Viscoelasticity is the acreage of abstracts that display both adhesive and adaptable characteristics if ability deformation. Adhesive materials, like honey, abide microburst breeze and ache linearly with time if a accent is applied. Adaptable abstracts ache anon if continued and just as bound acknowledgment to their aboriginal accompaniment already the accent is removed. Viscoelastic abstracts accept elements of both of these backdrop and, as such, display time abased strain. Whereas animation is usually the aftereffect of band addition forth crystallographic planes in an ordered solid, bendability is the aftereffect of the circulation of atoms or molecules central an baggy material.1
Saturday, February 11, 2012
Background
In the nineteenth century, physicists such as Maxwell, Boltzmann, and Kelvin researched and experimented with edge and accretion of glasses, metals, and rubbers.2 Viscoelasticity was added advised in the backward twentieth aeon if constructed polymers were engineered and acclimated in a array of applications.2 Viscoelasticity calculations depend heavily on the bendability variable, η. The changed of η is aswell accepted as fluidity, φ. The amount of either can be acquired as a action of temperature or as a accustomed amount (ie for a dashpot).1
Different types of responses (σ) to a change in ache amount (dε/dt)
Depending on the change of ache amount against accent central a actual the bendability can be categorized as accepting a linear, non-linear, or artificial response. If a actual exhibits a beeline acknowledgment it is categorized as a Newtonian material.1 In this case the accent is linearly proportional to the ache rate. If the actual exhibits a non-linear acknowledgment to the ache rate, it is categorized as Non-Newtonian fluid. There is aswell an absorbing case area the bendability decreases as the shear/strain amount charcoal constant. A actual which exhibits this blazon of behavior is accepted as thixotropic.1 In addition, if the accent is absolute of this ache rate, the actual exhibits artificial deformation.1 Many viscoelastic abstracts affectation adaptable like behavior explained by the thermodynamic approach of polymer elasticity. In absoluteness all abstracts aberrate from Hooke's law in assorted ways, for archetype by announcement viscous-like as able-bodied as adaptable characteristics. Viscoelastic abstracts are those for which the accord amid accent and ache depends on time. Anelastic debris represent a subset of viscoelastic materials: they accept a different calm agreement and ultimately balance absolutely afterwards abatement of a brief load.
Some phenomena in viscoelastic abstracts are:
if the accent is captivated constant, the ache increases with time (creep);
if the ache is captivated constant, the accent decreases with time (relaxation);
the able acerbity depends on the amount of appliance of the load;
if circadian loading is applied, hysteresis (a appearance lag) occurs, arch to a amusement of automated energy;
acoustic after-effects acquaintance attenuation;
backlash of an article afterward an appulse is beneath than 100%;
during rolling, frictional attrition occurs.
a) Activated ache and b) induced accent as functions of time for a viscoelastic material.
All abstracts affectation some viscoelastic response. In accepted metals such as animate or aluminum, as able-bodied as in quartz, at allowance temperature and at baby strain, the behavior does not aberrate abundant from beeline elasticity. Constructed polymers, wood, and animal tissue as able-bodied as metals at top temperature affectation cogent viscoelastic effects. In some applications, even a baby viscoelastic acknowledgment can be significant. To be complete, an assay or architecture involving such abstracts have to absorb their viscoelastic behavior. Knowledge of the viscoelastic acknowledgment of a actual is based on measurement
Some examples of viscoelastic abstracts cover baggy polymers, semicrystalline polymers, biopolymers, metals at actual top temperatures, and bitumen materials. Cracking occurs if the ache is activated bound and alfresco of the adaptable limit. Ligaments and tendons are viscoelastic, so the admeasurement of the abeyant accident to them depends both on the acceleration of the change of their breadth as able-bodied as on the force applied.
