Chapter 9

 

Modeling Material Failure

 

 

9.4 Energy methods in fracture mechanics

 

Energy methods provide additional insight into fracture, and also provide a foundation for a range of analytical and numerical methods in fracture mechanics.  In this section, we outline some of the most important results.

 

 

9.4.1 Definition of crack tip energy release rate for cracks in linear elastic solids

 

The crack tip energy release rate quantifies the rate of change of the potential energy of a cracked elastic solid as the crack grows.

 

To make this precise, consider an ideally elastic solid, subjected to some loading (applied tractions, displacements, or body forces).  Suppose the solid contains a crack (the figure shows a circular crack with radius a as a representative example).  Define the potential energy of the solid in the usual way (Sect 5.6.1) as

V(a)= V U(u) dV V ρ 0 b i u i dV 2 R t i u i dA MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xq qrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8 fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGa aiaabeqaaiWabaWaaaGcbaGaamOvaiaacIcacaWGHbGaaiykaiabg2 da9maapefabaGaamyvaiaacIcacaWH1bGaaiykaaWcbaGaamOvaaqa b0Gaey4kIipakiaadsgacaWGwbGaeyOeI0Yaa8quaeaacqaHbpGCda WgaaWcbaGaaGimaaqabaGccaWGIbWaaSbaaSqaaiaadMgaaeqaaOGa amyDamaaBaaaleaacaWGPbaabeaaaeaacaWGwbaabeqdcqGHRiI8aO GaamizaiaadAfacqGHsisldaWdrbqaaiaadshadaWgaaWcbaGaamyA aaqabaGccaWG1bWaaSbaaSqaaiaadMgaaeqaaOGaamizaiaadgeaaS qaaiabgkGi2oaaBaaameaacaaIYaaabeaaliaadkfaaeqaniabgUIi Ydaaaa@573B@

Suppose the crack increases in size, so that the crack advances a distance αδa(s) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHXoqycqaH0oazcaWGHbGaaiikai aadohacaGGPaaaaa@3941@  with loading kept fixed, where s measures position around the crack front. The principle of minimum potential energy (sect 5.6.2) shows that V(a+αδa)V(a) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOvaiaacI cacaWGHbGaey4kaSIaeqySdeMaeqiTdqMaamyyaiaacMcacqGHKjYO caWGwbGaaiikaiaadggacaGGPaaaaa@43F2@ , since the displacement field associated with V(a) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOvaiaacI cacaWGHbGaaiykaaaa@3900@  is a kinematically admissible field for the solid with a longer crack.  The energy release rate G(s) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4raiaacI cacaWGZbGaaiykaaaa@3903@  around the crack front is defined so that

C G(s)δa(s)ds = lim α0 V(a+αδa(s)) α MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa8quaeaaca WGhbGaaiikaiaadohacaGGPaGaeqiTdqMaamyyaiaacIcacaWGZbGa aiykaiaadsgacaWGZbaaleaacaWGdbaabeqdcqGHRiI8aOGaeyypa0 JaeyOeI0YaaCbeaeaaciGGSbGaaiyAaiaac2gaaSqaaiabeg7aHjab gkziUkaaicdaaeqaaOWaaSaaaeaacqGHciITcaWGwbGaaiikaiaadg gacqGHRaWkcqaHXoqycqaH0oazcaWGHbGaaiikaiaadohacaGGPaGa aiykaaqaaiabgkGi2kabeg7aHbaaaaa@5C31@

Energy release rate has units of N m 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOtaiaad2 gadaahaaWcbeqaaiabgkHiTiaaigdaaaaaaa@3A97@  (energy per unit area).

 

For the special case of a 2D slit crack with length a, the energy release rate is

G= V ¯ (a) a MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGhbGaeyypa0JaeyOeI0YaaSaaae aacqGHciITdaqdaaqaaiaadAfaaaGaaiikaiaadggacaGGPaaabaGa eyOaIyRaamyyaaaaaaa@3C72@

where V ¯ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaqdaaqaaiaadAfaaaaaaa@33B2@  is now the potential energy per unit out-of-plane distance.

 

 

 

9.4.2 Energy release rate as a fracture criterion

 

Phenomenological fracture (or fatigue) criteria can be based on energy release rate arguments as an alternative to the K based fracture criteria discussed earlier.

 

The argument is as follows.  Regardless of the actual mechanisms involved, crack propagation involves dissipation (or conversion) of energy.  A small amount of energy is required to create two new free surfaces (twice the surface energy per unit area of crack advance, to be precise).  In addition, there may be a complex process zone at the crack tip, where the material is plastically deformed; voids may be nucleated; there may be chemical reactions; and generally all hell breaks loose.  All these processes involve dissipation of energy.  We postulate, however, that the process zone remains self-similar during crack growth.  If this is the case, energy will be dissipated at a constant rate during crack growth.  The crack can only grow if the rate of change of potential energy is sufficient to provide this energy.

 

This leads to a fracture criterion of the form

G G C MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4raiabgw MiZkaadEeadaWgaaWcbaGaam4qaaqabaaaaa@3B4F@

for crack growth, where G C MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4ramaaBa aaleaacaWGdbaabeaaaaa@38BD@  is a property of the material. Unfortunately G C MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4ramaaBa aaleaacaWGdbaabeaaaaa@38BD@  is often referred to as the fracture toughness of a solid, just like K IC MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4samaaBa aaleaacaWGjbGaam4qaaqabaaaaa@398F@  defined earlier.  It is usually obvious from dimensional considerations which one is being used, but its an annoying source of confusion.  

