Chapter 9

 

Modeling Material Failure

 

 

9.3 Modeling failure by crack growth MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqabKqzahaeaa aaaaaaa8qacaWFtacaaa@3847@  linear elastic fracture mechanics

 

Phenomenological damage models are useful in design applications, but they have many limitations, including

 They require extensive experimental testing to calibrate the model for each application;

 They provide no insight into the relationship between a materials microstructure and its strength.

 

A more sophisticated approach is to model the mechanisms of failure directly.  Crack propagation through the solid, either as a result of fatigue, or by brittle or ductile fracture, is by far the most common cause of failure.  Consequently much effort has been devoted to developing techniques to predict the behavior of cracks in solids.  Below, we outline some of the most important results.

 

 

9.3.1 Crack tip fields in an isotropic, linear elastic solid.

 

Many of the techniques of fracture mechanics rely on the assumption that, if one gets sufficiently close to the tip of the crack, the stress, displacement and strain fields always have the same distribution, regardless of the geometry of the solid and how it is loaded.  The fields near a crack tip are a fundamental result in fracture mechanics.

 

The picture shows an infinitely large linear elastic solid, with Young’s modulus E and Poisson’s ratio ν MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8sk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaH9oGBaaa@348E@ , which contains a crack.  The solid is loaded at infinity.  Note that

 Crack tip fields are most conveniently expressed in terms of cylindrical-polar coordinates (r,θ,z) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8sk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaGGOaGaamOCaiaacYcacqaH4oqCca GGSaGaamOEaiaacMcaaaa@393B@  with origin at the crack tip;

 The displacement and stress near the crack tip can be characterized by three numbers K I , K II , K III MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8sk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGlbWaaSbaaSqaaiaadMeaaeqaaO GaaiilaiaadUeadaWgaaWcbaGaamysaiaadMeaaeqaaOGaaiilaiaa dUeadaWgaaWcbaGaamysaiaadMeacaWGjbaabeaaaaa@3C12@ , known as stress intensity factors. By definition

K I = lim r0 2πr σ 22 K II = lim r0 2πr σ 12 K III = lim r0 2πr σ 32 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4samaaBa aaleaacaWGjbaabeaakiabg2da9maaxababaGaciiBaiaacMgacaGG TbaaleaacaWGYbGaeyOKH4QaaGimaaqabaGcdaGcaaqaaiaaikdacq aHapaCcaWGYbaaleqaaOGaeq4Wdm3aaSbaaSqaaiaaikdacaaIYaaa beaakiaaykW6caaMcSUaaGPaRlaaykW6caaMcSUaaGPaRlaaykW6ca aMcSUaaGPaRlaaykW6caaMcSUaaGPaRlaadUeadaWgaaWcbaGaamys aiaadMeaaeqaaOGaeyypa0ZaaCbeaeaaciGGSbGaaiyAaiaac2gaaS qaaiaadkhacqGHsgIRcaaIWaaabeaakmaakaaabaGaaGOmaiabec8a WjaadkhaaSqabaGccqaHdpWCdaWgaaWcbaGaaGymaiaaikdaaeqaaO GaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7 caaMc8UaaGPaVlaadUeadaWgaaWcbaGaamysaiaadMeacaWGjbaabe aakiabg2da9maaxababaGaciiBaiaacMgacaGGTbaaleaacaWGYbGa eyOKH4QaaGimaaqabaGcdaGcaaqaaiaaikdacqaHapaCcaWGYbaale qaaOGaeq4Wdm3aaSbaaSqaaiaaiodacaaIYaaabeaaaaa@8CD2@

with the limit taken along θ=0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8sk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaH4oqCcqGH9aqpcaaIWaaaaa@364C@ .

 The stress intensity factors depend on the detailed shape of the solid, and the way that it is loaded. To calculate stress intensity factors, you need to find the full stress field in the solid, and then compute the limiting values in the definition.  These calculations can be difficult MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa aaaaaaa8qacaWFtacaaa@37E6@  you can try to find the solution in standard tables of stress intensity factors, or if this fails use a numerical method (such as FEM). A short table of stress intensity factors for various crack geometries can be found in Section 9.3.3, and FEM techniques are discussed in 9.3.4.

 Stress intensity factors have the bizarre units of N m 3/2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOtaiaad2 gadaahaaWcbeqaaiabgkHiTiaaiodacaGGVaGaaGOmaaaaaaa@3B01@ .

* The physical significance of the three stress intensity factors is illustrated in the picture below.  The `Mode I’ stress intensity factor K I MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8sk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGlbWaaSbaaSqaaiaadMeaaeqaaa aa@34A0@  quantifies the crack opening displacements and stresses; the `Mode II’ stress intensity factor characterizes in-plane shear displacements and stress; and the `Mode III’ stress intensity factor quantifies out-of-plane shear displacement of the crack faces and anti-plane shear stresses at the crack tip.

 

 

The stress field near the crack tip is

σ rr = K I 2πr ( 5 4 cos θ 2 1 4 cos 3θ 2 )+ K II 2πr ( 5 4 sin θ 2 + 3 4 sin 3θ 2 ) σ θθ = K I 2πr ( 3 4 cos θ 2 + 1 4 cos 3θ 2 ) K II 2πr ( 3 4 sin θ 2 + 3 4 sin 3θ 2 ) σ rθ = K I 2πr ( 1 4 sin θ 2 + 1 4 sin 3θ 2 )+ K II 2πr ( 1 4 cos θ 2 + 3 4 cos 3θ 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacqaHdp WCdaWgaaWcbaGaamOCaiaadkhaaeqaaOGaeyypa0ZaaSaaaeaacaWG lbWaaSbaaSqaaiaadMeaaeqaaaGcbaWaaOaaaeaacaaIYaGaeqiWda NaamOCaaWcbeaaaaGcdaqadaqaamaalaaabaGaaGynaaqaaiaaisda aaGaci4yaiaac+gacaGGZbWaaSaaaeaacqaH4oqCaeaacaaIYaaaai abgkHiTmaalaaabaGaaGymaaqaaiaaisdaaaGaci4yaiaac+gacaGG ZbWaaSaaaeaacaaIZaGaeqiUdehabaGaaGOmaaaaaiaawIcacaGLPa aacqGHRaWkdaWcaaqaaiaadUeadaWgaaWcbaGaamysaiaadMeaaeqa aaGcbaWaaOaaaeaacaaIYaGaeqiWdaNaamOCaaWcbeaaaaGcdaqada qaaiabgkHiTmaalaaabaGaaGynaaqaaiaaisdaaaGaci4CaiaacMga caGGUbWaaSaaaeaacqaH4oqCaeaacaaIYaaaaiabgUcaRmaalaaaba GaaG4maaqaaiaaisdaaaGaci4CaiaacMgacaGGUbWaaSaaaeaacaaI ZaGaeqiUdehabaGaaGOmaaaaaiaawIcacaGLPaaaaeaacqaHdpWCda WgaaWcbaGaeqiUdeNaeqiUdehabeaakiabg2da9maalaaabaGaam4s amaaBaaaleaacaWGjbaabeaaaOqaamaakaaabaGaaGOmaiabec8aWj aadkhaaSqabaaaaOWaaeWaaeaadaWcaaqaaiaaiodaaeaacaaI0aaa aiGacogacaGGVbGaai4CamaalaaabaGaeqiUdehabaGaaGOmaaaacq GHRaWkdaWcaaqaaiaaigdaaeaacaaI0aaaaiGacogacaGGVbGaai4C amaalaaabaGaaG4maiabeI7aXbqaaiaaikdaaaaacaGLOaGaayzkaa GaeyOeI0YaaSaaaeaacaWGlbWaaSbaaSqaaiaadMeacaWGjbaabeaa aOqaamaakaaabaGaaGOmaiabec8aWjaadkhaaSqabaaaaOWaaeWaae aadaWcaaqaaiaaiodaaeaacaaI0aaaaiGacohacaGGPbGaaiOBamaa laaabaGaeqiUdehabaGaaGOmaaaacqGHRaWkdaWcaaqaaiaaiodaae aacaaI0aaaaiGacohacaGGPbGaaiOBamaalaaabaGaaG4maiabeI7a XbqaaiaaikdaaaaacaGLOaGaayzkaaaabaGaeq4Wdm3aaSbaaSqaai aadkhacqaH4oqCaeqaaOGaeyypa0ZaaSaaaeaacaWGlbWaaSbaaSqa aiaadMeaaeqaaaGcbaWaaOaaaeaacaaIYaGaeqiWdaNaamOCaaWcbe aaaaGcdaqadaqaamaalaaabaGaaGymaaqaaiaaisdaaaGaci4Caiaa cMgacaGGUbWaaSaaaeaacqaH4oqCaeaacaaIYaaaaiabgUcaRmaala aabaGaaGymaaqaaiaaisdaaaGaci4CaiaacMgacaGGUbWaaSaaaeaa caaIZaGaeqiUdehabaGaaGOmaaaaaiaawIcacaGLPaaacqGHRaWkda WcaaqaaiaadUeadaWgaaWcbaGaamysaiaadMeaaeqaaaGcbaWaaOaa aeaacaaIYaGaeqiWdaNaamOCaaWcbeaaaaGcdaqadaqaamaalaaaba GaaGymaaqaaiaaisdaaaGaci4yaiaac+gacaGGZbWaaSaaaeaacqaH 4oqCaeaacaaIYaaaaiabgUcaRmaalaaabaGaaG4maaqaaiaaisdaaa Gaci4yaiaac+gacaGGZbWaaSaaaeaacaaIZaGaeqiUdehabaGaaGOm aaaaaiaawIcacaGLPaaaaaaa@D47A@

Equivalent expressions in rectangular coordinates are

σ 11 = K I 2πr cos θ 2 ( 1sin θ 2 sin 3θ 2 ) K II 2πr sin θ 2 ( 2+cos θ 2 cos 3θ 2 ) σ 22 = K I 2πr cos θ 2 ( 1+sin θ 2 sin 3θ 2 )+ K II 2πr cos θ 2 sin θ 2 cos 3θ 2 σ 12 = K I 2πr cos θ 2 sin θ 2 cos 3θ 2 + K II 2πr cos θ 2 ( 1sin θ 2 sin 3θ 2 ) σ 31 = K III 2πr sinθ/2 σ 32 = K III 2πr cosθ/2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacqaHdp WCdaWgaaWcbaGaaGymaiaaigdaaeqaaOGaeyypa0ZaaSaaaeaacaWG lbWaaSbaaSqaaiaadMeaaeqaaaGcbaWaaOaaaeaacaaIYaGaeqiWda NaamOCaaWcbeaaaaGcciGGJbGaai4BaiaacohadaWcaaqaaiabeI7a XbqaaiaaikdaaaWaaeWaaeaacaaIXaGaeyOeI0Iaci4CaiaacMgaca GGUbWaaSaaaeaacqaH4oqCaeaacaaIYaaaaiGacohacaGGPbGaaiOB amaalaaabaGaaG4maiabeI7aXbqaaiaaikdaaaaacaGLOaGaayzkaa GaeyOeI0YaaSaaaeaacaWGlbWaaSbaaSqaaiaadMeacaWGjbaabeaa aOqaamaakaaabaGaaGOmaiabec8aWjaadkhaaSqabaaaaOGaci4Cai aacMgacaGGUbWaaSaaaeaacqaH4oqCaeaacaaIYaaaamaabmaabaGa aGOmaiabgUcaRiGacogacaGGVbGaai4CamaalaaabaGaeqiUdehaba GaaGOmaaaaciGGJbGaai4BaiaacohadaWcaaqaaiaaiodacqaH4oqC aeaacaaIYaaaaaGaayjkaiaawMcaaaqaaiabeo8aZnaaBaaaleaaca aIYaGaaGOmaaqabaGccqGH9aqpdaWcaaqaaiaadUeadaWgaaWcbaGa amysaaqabaaakeaadaGcaaqaaiaaikdacqaHapaCcaWGYbaaleqaaa aakiGacogacaGGVbGaai4CamaalaaabaGaeqiUdehabaGaaGOmaaaa daqadaqaaiaaigdacqGHRaWkciGGZbGaaiyAaiaac6gadaWcaaqaai abeI7aXbqaaiaaikdaaaGaci4CaiaacMgacaGGUbWaaSaaaeaacaaI ZaGaeqiUdehabaGaaGOmaaaaaiaawIcacaGLPaaacqGHRaWkdaWcaa qaaiaadUeadaWgaaWcbaGaamysaiaadMeaaeqaaaGcbaWaaOaaaeaa caaIYaGaeqiWdaNaamOCaaWcbeaaaaGcciGGJbGaai4Baiaacohada WcaaqaaiabeI7aXbqaaiaaikdaaaGaci4CaiaacMgacaGGUbWaaSaa aeaacqaH4oqCaeaacaaIYaaaaiGacogacaGGVbGaai4Camaalaaaba GaaG4maiabeI7aXbqaaiaaikdaaaaabaGaeq4Wdm3aaSbaaSqaaiaa igdacaaIYaaabeaakiabg2da9maalaaabaGaam4samaaBaaaleaaca WGjbaabeaaaOqaamaakaaabaGaaGOmaiabec8aWjaadkhaaSqabaaa aOGaci4yaiaac+gacaGGZbWaaSaaaeaacqaH4oqCaeaacaaIYaaaai GacohacaGGPbGaaiOBamaalaaabaGaeqiUdehabaGaaGOmaaaaciGG JbGaai4BaiaacohadaWcaaqaaiaaiodacqaH4oqCaeaacaaIYaaaai abgUcaRmaalaaabaGaam4samaaBaaaleaacaWGjbGaamysaaqabaaa keaadaGcaaqaaiaaikdacqaHapaCcaWGYbaaleqaaaaakiGacogaca GGVbGaai4CamaalaaabaGaeqiUdehabaGaaGOmaaaadaqadaqaaiaa igdacqGHsislciGGZbGaaiyAaiaac6gadaWcaaqaaiabeI7aXbqaai aaikdaaaGaci4CaiaacMgacaGGUbWaaSaaaeaacaaIZaGaeqiUdeha baGaaGOmaaaaaiaawIcacaGLPaaaaeaacqaHdpWCdaWgaaWcbaGaaG 4maiaaigdaaeqaaOGaeyypa0JaeyOeI0YaaSaaaeaacaWGlbWaaSba aSqaaiaadMeacaWGjbGaamysaaqabaaakeaadaGcaaqaaiaaikdacq aHapaCcaWGYbaaleqaaaaakiGacohacaGGPbGaaiOBaiabeI7aXjaa c+cacaaIYaGaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaG PaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaM c8UaaGPaVlaaykW7caaMc8UaaGPaVlabeo8aZnaaBaaaleaacaaIZa GaaGOmaaqabaGccqGH9aqpdaWcaaqaaiaadUeadaWgaaWcbaGaamys aiaadMeacaWGjbaabeaaaOqaamaakaaabaGaaGOmaiabec8aWjaadk haaSqabaaaaOGaci4yaiaac+gacaGGZbGaeqiUdeNaai4laiaaikda aaaa@1CE9@

