The quasiconvex envelope of conformally invariant planar energy functions in isotropic hyperelasticity

We consider conformally invariant energies $W$ on the group $\operatorname{GL}^+(2)$ of $2\times2$-matrices with positive determinant, i.e. $W\colon\operatorname{GL}^+(2)\to\mathbb{R}$ such that \[W(AFB) = W(F) \qquad\text{for all }\; A,B\in\{aR\in\operatorname{GL}^+(2) \,|\, a\in(0,\infty)\,,\; R\in\operatorname{SO}(2)\}\,,\] where $\operatorname{SO}(2)$ denotes the special orthogonal group, and provide an explicit formula for the (notoriously difficult to compute) quasiconvex envelope of these functions. Our results, which are based on the representation $W(F)=h(\frac{\lambda_1}{\lambda_2})$ of $W$ in terms of the singular values $\lambda_1,\lambda_2$ of $F$, are applied to a number of example energies in order to demonstrate the convenience of the eigenvalue-based expression compared to the more common representation in terms of the distortion $\mathbb{K}:=\frac12\frac{\lVert F\rVert^2}{\det F}$. Special cases of our results can be obtained from earlier works by Astala et al. and Yan.


Introduction
A recent contribution [42]  i.e. left-and right-invariance under the special orthogonal group SO (2) and invariance under scaling. In nonlinear elasticity theory, where F = ∇ϕ represents the so-called deformation gradient of a deformation ϕ, the former two invariances correspond to the objectivity and isotropy of W , respectively. In this context, an 1 Note that this invariance property needs to be distinguished from the concept of (nearly) conformal energies [36,63], i.e. functions W ≥ 0 such that W (F ) = 0 if and only if F ∈ CSO(2), e.g. W (F ) = F 2 − 2 det F . Instead of invariances of the argument, these energies are characterized by a global "potential well" containing the unbounded set CSO(2) and can merely be considered "conformally invariant in F = 1".
In a planar minimization problem subject to the homeomorphic boundary condition ϕ| ∂Ω = ϕ 0 , the 2-harmonic Dirichlet energy I(ϕ) = Ω 1 2 ∇ϕ 2 dx is sometimes referred to as a conformal energy as well. Indeed, and equality holds if and only if ϕ is conformal, due to Hadamard's inequality and the fact that det ∇ϕ is a null Lagrangian. However, the energy density W (F ) = 1 2 F 2 is neither conformally invariant in the sense of (1.1) nor (nearly) conformal in the above sense. energy W satisfying W (aF ) = W (F ) is more commonly known as isochoric, and is often additively coupled [55,18] with a volumetric energy term of the form f (det F ) for some convex function f : (0, ∞) → R.
In this contribution, we consider the quasiconvex envelopes of conformally invariant energies on GL + (2). Based on our previous results, we provide an explicit formula that allows for a direct computation of the quasiconvex (as well as the rank-one convex and polyconvex) envelope for this class of functions. We also discuss different ways of expressing conformally invariant energies, including representations based on the singular values of F , i.e. the eigenvalues of √ F T F , in order to highlight the difficulties which arise from focusing on the seemingly more simple representation in terms of the distortion function K.
Our main result (Theorem 3.3) has been tested against a numerical algorithm for computing the polyconvex envelope [14] for a range of parameters, yielding agreement up to computational precision. In two special cases, we show that our results completely match previous developments of Astala, Iwaniec, and Martin [6] and Yan [65,64]. We also present direct finite element simulations of the microstructure using a trustregion-multigrid method [19,56] which show consistent results. In the appendix, we answer two questions by Adamowicz [1], and discuss a related relaxation result by Dacorogna and Koshigoe [21].

