On the existence of minimisers for strain-gradient single-crystal plasticity
aa r X i v : . [ m a t h . A P ] J a n ON THE EXISTENCE OF MINIMISERS FOR STRAIN-GRADIENTSINGLE-CRYSTAL PLASTICITY
KEITH ANGUIGE, PATRICK DONDL AND MARTIN KRUˇZ´IK
Abstract.
We prove the existence of minimisers for a family of models related to the single-slip-to-single-plane relaxation of single-crystal, strain-gradient elastoplasticity with L p -hardeningpenalty. In these relaxed models, where only one slip-plane normal can be activated at each ma-terial point, the main challenge is to show that the energy of geometrically necessary dislocationsis lower-semicontinuous along bounded-energy sequences which satisfy the single-plane condition,meaning precisely that this side condition should be preserved in the weak L p -limit. This is donewith the aid of an ‘exclusion’ lemma of Conti & Ortiz, which essentially allows one to put a lowerbound on the dislocation energy at interfaces of (single-plane) slip patches, thus precluding finephase-mixing in the limit. Furthermore, using div-curl techniques in the spirit of Mielke & M¨uller,we are able to show that the usual multiplicative decomposition of the deformation gradient intoplastic and elastic parts interacts with weak convergence and the single-plane constraint in such away as to guarantee lower-semicontinuity of the (polyconvex) elastic energy, and hence the totalelasto-plastic energy, given sufficient ( p >
2) hardening, thus delivering the desired result. Introduction and main results
Plastic deformation in crystals has long been known to be mediated by the motion of crystalimperfections, or dislocations , through the material. Each such dislocation travels predominantlyon a given crystallographic plane, a so-called slip-plane , in the direction of a fixed
Burgers’vector , where the Burgers’ vectors are also determined by the crytallographic structure. As aconsequence of this, the plastic strain can be thought of as the product of a number of simple-shear deformations, each with a given crystallographically determined shear normal and sheardirection.In numerous experiments, lamination-type microstructures with alternating layers of slip-system activity have been observed [35, 24, 5, 12], and this effect, which will be a central con-sideration in what follows, is widely believed to be a consequence of cross-hardening [34, 12],the phenomenon whereby activity in one slip system at a given point suppresses activity in allother slip systems at that point. A microscopic explanation for the effect is that it arises from theformation of energetically favorable (and sessile ) dislocation products when two dislocations fromdifferent slip-planes meet – these
Lomer-Cottrell locks have been observed in experiments [28],and also studied in detail in atomistic simulations [37]. The idea is that, in order to continuethe plastic deformation with activity in more than one slip-plane, either these locks have to bebroken or new dislocation loops must be formed, thus necessitating an increased energy input forplastic deformations involving activity in multiple slip-systems.In their seminal article [34], Ortiz and Repetto proposed a method for reproducing experi-mentally observed sub-grain pattern formation in plastically deforming crystals. The basic idea,which we will follow in this paper, is to model plastic evolution by an incremental time-steppingprocedure, where in each step the sum of an elastic energy and a stored plastic energy is min-imised. They then propose a non-convex stored energy in order to account for cross-hardening,which leads to the formation of microstructure.
Mathematics Subject Classification.
Key words and phrases.
Existence of minimizers, single-crystal plasticity, cross-hardening, geometrically neces-sary dislocations, strain-gradient.
As in [1, 2], we will introduce the aforementioned non-convexity generated by cross-hardeninginto this framework in the simplest possible way, namely by enforcing a hard single-slip condition,such that at each material point any finite-energy plastic deformation has to occur in single slip.We then relax this single-slip condition to a single-plane condition (namely that only one slip-planecan be active at each point) with the aid of a laminated microstructure – similar side-conditionshave been considered by several other authors; see, for example [13, 14]. In [12] it was shown thatthe predicted laminate microstructure arising from this assumption of infinite cross hardeningdoes indeed match experimental results, while evolutionary models of such laminate structureshave been analysed in [16, 22].Here, we continue our previous investigations [1, 2], which focused on optimal energy scalingsand relaxation of the single-slip condition to a (still non-convex) single-plane condition, by lookingat the existence question for a class of incremental minimisation problems which arise in the waydescribed above. Our main goal is to extend the existence result of Mielke and M¨uller [31] forfinite, multiplicative strain-gradient elasto-plasticity to the single-crystal, single-plane case. Itturns out that the single-plane restriction on the plastic slip allows one nicely to control theinverse of the plastic deformation, and, with the aid of a sufficiently strong div-curl lemma, theweak continuity of the minors of the elastic deformation – as a consequence, this allows for moregeneral (and realistic) material parameters than in [31]. On the other hand, proving that the non-convex single-plane constraint is preserved along energy-minimising sequences, which is the keylower-semicontinuity property, requires considerable additional analysis. In particular, we haveto adapt an exclusion Lemma of [8], originally designed for a 2-d problem, to our 3-d setting insuch a way as to exclude any further phase mixing in our ostensibly relaxed single-plane models.The family of single-plane incremental energies treated here is related to the single-plane re-laxation of the single-slip model discussed in [1, 2], and contains a strain-gradient penalisationof geometrically necessary dislocations, along with an L p -stored-energy term. We examine thisrelationship later on, and, while no explicit formula for the relaxation seems to be available ingeneral, we are nevertheless able to write down matching upper and lower bounds for the re-laxed single-plane energy functional which ensure the applicability of our main result, namely thefollowing. Theorem 1.
Let Ω ⊂ R be a bounded, Lipschitz domain, and let p > . Let m j , for j = 1 , . . . N ,be a family of slip normals, and s j ∈ m ⊥ j the corresponding slip vectors. Further suppose that W el is a polyconvex, frame-indifferent elastic-energy density satisfying the growth condition W el ( A ) ≥ − c + c | A | q with q > p/ ( p − , and that the single-plane plastic energy, E pl , is w.l.s.c. in ( L p (Ω)) N , and satisfies E pl (cid:0) { s j } Nj =1 (cid:1) ≥ C N X j =1 k s j k pL p (Ω) + N X j =1 Z Ω (cid:12)(cid:12)(cid:12) ∇ m ⊥ j s j (cid:12)(cid:12)(cid:12) , for some constant C > .Then, writing E el (cid:16) y, { s j } Nj =1 (cid:17) := R Ω W el (cid:16) ∇ y F − (cid:17) dx , where the plastic-deformation tensoris given by F pl = Id + P Nj =1 s j ⊗ m j and y ( x ) is the deformation, the single-plane elastoplasticenergy E (cid:0) y, { s j } Nj =1 (cid:1) = E el (cid:16) y, { s j } Nj =1 (cid:17) + E pl (cid:16) { s j } Nj =1 (cid:17) : | s j | | s i | = 0 a.e., for i = j + ∞ : otherwise , admits a minimiser (cid:16) y ∗ , { s ∗ j } Nj =1 (cid:17) in the class ( W ,r (Ω)) × ( L p (Ω)) N , where r = p + q . N THE EXISTENCE OF MINIMISERS FOR STRAIN-GRADIENT SINGLE-CRYSTAL PLASTICITY 3
Remark . We recall that W el is said to be polyconvex [4] if it can be written as a convexfunction of the deformation gradient, its cofactor matrix, and its determinant, i.e., W el ( F ) = h ( F, Cof F, det F ) for all F and some convex function h . Moreover, frame-indifference means that W el ( F ) = W el ( RF ) for all F ∈ R × and every proper rotation R . Strictly speaking, we oughtto impose additional non-interpenetration conditions of the form W el ( F ) → + ∞ as det F → + ,and W el ( F ) = + ∞ if det F ≤
