The concept of hyperelastic deformations of bi-conformal energy is developed as an extension of quasiconformality. These deformations are homeomorphisms $$\, h :{\mathbb {X}} \xrightarrow []{{}_{\!\!\text{ onto }\,\,\!\!}}{\mathbb {Y}}\,$$ h : X → onto Y between domains $$\,{\mathbb {X}}, {\mathbb {Y}} \subset {\mathbb {R}}^n\,$$ X , Y ⊂ R n of the Sobolev class $$\,{\mathscr {W}}^{1,n}_{\text{ loc }} ({\mathbb {X}}, {\mathbb {Y}})\,$$ W loc 1 , n ( X , Y ) whose inverse $$\, f {\mathop {=\!\!=}\limits ^{\text{ def }}} h^{-1} :{\mathbb {Y}} \xrightarrow []{{}_{\!\!\text{ onto }\,\,\!\!}}{\mathbb {X}}\,$$ f = = def h - 1 : Y → onto X also belongs to $$\,{\mathscr {W}}^{1,n}_{\text{ loc }}({\mathbb {Y}}, {\mathbb {X}})\,$$ W loc 1 , n ( Y , X ) . Thus the paper opens new topics in Geometric Function Theory (GFT) with connections to mathematical models of Nonlinear Elasticity (NE). In seeking differences and similarities with quasiconformal mappings we examine closely the modulus of continuity of deformations of bi-conformal energy. This leads us to a new characterization of quasiconformality. Specifically, it is observed that quasiconformal mappings behave locally at every point like radial stretchings; if a quasiconformal map $$\,h\,$$ h admits a function $$\,\phi \,$$ ϕ as its optimal modulus of continuity at a point $$\,x_\circ \, $$ x ∘ , then $$\,f = h^{-1}\,$$ f = h - 1 admits the inverse function $$\, \psi = \phi ^{-1}\,$$ ψ = ϕ - 1 as its modulus of continuity at $$\, y_\circ = h(x_\circ ) \,$$ y ∘ = h ( x ∘ ) . That is to say, a poor (possibly harmful) continuity of $$\,h\,$$ h at a given point $$\,x_\circ \,$$ x ∘ is always compensated by a better continuity of $$\,f\,$$ f at $$\,y_\circ \,$$ y ∘ , and vice versa. Such a gain/loss property, seemingly overlooked by many authors, is actually characteristic of quasiconformal mappings. It turns out that the elastic deformations of bi-conformal energy are very different in this respect. Unexpectedly, such a map may have the same optimal modulus of continuity as its inverse deformation. In line with Hooke’s Law, when trying to restore the original shape of the body (by the inverse transformation), the modulus of continuity may neither be improved nor become worse. However, examples to confirm this phenomenon are far from being obvious; indeed, elaborate computations are on the way. We eventually hope that our examples will gain an interest in the materials science, particularly in mathematical models of hyperelasticity.

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双共形能量的变形与拟共形的新表征

双共形能量的超弹性变形的概念是作为准共形的扩展而发展起来的。这些变形是同胚 $$\, h :{\mathbb {X}} \xrightarrow []{{}_{\!\!\text{ 到 }\,\,\!\!}}{\mathbb {Y }}\,$$ h : X → 到域之间的 Y $$\,{\mathbb {X}}, {\mathbb {Y}} \subset {\mathbb {R}}^n\,$$ X , Sobolev 类的 Y ⊂ R n $$\,{\mathscr {W}}^{1,n}_{\text{ loc }} ({\mathbb {X}}, {\mathbb {Y}}) \,$$ W loc 1 , n ( X , Y ) 其逆 $$\, f {\mathop {=\!\!=}\limits ^{\text{ def }}} h^{-1} ： {\mathbb {Y}} \xrightarrow []{{}_{\!\!\text{ 到 }\,\,\!\!}}{\mathbb {X}}\,$$ f == def h - 1 : Y → 到 X 也属于 $$\,{\mathscr {W}}^{1,n}_{\text{ loc }}({\mathbb {Y}}, {\mathbb {X }})\,$$ W loc 1 , n ( Y , X ) 。因此，本文在几何函数理论 (GFT) 中开辟了与非线性弹性 (NE) 数学模型相关联的新主题。在寻找与拟共形映射的异同时，我们仔细检查了双共形能量变形的连续性模量。这使我们对拟共形性进行了新的表征。具体来说，可以观察到拟共形映射在每个点都表现得像径向拉伸一样局部；如果拟共形映射 $$\,h\,$$h 承认函数 $$\,\phi \,$$ ϕ 作为其在点 $$\,x_\circ \, $$ x ∘ 处的最优连续模，则 $$\,f = h^{-1}\,$$ f = h - 1 承认反函数 $$\, \psi = \phi ^{-1}\,$$ ψ = ϕ - 1作为其在 $$\ 处的连续性模数，y_\circ = h(x_\circ ) \,$$ y ∘ = h ( x ∘ ) 。也就是说，$$\,h\ 的差（可能有害）连续性，$$ h 在给定点 $$\,x_\circ \,$$ x ∘ 总是通过 $$\,f\,$$ f 在 $$\,y_\circ \,$$ 的更好连续性来补偿y ∘ ，反之亦然。许多作者似乎忽略了这种增益/损失特性，实际上是拟共形映射的特征。事实证明，双共形能量的弹性变形在这方面是非常不同的。出乎意料的是，这样的地图可能具有与其逆变形相同的最佳连续性模数。根据虎克定律，当试图恢复物体的原始形状时（通过逆变换），连续性模量可能既不会提高也不会变差。然而，证实这一现象的例子远非显而易见。事实上，精细的计算正在进行中。我们最终希望我们的例子能够引起对材料科学的兴趣，