while the displacements can be calculated by integrating the strains, with the result

u 1 = K I μ r 2π [ 12ν+ sin 2 θ 2 ]cos θ 2 + K II μ r 2π [ 22ν+ cos 2 θ 2 ]sin θ 2 u 2 = K I μ r 2π [ 22ν cos 2 θ 2 ]sin θ 2 + K II μ r 2π [ 1+2ν+ sin 2 θ 2 ]cos θ 2 u 3 = K III μ 2r π sinθ/2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacaWG1b WaaSbaaSqaaiaaigdaaeqaaOGaeyypa0ZaaSaaaeaacaWGlbWaaSba aSqaaiaadMeaaeqaaaGcbaGaeqiVd0gaamaakaaabaWaaSaaaeaaca WGYbaabaGaaGOmaiabec8aWbaaaSqabaGcdaWadaqaaiaaigdacqGH sislcaaIYaGaeqyVd4Maey4kaSIaci4CaiaacMgacaGGUbWaaWbaaS qabeaacaaIYaaaaOWaaSaaaeaacqaH4oqCaeaacaaIYaaaaaGaay5w aiaaw2faaiGacogacaGGVbGaai4CamaalaaabaGaeqiUdehabaGaaG OmaaaacqGHRaWkdaWcaaqaaiaadUeadaWgaaWcbaGaamysaiaadMea aeqaaaGcbaGaeqiVd0gaamaakaaabaWaaSaaaeaacaWGYbaabaGaaG Omaiabec8aWbaaaSqabaGcdaWadaqaaiaaikdacqGHsislcaaIYaGa eqyVd4Maey4kaSIaci4yaiaac+gacaGGZbWaaWbaaSqabeaacaaIYa aaaOWaaSaaaeaacqaH4oqCaeaacaaIYaaaaaGaay5waiaaw2faaiGa cohacaGGPbGaaiOBamaalaaabaGaeqiUdehabaGaaGOmaaaaaeaaca WG1bWaaSbaaSqaaiaaikdaaeqaaOGaeyypa0ZaaSaaaeaacaWGlbWa aSbaaSqaaiaadMeaaeqaaaGcbaGaeqiVd0gaamaakaaabaWaaSaaae aacaWGYbaabaGaaGOmaiabec8aWbaaaSqabaGcdaWadaqaaiaaikda cqGHsislcaaIYaGaeqyVd4MaeyOeI0Iaci4yaiaac+gacaGGZbWaaW baaSqabeaacaaIYaaaaOWaaSaaaeaacqaH4oqCaeaacaaIYaaaaaGa ay5waiaaw2faaiGacohacaGGPbGaaiOBamaalaaabaGaeqiUdehaba GaaGOmaaaacqGHRaWkdaWcaaqaaiaadUeadaWgaaWcbaGaamysaiaa dMeaaeqaaaGcbaGaeqiVd0gaamaakaaabaWaaSaaaeaacaWGYbaaba GaaGOmaiabec8aWbaaaSqabaGcdaWadaqaaiabgkHiTiaaigdacqGH RaWkcaaIYaGaeqyVd4Maey4kaSIaci4CaiaacMgacaGGUbWaaWbaaS qabeaacaaIYaaaaOWaaSaaaeaacqaH4oqCaeaacaaIYaaaaaGaay5w aiaaw2faaiGacogacaGGVbGaai4CamaalaaabaGaeqiUdehabaGaaG OmaaaaaeaacaWG1bWaaSbaaSqaaiaaiodaaeqaaOGaeyypa0ZaaSaa aeaacaWGlbWaaSbaaSqaaiaadMeacaWGjbGaamysaaqabaaakeaacq aH8oqBaaWaaOaaaeaadaWcaaqaaiaaikdacaWGYbaabaGaeqiWdaha aaWcbeaakiGacohacaGGPbGaaiOBaiabeI7aXjaac+cacaaIYaaaaa a@BA46@

Note that the formulas for in-plane displacement components u 1 , u 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWG1bWaaSbaaSqaaiaaigdaaeqaaO GaaiilaiaadwhadaWgaaWcbaGaaGOmaaqabaaaaa@3743@  are valid for plane strain deformation only.

 

 

9.3.2 The assumptions and application of phenomenological linear elastic fracture mechanics

 

The objective of linear elastic fracture mechanics is to predict the critical loads that will cause a crack in a solid to grow.  For applications involving fatigue or dynamic fracture, the rate and direction of crack growth are also of interest.

 

The phenomenological theory is based on the following qualitative argument.  Consider a crack in a reasonably brittle, isotropic solid.  If the solid is ideally elastic, we expect the asymptotic solution listed in the preceding section to become progressively more accurate as we approach the crack tip.  Away from the crack tip, the fields are influenced by the geometry of the solid and boundary conditions, and the asymptotic crack tip field is not accurate.  In practice, the asymptotic field will also not give an accurate representation of the stress fields very close to the crack tip. The crack may not be perfectly sharp at its tip, and if it were, no solid could withstand the infinite stress predicted by the asymptotic linear elastic solution.  We therefore anticipate that in practice the linear elastic solution will not be accurate very close to the crack tip itself, where material nonlinearity and other effects play an important role.  So the true stress and strain distributions will have 3 general regions

1.      Close to the crack tip, there will be a process zone, where the material suffers irreversible damage. 

2.      A bit further from the crack tip, there will be a region where the linear elastic asymptotic crack tip field might be expected to be accurate.  This is known as the `region of K dominance’

3.      Far from the crack tip the stress field depends on the geometry of the solid and boundary conditions.

 

Material failure (crack growth or fatigue) is a consequence of the ugly stuff that goes on in the process zone.  Linear elastic fracture mechanics postulates that one doesn’t need to understand this ugly stuff in detail, since the fields in the process zone are likely to be controlled mainly by the fields in the region of K dominance.  The fields in this region depend only on the three stress intensity factors K I , K II , K III MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4samaaBa aaleaacaWGjbaabeaakiaacYcacaaMc8UaaGPaVlaadUeadaWgaaWc baGaamysaiaadMeaaeqaaOGaaiilaiaaykW7caaMc8UaaGPaVlaadU eadaWgaaWcbaGaamysaiaadMeacaWGjbaabeaaaaa@47F0@ .  Therefore, the state in the process zone can be characterized by the three stress intensity factors.

 

If this is true, the conditions for crack growth, or the rate of crack growth, will be only a function of stress intensity factor and nothing else.  We can measure the critical value of K I , K II , K III MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4samaaBa aaleaacaWGjbaabeaakiaacYcacaaMc8UaaGPaVlaadUeadaWgaaWc baGaamysaiaadMeaaeqaaOGaaiilaiaaykW7caaMc8UaaGPaVlaadU eadaWgaaWcbaGaamysaiaadMeacaWGjbaabeaaaaa@47F0@  required to cause the crack to grow in a standard laboratory test, and use this as a measure of the resistance of the solid to crack propagation.  For fatigue tests, we can measure crack growth rate as a function of K I , K II , K III MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4samaaBa aaleaacaWGjbaabeaakiaacYcacaaMc8UaaGPaVlaadUeadaWgaaWc baGaamysaiaadMeaaeqaaOGaaiilaiaaykW7caaMc8UaaGPaVlaadU eadaWgaaWcbaGaamysaiaadMeacaWGjbaabeaaaaa@47F0@  or their history, and characterize the relationship using appropriate phenomenological laws.

 

Having characterized the material, we can then estimate the safety of a structure or component that containing a crack.  To do so, calculate the stress intensity factors for the crack in the structure, and then use our phenomenological fracture or fatigue laws to decide whether or not the crack will grow.

 

For example, the fracture criterion under mode I loading is written

K I K IC MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4samaaBa aaleaacaWGjbaabeaakiabgwMiZkaadUeadaWgaaWcbaGaamysaiaa doeaaeqaaaaa@3D29@

for crack growth, where K IC MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4samaaBa aaleaacaWGjbGaam4qaaqabaaaaa@398F@  is the critical stress intensity factor for the onset of fracture.  The critical stress intensity factor is referred to as the fracture toughness of the solid.

 

Experimentally, it is found that this approach works quite well, provided that the assumptions inherent in linear elastic fracture mechanics are satisfied.

 

Careful tests have established the following conditions for the applicability of linear elastic fracture mechanics.

1.      All characteristic specimen dimensions must exceed 25 times the expected plastic zone size at the crack tip;

2.      For plane strain conditions at the crack tip the specimen thickness must exceed at least the plastic zone size.

 

For a material with yield stress Y  loaded in Mode I with stress intensity factor K I MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4samaaBa aaleaacaWGjbaabeaaaaa@37C0@  the plastic zone size can be estimated as

r p 2.5 ( K I Y ) 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOCamaaBa aaleaacaWGWbaabeaakiablYJi6iaaikdacaGGUaGaaGynamaabmaa baWaaSaaaeaacaWGlbWaaSbaaSqaaiaadMeaaeqaaaGcbaGaamywaa aaaiaawIcacaGLPaaadaahaaWcbeqaaiaaikdaaaaaaa@41A9@

 

 

Practical application of linear elastic fracture mechanics to in design

 

To apply LEFM in a design application, you need to be able to: 

1.      Design a laboratory specimen that can induce a prescribed stress intensity factor at a crack tip

2.      Measure the critical stress intensity factors that cause fracture in the laboratory specimen, or measure fatigue crack growth rates as a function of static or cyclic stress intensity

3.      Estimate the anticipated size and location of cracks in your structure or component

4.      Calculate the stress intensity factors for the cracks in your structure or component under anticipated loading conditions

5.      Combine the results of steps 2 and 4 to predict the behavior of the cracks in the structure of interest, and make appropriate design recommendations.

 

These steps are outlined in more detail below.

 

 

9.3.3 Calculating stress intensity factors

 

Calculating stress intensity factors is a critical step in fracture mechanics.  Various techniques can be used to do this, including

1.      Solve the full linear elastic boundary value problem for the specimen or component containing a crack, and deduce stress intensities from the asymptotic behavior of the stress field near the crack tips;

2.      Attempt to deduce stress intensity factors directly using energy methods or path independent integrals, to be discussed in Section 9.4;

3.      Look up the solution you need in tables;

4.      Use a numerical method MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa aaaaaaa8qacaWFtacaaa@37E6@  boundary integral equation methods are particularly effective for crack problems, but FEM can be used too.

