ N i= B x i. Due to this nonlinear quadratic constitutive stress-strain relation, the model case still contains the core radius and, therefore, it is only partially satisfactory. Nevertheless, it sets the stage for the subsequent analysis of more general energy densities satisfying the mixed growth conditions.. The strategy to analyse both the model case and the general case is to rigorously reduce to a linear problem, in the spirit of [2], and then to apply the convergence results obtained in the linear setting by Cermelli and Leoni [7], and by Garroni, Leoni and Ponsiglione [6]. This linearisation step is highly non-trivial requiring in particular, for a given strain, a fine estimate of the global deviation from being a rotation in terms of the local deviation see the Rigidity Estimate [5, Theorem 3.]. Moreover, since the mixed growth assumptions. on W introduce further technical difficulties due to the weak regularity of the admissible strains, to focus on the linearisation step and to illustrate it in a clear way we start our analysis with the model case.5.

4 4 L. SCARDIA AND C.I. ZEPPIERI In Section 3 we study the asymptotic behaviour of the energy functionals defined in.5 via Γ-convergence see [] for a comprehensive introduction to this variational convergence. The linearisation procedure for this functional is possible thanks to a uniform rigidity estimate in non-simply connected domains with small holes, like Ω see Lemma 3.. Indeed, by virtue of this estimate we can prove a convergence result for suitably renormalised sequences of strains with equi-bounded energy Proposition 3.5 as well as a rigorous secondorder Taylor expansion of the energy around an equilibrium configuration, on two different scales: the meso-scale of the strain, and the micro-scale of the dislocation measure in.3. This two-scale linearisation is achieved by means of a careful partition of the domain Ω into disoint annuli with fixed outer radii around each dislocation and the rest of the domain, which is connected. Then, each annulus surrounding a dislocation is in turn split up into annuli via a suitable dyadic decomposition and a delicate energy estimate is performed see Proposition 3.. The limiting macroscopic functional see Theorem 3.9 and Corollary 3.0 has the same form as the Γ-limit obtained in [6, Theorem 5] by Garroni, Leoni and Ponsiglione in the linear setting compare also with [7], in the subcritical regime. Then, since.5 can be seen as a nonlinear counterpart of the model introduced in [7] by Cermelli and Leoni, the Γ-converge result established in Theorem 3.9 ustifies the usual linear approximation of the energy far from the defects. Using the techniques developed to study the case model, in Section 4 we analyse the asymptotic behaviour of the general nonlinear functionals defined in.4 with W satisfying.. This is the most physically relevant part of the paper, since the model does not suffer from the same deficiency of the linear theory and of the nonlinear quadratic case treated in Section 3, where an ad hoc cut-off radius around each dislocation needs to be introduced. From the mathematical point of view, a substantial difference with the nonlinear quadratic case is the proof of the compactness result Proposition 4.4. This relies on a version of the Rigidity Estimate valid in the case of mixed growth conditions which has been proved in [22] see also [8]. Moreover, in this case the linearisation procedure shows some additional difficulties, as. guarantees, a priori, a weaker regularity of the strain field if compared with the quadratic model. Eventually, the Γ-limit of.4 is the same line-tension model obtained in the nonlinear quadratic case Theorem 4.6. Therefore the Γ-convergence results Theorem 3.9 and Theorem 4.6 can be summarized as follows: The two functionals defined in.5 and.4 and extended to L 2 Ω; R 2 2 and L p Ω; R 2 2, respectively Γ-converge to a limiting energy E : L 2 Ω; R 2 2 SO2 [0,+ ] of the form Cβ: β dx + ψr Tˆbi if Curlβ = 0, Eβ,R := 2 Ω i= + otherwise in L 2 Ω; R 2 2 SO2, where C = 2 W F 2 I and ψ is defined through a suitable cell formula. We remark that the second term of the Γ-limit E depends explicitly on the transpose of a rotation R, which is acting on the Burgers directions ˆb,...,ˆb N to bring the system back in the reference configuration. The presence of a rotation in the limit energy is genuinely nonlinear: The Γ-limit of quadratic energies derived in [6] does not contain any rotation. However, in the

5 LINE-TENSION MODEL FOR PLASTICITY AS THE Γ-LIMIT OF A NONLINEAR DISLOCATION ENERGY 5 case of a nonlinear isotropic strain energy namely WF = WF R for every rotation R, the dependency on the rotation in the limit energy E is dropped. In conclusion, our result provides a rigorous ustification of a line-tension model by showing that such a model can be derived without necessarily resorting to the introduction of semidiscrete models containing an ad hoc cut-off radius. 2. Notation and Setting of the problem In this section we introduce the two models we are going to study. Namely, we define two nonlinear dislocation energies associated to the elastic part of the deformation strain in presence of a finite system of fixed edge dislocations. We restrict our analysis to the case of plane isotropic elasticity, so that straight edge dislocations orthogonal to the plane of strain can be modeled by point defects in R The reference configuration. Let Ω R 2 be a simply connected, bounded, Lipschitz domain. Let N denote the number of dislocations, and x,...,x N their positions in Ω. Every dislocation is characterized by its position and by the Burgers vector, which represents the circulation of the deformation strain close to the dislocation. The size of the Burgers vector or, equivalently, the interatomic distance in the discrete lattice is denoted by > 0. Let S S denote the set of admissible directions for the Burgers vectors; therefore, the Burgers vectors associated with the system of dislocations located at x,...,x N Ω can be written as ˆb,...,ˆb N, where ˆb i S, for every i =,...,N The dislocation energy density. Let W : R 2 2 [0,+ ] satisfy the usual assumption of nonlinear elasticity; i.e., W has a single well at SO2, where SO2 := {R R 2 2 : R T R = I,det R = } denotes the set of rotations in R 2 2. Since, as previously stated, we are going to analyse the asymptotic behaviour of two different nonlinear dislocation energies, which are in particular defined through energy densities satisfying two different growth conditions, here we first list the common assumptions to the two models, then we specify the two different growth conditions g-2 and g-p. The common assumptions on W are the following: i W C 0 R 2 2, W C 2 in a neighbourhood of SO2; ii WI = 0 stress-free reference configuration; iii WRF = WF for every F R 2 2 and every R SO2 frame indifference. The two models differ in their growth conditions as follows: g-2 there exists two constants C,C 2 > 0 such that for every F R 2 2 C dist 2 F,SO2 WF C 2 dist 2 F,SO2; g-p there exist < p < 2 and two constants C,C 2 > 0 such that for every F R 2 2 C dist 2 F,SO2 F p + WF C 2 dist 2 F,SO2 F p +.

