A Cosserat Point Element (CPE) for the Numerical Solution of Problems in Finite Elasticity

Part of the Advances in Mechanics and Mathematics book series (AMMA, volume 21)


The theory of a Cosserat Point is a special continuum theory that characterizes the motion of a small material region which can be modeled as a point with finite volume. This theory has been used to develop a 3-D eight-noded brick Cosserat Point Element (CPE) to formulate the numerical solution of dynamical problems in finite elasticity. The kinematics of the CPE are characterized by eight director vectors which are functions of time only. Also, the kinetics of the CPE are characterized by balance laws which include: conservation of mass, balances of linear and angular momentum, as well as balances of director momentum. The main difference between the standard Bubnov–Galerkin and the Cosserat approaches is the way that they each develop constitutive equations. In the direct Cosserat approach, the kinetic quantities are given by derivatives of a strain energy function that models the CPE as a structure and that characterizes resistance to all models of deformation. A generalized strain energy function has been developed which yields a CPE that is truly a robust user friendly element for nonlinear elasticity that can be used with confidence for 3-D problems as well as for problems of thin shells and rods.


Deformable Body Strain Energy Function Director Vector Finite Elasticity Cosserat Point Element 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Antman, S.S.: The Theory of Rods. Handbuch der Physik, vol. VIa/2. Springer, Berlin (1972) Google Scholar
  2. 2.
    Antman, S.S.: Nonlinear Problems of Elasticity. Springer, New York (1995) MATHGoogle Scholar
  3. 3.
    Cohen, H.: Pseudo-rigid bodies. Utilitas Math. 20, 221–247 (1981) MATHMathSciNetGoogle Scholar
  4. 4.
    Cohen, H., Muncaster, R.G.: The dynamics of pseudo-rigid bodies: general structure and exact solutions. J. Elast. 14, 127–154 (1984) MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Cohen, H., Muncaster, R.G.: Theory of Pseudo-Rigid Bodies. Springer, Berlin (1984) Google Scholar
  6. 6.
    Cosserat, E., Cosserat, F.: Théorie des corps déformables. Hermann et Fils, Paris (1909). vi+226pp. = Appendix, pp. 953–1173 of Chwolson’s Traite de Physique, 2nd edn., Paris. Also in translated form: “Theory of deformable bodies”, NASA TT F-11, 561 (Washington, DC). Clearinghouse for US Federal Scientific and Technical Information, Springfield, Virginia Google Scholar
  7. 7.
    Green, A.E., Naghdi, P.M., Wenner, M.L.: On the theory of rods I: Derivations from the three-dimensional equations. Proc. R. Soc. Lond. A 337, 451–483 (1974) MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Green, A.E., Naghdi, P.M., Wenner, M.L.: On the theory of rods II: Developments by direct approach. Proc. R. Soc. Lond. A 337, 485–507 (1974) MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Jabareen, M., Rubin, M.B.: An improved 3-D Cosserat brick element for irregular shaped elements. Comput. Mech. 40, 979–1004 (2007) MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Jabareen, M., Rubin, M.B.: A generalized Cosserat point element (CPE) for isotropic nonlinear elastic materials including irregular 3-D brick and thin structures. United States Patent and Trademark Office, Serial No. 61/038 (2008) Google Scholar
  11. 11.
    Jabareen, M., Rubin, M.B.: A generalized Cosserat point element (CPE) for isotropic nonlinear elastic materials including irregular 3-D brick and thin structures. J. Mech. Mater. Struct. 3, 1465–1498 (2008) CrossRefGoogle Scholar
  12. 12.
    Loehnert, S., Boerner, E.F.I., Rubin, M.B., Wriggers, P.: Response of a nonlinear elastic general Cosserat brick element in simulations typically exhibiting locking and hourglassing. Comput. Mech. 36, 255–265 (2005) MATHCrossRefGoogle Scholar
  13. 13.
    Muncaster, R.G.: Invariant manifolds in mechanics I: Zero-dimensional elastic bodies with directors. Arch. Ration. Mech. Anal. 84, 353–373 (1984) MATHMathSciNetGoogle Scholar
  14. 14.
    Nadler, B., Rubin, M.B.: A new 3-D finite element for nonlinear elasticity using the theory of a Cosserat point. Int. J. Solids Struct. 40, 4585–4614 (2003) MATHCrossRefGoogle Scholar
  15. 15.
    Naghdi, P.M.: The theory of shells and plates. In: Truesdell, C. (ed.) S. Flugge’s Handbuch der Physik, vol. VIa/2, pp. 425–640. Springer, Berlin (1972) Google Scholar
  16. 16.
    Rubin, M.B.: On the theory of a Cosserat point and its application to the numerical solution of continuum problems. J. Appl. Mech. 52, 368–372 (1985) MATHCrossRefGoogle Scholar
  17. 17.
    Rubin, M.B.: On the numerical solution of one-dimensional continuum problems using the theory of a Cosserat point. J. Appl. Mech. 52, 373–378 (1985) MATHCrossRefGoogle Scholar
  18. 18.
    Rubin, M.B.: Numerical solution of two- and three-dimensional thermomechanical problems using the theory of a Cosserat point. J. Math. Phys. (ZAMP) 46, S308–S334 (1995). In Theoretical, Experimental, and Numerical Contributions to the Mechanics of Fluids and Solids, edited by J. Casey and M.J. Crochet, Brikhäuser, Basel, 1995 MATHGoogle Scholar
  19. 19.
    Rubin, M.B.: Cosserat Theories: Shells, Rods and Points. Solid Mechanics and its Applications, vol. 79. Kluwer Academic, Dordrecht (2000) MATHGoogle Scholar
  20. 20.
    Slawianowski, J.J.: Analytical mechanics of finite homogeneous strains. Arch. Mech. 26, 569–587 (1974) MATHMathSciNetGoogle Scholar
  21. 21.
    Slawianowski, J.J.: Newtonian dynamics of homogeneous strains. Arch. Mech. 26, 569–587 (1975) MathSciNetGoogle Scholar
  22. 22.
    Slawianowski, J.J.: The mechanics of the homogeneously-deformable body. Dynamical models with high symmetries. Z. Angew. Math. Mech. 62, 229–240 (1982) MATHCrossRefGoogle Scholar
  23. 23.
    Taylor, R.L.: FEAP—A Finite Element Analysis Program, Version 7.5. University of California, Berkeley (2005) Google Scholar
  24. 24.
    Wozniak, C.Z.: Basic concepts of the theory of discrete elasticity. Bull. Acad. Pol. Sci. 19, 753–758 (1971) MATHMathSciNetGoogle Scholar
  25. 25.
    Wozniak, C.Z.: Equations of motion and laws of conservation in the discrete elasticity. Arch. Mech. 25, 155–163 (1973) MATHMathSciNetGoogle Scholar
  26. 26.
    Wozniak, C.Z.: Discrete elastic Cosserat media. Arch. Mech. 25, 119–136 (1973) MATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Faculty of Civil and Environmental EngineeringTechnion-Israel Institute of TechnologyHaifaIsrael
  2. 2.Faculty of Mechanical EngineeringTechnion-Israel Institute of TechnologyHaifaIsrael

Personalised recommendations