American Institute of Mathematical Sciences

Advanced
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), 241-257. 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

[1]

Alexander V. Bobylev, Sergey V. Meleshko. On group symmetries of the hydrodynamic equations for rarefied gas. Kinetic & Related Models, 2021, 14 (3) : 469-482. doi: 10.3934/krm.2021012
[2]

Zhiping Fan, Duanzhi Zhang. Minimal period solutions in asymptotically linear Hamiltonian system with symmetries. Discrete & Continuous Dynamical Systems, 2021, 41 (5) : 2095-2124. doi: 10.3934/dcds.2020354
[3]

Mahesh Nerurkar. Forced linear oscillators and the dynamics of Euclidean group extensions. Discrete & Continuous Dynamical Systems - S, 2016, 9 (4) : 1201-1234. doi: 10.3934/dcdss.2016049
[4]

Roman Šimon Hilscher. On general Sturmian theory for abnormal linear Hamiltonian systems. Conference Publications, 2011, 2011 (Special) : 684-691. doi: 10.3934/proc.2011.2011.684
[5]

G. R. Cirmi, S. Leonardi. Higher differentiability for solutions of linear elliptic systems with measure data. Discrete & Continuous Dynamical Systems, 2010, 26 (1) : 89-104. doi: 10.3934/dcds.2010.26.89
[6]

Simona Fornaro, Stefano Lisini, Giuseppe Savaré, Giuseppe Toscani. Measure valued solutions of sub-linear diffusion equations with a drift term. Discrete & Continuous Dynamical Systems, 2012, 32 (5) : 1675-1707. doi: 10.3934/dcds.2012.32.1675
[7]

Pablo Ochoa. Approximation schemes for non-linear second order equations on the Heisenberg group. Communications on Pure & Applied Analysis, 2015, 14 (5) : 1841-1863. doi: 10.3934/cpaa.2015.14.1841
[8]

Axel Kohnert, Johannes Zwanzger. New linear codes with prescribed group of automorphisms found by heuristic search. Advances in Mathematics of Communications, 2009, 3 (2) : 157-166. doi: 10.3934/amc.2009.3.157
[9]

Harish Garg, Kamal Kumar. Group decision making approach based on possibility degree measure under linguistic interval-valued intuitionistic fuzzy set environment. Journal of Industrial & Management Optimization, 2020, 16 (1) : 445-467. doi: 10.3934/jimo.2018162
[10]

Xiaomei Feng, Zhidong Teng, Fengqin Zhang. Global dynamics of a general class of multi-group epidemic models with latency and relapse. Mathematical Biosciences & Engineering, 2015, 12 (1) : 99-115. doi: 10.3934/mbe.2015.12.99
[11]

Yasemin Şengül. Viscoelasticity with limiting strain. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 57-70. doi: 10.3934/dcdss.2020330
[12]

Raffaele D’Ambrosio, Giuseppe De Martino, Beatrice Paternoster. A symmetric nearly preserving general linear method for Hamiltonian problems. Conference Publications, 2015, 2015 (special) : 330-339. doi: 10.3934/proc.2015.0330
[13]

C. D. Ahlbrandt, A. C. Peterson. A general reduction of order theorem for discrete linear symplectic systems. Conference Publications, 1998, 1998 (Special) : 7-18. doi: 10.3934/proc.1998.1998.7
[14]

Andrew J. Steyer, Erik S. Van Vleck. Underlying one-step methods and nonautonomous stability of general linear methods. Discrete & Continuous Dynamical Systems - B, 2018, 23 (7) : 2859-2877. doi: 10.3934/dcdsb.2018108
[15]

Song-Mei Huan, Xiao-Song Yang. On the number of limit cycles in general planar piecewise linear systems. Discrete & Continuous Dynamical Systems, 2012, 32 (6) : 2147-2164. doi: 10.3934/dcds.2012.32.2147
[16]

Mahamadi Warma. Semi linear parabolic equations with nonlinear general Wentzell boundary conditions. Discrete & Continuous Dynamical Systems, 2013, 33 (11&12) : 5493-5506. doi: 10.3934/dcds.2013.33.5493
[17]

Ingrid Daubechies, Gerd Teschke, Luminita Vese. Iteratively solving linear inverse problems under general convex constraints. Inverse Problems & Imaging, 2007, 1 (1) : 29-46. doi: 10.3934/ipi.2007.1.29
[18]

Angelamaria Cardone, Zdzisław Jackiewicz, Adrian Sandu, Hong Zhang. Construction of highly stable implicit-explicit general linear methods. Conference Publications, 2015, 2015 (special) : 185-194. doi: 10.3934/proc.2015.0185
[19]

Kai Wang, Deren Han. On the linear convergence of the general first order primal-dual algorithm. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021134
[20]

Wen-Xiu Ma. Conservation laws by symmetries and adjoint symmetries. Discrete & Continuous Dynamical Systems - S, 2018, 11 (4) : 707-721. doi: 10.3934/dcdss.2018044

2020 Impact Factor: 0.857

Tools

Metrics

  • PDF downloads (31)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences