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Created page with "{{Language_sel|LANG=ger|ARTIKEL=Crack Tip Opening Displacement-Konzept}} {{PSM_Infobox}} Crack tip opening displacement concept (CTOD) __FORCETOC__ 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 a..."

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{{Language_sel|LANG=ger|ARTIKEL=Crack Tip Opening Displacement-Konzept}}
{{PSM_Infobox}}
Crack tip opening displacement concept (CTOD)
__FORCETOC__
CTOD concept

==On the diversity of terms==

The crack tip opening displacement (CTOD) concept of yield [[Fracture Mechanics | fracture mechanics]] was derived by Wells using the [[Crack Model according to DUGDALE | 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 [[Material Parameter | 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 | 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

{|
|-
|width="20px"|
|width="500px" | <math>\delta =\frac{8 \ \sigma_{F} \ a}{\pi \ E} \ ln\left ( sec\frac{\pi}{2} \right )\frac{\sigma}{\sigma_{F}} </math>
|(1)
|}

{|
|-
|width="20px"|
|width="500px" | <math>sec=\frac{Hypothenuse}{Ankathete}=\frac{1}{cos\ \alpha}</math>
|
|}

with ''σ''F (yield stress) and yield point ''R''e

{|
|-
|width="20px"|
|width="500px" | <math>\sigma_{F}=\frac{R_{e}+R_{m}}{2} </math>
|
|}

{|
|-
|Rm   –
|width="4px" |
|Tensile strength
|}

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

{|
|-
|width="20px"|
|width="500px" | <math>\delta=\frac{\pi \ \sigma^2 \ a}{E \ \sigma_{F}} </math>
|(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]:

{|
|-
|width="20px"|
|width="500px" | <math>K_{IC}^{COD}=(m \ \sigma_{F} \ E \ \delta_{c})^{\frac{1}{2}} </math>
|(3)
|}

with
{|
|-
|m = 1:
|Plane stress state
|-
|m = 2:
|Plane strain state
|}

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

{| border="1px" style="border-collapse:collapse"
!! style="width:150px; background:#DCDCDC" | Material
!! style="width:150px; background:#DCDCDC" | Test speed
!! style="width:150px; background:#DCDCDC" | Test temperature
!! style="width:150px; background:#DCDCDC" | m
|-
|width="100px"|1 PP
|width="150px"|vT = 8.33 * 10-4 ms-1
|width="100px"|T = 233 K
|width="100px"|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 [[Material Value | 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 | Crack model according to DUGDALE]]
*[[Fracture Mechanics | Fracture mechanics]]
*[[Plastic Zone | Plastic zone]]
*[[Fracture Process Zone | Fracture process zone]]

'''References'''

{|
|-valign="top"
|[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
|-valign="top"
|[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
|-valign="top"
|[3]
|[[Blumenauer, Horst|Blumenauer, H.]], Pusch, G.: Technische Bruchmechanik. Deutscher Verlag für Grundstoffindustrie, Leipzig Stuttgart (1993) 3rd Edition, p. 19, (ISBN 3-342-00659-5; see [[AMK-Büchersammlung | AMK Library]] under E 29-3)
|-valign="top"
|[4]
|Anderson, T. L.: Fracture Mechanics; Fundamentals and Applications. CRC Press, Boca Rat (2005) 3rd edition, (ISBN 978-0849342608; see [[AMK-Büchersammlung | AMK Library]] under E 8-2), DOI: [https://www.taylorfrancis.com/books/mono/10.1201/9781420058215/fracture-mechanics-ted-anderson-anderson https://www.taylorfrancis.com/books/mono/10.1201/9781420058215/fracture-mechanics-ted-anderson-anderson]
|-valign="top"
|[5]
|[[Grellmann,_Wolfgang|Grellmann, W.]], [[Seidler,_Sabine|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-Büchersammlung | AMK Library]] under A 22)
|}

[[Category:Fracture Mechanics]]

Crack Tip Opening Displacement Concept (CTOD) - Revision history