Plastic Zone

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Plastic zone

Modelling of the plastic zone (dog-bone model)

Linear-elastic or purely elastic deformation behaviour is an idealised model. This material behaviour is described by linear-elastic fracture mechanics (LEBM), which is based on the continuum-mechanical Griffith crack model. In this context, the actual processes of crack propagation in macroscopic regions at the tip of a crack are initially disregarded. Blumenauer [1] describes how, even in the case of macroscopically brittle material behaviour, mechanical strain can lead to stress overloading at the crack tip, resulting in plastic deformations occurring in front of the crack tip. Depending on the microstructure of the material, plastic zones of varying shape and size form, which must be taken into account in fracture mechanics testing when determining the corresponding 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, which can be attributed to the stress intensity factor decreasing from the interior to the exterior [2, 3].

Fig. 1:

Idealised shape of the plastic zone in small-scale yielding (‘dog bone’ shape)

The transition from the 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 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 in a test specimen or the component dimensions. This is taken into account in the determination of the characteristic values using the so-called effective crack length a_eff, as given by Eq. (1).

$a_{eff}=a+r_{pl}\!$

(1)

This formally completes the transition from linear-elastic fracture mechanics (LEBM) to LEBM with small-scale yielding.

Equation (2) for determining the stress intensity factor is:

$K_{eff}=\sigma (\pi \ a_{eff})^{\frac {1}{2}}f({\frac {a_{eff}}{W}})$

(2)

The size of the plastic zone in the ligament can be roughly estimated by substituting the yield stress R_e for the stress in the SNEDDON equations.

$r_{pl}={\frac {1}{2\pi }}\left({\frac {K_{I}}{R_{e}}}\right)^{2}$ for plane stress state

(3)

$r_{pl}={\frac {1}{2\pi }}\left({\frac {K_{I}}{R_{e}}}\right)^{2}\left(1-2\nu \right)^{2}$ for plane strain state

(4)

Example of a plastic zone for PVC-C

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

$r_{pl}=4.3\ {\frac {K_{I}^{2}}{E\ R_{e}}}$

(5)

Figure 2 shows the deformation zone that forms in front of the crack tip during the instrumented Charpy impact test on SENB-specimens subjected to an impact load. After the notch was cut, the specimens were polished and gold-sputtered. The cracks indicate the extent of the deformation region.

Fig. 2:

Deformation zone at the crack tip of a post-chlorinated polyvinyl chloride material (abbreviation: PVC-C). Overall view (a) and detail of the crack tip (b) of a SENB-specimen following sudden loading (KI < KId), polished and gold-sputtered

See also

• Crack model according to DUGDALE
• Crack model according to IRWIN und Mc CLINTOCK
• Fracture mirror
• Fracture process zone
• Fracture surface
• Crack

References

[1]	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)
[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
[3]	Williams, J. G.: Fracture Mechanics of Polymers. Ellis Horwood Ltd., Publisher (1984) (ISBN 978-0470-20013-1; see AMK-Library under E 42)
[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)
[5]	Grellmann, W.; 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)