 

 

9.4.3 Relation between energy release rate and stress intensity factor

 

The energy release rate G is closely related to the stress intensity factors defined in Sect 9.3.  Specifically, for an isotropic, linear elastic solid with Young’s modulus E MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGfbaaaa@3390@  and Poisson’s ratio ν MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaH9oGBaaa@347E@  the energy release rate is related to stress intensity factors by

G= 1 ν 2 E ( K I 2 + K II 2 )+ 1+ν E K III 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4raiabg2 da9maalaaabaGaaGymaiabgkHiTiabe27aUnaaCaaaleqabaGaaGOm aaaaaOqaaiaadweaaaWaaeWaaeaacaWGlbWaa0baaSqaaiaadMeaae aacaaIYaaaaOGaey4kaSIaam4samaaDaaaleaacaWGjbGaamysaaqa aiaaikdaaaaakiaawIcacaGLPaaacqGHRaWkdaWcaaqaaiaaigdacq GHRaWkcqaH9oGBaeaacaWGfbaaaiaadUeadaqhaaWcbaGaamysaiaa dMeacaWGjbaabaGaaGOmaaaaaaa@4F8B@

HEALTH WARNING: The result relating G to K I MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGlbWaaSbaaSqaaiaadMeaaeqaaa aa@3490@  and K II MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGlbWaaSbaaSqaaiaadMeacaWGjb aabeaaaaa@355E@  is valid only for plane strain deformation at the crack tip.

 

 

Derivation A neat argument due to Irwin provides the connection.

 

A crack of length a can be regarded as a crack with a+δa MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyaiabgU caRiabes7aKjaadggaaaa@3B50@  which is being pinched closed by an appropriate distribution of traction acting on the crack faces between x 1 =δa MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaBa aaleaacaaIXaaabeaakiabg2da9iabgkHiTiabes7aKjaadggaaaa@3D69@  and x 1 =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaBa aaleaacaaIXaaabeaakiabg2da9iaaicdaaaa@39A4@ .  We can therefore calculate the change in potential energy as the crack propagates by distance δa MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiTdqMaam yyaaaa@3881@  by computing the work done as these tractions are progressively relaxed to zero.   To this end, note that

1.      The tractions that pinch the crack tip closed can be calculated from the asymptotic crack tip field (Sect 9.3.1)

t 1 = K II 2π(δa+ x 1 ) t 2 = K I 2π(δa+ x 1 ) t 3 = K III 2π(δa+ x 1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiDamaaBa aaleaacaaIXaaabeaakiabg2da9maalaaabaGaam4samaaBaaaleaa caWGjbGaamysaaqabaaakeaadaGcaaqaaiaaikdacqaHapaCcaGGOa GaeqiTdqMaamyyaiabgUcaRiaadIhadaWgaaWcbaGaaGymaaqabaGc caGGPaaaleqaaaaakiaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaG PaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caWG 0bWaaSbaaSqaaiaaikdaaeqaaOGaeyypa0ZaaSaaaeaacaWGlbWaaS baaSqaaiaadMeaaeqaaaGcbaWaaOaaaeaacaaIYaGaeqiWdaNaaiik aiabes7aKjaadggacqGHRaWkcaWG4bWaaSbaaSqaaiaaigdaaeqaaO GaaiykaaWcbeaaaaGccaaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaa ykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaG PaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaM c8UaamiDamaaBaaaleaacaaIZaaabeaakiabg2da9maalaaabaGaam 4samaaBaaaleaacaWGjbGaamysaiaadMeaaeqaaaGcbaWaaOaaaeaa caaIYaGaeqiWdaNaaiikaiabes7aKjaadggacqGHRaWkcaWG4bWaaS baaSqaaiaaigdaaeqaaOGaaiykaaWcbeaaaaaaaa@99D3@

(equal and opposite tractions must act on the lower crack face).

2.      As the crack is allowed to open, the upper crack face displaces by

Δ u 1 = 2( 1 ν 2 ) E K II 2 x 1 π Δ u 2 = 2( 1 ν 2 ) E K I 2 x 1 π Δ u 3 = 2( 1+ν ) E K III 2 x 1 π MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeuiLdqKaam yDamaaBaaaleaacaaIXaaabeaakiabg2da9maalaaabaGaaGOmamaa bmaabaGaaGymaiabgkHiTiabe27aUnaaCaaaleqabaGaaGOmaaaaaO GaayjkaiaawMcaaaqaaiaadweaaaGaam4samaaBaaaleaacaWGjbGa amysaaqabaGcdaGcaaqaamaalaaabaGaeyOeI0IaaGOmaiaadIhada WgaaWcbaGaaGymaaqabaaakeaacqaHapaCaaGaaGPaVdWcbeaakiaa ykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaG PaVlaaykW7caaMc8UaaGPaVlabfs5aejaadwhadaWgaaWcbaGaaGOm aaqabaGccqGH9aqpdaWcaaqaaiaaikdadaqadaqaaiaaigdacqGHsi slcqaH9oGBdaahaaWcbeqaaiaaikdaaaaakiaawIcacaGLPaaaaeaa caWGfbaaaiaadUeadaWgaaWcbaGaamysaaqabaGcdaGcaaqaamaala aabaGaeyOeI0IaaGOmaiaadIhadaWgaaWcbaGaaGymaaqabaaakeaa cqaHapaCaaaaleqaaOGaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7ca aMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaa ykW7caaMc8UaaGPaVlaaykW7cqqHuoarcaWG1bWaaSbaaSqaaiaaio daaeqaaOGaeyypa0ZaaSaaaeaacaaIYaWaaeWaaeaacaaIXaGaey4k aSIaeqyVd4gacaGLOaGaayzkaaaabaGaamyraaaacaWGlbWaaSbaaS qaaiaadMeacaWGjbGaamysaaqabaGcdaGcaaqaamaalaaabaGaeyOe I0IaaGOmaiaadIhadaWgaaWcbaGaaGymaaqabaaakeaacqaHapaCaa aaleqaaaaa@A01E@

where we have assumed plane strain deformation.

