Geometry of Logarithmic Strain Measures in Solid Mechanics

We consider the two logarithmic strain measures omega(iso) = parallel to dev(n) log U parallel to = parallel to dev(n) log root F-T F parallel to and omega(vol) = parallel to tr(log U)parallel to = parallel to trlog root F-T F)parallel to = parallel to log(det U)parallel to, which are isotropic invariants of the Hencky strain tensor , and show that they can be uniquely characterized by purely geometric methods based on the geodesic distance on the general linear group . Here, is the deformation gradient, U = root F-T F is the right Biot-stretch tensor, denotes the principal matrix logarithm, parallel to.parallel to is the Frobenius matrix norm, is the trace operator and dev(n) X = X - 1/n tr(X) . 1 is the n-dimensional deviator of X is an element of R-nxn. This characterization identifies the Hencky (or true) strain tensor as the natural nonlinear extension of the linear (infinitesimal) strain tensor epsilon = sym del u, which is the symmetric part of the displacement gradient del u, and reveals a close geometric relation between the classical quadratic isotropic energy potential mu parallel to dev(n) sym del u parallel to(2) + kappa/2 [tr(sym del u)](2) = mu parallel to dev(n) epsilon parallel to(2) + kappa/2 [tr(epsilon)](2) in linear elasticity and the geometrically nonlinear quadratic isotropic Hencky energy mu parallel to dev(n) log U parallel to(2) + kappa/2 [tr(log U)](2) = mu omega(2)(iso) + kappa/2 omega(2)(vol), where is the shear modulus and denotes the bulk modulus. Our deduction involves a new fundamental logarithmic minimization property of the orthogonal polar factor , where is the polar decomposition of . 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.