0. However, these conditions will play no particular role in ouranalysis, and we therefore prefer to overlook them here.
Remark . The non-convex side condition on the plastic slip states that, at almost every point,at most one slip-plane normal can be active, while the lower-bound on E pl ensures that E issufficiently coercive to ensure the weak-closedness of the single-plane condition along minimisingsequences. Remark . Since s j ∈ m ⊥ j , we have F − ( x ) = Id − s j ( x ) ⊗ m j ( x ) almost everywhere for finite-energytest functions, which will simplify the analysis of minors of the elastic deformation in the sequel. Remark . The quantities (cid:12)(cid:12)(cid:12) ∇ m ⊥ j s j (cid:12)(cid:12)(cid:12) appearing in the lower bound on E pl are to be interpreted asmeasures.We also prove the following analogue of the above result for linearised elasticity, with a weakerrequirement on the hardening. Theorem 2.
Let Ω ⊂ R be a bounded, Lipschitz domain, and let p > . Let m j , for j = 1 , . . . N ,be a family of slip normals, and s j ∈ m ⊥ j the corresponding slip vectors. Further suppose that thesingle-plane plastic energy, E pl , is w.l.s.c. in ( L p (Ω)) N , and satisfies E pl (cid:0) { s j } Nj =1 (cid:1) ≥ C N X j =1 k s j k pL p (Ω) + N X j =1 Z Ω (cid:12)(cid:12)(cid:12) ∇ m ⊥ j s j (cid:12)(cid:12)(cid:12) , for some constant C > .Then, assuming a linearised elastic energy of the form E el (cid:0) u, { s j } Nj =1 (cid:1) := Z Ω | ( ∇ u − β ) sym | dx, where the plastic-distortion tensor is given by β = P Nj =1 s j ⊗ m j and u is the displacement, thesingle-plane elastoplastic energy E lin (cid:0) u, { s j } Nj =1 (cid:1) = E el (cid:16) u, { s j } Nj =1 (cid:17) + E pl (cid:16) { s j } Nj =1 (cid:17) : | s j | | s i | = 0 a.e., for i = j + ∞ : otherwise , admits a minimiser (cid:16) u ∗ , { s ∗ j } Nj =1 (cid:17) in the class ( W ,p (Ω)) × ( L p (Ω)) N .Remark . Unfortunately, one cannot expect an analogous existence result for p = 1, correspond-ing to rate-independent dissipation, even in the case of linearised elasticity, due to the possibilityof slip concentration (formation of singular measures) along minimising sequences, and the dif-ficulty of reconciling this with an intuitive interpretation of the single-plane side condition formeasures – see Example 2 at the end of the paper.The force of our existence results is that one should always seek to relax a non-convex single-slip condition in single-crystal strain-gradient plasticity to a single-plane condition. In this way,one obtains a well-posed model which, in particular, is not plagued by the kind of fine oscillationswhich have often been observed in simulations of single-slip models. It is also worth emphasisingthat both the single-plane condition and the regularising penalisation of geometrically necessarydislocations which appear in the definition of E , acting in concert, are essential ingredients inproving these results. KEITH ANGUIGE AND MARTIN KRUˇZ´IK
The article is organised as follows. In Section 2, we introduce a non-convex (single-slip) modelfor single-crystal, strain-gradient plasticity, including a very particular penalisation of geometri-cally necessary dislocations which prevents the cancellation of dislocations at collinear-slip-patchinterfaces. In Section 3, we discuss how to relax the single-slip condition (the source of thenon-convexity) in our model to a single-plane condition, resulting in a family of models to whichTheorem 1 or 2 applies. The mathematical heart of the paper is Section 4, in which the proof ofTheorems 1 and 2 is to be found – this consists of several lemmas which take care not only of thelower semi-continuity of the plastic and elastic parts of the single-plane energy, but also of thepreservation of the single-plane condition along minimising sequences.2.
A model for strain-gradient plasticity with cross hardening
We now introduce, term-by-term, the elements of our continuum crystal-plasticity model. Mod-ulo a very specific choice for the penalisation of geometrically necessary dislocations, which isessential for handling the non-convex slip conditions, the model ingredients are standard fare inthe continuum-plasticity literature. Note that a longer version of this discussion appeared in thereview [3].2.1.
Plastic deformation.
We consider an elasto-plastic body with its reference configurationΩ ⊂ R and a sufficiently smooth deformation y : Ω → R , satisfying suitable boundary conditions – for example, Dirichlet conditions on a subset of ∂ Ω. Wethen define the deformation gradient F = ∇ y , noting that its row-wise curl necessarily vanishes.Now we make the assumption that this deformation gradient can be decomposed into a prod-uct of plastic shears due to an atomistic rearrangement, F pl , which generates an intermediateconfiguration, followed by an elastic deformation, F el . We remark that the validity of this mul-tiplicative decomposition is still a matter for debate – see [9, 36] for two recent contributions tothe discussion.With this Lee-Liu decomposition [29] in hand, we can thus identify an elastic energy for ourcrystalline specimen depending only on the elastic strain, F el = F F − . A further assumption isthat the plastic energy, in the sense of the implicit time discretisation (see above), can be writtenas a function of the incremental change in plastic strain. Thus, restricting ourselves to the firstsuch time-step, we see that the incremental deformation requires an energy input of the form Z Ω W el ( F F − ) d x + E pl ( F pl ) , for a suitable frame-indifferent elastic energy density W el , and a plastic energy E pl . This plasticenergy contains a p -hardening term (or dissipation, in the case p = 1) that penalises the L p -normof the plastic shear undergone by the crystal. In the following E pl will furthermore be allowed todepend, amongst other things, on a (possibly singular) measure-valued functional of F pl whichtakes account of geometrically necessary dislocations.2.2. Cross-hardening.
Cross-hardening (or latent hardening) [38, 25, 15] describes the phenom-enon whereby shear in one slip system suppresses activity in other slip systems at the same pointin the crystal. This leads to a loss of convexity in the plastic energy E pl introduced above [34]– roughly speaking, E pl ( F pl ) will be locally minimal if F pl is a simple shear in one of the givenslip systems of the crystal, and, for general boundary constraints, one will have to dip into morethan one of these local energy wells to minimise the energy globally.Here, we will make the simplifying assumption of infinite cross hardening, meaning that F pl is required to be in single slip at each point. In line with this, it is thus assumed that the N THE EXISTENCE OF MINIMISERS FOR STRAIN-GRADIENT SINGLE-CRYSTAL PLASTICITY 5 crystallographic structure admits a set of slip-plane normals M = { m j } Nj =1 , each with a givenset of Burgers’ vectors B j = { b ij } K ( j ) i =1 , and that F pl takes the form. F pl = Id + N X j =1 K ( j ) X i =1 c ij m j ⊗ b ij , (1)subject to the following Single-Slip Condition (SSC) : c ij ( x ) c kl ( x ) = 0 for a . e . x ∈ Ω , if i = k or j = l. (2)Note that, under this condition, the product of simple shears assumed above simplifies imme-diately, such that there is at most one non-zero factor at almost every point, thus justifying therepresentation as a sum and still ensuring that det F pl = 1 almost everywhere. Furthermore, theplastic hardening can be written in terms of the slip coefficients c ij .2.3. Geometrically necessary dislocations.
