Infinite-order laminatesin a model in crystal plasticity

January 9, 2008

Nathan Albin1, Sergio Conti2, and Georg Dolzmann3

1 Applied and Computational MathematicsCalifornia Institute of Technology, Pasadena, CA 91101, USA

2 Fachbereich Mathematik, Universitat Duisburg-Essen47057 Duisburg, Germany

3 NWF-I Mathematik, Universitat Regensburg93040 Regensburg, Germany

We consider a geometrically nonlinear model for crystal plas-ticity in two dimensions, with two active slip systems andrigid elasticity. We prove that the rank-one convex envelopeof the condensed energy density is obtained by infinite-orderlaminates, and express it explicitly via the 2F1 hypergeometricfunction. We also determine the polyconvex envelope, leadingto an upper and a lower bound on the quasiconvex envelope.The two bounds differ by less than 2%.

1 Introduction

Plastic deformation of single crystals leads to the spontaneous formation ofmicrostructures, which largely influence the macroscopic material response.Possible origins for microstructure are both the interplay of different slipsystems and geometrical effects, i.e., the interplay of one slip system withrotations. Recent progress in the analysis of plastic microstructure has beenlargely based on variational formulations, starting from the work by Ortiz andRepetto [30], see also [25, 8]. This is admissible if one assumes monotonicity,leading to the so-called deformation theory of plasticity; or more in generalfor short time intervals (after discretization in time). After minimizing inthe internal variables one obtains a variational integral of the form

∫

Ω

W (∇u)dx

possibly complemented by additional external forces and boundary condi-tions. Here W contains both energetic and dissipative terms, see [30, 8, 26]for details.

1

The discrete nature of crystalline slip systems makes the energy densityW not convex, which in turn leads to the spontaneous formation of mi-crostructures. The theory of relaxation shows that the macroscopic materialbehavior can be studied by replacing W with its quasiconvex envelope, whichis defined as the largest quasiconvex function not larger than W ,

W qc(F ) = supV (F ) : V quasiconvex, V (G) ≤W (G) for all G ∈ Rn×n .

(1.1)We recall that a function W : R

n×n → R ∪ ∞ is quasiconvex if

W (F ) ≤ 1

|Ω|

∫

Ω

W (F + ∇ϕ) dx for all ϕ ∈W 1,∞0 (Ω; Rn) (1.2)

(whenever the integral exists) for all bounded, open, nonempty sets Ω ⊂ Rn

such that |∂Ω| = 0 , see [27, 5, 19, 13, 28, 15]. This definition corresponds tooptimizing locally (i.e., at any material point) over all possible microstruc-tures, which are here described by all possible Lipschitz continuous functionsϕ which vanish on the boundary. The definition is clearly very implicit, andthus difficult to handle directly; therefore it is in practice often replaced byeither of the two concepts of rank-one convexity and polyconvexity. A func-tion W : R

n×n → R ∪ ∞ is rank-one convex if it is convex along rank-onelines, in the sense that t 7→ W (F + ta ⊗ b) is convex for all a, b ∈ R

n. Afunction W : R

n×n → R∪∞ is polyconvex if it can be written as a convexfunction of its argument and its minors, i.e., for n = 2, if there is a convexfunction g : R

5 → R ∪ ∞ such that W (F ) = g(F, detF ).In a geometrically linear setting, quasiconvexity often reduces to the much

simpler concept of convexity. Assuming convex potentials, a very satisfactorygeneral theory can be obtained using methods from convex analysis and thetheory of functions of bounded deformation, see, e.g., [32]. Similar simpli-fications hold in the case of microstructure formation; indeed, for a modelof crystal plasticity without hardening the quasiconvex envelope W qc turnsout to be convex [11]. The analysis in [11] involved a realistic number ofslip systems (e.g., the result included the case of the 12 slip systems with fccsymmetry, appropriate for many metals) but used in a substantial way thelinear treatment of rotations, as well as the convexity of the relaxed problem.

In a finite-deformation context, it is well known that convexity contrastswith invariance under rotations. Whereas abstract theory shows that quasi-convexity is the appropriate concept, in practice this turns out to be muchmore difficult to handle. Ortiz and Repetto have shown that energy densi-ties describing a system with a single slip system in finite deformation lackquasiconvexity, and therefore lead to spontaneous microstructure formation

2

in the form of laminates, a fact known as geometric softening [30]. A two-dimensional energy density with a single slip system with linear hardeningand with a polyconvex elastic part was proposed and shown also to lack qua-siconvexity in [8]. In [12] an explicit formula for the quasiconvex envelope ofa simplification of that model, based on rigid elasticity and no self-hardening,was obtained. Work has also been devoted to numerical approximations; inparticular, in [7] the model from [8] was studied numerically. An approxi-mate numerical relaxation for the same model was obtained and integratedin a macroscopic finite-element computation in [24]. A finer analysis of thequasiconvex envelope of the same energy density is now under way, prelim-inary results are presented in [6]. In all these works, containing a propertreatment of rotations, only a single slip system was considered.

We study here for the first time the interplay among several slip systemswithin a geometrically nonlinear model formulation. Precisely, we focus on amodel with two slip systems in two dimensions, with rigid elasticity and noself-hardening. Our model,

W (F ) =

|γ| if F = Q(Id + γe1 ⊗ e2) for some γ ∈ R , Q ∈ SO(2) ,

|γ| if F = Q(Id + γe2 ⊗ e1) for some γ ∈ R , Q ∈ SO(2) ,

∞ otherwise ,

(1.3)

is a direct generalization to two slip systems of the one considered in [12].We explicitly determine the rank-one convex and the polyconvex envelopeof W . The envelopes are defined in analogy to (1.1) as the largest rank-oneconvex (polyconvex) functions not larger than W .

Theorem 1.1. The rank-one convex envelope W rc of W defined in (1.3) isgiven by

W rc(F ) =

(λ2 − λ1)(F ) if detF = 1,min|Fe1|, |Fe2| ≤ 1,

ψ(|Fe1|, |Fe2|) if detF = 1, 1 ≤ |Fe1| ≤ |Fe2|,ψ(|Fe2|, |Fe1|) if detF = 1, 1 ≤ |Fe2| ≤ |Fe1|,∞ if detF 6= 1,

(1.4)

where

ψ(α, β) =

∫ α

1

2s2

√s4 − 1

ds+1

α

(√α2β2 − 1 −

√α4 − 1

). (1.5)

Here λ1(F ) and λ2(F ) denote the singular values of F , i.e., the orderedeigenvalues of U in the polar decomposition F = QU , Q ∈ SO(2), U = UT .

3

They are identified uniquely by the conditions

λ21(F ) + λ2

2(F ) = |F |2 , λ1(F )λ2(F ) = detF , λ2 ≥ |λ1| . (1.6)

Theorem 1.1 is proven in Section 2 (upper bound) and Section 3 (lowerbound).

Remark 1.2. Using standard formulas for hypergeometric functions (seefor example Chapter 15 of [1]) one finds the following representation of theintegral above in terms of Gauss’ hypergeometric function 2F1:

ψ(α, α) =

∫ α

1

2s2

√s4 − 1

ds = 2α 2F1

(−1

4,1

2;3

4;α−4

)− 2

√π Γ(

34

)

Γ(

14

) .

The hypergeometric function 2F1 is defined through its power series via therising factorial (k)n = k(k + 1)(k + 2) · · · (k + n− 1) as

2F1(a, b; c; z) =∞∑

n=0

(a)n(b)n

(c)n

· zn

n!.