6 6 L. SCARDIA AND C.I. ZEPPIERI We observe that g-p requires that the energy density W satisfies a more restrictive bound from above than the one in g-2. This additional requirement ensures that the dislocation cores have finite energy and is used in the proof of the lim sup-inequality in Theorem 4.6. The upper bound for the energy density W, though, is unsatisfactory, since it rules out the physically relevant conditions that the deformations are orientation-preserving and that the energy blows up if the body is compressed to zero volume The model case. The energy density W satisfies assumptions i-iii and the quadratic growth condition g-2. Due to g-2 the strain energy associated to a deformation is singular at the dislocations, and therefore it is well defined only in the domain Ω obtained removing from Ω a disc of radius around each dislocation x,...,x N. More precisely, we set N Ω := Ω \ B x i. 2. i= The effect of the presence of dislocations can be measured in terms of a dislocation density, which represents the failure of the condition of being a gradient for the strain. In the case of a finite number of point defects this dislocation density reads as µ := ˆbi δ xi, 2.2 i= where δ xi is the Dirac mass centred in x i. Then, the class of the admissible strains associated with µ is defined by all functions β L 2 Ω ; R 2 2 satisfying Curlβ = 0 in Ω and β t ds = ˆb i, for i =,...,N, B x i where the equality Curlβ = 0 is intended in the sense of distributions. The vector t above denotes the oriented tangent vector 2 to B x i and the integrand β t is intended in the sense of traces see [3, Theorem 2, pag. 204]. The scaled elastic energy corresponding to an admissible strain β is defined by 2 Wβdx. 2.3 Ω In what follows it is useful to extend the admissible strains β to the whole domain Ω. There are different possible extensions compatible with our model. Here we decide to consider β = I in the discs B x i, for i =,...,N. Therefore, from now on the class of admissible strains associated with the measure µ in 2.2 is given by { AS 2 := β L 2 Ω; R 2 2 : β I in N i= B x i, Curlβ = 0 in Ω, B x i β t ds = ˆb i, for i =,...,N }. 2.4 For a matrix β R 2 2, Curl β is the vector field of R 2 defined as Curl β = β 2 2β, β 22 2β 2. 2 We choose t = ν to be a counterclockwise π/2-rotation of the outward normal ν to B.

7 LINE-TENSION MODEL FOR PLASTICITY AS THE Γ-LIMIT OF A NONLINEAR DISLOCATION ENERGY 7 Since from now on β is extended by the identity outside Ω, by ii we can rewrite 2.3 as E β := 2 Wβdx, 2.5 and we also define the elastic energy induced by the measure µ as F µ := inf β AS 2 Ω E β. 2.6 Moreover, in view of 2.4 we may also extend E to the space L 2 Ω; R 2 2. We set { E 2 E β := β if β AS 2, + otherwise in L 2 Ω; R The general case. In this case W satisfies assumptions i-iii and the mixed growth conditions g-p. Let µ be as in 2.2; we define the class of admissible strains associated to the dislocation density µ as follows AS p := { β L p Ω; R 2 2 : Curlβ = µ in Ω }. 2.8 Then, the strain energy corresponding to β L p Ω; R 2 2 is given by E p β := 2 Wβdx if β AS p, Ω + otherwise in L p Ω; R The model case: Quadratic growth conditions In this section we study the asymptotic behaviour, via Γ-convergence, of the sequence of functionals defined in 2.7 hence under the assumptions i-iii and g-2 on W. We stress once more that the dislocation energy 2.7 is a model case for the more general nonlinear energy we will consider in Section 4. In fact, it shares with the general case some difficulties that are due do the common nonlinear nature of the energies. On the other hand, due to the quadratic growth conditions g-2, it serves as an intermediate step between linear models for dislocations and the more general nonlinear model defined by Compactness. This subsection is devoted to prove a compactness result for sequences β AS 2 with equi-bounded energy E 2. To this purpose, we start proving a suitable version of the Friesecke, James and Müller Rigidity Estimate [5, Theorem 3.] in a domain with small holes see [23, Section 4] for analogous results in the linear setting. Since the Rigidity Estimate holds true in any space dimension n 2 and for any exponent q,+ see [5, Theorem 3.] and [0, Section 2.4], we think worth proving the following lemma in this more general setting. With a little abuse of notation, we denote by Ω the n-dimensional analogue of 2..

