Viscoelasticity is a material property that combines both viscous and elastic characteristics. Many materials have such viscoelastic properties, especially materials that consist of large molecules. Polymers are viscoelastic because their macromolecules can have temporary entanglements with neighbouring molecules, causing elastic properties. After some time these entanglements disappear and the macromolecules flow into other positions where new entanglements occur (viscous properties).
A viscoelastic material shows elastic properties on short time scales and viscous properties on long time scales. These materials exhibit behavior that depends on the time and rate of applied forces, allowing them to both store and dissipate energy.
Viscoelasticity has been studied since the 19th century by researchers such as James Clerk Maxwell, Ludwig Boltzmann, and Lord Kelvin.
Several models are available for the mathematical description of the viscoelastic properties of a substance:
Constitutive models of linear viscoelasticity assume a linear relationship between stress and strain. These models are valid for relatively small deformations only. Constitutive models of non-linear viscoelasticity are based on a more realistic non-linear relationship between stress and strain. These models are valid for relatively large deformations.
The viscoelastic properties of polymers are highly temperature dependent. From low to high temperature the material can be in the glass phase, rubber phase or the melt phase. These phases have a very strong effect on the mechanical and viscous properties of the polymers.
Typical viscoelastic properties are:
A time-dependent stress in the polymer under constant deformation (strain). A time dependent strain in the polymer under constant stress. A time and temperature dependent stiffness of the polymer. Viscous energy loss during deformation of the polymer in the glass or rubber phase (hysteresis). A strain rate-dependent viscosity of the molten polymer. An ongoing deformation of a polymer in the glass phase at constant load (creep).
The viscoelasticity properties are measured with various techniques, such as tensile testing, dynamic mechanical analysis, shear rheometry and extensional rheometry.
Background In the nineteenth century, physicists such as James Clerk Maxwell, Ludwig Boltzmann, and Lord Kelvin researched and experimented with creep and recovery of glasses, metals, and rubbers. Viscoelasticity was further examined in the late twentieth century when synthetic polymers were engineered and used in a variety of applications.McCrum, Buckley, and Bucknell (2003): "Principles of Polymer Engineering," 117-176.
Viscoelasticity calculations depend heavily on the viscosity variable, η. The inverse of η is also known as fluidity, φ. The value of either can be derived as a function of temperature or as a given value (i.e. for a dashpot).Meyers and Chawla (1999): "Mechanical Behavior of Materials", 98-103.
Depending on the change of strain rate versus stress inside a material, the viscosity can be categorized as having a linear, non-linear, or plastic response:
When a material exhibits a linear response it is categorized as a Newtonian material. In this case the stress is linearly proportional to the strain rate. If the material exhibits a non-linear response to the strain rate, it is categorized as non-Newtonian fluid. There is also an interesting case where the viscosity decreases as the shear/strain rate remains constant. A material which exhibits this type of behavior is known as thixotropic. In addition, when the stress is independent of this strain rate, the material exhibits plastic deformation. Many viscoelastic materials exhibit rubber like behavior explained by the thermodynamic theory of polymer elasticity.
Some examples of viscoelastic materials are amorphous polymers, semicrystalline polymers, biopolymers, metals at very high temperatures, and bitumen materials. Cracking occurs when the strain is applied quickly and outside of the elastic limit. Ligaments and tendons in the human body are viscoelastic, so the extent of the potential damage to them depends on both the rate of the change of their length and the force applied.
A viscoelastic material has the following properties: hysteresis is seen in the stress–strain curve stress relaxation occurs: step constant strain causes decreasing stress creep occurs: step constant stress causes increasing strain its stiffness depends on the strain rate or the stress rate
Elastic versus viscoelastic behavior
Unlike purely elastic substances, a viscoelastic substance has an elastic component and a viscous component. The viscosity of a viscoelastic substance gives the substance a strain rate dependence on time. Purely elastic materials do not dissipate energy (heat) when a load is applied, then removed. However, a viscoelastic substance dissipates energy when a load is applied, then removed. Hysteresis is observed in the stress–strain curve, with the area of the loop being equal to the energy lost during the loading cycle. Since viscosity is the resistance to thermally activated plastic deformation, a viscous material loses energy through a loading cycle. Plastic deformation results in lost energy, which is uncharacteristic of a purely elastic material's reaction to a loading cycle.
Specifically, viscoelasticity is a molecular rearrangement. When a stress is applied to a viscoelastic material such as a polymer, parts of the long polymer chain change positions. This movement or rearrangement is called creep. Polymers remain a solid material even when these parts of their chains are rearranging to accommodate the stress, and as this occurs, it creates a back stress in the material. When the back stress is the same magnitude as the applied stress, the material no longer creeps. When the original stress is taken away, the accumulated back stresses causes the polymer to return to its original form. The material creeps, which gives the prefix visco-, and the material fully recovers, which gives the suffix -elasticity.
