Abstract The effective yield set of ionic polycrystals is characterized by means ofvariational principles in L∞ associated to supremal functionals acting on matrix-valued divergence-free fields.
This paper is motivated in part by recent work in connection with the mathematicalderivation of various models related to polycrystal plasticity and the characterization
F. AbdullayevDepartment of Mathematical Sciences, Worcester Polytechnic Institute,100 Institute Road, Worcester, MA 01609, USAe-mail: [email protected]
M. Bocea (B)Department of Mathematics and Statistics, Loyola University Chicago,1032 W. Sheridan Road, Chicago, IL 60660, USAe-mail: [email protected]
M. MihailescuDepartment of Mathematics, University of Craiova, 200585 Craiova, Romaniae-mail: [email protected]
M. MihailescuResearch Group of the Project PN-II-ID-PCE-2011-3-0075, “Simion Stoilow” Instituteof Mathematics of the Romanian Academy, P.O. Box 1-764, 014700 Bucharest, Romania
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Appl Math Optim
of the effective yield of a polycrystal (see e.g., [1,4,3,12–15,18]). Polycrystals arecollections of grains, or single crystals, which are bonded together in different ori-entations. The yield of a single crystal is described by a closed convex subset K ofM
3×3sym , the space of symmetric 3×3 real matrices. Yield in a crystalline solid is associ-
ated with a finite number of slip systems, determined by pairs (nk, mk) of orthogonalvectors: nk is the normal to the slip plane, and mk is the direction of slip. Assum-ing that there are s slip systems present in the polycrystal, a typical yield set has theform
K ={
A ∈ M3×3sym : 〈A, μk〉 ≤ τk, k = 1, . . . , s
},
where μk := 12 (mk ⊗ nk + nk ⊗ mk) is the kth slip tensor, and τk is the critical shear
stress corresponding to the kth slip system (nk, mk), k = 1, . . . , s. The orientationsof the grains in a polycrystal occupying a domain � ⊂ R
3 are described through apiecewise constant function R : � → SO(3), where for each point x ∈ � the rotationR(x) indicates the orientation of the grain which contains that point. If K is the yieldset of the basic crystal, the stress in the polycrystal occupying � is constrained tosatisfy
σ(x) ∈ R(x)K RT (x), x ∈ �. (1.1)
The set of all average stresses σ := 1|�|∫�
σ(x)dx, where σ obeys the constraint(1.1), together with the equilibrium equation
Div σ = 0 in �, (1.2)
is called the effective yield set of the polycrystal:
Keff :=⎧⎨⎩σ := 1
|�|∫
�
σ(x)dx : (1.1) and (1.2) hold
⎫⎬⎭ . (1.3)
The definition of Keff is the usual one in the polycrystal plasticity literature (see,e.g., Sect. 2 in [18], and references therein). The rigorous justification of the fact thatthis accurately describes the macroscopic behavior of the polycrystal follows fromthe homogenization theory (see, e.g., [17]). The solutions of the equilibrium prob-lems at the (microscopic) length scale ε > 0 of the individual grains converge, asε → 0, to a solution of the equilibrium problem considered on the larger (macro-scopic) scale. In the traditional model of polycrystal plasticity, the latter comes froma degenerate variational principle governed by an effective energy Ehom, obtained viahomogenization, that is equal to zero at matrices corresponding to stresses which,in addition to solving the equilibrium equation, Div σ = 0, satisfy the constraint(1.1) at every point in the domain � occupied by the polycrystal, and it is equal toinfinity otherwise. Hence, the definition of Keff that is currently used in the litera-ture on the subject (and which we have also adopted in this paper) coincides with
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Appl Math Optim
how one would formally need to define the yield set of the polycrystal, namelyset it equal to the domain of the effective energy obtained via homogenization:Dom(Ehom) := {σ ∈ M
details regarding the homogenization procedure from a deformation based (gradientfields) point of view we refer to the paper by Kohn and Little [18]. In the divergence-free case, a different derivation of the model, based on power-law regularization, isgiven in [1].
