Infinite-order laminates in a model in crystal plasticity

Infinite-order laminates in a model in crystal plasticity January 9, 2008 Nathan Albin1 , Sergio Conti2 , and Georg Dolzmann3 1

Applied and Computational Mathematics California Institute of Technology, Pasadena, CA 91101, USA 2 Fachbereich Mathematik, Universit¨at Duisburg-Essen 47057 Duisburg, Germany 3 NWF-I Mathematik, Universit¨at Regensburg 93040 Regensburg, Germany We consider a geometrically nonlinear model for crystal plasticity in two dimensions, with two active slip systems and rigid elasticity. We prove that the rank-one convex envelope of the condensed energy density is obtained by infinite-order laminates, and express it explicitly via the 2 F1 hypergeometric function. We also determine the polyconvex envelope, leading to 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 of microstructures, which largely influence the macroscopic material response. Possible origins for microstructure are both the interplay of different slip systems and geometrical effects, i.e., the interplay of one slip system with rotations. Recent progress in the analysis of plastic microstructure has been largely based on variational formulations, starting from the work by Ortiz and Repetto [30], see also [25, 8]. This is admissible if one assumes monotonicity, leading to the so-called deformation theory of plasticity; or more in general for short time intervals (after discretization in time). After minimizing in the internal variables one obtains a variational integral of the form Z W (∇u)dx Ω

possibly complemented by additional external forces and boundary conditions. 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 density W not convex, which in turn leads to the spontaneous formation of microstructures. The theory of relaxation shows that the macroscopic material behavior can be studied by replacing W with its quasiconvex envelope, which is defined as the largest quasiconvex function not larger than W , W qc (F ) = sup{V (F ) : V quasiconvex, V (G) ≤ W (G) for all G ∈ Rn×n } . (1.1) n×n We recall that a function W : R → R ∪ {∞} is quasiconvex if Z 1 (1.2) W (F + ∇ϕ) dx for all ϕ ∈ W01,∞ (Ω; Rn ) W (F ) ≤ |Ω| Ω (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 to optimizing locally (i.e., at any material point) over all possible microstructures, which are here described by all possible Lipschitz continuous functions ϕ which vanish on the boundary. The definition is clearly very implicit, and thus difficult to handle directly; therefore it is in practice often replaced by either of the two concepts of rank-one convexity and polyconvexity. A function W : Rn×n → R ∪ {∞} is rank-one convex if it is convex along rank-one lines, in the sense that t 7→ W (F + ta ⊗ b) is convex for all a, b ∈ Rn . A function W : Rn×n → R ∪ {∞} is polyconvex if it can be written as a convex function of its argument and its minors, i.e., for n = 2, if there is a convex function g : R5 → R ∪ {∞} such that W (F ) = g(F, det F ). In a geometrically linear setting, quasiconvexity often reduces to the much simpler concept of convexity. Assuming convex potentials, a very satisfactory general theory can be obtained using methods from convex analysis and the theory of functions of bounded deformation, see, e.g., [32]. Similar simplifications hold in the case of microstructure formation; indeed, for a model of crystal plasticity without hardening the quasiconvex envelope W qc turns out to be convex [11]. The analysis in [11] involved a realistic number of slip systems (e.g., the result included the case of the 12 slip systems with fcc symmetry, appropriate for many metals) but used in a substantial way the linear treatment of rotations, as well as the convexity of the relaxed problem. In a finite-deformation context, it is well known that convexity contrasts with invariance under rotations. Whereas abstract theory shows that quasiconvexity is the appropriate concept, in practice this turns out to be much more difficult to handle. Ortiz and Repetto have shown that energy densities describing a system with a single slip system in finite deformation lack quasiconvexity, and therefore lead to spontaneous microstructure formation 2

in the form of laminates, a fact known as geometric softening [30]. A twodimensional energy density with a single slip system with linear hardening and with a polyconvex elastic part was proposed and shown also to lack quasiconvexity in [8]. In [12] an explicit formula for the quasiconvex envelope of a simplification of that model, based on rigid elasticity and no self-hardening, was obtained. Work has also been devoted to numerical approximations; in particular, in [7] the model from [8] was studied numerically. An approximate numerical relaxation for the same model was obtained and integrated in a macroscopic finite-element computation in [24]. A finer analysis of the quasiconvex envelope of the same energy density is now under way, preliminary results are presented in [6]. In all these works, containing a proper treatment of rotations, only a single slip system was considered. We study here for the first time the interplay among several slip systems within a geometrically nonlinear model formulation. Precisely, we focus on a model with two slip systems in two dimensions, with rigid elasticity and no self-hardening. Our model,   |γ| if F = Q(Id + γe1 ⊗ e2 ) for some γ ∈ R , Q ∈ SO(2) , W (F ) = |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 envelope of W . The envelopes are defined in analogy to (1.1) as the largest rank-one convex (polyconvex) functions not larger than W . Theorem 1.1. The rank-one convex envelope W rc of W defined in (1.3) is given by  (λ2 − λ1 )(F ) if det F = 1, min{|F e1 |, |F e2|} ≤ 1,    ψ(|F e |, |F e |) if det F = 1, 1 ≤ |F e | ≤ |F e |, 1 2 1 2 W rc (F ) = (1.4)  ψ(|F e |, |F e |) if det F = 1, 1 ≤ |F e | ≤ |F e |, 2 1 2 1    ∞ if det F 6= 1, where

ψ(α, β) =

Z

1

α

 √ 1 p 2 2 2s2 4 √ ds + α β −1− α −1 . α s4 − 1

(1.5)

Here λ1 (F ) and λ2 (F ) denote the singular values of F , i.e., the ordered eigenvalues of U in the polar decomposition F = QU, Q ∈ SO(2), U = U T . 3

They are identified uniquely by the conditions λ21 (F ) + λ22 (F ) = |F |2 ,

λ1 (F )λ2(F ) = det F ,

λ2 ≥ |λ1 | .