We also determine the polyconvex envelope, as given by the followingtheorem proved in Section 5.

Theorem 1.3. The polyconvex envelope W pc of W defined in (1.3) is givenby

W pc(F ) = maxθ∈[0,π/2]

√|F |2 + 2|Fe1 · Fe2| sin(2θ) + 2 cos(2θ) − 2 cos θ . (1.7)

Further, we show that these two bounds give a bound on the quasiconvexenvelope, in the sense that

W rc(F ) ≤W qc(F ) ≤W pc(F ) for all F ∈ R2×2 . (1.8)

The two bounds however differ, even if the difference is quantitatively rathersmall; details are discussed in Section 6 below.

The proof of Theorem 1.1 is as usual based on the construction of an ap-propriate laminate which realizes the relaxed energy; physically, this laminategives an indication for the expected microstructure. Almost all examples ofquasiconvex envelopes use the construction of suitable finite-order laminates,see, e.g., [22, 16, 14]. In this case, however, the appropriate laminate is notonly partially supported at infinity (if one considers an appropriate comple-tion of the domain, as is usual in problems with linear growth), but it cannotbe a finite-order laminate. Precisely, no finite-order lamination convex enve-lope W lc,n agrees with the rank-one convex envelope W rc (see Section 4 fora precise definition of the lamination convex envelope W lc,n).

4

Theorem 1.4. Let F be such detF = 1 < min|Fe1|, |Fe2|. Then

W rc(F ) < W lc,n(F )

for all n ∈ N. For all other F and all n ≥ 1, one has W rc(F ) = W lc,n(F ).

Mechanically, this means that slip concentration is necessary, a featurealready observed in a geometrically linear context in [11], and that structureson many different scales will coexist. Analytically, analogous unbounded lam-inates have been used in [17, 18] for constructing critical solutions to ellipticequations, and in [10] to give a rank-one convex function on diagonal matri-ces with locally unbounded Hessian, as well as to obtain a simple derivationof a classical counterexample to Korn’s inequality in L1 by Ornstein.

2 Upper bound on the rank-one convex en-

velope

In this section we prove the upper bound of Theorem 1.1. Precisely, we provethe following lemma.

Lemma 2.1. Let W rc be the rank-one convex envelope of W defined in (1.3),and

Φ(F ) =

(λ2 − λ1)(F ) if detF = 1,min|Fe1|, |Fe2| ≤ 1,

ψ(|Fe1|, |Fe2|) if detF = 1, 1 ≤ |Fe1| ≤ |Fe2|,ψ(|Fe2|, |Fe1|) if detF = 1, 1 ≤ |Fe2| ≤ |Fe1|,∞ if detF 6= 1,

(2.1)

where

ψ(α, β) =

∫ α

1

2s2

√s4 − 1

ds+1

α

(√α2β2 − 1 −

√α4 − 1

).

Then W rc(F ) ≤ Φ(F ) for all F ∈ R2×2.

We first prove that W rc is finite on the smooth manifold Σ = F ∈ R2×2 :

detF = 1 of matrices with determinant one.

Lemma 2.2. Let W be given by (1.3). Then

W rc(F ) ≤ |Fe1| + |Fe2| + 1 for all F ∈ Σ .

5

Proof. We use an approximation by two-well problems. Let γ ∈ R, anddefine

Kγ = SO(2)Aγ ∪ SO(2)Bγ , Aγ = Id + γe1 ⊗ e2, Bγ = Id + γe2 ⊗ e1 .

The set Kγ corresponds to the well-studied “two-well problem”, and its rank-one convex hull Krc

γ is explicitly known [3, 31]. In order to characterize Krcγ it

suffices to choose a pair v, w ∈ R2 of linearly independent vectors such that

|Aγv| = |Bγv|, |Aγw| = |Bγw|, in our case v = (1, 1), w = (1,−1). Then

Krcγ = F ∈ R

2×2 : detF = 1 , |Fv| ≤ |Aγv| , |Fw| ≤ |Aγw| .This implies that

W rc(F ) ≤ W (Aγ) = W (Bγ) = |γ| for all F ∈ Krcγ .

Now let F ∈ Σ. We assert that if γ = |Fe1|+ |Fe2|+1 then F ∈ Krcγ . Indeed,

we have

|Fw|2 = |Fe1 − Fe2|2 ≤ (|Fe1| + |Fe2|)2

= (γ − 1)2 ≤ (γ − 1)2 + 1 = |Aγw|2 ,and similarly

|Fv|2 = |Fe1 + Fe2|2 ≤ (|Fe1| + |Fe2|)2

= (γ − 1)2 ≤ (γ + 1)2 + 1 = |Aγv|2 ,which proves the assertion.

The proof of Lemma 2.1 is based on proving a bound on the derivativeof W rc along certain lines. We start by proving that W rc is locally Lipschitzon Σ, and hence that it is differentiable almost everywhere. This is well-known for finite-valued rank-one convex functions [13, 4]; we now show howthe argument can be generalized to the case that W rc is finite only on Σ.The key observation is that separately convex (i.e., convex independentlyin each variable) functions are locally Lipschitz. We report here this resultin the quantitative version proven by Ball, Kirchheim and Kristensen [4,Lemma 2.2].

Lemma 2.3 (Lemma 2.2 of [4]). Let ξ0 ∈ Rn and r > 0. If f : B2r(ξ0) → R

is separately convex then

Lip(f ;Br(ξ0)) ≤n

rosc(f ;B2r(ξ0))

where for any S ⊂ Rn,

osc(f ;S) = sup|f(ξ) − f(η)| : ξ, η ∈ S .

6

Using this result, we show that W rc is locally Lipschitz on Σ. Exactly thesame argument applies to any rank-one convex function which is finite-valuedon Σ.

Lemma 2.4. Let W be given in (1.3), and let F0 ∈ Σ. Then there existc, r > 0 such that for all F1, F2 ∈ Br(F0) ∩ Σ

∣∣W rc(F1) −W rc(F2)∣∣ ≤ c|F1 − F2| .

Proof. Define the map g : R3 → Σ by

g(x, y, z) = F0(Id + xR1)(Id + yR2)(Id + zR3)

where

R1 = e1 ⊗ e2, R2 = e2 ⊗ e1, R3 = (e1 + e2) ⊗ (e1 − e2),

and the map f : R3 → R by f = W rc g (f is finite-valued by Lemma 2.2).

Note that for x1, x2, y, z ∈ R

rank(g(x1, y, z) − g(x2, y, z)) ≤ 1

and similarly in the other two coordinate directions. Since W rc is rank-oneconvex, f is separately convex.

Also note that g is a diffeomorphism in a neighborhood of the origin.Indeed, the partial derivatives in x, y and z are F0R1, F0R2 and F0R3,respectively, and span the tangent plane to Σ at F0. Thus, by the implicitfunction theorem, there exists an r > 0 such that g is invertible in Br(F0)∩Σand on this set, W rc = f g−1.

Since W is non-negative, it follows that W rc is also non-negative. Onthe bounded set S = g−1(Br(F0)∩Σ), this observation together with the in-equality given in Lemma 2.2 implies that the oscillation osc(f ;S) is bounded.Thus, by Lemma 2.3, f is Lipschitz in a neighborhood of the origin. NearF0, W

rc|Σ is the composition of two Lipschitz functions and is therefore Lip-schitz.