 

Analytical solutions to some crack problems

 

Calculating stress intensity factors for a crack in a structure or component involves the solution of a standard linear elastic boundary value problem.  Once the stresses have been computed, the stress intensity factor is deduced from the definitions given in Section 9.3.1  Exact solutions are known for a few simple geometries.  A couple of examples are

 

2D Slit crack in an infinite solid The figure shows a 2D crack with length 2a in an infinite solid, which is subjected to a uniform state of stress σ 22 , σ 12 , σ 32 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHdpWCdaqhaaWcbaGaaGOmaiaaik daaeaacqGHEisPaaGccaGGSaGaeq4Wdm3aa0baaSqaaiaaigdacaaI YaaabaGaeyOhIukaaOGaaiilaiabeo8aZnaaDaaaleaacaaIZaGaaG Omaaqaaiabg6HiLcaaaaa@42C5@  at infinity.  The complex variable solution to this problem can be found in Section 5.3.  The solution is most conveniently expressed in terms of the polar coordinates (r,θ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaGGOaGaamOCaiaacYcacqaH4oqCca GGPaaaaa@377C@  centered at the origin, together with the auxiliary angles and distances r 1 , θ 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGYbWaaSbaaSqaaiaaigdaaeqaaO GaaiilaiabeI7aXnaaBaaaleaacaaIXaaabeaaaaa@37FB@  and r 2 , θ 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGYbWaaSbaaSqaaiaaikdaaeqaaO GaaiilaiabeI7aXnaaBaaaleaacaaIYaaabeaaaaa@37FD@  shown in the figure.  When evaluating the formulas, the angles θ 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8sk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaH4oqCdaWgaaWcbaGaaGymaaqaba aaaa@3573@  and θ 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8sk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaH4oqCdaWgaaWcbaGaaGOmaaqaba aaaa@3574@  must lie in the ranges π θ 1 π0 θ 2 2π MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOeI0Iaeq iWdaNaeyizImQaeqiUde3aaSbaaSqaaiaaigdaaeqaaOGaeyizImQa eqiWdaNaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVl aaykW7caaMc8UaaGPaVlaaykW7caaIWaGaeyizImQaeqiUde3aaSba aSqaaiaaikdaaeqaaOGaeyizImQaaGOmaiabec8aWbaa@5BB3@ , respectively.  The complete displacement and stress fields in the solid are

u 1 = (1+ν) σ 22 r 1 r 2 4E { 4(12ν)cos( θ 1 + θ 2 )/2 4r(1ν) r 1 r 2 cosθ 2 r 2 r 1 r 2 ( cos( θ 1 + θ 2 )/2cos(2θ θ 1 /2 θ 2 /2) ) } + (1+ν) σ 12 r 1 r 2 E { 2(1ν)sin( θ 1 + θ 2 )/22(1ν) r r 1 r 2 sinθ + r 2 r 1 r 2 sinθcos(θ θ 1 /2 θ 2 /2) } u 2 = (1+ν) σ 22 r 1 r 2 4E { 8(1ν)sin( θ 1 + θ 2 )/2+ 4νr r 1 r 2 sinθ 2 r 2 r 1 r 2 ( sin( θ 1 + θ 2 )/2+sin(2θ θ 1 /2 θ 2 /2) ) } + (1+ν) σ 12 r 1 r 2 E { (12ν)cos( θ 1 + θ 2 )/2+2(1ν) r r 1 r 2 cosθ r 2 r 1 r 2 sinθsin(θ θ 1 /2 θ 2 /2) } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakqaabeqaaiaadwhadaWgaaWcbaGaaGymaa qabaGccqGH9aqpdaWcaaqaaiaacIcacaaIXaGaey4kaSIaeqyVd4Ma aiykaiabeo8aZnaaDaaaleaacaaIYaGaaGOmaaqaaiabg6HiLcaakm aakaaabaGaamOCamaaBaaaleaacaaIXaaabeaakiaadkhadaWgaaWc 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c8UaaGPaVlaaykW7caaMc8UaaGPaVpaaciaabaGaaGPaVlaaykW7ca aMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaa ykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7cqGHsislda WcaaqaaiaaikdacaWGYbWaaWbaaSqabeaacaaIYaaaaaGcbaGaamOC amaaBaaaleaacaaIXaaabeaakiaadkhadaWgaaWcbaGaaGOmaaqaba aaaOWaaeWaaeaaciGGZbGaaiyAaiaac6gacaGGOaGaeqiUde3aaSba aSqaaiaaigdaaeqaaOGaey4kaSIaeqiUde3aaSbaaSqaaiaaikdaae qaaOGaaiykaiaac+cacaaIYaGaey4kaSIaci4CaiaacMgacaGGUbGa aiikaiaaikdacqaH4oqCcqGHsislcqaH4oqCdaWgaaWcbaGaaGymaa qabaGccaGGVaGaaGOmaiabgkHiTiabeI7aXnaaBaaaleaacaaIYaaa beaakiaac+cacaaIYaGaaiykaaGaayjkaiaawMcaaaGaayzFaaaaba GaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7 cqGHRaWkdaWcaaqaaiaacIcacaaIXaGaey4kaSIaeqyVd4Maaiykai abeo8aZnaaDaaaleaacaaIXaGaaGOmaaqaaiabg6HiLcaakmaakaaa baGaamOCamaaBaaaleaacaaIXaaabeaakiaadkhadaWgaaWcbaGaaG OmaaqabaaabeaaaOqaaiaadweaaaGaaGPaVpaaceaabaGaaiikaiaa igdacqGHsislcaaIYaGaeqyVd4MaaiykaiGacogacaGGVbGaai4Cai aacIcacqaH4oqCdaWgaaWcbaGaaGymaaqabaGccqGHRaWkcqaH4oqC 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σ 11 = σ 22 r r 1 r 2 { cos( θ θ 1 /2 θ 2 /2 )1 a 2 r 1 r 2 sinθsin3( θ 1 + θ 2 )/2 } + σ 12 r r 1 r 2 { 2sin( θ θ 1 /2 θ 2 /2 ) a 2 r 1 r 2 sinθcos3( θ 1 + θ 2 )/2 } σ 22 = σ 22 r r 1 r 2 { cos( θ θ 1 /2 θ 2 /2 )+ a 2 r 1 r 2 sinθsin3( θ 1 + θ 2 )/2 }+ σ 12 r r 1 r 2 a 2 r 1 r 2 sinθcos3( θ 1 + θ 2 )/2 σ 12 = σ 22 r r 1 r 2 a 2 r 1 r 2 sinθcos3( θ 1 + θ 2 )/2+ σ 12 r r 1 r 2 { cos( θ θ 1 /2 θ 2 /2 )+ a 2 r 1 r 2 sinθsin3( θ 1 + θ 2 )/2 } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakqaabeqaaiabeo8aZnaaBaaaleaacaaIXa GaaGymaaqabaGccqGH9aqpdaWcaaqaaiabeo8aZnaaDaaaleaacaaI YaGaaGOmaaqaaiabg6HiLcaakiaadkhaaeaadaGcaaqaaiaadkhada WgaaWcbaGaaGymaaqabaGccaWGYbWaaSbaaSqaaiaaikdaaeqaaaqa baaaaOWaaiWaaeaaciGGJbGaai4BaiaacohadaqadaqaaiabeI7aXj abgkHiTiabeI7aXnaaBaaaleaacaaIXaaabeaakiaac+cacaaIYaGa eyOeI0IaeqiUde3aaSbaaSqaaiaaikdaaeqaaOGaai4laiaaikdaai aawIcacaGLPaaacqGHsislcaaIXaGaeyOeI0YaaSaaaeaacaWGHbWa aWbaaSqabeaacaaIYaaaaaGcbaGaamOCamaaBaaaleaacaaIXaaabe aakiaadkhadaWgaaWcbaGaaGOmaaqabaaaaOGaci4CaiaacMgacaGG UbGaeqiUdeNaci4CaiaacMgacaGGUbGaaG4mamaabmaabaGaeqiUde 3aaSbaaSqaaiaaigdaaeqaaOGaey4kaSIaeqiUde3aaSbaaSqaaiaa ikdaaeqaaaGccaGLOaGaayzkaaGaai4laiaaikdaaiaawUhacaGL9b aaaeaacaaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8Ua aGPaVlabgUcaRmaalaaabaGaeq4Wdm3aa0baaSqaaiaaigdacaaIYa aabaGaeyOhIukaaOGaamOCaaqaamaakaaabaGaamOCamaaBaaaleaa caaIXaaabeaakiaadkhadaWgaaWcbaGaaGOmaaqabaaabeaaaaGcda GadaqaaiaaikdaciGGZbGaaiyAaiaac6gadaqadaqaaiabeI7aXjab gkHiTiabeI7aXnaaBaaaleaacaaIXaaabeaakiaac+cacaaIYaGaey OeI0IaeqiUde3aaSbaaSqaaiaaikdaaeqaaOGaai4laiaaikdaaiaa wIcacaGLPaaacqGHsisldaWcaaqaaiaadggadaahaaWcbeqaaiaaik daaaaakeaacaWGYbWaaSbaaSqaaiaaigdaaeqaaOGaamOCamaaBaaa leaacaaIYaaabeaaaaGcciGGZbGaaiyAaiaac6gacqaH4oqCciGGJb Gaai4BaiaacohacaaIZaWaaeWaaeaacqaH4oqCdaWgaaWcbaGaaGym aaqabaGccqGHRaWkcqaH4oqCdaWgaaWcbaGaaGOmaaqabaaakiaawI cacaGLPaaacaGGVaGaaGOmaaGaay5Eaiaaw2haaaqaaiabeo8aZnaa BaaaleaacaaIYaGaaGOmaaqabaGccqGH9aqpdaWcaaqaaiabeo8aZn aaDaaaleaacaaIYaGaaGOmaaqaaiabg6HiLcaakiaadkhaaeaadaGc aaqaaiaadkhadaWgaaWcbaGaaGymaaqabaGccaWGYbWaaSbaaSqaai aaikdaaeqaaaqabaaaaOWaaiWaaeaaciGGJbGaai4Baiaacohadaqa daqaaiabeI7aXjabgkHiTiabeI7aXnaaBaaaleaacaaIXaaabeaaki aac+cacaaIYaGaeyOeI0IaeqiUde3aaSbaaSqaaiaaikdaaeqaaOGa ai4laiaaikdaaiaawIcacaGLPaaacqGHRaWkdaWcaaqaaiaadggada ahaaWcbeqaaiaaikdaaaaakeaacaWGYbWaaSbaaSqaaiaaigdaaeqa aOGaamOCamaaBaaaleaacaaIYaaabeaaaaGcciGGZbGaaiyAaiaac6 gacqaH4oqCciGGZbGaaiyAaiaac6gacaaIZaWaaeWaaeaacqaH4oqC daWgaaWcbaGaaGymaaqabaGccqGHRaWkcqaH4oqCdaWgaaWcbaGaaG OmaaqabaaakiaawIcacaGLPaaacaGGVaGaaGOmaaGaay5Eaiaaw2ha aiaaykW7caaMc8Uaey4kaSYaaSaaaeaacqaHdpWCdaqhaaWcbaGaaG ymaiaaikdaaeaacqGHEisPaaGccaWGYbaabaWaaOaaaeaacaWGYbWa aSbaaSqaaiaaigdaaeqaaOGaamOCamaaBaaaleaacaaIYaaabeaaae qaaaaakmaalaaabaGaamyyamaaCaaaleqabaGaaGOmaaaaaOqaaiaa dkhadaWgaaWcbaGaaGymaaqabaGccaWGYbWaaSbaaSqaaiaaikdaae qaaaaakiGacohacaGGPbGaaiOBaiabeI7aXjGacogacaGGVbGaai4C aiaaiodadaqadaqaaiabeI7aXnaaBaaaleaacaaIXaaabeaakiabgU caRiabeI7aXnaaBaaaleaacaaIYaaabeaaaOGaayjkaiaawMcaaiaa c+cacaaIYaaabaGaeq4Wdm3aaSbaaSqaaiaaigdacaaIYaaabeaaki abg2da9maalaaabaGaeq4Wdm3aa0baaSqaaiaaikdacaaIYaaabaGa eyOhIukaaOGaamOCaaqaamaakaaabaGaamOCamaaBaaaleaacaaIXa aabeaakiaadkhadaWgaaWcbaGaaGOmaaqabaaabeaaaaGcdaWcaaqa aiaadggadaahaaWcbeqaaiaaikdaaaaakeaacaWGYbWaaSbaaSqaai aaigdaaeqaaOGaamOCamaaBaaaleaacaaIYaaabeaaaaGcciGGZbGa aiyAaiaac6gacqaH4oqCciGGJbGaai4BaiaacohacaaIZaWaaeWaae aacqaH4oqCdaWgaaWcbaGaaGymaaqabaGccqGHRaWkcqaH4oqCdaWg aaWcbaGaaGOmaaqabaaakiaawIcacaGLPaaacaGGVaGaaGOmaiaayk W7caaMc8UaaGPaVlabgUcaRmaalaaabaGaeq4Wdm3aa0baaSqaaiaa igdacaaIYaaabaGaeyOhIukaaOGaamOCaaqaamaakaaabaGaamOCam aaBaaaleaacaaIXaaabeaakiaadkhadaWgaaWcbaGaaGOmaaqabaaa beaaaaGcdaGadaqaaiGacogacaGGVbGaai4CamaabmaabaGaeqiUde NaeyOeI0IaeqiUde3aaSbaaSqaaiaaigdaaeqaaOGaai4laiaaikda cqGHsislcqaH4oqCdaWgaaWcbaGaaGOmaaqabaGccaGGVaGaaGOmaa GaayjkaiaawMcaaiabgUcaRmaalaaabaGaamyyamaaCaaaleqabaGa aGOmaaaaaOqaaiaadkhadaWgaaWcbaGaaGymaaqabaGccaWGYbWaaS baaSqaaiaaikdaaeqaaaaakiGacohacaGGPbGaaiOBaiabeI7aXjGa cohacaGGPbGaaiOBaiaaiodadaqadaqaaiabeI7aXnaaBaaaleaaca aIXaaabeaakiabgUcaRiabeI7aXnaaBaaaleaacaaIYaaabeaaaOGa ayjkaiaawMcaaiaac+cacaaIYaaacaGL7bGaayzFaaaaaaa@6AE9@