(1.6)

Theorem 1.1 is proven in Section 2 (upper bound) and Section 3 (lower bound). Remark 1.2. Using standard formulas for hypergeometric functions (see for example Chapter 15 of [1]) one finds the following representation of the integral above in terms of Gauss’ hypergeometric function 2 F1 :
√   Z α 2 π Γ 43 2s2 1 1 3 −4
. √ ψ(α, α) = − ds = 2α 2 F1 − , ; ; α 4 2 4 Γ 41 s4 − 1 1

The hypergeometric function 2 F1 is defined through its power series via the rising factorial (k)n = k(k + 1)(k + 2) · · · (k + n − 1) as ∞ X (a)n (b)n z n · . 2 F1 (a, b; c; z) = (c)n n! n=0

We also determine the polyconvex envelope, as given by the following theorem proved in Section 5. Theorem 1.3. The polyconvex envelope W pc of W defined in (1.3) is given by p |F |2 + 2|F e1 · F e2 | sin(2θ) + 2 cos(2θ) − 2 cos θ . (1.7) W pc (F ) = max θ∈[0,π/2]

Further, we show that these two bounds give a bound on the quasiconvex envelope, 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 rather small; details are discussed in Section 6 below. The proof of Theorem 1.1 is as usual based on the construction of an appropriate laminate which realizes the relaxed energy; physically, this laminate gives an indication for the expected microstructure. Almost all examples of quasiconvex envelopes use the construction of suitable finite-order laminates, see, e.g., [22, 16, 14]. In this case, however, the appropriate laminate is not only partially supported at infinity (if one considers an appropriate completion of the domain, as is usual in problems with linear growth), but it cannot be a finite-order laminate. Precisely, no finite-order lamination convex envelope W lc,n agrees with the rank-one convex envelope W rc (see Section 4 for a precise definition of the lamination convex envelope W lc,n ). 4

Theorem 1.4. Let F be such det F = 1 < min{|F e1 |, |F e2|}. 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 feature already observed in a geometrically linear context in [11], and that structures on many different scales will coexist. Analytically, analogous unbounded laminates have been used in [17, 18] for constructing critical solutions to elliptic equations, and in [10] to give a rank-one convex function on diagonal matrices with locally unbounded Hessian, as well as to obtain a simple derivation of a classical counterexample to Korn’s inequality in L1 by Ornstein.

2

Upper bound on the rank-one convex envelope

In this section we prove the upper bound of Theorem 1.1. Precisely, we prove the following lemma. Lemma 2.1. Let W rc be the rank-one convex envelope of W defined in (1.3), and  (λ2 − λ1 )(F ) if det F = 1, min{|F e1 |, |F e2|} ≤ 1,    ψ(|F e |, |F e |) if det F = 1, 1 ≤ |F e | ≤ |F e |, 1 2 1 2 Φ(F ) = (2.1)  ψ(|F e2 |, |F e1 |) if det F = 1, 1 ≤ |F e2 | ≤ |F e1 |,    ∞ if det F 6= 1, where

ψ(α, β) =

Z

α 1

√

 √ 1 p 2 2 2s2 4 ds + α β −1− α −1 . α s4 − 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 : det F = 1} of matrices with determinant one. Lemma 2.2. Let W be given by (1.3). Then W rc(F ) ≤ |F e1 | + |F e2 | + 1 5

for all F ∈ Σ .

Proof. We use an approximation by two-well problems. Let γ ∈ R, and define 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 rankone convex hull Kγrc is explicitly known [3, 31]. In order to characterize Kγrc it suffices to choose a pair v, w ∈ R2 of linearly independent vectors such that |Aγ v| = |Bγ v|, |Aγ w| = |Bγ w|, in our case v = (1, 1), w = (1, −1). Then Kγrc = {F ∈ R2×2 : det F = 1 , |F v| ≤ |Aγ v| , |F w| ≤ |Aγ w|} . This implies that W rc(F ) ≤ W (Aγ ) = W (Bγ ) = |γ|

for all F ∈ Kγrc .

Now let F ∈ Σ. We assert that if γ = |F e1 |+|F e2 |+1 then F ∈ Kγrc . Indeed, we have |F w|2 = |F e1 − F e2 |2 ≤ (|F e1 | + |F e2 |)2 = (γ − 1)2 ≤ (γ − 1)2 + 1 = |Aγ w|2 , and similarly |F v|2 = |F e1 + F e2 |2 ≤ (|F e1| + |F e2 |)2 = (γ − 1)2 ≤ (γ + 1)2 + 1 = |Aγ v|2 , which proves the assertion. The proof of Lemma 2.1 is based on proving a bound on the derivative of W rc along certain lines. We start by proving that W rc is locally Lipschitz on Σ, and hence that it is differentiable almost everywhere. This is wellknown for finite-valued rank-one convex functions [13, 4]; we now show how the 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 independently in each variable) functions are locally Lipschitz. We report here this result in 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 n Lip(f ; Br (ξ0 )) ≤ osc(f ; B2r (ξ0 )) r 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 the same argument applies to any rank-one convex function which is finite-valued on Σ. Lemma 2.4. Let W be given in (1.3), and let F0 ∈ Σ. Then there exist c, r > 0 such that for all F1 , F2 ∈ Br (F0 ) ∩ Σ rc W (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-one convex, 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 F0 R1 , F0 R2 and F0 R3 , respectively, and span the tangent plane to Σ at F0 . Thus, by the implicit function theorem, there exists an r > 0 such that g is invertible in Br (F0 ) ∩Σ and on this set, W rc = f ◦ g −1. Since W is non-negative, it follows that W rc is also non-negative. On the bounded set S = g −1(Br (F0 ) ∩ Σ), this observation together with the inequality 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. Near F0 , W rc |Σ is the composition of two Lipschitz functions and is therefore Lipschitz. Proof of Lemma 2.1. If det F 6= 1 there is nothing to prove, hence we can assume det F = 1. If |F e1 | ≤ 1 the result follows from [12]; for convenience of the reader we give here a short, self-contained proof. The key observation is that the function p G 7→ (λ2 − λ1 )(G) = |G|2 − 2 det G 7