Proof of Lemma 2.1. If detF 6= 1 there is nothing to prove, hence we canassume detF = 1. If |Fe1| ≤ 1 the result follows from [12]; for convenienceof the reader we give here a short, self-contained proof. The key observationis that the function

G 7→ (λ2 − λ1)(G) =√|G|2 − 2 detG

7

has a particularly simple form on rank-one lines originating from the identity,namely, if a ∈ R

2, |a| = 1, then

(λ2 − λ1)(Id + ta⊥ ⊗ a) = |t|

(here and below a⊥ = (−a2, a1)). Further, for all matrices Gγ = Id+γe2⊗e1we immediately obtain (λ2−λ1)(Gγ) = |γ| = W (Gγ). These two observationsimply the assertion for all matrices of the form Ft = Q(Id + ta ⊗ a⊥) withQ ∈ SO(2) and |Fte1| ≤ 1. It remains to show that any matrix F withdetF = 1 and |Fe1| ≤ 1 can be written in this form. In order to do so,fix F ∈ Σ and let a ∈ R

2 be such that |a| = |Fa| = 1. Such an a exists,since detF = 1 and |Fe1| ≤ 1, implying |Fe2| ≥ 1. Let Q ∈ SO(2) besuch that QTFa = a. Let c1,2 ∈ R be defined by QTFa⊥ = c1a + c2a

⊥.Since 1 = detF = QTFa ∧ QTFa⊥, we have c2 = 1 (we identify v ∧ w withthe scalar (v ∧ w) · (e1 ∧ e2) = v1w2 − v2w1, for any v, w ∈ R

2). HenceQTF = Id + c1a ⊗ a⊥, which proves the assertion with t = c1. The case|Fe2| ≤ 1 is analogous.

Now we turn to the case of matrices in the set

Σ+ = F ∈ R2×2 : detF = 1 , |Fe1| > 1 , |Fe2| > 1 .

We first prove the following fact which provides the basic step of our con-struction. Consider the rank-one line along the coordinate axis |Fe1| = constgiven by

Ft = F (Id + te1 ⊗ e2) . (2.2)

We assert that

W rc(Ft) ≤W rc(F ) + |t| |Fe1| for all t ∈ R (2.3)

and that the same statement holds after interchanging the indices. To prove(2.3) we remark that Fte1 = Fe1 and Fte2 = Fe2 + tFe1, which implies

|Fte2|2 = |Fe2|2 + 2tFe1 · Fe2 + t2|Fe1|2 . (2.4)

For t ∈ R and k ∈ N \ 0, the fact that W rc is rank-one convex implies that

W rc(Ft) ≤k − 1

kW rc(F ) +

1

kW rc(Fkt) . (2.5)

In the limit as k → ∞, |Fkte1|/k = |Fe1|/k → 0 and (2.4) shows that|Fkte2|/k → |t| |Fe1|. Thus, by Lemma 2.2, we have

lim supk→∞

W rc(Fkt)

k≤ lim

k→∞

|Fkte1| + |Fkte2| + 1

k= |t| |Fe1| .

8

Taking k → ∞ in (2.5) proves (2.3). The analogous inequality follows whenthe roles of e1 and e2 are reversed.

We apply this assertion once in each of the two directions |Fei| = constto a matrix of the form

G(F, s, t) = F (Id + te1 ⊗ e2)(Id + se2 ⊗ e1) (2.6)

and obtain in view of

|F (Id + te1 ⊗ e2)e2| = |Fe2 + tFe1| =√|Fe2|2 + 2tFe1 · Fe2 + t2|Fe1|2 ,

that

W rc(G(F, s, t)) ≤W rc(F ) + |t| |Fe1|+ |s|

√|Fe2|2 + 2tFe1 · Fe2 + t2|Fe1|2 .

(2.7)

We use this estimate to show W rc ≤ Φ.

Case 1: The columns of F have equal length. Let F ∈ Σ with |Fe1| =|Fe2| = α > 1. Fix ǫ > 0 and define Fǫ = G(F, s(ǫ), t(ǫ)) where s(ǫ) and t(ǫ)are real numbers chosen so that

|Fǫe1| = |Fǫe2| = α + ǫ .

That is, we choose t and s to solve

(1 + st)2α2 + 2s(1 + st)Fe1 · Fe2 + s2α2 = (α+ ǫ)2

α2 + 2tFe1 · Fe2 + t2α2 = (α+ ǫ)2 .

Without loss of generality, we may assume Fe1 · Fe2 ≥ 0 (otherwise taket and s negative in what follows). Then we take t = t(ǫ), s = s(ǫ) as thepositive roots of the polynomials above. By differentiating in ǫ, it followsthat

t(ǫ) =α√α4 − 1

ǫ+ o(ǫ), s(ǫ) =α√α4 − 1

ǫ+ o(ǫ) .

Then by (2.7)

W rc(Fǫ) ≤W rc(F ) + |t(ǫ)| |Fe1| + |s(ǫ)| |Fe2| + o(ǫ)

= W rc(F ) +2α2

√α4 − 1

ǫ+ o(ǫ) .

Taking ǫ to zero, we see that

lim supǫ→0+

W rc(Fǫ) −W rc(F )

ǫ≤ 2α2

√α4 − 1

. (2.8)

9

It is now convenient to interpret the values of W rc as a function of α =|Fe1| = |Fe2|. We consider the case Fe1 · Fe2 ≥ 0; the remaining case isanalogous. For α ≥ 1 we define

H(α) =

(√(α2 + 1)/2

√(α2 − 1)/2√

(α2 − 1)/2√

(α2 + 1)/2

).

(In the case Fe1 · Fe2 ≤ 0, the off-diagonal elements are defined to be neg-ative.) We define the function f : [1,∞) → R by f(α) = W rc(H(α)). Wefirst show that W rc(F ) = f(|Fe1|) for all F ∈ Σ such that |Fe1| = |Fe2| andFe1 ·Fe2 ≥ 0. To see this, it suffices to observe that by polar decomposition,there exists a rotation Q ∈ SO(2) with F = QH . Since the two columnshave equal length α, and since their scalar product is positive, H must coin-cide with H(α). By the rotational invariance of W , which implies the sameinvariance for W rc, we obtain W rc(F ) = W rc(H) = f(α).

It remains to determine the function f . By the definition of H and byLemma 2.4, f is locally Lipschitz on (1,∞) and is therefore almost every-where differentiable. Note that Fe1 · Fe2 = He1 ·He2 =

√α4 − 1 > 0 so for

sufficiently small ǫ > 0, the matrix Fǫ in (2.8) satisfies Fǫe1 · Fǫe2 ≥ 0 andtherefore W rc(Fǫ) = f(|Fǫe1|) = f(α+ ǫ). Thus, (2.8) implies that

f ′(α) ≤ 2α2

√α4 − 1

a.e. α ∈ (1,∞) .

Let 1 < α0 < α. By the fundamental theorem of calculus, we have

f(α) = f(α0) +

∫ α

α0

f ′(ζ) dζ ≤ f(α0) +

∫ α

α0

2ζ2

√ζ4 − 1

dζ .

But f is continuous on [1,∞) with f(1) = W rc(Id) = W (Id) = 0. Moreover,the integral on the right-hand side converges as α0 → 1. Taking the limitα0 → 1 in the foregoing inequality proves W rc ≤ Φ for F ∈ Σ+∩F ∈ R

2×2 :|Fe1| = |Fe2|.