σ 32 = σ 23 r r 1 r 2 cos(θ θ 1 /2 θ 2 /2) σ 31 = σ 23 r r 1 r 2 sin(θ θ 1 /2 θ 2 /2) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8sk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHdpWCdaWgaaWcbaGaaG4maiaaik daaeqaaOGaeyypa0ZaaSaaaeaacqaHdpWCdaqhaaWcbaGaaGOmaiaa iodaaeaacqGHEisPaaGccaWGYbaabaWaaOaaaeaacaWGYbWaaSbaaS qaaiaaigdaaeqaaOGaamOCamaaBaaaleaacaaIYaaabeaaaeqaaaaa kiGacogacaGGVbGaai4CaiaacIcacqaH4oqCcqGHsislcqaH4oqCda WgaaWcbaGaaGymaaqabaGccaGGVaGaaGOmaiabgkHiTiabeI7aXnaa BaaaleaacaaIYaaabeaakiaac+cacaaIYaGaaiykaiaaykW7caaMc8 UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7 caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8Uaeq4Wdm 3aaSbaaSqaaiaaiodacaaIXaaabeaakiabg2da9maalaaabaGaeq4W dm3aa0baaSqaaiaaikdacaaIZaaabaGaeyOhIukaaOGaamOCaaqaam aakaaabaGaamOCamaaBaaaleaacaaIXaaabeaakiaadkhadaWgaaWc baGaaGOmaaqabaaabeaaaaGcciGGZbGaaiyAaiaac6gacaGGOaGaeq iUdeNaeyOeI0IaeqiUde3aaSbaaSqaaiaaigdaaeqaaOGaai4laiaa ikdacqGHsislcqaH4oqCdaWgaaWcbaGaaGOmaaqabaGccaGGVaGaaG OmaiaacMcaaaa@896F@

The stress intensity factors are easily computed to be

K I = σ 22 πa K II = σ 12 πa MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4samaaBa aaleaacaWGjbaabeaakiabg2da9iabeo8aZnaaDaaaleaacaaIYaGa aGOmaaqaaiabg6HiLcaakmaakaaabaGaeqiWdaNaamyyaaWcbeaaki aaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8Ua aGPaVlaaykW7caaMc8Uaam4samaaBaaaleaacaWGjbGaamysaaqaba GccqGH9aqpcqaHdpWCdaqhaaWcbaGaaGymaiaaikdaaeaacqGHEisP aaGcdaGcaaqaaiabec8aWjaadggaaSqabaaaaa@5DC3@        K III = σ 32 πa MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4samaaBa aaleaacaWGjbGaamysaiaadMeaaeqaaOGaeyypa0Jaeq4Wdm3aa0ba aSqaaiaaiodacaaIYaaabaGaeyOhIukaaOWaaOaaaeaacqaHapaCca WGHbaaleqaaaaa@4315@

 

Penny shaped crack in an infinite solid The figure shows a circular crack with radius a in an infinite solid, subjected to uniaxial tension at infinity.  The displacement field, in cylindrical-polar coordinates, are

u r = νσr E + (1+ν)σr πE { (12ν)( a ρ 2 2 a 2 ρ 2 2 sin 1 a ρ 2 )+ 2 a 2 | z | a 2 ρ 1 2 ρ 2 2 ( ρ 2 2 ρ 1 2 ) } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWG1bWaaSbaaSqaaiaadkhaaeqaaO Gaeyypa0JaeyOeI0YaaSaaaeaacqaH9oGBcqaHdpWCcaWGYbaabaGa amyraaaacqGHRaWkdaWcaaqaaiaacIcacaaIXaGaey4kaSIaeqyVd4 Maaiykaiabeo8aZjaadkhaaeaacqaHapaCcaWGfbaaamaacmaabaGa aiikaiaaigdacqGHsislcaaIYaGaeqyVd4MaaiykamaabmaabaWaaS aaaeaacaWGHbWaaOaaaeaacqaHbpGCdaqhaaWcbaGaaGOmaaqaaiaa ikdaaaGccqGHsislcaWGHbWaaWbaaSqabeaacaaIYaaaaaqabaaake aacqaHbpGCdaqhaaWcbaGaaGOmaaqaaiaaikdaaaaaaOGaeyOeI0Ia ci4CaiaacMgacaGGUbWaaWbaaSqabeaacqGHsislcaaIXaaaaOWaaS aaaeaacaWGHbaabaGaeqyWdi3aaSbaaSqaaiaaikdaaeqaaaaaaOGa ayjkaiaawMcaaiabgUcaRiaaykW7daWcaaqaaiaaikdacaWGHbWaaW baaSqabeaacaaIYaaaaOWaaqWaaeaacaWG6baacaGLhWUaayjcSdWa aOaaaeaacaWGHbWaaWbaaSqabeaacaaIYaaaaOGaeyOeI0IaeqyWdi 3aa0baaSqaaiaaigdaaeaacaaIYaaaaaqabaaakeaacqaHbpGCdaqh aaWcbaGaaGOmaaqaaiaaikdaaaGcdaqadaqaaiabeg8aYnaaDaaale aacaaIYaaabaGaaGOmaaaakiabgkHiTiabeg8aYnaaDaaaleaacaaI XaaabaGaaGOmaaaaaOGaayjkaiaawMcaaaaaaiaawUhacaGL9baaca aMc8UaaGPaVdaa@8381@

u z = σz E + 2(1+ν) πE { 2(1ν)( z | z | a 2 ρ 1 2 z sin 1 a ρ 2 )+z( sin 1 a ρ 2 ρ 2 2 a 2 ρ 2 2 ρ 1 2 ) } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWG1bWaaSbaaSqaaiaadQhaaeqaaO Gaeyypa0ZaaSaaaeaacqaHdpWCcaWG6baabaGaamyraaaacqGHRaWk daWcaaqaaiaaikdacaGGOaGaaGymaiabgUcaRiabe27aUjaacMcaae aacqaHapaCcaWGfbaaamaacmaabaGaaGOmaiaacIcacaaIXaGaeyOe I0IaeqyVd4MaaiykamaabmaabaWaaSaaaeaacaWG6baabaWaaqWaae aacaWG6baacaGLhWUaayjcSdaaamaakaaabaGaamyyamaaCaaaleqa baGaaGOmaaaakiabgkHiTiabeg8aYnaaDaaaleaacaaIXaaabaGaaG OmaaaaaeqaaOGaeyOeI0IaamOEaiGacohacaGGPbGaaiOBamaaCaaa leqabaGaeyOeI0IaaGymaaaakmaalaaabaGaamyyaaqaaiabeg8aYn aaBaaaleaacaaIYaaabeaaaaaakiaawIcacaGLPaaacqGHRaWkcaWG 6bWaaeWaaeaaciGGZbGaaiyAaiaac6gadaahaaWcbeqaaiabgkHiTi aaigdaaaGcdaWcaaqaaiaadggaaeaacqaHbpGCdaWgaaWcbaGaaGOm aaqabaaaaOGaeyOeI0YaaSaaaeaadaGcaaqaaiabeg8aYnaaDaaale aacaaIYaaabaGaaGOmaaaakiabgkHiTiaadggadaahaaWcbeqaaiaa ikdaaaaabeaaaOqaaiabeg8aYnaaDaaaleaacaaIYaaabaGaaGOmaa aakiabgkHiTiabeg8aYnaaDaaaleaacaaIXaaabaGaaGOmaaaaaaaa kiaawIcacaGLPaaaaiaawUhacaGL9baaaaa@7C3C@

ρ 1 = 1 2 ( ( a+r ) 2 + z 2 ( ar ) 2 + z 2 ) ρ 2 = 1 2 ( ( a+r ) 2 + z 2 + ( ar ) 2 + z 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakqaabeqaaiabeg8aYnaaBaaaleaacaaIXa aabeaakiabg2da9maalaaabaGaaGymaaqaaiaaikdaaaWaaeWaaeaa daGcaaqaamaabmaabaGaamyyaiabgUcaRiaadkhaaiaawIcacaGLPa aadaahaaWcbeqaaiaaikdaaaGccqGHRaWkcaWG6bWaaWbaaSqabeaa caaIYaaaaaqabaGccqGHsisldaGcaaqaamaabmaabaGaamyyaiabgk HiTiaadkhaaiaawIcacaGLPaaadaahaaWcbeqaaiaaikdaaaGccqGH RaWkcaWG6bWaaWbaaSqabeaacaaIYaaaaaqabaaakiaawIcacaGLPa aacaaMc8oabaGaeqyWdi3aaSbaaSqaaiaaikdaaeqaaOGaeyypa0Za aSaaaeaacaaIXaaabaGaaGOmaaaadaqadaqaamaakaaabaWaaeWaae aacaWGHbGaey4kaSIaamOCaaGaayjkaiaawMcaamaaCaaaleqabaGa aGOmaaaakiabgUcaRiaadQhadaahaaWcbeqaaiaaikdaaaaabeaaki abgUcaRmaakaaabaWaaeWaaeaacaWGHbGaeyOeI0IaamOCaaGaayjk aiaawMcaamaaCaaaleqabaGaaGOmaaaakiabgUcaRiaadQhadaahaa WcbeqaaiaaikdaaaaabeaaaOGaayjkaiaawMcaaaaaaa@6448@

The displacement of the upper crack face can be found by setting r<a,z=0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8sk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGYbGaeyipaWJaamyyaiaacYcaca aMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaadQhacqGH9aqpcaaIWaaa aa@40DD@  in these expressions, which gives

u z = 4(1 ν 2 )σ πE a 2 r 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWG1bWaaSbaaSqaaiaadQhaaeqaaO Gaeyypa0ZaaSaaaeaacaaI0aGaaiikaiaaigdacqGHsislcqaH9oGB daahaaWcbeqaaiaaikdaaaGccaGGPaGaeq4WdmhabaGaeqiWdaNaam yraaaadaGcaaqaaiaadggadaahaaWcbeqaaiaaikdaaaGccqGHsisl caWGYbWaaWbaaSqabeaacaaIYaaaaaqabaaaaa@4575@

The stress intensity factor can be found directly from the displacement of the crack faces.  The asymptotic formulas in 9.3.1 show that

K I = lim ra E u z (r) 4(1 ν 2 ) 2π (ar) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGlbWaaSbaaSqaaiaadMeaaeqaaO Gaeyypa0ZaaCbeaeaaciGGSbGaaiyAaiaac2gaaSqaaiaadkhacqGH sgIRcaWGHbaabeaakmaalaaabaGaamyraiaadwhadaWgaaWcbaGaam OEaaqabaGccaGGOaGaamOCaiaacMcaaeaacaaI0aGaaiikaiaaigda cqGHsislcqaH9oGBdaahaaWcbeqaaiaaikdaaaGccaGGPaaaamaaka aabaWaaSaaaeaacaaIYaGaeqiWdahabaGaaiikaiaadggacqGHsisl caWGYbGaaiykaaaaaSqabaaaaa@4F06@

which shows that K I =2σ a/π MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa aiaabeqaaiqacaWaaaGcbaGaam4samaaBaaaleaacaWGjbaabeaaki abg2da9iaaikdacqaHdpWCdaGcaaqaaiaadggacaGGVaGaeqiWdaha leqaaaaa@3999@

 

It is not always necessary to solve the full linear elastic boundary value problem in order to compute stress intensity factors.  Energy methods, or the application of path independent integrals, can sometimes be used to obtain stress intensity factors directly.  These techniques will be discussed in more detail in Section 9.4.

