Geometry of Logarithmic Strain Measures in Solid Mechanics

• Published in: Archive for rational mechanics and analysis 2016-11, Vol.222 (2), p.507-572
• Main Authors: Neff, Patrizio, Eidel, Bernhard, Martin, Robert J.
• Format: Article
• Language: English
• Subjects: Bulk modulus; Classical Mechanics; Complex Systems; Deduction; Deformation; Elasticity; Fluid- and Aerodynamics; Geodesy; Invariants; Mathematical analysis; Mathematical and Computational Physics; Matrices (mathematics); Nonlinearity; Physics; Physics and Astronomy; Shear modulus; Solid mechanics; Strain; Tensors; Texts; Theoretical
• ISSN: 0003-9527; 1432-0673
• DOI: 10.1007/s00205-016-1007-x

We consider the two logarithmic strain measures ω iso = | | dev n log U | | = | | dev n log F T F | | and ω vol = | tr ( log U ) = | tr ( log F T F ) | = | log ( det U ) | , 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 GL ( n ) . Here, F is the deformation gradient, U = F T F is the right Biot-stretch tensor, log denotes the principal matrix logarithm, ‖ · ‖ is the Frobenius matrix norm, tr is the trace operator and dev n X = X - 1 n tr ( X ) · 1 is the n -dimensional deviator of X ∈ R n × n . This characterization identifies the Hencky (or true) strain tensor as the natural nonlinear extension of the linear (infinitesimal) strain tensor ε = sym ∇ u , which is the symmetric part of the displacement gradient ∇ u , and reveals a close geometric relation between the classical quadratic isotropic energy potential μ ‖ dev n sym ∇ u ‖ 2 + κ 2 [ tr ( sym ∇ u ) ] 2 = μ ‖ dev n ε ‖ 2 + κ 2 [ tr ( ε ) ] 2 in linear elasticity and the geometrically nonlinear quadratic isotropic Hencky energy μ ‖ dev n log U ‖ 2 + κ 2 [ tr ( log U ) ] 2 = μ ω iso 2 + κ 2 ω vol 2 , where μ is the shear modulus and κ 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.