View source for Rubber Elasticity

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<span style="font-size:1.2em;font-weight:bold;">Rubber elasticity; entropy elasticity</span>
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==Definition==

The term rubber elasticity, or the thermodynamically more accurate term [[Entropy Elasticity|entropy elasticity]] used in physics, refers to the resistance of rubber-like materials ([[Elastomers|elastomers]]) to [[Deformation|deformation]] [1, 2].

==Description of deformation==

When rubber-elastic materials are deformed and oriented, chain segments are moved from their equilibrium position into a less favourable entropic state, i.e. under load, they transition into a more ordered structure. Due to the weak [[Cross-linking Elastomers|cross-linking]], chain segments cannot slide completely against each other, which means that the entropically elastic bodies are reversibly deformable by very large amounts (several hundred percent).

When the load is removed, the chain segments return from an ordered to a disordered state, i.e. the entropy increases again. Thermodynamically, rubber elasticity is associated with a reduction in entropy in the deformed state.

From a molecular point of view, the [[Deformation|deformation]] forces an [[Tensile Test Residual Stresses Orientations|orientation]] in parts of the molecule, i.e. a change in conformation. Even non-cross-linked macromolecules exhibit entropy-elastic behaviour under certain [[Stress|loads]]. In this case, the interlocking and entanglement during rapid deformation act like cross-links, but the entropy-elastic behaviour of these [[Material & Werkstoff|materials]] is overlaid by a more or less pronounced flow.

==The thermodynamics of rubber elasticity==

The temperature dependence of the elastomer material group is of crucial importance. The [[Glass Transition Temperature|glass transition temperature]] ''T''<sub>G</sub> of [[Elastomers|elastomers]] is well below normal operating temperatures, i.e. within the operating temperature range, the chain mobility is so high that the stress during deformation occurs practically without delay.

From a thermodynamic point of view, rubber elasticity is essentially based on a decrease in entropy ''S'' in the general equation for the change in free energy <math style="height:12px; vertical-align:-12%;">F = U - T S</math> at a given strain.

The free energy ''F'' in the [[Tensile Test|tensile test]] can be calculated using '''Eq. (1)'''
{|
|-
|width="20px"|
|width="500px"|<math>dF= -S \cdot dT - p \cdot dV+f \cdot dl</math>
|(1)
|}

where
{|
|-
|width="150"|''F'' – free energy
|width="150"|''T'' – temperature
|-
|''S'' – entropy
|''V'' – volume
|-
|''p'' – pressure
|d''l'' – change in length
|-
|''f'' – tensile force
|}

can be calculated.

The measured force is

{|
|-
|width="20px"|
|width="500px"|<math>f =\left ( \frac{d F}{d l} \right )_{T,V}</math>
|(2)
|}

Since, in the case of a reversible isothermal process, the change in energy is equal to the mechanical work d''W''/d''l'' applied by external forces, '''Eq. (3)''' applies.

{|
|-
|width="20px"|
|width="500px"|<math>\frac{d W}{d l}=\frac{d F}{d l}=-T \cdot \frac{d S}{d l}</math>
|(3)
|}

A statistical analysis of the relationship ultimately leads to the conclusion that, in the case of a network with ''N'' chains per unit volume, '''Eq. (4)''' applies to the mechanical work ''W'' [3‒6]:

{|
|-
|width="20px"|
|width="500px"|<math>W= -T \cdot d S =\frac{1}{2}N\cdot k\cdot T\cdot \left (  \lambda^{2}_{1} + \lambda^{2}_{2} + \lambda^{2}_{3} - 3 \right )</math>
|(4)
|}

with
{|
|-
|width="300"|''W'' – mechanical work
|-
|''k'' – Boltzmann constant ''k'' = 1.38∙10<sup>23</sup> JK<sup>-1</sup>
|-
|''&lambda;'' – [[Deformation|deformation]] (strain ratio)
|}

The parameter ''NkT'' is defined as the [[Shear Modulus|shear modulus]] ''G''. Taking into account the molar mass ''M''<sub>c</sub> between the cross-linking points, the calculation '''Eq. (5)''' for the shear modulus ''G'' is obtained with ''R'' = ''k'' ∙ ''N''<sub>L</sub> and ''N'' = ''ρ'' / ''M''<sub>c</sub> ∙ ''N''<sub>L</sub>:

