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{{Language_sel|LANG=ger|ARTIKEL=Plastische Zone}}
{{PSM_Infobox}}
Plastic zone
__FORCETOC__

==Modelling of the plastic zone (dog-bone model)==

Linear-elastic or purely [[Deformation#Elastic deformation|elastic deformation behaviour]] is an idealised model. This material behaviour is described by [[Fracture Mechanics#Linear-elastic fracture mechanics|linear-elastic fracture mechanics (LEBM)]], which is based on the [[Continuum Mechanics|continuum-mechanical]] [[Crack Model according to GRIFFITH|Griffith crack model]]. In this context, the actual processes of [[Crack Propagation|crack propagation]] in macroscopic regions at the tip of a [[Crack|crack]] are initially disregarded. [[Blumenauer, Horst|Blumenauer]] [1] describes how, even in the case of macroscopically brittle material behaviour, mechanical strain can lead to stress overloading at the [[Crack|crack tip]], resulting in [[Deformation#Plastic deformation|plastic deformations]] occurring in front of the crack tip. Depending on the [[Microscopic Structure|microstructure]] of the [[Material & Werkstoff|material]], plastic zones of varying shape and size form, which must be taken into account in [[Fracture Mechanical Testing|fracture mechanics testing]] when determining the corresponding [[Material Value|characteristic values]].

In the idealised form of a plastic zone shown in '''Fig. 1''', the extent of deformation is smaller at the interior and on the [[Surface|surface]], which can be attributed to the [[Fracture Mechanics|stress intensity factor]] decreasing from the interior to the exterior [2, 3].

[[File:Plastic_Zone_Fig-1.jpg]]
{|
|- valign="top"
|width="50px"|'''Fig. 1''':
|width="600px" |Idealised shape of the plastic zone in [[Fracture Mechanics#Linear-elastic fracture mechanics with small-scale yielding|small-scale yielding]] (‘dog bone’ shape)
|}

The transition from the [[Plane Stress and Strain State|plane stress state]] to the plane strain state within the material results in the so-called ‘dog bone’ shape of the plastic zone, as derived by [[Crack Model according to IRWIN and Mc CLINTOCK|IRWIN and Mc CLINTOCK]].

==Taking the plastic zone into account when determining fracture mechanics values==

The extension of this plastic zone must be relatively small compared to the [[Initial Crack Length|initial crack length]] in a [[Specimen|test specimen]] or the [[Plastic Component|component]] dimensions. This is taken into account in the determination of the [[Material Value|characteristic values]] using the so-called [[Effective Crack Length|effective crack length]] ''a''eff, as given by '''Eq. (1)'''.

{|
|-
|width="20px"|
|width="500px" | <math>a_{eff}=a+r_{pl} \! </math>
|(1)
|}

This formally completes the transition from [[Fracture Mechanics#Linear-elastic fracture mechanics|linear-elastic fracture mechanics (LEBM)]] to [[Fracture Mechanics#Linear-elastic fracture mechanics with small-scale yielding|LEBM with small-scale yielding]].

'''Equation (2)''' for determining the [[Fracture Mechanics|stress intensity factor]] is:

{|
|-
|width="20px"|
|width="500px" | <math>K_{eff}= \sigma (\pi \ a_{eff})^{\frac{1}{2}}f(\frac{a_{eff}}{W})</math>
|(2)
|}

The size of the plastic zone in the ligament can be roughly estimated by substituting the [[Yield Stress|yield stress]] ''R''e for the stress in the SNEDDON equations.

{|
|-
|width="20px"|
|width="500px" | <math>r_{pl}=\frac{1}{2\pi}\left ( \frac{K_{I}}{R_{e}} \right )^2</math> for plane stress state
|(3)
|}

{|
|-
|width="20px"|
|width="500px" | <math>r_{pl}=\frac{1}{2\pi}\left ( \frac{K_{I}}{R_{e}} \right )^2 \left ( 1-2\nu \right )^2</math> for plane strain state
|(4)
|}

==Example of a plastic zone for PVC-C==

The formation of the plastic zone in [[Plastics|plastics]] has been shown to depend on the [[Microscopic Structure|microstructure]]; for example, for a post-chlorinated polyvinyl chloride material ([[Plastics – Symbols and Abbreviated Terms|abbreviation]]: PVC-C or PVCC) [4, 5], this is described by '''Eq. (5)'''.

{|
|-
|width="20px"|
|width="500px" | <math>r_{pl}=4.3 \ \frac{K_{I}^2}{E \ R_{e}}</math>
|(5)
|}

'''Figure 2''' shows the deformation zone that forms in front of the [[Crack|crack tip]] during the [[Instrumented Charpy Impact Test|instrumented Charpy impact test]] on [[SENB-Specimen|SENB-specimens]] subjected to an impact load. After the [[Notch|notch]] was cut, the specimens were [[Plastography#Grinding and polishing specimens|polished]] and gold-sputtered. The [[Crack|cracks]] indicate the extent of the deformation region.

[[File:plastische_zone2.jpg]]
{|
|- valign="top"
|width="50px"|'''Fig. 2''':
|width="600px" |Deformation zone at the crack tip of a post-chlorinated polyvinyl chloride material ([[Plastics – Symbols and Abbreviated Terms|abbreviation]]: PVC-C). Overall view (a) and detail of the crack tip (b) of a [[SENB-Specimen|SENB-specimen]] following sudden loading (''K''I < ''K''Id), polished and gold-sputtered
|}

==See also==

* [[Crack Model according to DUGDALE|Crack model according to DUGDALE]]
* [[Crack Model according to IRWIN and Mc CLINTOCK|Crack model according to IRWIN und Mc CLINTOCK]]
* [[Fracture Mirror|Fracture mirror]]
* [[Fracture Process Zone|Fracture process zone]]
* [[Fracture Surface|Fracture surface]]
* [[Crack]]

==References==

{|
|-valign="top"
|[1]
|[[Blumenauer, Horst|Blumenauer, H.]], Pusch, G.: Technische Bruchmechanik, Deutscher Verlag für Grundstoffindustrie, Leipzig Stuttgart (1993), (ISBN 3-342-00659-5; see [[AMK-Library]] under E 29-3)
|-valign="top"
|[2]
|Anderson, T. L.: Fracture Mechanics. Fundamentals and Applications. 3rd Ed., CRC Press Boca Raton (2005) (ISBN 978-0-8493-1656-2; see [[AMK-Library]] under E 8-2) https://doi.org/10.1201/9781315370293
|-valign="top"
|[3]
|[[Williams, James Gordon|Williams, J. G.]]: Fracture Mechanics of Polymers. Ellis Horwood Ltd., Publisher (1984) (ISBN 978-0470-20013-1; see [[AMK-Library]] under E 42)
|-valign="top"
|[4]
|Jungbluth, M.: Untersuchungen zum Verformungs- und Bruchverhalten von PVCC-Werkstoffen. PhD thesis, Technische Hochschule Leuna-Merseburg (see [[AMK-Library]] under B 1-1) (Bibliographic abstract/Content)
|-valign="top"
|[5]
|[[Grellmann,_Wolfgang|Grellmann, W.]]; [[Seidler,_Sabine|Seidler, S.]] (Eds.): Kunststoffprüfung. Carl Hanser, Munich (2025) 4th Edition, pp. 245/246 (ISBN 978-3-446-44718-9; E-Book: ISBN 978-3-446-48105-3; see [[AMK-Library]] under A 23)
|}

Plastic Zone - Revision history