A strain-gradient penalty term is sometimes in-cluded in models of crystal plasticity, since an argument can be made that the surface where twodifferently sheared subdomains meet admits a density of geometrically necessary dislocations.In [7] it is proposed that the correct term for this density of geometrically necessary dislocationsmust be 1det F pl (Curl F pl ) F Tpl , (3)(see also [31] for a brief discussion of this matter). Here, the expression Curl denotes the row-wisecurl of a matrix.Considering the fact that our single-slip side condition yields a very specific form of F pl , it iseasy to see that both the volumetric term and multiplication with F T pl in (3) are equal to identity.The GND-density therefore reduces to the simpler formCurl F pl . (4)The expression above, however, does not account for sessile dislocations at boundaries betweenabutting subdomains deformed in collinear slip due to cancellation of dislocations with oppositesign. This is in disagreement with simulations by Devincre et. al. [10, 11] who observe that thesecancellations are in practice not complete, such that a density of dislocations remains on thesurface between the subdomains – for a discussion of this matter in a simplified scalar model,see [8, Chapter 4]. In order to exclude such collinear cancellations, we thus introduce a non-standard (possibly singular) measure for the dislocation density, which, defining the single-planeslips s j = P K ( j ) i =1 c ij b ij , concretely takes the form G (cid:0) { s j } Nj =1 (cid:1) = N X j =1 |∇ m ⊥ j s j | , (5)i.e., for the j -th slip normal, we take the length of the planar-gradient vector (gradient orthogonalto the respective m j ) of the plastic slip s j , regardless of any activity in the other slip planes, andthen sum over j .2.4. The model.
To summarise, the geometrically nonlinear elasto-plastic energy we use tomodel the phenomena above in the case of L p -hardening, is taken to be E p ( y, { c ij } ) = R Ω W el ( F el ) d x + σ R Ω G ( { s j } ) + τ P K ( j ) i =1 P Nj =1 R Ω | c ij | p d x : if (SSC) holds , + ∞ : otherwise , (6) KEITH ANGUIGE AND MARTIN KRUˇZ´IK where W el satisfies the growth, convexity and indifference conditions appearing in Theorem 1,while the corresponding geometrically linear version of the model, with β = P Nj =1 s j ⊗ m j , is E p lin ( y, { c ij } ) = R Ω | ( ∇ y − β ) sym | d x + σ R Ω G ( { s j } )+ τ P K ( j ) i =1 P Nj =1 R Ω | c ij | p d x : if (SSC) holds , + ∞ : otherwise , (7)As we showed in [1], neither of these energy functionals (or indeed their analogues with, in-stead, a finite hardening matrix) are weakly lower-semicontinuous, due to the possibility of fineoscillations between multiple Burgers vectors with the same slip normal, and this leads to thesearch for a relaxed functional and a corresponding existence theorem.2.5. Unique decomposition of F pl into single-plane slips. Under certain conditions on thefamily of slip-normals m j , there is in fact a one-to-one correspondence between F pl = Id + P Nj =1 s j ⊗ m j and the slips s j . Thus, suppose N ∈ { , , , } , and let M = { m j } Nj =1 be acollection of slip-plane normals with the property that any collection of three or fewer vectors in M is linearly independent .For each j ∈ { , . . . , N } , denote by B j the space of matrices spanned by { s ⊗ m j : s ⊥ m j } .The following Proposition shows that the decomposition of any traceless matrix into a linearcombination of active slips, i.e. elements of B j , is unique. Proposition 3.
Suppose that β = P Nj =1 s j ⊗ m j , where s j ∈ R , s j ⊥ m j and m j ∈ M . Then,if the independence-condition on M given above holds, this slip-plane decomposition (i.e. thedetermination of the s j ) is unique.Proof. For
N <
4, the result holds trivially. Suppose, therefore, that N = 4, and that the claimis false. Then there exist s j ( ⊥ m j ), for j = 1 , . . . ,
4, not all zero, such that X j =1 s j ⊗ m j = 0 . (8)If we now take the scalar product of (8) with some m k , where k ∈ { , . . . , } , on the first factorin the dyadic product, we get 0 = X j =1 ( s j · m k ) m j = X j = k ( s j · m k ) m j . By the linear independence of any three m j , we have s j · m k = 0 for all j = k . Since k wasarbitrary, we have s j = 0 for j = 1 , . . . ,
4, a contradiction. (cid:3) Single-plane relaxation of the single-slip condition
We now investigate the single-slip-to-single-plane relaxation of (6). In other words, instead oftaking the single-slip condition : F pl ( x ) = Id + s ( x ) ⊗ m ( x ) , (9) Note that this condition is satisfied by the four slip planes of the f.c.c. crystal structure, as well as by thelow-temperature slip modes (on the basal and prismatic planes) of h.c.p. crystals.
N THE EXISTENCE OF MINIMISERS FOR STRAIN-GRADIENT SINGLE-CRYSTAL PLASTICITY 7 with m ( x ) = m j ( x ) ∈ M and s ( x ) ∈ K ( j ( x )) [ i =1 Span { b ij ( x ) } (10)almost everywhere, we simply enforce the single-plane condition , also referred to as the re-laxed slip condition (RSC) , which says that (9) holds subject to m ( x ) = m j ( x ) ∈ M and s ( x ) ∈ Span K ( j ( x )) [ i =1 { b ij ( x ) } , (11)and look for an optimal (with respect to the energy (6) or (7)) single-slip approximation to agiven single-plane test function.Suppose, then, that we have a displacement u ∈ W ,q on Ω ⊂ R , q ≥
1, satisfying a Dirichletcondition on a part of ∂ Ω, and a relaxed plastic strain F pl satisfying (9) and (11), and for eachslip-plane normal, m j , suppose we make a fixed choice of two admissible Burgers vectors, b j , b j ,such that F pl = N X j =1 s j ⊗ m j and s j = X i =1 c ij b ij . (12)With such a choice in hand, we aim to identify the single-plane relaxation of the single-slipenergy (6) by weakly approximating F pl with single-slip laminates, such that the slip directionalternates between b j and b j as we pass from slice to slice. In the case of L -hardening, thecorrect single-plane energy was already calculated explicitly in [1], modulo a technical issue re-lating to the smoothness of F pl , while for hardening with p > L -hardening. Suppose p = 1. Then, for any admissible selection of Burgers vectors asabove, the following functional is, morally, the relaxation of the energy E : E (cid:0) y, { s j } Nj =1 (cid:1) = R Ω W el ( F el ) d x + σ R Ω G lam (cid:16) { s j } Nj =1 (cid:17) + τ R Ω | F pl | lam d x : (RSC) holds , + ∞ : otherwise , (13)where the laminated curl is defined by G lam (cid:0) { s j } Nj =1 (cid:1) = N X j =1 2 X i =1 |∇ m ⊥ j c ij | , (14)and the laminated hardening by | F pl | lam = N X j =1 2 X i =1 | c ij | . (15)The justification for taking (13) as our expression for the relaxed energy is, as we showed in[1], that smooth, relaxed slips s j can be approximated by laminated single-slips s nj , such that Z Ω G ( { s nj } Nj =1 ) → Z Ω G lam ( { s j } Nj =1 ) , (16)and N X j =1 2 X i =1 Z Ω | c nij | d x → Z Ω | F pl | lam d x, (17)as n → ∞ , and, moreover, that W el behaves continuously under lamination of F pl .In particular, our main theorem from [1] is KEITH ANGUIGE AND MARTIN KRUˇZ´IK
Theorem 4.
