Chapter 9

 

Modeling Material Failure

 

 

9.5 Plastic fracture mechanics

 

Thus far we have avoided discussing the complicated material behavior in the process zone near the crack tip.  This is acceptable as long as the process zone is small compared with the specimen dimensions, and a clear zone of K dominance is established around the crack tip.  In some structures, however, the materials are so tough and ductile that the plastic zone near the crack tip is huge MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa aaaaaaa8qacaWFtacaaa@37E6@  and comparable to specimen dimensions.  Linear elastic fracture mechanics cannot be used under these conditions.  Instead, we adopt a framework based on plastic solutions to crack tip fields. 

 

In this section, we address three issues:

1.      The size of the plastic zone at the crack tip is estimated

2.      The asymptotic fields near the crack tip in a plastic material are calculated

3.      A phenomenological framework for predicting fracture in plastic solids is outlined.

 

9.5.1 Dugdale-Barenblatt cohesive zone model of yield at a crack tip

 

The simplest estimate of the size of the plastic zone at a crack tip can be obtained using Dugdale & Barenblatt’s cohesive zone model, which gives the plastic zone size at the tip of a crack in a thin sheet (deforming under conditions of plane stress) as

r p π 8 ( K I Y ) 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOCamaaBa aaleaacaWGWbaabeaakiabgIKi7oaalaaabaGaeqiWdahabaGaaGio aaaadaqadaqaamaalaaabaGaam4samaaBaaaleaacaWGjbaabeaaaO qaaiaadMfaaaaacaGLOaGaayzkaaWaaWbaaSqabeaacaaIYaaaaaaa @42A3@

where K I MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8sk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGlbWaaSbaaSqaaiaadMeaaeqaaa aa@34A0@  is the crack tip stress intensity factor and Y is the material yield stress. 

 

This estimate is derived as follows.  Consider a crack of length 2a in an elastic-perfectly plastic material with elastic constants E,ν MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyraiaacY cacqaH9oGBaaa@3928@  and yield stress Y.  We assume that the specimen is a thin sheet, with thickness much less than crack length, so that a state of plane stress is developed in the solid.  We anticipate that there will be a region near each crack tip where the material deforms plastically.  The Mises equivalent stress 3 S ij S ij /2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaOaaaeaaca aIZaGaam4uamaaBaaaleaacaWGPbGaamOAaaqabaGccaWGtbWaaSba aSqaaiaadMgacaWGQbaabeaakiaac+cacaaIYaaaleqaaaaa@3F1A@  should not exceed yield in this region.   It’s hard to find a solution with stresses at yield everywhere in the plastic zone, but we can easily construct an approximate solution where the stress along the line of the crack satisfies the yield condition, using the ‘cohesive zone’ model illustrated in the picture.  

 

Let r p MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOCamaaBa aaleaacaWGWbaabeaaaaa@3915@  denote the length of the cohesive zone at each crack tip.  To construct an appropriate solution we extend the crack in both directions to put fictitious crack tips at x 1 =±( a+ r p ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaBa aaleaacaaIXaaabeaakiabg2da9iabgglaXoaabmaabaGaamyyaiab gUcaRiaadkhadaWgaaWcbaGaamiCaaqabaaakiaawIcacaGLPaaaaa a@4152@ , and distribute tractions of magnitude Y MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamywaaaa@37DB@  over the crack flanks from x 1 =a MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaBa aaleaacaaIXaaabeaakiabg2da9iaadggaaaa@3AD7@  to x 1 =a+ r p MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaBa aaleaacaaIXaaabeaakiabg2da9iaadggacqGHRaWkcaWGYbWaaSba aSqaaiaadchaaeqaaaaa@3DD1@ , and similarly at the other crack tip..  Evidently, the stress then satisfies σ 22 =Y MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aaS baaSqaaiaaikdacaaIYaaabeaakiabg2da9iaadMfaaaa@3C52@  along the line of the crack just ahead of each crack.  

 

We can use point force solution given in the table in Section 9.3.3 to compute the stress intensity factor at the fictitious crack tip. Omitting the tedious details of evaluating the integral, we find that

K I * =( σY ) π(a+ r p ) + 2Y π π(a+ r p ) sin 1 a a+ r p MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4samaaDa aaleaacaWGjbaabaGaaiOkaaaakiabg2da9maabmaabaGaeq4WdmNa eyOeI0IaamywaaGaayjkaiaawMcaamaakaaabaGaeqiWdaNaaiikai aadggacqGHRaWkcaWGYbWaaSbaaSqaaiaadchaaeqaaOGaaiykaaWc beaakiabgUcaRmaalaaabaGaaGOmaiaadMfaaeaacqaHapaCaaWaaO aaaeaacqaHapaCcaGGOaGaamyyaiabgUcaRiaadkhadaWgaaWcbaGa amiCaaqabaGccaGGPaaaleqaaOGaci4CaiaacMgacaGGUbWaaWbaaS qabeaacqGHsislcaaIXaaaaOWaaSaaaeaacaWGHbaabaGaamyyaiab gUcaRiaadkhadaWgaaWcbaGaamiCaaqabaaaaaaa@5BBD@