8 8 L. SCARDIA AND C.I. ZEPPIERI Lemma 3. Rigidity with holes. Let < q < +, let n 2 and let Ω be a bounded Lipschitz domain of R n. There exists a constant C = CΩ,n,q > 0 with the following property: Let > 0 be sufficiently small, then for every u W,q Ω ; R n there is an associated rotation R SOn such that u R L 2 Ω ;R n n C dist u,son L 2 Ω. 3. Proof. Throughout the proof C is a positive constant independent of. We divide the proof into two steps. Step : Extension to Ω. In this step we extend u W,q Ω ; R n to a function ũ W,q Ω; R n satisfying dist q ũ,sondx C Ω dist q u,sondx, Ω 3.2 for some C > 0. To this end, for every i =,...,N, we apply the Rigidity Estimate [5, Theorem 3.] in B 2 x i \ B x i to find a constant C > 0 which is independent of thanks to the invariance of the rigidity estimate under uniform scaling of the domain and N rotations R i SOn, i =,...,N, such that u R i q dx C dist q u,sondx. 3.3 B 2 x i \B x i B 2 x i \B x i For i =,...,N, let û i be the restriction of u to B 2 x i \ B x i and set û R i := û i R i x, i =,...,N. Then, consider the functions v R i W,q B 2 x i \ B x i ; R n defined as Notice that for every i =,...,N v R i y := n q q û R i y, i =,...,N. B 2 x i \B x i v R i q dy = B 2 x i \B x i û R i q dx. 3.4 Appealing to the extension result for Sobolev functions [, Lemma 2.6], for every i =,...,N we find T i v R i W,q B 2 x i ; R n such that T i v R i v R i in B 2 x i \ B x i and T i v R i q dy C v R i q dy, 3.5 B 2 x i B 2 x i \B x i with C depending on q and n. Eventually, we define the functions v i W,q B 2 x i ; R n as Notice that, by definition Hence, if we set B 2 x i v i x := q n q T i v R i v i q dx = B 2 x i ṽ i := v i + R i x, x, for i =,...,N. T i v R i q dy, i =,...,N. 3.6 for i =,...,N,

9 LINE-TENSION MODEL FOR PLASTICITY AS THE Γ-LIMIT OF A NONLINEAR DISLOCATION ENERGY 9 it is immediate to check that ṽ i W,q B 2 x i ; R n and ṽ i û i u in B 2 x i \ B x i. Moreover, combining 3.3, 3.4, and 3.6 we find ṽ i R i q dx = v i q dx therefore C B 2 x i B 2 x i B 2 x i \B x i û R i q dx C dist q ṽ i,sondx C B 2 x i B 2 x i \B x i B 2 x i \B x i for every i =,...,N. Finally, we define { ṽ i in B x i, i =,...,N ũ := u in Ω. dist q u,sondx, dist q u,sondx, 3.7 Clearly, ũ extends u to Ω and ũ W,q Ω; R n. Moreover, since 3.7 entails 3.2, the first step is achieved. Step 2: Rigidity estimate. Now we apply the rigidity estimate to the function ũ W,q Ω; R n constructed in the previous step. This provides us with a constant C > 0 and a rotation R SOn with the property that ũ R q dx C dist q ũ,sondx. 3.8 Ω Therefore, in view of 3.2 we have u R q dx ũ R q dx Ω Ω C dist q ũ,sondx C dist q u,sondx, Ω Ω and the claim is proved. Lemma 3. is a key tool to establish a compactness result for sequences of strains with equi-bounded energy E 2. As E 2 is defined in Ω, which contains a finite number of holes with small radius, the relevance of Lemma 3. is clear. Nevertheless, this lemma cannot be directly applied to a sequence of strains β AS 2, as it is not a sequence of gradients. Then, we achieve the compactness result Proposition 3.5 by exploiting the specific singularity of the strains belonging to AS 2, and by applying Lemma 3. to a new curl-free field β β, with β suitably chosen. Another possible strategy to prove compactness is suggested by observing that β are in fact gradients in a suitable simply connected subset of Ω obtained removing from Ω a finite number of segments. Therefore, a compactness result can be as well a consequence of a variant of Lemma 3. for domains with holes and cuts. Since this alternative approach is suitable for more general types of singularities, we find it interesting to discuss here at least a special case, which is, moreover, an easy consequence of Lemma 3.. Ω

10 0 L. SCARDIA AND C.I. ZEPPIERI We consider the case Ω = B R 2, where B is the unit disc centred at the origin. We assume that there is a single dislocation located at the origin, so that the dislocation density µ 2.2 reads as µ = ˆbδ 0, with ˆb S. Let β AS, where AS is defined as { } AS := β L 2 B ; R 2 2 : β I in B, Curlβ = 0 in B \ B, β t ds = ˆb, 3.9 B in analogy with 2.4. We cut the annulus Ω = B \ B with the segment L := {z,0: < z < }; in this way we obtain the simply connected set B \ B \ L. Since Curlβ = 0 in B \ B \L by definition, there exists a function u H B \ B \L ; R 2 such that u = β in B \ B \ L. At this point we prove a modified version of Lemma 3., namely a rigidity estimate with a uniform constant in a set with a hole and a cut. Corollary 3.2 below will allow us to prove this result. We first set some notation. Let ϑ 0,2π; we denote by S ϑ the open sector of B \ B of angle ϑ; i.e., S ϑ := {r,θ: < r <, 0 < θ < ϑ}. 3.0 Corollary 3.2. Let < q < + and let S ϑ be as in 3.0. There exists a constant C = Cq,ϑ > 0 with the following property: Let > 0 be sufficiently small, then for every u W,q S ϑ; R 2 there is an associated rotation R SO2 such that u R L q S ϑ;r 2 2 C dist u,so2 L q S ϑ. 3. Proof. The proof follows exactly the line of that of Lemma 3.. We have the following uniform rigidity estimate in B \ B \ L. Proposition 3.3 Rigidity with a hole and a cut. Let < q < +. There exists a constant C = Cq > 0 with the following property: Let > 0 be sufficiently small, then for every u W,q B \ B \ L ; R 2 there is an associated rotation R SO2 such that u R L q B \B \L ;R 2 2 C dist u,so2 L q B \B \L. 3.2 Proof. Let u W,q B \ B \ L ; R 2. Define S := S π/2 and let S 2,S3,S4 be the sets obtained through a rotation of S as in Figure. Let R,R 2,R 3,R 4 SO2 be the constant rotations provided by Corollary 3.2; i.e., u R i q dx C dist q u,so2dx, 3.3 S i for some C > 0 and for every i =,...,4. We show that 3.2 holds true with R = R 2. To do this it is enough to prove that S R i 2 R i q C dist q u,so2dx, for i =,3,4. B \B \L We start considering S R R 2 q. To this end, we introduce the set S,2 := S π see Figure and, appealing to Corollary 3.2, the corresponding rotation R,2 SO2; i.e., the constant rotation matrix such that S,2 u R,2 q dx C S i S,2 dist q u,so2dx, 3.4

