Crack tip opening displacement: Difference between revisions

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[[File:Crack tip opening displacement.svg|thumb|Diagram of crack tip opening displacement (CTOD)]]

'''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, interceptintersect 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 characterisecharacterize 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|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.|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>

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"/>

<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" />

== 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=3-43–4 |pages=371-385371–385 |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.|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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== 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 |refpages=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]].

=== 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.|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>

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== Testing ==

A CTOD test is usually done on materials that undergo plastic deformation prior to failure. The testing material more or less resembles the original one, although dimensions can be reduced proportionally. Loading is done to resemble the expected load. More than 3 tests are done to minimize any experimental deviations. The dimensions of the testing material must maintain proportionality. The specimen is placed on the work table and a notch is created exactly at the centre. The crack should be generated such that the defect length is about half the depth. The load applied on the specimen is generally a three-point bending load. A type of [[strain gauge]] called a crack-mouth clip gage is used to measure the crack opening.<ref Crackname=":0" /> The crack tip plastically deforms until a critical point after which a cleavage crack is initiated that may lead to either partial or complete failure. The critical load and strain gauge measurements at the load are noted and a graph is plotted. The crack tip opening can be calculated from the length of the crack and opening at the mouth of the notch. According to the material used, the fracture can be brittle or ductile which can be concluded from the graph.

Standards for CTOD testing can be found in the ASTM E1820 - 20a code.<ref>{{Cite journal|last=E08 Committee|title=Test Method for Measurement of Fracture Toughness|url=http://www.astm.org/cgi-bin/resolver.cgi?E1820-20A|language=en|doi=10.1520/e1820-20a|url-access=subscription}}</ref>

=== Laboratory measurement ===