# Viscoelasticity

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Viscoelasticity is the property of materials that exhibit both viscous and elastic characteristics when undergoing deformation. Viscous materials, like water, resist shear flow and strain linearly with time when a stress is applied. Elastic materials strain when stretched and immediately return to their original state once the stress is removed.

The viscosity of a fluid is a measure of its resistance to deformation at a given rate. For liquids, it corresponds to the informal concept of "thickness": for example, syrup has a higher viscosity than water.

In physics, elasticity is the ability of a body to resist a distorting influence and to return to its original size and shape when that influence or force is removed. Solid objects will deform when adequate forces are applied to them. If the material is elastic, the object will return to its initial shape and size when these forces are removed. Hooke's law states that the force should be proportional to the extension. The physical reasons for elastic behavior can be quite different for different materials. In metals, the atomic lattice changes size and shape when forces are applied. When forces are removed, the lattice goes back to the original lower energy state. For rubbers and other polymers, elasticity is caused by the stretching of polymer chains when forces are applied.

In materials science, deformation refers to any changes in the shape or size of an object due to

## Contents

Viscoelastic materials have elements of both of these properties and, as such, exhibit time-dependent strain. Whereas elasticity is usually the result of bond stretching along crystallographic planes in an ordered solid, viscosity is the result of the diffusion of atoms or molecules inside an amorphous material. [1]

A chemical bond is a lasting attraction between atoms, ions or molecules that enables the formation of chemical compounds. The bond may result from the electrostatic force of attraction between oppositely charged ions as in ionic bonds or through the sharing of electrons as in covalent bonds. The strength of chemical bonds varies considerably; there are "strong bonds" or "primary bonds" such as covalent, ionic and metallic bonds, and "weak bonds" or "secondary bonds" such as dipole–dipole interactions, the London dispersion force and hydrogen bonding.

## Background

In the nineteenth century, physicists such as Maxwell, Boltzmann, and 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. [2] 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). [1]

James Clerk Maxwell was a Scottish scientist in the field of mathematical physics. His most notable achievement was to formulate the classical theory of electromagnetic radiation, bringing together for the first time electricity, magnetism, and light as different manifestations of the same phenomenon. Maxwell's equations for electromagnetism have been called the "second great unification in physics" after the first one realised by Isaac Newton.

Ludwig Eduard Boltzmann was an Austrian physicist and philosopher whose greatest achievement was in the development of statistical mechanics, which explains and predicts how the properties of atoms determine the physical properties of matter.

William Thomson, 1st Baron Kelvin, was an Ulster Scots Irish mathematical physicist and engineer who was born in Belfast in 1824. At the University of Glasgow he did important work in the mathematical analysis of electricity and formulation of the first and second laws of thermodynamics, and did much to unify the emerging discipline of physics in its modern form. He worked closely with mathematics professor Hugh Blackburn in his work. He also had a career as an electric telegraph engineer and inventor, which propelled him into the public eye and ensured his wealth, fame and honour. For his work on the transatlantic telegraph project he was knighted in 1866 by Queen Victoria, becoming Sir William Thomson. He had extensive maritime interests and was most noted for his work on the mariner's compass, which previously had limited reliability.

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. [1] Many viscoelastic materials exhibit rubber like behavior explained by the thermodynamic theory of polymer elasticity.

With regard to materials science, a material is said to be "Newtonian" if it exhibits a linear relationship between stress and strain rate.

A non-Newtonian fluid is a fluid that does not follow Newton's law of viscosity, i.e. constant viscosity independent of stress. In non-Newtonian fluids, viscosity can change when under force to either more liquid or more solid. Ketchup, for example, becomes runnier when shaken and is thus a non-Newtonian fluid. Many salt solutions and molten polymers are non-Newtonian fluids, as are many commonly found substances such as custard, honey, toothpaste, starch suspensions, corn starch, paint, blood, and shampoo.

