The Quasiconvex Envelope of Conformally Invariant Planar Energy Functions in Isotropic Hyperelasticity

Abstract

We consider conformally invariant energies W on the group $${{\,\mathrm{GL}\,}}^{\!+}(2)$$ GL + ( 2 ) of $$2\times 2$$ 2 × 2 -matrices with positive determinant, i.e., $$W:{{\,\mathrm{GL}\,}}^{\!+}(2)\rightarrow {\mathbb {R}}$$ W : GL + ( 2 ) → R such that $$\begin{aligned} W(A\, F\, B) = W(F) \quad \text {for all }\; A,B\in \{a\, R\in {{\,\mathrm{GL}\,}}^{\!+}(2) \,|\,a\in (0,\infty ),\; R\in {{\,\mathrm{SO}\,}}(2)\}, \end{aligned}$$ W ( A F B ) = W ( F ) for all A , B ∈ { a R ∈ GL + ( 2 ) | a ∈ ( 0 , ∞ ) , R ∈ SO ( 2 ) } , where $${{\,\mathrm{SO}\,}}(2)$$ SO ( 2 ) denotes the special orthogonal group and provides an explicit formula for the (notoriously difficult to compute) quasiconvex envelope of these functions. Our results, which are based on the representation $$W(F)=h\bigl (\frac{\lambda _1}{\lambda _2}\bigr )$$ W ( F ) = h ( λ 1 λ 2 ) of W in terms of the singular values $$\lambda _1,\lambda _2$$ λ 1 , λ 2 of F, are applied to a number of example energies in order to demonstrate the convenience of the singular-value-based expression compared to the more common representation in terms of the distortion $${\mathbb {K}}:=\frac{1}{2}\frac{\Vert F \Vert ^2}{\det F}$$ K : = 1 2 ‖ F ‖ 2 det F . Applying our results, we answer a conjecture by Adamowicz (in: Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Serie IX. Matematica e Applicazioni, vol 18(2), pp 163, 2007) and discuss a connection between polyconvexity and the Grötzsch free boundary value problem. Special cases of our results can also be obtained from earlier works by Astala et al. (Elliptic partial differential equations and quasiconformal mappings in the plane, Princeton University Press, Princeton, 2008) and Yan (Trans Am Math Soc 355(12):4755–4765, 2003). Since the restricted domain of the energy functions in question poses additional difficulties with respect to the notion of quasiconvexity compared to the case of globally defined real-valued functions, we also discuss more general properties related to the $$W^{1,p}$$ W 1 , p -quasiconvex envelope on the domain $${{\,\mathrm{GL}\,}}^{\!+}(n)$$ GL + ( n ) which, in particular, ensure that a stricter version of Dacorogna’s formula is applicable to conformally invariant energies on $${{\,\mathrm{GL}\,}}^{\!+}(2)$$ GL + ( 2 ) .

Quasiconvex relaxation of isotropic functions in incompressible planar hyperelasticity

“…Another result similar to theorem 3.5 has previously been obtained [24] for the so-called conformally invariant functions on GL + (2), i.e. any W :…”

Sharp Rank-One Convexity Conditions in Planar Isotropic Elasticity for the Additive Volumetric-Isochoric Split

“…W is rank-one convex if and only if φ is convex and nondecreasing. (1.8) These results also allow for an explicit calculation of the quasiconvex relaxation for conformally invariant and incompressible isotropic planar hyperelasticity [37,38].…”

“…Here, K ≥ 1 is the so-called distortion function, which plays an important role in the theory of quasiconformal mappings [2,38]. Note that K ≡ 1 if and only if ϕ is conformal.…”

“…In a recent contribution [34,38], we have been able to classify all planar, isotropic rank-one convex functions W : GL + (2) → R which are isochoric (conformally invariant), i.e., satisfy W (AF B) = W (F ) for all A, B ∈ a R ∈ GL + (2) | a ∈ (0, ∞) , R ∈ SO (2) . (1.6) For these energies we have the simple characterization 1…”

A rank-one convex, nonpolyconvex isotropic function on with compact connected sublevel sets

“…If , and therefore , is objective and isotropic, then any function of the form (3.2) can be expressed as [25] for all with singular values and with uniquely defined functions and . Here, is the linear distortion (or dilation ) function, which plays an important role in the theory of conformal and quasiconformal mappings [2] as well as for the characterization of rank-one convexity, quasiconvexity and polyconvexity of isochoric planar energy functions [25, 27]. More specifically, the isochoric part of an energy of the form (3.2) is rank-one convex, quasiconvex and polyconvex if and only if the function given by (3.3) is nondecreasing and convex [25, theorem 3.3].…”