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

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.
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), 241-257. 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), 225-227. 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), 452-455.  Google Scholar

[6]

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.  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), 28-42. doi: 10.1016/j.laa.2014.02.012.  Google Scholar

[8]

J. Lee, Introduction to Smooth Manifolds, Springer-Verlag, New York, 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), 323-357. doi: 10.3934/jgm.2016010.  Google Scholar

[10]

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.  Google Scholar

[11]

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.  Google Scholar

[12]

P. Neff, B. Eidel and R. Martin, Geometry of logarithmic strain measures in solid mechanics, Arch. Rat. Mech. Anal., 222 (2016), 507-572. 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), 209-217. 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), 1132-1154. 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), 241-257. 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), 225-227. 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), 452-455.  Google Scholar

[6]

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.  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), 28-42. doi: 10.1016/j.laa.2014.02.012.  Google Scholar

[8]

J. Lee, Introduction to Smooth Manifolds, Springer-Verlag, New York, 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), 323-357. doi: 10.3934/jgm.2016010.  Google Scholar

[10]

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.  Google Scholar

[11]

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.  Google Scholar

[12]

P. Neff, B. Eidel and R. Martin, Geometry of logarithmic strain measures in solid mechanics, Arch. Rat. Mech. Anal., 222 (2016), 507-572. 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), 209-217. 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), 1132-1154. 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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