Viscoplasticity

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 elastic response of viscoplastic materials can be represented in one-dimension by Hookean spring elements.

[4] 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.

For polymers, wood, and bitumen, the theory of viscoplasticity is required to describe behavior beyond the limit of elasticity or viscoelasticity.

In viscoplasticity, the development of a mathematical model heads back to 1910 with the representation of primary creep by Andrade's law.

[9] In 1929, Norton[10] developed a one-dimensional dashpot model which linked the rate of secondary creep to the stress.

This model provided a relation between the deviatoric stress and the strain rate for an incompressible Bingham solid[15] 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[16] 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.

For a qualitative analysis, several characteristic tests are performed to describe the phenomenology of viscoplastic materials.

Some examples of these tests are [9] One consequence of yielding is that as plastic deformation proceeds, an increase in stress is required to produce additional strain.

To obtain the stress–strain behavior shown in blue in the figure, the material is initially loaded at a strain rate of 0.1/s.

Creep is the tendency of a solid material to slowly move or deform permanently under constant stresses.

Creep tests measure the strain response due to a constant stress as shown in Figure 3.

The classical creep curve represents the evolution of strain as a function of time in a material subjected to uniaxial stress at a constant temperature.

The creep test, for instance, is performed by applying a constant force/stress and analyzing the strain response of the system.

In fact, these tests characterize the viscosity and can be used to determine the relation which exists between the stress and the rate of viscoplastic strain.

The residual value that is reached when the stress has plateaued at the end of a relaxation test corresponds to the upper limit of elasticity.

It is important to note that relaxation tests are extremely difficult to perform because maintaining the condition

If we assume that plastic flow is isochoric (volume preserving), then the above relation can be expressed in the more familiar form[21]

The responses for strain hardening, creep, and relaxation tests of such material are shown in Figure 6.

In the first situation, the sliding friction element and the dashpot are arranged in parallel and then connected in series to the elastic spring as shown in Figure 7.

The sliding element represents a constant yielding stress when the elastic limit is exceeded irrespective of the strain.

The responses for strain hardening, creep, and relaxation tests of such material are shown in Figure 8.

This implies that the yield stress in the sliding element increases with strain and the model may be expressed in generic terms as

This model is adopted when metals and alloys are at medium and higher temperatures and wood under high loads.

The responses for strain hardening, creep, and relaxation tests of such a material are shown in Figure 9.

In other situations, the yield stress model provides a direct means of computing the plastic strain rate.

is the component of the flow stress due to intrinsic barriers to thermally activated dislocation motion and dislocation-dislocation interactions,

is the component of the flow stress due to microstructural evolution with increasing deformation (strain hardening), (

Figure 1. Elements used in one-dimensional models of viscoplastic materials.
Figure 2. Stress–strain response of a viscoplastic material at different strain rates. The dotted lines show the response if the strain-rate is held constant. The blue line shows the response when the strain rate is changed suddenly.
Figure 3a. Creep test
Figure 3b. Strain as a function of time in a creep test
Figure 4. a) Applied strain in a relaxation test and b) induced stress as functions of time over a short period for a viscoplastic material.
Figure 5. Norton-Hoff model for perfectly viscoplastic solid
Figure 6: The response of perfectly viscoplastic solid to hardening, creep and relaxation tests
Figure 7. The elastic perfectly viscoplastic material.
Figure 8. The response of elastic perfectly viscoplastic solid to hardening, creep and relaxation tests.
Figure 9. The response of elastoviscoplastic hardening solid to hardening, creep and relaxation tests.