Thixotropy is a time-dependent shear thinning property. Certain gels or fluids that are thick or viscous under static conditions will flow over time when shaken, agitated, sheared or otherwise stressed. They then take a fixed time to return to a more viscous state. Some non-Newtonian pseudoplastic fluids show a time-dependent change in viscosity; the longer the fluid undergoes shear stress, the lower its viscosity. A thixotropic fluid is a fluid which takes a finite time to attain equilibrium viscosity when introduced to a steep change in shear rate. Some thixotropic fluids return to a gel state almost instantly, such as ketchup, and are called pseudoplastic fluids. Others such as yogurt take much longer and can become nearly solid. Many gels and colloids are thixotropic materials, exhibiting a stable form at rest but becoming fluid when agitated. Thixotropy arises because particles or structured solutes require time to organize. An excellent overview of thixotropy has been provided by Mewis and Wagner.

Some examples of viscoelastic materials include 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 are viscoelastic, so the extent of the potential damage to them depends both on the rate of the change of their length as well as on the force applied.[ citation needed ]

A ligament is the fibrous connective tissue that connects bones to other bones. It is also known as articular ligament, articular larua, fibrous ligament, or true ligament. Other ligaments in the body include the:

A tendon or sinew is a tough band of fibrous connective tissue that usually connects muscle to bone and is capable of withstanding tension.

A viscoelastic material has the following properties:

Hysteresis is the dependence of the state of a system on its history. For example, a magnet may have more than one possible magnetic moment in a given magnetic field, depending on how the field changed in the past. Plots of a single component of the moment often form a loop or hysteresis curve, where there are different values of one variable depending on the direction of change of another variable. This history dependence is the basis of memory in a hard disk drive and the remanence that retains a record of the Earth's magnetic field magnitude in the past. Hysteresis occurs in ferromagnetic and ferroelectric materials, as well as in the deformation of rubber bands and shape-memory alloys and many other natural phenomena. In natural systems it is often associated with irreversible thermodynamic change such as phase transitions and with internal friction; and dissipation is a common side effect.

The relationship between the stress and strain that a particular material displays is known as that particular material's stress–strain curve. It is unique for each material and is found by recording the amount of deformation (strain) at distinct intervals of a variety of loadings (stress). These curves reveal many of the properties of a material.

In materials science, stress relaxation is the observed decrease in stress in response to strain generated in the structure. This is primarily due to keeping the structure in a strained condition for some finite interval of time and hence causing some amount of plastic strain. This should not be confused with creep, which is a constant state of stress with an increasing amount of strain.

## 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 loses 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 will lose 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. [1]

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 in order to accompany 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 will cause 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. [2]

## Types

Linear viscoelasticity is when 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:

${\displaystyle \varepsilon (t)={\frac {\sigma (t)}{E_{\text{inst,creep}}}}+\int _{0}^{t}K(t-t^{\prime }){\dot {\sigma }}(t^{\prime })dt^{\prime }}$

or

${\displaystyle \sigma (t)=E_{\text{inst,relax}}\varepsilon (t)+\int _{0}^{t}F(t-t^{\prime }){\dot {\varepsilon }}(t^{\prime })dt^{\prime }}$

where

• t is time
• ${\displaystyle \sigma (t)}$ is stress
• ${\displaystyle \varepsilon (t)}$ is strain
• ${\displaystyle E_{\text{inst,creep}}}$ and ${\displaystyle E_{\text{inst,relax}}}$ are instantaneous elastic moduli for creep and relaxation
• K(t) is the creep function
• F(t) is the relaxation function

Linear viscoelasticity is usually applicable only for small deformations.

Nonlinear viscoelasticity is when the function is not separable. It usually happens when the deformations are large or if the material changes its properties under deformations.

An anelastic material is a special case of a viscoelastic material: an anelastic material will fully recover to its original state on the removal of load.

## 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 lag.
• 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:

${\displaystyle G=G'+iG''}$

where ${\displaystyle i^{2}=-1}$; ${\displaystyle G'}$ is the storage modulus and ${\displaystyle G''}$ is the loss modulus:

${\displaystyle G'={\frac {\sigma _{0}}{\varepsilon _{0}}}\cos \delta }$
${\displaystyle G''={\frac {\sigma _{0}}{\varepsilon _{0}}}\sin \delta }$

where ${\displaystyle \sigma _{0}}$ and ${\displaystyle \varepsilon _{0}}$ are the amplitudes of stress and strain respectively, and ${\displaystyle \delta }$ 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, [3] 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 voltage, and strain rate by current. The elastic modulus of a spring is analogous to a circuit's capacitance (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:

