GEOMETRIC RIGIDITY FOR INCOMPATIBLE FIELDS AND AN APPLICATION TO STRAIN-GRADIENT PLASTICITY

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1 GEOMETRIC RIGIDITY FOR INCOMPATIBLE FIELDS AND AN APPLICATION TO STRAIN-GRADIENT PLASTICITY STEFAN MÜLLER, LUCIA SCARDIA, AND CATERINA IDA ZEPPIERI Abstract. In this paper we show that a strain-gradient plasticity model arises as the Γ-limit of a nonlinear semi-discrete dislocation energy. We restrict our analysis to the case of plane elasticity, so that edge dislocations can be modelled as point singularities of the strain field. A key ingredient in the derivation is the extension of the rigidity estimate [0, Theorem 3.] to the case of fields β : U R 2 R 2 2 with nonzero curl. We prove that the L 2 -distance of β from a single rotation matrix is bounded up to a multiplicative constant) by the L 2 -distance of β from the group of rotations in the plane, modulo an error depending on the total mass of Curl β. This reduces to the classical rigidity estimate in the case Curl β = 0. Keywords: Γ-convergence, rigidity estimate, nonlinear plane elasticity, edge dislocations, strain-gradient plasticity Mathematics Subject Classification: 49J45, 58K45, 74C05.. Introduction The permanent or plastic) deformations of metals rely on the presence of many types of defects in their atomic structure. Dislocations are one type of such defects and they play a prominent role in the so-called plastic slip, the relative slip of atomic layers that alters permanently the lattice structure of a metal. For this reason there is an increasing interest and effort in the derivation of plasticity models from dislocation models, both in the mathematical and in the mechanical engineering communities see e.g. [4, 6,, 2, 3, 5, 6]). Clearly, the large freedom in the choice of the dislocation model has a strong influence on the method of derivation and on the resulting plasticity theories, and therefore requires some care. In most of the cases the starting point is a semi-discrete mesoscopic) dislocation model in which the dislocations are modelled individually, while the underlying atomic lattice is averaged out. This simplification is supported by the fact that at low strains the interatomic distance of the order of few tenth of a nanometer) is much smaller than the typical distance between two dislocations few microns). For straight and parallel edge dislocations the natural setting is that of plane elasticity. Indeed in this case only the two components of the strain on the slip plane are relevant and the positions of the dislocations are completely identified by the intersection of the dislocation lines with an orthogonal plane; i.e., by their trace on a two-dimensional domain. Moreover, in semi-discrete models the dislocation energy is usually assumed to be quadratic see, e.g., [4, 9,, 9]). More precisely, the energy is given by Cβ : β dx,.) 2 εµ)

2 2 S. MÜLLER, L. SCARDIA, AND C.I. ZEPPIERI where C R 4 4 is the elasticity tensor, β : R 2 2 denotes the elastic part of the strain of a planar deformation, and ε µ) is obtained from by removing discs of radius ε > 0, the so-called core regions, around each dislocation, on which the measure µ the dislocation density) is supported. The dislocation density µ is a measure of the amount of disturbance in the lattice due to the presence of dislocations, and is related to the incompatibility of the strain β; i.e., to Curlβ. Notice that in this linear setting the ε-regularisation of the energy.), although not ideal, is necessary to prevent the blow-up of the energy at the dislocations. Moreover, also the assumption of a linear relation between stress and strain, which is equivalent to assuming small deformations, is debatable. Indeed, few atoms away from a dislocation the use of the quadratic energy.) is justified, since the presence of dislocations causes a very local lattice distortion. However, this description is not satisfactory close to the dislocations, where the strains are too large for the linear approximation to hold. Moreover, in presence of a large number of dislocations, the question of reducing to the small-strains case is more subtle. Considering a more general, nonlinear dislocation energy is therefore desirable. This general principle triggered the analysis done in [20], where the authors considered a nonlinear dislocation energy of the form W β) dx.2) where the energy density W : R 2 2 [0, + ) satisfies the usual assumptions of nonlinear elasticity e.g. stress-free reference configuration and frame indifference). In addition, W is required to satisfy mixed growth conditions considered also in e.g. [7]) ensuring that far from dislocations the energy is essentially quadratic; i.e., W β) dist 2 β, SO2)), whereas close to the defects W β) β p, for some p, 2). Therefore W β) is integrable also close to the dislocations, thus the ε-regularization needed in the linear case is no longer necessary. In [20] the authors considered the case of a finite number N of fixed edge dislocations located at points x,..., x N with Burgers vectors εˆb,..., εˆb N, where ˆb i = and ε > 0 is proportional to the interatomic spacing, and analysed the asymptotic behaviour of the scaled energies ε 2 W β) dx.3) log ε in the limit as ε tends to zero, by Γ-convergence. In.3), the strain β and the dislocation density which is encoded in the measure µ = N i= εˆb i δ xi are coupled via the relation Curl β = µ. In [20] it was shown that the energies.3) give rise in the limit to the line-tension plasticity model described by N C u: u dx + ψr Tˆbi ),.4) 2 where C = 2 W F 2 I), ψ is given in terms of an asymptotic cell formula, u is the limit of a sequence of suitably renormalized strains, and R is a rotation whose presence is characteristic of the nonlinear setting. Hence, although in the ε-energy.3) the strain β and the dislocation density µ are coupled, their limit objects are decoupled in the limit energy.4), that depends on a curl-free strain u. The decoupling is typical of this dilute regime see also [4, ]) and is due to the fact that the strains β live on a scale ε log ε while the dislocation densities µ on the smaller scale ε. As a consequence, in this regime, the limit procedure yields a pair of i=

3 GEOMETRIC RIGIDITY FOR INCOMPATIBLE FIELDS AND STRAIN-GRADIENT PLASTICITY 3 macroscopic decoupled variables and therefore two corresponding decoupled terms in the limit energy. In analogy with the linear case [], in order to overcome the degeneracy of the dilute regime, in the present paper we consider a scaling of the nonlinear energy.2) of order ε 2 log ε 2. Loosely speaking, considering this different scaling corresponds to considering a system of log ε dislocations. Then, our aim is to derive in the limit as ε tends to zero a strain-gradient model for plasticity; i.e., a model in which the energy depends on an incompatible field, and in which elastic energy and dislocation energy are coupled. From a mathematical point of view the transition between a finite and an infinite number of defects is highly nontrivial. Indeed in the linear case it required a rather sophisticated tool, namely a Korn-type inequality for fields with nonzero curl, see [, Theorem ]. Analogously, in our nonlinear setting, it requires an extension of the rigidity estimate [0, Theorem 3.] to the case of incompatible fields. More precisely, in Theorem 3.3 we prove that if R 2 is open, bounded, simply connected, and with Lipschitz boundary, then for every β L 2 ; R 2 2 ) whose curl is a measure with bounded total variation there exists a constant rotation R SO2) such that β R L 2 ;R 2 2 ) C distβ, SO2)) L 2 ) + Curl β ) ),.5) for some C > 0 depending only on. Notice that.5) clearly reduces to the classical rigidity estimate when Curl β = 0. The above generalised rigidity estimate is one of the main results of this paper and would appear to be widely applicable. The key ingredients of the proof of.5) are an L p + L q rigidity estimate recently proved in [5] and a fine regularity result for two-dimensional L -vector fields with divergence in H 2 proved in [3] see also [2]). Coming back to our model, in the present paper we treat the case of infinitely many dislocations in the nonlinear setting, although under more restrictive coerciveness assumptions on the energy density W than in [20]. More precisely, the dislocation energy in our model is given by W β) dx,.6) εµ) where the nonlinear energy density W behaves essentially as dist 2 β, SO2)) see Section 2 for more details), and the strain β satisfies at any dislocation point x i the incompatibility condition β t ds = ε ˆb i, B εx i ) where, as above, ˆb i is the direction of the Burgers vector associated to the dislocation located at x i, and ε is proportional to the interatomic distance. Unlike the case of fixed dislocations studied in [20], where the dislocation density µ was constant up to an ε-scaling) and the energy.2) depended only on the strain β, in the present case the distribution of dislocations µ = N i= εˆb i δ xi is a variable of the problem, and therefore the dislocation energy.6) depends on both β and µ. Notice that, due to the quadratic growth of the energy density, also in this nonlinear setting the ε-regularization of the dislocation energy is needed, as in the linear case []. In order to obtain in the limit a strain-gradient model we consider the rescaled functionals ε 2 log ε 2 εµ) W β) dx,.7)