The key issue in polycrystal plasticity is to understand the structure of the effectiveyield set Keff , when the yield set K of the basic crystal is known, and when someinformation on the shapes and orientations of the grains present in the polycrystalis given. A similar problem arises in the analysis of models of dielectric breakdownand electrical resistivity, where an effective yield (strength) set is defined similarly,with a suitable modification of the pointwise constraint, and with (1.2) replaced bythe requirement that the field σ : � → R
3 be either curl-free or divergence-free,respectively (see [1,13,12]). For example, in [12], the pointwise constraint readsσ(x) ∈ R(x)K . The reason for the difference when compared to (1.1) is that [12] isconcerned with two model problems, antiplane shear and plane stress, correspondingto gradient vector fields and divergence-free vector fields, respectively. For example,in the (two-dimensional, for simplicity) antiplane shear model, there are four basic
slip systems with slip tensors ±μ(1),±μ(2), where μ(1) = 12 (
−→i ⊗ −→
k + −→k ⊗ −→
i )
and μ(2) = 12 (j ⊗ −→
k + −→k ⊗ −→
j ), and with critical stresses equal to ±M and ±1,respectively. The stress takes the particular form
σ(x) =⎛⎜⎝
0 0 σ13(x)
0 0 σ23(x)
σ13(x) σ23(x) 0
⎞⎟⎠ , x ∈ �.
Thus, σ can be identified with a vector field 〈σ13, σ23〉 in the plane. This is a simpli-fication of the polycrystal plasticity setting which we consider in this paper, wherethe stresses are divergence-free tensor fields (matrix-valued, divergence-free on everyrow). In the particular cases considered by Garroni and Kohn in [12], the pointwiseconstraint needs to be adapted to the fact that the stress σ(x) is assumed to be athree-dimensional vector at any point x ∈ �, so in their work the constraint becomesσ(x) ∈ R(x)K ⊂ R
3.
When a direct description of the effective yield set is not available, the commonapproach has been to study the so-called Sachs and Bishop–Hill–Taylor bounds, whichare the natural inner and outer bounds for this set (see [12,14,15,18]).
During the last decade the issues described above have been undertaken in theframework of �-convergence. The first work in this direction is due to Garroni et al.[13], who gave a mathematical derivation of first-failure dielectric breakdown mod-els as limiting cases (via �-convergence) of various power-law models, leading to anefficient variational characterization of the effective yield set by means of variationalprinciples associated to the limiting functionals. Bocea and Nesi [1] have consid-ered the corresponding problems in the framework of A-quasiconvexity, allowing formore general linear differential constraints on the underlying fields. In particular, the
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Appl Math Optim
analysis in [1] leads to variational characterizations of the yield (strength) set in theframework of electrical resistivity, where the underlying fields are divergence-free.More recently these results have been extended in several directions (see, e.g., [2,4]).First, it turns out that one can consider as a starting point more flexible power-lawmodels where the exponent in the power-law regularization is allowed to depend onthe point x ∈ �. Second, the power-law functionals can be adapted to treat situa-tions where the underlying fields take values in stress space M
3×3sym , are divergence
free, and where several (depending on the number of slip systems present in the basiccrystal) distinct pointwise constraints are simultaneously verified. This is the case insome two-dimensional polycrystal plasticity models, such as antiplane shear and planestress.
The aim of this paper is to propose an approach to the analysis of a three-dimensionalmodel of polycrystal plasticity for which the work mentioned above does not apply.The focus will be on polycrystalline materials whose individual grains (crystallites)are assumed to be ionic crystals. This class of crystals was introduced in the cele-brated work of Hutchinson [16], and it is representative in the modelling of crystallinematerials exhibiting a deficient supply of slip systems.
The plan of the paper is as follows: In Sect. 2 we recall the definition of �-convergence and we provide a brief review of A-quasiconvexity. Section 3 is devotedto the variational characterization of the effective yield set in Hutchinson’s model.Finally, in Sect. 4 of the paper we prove a �-convergence result for the class of supre-mal functionals involved in the characterization of the effective yield set.
2 Preliminaries
We first recall the definition of �-convergence [9,10] in metric spaces. A thoroughintroduction to the subject may be found in [8] (see also [5,6]).
Definition 1 Let X be a metric space. A sequence {In} of functionals In : X → R :=R∪{+∞} is said to �(X)-converge to I : X → R (we write �(X)− limn→∞ In = I )if
(i) for every u ∈ X and {un} ⊂ X such that un → u in X , we have
I (u) ≤ lim infn→∞ In(un);
(ii) for every u ∈ X there exists a recovery sequence {un} ⊂ X such that un → u inX , and
I (u) ≥ lim supn→∞
In(un).
