Abstract
We consider the two logarithmic strain measures
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 \({{\rm GL}(n)}\). Here, \({F}\) is the deformation gradient, \({U=\sqrt{F^TF}}\) is the right Biot-stretch tensor, \({\log}\) denotes the principal matrix logarithm, \({\| \cdot \|}\) is the Frobenius matrix norm, \({\rm tr}\) is the trace operator and \({{\text dev}_n X = X- \frac{1}{n} \,{\text tr}(X)\cdot {\mathbb{1}}}\) 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 \({\varepsilon={\text sym}\nabla u}\), 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
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 \({F=RU}\) 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.
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Communicated by M. Ortiz
In memory of Giuseppe Grioli (*10.4.1912 – †4.3.2015), a true paragon of rational mechanics.
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Neff, P., Eidel, B. & Martin, R.J. Geometry of Logarithmic Strain Measures in Solid Mechanics. Arch Rational Mech Anal 222, 507–572 (2016). https://doi.org/10.1007/s00205-016-1007-x
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DOI: https://doi.org/10.1007/s00205-016-1007-x