11 LINE-TENSION MODEL FOR PLASTICITY AS THE Γ-LIMIT OF A NONLINEAR DISLOCATION ENERGY B \ B \ L S 2 S S 3 S 4 S,2 S 2,3 S 3,4 Figure. Different coverings of B \ B \ L. for some C > 0. Notice that S S 2 S,2 B \ B \ L. As S = S 2, we immediately deduce R R 2 q dx C R R,2 q dx + R 2 R,2 q dx. 3.5 S S 2 S Now we estimate only the first term in the right end side of 3.5, the other being analogous. Combining 3.3 and 3.4 we get R R,2 q dx C u R q dx + u R,2 q dx S C S S dist q u,so2dx + C S,2 S dist q u,so2dx. Then, considering the sets S 2,3 and S 3,4 as in Figure and noticing that S 2,3,S 3,4 B \ B \L we can easily proceed as above to estimate S 3 R 2 R 3 q and S 3 R 3 R 4 q and therefore also S 4 R 2 R 4 q. Remark 3.4 Heuristics for the scaling. We note that the definition of the class of admissible deformations 2.4 ensures that the strain energy Ω Wβdx associated to an admissible deformation β is bounded from below by 2 up to a multiplicative constant, which ustifies the scaling in 2.5. Here we prove this bound in the special case Ω = B, assuming that there is a single dislocation located at the centre of the disc; i.e., µ = ˆb δ 0. Let β AS, where AS is defined in 3.9. We cut the annulus B \ B with L and we consider a function u H B \ B \ L ; R 2 such that u = β in B \ B \ L. Then, Proposition 3.3 with

12 2 L. SCARDIA AND C.I. ZEPPIERI q = 2 applied to u provides us with a constant C > 0 and a rotation R SO2 such that β R 2 dx C dist 2 β,so2dx C Wβdx, B \B B \B B the last inequality being a consequence of the assumption g-2 on W. Moreover, since β AS we find β R 2 2 dx B \B 2πr β R t ds dr B r 2 = 2πr β t ds 2 dr = B r 2πr ˆb 2 dr = 2. 2π We now go back to the case of N dislocations in a generic domain Ω. Appealing to Lemma 3., we are in a position to prove a compactness result for suitably renormalised sequences of admissible strains β. The renormalisation factor for strains β with equi-bounded energy is dictated by the scaling of the energy and by the quadratic growth condition g-2 on the energy density, and is. Then, since the natural scaling for the dislocation density µ is, the effect of the renormalisation of the strains is that the admissibility condition valid for β disappears in the limit. More precisely, we find that the limit strains β are always gradients; i.e., Curl β = 0 cf. [6, Theorem 5 i]. Proposition 3.5 Compactness. Let 0 and let β L 2 Ω; R 2 2 be a sequence such that sup E 2 β < +. Then there exist a sequence of constant rotations R SO2 and a function β L 2 Ω; R 2 2 with Curlβ = 0 such that up to subsequences R T β I log β in L 2 Ω; R Proof. Let β L 2 Ω; R 2 2 be as in the statement; therefore, in view of assumption g-2 on W, we have Ω dist 2 β,so2dx M C 2 log, 3.7 for every. In R 2 \ {x,...,x N } we define the function η := i J 2πˆb x x i x x i 2, i= where J is the clockwise rotation of π/2; then we set β := η χ Ω. Clearly β is defined in the whole Ω; moreover it is immediate to check that β 2 dx C 2 log. 3.8 Ω By construction we have Curlβ β = 0 in Ω and B x i β β t ds = 0, for every i =,...,N. Hence, there exists u H Ω ; R 2 such that β β = u in Ω. Then,

13 LINE-TENSION MODEL FOR PLASTICITY AS THE Γ-LIMIT OF A NONLINEAR DISLOCATION ENERGY3 Lemma 3. with q = n = 2 provides us with a constant C > 0 independent of, and a sequence R SO2 such that β β R 2 dx = u R 2 dx Ω Ω C dist 2 u,so2dx = C dist 2 β β,so2dx Ω Ω C dist 2 β,so2dx + C β 2 dx. Ω Ω Thus, appealing to 3.7 and 3.8, the previous estimate yields β R Ω 2 2 log dx C, for every. Finally, recalling that β I in N i= B x i and by the boundedness of R, we deduce that, up to subsequences, R T β R log = RT β I log β in L 2 Ω; R 2 2. Now we prove that Curlβ = 0 in Ω, in the sense of distributions. To this end, let φ C0 Ω and let φ H0 Ω be a sequence converging to φ uniformly and strongly in H 0 Ω and such that φ φx i in B x i, for i =,...,N. Then we have Curlβ,φ = lim 0 log Curl RT β I,φ = lim 0 log Curl RT β,φ = lim 0 N i= φx ir T ˆb i log = 0. In view of Proposition 3.5 we give the following notion of L 2 -convergence for sequences of admissible strains β. Definition 3.6. A sequence β AS 2 converges to a pair β,r L 2 Ω; R 2 2 SO2 if there exists a sequence of rotations R SO2 such that R T β I β in L 2 Ω; R 2 2 and R R Γ-convergence. The compactness result proved in Proposition 3.5 and Definition 3.6 suggest that the Γ-limit of the energies E 2 is a function of a pair: A gradient β and a rotation R, representing, respectively, the macroscopic strain and the rotation acting on the Burgers directions ˆb,...,ˆb N to bring the system back in the reference configuration.

