Maxwell model#Effect of a sudden deformation

{{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.{{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 }} It shows viscous flow on the long timescale, but additional elastic resistance to fast deformations.{{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}} It is named for James Clerk Maxwell who proposed the model in 1867.{{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 }}{{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 }} 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

Image:Maxwell diagram.svg

The Maxwell model is represented by a purely viscous damper and a purely elastic spring connected in series,{{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 }} as shown in the diagram. If, instead, we connect these two elements in parallel, we get the generalized model of a solid Kelvin–Voigt material.

In Maxwell configuration, under an applied axial stress, the total stress, \sigma_\mathrm{Total} and the total strain, \varepsilon_\mathrm{Total} can be defined as follows:

:\sigma_\mathrm{Total}=\sigma_{\rm D} = \sigma_{\rm S}

:\varepsilon_\mathrm{Total}=\varepsilon_{\rm D}+\varepsilon_{\rm S }

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:

:\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}

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:

:\frac {1} {E} \frac {d\sigma} {dt} + \frac {\sigma} {\eta} = \frac {d\varepsilon} {dt}

or, in dot notation:

:\frac {\dot {\sigma}} {E} + \frac {\sigma} {\eta}= \dot {\varepsilon}

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

Image:Maxwell deformation.PNG

If a Maxwell material is suddenly deformed and held to a strain of \varepsilon_0, then the stress decays on a characteristic timescale of \frac{\eta}{E}, known as the relaxation time. The phenomenon is known as stress relaxation.

The picture shows dependence of dimensionless stress \frac {\sigma(t)} {E\varepsilon_0}

upon dimensionless time \frac{E}{\eta} t:

If we free the material at time t_1, then the elastic element will spring back by the value of

:\varepsilon_\mathrm{back} = -\frac {\sigma(t_1)} E = \varepsilon_0 \exp \left(-\frac{E}{\eta} t_1\right).

Since the viscous element would not return to its original length, the irreversible component of deformation can be simplified to the expression below:

:\varepsilon_\mathrm{irreversible} = \varepsilon_0 \left[1- \exp \left(-\frac{E}{\eta} t_1\right)\right].

Effect of a sudden stress

If a Maxwell material is suddenly subjected to a stress \sigma_0, then the elastic element would suddenly deform and the viscous element would deform with a constant rate:

:\varepsilon(t) = \frac {\sigma_0} E + t \frac{\sigma_0} \eta

If at some time t_1 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:

:\varepsilon_\mathrm{reversible} = \frac {\sigma_0} E,

:\varepsilon_\mathrm{irreversible} = t_1 \frac{\sigma_0} \eta.

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 \dot{\epsilon}then the stress increases, reaching a constant value of

\sigma=\eta \dot{\varepsilon}

In general

\sigma (t)=\eta \dot{\varepsilon}(1- e^{-Et/\eta})


Dynamic modulus

Image:Maxwell relax spectra.PNG

The complex dynamic modulus of a Maxwell material would be:

: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}

Thus, the components of the dynamic modulus are :

: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

and

: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

The picture shows relaxational spectrum for Maxwell material. The relaxation time constant is \tau \equiv \eta / E .

border="1" cellspacing="0"

| Blue curve

dimensionless elastic modulus \frac {E_1} {E}
Pink curvedimensionless modulus of losses \frac {E_2} {E}
Yellow curvedimensionless apparent viscosity \frac {E_2} {\omega \eta}
X-axisdimensionless frequency \omega\tau.

See also

References