4 4 S. MÜLLER, L. SCARDIA, AND C.I. ZEPPIERI and analyse their asymptotic behaviour via Γ-convergence as ε tends to zero. As in [20] also here the key idea is to rigorously reduce to the linear setting in the spirit of [7]. To this end it is necessary to show that, in particular, sequences of strains with equibounded energies converge to constant rotations of the plane minimisers of the nonlinear energy). In the case of an infinite number of dislocations, however, the compactness of the strains does not follow from the corresponding result in [20]. It follows instead from the generalised rigidity estimate.5), which allows us to perform a second order Taylor expansion of the energy around a rotation, and to get a quadratic functional in terms of a renormalised strain. At this point the final step of our approach is to apply previous results known for linear energies to the linearised functional. Then, as in [], the Γ-limit is a strain-gradient plasticity energy see Theorem 4.6) and has the form Cβ : β dx + ϕr, Curl β) dx, 2 where β is the limit of suitably scaled strains and R SO2) is the limit of the sequence of constant rotations provided by the generalised rigidity estimate see Proposition 4.3). Concerning the densities of the two terms in the energy, the elasticity tensor C equals 2 W I), while F 2 the plastic energy density ϕ is defined in terms of an asymptotic cell formula and is such that ϕr, ) is positively -homogeneous and convex. This paper is organised as follows: Section 2 is devoted to the introduction of the necessary notation and to the definition of the mesoscopic dislocation model. Then, the two main results, namely the generalised rigidity estimate and the Γ-convergence result, are treated in Sections 3 and 4, respectively. 2. Notation and setting of the problem In this section we introduce the nonlinear mesoscopic dislocation energy associated to the elastic part of the) deformation strain in presence of a system of straight and parallel edge dislocations. In this setting the dislocations are modelled by points in the plane. Let R 2 be a simply connected, bounded, Lipschitz domain representing a horizontal section of an infinite cylindrical crystal. Let S := {b, b 2 } be a set of admissible renormalised) Burgers vectors for the crystal; i.e., b, b 2 R 2 are two linearly independent vectors depending on the crystalline structure, e.g., for a square lattice S = {e, e 2 }. We also consider S := Span Z S, the span of S with integer coefficients; i.e., the set of renormalised) Burgers vectors for multiple dislocations. Every dislocation is then characterised by a point x i and by a vector ξ i S. For the given crystal, let ε > 0 denote the interatomic distance. We assume that the distance between two distinct dislocations is bounded from below in terms of an intermediate scale ρ ε ε, with ρ ε 0 as ε 0. This assumption implies that dislocations are well separated with respect to the atomic spacing ε); i.e., there is a scale separation between ε, the scale of the atomic lattice, and the scale of the dislocations distribution, represented by ρ ε. We refer the reader to the recent paper [9] where the assumption that dislocations are well-separated is overcome in the case of a quadratic energy of type.) and for finite number of defects. Here we also require that cf. [])

5 GEOMETRIC RIGIDITY FOR INCOMPATIBLE FIELDS AND STRAIN-GRADIENT PLASTICITY 5 ) lim ε 0 ρ ε /ε s = +, for every fixed s 0, ); 2) lim ε 0 log ε ρ 2 ε = 0. Under this assumptions on the hard-core scale ρ ε, we will show that in the limit the energy can be decomposed into two contributions: a self energy, which is concentrated in the hard-core regions B ρε x i ), and an interaction energy, which is essentially all stored outside the union of the hard-core regions. We define the class X ε of the admissible dislocation densities as X ε := { µ M; R 2 ): µ = M ε ξ i δ xi, M N, B ρε x i ), i= } x j x k 2ρ ε for every j k, ξ i S, 2.) where M; R 2 ) denotes the space of vector-valued Radon measures on and, for every i, δ xi denotes the Dirac mass centred at x i. For a given measure µ X ε and r > 0 we define r µ) := \ B r x i ). 2.2) x i suppµ) The class of admissible strains associated with any µ X ε is given by those β L 2 ε µ); R 2 2 ) satisfying Curl β = 0 in ε µ) and β t ds = ε ξ i, for i =,..., M, B εx 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 ε x i ) and the integrand β t is intended in the sense of traces see [8, Theorem 2, pag. 204]). Then, in this mathematical setting, an admissible µ measures the failure of the condition of being a gradient for the strain β and the presence of dislocations can be detected by looking at the topological singularities of β. Let SO2) := {R R 2 2 : R T R = I, det R = } be the set of rotations in R 2 2. Moreover, for F R 2 2 we denote the Euclidean nom by F = tr F T F. The elastic energy density W : R 2 2 [0, + ) satisfies the usual assumptions of nonlinear elasticity, namely i) W C 0 R 2 2 ), W C 2 in a neighbourhood of SO2); ii) the reference configuration is stress-free; i.e., W I) = 0; iii) W is frame indifferent; i.e., W RF ) = W F ) for every F R 2 2 and R SO2). Moreover, W satisfies the following growth condition: 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 ε.

6 6 S. MÜLLER, L. SCARDIA, AND C.I. ZEPPIERI iv) there exist two constants C, C 2 > 0 such that for every F R 2 2 C dist 2 F, SO2)) W F ) C 2 dist 2 F, SO2)), where distf, SO2)) := inf R SO2) F R. The lower bound in iv) states that the energy wells are non-degenerate and is widely used in nonlinear elasticity. The upper bound, though, is rather restrictive as it rules out the physically relevant case of orientation-preserving deformations as well as the blow-up of the energy for a total compression. Although partially unsatisfactory, the upper bound in iv) is heavily used in the proof of the Γ-convergence result Theorem 4.6) to guarantee that the energy along the recovery sequence is linear, up to a small error. Due to the quadratic growth iv) the energy associated to an admissible pair µ, β) is well defined only away from the dislocations, as in the linear case, namely in the domain ε µ): W β) dx. εµ) In what follows it is useful to extend the admissible strains β to the whole domain. There are different possible extensions compatible with our model. Here we decide to consider β = I in the discs B ε x i ). Therefore, from now on the class of admissible strains associated with a measure µ X ε is given by AS ε µ) := { β L 2 ; R 2 2 ): β I in M i= B ε x i ), Curl β = 0 in ε µ), B εx i ) β t ds = ε ξ i, for i =,..., M }. 2.3) By ii) we can rewrite the energy associated to an extended) admissible strain β AS ε µ) as E ε µ, β) := W β) dx. For our purposes, as in the linear case [], the relevant scaling for the energy is ε 2 log ε 2 ; therefore we consider the scaled nonlinear dislocation energy given by E ε µ, β) := ε 2 log ε 2 E εµ, β) if µ X ε, β AS ε µ), 2.4) + otherwise in M; R 2 ) L 2 ; R 2 2 ). Then, as in [], we notice that this is the only scaling of the energy for which the strain β and the measure µ are of the same order in ε. This results into a coupling of their limit rescaled objects, and therefore to a strain-gradient plasticity model Theorem 4.6). 3. Rigidity estimate for fields with prescribed curl In this section we prove a generalised rigidity estimate for vector fields with nonzero curl. This result provides a quantitative estimate of the distance of a two-dimensional matrix-valued field from a constant rotation in terms of its distance from the set of rotations of the plane, like the classical rigidity estimate [0] in two dimensions, with an additional term depending on the total mass of the curl.