2.1 A-Quasiconvexity
Let N , d, l ∈ N be given, � be an open, bounded domain in RN , 1 < p < ∞, and let
p′ be the Hölder conjugate exponent of p, that is, 1/p + 1/p′ = 1. Given a family of
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Appl Math Optim
linear operators A(1), A(2), . . . , A(N ) ∈ Lin(Rd; Rl), consider the differential operator
A : L p(�; Rd) → W −1,p(�; R
l) defined by
Av :=N∑
i=1
A(i) ∂v
∂xi, (2.1)
that is,
〈Av, u〉 :=⟨
N∑i=1
A(i) ∂v
∂xi, u
⟩= −
N∑i=1
∫
�
A(i)v∂u
∂xidx for all u ∈ W 1,p′
0 (�; Rl).
(2.2)
Here W −1,p(�; Rl) stands for the dual of W 1,p′
0 (�; Rl).
Definition 2 The operator A satisfies the constant rank property if there exists r ∈ N
such that
rank (A(w)) = r for all w = (w1, . . . , wN ) ∈ SN−1, (2.3)
where
A(w) :=N∑
i=1
wi A(i) ∈ Lin(R
d ; Rl)
.
The constant rank property was introduced by Murat and Tartar in connection to thetheory of compensated compactness (see, e.g., [19–21]). We note that this restrictionstill allows the treatment of a broad class of differential constraints encountered inapplications. Among these, we mention curl free fields (gradients and partial gradi-ents), divergence free fields, higher order gradients, symmetrized gradients, and fieldswhich satisfy Maxwell’s equations.
For the applications that we discuss in this paper A will be the divergence operatoracting on fields which take values in the space of symmetric N × N matrices. Givena function U ∈ L p
(�; M
N×N), the differential operator A is given by
AU := Div U =
⎛⎜⎜⎜⎜⎝
div U (1)
div U (2)
...
div U (N )
⎞⎟⎟⎟⎟⎠
,
where, for i = 1, . . . , N , U (i)(x) := (Ui1(x), Ui2(x), . . . , Ui N (x)) stands for the i throw of the matrix U (x), x ∈ �. Thus, if we take d = N 2, l = N , and we define, fori, k = 1, . . . , N and j = 1, . . . , N 2,
A(k)i j =
{δi( j−(k−1)N ) if (k − 1)N + 1 ≤ j ≤ k N
0 else,
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Appl Math Optim
the differential constraint AU = 0 can be written in the form (see (2.1))
N∑k=1
A(k) ∂U
∂xk= 0.
Note that the constant rank condition (2.3) is satisfied since for every w =(w1, . . . , wN ) ∈ SN−1 we have
ker(A(w)) ={
V ∈ MN×N : wV = 0
},
and thus dim(ker A(w)) = N 2 − N .
We now recall the definition of A-quasiconvexity, introduced in Fonseca and Müller[11] (see also [7]).
Definition 3 A function g : Rd → R is said to be A-quasiconvex if
g(A) ≤∫
Q
g(A + w(x))dx
for all A ∈ Rd , and all Q-periodic w ∈ C∞(Q; R
d) such that Aw = 0 and∫Q w(x)dx = 0, where Q = (0, 1)N is the unit cube in R
N .
By Jensen’s inequality, convex functions are A-quasiconvex. It is shown in [11]that if A satisfies the constant rank property (2.3), � ⊂ R
N is an open, bounded set,(u, v) : � → R
m × Rd is measurable, and g : � × R
m × Rd → R is a normal
integrand satisfying suitable growth assumptions, then A-quasiconvexity of g(x, u, ·)is a necessary and sufficient condition for the sequential lower semicontinuity ofintegral functionals of the form
(u, v) →∫
�
g(x, u(x), v(x))dx
along sequences such that un → u in measure, vn ⇀ v weakly in L p, and Avn → 0in W −1,p. We will only need to use the following result from [11].
Proposition 1 (see [11, Theorem 3.7]) Let 1 ≤ p ≤ +∞ and suppose that g :� × R
m × Rd → [0,+∞) is a normal integrand such that z → g(x, u, z) is A-
quasiconvex and continuous for a.e. x ∈ � and all u ∈ Rd . If 1 ≤ p < +∞, assume
further that there exists a locally bounded function a : � × Rd → [0,+∞) such that
0 ≤ g(x, u, v) ≤ a(x, u)(1 + |v|p),
for a.e. x ∈ �, and all (u, v) ∈ Rm × R
d . If
un → u in measure, vn ⇀ v weakly (weakly∗ if p = ∞) in L p(�; Rd),
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Appl Math Optim
and
Avn → 0 in W −1,p(�; R
l)
(Avn = 0 if p = ∞),
then
∫
�
g(x, u(x), v(x))dx ≤ lim infn→∞
∫
�
g(x, un(x), vn(x))dx .
