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Characterization of polyconvex isotropic functions

David Wiedemann, Malte A. Peter

TL;DR

The paper addresses the challenge of characterizing polyconvexity for isotropic energies on $\mathbb{R}^{d\times d}$ with $d\in\{2,3\}$ by reducing the problem to the diagonal setting. It introduces signed singular values and the $\Pi(d)$-invariance to identify isotropic energies with vector-valued functions on $\mathbb{R}^d$, and then proves an equivalence: polyconvexity of the matrix function $W$ is equivalent to polyconvexity of the associated vector function $\Phi$, with a corresponding convex envelope defined on the smaller minor vector $\mathcal{m}(\nu)$. This yields a concrete dimension reduction from $K_2=5$, $K_3=19$ minors to $k_2=3$, $k_3=7$ minors, and provides a new sufficient criterion for polyconvexity in terms of elementary symmetric polynomials of signed singular values or invariants of the stretch tensor. The approach integrates polyconvex conjugation within the vector setting, enabling construction of isotropic polyconvex energies and envelopes, with clear implications for elasticity modeling and energy design."

Abstract

Polyconvexity is an important concept in the analysis of energies related to elasticity. A function $f \colon \R^{d\times d} \to \R$ is called polyconvex if it can be written as a convex function in the minors of the argument. We show that for isotropic functions it suffices to consider diagonal matrices. For $d=3$, this leads to a dimension reduction for the convex representative of $f$ from $\R^{19}$ to $\R^7$. Moreover, we present a new result for the polyconvexity of functions formulated in the principal invariant of the left or right stretch tensor.

Characterization of polyconvex isotropic functions

TL;DR

The paper addresses the challenge of characterizing polyconvexity for isotropic energies on with by reducing the problem to the diagonal setting. It introduces signed singular values and the -invariance to identify isotropic energies with vector-valued functions on , and then proves an equivalence: polyconvexity of the matrix function is equivalent to polyconvexity of the associated vector function , with a corresponding convex envelope defined on the smaller minor vector . This yields a concrete dimension reduction from , minors to , minors, and provides a new sufficient criterion for polyconvexity in terms of elementary symmetric polynomials of signed singular values or invariants of the stretch tensor. The approach integrates polyconvex conjugation within the vector setting, enabling construction of isotropic polyconvex energies and envelopes, with clear implications for elasticity modeling and energy design."

Abstract

Polyconvexity is an important concept in the analysis of energies related to elasticity. A function is called polyconvex if it can be written as a convex function in the minors of the argument. We show that for isotropic functions it suffices to consider diagonal matrices. For , this leads to a dimension reduction for the convex representative of from to . Moreover, we present a new result for the polyconvexity of functions formulated in the principal invariant of the left or right stretch tensor.
Paper Structure (12 sections, 21 theorems, 63 equations)

This paper contains 12 sections, 21 theorems, 63 equations.

Table of Contents

  1. Introduction
  2. Main result
  3. Motivation and overview on the literature
  4. Singed singular values and isotropy
  5. Singed singular value decomposition
  6. Isotropic and O(d)xO(d)-times invariant functions
  7. Dimension reduction for convex functions
  8. Polyconvexity and isotropy
  9. (Poly)convex conjugation
  10. Polyconvex conjugation for Pid-invariant functions
  11. Some remarks on polyconvexity with respect to matrix invariants
  12. Conclusion

Key Result

Theorem 1.4

Let $d \in \{2,3\}$ and $W \colon \mathbb{R}^{d \times d} \to \mathbb{R}_\infty$ be isotropic and $\Phi \colon \mathbb{R}^{d } \to \mathbb{R}_\infty$ be given by eq:Phi=Wdiag. Then, the following statements are equivalent: (i) $W$ is lower semicontinuous polyconvex, (ii) there exists a lower semicon (iii) $\Phi$ is lower semicontinuous polyconvex (in the sense of Definition def:PolyVector).

Theorems & Definitions (41)

  • Definition 1.1: Polyconvexity for functions on $\mathbb{R}^{d \times d}$
  • Definition 1.2: Isotropic functions
  • Definition 1.3: Polyconvexity for functions on $\mathbb{R}^{d}$
  • Theorem 1.4: Dimension reduction for polyconvexity of isotropic functions
  • Corollary 1.5: Dimension reduction for polyconvexity of finite isotropic functions
  • Remark 1.6: On the lower semicontinuity of the convex representative
  • Theorem 2.1: Polyconvexity in terms of the signed singular values
  • Remark 2.2: Alternative definition of isotropy
  • Definition 2.3: $\Pi(d)$-invariance
  • Lemma 2.4: Identification of isotropic and $\Pi(d)$-invariant functions
  • ...and 31 more