7 GEOMETRIC RIGIDITY FOR INCOMPATIBLE FIELDS AND STRAIN-GRADIENT PLASTICITY 7 Before proving the desired result, for the reader s convenience we state here a variant of the Rigidity Estimate recently proved in [5]. To this end, we first recall some useful notation. Let U R n be a measurable set. We denote by L 2, U; R m ) the space of weak-l 2 functions; i.e., f L 2, U; R m ) if and only if f is measurable and there exists a constant C > 0 such that We also set L n { x U : fx) > λ } ) C2, for every λ > 0. λ2 f L 2, U;R m ) := inf { C > 0: λ L n { f > λ } ) /2 C, λ > 0 }. Notice that L 2, U;R m ) is not a norm but only a quasi-norm since the Minkowski Inequality holds only in the following form f + g L 2, U;R m ) 2 f L 2, U;R m ) + 2 g L 2, U;R m ). If f L 2 U; R m ) then clearly f L 2, U; R m ) and f L 2, U;R m ) f L 2 U;R m ); but L 2 U; R m ) L 2, U; R m ) as, for example, / x n/2 belongs to L 2, U) but not to L 2 U). We are now ready to recall the weak rigidity estimate see [5]). Theorem 3. L 2, -rigidity). Let U be a bounded Lipschitz domain of R n. There exists a constant C = CU) > 0 with the following property: For every u L U; R n ) such that u L 2, U; R n n ) there is an associated rotation R SOn) such that u R L 2, U;R n n ) C dist u, SOn)) L 2, U). 3.) We prove a technical result we use in what follows. Proposition 3.2. Let g : R R be a bounded function such that gt) γ t α, for some γ > 0 and for some α >. Let U R n be a measurable set; if θ L 2, U) then g θ L 2 U) and where M := max{ g L U), γ 2 α / 4 α) ) /2 }. Proof. We have U gθ) 2 dx = g θ L 2 U) M θ L 2, U), 3.2) {x: θ >} gθ) 2 dx + gθ) 2 dx. 3.3) {x: θ } The first term in the right hand side of 3.3) can be easily estimated appealing to the boundedness of g, in fact gθ) 2 g 2 L U) Ln {x: θ > }) g 2 L U) θ 2 L 2, U). 3.4) {x: θ >} For the second term in 3.3) we proceed as follows. Using the growth assumption on g we find gθ) 2 dx γ 2 θ 2α dx. 3.5) For δ 0, /2] we have {x: δ< θ 2δ} {x: θ } {x: 0< θ } θ 2α dx 4 α δ 2α L n {x: θ > δ}) 4 α δ 2α ) θ 2 L 2, U). 3.6)

8 8 S. MÜLLER, L. SCARDIA, AND C.I. ZEPPIERI Therefore using 3.6) with δ = /2 k and k N we get θ 2α dx = θ 2α dx {x: 0< θ } k {x: 2 k < θ 2 k } 4 α 4 α )k θ 2 L 2, U) k 0 4 α = 4 α) θ 2 L 2, U). Finally, combining the last inequality with 3.3)-3.5) entails the thesis. The following theorem is the main result of this section. It states that in dimension two the rigidity estimate holds true also for vector fields with nonvanishing curl, modulo an error depending on the total mass of the curl. This result is the nonlinear counterpart of the generalised Korn Inequality proved in [, Theorem ]. Theorem 3.3 Generalised Rigidity Estimate). Let R 2 be open, bounded, simply connected, and Lipschitz. There exists a constant C = C) > 0 with the following property: For every β L 2 ; R 2 2 ) with µ := Curl β M b ; R 2 ) there is an associated rotation R SO2) such that β R L 2 ;R 2 2 ) C distβ, SO2)) L 2 ) + µ ) ). 3.7) Proof. Set δ := distβ, SO2)) L 2 ) + µ ). Notice that for i =, 2 µ i = curlβ T e i ) = divjβ T e i )), with J := 0 0 therefore µ H ; R 2 ) and there exists a unique solution to the following problem: { v = µ in, v H 0 ; R2 ). By classical regularity theory for linear elliptic systems with measure data see e.g. [8] and references therein) we have v L 2, ;R 2 2 ) C µ ). 3.9) Let β := vj; in view of 3.8) we have that Curl β = µ. Hence, Curl β β) = 0 in, which implies the existence of u H ; R 2 ) such that β β = u a.e. in. Then we have dist u, SO2)) = distβ β, SO2)) distβ, SO2)) + β This implies, by 3.9) and by the definition of δ, that ) ; 3.8) = distβ, SO2)) + v. 3.0) dist u, SO2)) L 2, ) Cδ. Then, Theorem 3. provides us with a constant C > 0 and a constant rotation R SO2) such that u R L 2, ;R 2 2 ) Cδ,

9 GEOMETRIC RIGIDITY FOR INCOMPATIBLE FIELDS AND STRAIN-GRADIENT PLASTICITY 9 and as a consequence β R L 2, ;R 2 2 ) Cδ. 3.) Without loss of generality we may assume that R = I otherwise we consider R T β). Let ϑ: [ π, π) be a measurable function such that the corresponding rotation ) cos ϑ sin ϑ Rϑ) = sin ϑ cos ϑ satisfies βx) Rϑx)) = distβx), SO2)) for a.e. x. Then, 3.) yields I Rϑ) L 2, ;R 2 2 ) Cδ. 3.2) Since I Rϑ) ϑ /2 for a.e. x, by 3.2) we deduce that ϑ L 2, ) Cδ. 3.3) We now consider the linearisation of the rotation Rϑ) around zero, namely ) ϑ R lin ϑ) :=. ϑ Appealing to Proposition 3.2 with gt) = cos t, or gt) = sin t t, from 3.3) we derive the two following bounds cos ϑ L 2 ) Cδ and sin ϑ ϑ L 2 ) Cδ; therefore Rϑ) R lin ϑ) L 2 ;R 2 2 ) Cδ. Since β R lin ϑ) L 2 ;R 2 2 ) distβ, SO2)) L 2 ) + Rϑ) R lin ϑ) L 2 ;R 2 2 ) Cδ, we have β = R lin ϑ) + h, with h L 2 ;R 2 2 ) Cδ. Then by the definition of R lin we deduce Curl β = ϑ + Curl h, which in its turn implies divcurl β) ) = divcurl h) ), 3.4) where, for a vector a R 2 we use the notation a := Ja. Hence we have divcurl β) ) H 2 ) C h L 2 ;R 2 2 ) Cδ. 3.5) By [3, Theorem 3. and Remark 3.3] see also [2]) if f L ; R 2 ) is a vector field satisfying divf H 2 ), then f also belongs to H ; R 2 ) and the following estimate holds true f H ;R 2 ) C divf H 2 ) + f L ;R 2 )). This estimate clearly extends by density to measures with bounded total variation. Thus, by applying the previous estimate with f = Curl β), by virtue of 3.5) we have Curl β) H ;R 2 ) divcurl β) ) H 2 ) + Curl β) ) Cδ. 3.6) Eventually, recalling that v solves 3.8), by 3.6) we deduce v L 2 ;R 2 2 ) Cδ,