 

Vast numbers of crack problems have been solved to catalog stress intensity factors in various geometries of interest.  Two excellent (but expensive) sources of such solutions are Tada’s Handbook of Stress Intensity Factors, and Murakami “Stress intensity factors handbook,” Pergamon Press, New York (1987).  A few important (and relatively simple) results are listed below.

 

 

A short table of stress intensity factors

K I = σ 22 πa K II = σ 12 πa K III = σ 32 πa MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4samaaBa aaleaacaWGjbaabeaakiabg2da9iabeo8aZnaaDaaaleaacaaIYaGa aGOmaaqaaiabg6HiLcaakmaakaaabaGaeqiWdaNaamyyaaWcbeaaki aaykW7caaMc8UaaGPaVlaadUeadaWgaaWcbaGaamysaiaadMeaaeqa aOGaeyypa0Jaeq4Wdm3aa0baaSqaaiaaigdacaaIYaaabaGaeyOhIu kaaOWaaOaaaeaacqaHapaCcaWGHbaaleqaaOGaaGPaVlaaykW7caaM c8Uaam4samaaBaaaleaacaWGjbGaamysaiaadMeaaeqaaOGaeyypa0 Jaeq4Wdm3aa0baaSqaaiaaiodacaaIYaaabaGaeyOhIukaaOWaaOaa aeaacqaHapaCcaWGHbaaleqaaaaa@622E@

K I =2 σ a/π MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa aiaabeqaaiqacaWaaaGcbaGaam4samaaBaaaleaacaWGjbaabeaaki abg2da9iaaikdacqaHdpWCdaahaaWcbeqaaiabg6HiLcaakmaakaaa baGaamyyaiaac+cacqaHapaCaSqabaaaaa@3B41@

K I = 2 F 2 2πb K II = 2 F 1 2πb K III = 2 F 3 2πb MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4samaaBa aaleaacaWGjbaabeaakiabg2da9maalaaabaGaaGOmaiaadAeadaWg aaWcbaGaaGOmaaqabaaakeaadaGcaaqaaiaaikdacqaHapaCcaWGIb aaleqaaaaakiaaykW7caaMc8Uaam4samaaBaaaleaacaWGjbGaamys aaqabaGccqGH9aqpdaWcaaqaaiaaikdacaWGgbWaaSbaaSqaaiaaig daaeqaaaGcbaWaaOaaaeaacaaIYaGaeqiWdaNaamOyaaWcbeaaaaGc caaMc8UaaGPaVlaaykW7caWGlbWaaSbaaSqaaiaadMeacaWGjbGaam ysaaqabaGccqGH9aqpdaWcaaqaaiaaikdacaWGgbWaaSbaaSqaaiaa iodaaeqaaaGcbaWaaOaaaeaacaaIYaGaeqiWdaNaamOyaaWcbeaaaa aaaa@5BCC@

 

K I =1.1215σ πa MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4samaaBa aaleaacaWGjbaabeaakiabg2da9iaaigdacaGGUaGaaGymaiaaikda caaIXaGaaGynaiabeo8aZnaakaaabaGaeqiWdaNaamyyaaWcbeaaaa a@42B6@

K I = F 2 πa f( b a ) K II = F 1 πa f( b a ) K III = F 3 πa f( b a )f(ξ)= (1+ξ)/(1ξ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacaWGlb WaaSbaaSqaaiaadMeaaeqaaOGaeyypa0ZaaSaaaeaacaWGgbWaaSba aSqaaiaaikdaaeqaaaGcbaWaaOaaaeaacqaHapaCcaWGHbaaleqaaa aakiaadAgadaqadaqaamaalaaabaGaamOyaaqaaiaadggaaaaacaGL OaGaayzkaaGaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaG PaVlaaykW7caWGlbWaaSbaaSqaaiaadMeacaWGjbaabeaakiabg2da 9maalaaabaGaamOramaaBaaaleaacaaIXaaabeaaaOqaamaakaaaba GaeqiWdaNaamyyaaWcbeaaaaGccaWGMbWaaeWaaeaadaWcaaqaaiaa dkgaaeaacaWGHbaaaaGaayjkaiaawMcaaiaaykW7caaMc8UaaGPaVd qaaiaadUeadaWgaaWcbaGaamysaiaadMeacaWGjbaabeaakiabg2da 9maalaaabaGaamOramaaBaaaleaacaaIZaaabeaaaOqaamaakaaaba GaeqiWdaNaamyyaaWcbeaaaaGccaWGMbWaaeWaaeaadaWcaaqaaiaa dkgaaeaacaWGHbaaaaGaayjkaiaawMcaaiaaykW7caaMc8UaaGPaVl aaykW7caWGMbGaaiikaiabe67a4jaacMcacqGH9aqpdaGcaaqaaiaa cIcacaaIXaGaey4kaSIaeqOVdGNaaiykaiaac+cacaGGOaGaaGymai abgkHiTiabe67a4jaacMcaaSqabaaaaaa@832B@

K I = 4 F 2 (2πb) 3/2 f( x 3 b ) K II = 4 F 1 (2πb) 3/2 f( x 3 b ) K III = 4 F 3 (2πb) 3/2 f( x 3 b )f(ξ)=1/( 1+ ξ 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacaWGlb WaaSbaaSqaaiaadMeaaeqaaOGaeyypa0ZaaSaaaeaacaaI0aGaamOr amaaBaaaleaacaaIYaaabeaaaOqaaiaacIcacaaIYaGaeqiWdaNaam OyaiaacMcadaahaaWcbeqaaiaaiodacaGGVaGaaGOmaaaaaaGccaWG MbWaaeWaaeaadaWcaaqaaiaadIhadaWgaaWcbaGaaG4maaqabaaake aacaWGIbaaaaGaayjkaiaawMcaaiaaykW7caaMc8UaaGPaVlaaykW7 caaMc8UaaGPaVlaaykW7caWGlbWaaSbaaSqaaiaadMeacaWGjbaabe aakiabg2da9maalaaabaGaaGinaiaadAeadaWgaaWcbaGaaGymaaqa baaakeaacaGGOaGaaGOmaiabec8aWjaadkgacaGGPaWaaWbaaSqabe aacaaIZaGaai4laiaaikdaaaaaaOGaamOzamaabmaabaWaaSaaaeaa caWG4bWaaSbaaSqaaiaaiodaaeqaaaGcbaGaamOyaaaaaiaawIcaca GLPaaacaaMc8UaaGPaVlaaykW7aeaacaWGlbWaaSbaaSqaaiaadMea caWGjbGaamysaaqabaGccqGH9aqpdaWcaaqaaiaaisdacaWGgbWaaS baaSqaaiaaiodaaeqaaaGcbaGaaiikaiaaikdacqaHapaCcaWGIbGa aiykamaaCaaaleqabaGaaG4maiaac+cacaaIYaaaaaaakiaadAgada qadaqaamaalaaabaGaamiEamaaBaaaleaacaaIZaaabeaaaOqaaiaa dkgaaaaacaGLOaGaayzkaaGaaGPaVlaaykW7caaMc8UaaGPaVlaayk W7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaadAgacaGGOaGaeqOV dGNaaiykaiabg2da9iaaigdacaGGVaWaaeWaaeaacaaIXaGaey4kaS IaeqOVdG3aaWbaaSqabeaacaaIYaaaaaGccaGLOaGaayzkaaaaaaa@9A35@

 

 

 

Calculating stress intensity factors for cracks in nonuniform stress fields

 

The solutions for cracks loaded by point forces acting on their faces are particularly useful, because they allow you to calculate stress intensity factors for a crack in an arbitrary stress field using a simple superposition argument.  The procedure works like this. 

1.      We start by computing the stress field in a solid without a crack in it.  This solution satisfies all boundary conditions except that the crack faces are subject to tractions

2.      We could correct solution by applying pressure (and shear) to the crack faces that are just sufficient to remove the unwanted tractions.

3.      If we know the stress intensity factors induced by point forces acting on the crack faces, we can superpose an appropriate distribution of point forces on the crack faces calculate stress intensity factors induced by the corrective pressure distribution.

 

As an example, suppose that we want to calculate stress intensity factors for a crack in a linearly varying stress field (such as would be induced by bending a beam, for example), as illustrated in the figure to the right.

 

1.      In the uncracked solid, the stress field is  σ 22 = σ 0 x 1 /L MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHdpWCdaWgaaWcbaGaaGOmaiaaik daaeqaaOGaeyypa0Jaeq4Wdm3aaSbaaSqaaiaaicdaaeqaaOGaamiE amaaBaaaleaacaaIXaaabeaakiaac+cacaWGmbaaaa@3D62@

2.      The traction acting along the line of the crack is p= σ 0 x 1 /L MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiCaiabg2 da9iabeo8aZnaaBaaaleaacaaIWaaabeaakiaadIhadaWgaaWcbaGa aGymaaqabaGccaGGVaGaamitaaaa@3E16@  The sign convention for p is that a positive p acts downwards on the upper crack face, and upwards on the lower crack face

3.      To remove the traction from the crack faces, we must superpose an equal and opposite distribution of point forces on the crack faces.  The stress intensity factor induced at the left (L) and right (R) crack tips are

K I R = 1 πa a a ( σ 0 x 1 /L) a+ x 1 a x 1 d x 1 K I L = 1 πa a a ( σ 0 x 1 /L) a x 1 a+ x 1 d x 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4samaaDa aaleaacaWGjbaabaGaamOuaaaakiabg2da9maalaaabaGaaGymaaqa amaakaaabaGaeqiWdaNaamyyaaWcbeaaaaGcdaWdXbqaaiaacIcacq aHdpWCdaWgaaWcbaGaaGimaaqabaGccaWG4bWaaSbaaSqaaiaaigda aeqaaOGaai4laiaadYeacaGGPaWaaOaaaeaadaWcaaqaaiaadggacq GHRaWkcaWG4bWaaSbaaSqaaiaaigdaaeqaaaGcbaGaamyyaiabgkHi TiaadIhadaWgaaWcbaGaaGymaaqabaaaaaqabaaabaGaeyOeI0Iaam yyaaqaaiaadggaa0Gaey4kIipakiaadsgacaWG4bWaaSbaaSqaaiaa igdaaeqaaOGaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaG PaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaM c8UaaGPaVlaaykW7caWGlbWaa0baaSqaaiaadMeaaeaacaWGmbaaaO Gaeyypa0ZaaSaaaeaacaaIXaaabaWaaOaaaeaacqaHapaCcaWGHbaa leqaaaaakmaapehabaGaaiikaiabeo8aZnaaBaaaleaacaaIWaaabe aakiaadIhadaWgaaWcbaGaaGymaaqabaGccaGGVaGaamitaiaacMca daGcaaqaamaalaaabaGaamyyaiabgkHiTiaadIhadaWgaaWcbaGaaG ymaaqabaaakeaacaWGHbGaey4kaSIaamiEamaaBaaaleaacaaIXaaa beaaaaaabeaaaeaacqGHsislcaWGHbaabaGaamyyaaqdcqGHRiI8aO GaamizaiaadIhadaWgaaWcbaGaaGymaaqabaaaaa@8DB8@

Evaluating the integrals gives

K I R = 1 2 ( σ 0 /L)a πa K I L = 1 2 ( σ 0 /L)a πa MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4samaaDa aaleaacaWGjbaabaGaamOuaaaakiabg2da9maalaaabaGaaGymaaqa aiaaikdaaaGaaiikaiabeo8aZnaaBaaaleaacaaIWaaabeaakiaac+ cacaWGmbGaaiykaiaadggadaGcaaqaaiabec8aWjaadggaaSqabaGc caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVl aaykW7caaMc8UaaGPaVlaaykW7caWGlbWaa0baaSqaaiaadMeaaeaa caWGmbaaaOGaeyypa0JaeyOeI0YaaSaaaeaacaaIXaaabaGaaGOmaa aacaGGOaGaeq4Wdm3aaSbaaSqaaiaaicdaaeqaaOGaai4laiaadYea caGGPaGaamyyamaakaaabaGaeqiWdaNaamyyaaWcbeaaaaa@674C@

 

Actually, this solution is not quite right - note that the stress intensity factor at the left crack tip is predicted to be negative.  This cannot be correct MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa aaaaaaa8qacaWFtacaaa@37E6@  from the asymptotic stress field we know that if the stress intensity factor is negative, the crack faces must overlap behind the crack tip (the displacement jump is negative). 

