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December  2016, 8(4): 437-460. doi: 10.3934/jgm.2016015

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

Raz Kupferman 1, and  Asaf Shachar 1,

1. 

Institute of Mathematics, The Hebrew University, Jerusalem 91904, Israel

Received  March 2016 Revised  August 2016 Published  November 2016

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.
Keywords: strain measure, General linear group, symmetries..
Mathematics Subject Classification: Primary: 53Zxx; Secondary: 74Bx.
Citation: Raz Kupferman, Asaf Shachar. On strain measures and the geodesic distance to $SO_n$ in the general linear group. Journal of Geometric Mechanics, 2016, 8 (4) : 437-460. doi: 10.3934/jgm.2016015
References:
[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.  doi: 10.1016/j.geomphys.2014.08.009.  Google Scholar

[2]

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

[3]

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

[4]

S. Goette, Existence of $SO(n)$-isotropic inner products which are not $O(n)$-isotropic,, MathOverflow, ().   Google Scholar

[5]

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

[6]

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

[7]

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.  doi: 10.1016/j.laa.2014.02.012.  Google Scholar

[8]

J. Lee, Introduction to Smooth Manifolds,, Springer-Verlag, (2003).  doi: 10.1007/978-0-387-21752-9.  Google Scholar

[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.  doi: 10.3934/jgm.2016010.  Google Scholar

[10]

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

[11]

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

[12]

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

[13]

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.  doi: 10.1016/j.ijengsci.2014.02.026.  Google Scholar

[14]

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.  doi: 10.1137/130909949.  Google Scholar

[15]

A. Shahar, How to show the space of inverse-invariant metrics on a lie group is infinite dimensional?,, Mathematics Stack Exchange, ().   Google Scholar

[16]

Tomás, Geodesic distance from point to manifold,, Mathematics Stack Exchange, ().   Google Scholar

show all references

References:
[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.  doi: 10.1016/j.geomphys.2014.08.009.  Google Scholar

[2]

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

[3]

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

[4]

S. Goette, Existence of $SO(n)$-isotropic inner products which are not $O(n)$-isotropic,, MathOverflow, ().   Google Scholar

[5]

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

[6]

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

[7]

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.  doi: 10.1016/j.laa.2014.02.012.  Google Scholar

[8]

J. Lee, Introduction to Smooth Manifolds,, Springer-Verlag, (2003).  doi: 10.1007/978-0-387-21752-9.  Google Scholar

[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.  doi: 10.3934/jgm.2016010.  Google Scholar

[10]

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

[11]

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

[12]

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

[13]

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.  doi: 10.1016/j.ijengsci.2014.02.026.  Google Scholar

[14]

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.  doi: 10.1137/130909949.  Google Scholar

[15]

A. Shahar, How to show the space of inverse-invariant metrics on a lie group is infinite dimensional?,, Mathematics Stack Exchange, ().   Google Scholar

[16]

Tomás, Geodesic distance from point to manifold,, Mathematics Stack Exchange, ().   Google Scholar

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