Case 2: The columns of F have different lengths. Without loss of gener-ality, we may assume 1 < |Fe1| = α < β = |Fe2|. We estimate W rc(F ) by alamination in the direction e1 ⊗ e2 supported on one matrix with columns ofequal length and the other at infinity, in the sense of (2.5). Define Ft as in(2.2). Then |Fte1| = α for all t and

|Fte2| =√β2 + 2tFe1 · Fe2 + t2α2 .

10

We choose t so that this quantity equals α. Again, we may assume thatFe1 · Fe2 ≥ 0 since otherwise we can change the sign of t. Thus, we choose

t = − 1

α2

(√α2β2 − 1 −

√α4 − 1

).

Applying (2.3) once again, we see that

W rc(F ) ≤W rc(Ft) + |t||Fte2| ≤ ψ(α, α) + |t|α ,

which implies W rc(F ) ≤ ψ(α, β) = Φ(F ).

3 Lower bound on the rank-one convex enve-

lope

In this section, we prove that the upper bound forW rc derived in the previoussection is actually rank-one convex and is therefore a lower bound. Thisimplies the desired characterization of W rc.

Lemma 3.1. The function Φ given by (2.1) is rank-one convex.

Proof. We consider for fixed F ∈ R2×2 and a, b ∈ R

2, the function

φ(t) = Φ(Ft), Ft = F + tFa⊗ b .

It suffices to show that for any F , a, b the function φ is convex in a neigh-borhood of 0. Since

det(Ft) = detF (1 + ta · b),and Φ is finite only on Σ, we see that φ is either finite for all t or finite for atmost one value of t. Thus, we may restrict our attention to the case wheredetF = 1 and b = a⊥.

We first consider separately the cases

F ∈ Σ1 =F ∈ R

2×2 : detF = 1, |Fe1| < 1 < |Fe2|,

F ∈ Σ2 =F ∈ R

2×2 : detF = 1, |Fe2| < 1 < |Fe1|,

F ∈ Σ3 =F ∈ R

2×2 : detF = 1, 1 < |Fe1| < |Fe2|,

F ∈ Σ4 =F ∈ R

2×2 : detF = 1, 1 < |Fe2| < |Fe1|

(see Figure 1). Once we have done this, it will remain only to check smooth-ness across the shared boundaries of these sets. Suppose first that F ∈

11

Figure 1: Decomposition of the domain in the proof of Lemma 3.1.

Σ1 ∪ Σ2. Then the same is true for Ft with t in a neighborhood of 0. SinceΦ coincides on this domain with the convex function

(λ2 − λ1)(F ) =√|F |2 − 2 detF =

√(F11 − F22)2 + (F12 + F21)2 (3.1)

we conclude that φ is convex on this set.Consider next F ∈ Σ3, θ ∈ R, a = (cos θ, sin θ), b = a⊥ = (− sin θ, cos θ).

We shall show that for any such matrix φ is twice differentiable at the originand φ′′(t)|t=0 ≥ 0. If this holds for all F ∈ Σ3, then φ necessarily hasnonnegative second derivative in a neighborhood of the origin and thereforeit is convex in a neighborhood of the origin.

To prove the assertion it is useful to introduce the variables

ξ(t) = |Fte1|2, η(t) = |Fte2|2 .

Starting from φ(t) = ψ(ξ1/2(t), η1/2(t)), we compute

dφ

dt=∂ψ

∂α

1

2ξ1/2

dξ

dt+∂ψ

∂β

1

2η1/2

dη

dt,

with ∇ψ evaluated at α = ξ1/2(t), β = η1/2(t). Differentiating a second time,and rearranging terms,

d2φ

dt2=

(1

4α2

∂2ψ

∂α2− 1

4α3

∂ψ

∂α

)(dξ

dt

)2

+

(1

4β2

∂2ψ

∂β2− 1

4β3

∂ψ

∂β

)(dη

dt

)2

+

(1

2αβ

∂2ψ

∂α∂β

)(dξ

dt

dη

dt

)+

(1

2α

∂ψ

∂α

)(d2ξ

dt2

)+

(1

2β

∂ψ

∂β

)(d2η

dt2

).

12

Again, derivatives of ψ are evaluated at α = ξ1/2(t), β = η1/2(t).From the definition of ξ we obtain

dξ

dt

∣∣∣t=0

= 2(Fe1 · Fa)(a⊥ · e1) = −2|Fe1|2 cos θ sin θ − 2(Fe1 · Fe2) sin2 θ ,

d2ξ

dt2

∣∣∣t=0

= 2|Fa|2 (a⊥ · e1)2

= 2|Fe1|2 cos2 θ sin2 θ + 4(Fe1 · Fe2) cos θ sin3 θ + 2|Fe2|2 sin4 θ .

Analogously

dη

dt

∣∣∣t=0

= 2|Fe2|2 cos θ sin θ + 2(Fe1 · Fe2) cos2 θ ,

d2η

dt2

∣∣∣t=0

= 2|Fe1|2 cos4 θ + 4(Fe1 · Fe2) cos3 θ sin θ + 2|Fe2|2 cos2 θ sin2 θ .

From (1.5) we compute

∂ψ

∂α= δ +

1

α2γ,

∂ψ

∂β=αβ

γ, (3.2)

where γ = (α2β2 − 1)1/2 and δ = (1 − α−4)1/2. Differentiating once again,

∂2ψ

∂α2=

2

α5δ− 3α2β2 − 2

α3γ3,

∂2ψ

∂β2= − α

γ3,

∂2ψ

∂α∂β= − β

γ3.

At this point we have all ingredients to evaluate d2φ/dt2|t=0. By the aboveexpressions it is clear that it is a homogeneous polynomial of fourth order incos θ and sin θ, therefore one only has to evaluate the five coefficients. Thecoefficients can be expressed in terms of |Fe1| = α, |Fe2| = β, Fe1 ·Fe2 = γ,and (1 − α−4)1/2 = δ. A direct computation that uses γ2 = α2β2 − 1 showsthat

d2φ

dt2

∣∣∣t=0

=sin2 θ

|Fe1|5√

|Fe1|4 − 1

(c1 cos2 θ + c2 cos θ sin θ + c3 sin2 θ

), (3.3)

where

c1 = 2|Fe1|4 ,c2 = 4|Fe1|2

(√|Fe1|2|Fe2|2 − 1 −

√|Fe1|4 − 1

),

c3 =(√

|Fe1|2|Fe2|2 − 1 −√|Fe1|4 − 1

)(2√|Fe1|2|Fe2|2 − 1 −

√|Fe1|4 − 1

).

13

Rearranging terms and minimizing the parenthesis over θ we obtain

d2φ

dt2

∣∣∣t=0

=sin2 θ

2|Fe1|5√|Fe1|4 − 1

((c1 − c3) cos 2θ + c2 sin 2θ + c1 + c3)

≥ sin2 θ

2|Fe1|5√

|Fe1|4 − 1

(c1 + c3 −

√c22 + (c1 − c3)2

)

=sin2 θ

2|Fe1|5√|Fe1|4 − 1

(c1 + c3 −

√(c1 + c3)2 − (4c1c3 − c22)

).