Suppose that W el : M × [0 , ∞ ) is continuous and satisfies the q -growth condition − k + k | F | q ≤ W el ( F ) ≤ K + K | F | q , (18) for some positive constants k , k , K , K and q ≥ . Suppose, furthermore, that we have aLipschitz domain Ω ⊂ R and a test function ( u, { s j } Nj =1 ) defined on Ω , such that u ∈ W ,q satisfies a Dirichlet condition on a Lipschitz subset of ∂ Ω , F pl satisfies (RSC) with the j -thslip normal active only on Ω j ⊂ Ω , and the relaxed energy (13) is finite. Assume that the sets { Ω j } Nj =1 , on which F pl = Id + s ⊗ m j , satisfy the regularity condition H ( ∂ Ω j \ F Ω j ) = 0 .Then, for each ε > , there exists a pair of test functions ( u ε , s ε ) satisfying the same Dirichletcondition and (SSC) , such that u ε ⇀ u ∈ W , , s ε ⇀ s ∈ L and E ( u ε , s ε ) ≤ E ( u, s ) + ε. (19) Remark . The same approximation result also holds for linearised elasticity
Remark . In cases where Proposition 3 applies, we could equally well write E as a function of F pl or of the c j , c j .Unfortunately, despite this nice characterisation of the l.s.c. envelope, an existence result for(13) remains elusive, since, in particular, L -control of the plastic slip is not sufficient to enforceweak-continuity of determinants of the elastic deformation in the case of nonlinear elasticity.Moreover, even for linearised elasticity, there is a problem with the relaxed side condition (RSC) for p = 1, since one does not have enough coercivity to prevent slip concentration along minimisingsequences (see Example 2). In order to force the existence of minimisers, one can, however, addto E a small, admittedly somewhat ad hoc , penalty which is bounded below by ǫ P Ni =1 k s i k pp (with p > ǫ >
0) to E , such that the resulting energy satisfies the conditions of Theorem1. For convenience, we now state this result separately. Theorem 5.
Let Ω ⊂ R be a bounded, Lipschitz domain, and suppose p > , ǫ > . Let m j ,for j = 1 , . . . N , be a family of slip normals, and s j ∈ m ⊥ j the corresponding slip vectors. Furthersuppose that W el is a polyconvex, frame-indifferent elastic-energy density satisfying the growthcondition W el ( A ) ≥ − c + c | A | q with q > p/ ( p − , and that we are given a functional F ǫ ( { s j } Nj =1 ) which is w.l.s.c. in ( L p ) N and bounded belowaccording to F ǫ ( { s j } Nj =1 ) ≥ ǫ P Ni =1 k s j k pp , for some ǫ > .Then, writing E el (cid:16) y, { s j } Nj =1 (cid:17) := R Ω W el (cid:16) ∇ y F − (cid:17) d x , where the plastic-deformation tensoris given by F pl = Id + P Nj =1 s j ⊗ m j , the regularised single-plane elastoplastic energy with L -hardening, E (cid:0) y, { s j } Nj =1 (cid:1) = E (cid:16) y, { s j } Nj =1 (cid:17) + F ǫ : | s j | | s i | = 0 a.e., for i = j + ∞ : otherwise , admits a minimiser (cid:16) y ∗ , { s ∗ j } Nj =1 (cid:17) in the class ( W ,r (Ω)) × ( L p (Ω)) N , where r = p + q .Remark . Proving Theorem 5 is equivalent to proving Theorem 1, since G lam ( { s j } ) is semi-normequivalent to G ( { s j } ), and | F pl | lam is norm-equivalent to | F pl | , for fixed b j , b j , as we showed in([2], Prop. 2.1). Remark . Geometrically linear elasticity also works here – simply replace W ,r with W ,p in thestatement of the result. N THE EXISTENCE OF MINIMISERS FOR STRAIN-GRADIENT SINGLE-CRYSTAL PLASTICITY 9 L p -hardening, p > . We now derive upper and lower bounds for the l.s.c. envelopeof the single-slip energy, E p , such that the upper bound is obtained by approximating a givensingle-plane test funtion with single-slip laminates. For the lamination procedure we needn’t payattention to the elastic energy, since it is appropriately continuous with respect to the single-slipapproximation which we use – see the proof of Theorem 5.1 in [1].3.2.1. Lower bound.
First of all, the functional E lb ( y, { s j } ) := R Ω W el ( F el ) d x + σ R Ω G lam ( { s j } ) + τ P Nj =1 R Ω ( | c j | + | c j | ) p d x : (RSC) holds , + ∞ : otherwise(20)is a good lower bound on the l.s.c. envelope of E , in the sense that • It agrees with E p on the set of smooth single-slip test functions. • It is W ,r × L p -weakly l.s.c on the set of single-plane test functions, due to the convexityof the hardening term and the definition of the laminated curl as a total variation, andalso the fact that Lemma 7 (below) takes care of the weak continuity of the minors of F el . • (RSC) is preserved along weakly converging bounded-energy sequences – see Lemma 8. Remark . Note that the two plastic contributions to this lower bound can be arrived at byseparately optimising the hardening energy (pointwise), and then the dislocation energy via ap-proximation of a single-plane test function with single-slip laminates in two different ways: ingeneral, one cannot reach this lower bound by optimising the total plastic energy with suchlaminates in one fell swoop, in contrast to the L -case – see below.3.2.2. Upper bounds.