Conformal and quasiconformal mappings
Energy functions of the form (1.1) are intrinsically linked to conformal geometry and geometric function theory [6]. A mapping ϕ : Ω → R 2 is called conformal if and only if ∇ϕ(x) ∈ CSO(2) on Ω or, equivalently, where 1 ∈ GL + (2) denotes the identity matrix. If R 2 is identified with the complex plane C, then ϕ is conformal if and only if ϕ : Ω ⊂ C → C is holomorphic and the derivative is non-zero everywhere. Although the Riemann mapping theorem states that any non-empty, simply connected open planar domain can be mapped conformally to the unit disc, conformal mappings exhibit aspects of rigidity [25] that make them too restrictive for many interesting applications. In particular, since the Riemann mapping is uniquely determined by prescribing the function value for three points, conformal mappings are not able to satisfy arbitrary boundary conditions.
A significantly larger and more flexible class is given by the so-called quasiconformal mappings, i.e. functions ϕ : Ω → R 2 that satisfy the uniform bound where K denotes the distortion function [33,7] or outer distortion [34] K : GL + (2) → R , K(F ) := 1 2 Due to Hadamard's inequality, K(F ) ≥ 1 for all F ∈ GL + (2). In particular, if (1.3) is satisfied with L = 1, then K(∇ϕ) ≡ 1, which implies that ϕ is conformal. The classical Grötzsch free boundary value problem [30] (cf. Appendix B) is to find and characterize quasiconformal mappings of rectangles into rectangles that minimize the maximal distortion K ∞ and map faces to corresponding faces, i.e. to solve the minimization problem A much more involved problem has been solved by Teichmüller [61,2]. The classical Teichmüller problem (cf. Appendix B) is to find and characterize quasiconformal solutions to for 0 < a < 1. Computational approaches for calculating extremal quasiconformal mappings (with direct applications in engineering) are discussed, e.g., in [62]. However, the analytical difficulties posed by this problem also motivate the study of integral generalizations of (1.6), i.e.
where Ψ : [1, ∞) → [0, ∞) is assumed to be strictly increasing. Further generalizing the domain, boundary condition and additional constraints, we obtain a more classical problem in the calculus of variations: the existence and uniqueness of mappings between planar domains with prescribed boundary values that minimize certain integral functions of K, i.e. the minimization problem for given Ψ : [1, ∞) → R and ϕ 0 : Ω → R 2 . Since K(aR∇ϕ) = K(∇ϕaR) = K(∇ϕ) for all a > 0 and all R ∈ SO(2), the distortion function K is conformally invariant, and indeed every conformally invariant energy W on GL + (2) can be expressed in the form W (F ) = Ψ(K(F )), see [42]. However, the mapping F → K(F ) is non-convex. Without additional restrictions on Ψ, it is therefore difficult to establish results regarding the existence or regularity of minimizers. It is generally believed [6, Conjecture 21.2.1, p. 599] that for "well-behaved" functions Ψ, e.g. if Ψ is smooth, strictly increasing and convex, any solution 2 to the minimization problem (1.7) is a C 1,α -diffeomorphism; this would contrast typical regularity results for more general problems in the calculus of variations (including nonlinear elasticity), where only partial regularity (e.g. C 1,α up to a set of measure zero) can be expected.
In this contribution, we are interested in cases where Ψ is not well behaved in the above sense; more specifically, we allow for some lack of convexity and monotonicity of Ψ. Our results demonstrate that the common representation W (F ) = Ψ(K(F )) of an arbitrary conformally invariant function W on GL + (2) is neither ideal nor "natural" as far as convexity properties of W are concerned. Instead, by introducing the linear distortion (or (large) dilatation [62]) where F op = sup ξ =1 F ξ R 2 denotes the operator norm (i.e. the largest singular value) of F , we can equivalently express any conformally invariant energy W as W (F ) = h(K(F )). This representation turns out to be much more convenient and suitable with respect to convexity properties of W . 3 In particular, our results (cf. Remark 3.5) will allow us to easily generalize a consequence of a theorem by Astala, Iwaniec and Martin [6, Theorem 21.1.3, p. 591], stating that for F 0 ∈ GL + (2) and Ω = B 1 (0) and any strictly increasing Ψ : (1.8) Note that the corresponding minimization problem has no solution unless F 0 ∈ CSO(2). This result can be expressed as a specific relaxation statement, namely that for these W (F ) = Ψ(K(F )), the quasiconvex envelope QW of W is given by QW (F ) ≡ Ψ(1). Quasiconvex envelopes arise naturally in the calculus of variations, representing the energetic response of a minimization problem without minimizers for linearly homogeneous boundary values under the presence of microstructure, i.e. [20,59,53,58] QW for any domain Ω ⊂ R 2 with Lebesgue measure |Ω|. If QW (F 0 ) < W (F 0 ) for some F 0 ∈ GL + (2), then the equilibrium state of the homogeneous deformation ϕ(x) = F 0 x is unstable: the material shows an energetic preference to develop finer and finer spatially modulated deformations (in engineering applications these are typically shear bands) at fixed averaged deformation F 0 x. In this case, there are infimizing sequences with highly oscillating gradients which converge weakly (presuming appropriate coercivity conditions), but the weak limit is not a minimizer. In this respect, our main result also answers a particular case of Iwaniec's Question 3 from [35]: "Which [functions of the] distortion [Ψ(K(∇ϕ))] are weakly lower semi-continuous in W 1,2 (Ω, R 2 )?"
The importance of quasiconvexity stems from the fact that quasiconvexity of W is essentially equivalent to the weak lower semi-continuity of Ω W (∇ϕ(x))dx [46]. The problem whereas for dimension n ≥ 3, it is well known that the corresponding convexity properties are not equivalent; Sverak famously showed that rank-one convexity does not imply quasiconvexity with a counterexample consisting of a non-isotropic, non-objective polynomial of order four [60]. In the two-dimensional case discussed here, however, the question whether rank-one convexity is equivalent to quasiconvexity, known as the remaining part of Morrey's conjecture [46], is still unanswered [46,5] and is considered one of the major open problems in the calculus of variations [12,11,48].
In order to state criteria for the above convexity properties in the special case of conformally invariant functions on GL + (2), we consider a number of different representations available to express such functions.
Conversely, if the requirements (2.3) are satisfied for otherwise arbitrary functions g : Note that h is already uniquely determined by its values on [1, ∞) and recall that K ≥ 1, with K(∇ϕ) = 1 if and only if ϕ is conformal.
The following proposition summarizes the main results from [42] and completely characterizes the generalized convexity of conformally invariant functions on GL + (2).
Then the following are equivalent: Note that in terms of the representation function h, the convexity criteria can be expressed in a remarkably simple way, especially when compared to vii), i.e. the representation in terms of the classical distortion K. In particular, while monotonicity and convexity of Ψ are sufficient for the considered properties, 4 convexity of the energy with respect to K is not a necessary condition; for example, if W : GL + (2) → R is given by where 1 denotes the identity matrix. This energy W can be expressed in the form (2.2) with Since h : (0, ∞) → R is convex, the planar isochoric St. Venant-Kirchhoff energy is quasiconvex according to Proposition 2.2, while, e.g. the non-conformally-invariant term