{|
|-
|width="20px"|
|width="500px"|<math>G= N \cdot k \cdot T= \frac{\rho \cdot R \cdot T}{M_{c}}</math>
|(5)
|}

with
{|
|-
|width="300"|''R'' – gas constant
|-
|''N''<sub>L</sub> – Loschmidt number
|-
|''&rho;'' – [[Density|density]] of the [[Material & Werkstoff|material]]
|}

==The uniaxial tensile test==

In the case of [[Uniaxial Stress State|uniaxial]] tensile loading, a deformation <math style="height:15px; vertical-align:-15%;">\lambda_{1}=\lambda\!</math> is applied, and the lateral deformations are then <math style="height:20px; vertical-align:-15%;">\lambda_{2}=\lambda_{3}=\lambda^{-\frac{1}{2}}\!</math> (at constant volume, i. e. <math style="height:15px; vertical-align:-15%;">\lambda_{1}\cdot\lambda_{2}\cdot\lambda_{3}=1</math>). The following '''Eq. (6)''' then applies to the stress applied to a test specimen (see also: [[Test Piece|test piece]])

{|
|-
|width="20px"|
|width="500px"|<math>\sigma = \frac{d W}{d\lambda}=-T\cdot\frac{d S}{d \lambda }= G \ \left ( \lambda -\frac{1}{\lambda ^{2}} \right )</math>
|(6)
|},

which in many cases corresponds quite well with experiments. At higher degrees of deformation ''λ'' > 1.5, the empirical MOONEY-RIVLIN Eq. (7) describes the behaviour of [[Elastomers|elastomers]] more precisely. The following then applies

{|
|-
|width="20px"|
|width="500px"|<math>\sigma= C_{1} \cdot \left ( \lambda^{2} -\frac{1}{\lambda } \right )+C_{2} \cdot \left ( \lambda -\frac{1}{\lambda ^{2}} \right )</math>
|(7)
|}

with ''C''<sub>1</sub> and ''C''<sub>2</sub> being elasticity constants.

From '''Eq. (5)''', it can be concluded that the [[Shear Modulus|shear modulus]] increases with the number of [[Cross-linking Elastomers|cross-links]] or the cross-link density.

==See also==
* [[Elasticity]]
* [[Entropy Elasticity|Entropy elasticity]]
* [[Cross-linking Elastomers|Cross-linking elastomers]]
* [[Glass Transition Temperature|Glass transition temperature]]
* [[Elastomers]]

==References==

{|
|-valign="top"
|[1]
|Guth, E., Mark, H.: Zur statistischen Theorie der Kautschukelastizität. Zeitschrift Elektrochemie 43 (1937) 683−686 
|-valign="top"
|[2]
|Schmiedel, H. (Ed.): Handbuch der Kunststoffprüfung. Carl Hanser, Munich Vienna (1992) pp. 58‒59 (ISBN 3-446-16336-0; see AMK-Library under A 3) 
|-valign="top"
|[3]
|[[Kausch,_Hans-Henning|Kausch, H. H.]], Stalder, B., Barbezat, M.: Introduction aux matieres plastiques. Lausanne: Ecole Polytechnique Federale de Lausanne (1986) 
|-valign="top"
|[4]
|Batzer, H. (Ed.): Polymere Werkstoffe, Vol. I, Chemie und Physik. Stuttgart, G. Thieme Publishing (1985) (ISBN 9783136481011; [[AMK-Library]] under G 17) 
|-valign="top"
|[5]
|Aklonis, J. J., MacKnight, W. J., Shen, M. (Eds.): Introduction to Polymer Viscoelasticity. New York: J Willey & Son. Inc. (1972), DOI: http://onlinelibrary.wiley.com/doi/10.1002/pol.1973.130110114/abstract 
|-valign="top"
|[6]
|Paufler, P., Schulze, G. E. R.: Physikalische Grundlagen mechanischer Festkörpereigenschaften, Vol. 1. Berlin: Akademie Publishing (1978) 
|}

==Weblink==

* Wikipedia – The free encyclopedia: [https://de.wikipedia.org/wiki/Gummielastizität Gummielastizität] (accass: 07.08.2025)

[[Category:Elastomers]]