page_title: Viscoplasticity
section_title: Perfectly viscoplastic solid (Norton-Hoff model)
hierarchical_section_title: Viscoplasticity / Rheological models of viscoplasticity / Perfectly viscoplastic solid (Norton-Hoff model)
caption_attribution_description: English: This was performed on Power Point Presentation.
context_page_description: Viscoplasticity is a theory in continuum mechanics that describes the rate-dependent inelastic behavior of solids. Rate-dependence in this context means that the deformation of the material depends on the rate at which loads are applied. The inelastic behavior that is the subject of viscoplasticity is plastic deformation which means that the material undergoes unrecoverable deformations when a load level is reached. Rate-dependent plasticity is important for transient plasticity calculations. The main difference between rate-independent plastic and viscoplastic material models is that the latter exhibit not only permanent deformations after the application of loads but continue to undergo a creep flow as a function of time under the influence of the applied load.
  The elastic response of viscoplastic materials can be represented in one-dimension by Hookean spring elements. Rate-dependence can be represented by nonlinear dashpot elements in a manner similar to viscoelasticity. Plasticity can be accounted for by adding sliding frictional elements as shown in Figure 1.
context_section_description: In a perfectly viscoplastic solid, also called the Norton-Hoff model of viscoplasticity, the stress (as for viscous fluids) is a function of the rate of permanent strain.  The effect of elasticity is neglected in the model, i.e.,  and hence there is no initial yield stress, i.e., .  The viscous dashpot has a response given by
  where  is the viscosity of the dashpot.  In the Norton-Hoff model the viscosity  is a nonlinear function of the applied stress and is given by
  where  is a fitting parameter, λ is the kinematic viscosity of the material and .  Then the viscoplastic strain rate is given by the relation
  In one-dimensional form, the Norton-Hoff model can be expressed as
  When  the solid is viscoelastic.
  If we assume that plastic flow is isochoric (volume preserving), then the above relation can be expressed in the more familiar form
  where  is the deviatoric stress tensor,  is the von Mises equivalent strain rate, and  are material parameters.  The equivalent strain rate is defined as
  These models can be applied in metals and alloys at temperatures higher than two thirds of their absolute melting point (in kelvins) and polymers/asphalt at elevated temperature.  The responses for strain hardening, creep, and relaxation tests of such material are shown in Figure 6.
caption: Figure 6: The response of perfectly viscoplastic solid to hardening, creep and relaxation tests.
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