• Robert M. Hackett


An objective stress measure is one that ensures that stress-strain responses are not affected by superposed rigid-body rotations. This means that they should be invariant to observers in different frames of reference. For example, if frame one is fixed, while frame two is rotating with respect to frame one, the stress response obtained in both frames using the same constitutive equation should obey the transformation that rotates frame one to the orientation of frame two. Single-based second-order deformation tensors and strain tensors are objective. Two-point second-order tensors such as the deformation gradient are also objective, even though they transform like vectors and not like second-order tensors, because one of the indices of the tensor describes the material coordinates which are independent of the observer. Objective tensors are suitable for describing material response and for the development of incremental constitutive laws, since they are independent of the observer; however, objective tensors usually do not preserve their objectivity through time differentiation. A frequently encountered non-objective tensor is the spatial velocity gradient tensor, while the rate-of-deformation tensor is objective. It consequently can be used in the formulation of spatial rate-constitutive laws. A strain-energy function is objective if and only if the balance of angular momentum condition holds. A numerical example is presented in order to illustrate some of the important objectivity determinations.


Objectivity Rigid-body rotations Velocity gradient tensor Rate-of-deformation tensor Strain-energy function Numerical example 


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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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