10 0 S. MÜLLER, L. SCARDIA, AND C.I. ZEPPIERI and therefore, by 3.0), dist u, SO2)) L 2 ) Cδ. Hence the classical Rigidity Estimate [0, Theorem 3.] provides us with a constant C > 0 and with a constant rotation R SO2) such that thus and the thesis is achieved. u R L 2 ;R 2 2 ) Cδ, β R L 2 ;R 2 2 ) Cδ 4. Γ-convergence of the nonlinear dislocation energy In this section we study the asymptotic behaviour of the scaled energies E ε, defined in 2.4), as ε tends to zero. In the spirit of the Γ-convergence analysis performed in [, 20], we show that a linearisation takes place in the limit and that the limit energy is a macroscopic strain-gradient model for plasticity, namely there is a nontrivial interplay between the interaction and the self energy. 4.. Cell formula for the limit self energy. For the definitions and results contained in this subsection we refer the reader to [, Section 6]. For later reference, it is convenient to introduce a new class of admissible scaled) strains. For 0 < r < r 2 < and ξ R 2 we define { } AS r,r 2 ξ) := η L 2 B r2 \ B r ): Curl η = 0 in B r2 \ B r, η t ds = ξ, B r where B r denotes the disc of radius r centred at 0. In the special case r 2 = we will simply write AS r ξ) instead of AS r,ξ). We also set { } ψξ, δ) := min Cη : η dx, η AS δ ξ), 4.) 2 B \B δ where C = 2 W F 2 I). We recall the following fundamental result see [, Corollary 6, Remark 7]). Proposition 4.. Let ξ R 2 and δ 0, ), and let ψξ, δ) be as in 4.). Then for every ξ R 2 ψξ, δ) lim δ 0 log δ = ˆψξ), where ˆψ : R 2 [0, + ) is defined by ˆψξ) := lim δ 0 log δ and η 0 : R 2 R 2 2 is a distributional solution to { Curl η = ξ δ0 in R 2, Div Cη = 0 in R 2. Cη 0 : η 0 dx, 4.2) 2 B \B δ

11 GEOMETRIC RIGIDITY FOR INCOMPATIBLE FIELDS AND STRAIN-GRADIENT PLASTICITY Remark 4.2. Assume that ρ ε satisfies ) and 2), from [, Proposition 8] it follows that the function ψ ε : R 2 [0, + ) defined as { ψ ε ξ) := } log ε min Cη : η dx, η AS ε,ρε ξ), 4.3) 2 B ρε \B ε satisfies ψ ε ξ) = ψξ, ε) + o)), log ε where o) 0 as ε 0, uniformly with respect to ξ. pointwise as ε 0 to ˆψ given by 4.2). Then, in particular, ψ ε converges We are now in a position to define the density ϕ : SO2) R 2 [0, + ) of the self-energy through the following relaxation procedure: { M } M ϕr, ξ) := min λ k ˆψR T ξ k ) : λ k ξ k = ξ, M N, λ k 0, ξ k S. 4.4) k= k= It follows from the above definition that the function ϕ is positively -homogeneous and convex see also [, Remark 9]) Compactness. In the next proposition we prove a compactness result for sequences of pairs µ ε, β ε ) with equibounded energy E ε by means of the generalised Rigidity Estimate Theorem 3.3. Proposition 4.3 Compactness). Let ε j 0 and let µ j, β j ) M; R 2 ) L 2 ; R 2 2 ) be a sequence such that sup j E εj µ j, β j ) < +. Then there exist a sequence of constant rotations R j ) SO2), a measure µ H ; R 2 ) M; R 2 ), and a function β L 2 ; R 2 2 ) such that, up to subsequences, µ j ε j log ε j µ in M; R 2 ), 4.5) R T j β j I ε j log ε j moreover, Curl β = R T µ, where R := lim j + R j. β in L 2 ; R 2 2 ); 4.6) Proof. Let µ j, β j ) M; R 2 ) L 2 ; R 2 2 ) be a sequence such that E εj µ j, β j ) C for some positive constant C independent of j, where M j µ j = ε j ξ i,j δ xi,j, with ξ i,j S, x i,j such that B ρεj x i,j ) and x i,j x k,j 2ρ εj for every i k. The proof of the compactness is divided into three steps. Step. Weak convergence of the scaled dislocation measures. i=

12 2 S. MÜLLER, L. SCARDIA, AND C.I. ZEPPIERI We first show that the sequence µ j /ε j log ε j ) is uniformly bounded in mass. We claim that ε j log ε j µ j ) = log ε j M j ξ i,j C. 4.7) Let s 0, ) and δ 0, ) be fixed parameters. From the bound on the energy it follows that, for j sufficiently large, C ε 2 j log ε j 2 W β j ) dx εj µ j ) M j ε 2 j log ε j 2 W β j ) dx, 4.8) i= i= B ρεj x i,j )\B δε s j x i,j ) where in the last inequality we used the assumption B ρεj x i,j ) B ρεj x k,j ) = for i k. For every i {,..., M j } we decompose the annulus B ρεj x i,j ) \ B δε s j x i,j ) centred at x i,j into dyadic annuli with constant ratio δ 0, ) between inner and outer radii. More precisely, the annuli are defined as C k,i j := B ρεj δ k x i,j) \ B ρεj δ kx i,j), 4.9) and we consider only those corresponding to k =,..., k εj, where Notice that ρ εj δ k εj k εj := k εj + and k εj := s log ε j log δ log ρ ε j log δ. 4.0) ρ εj δ kε j + = δε s j ; therefore, for every i {,..., M j} we have ε 2 j log ε j 2 W β j ) dx B ρεj x i,j )\B δε sx i,j ) j C k,i j log ε j 2 k εj k= C k,i j W β j ) ε 2 j dx. 4.) Arguing as in [20, Proposition 3.] we deduce that for every j, i, and k the following estimate holds true W β j ) ε 2 dx ψr T ξ i,j, δ) σ j, 4.2) j where ψ, δ) is defined as in 4.) and σ j is a nonnegative infinitesimal sequence as j +. Combining 4.8), 4.), and 4.2) we find that for every δ 0, ) C ε 2 j log ε j 2 W β j ) dx εj µ j ) log ε j log ε j M j i= M j i= log ε j 2 M j k εj ψr T ) ξ i,j, δ) σ j i= k= k εj ψr T ) ξ i,j, δ) σ j log ε j s log ρ ) ε j ψr T ξ i,j, δ) log ε j log δ σ ) j, log δ where in the last inequality we used the relation k εj = k εj + k εj.

13 GEOMETRIC RIGIDITY FOR INCOMPATIBLE FIELDS AND STRAIN-GRADIENT PLASTICITY 3 Now, as a first step we let δ 0 in the previous estimate; this leads to M j C ε 2 j log ε j 2 W β j ) dx s log ρ ) ε j ˆψR T ξ i,j ), 4.3) log ε j log ε j εj µ j ) with ˆψ defined as in 4.2). By assumption ρ εj ε j, hence 4.3) entails that for j sufficiently large M j s C ε 2 j log ε j 2 W β j ) dx εj µ j ) log ε j 2 ˆψR T ξ i,j ). 4.4) Since the function ˆψ is 2-homogeneous being the pointwise limit of 2-homogeneous functions, cf. 4.2)), we have in particular that for every i {,..., M j }, R ˆψR T ξ i,j ) = ξ i,j 2 ˆψ T ) ξ i,j R T. ξ i,j Set c := inf ξ = ˆψξ), from 4.4) we finally deduce 2C s c log ε j M j ξ i,j 2 i= i= c log ε j i= M j ξ i,j, 4.5) where the last inequality follows from the fact that since ξ i,j S = span Z S, ξ i,j are bounded away from zero. Therefore the claim 4.7) follows. Step 2. Weak convergence of the scaled strains. In view of the growth condition iv) we have Cε 2 j log ε j 2 C W β j ) dx C dist 2 β j, SO2)) dx. 4.6) The idea is to apply the generalised rigidity estimate Theorem 3.3 to a suitable modification of β j. The estimate cannot be applied directly to the strains β j since it is not clear whether the crucial bound Curl β j ) Cε j log ε j holds true. Indeed, on the one hand the total variation of the measure µ j is bounded by Cε j log ε j by Step ; on the other hand, however, Curl β j is related to the measure µ j, but it is not exactly µ j and therefore it does not necessarily satisfy the same bound. To overcome this problem, in what follows we construct new strains β j satisfying Curl β j ) = µ j ), and hence the crucial bound. For every x i,j in the support of µ j, set C i,j := B 2εj x i,j ) \ B εj x i,j ) and consider the function K i,j : C i,j R 2 2 defined as follows i= K i,j x) := ε j 2π ξ i,j J x x i,j x x i,j 2, where J, as above, is the clockwise rotation of π/2. Then we have K i,j 2 dx C dist 2 β j, SO2)) dx. 4.7) C i,j C i,j Indeed, a straightforward calculation gives C i,j K i,j 2 dx = Cε 2 j ξ i,j 2,

