Incrementation: Part Four

  • Robert M. Hackett


Incremental constitutive relationships are considered. Given rate relationships, corresponding incremental constitutive relationships can be written. While the rate equations are exact, the incremental equations are approximate and correct only to the first order. The increment of a second-order tensor function of the deformation gradient can be written formally as a Taylor series and expanded to yield, respectively, fourth, sixth, … order tensors with Cartesian components. This procedure is employed to develop the sixth-order “first elasticity tensor.” A very significant numerical example is presented to demonstrate the utilization of the first elasticity tensor. A solution of the simple shear problem, in which the incremental first Piola-Kirchhoff stress tensor is evaluated as the product of the first elasticity tensor and the incremental deformation gradient and summed at each time step, is obtained, first using the fourth-order first elasticity tensor and then using the sixth-order first elasticity tensor. The enhanced convergence resulting from the use of the sixth-order first elasticity tensor is demonstrated.


Taylor series Fourth-order first elasticity tensor Sixth-order first elasticity tensor Incremental first Piola-Kirchhoff stress tensor Incremental deformation gradient Numerical example 


  1. Ogden RW (1997) Non-linear elastic deformations. Dover, New YorkGoogle 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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