Maxwell material explained

A Maxwell material is the most simple model viscoelastic material showing properties of a typical liquid. It shows viscous flow on the long timescale, but additional elastic resistance to fast deformations.^[1] It is named for James Clerk Maxwell who proposed the model in 1867.^{[2] [3]} 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.

Definition

The Maxwell model is represented by a purely viscous damper and a purely elastic spring connected in series,^[4] 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 _{Total} and the total strain, \varepsilon _{Total} can be defined as follows:\sigma _{Total=\sigma} _{\rm} = \sigma _{\rm} \varepsilon _{Total=\varepsilon} _{\rm} + \varepsilon _{\rm}

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:

(d\varepsilon _{Total})/(dt) = (d\varepsilon _{\rm})/(dt) + (d\varepsilon _{\rm})/(dt) = (\sigma)/(η) + (1)/(E) (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:

(1)/(E) (d\sigma)/(dt) + (\sigma)/(η) = (d\varepsilon)/(dt)

or, in dot notation:

(• \sigma)/(E) + (\sigma)/(η) = • \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

If a Maxwell material is suddenly deformed and held to a strain of

\varepsilon _{0} , then the stress decays on a characteristic timescale of (η)/(E) , known as the relaxation time. The phenomenon is known as stress relaxation.

The picture shows dependence of dimensionless stress

(\sigma(t))/(E\varepsilon _{0}) upon dimensionless time (E)/(η) t :

If we free the material at time

t _{1} , then the elastic element will spring back by the value of\varepsilon _{back} =- (\sigma(t _{1)})/(E) = \varepsilon _{0} \exp \left( - (E)/(η) 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 _{irreversible} = \varepsilon _{0} \left[ 1- \exp \left( - (E)/(η) 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) = (\sigma _{0})/(E) + t (\sigma _{0})/(η)

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 _{reversible} = (\sigma _{0})/(E,) \varepsilon _{irreversible} = t _{1} (\sigma _{0})/(η .)

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

• \epsilon then the stress increases, reaching a constant value of \sigma =η • \varepsilon

In general

\sigma (t) =η • \varepsilon ( 1- e ^{-Et/η} )

Dynamic modulus

The complex dynamic modulus of a Maxwell material would be:

E ^{*(\omega)} = (1)/(1 /E - i/(\omega η )) = (E η ^{2} \omega ^{2} + i \omega E ^{2η})/(η ^{2} \omega ^{2} + E ^{2})

Thus, the components of the dynamic modulus are :

E _{1(\omega)} = (E η ^{2} \omega ^{2})/(η ^{2} \omega ^{2} + E ^{2}) = (( η /E) ^{2\omega} ^{2})/(( η /E) ^{2} \omega ^{2} +1) E = (\tau ^{2\omega} ^{2})/(\tau ^{2} \omega ^{2} +1) E

and

E _{2(\omega)} = (\omega E ^{2η})/(η ^{2} \omega ^{2} + E ^{2}) = (( η /E)\omega)/(( η /E) ^{2} \omega ^{2} +1) E = (\tau\omega)/(\tau ^{2} \omega ^{2} +1) E The picture shows relaxational spectrum for Maxwell material. The relaxation time constant is \tau \equiv η / E .

Blue curve	dimensionless elastic modulus (E _{1})/(E)
Pink curve	dimensionless modulus of losses (E _{2})/(E)
Yellow curve	dimensionless apparent viscosity (E _{2})/(\omega η)
X-axis	dimensionless frequency \omega\tau .

See also

• Burgers material
• Generalized Maxwell model
• Kelvin–Voigt material
• Oldroyd-B model
• Standard linear solid model
• Upper-convected Maxwell model

Notes and References

1. Book: Roylance, David. Engineering Viscoelasticity. 2001. Massachusetts Institute of Technology. Cambridge, MA 02139. 8–11.
2. Boyaval . Sébastien . 2021-05-01 . Viscoelastic flows of Maxwell fluids with conservation laws . ESAIM: Mathematical Modelling and Numerical Analysis . en . 55 . 3 . 807–831 . 10.1051/m2an/2020076 . 0764-583X. 2007.16075 .
3. 1867-12-31 . IV. On the dynamical theory of gases . Philosophical Transactions of the Royal Society of London . en . 157 . 49–88 . 10.1098/rstl.1867.0004 . 0261-0523.
4. Book: Christensen, R. M. Theory of Viscoelasticity. registration. 1971. Academic Press. London, W1X6BA. 16–20. 9780121742508 .