Crack tip opening displacement: Difference between revisions

Revision as of 07:30, 29 June 2014

Latest revision as of 01:08, 19 July 2025

(48 intermediate revisions by 20 users not shown)

Line 1:

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

{{multiple issues|

'''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]].

{{Technical|date=April 2013}}

{{Copy edit|date=April 2013}}

}}

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>

[[Fracture_mechanics#Linear_elastic_fracture_mechanics|LEFM]] (Linear elastic [[fracture mechanics]]) is not valid. This assumption was proven when a unique relationship between Crack Tip Opening Displacement (CTOD) and the [[J integral]] was established.<ref>T L Anderson, "Fracture Mechanics Fundamentals and Applications", CRC Press 2005</ref> In the case of the ductile materials, the crack faces move prior to fracture and create a blunt crack tip. This crack opening can be used as a measure of the toughness of the material. This parameter is known as '''Crack Tip Opening Displacement''' or commonly in the form of abbreviation '''CTOD'''.

<ref>T. L. Andreson, Frcature Mechanics: Fundamentals and Applications, Taylor and Fransis Pub, pp103</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>

==Definition==

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.

There are a number of alternative definitions of Crack Tip opening displacement. The most common definition is the displacement at the original [[fracture|crack]] tip and the 90° intercept. These are equivalent if the [[fracture|crack]] blunts in a semicircle.

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" />

== Relationship between K and CTOD ==

== History ==

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

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 being considered as a measure of fracture toughness. TodayThe COD was originally independently proposed by [[Alan Cottrell]] and A. A. Wells.<ref>A. A. Wells, this''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 isbecame 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.

CTOD can be expressed in terms of K as <math> \frac{K^2}{\sigma_y E} </math>

== Use as a design parameter ==

<ref>T. L. Andreson, Frcature Mechanics: Fundamentals and Applications, Taylor and Fransis Pub, pp104-105</ref>

CTOD is a single parameter that accommodates crack 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.

In case of the small scale yielding conditions the same can be expressed as

<ref>T. L. Andreson, Frcature Mechanics: Fundamentals and Applications, Taylor and Fransis Pub, pp107-108</ref>

== Relation with other crack tip parameters ==

==Relationship between J-integral and CTOD ==

== 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 |pages=104–105}}</ref>

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

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

For Linear Elastic Conditions, the relationship between CTOD and G is given by Equation

<ref>T. L. Andreson, Frcature Mechanics: Fundamentals and Applications, Taylor and Fransis Pub, pp120-122</ref>

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]].

between CTOD and J in the limit of Small-Scale Yielding.

== Relationship between= 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>

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

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

=== CTODJ-integral testingand 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>

Crack Tip Opening Displacement test is usually done on materials that undergoes plastic deformation prior to failure. The testing material should more or less resemble the original one though dimensions can be reduced proportionally. Loading is also done so as to resemble the actual load expected. More than 3 testing are done so as to ensure any experimental deviations are minimum. There is also an inter relationship between the dimensions of the testing material to ensure proportionality is maintained throughout 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 length of defect reaches a value of 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 which may lead to either partial or complete failure. The critical load and strain gauge measurements at the load is noted and a graph is plotted. Crack tip opening can be calculated from the length of the crack and opening at the mouth of the notch. According to material used fracture can be brittle or ductile which can be concluded from graph plotted

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

===Laboratory measurement===

== Testing ==

Early experiments used a flat paddle-shaped gauge that was inserted into the crack; as the crack opened, the paddle gauge rotated, and an electronic signal was 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>

Crack Tip OpeningA DisplacementCTOD test is usually done on materials that undergoesundergo plastic deformation prior to failure. The testing material should more or less resembleresembles the original one, thoughalthough dimensions can be reduced proportionally. Loading is also done so as to resemble the actualexpected load expected. More than 3 testingtests are done so as to ensureminimize any experimental deviations are minimum. There is also an inter relationship between theThe dimensions of the testing material tomust ensuremaintain proportionality is maintained throughout. 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 length of defect reacheslength a value ofis 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 whichthat may lead to either partial or complete failure. The critical load and strain gauge measurements at the load isare noted and a graph is plotted. CrackThe 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 graphthe plottedgraph.

==Application==

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

Examination of fractured test specimens led to the observation that the crack faces had moved apart prior to fracture, due to 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 the opening at the crack tip being considered as a measure of fracture toughness. Today, this parameter is 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.

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>

== Advantages and disadvantages of using CTOD as a design parameter==

*'''Advantages'''

** Single parameter which takes care for the cracktip plasticity also

** Easy to measure when compared with the similar techniques like J integral etc.

** A fracture parameter which has more physical meaning than the rest

* '''Disadvantages'''

**The equavalance of CTOD and J integral is proved only for non-linear materials, and not for plastic materials

**Hard to expand the concept of CTOD for large deformation conditions

**It is easier to calculate J-integral in case of a design process using FEM techniques

=== Laboratory measurement ===

==References==

Early experiments used a flatflat, paddle-shaped gauge that was inserted into the crack; as the crack openedopens, the paddle gauge rotated,rotates and an electronic signal wasis sent to an x–y plotter. This method was inaccurate, however, because it was difficultdifficult 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>

== References ==

<references/>