14 4 S. MÜLLER, L. SCARDIA, AND C.I. ZEPPIERI while a scaling argument see [20, Proposition 3.3 and Remark 3.4]) shows that Cε 2 j ξ i,j 2 dist 2 β j, SO2)) dx, C i,j and hence 4.7). By construction Curl β j K i,j ) = 0 in C i,j and B εx i,j ) β j K i,j ) t ds = 0, for every i =,..., M j. Hence, there exist M j functions u i,j H C i,j ; R 2 ) such that β j K i,j = u i,j in C i,j, for every i =,..., M j. In view of 4.7) we have dist 2 u i,j, SO2)) dx C dist 2 β j, SO2))+ K i,j 2) dx C dist 2 β j, SO2)) dx, C i,j C i,j C i,j therefore the classical rigidity estimate applied to u i,j provides us with a constant rotation R i,j SO2) such that u i,j R i,j 2 dx C C i,j dist 2 u i,j, SO2)) dx C C i,j dist 2 β j, SO2)) dx, C i,j for some C > 0 independent of j. By standard extension arguments, there exists a function v i,j H B 2εj x i,j ); R 2 ) such that v i,j u i,j R i,j in C i,j and v i,j 2 dx C B 2εj x i,j ) u i,j R i,j 2 dx C C i,j dist 2 β j, SO2)) dx. C i,j 4.8) Now define the field β j : R 2 2 as { β j in εj µ j ), β j := v i,j + R i,j in B εj x i,j ) for i =,..., M j. By 4.8) we get dist 2 β j, SO2)) dx C M j dist 2 β j, SO2)) dx + i= dist 2 β j, SO2)) dx Cε 2 j log ε j 2. B εj x i,j ) v i,j 2 Moreover, by construction Curl β j ) = µ j ); then we are in a position to apply Theorem 3.3 to β j to deduce the existence of a sequence of constant rotations R j SO2) such that β j R j 2 C dist 2 β j, SO2)) + Curl β j )) 2) Cε 2 j log ε j 2, 4.9) for some C > 0 independent of j. By the definition of β j and by 4.9) we deduce that β j R j 2 = β j R j 2 β j R j 2 Cε 2 j log ε j 2, εj µ j ) εj µ j )

15 GEOMETRIC RIGIDITY FOR INCOMPATIBLE FIELDS AND STRAIN-GRADIENT PLASTICITY 5 and therefore εj µ j ) β j R j 2 ε 2 j log ε dx C, 2 j for every j. Finally, recalling that β j I in M j i= B ε j x i,j ) and that from Step we have the bound M j C log ε j, we deduce that, up to subsequences, R T j β j R j ε j log ε j = RT j β j I ε j log ε j β in L 2 ; R 2 2 ). Step 3. The limit measure µ belongs to H ; R 2 ) and Curl β = R T µ. Let φ C0 ) and let φ j) H0 ) be a sequence converging to φ uniformly and strongly in H0 ) and such that φ j φx i,j ) in B εj x i,j ), for every x i,j in the support of µ j. Then we have φ dµ = lim φ j dµ j = lim j + ε j log ε j j + ε j log ε j Curl β j, φ j = lim j + ε j log ε j Curl β j R j ), φ j = lim β j R j )J φ j dx j + ε j log ε j = RβJ φ dx = Curl Rβ), φ = R Curl β, φ, from which we deduce the admissibility condition Curl β = R T µ. Finally, since in Step 2 we proved that β L 2 ; R 2 2 ), we immediately get that µ belongs to H ; R 2 ). Remark 4.4. Notice that in Step of Proposition 4.3, we show that the number of dislocations M j corresponding to a pair µ j, β j ) with equibounded energy is such that M j C log ε j see 4.5)). In view of Proposition 4.3, it is convenient to give the following notion of convergence for sequences of pairs µ ε, β ε ). Definition 4.5. A pair of sequences µ ε, β ε ) M; R 2 ) L 2 ; R 2 2 ) is said to converge to a triplet µ, β, R) M; R 2 ) L 2 ; R 2 2 ) SO2) if there exists a sequence of rotations R ε ) SO2) such that R T ε β ε I ε log ε ε log ε µ ε µ in M; R 2 ), 4.20) β in L 2 ; R 2 2 ), and R ε R. 4.2) 4.3. Γ-convergence result. We are now in a position to state and prove the main result of this section, namely a Γ-convergence result for the scaled functionals E ε. In what follows we additionally assume that has C, boundary. Notice however that the higher regularity of will be used only in the proof of the limsup inequality.

16 6 S. MÜLLER, L. SCARDIA, AND C.I. ZEPPIERI Theorem 4.6. The energy functionals E ε defined in 2.4) Γ-converge with respect to the convergence of Definition 4.5 to the functional E defined on M; R 2 ) L 2 ; R 2 2 ) SO2) by Cβ : β dx + ϕ R, dµ ) d µ if µ H ; R 2 ) M; R 2 ) 2 d µ Eµ, β, R) := and Curl β = R T µ, 4.22) + otherwise, where C = 2 W F 2 I) and ϕ is as in 4.4). Specifically, the following two inequalities hold true. Γ-liminf inequality: For every µ, β, R) H ; R 2 ) M; R 2 ) ) L 2 ; R 2 2 ) SO2) with Curl β = R T µ, and for every sequence µ ε, β ε ) M; R 2 ) L 2 ; R 2 2 ) converging to µ, β, R) in the sense of Definition 4.5, we have lim inf ε 0 E ε µ ε, β ε ) Eµ, β, R). Γ-limsup inequality: For every µ, β, R) H ; R 2 ) M; R 2 ) ) L 2 ; R 2 2 ) SO2) with Curl β = R T µ, there exists a sequence µ ε, β ε ) M; R 2 ) L 2 ; R 2 2 ) converging to µ, β, R) in the sense of Definition 4.5, such that lim sup E ε µ ε, β ε ) Eµ, β, R). ε 0 Remark 4.7. Notice that 4.4) can be equivalently rewritten as { M } M ϕr, ξ) = min λ k ˆψζk ) : λ k ζ k = R T ξ, M N, λ k 0, ζ k R T S, k= k= hence ϕ depends on ξ only in terms of R T ξ; therefore ϕr, ξ) = ϕr, R T ξ) for some function ϕ. Then, since the limit strains β satisfy the condition Curl β = R T µ, thanks to the -homogeneity of ϕr, ) the Γ-limit E can be expressed in terms of β and Curl β in the following way Eµ, β, R) = 2 Cβ : β dx + ϕ In particular, if Curl β L ; R 2 ) we have Eµ, β, R) = Cβ : β dx + 2 R, dcurl β d Curl β ) d Curl β. ϕr, Curl β) dx. Proof of Theorem 4.6. Γ-liminf inequality. Let µ, β, R) H ; R 2 ) M; R 2 ) ) L 2 ; R 2 2 ) SO2) and µ ε, β ε ) M; R 2 ) L 2 ; R 2 2 ) be as in the statement and assume that lim inf ε 0 E ε µ ε, β ε ) = lim ε 0 E ε µ ε, β ε ). Suppose moreover that E ε µ ε, β ε ) C for every ε > 0 otherwise there is nothing to prove). This implies in particular that µ ε, β ε ) X ε AS ε µ ε ), where X ε and AS ε µ ε ) are defined in 2.) and 2.3), respectively.

