Crack Tip Opening Displacement Concept (CTOD)

Sprachauswahl/Language selection

Dieser Artikel ist auch auf Deutsch verfügbar Crack Tip Opening Displacement-Konzept

A service provided by
Polymer Service GmbH Merseburg
Tel.: +49 3461 30889-50 E-Mail: info@psm-merseburg.de Web: https://www.psm-merseburg.de
Our further education offers: https://www.psm-merseburg.de/weiterbildung
PSM on Wikipedia: https://de.wikipedia.org/wiki/Polymer Service Merseburg

Crack tip opening displacement concept (CTOD)

CTOD concept

On the diversity of terms

The crack tip opening displacement (CTOD) concept of yield fracture mechanics was derived by Wells using the crack model according to DUGDALE [1, 2]. It is often abbreviated as the COD concept; sometimes the term COS (crack opening stretch) is also used [3]. The critical crack opening δ_c or crack opening displacement, which represents a measure of the widening of a crack, is specified as a parameter.

Fundamentals of the concept

The concept is based on the assumption that in ductile material behaviour the fracture process is not controlled by a critical stress intensity (see fracture mechanics), but by a critical deformation at the crack tip.

The DUGDALE model yields the following relationship (Eq. 1) between the crack opening δ, the crack length a and the stress σ for the plane-stress state

$\delta ={\frac {8\ \sigma _{F}\ a}{\pi \ E}}\ ln\left(sec{\frac {\pi }{2}}\right){\frac {\sigma }{\sigma _{F}}}$

(1)

$sec={\frac {Hypothenuse}{Ankathete}}={\frac {1}{cos\ \alpha }}$

with σ_F (yield stress) and yield point R_e

$\sigma _{F}={\frac {R_{e}+R_{m}}{2}}$

Rm   –

Tensile strength

A derivation of this equation with the valid approximation σ/σ_F < 0.6 leads to the following equation (2)

$\delta ={\frac {\pi \ \sigma ^{2}\ a}{E\ \sigma _{F}}}$

(2)

In contrast to linear elastic fracture mechanics (LEFM), in which a critical stress intensity is determined, this concept involves a critical strain δ_c at the notch base.

The fracture process is therefore controlled by a critical plastic deformation. The following simple relationship exists with the LEFM concept (Eq. 3) [3, 4]:

$K_{IC}^{COD}=(m\ \sigma _{F}\ E\ \delta _{c})^{\frac {1}{2}}$

(3)

with

m = 1:	Plane stress state
m = 2:	Plane strain state

Experimental results for plastics from our own research work [5]

Material	Test speed	Test temperature	m
1 PP	vT = 8.33 * 10-4 ms-1	T = 233 K	m = 0.7
2 PE-HD	vH = 1 ms-1	T = 293 K	m = 2.28
3 PVCC	vH = 0.75 ms-1	T = 293 K	m = 2

The reasons for the deviations between K_IC and δ_IC from this relationship are, for example, that a smaller δ_c-value is determined at the main crack when crack branching occurs than would result without branching.

The use of δ_c as a characteristic value for elastic-plastic fracture behaviour is only possible if the crack immediately begins to propagate unstably. However, this is not the case with ductile material behaviour; instead, stable crack propagation initially occurs, which only leads to fracture or can change to unstable crack propagation with a further increase in load. The start of stable crack propagation is determined by a δ_i value.

See also

• Crack model according to DUGDALE
• Fracture mechanics
• Plastic zone
• Fracture process zone

References

[1]	Wells, A. A.: Unstable Crack Propagation in Metals: Cleavage and Fast Fracture. In: Proceedings of the Crack Propagation Symposium: Cranfield (England) September 1961, Vol. 1, No. 84
[2]	Dugdale, D. S.: Yielding of Steel Sheets Containing Slits. J. Mech. Phys. Solids 8 (1960) 2, 100–104, DOI: https://doi.org/10.1016/0022-5096(60)90013-2
[3]	Blumenauer, H., Pusch, G.: Technische Bruchmechanik. Deutscher Verlag für Grundstoffindustrie, Leipzig Stuttgart (1993) 3rd Edition, p. 19, (ISBN 3-342-00659-5; see AMK Library under E 29-3)
[4]	Anderson, T. L.: Fracture Mechanics; Fundamentals and Applications. CRC Press, Boca Rat (2005) 3rd edition, (ISBN 978-0849342608; see AMK Library under E 8-2), DOI: https://www.taylorfrancis.com/books/mono/10.1201/9781420058215/fracture-mechanics-ted-anderson-anderson
[5]	Grellmann, W., Seidler, S. (Eds.): Polymer Testing. Carl Hanser, Munich (2022) 3rd Edition, p. 236–239 ((ISBN 978-1-56990-806-8; e-book ISBN 978-1-56990-807-5; see AMK Library under A 22)