Let � ⊂ RN be a bounded, open domain, with sufficiently smooth boundary, and
let s ∈ N be a positive integer. For i = 1, 2, . . . , s, consider Carathéodory integrandsfi : � × R
d → [0,+∞) such that
fi (x, ·) is A-quasiconvex for a.e. x ∈ �. (2.4)
Assume that there exists a constant C > 0 such that for every i ∈ {1, 2, . . . , s} wehave
fi (x, v) ≤ C (1 + |v|) for a.e. x ∈ �, and all v ∈ Rd . (2.5)
Further, we assume that there exists a constant c > 0 such that
s∑i=1
fi (x, v) ≥ c|v| for a.e. x ∈ �, and all v ∈ Rd . (2.6)
The effective yield set of a polycrystal can be characterized in several models ofpolycrystal plasticity by means of variational principles in L∞ associated to �-limitsof certain power-law functionals. Indeed, it is shown in [4, Theorem 5] that for suitablechoices of the positive integers N , d, s, the differential operator A, and of the functionsfi (i = 1, . . . , s) satisfying the conditions (2.4), (2.5), and (2.6), we have
Keff ={η ∈ R
d : f effs,∞(η) ≤ 1
}, (2.7)
where
f effs,∞(η) := inf
{max
i∈{1,...,s} ess supx∈�
fi (x, w(x) + η) : w ∈ L∞ (�; R
d)
,
∫
�
w(x) dx = 0, Aw = 0
}.
We will see in the following section that this result is not directly applicable to Hutchin-son’s model of ionic polycrystals, which is our focus here. It turns out (see Theorem 1below) that in this case the effective yield set can be described in a similar way bymeans of variational principles adapted to this setting.
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Appl Math Optim
3 Ionic Polycrystals
The focus of this section is to characterize the effective yield set for ionic polycrystals,introduced by Hutchinson in [16]. In this model each individual grain has two differenttypes of slip systems with critical stresses ±τA and ±τB , which leads to a yield set Kof the form
K ={η=(ηi j
)∈M3×3sym : |ηi i − η j j |≤τA, |ηi j |≤τB, for all i, j ∈{1, 2, 3}, i �= j
}.
(3.1)
Let R : � → SO(3) be a piecewise constant rotation field, given by
R(x) =⎛⎜⎝
cos θ(x) − sin θ(x) 0
sin θ(x) cos θ(x) 0
0 0 1
⎞⎟⎠, (3.2)
where θ(x) is the angle of rotation describing the orientation of the grain which containsthe point x ∈ � in the polycrystal occupying the region � ⊂ R
3. After computations,the pointwise constraint (1.1) on the stress field σ : � → M
3×3sym becomes
⎛⎜⎜⎜⎜⎝
σ11 cos2 θ+σ12 sin 2θ+σ22 sin2 θ σ22−σ112 sin 2θ+σ12 cos 2θ σ13 cos θ+σ23 sin θ
σ22−σ112 sin 2θ+σ12 cos 2θ σ11 sin2 θ−σ12 sin 2θ+σ22 cos2 θ −σ13 sin θ+σ23 cos θ
σ13 cos θ+σ23 sin θ −σ13 sin θ+σ23 cos θ σ33
⎞⎟⎟⎟⎟⎠
(x)∈ K ,
where σi j (i, j ∈ {1, 2, 3}) are the components of the stress field. Taking into accountthe specific form (3.1) of the yield set, this can be written in the form
σ(x) ∈{η ∈ M
3×3sym : fi (x, η) ≤ 1 for i ∈ {1, . . . , 6}
}, (3.3)
where the functions fi : � × M3×3sym → [0,+∞)(i = 1, . . . , 6) are given by the
