In
materials science and
continuum mechanics
Continuum mechanics is a branch of mechanics that deals with the mechanical behavior of materials modeled as a continuous mass rather than as discrete particles. The French mathematician Augustin-Louis Cauchy was the first to formulate such m ...
, viscoelasticity is the property of
materials
Material is a substance or mixture of substances that constitutes an object. Materials can be pure or impure, living or non-living matter. Materials can be classified on the basis of their physical and chemical properties, or on their geologic ...
that exhibit both
viscous
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.
Viscosity quantifies the inte ...
and
elastic
Elastic is a word often used to describe or identify certain types of elastomer, elastic used in garments or stretchable fabrics.
Elastic may also refer to:
Alternative name
* Rubber band, ring-shaped band of rubber used to hold objects togeth ...
characteristics when undergoing
deformation
Deformation can refer to:
* Deformation (engineering), changes in an object's shape or form due to the application of a force or forces.
** Deformation (physics), such changes considered and analyzed as displacements of continuum bodies.
* Defor ...
. Viscous materials, like water, resist
shear flow The term shear flow is used in solid mechanics as well as in fluid dynamics. The expression ''shear flow'' is used to indicate:
* a shear stress over a distance in a thin-walled structure (in solid mechanics);Higdon, Ohlsen, Stiles and Weese (1960) ...
and
strain
Strain may refer to:
Science and technology
* Strain (biology), variants of plants, viruses or bacteria; or an inbred animal used for experimental purposes
* Strain (chemistry), a chemical stress of a molecule
* Strain (injury), an injury to a mu ...
linearly with time when a
stress
Stress may refer to:
Science and medicine
* Stress (biology), an organism's response to a stressor such as an environmental condition
* Stress (linguistics), relative emphasis or prominence given to a syllable in a word, or to a word in a phrase ...
is applied. Elastic materials strain when stretched and immediately return to their original state once the stress is removed.
Viscoelastic materials have elements of both of these properties and, as such, exhibit time-dependent strain. Whereas elasticity is usually the result of
bond
Bond or bonds may refer to:
Common meanings
* Bond (finance), a type of debt security
* Bail bond, a commercial third-party guarantor of surety bonds in the United States
* Chemical bond, the attraction of atoms, ions or molecules to form chemica ...
stretching along
crystallographic planes in an ordered solid, viscosity is the result of the diffusion of atoms or molecules inside an
amorphous
In condensed matter physics and materials science, an amorphous solid (or non-crystalline solid, glassy solid) is a solid that lacks the long-range order that is characteristic of a crystal.
Etymology
The term comes from the Greek ''a'' ("wi ...
material.
[Meyers and Chawla (1999): "Mechanical Behavior of Materials", 98-103.]
Background
In the nineteenth century, physicists such as
Maxwell,
Boltzmann
Ludwig Eduard Boltzmann (; 20 February 1844 – 5 September 1906) was an Austrian physicist and philosopher. His greatest achievements were the development of statistical mechanics, and the statistical explanation of the second law of thermodyn ...
, and
Kelvin
The kelvin, symbol K, is the primary unit of temperature in the International System of Units (SI), used alongside its prefixed forms and the degree Celsius. It is named after the Belfast-born and University of Glasgow-based engineer and phy ...
researched and experimented with
creep and recovery of
glass
Glass is a non-crystalline, often transparent, amorphous solid that has widespread practical, technological, and decorative use in, for example, window panes, tableware, and optics. Glass is most often formed by rapid cooling (quenching) of ...
es,
metal
A metal (from Greek μέταλλον ''métallon'', "mine, quarry, metal") is a material that, when freshly prepared, polished, or fractured, shows a lustrous appearance, and conducts electricity and heat relatively well. Metals are typicall ...
s, and
rubber
Rubber, also called India rubber, latex, Amazonian rubber, ''caucho'', or ''caoutchouc'', as initially produced, consists of polymers of the organic compound isoprene, with minor impurities of other organic compounds. Thailand, Malaysia, and ...
s. 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
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.
Viscosity quantifies the inte ...