3.      The total work done as the tractions are relaxed quasi-statically to zero is

Gδa= δa 0 ( t 1 Δ u 1 + t 2 Δ u 2 + t 3 Δ u 3 )d x 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4raiabes 7aKjaadggacqGH9aqpdaWdXbqaaiaacIcacaWG0bWaaSbaaSqaaiaa igdaaeqaaOGaeuiLdqKaamyDamaaBaaaleaacaaIXaaabeaakiabgU caRiaadshadaWgaaWcbaGaaGOmaaqabaGccqqHuoarcaWG1bWaaSba aSqaaiaaikdaaeqaaOGaey4kaSIaamiDamaaBaaaleaacaaIZaaabe aakiabfs5aejaadwhadaWgaaWcbaGaaG4maaqabaGccaGGPaGaamiz aiaadIhadaWgaaWcbaGaaGymaaqabaaabaGaeyOeI0IaeqiTdqMaam yyaaqaaiaaicdaa0Gaey4kIipaaaa@578B@

(the work done by tractions acting on the upper crack face per unit length is t i Δ u i /2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiDamaaBa aaleaacaWGPbaabeaakiabfs5aejaadwhadaWgaaWcbaGaamyAaaqa baGccaGGVaGaaGOmaaaa@3D06@ , and there are two crack faces).

4.      Evaluating the integrals gives

G= 1 ν 2 E ( K I 2 + K II 2 )+ 1+ν E K III 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4raiabg2 da9maalaaabaGaaGymaiabgkHiTiabe27aUnaaCaaaleqabaGaaGOm aaaaaOqaaiaadweaaaWaaeWaaeaacaWGlbWaa0baaSqaaiaadMeaae aacaaIYaaaaOGaey4kaSIaam4samaaDaaaleaacaWGjbGaamysaaqa aiaaikdaaaaakiaawIcacaGLPaaacqGHRaWkdaWcaaqaaiaaigdacq GHRaWkcqaH9oGBaeaacaWGfbaaaiaadUeadaqhaaWcbaGaamysaiaa dMeacaWGjbaabaGaaGOmaaaaaaa@4F8B@

The same result can be obtained by applying crack tip energy flux integrals, to be discussed below.

 

 

 

9.4.4 Relation between energy release rate and compliance

 

Energy release rate is related to the compliance of a structure or specimen, as follows.  Consider the compact tension specimen shown in the picture.  Suppose that the specimen is subjected to a load P, which causes the point of application of the load to displace by a distance x 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaBa aaleaacaaIWaaabeaaaaa@38E0@  in a direction parallel to the load. The compliance of the specimen is defined as

C= x 0 P MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4qaiabg2 da9maalaaabaGaamiEamaaBaaaleaacaaIWaaabeaaaOqaaiaadcfa aaaaaa@3B9D@

As the crack grows, the compliance of the specimen always increases, so C is a function of crack length.  The energy release rate is related to compliance C by

G= 1 2 P 2 B dC da MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4raiabg2 da9maalaaabaGaaGymaaqaaiaaikdaaaWaaSaaaeaacaWGqbWaaWba aSqabeaacaaIYaaaaaGcbaGaamOqaaaadaWcaaqaaiaadsgacaWGdb aabaGaamizaiaadggaaaaaaa@4085@

This formula applies to any structure or component, not just to compact tension specimens.  The formula is useful for two reasons:

(i) It can be used to measure energy release rate in an experiment.  All you need to do is to measure the crack length as it grows, and at the same time measure the compliance of your specimen.

(ii) It can be used to calculate stress intensity factors, as outlined in the next section.

 

Derivation: This result can be derived by calculating the change in energy of the system as the crack grows.  Note that

1.      The load P induces a total strain energy Φ= 1 2 x 0 P= 1 2 C P 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeuOPdyKaey ypa0ZaaSaaaeaacaaIXaaabaGaaGOmaaaacaWG4bWaaSbaaSqaaiaa icdaaeqaaOGaamiuaiabg2da9maalaaabaGaaGymaaqaaiaaikdaaa Gaam4qaiaadcfadaahaaWcbeqaaiaaikdaaaaaaa@42D9@  in the specimen.  To see this, note that the the solid is elastic and so behaves like a linear spring MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa aaaaaaa8qacaWFtacaaa@37E6@  this is just the formula for the energy in a spring.

2.      Now, suppose that the crack extends by a distance δa MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiTdqMaam yyaaaa@3988@ .  During crack growth, the load increases to P+δP MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiuaiabgU caRiabes7aKjaadcfaaaa@3B2E@  and displaces to x 0 +δx MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaBa aaleaacaaIWaaabeaakiabgUcaRiabes7aKjaadIhaaaa@3C6E@ .  In addition, the strain energy changes to Φ+δΦ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeuOPdyKaey 4kaSIaeqiTdqMaeuOPdyeaaa@3C78@ , while the compliance increases to C+δC MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4qaiabgU caRiabes7aKjaadoeaaaa@3B14@ .