Generalized convexity properties and convex envelopes
For each of the convexity properties listed in the previous section, we can consider the corresponding envelope of an arbitrary energy function W : Since polyconvexity implies quasiconvexity, which in turn implies rank-one convexity (cf. [8, Theorem 3.3] for the case of isotropic functions on GL + (n)), it is easy to see that CW (F ) ≤ P W (F ) ≤ QW (F ) ≤ RW (F ). 5 However, while a number of numerical methods are available to approximate the rank-one convex envelope RW [23,13,52] as well as the polyconvex envelope P W [22,39,14,4], it is difficult to analytically compute RW , P W or the quasiconvex envelope QW of a given energy W in general, although explicit representations have been found for a number of particular functions, including the St. Venant-Kirchhoff energy [41] and several challenging problems encountered in engineering applications [17,3]. Further examples can be found in [20,Chapter 6].
More general methods for computing the quasiconvex envelope are often based on the observation that RW = P W and thus RW = QW for certain classes of energy functions W . In many such cases, even the equality RW = CW holds [21,54], i.e. the generalized convex envelopes are all identical to the classical convex envelope of W , cf. Appendix C.
Yan [63] showed that non-constant rank-one convex conformal energy functions (cf. Footnote 1 for the distinction between conformally invariant and conformal energy functions) defined on all of R n×n for n ≥ 3 must grow at least with power n 2 , which implies that the quasiconvex envelope of a conformal energy W on R 3×3 must be constant if W exhibits sublinear growth. 6 The results given in the following show that an analogous property holds for conformally invariant energies on GL + (2).
In order to apply Proposition 2.2 to the computation of generalized convex envelopes, the following invariance property of the rank-one convex envelope will be required. Proof. It is well known that the left-and right-SO(2)-invariance is preserved by the rank-one convex envelope [16,21,40], so due to the characterization (1.2) of conformal invariance it remains to show that RW (aF ) = RW (F ) for all a > 0 and all F ∈ GL + (2).
We use the characterization RW (F ) = lim k→∞ R k W (F ) of the rank-one convex envelope [20, p. 202], where R 0 W (F ) = W (F ) and Then, since taF 1 + (1 − t)aF 2 = aF and rank(aF 1 − aF 2 ) = 1, Remark 3.2. By direct computation, it is easy to see that QW is conformally invariant if W : GL + (n) → R is conformally invariant. Similar to the rank-one convex envelope it is well known [16,21,40] that the leftand right-SO(2)-invariance is preserved by QW , and the equality can be obtained by utilizing the scaling invariance W (aF ) = W (F ) for all a > 0 and F ∈ GL + (n).