Since |Fe2| > |Fe1|, each ci > 0. Moreover,

4c1c3 − c22 = 8|Fe1|4√|Fe1|4 − 1

(√|Fe1|2|Fe2|2 − 1 −

√|Fe1|4 − 1

)> 0 ,

so φ′′(t)|t=0 ≥ 0 and we have proved that Φ is locally rank-one convex inΣ3. The case F ∈ Σ4 is analogous and follows by interchanging indices. Forlater reference we observe that all inequalities are strict, unless sin θ = 0, i.e.,Ft = F (Id + te1 ⊗ e2), and in particular that

if F ∈ Σ3 and sin θ 6= 0 thend2φ

dt2

∣∣∣∣t=0

> 0 . (3.4)

To finish the proof that Φ is rank-one convex, we need to check the followingremaining cases on Σ corresponding to the intersection of the boundaries ofdifferent domains,

F ∈ (∂Σ1 ∩ ∂Σ3) ∪ (∂Σ3 ∩ ∂Σ4) ∪ (∂Σ4 ∩ ∂Σ2)

(see Figure 1). In fact, we will show that Φ is C1 in the set

F ∈ R2×2 : detF = 1, F /∈ SO(2) .

A final calculation at the identity completes the proof.To check the smoothness, it is convenient to use as before the variables

α = |Fe1|, β = |Fe2|. First consider the intersection (∂Σ1 ∩ ∂Σ3) \ SO(2).By construction, Φ is continuous on Σ. Moreover, the normal derivative tothe boundary in Σ1 is

∂

∂α

√α2 + β2 − 2

∣∣∣α=1

=1√β2 − 1

,

while in Σ3, recalling (3.2) the normal derivative is

∂

∂αψ(α, β)

∣∣∣α=1

=1

α2

(1√

α2β2 − 1+√α4 − 1

) ∣∣∣α=1

=1√β2 − 1

.

14

A similar computation shows smoothness at the intersection (∂Σ2 ∩ ∂Σ4) \SO(2). The smoothness across (∂Σ3 ∩ ∂Σ4) \ SO(2) follows again from (3.2)since

∂

∂αψ(α, β)

∣∣α=β

=α2

√α4 − 1

=∂

∂βψ(α, β)

∣∣α=β

.

Finally, we need to check rank-one convexity in the case F ∈ SO(2), soFt = R(Id + ta⊗ a⊥) for some R ∈ SO(2). Then for i = 1, 2,

|Ftei|2 = |ei + t(a⊥ · ei)a|2 = 1 + 2t(a⊥ · ei)(a · ei) + t2(a⊥ · ei)2 .

Since the coefficient or the linear term (a⊥ · ei)(a · ei) is the product of twofactors with opposite signs for i = 1, 2, it follows that for t sufficiently smallmin|Fte1|, |Fte2| ≤ 1, hence Ft ∈ Σ1 ∪ Σ2. Hence φ(t) = Φ(Ft) is definedthrough the convex function in (3.1) and thus convex on this interval.

Proof of Theorem 1.1. Lemma 2.1 proves that W rc ≤ Φ. To prove the con-verse inequality, we observe that by Lemma 3.1 Φ is rank-one convex, andby construction Φ ≤W , hence Φ ≤W rc. This concludes the proof.

4 Infinite-order laminates

Given a function V : R2×2 → [0,∞], its n-th lamination convex envelope

V lc,n is defined inductively by V lc,0 = V , and

V lc,n+1(F ) = infλV lc,n(F1) + (1 − λ)V lc,n(F2) : (4.1)

λ ∈ [0, 1] , rank(F1 − F2) ≤ 1 , λF1 + (1 − λ)F2 = F.

The lamination envelopes are often used as approximations to the rank-oneconvex envelope (and, as a consequence, of the quasiconvex one); in manycases it turns out that a lamination envelope of relatively low order coincideswith the rank-one and quasiconvex envelopes. We show that this is not thecase here, and that no finite-order lamination-convex envelope coincides withthe rank-one convex envelope, in the sense that

W rc(F ) < W lc,n(F )

for all matrices F with detF = 1 < min|Fe1|, |Fe2| (see Theorem 1.4).The key difficulty in the proof is that the set over which the infimum in (4.1)is taken is unbounded, and the construction of Section 2 shows that it isexactly pairs with one of (F1, F2) diverging at infinity which are relevant.

15

We restrict our attention to a compact set by introducing a cutoff M , whichwill then be chosen large enough. Given M > 1 we define

VM(F ) =

W (F ) if detF = 1,max|Fe1|, |Fe2| < M ,

W rc(F ) if detF = 1,max|Fe1|, |Fe2| = M ,

∞ otherwise

(4.2)

(see Figure 2). We observe that VM is finite only on a compact set, and thatit is lower semicontinuous. We first show that these properties are stableunder the passage to the lamination envelopes.

Lemma 4.1. Let Ψ : R2×2 → [0,∞] be lower semicontinuous and such that

KΨ = Ψ−1([0,∞[) is compact. Then all its lamination convex envelopes Ψlc,n

have the same property, i.e., Kn = (Ψlc,n)−1([0,∞[) is compact and Ψlc,n islower semicontinuous for all n. Further, all infima in the definition of thelamination convex envelope are minima.

Proof. It clearly suffices to prove the assertion for n = 1, and then proceedby induction. Let Φ = Ψlc,1. By definition,

Φ(F ) = infλΨ(F1) + (1 − λ)Ψ(F2) :

λ ∈ [0, 1], rank(F1 − F2) ≤ 1 , λF1 + (1 − λ)F2 = F .

There is nothing to show if Φ(F ) = ∞. Assume thus Φ(F ) < ∞, i.e.,F ∈ K1 = KΦ. Then we can assume F1,2 ∈ KΨ in the infimum above. Bycompactness of [0, 1] ×KΨ × KΨ, and lower semicontinuity, the infimum isactually a minimum.

We now show that if F i → F , and F i ∈ KΦ, then

F ∈ KΦ and Φ(F ) ≤ lim infi→∞

Φ(F i) .

This will imply that KΦ is closed and that Φ is lower semicontinuous on it.Let F i be a sequence with the foregoing properties and let F i

1,2, λi be such

thatΦ(F i) = λiΨ(F i

1) + (1 − λi)Ψ(F i2) (4.3)

and

λi ∈ [0, 1], rank(F i1 − F i

2) ≤ 1 , λiF i1 + (1 − λi)F i

2 = F i . (4.4)

By compactness we can, after extracting a subsequence, assume that F i1 →

F1 ∈ KΨ, F i2 → F2 ∈ KΨ, λi → λ ∈ [0, 1]; all conditions in (4.4) automati-

cally hold for the limit. In particular Φ(F ) ≤ λΨ(F1) + (1 − λ)Ψ(F2) < ∞

16

Figure 2: Restriction to a compact domain as in Formula (4.2). The dashedline represents the vertical rank-one line in the proof of Lemma 4.2.

and F ∈ KΦ. Passing to the limit in (4.3) we obtain Φ(F ) ≤ lim inf Φ(F i).This proves the assertion.

Finally, observe that if F ∈ KΦ and F1, F2 are as above, then |F | ≤max|F1|, |F2|, hence KΦ is bounded and compact.

Lemma 4.2. Let M > 1, VM as in (4.2). Then for all n ∈ N and all F ∈ Σsuch that 1 < |Fe1|, |Fe2| < M one has

W rc(F ) < V lc,nM (F ) .

