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Our results, which are based on the representation$$W(F)=h\\bigl (\\frac{\\lambda _1}{\\lambda _2}\\bigr )$$<\/jats:tex-math>W<\/mml:mi>(<\/mml:mo>F<\/mml:mi>)<\/mml:mo><\/mml:mrow>=<\/mml:mo>h<\/mml:mi>(<\/mml:mo><\/mml:mrow>\u03bb<\/mml:mi>1<\/mml:mn><\/mml:msub>\u03bb<\/mml:mi>2<\/mml:mn><\/mml:msub><\/mml:mfrac>)<\/mml:mo><\/mml:mrow><\/mml:mrow><\/mml:math><\/jats:alternatives><\/jats:inline-formula>ofW<\/jats:italic>in terms of the singular values$$\\lambda _1,\\lambda _2$$<\/jats:tex-math>\u03bb<\/mml:mi>1<\/mml:mn><\/mml:msub>,<\/mml:mo>\u03bb<\/mml:mi>2<\/mml:mn><\/mml:msub><\/mml:mrow><\/mml:math><\/jats:alternatives><\/jats:inline-formula>ofF<\/jats:italic>, 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$${\\mathbb {K}}:=\\frac{1}{2}\\frac{\\Vert F \\Vert ^2}{\\det F}$$<\/jats:tex-math>K<\/mml:mi>:<\/mml:mo>=<\/mml:mo>1<\/mml:mn>2<\/mml:mn><\/mml:mfrac>\u2016<\/mml:mo>F<\/mml:mi>\u2016<\/mml:mo><\/mml:mrow>2<\/mml:mn><\/mml:msup>det<\/mml:mo>F<\/mml:mi><\/mml:mrow><\/mml:mfrac><\/mml:mrow><\/mml:math><\/jats:alternatives><\/jats:inline-formula>. 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 Gr\u00f6tzsch 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\u20134765, 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$$W^{1,p}$$<\/jats:tex-math>W<\/mml:mi>1<\/mml:mn>,<\/mml:mo>p<\/mml:mi><\/mml:mrow><\/mml:msup><\/mml:math><\/jats:alternatives><\/jats:inline-formula>-quasiconvex envelope on the domain$${{\\,\\mathrm{GL}\\,}}^{\\!+}(n)$$<\/jats:tex-math>GL<\/mml:mi><\/mml:mrow><\/mml:mrow>+<\/mml:mo><\/mml:mrow><\/mml:msup>(<\/mml:mo>n<\/mml:mi>)<\/mml:mo><\/mml:mrow><\/mml:mrow><\/mml:math><\/jats:alternatives><\/jats:inline-formula>which, in particular, ensure that a stricter version ofDacorogna\u2019s formula<\/jats:italic>is applicable to conformally invariant energies on$${{\\,\\mathrm{GL}\\,}}^{\\!+}(2)$$<\/jats:tex-math>GL<\/mml:mi><\/mml:mrow><\/mml:mrow>+<\/mml:mo><\/mml:mrow><\/mml:msup>(<\/mml:mo>2<\/mml:mn>)<\/mml:mo><\/mml:mrow><\/mml:mrow><\/mml:math><\/jats:alternatives><\/jats:inline-formula>.<\/jats:p>","DOI":"10.1007\/s00332-020-09639-4","type":"journal-article","created":{"date-parts":[[2020,7,21]],"date-time":"2020-07-21T16:05:20Z","timestamp":1595347520000},"page":"2885-2923","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":7,"title":["The Quasiconvex Envelope of Conformally Invariant Planar Energy Functions in Isotropic Hyperelasticity"],"prefix":"10.1007","volume":"30","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-4443-1641","authenticated-orcid":false,"given":"Robert J.","family":"Martin","sequence":"first","affiliation":[]},{"given":"Jendrik","family":"Voss","sequence":"additional","affiliation":[]},{"given":"Ionel-Dumitrel","family":"Ghiba","sequence":"additional","affiliation":[]},{"given":"Oliver","family":"Sander","sequence":"additional","affiliation":[]},{"given":"Patrizio","family":"Neff","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2020,7,21]]},"reference":[{"key":"9639_CR1","doi-asserted-by":"crossref","unstructured":"Adamowicz, T.: The Gr\u00f6tzsch problem in higher dimensions. 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