3.      The energy released during crack advance is equal to the decrease in potential energy of the system, so that

GBδa=δV=[ (Φ+δΦ)ΦPδx ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4raiaayk W7caWGcbGaaGPaVlabes7aKjaadggacqGH9aqpcqGHsislcqaH0oaz caWGwbGaeyypa0JaeyOeI0YaamWaaeaacaGGOaGaeuOPdyKaey4kaS IaeqiTdqMaeuOPdyKaaiykaiabgkHiTiabfA6agjabgkHiTiaadcfa cqaH0oazcaWG4baacaGLBbGaayzxaaaaaa@5428@

4.      Note that

Φ+δΦ= 1 2 ( C+δC ) ( P+δP ) 2 1 2 C P 2 +CPδP+ 1 2 δC P 2 δx=(C+δC)( P+δP )CPCδP+PδC MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacqqHMo GrcqGHRaWkcqaH0oazcqqHMoGrcqGH9aqpdaWcaaqaaiaaigdaaeaa caaIYaaaamaabmaabaGaam4qaiabgUcaRiabes7aKjaadoeaaiaawI cacaGLPaaadaqadaqaaiaadcfacqGHRaWkcqaH0oazcaWGqbaacaGL OaGaayzkaaWaaWbaaSqabeaacaaIYaaaaOGaeyisIS7aaSaaaeaaca aIXaaabaGaaGOmaaaacaWGdbGaamiuamaaCaaaleqabaGaaGOmaaaa kiabgUcaRiaadoeacaWGqbGaeqiTdqMaamiuaiabgUcaRmaalaaaba GaaGymaaqaaiaaikdaaaGaeqiTdqMaam4qaiaadcfadaahaaWcbeqa aiaaikdaaaaakeaacqaH0oazcaWG4bGaeyypa0Jaaiikaiaadoeacq GHRaWkcqaH0oazcaWGdbGaaiykamaabmaabaGaamiuaiabgUcaRiab es7aKjaadcfaaiaawIcacaGLPaaacqGHsislcaWGdbGaamiuaiabgI Ki7kaadoeacqaH0oazcaWGqbGaey4kaSIaamiuaiabes7aKjaadoea aaaa@772B@

5.      Substituting these results into the expression in step (3) and simplifying shows that

GBδa= 1 2 P 2 δC= 1 2 P 2 dC da δa MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4raiaayk W7caWGcbGaaGPaVlabes7aKjaadggacqGH9aqpdaWcaaqaaiaaigda aeaacaaIYaaaaiaadcfadaahaaWcbeqaaiaaikdaaaGccqaH0oazca WGdbGaeyypa0ZaaSaaaeaacaaIXaaabaGaaGOmaaaacaWGqbWaaWba aSqabeaacaaIYaaaaOWaaSaaaeaacaWGKbGaam4qaaqaaiaadsgaca WGHbaaaiabes7aKjaadggaaaa@4F63@

The energy release rate therefore is related to compliance by

G= 1 2 P 2 B dC da MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4raiabg2 da9maalaaabaGaaGymaaqaaiaaikdaaaWaaSaaaeaacaWGqbWaaWba aSqabeaacaaIYaaaaaGcbaGaamOqaaaadaWcaaqaaiaadsgacaWGdb aabaGaamizaiaadggaaaaaaa@4085@

 

 

 

9.4.5 Calculating stress intensity factors using compliance

 

The relation between compliance and energy release rate can be used to determine energy release rates, and sometimes also stress intensity factors, for structures whose rate of change of compliance with crack length can be easily determined.  One example is the cantilever beam specimen shown in the figure.  The mode I stress intensity factor for this specimen can be derived as

K I = 2 3 1 ν 2 Pa B h 3/2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4samaaBa aaleaacaWGjbaabeaakiabg2da9maalaaabaGaaGOmamaakaaabaGa aG4maaWcbeaaaOqaamaakaaabaGaaGymaiabgkHiTiabe27aUnaaCa aaleqabaGaaGOmaaaaaeqaaaaakmaalaaabaGaamiuaiaadggaaeaa caWGcbGaamiAamaaCaaaleqabaGaaG4maiaac+cacaaIYaaaaaaaaa a@45C0@

 

Derivation This result is derived by first calculating the compliance of the solid; then using the formula to deduce the energy release rate, and finally using the relationship between stress intensity factor and energy release rate.  To proceed,

1.      Note that the deflection d of the loaded point can be calculated by visualizing the specimen as two cantilever beams, length a, width B and height h, clamped on their right hand end and subjected to a load P at their left hand ends.  From elementary beam theory, the deflection is

d=2 a 3 P 3E(B h 3 /12) =8 a 3 P EB h 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamizaiabg2 da9iaaikdadaWcaaqaaiaadggadaahaaWcbeqaaiaaiodaaaGccaWG qbaabaGaaG4maiaadweacaGGOaGaamOqaiaadIgadaahaaWcbeqaai aaiodaaaGccaGGVaGaaGymaiaaikdacaGGPaaaaiabg2da9iaaiIda daWcaaqaaiaadggadaahaaWcbeqaaiaaiodaaaGccaWGqbaabaGaam yraiaadkeacaWGObWaaWbaaSqabeaacaaIZaaaaaaaaaa@4C08@

where E is the Young’s modulus of the specimen.