It is easy check that each fi (i ∈ {1, . . . , 6}) satisfies (2.4) and (2.5). However, thecoercivity condition (2.6) does not hold. This is precisely because in Hutchinson’smodel we are dealing with a deficient supply of slip systems. Since (2.6) is a keyhypothesis in the proof of the characterization (2.7) of the yield set in [4], the variationalcharacterization of the effective yield set for the model under consideration here doesnot follow from the analysis in that paper. Our strategy to overcoming this drawbackis to modify the yield set of the basic crystal by imposing additional constraints, andthen to show that the effective yield set can in fact be completely characterized bymeans of a family of variational principles parametrized by the corresponding criticalshear stresses. Precisely, for each m ∈ N, we introduce the modified yield sets
K (m) ={η ∈ M
3×3sym : |ηi i − η j j | ≤ τA, |ηi j | ≤ τB, for all i, j ∈ {1, 2, 3},
×i �= j, |tr(η)| ≤ m}
, (3.10)
where tr(η) stands for the trace of the matrix η ∈ M3×3sym . If R is the rotation field
defined by (3.2), the pointwise constraint on the stress field σ : � → M3×3sym acting on
the polycrystal occupying the domain � and whose individual grains have yield setK (m) reads:
σ(x) ∈ R(x)K (m) RT (x), x ∈ �. (3.11)
It is easy to see that (3.11) can be written in a form similar to (3.3), that is,
σ(x) ∈{η ∈ M
3×3sym : fi (x, η) ≤ 1 for i ∈ {1, . . . , 7}
}, (3.12)
where fi : � × M3×3sym → [0,+∞) (i = 1, . . . , 6) are defined as before, and with
f7 : � × M3×3sym → [0,+∞) given by
f7(x, η) := 1
m|η11 + η22 + η33| . (3.13)
It is easy immediate that f7 satisfies our hypotheses (2.4) and (2.5). We claim that(2.6) also holds (with s = 7), that is, there exists c > 0 such that
7∑i=1
fi (x, η) ≥ c|η| for a.e. x ∈ � and all η ∈ M3×3sym . (3.14)
The computations are elementary, but we include them below for the convenienceof the reader. First, note that we have
|η12| =∣∣∣∣(
η12 cos 2θ+ η22−η11
2sin 2θ
)cos 2θ+
(η12 sin 2θ+ η11−η22
2cos 2θ
)sin 2θ
∣∣∣∣
≤∣∣∣∣η12 cos 2θ + η22 − η11
2sin 2θ
∣∣∣∣+∣∣∣∣η12 sin 2θ + η11 − η22
2cos 2θ
∣∣∣∣ .
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Appl Math Optim
Thus,
|η12| ≤ τB f4(x, η) + τA
2f1(x, η) for a.e. x ∈ � and all η ∈ M
3×3sym . (3.15)
Similarly,
|η13| = |(η13 cos θ + η23 sin θ) cos θ − (η23 cos θ − η13 sin θ) sin θ |≤ |η13 cos θ + η23 sin θ | + |η23 cos θ − η13 sin θ |,
and
|η23| = |(η23 cos θ − η13 sin θ) cos θ + (η13 cos θ + η23 sin θ) sin θ |≤ |η23 cos θ − η13 sin θ | + |η13 cos θ + η23 sin θ |.
Hence,
|η13| ≤ τB ( f5(x, η) + f6(x, η)) , (3.16)
and
|η23| ≤ τB ( f5(x, η) + f6(x, η)) , (3.17)
for a.e. x ∈ �, and all η ∈ M3×3sym . Since
|η11−η22| ≤ |(η11−η22) cos 2θ+2η12 sin 2θ |+2
∣∣∣∣η12 cos 2θ+ (η22−η11)
2sin 2θ
∣∣∣∣ ,
we have
|η11 − η22| ≤ τA f1(x, η) + 2τB f4(x, η) for a.e. x ∈ � and all η ∈ M3×3sym .
|η11+η22−2η33| ≤ τA f2(x, η)+τA f3(x, η) for a.e. x ∈� and all η∈M3×3sym .
(3.19)
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Appl Math Optim
Taking into account (3.13), we obtain that
|η11 + η22| ≤ 2m
3f7(x, η) + τA
3f2(x, η) + τA
3f3(x, η), (3.20)
for a.e. x ∈ �, and all η ∈ M3×3sym . In view of (3.18) and (3.20), we find
|η11|+|η22| ≤ 2m
3f7(x, η)+ τA
3f2(x, η)+ τA
3f3(x, η)+τA f1(x, η)+2τB f4(x, η)
(3.21)
for a.e. x ∈ �, and all η ∈ M3×3sym . Further, since
|η33| ≤ 1
3|η11 + η22 − 2η33| + 1
3|η11 + η22 + η33| ,
(3.13) and (3.19) give
|η33| ≤ τA
3f2(x, η) + τA
3f3(x, η) + m
3f7(x, η) for a.e. x ∈ � and all η ∈ M
3×3sym .
