New Thoughts in Nonlinear Elasticity Theory via Hencky’s Logarithmic Strain Tensor

Abstract

We consider the two logarithmic strain measures $\omega _{\mathrm {iso}}= ||{{\mathrm{dev}}}_n \log U ||$ and $\omega _{\mathrm {vol}}= |{{\mathrm{tr}}}(\log U) |$, which are isotropic invariants of the Hencky strain tensor $\log U = \log (F^TF)$, and show that they can be uniquely characterized by purely geometric methods based on the geodesic distance on the general linear group ${{\mathrm{GL}}}(n)$. Here, F is the deformation gradient, $U=\sqrt{F^TF}$ is the right Biot-stretch tensor, $\log$ denotes the principal matrix logarithm, $||\,.\, ||$ is the Frobenius matrix norm, ${{\mathrm{tr}}}$ is the trace operator and is the n-dimensional deviator of $X\in \mathbb {R}^{n\times n}$. This characterization identifies the Hencky (or true) strain tensor as the natural nonlinear extension of the linear (infinitesimal) strain tensor $\varepsilon ={{\mathrm{sym}}}\nabla u$, which is the symmetric part of the displacement gradient $\nabla u$, and reveals a close geometric relation between the classical quadratic isotropic energy potential in linear elasticity and the geometrically nonlinear quadratic isotropic Hencky energy. Our deduction involves a new fundamental logarithmic minimization property of the orthogonal polar factor R, where $F=RU$ is the polar decomposition of F.