Stress Measures

  • Robert M. Hackett


Within the framework of hyperelasticity, there are as many different stress measures as there are strain measures. The second Piola-Kirchhoff stress tensor, a Lagrangian formulation, is the most significant of the stress measures. The formulation and steps for computing it are presented in terms of the Mooney-Rivlin strain-energy function model. The Cauchy stress tensor, a Eulerian formulation, is obtained directly from the second Piola-Kirchhoff stress tensor. The first Piola-Kirchhoff stress tensor, a Eulerian-Lagrangian two-point tensor, is also obtained directly from the second Piola-Kirchhoff stress tensor. The transpose of the first Piola-Kirchhoff stress tensor is the so-called nominal stress tensor. Both the first Piola-Kirchhoff stress tensor and the nominal stress tensor are widely used in the field of hyperelasticity. The Kirchhoff stress tensor (weighted Cauchy stress tensor) is related to the Cauchy stress tensor through a multiplication by the Jacobian (the determinant of the deformation gradient). The Biot stress, a Lagrangian-based stress tensor, is also an important stress measure. Only somewhat recently has it been recognized that the Biot stress tensor is helpful in the understanding of certain fundamental problems in elasticity theory. Four detailed numerical examples are presented, two of which employ a uniaxial elongation model in comparing/contrasting the solutions of incompressible and nearly incompressible material formulations.


Piola-Kirchhoff stress tensors Cauchy stress tensor Nominal stress tensor Kirchhoff stress tensor Biot stress tensor Numerical examples 


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Authors and Affiliations

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

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