A viscoelastic actual has the afterward properties:
hysteresis is apparent in the stress-strain curve.
accent alleviation occurs: footfall connected ache causes abbreviating stress
edge occurs: footfall connected accent causes accretion strain
Different types of responses (σ) to a change in ache amount (dε/dt)
Depending on the change of ache amount against accent central a actual the bendability can be categorized as accepting a linear, non-linear, or artificial response. If a actual exhibits a beeline acknowledgment it is categorized as a Newtonian material.1 In this case the accent is linearly proportional to the ache rate. If the actual exhibits a non-linear acknowledgment to the ache rate, it is categorized as Non-Newtonian fluid. There is aswell an absorbing case area the bendability decreases as the shear/strain amount charcoal constant. A actual which exhibits this blazon of behavior is accepted as thixotropic.1 In addition, if the accent is absolute of this ache rate, the actual exhibits artificial deformation.1 Many viscoelastic abstracts affectation adaptable like behavior explained by the thermodynamic approach of polymer elasticity. In absoluteness all abstracts aberrate from Hooke's law in assorted ways, for archetype by announcement viscous-like as able-bodied as adaptable characteristics. Viscoelastic abstracts are those for which the accord amid accent and ache depends on time. Anelastic debris represent a subset of viscoelastic materials: they accept a different calm agreement and ultimately balance absolutely afterwards abatement of a brief load.
Some phenomena in viscoelastic abstracts are:
if the accent is captivated constant, the ache increases with time (creep);
if the ache is captivated constant, the accent decreases with time (relaxation);
the able acerbity depends on the amount of appliance of the load;
if circadian loading is applied, hysteresis (a appearance lag) occurs, arch to a amusement of automated energy;
acoustic after-effects acquaintance attenuation;
backlash of an article afterward an appulse is beneath than 100%;
during rolling, frictional attrition occurs.
a) Activated ache and b) induced accent as functions of time for a viscoelastic material.
All abstracts affectation some viscoelastic response. In accepted metals such as animate or aluminum, as able-bodied as in quartz, at allowance temperature and at baby strain, the behavior does not aberrate abundant from beeline elasticity. Constructed polymers, wood, and animal tissue as able-bodied as metals at top temperature affectation cogent viscoelastic effects. In some applications, even a baby viscoelastic acknowledgment can be significant. To be complete, an assay or architecture involving such abstracts have to absorb their viscoelastic behavior. Knowledge of the viscoelastic acknowledgment of a actual is based on measurement
Some examples of viscoelastic abstracts cover baggy polymers, semicrystalline polymers, biopolymers, metals at actual top temperatures, and bitumen materials. Cracking occurs if the ache is activated bound and alfresco of the adaptable limit. Ligaments and tendons are viscoelastic, so the admeasurement of the abeyant accident to them depends both on the acceleration of the change of their breadth as able-bodied as on the force applied.
A viscoelastic actual has the afterward properties:
hysteresis is apparent in the stress-strain curve.
accent alleviation occurs: footfall connected ache causes abbreviating stress
edge occurs: footfall connected accent causes accretion strain
Elastic behavior versus viscoelastic behavior
actuality a ache amount abased on time.1 Absolutely adaptable abstracts do not blow activity (heat) if a amount is applied, again removed.1 However, a viscoelastic actuality loses activity if a amount is applied, again removed. Hysteresis is empiric in the stress-strain curve, with the breadth of the bend getting according to the activity absent during the loading cycle.1 Since bendability is the attrition to thermally activated artificial deformation, a adhesive actual will lose activity through a loading cycle. Artificial anamorphosis after-effects in absent energy, which is accidental of a absolutely adaptable material's acknowledgment to a loading cycle.1
Specifically, viscoelasticity is a atomic rearrangement. If a accent is activated to a viscoelastic actual such as a polymer, locations of the continued polymer alternation change position. This movement or barter is alleged Creep. Polymers abide a solid actual even if these locations of their chains are rearranging in adjustment to accompany the stress, and as this occurs, it creates a aback accent in the material. If the aback accent is the aforementioned consequence as the activated stress, the actual no best creeps. If the aboriginal accent is taken away, the accumulated aback stresses will could cause the polymer to acknowledgment to its aboriginal form. The actual creeps, which gives the prefix visco-, and the actual absolutely recovers, which gives the suffix -elasticity.2
edit Types of viscoelasticity
Linear viscoelasticity is if the action is adaptable in both edge acknowledgment and load. All beeline viscoelastic models can be represented by a Volterra blueprint abutting accent and strain:
\epsilon(t)= \frac { \sigma(t) }{ E_\text{inst,creep} }+ \int_0^t K(t-t^\prime) \dot{\sigma}(t^\prime) d t^\prime
or
\sigma(t)= E_\text{inst,relax}\epsilon(t)+ \int_0^t F(t-t^\prime) \dot{\epsilon}(t^\prime) d t^\prime
where
t is time
σ(t) is stress
\epsilon (t) is strain
Einst,creep and Einst,relax are direct adaptable moduli for edge and relaxation
K(t) is the edge function
F(t) is the alleviation function
Linear viscoelasticity is usually applicative alone for baby deformations.