has a particularly simple form on rank-one lines originating from the identity, namely, if a ∈ R2 , |a| = 1, then (λ2 − λ1 )(Id + ta⊥ ⊗ a) = |t| (here and below a⊥ = (−a2 , a1 )). Further, for all matrices Gγ = Id + γe2 ⊗ e1 we immediately obtain (λ2 −λ1 )(Gγ ) = |γ| = W (Gγ ). These two observations imply the assertion for all matrices of the form Ft = Q(Id + ta ⊗ a⊥ ) with Q ∈ SO(2) and |Ft e1 | ≤ 1. It remains to show that any matrix F with det F = 1 and |F e1 | ≤ 1 can be written in this form. In order to do so, fix F ∈ Σ and let a ∈ R2 be such that |a| = |F a| = 1. Such an a exists, since det F = 1 and |F e1 | ≤ 1, implying |F e2 | ≥ 1. Let Q ∈ SO(2) be such that QT F a = a. Let c1,2 ∈ R be defined by QT F a⊥ = c1 a + c2 a⊥ . Since 1 = det F = QT F a ∧ QT F a⊥ , we have c2 = 1 (we identify v ∧ w with the scalar (v ∧ w) · (e1 ∧ e2 ) = v1 w2 − v2 w1 , for any v, w ∈ R2 ). Hence QT F = Id + c1 a ⊗ a⊥ , which proves the assertion with t = c1 . The case |F e2 | ≤ 1 is analogous. Now we turn to the case of matrices in the set Σ+ = {F ∈ R2×2 : det F = 1 , |F e1| > 1 , |F e2| > 1} . We first prove the following fact which provides the basic step of our construction. Consider the rank-one line along the coordinate axis |F e1 | = const given by Ft = F (Id + te1 ⊗ e2 ) . (2.2) We assert that

W rc(Ft ) ≤ W rc (F ) + |t| |F e1 |

for all t ∈ R

(2.3)

and that the same statement holds after interchanging the indices. To prove (2.3) we remark that Ft e1 = F e1 and Ft e2 = F e2 + tF e1 , which implies |Ft e2 |2 = |F e2 |2 + 2tF e1 · F e2 + t2 |F e1 |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 rc 1 W (F ) + W rc(Fkt ) . k k

(2.5)

In the limit as k → ∞, |Fkt e1 |/k = |F e1 |/k → 0 and (2.4) shows that |Fkt e2 |/k → |t| |F e1 |. Thus, by Lemma 2.2, we have lim sup k→∞

|Fkt e1 | + |Fkt e2 | + 1 W rc (Fkt ) ≤ lim = |t| |F e1| . k→∞ k k 8

Taking k → ∞ in (2.5) proves (2.3). The analogous inequality follows when the roles of e1 and e2 are reversed. We apply this assertion once in each of the two directions |F ei | = const to 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 | = |F e2 + tF e1 | = that

p |F e2 |2 + 2tF e1 · F e2 + t2 |F e1 |2 ,

W rc (G(F, s, t)) ≤ W rc(F ) + |t| |F e1| p + |s| |F e2 |2 + 2tF e1 · F e2 + t2 |F e1 |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 |F e1 | = |F e2 | = 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 |Fǫ e1 | = |Fǫ e2 | = α + ǫ . That is, we choose t and s to solve (1 + st)2 α2 + 2s(1 + st)F e1 · F e2 + s2 α2 = (α + ǫ)2 α2 + 2tF e1 · F e2 + t2 α2 = (α + ǫ)2 . Without loss of generality, we may assume F e1 · F e2 ≥ 0 (otherwise take t and s negative in what follows). Then we take t = t(ǫ), s = s(ǫ) as the positive roots of the polynomials above. By differentiating in ǫ, it follows that α α ǫ + o(ǫ), s(ǫ) = √ ǫ + o(ǫ) . t(ǫ) = √ α4 − 1 α4 − 1 Then by (2.7) W rc (Fǫ ) ≤ W rc (F ) + |t(ǫ)| |F e1| + |s(ǫ)| |F e2| + o(ǫ) 2α2 = W rc (F ) + √ ǫ + o(ǫ) . α4 − 1 Taking ǫ to zero, we see that lim sup ǫ→0+

W rc (Fǫ ) − W rc (F ) 2α2 ≤√ . ǫ α4 − 1 9

(2.8)

It is now convenient to interpret the values of W rc as a function of α = |F e1 | = |F e2 |. We consider the case F e1 · F e2 ≥ 0; the remaining case is analogous. For α ≥ 1 we define p p  2 + 1)/2 2 − 1)/2 (α (α p H(α) = p 2 . (α − 1)/2 (α2 + 1)/2

(In the case F e1 · F e2 ≤ 0, the off-diagonal elements are defined to be negative.) We define the function f : [1, ∞) → R by f (α) = W rc (H(α)). We first show that W rc (F ) = f (|F e1|) for all F ∈ Σ such that |F e1 | = |F e2 | and F e1 · F e2 ≥ 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 columns have equal length α, and since their scalar product is positive, H must coincide with H(α). By the rotational invariance of W , which implies the same invariance 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 by Lemma 2.4, f is locally Lipschitz on (1, ∞) and is therefore √ almost everywhere differentiable. Note that F e1 · F e2 = He1 · He2 = α4 − 1 > 0 so for sufficiently small ǫ > 0, the matrix Fǫ in (2.8) satisfies Fǫ e1 · Fǫ e2 ≥ 0 and therefore W rc(Fǫ ) = f (|Fǫ e1 |) = 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 Z α Z α 2ζ 2 ′ p dζ . f (α) = f (α0 ) + f (ζ) dζ ≤ f (α0 ) + ζ4 − 1 α0 α0

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 limit α0 → 1 in the foregoing inequality proves W rc ≤ Φ for F ∈ Σ+ ∩ {F ∈ R2×2 : |F e1 | = |F e2 |}. Case 2: The columns of F have different lengths. Without loss of generality, we may assume 1 < |F e1 | = α < β = |F e2 |. We estimate W rc(F ) by a lamination in the direction e1 ⊗ e2 supported on one matrix with columns of equal length and the other at infinity, in the sense of (2.5). Define Ft as in (2.2). Then |Ft e1 | = α for all t and p |Ft e2 | = β 2 + 2tF e1 · F e2 + t2 α2 . 10

We choose t so that this quantity equals α. Again, we may assume that F e1 · F e2 ≥ 0 since otherwise we can change the sign of t. Thus, we choose  √ 1 p 2 2 α β − 1 − α4 − 1 . t=− 2 α Applying (2.3) once again, we see that

W rc (F ) ≤ W rc (Ft ) + |t||Ft e2 | ≤ ψ(α, α) + |t|α , which implies W rc(F ) ≤ ψ(α, β) = Φ(F ).