The * on the stress intensity factor is introduced to emphasize that this is not the true crack tip stress intensity factor (which is of course K I =σ πa MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4samaaBa aaleaacaWGjbaabeaakiabg2da9iabeo8aZnaakaaabaGaeqiWdaNa amyyaaWcbeaaaaa@3E58@  ), but the stress intensity factor at the fictitious crack tip. The stresses must remain bounded just ahead of the fictitious crack tip, so that r p MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOCamaaBa aaleaacaWGWbaabeaaaaa@3915@  must be chosen to satisfy K I * =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4samaaDa aaleaacaWGjbaabaGaaiOkaaaakiabg2da9iaaicdaaaa@3A39@ .  This gives

r p = a sin( π[1σ/Y]/2 ) a π 2 8 ( σ Y ) 2 a MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOCamaaBa aaleaacaWGWbaabeaakiabg2da9maalaaabaGaamyyaaqaaiGacoha caGGPbGaaiOBamaabmaabaGaeqiWdaNaai4waiaaigdacqGHsislcq aHdpWCcaGGVaGaamywaiaac2facaGGVaGaaGOmaaGaayjkaiaawMca aaaacqGHsislcaWGHbGaeyisIS7aaSaaaeaacqaHapaCdaahaaWcbe qaaiaaikdaaaaakeaacaaI4aaaamaabmaabaWaaSaaaeaacqaHdpWC aeaacaWGzbaaaaGaayjkaiaawMcaamaaCaaaleqabaGaaGOmaaaaki aadggaaaa@567D@

Its more sensible to express this in terms of stress intensity factor

r p π 8 ( K I Y ) 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOCamaaBa aaleaacaWGWbaabeaakiabgIKi7oaalaaabaGaeqiWdahabaGaaGio aaaadaqadaqaamaalaaabaGaam4samaaBaaaleaacaWGjbaabeaaaO qaaiaadMfaaaaacaGLOaGaayzkaaWaaWbaaSqabeaacaaIYaaaaaaa @4293@

This estimate turns out to be remarkably accurate for plane stress conditions, where more detailed calculations give

r p = 1 π ( K IC Y ) 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOCamaaBa aaleaacaWGWbaabeaakiabg2da9maalaaabaGaaGymaaqaaiabec8a WbaadaqadaqaamaalaaabaGaam4samaaBaaaleaacaWGjbGaam4qaa qabaaakeaacaWGzbaaaaGaayjkaiaawMcaamaaCaaaleqabaGaaGOm aaaaaaa@42A9@

For plane strain the plastic zone is smaller: detailed calculations show that the plastic zone size is

r p = 1 3π ( K IC Y ) 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOCamaaBa aaleaacaWGWbaabeaakiabg2da9maalaaabaGaaGymaaqaaiaaioda cqaHapaCaaWaaeWaaeaadaWcaaqaaiaadUeadaWgaaWcbaGaamysai aadoeaaeqaaaGcbaGaamywaaaaaiaawIcacaGLPaaadaahaaWcbeqa aiaaikdaaaaaaa@4366@

 

 

9.5.2 Hutchinson-Rice-Rosengren (HRR) crack tip fields for stationary crack in a power law hardening solid

 

The HRR fields are an exact solution to the stress, strain and displacement fields near a crack tip in a power-law strain hardening, rigid plastic material, which is subjected to monotonically increasing stress at infinity.  The model is based on the following assumptions:

1.      The solid is infinitely large, and contains an infinitely long crack with its tip at the origin

2.      The material is a rigid plastic, strain hardening solid with uniaxial stress -v- strain curve

σ= σ 0 (ε/ ε 0 ) 1/n MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHdpWCcqGH9aqpcqaHdpWCdaWgaa WcbaGaaGimaaqabaGccaGGOaGaeqyTduMaai4laiabew7aLnaaBaaa leaacaaIWaaabeaakiaacMcadaahaaWcbeqaaiaaigdacaGGVaGaam OBaaaaaaa@411A@

where σ 0 , ε 0 ,n MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHdpWCdaWgaaWcbaGaaGimaaqaba GccaGGSaGaeqyTdu2aaSbaaSqaaiaaicdaaeqaaOGaaiilaiaad6ga aaa@3A63@  are material properties, with n>1.

The HRR solution shows that the stress, strain and displacement fields at a point (r,θ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaGGOaGaamOCaiaacYcacqaH4oqCca GGPaaaaa@377C@  in the solid can be calculated from functions of the form

σ ij = σ 0 ( J σ 0 ε 0 r ) 1/(n+1) Σ ij (θ,n) ε ij = ε 0 ( J σ 0 ε 0 r ) n/(n+1) E ij (θ,n) u i = J σ 0 ( σ 0 ε 0 r J ) 1/(n+1) U i (θ,n) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacqaHdp WCdaWgaaWcbaGaamyAaiaadQgaaeqaaOGaeyypa0Jaeq4Wdm3aaSba aSqaaiaaicdaaeqaaOWaaeWaaeaadaWcaaqaaiaadQeaaeaacqaHdp WCdaWgaaWcbaGaaGimaaqabaGccqaH1oqzdaWgaaWcbaGaaGimaaqa baGccaWGYbaaaaGaayjkaiaawMcaamaaCaaaleqabaGaaGymaiaac+ cacaGGOaGaamOBaiabgUcaRiaaigdacaGGPaaaaOGaeu4Odm1aaSba aSqaaiaadMgacaWGQbaabeaakiaacIcacqaH4oqCcaGGSaGaamOBai aacMcaaeaacqaH1oqzdaWgaaWcbaGaamyAaiaadQgaaeqaaOGaeyyp a0JaeqyTdu2aaSbaaSqaaiaaicdaaeqaaOWaaeWaaeaadaWcaaqaai aadQeaaeaacqaHdpWCdaWgaaWcbaGaaGimaaqabaGccqaH1oqzdaWg aaWcbaGaaGimaaqabaGccaWGYbaaaaGaayjkaiaawMcaamaaCaaale qabaGaamOBaiaac+cacaGGOaGaamOBaiabgUcaRiaaigdacaGGPaaa aOGaamyramaaBaaaleaacaWGPbGaamOAaaqabaGccaGGOaGaeqiUde Naaiilaiaad6gacaGGPaaabaGaamyDamaaBaaaleaacaWGPbaabeaa kiabg2da9maalaaabaGaamOsaaqaaiabeo8aZnaaBaaaleaacaaIWa aabeaaaaGcdaqadaqaamaalaaabaGaeq4Wdm3aaSbaaSqaaiaaicda aeqaaOGaeqyTdu2aaSbaaSqaaiaaicdaaeqaaOGaamOCaaqaaiaadQ eaaaaacaGLOaGaayzkaaWaaWbaaSqabeaacaaIXaGaai4laiaacIca caWGUbGaey4kaSIaaGymaiaacMcaaaGccaWGvbWaaSbaaSqaaiaadM gaaeqaaOGaaiikaiabeI7aXjaacYcacaWGUbGaaiykaaaaaa@8E06@