Main result on the quasiconvex envelope
We can now state our main result.

2)
where C m h : [1, ∞) → R denotes the monotone-convex envelope given by Proof. Let w(F ) := C m h λ1 λ2 . Due to the convexity and monotonicity of C m h and Proposition 2.2, the mapping w : GL + (2) → R is polyconvex. Therefore, since for all F ∈ GL + (2) with singular values λ 1 ≥ λ 2 . Due to the rank-one convexity of RW and Proposition 2.2, the function h is convex and non-decreasing. Since as well, we find h(t) ≤ C m h(t) for all t ∈ [1, ∞) and thus  For this special case, we directly recover the earlier result (1.8) originally due to Astala, Iwaniec, and Martin [6].
Remark 3.6. The monotone-convex envelope of h : [1, ∞) → R can also be obtained by "reflecting" the graph of the function at x = 1 and taking the classical convex envelope: if h : R → R denotes the extension of h to R defined by Figure 1 and Appendix A.

Specific relaxation examples and numerical simulations
Theorem 3.3 can be used to explicitly compute the quasiconvex envelope for a substantial class of functions. In the following, a number of explicit relaxation examples will be considered and some of our analytical results will be compared to numerical simulations.

The deviatoric Hencky energy
First, consider the (planar) deviatoric Hencky strain energy [32,49] where dev n X := X − 1 n tr(X) · 1 is the deviatoric (trace-free) part of X ∈ R n×n and log U denotes the principal matrix logarithm of the right stretch tensor U := √ F T F . The energy W dH can be expressed as Since the representing function h : [1, ∞) → R with h(t) = log 2 (t) is monotone, we find and thus RW dH = QW dH = P W dH ≡ 0 .
Note that due to the sublinear growth of h (or, equivalently, of the representation K → arccosh 2 (K)), this result can also be obtained by eq. (1.8), cf. Remark 3.5. Interestingly, the deviatoric Hencky strain energy itself is directly related to the conformal group CSO(n): Let dist geod (·, ·) denote the geodesic distance on the Lie group GL + (n) with respect to the canonical leftinvariant Riemannian metric [44,45]. Then the distance of F ∈ GL + (n) to the special orthogonal group SO(n) ⊂ GL + (n) is given by [49,Theorem 3.3] dist 2 geod (F, SO(n)) = min The deviatoric Hencky strain energy can therefore be characterized by the equality where ( * ) holds due to the left-invariance of the metric.