17 GEOMETRIC RIGIDITY FOR INCOMPATIBLE FIELDS AND STRAIN-GRADIENT PLASTICITY 7 Arguing as in [20, Proposition 3.], we decompose the energy into a contribution far from the dislocations, in ρε µ ε ) := \ M i= B ρ ε x i ), and a contribution close to the dislocations; i.e., E ε µ ε, β ε ) = We treat the two contributions separately. M ε 2 log ε 2 W β ε ) dx + ρε µ ε) ε 2 log ε 2 W β ε ) dx i= B ρε x i ) M =: E ε µ ε, β ε ; ρε µ ε )) + E ε µ ε, β ε ; B ρε x i )). 4.23) Lower bound far from the dislocations. For the energy contribution far from the dislocations we perform a linearisation of the energy at scale ε log ε around the identity matrix. By a Taylor expansion of order two we get W I + F ) = 2 CF : F + σf ), where σf )/ F 2 0 as F 0. Then, setting ωt) := sup F t σf ), we have with ωt)/t 2 0 as t 0. Now, let i= W I + ε log ε F ) 2 ε2 log ε 2 CF : F ωε log ε F ), 4.24) G ε := RT ε β ε I, ε log ε and define the characteristic function { if x ρε µ χ ε := ε ) and G ε ε /2 0 otherwise in. 4.25) By the boundedness of G ε ) in L 2 ; R 2 2 ) and in view of the definition of ρ ε that ensures that ρε µ ε ) has asymptotically full measure, it easily follows that χ ε boundedly in measure. Therefore, from 4.2) we deduce that G ε := χ ε G ε β in L 2 ; R 2 2 ). 4.26) Using the frame indifference of W and 4.24) we get E ε µ ε, β ε ; ρε µ ε )) ε 2 log ε 2 χ ε W β ε ) dx = ε 2 log ε 2 χ ε W Rε T β ε ) dx = ε 2 log ε 2 χ ε W I + ε log ε G ε ) dx 2 C G ε : G ωε log ε G ε ) ) ε χ ε ε 2 log ε 2 dx. 4.27) Then, the first term in 4.27) is lower semicontinuous with respect to the convergence 4.26). On the other hand, the second term converges to zero, which can be easily seen multiplying its numerator and denominator by G ε 2. Indeed, G ε 2 χ ε ωε log ε G ε )/ε log ε G ε ) 2 is

18 8 S. MÜLLER, L. SCARDIA, AND C.I. ZEPPIERI the product of a bounded sequence in L ) and a sequence tending to zero in L ), since ε log ε G ε ε /2 log ε whenever χ ε 0. Combining these two facts, we eventually obtain lim inf E ε µ ε, β ε ; ρε µ ε )) Cβ : β dx. 4.28) ε 0 2 Lower bound close to the dislocations. We are going to estimate the energy contribution E ε µ ε, β ε ; B ρε x i )), for i =,..., M. For brevity, we write Br i instead of B r x i ) for every r > 0. Let s 0, ) and δ 0, ) be fixed and independent of ε; then, for small enough ε, E ε µ ε, β ε ; Bρ i ε ) = B i ρε \Bi ε k ε W β ε ) dx W β ε ) dx Bρε i \Bi δε s k= C k,i ε W β ε ) dx, where Cε k,i and k ε are defined as in 4.9) and 4.0), respectively. Proceeding as in the proof of Proposition 4.3, Step, we prove that, as in 4.4), M i= E ε µ ε, β ε ; B i ρ ε ) log ε M i= s log ρ ) ε log ε By formula 4.4) and by the definition of µ ε it follows that log ε M ˆψR T ξ i ) i= ˆψR T ξ i ). 4.29) ϕ R, d µ ) ε d µ ε, 4.30) d µ ε where µ ε := µ ε /ε log ε ). Notice that 4.20) entails that µ ε µ in M; R 2 ). Since Span R S = R 2, the convex -homogeneous function ϕ is finite on R 2 and therefore continuous. Then, invoking Reshetnyak s lower-semicontinuity Theorem, 4.29) and 4.30) give lim inf ε 0 M i= E ε µ ε, β ε ; Bρ i ε ) s ϕ R, dµ ) d µ. 4.3) d µ Hence the lower bound for the energy follows from 4.28), 4.3), and from the arbitrariness of s, which can be taken arbitrarily close to. Γ-limsup inequality. Let µ, β, R) H ; R 2 ) M; R 2 ) ) L 2 ; R 2 2 ) SO2) be such that Curl β = R T µ. By standard density arguments we can assume that µ, β) W, ; R 2 ) L ; R 2 2 ). We divide the proof into three steps. Step. µ = ξ dx with ξ R 2. Given ξ R 2 and β L ; R 2 2 ) with Curl β = R T ξ dx, we are going to construct a sequence µ ε, β ε ) X ε AS ε µ ε ) converging to µ, β) in the sense of Definition 4.5 and such that lim sup E ε µ ε, β ε ) Cβ : β dx + ϕ R, ξ) dx. 4.32) ε 0 2

19 GEOMETRIC RIGIDITY FOR INCOMPATIBLE FIELDS AND STRAIN-GRADIENT PLASTICITY 9 Let ˆψ be as in 4.2); by 4.4) there exist M N, ξ,..., ξ M S, and λ k 0, with k =,..., M, such that ξ = M k= λ kξ k and M ϕr, ξ) = λ k ˆψR T ξ k ). 4.33) Set Λ := k= M λ k, r ε := k= 2 Λ log ε ; notice that r ε ρ ε. By [, Lemma 4] there exists a sequence of admissible measures defined as M M ε k µ ε := ε ξ k µ k ε, where µ k ε := δ xi,ε, k= with the {x i,ε } such that B rε x i,ε ), x i,ε x j,ε 2r ε for every i j, satisfying µ ε ε log ε µ k ε log ε i= µ weakly in M; R 2 ) 4.34) λ k dx weakly in M). 4.35) We show that µ ε converges to µ. For what follows it is useful to combine the two summations in the definition of µ ε into just one sum and to rewrite it as M ε µ ε = ε ξ i,ε δ xi,ε ; we also introduce the auxiliary measures µ rε ε := πr 2 ε M ε ε ξ i,ε χ Brε x i,ε ) dx, i= Appealing again to [, Lemma 4] we get µ rε ε ε log ε µ strongly in H ; R 2 ), i= ˆµ rε ε := 2πr ε ˆµ rε ε ε log ε M ε ε ξ i,ε H B rε x i,ε ). To define a recovery sequence for β we first introduce the auxiliary strains K ε := M ε i= i= µ weakly in M; R 2 ). 4.36) K ξ i,ε i,ε χ B rε x i,ε ) with i,ε Kξ i,ε x) := ε 2πrε 2 R T ξ i,ε Jx x i,ε ), where J is the clockwise rotation of π/2. Notice that Curl K ε = R T µ rε ε R T ˆµ rε ε. Now, for every i =,..., M ε, let ηi ε : R2 R 2 2 be a distributional solution of { Curl η = R T ξ i,ε δ 0 in R 2, Div Cη = 0 in R 2,