${\displaystyle \sigma =E\varepsilon }$

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,

${\displaystyle \sigma =\eta {\frac {d\varepsilon }{dt}}}$

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 rates. For high stress states/short time periods, the time derivative components of the stress–strain relationship dominate. A dashpot resists changes in length, and in a high stress state it can be approximated as a rigid rod. Since a rigid rod cannot be stretched past its original length, no strain is added to the system. [4]

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 will contribute to the total strain in the system. [4]

### 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:

${\displaystyle \sigma +{\frac {\eta }{E}}{\dot {\sigma }}=\eta {\dot {\varepsilon }}}$

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 will increase linearly with time. However, polymers for the most part show the strain rate to be decreasing with time. [2]

Applications to soft solids: thermoplastic polymers in the vicinity of their melting temperature, fresh concrete (neglecting its aging), numerous metals at a temperature close to their melting point.

### Kelvin–Voigt model

The Kelvin–Voigt model, also known as the Voigt model, consists of a Newtonian damper and Hookean elastic spring connected in parallel, as shown in the picture. It is used to explain the creep behaviour of polymers.

The constitutive relation is expressed as a linear first-order differential equation:

${\displaystyle \sigma =E\varepsilon +\eta {\dot {\varepsilon }}}$

This model represents a solid undergoing reversible, viscoelastic strain. Upon application of a constant stress, the material deforms at a decreasing rate, asymptotically approaching the steady-state strain. When the stress is released, the material gradually relaxes to its undeformed state. At constant stress (creep), the model is quite realistic as it predicts strain to tend to σ/E as time continues to infinity. Similar to the Maxwell model, the Kelvin–Voigt model also has limitations. The model is extremely good with modelling creep in materials, but with regards to relaxation the model is much less accurate. [5]

Applications: organic polymers, rubber, wood when the load is not too high.

### Standard linear solid model

The standard linear solid model, also known as the Zener model, consists of two springs and a dashpot. It is the simplest model that describes both the creep and stress relaxation behaviors of a viscoelastic material properly. For this model, the governing constitutive relations are:

Maxwell representationKelvin representation
${\displaystyle \sigma +{\frac {\eta }{E_{2}}}{\dot {\sigma }}=E_{1}\varepsilon +{\frac {\eta (E_{1}+E_{2})}{E_{2}}}{\dot {\varepsilon }}}$${\displaystyle \sigma +{\frac {\eta }{E_{1}+E_{2}}}{\dot {\sigma }}={\frac {E_{1}E_{2}}{E_{1}+E_{2}}}\varepsilon +{\frac {E_{1}\eta }{E_{1}+E_{2}}}{\dot {\varepsilon }}}$

Under a constant stress, the modeled material will instantaneously deform to some strain, which is the instantaneous elastic portion of the strain. After that it will continue to deform and asymptotically approach a steady-state strain, which is the retarded elastic portion of the strain. Although the Standard Linear Solid Model is more accurate than the Maxwell and Kelvin–Voigt models in predicting material responses, mathematically it returns inaccurate results for strain under specific loading conditions.

### Burgers model

The Burgers model combines the Maxwell and Kelvin–Voigt models in series. The constitutive relation is expressed as follows:

${\displaystyle \sigma +\left({\frac {\eta _{1}}{E_{1}}}+{\frac {\eta _{2}}{E_{1}}}+{\frac {\eta _{2}}{E_{2}}}\right){\dot {\sigma }}+{\frac {\eta _{1}\eta _{2}}{E_{1}E_{2}}}{\ddot {\sigma }}=\eta _{2}{\dot {\varepsilon }}+{\frac {\eta _{1}\eta _{2}}{E_{1}}}{\ddot {\varepsilon }}}$

This model incorporates viscous flow into the standard linear solid model, giving a linearly increasing asymptote for strain under fixed loading conditions.

### Generalized Maxwell model

The Generalized Maxwell model, also known as the Wiechert model, is the most general form of the linear model for viscoelasticity. It takes into account that the relaxation does not occur at a single time, but at a distribution of times. Due to molecular segments of different lengths with shorter ones contributing less than longer ones, there is a varying time distribution. The Wiechert model shows this by having as many spring–dashpot Maxwell elements as are necessary to accurately represent the distribution. The figure on the right shows the generalised Wiechert model. [6] Applications: metals and alloys at temperatures lower than one quarter of their absolute melting temperature (expressed in K).

