{{Short description|Model of viscoelastic material}} {{Use dmy dates|date=November 2017}}
A '''Maxwell model''' is the most simple model viscoelastic material showing properties of a typical liquid.<ref>{{cite book |doi=10.1016/bs.aams.2022.09.003 |quote=The Maxwell model is a type of simplest and basic mathematical model to describe the mechanics characteristic of viscoelastic solid material. |chapter=Mechanics constitutive models for viscoelastic solid materials: Development and a critical review |title=Advances in Applied Mechanics |date=2023 |last1=Zhou |first1=Xiaoqiang |last2=Yu |first2=Daoyuan |last3=Barrera |first3=Olga |volume=56 |pages=189–321 |isbn=978-0-323-99248-0 }}</ref> It shows viscous flow on the long timescale, but additional elastic resistance to fast deformations.<ref name=roylance_EV>{{cite report |last1=Roylance |first1=David |date=24 October 2001 |title=Engineering Viscoelasticity |pages=8–11 |url=http://web.mit.edu/course/3/3.11/www/modules/visco.pdf }}{{self-published inline|date=May 2025}}</ref> It is named for James Clerk Maxwell who proposed the model in 1867.<ref>{{cite journal |last1=Boyaval |first1=Sébastien |title=Viscoelastic flows of Maxwell fluids with conservation laws |journal=ESAIM: Mathematical Modelling and Numerical Analysis |date=May 2021 |volume=55 |issue=3 |pages=807–831 |doi=10.1051/m2an/2020076 |arxiv=2007.16075 }}</ref><ref>{{cite journal |title=IV. On the dynamical theory of gases |journal=Philosophical Transactions of the Royal Society of London |date=31 December 1867 |volume=157 |pages=49–88 |doi=10.1098/rstl.1867.0004 }}</ref> It is also known as a Maxwell fluid. A generalization of the scalar relation to a tensor equation lacks motivation from more microscopic models and does not comply with the concept of material objectivity. However, these criteria are fulfilled by the Upper-convected Maxwell model.{{fact|date=May 2025}}
== Definition ==
thumb|300x300px|Diagram of a Maxwell material
The Maxwell model is represented by a purely viscous damper and a purely elastic spring connected in series,<ref name="christensen">{{cite book |doi=10.1016/B978-0-12-174252-2.50005-3 |chapter=Viscoelastic Stress Strain Constitutive Relations |title=Theory of Viscoelasticity |date=1982 |last1=Christensen |first1=R.M. |pages=1–34 |isbn=978-0-12-174252-2 }}</ref> as shown in the diagram. If, instead, we connect these two elements in parallel,<ref name="christensen" /> we get the generalized model of a solid Kelvin–Voigt material.
In Maxwell configuration, under an applied axial stress, the total stress, <math>\sigma_\mathrm{Total}</math> and the total strain, <math>\varepsilon_\mathrm{Total}</math> can be defined as follows:<ref name="roylance_EV" />
:<math>\sigma_\mathrm{Total}=\sigma_{\rm D} = \sigma_{\rm S}</math> :<math>\varepsilon_\mathrm{Total}=\varepsilon_{\rm D}+\varepsilon_{\rm S }</math>
where the subscript D indicates the stress–strain in the damper and the subscript S indicates the stress–strain in the spring. Taking the derivative of strain with respect to time, we obtain:
:<math>\frac {d\varepsilon_\mathrm{Total}} {dt} = \frac {d\varepsilon_{\rm D}} {dt} + \frac {d\varepsilon_{\rm S}} {dt} = \frac {\sigma} {\eta} + \frac {1} {E} \frac {d\sigma} {dt}</math>
where ''E'' is the elastic modulus and ''η'' is the material coefficient of viscosity. This model describes the damper as a Newtonian fluid and models the spring with Hooke's law.
In a Maxwell material, stress ''σ'', strain ''ε'' and their rates of change with respect to time ''t'' are governed by equations of the form:<ref name=roylance_EV />
:<math>\frac {1} {E} \frac {d\sigma} {dt} + \frac {\sigma} {\eta} = \frac {d\varepsilon} {dt}</math>
or, in dot notation:
:<math>\frac {\dot {\sigma}} {E} + \frac {\sigma} {\eta}= \dot {\varepsilon}</math>
The equation can be applied either to the shear stress or to the uniform tension in a material. In the former case, the viscosity corresponds to that for a Newtonian fluid. In the latter case, it has a slightly different meaning relating stress and rate of strain.