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
A dashpot, also known as a damper, is a mechanical device that resists motion via viscous friction. The resulting force is proportional to the velocity, but acts in the opposite direction, slowing the motion and absorbing energy. It is commonly us ...
).
[
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 With regard to materials science, a material is said to be "Newtonian" if it exhibits a linear relationship between stress and strain rate.
See also
*Stress
* Strain
*Classical mechanics
Classical mechanics is a physical theory describing ...
. 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
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 ex ...
. 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
Thixotropy is a time-dependent shear thinning property. Certain gels or fluids that are thick or viscous under static conditions will flow (become thinner, less viscous) over time when shaken, agitated, shear-stressed, or otherwise stressed ( ...
. In addition, when the stress is independent of this strain rate, the material exhibits plastic deformation.[ Many viscoelastic materials exhibit ]rubber
Rubber, also called India rubber, latex, Amazonian rubber, ''caucho'', or ''caoutchouc'', as initially produced, consists of polymers of the organic compound isoprene, with minor impurities of other organic compounds. Thailand, Malaysia, and ...
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. Ligament
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:
* Peritoneal li ...
s and tendon
A tendon or sinew is a tough, high-tensile-strength band of dense fibrous connective tissue that connects muscle to bone. It is able to transmit the mechanical forces of muscle contraction to the skeletal system without sacrificing its ability ...
s 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 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 hence causing some ...
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
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.
Viscosity quantifies the inte ...
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 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.[
Specifically, viscoelasticity is a molecular rearrangement. When a stress is applied to a viscoelastic material such as a ]polymer
A polymer (; Greek '' poly-'', "many" + ''-mer'', "part")
is a substance or material consisting of very large molecules called macromolecules, composed of many repeating subunits. Due to their broad spectrum of properties, both synthetic a ...
, 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.[
]
Linear viscoelasticity and Nonlinear viscoelasticity
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 In mathematics, the Volterra integral equations are a special type of integral equations. They are divided into two groups referred to as the first and the second kind.
A linear Volterra equation of the first kind is
: f(t) = \int_a^t K(t,s)\,x(s) ...
connecting stress
Stress may refer to:
Science and medicine
* Stress (biology), an organism's response to a stressor such as an environmental condition
* Stress (linguistics), relative emphasis or prominence given to a syllable in a word, or to a word in a phrase ...
and strain
Strain may refer to:
Science and technology
* Strain (biology), variants of plants, viruses or bacteria; or an inbred animal used for experimental purposes
* Strain (chemistry), a chemical stress of a molecule
* Strain (injury), an injury to a mu ...
:
or
where
* is time
* is stress
Stress may refer to:
Science and medicine
* Stress (biology), an organism's response to a stressor such as an environmental condition
* Stress (linguistics), relative emphasis or prominence given to a syllable in a word, or to a word in a phrase ...
* is strain
Strain may refer to:
Science and technology
* Strain (biology), variants of plants, viruses or bacteria; or an inbred animal used for experimental purposes
* Strain (chemistry), a chemical stress of a molecule
* Strain (injury), an injury to a mu ...
* and are instantaneous elastic moduli
An elastic modulus (also known as modulus of elasticity) is the unit of measurement of an object's or substance's resistance to being deformed elastically (i.e., non-permanently) when a stress is applied to it. The elastic modulus of an object is ...
for creep and relaxation
* is the creep function
* is the relaxation function
Linear viscoelasticity is usually applicable only for small deformation
Deformation can refer to:
* Deformation (engineering), changes in an object's shape or form due to the application of a force or forces.
** Deformation (physics), such changes considered and analyzed as displacements of continuum bodies.
* Defor ...
s.
Nonlinear viscoelasticity is when the function is not separable. It usually happens when the deformation
Deformation can refer to:
* Deformation (engineering), changes in an object's shape or form due to the application of a force or forces.
** Deformation (physics), such changes considered and analyzed as displacements of continuum bodies.
* Defor ...
s 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 will fully recover 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
Dynamic mechanical analysis (abbreviated DMA) 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, ...
, 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 Dynamic modulus (sometimes complex modulusThe Open University (UK), 2000. ''T838 Design and Manufacture with Polymers: Solid properties and design'', page 30. Milton Keynes: The Open University.) is the ratio of stress to strain under ''vibratory c ...