## Prony series

In a one-dimensional relaxation test, the material is subjected to a sudden strain that is kept constant over the duration of the test, and the stress is measured over time. The initial stress is due to the elastic response of the material. Then, the stress relaxes over time due to the viscous effects in the material. Typically, either a tensile, compressive, bulk compression, or shear strain is applied. The resulting stress vs. time data can be fitted with a number of equations, called models. Only the notation changes depending of the type of strain applied: tensile-compressive relaxation is denoted ${\displaystyle E}$, shear is denoted ${\displaystyle G}$, bulk is denoted ${\displaystyle K}$. The Prony series for the shear relaxation is

${\displaystyle G(t)=G_{\infty }+\Sigma _{i=1}^{N}G_{i}\exp(-t/\tau _{i})}$

where ${\displaystyle G_{\infty }}$ is the long term modulus once the material is totally relaxed, ${\displaystyle \tau _{i}}$ are the relaxation times (not to be confused with ${\displaystyle \tau _{i}}$ in the diagram); the higher their values, the longer it takes for the stress to relax. The data is fitted with the equation by using a minimization algorithm that adjust the parameters (${\displaystyle G_{\infty },G_{i},\tau _{i}}$) to minimize the error between the predicted and data values. [7]

An alternative form is obtained noting that the elastic modulus is related to the long term modulus by

${\displaystyle G(t=0)=G_{0}=G_{\infty }+\Sigma _{i=1}^{N}G_{i}}$

Therefore,

${\displaystyle G(t)=G_{0}-\Sigma _{i=1}^{N}G_{i}[1-\exp(-t/\tau _{i})]}$

This form is convenient when the elastic shear modulus ${\displaystyle G_{0}}$ is obtained from data independent from the relaxation data, and/or for computer implementation, when it is desired to specify the elastic properties separately from the viscous properties, as in. [8]

A creep experiment is usually easier to perform than a relaxation one, so most data is available as (creep) compliance vs. time. [9] Unfortunately, there is no known closed form for the (creep) compliance in terms of the coefficient of the Prony series. So, if one has creep data, it is not easy to get the coefficients of the (relaxation) Prony series, which are needed for example in. [8] An expedient way to obtain these coefficients is the following. First, fit the creep data with a model that has closed form solutions in both compliance and relaxation; for example the Maxwell-Kelvin model (eq. 7.18-7.19) in [10] or the Standard Solid Model (eq. 7.20-7.21) in [10] (section 7.1.3). Once the parameters of the creep model are known, produce relaxation pseudo-data with the conjugate relaxation model for the same times of the original data. Finally, fit the pseudo data with the Prony series.

## Effect of temperature on viscoelastic behavior

The secondary bonds of a polymer constantly break and reform due to thermal motion. Application of a stress favors some conformations over others, so the molecules of the polymer will gradually "flow" into the favored conformations over time. [11] Because thermal motion is one factor contributing to the deformation of polymers, viscoelastic properties change with increasing or decreasing temperature. In most cases, the creep modulus, defined as the ratio of applied stress to the time-dependent strain, decreases with increasing temperature. Generally speaking, an increase in temperature correlates to a logarithmic decrease in the time required to impart equal strain under a constant stress. In other words, it takes less work to stretch a viscoelastic material an equal distance at a higher temperature than it does at a lower temperature.

Extreme cold temperatures can cause viscoelastic materials to change to the glass phase and become brittle. For example, exposure of pressure sensitive adhesives to extreme cold (dry ice, freeze spray, etc.) causes them to lose their tack, resulting in debonding.

## Viscoelastic creep

When subjected to a step constant stress, viscoelastic materials experience a time-dependent increase in strain. This phenomenon is known as viscoelastic creep.

At a time ${\displaystyle t_{0}}$, a viscoelastic material is loaded with a constant stress that is maintained for a sufficiently long time period. The material responds to the stress with a strain that increases until the material ultimately fails, if it is a viscoelastic liquid. If, on the other hand, it is a viscoelastic solid, it may or may not fail depending on the applied stress versus the material's ultimate resistance. When the stress is maintained for a shorter time period, the material undergoes an initial strain until a time ${\displaystyle t_{1}}$, after which the strain immediately decreases (discontinuity) then gradually decreases at times ${\displaystyle t>t_{1}}$ to a residual strain.

