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Energy Elasticity: Difference between revisions

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Created page with "{{Language_sel|LANG=ger|ARTIKEL=Energieelastizität}} {{PSM_Infobox}} <span style="font-size:1.2em;font-weight:bold;">Energy elasticity</span> __FORCETOC__ ==Structural causes of energy elasticity== The structural cause of energy-elastic behaviour is the change in the average atomic distances and bond angles when mechanical stresses are applied. The mechanical work required to do this is stored in the form of potential energy (increase in internal energy) an..."
 
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{{Language_sel|LANG=ger|ARTIKEL=Energieelastizität}}
{{Language_sel|LANG=ger|ARTIKEL=Entropieelastizität}}
{{PSM_Infobox}}
{{PSM_Infobox}}
<span style="font-size:1.2em;font-weight:bold;">Energy elasticity</span>
<span style="font-size:1.2em;font-weight:bold;">Entropy elasticity</span>
__FORCETOC__
__FORCETOC__


==Structural causes of energy elasticity==
==Physical fundamentals==


The structural cause of energy-elastic behaviour is the change in the average atomic distances and bond angles when mechanical [[Stress|stresses]] are applied. The mechanical work required to do this is stored in the form of potential energy (increase in internal energy) and is recovered completely and immediately when the stress is removed (1st law of thermodynamics) [1]. Due to its structural causes, energy-elastic behaviour is limited to the range of small deformations. Here, a linear relationship between stress and strain is observed, which is described by [[HOOKE's Law|HOOKE's law]].
Entropy elasticity expresses the tendency of macromolecules to return to the entropically most favourable state, the tangled state, after deformation [1]. When a flexible-chain [[Polymer|polymer]] [[Material & Werkstoff|material]] is subjected to mechanical [[Stress|stress]], the macromolecules align themselves in the field of stress. The molecular state of order is accompanied by a reduction in the entropy of the system. If the irreversible slippage of the chain segments can be prevented, for example by cross-linking, the macromolecules strive to maximise entropy when the stress is relieved (2nd law of thermodynamics). They transition to the disordered state of equilibrium in a timeless manner.


==HOOKE's law for energy elastic behaviour==
Entropy-elastic behaviour is observed up to large strains of several hundred percent. The relationship between stress and deformation is non-linear. Simple continuum mechanical considerations as well as molecular statistical models [2] lead to a relationship of the form in the case of [[Uniaxial Stress State|uniaxial stress]]:
 
For the simple case of [[Uniaxial Stress State|uniaxial]] tensile stress, '''Eq. (1)''' applies:


{|
{|
|-
|-
|width="20px"|
|width="20px"|
|width="500px" | <math>\sigma\,=\,E\cdot \varepsilon</math>.
|width="500px" | <math>\sigma\,=\, \frac{E}{3}\cdot \left( \lambda- \lambda^{-2} \right).</math>
|(1)
|(1)
|}
|}


The proportionality constant between stress and strain is referred to as the [[Elastic Modulus|modulus of elasticity]] ''E''. It is related to the binding forces in the material. Alternatively, the compliance ''C'' can also be determined ('''Eq. 2'''):
==Derivation of material parameters==
 
The [[Elastic Modulus|modulus of elasticity ''E'']] as a [[Material Parameter|material parameter]] is determined by the cross-linking density ''N'' or the average molar mass between the cross-linking nodes of the [[Polymer|polymer]] ''M''<sub>C</sub>. It also depends on the temperature ''T'', the Boltzmann constant ''k'', the universal gas constant ''R'' and the [[Density|density]]:


{|
{|
|-
|-
|width="20px"|
|width="20px"|
|width="500px" | <math>\varepsilon\,=\,C\cdot \sigma</math>.
|width="500px" | <math>E\,=\, 3NkT \,=\, \frac{3 \rho}{\overline {M}_C}RT.</math>
|(2)
|(2)
|}
|}


In addition to the change in length, a [[Specimen|test specimen]] under [[Tensile Test|tensile stress]] also undergoes a reduction in cross-section if it is in a [[Plane Stress and Strain State|plane stress state]] due to its geometry. The magnitude of this cross-sectional change is described by the transverse contraction coefficient ([[Poisson's Ratio|Poisson's ratio]]) ''&nu;''. It expresses the ratio of the strain in the transverse direction (''&epsilon;''<sub>y</sub>, ''&epsilon;''<sub>z</sub>) and longitudinal direction (&epsilon;<sub>x</sub>). For uniaxial stress, '''Eq. (3)''' applies:
With the help of '''Eq. (1)''', essential phenomena of the mechanical behaviour of rubber vulcanisates can be represented. However, their quantitative validity is often limited to strains below 100 %. For this reason, the simple theory of [[Rubber Elasticity|rubber elasticity]] has undergone a series of further developments, which are reported on in [3], for example.
 
