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Kinematics of Material Continua

  • A. C. Eringen
  • G. A. Maugin

Abstract

This chapter is concerned with the kinematics of deformable and fluent bodies. Here we collect the most essential kinematic notions relevant to the later chapters. The notion of material continua with charges is discussed in Section 1.2. The motion and deformation gradients introduced in Section 1.3 are fundamental concepts, upon which the mechanics of deformable continua are built. Section 1.4 discusses strain measures. By means of the polar decomposition theorem, one is led to the concepts of finite rotation and finite strain in Section 1.5. This concept is also fundamental to an understanding of the constitutive theory. Infinitesimal strains and rotations are presented in Section 1.6, and area and volume changes upon deformation are presented in Section 1.7. Section 1.8 discusses the integrability conditions of the strain tensor, i.e., given a set of strain tensors, and the conditions that lead to a single-valued displacement field corresponding to these strains.

Keywords

Constitutive Theory Material Derivative Discontinuity Line Rotation Tensor Spatial Frame 
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References

  1. 4.
    See Truesdell and Noll [1965].Google Scholar
  2. 5.
    See Eringen [1975a, p. 37].Google Scholar
  3. 6.
    For example, Eringen [1967, pp. 28–41] and [1980, Sect. 1.11].Google Scholar
  4. 8.
    See Eringen [1967, pp. 71–72] and [1975a, p. 40].Google Scholar
  5. 9.
    See Eringen [1967, p. 58] and [1980, Sect. 1.14 and Appendix C6] or [1975a, p. 42].Google Scholar
  6. 10.
    See Eringen [1971b, p. 134].Google Scholar
  7. 11.
    See Eringen [1975a, p. 48].Google Scholar
  8. 12.
    In that case, it can be shown that the eα are none other than, at each time, the unit eigenvectors of the deformation rate tensor. Then (1.12.2)2 expresses Gosiewski’s theorem (see Eringen [1975a, Theorem 2, p. 52]).Google Scholar
  9. 13.
    See Chapter 9 (Vol. II).Google Scholar
  10. 14.
    See Eringen [1967, p. 428] and [1980, Appendix A2].Google Scholar
  11. 15.
    See Eringen [1967, p. 428] and [1980, p. 524].Google Scholar

Copyright information

© Springer-Verlag New York Inc. 1990

Authors and Affiliations

  • A. C. Eringen
    • 1
  • G. A. Maugin
    • 2
  1. 1.Princeton UniversityPrincetonUSA
  2. 2.Laboratoire de Modélisation en MécaniqueUniversité Pierre et Marie Curie et C.N.R.S.Paris 05France

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