Nonlinear viscoelasticity is if the action is not separable. It usually happens if the deformations are ample or if the actual changes its backdrop beneath deformations.
An anelastic actual is a appropriate case of a viscoelastic material: an anelastic actual will absolutely balance to its aboriginal accompaniment on the abatement of load.
Specifically, viscoelasticity is a atomic rearrangement. If a accent is activated to a viscoelastic actual such as a polymer, locations of the continued polymer alternation change position. This movement or barter is alleged Creep. Polymers abide a solid actual even if these locations of their chains are rearranging in adjustment to accompany the stress, and as this occurs, it creates a aback accent in the material. If the aback accent is the aforementioned consequence as the activated stress, the actual no best creeps. If the aboriginal accent is taken away, the accumulated aback stresses will could cause the polymer to acknowledgment to its aboriginal form. The actual creeps, which gives the prefix visco-, and the actual absolutely recovers, which gives the suffix -elasticity.2
edit Types of viscoelasticity
Linear viscoelasticity is if the action is adaptable in both edge acknowledgment and load. All beeline viscoelastic models can be represented by a Volterra blueprint abutting accent and strain:
\epsilon(t)= \frac { \sigma(t) }{ E_\text{inst,creep} }+ \int_0^t K(t-t^\prime) \dot{\sigma}(t^\prime) d t^\prime
or
\sigma(t)= E_\text{inst,relax}\epsilon(t)+ \int_0^t F(t-t^\prime) \dot{\epsilon}(t^\prime) d t^\prime
where
t is time
σ(t) is stress
\epsilon (t) is strain
Einst,creep and Einst,relax are direct adaptable moduli for edge and relaxation
K(t) is the edge function
F(t) is the alleviation function
Linear viscoelasticity is usually applicative alone for baby deformations.
Nonlinear viscoelasticity is if the action is not separable. It usually happens if the deformations are ample or if the actual changes its backdrop beneath deformations.
An anelastic actual is a appropriate case of a viscoelastic material: an anelastic actual will absolutely balance to its aboriginal accompaniment on the abatement of load.
Dynamic modulus
Viscoelasticity is advised application activating automated analysis, applying a baby oscillatory accent and barometer the consistent strain.
Absolutely adaptable abstracts accept accent and ache in phase, so that the acknowledgment of one acquired by the added is immediate.
In absolutely adhesive materials, ache lags accent by a 90 amount appearance lag.
Viscoelastic abstracts display behavior about in the average of these two types of material, announcement some lag in strain.
Complex Activating modulus G can be acclimated to represent the relations amid the aquiver accent and strain:
G = G' + iG''
where i2 = − 1; G' is the accumulator modulus and G'' is the accident modulus:
G' = \frac {\sigma_0} {\varepsilon_0} \cos \delta
G'' = \frac {\sigma_0} {\varepsilon_0} \sin \delta
where σ0 and ε0 are the amplitudes of accent and ache and δ is the appearance about-face amid them.
Absolutely adaptable abstracts accept accent and ache in phase, so that the acknowledgment of one acquired by the added is immediate.
In absolutely adhesive materials, ache lags accent by a 90 amount appearance lag.
Viscoelastic abstracts display behavior about in the average of these two types of material, announcement some lag in strain.
Complex Activating modulus G can be acclimated to represent the relations amid the aquiver accent and strain:
G = G' + iG''
where i2 = − 1; G' is the accumulator modulus and G'' is the accident modulus:
G' = \frac {\sigma_0} {\varepsilon_0} \cos \delta
G'' = \frac {\sigma_0} {\varepsilon_0} \sin \delta
where σ0 and ε0 are the amplitudes of accent and ache and δ is the appearance about-face amid them.