 

With a bit of cunning, we can fix this problem.  The cause of the error in the quick estimate is that we removed tractions from the entire crack MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa aaaaaaa8qacaWFtacaaa@37E6@  this was a mistake; we should only have removed tractions from parts of the crack faces that open up.   So let’s suppose that the crack closes at x 1 =b MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaBa aaleaacaaIXaaabeaakiabg2da9iabgkHiTiaadkgaaaa@3ABE@ , and put the left hand crack tip there.  The stress intensity factors are then

K I R = 1 π(a+b)/2 b a ( σ 0 x 1 /L) b+ x 1 a x 1 d x 1 K I L = 1 π(a+b)/2 b a ( σ 0 x 1 /L) a x 1 b+ x 1 d x 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacaWGlb Waa0baaSqaaiaadMeaaeaacaWGsbaaaOGaeyypa0ZaaSaaaeaacaaI XaaabaWaaOaaaeaacqaHapaCcaGGOaGaamyyaiabgUcaRiaadkgaca GGPaGaai4laiaaikdaaSqabaaaaOWaa8qCaeaacaGGOaGaeq4Wdm3a aSbaaSqaaiaaicdaaeqaaOGaamiEamaaBaaaleaacaaIXaaabeaaki aac+cacaWGmbGaaiykamaakaaabaWaaSaaaeaacaWGIbGaey4kaSIa amiEamaaBaaaleaacaaIXaaabeaaaOqaaiaadggacqGHsislcaWG4b WaaSbaaSqaaiaaigdaaeqaaaaaaeqaaaqaaiabgkHiTiaadkgaaeaa caWGHbaaniabgUIiYdGccaWGKbGaamiEamaaBaaaleaacaaIXaaabe aaaOqaaiaadUeadaqhaaWcbaGaamysaaqaaiaadYeaaaGccqGH9aqp daWcaaqaaiaaigdaaeaadaGcaaqaaiabec8aWjaacIcacaWGHbGaey 4kaSIaamOyaiaacMcacaGGVaGaaGOmaaWcbeaaaaGcdaWdXbqaaiaa cIcacqaHdpWCdaWgaaWcbaGaaGimaaqabaGccaWG4bWaaSbaaSqaai aaigdaaeqaaOGaai4laiaadYeacaGGPaWaaOaaaeaadaWcaaqaaiaa dggacqGHsislcaWG4bWaaSbaaSqaaiaaigdaaeqaaaGcbaGaamOyai abgUcaRiaadIhadaWgaaWcbaGaaGymaaqabaaaaaqabaaabaGaeyOe I0IaamOyaaqaaiaadggaa0Gaey4kIipakiaadsgacaWG4bWaaSbaaS qaaiaaigdaaeqaaaaaaa@7CAA@

This gives

K I L = ( σ 0 /L) 2π 8 (a+b) ( a 2 2ab3 b 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4samaaDa aaleaacaWGjbaabaGaamitaaaakiabg2da9maalaaabaGaaiikaiab eo8aZnaaBaaaleaacaaIWaaabeaakiaac+cacaWGmbGaaiykamaaka aabaGaaGOmaiabec8aWbWcbeaaaOqaaiaaiIdadaGcaaqaaiaacIca caWGHbGaey4kaSIaamOyaiaacMcaaSqabaaaaOWaaeWaaeaacaWGHb WaaWbaaSqabeaacaaIYaaaaOGaeyOeI0IaaGOmaiaadggacaWGIbGa eyOeI0IaaG4maiaadkgadaahaaWcbeqaaiaaikdaaaaakiaawIcaca GLPaaaaaa@5232@

for the stress intensity factor at the left hand crack tip.  The stress must be bounded at x 1 =b MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaBa aaleaacaaIXaaabeaakiabg2da9iabgkHiTiaadkgaaaa@3BC5@  where the crack faces touch, so that K I L =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4samaaDa aaleaacaWGjbaabaGaamitaaaakiabg2da9iaaicdaaaa@3B63@ .  This gives b=a/3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOyaiabg2 da9iaadggacaGGVaGaaG4maaaa@3B40@ .  The stress intensity factor at the right hand crack tip then follows as

MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcaa@35F5@ K I R = 2( σ 0 /L)a 6πa 9 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4samaaDa aaleaacaWGjbaabaGaamOuaaaakiabg2da9maalaaabaGaaGOmaiaa cIcacqaHdpWCdaWgaaWcbaGaaGimaaqabaGccaGGVaGaamitaiaacM cacaWGHbWaaOaaaeaacaaI2aGaeqiWdaNaamyyaaWcbeaaaOqaaiaa iMdaaaaaaa@463C@

This is not very different to our earlier estimate.  This illustrates a general feature of the field of fracture mechanics.  There are many opportunities to do clever things, but often the results of all the cleverness are irrelevant.

 

 

9.3.4 Calculating stress intensity factors using finite element analysis

 

For solids with a complicated geometry, finite element methods (or boundary element methods) are the only way to calculate stress intensity factors.  It is conceptually very straightforward to calculate stress intensities using finite elements MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa aaaaaaa8qacaWFtacaaa@37E6@  you just need to solve a routine linear elastic boundary value problem to determine the stress field in the solid, and then deduce the stress intensity factors by taking the limits given in Section 8.3.1.

 

Unfortunately this is easier said than done.  The problem is that the stress and strain fields at a crack tip are infinite, and so standard finite element methods have problems calculating the stresses accurately.  Two special procedures have been developed to help deal with this:

  1. Special crack tip elements are available to approximate the singular strains at a crack tip;
  2. Special techniques are available to calculate stress intensity factors from stresses far from the crack tip (where they should be accurate) instead of using the formal definition.

These methods can both give very accurate values for stress intensity factors MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa aaaaaaa8qacaWFtacaaa@37E6@  and can be used together to obtain the best results.

 

Crack tip elements

 

A very simple procedure can be used to approximate the strain singularity at a crack tip.

1.      The solid near the crack tip must be meshed with quadratic elements (8 noded quadrilaterals or 6 noded triangles in 2D, or 20 noded bricks/10 noded tetrahedral in 3D). 

2.      The elements connected to the crack tip must be quadrilateral or brick elements

3.      One side of each element connected to the crack tip is collapsed to make the three nodes on the side coincident, as shown in the figure to the right

4.      The mid-side nodes on the elements connected to the crack tip are shifted to ¼ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa aaaaaaa8qacaWF8caaaa@386F@  point positions, as shown in the figure

5.      If the coincident nodes a,b,c on each crack-tip element are constrained to move together, this procedure generates a ( r ) 1/2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaqadaqaaiaadkhaaiaawIcacaGLPa aadaahaaWcbeqaaiabgkHiTiaaigdacaGGVaGaaGOmaaaaaaa@388A@  singularity in strain at the crack tip (good for linear elastic problems).  If the nodes are permitted to move independently, a r 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGYbWaaWbaaSqabeaacqGHsislca aIXaaaaaaa@3592@  singularity in strain is produced (good for problems involving crack tip plasticity)

 

 

Calculating stress intensity factors using path independent integrals

 

Energy methods in fracture mechanics are discussed in detail in Section 9.4  Two crucial results emerge from this analysis

1.      The ‘energy release rate’ for a mode I crack in a linear elastic solid with Young’s modulus E and Poisson’s ratio ν MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaH9oGBaaa@347E@  is related to the mode I stress intensity factor by

G= 1 ν 2 E K I 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4raiabg2 da9maalaaabaGaaGymaiabgkHiTiabe27aUnaaCaaaleqabaGaaGOm aaaaaOqaaiaadweaaaGaam4samaaDaaaleaacaWGjbaabaGaaGOmaa aaaaa@4083@

2.      The energy release rate for a crack can be calculated by evaluating the following line integral for any contour that starts on one crack face and ends on the other

G= Γ (W δ j1 σ ij u i x 1 ) m j ds MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4raiabg2 da9maapefabaGaaiikaiaadEfacqaH0oazdaWgaaWcbaGaamOAaiaa igdaaeqaaOGaeyOeI0Iaeq4Wdm3aaSbaaSqaaiaadMgacaWGQbaabe aakmaalaaabaGaeyOaIyRaamyDamaaBaaaleaacaWGPbaabeaaaOqa aiabgkGi2kaadIhadaWgaaWcbaGaaGymaaqabaaaaOGaaiykaiaad2 gadaWgaaWcbaGaamOAaaqabaGccaWGKbGaam4CaaWcbaGaeu4KdCea beqdcqGHRiI8aaaa@51DE@

where W= σ ij ε ij /2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGxbGaeyypa0Jaeq4Wdm3aaSbaaS qaaiaadMgacaWGQbaabeaakiabew7aLnaaBaaaleaacaWGPbGaamOA aaqabaGccaGGVaGaaGOmaaaa@3DA7@  is the strain energy density, σ ij MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHdpWCdaWgaaWcbaGaamyAaiaadQ gaaeqaaaaa@3692@  is the stress field, u i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWG1bWaaSbaaSqaaiaadMgaaeqaaa aa@34DA@  is the displacement field, m i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGTbWaaSbaaSqaaiaadMgaaeqaaa aa@34D2@  is a unit vector normal to Γ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqqHtoWraaa@342E@ , and the e 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHLbWaaSbaaSqaaiaaigdaaeqaaa aa@349B@  basis vector is parallel to the direction of crack propagation.

 

These results are ideally suited for FEM calculations.  The path independent integral can be calculated for a contour far from the crack tip, where the stresses are accurate, and then the relationship between G and K I MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGlbWaaSbaaSqaaiaadMeaaeqaaa aa@3490@  can be used to deduce the stress intensity factors.  Analogous, but rather more complex, procedures exist to extract all three components of stress intensity factor, as well as to compute stress intensity factors for 3D cracks, where the stress intensity factor is a function of position on the crack front.

 

 

9.3.5 Measuring fracture toughness

 

For structural applications, standard testing techniques are available to measure material properties for fracture applications.  Two standard test specimen geometries are shown below.Stress intensity factors for these specimens have been carefully computed as a function of crack length and the results fit by curves, as outlined below

 

 Compact tension specimen:

K I = P B π W { 16.7 ( a W ) 1/2 104.7 ( a W ) 3/2 +369.9 ( a W ) 5/2 573.8 ( a W ) 7/2 +360.5 ( a W ) 9/2 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacaWGlb WaaSbaaSqaaiaadMeaaeqaaOGaeyypa0ZaaSaaaeaacaWGqbaabaGa amOqaaaadaGcaaqaamaalaaabaGaeqiWdahabaGaam4vaaaaaSqaba GcdaGabaqaaiaaigdacaaI2aGaaiOlaiaaiEdadaqadaqaamaalaaa baGaamyyaaqaaiaadEfaaaaacaGLOaGaayzkaaWaaWbaaSqabeaaca aIXaGaai4laiaaikdaaaGccqGHsislcaaIXaGaaGimaiaaisdacaGG UaGaaG4namaabmaabaWaaSaaaeaacaWGHbaabaGaam4vaaaaaiaawI cacaGLPaaadaahaaWcbeqaaiaaiodacaGGVaGaaGOmaaaakiabgUca RiaaiodacaaI2aGaaGyoaiaac6cacaaI5aWaaeWaaeaadaWcaaqaai aadggaaeaacaWGxbaaaaGaayjkaiaawMcaamaaCaaaleqabaGaaGyn aiaac+cacaaIYaaaaaGccaGL7baaaeaacaaMc8UaaGPaVlaaykW7ca aMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaa ykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaG PaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaM c8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaayk W7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPa VlaaykW7daGacaqaaiabgkHiTiaaiwdacaaI3aGaaG4maiaac6caca aI4aWaaeWaaeaadaWcaaqaaiaadggaaeaacaWGxbaaaaGaayjkaiaa wMcaamaaCaaaleqabaGaaG4naiaac+cacaaIYaaaaOGaey4kaSIaaG 4maiaaiAdacaaIWaGaaiOlaiaaiwdadaqadaqaamaalaaabaGaamyy aaqaaiaadEfaaaaacaGLOaGaayzkaaWaaWbaaSqabeaacaaI5aGaai 4laiaaikdaaaaakiaaw2haaaaaaa@B7FE@

 

 Three point bend specimen.

