Geometry of logarithmic strain measures in solid mechanics

Abstract

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