2.      The compliance follows as

C= d P =8 a 3 EB h 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4qaiabg2 da9maalaaabaGaamizaaqaaiaadcfaaaGaeyypa0JaaGioamaalaaa baGaamyyamaaCaaaleqabaGaaG4maaaaaOqaaiaadweacaWGcbGaam iAamaaCaaaleqabaGaaG4maaaaaaaaaa@41B3@

3.      The energy release rate formula in Sect 9.4.4 gives

G= 1 2 P 2 B dC da =12 P 2 a 2 E B 2 h 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4raiabg2 da9maalaaabaGaaGymaaqaaiaaikdaaaWaaSaaaeaacaWGqbWaaWba aSqabeaacaaIYaaaaaGcbaGaamOqaaaadaWcaaqaaiaadsgacaWGdb aabaGaamizaiaadggaaaGaeyypa0JaaGymaiaaikdadaWcaaqaaiaa dcfadaahaaWcbeqaaiaaikdaaaGccaWGHbWaaWbaaSqabeaacaaIYa aaaaGcbaGaamyraiaadkeadaahaaWcbeqaaiaaikdaaaGccaWGObWa aWbaaSqabeaacaaIZaaaaaaaaaa@4B0E@

4.      By symmetry, the crack must be loaded in pure mode I.  We can therefore deduce the stress intensity factor using the relation

G= 1 ν 2 E K I 2 K I = 2 3 1 ν 2 Pa B h 3/2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4raiabg2 da9maalaaabaGaaGymaiabgkHiTiabe27aUnaaCaaaleqabaGaaGOm aaaaaOqaaiaadweaaaGaam4samaaDaaaleaacaWGjbaabaGaaGOmaa aakiabgkDiElaadUeadaWgaaWcbaGaamysaaqabaGccqGH9aqpdaWc aaqaaiaaikdadaGcaaqaaiaaiodaaSqabaaakeaadaGcaaqaaiaaig dacqGHsislcqaH9oGBdaahaaWcbeqaaiaaikdaaaaabeaaaaGcdaWc aaqaaiaadcfacaWGHbaabaGaamOqaiaadIgadaahaaWcbeqaaiaaio dacaGGVaGaaGOmaaaaaaaaaa@51AD@

 

 

9.4.6 Integral expressions for energy flux to a crack tip

 

In this section we outline a way to compute the energy release rate for a crack, which applies not only to linear elastic solids under quasi-static loading conditions, but is completely independent of the constitutive response of the solid, and also applies under dynamic loading (it is restricted to small strains, however).  The approach will be to find an expression for the flux of energy through a cylindrical surface Γ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeu4KdCeaaa@375E@  enclosing the crack tip, which moves with the crack.  We will get the energy release rate by shrinking the surface down onto the crack tip.

 

Energy flux across a surface in a solid: We first derive a formula that can be used to calculate the flux of kinetic and potential energy across a surface in a deformable solid.  To this end,

 Consider an arbitrary surface S, which encloses some volume V in a solid.  The surface need not necessarily be a material surface MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa aaaaaaa8qacaWFtacaaa@37E6@  it could move with respect to the solid.  We will denote the velocity of S  (with respect to a fixed origin) by v j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWG2bWaaSbaaSqaaiaadQgaaeqaaa aa@34DC@

 Assume that the solid is free of body forces, for simplicity. 

 Let [ u i , ε ij , σ ij ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaai4waiaadw hadaWgaaWcbaGaamyAaaqabaGccaGGSaGaeqyTdu2aaSbaaSqaaiaa dMgacaWGQbaabeaakiaacYcacqaHdpWCdaWgaaWcbaGaamyAaiaadQ gaaeqaaOGaaiyxaaaa@43DB@  denote the displacement, (infinitesimal) strain and stress field in the solid, and let u ˙ i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmyDayaaca WaaSbaaSqaaiaadMgaaeqaaaaa@3813@  denote the velocity of a material point with respect to a fixed origin.

 Let T=ρ u ˙ i u ˙ i /2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamivaiabg2 da9iabeg8aYjqadwhagaGaamaaBaaaleaacaWGPbaabeaakiqadwha gaGaamaaBaaaleaacaWGPbaabeaakiaac+cacaaIYaaaaa@4059@  denote the kinetic energy of a material particle in the solid

 Let W ˙ = σ ij ε ˙ ij = σ ij u ˙ i x j MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabm4vayaaca Gaeyypa0Jaeq4Wdm3aaSbaaSqaaiaadMgacaWGQbaabeaakiqbew7a LzaacaWaaSbaaSqaaiaadMgacaWGQbaabeaakiabg2da9iabeo8aZn aaBaaaleaacaWGPbGaamOAaaqabaGcdaWcaaqaaiabgkGi2kqadwha gaGaamaaBaaaleaacaWGPbaabeaaaOqaaiabgkGi2kaadIhadaWgaa WcbaGaamOAaaqabaaaaaaa@4C78@  denote the rate of work done by stresses at a point in the solid

 Define the rate of change of mechanical energy density at an arbitrary point in the solid as ψ ˙ = W ˙ + T ˙ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqiYdKNbai aacqGH9aqpceWGxbGbaiaacqGHRaWkceWGubGbaiaaaaa@3C83@ , and let ψ= t ψ ˙ dt MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHipqEcqGH9aqpdaWdXbqaaiqbeI 8a5zaacaGaamizaiaadshaaSqaaiabgkHiTiabg6HiLcqaaiaadsha a0Gaey4kIipaaaa@3F13@

 Denote the total energy within V as Ψ= V ψdV MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqqHOoqwcqGH9aqpdaWdrbqaaiabeI 8a5jaadsgacaWGwbaaleaacaWGwbaabeqdcqGHRiI8aaaa@3C12@