(3.22)
Overall, (3.15), (3.16), (3.17), (3.21), and (3.22) give
|η| ≤ |η11| + |η22| + |η33| + √2|η12| + √
2|η13| + √2|η23|
≤ m f7(x, η) + 2√
2τB( f6(x, η) + f5(x, η)) + (2 + √2)τB f4(x, η)
+ 2τA
3( f3(x, η) + f2(x, η)) +
(1 +
√2
2
)τA f1(x, η)
≤ max
{m, (2 + √
2)τB,
(1 +
√2
2
)τA
}7∑
i=1
fi (x, η),
for a.e. x ∈ � and all η ∈ M3×3sym .
Thus, (3.14) holds, with c =(
max{
m, (2 + √2)τB ,
(1 +
√2
2
)τA
})−1.
The remainder of this section is devoted to the variational characterization of theeffective yield set in Hutchinson’s model. To simplify the presentation, we will workin � = Q = (0, 1)3 - the unit cube in R
3. The Definition (1.3) of the effective yieldset becomes
Keff :=
⎧⎪⎨⎪⎩
σ :=∫
Q
σ(x)dx : Div σ = 0 in Q, and σ(x) ∈ R(x)K RT (x), x ∈ Q
⎫⎪⎬⎪⎭
,
(3.23)
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Appl Math Optim
where K is defined by (3.1). We have already established (see (3.3)) that the pointwiseconstraint on the stress (1.1) may be written in the form
σ(x) ∈{η ∈ M
3×3sym : fi (x, η) ≤ 1 for all i = 1, . . . , s
}, x ∈ Q, (3.24)
where s = 6, and where fi : Q × M3×3sym → R (i = 1, . . . , 6) are defined by
the formulas (3.4) through (3.9). These are Carathéodory integrands satisfying ourhypotheses (2.4) and (2.5). However, the coercivity condition (2.6) does not hold(with s = 6), which makes the characterization (2.7) inapplicable. It is worth notingthat if the yield set of the basic crystal is modified to be K (m), given by (3.10), ratherthan K (given by (3.1)), then (1.1) may be written in the form (3.24) with s = 7, wherethe additional function f7 (which depends on m) is defined in (3.13). In view of ourcomputations above, (2.6) does hold in this case, and thus (see [4]) the effective yieldset of the modified polycrystal admits the variational characterization
K (m)eff =
{η ∈ M
3×3sym : f m,eff∞ (η) ≤ 1
}, (3.25)
where
f m,eff∞ (η) := inf
⎧⎪⎨⎪⎩
maxi∈{1,...,7} ess sup
x∈�
fi (x, w(x) + η) : w ∈ L∞ (Q; M
3×3sym
),
∫
Q
w(x) dx = 0, Div w = 0
⎫⎪⎬⎪⎭
.
Note that the dependence on m above is realized through f7 only.The next result gives a characterization of the effective yield set of a ionic polycrystal
in terms of the family of variational principles{
f m,eff∞}
defined above.
Theorem 1 Let K and Keff be given by (3.1) and (3.23), respectively. Then
Keff ={η ∈ M
3×3sym : ∃ m ∈ N s.t. f m,eff∞ (η) ≤ 1
}. (3.26)
Proof First, note that in view of (3.24), with fi : Q × M3×3sym → R (i = 1, . . . , 6)
defined by (3.4)–(3.9), we have
Keff ={η ∈ M
3×3sym : there exists σ ∈ L∞ (
Q; M3×3sym
)such that η =
∫
Q
σ(x)dx,
Div σ = 0 in Q, fi (x, σ (x)) ≤ 1 for a.e. x ∈ Q, i = 1, . . . , 6
}.
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Appl Math Optim
Equivalently,
Keff ={η ∈ M
3×3sym : there exists σ ∈ L∞ (
Q; M3×3sym
)such that
∫
Q
σ(x)dx = 0,
Div σ = 0 in Q, fi (x, σ (x) + η) ≤ 1 for a.e. x ∈ Q, i = 1, . . . , 6
}.
(3.27)
Let η ∈ Keff . There exists σ ∈ L∞(Q; M3×3sym ) such that
∫Q σ(x)dx = 0, Div σ = 0
in Q, and with fi (x, σ (x) + η) ≤ 1 for LN −a.e. x ∈ Q, and all i = 1, . . . , 6. Thus,
ess supx∈Q
fi (x, σ (x) + η) ≤ 1 for every i ∈ {1, . . . , 6}.
