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.
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