Oluschinski: Created page with "{{Language_sel|LANG=ger|ARTIKEL=Ebener Spannungszustand}} {{PSM_Infobox}} Plane stress and strain state FORCETOC ==Generalised HOOKE's Law== Assuming isotropic material properties in all spatial and axial directions, the generalised HOOKE's law ('''Eq. 1''') is as follows: {| |- |width="20px"| |width="500px" | $\epsilon_{11}=\frac{\sigma_{11}}{E}-\frac{\nu}{E}(\si..."$

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Created page with "{{Language_sel|LANG=ger|ARTIKEL=Ebener Spannungszustand}} {{PSM_Infobox}} Plane stress and strain state __FORCETOC__ ==Generalised HOOKE's Law== Assuming isotropic material properties in all spatial and axial directions, the generalised HOOKE's law ('''Eq. 1''') is as follows: {| |- |width="20px"| |width="500px" | <math>\epsilon_{11}=\frac{\sigma_{11}}{E}-\frac{\nu}{E}(\si..."

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{{Language_sel|LANG=ger|ARTIKEL=Ebener Spannungszustand}}
{{PSM_Infobox}}
Plane stress and strain state 
__FORCETOC__

==Generalised HOOKE's Law==

Assuming isotropic [[Material & Werkstoff|material]] properties in all spatial and axial directions, the generalised [[HOOKE's Law|HOOKE's law]] ('''Eq. 1''') is as follows:

{|
|-
|width="20px"|
|width="500px" | <math>\epsilon_{11}=\frac{\sigma_{11}}{E}-\frac{\nu}{E}(\sigma_{22}+\sigma_{33})</math> <math>\epsilon_{22}=\frac{\sigma_{22}}{E}-\frac{\nu}{E}(\sigma_{33}+\sigma_{11})</math> <math>\epsilon_{33}=\frac{\sigma_{33}}{E}-\frac{\nu}{E}(\sigma_{11}+\sigma_{22})</math>
|(1)
|}

It can be seen that the strain in the respective axial direction is primarily caused by the stress in the same axis, but an additional strain component from the other stress directions is added, depending on [[Poisson's Ratio|Poisson's ratio]] ''ν'' and the [[Elastic Modulus|modulus of elasticity]] ''E''. In the case of [[Uniaxial Stress State|uniaxial]] stress in a [[Tensile Test|tensile]] or [[Compression Test|compression]] test, in addition to the longitudinal strain ''ε''x, there is also transverse strain ''ε''y and ''ε''z, i.e. in the width and thickness directions of the [[Specimen|test specimen]], if a plane stress state is present. This is usually the case when a sufficiently slender test specimen is used, which allows the transverse strain to be measured in the width or thickness direction ('''Fig. 1''').

[[File:edz1.jpg]]
{|
|- valign="top"
|width="50px"|'''Fig. 1''':
|width="600px" |[[Deformation]] of the [[Specimen|test specimen]] under load in a plane stress state
|}

In addition to the normal stress ''σ''x ('''Eq. 2'''), loading causes longitudinal strain ''ε''x or ''ε''L ('''Eq. 3''') and transverse strain in the width and thickness directions ('''Eq. 4''').

{|
|-
|width="20px"|
|width="500px" | <math>\sigma_{x}=\frac{F_{x}}{A_{0}}</math>
|(2)
|}

{|
|-
|width="20px"|
|width="500px" | <math>\epsilon_{x}=\epsilon_{L}=\frac{L-L_{0}}{L_{0}}=\frac{\Delta L}{L_{0}}</math>
|(3)
|}

{|
|-
|width="20px"|
|width="500px" | <math>\epsilon_{y}=\epsilon_{q}=\frac{B-B_{0}}{B_{0}}=\frac{\Delta B}{B_{0}}</math>
|(4)
|}

In the case of isotropy and a homogeneous material state, the strain in the width direction ''ε''y and in the thickness direction ''ε''z ('''Eq. 5''') are identical, with the absolute value being specified for both transverse strains. The stresses ''σ''y and ''σ''z are equal to zero in the plane stress state.

{|
|-
|width="20px"|
|width="500px" | <math>\epsilon_{z}=\epsilon_{q}=\frac{H_{0}-H}{H_{0}}=\frac{\Delta H}{H_{0}}</math>
|(5)
|}

==Determination of Poisson's ratio==

The [[Poisson's Ratio|Poisson's ratio]], also known as the transverse contraction number, is then calculated according to Eq. (6):

{|
|-
|width="20px"|
|width="500px" | <math>\nu=\nu_{B}=\nu_{H}=\frac{\epsilon_{y}}{\epsilon_{L}}=\frac{\epsilon_{z}}{\epsilon_{L}}=\frac{\epsilon_{q}}{\epsilon_{L}}</math>
|(6)
|}

If, instead of the slim test specimens, thicker and/or wider prismatic [[Specimen|test specimens]] are used as shown in '''Fig. 2''', no measurable strain signal will be obtained in either the thickness or width direction, as the test specimen is now in a plane strain state.

[[File:edz2.jpg]]
{|
|- valign="top"
|width="50px"|'''Fig. 2''':
|width="600px" |Deformation of the test specimen under load in a plane state of strain
|}

The resulting longitudinal strain ''ε''x should be identical, which is why an increased normal stress ''σ''x is required due to the larger cross-section ''A''0. To explain this situation, we can assume an identical geometry as in '''Fig. 1''', whereby a possible change in the width of the [[Specimen|test specimen]] is prevented by lateral abutments ('''Fig. 3'''). If these supports could also be equipped with a large number of load cells (see: [[Electro-mechanical Force Transducer|electro-mechanical force transducer]] and [[Piezoelectric Force Transducer|piezoelectric force transducer]]), the forces in the width and thickness directions of the test specimen could be measured. Under load, this would result in an identical longitudinal strain ''ε''x according to '''Eq. (3)''' and a higher normal stress ''σ''x. In this case, Poisson's ratio ''ν'' can no longer be calculated because the strains ''ε''y and ''ε''z are equal to zero. Since forces in the y and z directions would be measured in this simulated state, stresses also arise in these directions, the magnitude of which can be calculated according to '''Eq. (1)''' [1].

[[File:edz3.jpg]]
{|
|- valign="top"
|width="50px"|'''Fig. 3''':
|width="600px" |Simulation of the slender test specimen in a plane strain state
|}

==See also==

* [[Uniaxial Stress State|Uniaxial stress state]]
* [[Energy Elasticity|Energy elasticity]]
* [[Crack Model according to IRWIN and Mc CLINTOCK|Crack model according to IRWIN and Mc CLINTOCK]]
* [[Plastic Zone|Plastic zone]]
* [[Plastic Hinge Model|Plastic hinge model]]

==References==

{|
|-valign="top"
|[1]
|[[Bierögel,_Christian|Bierögel, C.]]: Quasi-Static Test Methods. In: [[Grellmann,_Wolfgang|Grellmann, W.]], [[Seidler,_Sabine|Seidler, S.]] (Eds.): Polymer Testing. Carl Hanser, Munich (2022) 3rd Edition, pp. 101–143 (ISBN 978-1-56990-806-8; E-Book: ISBN 978-1-56990-807-3; see [[AMK-Library]] under A 23)
|}

[[Category:Fracture Mechanics]]
[[Category:Deformation]]

Plane Stress and Strain State - Revision history