 Define the work flux vector as ω j = σ ij u ˙ i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyYdC3aaS baaSqaaiaadQgaaeqaaOGaeyypa0Jaeq4Wdm3aaSbaaSqaaiaadMga caWGQbaabeaakiqadwhagaGaamaaBaaaleaacaWGPbaabeaaaaa@40E8@

 

The energy flux across S can be calculated in terms of these quantities as follows:

dΨ dt = d dt V ψdV = S ( ω j +ψ v j ) m j dA MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca WGKbGaeuiQdKfabaGaamizaiaadshaaaGaeyypa0ZaaSaaaeaacaWG KbaabaGaamizaiaadshaaaWaa8quaeaacqaHipqEcaWGKbGaamOvaa WcbaGaamOvaaqab0Gaey4kIipakiabg2da9maapefabaGaaiikaiab eM8a3naaBaaaleaacaWGQbaabeaakiabgUcaRiabeI8a5jaadAhada WgaaWcbaGaamOAaaqabaGccaGGPaGaamyBamaaBaaaleaacaWGQbaa beaakiaadsgacaWGbbaaleaacaWGtbaabeqdcqGHRiI8aaaa@5722@

The right hand side of this expression denotes the energy flux across the surface; the left hand side is the rate of change of the total energy within V.  The two are equal by energy conservation, as shown below.

 

Derivation:

1.      Begin by showing that the energy flux vector and the rate of change of mechanical energy density are related by

ω j / x j = ψ ˙ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabgkGi2kabeM8a3naaBaaaleaacaWGQb aabeaakiaac+cacqGHciITcaWG4bWaaSbaaSqaaiaadQgaaeqaaOGa eyypa0JafqiYdKNbaiaaaaa@3BD8@

To see this, note that

ψ ˙ = W ˙ + T ˙ = σ ij u ˙ i x j +ρ u ¨ i u ˙ i = σ ij u ˙ i x j + σ ij x j u ˙ i = x j ( σ ij u ˙ i )= ω j x j MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqiYdKNbai aacqGH9aqpceWGxbGbaiaacqGHRaWkceWGubGbaiaacqGH9aqpcqaH dpWCdaWgaaWcbaGaamyAaiaadQgaaeqaaOWaaSaaaeaacqGHciITce WG1bGbaiaadaWgaaWcbaGaamyAaaqabaaakeaacqGHciITcaWG4bWa aSbaaSqaaiaadQgaaeqaaaaakiabgUcaRiabeg8aYjqadwhagaWaam aaBaaaleaacaWGPbaabeaakiqadwhagaGaamaaBaaaleaacaWGPbaa beaakiabg2da9iabeo8aZnaaBaaaleaacaWGPbGaamOAaaqabaGcda WcaaqaaiabgkGi2kqadwhagaGaamaaBaaaleaacaWGPbaabeaaaOqa aiabgkGi2kaadIhadaWgaaWcbaGaamOAaaqabaaaaOGaey4kaSYaaS aaaeaacqGHciITcqaHdpWCdaWgaaWcbaGaamyAaiaadQgaaeqaaaGc baGaeyOaIyRaamiEamaaBaaaleaacaWGQbaabeaaaaGcceWG1bGbai aadaWgaaWcbaGaamyAaaqabaGccqGH9aqpdaWcaaqaaiabgkGi2cqa aiabgkGi2kaadIhadaWgaaWcbaGaamOAaaqabaaaaOWaaeWaaeaacq aHdpWCdaWgaaWcbaGaamyAaiaadQgaaeqaaOGabmyDayaacaWaaSba aSqaaiaadMgaaeqaaaGccaGLOaGaayzkaaGaeyypa0ZaaSaaaeaacq GHciITcqaHjpWDdaWgaaWcbaGaamOAaaqabaaakeaacqGHciITcaWG 4bWaaSbaaSqaaiaadQgaaeqaaaaaaaa@7DC9@

where we have used the linear and angular momentum balance equations σ ij / x i =ρ u ¨ σ ij = σ ji MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOaIyRaeq 4Wdm3aaSbaaSqaaiaadMgacaWGQbaabeaakiaac+cacqGHciITcaWG 4bWaaSbaaSqaaiaadMgaaeqaaOGaeyypa0JaeqyWdiNabmyDayaada GaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7cqaHdpWCdaWgaaWcbaGa amyAaiaadQgaaeqaaOGaeyypa0Jaeq4Wdm3aaSbaaSqaaiaadQgaca WGPbaabeaaaaa@549C@ .

 

2.      Now, integrate both sides of this equation over the volume V and apply the divergence theorem to see that

V ω j x j dV = V ψ ˙ dV S ω j m j dA = V ψ ˙ dV MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa8quaeaada WcaaqaaiabgkGi2kabeM8a3naaBaaaleaacaWGQbaabeaaaOqaaiab gkGi2kaadIhadaWgaaWcbaGaamOAaaqabaaaaOGaamizaiaadAfaaS qaaiaadAfaaeqaniabgUIiYdGccqGH9aqpdaWdrbqaaiqbeI8a5zaa caGaamizaiaadAfaaSqaaiaadAfaaeqaniabgUIiYdGccaaMc8UaaG PaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlabgkDiElaa ykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVpaapefabaGaeqyYdC 3aaSbaaSqaaiaadQgaaeqaaOGaamyBamaaBaaaleaacaWGQbaabeaa kiaadsgacaWGbbaaleaacaWGtbaabeqdcqGHRiI8aOGaeyypa0Zaa8 quaeaacuaHipqEgaGaaiaadsgacaWGwbaaleaacaWGwbaabeqdcqGH RiI8aaaa@7551@