14 4 L. SCARDIA AND C.I. ZEPPIERI For later references, it is convenient to introduce a new class of admissible scaled strains. For 0 < r < r 2 < and ξ S we define { } AS r,r 2 ξ := η L 2 B r2 \ B r : Curlη = 0 in B r2 \ B r, η t ds = ξ. B r In the special case r 2 = we will simply write AS r ξ instead of AS r,ξ. We also set the following notation: and C δ := B \ B δ, L δ := {z,0: δ < z < }, Cδ := C δ \ L δ, { } ψξ,δ := min Cη : η dx, η AS δ ξ 2 C { δ } = min C v : v dx, v H 2 C δ ; R 2, [v] = ξ on L δ, 3.20 ec δ where C = 2 W F 2 I and [v] is the ump of v. We recall the following fundamental result see [6, Corollary 6, Remark 7]. Proposition 3.7. For ξ S and δ 0,, let ψ δ ξ := ψξ,δ log δ, with ψξ,δ as in Then, the functions ψ δ converge pointwise to the function ψ: S R + defined by ψξ := lim Cη 0 : η 0 dx, 3.2 δ 0 log δ 2 C δ where η 0 : R 2 R 2 2 is a distributional solution of { Curlη = ξ δ 0 in R 2, Div Cη = 0 in R 2. Remark 3.8. Let 0 < δ < r < be fixed and let ψ δ be defined through the following minimisation problem { ψ δ ξ := } log δ min Cη : η dx, η AS δ,r ξ. 2 B r\b δ Then ψ δ = ψ δ + oδ, as δ 0 see [6, Proposition 8]. The following theorem is the main result of this section. Theorem 3.9 Γ-convergence. The functionals E 2 defined in 2.7 Γ-converge with respect to the convergence of Definition 3.6 to the functional E defined on L 2 Ω; R 2 2 SO2 by Cβ: β dx + ϕ b R if Curlβ = 0, Eβ,R := 2 Ω + otherwise in L 2 Ω; R 2 2 SO2,

15 LINE-TENSION MODEL FOR PLASTICITY AS THE Γ-LIMIT OF A NONLINEAR DISLOCATION ENERGY5 where C = 2 W F 2 I, ϕ b R := N i= ψrtˆbi, with ψ as in 3.2 and b := ˆb,...,ˆb N. As an immediate consequence of Theorem 3.9, we can deduce the following convergence result for the elastic energy induced by the dislocation measure µ. Corollary 3.0 Convergence of F µ. Let µ be as in 2.2. The following convergence holds true for the sequence F µ defined in 2.6 lim F µ = inf ϕ br R SO2 Proof. By virtue of Proposition 3.5 and Theorem 3.9, 3.22 is a straightforward consequence of the fundamental property of Γ-convergence. To shorten the notation, in what follows we always write B i s for s > 0 in place of B s x i. Proposition 3. Γ-lim inf inequality. For every β L 2 Ω; R 2 2 with Curlβ = 0, for every R SO2, and for every sequence β L 2 Ω; R 2 2 converging to β,r in the sense of Definition 3.6, we have lim inf E 2 β Eβ,R. Proof. Let β L 2 Ω; R 2 2 with Curlβ = 0, let R SO2, and let β AS 2 with equi-bounded energy E 2 β such that R T β I be a sequence β in L 2 Ω; R 2 2, 3.23 for some sequence of constant rotations R SO2 such that lim R = R. We study separately the asymptotic behavior of the energy concentrated in regions surrounding the dislocations and of the energy diffused in the remaining part of the domain, far from the dislocations. To this end, let r min > 0 denote the minimum distance between two dislocations; i.e., r min := min{ x i x,i, =,...,N,i }, and let r min > 0 denote the distance between the set {x,...,x N } and Ω; i.e., r min := min i N distx i, Ω. Let 0 < r < min{r min,r min}/2 and define Ω r := Ω \ N i= Bi r; we have E 2 β = 2 Ω r Wβ dx + =: E 2 β ;Ω r + We divide the proof into two main steps. i= 2 i= E 2 β ;Br i \ B. i B i r \Bi Wβ dx Step : Lower bound far from the core-regions. The idea is to linearise the energy density W around the identity. By a Taylor expansion of order two we get WI + F = 2 CF : F + σf, where C := 2 W F I 2 and σf/ F 2 0 as F 0. Setting ωt := sup F t σf, we have WI + F 2 2 CF : F ω F, 3.24

16 6 L. SCARDIA AND C.I. ZEPPIERI with ωt/t 2 0 as t 0. Let and define the characteristic function χ := G := RT β I, { if G /2 0 otherwise in Ω By the boundedness of G in L 2 Ω; R 2 2 it easily follows that χ boundedly in measure. Therefore, in view of 3.23 we deduce that Ω r G := χ G β in L 2 Ω; R Using the frame indifference of W and 3.24 we get E 2 β ;Ω r 2 χ Wβ dx Ω r = 2 χ WR T β dx Ω r = 2 χ WI + G dx Ω r 2 C G : G ω G χ 2 dx Then, the first term in 3.27 is lower semicontinuous with respect to the convergence On the other hand, the second term converges to zero, which can be easily seen multiplying its numerator and denominator by G 2. Indeed, G 2 χ ω G / G 2 is the product of a bounded sequence in L Ω and a sequence tending to zero in L Ω, since G /2 whenever χ 0. Combining these two facts, we eventually obtain lim inf for every 0 < r < min{r min,r min}/2. E 2 β ;Ω r 2 Step 2: Lower bound close to the core-regions. Ω r Cβ : β dx, 3.28 The idea is to divide the annulus Br i \ Bi for i =,...,N into dyadic annuli in order to rewrite E β ;Br i \ B i as the sum of contributions. Then, for each of these contributions we provide a linearisation argument analogous to that performed in Step. Finally, we conclude by means of the Γ-convergence results established in [6], in the linear framework. By 3.23 we have that β R 2 dx C 2, 3.29 for some C > 0 and for every sufficiently small > 0. Ω