where Σ ij , E ij , U i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeu4Odm1aaS baaSqaaiaadMgacaWGQbaabeaakiaacYcacaWGfbWaaSbaaSqaaiaa dMgacaWGQbaabeaakiaacYcacaWGvbWaaSbaaSqaaiaadMgaaeqaaa aa@40C5@  are dimensionless functions of the angle θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiUdehaaa@37AC@  and the hardening index n only, and J is the value of the (path independent) J integral

J= C (W( σ e ) δ j1 σ ij u i x 1 ) m j ds MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOsaiabg2 da9maapefabaGaaiikaiaadEfacaGGOaGaeq4Wdm3aaSbaaSqaaiaa dwgaaeqaaOGaaiykaiabes7aKnaaBaaaleaacaWGQbGaaGymaaqaba GccqGHsislcqaHdpWCdaWgaaWcbaGaamyAaiaadQgaaeqaaOWaaSaa aeaacqGHciITcaWG1bGaaCjaVpaaBaaaleaacaWGPbaabeaaaOqaai abgkGi2kaadIhadaWgaaWcbaGaaGymaaqabaaaaOGaaiykaiaad2ga daWgaaWcbaGaamOAaaqabaGccaWGKbGaam4CaaWcbaGaam4qaaqab0 Gaey4kIipaaaa@5705@

where

W= n n+1 σ 0 ε 0 ( 3 S ij S ij 2 σ 0 ) n+1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4vaiabg2 da9maalaaabaGaamOBaaqaaiaad6gacqGHRaWkcaaIXaaaaiabeo8a ZnaaBaaaleaacaaIWaaabeaakiabew7aLnaaBaaaleaacaaIWaaabe aakmaabmaabaWaaSaaaeaacaaIZaGaam4uamaaBaaaleaacaWGPbGa amOAaaqabaGccaWGtbWaaSbaaSqaaiaadMgacaWGQbaabeaaaOqaai aaikdacqaHdpWCdaWgaaWcbaGaaGimaaqabaaaaaGccaGLOaGaayzk aaWaaWbaaSqabeaacaWGUbGaey4kaSIaaGymaaaakiaaykW7caaMc8 oaaa@5334@

with S ij = σ ij σ kk δ ij /3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGtbWaaSbaaSqaaiaadMgacaWGQb aabeaakiabg2da9iabeo8aZnaaBaaaleaacaWGPbGaamOAaaqabaGc cqGHsislcqaHdpWCdaWgaaWcbaGaam4AaiaadUgaaeqaaOGaeqiTdq 2aaSbaaSqaaiaadMgacaWGQbaabeaakiaac+cacaaIZaaaaa@447B@ W can be interpreted as the total work done in loading the material up to a stress σ ij MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHdpWCdaWgaaWcbaGaamyAaiaadQ gaaeqaaaaa@3692@  under monotonically increasing, proportional loading.

 

These results are important for two reasons:

1.      They show that the magnitudes of the stress, strain and displacement near the crack tip are characterized by J.  Thus, in highly plastic materials, J can replace K as the fracture criterion.

2.      They illustrate the nature of the stress and strain fields near the crack tip.  In particular, they show that the stress has a r (n+1) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGYbWaaWbaaSqabeaacqGHsislca GGOaGaamOBaiabgUcaRiaaigdacaGGPaaaaaaa@38C0@  singularity MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa aaaaaaa8qacaWFtacaaa@37E6@  for n=1 (a linear stress-strain curve) we recover the square root singularity found in elastic materials; while for a perfectly plastic solid ( n MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGUbGaeyOKH4QaeyOhIukaaa@3717@  ) the stress is constant.  In contrast, the strains have a square root singularity for n=1 and an r 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGYbWaaWbaaSqabeaacqGHsislca aIXaaaaaaa@3592@  singularity for n MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGUbGaeyOKH4QaeyOhIukaaa@3717@

 

 

Derivation  The HRR solution is derived by solving the following governing equations for displacements u i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWG1bWaaSbaaSqaaiaadMgaaeqaaa aa@34DA@ , strains ε ij MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaH1oqzdaWgaaWcbaGaamyAaiaadQ gaaeqaaaaa@3676@  and stresses σ ij MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHdpWCdaWgaaWcbaGaamyAaiaadQ gaaeqaaaaa@3692@  

 Strain-displacement relation ε ij =( u i / x j + u j / x i )/2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyTdu2aaS baaSqaaiaadMgacaWGQbaabeaakiabg2da9iaacIcacqGHciITcaWG 1bWaaSbaaSqaaiaadMgaaeqaaOGaai4laiabgkGi2kaadIhadaWgaa WcbaGaamOAaaqabaGccqGHRaWkcqGHciITcaWG1bWaaSbaaSqaaiaa dQgaaeqaaOGaai4laiabgkGi2kaadIhadaWgaaWcbaGaamyAaaqaba GccaGGPaGaai4laiaaikdaaaa@4EE5@