Our first (and coarsest) upper bound can be obtained by weakly approxi-mating a given single-plane slip, β = P i,j c ij b ij ⊗ m j , with alternating flat slices of single-slip inthe b j and b j -directions, as we did in [1] for the p = 1 case. Here, however, the relative thickness(weight) of the alternate slices in a bi-layer is allowed to vary from one bi-layer to the next – inthe L -case the laminated plastic energy is indifferent to the weighting, due to the 1-homogeneityof the hardening and the curl, but for p > β by filling each Ω j with a stack of bi-layers, each parallel to m ⊥ j , and having thickness n , n ∈ N , and then definingon each successive bi-layer an alternating (as we move in the m j -direction), single-slip β jn , by β jn = λ − j c j b j ⊗ m j : top slice (cid:16) of thickness λ j n (cid:17) (1 − λ j ) − c j b j ⊗ m j : bottom slice (cid:16) of thickness (1 − λ j )2 n (cid:17) , (21)where the c ij are evaluated on the dividing-plane of the bi-layer in (21), and, for definiteness, the( n + 1)-th laminate is obtained from the n -th by bisecting each of the bi-layers along a slip plane.The scaling with λ j ( t j ) ∈ (0 ,
1) (resp. (1 − λ j ( t j ))) guarantess that β jn ⇀ β j in L p (Ω j ) as n → ∞ .Here, t j represents a coordinate running in the m j -direction, and the weighting function λ j ( · ) isassumed continuous.With this definition, we get, with abuse of notation E p ( y n , β n ) → Z Ω W el ( F el ( y, β )) d x + τ N X j =1 Z Ω j | c j | p λ p − j + | c j | p (1 − λ j ) p − ! d x + σ Z Ω G lam ( F pl ( β )) , (22)as n → ∞ , for an appropriate zig-zag perturbation, y n , of y which accommodates the lamination– see Theorem 5.1 of [1] for details, and for the convergence of the elastic energy under such perturbations. Note also that the limiting curl of β n is independent of λ ( · ) (where λ | Ω j = λ j ), byhomogeneity, and that this limit is just the laminated curl that appears in the p = 1 problem.Next, elementary calculus allows us to optimise each λ j ( t j ) in (22), which results in λ opt j ( t j ) = k c j k L p (cid:16) Ω tjj (cid:17) k c j k L p (cid:16) Ω tjj (cid:17) + k c j k L p (cid:16) Ω tjj (cid:17) , where Ω t j j is a 2-d slice through Ω j at the level t j .Thus, inserting these optimal λ j into the right-hand side of (22) shows that E (1)ub ( y, { s j } ) := R Ω W el ( F el ( y, { s j } ) d x + τ P Nj =1 R (cid:18) k c j k L p (cid:16) Ω tjj (cid:17) + k c j k L p (cid:16) Ω tjj (cid:17) (cid:19) p dt j + σ R Ω G lam ( { s j } ) : (RSC) holds , + ∞ : otherwise (23)is an upper bound for the sought-after relaxation. Remark . This upper bound also holds for test functions which satisfy just the mild slip-patch-regularity condition of Theorem 5.1 in [1], since we showed there that one can mollifya single-plane β strongly continuously in L p , such that the laminated curl essentially does notincrease. Remark . If c j ∝ c j globally on Ω j , for each j = 1 , . . . N , then E lb ( y, { s j } ) = E (1)ub ( y, { s j } ), byCauchy-Schwarz, so we know precisely what the relaxation looks like for such nice test functions(provided they are reasonably smooth). Remark . In uniform-shear experiments of the type described in [1], one expects minimisingslips to point in the direction of the overall shear everywhere (and uniqueness of minimisers fora strictly convex hardening penalty, along with appropriate reflection symmetry of the energy,proves this, at least for geometrically linear elasticity). Thus, in such cases, it looks as thoughthe proportionality condition just mentioned can be taken to hold w.l.o.g. when looking forminimisers, and hence one might conjecture that E lb is the correct relaxed energy.We can obtain a better upper bound by allowing the bi-layer weighting λ , now assumed tobe a Lipschitz function of x ∈ Ω, to be non-constant in activated slip-planes. Using the co-areaformula to take care of the curl generated across the (in general) undulating dividing-surfaces ofthe bi-layers, we obtain the following laminated energy in the limit n → ∞ : I ( y, { s j } , λ ) := Z Ω W el ( F el ) d x + N X j =1 ( τ Z Ω j | c j | p λ p − j + | c j | p (1 − λ j ) p − d x + σ Z Ω j (cid:12)(cid:12)(cid:12) ∇ m ⊥ j c j − c j ∇ m ⊥ j ln λ j (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) c j ∇ m ⊥ j ln λ j (cid:12)(cid:12)(cid:12) d x + σ Z Ω j (cid:12)(cid:12)(cid:12) ∇ m ⊥ j c j − c j ∇ m ⊥ j ln(1 − λ j ) (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) c j ∇ m ⊥ j ln(1 − λ j ) (cid:12)(cid:12)(cid:12) d x ) . (24)Here, assuming λ ( x ) to be Lipschitz continuous and bounded away from 0 and 1, along with β ∈ C , allows one to handle the displacement-perturbation as in [1]: that is to say, the elasticenergy is still continuous with respect to lamination.Clearly, by the triangle inequality, the curl-type contribution to (24) is no smaller than thelaminated curl, R Ω G lam ( { s j } ), and an easy calculation shows that the laminated curl is reachediff at each point of Ω j either ∇ m ⊥ j λ j vanishes or ∇ m ⊥ j c j and ∇ m ⊥ j c j point in opposite directions, N THE EXISTENCE OF MINIMISERS FOR STRAIN-GRADIENT SINGLE-CRYSTAL PLASTICITY 11 the latter of which will not hold for general test functions, of course. Choosing λ j constant on slipplanes minimises the curl contribution, but the corresponding hardening contribution will then,in general, be some way from the pointwise-optimal hardening energy which appears in the lowerbound (20). Thus, for p >
1, there is a trade-off between the hardening and dislocation energyof an undulating laminate, and one would have to be supremely optimistic to expect an explicitformula for the optimal energy.One might conjecture that the upper-bound E (2)ub ( y, { s j } ) := inf { I ( y, { s j } , λ ) : λ ∈ Lip(Ω) , < λ ( x ) < } (25)is in fact the relaxation we’re looking for, at least for sufficiently smooth s j . We have not, however,been able to prove the convexity of this uncountable infimum .By construction, if E p rel ( y, F pl ( β )) denotes the single-slip-to-single-plane relaxation of E (i.e., E p rel ( x ) = inf { lim inf x n ⇀x E ( x n ) } , with the infimum taken over all single-plane sequences weaklyconverging to a given single-plane x ), then we have E lb ( y, { s j } ) ≤ E p rel ( y, { s j } ) ≤ E (2)ub ( y, { s j } ) ≤ E (1)ub ( y, { s j } ) , (26)on single-plane test functions, so that, in particular, E p rel , extended to + ∞ when (RSC) is violated,satisfies the conditions of Theorem 1. Moreover, the last inequality is in general strict, by theremarks above and the following example. Example . Here is an essentially 1-d example of a single-plane β for which the optimal flatlamination can be bettered by a sigmoidal one for p = 2.Let Ω = (0 , × ( − X, X ) × (0 , ∈ R , with active slip-normal pointing in the x -directioneverywhere, and for the material parameters let σ = τ = 1. Denote the single-slip strain by β = c b + c b , such that c ( x ) = x < ǫ : x ≥ , c ( x ) = ǫ : x <
01 : x ≥ , for positive constants ǫ ≪ X ≫ λ = everywhere, andthe plastic part of E (1)ub ( u, F pl ( β )) is readily calculated to be 2(2 X (1 + ǫ ) + (1 − ǫ )).By choosing, instead, an appropriate non-constant λ , and X large enough, we can almost halvethe plastic energy, coming close to the plastic part of E lb ( u, { s j } ) (note that this is a case where ∇ c and ∇ c point in opposite directions everywhere).Specifically, we choose our Lipschitz λ to be λ ( x ) = ǫ : x ≤ − − < x < ǫ ǫ : x ≥ . Now, the L -part of the plastic energy in I ( u, { s j } , λ ) is calculated to be 2 { ( X − ǫ ) + H ( ǫ ) } , for some H ( ǫ ) ∈ [1 + ǫ + 2 ǫ , ǫ (1 + ǫ )], while the curl part is of the form H ( ǫ )(i.e. independent of X ). Thus, the total plastic energy of the jagged laminate may be written as2( X − ǫ ) + 2 H ( ǫ ) + H ( ǫ ). For X large and ǫ small, this is roughly one-half of the plasticenergy of the optimal flat laminate, as claimed.4. Existence of minimisers
We will use the direct method of the calculus of variations to prove the existence of minimisersfor any single-plane energy which satisfies the requirements stated in Theorem 1.