{|
|-
|width="20px"|
|width="500px" | <math>v\,=\,-\frac{\varepsilon_y}{\varepsilon_x}\,=\,-\frac{\varepsilon_z}{\varepsilon_x}</math>.
|(3)
|}


In the case of shear stress, Hooke's law applies as follows '''Eq. (4)''', where ''G'' denotes the [[Shear Modulus|shear modulus]], ''&tau;'' the corresponding shear stress and ''&gamma;'' the shear.
Entropy elasticity is not limited to covalently cross-linked [[Polymer|polymers]]. It also plays an important role above the [[Glass Transition Temperature|glass transition temperature]] in amorphous and semi-crystalline [[Thermoplastic Material|thermoplastics]] with sufficiently high molecular weight. Here, physical entanglements (see: polymer & structure) and interlockings take on the role of temporary cross-linking points [4–6].
 
{|
|-
|width="20px"|
|width="500px" | <math>\tau\,=\,G\cdot \gamma</math>
|(4)
|}
 
==Relationships between elastic constants==
 
With the [[Poisson's Ratio|Poisson's ratio]] ''&nu;'', which indicates the ratio between transverse strain ''&epsilon''<sub>q</sub> and longitudinal strain ''&epsilon''<sub>l</sub> as an absolute value according to '''Eq. (5)'''
 
{|
|-
|width="20px"|
|width="500px" | <math>\nu\,=\,\frac{\varepsilon_q}{\varepsilon_l}</math>
|(5)
|}
 
the relationship between the [[Elastic Modulus|modulus of elasticity]] and the [[Shear Modulus|shear modulus]] for small elastic deformations is obtained as
 
{|
|-
|width="20px"|
|width="500px" | <math>E\,=\,2\cdot \left(1+\nu \right) \cdot G</math>
|(6)
|}
 
In the case of incompressibility, as with rubber, the upper limit of ''&nu;'' = 0.5, whereby a Poisson's ratio of around 0.3 is recorded for most [[Plastics|plastics]] due to volume effects occurring under [[Tensile Test|tensile stress]] [2]. Assuming [[Multiaxial Stress State|multiaxial]] compression on all sides (hydrostatic stress), the compression modulus can be calculated as a further elastic constant according to '''Eq. (7)''':
 
{|
|-
|width="20px"|
|width="500px" | <math>K\,=\,\frac{E}{3\left (1-2\mu\right)}</math>
|(7)
|}
 
The equations given here apply only to [[Deformation#Elastic behaviour|ideal elastic behaviour]] with [[Deformation|deformation]] that is very small in relation to the geometric dimensions of the test specimens used.


==See also==
==See also==


* [[HOOKE´s Law|HOOKE´s law]]
* [[Elasticity]]
* [[Linear-viscoelastic Behaviour|Linear-viscoelastic behaviour]]
* [[Rubber Elasticity|Rubber elasticity]]
* [[Elastic Modulus|Elastic modulus]]
* [[Cross-linking Elastomers|Cross-linking elastomers]]
* [[Glass Transition Temperarure|Glass transition temperarure]]
* [[Deformation]]
* [[Deformation]]
* [[Thermoelastic Effect|Thermoelastic effect]]


 
==References==
'''References'''


