View source for MAXWELL Model

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<span style="font-size:1.2em;font-weight:bold;">MAXWELL model</span>
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==Mechanical analogy models==

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

However, [[Viscoelastic Material Behaviour|linear viscoelasticity]] is only precisely defined for the range of infinitesimally small [[Stress|stresses]]. The [[Material & Werkstoff|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|HOOKE's law]]), while the viscous component (NEWTON' s damper: '''Fig. 1b''') generally causes time-dependent, unlimited, irreversible [[Deformation|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|viscoelastic material behaviour]] can therefore be modelled by combining two or more of these elements [1].

[[File:Maxwell-Modell1.png|450px]]
{| 
|- valign="top"
|width="50px"|'''Fig. 1''': 
|width="600px"|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 Elasticity|energy-elastic behaviour]] between spring force and elongation or stress and strain, a complete description is possible using [[HOOKE's Law|HOOKE's law]] ('''Eq. (1)''').

{|
|-
|width="20px"|
|width="500px"|<math>\sigma = E \cdot \varepsilon</math>
|(1)
|}

The damper for NEWTONIAN fluids describes purely [[Deformation#Viscous deformation|viscous]] behaviour, which depends significantly on the [[Viscosity|viscosity]] ''η'' ('''Eq. (2)'''). At constant viscosity, the resulting stress at the damper is determined solely by the deformation speed or the [[Strain Rate Basics|strain rate]] d''ε''/d''t''.

{|
|-
|width="20px"|
|width="500px"|<math>\sigma = \eta \cdot \frac{\text{d}\varepsilon}{\text{d} t}</math>
|(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 Plastics|relaxation behaviour of plastics]] [2], which is based on the addition of elastic and viscous deformation components (see: [[Deformation|deformation]]) ('''Eq. (3)''').

[[File:Maxwell-Modell2.png|300px]]
{| 
|- valign="top"
|width="50px"|'''Fig. 2''': 
|width="600px"|Deformation behaviour of the MAXWELL model (relaxation behaviour)
|}

{|
|-
|width="20px"|
|width="500px"|<math>\frac{\text{d}\varepsilon}{\text{d}t}= \frac{\text{d}\varepsilon_{1}}{\text{d}t} + \frac{\text{d}\varepsilon_{2}}{\text{d}t}</math>
|(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.

{|
|-
|width="20px"|
|width="500px"|<math>\sigma = \eta \cdot  \frac{\text{d}\varepsilon}{\text{d}t} + \frac{\eta }{E}\cdot \frac{\text{d}\sigma}{\text{d}t}</math>
|(4)
|}

==Description of the relaxation behaviour of plastics==

For relaxation, ''ε'' = ''ε''<sub>0</sub> = const., so that d''ε''/d''t'' = 0 and the stress ''σ'' depends only on time. Integration then yields '''Eq. (5)''', which allows a simple description of the [[Relaxation Behaviour Determination|relaxation behaviour]] of [[Plastics|plastics]] ('''Fig. 3''').

{|
|-
|width="20px"|
|width="500px"|<math>\sigma \left ( t \right )= \sigma_{0} \cdot e^{-\frac{t}{\tau_{rel}}}</math>
|(5)
|}

The quotient ''η''/''E'' corresponds to the time constant or relaxation time ''τ''<sub>rel</sub>, which indicates the time after which the stress ''σ'' has decreased to the e-th part of the initial stress ''σ''<sub>0</sub>. With a relaxation time, the MAXWELL model is insufficient to describe the complex [[Relaxation Behaviour Determination|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.

[[File:Maxwell-Modell3.png|300px]]
{| 
|- valign="top"
|width="50px"|'''Fig. 3''': 
|width="600px"|Time behaviour of the MAXWELL model (relaxation behaviour)
|}

==See also==

* [[Linear-viscoelastic Behaviour|Linear-viscoelastic behaviour]]
* [[Viscoelastic Material Behaviour|Viscoelastic material behaviour]]
* [[HOOKE's Law|HOOKE's law]]
* [[Relaxation Plastics|Relaxation plastics]]
* [[Relaxation Behaviour Determination|Relaxation behaviour determination]]

==References==

{|
|-valign="top"
|[1]
|Lüpke, T.: Fundamental principles of mechanical behavior. In: [[Grellmann,_Wolfgang|Grellmann, W.]], [[Seidler,_Sabine|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) 
|-valign="top"
|[2]
|[[Bierögel, Christian|Bierögel, C.]]: Bend test on polymers. In: [https://www.researchgate.net/profile/Wolfgang-Grellmann 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) 
|}

[[Category:Deformation]]