17 LINE-TENSION MODEL FOR PLASTICITY AS THE Γ-LIMIT OF A NONLINEAR DISLOCATION ENERGY7 Fix δ 0,/2; for every i =,...,N, we divide Br i \ B i into dyadic annuli C k,i := B i \B i, and we consider only those annuli C k,i corresponding to the indices k =,..., k rδ k rδ k, where k := k and k := ρ log log δ, 3.30 for some fixed ρ 0, t denotes the integer part of t R. Notice that the smallest inner radius of the dyadic annuli, namely rδ k, is much bigger than ; indeed, Therefore, we have r δ k r δ k = r δ ρlog log δ = r δ log δ ρ = r ρ. 3.3 E 2 β ;B i r \ B i k k= C k,i Wβ 2 dx, 3.32 for every i =,...,N. The main point of this step is proving a lower bound uniform in k for each term in the sum in Let ψr Tˆbi,δ be as in 3.20 with ξ = R Tˆbi, for i =,...,N. We claim that there exists a positive sequence σ, infinitesimal for 0, such that Wβ C k,i 2 dx ψr Tˆbi,δ σ, 3.33 for every i =,...,N, for every k =,..., k, and for every > 0. We establish 3.33 arguing by contradiction. If 3.33 does not hold true, then there exists a sequence of positive numbers 0 as + such that, for every positive infinitesimal sequence ς there exist an index i {,...,N} and an index k {,..., k } such that Wβ C k,i 2 dx < ψr Tˆbi,δ ς, 3.34 for every N, where we set β := β for brevity. By assumption g-2 on W, 3.34 yields in particular C k,i dist 2 β,so2 dx < CψR Tˆbi,δ 2. Therefore, Proposition 3.3 gives the existence of a constant C > 0 independent of i,k and δ and a sequence of constant rotations R SO2 such that C k,i β R 2 dx CψR Tˆbi,δ Set R := R where R is the sequence satisfying We have lim + R = lim + R = R. 3.36

18 8 L. SCARDIA AND C.I. ZEPPIERI Indeed, the following estimate holds true R R 2 2 πr 2 δ 2k R β 2 dx + R β 2 dx δ 2 C k,i C k,i 2ψR 2 log C Tˆbi rδ k,δ + C rδ k, where in the last inequality we used 3.35 and the fact that β satisfies Then, since by 3.3 we have rδ k ρ 3.37 rδ k r for ρ > 0, we infer Set η := RT β I ; we cut the annulus C k,i with the segment L k,i := {x i + z,0: rδ k < z < rδ k }, thus obtaining the simply connected set C k,i \ L k,i. Then, let v H C k,i \ L k,i ; R 2 denote the sequence with zero average such that η = v in C k,i \ L k,i. Notice that [v ] = R T ˆb i on L k,i. Moreover, in view of 3.35 we deduce that 2 R v 2 dx CψR Tˆbi,δ 2, C k,i \L k,i and therefore, since the multiplication by a rotation preserves the norm, C k,i \L k,i v 2 dx CψR Tˆbi,δ. Then, setting ṽ x := v rδ k x x i we immediately get that ṽ is bounded in L 2 C δ ; R 2 2 uniformly in. The latter combined with e Cδ ṽ dx = 0, yields ṽ ṽ in H C δ ; R Moreover, since on L δ [ṽ ] = R Tˆb i R Tˆbi, as +, it follows that [ṽ] = R Tˆbi on L δ. We are going to linearise the energy density W around the identity, in analogy to what we did in Step. For the convenience of the reader we define λ k := /rδ k and we follow closely the steps leading to formula 3.27, with log replaced by λ k. We notice that 3.37 provides a bound independent of k for the sequence λ k, which is infinitesimal for +. First of all, we define the sequence χ of characteristic functions χ := { if ṽ ρ/2 0 otherwise in C δ By the boundedness of ṽ in L 2 C δ ; R 2 2 it follows that χ boundedly in measure so that, by 3.38, ṽ χ ṽ in L 2 C δ ; R 2 2. Using the frame indifference of W we may

19 LINE-TENSION MODEL FOR PLASTICITY AS THE Γ-LIMIT OF A NONLINEAR DISLOCATION ENERGY9 perform a second order Taylor expansion of the energy density around the identity, obtaining Wβ WI + v WI + λ k C k,i 2 dx χ C k,i \L k,i 2 dx = χ ṽ ec δ λ k dx 2 2 C ωλ k ṽ χ : ṽ χ ṽ χ λ k dx, ec δ where ω is defined as in Step. The first term in 3.40 is lower semicontinuous with respect to the L 2 C δ ; R 2 2 -convergence, therefore there exists a positive infinitesimal sequence ς such that ec δ 2 C ṽ χ : ṽ χ dx ec δ 2 C ṽ : ṽ dx ς. 3.4 Moreover, the second term in 3.40 converges to zero as +. In fact, we can rewrite its integrand as ωλ k χ ṽ λ k = ṽ 2 ωλ k χ ṽ 2 λ k ṽ, 2 which is the product of a bounded sequence in L Cδ and a sequence converging to zero in L Cδ, since λ k ṽ C ρ/2 for every k, when χ 0. Therefore, setting ς 2 ωλ k := sup χ ṽ k {,..., k } ec δ λ k dx, and combining 3.40, 3.4 and 3.42 we have Wβ C k,i 2 dx C ṽ : ṽ dx ς 2 ec δ for every, where we set ς := ς + ς2. Finally, taking the infimum over all the ṽ H C δ ; R 2 with [ṽ] = R Tˆbi on L δ, and recalling 3.20, we get Wβ C k,i 2 dx ψr Tˆbi,δ ς for every, and thus the contradiction, since ς is infinitesimal as +. Once 3.33 is proved, by 3.32 and 3.30 we have lim inf E 2 β ;Br i \ Bi lim inf k ψr Tˆbi,δ σ k= ρ lim inf log δ ψrtˆbi,δ σ = ρ log δ ψrtˆbi,δ = ρψ δ R Tˆbi,