Proof. For simplicity we write V for VM in this proof. Since W rc ≤ V andW rc is rank-one convex, it is clear that W rc ≤ V rc ≤ V lc,n everywhere.

To prove that the inequality is strict, we proceed by contradiction. Letn be the smallest number such that there is an F satisfying the assumptionsin the lemma with W rc(F ) = V lc,n(F ), and assume without loss of gener-ality that |Fe1| ≤ |Fe2| (i.e., F ∈ Σ3 in the notation of Section 3). SinceV lc,0(F ) = V (F ) = ∞, obviously n > 0. Then V lc,n is defined by (4.1). ByLemma 4.1 the infimum is actually a minimum. Let F1, F2, λ be such that

V lc,n(F ) = λV lc,n−1(F1) + (1 − λ)V lc,n−1(F2)

with λ ∈ [0, 1], rank(F1 − F2) = 1, λF1 + (1 − λ)F2 = F . By minimality ofn, it follows that λ ∈ (0, 1). Since W rc(F ) is rank-one convex, we obtain

V lc,n(F ) = W rc(F ) ≤λW rc(F1) + (1 − λ)W rc(F2)

≤λV lc,n−1(F1) + (1 − λ)V lc,n−1(F2) = V lc,n(F ) .

17

Therefore all inequalities must be equalities. By (3.4) the first inequality canbe an equality only if sin θ = 0 in the rank-one direction, i.e., if |F1e1| =|Fe1| = |F2e1|. Further, by the same argument, the segment [F1, F2] cannotcross the set G : |Ge1| = |Ge2|. At the same time, by minimality of nthe second inequality is strict if 1 < |F1,2e2| < M . They cannot both equalM , since ψ is strictly increasing in its arguments. Analogously, the case|F1,2e2| ≤ 1 is ruled out since by convexity the same would hold for |Fe2|.Therefore, possibly after relabeling,

|F1e2| ≤ 1 < M = |F2e2| .

By continuity, this implies that [F1, F2]∩ G : |Ge1| = |Ge2| is nonempty, acontradiction (see Figure 2).

Proof of Theorem 1.4. Let F0 be a matrix as given in the statement, and letM = |F0| + 1. We first assert that

V lc,nM (F ) ≤W lc,n(F ) (4.5)

for all n and all F ∈ S, where S = F : 1 < |Fe1| < M, 1 < |Fe2| < M.The assertion is proven by induction. For n = 0, both equal ∞. Assume theinequality to hold for n. For notational convenience we rewrite the definitionof the lamination convex envelope as

V lc,n+1(F ) = infℓ(F )|ℓ : R

2×2 → R linear, there are F1, F2 ∈ R2×2

such that ℓ(F1) = V lc,n(F1), ℓ(F2) = V lc,n(F2),

rank(F1 − F2) ≤ 1 , F ∈ [F1, F2]

(4.6)

and the same for W (with the convention inf ∅ = ∞). Fix one matrix F ∈S ∩Σ. If W lc,n+1(F ) = ∞ there is nothing to prove. Otherwise, let ℓ, F1, F2

be as in the definition of W lc,n+1; clearly F1,2 ∈ Σ. It remains to show that

V lc,n+1M (F ) ≤ ℓ(F ) . (4.7)

The key property will be that VM = W rc on Σ ∩ ∂S, which immediatelyimplies V lc,n

M = W rc on Σ ∩ ∂S for all n.We define F ′

1 = F1 if F1 ∈ S, and F ′1 to be one element of ∂S ∩ [F, F1]

otherwise, and analogously F ′2. Let k : R

2×2 → R be linear and such thatk(F ′

i ) = V lc,nM (F ′

i ), for i = 1, 2. It suffices to show that

k(F ′i ) = V lc,n

M (F ′i ) ≤ ℓ(F ′

i ) , i = 1, 2 . (4.8)

Indeed, if (4.8) holds then by linearity the same holds in the segment [F ′1, F

′2]

which contains F , and hence V lc,n+1M (F ) ≤ k(F ) ≤ ℓ(F ) and (4.7) follows.

18

To prove (4.8), we distinguish to cases. If F ′i ∈ S then F ′

i = Fi and thisfollows from the inductive assumption. Otherwise, F ′

i ∈ ∂S ∩ [F1, F2] andtherefore

k(F ′i ) = V lc,n

M (F ′i ) = W rc(F ′

i ) = V lc,n+1M (F ′

i ) ≤ ℓ(F ′i ) .

This proves the assertion (4.8) and therefore (4.5).Recalling Lemma 4.2 we obtain

W rc(F0) < V lc,nM (F0) ≤W lc,n(F0) ,

which concludes the proof.

5 Polyconvex envelope

In this section, we prove Theorem 1.3 by constructing the polyconvex enve-lope W pc of the energy W defined in (1.3). The polyconvex envelope of afunction W is defined by

W pc(F ) = supV (F ) : V polyconvex, V (G) ≤ W (G) for all G ∈ Rn×n .

(5.1)It is well-known that for finite-valued energy densities W it suffices to takethe supremum over polyaffine functions V , see, e.g., [13, Sect. 5.1.1.2]. Wefirst verify that the same is true in the case of interest here where V isfinite only on Σ. The following lemma shows that it suffices to consider thesmaller class of all affine functions. We state the result in a more generalsetting which does not rely on special properties of W .

Lemma 5.1. Let V : R2×2 → [0,∞] be such that V pc(F ) <∞ if and only if

detF = 1. Then for all F with detF = 1 we have

V pc(F ) = supℓ(F ) : ℓ ∈ P, ℓ(G) ≤ V (G) for all G ∈ R

2×2, (5.2)

where P denotes the class of affine functions

P = ℓ : R2×2 → R, ℓ(F ) = F : G+ β , G ∈ R

2×2, β ∈ R .

We recall that F : G = TrF TG =∑

ij FijGij .

Proof. Since affine functions are polyconvex, the supremum in (5.2) is takenover a smaller class than in (5.1), hence it is less than or equal to V pc.Therefore we only need to show that for any F ∈ Σ there exists an ℓ ∈ Pwith ℓ(F ) = V pc(F ), ℓ ≤ V .

19

Let ψ : R5 → [0,∞] be defined by

ψ(x) = supg(x), g : R5 → [0,∞], g convex, g(F, detF ) ≤ V (F ) ∀F

(we identify R2×2 × R with R

5). As the pointwise supremum over a classof convex functions, ψ is convex, and ψ(F, detF ) = V pc(F ). We show nextthat there is a convex function h : R

2×2 → R such that the function ψ canbe represented as

ψ(F, t) =

h(F ) if t = 1

∞ otherwise,

for all F ∈ R2×2, t ∈ R. The fact that ψ(F, t) = ∞ for t 6= 1 follows from

the fact that

gd(F, t) =

0 if t = 1

∞ otherwise,

belongs to the class of functions in the definition of ψ, hence ψ ≥ gd. Toprove the existence of h we define

h(F ) = ψ(F, 1) .

The convexity of h follows from the convexity of ψ, and it remains to showthat h is finite-valued. Since h(F ) = V pc(F ) < ∞ for all matrices F withdetF = 1, it suffices to prove that the convex hull of Σ = F ∈ R

2×2 :detF = 1 coincides with R

2×2. To do this, fix F ∈ R2×2. If detF < 1,

we consider the line t 7→ Ft = F + tId. Obviously detFt is continuous, andlimt→±∞ detFt = ∞. Therefore there are two values t− < 0 < t+ such thatdetFt± = 1. This implies that F belongs to the convex hull of Ft− , Ft+ ⊂ Σ.