 Stress equilibrium σ ij /x i =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqGHciITcqaHdpWCdaWgaaWcbaGaam yAaiaadQgaaeqaaOGaai4laiabgkGi2kaadIhacaWLa8+aaSbaaSqa aiaadMgaaeqaaOGaeyypa0JaaGimaaaa@3F84@

 Boundary conditions σ 22 = σ 12 =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHdpWCdaWgaaWcbaGaaGOmaiaaik daaeqaaOGaeyypa0Jaeq4Wdm3aaSbaaSqaaiaaigdacaaIYaaabeaa kiabg2da9iaaicdaaaa@3C6D@  on x 2 =0 x 1 <0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWG4bWaaSbaaSqaaiaaikdaaeqaaO Gaeyypa0JaaGimaiaaykW7caaMc8UaamiEamaaBaaaleaacaaIXaaa beaakiabgYda8iaaicdaaaa@3D37@

 The stress-strain relation for a power-law hardening rigid plastic material subjected to monotonically increasing, proportional loading (this means that material particles are subjected to stresses and strains whose principal axes don’t rotate during loading) can be expressed as

ε ij = ε 0 ( σ e σ 0 ) n 3 2 S ij σ e S ij = σ 0 ( ε e ε 0 ) 1/n 2 3 ε ij ε 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyTdu2aaS baaSqaaiaadMgacaWGQbaabeaakiabg2da9iabew7aLnaaBaaaleaa caaIWaaabeaakmaabmaabaWaaSaaaeaacqaHdpWCdaWgaaWcbaGaam yzaaqabaaakeaacqaHdpWCdaWgaaWcbaGaaGimaaqabaaaaaGccaGL OaGaayzkaaWaaWbaaSqabeaacaWGUbaaaOWaaSaaaeaacaaIZaaaba GaaGOmaaaadaWcaaqaaiaadofadaWgaaWcbaGaamyAaiaadQgaaeqa aaGcbaGaeq4Wdm3aaSbaaSqaaiaadwgaaeqaaaaakiaaykW7caaMc8 UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7 caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVl aaykW7caaMc8Uaam4uamaaBaaaleaacaWGPbGaamOAaaqabaGccqGH 9aqpcqaHdpWCdaWgaaWcbaGaaGimaaqabaGcdaqadaqaamaalaaaba GaeqyTdu2aaSbaaSqaaiaadwgaaeqaaaGcbaGaeqyTdu2aaSbaaSqa aiaaicdaaeqaaaaaaOGaayjkaiaawMcaamaaCaaaleqabaGaaGymai aac+cacaWGUbaaaOWaaSaaaeaacaaIYaaabaGaaG4maaaadaWcaaqa aiabew7aLnaaBaaaleaacaWGPbGaamOAaaqabaaakeaacqaH1oqzda WgaaWcbaGaaGimaaqabaaaaaaa@8503@

where ε 0 , σ 0 ,n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyTdu2aaS baaSqaaiaaicdaaeqaaOGaaiilaiabeo8aZnaaBaaaleaacaaIWaaa beaakiaacYcacaWGUbaaaa@3E9A@  are material constants and S ij = σ ij σ kk δ ij /3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4uamaaBa aaleaacaWGPbGaamOAaaqabaGccqGH9aqpcqaHdpWCdaWgaaWcbaGa amyAaiaadQgaaeqaaOGaeyOeI0Iaeq4Wdm3aaSbaaSqaaiaadUgaca WGRbaabeaakiabes7aKnaaBaaaleaacaWGPbGaamOAaaqabaGccaGG VaGaaG4maaaa@48B2@  is the deviatoric stress and σ e = 3 S ij S ij /2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHdpWCdaWgaaWcbaGaamyzaaqaba GccqGH9aqpdaGcaaqaaiaaiodacaWGtbWaaSbaaSqaaiaadMgacaWG QbaabeaakiaadofadaWgaaWcbaGaamyAaiaadQgaaeqaaOGaai4lai aaikdaaSqabaaaaa@3ECC@  is the Von Mises effective stress.  Of course, we don’t know a priori that material elements ahead of a crack tip experience proportional loading, but this can be verified after the solution has been found.  It is helpful to note that under proportional loading, the rigid plastic material is indistinguishable from an elastic material with strain energy potential

W= n n+1 σ 0 ε 0 ( σ e σ 0 ) n+1 = n n+1 σ 0 ε 0 ( ε e ε 0 ) (1+n)/n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4vaiabg2 da9maalaaabaGaamOBaaqaaiaad6gacqGHRaWkcaaIXaaaaiabeo8a ZnaaBaaaleaacaaIWaaabeaakiabew7aLnaaBaaaleaacaaIWaaabe aakmaabmaabaWaaSaaaeaacqaHdpWCdaWgaaWcbaGaamyzaaqabaaa keaacqaHdpWCdaWgaaWcbaGaaGimaaqabaaaaaGccaGLOaGaayzkaa WaaWbaaSqabeaacaWGUbGaey4kaSIaaGymaaaakiaaykW7caaMc8Ua aGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7ca aMc8UaaGPaVlaaykW7cqGH9aqpdaWcaaqaaiaad6gaaeaacaWGUbGa ey4kaSIaaGymaaaacqaHdpWCdaWgaaWcbaGaaGimaaqabaGccqaH1o qzdaWgaaWcbaGaaGimaaqabaGcdaqadaqaamaalaaabaGaeqyTdu2a aSbaaSqaaiaadwgaaeqaaaGcbaGaeqyTdu2aaSbaaSqaaiaaicdaae qaaaaaaOGaayjkaiaawMcaamaaCaaaleqabaGaaiikaiaaigdacqGH RaWkcaWGUbGaaiykaiaac+cacaWGUbaaaOGaaGPaVdaa@77EC@

The J integral must then be path independent.