3.      Next note that the total rate of change of ψ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHipqEaaa@3494@  within the volume V bounded by S can be expressed as

d dt V ψdV = V ψ ˙ dV + S ψ v j m j dA MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca WGKbaabaGaamizaiaadshaaaWaa8quaeaacqaHipqEcaWGKbGaamOv aaWcbaGaamOvaaqab0Gaey4kIipakiabg2da9maapefabaGafqiYdK NbaiaacaWGKbGaamOvaaWcbaGaamOvaaqab0Gaey4kIipakiabgUca RmaapefabaGaeqiYdKNaamODamaaBaaaleaacaWGQbaabeaakiaad2 gadaWgaaWcbaGaamOAaaqabaGccaWGKbGaamyqaaWcbaGaam4uaaqa b0Gaey4kIipaaaa@5431@

Here, the first term on the right represents the rate of change due to the time derivative of ψ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHipqEaaa@3494@  within V, while the second term represents the flux of energy crossing S as the surface moves with velocity v j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWG2bWaaSbaaSqaaiaadQgaaeqaaa aa@34DC@ .

4.      Combining (2) and (3) shows that 

S ω j m j dA = d dt V ψdV S ψ v j m j dA S ( ω j +ψ v j ) m j dA = d dt V ψdV MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa8quaeaacq aHjpWDdaWgaaWcbaGaamOAaaqabaGccaWGTbWaaSbaaSqaaiaadQga aeqaaOGaamizaiaadgeaaSqaaiaadofaaeqaniabgUIiYdGccqGH9a qpdaWcaaqaaiaadsgaaeaacaWGKbGaamiDaaaadaWdrbqaaiabeI8a 5jaadsgacaWGwbaaleaacaWGwbaabeqdcqGHRiI8aOGaeyOeI0Yaa8 quaeaacqaHipqEcaWG2bWaaSbaaSqaaiaadQgaaeqaaOGaamyBamaa BaaaleaacaWGQbaabeaakiaadsgacaWGbbaaleaacaWGtbaabeqdcq GHRiI8aOGaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPa VlabgkDiElaaykW7caaMc8UaaGPaVlaaykW7daWdrbqaaiaacIcacq aHjpWDdaWgaaWcbaGaamOAaaqabaGccqGHRaWkcqaHipqEcaWG2bWa aSbaaSqaaiaadQgaaeqaaOGaaiykaiaad2gadaWgaaWcbaGaamOAaa qabaGccaWGKbGaamyqaaWcbaGaam4uaaqab0Gaey4kIipakiabg2da 9maalaaabaGaamizaaqaaiaadsgacaWG0baaamaapefabaGaeqiYdK NaamizaiaadAfaaSqaaiaadAfaaeqaniabgUIiYdaaaa@854B@

The term on the right hand side clearly represents the total rate of change of mechanical energy in V.  Consequently, the term on the left hand side must represent the mechanical energy flux across S MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4uaaaa@36CE@ .  This is the result we need.

 

Energy flux to a crack tip. We can use the energy flux integral to obtain an expression for the energy flux to a crack tip.  Suppose the crack tip runs with steady speed v in the x 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaBa aaleaacaaIXaaabeaaaaa@38E1@  direction.  Let Γ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeu4KdCeaaa@3865@  denote a cylindrical surface enclosing the crack tip, which moves with the crack tip.  The energy flux through Γ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeu4KdCeaaa@3865@  follows as

dΨ dt = Γ ( ω j +ψv δ j1 ) m j dA = Γ ( ω j +(T+W)v δ j1 ) m j dA MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca WGKbGaeuiQdKfabaGaamizaiaadshaaaGaeyypa0Zaa8quaeaacaGG OaGaeqyYdC3aaSbaaSqaaiaadQgaaeqaaOGaey4kaSIaeqiYdKNaam ODaiabes7aKnaaBaaaleaacaWGQbGaaGymaaqabaGccaGGPaGaamyB amaaBaaaleaacaWGQbaabeaakiaadsgacaWGbbaaleaacqqHtoWrae qaniabgUIiYdGccqGH9aqpdaWdrbqaaiaacIcacqaHjpWDdaWgaaWc baGaamOAaaqabaGccqGHRaWkcaGGOaGaamivaiabgUcaRiaadEfaca GGPaGaamODaiabes7aKnaaBaaaleaacaWGQbGaaGymaaqabaGccaGG PaGaamyBamaaBaaaleaacaWGQbaabeaakiaadsgacaWGbbaaleaacq qHtoWraeqaniabgUIiYdaaaa@6585@

where

W= t W ˙ dt= t σ ij ε ˙ ij dt= 0 ε ij σ ij d ε ij MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadEfacqGH9a qpdaWdXbqaaiqadEfagaGaaaWcbaGaeyOeI0IaeyOhIukabaGaamiD aaqdcqGHRiI8aOGaamizaiaadshacqGH9aqpdaWdXbqaaiabeo8aZn aaBaaaleaacaWGPbGaamOAaaqabaGccuaH1oqzgaGaamaaBaaaleaa caWGPbGaamOAaaqabaaabaGaeyOeI0IaeyOhIukabaGaamiDaaqdcq GHRiI8aOGaamizaiaadshacqGH9aqpdaWdXbqaaiabeo8aZnaaBaaa leaacaWGPbGaamOAaaqabaGccaWGKbGaeqyTdu2aaSbaaSqaaiaadM gacaWGQbaabeaaaeaacaaIWaaabaGaeqyTdu2aaSbaaWqaaiaadMga caWGQbaabeaaa0Gaey4kIipaaaa@6120@

is the net work done on the solid per unit volume by stresses, and T=ρ u ˙ i u ˙ i /2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamivaiabg2 da9iabeg8aYjqadwhagaGaamaaBaaaleaacaWGPbaabeaakiqadwha gaGaamaaBaaaleaacaWGPbaabeaakiaac+cacaaIYaaaaa@4059@  is the kinetic energy density. The energy flux to the crack tip follows by taking the limit as Γ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeu4KdCeaaa@375E@  shrinks down onto the crack tip.

