Viscoelasticity at large strains

The Maxwell body of a linear-elastic Hookean spring in series with a Newtonian dashpot is the simplest rheological model for geological deformation. It has been employed to describe many deformation processes.

However current Maxwell models display well-known errors when the associated strains and, importantly, rotations are large. Such conditions are often met in Earth sciences. Large rotations pose a mathematical challenge when elasticity is considered in the rheology of highly transformed materials as one requires an objective formulation of the stress rate (time derivative of stress).

In a new publication, Schrank et al. introduce a new large-strain model for Maxwell viscoelasticity with a logarithmic co-rotational stress rate (the ‘FT model’). An analysis of homogeneous isothermal simple shear with the FT model compared
to a classic small-strain formulation (the ‘SS model’) and a model using the classic Jaumann stress rate (the ‘MJ model’) leads to the following key conclusions:

  • At W  ≤ 0.1, all models yield essentially identical results.
  • At larger W, the models show increasing differences for γ > 0.5. The SS model overestimates shear stresses compared to the FT model while the MJ model exhibits an oscillatory response underestimating the FT model.
  • The MJ model violates the self-consistency condition resulting in stress oscillations and should be disregarded. It does not deliver truly elastic behaviour.
  • In the intermediate-W regime, the shear-stress overestimates of the SS model may constitute acceptable errors if energy consistency is not important. If energy consistency is desirable, the SS model should not be used at W ≥ 0.3.
  • In the high-W regime, stresses in the SS model become unacceptably
    large. The FT model should be used in this domain.

The FT model constitutes a physically consistent Maxwell model for large non-coaxial deformations, even at high Weissenberg numbers (W). It overcomes the conceptual limitations of the SS model, which is limited to small transformations, not objective and not self-consistent. It also solves the problem of the energetically aberrant oscillations of the MJ model.

 

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