Minimising sequences.Proposition 6.
Assume that the conditions of Theorem 1 are satisfied, and consider a finite-energy minimising sequence ( y k , { s j } k ) . We then have, up to taking a subsequence, { s j } k ⇀ s j in L p and y k ⇀ y in W ,r , for some s j and y , where r = p + q > .Proof. The first part of the claim is immediate by inspection, while the second follows from (cid:13)(cid:13)(cid:13) ∇ yF − (cid:13)(cid:13)(cid:13) L q ≥ k∇ y k L r k F pl k L p , which in turn follows from H¨older’s inequality, since F − = Id − P Nj =1 s j ⊗ m j almost everywherealong the minimising sequence, by the single-plane condition – see [31]. (cid:3) Weak convergence of minors.Lemma 7. If p > and q > pp − , then for a minimising sequence as in Proposition 6, we have,up to a subsequence, M , , (( F el ) k ) := M , , ( ∇ y k ( F − ) k ) ⇀ M , , ( ∇ yF − ) in L (Ω) , i.e., each sequence of minors converges weakly to the minor of the limiting elasticdeformation along the minimising sequence.Proof. In what follows, we will often drop sequence subscripts for notational convenience, and r will be determined by the model parameters p and q as in Proposition 6.Along our energy-bounded minimising sequence, the plastic-strain tensor has the form F pl = Id + N X j =1 s j ( x ) ⊗ m j = P Nj =1 s j ( x ) m j P Nj =1 s j ( x ) m j P Nj =1 s j ( x ) m j P Nj =1 s j ( x ) m j P Nj =1 s j ( x ) m j P Nj =1 s j ( x ) m j P Nj =1 s j ( x ) m j P Nj =1 s j ( x ) m j P Nj =1 s j ( x ) m j , with s j ⊥ m j and (cid:12)(cid:12) s i (cid:12)(cid:12) (cid:12)(cid:12) s j (cid:12)(cid:12) = 0 for i = j , a.e. in Ω.Moreover, due to the side condition and s j ⊥ m j , we have F − = Id − N X j =1 s j ( x ) ⊗ m j , a.e. in Ω . In order to prove convergence of minors, we need to control the integrability of the entries in theelastic-strain tensor, which reads F el = ∇ yF − = y , − P Nj =1 m j P k =1 s jk y ,k y , − P Nj =1 m j P k =1 s jk y ,k y , − P Nj =1 m j P k =1 s jk y ,k y , − P Nj =1 m j P k =1 s jk y ,k y , − P Nj =1 m j P k =1 s jk y ,k y , − P Nj =1 m j P k =1 s jk y ,k y , − P Nj =1 m j P k =1 s jk y ,k y , − P Nj =1 m j P k =1 s jk y ,k y , − P Nj =1 m j P k =1 s jk y ,k , such that all of the matrix entries are in L q , due to the growth condition on the elastic energydensity, W el . The × -minors. The ‘bad’ terms appearing in the 1 × M := ∇ y i · s j , such that, by the argument of Proposition 6, and another application of H¨older’sinequality, M ∈ L z , with z given by z = r + p = p + q < M is thus equi-integrable alongthe minimising sequence. N THE EXISTENCE OF MINIMISERS FOR STRAIN-GRADIENT SINGLE-CRYSTAL PLASTICITY 13
We now apply a div-curl argument to M . Clearly, ∇ y is curl-free, and with the aid of anorthonormal basis O j = ( m j , e j , e j ) adapted to the j -th slip normal, we have | div s j | = | ∂ m j ˜ s jm + ∂ e j ˜ s j + ∂ e j ˜ s j | = | ∂ e j ˜ s j + ∂ e j ˜ s j |≤ |∇ m j ⊥ s j | (27) ≤ C, (28)where ˜ s ji are the components of s j w.r.t. O j , and C is a constant.Thus, passing to a subsequence, we get div s jk → div s j ∈ W − , for some s j as k → ∞ , sinceweak- ∗ convergence of Radon measures implies strong convergence in W − , (Ω) = ( W , ∞ ) ∗ (Ω),and we may thus appeal to the div-curl Theorem of [6] to get the required convergence of the1 × The × -minors. For the 2 × ∇ y · F − ) = cof ∇ y · cof F − (29)= (cof ∇ y ) · F T pl , (30)a.e., since det F − = 1, a.e..Thus, since cof ( ∇ y · F − ) ∈ L q , by our assumptions on W pl , we have, again by [31], cof ∇ y ∈ L r .In other words, the 2 × ∇ y are also in L r .Now, according to Lemma 2.4 of [31], the 2 × ∇ y · F − can be written as det H det F pl (=det H ), for a 3 × H which consists of two rows of ∇ y , and one of F pl : for example,something of the form H = y , y , y , y , y , y , P Nj =1 s j ( x ) m j P Nj =1 s j ( x ) m j P Nj =1 s j ( x ) m j . Expanding the determinant of this example about the 3rd row, we see by the standard div-curllemma that the 2 × L to the correctlimit. Since they are bounded in L r , by the above, we also have these minors converging weaklyin L r to the correct limit, along a subsequence.Next note that det H = η · ξ , where η is a vector of 2 × ∇ y , and ξ = N X j =1 s j ( x ) m j , N X j =1 s j ( x ) m j , N X j =1 s j ( x ) m j T . Thus, div η = 0, while curl( s j ( m j , m j , m j ) T ) takes the form (0 , ∂ e j s j , − ∂ e j s j ) in a frame O j asabove, which is once again a measure dominated by the dislocation energy. Taking q > pp − ensures that ξ and η converge in conjugate Lebesgue spaces, and hence all the conditions of thediv-curl Theorem in [6] are satisfied, giving the required convergence along a subsequence. The × -minor. This is just det ∇ y . We can expand this 3 × × L r , as above. By our stated assumption on p and q , this ensures that the two vectors convergeweakly in conjugate Lebesque spaces, and so once more we may apply [6] to get the requiredconvergence along a suitable subsequence. (cid:3) Preservation of the single-plane condition under weak- L p convergence. Considernow a sequence { s kj } , j = 1 , . . . , N , k ∈ N , of relaxed slips with the following properties:(P1) For each k , { s kj } Nj =1 satisfies our single-plane condition , i.e., s kj ( x ) ∈ m ⊥ j and | s ki ( x ) || s kj ( x ) | = 0 for a.e. x ∈ Ω , i = j. (P2) s kj ⇀ s j in L p , p ≥
1, for some s j , as k → ∞ .(P3) The density of geometrically necessary dislocations is uniformly summable, i.e., G ( { s kj } Nj =1 ) ≤ K, for all k ∈ N and some K < ∞ , where G ( · ) is the modified curl defined in (5).Our closedness result is then the following. Lemma 8.