{|
{|
|-valign="top"
|-valign="top"
|[1]
|[1]
|Lüpke, Th.: Material Behavior and Constitutive Equations. In: [[Grellmann, Wolfgang|Grellmann, W.]], [[Seidler, Sabine|Seidler, S.]] (Eds.): Polymer Testing. Carl Hanser, Munich, (2022) 3rd Edition, pp. 75–77 (ISBN 978-1-56990-806-8; E-Book: ISBN 978-1-56990-807-5; see [[AMK-Büchersammlung|AMK-Library]] under A 22)
|Lüpke, Th.: Material Behavior and Constitutive Equations. In: [[Grellmann,_Wolfgang|Grellmann, W.]], [[Seidler,_Sabine|Seidler, S.]] (Eds.): Polymer Testing. Carl Hanser, Munich, (2022) 3rd Edition, pp. 75–77 (ISBN 978-1-56990-806-8; E-Book: ISBN 978-1-56990-807-5; see [[AMK-Library]] under A 22)  
|-valign="top"
|-valign="top"
|[2]
|[2]
|Wehrstedt, A.: Neues auf dem Gebiet der Werkstoffprüfung. In: Frenz, H., Wehrstedt, A. (Eds.): Kennwertermittlung für die Praxis. Wiley VCH (2003) pp. 1–12, (ISBN 3-527-30674-9; see [[AMK-Büchersammlung|AMK-Library]] under M 10)
|Treloar, L. R. G.: The Physics of Rubber Elasticity. Clarendon Press, Oxford (1975)
|-valign="top"
|[3]
|Stavemann, A. J.: Properties of phantom networks and real networks. Adv. Polym. Sci. 44 (1982) 73–101 DOI: https://doi.org/10.1007/3-540-11471-8_3
|-valign="top"
|[4]
|Erman, B., Mark, J. E.: Structure and Properties of Rubberlike Networks. Oxford University Press, New York (1997) (ISBN 978-0-1950-8237-1)
|-valign="top"
|[5]
|Termonia, Y., Smith, P.: Kinetic model for tensile deformation of polymers. Macromolecules 20 (1987) 835–838 DOI: https://doi.org/10.1021/MA00170A023
|-valign="top"
|[6]
|Bensason, S., Stepanov, E. V., Chum, S., Hiltner, A., Baer, E.: Deformation of elastomeric ethylene-octene copolymers. Macromolecules 30 (1997) 2436–2444 DOI: https://doi.org/10.1021/ma961685j
|}
|}


[[Category:Deformation]]
[[Category:Deformation]]
[[Category:Elastomers]]

Latest revision as of 13:44, 3 September 2026

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Entropy elasticity


Physical fundamentals

Entropy elasticity expresses the tendency of macromolecules to return to the entropically most favourable state, the tangled state, after deformation [1]. When a flexible-chain polymer material is subjected to mechanical stress, the macromolecules align themselves in the field of stress. The molecular state of order is accompanied by a reduction in the entropy of the system. If the irreversible slippage of the chain segments can be prevented, for example by cross-linking, the macromolecules strive to maximise entropy when the stress is relieved (2nd law of thermodynamics). They transition to the disordered state of equilibrium in a timeless manner.

Entropy-elastic behaviour is observed up to large strains of several hundred percent. The relationship between stress and deformation is non-linear. Simple continuum mechanical considerations as well as molecular statistical models [2] lead to a relationship of the form in the case of uniaxial stress:

(1)

Derivation of material parameters

The modulus of elasticity E as a material parameter is determined by the cross-linking density N or the average molar mass between the cross-linking nodes of the polymer MC. It also depends on the temperature T, the Boltzmann constant k, the universal gas constant R and the density:

(2)

With the help of Eq. (1), essential phenomena of the mechanical behaviour of rubber vulcanisates can be represented. However, their quantitative validity is often limited to strains below 100 %. For this reason, the simple theory of rubber elasticity has undergone a series of further developments, which are reported on in [3], for example.

Entropy elasticity is not limited to covalently cross-linked polymers. It also plays an important role above the glass transition temperature in amorphous and semi-crystalline thermoplastics with sufficiently high molecular weight. Here, physical entanglements (see: polymer & structure) and interlockings take on the role of temporary cross-linking points [4–6].

See also

References

[1] Lüpke, Th.: Material Behavior and Constitutive Equations. In: Grellmann, W., Seidler, S. (Eds.): Polymer Testing. Carl Hanser, Munich, (2022) 3rd Edition, pp. 75–77 (ISBN 978-1-56990-806-8; E-Book: ISBN 978-1-56990-807-5; see AMK-Library under A 22)
[2] Treloar, L. R. G.: The Physics of Rubber Elasticity. Clarendon Press, Oxford (1975)
[3] Stavemann, A. J.: Properties of phantom networks and real networks. Adv. Polym. Sci. 44 (1982) 73–101 DOI: https://doi.org/10.1007/3-540-11471-8_3
[4] Erman, B., Mark, J. E.: Structure and Properties of Rubberlike Networks. Oxford University Press, New York (1997) (ISBN 978-0-1950-8237-1)
[5] Termonia, Y., Smith, P.: Kinetic model for tensile deformation of polymers. Macromolecules 20 (1987) 835–838 DOI: https://doi.org/10.1021/MA00170A023
[6] Bensason, S., Stepanov, E. V., Chum, S., Hiltner, A., Baer, E.: Deformation of elastomeric ethylene-octene copolymers. Macromolecules 30 (1997) 2436–2444 DOI: https://doi.org/10.1021/ma961685j
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