3

Lower bound on the rank-one convex envelope

In this section, we prove that the upper bound for W rc derived in the previous section is actually rank-one convex and is therefore a lower bound. This implies 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 ∈ R2 , the function φ(t) = Φ(Ft ),

Ft = F + tF a ⊗ b .

It suffices to show that for any F , a, b the function φ is convex in a neighborhood of 0. Since det(Ft ) = det F (1 + ta · b), and Φ is finite only on Σ, we see that φ is either finite for all t or finite for at most one value of t. Thus, we may restrict our attention to the case where det F = 1 and b = a⊥ . We first consider separately the cases  F ∈ Σ1 = F ∈ R2×2 : det F = 1, |F e1 | < 1 < |F e2 | ,  F ∈ Σ2 = F ∈ R2×2 : det F = 1, |F e2 | < 1 < |F e1 | ,  F ∈ Σ3 = F ∈ R2×2 : det F = 1, 1 < |F e1 | < |F e2 | ,  F ∈ Σ4 = F ∈ R2×2 : det F = 1, 1 < |F e2 | < |F e1 | (see Figure 1). Once we have done this, it will remain only to check smoothness 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. Since Φ coincides on this domain with the convex function p p (λ2 − λ1 )(F ) = |F |2 − 2 det F = (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 origin and φ′′ (t)|t=0 ≥ 0. If this holds for all F ∈ Σ3 , then φ necessarily has nonnegative second derivative in a neighborhood of the origin and therefore it is convex in a neighborhood of the origin. To prove the assertion it is useful to introduce the variables ξ(t) = |Ft e1 |2 ,

η(t) = |Ft e2 |2 .

Starting from φ(t) = ψ(ξ 1/2 (t), η 1/2 (t)), we compute ∂ψ 1 dξ ∂ψ 1 dη dφ = + , dt ∂α 2ξ 1/2 dt ∂β 2η 1/2 dt with ∇ψ evaluated at α = ξ 1/2 (t), β = η 1/2 (t). Differentiating a second time, and rearranging terms,   2    2  d2 φ dξ dη 1 ∂2ψ 1 ∂ψ 1 ∂ψ 1 ∂2ψ = − 3 − 3 + 2 2 2 2 2 dt 4α ∂α 4α ∂α dt 4β ∂β 4β ∂β dt     2    2   2 dξ dη 1 ∂ψ dξ dη 1 ∂ψ 1 ∂ ψ + + . + 2 2αβ ∂α∂β dt dt 2α ∂α dt 2β ∂β dt2 12

Again, derivatives of ψ are evaluated at α = ξ 1/2 (t), β = η 1/2 (t). From the definition of ξ we obtain dξ = 2(F e1 · F a)(a⊥ · e1 ) = −2|F e1 |2 cos θ sin θ − 2(F e1 · F e2 ) sin2 θ , dt t=0 d2 ξ = 2|F a|2 (a⊥ · e1 )2 2 dt t=0 = 2|F e1 |2 cos2 θ sin2 θ + 4(F e1 · F e2 ) cos θ sin3 θ + 2|F e2 |2 sin4 θ . Analogously dη = 2|F e2 |2 cos θ sin θ + 2(F e1 · F e2 ) cos2 θ , dt t=0 d2 η = 2|F e1 |2 cos4 θ + 4(F e1 · F e2 ) cos3 θ sin θ + 2|F e2 |2 cos2 θ sin2 θ . dt2 t=0

From (1.5) we compute

∂ψ 1 =δ+ 2 , ∂α α γ

∂ψ αβ = , ∂β γ

(3.2)

where γ = (α2 β 2 − 1)1/2 and δ = (1 − α−4 )1/2 . Differentiating once again, ∂2ψ 2 3α2β 2 − 2 = − , ∂α2 α5 δ α3 γ 3

∂2ψ α =− 3, 2 ∂β γ

∂2ψ β =− 3. ∂α∂β γ

At this point we have all ingredients to evaluate d2 φ/dt2 |t=0 . By the above expressions it is clear that it is a homogeneous polynomial of fourth order in cos θ and sin θ, therefore one only has to evaluate the five coefficients. The coefficients can be expressed in terms of |F e1 | = α, |F e2 | = β, F e1 · F e2 = γ, and (1 − α−4 )1/2 = δ. A direct computation that uses γ 2 = α2 β 2 − 1 shows that
d2 φ sin2 θ 2 2 p c cos θ + c cos θ sin θ + c sin θ , (3.3) = 1 2 3 dt2 t=0 |F e1 |5 |F e1 |4 − 1 where

c1 = 2|F e1 |4 , p  p 2 2 2 4 |F e1 | |F e2 | − 1 − |F e1 | − 1 , c2 = 4|F e1 | p  p  p p c3 = |F e1 |2 |F e2 |2 − 1 − |F e1 |4 − 1 2 |F e1 |2 |F e2 |2 − 1 − |F e1 |4 − 1 . 13

Rearranging terms and minimizing the parenthesis over θ we obtain d2 φ sin2 θ p ((c1 − c3 ) cos 2θ + c2 sin 2θ + c1 + c3 ) = dt2 t=0 2|F e1|5 |F e1 |4 − 1   q sin2 θ 2 2 p c1 + c3 − c2 + (c1 − c3 ) ≥ 2|F e1 |5 |F e1 |4 − 1   q sin2 θ 2 2 p c1 + c3 − (c1 + c3 ) − (4c1 c3 − c2 ) . = 2|F e1|5 |F e1 |4 − 1

Since |F e2 | > |F e1 |, each ci > 0. Moreover,  p p p |F e1 |2 |F e2 |2 − 1 − |F e1 |4 − 1 > 0 , 4c1 c3 − c22 = 8|F e1 |4 |F e1 |4 − 1

so φ′′ (t)|t=0 ≥ 0 and we have proved that Φ is locally rank-one convex in Σ3 . The case F ∈ Σ4 is analogous and follows by interchanging indices. For later reference we observe that all inequalities are strict, unless sin θ = 0, i.e., Ft = F (Id + te1 ⊗ e2 ), and in particular that d2 φ if F ∈ Σ3 and sin θ 6= 0 then > 0. (3.4) dt2 t=0

