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 Open image in new window 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.