On the geometric character of stress in continuum mechanics

Zeitschrift fur Angewandte Mathematik und Physik

Volumn 58, Issue 5, 2007, Pages 843-856

  Kanso, Eva ^a   Arroyo, Marino ^b   Tong, Yiying ^c   Yavari, Arash ^d   Marsden, Jerrold G ^c   Desbrun, Mathieu ^c  

^a UNIVERSITY OF SOUTHERN CALIFORNIA   (United States)

^b UNIVERSITAT POLITÈCNICA DE CATALUNYA   (Spain)

^c CALIFORNIA INSTITUTE OF TECHNOLOGY   (United States)

^d GEORGIA INSTITUTE OF TECHNOLOGY   (United States)

Author keywords

Continuum mechanics; Differential forms; Elasticity; Stress tensor

Indexed keywords

EID: 34547871348     PISSN: 00442275     EISSN: None     Source Type: Journal    
DOI: 10.1007/s00033-007-6141-8     Document Type: Article

References (23)

1. Abraham, R., J.E. Marsden and T. Ratiu, Manifolds, Tensor Analysis, and Applications, Springer-Verlag, New York 1988.
2. Cartan, E., On Manifolds with an Affine Connection and the Theory of General Relativity. Bibliopolis, Napoli 1995.
3. Casey, J. and P. Papadopoulos, Material transport of sets and fields, Math. Mech. Sol. 7 (2002), 647-676.
4. Desbrun, M., A. N. Hirani and J. E. Marsden, Discrete exterior calculus for variational problems in computer graphics, Proc. CDC, Hawaii 2003.
5. Desbrun, M., E. Kanso and Y. Tong, Discrete differential forms for computational modeling, Chapter in "Discrete Differential Geometry" Course Notes, ACM SIGGRAPH 2005.
6. Di Carlo, A., A non-standard format for continuum mechanics, Contemporary Research in the Mechanics and Mathematics of Materials, R.C. Batra and M.F. Beatty, CIMNE, Barcelona 1996.
7. Eyck, T. and Lew, A., Discontinuous Galerkin methods for nonlinear elasticity, IJNME 67 (2006), 1204-1243.
8. Flanders, H., Differential Forms and Application to Physical Sciences, Dover 1990.
9. Frankel, T., The Geometry of Physics, Cambridge University Press, United Kingdom 1997 (Second Edition 2004).
10. Frölicher, A. and Nijenhuis, A., Theory of vector-valued differential forms. I. Derivations of the graded ring of differential forms, Indagationes Mathematics 18 (1956), 338-359.
11. Green, A. E. and Rivilin, R. S., On Cauchy's equations of motion, ZAMP 15 (1964), 290-293.
12. Hauret, P. and E. Kuhl, and M. Ortiz, Diamond elements: A finite-element/discrete-mechanics approximation scheme with guaranteed optimal convergence in incompressible elasticity, IJNME (2006), to appear.
13. Hehl, F. W. and McCrea, J. D., Bianchi identities and the automatic conservation of energy-momentum and angular momentum in general-relativistic field theories., Foundation of Physics 16 (1986), 267-293.
14. Hirani, A. N., Discrete Exterior Calculus, Ph.D. thesis, California Institute of Technology 2003.
15. Lewis, D., and J. C. Simo, Conservative Algorithms for the Dynamics of Hamiltonian Systems on Lie Groups, Nonlinear Science 4 (1994), 253.
16. Madsen, I.H., and J. Tornehave, From Calculus to Cohomology : De Rham Cohomology and Characteristic Classes, Cambridge University Press 1997.
17. Marsden, J. E. and T. J. R. Hughes, Mathematical Foundations of Elasticity, Dover, New York 1983.
18. Segev R. and G. Rodnay, Cauchy's theorem on manifolds, Journal of Elasticity 56 (1999), 129-144.
19. Simo, J. C. and N. Tarnow, The discrete energy momentum method: conserving algorithms for nonlinear elastodynamics, Zeitschrift für Angewandte Mathematik und Physik (ZAMP) 43 (1992), 757-792.
20. Simo, J. C., N. Tarnow and K. K. Wong, Exact energy-momentum conserving algorithms and symplectic schemes for nonlinear dynamics, Computer Methods in Applied Mechanics and Engineering 100 (1992), 63-116.
21. Vargas, J. G. and D. G. Torr, Conservation of vector-valued forms and the question of the existence of gravitational energy-momentum in general relativity. General Relativity and Gravitation, 23 (1991), 713-732.
22. West, M. and J. E. Marsden, Discrete mechanics and variational integrators, Acta Numerica (2001), 357-514.
23. Yavari, A., J. E. Marsden and M. Ortiz, On the spatial and material covariant balance laws in elasticity, Journal of Mathematical Physics 47 (2006), 1-53.