To finish the proof that Φ is rank-one convex, we need to check the following remaining cases on Σ corresponding to the intersection of the boundaries of different domains, F ∈ (∂Σ1 ∩ ∂Σ3 ) ∪ (∂Σ3 ∩ ∂Σ4 ) ∪ (∂Σ4 ∩ ∂Σ2 ) (see Figure 1). In fact, we will show that Φ is C 1 in the set {F ∈ R2×2 : det F = 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 = |F e1 |, β = |F e2 |. First consider the intersection (∂Σ1 ∩ ∂Σ3 ) \ SO(2). By construction, Φ is continuous on Σ. Moreover, the normal derivative to the boundary in Σ1 is ∂ p 2 1 α + β 2 − 2 =p , ∂α α=1 β2 − 1

while in Σ3 , recalling (3.2) the normal derivative is ! √ 1 1 1 ∂ 4 + α −1 = 2 p . ψ(α, β) =p α=1 ∂α α α=1 α2 β 2 − 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 ∂ α2 ∂ √ = ψ(α, β) α=β = ψ(α, β) α=β . ∂α ∂β α4 − 1 Finally, we need to check rank-one convexity in the case F ∈ SO(2), so Ft = R(Id + ta ⊗ a⊥ ) for some R ∈ SO(2). Then for i = 1, 2, |Ft ei |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 two factors with opposite signs for i = 1, 2, it follows that for t sufficiently small min{|Ft e1 |, |Ft e2 |} ≤ 1, hence Ft ∈ Σ1 ∪ Σ2 . Hence φ(t) = Φ(Ft ) is defined through 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 converse inequality, we observe that by Lemma 3.1 Φ is rank-one convex, and by 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 ) = inf λV 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-one convex envelope (and, as a consequence, of the quasiconvex one); in many cases it turns out that a lamination envelope of relatively low order coincides with the rank-one and quasiconvex envelopes. We show that this is not the case here, and that no finite-order lamination-convex envelope coincides with the rank-one convex envelope, in the sense that W rc (F ) < W lc,n (F ) for all matrices F with det F = 1 < min{|F e1 |, |F e2|} (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 is exactly 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, which will then be chosen large enough. Given M > 1 we define   if det F = 1, max{|F e1|, |F e2 |} < M , W (F ) rc VM (F ) = W (F ) if det F = 1, max{|F e1|, |F e2 |} = M , (4.2)   ∞ otherwise

(see Figure 2). We observe that VM is finite only on a compact set, and that it is lower semicontinuous. We first show that these properties are stable under 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 is lower semicontinuous for all n. Further, all infima in the definition of the lamination convex envelope are minima. Proof. It clearly suffices to prove the assertion for n = 1, and then proceed by induction. Let Φ = Ψlc,1 . By definition, Φ(F ) = inf{λΨ(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. By compactness of [0, 1] × KΨ × KΨ , and lower semicontinuity, the infimum is actually a minimum. We now show that if F i → F , and F i ∈ KΦ , then F ∈ KΦ

and

Φ(F ) ≤ lim inf Φ(F i ) . i→∞

This will imply that KΦ is closed and that Φ is lower semicontinuous on it. i Let F i be a sequence with the foregoing properties and let F1,2 , λi be such that Φ(F i ) = λi Ψ(F1i) + (1 − λi )Ψ(F2i ) (4.3) and λi ∈ [0, 1],

rank(F1i − F2i ) ≤ 1 ,

λi F1i + (1 − λi )F2i = F i .

(4.4)

By compactness we can, after extracting a subsequence, assume that F1i → F1 ∈ KΨ , F2i → F2 ∈ KΨ , λi → λ ∈ [0, 1]; all conditions in (4.4) automatically 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 dashed line 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 ∈ Σ such that 1 < |F e1 |, |F e2 | < M one has W rc (F ) < VMlc,n (F ) . Proof. For simplicity we write V for VM in this proof. Since W rc ≤ V and W 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. Let n be the smallest number such that there is an F satisfying the assumptions in the lemma with W rc(F ) = V lc,n (F ), and assume without loss of generality that |F e1 | ≤ |F e2 | (i.e., F ∈ Σ¯3 in the notation of Section 3). Since V lc,0 (F ) = V (F ) = ∞, obviously n > 0. Then V lc,n is defined by (4.1). By Lemma 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 of n, 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 can be an equality only if sin θ = 0 in the rank-one direction, i.e., if |F1 e1 | = |F e1 | = |F2 e1 |. Further, by the same argument, the segment [F1 , F2 ] cannot cross the set {G : |Ge1 | = |Ge2 |}. At the same time, by minimality of n the second inequality is strict if 1 < |F1,2 e2 | < M. They cannot both equal M, since ψ is strictly increasing in its arguments. Analogously, the case |F1,2 e2 | ≤ 1 is ruled out since by convexity the same would hold for |F e2 |. Therefore, possibly after relabeling, |F1 e2 | ≤ 1 < M = |F2 e2 | . By continuity, this implies that [F1 , F2 ] ∩ {G : |Ge1 | = |Ge2 |} is nonempty, a contradiction (see Figure 2). Proof of Theorem 1.4. Let F0 be a matrix as given in the statement, and let M = |F0 | + 1. We first assert that VMlc,n (F ) ≤ W lc,n (F )

(4.5)

for all n and all F ∈ S, where S = {F : 1 < |F e1 | < M, 1 < |F e2 | < M}. The assertion is proven by induction. For n = 0, both equal ∞. Assume the inequality to hold for n. For notational convenience we rewrite the definition of the lamination convex envelope as  V lc,n+1 (F ) = inf ℓ(F )|ℓ : R2×2 → 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 VMlc,n+1 (F ) ≤ ℓ(F ) .

(4.7)

The key property will be that VM = W rc on Σ ∩ ∂S, which immediately implies VMlc,n = W rc on Σ ∩ ∂S for all n. ¯ and F ′ to be one element of ∂S ∩ [F, F1 ] We define F1′ = F1 if F1 ∈ S, 1 otherwise, and analogously F2′ . Let k : R2×2 → R be linear and such that k(Fi′ ) = VMlc,n (Fi′ ), for i = 1, 2. It suffices to show that k(Fi′ ) = VMlc,n (Fi′ ) ≤ ℓ(Fi′ ) ,

i = 1, 2 .

(4.8)

Indeed, if (4.8) holds then by linearity the same holds in the segment [F1′ , F2′ ] which contains F , and hence VMlc,n+1(F ) ≤ k(F ) ≤ ℓ(F ) and (4.7) follows. 18

To prove (4.8), we distinguish to cases. If Fi′ ∈ S then Fi′ = Fi and this follows from the inductive assumption. Otherwise, Fi′ ∈ ∂S ∩ [F1 , F2 ] and therefore k(Fi′ ) = VMlc,n (Fi′ ) = W rc (Fi′) = VMlc,n+1 (Fi′ ) ≤ ℓ(Fi′) . This proves the assertion (4.8) and therefore (4.5). Recalling Lemma 4.2 we obtain W rc(F0 ) < VMlc,n (F0 ) ≤ W lc,n (F0 ) , which concludes the proof.

5

Polyconvex envelope

In this section, we prove Theorem 1.3 by constructing the polyconvex envelope W pc of the energy W defined in (1.3). The polyconvex envelope of a function W is defined by W pc (F ) = sup{V (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 take the supremum over polyaffine functions V , see, e.g., [13, Sect. 5.1.1.2]. We first verify that the same is true in the case of interest here where V is finite only on Σ. The following lemma shows that it suffices to consider the smaller class of all affine functions. We state the result in a more general setting 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 det F = 1. Then for all F with det F = 1 we have  V pc (F ) = sup ℓ(F ) : ℓ ∈ P, ℓ(G) ≤ V (G) for all G ∈ R2×2 , (5.2) where P denotes the class of affine functions

P = {ℓ : R2×2 → R, ℓ(F ) = F : G + β , G ∈ R2×2 , β ∈ R} . P We recall that F : G = Tr F T G = ij Fij Gij .

Proof. Since affine functions are polyconvex, the supremum in (5.2) is taken over 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 ∈ P with ℓ(F ) = V pc (F ), ℓ ≤ V . 19

Let ψ : R5 → [0, ∞] be defined by ψ(x) = sup{g(x), g : R5 → [0, ∞], g convex, g(F, det F ) ≤ V (F ) ∀ F } (we identify R2×2 × R with R5 ). As the pointwise supremum over a class of convex functions, ψ is convex, and ψ(F, det F ) = V pc (F ). We show next that there is a convex function h : R2×2 → R such that the function ψ can be represented as ( h(F ) if t = 1 ψ(F, t) = ∞ otherwise, for all F ∈ R2×2 , t ∈ R. The fact that ψ(F, t) = ∞ for t 6= 1 follows from the fact that ( 0 if t = 1 gd (F, t) = ∞ otherwise, belongs to the class of functions in the definition of ψ, hence ψ ≥ gd . To prove 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 show that h is finite-valued. Since h(F ) = V pc (F ) < ∞ for all matrices F with det F = 1, it suffices to prove that the convex hull of Σ = {F ∈ R2×2 : det F = 1} coincides with R2×2 . To do this, fix F ∈ R2×2 . If det F < 1, we consider the line t 7→ Ft = F + tId. Obviously det Ft is continuous, and limt→±∞ det Ft = ∞. Therefore there are two values t− < 0 < t+ such that det Ft± = 1. This implies that F belongs to the convex hull of {Ft− , Ft+ } ⊂ Σ. If det F > 1 one proceeds analogously with Fet = 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, det G) = 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 wellknown fact that for any F ∈ R2×2 one has max F : G = (λ2 + λ1 )(F ) .

G∈SO(2)

20

(5.3)

Here λ1 and λ2 are the singular values of F , defined by the conditions (λ21 + λ22 )(F ) = |F |2 , (λ1 λ2 )(F ) = det F , λ2 ≥ |λ1 |. We further observe that if F = QH, with Q ∈ SO(2) and H = H T , then (λ2 + λ1 )(F ) = | Tr H| . Multiplying both F and G in (5.3) by e1 ⊗ e1 − e2 ⊗ e2 we obtain max

G∈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 ( max{ϕF (G) : G ∈ R2×2 , |Ge1 | ≤ 1, |Ge2 | ≤ 1} if det F = 1 , W pc (F ) = ∞ 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 the corresponding affine function ℓ satisfies ℓ ≤ W . Then necessarily for all F = 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 → ±∞, we see that (5.7) implies |Qei · Ge⊥ i | ≤ 1 for all Q ∈ SO(2), i ∈ {1, 2}. This is equivalent to the conditions |Ge1 | ≤ 1 ,

|Ge2 | ≤ 1 .

(5.8)

For γ = 0 instead (5.7) reduces to β + max Q : G ≤ 0 , Q∈SO(2)

(5.9)

p which by (5.3) is equivalent to β ≤ − |G|2 + 2 det G = −(λ1 + λ2 )(G). By linearity 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 compatible with (5.9), i.e., −(λ1 + λ2 )(G). This concludes the proof. 21

Lemma 5.3. The result of Lemma 5.2 is equivalent to ( max{ϕF (G) : G ∈ R2×2 , |Ge1 | = |Ge2 | = 1} W pc (F ) = ∞

if det F = 1 , else . (5.10)

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

G∈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 exists G ∈ K such that ϕF (G) = max{ϕF (H), H ∈ K}. In the following argument we distinguish several cases depending on properties of G. If ϕF (G) = 0, then by (5.12) there exists also a matrix with the asserted properties that realizes the maximum. This in particular treats the case G = 0. Assume next that G 6= 0, with both columns of length less than one and ϕF (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 matrix H and Q ∈ SO(2) such that G = QH. The (signed) singular values of G are the eigenvalues of H, up to a global sign, and therefore (λ1 + λ2 )(G) = (λ1 + λ2 )(H) = | Tr H|. Consider for t ∈ R the matrices Gt = QHt = Q(H + te1 ⊗ e1 ) . It is clear that |Gt e2 | = |Ge2 | for all t ∈ R, hence by continuity for small t we have Gt ∈ K. Consider now the function t 7→ ϕF (Gt ). Since Ht = H +te1 ⊗e1 is symmetric, and Q ∈ SO(2), we have (λ1 + λ2 )(QHt ) = (λ1 + λ2 )(Ht ) = | Tr Ht | = | Tr H + t| . 22

We now distinguish two cases. If Tr H = 0, then the fact that |He2 | = 1 and H T = 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 and t 7→ ϕF (Gt ) is affine on I; its endpoints satisfy either |Gt e1 | = 1 or Tr Ht = 0. If ϕF attains its maximum in the interior of the interval, the coefficient of the linear term must vanish and the function ϕ(Gt ) is constant for t ∈ I. The endpoints belong to K ′ and hence in any case the maximum on K coincides with 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 a symmetric matrix with both columns of length one, it necessarily has the form     cos θ sin θ cos θ sin θ . or H = H= sin θ cos θ sin θ − cos θ

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

Q∈SO(2)

max Tr QT F H − 2| cos θ|

Q∈SO(2)

= (λ1 + λ2 )(F H) − 2| cos θ| . Thus (λ1 + λ2 )(F H) = = =

p

p

p

|F H|2 + 2 det(F H)

|F He1 |2 + |F He2|2 + 2 det H

|F e1 |2 + |F e2 |2 + 2F e1 · F e2 sin(2θ) + 2 cos(2θ) .