Let m := [3(‖σ‖L∞(Q;M3×3sym )
+ |η|)] + 1, where [x] stands for the integer part of thereal number x . For a.e. x ∈ Q, we have
|tr(σ (x) + η)| ≤3∑
i=1
|σi i (x) + ηi i | ≤ 3
(‖σ‖
L∞(
Q;M3×3sym
) + |η|)
< m,
and thus, ess supx∈Q f7(x, σ (x) + η) ≤ 1. Overall, we have obtained that
maxi∈{1,...,7} ess supx∈� fi (x, σ (x) + η) ≤ 1, which gives that f m,eff∞ (η) ≤ 1.
Conversely, let η ∈ M3×3sym be such that there exists m ∈ N with f m,eff∞ (η) ≤ 1.
Since
f m,eff∞ (η)
= inf
⎧⎪⎨⎪⎩
max
{ess sup
x∈�
fi (x, σ (x)+η) (i ∈ {1, . . . , 6}), 1
mess sup
x∈�
|tr (σ (x)+η)|}
:
σ ∈ L∞ (Q; M
3×3sym
),
∫
Q
σ(x) dx = 0, Div σ = 0 in Q
⎫⎪⎬⎪⎭
,
there exists a sequence {σm,n}n∈N ⊆ L∞(Q; M3×3sym ) such that Div σm,n = 0 in Q,∫
Q σm,n(x)dx = 0 for all n ∈ N, and
max
{ess sup
x∈�
fi (x, σm,n(x) + η) (i ∈ {1, . . . , 6}), 1
mess sup
x∈�
|tr (σm,n(x) + η)|}
→ f m,eff∞ (η) (3.28)
as n → ∞.
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Appl Math Optim
The coercivity condition (2.6) implies that the sequence {σm,n}n∈N is bounded inL∞(Q; M
3×3sym ). Thus, there exists a subsequence of {σm,n}n∈N (not relabelled) and
σm ∈ L∞(Q; M3×3sym ) such that σm,n ⇀ σm weakly* in L∞(Q; M
3×3sym ) as n → ∞,
with Div σm = 0, and∫
Q σm(x)dx = 0. Let x ∈ Q be a Lebesgue point for each ofthe fi (·, σm(·) + η), i = 1, . . . , 6. By Proposition 1 we deduce that for sufficientlysmall r > 0 we have∫
B(x,r)
fi (y, σm(y) + η)dy ≤ lim infn→∞
∫
B(x,r)
fi (y, σm,n(y) + η)dy, i = 1, . . . , 6.
The integral on the right hand side is bounded above by
|B(x, r)| max
{ess sup
x∈�
fi (x, σm,n(x)+η) (i ∈ {1, . . . , 6}), 1
mess sup
x∈�
|tr (σm,n(x)+η)|},
and we deduce by (3.28) that
1
|B(x, r)|∫
B(x,r)
fi (y, σm(y) + η)dy ≤ f m,eff∞ (η) ≤ 1, i = 1, . . . , 6. (3.29)
Letting r → 0+, since almost every point x ∈ Q is a Lebesgue point for allfi (·, σm(·) + η), i = 1, . . . , 6, we have that fi (x, σm(x) + η) ≤ 1 for a.e.x ∈ Q, i = 1, . . . , 6. Taking (3.27) into account, we conclude that η ∈ Keff . ��
4 A �-Convergence Result
In this section we prove a �-convergence result for the class of supremal functionalsgoverning the variational principles f m,eff∞ as the parameter m tends to ∞.
Theorem 2 Let s be a positive integer, and for i = 1, 2, . . . , s, let fi : � × Rd →
[0,+∞) be Carathéodory integrands satisfying (2.4), (2.5), and (2.6). Consider thesequence {Fm} of functionals Fm : L∞(�; R
d) → [0,+∞] defined by
Fm(w)
=
⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩
max
{ess sup
x∈�
fi (x, w(x)) (i ∈ {1, . . . , s − 1}),1m ess sup
x∈�
fs(x, w(x))
}if w ∈ L∞ (
�; Rd) ∩ kerA
+∞ otherwise,
and let F∞ : L∞(�; Rd) → [0,+∞] be defined by
F∞(w) =⎧⎨⎩
maxi∈{1,...,s−1} ess sup
x∈�
fi (x, w(x)) if w ∈ L∞(�; Rd) ∩ kerA
+∞ otherwise.