20 20 S. MÜLLER, L. SCARDIA, AND C.I. ZEPPIERI where C := 2 W F 2 I). In polar coordinates the planar strain η ε i has the form η ε i r, θ) = r Γ R T ξ i,ε θ), 4.37) where the function Γ R T ξ ε,i depends on R, ξ i,ε and on the elasticity tensor C, and satisfies the bound Γ R T ξ i,ε θ) C for every θ [0, 2π) see e.g. []). Let ˆη i εx) := ηε i x x ε,i), and let ˆη ε be defined as M ε ˆη ε := ε ˆη i ε χ B i rε. Notice that Curl ˆη ε = R T µ ε ˆµ rε ε ). i= We define the recovery sequence β ε as β ε := R I + ε log ε β + ˆη ε K ε + β ) ε χ εµ ε) + I χ Mε, i= Bi ε where β ε := w ε J and w ε is the solution to the following system { w ε = ε log ε R T µ R T µ rε ε in, w ε H0 ; R2 ). Notice that β ε AS ε µ ε ). In fact β ε I in Mε i= Bi ε; moreover Curl β ε ) ε µ ε ) = ε log ε µ + µ ε ˆµ rε ε µ rε ε + ˆµ rε ε ε log ε µ + µ rε ε ) ε µ ε ) = 0, 4.38) and by construction β ε satisfies the circulation condition on Mε i= Bi ε. Now we prove that β ε converges to β in the sense of Definition 4.5 with R ε = R; i.e., we show that RT β ε I ε log ε β weakly in L 2 ; R 2 2 ). To this end we prove the following convergence properties: a) b) c) ˆη ε χ εµ ε) 0 ε log ε weakly in L 2 ; R 2 2 ); K ε ε log ε 0 strongly in L2 ; R 2 2 ); w ε ε log ε 0 strongly in L2 ; R 2 2 ). Clearly a), b), and c) imply the desired convergence for the sequence β ε. ˆη To prove a), we first notice that the sequence ε ε log ε has bounded L2 -norm in ε µ ε ). Indeed, since ˆη ε i C x x i,ε for i =,..., M ε, we have ε 2 log ε 2 ˆη ε 2 M ε dx = log ε 2 ˆη ε i 2 M ε C dx log ε 2 x x i,ε 2 dx εµ ε) C log ε 2 i= Brε i \Bi ε M ε rε i= ε i= B i rε \Bi ε dr r M εlog r ε log ε) log ε 2 C. 4.39)

21 GEOMETRIC RIGIDITY FOR INCOMPATIBLE FIELDS AND STRAIN-GRADIENT PLASTICITY 2 Moreover, the L 2 -norm of ε 2 log ε 2 ˆη ε 2 dx ρε µ ε) ˆη ε ε log ε is concentrated in Mε i= B i ρε \ B i ε) ; in fact, similarly as above, C log ε 2 M ε rε i= ρ ε dr r M εlog r ε log ρ ε ) log ε 2 0, 4.40) as ε 0. Then since the measure of the set Mε i= B i ρε \ B i ε) tends to zero as ε 0, the two properties 4.39) and 4.40) entail a). Concerning b), we have that ε 2 log ε 2 K ε 2 dx = C log ε 2 M ε i= B i rε M ε 0, log ε 2 4π 2 rε 4 ξ i,ε 2 x x ε,i 2 dx as ε 0, which proves the claim. Finally, the convergence c) follows from the estimate directly implied by 4.38)) w ε ε log ε C µ µrε ε L 2 ;R 2 2 ) ε log ε H ;R 2 ) and from 4.36). For what follows it is convenient to notice that, by construction we also have R T ) β ε I β t 0 strongly in H /2 ). 4.4) ε log ε Now it remains to prove 4.32). Recalling that W I) = 0, we have E ε µ ε, β ε ) = ε 2 log ε 2 W β ε ) dx = ε 2 log ε 2 W β ε ) dx = ε 2 log ε 2 ε sµ ε) W β ε ) dx + εµ ε) ε 2 log ε 2 M ε i= B i ε s \B i ε W β ε ) dx =: I ε + I 2 ε, where s 0, ) is arbitrarily fixed. We are going to perform a linearisation for the term Iε around the identity matrix. Appealing to i), ii), and to the frame-indifference of W iii) we have I ε = = + ε 2 log ε 2 ε 2 log ε 2 2 ε 2 log ε 2 W I + ε log ε β + ˆη ε ε s ) ε log ε β + ˆη ε ε s ) C ε s ) σ where σf ) F 2 0, as F 0. ε log ε β + ˆη ε K ε + β ε ) K ε + β ) ε : ε log ε β + ˆη ε K ε + β ε ) dx K ε + β ε ) dx, 4.42)

22 22 S. MÜLLER, L. SCARDIA, AND C.I. ZEPPIERI We claim that lim sup ε 0 ε 2 log ε 2 2 Cβ : β dx + 2 ε s ) C ε log ε β + ˆη ε K ε + β ) ε : ε log ε β + ˆη ε K ε + β ε ) dx ϕr, ξ) dx, 4.43) where ϕ is defined in 4.4). We trivially have ε 2 log ε 2 C ε log ε β : ε log ε β dx 2 ε s ) 2 Cβ : β dx. Then we notice that the mixed products in the left-hand side of 4.43) converge to zero as ε 0 by a), b), and c), as well as the quadratic terms involving K ε and β ε. Moreover, ε 2 log ε 2 Cˆη ε : ˆη ε dx = 2 ε s ) ε 2 log ε 2 Cˆη ε : ˆη ε dx + o) 2 ε s ) \ ρε µ ε) M ε = log ε 2 Cˆη ε i : ˆη 2 εdx i + o), Bρε i \Bi ε s as ε 0 since we showed in 4.40) that the L 2 ˆη -norm of ε ε log ε is concentrated outside ρε µ ε ). By 4.3) we have that for i =,..., M ε Cˆη ε i : ˆη log ε 2 εdx i ψ ε R T ξ i,ε ) + o)), Bρε i \Bi ε s and this leads to ε 2 log ε 2 2 Moreover, by 4.33) and 4.35), lim ε 0 ε s ) Cˆη ε : ˆη ε dx log ε log ε M ε i= = i= M ε ψ ε R T ξ i,ε ) + o), as ε ) i= ψ ε R T ξ i,ε ) = lim ε 0 log ε M λ k ˆψR T ξ k ) = k= Thus, combining 4.44) and 4.45) gives lim ε 0 ε 2 log ε 2 Cˆη ε : ˆη ε dx 2 ε s ) M µ k ε ) ψ ε R T ξ k ) k= ϕr, ξ) dx. 4.45) ϕr, ξ) dx. We now prove that the remainder term in Iε tends to zero as ε 0. We notice that, if x ε sµ), then ˆη ε x) sup ε χ B i rε x) ˆη εx) i Cε s ; 4.46) i=,...,m ε