Therefore F : (QH) − | Tr H| for the matrix under consideration coincides with p e = |F |2 + 2F e1 · F e2 sin(2θ) + 2 cos(2θ) − 2| cos θ| . ψ(θ) Finally, we compute

e ψ(π/2) =

p

|F |2 − 2 = λ2 (F ) − λ1 (F ) ,

which proves that the maximum of ψe is always larger than λ2 (F ) − λ1 (F ), hence the matrices in O(2) can also be neglected. Since ψe is π-periodic, we can 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 the proof. 23

6

Bounds on the quasiconvex envelope

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

6.1

Analytical bounds

In the situation at hand it turns out that the rank-one convex envelope does give a bound on the quasiconvex one. Proposition 6.1. For all matrices F ∈ R2×2 , and with the notation above, we have W pc (F ) ≤ W qc (F ) ≤ W rc(F ) . Proof. The lower bound W pc ≤ W qc holds for generic extended-valued functions [28, Lemma 4.3], hence there is nothing to prove. The upper bound W qc (F ) ≤ W rc (F ) holds in general for finite-valued functions W , but not necessarily for extended-valued ones, see, e.g., [5] for an example where it does not. We show next that W qc (F ) < ∞ for all matrices F with det F = 1. If the assertion 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 on W rc and the statement follows. The assertion can be proven using the method of convex integration for ˇ ak [29], see for example Lipschitz mappings developed by M¨ uller and Sver´ [14, 12, 2]. Instead of constructing a suitable in-approximation, we shall argue that it suffices to apply two known consequences of the result by M¨ uller ˇ and Sver´ak. We use different constructions in the different parts of the domain. If |F e1 | ≤ 1, then the assertion follows from Theorem 1 of [12], and analogously if |F e2 | ≤ 1. Consider now a matrix F with det F = 1 and |F e1 |, |F e2 | > 1. Then, arguing as in the proof of Lemma 2.1, we find γ ∈ R such that F is in the 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) = F x on the boundary. Therefore W qc (F ) ≤ |γ|, and the assertion is proven. 24

We remark that, even if one R knew the quasiconvex envelope, R this would not immediately prove that W qc (Du)dx is the relaxation of W (Du)dx. Indeed, standard theorems apply only for finite-valued energy densities with p-growth conditions, p > 1. In the case of linear growth p = 1 the variational integral needs to be augmented by additional terms involving the recession function of W qc and the singular part of the distributional gradient Du, in the sense of BV functions [23, 20, 21]. In this case additional difficulties are expected from the constraint on the determinant.

6.2

Numerical approximations to the quasiconvex envelope

We discuss in this section the numerical difference between several approximations to W qc we have obtained. Since plotting the absolute values would result in indistinguishable curves, we display instead the relative distance from 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, the second 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 the worst one (i.e., from the highest to the lowest). The worst approximation, labeled “2-well approximation” is an upper bound based on a refinement of Lemma 2.2. In the notation of the proof of Lemma 2.2, we determine for each F the smallest γ such that F ∈ Kγrc (which amounts 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 less than 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 |Ft e2 | = 1 (again, t is the solution of a quadratic equation). This gives the bound W rc (F ) ≤ W (Ft ) + |t| |F e1|. The minimum between this bound and the one obtained swapping the indices 1 and 2 is labeled “1st order laminate” in the figure. The third bound, labelled “2nd order laminate,” arises from (2.7). Given F , let θ ∈ [0, ∞) and define   1 ±θ , Fθ = 0 1 25

0.05 2-well approximation 1st order laminate 2nd order laminate Wrc

0.045 0.04

relative distance from Wpc

0.035 0.03 0.025 0.02 0.015 0.01 0.005 0 1

2

3

4

5

6

|Fe1| = |Fe2| = t

Figure 3: A comparison between four upper bounds for W qc and the lower bound W pc on matrices |F e1 | = |F e2 |. The curves represent the relative distance between a given upper bound and the lower bound. From highest to lowest, the curves correspond to (1) a bound using an estimate by the two-well problem, (2) a bound based on a simple laminate with one support at infinity, (3) a bound based on a second-order laminate whose support contains two matrices at infinity in the sense of (2.5), (4) the rank-one convex envelope W rc generated by infinite-rank laminates with support on SO(2) and at infinity. where the sign in front of θ is chosen to the same as that of F e1 · F e2 . Since any F ∈ Σ is uniquely determined, up to left-multiplication by a rotation, by the lengths of its columns and the sign of the inner product of its columns, it is not difficult to show that there exists a rotation Q ∈ SO(2) and numbers s, t ∈ R such that F = G(QFθ , s, t). Then by (2.7) and the rotational invariance of W , we find p 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 associated quadratic equations, we arrive at an upper bound for W rc (F ) for every θ. We report the best of these bounds, obtained from numerically optimizing in θ. 26

0.05 2-well approximation 1st order laminate 2nd order laminate Wrc

relative distance from Wpc

0.04

0.03

0.02

0.01

0 1

1.5

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

2.5

3

Figure 4: Comparison between the bounds, as in Figure 3, but along the line {|F e1 | + |F e2 | = 4}. The midpoint corresponds to the point at t = 2 of Figure 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 relative error of about 1.7%.