Viscoelastic creep data can be presented by plotting the creep modulus (constant applied stress divided by total strain at a particular time) as a function of time. [12] Below its critical stress, the viscoelastic creep modulus is independent of stress applied. A family of curves describing strain versus time response to various applied stress may be represented by a single viscoelastic creep modulus versus time curve if the applied stresses are below the material's critical stress value.

Viscoelastic creep is important when considering long-term structural design. Given loading and temperature conditions, designers can choose materials that best suit component lifetimes.

## Measurement

Though there are many instruments that test the mechanical and viscoelastic response of materials, broadband viscoelastic spectroscopy (BVS) and resonant ultrasound spectroscopy (RUS) are more commonly used to test viscoelastic behavior because they can be used above and below ambient temperatures and are more specific to testing viscoelasticity. These two instruments employ a damping mechanism at various frequencies and time ranges with no appeal to time–temperature superposition. Using BVS and RUS to study the mechanical properties of materials is important to understanding how a material exhibiting viscoelasticity will perform. [13]

## Related Research Articles

Young's modulus or Young modulus is a mechanical property that measures the stiffness of a solid material. It defines the relationship between stress and strain in a material in the linear elasticity regime of a uniaxial deformation.

Dynamic mechanical analysis is a technique used to study and characterize materials. It is most useful for studying the viscoelastic behavior of polymers. A sinusoidal stress is applied and the strain in the material is measured, allowing one to determine the complex modulus. The temperature of the sample or the frequency of the stress are often varied, leading to variations in the complex modulus; this approach can be used to locate the glass transition temperature of the material, as well as to identify transitions corresponding to other molecular motions.

A Maxwell material is a viscoelastic material having the properties both of elasticity and viscosity. It is named for James Clerk Maxwell who proposed the model in 1867. It is also known as a Maxwell fluid.

Hemorheology, also spelled haemorheology, or blood rheology, is the study of flow properties of blood and its elements of plasma and cells. Proper tissue perfusion can occur only when blood's rheological properties are within certain levels. Alterations of these properties play significant roles in disease processes. Blood viscosity is determined by plasma viscosity, hematocrit and mechanical properties of red blood cells. Red blood cells have unique mechanical behavior, which can be discussed under the terms erythrocyte deformability and erythrocyte aggregation. Because of that, blood behaves as a non-Newtonian fluid. As such, the viscosity of blood varies with shear rate. Blood becomes less viscous at high shear rates like those experienced with increased flow such as during exercise or in peak-systole. Therefore, blood is a shear-thinning fluid. Contrarily, blood viscosity increases when shear rate goes down with increased vessel diameters or with low flow, such as downstream from an obstruction or in diastole. Blood viscosity also increases with increases in red cell aggregability.

In materials science, creep is the tendency of a solid material to move slowly or deform permanently under the influence of persistent mechanical stresses. It can occur as a result of long-term exposure to high levels of stress that are still below the yield strength of the material. Creep is more severe in materials that are subjected to heat for long periods and generally increases as they near their melting point.

A Kelvin–Voigt material, also called a Voigt material, is a viscoelastic material having the properties both of elasticity and viscosity. It is named after the British physicist and engineer Lord Kelvin and after German physicist Woldemar Voigt.

A Burgers material is a viscoelastic material which consists of a Maxwell material and a Kelvin material in series. It is named after the Dutch physicist Johannes Martinus Burgers.

Dynamic modulus is the ratio of stress to strain under vibratory conditions. It is a property of viscoelastic materials.

Nanoindentation is a variety of indentation hardness tests applied to small volumes. Indentation is perhaps the most commonly applied means of testing the mechanical properties of materials. The nanoindentation technique was developed in the mid-1970s to measure the hardness of small volumes of material.

The Ramberg–Osgood equation was created to describe the non linear relationship between stress and strain—that is, the stress–strain curve—in materials near their yield points. It is especially useful for metals that harden with plastic deformation, showing a smooth elastic-plastic transition.