23 and GEOMETRIC RIGIDITY FOR INCOMPATIBLE FIELDS AND STRAIN-GRADIENT PLASTICITY 23 K ε x) sup i=,...,m ε χ Brε i x) ε 2πrε 2 ξ i,ε Jx x i,ε ) C ε. 4.47) r ε Moreover, in view of the regularity of, by standard regularity theory for elliptic partial differential equations, the following estimate for w ε holds true for every < p < see, e.g. [4, Lemma 9.7]): w ε W 2,p ;R 2 ) C ε log ε µ µ rε ε L p ;R 2 ), 4.48) where the constant C depends on p and. Notice also that ε log ε µ µ rε ε L p ;R 2 ) Cε log ε, for every < p <. By the Sobolev Imbedding Theorem the estimate above together with 4.48) imply w ε C,α ;R 2 ) Cε log ε, for every α 0, ). Therefore, in particular, w ε L ;R 2 2 ) Cε log ε. 4.49) Then 4.46), 4.47), 4.49), and the boundedness of β entail ε log ε β + ˆη ε K ε + β ε C ε log ε + ε s) + ε ) + ε log ε r ε Hence, setting χ ε := χ ε s ) and ωt) := sup F t σf ), we have that σ ε log ε β + ˆη ε K ε + εs) β ) ε ε 2 log ε 2 dx ω ε log ε β + ˆη ε K ε + β ) ε ε log ε β + ˆηε K ε + β ε 2 χ ε ε log ε β + ˆη ε K ε + β ε 2 ε 2 log ε 2 in ε sµ ε ). 4.50) and the limit for ε 0 of the expression above is zero as the integrand is the product of a sequence converging to zero uniformly by 4.50)) and a bounded sequence in L ). Therefore, combining this fact with 4.42) and 4.43), we get dx, lim sup Iε Eµ, β, R). 4.5) ε 0 Finally, we deal with Iε 2. Using the quadratic upper bound on W iv) we deduce Iε 2 M ε ε 2 log ε 2 C ε log ε β + ˆη ε K ε + β 2 ε dx. 4.52) i= B i ε s \B i ε Similarly as before, due to a), b), and c) the mixed products in 4.52) converge to zero, as well as the quadratic terms involving K ε and β ε. In addition, M ε β 2 dx β L ;R 2 2 ) M ε ε 2s ε 2) 0, as ε 0 i= B i ε s \B i ε

24 24 S. MÜLLER, L. SCARDIA, AND C.I. ZEPPIERI and log ε 2 M ε i= B i ε s \B i ε ˆη i ε 2 dx C M ε s ε log ε 2 dr C s); ε r therefore lim sup Iε 2 C s). 4.53) ε 0 Then, gathering 4.5) and 4.53) leads to the final estimate lim sup E ε µ ε, β ε ) Eµ, β, R) + C s), ε 0 which entails the claim 4.32), by the arbitrariness of s 0, ). Step 2. µ = L l= ξl dx l, where ξ l R 2 and l are Lipschitz pair-wise disjoint domains such that = L l= l and \ L l= l = 0. For every l =,..., L, let A l,k ) k N be an increasing sequence of regular domains such that A l,k l and l \ A l,k k. For fixed k N and for every l =,..., L, we can argue as in Step to construct µ l,k ε and, recovery sequences for the energy in A l,k, relative to µ l,k := µ A l,k and β l,k := β A l,k, β l,k ε respectively. Then, we define the sequences µ k ε, β k ε ) in the whole domain as µ k ε := L l= µ l,k ε χ A l,k, βk ε := L l= β l,k ε χ A l,k + R χ \A k. where A k := L l= Al,k. Notice that by construction µ k ε, β k ε ) converges in to µ k, β k ) := µχ A k, βχ A k + Rχ \A k) in the sense of Definition 4.5. Moreover, µ k ε is admissible in while, in general, the sequence β ε k does not belong to AS ε µ k ε) since Curl β ε k 0 in ε µ k ε). In fact, Curl β ε k may have a contribution concentrated on A k. ) On the other hand, since by 4.4), R T βε l,k I ε log ε β l,k t 0 strongly in H /2 A l,k ) for every l =,..., L, we may deduce that Curl β ε k ε µ k ε) L R T βε l,k I β l,k 0, 4.54) ε log ε ε log ε H /2 A l,k ) H ;R 2 ) l= as ε tends to zero. Then, invoking the regularity of and 4.54) we can argue as in Step and add to β ε k a suitably chosen vanishing sequence, so that the resulting sequence βε k satisfies Curl βε k ε µ k ε) = 0 and it is still a recovery sequence for µ k, β k, R). Hence in particular we have lim sup E ε µ k ε, βε k ) ) Cβ k : β k dx + ϕ R, dµk ε 0 2 d µ k d µ k. Moreover, taking into account that, for k +, β k converges to β strongly in L 2 ; R 2 2 ) and µ k converges to µ strongly in measure, we get lim sup k + lim sup E ε µ k ε, βε k ) Eµ, β, R). ε 0

25 GEOMETRIC RIGIDITY FOR INCOMPATIBLE FIELDS AND STRAIN-GRADIENT PLASTICITY 25 Then, invoking a standard diagonalization argument we can find an increasing sequence k ε ) N, with k ε + as ε 0, such that lim sup E ε µ kε ε, βε kε ) Eµ, β, R). ε 0 Finally, it is easy to check that µ ε, β ε ) := µ kε ε, β kε ε ) is the desired recovery sequence. Step 3. µ W, ; R 2 ). We can argue as in [, Theorem 2] Step 3 to reduce to the case of locally constant measures, namely to Step 2 in this proof. Acknowledgements The authors thank Sergio Conti and Marcello Ponsiglione for interesting discussions. C.Z. wishes to thank the Institute for Applied Mathematics of the University of Bonn, where most of this work was carried out. This work was partially supported by the Deutsche Forschungsgemeinschaft through the Forschergruppe 797 and the Sonderforschungsbereich 6. References [] Bacon D.J., Barnett D.M., and Scattergood R.O.: Anisotropic continuum theory of lattice defects. Progress in Materials Science, ), [2] Bourgain J., Brezis H.: New estimates for elliptic equations and Hodge-type systems. J. Eur. Math. Soc., ), [3] Brezis H., van Schaftingen J.: Boundary estimates for elliptic systems with L -data. Calc. Var., ), [4] Cermelli P., Leoni G.: Renormalized energy and forces on dislocations. SIAM J. Math. Anal., 37/4 2005), [5] Conti S., Dolzmann G., Müller S.: Korn s second inequality and geometric rigidity with mixed growth conditions. Calc. Var. Partial Differential Equations, to appear. [6] Conti S., Garroni A., Müller S.: Singular kernels, multiscale decomposition of microstructure, and dislocation models. Arch. Ration. Mech. Anal., 99 20), no. 3, [7] Dal Maso G., Negri M., Percivale D.: Linearized elasticity as Γ-limit of finite elasticity. Set-Valued Anal., 02002), no. 2-3, [8] Dautray R., Lions J.-L.: Mathematical Analysis and Numerical Methods for Sciences and Technology. Vol. 3, Springer, Berlin, 988. [9] De Luca L., Garroni A., Ponsiglione M.: Γ-convergence analysis of systems of edge dislocations: the self energy regime. Arch. Ration. Mech. Anal., ), no. 3, [0] Friesecke G., James R.D., Müller S.: A theorem on geometric rigidity and the derivation of nonlinear plate theory from three-dimensional elasticity. Comm. Pure Appl. Math., ), [] Garroni A., Leoni G., Ponsiglione M.: Gradient theory for plasticity via homogenization of discrete dislocations. J. Eur. Math. Soc., 2 200), [2] Garroni A., Müller S.: Γ-limit of a phase-field model of dislocations. SIAM J. Math. Anal ), [3] Garroni A., Müller S.: A variational model for dislocations in the line tension limit. Arch. Ration. Mech. Anal ), [4] Gilbarg D., Trudinger N.S.: Elliptic Partial Differential Equations. Springer-Verlag, Berlin, 998. [5] Groma I.: Link between the microscopic and mesoscopic length-scale description of the collective behavior of dislocations. Phys. Rev. B, 56/0 997),