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Viscoelastic materials and Viscoelastic Models

Viscoelastic materials are materials that have both viscose and elastic properties. When energy is applied to a completely elastic (ideal elastic) material, the energy will be stored as potential energy. When the energy is released, the material will come to its original position using the stored energy. When the energy is applied to a complete viscose (ideal viscose) material, the supplied energy will dissipate as heat.

Polymers are viscoelastic materials. Therefore, when a polymer is applied energy, a part of the energy will be stored due to the elastic components and other parts of the energy will be dissipated as heat due to the viscose components.

Viscoelastic Models

Several models have been introduced to explain the viscose and elastic behavior of viscoelastic materials. Ideal viscose materials are described using dashpots and ideal elastic materials are described using springs. By combining dashpots and springs together, viscoelastic models have been developed. A good viscoelastic model should be able to explain creep, relaxation, and recovery using all the components (spring and dashpot) of the model.

Creep

Creep is the strain that occurs as a function of time at constant stress. During the primary stage strain rate decreases with time. However overall strain will improve. Then it comes to the secondary phase. In the secondary phase, strain is constant. After a certain period, strain is improved rapidly and ruptured.

Strain time curve for viscoelastic materials
Figure 01: Strain time curve for viscoelastic materials

Stress relaxation

Stress relaxation is the decrease of stress at the constant strain. After the stress is removed, the stored energy on the elastic components is released. Due to the viscose components strain remains constant. If the material has more viscose components, it has high-stress relaxation.

Recovery

Recover is the process that the material comes back to its original state after the stress is removed.

Creep and recovery for viscoelastic materials
Figure 02: Creep and recovery for viscoelastic materials

Pure viscose and pure elastic models

The behavior of pure viscose material and pure elastic material can be described using a dashpot and a spring respectively.

Pure viscous and pure elastic models
Figure 03: Pure viscous and pure elastic models

For the dashpot,

Viscoelastic models for polymers eq 01

for the spring,

Viscoelastic models for polymers eq 02

Where,

  • σ = stress
  • η = viscosity of the liquid
  • ε̇ = strain rate
  • k = young’s modulus

Kelvin model

Kelvin model also known as the Voigt model is the simplest viscoelastic model. To describe the viscose and elastic components, a pure viscose dashpot and pure elastic spring is placed parallel to each other.

Kelvin model
Figure 04: Kelvin model

When applying stress both the spring and the dashpot undergo the same elongation. The total stress will be divided into spring and dashpot.

Viscoelastic models for polymers eq 03

Energy is stored in spring. After the stress is released, only the spring will undergo relaxation. Dashpot is not involved in relaxation. Therefore, this model is not suitable for describing relaxation. But it is used to describe the creep.

Maxwell model

In the Maxwell model, a pure dashpot and pure spring are placed in the serial.

Maxwell model
Figure 05: Maxwell model

When applying stress, there will be a sudden deformation due to the sudden response of the spring. And there will be further strain due to the deformation of the dashpot.

Viscoelastic models for polymers eq 04

The strain rate of the spring,

Viscoelastic models for polymers eq 05
  • Where σ̇ is the stress rate.

The strain rate of the dashpot,

Viscoelastic models for polymers eq 06

Therefore, the total strain is given by,

Viscoelastic models for polymers eq 07

Once the force is removed, the spring will come to its original state. The dashpot will not recover. The deformation takes place of each component in the model separately. Deformation and recovery of the spring will not be affected by the dashpot. Therefore, the creep cannot be described by this model.

Burger model

The burger model is a combination of both the maxwell and kelvin models. It describes the viscoelastic behavior (creep, relaxation, and recovery) of non-Newtonian fluids. In this model, the Kelvin unit is placed in between the maxwell unit.

Burger model
Figure 06: Burger model

Once we apply stress, the F1 spring will come to its maximum strain within a very short time. This explains the elastic domain of the materials. When we further improve the stress, F2 and D2 start to strain. This represents the viscoelastic behavior. as we improve the stress further D3 dashpot starts deforming permanently. This shows the viscose behavior of the materials.

Stress recovery curve of the burger model
Figure 07: Stress recovery curve of Burger model
  • ε1 = Pure elastic strain (Spring F1)
  • ε2 = Viscoelastic strain (Parallel connection of Spring F2 and damper D2)
  • ε3 = Pure viscous strain (Damper D3)
  • εmax = Maximum strain (ε1 + ε2 + ε3)
  • εE = Elastic recovery (Spring F1)
  • εVE = Viscoelastic recovery (Spring F2 and damper D2)
  • εV = Viscous deformation (Damper D3)

During the recovery process, the F1 spring recovers suddenly. then there is a slow recovery from the middle Kelvin unit. Afterward, there is a permanent deformation due to the damper D3.


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References and Attributes

Figures:

The cover image was designed using an image by Sander van der Wel, licensed under CC BY-SA 2.0, via Wikimedia Commons


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