Assume we have a sequence { s j } k , k ∈ N , satisfying properties (P1)-(P3) above. Thenthe limit { s j } satisfies the single-plane condition, (P1).Proof. First of all, we clearly have s j ( x ) ⊥ m j , for a.e x ∈ Ω, by weak convergence. For acontradiction, assume that there exists a measurable set S ⊂ Ω with | S | >
0, such that on S atleast two of the s j are non-vanishing. Without loss of generality, we can take these to be s and s . By approximation of measurable sets by closed sets from the inside, there exists a set S ′ ⊂ S and a δ >
0, such that on S ′ we have | s | , | s | ≥ δ and | S ′ | ≥ δ .Now introduce coordinates ( x , x , x ) which are adapted to the (not necessarily orthonormal)frame ( m , m , m × m ). By approximation of measurable sets by open sets from the outside, fora given ǫ >
0, we can find a finite collection of (open) parallelepipeds, aligned with the coordinatemesh, the union of which we denote by V , such that Z V \ S ′ | s | + | s | d x ≤ ǫ and S ′ ⊂ V. (31)Note that we can always assume V ⊂⊂ Ω, by subtracting a thin collar-neighbourhood of theLipschitz boundary, ∂ Ω, if necessary,.We now fill V with finitely many non-overlapping parallelepipeds { C i } Li =1 of edge-length nolarger than an arbitrary l >
0, once more aligned with the x i -coordinate mesh. By the assumedweak convergence of s kj , we havelim inf k →∞ Z C i ∩ S ′ (cid:12)(cid:12)(cid:12) s kj (cid:12)(cid:12)(cid:12) d x ≥ Z C i ∩ S ′ | s j | d x, j = 1 , . (32)Moreover, by Lemma 9 (below), there exists a geometric constant c > G ( { s kj } ) ≥ c Z V (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∂ (cid:12)(cid:12) s k (cid:12)(cid:12) ∂x (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∂ (cid:12)(cid:12) s k (cid:12)(cid:12) ∂x (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) d x, and thus, using Lemma 10 (even further below), G ( { s kj } ) ≥ cl X i min (cid:26)Z C i ∩ S ′ (cid:12)(cid:12)(cid:12) s k (cid:12)(cid:12)(cid:12) d x, Z C i ∩ S ′ (cid:12)(cid:12)(cid:12) s k (cid:12)(cid:12)(cid:12) d x (cid:27)! . Now, keeping l fixed, pick an arbitrary ˆ ǫ >
0, and choose M (ˆ ǫ, l ) large enough such that, for all i and for j = 1 ,
2, we have Z C i ∩ S ′ (cid:12)(cid:12)(cid:12) s kj (cid:12)(cid:12)(cid:12) d x ≥ Z C i ∩ S ′ | s j | d x − ˆ ǫlLc N THE EXISTENCE OF MINIMISERS FOR STRAIN-GRADIENT SINGLE-CRYSTAL PLASTICITY 15 for any k ≥ M . We thus obtain for any k large enough that G ( { s kj } ) ≥ cl X i min (cid:26)Z C i ∩ S ′ | s | d x, Z C i ∩ S ′ | s | d x (cid:27)! − ˆ ǫ ≥ cl δ (cid:12)(cid:12) S ′ (cid:12)(cid:12) − ˆ ǫ, by (31).Finally, by taking ˆ ǫ and l small enough, keeping δ > S ′ fixed, we can thus make the curlarbitrarily large, which is a contradiction to energy boundedness. (cid:3) Lemma 9.
Consider an open set V ⊂⊂ Ω . We then have, for any { s j } on Ω , G ( { s j } ) | V ≥ c Z V (cid:12)(cid:12)(cid:12)(cid:12) ∂ | s | ∂x (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12) ∂ | s | ∂x (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12) ∂ | s | ∂x (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12) ∂ | s | ∂x (cid:12)(cid:12)(cid:12)(cid:12) d x, (33) for some geometric constant c > , whereby the coordinates x , x and x are adapted to m , m and m × m , respectively.Proof. First assume the s i are smooth. Next, it is convenient to calculate the row-wise curl of β (as above) by first transforming from the ( m , m , m × m )-frame to an orthonormal one given byfirst shearing m in the ( m , m )-plane such that it becomes orthogonal to m , and then shearing m to make it orthogonal to the ( m , m )-plane. The corresponding transformation matrix isthus of the form M = φ cos ψ φ sin ψ φ θ
00 sin θ
00 0 1 , (34)for some angles θ, φ and ψ .Applying this to s ⊗ m , and then taking the 3-component of each of the row-curls, gives apointwise estimate which leads to Z V (cid:12)(cid:12)(cid:12) ∇ m ⊥ s (cid:12)(cid:12)(cid:12) ≥ c Z V (cid:12)(cid:12) ∂ x s (cid:12)(cid:12) + (cid:12)(cid:12) ∂ x (cid:0) s + c s (cid:1)(cid:12)(cid:12) + (cid:12)(cid:12) ∂ x (cid:0) s + c s + c s (cid:1)(cid:12)(cid:12) d x, (35)for some c i ( θ, φ, ψ ) >
0, where the s i are components of s in the original ( m , m , m × m )-frame. An entirely analogous inequality can be obtained for curl( s ⊗ m ) by making the subscriptswitch 1 ↔ s ⊗ m and s ⊗ m wealso get analogous inequalities with x - rather than x -derivatives everywhere on the rhs (in bothcases). Repeated application of the triangle inequality now gives (33).Finally, we remove the assumption of smoothness on { s j } . Thus, by the first part of the proofof Proposition 3.3 in [2] (eq. 3.23, in particular, which doesn’t depend on slip-patch-boundaryregularity), we can mollify the s i to get smooth ( s i ) ǫ such that k s j − ( s j ) ǫ k L p ( V ) ≤ ǫ and Z V G ( s j ) | ≥ Z V G ( s j ) ǫ | − ǫ. (36)Then we apply (33) to ( s j ) ǫ and appeal to the L p -lower semicontinuity of all the derivative termson the right-hand side as ǫ → (cid:3) Remark . One also obtains inequalities analogous to (33) by replacing the absolute values of s and s with any of their components, which will be useful below. Lemma 10.
