Oluschinski: Created page with "{{Language_sel|LANG=ger|ARTIKEL=Energieelastizität}} {{PSM_Infobox}} Energy elasticity 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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Created page with "{{Language_sel|LANG=ger|ARTIKEL=Energieelastizität}} {{PSM_Infobox}} Energy elasticity __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}}
{{PSM_Infobox}}
Energy elasticity
__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 [[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]].

==HOOKE's law for energy elastic behaviour==

For the simple case of [[Uniaxial Stress State|uniaxial]] tensile stress, '''Eq. (1)''' applies:

{|
|-
|width="20px"|
|width="500px" | <math>\sigma\,=\,E\cdot \varepsilon</math>.
|(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'''):

{|
|-
|width="20px"|
|width="500px" | <math>\varepsilon\,=\,C\cdot \sigma</math>.
|(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]]) ''ν''. It expresses the ratio of the strain in the transverse direction (''ε''y, ''ε''z) and longitudinal direction (εx). For uniaxial stress, '''Eq. (3)''' applies:

{|
|-
|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]], ''τ'' the corresponding shear stress and ''γ'' the shear.

{|
|-
|width="20px"|
|width="500px" | <math>\tau\,=\,G\cdot \gamma</math>
|(4)
|}

==Relationships between elastic constants==

With the [[Poisson's Ratio|Poisson's ratio]] ''ν'', which indicates the ratio between transverse strain ''&epsilon''q and longitudinal strain ''&epsilon''l 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 ''ν'' = 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==

* [[HOOKE´s Law|HOOKE´s law]]
* [[Linear-viscoelastic Behaviour|Linear-viscoelastic behaviour]]
* [[Elastic Modulus|Elastic modulus]]
* [[Deformation]]

'''References'''

{|
|-valign="top"
|[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)
|-valign="top"
|[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)
|}

[[Category:Deformation]]

Energy Elasticity - Revision history