 

The equilibrium condition may be satisfied through an Airy stress function ϕ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqy1dygaaa@38C5@ , generating stresses in the usual way as

σ rr = 1 r ϕ r + 1 r 2 2 ϕ θ 2 σ θθ = 2 ϕ r 2 σ rθ = r ( 1 r ϕ θ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa aiaabeqaaiqacaWaaaGcbaGaeq4Wdm3aaSbaaSqaaiaadkhacaWGYb aabeaakiabg2da9maalaaabaGaaGymaaqaaiaadkhaaaWaaSaaaeaa cqGHciITcqaHvpGzaeaacqGHciITcaWGYbaaaiabgUcaRmaalaaaba GaaGymaaqaaiaadkhadaahaaWcbeqaaiaaikdaaaaaaOWaaSaaaeaa cqGHciITdaahaaWcbeqaaiaaikdaaaGccqaHvpGzaeaacqGHciITcq aH4oqCdaahaaWcbeqaaiaaikdaaaaaaOGaaGPaVlaaykW7caaMc8Ua aGPaVlaaykW7caaMc8Uaeq4Wdm3aaSbaaSqaaiabeI7aXjabeI7aXb qabaGccqGH9aqpdaWcaaqaaiabgkGi2oaaCaaaleqabaGaaGOmaaaa kiabew9aMbqaaiabgkGi2kaadkhadaahaaWcbeqaaiaaikdaaaaaaO GaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7cqaHdpWCdaWgaaWcbaGa amOCaiabeI7aXbqabaGccqGH9aqpcqGHsisldaWcaaqaaiabgkGi2c qaaiabgkGi2kaadkhaaaWaaeWaaeaadaWcaaqaaiaaigdaaeaacaWG YbaaamaalaaabaGaeyOaIyRaeqy1dygabaGaeyOaIyRaeqiUdehaaa GaayjkaiaawMcaaaaa@7BE5@

The solution can be derived from an Airy function that has a separable form

ϕ= r α f( θ )+... MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqy1dyMaey ypa0JaamOCamaaCaaaleqabaGaeqySdegaaOGaamOzamaabmaabaGa eqiUdehacaGLOaGaayzkaaGaey4kaSIaaiOlaiaac6cacaGGUaaaaa@43BA@

where the power α MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySdegaaa@389C@  and f( θ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzamaabm aabaGaeqiUdehacaGLOaGaayzkaaaaaa@3B27@  are to be determined.

 

The strength of the singularity α MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySdegaaa@389C@  can be determined using the J integral. Evaluating the integral around a circular contour radius r enclosing the crack tip we obtain

J= C (W δ j1 σ ij u i x 1 ) m j ds = π π (W δ j1 σ ij u i x 1 ) m j rdθ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOsaiabg2 da9maapefabaGaaiikaiaadEfacqaH0oazdaWgaaWcbaGaamOAaiaa igdaaeqaaOGaeyOeI0Iaeq4Wdm3aaSbaaSqaaiaadMgacaWGQbaabe aakmaalaaabaGaeyOaIyRaamyDamaaBaaaleaacaWGPbaabeaaaOqa aiabgkGi2kaadIhadaWgaaWcbaGaaGymaaqabaaaaOGaaiykaiaad2 gadaWgaaWcbaGaamOAaaqabaGccaWGKbGaam4CaaWcbaGaam4qaaqa b0Gaey4kIipakiabg2da9maapehabaGaaiikaiaadEfacqaH0oazda WgaaWcbaGaamOAaiaaigdaaeqaaOGaeyOeI0Iaeq4Wdm3aaSbaaSqa aiaadMgacaWGQbaabeaakmaalaaabaGaeyOaIyRaamyDamaaBaaale aacaWGPbaabeaaaOqaaiabgkGi2kaadIhadaWgaaWcbaGaaGymaaqa baaaaOGaaiykaiaad2gadaWgaaWcbaGaamOAaaqabaGccaWGYbGaam izaiabeI7aXbWcbaGaeyOeI0IaeqiWdahabaGaeqiWdahaniabgUIi Ydaaaa@7033@

For the J integral to be path independent, it must be independent of r and therefore W must be of order r 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOCamaaCa aaleqabaGaeyOeI0IaaGymaaaaaaa@39C9@ .  The Airy function gives stresses of order r α2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOCamaaCa aaleqabaGaeqySdeMaeyOeI0IaaGOmaaaaaaa@3B69@ , and the corresponding strain energy density would have order r (n+1)(α2) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOCamaaCa aaleqabaGaaiikaiaad6gacqGHRaWkcaaIXaGaaiykaiaacIcacqaH XoqycqGHsislcaaIYaGaaiykaaaaaaa@40AB@ .  Consequently, for a path independent J, we must have (n+1)(α2)=1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaad6 gacqGHRaWkcaaIXaGaaiykaiaacIcacqaHXoqycqGHsislcaaIYaGa aiykaiabg2da9iabgkHiTiaaigdaaaa@4235@ , so

α= 2n+1 n+1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySdeMaey ypa0ZaaSaaaeaacaaIYaGaamOBaiabgUcaRiaaigdaaeaacaWGUbGa ey4kaSIaaGymaaaaaaa@3F8E@

Note for a linear material (n=1), we find α=3/2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySdeMaey ypa0JaaG4maiaac+cacaaIYaaaaa@3BCE@ , which corresponds to the expected square-root stress singularity.