Linear viscoelasticity and nonlinear viscoelasticity Linear viscoelasticity is behavior in which the function is separable in both creep response and load. All linear viscoelastic models can be represented by a Volterra equation connecting stress and strain:
or
where is time is stress is strain and are instantaneous elastic moduli for creep and relaxation is the creep function is the relaxation function
Linear viscoelasticity is usually applicable only for small deformations.
In nonlinear viscoelasticity, the function is not separable. It usually happens when the deformations are large or if the material changes its properties under deformations. Nonlinear viscoelasticity also elucidates observed phenomena such as normal stresses, shear thinning, and extensional thickening in viscoelastic fluids.
An anelastic material is a special case of a viscoelastic material: an anelastic material fully recovers to its original state on the removal of load.
When distinguishing between elastic, viscous, and forms of viscoelastic behavior, it is helpful to reference the time scale of the measurement relative to the relaxation times of the material being observed, known as the Deborah number (De) where:
where is the relaxation time of the material is time
Dynamic modulus
Viscoelasticity is studied using dynamic mechanical analysis, applying a small oscillatory stress and measuring the resulting strain. Purely elastic materials have stress and strain in phase, so that the response of one caused by the other is immediate. In purely viscous materials, strain lags stress by a 90-degree phase. Viscoelastic materials exhibit behavior somewhere in the middle of these two types of material, exhibiting some lag in strain.
A complex dynamic modulus G can be used to represent the relations between the oscillating stress and strain:
where ; is the storage modulus and is the loss modulus:
where and are the amplitudes of stress and strain respectively, and is the phase shift between them.
Constitutive models of linear viscoelasticity
Viscoelastic materials, such as amorphous polymers, semicrystalline polymers, biopolymers and even the living tissue and cells, can be modeled in order to determine their stress and strain or force and displacement interactions as well as their temporal dependencies. These models, which include the Maxwell model, the Kelvin–Voigt model, the standard linear solid model, and the Burgers model, are used to predict a material's response under different loading conditions.
Viscoelastic behavior has elastic and viscous components modeled as linear combinations of springs and dashpots, respectively. Each model differs in the arrangement of these elements, and all of these viscoelastic models can be equivalently modeled as electrical circuits.
In an equivalent electrical circuit, stress is represented by current, and strain rate by voltage. The elastic modulus of a spring is analogous to the inverse of a circuit's inductance (it stores energy) and the viscosity of a dashpot to a circuit's resistance (it dissipates energy).
The elastic components, as previously mentioned, can be modeled as springs of elastic constant E, given the formula:
where σ is the stress, E is the elastic modulus of the material, and ε is the strain that occurs under the given stress, similar to Hooke's law.
The viscous components can be modeled as dashpots such that the stress–strain rate relationship can be given as,
where σ is the stress, η is the viscosity of the material, and dε/dt is the time derivative of strain.
The relationship between stress and strain can be simplified for specific stress or strain rates. For high stress or strain rates/short time periods, the time derivative components of the stress–strain relationship dominate. In these conditions it can be approximated as a rigid rod capable of sustaining high loads without deforming. Hence, the dashpot can be considered to be a "short-circuit".Van Vliet, Krystyn J. (2006). "3.032 Mechanical Behavior of Materials"
Conversely, for low stress states/longer time periods, the time derivative components are negligible and the dashpot can be effectively removed from the system – an "open" circuit. As a result, only the spring connected in parallel to the dashpot contributes to the total strain in the system.
Maxwell model
The Maxwell model can be represented by a purely viscous damper and a purely elastic spring connected in series, as shown in the diagram. The model can be represented by the following equation:
Under this model, if the material is put under a constant strain, the stresses gradually relax. When a material is put under a constant stress, the strain has two components. First, an elastic component occurs instantaneously, corresponding to the spring, and relaxes immediately upon release of the stress. The second is a viscous component that grows with time as long as the stress is applied. The Maxwell model predicts that stress decays exponentially with time, which is accurate for most polymers. One limitation of this model is that it does not predict creep accurately. The Maxwell model for creep or constant-stress conditions postulates that strain increases linearly with time. However, polymers for the most part show the strain rate to be decreasing with time.
This model can be applied to soft solids: thermoplastic polymers in the vicinity of their melting temperature, fresh concrete (neglecting its aging), and numerous metals at a temperature close to their melting point.
The equation introduced here, however, lacks a consistent derivation from more microscopic model and is not observer independent. The Upper-convected Maxwell model is its sound formulation in terms of the Cauchy stress tensor and constitutes the simplest tensorial constitutive model for viscoelasticity (see e.g. or ).