Acknowledgements This work was performed while NA was at the Universit¨at Duisburg-Essen supported by the National Science Foundation through the Mathematical Sciences Postdoctoral Research Fellowship Award #0603611. The work of SC was supported by the Deutsche Forschungsgemeinschaft through the Schwerpunktprogramm 1253 “Optimization with Partial Differential Equations”, project CO 304/2-1. The work of GD was supported by the National Science Foundation through grant DMS 0405853.

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References [1] M. Abramowitz and I. A. Stegun, Handbook of mathematical functions with formulas, graphs, and mathematical tables, National Bureau of Standards Applied Mathematics Series, vol. 55, U.S. Government Printing Office, Washington, D.C., 1964. [2] J. Adams, S. Conti, A. DeSimone, and G. Dolzmann, Relaxation of some transversally isotropic energies and applications to smectic A elastomers, Math. Mod. Meth. Appl. Sci. 18 (2008), 1–20. [3] J. M. Ball and R. D. James, Proposed experimental tests of a theory of fine microstructure and the two-well problem, Phil. Trans. R. Soc. Lond. A 338 (1992), 389–450. [4] J. M. Ball, B. Kirchheim, and J. Kristensen, Regularity of quasiconvex envelopes, Calc. Var. Partial Differential Equations 11 (2000), 333–359. [5] J. M. Ball and F. Murat, W 1,p -quasiconvexity and variational problems for multiple integrals, J. Funct. Anal. 58 (1984), 225–253. [6] S. Bartels, C. Carstensen, S. Conti, K. Hackl, U. Hoppe, and A. Orlando, Relaxation and the computation of effective energies and microstructures in solid mechanics, Analysis, Modeling and Simulation of Multiscale Problems (A. Mielke, ed.), Springer, 2006, pp. 197–224. [7] S. Bartels, C. Carstensen, K. Hackl, and U. Hoppe, Effective relaxation for microstructure simulations: algorithms and applications, Comp. Methods Appl. Mech. Engrg. 193 (2004), 5143–5175. [8] C. Carstensen, K. Hackl, and A. Mielke, Nonconvex potentials and microstructure in finite-strain plasticity, Proc. Roy. Soc. London, Ser. A 458 (2002), 299–317. [9] S. Conti, Quasiconvex functions incorporating volumetric constraints are rank-one convex, preprint (2007). [10] S. Conti, D. Faraco, and F. Maggi, A new approach to counterexamples to L1 estimates: Korn’s inequality, geometric rigidity, and regularity for gradients of separately convex functions, Arch. Rat. Mech. Anal. 175 (2005), 287–300. [11] S. Conti and M. Ortiz, Dislocation microstructures and the effective behavior of single crystals, Arch. Rat. Mech. Anal. 176 (2005), 103–147. 28

[12] S. Conti and F. Theil, Single-slip elastoplastic microstructures, Arch. Rat. Mech. Anal. 178 (2005), 125–148. [13] B. Dacorogna, Direct methods in the calculus of variations, SpringerVerlag, New York, 1989. [14] A. DeSimone and G. Dolzmann, Macroscopic response of nematic elastomers via relaxation of a class of SO(3)-invariant energies, Arch. Rat. Mech. Anal. 161 (2002), 181–204. [15] G. Dolzmann, Variational methods for crystalline microstructure - analysis and computation, Lecture Notes in Mathematics, no. 1803, SpringerVerlag, 2003. [16] H. Le Dret and A. Raoult, The quasiconvex envelope of the Saint Venant-Kirchhoff stored energy function, Proc. Roy. Soc. Edinburgh Sect. A 125 (1995), 1179–1192. [17] D. Faraco, Milton’s conjecture on the regularity of solutions to isotropic equations, Ann. I. H. Poincar´e 20 (2003), 889–909. [18]

, Tartar conjecture and Beltrami operators, Michigan Math. J. 52 (2004), 83–104.

[19] I. Fonseca, The lower quasiconvex envelope of the stored energy function for an elastic crystal, J. Math. pures et appl. 67 (1988), 175–195. [20] I. Fonseca and S. M¨ uller, Quasi-convex integrands and lower semiconti1 nuity in L , SIAM J. Math. Anal. 23 (1992), 1081–1098. [21]

, Relaxation of quasiconvex functionals in BV(Ω, Rp ) for integrands f (x, u, ∇u), Arch. Rational Mech. Anal. 123 (1993), 1–49.

[22] R. V. Kohn and G. Strang, Optimal design and relaxation of variational problems. I, II, III, Comm. Pure Appl. Math. 39 (1986), 113–137; 139– 182; 353–377. [23] G. Dal Maso, Integral representation on BV(Ω) of Γ-limits of variational integrals, Manuscripta Math. 30 (1979/80), 387–416. [24] C. Miehe, M. Lambrecht, and E. G¨ urses, Analysis of material instabilities in inelastic solids by incremental energy minimization and relaxation methods: evolving deformation microstructures in finite plasticity, J. Mech. Phys. Solids 52 (2004), 2725–2769. 29

[25] C. Miehe, J. Schotte, and M. Lambrecht, Homogeneization of inelastic solid materials at finite strains based on incremental minimiza tion principles. application to the texture analysis of polycrystals, J. Mech. Phys. Solids 50 (2002), 2123–2167. [26] A. Mielke, Energetic formulation of multiplicative elasto-plasticity using dissipation distances, Contin. Mech. Thermodyn. 15 (2003), 351–382. [27] C. B. Morrey, Quasi-convexity and the lower semicontinuity of multiple integrals, Pacific J. Math. 2 (1952), 25–53. [28] S. M¨ uller, Variational models for microstructure and phase transitions, in: Calculus of variations and geometric evolution problems (F. Bethuel et al., eds.), Springer Lecture Notes in Math. 1713, Springer-Verlag, 1999, pp. 85–210. ˇ ak, Convex integration with constraints and ap[29] S. M¨ uller and V. Sver´ plications to phase transitions and partial differential equations, J. Eur. Math. Soc. (JEMS) 1 (1999), 393–442. [30] M. Ortiz and E. A. Repetto, Nonconvex energy minimization and dislocation structures in ductile single crystals, J. Mech. Phys. Solids 47 (1999), 397–462. ˇ ak, On the problem of two wells, Microstructure and phase transi[31] V. Sver´ tion, IMA Vol. Math. Appl., vol. 54, Springer, New York, 1993, pp. 183– 189. [32] R. Temam, Mathematical problems in plasticity, Bordas, Paris, 1985.

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