20 20 L. SCARDIA AND C.I. ZEPPIERI for every i =,...,N. Then, appealing to Proposition 3.7, we pass to the limit on δ, and we get lim inf E 2 β ;Br i \ B i ρψr Tˆbi. Therefore, summing over i =,...,N we have lim inf i= E 2 Finally, combining 3.28 and 3.43 entails lim inf E 2 β lim inf 2 N β ;Br i \ Bi ρ ψr Tˆbi E 2 β ;Ω r + lim inf i= i= Ω r Cβ : β dx + ρϕ b R, therefore the lim inf-inequality is achieved letting r and ρ tend to zero. E 2 β ;B i r \ B i The following proposition states the lim sup-inequality for the Γ-limit. Proposition 3.2 Γ-lim sup inequality. Given β L 2 Ω; R 2 2 with Curlβ = 0 and R SO2, there exists a sequence β L 2 Ω; R 2 2 converging to β,r in the sense of Definition 3.6 such that lim sup E 2 β Eβ,R. Proof. Let β L 2 Ω; R 2 2 with Curlβ = 0 and let R SO2. By standard density arguments, it suffices to prove the claim for β L Ω; R 2 2. For every i =,...,N let η i : R 2 R 2 2 be a distributional solution of { Curlη = R Tˆbi δ 0 in R 2, Div Cη = 0 in R 2. In polar coordinates the planar strain η i has the form η i r,θ = r Γ R Tˆb i θ, 3.44 where the function Γ R depends on R, ˆb Tˆbi i and on the elasticity tensor C, and satisfies the bound Γ R θ C for every θ [0,2π see e.g. [3]. Moreover, by Proposition 3.7 and Remark 3.8 Tˆbi we have that for every r > 0 lim 2 Let ˆη i x := η i x x i. We assert that the following β := R I + β + B r\b Cη i : η i dx = ψr Tˆbi i= ˆη i χ Ω + Iχ S N i= Bi

21 LINE-TENSION MODEL FOR PLASTICITY AS THE Γ-LIMIT OF A NONLINEAR DISLOCATION ENERGY2 is a recovery sequence. Clearly β AS 2. Moreover, it satisfies 3.9 with R = R for every. Indeed, we have R T β I = β + i= ˆη i χ Ω + RT I χs N i= B ; i as the last term converges to zero strongly in L 2 Ω; R 2 2, it remains only to prove that the ˆη sequences i χ Ω converge to zero weakly in L 2 Ω; R 2 2 for every i =,...,N. These sequences are bounded in L 2 Ω; R 2 2 and converge to zero strongly in L Ω; R 2 2, hence the claim. To prove the lim sup-inequality for E 2 we first notice that, as β = I in N i= Bi, the energy contribution in N i= Bi is identically zero. Now we fix ρ 0, and we set Ω ρ := Ω \ N i= Bi. Then, in view of the frame ρ indifference of W, we have E 2 β = 2 W I + β + ˆη i dx Ω i= = 2 W I + β + ˆη i dx Ω ρ i= + 2 W I + β + ˆη i dx =: I + I 2. i= B i ρ \Bi We now estimate I and I. 2 Regarding I, by a Taylor expansion of order two of W around the identity we get I = Cβ : β dx + Cˆη i : ˆη i dx 2 Ω 2 ρ i= Ω ρ N + Cβ : ˆη i dx + Cˆη i : ˆη dx i= Ω ρ i,=,i Ω ρ σ β + N i= ˆη i + Ω 2 dx, ρ where σf/ F 2 0 as F 0. Recalling that β L Ω; R 2 2, by virtue of 3.44 we immediately get lim Ω ρ i= Cβ : ˆη i dx = 0, for every i =,...,N We also claim that for every i, =,...,N, with i, lim Cˆη i : ˆη dx = Ω ρ

22 22 L. SCARDIA AND C.I. ZEPPIERI Indeed, let r min and r min be as in the proof of Proposition 3. and let 0 < r < min{r min,r min}/2. Then ˆη i is bounded in Ω \ B i r for every i =,...,N. Therefore the claim follows as in We now show that the reminder in the Taylor expansion tends to zero as 0. Since by 3.44 β + ˆη i C + ρ in i= Ω ρ, setting χ := χ and ωt := sup Ω ρ F t σf, we have σ β + N i= lim ˆη i Ω 2 dx ρ ω β + N i= ˆη i lim χ Ω β + N i= ˆη β + N i= ˆη i 2 i 2 2 dx = In fact the above integrand is the product of a sequence converging to zero in L Ω and a bounded sequence in L Ω. Thus, combining 3.45, 3.46, 3.47 and 3.48, we get lim supi Eβ,R By the growth assumption on W, by the definition of ˆη i, and by the L Ω; R 2 2 -bound on β we find I 2 C β + ˆη k dx i= B i ρ \Bi k= C β 2 L Ω;R ρ 2 + ρ, then, as ρ <, we get Since lim sup in view of 3.49 and 3.50 we have E 2 lim supi 2 ρ β lim sup I + lim supi, 2 lim sup E 2 β Eβ,R + ρ, hence the thesis follows by the arbitrariness of ρ 0,. Remark 3.3. The Γ-convergence result stated in Theorem 3.9 can be extended with minor changes to the more general case of a dislocation density µ of the form cf. 2.2 µ := ˆbi δ x i, 3.5 i= under the assumption that x k x 2 for every k, where / s + as 0, for every fixed s 0,.

