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)]]

{{Technical|date=April 2013}}

'''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|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>

'''Crack tip opening displacement''' (CTOD) is the displacement at the original [[fracture|crack]] tip and the 90° intercept.

<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>

[[Fracture mechanics#Linear elastic fracture mechanics|LEFM]] (Linear elastic [[fracture mechanics]]) is not valid. This assumption was proven when a unique relationship between CTOD and the [[J integral]] was established.{{sfn|Anderson|2005}}

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.

== Relationship between K and CTOD ==

{{Expand section|date=May 2013|reason=[[hybrid PDF]], a variant of [[LibreOffice]] isn't mentioned}}

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" />

CTOD can be expressed in terms of K as <math> \frac{K^2}{\sigma_y E m} </math>{{sfn|Anderson|2005|pp=104-105}}

that in this case m=1 for plane stress and 2 for plane strain.

== TestingHistory ==

== Relationship between J-integral and CTOD ==

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",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.|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.

{{Expand section|date=May 2013|reason=[[hybrid PDF]], a variant of [[LibreOffice]] is not mentioned}}

== Use as a design parameter ==

For Linear Elastic Conditions, the relationship between CTOD and G is given by an equation{{sfn|Anderson|2005|pp=120-122}} between CTOD and J in the limit of Small-Scale Yielding.

CTOD is a single parameter that accommodates cracktipcrack tip plasticity. It is easy to measure when compared with techniques such as J integral. It is a fracture parameter that has more physical meaning than the rest.

However, the equivalence of CTOD and J integral is proven only for non-linear materials, but not for plastic materials. It is hard to expand the concept of CTOD for large deformations. It is easier to calculate J-integral in case of a design process using [[finite element method]] techniques.

== Relationship between G and CTOD ==

{{Expand section|date=May 2013|reason=[[hybrid PDF]], a variant of [[LibreOffice]] is not mentioned}}

== Relation with other crack tip parameters ==

Since J = G for linear elastic material behavior, these equations also describe the relationship between the G and CTOD also.

== Relationship between= 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=harv104–105}}</ref>

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

== Testing ==

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]].

A CTOD test is usually done on materials that undergoes 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 maintained 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 [[strain gauge]] is used to measure the crack opening. 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.

=== LaboratoryG measurementand 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>

Early experiments used a flat, paddle-shaped gauge that was inserted into the crack; as the crack opens, the paddle gauge rotates and an electronic signal is sent to an x–y plotter. This method was inaccurate, however, because it was difficult to reach the crack tip with the paddle gauge. Today, the displacement V at the crack mouth is measured and the CTOD is inferred by assuming that the specimen halves are rigid and rotate about a hinge point.<ref>B E Amstutz, M A Sutton, D S Dawicke"An Experimental study of CTOD for mode I/mode II stable crack growth in thin aluminium specimens", ASTM Special 1995</ref>

=== ApplicationJ-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>

{{Expand section|date=May 2013|reason=[[hybrid PDF]], a variant of [[LibreOffice]] isn't mentioned}}

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

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>J C Newman, M A James, U Zerbst, "Engineering Fracture mechanics", Elsevier 2003</ref> This observation led to considering the opening at the crack tip as a measure of fracture toughness. This parameter became known as CTOD. 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.

where the variable <math>d_n</math> is typically between 0.3 and 0.8.

== Use as a design parameter ==

== SourcesTesting ==

CTOD is a single parameter that accommodates cracktip plasticity. It is easy to measure when compared with techniques such as J integral. It is a fracture parameter that has more physical meaning than the rest.

A CTOD test is usually done on materials that undergoesundergo 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 maintainedmaintain 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.

However, the equivalence of CTOD and J integral is proven only for non-linear materials, but not for plastic materials. It is hard to expand the concept of CTOD for large deformations. It is easier to calculate J-integral in case of a design process using [[finite element method]] techniques.

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>

== References ==

=== Laboratory measurement ===

<references/>

Early experiments used a flat, paddle-shaped gauge that was inserted into the crack; as the crack opens, the paddle gauge rotates and an electronic signal is sent to an x–y plotter. This method was inaccurate, however, because it was difficult to reach the crack tip with the paddle gauge. Today, the displacement V at the crack mouth is measured and the CTOD is inferred by assuming that the specimen halves are rigid and rotate about a hinge point.<ref>B E Amstutz, M A Sutton, D S Dawicke"An Experimental study of CTOD for mode I/mode II stable crack growth in thin aluminium specimens", ASTM Special 1995</ref>

== Sources ==

== References ==

* {{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}}

<references/>

[[Category:Fracture mechanics]]