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

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

In this note, we provide an explicit formula for computing the quasiconvex envelope of any real-valued function W; SL(2) → ℝ with W(RF) = W(FR) = W(F) for all F ∈ SL(2) and all R ∈ SO(2), where SL(2) and SO(2) denote the special linear group and the special orthogonal group, respectively. In order to obtain our result, we combine earlier work by Dacorogna and Koshigoe on the relaxation of certain conformal planar energy functions with a recent result on the equivalence between polyconvexity and rank-one convexity for objective and isotropic energies in planar incompressible nonlinear elasticity.

A note on relaxation with constraints on the determinant

We consider multiple integrals of the Calculus of Variations of the form E(u) = ∫ W(x, u(x), Du(x)) dx where W is a Carathéodory function finite on matrices satisfying an orientation preserving or an incompressibility constraint of the type, det Du > 0 or det Du = 1, respectively. Under suitable growth and lower semicontinuity assumptions in the u variable we prove that the functional ∫ Wqc(x, u(x), Du(x)) dx is an upper bound for the relaxation of E and coincides with the relaxation if the quasiconvex envelope Wqc of W is polyconvex and satisfies p growth from below for p bigger then the ambient dimension. Our result generalises a previous one by Conti and Dolzmann [Arch. Rational Mech. Anal. 217 (2015) 413–437] relative to the case where W depends only on the gradient variable.

Relaxation of a model energy for the cubic to tetragonal phase transformation in two dimensions

We consider a two-dimensional problem in nonlinear elasticity which corresponds to the cubic-to-tetragonal phase transformation. Our model is frame invariant and the energy density is given by the squared distance from two potential wells. We obtain the quasiconvex envelope of the energy density and therefore the relaxation of the variational problem. Our result includes the constraint of positive determinant.

Infinite-order laminates in a model in crystal plasticity

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-1 convex envelope of the condensed energy density is obtained by infinite-order laminates, and express it explicitly via the 2F1 hypergeometric function. We also determine the polyconvex envelope, leading to upper and lower bounds on the quasiconvex envelope. The two bounds differ by less than 2%.