Crack tip opening displacement: Difference between revisions

Revision as of 17:23, 26 April 2023

Revision as of 18:49, 7 October 2024

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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>

where <math>\sigma_\text{ys}</math> is the [[Yield (engineering)|yield stress]], <math>a</math> is the crack length, <math>E</math> is the [[Young's modulus]] , and <math>\sigma^\infty</math> is the remote applied stress.

Under [[fatigue (material)|fatigue]] loading, the range of movement of the crack tip during a loading cycle <math>\Delta\delta_\text{t}</math> can be used for determining the rate of fatigue growth using a [[crack growth equation]]. The crack extension for a cycle <math>da/dN</math>, is typically of the order of <math>\Delta\delta_\text{t}</math>.<ref name="suresh04" />

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== Relation with other crack tip parameters ==

=== K and CTOD ===

CTOD can be expressed in terms of stress intensity factor <math>K</math> as: <ref name=":1">{{cite book |first=T. L. |last=Anderson |title=Fracture Mechanics: Fundamentals and Applications |edition=Third |url={{google books |plainurl=y |id=MxrtsC-ZooQC}}|date=24 June 2005|publisher=CRC Press|isbn=978-0-8493-1656-2 |pages=104–105}}</ref>

:<math>\delta_\text{t} = \frac{4}{\pi}\frac{K^2}{m\sigma_\text{y} E} </math>

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=== G and CTOD ===

CTOD can be related to the energy release rate G as: <ref name=":1" />

<math>\delta_t= \frac{4}{\pi} \frac{G}{\sigma_{y}}</math>

=== 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>