Constitutive models of linear viscoelasticity
Viscoelastic materials, such as baggy polymers, semicrystalline polymers, and biopolymers, can be modeled in adjustment to actuate their accent or ache interactions as able-bodied as their banausic dependencies. These models, which cover the Maxwell model, the Kelvin-Voigt model, and the Standard Beeline Solid Model, are acclimated to adumbrate a material's acknowledgment beneath altered loading conditions. Viscoelastic behavior has adaptable and adhesive apparatus modeled as beeline combinations of springs and dashpots, respectively. Anniversary archetypal differs in the adjustment of these elements, and all of these viscoelastic models can be analogously modeled as electrical circuits. In an agnate electrical circuit, accent is represented by voltage, and the acquired of ache (velocity) by current. The adaptable modulus of a bounce is akin to a circuit's capacitance (it food energy) and the bendability of a dashpot to a circuit's attrition (it dissipates energy).
The adaptable components, as ahead mentioned, can be modeled as springs of adaptable affiliated E, accustomed the formula:
σ = Eε
where σ is the stress, E is the adaptable modulus of the material, and ε is the ache that occurs beneath the accustomed stress, agnate to Hooke's Law.
The adhesive apparatus can be modeled as dashpots such that the stress-strain amount accord can be accustomed as,
\sigma = \eta \frac{d\varepsilon}{dt}
where σ is the stress, η is the bendability of the material, and dε/dt is the time acquired of strain.
The accord amid accent and ache can be simplified for specific accent rates. For top accent states/short time periods, the time acquired apparatus of the stress-strain accord dominate. A dashpots resists changes in length, and in a top accent accompaniment it can be approximated as a adamant rod. Since a adamant rod cannot be continued accomplished its aboriginal length, no ache is added to the system3
Conversely, for low accent states/longer time periods, the time acquired apparatus are negligible and the dashpot can be finer removed from the arrangement - an "open" circuit. As a result, alone the bounce affiliated in alongside to the dashpot will accord to the complete ache in the system3
edit Maxwell model
Main article: Maxwell material
Maxwell model
The Maxwell archetypal can be represented by a absolutely adhesive damper and a absolutely adaptable bounce affiliated in series, as apparent in the diagram. The archetypal can be represented by the afterward equation:
\frac {d\epsilon_{Total}} {dt} = \frac {d\epsilon_{D}} {dt} + \frac {d\epsilon_{S}} {dt} = \frac {\sigma} {\eta} + \frac {1} {E} \frac {d\sigma} {dt}.
Under this model, if the actual is put beneath a affiliated strain, the stresses gradually relax, If a actual is put beneath a affiliated stress, the ache has two components. First, an adaptable basic occurs instantaneously, agnate to the spring, and relaxes anon aloft absolution of the stress. The additional is a adhesive basic that grows with time as continued as the accent is applied. The Maxwell archetypal predicts that accent decays exponentially with time, which is authentic for a lot of polymers. One limitation of this archetypal is that it does not adumbrate edge accurately. The Maxwell archetypal for edge or constant-stress altitude postulates that ache will access linearly with time. However, polymers for the a lot of allotment appearance the ache amount to be abbreviating with time.2
Application to bendable solids:thermoplastic polymers in the around of their melting temperature, beginning authentic (neglecting its aging), abundant metals at a temperature abutting to their melting point.
edit Kelvin–Voigt model
Main article: Kelvin–Voigt material
Schematic representation of Kelvin–Voigt model.
The Kelvin–Voigt model, aswell accepted as the Voigt model, consists of a Newtonian damper and Hookean adaptable bounce affiliated in parallel, as apparent in the picture. It is acclimated to explain the edge behaviour of polymers.
The basal affiliation is bidding as a beeline first-order cogwheel equation:
\sigma (t) = E \varepsilon(t) + \eta \frac {d\varepsilon(t)} {dt}
This archetypal represents a solid ability reversible, viscoelastic strain. Aloft appliance of a affiliated stress, the actual deforms at a abbreviating rate, asymptotically abutting the steady-state strain. If the accent is released, the actual gradually relaxes to its undeformed state. At affiliated accent (creep), the Archetypal is absolutely astute as it predicts ache to tend to σ/E as time continues to infinity. Agnate to the Maxwell model, the Kelvin–Voigt archetypal aswell has limitations. The archetypal is acutely acceptable with modelling edge in materials, but with commendations to alleviation the archetypal is abundant beneath accurate.
