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On strain measures and the geodesic distance to $SO_n$ in the general linear group

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  • We consider various notions of strains---quantitative measures for the deviation of a linear transformation from an isometry. The main approach, which is motivated by physical applications and follows the work of [12], is to select a Riemannian metric on $GL_n$, and use its induced geodesic distance to measure the distance of a linear transformation from the set of isometries. We give a short geometric derivation of the formula for the strain measure for the case where the metric is left-$GL_n$-invariant and right-$O_n$-invariant. We proceed to investigate alternative distance functions on $GL_n$, and the properties of their induced strain measures. We start by analyzing Euclidean distances, both intrinsic and extrinsic. Next, we prove that there are no bi-invariant distances on $GL_n$. Lastly, we investigate strain measures induced by inverse-invariant distances.
    Mathematics Subject Classification: Primary: 53Zxx; Secondary: 74Bxx.


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  • [1]

    E. Andruchow, G. Larotonda, L. Recht and A. Varela, The left invariant metric in the general linear group, Journal of Geometry and Physics, 86 (2014), 241-257.doi: 10.1016/j.geomphys.2014.08.009.


    C. Boor, A naive proof of the representation theorem for isotropic, linear asymmetric stress-strain relations, J. Elast., 15 (1985), 225-227.doi: 10.1007/BF00041995.


    M. Do Carmo, Riemannian Geometry, Birkhäuser, 1992.doi: 10.1007/978-1-4757-2201-7.


    S. Goette, Existence of $SO(n)$-isotropic inner products which are not $O(n)$-isotropic, MathOverflow, URL http://mathoverflow.net/q/229549.


    G. Grioli, Una proprieta di minimo nella cinematica delle deformazioni finite, Bollettino dell'Unione Matematica Italiana, 2 (1940), 452-455.


    K. Hackl, A. Mielke and D. Mittenhuber, Dissipation distances in multiplicative elastoplasticity, in Analysis and Simulation of Multifield Problems, Springer, 12 (2003), 87-100.doi: 10.1007/978-3-540-36527-3_8.


    J. Lankeit, P. Neff and Y. Nakatsukasa, The minimization of matrix logarithms: On a fundamental property of the unitary polar factor, Lin. Alg. Appl., 449 (2014), 28-42.doi: 10.1016/j.laa.2014.02.012.


    J. Lee, Introduction to Smooth Manifolds, Springer-Verlag, New York, 2003.doi: 10.1007/978-0-387-21752-9.


    R. Martin and P. Neff, Minimal geodesics on $GL(n)$ for left-invariant, right-$O(n)$-invariant Riemannian metrics, J. Geom. Mech., 8 (2016), 323-357.doi: 10.3934/jgm.2016010.


    A. Mielke, Finite elastoplasticity lie groups and geodesics on sl (d), in Geometry, mechanics, and dynamics, Springer, 2002, 61-90.doi: 10.1007/0-387-21791-6_2.


    A. Mielke, Energetic formulation of multiplicative elasto-plasticity using dissipation distances, Continuum Mechanics and Thermodynamics, 15 (2003), 351-382.doi: 10.1007/s00161-003-0120-x.


    P. Neff, B. Eidel and R. Martin, Geometry of logarithmic strain measures in solid mechanics, Arch. Rat. Mech. Anal., 222 (2016), 507-572.doi: 10.1007/s00205-016-1007-x.


    P. Neff, J. Lankeit and A. Madeo, On Grioli's minimum property and its relation to Cauchy's polar decomposition, Int. J. Eng. Sci., 80 (2014), 209-217.doi: 10.1016/j.ijengsci.2014.02.026.


    P. Neff, Y. Nakatsukasa and A. Fischle, A logarithmic minimization property of the unitary polar factor in the spectral norm and the Frobenius matrix norm, SIAM J. Mat. Anal. Appl., 35 (2014), 1132-1154.doi: 10.1137/130909949.


    A. Shahar, How to show the space of inverse-invariant metrics on a lie group is infinite dimensional?, Mathematics Stack Exchange, URL http://math.stackexchange.com/q/1623388.


    Tomás, Geodesic distance from point to manifold, Mathematics Stack Exchange, URL http://math.stackexchange.com/q/243767.

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