MAXWELL Model

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MAXWELL model

Mechanical analogy models

The linear-viscoelastic behaviour of plastics can be approximated in the model by the mathematical combination of linear-elastic and linear-viscous processes (see also: deformation). In mechanics, mechanical or electrical analogy models are used for a better description. A spring is used for the elastic behaviour and a damper for the viscous behaviour (Figure 1).

However, linear viscoelasticity is only precisely defined for the range of infinitesimally small stresses. The material properties therefore depend only on time and not on the level of mechanical stress.

Spring and damper model of linear viscosity

The elastic component (HOOKE spring: Fig. 1a) induces spontaneous, limited, reversible deformation (see also: HOOKE's law), while the viscous component (NEWTON' s damper: Fig. 1b) generally causes time-dependent, unlimited, irreversible deformation. The viscous and elastic components vary in strength in different viscoelastic plastics, and the way in which the two components interact also differs. The viscoelastic material behaviour can therefore be modelled by combining two or more of these elements [1].

Fig. 1:

Spring model (a) and damper model (b) of linear viscoelasticity

When the spring is loaded, it spontaneously elongates and returns to its original length without delay when the load is removed. With linear energy-elastic behaviour between spring force and elongation or stress and strain, a complete description is possible using HOOKE's law (Eq. (1)).

$\sigma =E\cdot \varepsilon$

(1)

The damper for NEWTONIAN fluids describes purely viscous behaviour, which depends significantly on the viscosity η (Eq. (2)). At constant viscosity, the resulting stress at the damper is determined solely by the deformation speed or the strain rate dε/dt.

$\sigma =\eta \cdot {\frac {{\text{d}}\varepsilon }{{\text{d}}t}}$

(2)

The MAXWELL model

The series connection of the two elements (spring and damper) results in the MAXWELL model (Fig. 2). When loaded, the spring deforms immediately, followed by time-dependent and unlimited viscous deformation, which depends on viscosity. After the load is removed, only the spring moves back and the viscous component remains. As a result, there is a time-dependent, unlimited, irreversible deformation as with a liquid, but there is also a time-independent and reversible spontaneous elastic component as with a solid. The MAXWELL model represents the simplest model approach for describing the relaxation behaviour of plastics [2], which is based on the addition of elastic and viscous deformation components (see: deformation) (Eq. (3)).

Fig. 2:

Deformation behaviour of the MAXWELL model (relaxation behaviour)

${\frac {{\text{d}}\varepsilon }{{\text{d}}t}}={\frac {{\text{d}}\varepsilon _{1}}{{\text{d}}t}}+{\frac {{\text{d}}\varepsilon _{2}}{{\text{d}}t}}$

(3)

If the time derivative of Eq. (1) and Eq. (2) is substituted into Eq. (3), the mathematical description for the MAXWELL body (Eq. 4) is obtained.

$\sigma =\eta \cdot {\frac {{\text{d}}\varepsilon }{{\text{d}}t}}+{\frac {\eta }{E}}\cdot {\frac {{\text{d}}\sigma }{{\text{d}}t}}$

(4)

Description of the relaxation behaviour of plastics

For relaxation, ε = ε₀ = const., so that dε/dt = 0 and the stress σ depends only on time. Integration then yields Eq. (5), which allows a simple description of the relaxation behaviour of plastics (Fig. 3).

$\sigma \left(t\right)=\sigma _{0}\cdot e^{-{\frac {t}{\tau _{rel}}}}$

(5)

The quotient η/E corresponds to the time constant or relaxation time τ_rel, which indicates the time after which the stress σ has decreased to the e-th part of the initial stress σ₀. With a relaxation time, the MAXWELL model is insufficient to describe the complex relaxation behaviour of real plastics. Improved agreement between the model and the experiment is achieved by introducing a discrete relaxation time spectrum. In the analogue model, this can be achieved by connecting several MAXWELL elements in parallel.

Fig. 3:

Time behaviour of the MAXWELL model (relaxation behaviour)

See also

• Linear-viscoelastic behaviour
• Viscoelastic material behaviour
• HOOKE's law
• Relaxation plastics
• Relaxation behaviour determination

References

[1]	Lüpke, T.: Fundamental principles of mechanical behavior. In: Grellmann, W., Seidler, S. (Eds.): Polymer Testing. Carl Hanser, Munich (2022) 3rd Edition, pp. 79–82 (ISBN 978-1-56990-806-8; E-Book: ISBN 978-1-56990-807-5; see AMK-Library under A 22)
[2]	Bierögel, C.: Bend test on polymers. In: Grellmann, W., Seidler, S. (Eds.): Polymer Testing. Carl Hanser, Munich (2022) 3rd Edition, pp. 133–143 (ISBN 978-1-56990-806-8; E-Book: ISBN 978-1-56990-807-5; see AMK-Library under A 22)