The squared logarithm of K
Similarly, consider W log (F ) = (log K) 2 = log 2 1 2 Since h is again monotone on [1, ∞), we find and thus
In order to further investigate the behavior of this quasiconvex relaxation with finite element simulations, we choose the particular value k = 0.11 < 1 8 and consider the quasiconvex envelope QW (F ) of Using Maxwell's equal area rule [58, p. 319], we numerically compute the monotone-convex envelope of h up to five decimal digits: This explicit representation allows us to determine the set of all F ∈ GL + (2) with QW (F ) < W (F ), known as the binodal region [29,28]. In particular, the microstructure energy gap (cf. Figure 3) between h and Ch is maximal at λ1 λ2 ≈ 12.0186 =: x 0 with a value of ∆ ≈ 0.0221558. We therefore choose homogeneous Dirichlet boundary conditions given by such that det F 0 = 1, for the finite element simulation. The energy level of the homogeneous solution is whereas the infimum of the energy levels of the microstructure solutions is "maximal microstructure energy gap"∆ ) for k < 1 8 . Figure 4 shows two numerical simulations of the microstructure on triangle grids with different resolutions. The illustration shows the reference configuration, colored according to the value of the determinant of the deformation gradient (plotting K instead results in similar images). The energy level of the configuration on the left is 6.17149 on a grid with 294 912 vertices. Repeating the computation on a grid with one additional step of uniform refinement leads to the configuration on the right, which has an energy level of 6.16216.
Note that the values obtained for the energy level still differ significantly from the expected value of 6.13194. It is unclear whether the discrepancy is solely due to insufficient mesh resolution; further numerical investigations on more performant hardware are planned for the future. The expected energy level was, however, obtained numerically using a modification of an algorithm by Bartels [14] for computing the polyconvex envelope.

An energy function related to a result by Yan
Lastly, we consider the energy function which penalizes the deviation of the distortion K from a prescribed value L ≥ 1. According to Theorem 3.3, the quasiconvex envelope of W is given by Again, we want to further investigate the microstructure induced by W with numerical simulations on Ω = B 1 (0). For our calculations, we consider the case L = 2. At x 0 = λ1 λ2 = 1, the microstructure energy gap between between h and Ch is maximal with a value of ∆ ≈ 0.54308, hence we use homogeneous Dirichlet boundary values with F 0 = 1. The energy value of the homogeneous solution is whereas the energy level of the microstructure solution should, in the limit, approach inf I(ϕ) = inf We again compute the microstructure using finite element simulations. The microstructure exhibited by this example significantly differs from the previous one; in particular, no simple laminar structure can be observed at all ( Figure 6). As expected, we obtain deformations with K very close to the value 2 throughout the domain (Figure 7). The energy levels obtained numerically are also very close to the expected value of 0. Specifically, for meshes with 294 912 and 1 179 648 grid vertices, the obtained energy levels are 2.533 · 10 −3 and 1.369 · 10 −3 , respectively. In the following, we will discuss a close connection of the quasiconvex envelope (4.3) and the observed microstructure to an earlier result by Yan [65,64], which implies that the microstructure should approximately satisfy K(∇ϕ) = 2 almost everywhere on Ω.
In two remarkable contributions [65,64], Yan considered the Dirichlet problem  Since in the two-dimensional case Furthermore, recalling that K = 1 2 K + 1 K and letting L = 1 2 l + 1 l , Corollary 4.2 can equivalently be expressed in terms of the distortion K.
For K(F 0 ) = L, the infimum value zero is already realized by the homogeneous solution. For K(F 0 ) < L, although there is no homogeneous equilibrium solution, there exist a deformation ϕ ∈ W 1,2 (Ω; R 2 ) with ϕ| ∂Ω = F 0 x and K(∇ ϕ) = L due to Corollary 4.3. Then Ψ L (K(∇ ϕ)) = 0 and thus since Ψ L is monotone increasing and convex for K(F 0 ) ≥ L.

A The quasiconvex envelope for a class of conformal energies
The concept of monotone-convex envelopes is directly connected to an earlier result by Dacorogna and Koshigoe [21], who obtained an explicit relaxation result for a subclass of conformal energy functions.
The same result can be found in [59,Prop. 4.1]. Note that the convexity of the mapping F → g * * ( F 2 − 2 det F ) = g * * ( (λ 1 − λ 2 ) 2 ) follows directly [10] from the fact that g * * is convex and non-decreasing on [0, ∞). Furthermore, if g ≥ 0, then W of the form (A.1) is a conformal energy in the sense of Footnote 1. Cg(x) x 0 g(x) C g(x) x Figure 8: The monotone-convex envelope Cmg of g : [0, ∞) → R can be obtained via the convex envelope Cmg of the even extension g of g.
If g is continuous and bounded below, then based on [20,Theorem 2.43] it is easy to show that the monotone-convex envelope of g is exactly the restriction of g to [0, ∞): Similar to the geodesic distance considered in Section 4.1, the expression F 2 − 2 det F can be characterized as a measure of distance to the conformal group: 7 since the closure CSO(2) ∪ {0} of CSO(2) is a linear subspace 8 of R 2×2 with an orthonormal basis given by where ·, · denotes the canonical inner product on R 2×2 . Therefore, the energy functions considered in Lemma A.1 depend only on the Euclidean distance of F to CSO(2).

