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. In the figure ${\displaystyle E}$ is the modulus of elasticity, ${\displaystyle \lambda }$ is the viscosity parameter and ${\displaystyle N}$ is a power-law type parameter that represents non-linear dashpot ${\displaystyle [\sigma (\mathrm {d} \varepsilon /\mathrm {d} t)=\sigma =\lambda (\mathrm {d} \varepsilon /\mathrm {d} t)^{1/N}]}$. The sliding element can have a yield stress (${\displaystyle \sigma _{y}}$) that is strain rate dependent, or even constant, as shown in Figure 1c.

Viscoplasticity is usually modeled in three-dimensions using overstress models of the Perzyna or Duvaut-Lions types. In these models, the stress is allowed to increase beyond the rate-independent yield surface upon application of a load and then allowed to relax back to the yield surface over time. The yield surface is usually assumed not to be rate-dependent in such models. An alternative approach is to add a strain rate dependence to the yield stress and use the techniques of rate independent plasticity to calculate the response of a material.

For metals and alloys, viscoplasticity is the macroscopic behavior caused by a mechanism linked to the movement of dislocations in grains, with superposed effects of inter-crystalline gliding. The mechanism usually becomes dominant at temperatures greater than approximately one third of the absolute melting temperature. However, certain alloys exhibit viscoplasticity at room temperature (300 K). For polymers, wood, and bitumen, the theory of viscoplasticity is required to describe behavior beyond the limit of elasticity or viscoelasticity.

In general, viscoplasticity theories are useful in areas such as:

Research on plasticity theories started in 1864 with the work of Henri Tresca, Saint Venant (1870) and Levy (1871) on the maximum shear criterion. An improved plasticity model was presented in 1913 by Von Mises which is now referred to as the von Mises yield criterion. In viscoplasticity, the development of a mathematical model heads back to 1910 with the representation of primary creep by Andrade's law. In 1929, Norton developed a one-dimensional dashpot model which linked the rate of secondary creep to the stress. In 1934, Odqvist generalized Norton's law to the multi-axial case.

Concepts such as the normality of plastic flow to the yield surface and flow rules for plasticity were introduced by Prandtl (1924)^{[full citation needed]} and Reuss (1930). In 1932, Hohenemser and Prager proposed the first model for slow viscoplastic flow. This model provided a relation between the deviatoric stress and the strain rate for an incompressible Bingham solid However, the application of these theories did not begin before 1950, where limit theorems were discovered.

In 1960, the first IUTAM Symposium "Creep in Structures" organized by Hoff provided a major development in viscoplasticity with the works of Hoff, Rabotnov, Perzyna, Hult, and Lemaitre for the isotropic hardening laws, and those of Kratochvil, Malinini and Khadjinsky, Ponter and Leckie, and Chaboche for the kinematic hardening laws. Perzyna, in 1963, introduced a viscosity coefficient that is temperature and time dependent. The formulated models were supported by the thermodynamics of irreversible processes and the phenomenological standpoint. The ideas presented in these works have been the basis for most subsequent research into rate-dependent plasticity.