Crack tip opening displacement: Difference between revisions

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Latest revision as of 01:08, 19 July 2025

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'''Crack tip opening displacement''' ('''CTOD''') or <math>\delta_\text{t}</math> is the distance between the opposite faces of a [[fracture|crack]] tip at the 90° intercept position. The position behind the crack tip at which the distance is measured is arbitrary but commonly used is the point where two 45° lines, starting at the crack tip, intersect the crack faces.<ref name="suresh04">{{cite book |last1=Suresh |first1=S. |date=2004 |title=Fatigue of Materials |publisher=Cambridge University Press |isbn=978-0-521-57046-6}}</ref> The parameter is used in [[fracture mechanics]] to characterize the loading on a crack and can be related to other crack tip loading parameters such as the [[stress intensity factor]] <math>K</math> and the elastic-plastic [[J-integral]].

For [[plane stress]] conditions, the CTOD can be written as:<ref>{{Cite book|last=Janssen|first=Michael|url=https://www.worldcat.org/oclc/57491375|title=Fracture mechanics|date=2004|publisher=Spon Press|others=Zuidema, J. (Jan), Wanhill, R. J. H.|isbn=0-203-59686-2|edition=2nd|___location=London|pages=150|oclc=57491375}}</ref><ref name=":0">{{Cite book|last=Soboyejo|first=W. O.|url=http://worldcat.org/oclc/300921090|title=Mechanical properties of engineered materials|date=2003|publisher=Marcel Dekker|isbn=0-8247-8900-8|chapter=11.6.3 Plastic Zone Size|oclc=300921090}}</ref>

<math>\delta_\text{t} = \left(\frac{8\sigma_\text{ys}a}{\pi E}\right)\ln\left[\sec\left(\frac{\pi \sigma^\infty}{2\sigma_\text{ys}}\right)\right]</math>

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== History ==

Examination of fractured test specimens led to the observation that the crack faces had moved apart prior to fracture, due to the blunting of an initially sharp crack by plastic deformation. The degree of crack blunting increased in proportion to the toughness of the material.<ref>{{cite journal |first1=J. C. |last1=Newman Jr.|first2=M. A. |last2=James |first3=U. |last3=Zerbst |title=A review of the CTOA/CTOD fracture criterion |journal=Engineering Fracture Mechanics |volume=30 |issue=3–4 |pages=371–385 |publisher=Elsevier |date=2003|doi=10.1016/S0013-7944(02)00125-X }}</ref> This observation led to considering the opening at the crack tip as a measure of fracture toughness. The COD was originally independently proposed by [[Alan Cottrell]] and A. A. Wells.<ref>A. A. Wells, ''Crack Propagation Symposium'', Cranfield, (1961) 210</ref><ref>{{Cite book|last=Soboyejo|first=W. O.|url=http://worldcat.org/oclc/300921090|title=Mechanical properties of engineered materials|date=2003|publisher=Marcel Dekker|isbn=0-8247-8900-8|chapter=11.7.1 Crack Opening Displacement|oclc=300921090}}</ref> This parameter became known as CTOD. [[George Rankine Irwin|G. R. Irwin]] later postulated that crack-tip plasticity makes the crack behave as if it were slightly longer. Thus, estimation of CTOD can be done by solving for the displacement at the physical crack tip.

== Use as a design parameter ==

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=== J-integral and CTOD ===

The relationship between the CTOD and J is given by:<ref name="suresh04"/><ref>{{Cite book|last=Zehnder|first=Alan T.|url=https://www.worldcat.org/oclc/773034407|title=Fracture mechanics|date=3 January 2012 |isbn=978-94-007-2595-9|___location=Dordrecht|pages=172|oclc=773034407}}</ref>

:<math>\delta_\text{t} = d_n \frac{J}{\sigma_\text{y}}</math>