Strain Measures

  • Robert M. Hackett
Chapter

Abstract

Strain measures for hyperelastic materials must model the effect of finite deformations. They are single-based second-order tensors, either Eulerian or Lagrangian, and are defined in terms of the Cauchy-Green deformation tensors, which are derived from the deformation gradient. The Green-Lagrange strain tensor is Lagrangian based, while the Almansi strain tensor is Eulerian based. Both of these strain measures are described in detail. The Green-Lagrange strain tensor is in terms of the right Cauchy-Green deformation tensor, while the Almansi strain tensor is in terms of the left Cauchy-Green deformation tensor. The reduced invariants of the right and left Cauchy-Green deformation tensors, known as the invariants of the right and left Cauchy-Green distortion tensors, are introduced, and the derivation of the reduced invariants is presented and defined. Since the strain measures are derived from the deformation gradient, they are related, the relationship is easily demonstrated. An additional strain measure, one which is less commonly employed, is the Biot strain tensor. The different strain measures can be formally reduced to those of linear elastic systems, this being demonstrated.

Keywords

Strain measures Eulerian Lagrangian Cauchy-Green deformation Almansi Biot Invariant 

References

  1. Almansi E (1911) Sulle deformazioni finite dei solidi elastici isotropi, I and II. Rend Accad Lincei 20(1):705–744, 20(2):289–296MATHGoogle Scholar
  2. Biot MA (1939) Nonlinear theory of elasticity and the linearized case for a body under initial stress. Philos Mag 27(7):468–489CrossRefMATHGoogle Scholar
  3. Bonet J, Wood RD (2008) Nonlinear continuum mechanics for finite element analysis, 2nd edn. Cambridge University Press, CambridgeCrossRefMATHGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • Robert M. Hackett
    • 1
  1. 1.Department of Civil EngineeringThe University of MississippiUniversityUSA

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