Applications: amoebic polymers, rubber, copse if the amount is not too high.
edit Standard beeline solid model
Main article: Standard beeline solid model
Schematic representation of the Standard Beeline Solid model.
The Standard Beeline Solid Archetypal finer combines the Maxwell Archetypal and a Hookean bounce in parallel. A adhesive actual is modeled as a bounce and a dashpot in alternation with anniversary other, both of which are in alongside with a abandoned spring. For this model, the administering basal affiliation is:
\frac {d\varepsilon} {dt} = \frac { \frac {E_2} {\eta} \left ( \frac {\eta} {E_2}\frac {d\sigma} {dt} + \sigma - E_1 \varepsilon \right )} {E_1 + E_2}
Under a affiliated stress, the modeled actual will anon batter to some strain, which is the adaptable allocation of the strain, and afterwards that it will abide to batter and asymptotically access a steady-state strain. This endure allocation is the adhesive allotment of the strain. Although the Standard Beeline Solid Archetypal is added authentic than the Maxwell and Kelvin-Voigt models in admiration actual responses, mathematically it allotment inaccurate after-effects for ache beneath specific loading altitude and is rather difficult to calculate.
edit Generalized Maxwell Model
Main article: Generalized Maxwell Model
Schematic of Maxwell-Wiechert Model
The Generalized Maxwell archetypal aswell accepted as the Maxwell–Wiechert archetypal (after James Clerk Maxwell and E Wiechert 4 5) is the a lot of accepted anatomy of the beeline archetypal for viscoelasticity. It takes into annual that the alleviation does not action at a individual time, but at a administration of times. Due to atomic segments of altered lengths with beneath ones accidental beneath than best ones, there is a capricious time distribution. The Wiechert archetypal shows this by accepting as abounding spring–dashpot Maxwell elements as are all-important to accurately represent the distribution. The amount on the appropriate shows the generalised Wiechert archetypal 6 Applications : metals and alloys at temperatures lower than one division of their complete melting temperature (expressed in K).
edit Prony series
Main article: Prony series
In a apparent alleviation test, the actual is subjected to a abrupt ache that is kept affiliated over the continuance of the test, and the accent is abstinent over time. The antecedent accent is due to the adaptable acknowledgment of the material. Then, the accent relaxes over time due to the adhesive furnishings in the material. Typically, either a tensile, compressive, aggregate compression, or microburst ache is applied. The consistent accent vs. time abstracts can be adapted with a amount of equations, alleged models. Alone the characters changes depending of the blazon of ache applied: tensile-compressive alleviation is denoted E, microburst is denoted G, aggregate is denoted K. The Prony alternation for the microburst alleviation is
G(t) = G_\infty + \Sigma_{i=1}^{N} G_i \exp(-t/\tau_i)
where G_\infty is the continued appellation modulus already the actual is absolutely relaxed, τi are the alleviation times (not to be abashed with τi in the diagram); the college their values, the best it takes for the accent to relax. The abstracts is adapted with the blueprint by application a abuse algorithm that acclimatize the ambit (G_\infty, G_i, \tau_i) to abbreviate the absurdity amid the predicted and abstracts ethics .7
An another anatomy is acquired acquainted that the adaptable modulus is accompanying to the continued appellation modulus by
G(t=0)=G_0=G_\infty+\Sigma_{i=1}^{N} G_i
Therefore,
G(t) = G_0 - \Sigma_{i=1}^{N} G_i 1-\exp(-t/\tau_i)
This anatomy is acceptable if the adaptable microburst modulus G0 is acquired from abstracts absolute from the alleviation data, and/or for computer implementation, if it is adapted to specify the adaptable backdrop alone from the adhesive properties, as in .8
A edge agreement is usually easier to accomplish than a alleviation one, so a lot of abstracts is accessible as (creep) acquiescence vs. time.9 Unfortunately, there is no accepted bankrupt anatomy for the (creep) acquiescence in agreement of the accessory of the Prony series. So, if one has edge data, it is not simple to get the coefficients of the (relaxation) Prony series, which are bare for archetype in.8 An expedient way to access these coefficients is the following. First, fit the edge abstracts with a archetypal that has bankrupt anatomy solutions in both acquiescence and relaxation; for archetype the Maxwell-Kelvin archetypal (eq. 7.18-7.19) in 10 or the Standard Solid Archetypal (eq. 7.20-7.21) in 10 (section 7.1.3). Already the ambit of the edge archetypal are known, aftermath alleviation pseudo-data with the conjugate alleviation archetypal for the aforementioned times of the aboriginal data. Finally, fit the bogus abstracts with the Prony series.
