Fracture mechanics: Difference between revisions

[[File:Griffith-Riss-Zug.svg|thumb|right|A Griffith crack (flaw) of length <math>a</math> is in the middle<ref>{{Cite journal|last=McMeeking|first=Robert M.|date=May 2004|title=The energy release rate for a Griffith crack in a piezoelectric material|url=https://linkinghub.elsevier.com/retrieve/pii/S0013794403001358|journal=Engineering Fracture Mechanics|language=en|volume=71|issue=7–8|pages=1149–1163|doi=10.1016/S0013-7944(03)00135-8|access-date=2021-12-18|archive-date=2022-02-13|archive-url=https://web.archive.org/web/20220213000601/https://linkinghub.elsevier.com/retrieve/pii/S0013794403001358|url-status=live}}</ref><ref>{{Cite journal|last=Lenci|first=Stefano|date=2001|title=Analysis of a crack at a weak interface|url=http://link.springer.com/10.1023/A:1011041409243|journal=International Journal of Fracture|volume=108|issue=3|pages=275–290|doi=10.1023/A:1011041409243|s2cid=115306909|access-date=2021-12-18|archive-date=2023-04-17|archive-url=https://web.archive.org/web/20230417155849/https://link.springer.com/article/10.1023/A:1011041409243|url-status=live}}</ref> an infinity large material]]

Fracture mechanics was developed during World War I by English aeronautical engineer [[Alan Arnold Griffith|A. A. Griffith]] – thus the term '''Griffith crack''' – to explain the failure of brittle materials.<ref name="griffith21">{{Citation |last =Griffith |first =A. A. |author-link =Alan Arnold Griffith |title =The phenomena of rupture and flow in solids |journal =Philosophical Transactions of the Royal Society of London |series =A |volume =221 |issue =582–593 |pages =163–198 |year =1921 |doi =10.1098/rsta.1921.0006 |bibcode =1921RSPTA.221..163G |doi-access =free }}.</ref> Griffith's work was motivated by two contradictory facts:

''Griffith's work was largely ignored by the engineering community until the early 1950s. The reasons for this appear to be (a) in the actual structural materials the level of energy needed to cause fracture is orders of magnitude higher than the corresponding surface energy, and (b) in structural materials there are always some inelastic deformations around the crack front that would make the assumption of linear elastic medium with infinite stresses at the crack tip highly unrealistic.'' <ref name=Erdogan00>E. Erdogan (2000) ''Fracture Mechanics'', International Journal of Solids and Structures, 37, pp. 171–183.</ref>

Another significant achievement of Irwin and his colleagues was to find a method of calculating the amount of energy available for fracture in terms of the asymptotic stress and displacement fields around a crack front in a linear elastic solid.<ref name="Irwin57" /> This asymptotic expression for the stress field in mode I loading is related to the stress intensity factor <math>

</math> stress intensity, <math>K_c</math> the fracture toughness, and <math>\nu</math> is Poisson’sPoisson's ratio.

</math>, by:

Irwin was the first to observe that if the size of the plastic zone around a crack is small compared to the size of the crack, the energy required to grow the crack will not be critically dependent on the state of stress (the plastic zone) at the crack tip.<ref name="Erdogan00"/> In other words, a purely elastic solution may be used to calculate the amount of energy available for fracture.

Nevertheless, there must be some sort of mechanism or property of the material that prevents such a crack from propagating spontaneously. The assumption is, the plastic deformation at the crack tip effectively blunts the crack tip. This deformation depends primarily on the applied stress in the applicable direction (in most cases, this is the y-direction of a regular Cartesian coordinate system), the crack length, and the geometry of the specimen.<ref name="Purdue University">{{cite book|last1=Weisshaar|first1=Terry|title=Aerospace Structures- an Introduction to Fundamental Problems|date=July 28, 2011|publisher=Purdue University|location=West Lafayette, IN}}</ref> To estimate how this plastic deformation zone extended from the crack tip, Irwin equated the yield strength of the material to the far-field stresses of the y-direction along the crack (x direction) and solved for the effective radius. From this relationship, and assuming that the crack is loaded to the critical stress intensity factor, Irwin developed the following expression for the idealized radius of the zone of plastic deformation at the crack tip:

The same process as described above for a single event loading also applies and to cyclic loading. If a crack is present in a specimen that undergoes cyclic loading, the specimen will plastically deform at the crack tip and delay the crack growth. In the event of an overload or excursion, this model changes slightly to accommodate the sudden increase in stress from that which the material previously experienced. At a sufficiently high load (overload), the crack grows out of the plastic zone that contained it and leaves behind the pocket of the original plastic deformation. Now, assuming that the overload stress is not sufficiently high as to completely fracture the specimen, the crack will undergo further plastic deformation around the new crack tip, enlarging the zone of residual plastic stresses. This process further toughens and prolongs the life of the material because the new plastic zone is larger than what it would be under the usual stress conditions. This allows the material to undergo more cycles of loading. This idea can be illustrated further by the [http://www.afgrow.net/applications/dtdhandbook/sections/page5_2_1_0.aspx graph] of Aluminum with a center crack undergoing overloading events.<ref>{{cite web|title=Retardation|url=http://www.afgrow.net/applications/dtdhandbook/sections/page5_2_1_0.aspx|website=Handbook for Damage Tolerant Design|publisher=LexTech, Inc.|access-date=20 November 2016|archive-date=21 November 2016|archive-url=https://web.archive.org/web/20161121103334/http://www.afgrow.net/applications/dtdhandbook/sections/page5_2_1_0.aspx|url-status=live}}</ref>

When a significant region around a crack tip has undergone plastic deformation, other approaches can be used to determine the possibility of further crack extension and the direction of crack growth and branching. A simple technique that is easily incorporated into numerical calculations is the ''cohesive zone model'' method which is based on concepts proposed independently by [[G. I. Barenblatt|Barenblatt]]<ref name=baren>{{citation |title=The mathematical theory of equilibrium cracks in brittle fracture |author=Barenblatt, G. I. |journal=Advances in Applied Mechanics |volume=7 |pages=55–129 |year=1962 |doi=10.1016/s0065-2156(08)70121-2 |isbn=9780120020072 |url=https://hal.archives-ouvertes.fr/hal-03601989/file/barenblatt1962.pdf |access-date=2022-06-08 |archive-date=2023-04-17 |archive-url=https://web.archive.org/web/20230417155917/https://hal.science/hal-03601989/file/barenblatt1962.pdf |url-status=live }}</ref> and Dugdale<ref name=dug>{{citation|title=Yielding of steel sheets containing slits|author=Dugdale, D. S.|journal=Journal of the Mechanics and Physics of Solids|volume=8|number=2 |pages=100–104|year=1960|doi=10.1016/0022-5096(60)90013-2|bibcode = 1960JMPSo...8..100D |s2cid=136484892 }}</ref> in the early 1960s. The relationship between the Dugdale-Barenblatt models and Griffith's theory was first discussed by [[John R. Willis|Willis]] in 1967.<ref name=willis>{{citation|title=A comparison of the fracture criteria of Griffith and Barenblatt |author=Willis, J. R. |journal=Journal of the Mechanics and Physics of Solids|volume=15|number=3 |pages=151–162|year=1967 |doi=10.1016/0022-5096(67)90029-4|bibcode = 1967JMPSo..15..151W }}.</ref> The equivalence of the two approaches in the context of brittle fracture was shown by [[James R. Rice|Rice]] in 1968.<ref name=rice/>

'''Concrete fracture analysis''' is part of [[fracture mechanics]] that studies crack propagation and related failure modes in [[concrete]].<ref name="epfl">{{cite web |author= |title=Fracture Mechanics for Structural Concrete |url=http://ibeton.epfl.ch/Etudiant/et_doct-11/Courses/2011-11-23%20Topic%207%20Fracture.pdf |accessdate=13 April 2013}}</ref> As it is widely used in construction, fracture analysis and modes of reinforcement are an important part of the study of concrete, and different concretes are characterized in part by their fracture properties.<ref name="saouma">Lecture Notes in Fracture Mechanics by Victor E. Saouma</ref> Common fractures include the [[Concrete cone failure|cone-shaped fractures]] that form around anchors under tensile strength.

<math>\sigma</math> = <math>\tau</math> / √(1+{<math>d</math>/<math>\lambda</math><math>\delta</math>}),<ref name="bazant1" /><ref>Bažant, Z. P., and Pang, S.-D. (2006) “Mechanics"Mechanics based statistics of failure risk of quasibrittle structures and size effect on safety factors".” ''Proc. Nat'l Acad. Sci., USA'' 103 (25), pp.&nbsp;9434–9439</ref> </blockquote>

where <math>\sigma</math> = stress intensity factor, <math>\tau</math> = tensile strength, <math>d</math> = size of specimen, <math>\delta</math> = maximum aggregate size, and <math>\lambda</math> = an empirical constant.