Class of motions of elastic, symmetric Cosserat points: Existence, bifurcation, and stability

International Journal of Non-Linear Mechanics

Volumn 36, Issue 2, 2001, Pages 353-374

  Nordenholz, T R ^a   O'Reilly, O M ^b  

^a CALIFORNIA STATE UNIVERSITY MARITIME ACADEMY   (United States)

^b UNIVERSITY OF CALIFORNIA   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

BIFURCATION (MATHEMATICS); DEFORMATION; ELASTICITY; STRAIN; TENSORS;

COSSERAT POINTS; MOONEY-RIVLIN STRAIN ENERGY; ROUTHIAN REDUCTION;

MECHANICS;

EID: 0035285393     PISSN: 00207462     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0020-7462(00)00021-4     Document Type: Article

References (35)

1. Cohen H., Muncaster R.G. The dynamics of pseudo-rigid bodies: general structure and exact solutions. J. Elasticity. 14:1984;127.
2. Cohen H., Muncaster R.G. The Theory of Pseudo-rigid Bodies. 1988;Springer, New York.
3. Slawianowski J.J. The mechanics of the homogeneously-deformable body. Z. Angew. Math. Mech. 62:1982;229.
4. Nordenholz T.R., O'Reilly O.M. On steady motions of isotropic, elastic Cosserat points. IMA J. Appl. Math. 60:1998;55.
5. Rubin M.B. On the theory of a Cosserat point and its application to the numerical solution of continuum problems. J. Appl. Mech. 52:1985;368.
6. Green A.E., Naghdi P.M. A thermomechanical theory of a Cosserat point with application to composite materials. Quart. J. Mech. Appl. Math. 44:1991;335.
7. Varadi P.C., Lo G.-J., O'Reilly O.M., Papadopoulos P. A novel approach to vehicle system dynamics using the theory of a Cosserat point and its application to collision analyses of platooning vehicles. Vehicle System Dyn. 32:1999;85.
8. Rubin M.B., Gottlieb O. Numerical solutions of forced vibration and whirling of a non-linear string using the theory of a Cosserat point. J. Sound Vib. 197:1996;85.
9. O.M. O'Reilly, P.C. Varadi, Hoberman's sphere, Euler parameters and Lagrange's equations, J. Elasticity (2000) in press.
10. Rubin M.B. Numerical solution of two- and three-dimensional thermomechanical problems using the theory of a Cosserat point. Z. Angew. Math. Phys. 46:1995;S308.
11. Solberg J.M., Papadopoulos P. A simple finite element-based framework for the analysis of elastic pseudo-rigid bodies. Int. J. Numer. Methods Eng. 45:1999;1297.
12. Cohen H., Mac Sithigh G.P. Plane motions of elastic pseudo-rigid bodies. J. Elasticity. 21:1989;193.
13. Lewis D., Simo J.C. Nonlinear stability of rotating pseudo-rigid bodies. Proc. Roy. Soc. London A. 427:1990;281.
14. F. Gantmacher, Lectures in Analytical Mechanics, 2nd Printing, Mir, Moscow, 1975.
15. Karapetyan A.V., Rumyantsev V.V. Stability of conservative and dissipative systems. Appl. Mech. Soviet Rev. 1:1990;3.
16. Merkin D. Introduction to the Theory of Stability. 1997;Springer, New York.
17. Rumyantsev V.V. On the stability of steady motions. J. Appl. Math. Mech. (PMM). 30:1966;1090.
18. Simo J.C., Lewis D., Marsden J.E. Stability of relative equilibria, Part I: the reduced-energy momentum method. Arch. Rational Mech. Anal. 115:1991;15.
19. T.R. Nordenholz, The stability of steady motions of pseudo-rigid bodies, Doctoral Dissertation, University of California at Berkeley, 1998.
20. Noll W. Motions with constant stretch history. Arch. Rational Mech. Anal. 11:1962;97.
21. Truesdell C., Noll W. The Non-Linear Field Theories of Mechanics. 2nd edition:1992;Springer, Berlin.
22. Cohen H., Mac Sithigh G.P. Symmetric and asymmetric roto-deformations of a symmetrical, isotropic, elastic pseudo-rigid. Int. J. Non-Linear Mech. 27:1992;519.
23. V.I. Zubov, Methods of A.M. Lyapunov and their Application. P. Noordhoff, Groningen, 1964.
24. O'Reilly O.M., Varadi P.C. A unified treatment of constraints in the theory of a Cosserat point. Z. Angew. Math. Phys. 49:1998;205.
25. Carlson D.E., Tortorelli D.A. On hyperelasticity with internal constraints. J. Elasticity. 42:1996;91.
26. Casey J. Geometrical derivation of Lagrange's equations for a system of particles. Amer. J. Phys. 62:1994;836.
27. Casey J. On the advantages of a geometrical viewpoint in the derivation of Lagrange's equations for a rigid continuum. Z. Angew. Math. Phys. 46:1995;S805.
28. Chetayev N.G. The Stability of Motion. 1961;Pergamon Press, London.
29. Simo J.C., Posbergh T.A., Marsden J.E. Stability of coupled rigid bodies and geometrically exact rods: block diagonalization and the energy-momentum method. Phys. Rep. 193(6):1990;279.
30. Marsden J.E., Scheurle J. Lagrangian reduction and the double spherical pendulum. Z. Angew. Math. Phys. 44:1993;17.
31. Zombro B., Holmes P. Reduction, stability, instability and bifurcation in rotationally symmetric Hamiltonian systems. Dyn. Stability Systems. 8:1993;41.
32. Lewis D. Lagrangian block diagonalization. J. Dyn. Differential Equations. 4:1992;1.
33. E.J. Doedel, A.R. Champneys, T.F. Fairgrieve, Y. Kuznetsov, B. Sandstede, X. Wang, AUTO97: continuation and bifurcation software for ordinary differential equations (with HomCont), 1997.
34. Greenwood D.T. Principles of Dynamics. 1988;Prentice-Hall, Englewood Cliffs.
35. Casey J., Lam V.C. On the relative angular velocity vector. ASME J. Mech. Transmissions Automat. Des. 108:1986;399.