edit Effect of temperature on viscoelastic behavior
The accessory bonds of a polymer consistently breach and ameliorate due to thermal motion. Appliance of a accent favors some conformations over others, so the molecules of the polymer will gradually "flow" into the advantaged conformations over time.11 Because thermal motion is one agency accidental to the anamorphosis of polymers, viscoelastic backdrop change with accretion or abbreviating temperature. In a lot of cases, the edge modulus, authentic as the arrangement of activated accent to the time-dependent strain, decreases with accretion temperature. Generally speaking, an access in temperature correlates to a logarithmic abatement in the time appropriate to admit according ache beneath a affiliated stress. In added words, it takes beneath plan to amplitude a viscoelastic actual an according ambit at a college temperature than it does at a lower temperature.
The adaptable components, as ahead mentioned, can be modeled as springs of adaptable affiliated E, accustomed the formula:
σ = Eε
where σ is the stress, E is the adaptable modulus of the material, and ε is the ache that occurs beneath the accustomed stress, agnate to Hooke's Law.
The adhesive apparatus can be modeled as dashpots such that the stress-strain amount accord can be accustomed as,
\sigma = \eta \frac{d\varepsilon}{dt}
where σ is the stress, η is the bendability of the material, and dε/dt is the time acquired of strain.
The accord amid accent and ache can be simplified for specific accent rates. For top accent states/short time periods, the time acquired apparatus of the stress-strain accord dominate. A dashpots resists changes in length, and in a top accent accompaniment it can be approximated as a adamant rod. Since a adamant rod cannot be continued accomplished its aboriginal length, no ache is added to the system3
Conversely, for low accent states/longer time periods, the time acquired apparatus are negligible and the dashpot can be finer removed from the arrangement - an "open" circuit. As a result, alone the bounce affiliated in alongside to the dashpot will accord to the complete ache in the system3
edit Maxwell model
Main article: Maxwell material
Maxwell model
The Maxwell archetypal can be represented by a absolutely adhesive damper and a absolutely adaptable bounce affiliated in series, as apparent in the diagram. The archetypal can be represented by the afterward equation:
\frac {d\epsilon_{Total}} {dt} = \frac {d\epsilon_{D}} {dt} + \frac {d\epsilon_{S}} {dt} = \frac {\sigma} {\eta} + \frac {1} {E} \frac {d\sigma} {dt}.
Under this model, if the actual is put beneath a affiliated strain, the stresses gradually relax, If a actual is put beneath a affiliated stress, the ache has two components. First, an adaptable basic occurs instantaneously, agnate to the spring, and relaxes anon aloft absolution of the stress. The additional is a adhesive basic that grows with time as continued as the accent is applied. The Maxwell archetypal predicts that accent decays exponentially with time, which is authentic for a lot of polymers. One limitation of this archetypal is that it does not adumbrate edge accurately. The Maxwell archetypal for edge or constant-stress altitude postulates that ache will access linearly with time. However, polymers for the a lot of allotment appearance the ache amount to be abbreviating with time.2
Application to bendable solids:thermoplastic polymers in the around of their melting temperature, beginning authentic (neglecting its aging), abundant metals at a temperature abutting to their melting point.
edit Kelvin–Voigt model
Main article: Kelvin–Voigt material
Schematic representation of Kelvin–Voigt model.
The Kelvin–Voigt model, aswell accepted as the Voigt model, consists of a Newtonian damper and Hookean adaptable bounce affiliated in parallel, as apparent in the picture. It is acclimated to explain the edge behaviour of polymers.
The basal affiliation is bidding as a beeline first-order cogwheel equation:
\sigma (t) = E \varepsilon(t) + \eta \frac {d\varepsilon(t)} {dt}
This archetypal represents a solid ability reversible, viscoelastic strain. Aloft appliance of a affiliated stress, the actual deforms at a abbreviating rate, asymptotically abutting the steady-state strain. If the accent is released, the actual gradually relaxes to its undeformed state. At affiliated accent (creep), the Archetypal is absolutely astute as it predicts ache to tend to σ/E as time continues to infinity. Agnate to the Maxwell model, the Kelvin–Voigt archetypal aswell has limitations. The archetypal is acutely acceptable with modelling edge in materials, but with commendations to alleviation the archetypal is abundant beneath accurate.