 

We can now scale the governing equations as discussed in Section 7.2.13.   To this end, define normalized length, displacement, strain, stress and Airy function as

X i = σ 0 ε 0 x i /JR= σ 0 ε 0 r/J U i = u i σ 0 /J E ij =ε ij / ε 0 Σ ij = σ ij / σ 0 Φ=ϕ ( σ 0 ε 0 /J ) 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGybWaaSbaaSqaaiaadMgaaeqaaO Gaeyypa0Jaeq4Wdm3aaSbaaSqaaiaaicdaaeqaaOGaeqyTdu2aaSba aSqaaiaaicdaaeqaaOGaamiEamaaBaaaleaacaWGPbaabeaakiaac+ cacaWGkbGaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPa VlaadkfacqGH9aqpcqaHdpWCdaWgaaWcbaGaaGimaaqabaGccqaH1o qzdaWgaaWcbaGaaGimaaqabaGccaWGYbGaai4laiaadQeacaaMc8Ua aGPaVlaaykW7caaMc8UaaGPaVlaaykW7caWGvbWaaSbaaSqaaiaadM gaaeqaaOGaeyypa0JaamyDamaaBaaaleaacaWGPbaabeaakiabeo8a ZnaaBaaaleaacaaIWaaabeaakiaac+cacaWGkbGaaGPaVlaaykW7ca aMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caWGfbWaaSbaaSqa aiaadMgacaWGQbaabeaakiabg2da9iabew7aLjaaxcW7daWgaaWcba GaamyAaiaadQgaaeqaaOGaai4laiabew7aLnaaBaaaleaacaaIWaaa beaakiaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlabfo6atn aaBaaaleaacaWGPbGaamOAaaqabaGccqGH9aqpcqaHdpWCdaWgaaWc baGaamyAaiaadQgaaeqaaOGaai4laiabeo8aZnaaBaaaleaacaaIWa aabeaakiaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7 cqqHMoGrcqGH9aqpcqaHvpGzdaqadaqaaiabeo8aZnaaBaaaleaaca aIWaaabeaakiabew7aLnaaBaaaleaacaaIWaaabeaakiaac+cacaWG kbaacaGLOaGaayzkaaWaaWbaaSqabeaacaaIYaaaaaaa@ABF5@

With these definitions the governing equations reduce to

 Strain-displacement relation E ij =( U i / X j + U j / X i )/2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyramaaBa aaleaacaWGPbGaamOAaaqabaGccqGH9aqpcaGGOaGaeyOaIyRaamyv amaaBaaaleaacaWGPbaabeaakiaac+cacqGHciITcaWGybWaaSbaaS qaaiaadQgaaeqaaOGaey4kaSIaeyOaIyRaamyvamaaBaaaleaacaWG Qbaabeaakiaac+cacqGHciITcaWGybWaaSbaaSqaaiaadMgaaeqaaO Gaaiykaiaac+cacaaIYaaaaa@4D88@

 Stress equilibrium Σ ij /X i =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqGHciITcqqHJoWudaWgaaWcbaGaam yAaiaadQgaaeqaaOGaai4laiabgkGi2kaadIfacaWLa8+aaSbaaSqa aiaadMgaaeqaaOGaeyypa0JaaGimaaaa@3F25@

 Constitutive equation

E ij = ( Σ e ) n 3 2 Σ ij Σ e Σ ij = Σ ij Σ kk δ ij /3 Σ e = 3 Σ ij Σ ij /2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyramaaBa aaleaacaWGPbGaamOAaaqabaGccqGH9aqpdaqadaqaaiabfo6atnaa BaaaleaacaWGLbaabeaaaOGaayjkaiaawMcaamaaCaaaleqabaGaam OBaaaakmaalaaabaGaaG4maaqaaiaaikdaaaWaaSaaaeaacuqHJoWu gaqbamaaBaaaleaacaWGPbGaamOAaaqabaaakeaacqqHJoWudaWgaa WcbaGaamyzaaqabaaaaOGaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7 caaMc8Uafu4OdmLbauaadaWgaaWcbaGaamyAaiaadQgaaeqaaOGaey ypa0Jaeu4Odm1aaSbaaSqaaiaadMgacaWGQbaabeaakiabgkHiTiab fo6atnaaBaaaleaacaWGRbGaam4AaaqabaGccqaH0oazdaWgaaWcba GaamyAaiaadQgaaeqaaOGaai4laiaaiodacaaMc8UaaGPaVlaaykW7 caaMc8UaaGPaVlaaykW7caaMc8Uaeu4Odm1aaSbaaSqaaiaadwgaae qaaOGaeyypa0ZaaOaaaeaacaaIZaGaeu4Odm1aaSbaaSqaaiaadMga caWGQbaabeaakiabfo6atnaaBaaaleaacaWGPbGaamOAaaqabaGcca GGVaGaaGOmaaWcbeaaaaa@7B34@

 