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 A Burgers material is a viscoelastic material having the properties both of elasticity and viscosity. It is named after the Dutch physicist Johannes Martinus Burgers.
Overview Maxwell representation
Given that one Maxwell material has an elas ...
, 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
In materials science, strain rate is the change in strain ( deformation) of a material with respect to time.
The strain rate at some point within the material measures the rate at which the distances of adjacent parcels of the material change ...
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
In physics, Hooke's law is an empirical law which states that the force () needed to extend or compress a spring by some distance () scales linearly with respect to that distance—that is, where is a constant factor characteristic of ...
.
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"
/ref>
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.[
]
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
Relax may refer to:
Aviation
* Roland Z-120 Relax, a German ultralight aircraft design for the 120 kg class
Music Albums
* ''Relax'' (Blank & Jones album), 2003
* ''Relax'' (Das Racist album), 2011
Songs
* "Relax" (song), a 1983 song by Fran ...
. 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.[
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.
]
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:
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.
This model can be applied to organic polymers, rubber, and 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:
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 consists of either two Maxwell components in parallel or a Kelvin–Voigt component, a spring and a dashpot in series. For this model, the governing constitutive relations are:
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 necessary to accurately represent the distribution. The figure on the right shows the generalised Wiechert model.[Roylance, David (2001); "Engineering Viscoelasticity", 14–15]
Applications: metals and alloys at temperatures lower than one quarter of their absolute melting temperature (expressed in K).
Constitutive models for nonlinear viscoelasticity
Non-linear viscoelastic constitutive equations are needed to quantitatively account for phenomena in fluids like differences in normal stresses, shear thinning, and extensional thickening. Necessarily, the history experienced by the material is needed to account for time-dependent behavior, and is typically included in models as a history kernel K.
Second-order fluid
The second-order fluid is typically considered the simplest nonlinear viscoelastic model, and typically occurs in a narrow region of materials behavior occurring at high strain amplitudes and Deborah number between Newtonian fluids and other more complicated nonlinear viscoelastic fluids. The second-order fluid constitutive equation is given by:
where:
* is the identity tensor
* is the deformation tensor
* denote viscosity, and first and second normal stress coefficients, respectively
* denotes the upper-convected derivative of the deformation tensor where and is the material time derivative of the deformation tensor.
Upper-convected Maxwell model
The upper-convected Maxwell model incorporates nonlinear time behavior into the viscoelastic Maxwell model, given by:
where denotes the stress tensor.
Oldroyd-B model
The Oldroyd-B model is an extension of the Upper Convected Maxwell model The upper-convected Maxwell (UCM) model is a generalisation of the Maxwell material for the case of large deformations using the upper-convected time derivative. The model was proposed by James G. Oldroyd. The concept is named after James Clerk M ...
and is interpreted as a solvent filled with elastic bead and spring dumbbells.
The model is named after its creator James G. Oldroyd
James Gardner Oldroyd (25 April 1921 – 22 November 1982''The Times'' November 25, 1982 page 26 "Deaths") was a British mathematician and noted rheologist. He formulated the Oldroyd-B model to describe the viscoelastic behaviour of non-Newtonia ...
.
The model can be written as:
where:
* is the stress
Stress may refer to:
Science and medicine
* Stress (biology), an organism's response to a stressor such as an environmental condition
* Stress (linguistics), relative emphasis or prominence given to a syllable in a word, or to a word in a phrase ...
tensor
In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects related to a vector space. Tensors may map between different objects such as vectors, scalars, and even other tenso ...
;
* is the relaxation time;
* is the retardation time = ;
* is the Upper convected time derivative In continuum mechanics, including fluid dynamics, an upper-convected time derivative or Oldroyd derivative, named after James G. Oldroyd, is the rate of change of some tensor property of a small parcel of fluid that is written in the coordinate sy ...
of stress tensor:
* is the fluid velocity;
* is the total viscosity
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.
Viscosity quantifies the inte ...
composed of solvent and polymer components, ;
* is the deformation rate tensor or rate of strain tensor,