English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Tools annual

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools monthly

Global: Zendy Plus monthly

Presents the access of premium content as premium feature

Premium Content

Global: Zendy Plus annual

Global: Zendy Free Plan

It is well known that a twice-differentiable real-valued function$W:\operatorname{GL}^+(n)\rightarrow\mathbb{R}$ on the group$\operatorname{GL}^+(n)$ of invertible $n\times n-$matrices with positivedeterminant is rank-one convex if and only if it is Legendre-Hadamard elliptic.Many energy functions arising from interesting applications in isotropicnonlinear elasticity, however, are not necessarily twice differentiableeverywhere on $\operatorname{GL}^+(n)$, especially at points with non-simplesingular values.  Here, we show that if an isotropic function $W$ on $\operatorname{GL}^+(n)$is twice differentiable at each $F\in\operatorname{GL}^+(n)$ with simplesingular values and Legendre-Hadamard elliptic at each such $F$, then $W$ isalready rank-one convex under strongly reduced regularity assumptions. Inparticular, this generalization makes (local) ellipticity criteria accessibleas criteria for (global) rank-one convexity to a wider class of elastic energypotentials expressed in terms of ordered singular values. Our results are alsodirectly applicable to so-called conformally invariant energy functions. Wealso discuss a classical ellipticity criterion for the planar case by Knowlesand Sternberg which has often been used in the literature as a criterion forglobal rank-one convexity and show that for this purpose, it is stillapplicable under weakened regularity assumptions.

Rank-one convexity vs. ellipticity for isotropic functions