If detF > 1 one proceeds analogously with Ft = F + t(e1 ⊗ e1 − e2 ⊗ e2).This concludes the proof of the assertion.

Let now F ∈ Σ. Since h : R4 → R is convex, and h(F ) ∈ R, there is an

affine function ℓ ∈ P such that ℓ(F ) = h(F ) = V pc(F ), and ℓ ≤ h on R4.

The latter implies

ℓ(G) ≤ h(G) = ψ(G, 1) ≤ ψ(G, detG) = V pc(G)

for all G ∈ R2×2. This concludes the proof.

We now turn to the specific problem at hand. We first recall the well-known fact that for any F ∈ R

2×2 one has

maxG∈SO(2)

F : G = (λ2 + λ1)(F ) . (5.3)

20

Here λ1 and λ2 are the singular values of F , defined by the conditions (λ21 +

λ22)(F ) = |F |2, (λ1λ2)(F ) = detF , λ2 ≥ |λ1|. We further observe that ifF = QH , with Q ∈ SO(2) and H = HT , then

(λ2 + λ1)(F ) = |TrH| .

Multiplying both F and G in (5.3) by e1 ⊗ e1 − e2 ⊗ e2 we obtain

maxG∈O(2)\SO(2)

F : G = (λ2 − λ1)(F ) . (5.4)

Lemma 5.2. The polyconvex envelope W pc of the function W defined in(1.3) satisfies

W pc(F ) =

maxϕF (G) : G ∈ R

2×2, |Ge1| ≤ 1, |Ge2| ≤ 1 if detF = 1 ,

∞ else ,

(5.5)where

ϕF (G) = F : G− (λ1 + λ2)(G) . (5.6)

Proof. We start from (5.2), with V = W . Choose G, β such that thecorresponding affine function ℓ satisfies ℓ ≤ W . Then necessarily for allF = Q(Id + γei ⊗ e⊥i ), i ∈ 1, 2 one has

F : G+ β ≤ |γ| .

In other words,Q : G+ γQei ·Ge⊥i + β ≤ |γ| (5.7)

for all Q ∈ SO(2), γ ∈ R, i ∈ 1, 2. Considering the limits γ → ±∞, wesee that (5.7) implies |Qei · Ge⊥i | ≤ 1 for all Q ∈ SO(2), i ∈ 1, 2. This isequivalent to the conditions

|Ge1| ≤ 1 , |Ge2| ≤ 1 . (5.8)

For γ = 0 instead (5.7) reduces to

β + maxQ∈SO(2)

Q : G ≤ 0 , (5.9)

which by (5.3) is equivalent to β ≤ −√

|G|2 + 2 detG = −(λ1 + λ2)(G). Bylinearity it is immediate to see that (5.8) and (5.9) are equivalent to (5.7).Further, it is clear that it suffices to consider the largest value of β compatiblewith (5.9), i.e., −(λ1 + λ2)(G). This concludes the proof.

21

Lemma 5.3. The result of Lemma 5.2 is equivalent to

W pc(F ) =

maxϕF (G) : G ∈ R

2×2, |Ge1| = |Ge2| = 1 if detF = 1 ,

∞ else .

(5.10)

Proof. We only need to show that we can restrict the class of matrices in themaximum to those for which both columns have length one. Fix F ∈ Σ. Wehave to show that the maximum of ϕF on the set

K = G ∈ R2×2 : |Ge1| ≤ 1, |Ge2| ≤ 1 (5.11)

is attained on the set K ′ = G ∈ R2×2 : |Ge1| = |Ge2| = 1 ⊂ K.

Recalling (5.3) and (5.4) we see that

maxG∈O(2)\SO(2)

ϕF (G) = (λ2 − λ1)(F ) ≥ 0 . (5.12)

In order to conclude the proof, we note first that by continuity there existsG ∈ K such that ϕF (G) = maxϕF (H), H ∈ K. In the following argumentwe distinguish several cases depending on properties of G.

If ϕF (G) = 0, then by (5.12) there exists also a matrix with the assertedproperties that realizes the maximum. This in particular treats the caseG = 0.

Assume next that G 6= 0, with both columns of length less than one andϕF (G) 6= 0. By continuity, there exists a t > 1 such that tG ∈ K. But since

ϕF (tG) = tϕF (G)

it follows that the maximum was not attained at G, a contradiction.We finally consider the case that only one of the columns of G has length

less than one, i.e., we assume

|Ge1| < 1 = |Ge2| .We consider the polar decomposition of G, i.e., choose a symmetric matrixH and Q ∈ SO(2) such that G = QH . The (signed) singular values of Gare the eigenvalues of H , up to a global sign, and therefore (λ1 + λ2)(G) =(λ1 + λ2)(H) = |TrH|. Consider for t ∈ R the matrices

Gt = QHt = Q(H + te1 ⊗ e1) .

It is clear that |Gte2| = |Ge2| for all t ∈ R, hence by continuity for small t wehave Gt ∈ K. Consider now the function t 7→ ϕF (Gt). Since Ht = H+te1⊗e1is symmetric, and Q ∈ SO(2), we have

(λ1 + λ2)(QHt) = (λ1 + λ2)(Ht) = |TrHt| = |TrH + t| .

22

We now distinguish two cases. If TrH = 0, then the fact that |He2| = 1 andHT = H implies necessarily H ∈ O(2) \ SO(2) ⊂ K ′. Otherwise, there is a(maximal) closed segment I ⊂ R, 0 ∈ I, such that Gt ∈ K for all t ∈ I andt 7→ ϕF (Gt) is affine on I; its endpoints satisfy either |Gte1| = 1 or TrHt = 0.If ϕF attains its maximum in the interior of the interval, the coefficient of thelinear term must vanish and the function ϕ(Gt) is constant for t ∈ I. Theendpoints belong to K ′ and hence in any case the maximum on K coincideswith the maximum on K ′. This concludes the proof.

Proof of Theorem 1.3. Consider in Lemma 5.3 a generic G with |Ge1| =|Ge2| = 1, and let G = QH be its polar decomposition. Since H is asymmetric matrix with both columns of length one, it necessarily has theform

H =

(cos θ sin θsin θ − cos θ

)or H =

(cos θ sin θsin θ cos θ

).

In the first case, H ∈ O(2). Recalling (5.3) and (5.4), together with the factthat detF = 1 implies λ1(F ) > 0, we see that the supremum over all H ofthe first form is (λ2 − λ1)(F ). It remains to treat the second case. Since(λ1 + λ2)(G) = |TrH| = 2| cos θ|, we need to consider

maxQ∈SO(2)

F : (QH) − 2| cos θ| = maxQ∈SO(2)

TrQTFH − 2| cos θ|

= (λ1 + λ2)(FH) − 2| cos θ| .Thus

(λ1 + λ2)(FH) =√

|FH|2 + 2 det(FH)

=√

|FHe1|2 + |FHe2|2 + 2 detH

=√

|Fe1|2 + |Fe2|2 + 2Fe1 · Fe2 sin(2θ) + 2 cos(2θ) .