The model is usually applied to the case of small deformations. For the large deformations we should include some geometrical non-linearity. For the simplest way of generalizing the Maxwell model, refer to the upper-convected Maxwell model.
== Effect of a sudden deformation == <!-- Deleted image removed: right|thumb|400px|Dependence of dimensionless stress upon dimensionless time under constant strain -->
If a Maxwell material is suddenly deformed and held to a strain of <math>\varepsilon_0</math>, then the stress decays on a characteristic timescale of <math>\frac{\eta}{E}</math>, known as the relaxation time. The phenomenon is known as stress relaxation.
The picture shows dependence of dimensionless stress <math>\frac {\sigma(t)} {E\varepsilon_0} </math> upon dimensionless time <math>\frac{E}{\eta} t</math>:
If we free the material at time <math>t_1</math>, then the elastic element will spring back by the value of
:<math>\varepsilon_\mathrm{back} = -\frac {\sigma(t_1)} E = \varepsilon_0 \exp \left(-\frac{E}{\eta} t_1\right). </math>
Since the viscous element would not return to its original length, the irreversible component of deformation can be simplified to the expression below:
:<math>\varepsilon_\mathrm{irreversible} = \varepsilon_0 \left[1- \exp \left(-\frac{E}{\eta} t_1\right)\right]. </math>
==Effect of a sudden stress ==
If a Maxwell material is suddenly subjected to a stress <math>\sigma_0</math>, then the elastic element would suddenly deform and the viscous element would deform with a constant rate:
:<math>\varepsilon(t) = \frac {\sigma_0} E + t \frac{\sigma_0} \eta </math>
If at some time <math>t_1</math> we released the material, then the deformation of the elastic element would be the spring-back deformation and the deformation of the viscous element would not change:
:<math>\varepsilon_\mathrm{reversible} = \frac {\sigma_0} E, </math>
:<math>\varepsilon_\mathrm{irreversible} = t_1 \frac{\sigma_0} \eta. </math>
The Maxwell model does not exhibit creep since it models strain as linear function of time.
If a small stress is applied for a sufficiently long time, then the irreversible strains become large. Thus, Maxwell material is a type of liquid.
== Effect of a constant strain rate == If a Maxwell material is subject to a constant strain rate <math>\dot{\epsilon}</math>then the stress increases, reaching a constant value of
<math>\sigma=\eta \dot{\varepsilon} </math>
In general
<math>\sigma (t)=\eta \dot{\varepsilon}(1- e^{-Et/\eta}) </math> <br />
== Dynamic modulus == thumb|right|400px|Relaxational spectrum for Maxwell material The complex dynamic modulus of a Maxwell material would be:
:<math>E^*(\omega) = \frac 1 {1/E - i/(\omega \eta) } = \frac {E\eta^2 \omega^2 +i \omega E^2\eta} {\eta^2 \omega^2 + E^2} </math>
Thus, the components of the dynamic modulus are :
:<math>E_1(\omega) = \frac {E\eta^2 \omega^2 } {\eta^2 \omega^2 + E^2} = \frac {(\eta/E)^2\omega^2} {(\eta/E)^2 \omega^2 + 1} E = \frac {\tau^2\omega^2} {\tau^2 \omega^2 + 1} E </math>
and
:<math>E_2(\omega) = \frac {\omega E^2\eta} {\eta^2 \omega^2 + E^2} = \frac {(\eta/E)\omega} {(\eta/E)^2 \omega^2 + 1} E = \frac {\tau\omega} {\tau^2 \omega^2 + 1} E </math> The picture shows relaxational spectrum for Maxwell material. The relaxation time constant is <math> \tau \equiv \eta / E </math>. {| border="1" cellspacing="0" | Blue curve || dimensionless elastic modulus <math>\frac {E_1} {E}</math> |- | Pink curve || dimensionless modulus of losses <math>\frac {E_2} {E}</math> |- | Yellow curve || dimensionless apparent viscosity <math>\frac {E_2} {\omega \eta}</math> |- | X-axis || dimensionless frequency <math> \omega\tau</math>. |}
==See also== *Burgers material *Generalized Maxwell model *Kelvin–Voigt material *Oldroyd-B model *Standard linear solid model *Upper-convected Maxwell model
==References== {{reflist}}
{{DEFAULTSORT:Maxwell Material}} Category:Non-Newtonian fluids Category:Materials science Category:James Clerk Maxwell