123
Appl Math Optim
Then
(i) for every w ∈ L∞(�; Rd) and {wm} ⊂ L∞(�, R
d) such that wm∗⇀ w weakly*
in L∞(�; Rd) we have
F∞(w) ≤ lim infm→∞ Fm(wm); (4.1)
(ii) for every w ∈ L∞(�; Rd) there exists a sequence {wm} ⊂ L∞(�; R
d) such thatwm → w in L∞(�; R
d), and limm→∞ Fm(wm) = F∞(w).
In particular, �(L∞(�; Rd)) − limm→∞ Fm = F∞.
Proof Let w ∈ L∞(�; Rd) and {wm} ⊂ L∞(�; R
d) be such that wm∗⇀ w in
L∞(�; Rd).Without loss of generality, and after extracting a subsequence if necessary,
we may assume that
lim infm→∞ Fm(wm) = lim
m→∞ Fm(wm) < +∞.
Note that in view of our growth condition (2.5), we have that fi (·, w(·)) ∈ L1(�). Letx ∈ � be a Lebesgue point for fi (·, w(·)), i ∈ {1, . . . , s −1}. For any ball Br (x) ⊂ �
with sufficiently small radius we have, in view of Proposition 1,
∫
Br (x)
fi (y, w(y))dy ≤ lim infm→∞
∫
Br x)
fi (y, wm(y))dy
≤ lim infm→∞
∫
Br (x)
‖ fi (·, wm(·))‖L∞(�)dy,
for every i ∈ {1, . . . , s − 1}. Thus,
1
|Br (x)|∫
Br (x)
fi (y, w(y))dy ≤ lim infm→∞ ‖ fi (·, wm(·))‖L∞(�).
Since almost every x ∈ � is a Lebesque point for fi (·, w(·)), passing to the limitr → 0+ in the above inequality yields
fi (x, w(x)) ≤ lim infk→∞ ‖ fi (·, wm(·))‖L∞(�) for a.e. x ∈ �.
We deduce that
‖ fi (·, w(·))‖L∞(�) ≤ lim infm→∞ ‖ fi (·, wm(·))‖L∞(�),
for every i ∈ {1, . . . , s − 1}. Thus,
123
Appl Math Optim
‖ fi (·, w(·))‖L∞(�) ≤ lim infm→∞ max
{ess sup
x∈�
fi (x, wm(x)) (i ∈ {1, . . . , s − 1}),1
mess sup
x∈�
fs(x, wm(x))
}
= limm→∞ Fm(wm), for all i ∈ {1, . . . , s − 1}.
Hence, (4.1) holds.To prove (ii), let w ∈ L∞(�; R
d), and note that since we only need to prove that
lim supm→∞
Fm(wm) ≤ F∞(w),
we may assume, without loss of generality, that F∞(w) < +∞, and thus w ∈L∞(�; R
d). It is now easy to show that the constant sequence {wm} = {w} is arecovery sequence for the �-limit. Indeed, since by the growth condition (2.5) wehave fs(·, w(·)) ∈ L∞(�), it follows that
max
{ess sup
x∈�
fi (x, w(x)) (i ∈ {1, . . . , s − 1}), 1
mess sup
x∈�
fs(x, w(x))
}
= maxi∈{1,··· ,s−1} ess sup
x∈�
fi (x, w(x)),
for all m ∈ N sufficiently large. Thus,
limm→∞ Fm(wm) = lim
m→∞ Fm(w) = F∞(w),
which concludes the proof. ��It remains an open problem to determine whether the effective yield set for the
Hutchinson’s model can be characterized in terms of a variational principle involvingsupremal functionals which only depend on the mappings f1, f2, . . . , f6 as definedin the previous section. We conjecture that
η ∈ Keff if and only if f eff∞ (η) ≤ 1,
where f eff∞ (η) is given in terms of the �-limit, F∞, defined in the statement of Theo-rem 2 (with s = 6 and A = Div) by the formula
f eff∞ (η) := inf
{F∞(w(·)+η) : w ∈ L∞(�; M
3×3sym ),
∫
�
w(x) dx = 0, Div w = 0
}.
Acknowledgments The research of F. Abdullayev was partially funded by the National Science Foun-dation under Grant No. DMS-1156393. The research of M. Bocea was partially funded by the NationalScience Foundation under Grants No. DMS-0806789 and DMS-1156393. M. Mihailescu has been par-tially supported by the CNCS-UEFISCDI Grant No. PN-II-ID-PCE-2011-3-0075 “Analysis, Control andNumerical Approximations of Partial Differential Equations”.
123
Appl Math Optim
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