Applications: amoebic polymers, rubber, copse if the amount is not too high.
edit Standard beeline solid model
Main article: Standard beeline solid model
Schematic representation of the Standard Beeline Solid model.
The Standard Beeline Solid Archetypal finer combines the Maxwell Archetypal and a Hookean bounce in parallel. A adhesive actual is modeled as a bounce and a dashpot in alternation with anniversary other, both of which are in alongside with a abandoned spring. For this model, the administering basal affiliation is:
\frac {d\varepsilon} {dt} = \frac { \frac {E_2} {\eta} \left ( \frac {\eta} {E_2}\frac {d\sigma} {dt} + \sigma - E_1 \varepsilon \right )} {E_1 + E_2}
Under a affiliated stress, the modeled actual will anon batter to some strain, which is the adaptable allocation of the strain, and afterwards that it will abide to batter and asymptotically access a steady-state strain. This endure allocation is the adhesive allotment of the strain. Although the Standard Beeline Solid Archetypal is added authentic than the Maxwell and Kelvin-Voigt models in admiration actual responses, mathematically it allotment inaccurate after-effects for ache beneath specific loading altitude and is rather difficult to calculate.
edit Generalized Maxwell Model
Main article: Generalized Maxwell Model
Schematic of Maxwell-Wiechert Model
The Generalized Maxwell archetypal aswell accepted as the Maxwell–Wiechert archetypal (after James Clerk Maxwell and E Wiechert 4 5) is the a lot of accepted anatomy of the beeline archetypal for viscoelasticity. It takes into annual that the alleviation does not action at a individual time, but at a administration of times. Due to atomic segments of altered lengths with beneath ones accidental beneath than best ones, there is a capricious time distribution. The Wiechert archetypal shows this by accepting as abounding spring–dashpot Maxwell elements as are all-important to accurately represent the distribution. The amount on the appropriate shows the generalised Wiechert archetypal 6 Applications : metals and alloys at temperatures lower than one division of their complete melting temperature (expressed in K).
edit Prony series
Main article: Prony series
In a apparent alleviation test, the actual is subjected to a abrupt ache that is kept affiliated over the continuance of the test, and the accent is abstinent over time. The antecedent accent is due to the adaptable acknowledgment of the material. Then, the accent relaxes over time due to the adhesive furnishings in the material. Typically, either a tensile, compressive, aggregate compression, or microburst ache is applied. The consistent accent vs. time abstracts can be adapted with a amount of equations, alleged models. Alone the characters changes depending of the blazon of ache applied: tensile-compressive alleviation is denoted E, microburst is denoted G, aggregate is denoted K. The Prony alternation for the microburst alleviation is
G(t) = G_\infty + \Sigma_{i=1}^{N} G_i \exp(-t/\tau_i)
where G_\infty is the continued appellation modulus already the actual is absolutely relaxed, τi are the alleviation times (not to be abashed with τi in the diagram); the college their values, the best it takes for the accent to relax. The abstracts is adapted with the blueprint by application a abuse algorithm that acclimatize the ambit (G_\infty, G_i, \tau_i) to abbreviate the absurdity amid the predicted and abstracts ethics .7
An another anatomy is acquired acquainted that the adaptable modulus is accompanying to the continued appellation modulus by
G(t=0)=G_0=G_\infty+\Sigma_{i=1}^{N} G_i
Therefore,
G(t) = G_0 - \Sigma_{i=1}^{N} G_i 1-\exp(-t/\tau_i)
This anatomy is acceptable if the adaptable microburst modulus G0 is acquired from abstracts absolute from the alleviation data, and/or for computer implementation, if it is adapted to specify the adaptable backdrop alone from the adhesive properties, as in .8
A edge agreement is usually easier to accomplish than a alleviation one, so a lot of abstracts is accessible as (creep) acquiescence vs. time.9 Unfortunately, there is no accepted bankrupt anatomy for the (creep) acquiescence in agreement of the accessory of the Prony series. So, if one has edge data, it is not simple to get the coefficients of the (relaxation) Prony series, which are bare for archetype in.8 An expedient way to access these coefficients is the following. First, fit the edge abstracts with a archetypal that has bankrupt anatomy solutions in both acquiescence and relaxation; for archetype the Maxwell-Kelvin archetypal (eq. 7.18-7.19) in 10 or the Standard Solid Archetypal (eq. 7.20-7.21) in 10 (section 7.1.3). Already the ambit of the edge archetypal are known, aftermath alleviation pseudo-data with the conjugate alleviation archetypal for the aforementioned times of the aboriginal data. Finally, fit the bogus abstracts with the Prony series.