K I = 4P B π W { 1.6 ( a W ) 1/2 2.6 ( a W ) 3/2 +12.3 ( a W ) 5/2 21.2 ( a W ) 7/2 +21.8 ( a W ) 9/2 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacaWGlb WaaSbaaSqaaiaadMeaaeqaaOGaeyypa0ZaaSaaaeaacaaI0aGaamiu aaqaaiaadkeaaaWaaOaaaeaadaWcaaqaaiabec8aWbqaaiaadEfaaa aaleqaaOWaaiqaaeaacaaIXaGaaiOlaiaaiAdadaqadaqaamaalaaa baGaamyyaaqaaiaadEfaaaaacaGLOaGaayzkaaWaaWbaaSqabeaaca aIXaGaai4laiaaikdaaaGccqGHsislcaaIYaGaaiOlaiaaiAdadaqa daqaamaalaaabaGaamyyaaqaaiaadEfaaaaacaGLOaGaayzkaaWaaW baaSqabeaacaaIZaGaai4laiaaikdaaaGccqGHRaWkcaaIXaGaaGOm aiaac6cacaaIZaWaaeWaaeaadaWcaaqaaiaadggaaeaacaWGxbaaaa GaayjkaiaawMcaamaaCaaaleqabaGaaGynaiaac+cacaaIYaaaaaGc caGL7baaaeaadaGacaqaaiaaykW7caaMc8UaaGPaVlaaykW7caaMc8 UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7 caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVl aaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8Ua aGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7ca aMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaa ykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaG PaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaeyOeI0IaaGOmaiaa igdacaGGUaGaaGOmamaabmaabaWaaSaaaeaacaWGHbaabaGaam4vaa aaaiaawIcacaGLPaaadaahaaWcbeqaaiaaiEdacaGGVaGaaGOmaaaa kiabgUcaRiaaikdacaaIXaGaaiOlaiaaiIdadaqadaqaamaalaaaba GaamyyaaqaaiaadEfaaaaacaGLOaGaayzkaaWaaWbaaSqabeaacaaI 5aGaai4laiaaikdaaaaakiaaw2haaaaaaa@C9C5@

 

Various other test specimens exist.

 

Conducting a fracture test or fatigue test is (at least conceptually) straightforward MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa aaaaaaa8qacaWFtacaaa@37E6@  you make a specimen (for fracture tests a sharp crack is usually created by initiating a fatigue crack at the tip of a notch); and load it in a tensile testing machine.

 

 In principle, the fracture toughness can be determined by measuring the critical load when the crack starts to grow.  In practice it can be difficult to detect the onset of crack growth.  For this reason, the usual approach is to monitor the crack opening displacement δ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiTdqgaaa@379B@  during the test, then plot load as a function of crack opening displacement.  A typical result is illustrated in the picture on the right.

 

The load-CTOD curve ceases to be linear when the crack begins to grow. This point is hard to identify, so instead the convention is to draw a line with slope 5% lower than the initial Pδ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiuaiabgk HiTiabes7aKbaa@395D@  curve (the 5% secant line) and use the point where this line intersects the Pδ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiuaiabgk HiTiabes7aKbaa@395D@  as the fracture load. The plane strain fracture toughness of the material, K IC MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4samaaBa aaleaacaWGjbGaam4qaaqabaaaaa@3888@ , is deduced from the fracture load, using the calibration for the specimen.

 

 

After measurement, one must check that K IC MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4samaaBa aaleaacaWGjbGaam4qaaqabaaaaa@3888@  is within the limits required for K dominance in the specimen, following the rules in the preceding section.

 

 

 

9.3.6 Typical values for fracture toughness

 

A short table of toughness values (From Ashby & Jones, `Engineering Materials’, Pergammon, 1980) is given below.  The values are highly dependent on material composition and microstructure, however, so if you need accurate data you will need to measure the toughness of your materials yourself.

 

Material

Approximate fracture toughness, MN m 3/2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamytaiaad6 eacaWGTbWaaWbaaSqabeaacqGHsislcaaIZaGaai4laiaaikdaaaaa aa@3BD3@

Pressure vessel steel (HY100)

50-160

Mild steel

140

Titanium alloys

55-120

High carbon steel

30

Nickel, copper

>100

Aluminum and alloys

20-50

Co/WC metal matrix composites

14-16

Woods, perpendicular to grain

11-13

Concrete (steel reinforced)

10-15

Ceramics (Alumina, SiC)

3-5

ABS polystryrene

4

Nylon, polyproplyene

3

Glasses, rocks

1

Wood, parallel to grain

0.5-1

Concrete (unreinforced)

0.2

 

 

 

 

9.3.7 Stable Tearing MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqabKqzGfaeaa aaaaaaa8qacaWFtacaaa@37E7@  Kr curves and Crack Stability

 

In ideally brittle materials, fracture is a catastrophic event.  Once the load reaches the level required to trigger crack growth, the crack continues to propagate dynamically through the specimen.  In more ductile materials, a period of stable crack growth under steadily increasing load may occur prior to complete failure.  This behavior is particularly common in tearing of thin sheets of metals, but stable crack growth is observed in most materials MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa aaaaaaa8qacaWFtacaaa@37E6@  even polycrystalline ceramics.

 

Stable crack growth in metals usually occurs because a zone of plastically deformed material is left in the wake of the crack.  This deformed material tends to reduce the stresses at the crack tip.  In brittle polycrystalline ceramics, or in fiber reinforced brittle composites, the stable crack growth is caused by the formation of a `bridging zone’ behind the crack tip.  Some fibers, or grains, remain intact in the crack wake, and tend to hold the crack faces shut, increasing the apparent strength of the solid.

 

In some materials, the increase in load during stable crack growth is so significant that it’s worth accounting for the effect in design calculations.  The protective effect of the process zone in the crack wake is modeled by making the toughness of the material a function of the increase in crack length.  The apparent toughness is measured in the same way as K IC MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4samaaBa aaleaacaWGjbGaam4qaaqabaaaaa@398F@  - a pre-cracked specimen is subjected to progressively increasing load, and the crack length is monitored either optically or using compliance methods (more on this later).  A value of K I MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4samaaBa aaleaacaWGjbaabeaaaaa@38C7@  can be computed for the specimen using the calibrations MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa aaaaaaa8qacaWFtacaaa@37E6@  during crack growth it is assumed that K I MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4samaaBa aaleaacaWGjbaabeaaaaa@38C7@  is equal to the fracture toughness of the material.

 

 

The results are plotted in a `resistance curve’ or `R curve’ for the material.  The fracture toughness K IC MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4samaaBa aaleaacaWGjbGaam4qaaqabaaaaa@398F@  is the critical stress intensity factor required to initiate crack growth.  The variation of toughness with crack growth is denoted K r (Δa) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4samaaBa aaleaacaWGYbaabeaakiaacIcacqqHuoarcaWGHbGaaiykaaaa@3C9F@ .

 

The resistance curve is then used to predict the conditions necessary for unstable crack growth through the material.  To see how this is done

1.      Consider a large sample of material containing a slit crack of length 2a, subjected to stress σ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdmhaaa@37B9@ .  The stress intensity factor for this crack (from the table in sect 9.3.3) is K I =σ πa MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4samaaBa aaleaacaWGjbaabeaakiabg2da9iabeo8aZnaakaaabaGaeqiWdaNa amyyaaWcbeaaaaa@3E58@ .

2.      Crack growth begins when σ πa = K IC MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aaO aaaeaacqaHapaCcaWGHbaaleqaaOGaeyypa0Jaam4samaaBaaaleaa caWGjbGaam4qaaqabaaaaa@3F20@ .   Thereafter, there will be a period of stable crack growth, during which the applied stress increases.  During the period of stable growth the stress intensity factor must equal the apparent toughness

σ π(a+Δa) = K r ( Δa )σ= K r ( Δa ) π(a+Δa) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aaO aaaeaacqaHapaCcaGGOaGaamyyaiabgUcaRiabfs5aejaadggacaGG PaaaleqaaOGaeyypa0Jaam4samaaBaaaleaacaWGYbaabeaakmaabm aabaGaeuiLdqKaamyyaaGaayjkaiaawMcaaiaaykW7caaMc8UaaGPa VlaaykW7caaMc8UaaGPaVlabgkDiElaaykW7caaMc8UaaGPaVlaayk W7cqaHdpWCcqGH9aqpcaaMc8UaaGPaVpaalaaabaGaam4samaaBaaa leaacaWGYbaabeaakmaabmaabaGaeuiLdqKaamyyaaGaayjkaiaawM caaaqaamaakaaabaGaeqiWdaNaaiikaiaadggacqGHRaWkcqqHuoar caWGHbGaaiykaaWcbeaaaaaaaa@6BB8@

3.      The stress will continue to increase as long as K r (Δa) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGlbWaaSbaaSqaaiaadkhaaeqaaO Gaaiikaiabfs5aejaadggacaGGPaaaaa@3868@  increases more rapidly than π(a+Δa) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaGcaaqaaiabec8aWjaacIcacaWGHb Gaey4kaSIaeuiLdqKaamyyaiaacMcaaSqabaaaaa@3A0B@  with Δa MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqqHuoarcaWGHbaaaa@3512@ .  Catastrophic failure (unstable crack growth) will occur when continued crack growth is possible at constant or decreasing load.  The crack length at the point of unstable crack growth follows from the condition that

dσ dΔa =0a+Δa= K r 2(d K r /dΔa) d K r dΔa MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGPaVpaala aabaGaamizaiabeo8aZbqaaiaadsgacqqHuoarcaWGHbaaaiabg2da 9iaaicdacqGHshI3caWGHbGaey4kaSIaeuiLdqKaamyyaiabg2da9m aalaaabaGaam4samaaBaaaleaacaWGYbaabeaaaOqaaiaaikdacaGG OaGaamizaiaadUeadaWgaaWcbaGaamOCaaqabaGccaGGVaGaamizai abfs5aejaadggacaGGPaaaaiaaykW7caaMc8+aaSaaaeaacaWGKbGa am4samaaBaaaleaacaWGYbaabeaaaOqaaiaadsgacqqHuoarcaWGHb aaaaaa@5BE1@

4.      Substituting the crack length back into the fracture criterion σ π(a+Δa) = K r ( Δa ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aaO aaaeaacqaHapaCcaGGOaGaamyyaiabgUcaRiabfs5aejaadggacaGG PaaaleqaaOGaeyypa0Jaam4samaaBaaaleaacaWGYbaabeaakmaabm aabaGaeuiLdqKaamyyaaGaayjkaiaawMcaaaaa@46E7@  gives the critical stress at unstable fracture as

σ= 2 π K r d K r dΔa MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4WdmNaey ypa0ZaaOaaaeaadaWcaaqaaiaaikdaaeaacqaHapaCaaGaam4samaa BaaaleaacaWGYbaabeaakmaalaaabaGaamizaiaadUeadaWgaaWcba GaamOCaaqabaaakeaacaWGKbGaeuiLdqKaamyyaaaaaSqabaaaaa@4492@

 

 

9.3.8 Mixed Mode fracture criteria

 

Fracture toughness is almost always measured under mode I loading (except when measuring fracture toughness of a bi-material interface).  If a crack is subjected to combined mode I and mode II loading, a mixed mode fracture criterion is required.  There are several ways to construct mixed mode fracture criteria MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa aaaaaaa8qacaWFtacaaa@37E6@  the issue has been the subject of some quite heated arguments.  The criterion of maximum hoop stress is one example.  Recall that the crack tip hoop and shear stresses are

σ θθ = K I 2πr ( 3 4 cos θ 2 + 1 4 cos 3θ 2 ) K II 2πr ( 3 4 sin θ 2 + 3 4 sin 3θ 2 ) σ rθ = K I 2πr ( 1 4 sin θ 2 + 1 4 sin 3θ 2 )+ K II 2πr ( 1 4 cos θ 2 + 3 4 cos 3θ 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacqaHdp WCdaWgaaWcbaGaeqiUdeNaeqiUdehabeaakiabg2da9maalaaabaGa am4samaaBaaaleaacaWGjbaabeaaaOqaamaakaaabaGaaGOmaiabec 8aWjaadkhaaSqabaaaaOWaaeWaaeaadaWcaaqaaiaaiodaaeaacaaI 0aaaaiGacogacaGGVbGaai4CamaalaaabaGaeqiUdehabaGaaGOmaa aacqGHRaWkdaWcaaqaaiaaigdaaeaacaaI0aaaaiGacogacaGGVbGa ai4CamaalaaabaGaaG4maiabeI7aXbqaaiaaikdaaaaacaGLOaGaay zkaaGaeyOeI0YaaSaaaeaacaWGlbWaaSbaaSqaaiaadMeacaWGjbaa beaaaOqaamaakaaabaGaaGOmaiabec8aWjaadkhaaSqabaaaaOWaae WaaeaadaWcaaqaaiaaiodaaeaacaaI0aaaaiGacohacaGGPbGaaiOB amaalaaabaGaeqiUdehabaGaaGOmaaaacqGHRaWkdaWcaaqaaiaaio daaeaacaaI0aaaaiGacohacaGGPbGaaiOBamaalaaabaGaaG4maiab eI7aXbqaaiaaikdaaaaacaGLOaGaayzkaaaabaGaeq4Wdm3aaSbaaS qaaiaadkhacqaH4oqCaeqaaOGaeyypa0ZaaSaaaeaacaWGlbWaaSba aSqaaiaadMeaaeqaaaGcbaWaaOaaaeaacaaIYaGaeqiWdaNaamOCaa WcbeaaaaGcdaqadaqaamaalaaabaGaaGymaaqaaiaaisdaaaGaci4C aiaacMgacaGGUbWaaSaaaeaacqaH4oqCaeaacaaIYaaaaiabgUcaRm aalaaabaGaaGymaaqaaiaaisdaaaGaci4CaiaacMgacaGGUbWaaSaa aeaacaaIZaGaeqiUdehabaGaaGOmaaaaaiaawIcacaGLPaaacqGHRa WkdaWcaaqaaiaadUeadaWgaaWcbaGaamysaiaadMeaaeqaaaGcbaWa aOaaaeaacaaIYaGaeqiWdaNaamOCaaWcbeaaaaGcdaqadaqaamaala aabaGaaGymaaqaaiaaisdaaaGaci4yaiaac+gacaGGZbWaaSaaaeaa cqaH4oqCaeaacaaIYaaaaiabgUcaRmaalaaabaGaaG4maaqaaiaais daaaGaci4yaiaac+gacaGGZbWaaSaaaeaacaaIZaGaeqiUdehabaGa aGOmaaaaaiaawIcacaGLPaaaaaaa@A016@

