  We consider the two logarithmic strain measures\[\omega_{\rm
iso}=\|\mathrm{dev}_n\log U\|=\|\mathrm{dev}_n\log \sqrt{F^TF}\|\quad\text{ and
}\quad \omega_{\rm vol}=|\mathrm{tr}(\log
U)|=|\mathrm{tr}(\log\sqrt{F^TF})|\,,\]which are isotropic invariants of the
Hencky strain tensor $\log U$, 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 $\mathrm{dev}_n X$ 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
\[\mu\,\|\mathrm{dev}_n\mathrm{sym}\nabla
u\|^2+\frac{\kappa}{2}\,[\mathrm{tr}(\mathrm{sym}\nabla
u)]^2=\mu\,\|\mathrm{dev}_n\varepsilon\|^2+\frac{\kappa}{2}\,[\mathrm{tr}(\varepsilon)]^2\]in
linear elasticity and the geometrically nonlinear quadratic isotropic Hencky
energy\[\mu\,\|\mathrm{dev}_n\log U\|^2+\frac{\kappa}{2}\,[\mathrm{tr}(\log
U)]^2=\mu\,\omega_{\rm iso}^2+\frac\kappa2\,\omega_{\rm vol}^2\,,\]where $\mu$
is the shear modulus and $\kappa$ denotes the bulk modulus. Our deduction
involves a new fundamental logarithmic minimization property of the orthogonal
polar factor $R$, where $F=R\,U$ is the polar decomposition of $F$. We also
contrast our approach with prior attempts to establish the logarithmic Hencky
strain tensor directly as the preferred strain tensor in nonlinear isotropic
elasticity.