The standard linear solid (SLS), also known as the Zener model, is a method of modeling the behavior of a viscoelastic material using a linear combination of springs and dashpots to represent elastic and viscous components, respectively. Often, the simpler Maxwell model and the Kelvin–Voigt model are used. These models often prove insufficient, however; the Maxwell model does not describe creep or recovery, and the Kelvin–Voigt model does not describe stress relaxation. SLS is the simplest model that predicts both phenomena.

In continuum mechanics, the Lamé parameters are two material-dependent quantities denoted by λ and μ that arise in strain-stress relationships. In general, λ and μ are individually referred to as Lamé's first parameter and Lamé's second parameter, respectively. Other names are sometimes employed for one or both parameters, depending on context. For example, the parameter μ is referred to in fluid dynamics as the dynamic viscosity of a fluid; whereas in the context of elasticity, μ is called the shear modulus, and is sometimes denoted by G instead of μ. Typically the notation G is seen paired with the use of Young's modulus, and the notation μ is paired with the use of λ.

The time–temperature superposition principle is a concept in polymer physics and in the physics of glass-forming liquids. This superposition principle is used to determine temperature-dependent mechanical properties of linear viscoelastic materials from known properties at a reference temperature. The elastic moduli of typical amorphous polymers increase with loading rate but decrease when the temperature is increased. Curves of the instantaneous modulus as a function of time do not change shape as the temperature is changed but appear only to shift left or right. This implies that a master curve at a given temperature can be used as the reference to predict curves at various temperatures by applying a shift operation. The time-temperature superposition principle of linear viscoelasticity is based on the above observation.

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 viscous stress tensor is a tensor used in continuum mechanics to model the part of the stress at a point within some material that can be attributed to the strain rate, the rate at which it is deforming around that point.

Flow plasticity is a solid mechanics theory that is used to describe the plastic behavior of materials. Flow plasticity theories are characterized by the assumption that a flow rule exists that can be used to determine the amount of plastic deformation in the material.

Visco-elastic jets are the jets of viscoelastic fluids, i.e. fluids that disobey Newton’s law of Viscocity. A Viscoelastic fluid that returns to its original shape after the applied stress is released.

## References

1. Meyers and Chawla (1999): "Mechanical Behavior of Materials", 98-103.
2. McCrum, Buckley, and Bucknell (2003): "Principles of Polymer Engineering," 117-176.
3. Biswas, Abhijit; Manivannan, M.; Srinivasan, Mandyam A. (2015). "Multiscale Layered Biomechanical Model of the Pacinian Corpuscle". IEEE Transactions on Haptics. 8 (1): 31–42. doi:10.1109/TOH.2014.2369416. PMID   25398182.
4. Tanner, Roger I. (1988). Engineering Rheologu. Oxford University Press. p. 27. ISBN   0-19-856197-0.
5. Roylance, David (2001); "Engineering Viscoelasticity", 14-15
6. E. J. Barbero. Time-temperature-age Superposition Principle for Predicting Long-term Response of Linear Viscoelastic Materials, chapter 2 in Creep and fatigue in polymer matrix composites. Woodhead, 2011..
7. 1 2 Simulia. Abaqus Analysis User's Manual, 19.7.1 Time domain vicoelasticity, 6.10 edition, 2010
8. E. J. Barbero. Finite Element Analysis of Composite Materials. CRC Press, Boca Raton, Florida, 2007.
9. S.A. Baeurle, A. Hotta, A.A. Gusev, Polymer 47, 6243-6253 (2006).
10. Rosato, et al. (2001): "Plastics Design Handbook", 63-64.
11. Rod Lakes (1998). Viscoelastic solids. CRC Press. ISBN   0-8493-9658-1.
• Silbey and Alberty (2001): Physical Chemistry, 857. John Wiley & Sons, Inc.
• Alan S. Wineman and K. R. Rajagopal (2000): Mechanical Response of Polymers: An Introduction
• Allen and Thomas (1999): The Structure of Materials, 51.
• Crandal et al. (1999): An Introduction to the Mechanics of Solids 348
• J. Lemaitre and J. L. Chaboche (1994) Mechanics of solid materials
• Yu. Dimitrienko (2011) Nonlinear continuum mechanics and Large Inelastic Deformations, Springer, 772p