B Connections to the Grötzsch problem
Proposition 2.2 negatively answers a conjecture by Adamowicz [1, Conjecture 1], which (in the two-dimensional case) states that if a conformal energy W : GL + (2) → R with W (F ) = Ψ(K(F )) is polyconvex, then Ψ is non-decreasing and convex. A direct counterexample is given by W (F ) = λmax λ min , which is polyconvex due to criterion v) in Proposition 2.2 with h(t) = t, but the representation W (F ) = Ψ(K(F )) = e arccosh(K(F )) is not convex with respect to K(F ). Furthermore, criterion iv) in Proposition 2.2 reveals a direct connection between the so-called Grötzsch property and quasiconvexity in the two-dimensional case.  1 x 1 , . . . , a n an xn); here, the set A of admissible functions consists of all ϕ ∈ W 1,p loc (Q; Q ), p ≥ n with det ∇ϕ > 0 that satisfy the Grötzsch boundary conditions, i.e. map each (n − 1)-dimensional face of Q to the corresponding face of Q . 7 Note that the Euclidean distance can be considered a linearization of the geodesic distance and, unlike the latter, does not take into account the Lie group structure of either GL + (2) or CSO (2). For a detailed discussion of the relation between these distance measures and their applicability to the deformation gradient in nonlinear mechanics, see [49]. 8 More generally [57, p.24], the set [0, ∞) · SO(n) is convex for n ≥ 1.
Note that the boundary condition imposed in Definition B.1 does not require the admissible mappings to be affine at the boundary, since each of the faces can be mapped to the corresponding ones in an arbitrary (possibly non-linear) manner.
In the two-dimensional case, the representation of the energy in terms of the singular values allows us to infer the quasiconvexity from the Grötzsch property in a particularly straightforward way.
Proposition B.2. Let W : GL + (2) → R be conformally invariant and satisfy the Grötzsch property for all Q, Q . Then W is polyconvex.

C The convex envelope of conformally invariant planar energies
The quasiconvex envelopes computed in Section 4 are, in general, not convex, i.e. QW (F ) > CW (F ) for some F ∈ GL + (2). In fact, the following explicit computation shows that the convex envelope of any conformally invariant energy is necessarily constant.
Recall that the convex envelope CW of an energy W : M → R with a non-convex domain M ⊂ R n×n (e.g. M = GL + (2)) is defined as the restriction C W | M of the convex envelope C W of the function for all F ∈ GL + (2).
Proof. We only need to show that CW is constant on GL + (2). First, observe that the convex envelope of W is conformally invariant. 9 By the definition of convexity on GL + (2) employed here, CW must be the restriction of a convex function W : R 2×2 → R to GL + (2). Let b := W (0). Then for all F ∈ GL + (2) and t ∈ [−1, 1], we find W (tF ) = CW (tF ) = CW (F ) : t = 0 , b : t = 0 , thus CW (F ) = b due to the convexity of W .
As indicated in Section 3, analytical methods for finding generalized convex envelopes have often been based on the observation that RW = CW for certain classes of energy functions W and the subsequent computation of the classical convex envelope CW ; for example, this method is applicable to the St. Venant-Kirchhoff energy function [41] W SVK (F ) = µ 4 F T F − 1 2 + λ 8 tr(F T F − 1) 2 .
One of the most frequently cited examples of an isotropic and objective energy function W with RW = QW = P W = CW is the example of Kohn and Strang [37,38], where, in the R 2×2 -case [66,24],