Therefore F : (QH) − |TrH| for the matrix under consideration coincideswith

ψ(θ) =√

|F |2 + 2Fe1 · Fe2 sin(2θ) + 2 cos(2θ) − 2| cos θ| .Finally, we compute

ψ(π/2) =√

|F |2 − 2 = λ2(F ) − λ1(F ) ,

which proves that the maximum of ψ is always larger than λ2(F ) − λ1(F ),

hence the matrices in O(2) can also be neglected. Since ψ is π-periodic, wecan take θ ∈ [−π/2, π/2] and drop the absolute value on the last cosine.Since sin 2θ is odd, and all other terms are even, we can take θ ∈ [0, π/2]if we put the absolute value on the coefficient of sin 2θ. This concludes theproof.

23

6 Bounds on the quasiconvex envelope

We address here bounds on the quasiconvex envelope W qc, which are non-trivial since W takes the value ∞ on a large set, and for extended-valuedfunctions quasiconvexity does not automatically imply rank-one convexity.We also investigate numerically the quality of the approximations we havederived.

6.1 Analytical bounds

In the situation at hand it turns out that the rank-one convex envelope doesgive a bound on the quasiconvex one.

Proposition 6.1. For all matrices F ∈ R2×2, and with the notation above,

we haveW pc(F ) ≤W qc(F ) ≤ W rc(F ) .

Proof. The lower bound W pc ≤W qc holds for generic extended-valued func-tions [28, Lemma 4.3], hence there is nothing to prove.

The upper bound W qc(F ) ≤ W rc(F ) holds in general for finite-valuedfunctions W , but not necessarily for extended-valued ones, see, e.g., [5] foran example where it does not.

We show next that W qc(F ) <∞ for all matrices F with detF = 1. If theassertion holds, then Theorem 1.1 of [9] implies that W qc is rank-one convex.Since obviously W qc ≤W , we obtain that W qc constitutes a lower bound onW rc and the statement follows.

The assertion can be proven using the method of convex integration forLipschitz mappings developed by Muller and Sverak [29], see for example[14, 12, 2]. Instead of constructing a suitable in-approximation, we shallargue that it suffices to apply two known consequences of the result by Mullerand Sverak.

We use different constructions in the different parts of the domain. If|Fe1| ≤ 1, then the assertion follows from Theorem 1 of [12], and analogouslyif |Fe2| ≤ 1. Consider now a matrix F with detF = 1 and |Fe1|, |Fe2| > 1.Then, arguing as in the proof of Lemma 2.1, we find γ ∈ R such that F is inthe rank-one convex hull of the set

Kγ = SO(2)Aγ ∪ SO(2)Bγ , Aγ = Id + γe1 ⊗ e2, Bγ = Id + γe2 ⊗ e1 .

By Corollary 1.4 in [29] there is a Lipschitz mapping u ∈ W 1,∞((0, 1)2; R2)such that ∇u ∈ K a.e. and u(x) = Fx on the boundary. ThereforeW qc(F ) ≤|γ|, and the assertion is proven.

24

We remark that, even if one knew the quasiconvex envelope, this wouldnot immediately prove that

∫W qc(Du)dx is the relaxation of

∫W (Du)dx.

Indeed, standard theorems apply only for finite-valued energy densities withp-growth conditions, p > 1. In the case of linear growth p = 1 the variationalintegral needs to be augmented by additional terms involving the recessionfunction of W qc and the singular part of the distributional gradient Du, inthe sense of BV functions [23, 20, 21]. In this case additional difficulties areexpected from the constraint on the determinant.

6.2 Numerical approximations to the quasiconvex en-

velope

We discuss in this section the numerical difference between several approxi-mations to W qc we have obtained. Since plotting the absolute values wouldresult in indistinguishable curves, we display instead the relative distancefrom the lower bound W pc, defined, in the case of W rc for example, as

W rc(F ) −W pc(F )

W pc(F ).

We consider two different directions in the plane, in Figure 3 and Figure 4.The first one considers matrices in Σ with columns of equal lengths, thesecond one matrices such that the sum of the lengths of the columns is fixed.

We now discuss the different upper bounds to W qc, starting from theworst one (i.e., from the highest to the lowest).

The worst approximation, labeled “2-well approximation” is an upperbound based on a refinement of Lemma 2.2. In the notation of the proof ofLemma 2.2, we determine for each F the smallest γ such that F ∈ Krc

γ (whichamounts to solving a quadratic equation), and then estimate W rc(F ) ≤ |γ|.This simple computation results in a maximum relative error on W qc of lessthan 5%.

The next bound is derived directly from (2.3). Given F , we define Ft as in(2.2) and find the parameter t with smallest magnitude such that |Fte2| = 1(again, t is the solution of a quadratic equation). This gives the boundW rc(F ) ≤W (Ft) + |t| |Fe1|. The minimum between this bound and the oneobtained swapping the indices 1 and 2 is labeled “1st order laminate” in thefigure.

The third bound, labelled “2nd order laminate,” arises from (2.7). GivenF , let θ ∈ [0,∞) and define

Fθ =

(1 ±θ0 1

),

25

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0.05

1 2 3 4 5 6

rela

tive

dist

ance

from

Wpc

|Fe1| = |Fe2| = t

2-well approximation1st order laminate2nd order laminate

Wrc

Figure 3: A comparison between four upper bounds for W qc and the lowerbound W pc on matrices |Fe1| = |Fe2|. The curves represent the relativedistance between a given upper bound and the lower bound. From highestto lowest, the curves correspond to (1) a bound using an estimate by thetwo-well problem, (2) a bound based on a simple laminate with one supportat infinity, (3) a bound based on a second-order laminate whose supportcontains two matrices at infinity in the sense of (2.5), (4) the rank-one convexenvelope W rc generated by infinite-rank laminates with support on SO(2) andat infinity.

where the sign in front of θ is chosen to the same as that of Fe1 · Fe2. Sinceany F ∈ Σ is uniquely determined, up to left-multiplication by a rotation, bythe lengths of its columns and the sign of the inner product of its columns, itis not difficult to show that there exists a rotation Q ∈ SO(2) and numberss, t ∈ R such that F = G(QFθ, s, t). Then by (2.7) and the rotationalinvariance of W , we find

W rc(F ) ≤W (Fθ) + |t| + |s|√

(θ2 + 1)2 ± 2tθ + t2 .

By choosing s and t to have the smallest magnitude when solving the associ-ated quadratic equations, we arrive at an upper bound for W rc(F ) for everyθ. We report the best of these bounds, obtained from numerically optimizingin θ.

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0

0.01

0.02

0.03

0.04

0.05

1 1.5 2 2.5 3

rela

tive

dist

ance

from

Wpc

|Fe1| = t, |Fe2| = 4-t

2-well approximation1st order laminate2nd order laminate

Wrc

Figure 4: Comparison between the bounds, as in Figure 3, but along theline |Fe1| + |Fe2| = 4. The midpoint corresponds to the point at t = 2 ofFigure 3, the endpoints belong to the set where W = W rc.

The lowest curve in the figure is the rank-one convex envelope, W rc.Observe that as estimates of W qc, W rc and W pc have a maximum relativeerror of about 1.7%.

Acknowledgements

This work was performed while NA was at the Universitat Duisburg-Essensupported by the National Science Foundation through the Mathematical Sci-ences Postdoctoral Research Fellowship Award #0603611. The work of SCwas supported by the Deutsche Forschungsgemeinschaft through the Schw-erpunktprogramm 1253 “Optimization with Partial Differential Equations”,project CO 304/2-1. The work of GD was supported by the National ScienceFoundation through grant DMS 0405853.

27

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