In addition, the stresses are related to the normalized Airy function by

Σ rr = 1 R Φ R + 1 R 2 2 Φ θ 2 Σ θθ = 2 Φ R 2 Σ rθ = R ( 1 R Φ θ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa aiaabeqaaiqacaWaaaGcbaGaeu4Odm1aaSbaaSqaaiaadkhacaWGYb aabeaakiabg2da9maalaaabaGaaGymaaqaaiaadkfaaaWaaSaaaeaa cqGHciITcqqHMoGraeaacqGHciITcaWGsbaaaiabgUcaRmaalaaaba GaaGymaaqaaiaadkfadaahaaWcbeqaaiaaikdaaaaaaOWaaSaaaeaa cqGHciITdaahaaWcbeqaaiaaikdaaaGccqqHMoGraeaacqGHciITcq aH4oqCdaahaaWcbeqaaiaaikdaaaaaaOGaaGPaVlaaykW7caaMc8Ua aGPaVlaaykW7caaMc8Uaeu4Odm1aaSbaaSqaaiabeI7aXjabeI7aXb qabaGccqGH9aqpdaWcaaqaaiabgkGi2oaaCaaaleqabaGaaGOmaaaa kiabfA6agbqaaiabgkGi2kaadkfadaahaaWcbeqaaiaaikdaaaaaaO GaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7cqqHJoWudaWgaaWcbaGa amOCaiabeI7aXbqabaGccqGH9aqpcqGHsisldaWcaaqaaiabgkGi2c qaaiabgkGi2kaadkfaaaWaaeWaaeaadaWcaaqaaiaaigdaaeaacaWG sbaaamaalaaabaGaeyOaIyRaeuOPdyeabaGaeyOaIyRaeqiUdehaaa GaayjkaiaawMcaaaaa@7930@

while the expression for the J integral becomes

1= π π ( W ^ δ j1 Σ ij U i X 1 ) m j Rdθ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGymaiabg2 da9maapehabaGaaiikaiqadEfagaqcaiabes7aKnaaBaaaleaacaWG QbGaaGymaaqabaGccqGHsislcqqHJoWudaWgaaWcbaGaamyAaiaadQ gaaeqaaOWaaSaaaeaacqGHciITcaWGvbWaaSbaaSqaaiaadMgaaeqa aaGcbaGaeyOaIyRaamiwamaaBaaaleaacaaIXaaabeaaaaGccaGGPa GaamyBamaaBaaaleaacaWGQbaabeaakiaadkfacaWGKbGaeqiUdeha leaacqGHsislcqaHapaCaeaacqaHapaCa0Gaey4kIipaaaa@5611@

where W ^ =W/ σ 0 ε 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaaceWGxbGbaKaacqGH9aqpcaWGxbGaai 4laiabeo8aZnaaBaaaleaacaaIWaaabeaakiabew7aLnaaBaaaleaa caaIWaaabeaaaaa@3B87@ .  The only material parameter appearing in the scaled equations is n.  In addition, note that J has been eliminated from the equations, so the solution is independent of J.

 

The stresses can be derived from an Airy function

Φ= R (2n+1)/(n+1) f(θ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqqHMoGrcqGH9aqpcaWGsbWaaWbaaS qabeaacaGGOaGaaGOmaiaad6gacqGHRaWkcaaIXaGaaiykaiaac+ca caGGOaGaamOBaiabgUcaRiaaigdacaGGPaaaaOGaamOzaiaacIcacq aH4oqCcaGGPaaaaa@438F@

The scaling of displacements, strain and stress with load and material properties then follows directly from the definition of the normalized quantities. 

 

To compute the full expression for f(θ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGMbGaaiikaiabeI7aXjaacMcaaa a@36C0@  and hence to determine Σ ij , E ij , U i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeu4Odm1aaS baaSqaaiaadMgacaWGQbaabeaakiaacYcacaWGfbWaaSbaaSqaaiaa dMgacaWGQbaabeaakiaacYcacaWGvbWaaSbaaSqaaiaadMgaaeqaaa aa@40C5@  is a tedious and not especially straightforward exercise.  The governing equation for f MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzaaaa@36D1@  is obtained from the condition that the strain field must be compatible.  This requires

r 1 ε rr r r 2 2 ε rr θ 2 2 r 1 ε θθ r 2 ε θθ r 2 + r 1 2 ε rθ rθ + r 2 ε rθ θ =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOCamaaCa aaleqabaGaeyOeI0IaaGymaaaakmaalaaabaGaeyOaIyRaeqyTdu2a aSbaaSqaaiaadkhacaWGYbaabeaaaOqaaiabgkGi2kaadkhaaaGaey OeI0IaamOCamaaCaaaleqabaGaeyOeI0IaaGOmaaaakmaalaaabaGa eyOaIy7aaWbaaSqabeaacaaIYaaaaOGaeqyTdu2aaSbaaSqaaiaadk hacaWGYbaabeaaaOqaaiabgkGi2kabeI7aXnaaCaaaleqabaGaaGOm aaaaaaGccqGHsislcaaIYaGaamOCamaaCaaaleqabaGaeyOeI0IaaG ymaaaakmaalaaabaGaeyOaIyRaeqyTdu2aaSbaaSqaaiabeI7aXjab eI7aXbqabaaakeaacqGHciITcaWGYbaaaiabgkHiTmaalaaabaGaey OaIy7aaWbaaSqabeaacaaIYaaaaOGaeqyTdu2aaSbaaSqaaiabeI7a XjabeI7aXbqabaaakeaacqGHciITcaWGYbWaaWbaaSqabeaacaaIYa aaaaaakiabgUcaRiaadkhadaahaaWcbeqaaiabgkHiTiaaigdaaaGc daWcaaqaaiabgkGi2oaaCaaaleqabaGaaGOmaaaakiabew7aLnaaBa aaleaacaWGYbGaeqiUdehabeaaaOqaaiabgkGi2kaadkhacqGHciIT cqaH4oqCaaGaey4kaSIaamOCamaaCaaaleqabaGaeyOeI0IaaGOmaa aakmaalaaabaGaeyOaIyRaeqyTdu2aaSbaaSqaaiaadkhacqaH4oqC aeqaaaGcbaGaeyOaIyRaeqiUdehaaiabg2da9iaaicdaaaa@87BB@

Computing the stresses from the Airy function, deducing the strains using the constitutive law and substituting the results into this equation yields a fourth order nonlinear ODE for f, which must be solved subject to appropriate symmetry and boundary conditions.  The solution must be found numerically MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa aaaaaaa8qacaWFtacaaa@37E6@  details are given in Hutchinson Journal of the Mechanics and Physics of Solids, 16 13 (1968) and Rice and Rosengren ibid, 31.