The maximum hoop stress criterion postulates that a crack under mixed mode loading starts to propagate when the greatest value of hoop stress σ θθ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aaS baaSqaaiabeI7aXjabeI7aXbqabaaaaa@3C58@  reaches a critical magnitude, at which point the crack will branch at the angle for which σ θθ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aaS baaSqaaiabeI7aXjabeI7aXbqabaaaaa@3C58@  is greatest (or equivalently the angle for which σ rθ =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aaS baaSqaaiaadkhacqaH4oqCaeqaaOGaeyypa0JaaGimaaaa@3D63@  ).  The critical angle is plotted as a function of K II / K I MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4samaaBa aaleaacaWGjbGaamysaaqabaGccaGGVaGaam4samaaBaaaleaacaWG jbaabeaaaaa@3C1C@  below.  The asymptote for K II / K I MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4samaaBa aaleaacaWGjbGaamysaaqabaGccaGGVaGaam4samaaBaaaleaacaWG jbaabeaakiabgkziUkabg6HiLcaa@3F84@  is -70.7 degrees.  The resulting failure locus (the critical combination of K I / K IC MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGlbWaaSbaaSqaaiaadMeaaeqaaO Gaai4laiaadUeadaWgaaWcbaGaamysaiaadoeaaeqaaaaa@37DF@  and K II / K IC MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGlbWaaSbaaSqaaiaadMeacaWGjb aabeaakiaac+cacaWGlbWaaSbaaSqaaiaadMeacaWGdbaabeaaaaa@38AD@  that leads to failure) is also shown

       

All available criteria predict that, after branching, a crack will follow a path such that the local mode II stress intensity factor is zero.

 

 

9.3.9 Static fatigue crack growth

 

For a fatigue test, the crack length is measured (optically, or using compliance techniques) as a function of time or number of load cycles. Fatigue laws are deduced by plotting crack growth rate as a function of applied stress intensity factor.

 

Typical static fatigue data (e.g. for corrosion crack growth, or creep crack growth) behavior is shown on the right.

 

Most materials have a static fatigue threshold MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa aaaaaaa8qacaWFtacaaa@37E6@  a value of K I MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4samaaBa aaleaacaWGjbaabeaaaaa@38C7@  below which crack growth is undetectable.  Then there is a range where crack growth rate shows a power-law dependence on stress intensity factor of the form

da dt =C K I m MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca WGKbGaamyyaaqaaiaadsgacaWG0baaaiabg2da9iaadoeacaWGlbWa a0baaSqaaiaadMeaaeaacaWGTbaaaaaa@3F49@

where m is typically of order 5-10.  Finally, for values of K I MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4samaaBa aaleaacaWGjbaabeaaaaa@38C7@  approaching the fracture toughness, the crack growth rate increases drastically with K I MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4samaaBa aaleaacaWGjbaabeaaaaa@38C7@ .

 

This crack growth law can be used to derive the phenomenological static fatigue criterion outlined in Section 8.2.3.    Assume that at time t=0 the material contains a crack of initial length 2 a 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyamaaBa aaleaacaaIWaaabeaaaaa@38C9@ , and is subjected to a uniaxial stress σ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdmhaaa@38C0@ .  The stress will cause the crack to increase in length, until it becomes long enough to trigger brittle fracture.  The table in Sect 8.3.9 below shows that a crack of length 2a subjected to stress σ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdmhaaa@37B9@  the stress has a crack tip stress intensity factor K I =σ πa MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4samaaBa aaleaacaWGjbaabeaakiabg2da9iabeo8aZnaakaaabaGaeqiWdaNa amyyaaWcbeaaaaa@3E58@ .   Substituting into the static fatigue crack growth law and integrating gives the following expression for crack length as a function of time

2 m2 ( 1 a 0 m/21 1 a m/21 )=C π m/2 σ m dt MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca aIYaaabaGaamyBaiabgkHiTiaaikdaaaWaaeWaaeaadaWcaaqaaiaa igdaaeaacaWGHbWaa0baaSqaaiaaicdaaeaacaWGTbGaai4laiaaik dacqGHsislcaaIXaaaaaaakiabgkHiTmaalaaabaGaaGymaaqaaiaa dggadaahaaWcbeqaaiaad2gacaGGVaGaaGOmaiabgkHiTiaaigdaaa aaaaGccaGLOaGaayzkaaGaeyypa0Jaam4qaiabec8aWnaaCaaaleqa baGaamyBaiaac+cacaaIYaaaaOWaa8qaaeaacqaHdpWCdaahaaWcbe qaaiaad2gaaaGccaWGKbGaamiDaaWcbeqab0Gaey4kIipaaaa@5662@

where 2 a 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyamaaBa aaleaacaaIWaaabeaaaaa@38C9@  is the crack length at time t=0.   The solid will fracture when the crack tip stress intensity factor reaches the fracture toughness K IC MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4samaaBa aaleaacaWGjbGaam4qaaqabaaaaa@398F@ , so that the tensile strength at time t=0   and at time t  must satisfy

σ TS0 π a 0 = K IC σ TS πa = K IC MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aaS baaSqaaiaadsfacaWGtbGaaGimaaqabaGcdaGcaaqaaiabec8aWjaa dggadaWgaaWcbaGaaGimaaqabaaabeaakiaaykW7cqGH9aqpcaWGlb WaaSbaaSqaaiaadMeacaWGdbaabeaakiaaykW7caaMc8UaaGPaVlaa ykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaG PaVlabeo8aZnaaBaaaleaacaWGubGaam4uaaqabaGcdaGcaaqaaiab ec8aWjaadggaaSqabaGccqGH9aqpcaWGlbWaaSbaaSqaaiaadMeaca WGdbaabeaaaaa@60BF@

Eliminating the crack length and simplifying gives

σ TS = σ TS0 ( 1 (m2)πC K IC m2 2 σ TS0 m2 σ m dt ) 1/(m2) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aaS baaSqaaiaadsfacaWGtbaabeaakiabg2da9iabeo8aZnaaBaaaleaa caWGubGaam4uaiaaicdaaeqaaOWaaeWaaeaacaaIXaGaeyOeI0YaaS aaaeaacaGGOaGaamyBaiabgkHiTiaaikdacaGGPaGaeqiWdaNaam4q aiaadUeadaqhaaWcbaGaamysaiaadoeaaeaacaWGTbGaeyOeI0IaaG OmaaaaaOqaaiaaikdacqaHdpWCdaqhaaWcbaGaamivaiaadofacaaI WaaabaGaamyBaiabgkHiTiaaikdaaaaaaOWaa8qaaeaacqaHdpWCda ahaaWcbeqaaiaad2gaaaGccaWGKbGaamiDaaWcbeqab0Gaey4kIipa aOGaayjkaiaawMcaamaaCaaaleqabaGaaGymaiaac+cacaGGOaGaam yBaiabgkHiTiaaikdacaGGPaaaaaaa@6321@

Assuming that the operating stress is well below the fracture stress, we can approximate this by

σ TS = σ TS0 ( 1α ( σ/ σ TS0 ) m dt ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aaS baaSqaaiaadsfacaWGtbaabeaakiabg2da9iabeo8aZnaaBaaaleaa caWGubGaam4uaiaaicdaaeqaaOWaaeWaaeaacaaIXaGaeyOeI0Iaeq ySde2aa8qaaeaadaqadaqaaiabeo8aZjaac+cacqaHdpWCdaWgaaWc baGaamivaiaadofacaaIWaaabeaaaOGaayjkaiaawMcaamaaCaaale qabaGaamyBaaaakiaadsgacaWG0baaleqabeqdcqGHRiI8aaGccaGL OaGaayzkaaaaaa@5254@

which is the stress based static fatigue law of Sect 9.2.3.

 

 

 

9.3.10 Cyclic fatigue crack growth

 

Under cyclic loading, the crack is subjected to a cycle of mode I and mode II stress intensity factor. Most fatigue tests are performed under a steady cycle of pure mode I loading, as sketched in the figure. 

 

The results are usually displayed by plotting the crack growth per cycle da/dN MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGKbGaamyyaiaac+cacaWGKbGaam Otaaaa@3704@  as a function of the stress intensity factor range

Δ K I ={ K max K min K min 0 K max K min <0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeuiLdqKaam 4samaaBaaaleaacaWGjbaabeaakiabg2da9maaceaabaqbaeqabiqa aaqaaiaadUeadaWgaaWcbaGaciyBaiaacggacaGG4baabeaakiabgk HiTiaadUeadaWgaaWcbaGaciyBaiaacMgacaGGUbaabeaakiaaykW7 caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVl aaykW7caaMc8Uaam4samaaBaaaleaaciGGTbGaaiyAaiaac6gaaeqa aOGaeyyzImRaaGimaiaaykW7caaMc8oabaGaam4samaaBaaaleaaci GGTbGaaiyyaiaacIhaaeqaaOGaaGPaVlaaykW7caaMc8UaaGPaVlaa ykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaG PaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaM c8UaaGPaVlaaykW7caWGlbWaaSbaaSqaaiGac2gacaGGPbGaaiOBaa qabaGccqGH8aapcaaIWaGaaGPaVlaaykW7aaaacaGL7baaaaa@8F6D@

 

A typical result shows three regions. There is a fatigue threshold Δ K th MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeuiLdqKaam 4samaaBaaaleaacaWG0bGaamiAaaqabaaaaa@3B45@  below which crack growth is undetectable.  For modest loads, the crack growth rate obeys Paris law

da dN =C ( Δ K I ) n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca WGKbGaamyyaaqaaiaadsgacaWGobaaaiabg2da9iaadoeadaqadaqa aiabfs5aejaadUeadaWgaaWcbaGaamysaaqabaaakiaawIcacaGLPa aadaahaaWcbeqaaiaad6gaaaaaaa@4249@

where the index n is between 2 and 4.  As the maximum stress intensity factor approaches the fracture toughness of the material, the crack growth rate accelerates dramatically.

 

In the Paris law regime, the crack growth rate is only weakly sensitive to the mean value of stress intensity factor K m MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4samaaBa aaleaacaWGTbaabeaaaaa@38EB@ .  In the other two regimes,  K m MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4samaaBa aaleaacaWGTbaabeaaaaa@38EB@  has a noticeable effect - the fatigue threshold is reduced as K m MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4samaaBa aaleaacaWGTbaabeaaaaa@38EB@  increases, and the crack growth rate in regime III increases with K m MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4samaaBa aaleaacaWGTbaabeaaaaa@38EB@

 

 

 

9.3.11 Finding cracks in structures

 

Determining the length of pre-existing cracks in a component is often the most difficult part of applying fracture mechanics in practice.  For most practical applications you simply don’t know if your component will have a crack in it, and it will be expensive if you need to find out.  Your options are:

1.      Take a wild guess, based on microscopic examinations of representative samples of material.  Alternatively, you can specify the biggest flaw you are prepared to tolerate and insist that your material suppliers manufacture appropriately defect free materials.

2.      Conduct a proof test (popular e.g. with pressure vessel applications) wherein the structure or component is subjected to a load greatly exceeding the anticipated service load under controlled conditions.  If the fracture toughness of the material is known, you can then deduce the largest crack size that could be present in the structure without causing failure during proof testing.

3.      Use some kind of non-destructive test technique to attempt to detect cracks in your structure.  Examples of such techniques are ultrasound, where you look for echoes off crack surfaces; x-ray techniques; and inspection with optical microscopy.  If you detect a crack, most of these techniques will allow you to estimate the crack length.  If not, you have to assume for design purposes that your structure is crammed full of cracks that are just too short to be detected.