Angular Velocities, Twirls, Spins and Rotation Tensors in the Continuum Mechanics Revisited

Chapter
Part of the Advanced Structured Materials book series (STRUCTMAT, volume 89)

Abstract

In the classical continuum mechanics several quantities related to angular velocity of rotation are introduced. Examples include vorticity vector, twirl tensors and logarithmic spin. Furthermore the corresponding rotation tensors can be defined to capture the orientation of triads. All of these quantities are measures of accompanying rotational motion and can be related to the deformation and velocity gradient. Such relationships are crucial for constitutive modeling of material behavior. The aim of this contribution is to recall classical definitions of rotation-like quantities and to present several new relationships between them.

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References

  1. Altenbach H, Eremeyev V (2014) Strain rate tensors and constitutive equations of inelastic micropolar materials. International Journal of Plasticity 63:3 – 17Google Scholar
  2. Altenbach H, Naumenko K, L’vov G, Pilipenko S (2003a) Numerical estimation of the elastic properties of thin-walled structures manufactured from short-fiber-reinforced thermoplastics. Mechanics of composite materials 39(3):221–234Google Scholar
  3. Altenbach H, Naumenko K, Zhilin P (2003b) A micro-polar theory for binary media with application to phase-transitional flow of fiber suspensions. Continuum Mechanics and Thermodynamics 15:539–570Google Scholar
  4. Altenbach H, Naumenko K, Zhilin PA (2005) A direct approach to the formulation of constitutive equations for rods and shells. In: Pietraszkiewicz W, Szymczak C (eds) Shell Structures: Theory and Applications, Taylor & Francis, Leiden, pp 87–90Google Scholar
  5. Altenbach H, Naumenko K, Pylypenko S, Renner B (2007) Influence of rotary inertia on the fiber dynamics in homogeneous creeping flows. ZAMM-Journal of Applied Mathematics and Mechanics/Zeitschrift für Angewandte Mathematik und Mechanik 87(2):81–93Google Scholar
  6. Antman S (1995) Nonlinear Problems of Elasticity. Springer, BerlinGoogle Scholar
  7. Bertram A (2012) Elasticity and Plasticity of Large Deformations, 3rd edn. Springer, BerlinGoogle Scholar
  8. Eremeyev VA, Lebedev LP, Altenbach H (2013) Foundations of micropolar mechanics. Springer Science & Business MediaGoogle Scholar
  9. Libai A, Simmonds JG (1998) The Nonlinear Theory of Elastic Shells. Cambridge University Press, CambridgeGoogle Scholar
  10. Maugin G (2013) Continuum Mechanics Through the Twentieth Century: A Concise Historical Perspective. Solid Mechanics and Its Applications, SpringerGoogle Scholar
  11. Maugin GA (1992) The Thermomechanics of Plasticity and Fracture. Cambridge University Press, CambridgeGoogle Scholar
  12. Maugin GA (2014) Continuum Mechanics Through the Eighteenth and Nineteenth Centuries. SpringerGoogle Scholar
  13. Maugin GA (2016) Non-Classical Continuum Mechanics: A Dictionary. Advanced Structured Saterials, vol 51. SpringerGoogle Scholar
  14. Naumenko K, Altenbach H (2016) Modeling High Temperature Materials Behavior for Structural Analysis: Part I: Continuum Mechanics Foundations and Constitutive Models. Advanced structured materials, vol 28. SpringerGoogle Scholar
  15. Reinhardt W, Dubey R (1996) Coordinate-independent representation of spins in continuum mechanics. Journal of Elasticity 42(2):133–144Google Scholar
  16. Xiao H, Bruhns O, Meyers A (1997) Logarithmic strain, logarithmic spin and logarithmic rate. Acta Mechanica 124(1-4):89–105Google Scholar
  17. Zhilin PA (1996) A new approach to the analysis of free rotations of rigid bodies. ZAMM-Journal of Applied Mathematics and Mechanics/Zeitschrift für Angewandte Mathematik und Mechanik 76(4):187–204Google Scholar
  18. Zhilin PA (2001) Vektory i tensory vtorogo ranga v trekhmernom prostranstve (Vectors and second rank tensors in three-dimensional space, in Russ.). Nestor, St. PetersburgGoogle Scholar
  19. Zhong-Heng G, Lehmann T, Haoyun L, Man CS (1992) Twirl tensors and the tensor equation AX- XA= C. Journal of Elasticity 27(3):227–245Google Scholar

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© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Institut für MechanikOtto-von-Guericke-Universität MagdeburgMagdeburgGermany

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