Suppose we have a single-plane sequence { s j } k with uniformly bounded dislocation-and-hardening energy, with active slips s j , s j which mix in the limit on a covering of paral-lelepipeds V = ∪ i C i ⊃ S ′ , as in Theorem 8. Then, for sufficiently large j , there exists c > suchthat Z V (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∂ (cid:12)(cid:12)(cid:12) s j (cid:12)(cid:12)(cid:12) ∂x (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∂ (cid:12)(cid:12)(cid:12) s j (cid:12)(cid:12)(cid:12) ∂x (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) d x ≥ cl X i min (cid:26)Z C i ∩ S ′ (cid:12)(cid:12)(cid:12) s j (cid:12)(cid:12)(cid:12) d x, Z C i ∩ S ′ (cid:12)(cid:12)(cid:12) s j (cid:12)(cid:12)(cid:12) d x (cid:27) , (37) for coordinates x i aligned with the ( m , m , m × m ) -frame, provided the x - and x -coordinateextents of the C i are no greater than l , where c depends on the energy bound.Proof. Consider a parallelepiped P = Q × [0 , t ], with Q a parallelogram aligned with the x , x -coordinate mesh, having coordinate extents l and l which are dominated by a constant l . Tobe specific, and without loss of generality, let Q = { ( x , x ) : x ∈ (0 , l ) , x ∈ (0 , l ) } . Then thesingle-plane condition allows us to employ the argument of Lemma 4.3 in [8], which also worksfor non-orthogonal coordinates, on slices Q × { τ } . Thus, for each k and τ , we define the followingsubsets of Q : ω k,τ = n x ∈ (0 , l ) : s k ( x , x , τ ) = 0 for a.e. x ∈ (0 , l ) o , (38) ω k,τ = n x ∈ (0 , l ) : s k ( x , x , τ ) = 0 for a.e. x ∈ (0 , l ) o , (39)and, from the single-plane condition and basic measure theory, we conclude that for each k anda.e. τ ∈ [0 , t ] at least one of ω k,τ and ω k,τ must be a null set.For the sake of argument, assume that, for a given k and τ , ω k,τ is a null set. Then, for a.e. x ∈ (0 , l ), we have (cid:12)(cid:12)(cid:12) s k (cid:12)(cid:12)(cid:12) ( x , x , τ ) ≤ Z l ∂ (cid:12)(cid:12) s k (cid:12)(cid:12) ∂x ( x , x ′ , τ ) d x ′ . (40)Hence, integrating w.r.t. x and then x , we obtain Z Z Q (cid:12)(cid:12)(cid:12) s k (cid:12)(cid:12)(cid:12) d x ≤ l Z Z Q ∂ (cid:12)(cid:12) s k (cid:12)(cid:12) ∂x d x, (41)and, by a similar argument for the other possible case, we therefore see that, for a.e. τ ∈ (0 , t ),either Z Z Q (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∂ (cid:12)(cid:12) s k (cid:12)(cid:12) ∂x (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ( · , τ ) d x ≥ l Z Z Q (cid:12)(cid:12)(cid:12) s k (cid:12)(cid:12)(cid:12) ( · , τ ) d x, (42)or Z Z Q (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∂ (cid:12)(cid:12) s k (cid:12)(cid:12) ∂x (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ( · , τ ) d x ≥ l Z Z Q (cid:12)(cid:12)(cid:12) s k (cid:12)(cid:12)(cid:12) ( · , τ ) d x. (43)Define C ti = C i ∩ S ′ ∩ { x = t } , so that (42) implies Z V (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∂ (cid:12)(cid:12)(cid:12) s j (cid:12)(cid:12)(cid:12) ∂x (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∂ (cid:12)(cid:12)(cid:12) s j (cid:12)(cid:12)(cid:12) ∂x (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) d x ≥ l Z X i min (Z Z C ti | s k | d x, Z Z C ti | s k | d x ) d t. (44)Write a ki ( t ) = RR C ti (cid:12)(cid:12) s k (cid:12)(cid:12) d x and b ki ( t ) = RR C ti (cid:12)(cid:12) s k (cid:12)(cid:12) d x . Then, in order to apply (44), it sufficesto show that Z X i min n(cid:12)(cid:12)(cid:12) a ki (cid:12)(cid:12)(cid:12) , (cid:12)(cid:12)(cid:12) b ki (cid:12)(cid:12)(cid:12)o d t ≥ c X i min (cid:26)Z (cid:12)(cid:12)(cid:12) a ki (cid:12)(cid:12)(cid:12) d t, Z (cid:12)(cid:12)(cid:12) b ki (cid:12)(cid:12)(cid:12) d t (cid:27) , (45)for k large, and some constant c > s k and s k weakly converge in L p , with limits greater than some δ > S ′ . Let f ( t ) = L ( S ′ ∩ { x = t } ). Then ∃ T ⊂ R with | T | ≥ δ , such that f ( t ) ≥ δ on T , for some δ >
0, since R f ( t ) dt = | S ′ | >
0. By taking a fine mesh of manypoints in T , and noting that Lemma 9 (resp. Remark 14) gives control on the component-wisevariation of RR s k dx dx and RR s k dx in the t -direction, we see by weak convergence that P i min (cid:8)(cid:12)(cid:12) a ki (cid:12)(cid:12) , (cid:12)(cid:12) b ki (cid:12)(cid:12)(cid:9) > δδ / T , if k is large enough (we mayassume w.l.o.g. that one component of each of s and s is larger than δ on S ′ , then weakconvergence of RR s k dx dx and RR s k dx dx , together with the curl bound, prevent a ki and b ki from straying too close to zero). Thus, the left-hand side of (45) is greater than some C for k N THE EXISTENCE OF MINIMISERS FOR STRAIN-GRADIENT SINGLE-CRYSTAL PLASTICITY 17 large. Since the right-hand side of (45) is bounded by the (hardening) energy, we see that (45)does in fact hold, for a constant c which depends on the energy bound. (cid:3) Remark . While Lemma 8 still holds in the case p = 1, L -hardening is of course not sufficientto guarantee s kj ⇀ s j in L along (a subsequence of) a minimising sequence for an elastoplasticenergy of the form E .4.4. Proof of Theorem 1.
Putting together Lemmas 7-10 finally gives our main result, Theorem1, since we only need an upper bound on the dislocation and hardening energy along bounded-energy sequences to apply these results. Moreover, Theorem 2 also follows, since the strongerrequirement p >
Counter-example for p = 1 . We cannot expect an analogue of our main existence resultfor p = 1 rate-independent dissipation, even for geometrically linear elasticity, by virtue of thefollowing essentially 2-d example, which highlights the problem of slip concentration. Example . Suppose p = 1, and consider two infinitesimally thin slip-lines (i.e. slip-planes viewedside-on) crossing at right angles, and exiting a shear sample at free boundaries. Now, by anyreasonable interpretation of the relaxed side condition for measures, this construction should haveinfinite plastic energy. However, one can approximate these crossing slip lines weakly- ∗ in thespace of Radon measures by fattening both of the slip lines a little, while preserving the totalshear, and cutting off one of the resulting mollified shear bands just outside the region of overlap,such that ( RSC ) is now satisfied. The dislocation energy of such an approximation is just twicethe total shear on the cut-off shear band, and in this way one can therefore pass to the weak- ∗ limit of crossing slip lines with finite plastic energy. In other words, for p = 1 any reasonableinterpretation of ( RSC ) produces a single-plane energy which fails to be lower semi-continuous.
Acknowledgments
MK was supported by GA ˇCR through projects 14-15264S and 16-34894L. This research waspartly conducted when MK held the visiting Giovanni-Prodi professorship in the Institute ofMathematics, University of W¨urzburg. Its support and hospitality are gratefully acknowledged.PWD is partially supported by the German Scholars Organization / Carl-Zeiss-Stiftung via theWissenschaftler-R¨uckkehrprogramm.
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