Crack tip opening displacement: Difference between revisions

Revision as of 19:52, 29 June 2020

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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.|year=|isbn=0-203-59686-2|edition=2nd ed|___location=London|pages=150|oclc=57491375}}</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|year=|isbn=0-8247-8900-8|___location=|pages=|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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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" />

== 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=371-385371–385 |publisher=Elsevier |date=2003}}</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|year=|isbn=0-8247-8900-8|___location=|pages=|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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== Relation with other crack tip parameters ==

=== K and CTOD ===

CTOD can be expressed in terms of stress intensity factor <math>K</math> as<ref>{{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 |ref=harv |pp=104-105104–105}}</ref>

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

where <math>\sigma_\text{y}</math> is the yield strength, <math>E</math> is Young's modulus and <math>m=1</math> for [[plane stress]] and <math>m=2</math> for [[plane strain]].