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The Tulczyjew triple in mechanics on a Lie group
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 |
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. |
[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. |
[3] |
M. Do Carmo, Riemannian Geometry, Birkhäuser, 1992.
doi: 10.1007/978-1-4757-2201-7. |
[4] |
S. Goette, Existence of $SO(n)$-isotropic inner products which are not $O(n)$-isotropic,, MathOverflow, ().
|
[5] |
G. Grioli, Una proprieta di minimo nella cinematica delle deformazioni finite, Bollettino dell'Unione Matematica Italiana, 2 (1940), 452-455. |
[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. |
[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. |
[8] |
J. Lee, Introduction to Smooth Manifolds, Springer-Verlag, New York, 2003.
doi: 10.1007/978-0-387-21752-9. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[15] |
A. Shahar, How to show the space of inverse-invariant metrics on a lie group is infinite dimensional?,, Mathematics Stack Exchange, ().
|
[16] |
Tomás, Geodesic distance from point to manifold,, Mathematics Stack Exchange, ().
|
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. |
[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. |
[3] |
M. Do Carmo, Riemannian Geometry, Birkhäuser, 1992.
doi: 10.1007/978-1-4757-2201-7. |
[4] |
S. Goette, Existence of $SO(n)$-isotropic inner products which are not $O(n)$-isotropic,, MathOverflow, ().
|
[5] |
G. Grioli, Una proprieta di minimo nella cinematica delle deformazioni finite, Bollettino dell'Unione Matematica Italiana, 2 (1940), 452-455. |
[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. |
[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. |
[8] |
J. Lee, Introduction to Smooth Manifolds, Springer-Verlag, New York, 2003.
doi: 10.1007/978-0-387-21752-9. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[15] |
A. Shahar, How to show the space of inverse-invariant metrics on a lie group is infinite dimensional?,, Mathematics Stack Exchange, ().
|
[16] |
Tomás, Geodesic distance from point to manifold,, Mathematics Stack Exchange, ().
|
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