23 LINE-TENSION MODEL FOR PLASTICITY AS THE Γ-LIMIT OF A NONLINEAR DISLOCATION ENERGY23 If the number N of dislocations becomes increasingly large as 0 a different approach needs to be considered, which will be the subect of a forthcoming paper. 4. Beyond the model case: Mixed growth conditions In this section we study the asymptotic behaviour, via Γ-convergence, of the sequence of functionals defined in 2.9 hence under the assumptions i-iii and g-p on W. In this case the energy is quadratic for small strains and of order p,2 for big strains; i.e., quadratic far from the dislocations as in Section 3, and of order p in the core regions around each dislocation. 4.. Compactness. The compactness result proved in the case of the quadratic growth relies on a suitable version of the rigidity estimate in a domain with small holes Lemma 3., or in a domain with small holes and cuts Proposition 3.3. In this case we need a rigidity estimate in a domain with a cut, where the cut is a simple path through the dislocation points. Moreover, due to the mixed growth conditions g-p we make use of a variant of the rigidity estimate proved in [22] see also [8]. For the reader s convenience, here we recall the precise statement. Proposition 4.. [22, Proposition 2.3] Let p < 2, let n 2, and let U R n be a bounded Lipschitz domain. Then there exists a positive constant CU such that for each u W,p U; R n there exists R SOn such that u R 2 u p + dx CU dist 2 u,son u p + dx. 4. U We want to use Proposition 4. to prove a compactness result for sequences β AS p with equi-bounded energy E p. Proposition 4., though, cannot be directly applied to the sequence β as it is not a sequence of gradients. This problem can be overcome observing that β is a gradient in any simply connected subset of Ω \ {x,...,x N }, and suitably choosing one of such subsets in which Proposition 4. still holds true. The idea is very simple, in fact, in the case Ω = B s 0 R 2, for s > 0, with only one singularity located at 0. Indeed, we can ust cut the disc with a radius L to obtain the simply connected domain B s 0 \ L. Then, arguing as in Proposition 3.3 we easily derive the following result. Proposition 4.2. Let p < 2. There exists a constant C = Cs > 0 such that for every u W,p B s 0 \ L; R 2 there is an associated rotation R SO2 such that u R 2 u p + dx C dist 2 u,so2 u p + dx. 4.2 B s0\l U B s0\l Proof. The proof can be derived easily from that of Proposition 3.3 with = 0 and using Proposition 4.. Now we consider the general case of a simply connected, bounded Lipschitz domain Ω R 2 containing N singularity points for the strain β. In all that follows we denote by S a simple path through x,...,x N and such that Ω \ S is simply connected. We prove the following rigidity estimate in Ω \ S.

24 24 L. SCARDIA AND C.I. ZEPPIERI Proposition 4.3. Let p < 2. There exists C > 0 such that for every u W,p Ω \ S; R 2 there is an associated rotation R SO2 such that u R 2 u p + dx C dist 2 u,so2 u p + dx. 4.3 Ω\S Ω\S Proof. We first observe that there exists a bi-lipschitz transformation of Ω into a new domain ˆΩ which maps S into a segment L and such that ˆΩ \ L is simply connected. Then, we notice that the result in Proposition 4.2 can be easily extended to the case of a general Lipschitz domain with a straight cut. Indeed, the constant in 4. can be chosen independent of the domain for a finite number of sets that are bi-lipschitz images of a half disc. Therefore, the constant provided by Proposition 4.2 turns out to be invariant under bi-lipschitz transformation and since estimate 4.2 holds true for the domain ˆΩ \L, it is in turn true for its bi-lipschitz image Ω \ S. Notice that from estimate 4.3 we can deduce that there is a constant C p > 0 such that u R 2 C p u R p + dx C dist 2 u,so2 u p + dx 4.4 Ω\S We are now ready to prove a compactness result for strains with equi-bounded energy. Proposition 4.4 Compactness. Let < p < 2. Let 0 and let β L p Ω; R 2 2 be a sequence such that sup E p β < +. Then there exist a sequence of constant rotations R SO2 and a function β L 2 Ω; R 2 2 with Curlβ = 0 such that up to subsequences Ω\S R T β I log β in L p Ω; R Proof. Let β AS p be a sequence with equi-bounded energy E p. There exists u W,p Ω \ S; R 2 such that β = u in Ω \ S. Then, Proposition 4.3 together with 4.4 and assumption g-p on W guarantee the existence of a sequence of constant rotations R SO2 such that for some C > 0. Hence, if we set Ω\S u R 2 C p u R p + dx C 2 log, 4.6 G := RT β I log, estimate 4.6 easily yields the following bound G 2 G p C p log + 2 p 2 log dx C. 4.7 Ω

25 LINE-TENSION MODEL FOR PLASTICITY AS THE Γ-LIMIT OF A NONLINEAR DISLOCATION ENERGY25 We provide a partition of Ω considering the two sets A 2 and A p defined as follows } A 2 := {x Ω : G x 2 G x p C p log + 2 p 2 log, 4.8 } A p := {x Ω : G x 2 G x p > C p log + 2 p 2 log. 4.9 Therefore 4.7 can be rewritten as A 2 G 2 dx + C p A p G p log + 2 p 2 log dx C. 4.0 We claim that there exists a function β L p Ω; R 2 2 such that G β in L p Ω; R 2 2. In order to prove it, we need to show that the sequence G has equi-bounded L p Ω; R 2 2 -norm. Since by 4.0 we have that the L 2 A 2 ; R 2 2 -norm of G is bounded, it remains to provide an L p -bound for G in A p. This bound easily follows as we notice that 4.0 in particular implies G p dx C log 2 p. 4. A p Now we show that the limit function β is actually in L 2 Ω; R 2 2. Indeed, denoting by χ A 2 the characteristic function of the set A 2, 4.0 implies that the sequence G χ A 2 is equi-bounded in L 2 Ω; R 2 2 ; therefore it converges weakly in L 2 Ω; R 2 2 to a function ˆβ. Hence, it remains to prove that ˆβ = β. This follows since the set A 2 has asymptotically full measure as +, as 4.0 implies that A p C 2 log 0, as +. Therefore χ A 2 boundedly in measure and this yields G χ A 2 β in L p Ω; R 2 2, hence ˆβ = β a.e. in Ω. Finally, we prove that Curlβ = 0 in Ω in the sense of distributions. Let φ C0 Ω; then we have Curlβ,φ = lim 0 log Curl RT β I,φ = lim 0 log Curl RT β,φ = lim 0 N i= φx ir T ˆb i log = 0, which completes the proof. Before stating the Γ-convergence result, it is convenient to give the following definition of L p -convergence of admissible strains.