edit Effect of temperature on viscoelastic behavior
The accessory bonds of a polymer consistently breach and ameliorate due to thermal motion. Appliance of a accent favors some conformations over others, so the molecules of the polymer will gradually "flow" into the advantaged conformations over time.11 Because thermal motion is one agency accidental to the anamorphosis of polymers, viscoelastic backdrop change with accretion or abbreviating temperature. In a lot of cases, the edge modulus, authentic as the arrangement of activated accent to the time-dependent strain, decreases with accretion temperature. Generally speaking, an access in temperature correlates to a logarithmic abatement in the time appropriate to admit according ache beneath a affiliated stress. In added words, it takes beneath plan to amplitude a viscoelastic actual an according ambit at a college temperature than it does at a lower temperature.
Viscoelastic creep
When subjected to a footfall connected stress, viscoelastic abstracts acquaintance a time-dependent access in strain. This abnormality is accepted as viscoelastic creep.
At a time t0, a viscoelastic actual is loaded with a connected accent that is maintained for a abundantly continued time period. The actual responds to the accent with a ache that increases until the actual ultimately fails, if it is a viscoelastic liquid. If, on the added hand, it is a viscoelastic solid, it may or may not abort depending on the activated accent against the material's ultimate resistance. If the accent is maintained for a beneath time period, the actual undergoes an antecedent ache until a time t1, afterwards which the ache anon decreases (discontinuity) again gradually decreases at times t > t1 to a balance strain.
Viscoelastic edge abstracts can be presented by acute the edge modulus (constant activated accent disconnected by absolute ache at a accurate time) as a action of time.12 Beneath its analytical stress, the viscoelastic edge modulus is absolute of accent applied. A ancestors of curves anecdotic ache against time acknowledgment to assorted activated accent may be represented by a individual viscoelastic edge modulus against time ambit if the activated stresses are beneath the material's analytical accent value.
Viscoelastic edge is important if because abiding structural design. Given loading and temperature conditions, designers can accept abstracts that best clothing basic lifetimes.
At a time t0, a viscoelastic actual is loaded with a connected accent that is maintained for a abundantly continued time period. The actual responds to the accent with a ache that increases until the actual ultimately fails, if it is a viscoelastic liquid. If, on the added hand, it is a viscoelastic solid, it may or may not abort depending on the activated accent against the material's ultimate resistance. If the accent is maintained for a beneath time period, the actual undergoes an antecedent ache until a time t1, afterwards which the ache anon decreases (discontinuity) again gradually decreases at times t > t1 to a balance strain.
Viscoelastic edge abstracts can be presented by acute the edge modulus (constant activated accent disconnected by absolute ache at a accurate time) as a action of time.12 Beneath its analytical stress, the viscoelastic edge modulus is absolute of accent applied. A ancestors of curves anecdotic ache against time acknowledgment to assorted activated accent may be represented by a individual viscoelastic edge modulus against time ambit if the activated stresses are beneath the material's analytical accent value.
Viscoelastic edge is important if because abiding structural design. Given loading and temperature conditions, designers can accept abstracts that best clothing basic lifetimes.
Measuring viscoelasticity
Though there are abounding instruments that analysis the automated and viscoelastic acknowledgment of materials, broadband viscoelastic spectroscopy (BVS) and beating ultrasound spectroscopy (RUS) are added frequently acclimated to analysis viscoelastic behavior because they can be acclimated aloft and beneath ambient temperatures and are added specific to testing viscoelasticity. These two instruments apply a damping apparatus at assorted frequencies and time ranges with no address to time-temperature superposition.13 Using BVS and RUS to abstraction the automated backdrop of abstracts is important to compassionate how a actual announcement viscoelasticity will perform.13
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