 

 

9.5.3. Plastic fracture mechanics based on J

 

There are many situations (e.g. in design of pressure vessels, pipelines, etc) where the structure is purposely made from a tough, ductile material. Usually, one cannot apply LEFM to these structures, because a large plastic zone forms at the crack tip (the plastic zone is comparable to specimen dimensions, and there is no K dominant zone).  Some other approach is needed to design against fracture in these applications.

 

Two related approaches are used MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa aaaaaaa8qacaWFtacaaa@37E6@  one is based on the HRR crack tip field and uses J as a fracture criterion; the other uses the crack tip opening displacements as a fracture criterion.  Only the J based approach will be discussed here.

 

The most important conclusion from the HRR crack tip field is that the amplitude of stresses, strains and displacements near a crack tip in a plastically deforming solid scale in a predictable way with J.  Just as stress intensity factors quantify the stress and strain magnitudes in a linear elastic solid, J can be used as a parameter to quantify the state of stress in a plastic solid.

 

Phenomenological J based fracture mechanics is based on the same reasoning that is used to justify K based LEFM.  We postulate that we will observe three distinct regions in a plastically deforming specimen containing a crack,

 

1.      A process zone near the crack tip, with finite deformations and extensive material damge, where the asymptotic HRR field is not accurate

2.      A J dominant zone, outside the process zone, but small compared with specimen dimensions, where the HRR field accurately describes the deformation

3.      The remainder, where stress and strain fields are controlled by specimen geometry and loading.

 

 

As for LEFM, we hope that the process zone is controlled by the surrounding J dominant zone, so that crack tip loading conditions can be characterized by J.

 

J based fracture mechanics is applied in much the same way as LEFM.  We assume that crack growth starts when J reaches a critical value (for mode I plane strain loading this value is denoted J IC MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOsamaaBa aaleaacaWGjbGaam4qaaqabaaaaa@3887@  ).  The critical value must be measured experimentally for a given material, using standard test specimens.  To assess the safety of a structure or component containing a crack, one must calculate J and compare the predicted value to J IC MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOsamaaBa aaleaacaWGjbGaam4qaaqabaaaaa@3887@  - if J< J IC MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOsaiabgY da8iaadQeadaWgaaWcbaGaamysaiaadoeaaeqaaaaa@3A5A@  the structure is safe.

 

 

Practical application of J based fracture mechanics is somewhat more involved than LEFM.  Tests to measure   J IC MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOsamaaBa aaleaacaWGjbGaam4qaaqabaaaaa@3887@  are performed using standard test specimens MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa aaaaaaa8qacaWFtacaaa@37E6@  deeply cracked 3 or 4 point bend bars are often used.  Calibrations for the latter case are available in J. R. Rice, P. C. Paris and J. G. Merkle,. Progress in Flaw Growth and Fracture Toughness Testing, ASTM STP 536,. 231 (1973).

 

Calculating J for a specimen or component usually requires a full field FEM analysis.  Cataloging solutions to standard problems is much more difficult than for LEFM, because the results depend on the stress-strain behavior of the material.  Specifically, for a power-law solid containing a crack of length a and subjected to stress σ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdmhaaa@37A9@ , we expect that

J= σ 0 ε 0 a (σ/ σ 0 ) n+1 f(n,geometry) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOsaiabg2 da9iabeo8aZnaaBaaaleaacaaIWaaabeaakiabew7aLnaaBaaaleaa caaIWaaabeaakiaadggacaGGOaGaeq4WdmNaai4laiabeo8aZnaaBa aaleaacaaIWaaabeaakiaacMcadaahaaWcbeqaaiaad6gacqGHRaWk caaIXaaaaOGaamOzaiaacIcacaWGUbGaaiilaiaadEgacaWGLbGaam 4Baiaad2gacaWGLbGaamiDaiaadkhacaWG5bGaaiykaaaa@53C6@

For example, a slit crack of length 2a subjected to mode I loading with stress σ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdmhaaa@37B9@  has (approximately MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa aaaaaaa8qacaWFtacaaa@37E6@  see He & Hutchinson J. Appl. Mech 48 830 1981)

J= σ 0 ε 0 aπ n ( 3 σ σ 0 ) n+1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOsaiabg2 da9iabeo8aZnaaBaaaleaacaaIWaaabeaakiabew7aLnaaBaaaleaa caaIWaaabeaakiaadggacqaHapaCdaGcaaqaaiaad6gaaSqabaGcda qadaqaamaalaaabaWaaOaaaeaacaaIZaaaleqaaOGaeq4WdmhabaGa eq4Wdm3aaSbaaSqaaiaaicdaaeqaaaaaaOGaayjkaiaawMcaamaaCa aaleqabaGaamOBaiabgUcaRiaaigdaaaaaaa@4B85@

Finally, to apply the theory it is necessary to ensure that both test specimen and component satisfy conditions necessary for J dominance.  As a rough rule of thumb, if all characteristic specimen dimensions (crack length, etc)  exceed 200J/ ε 0 σ 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGOmaiaaic dacaaIWaGaamOsaiaac+cacqaH1oqzdaWgaaWcbaGaaGimaaqabaGc cqaHdpWCdaWgaaWcbaGaaGimaaqabaaaaa@3FEF@  J dominance is likely to be satisfied.