Polar Decomposition

  • Robert M. Hackett


Within the framework of hyperelastic materials, the polar decomposition theorem of Trusdell and Noll* occupies a position of primary importance. The polar decomposition theorem states that any deformation gradient tensor can be multiplicatively decomposed into the product of an orthogonal tensor, known as the rotation tensor, and a symmetric tensor called the right stretch tensor, or into the product of a symmetric tensor called the left stretch tensor and the same rotation tensor. The rotation tensor is a two-point Eulerian-Lagrangian second-order tensor, the right stretch tensor is a Lagrangian single-based second-order tensor and the left stretch tensor is single-order and Eulerian based. The relationship between right and left stretch tensors is derived through eigenvalue extraction. The polar decomposition theorem is the principal tool for studying finite deformations and the corresponding strains. The detailed formulation of the polar decomposition theorem is presented in terms of the right Cauchy-Green deformation tensor. A detailed numerical example is presented to demonstrate the polar decomposition theorem.


Polar decomposition Deformation gradient Stretch tensors Rotation tensor Multiplicative decomposition Eigenvalue Numerical example 


  1. Chung TJ (1988) Continuum mechanics. Prentice-Hall, Englewood CliffzbMATHGoogle Scholar
  2. Hill R (1970) Constitutive inequalities for isotropic elastic solids under finite strain. Proc R Soc Lond A314:457–72CrossRefzbMATHGoogle Scholar
  3. Seth BR (1964) Generalized strain measure with applications to physical problems. In: Reiner M, Abir D (eds) Second-order effects in elasticity, plasticity and fluid dynamics. Pergamon, Oxford, pp 162–172Google Scholar
  4. Truesdell C, Noll W (1965) The non-linear theories of mechanics. In: Flügge S (ed) Encyclopedia of physics, vol 3, 3rd edn. Springer, HeidelbergGoogle Scholar

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© 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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