 

Contour integral formula for energy release rate. To obtain an expression for the energy release rate, assume that the crack tip fields remain self-similar (i.e. an observer traveling with the crack tip sees a fixed state of strain and stress).  In addition, assume that the crack front is straight, and has length L in direction perpendicular to the plane of the figure.  Under these conditions u ˙ i =v u i / x 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmyDayaaca WaaSbaaSqaaiaadMgaaeqaaOGaeyypa0JaeyOeI0IaamODaiabgkGi 2kaadwhadaWgaaWcbaGaamyAaaqabaGccaGGVaGaeyOaIyRaamiEam aaBaaaleaacaaIXaaabeaaaaa@4393@ , and dΨ/dt=GLv MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamizaiabfI 6azjaac+cacaWGKbGaamiDaiabg2da9iaadEeacaWGmbGaamODaaaa @3EA1@ .  Consequently

G= 1 L lim Γ0 Γ ((T+W) δ j1 σ ij u i x 1 ) m j dA = lim C0 C ((T+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 da9maalaaabaGaaGymaaqaaiaadYeaaaWaaCbeaeaaciGGSbGaaiyA aiaac2gaaSqaaiabfo5ahjabgkziUkaaicdaaeqaaOWaa8quaeaaca GGOaGaaiikaiaadsfacqGHRaWkcaWGxbGaaiykaiabes7aKnaaBaaa leaacaWGQbGaaGymaaqabaGccqGHsislcqaHdpWCdaWgaaWcbaGaam yAaiaadQgaaeqaaOWaaSaaaeaacqGHciITcaWG1bWaaSbaaSqaaiaa dMgaaeqaaaGcbaGaeyOaIyRaamiEamaaBaaaleaacaaIXaaabeaaaa GccaGGPaGaamyBamaaBaaaleaacaWGQbaabeaakiaadsgacaWGbbaa leaacqqHtoWraeqaniabgUIiYdGccqGH9aqpdaWfqaqaaiGacYgaca GGPbGaaiyBaaWcbaGaam4qaiabgkziUkaaicdaaeqaaOWaa8quaeaa caGGOaGaaiikaiaadsfacqGHRaWkcaWGxbGaaiykaiabes7aKnaaBa aaleaacaWGQbGaaGymaaqabaGccqGHsislcqaHdpWCdaWgaaWcbaGa amyAaiaadQgaaeqaaOWaaSaaaeaacqGHciITcaWG1bWaaSbaaSqaai aadMgaaeqaaaGcbaGaeyOaIyRaamiEamaaBaaaleaacaaIXaaabeaa aaGccaGGPaGaamyBamaaBaaaleaacaWGQbaabeaakiaadsgacaWGZb aaleaacaWGdbaabeqdcqGHRiI8aaaa@8093@

where C is a contour enclosing the crack tip. (Equivalent results can be derived for general 3D cracks, but these details are omitted here).

 

This result is valid for any material response (including plastic materials), and applies to both static and dynamic conditions.

 

 

9.4.7 Rice’s J integral

 

The result derived in the preceding section becomes particularly useful if we make two further assumptions:

1.      Loading is quasi-static;

2.      The material is elastic.

In this case T=0 and W MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGxbaaaa@33A2@  is simply the strain energy density in the solid -  e.g. for a linear elastic solid with no thermal stress,

W= E 2( 1+ν ) ε ij ε ij + Eν 2( 1+ν )( 12ν ) ε jj ε kk MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadEfacqGH9aqpdaWcaaqaaiaadweaae aacaaIYaWaaeWaaeaacaaIXaGaey4kaSIaeqyVd4gacaGLOaGaayzk aaaaaiabew7aLnaaDaaaleaacaWGPbGaamOAaaqaaaaakiabew7aLn aaDaaaleaacaWGPbGaamOAaaqaaaaakiabgUcaRmaalaaabaGaamyr aiabe27aUbqaaiaaikdadaqadaqaaiaaigdacqGHRaWkcqaH9oGBai aawIcacaGLPaaadaqadaqaaiaaigdacqGHsislcaaIYaGaeqyVd4ga caGLOaGaayzkaaaaaiabew7aLnaaDaaaleaacaWGQbGaamOAaaqaaa aakiabew7aLnaaDaaaleaacaWGRbGaam4Aaaqaaaaaaaa@5656@

 

The expression for energy flux through a surface surrounding the crack tip reduces to

J= 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=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOsaiabg2 da9maapefabaGaaiikaiaadEfacqaH0oazdaWgaaWcbaGaamOAaiaa igdaaeqaaOGaeyOeI0Iaeq4Wdm3aaSbaaSqaaiaadMgacaWGQbaabe aakmaalaaabaGaeyOaIyRaamyDamaaBaaaleaacaWGPbaabeaaaOqa aiabgkGi2kaadIhadaWgaaWcbaGaaGymaaqabaaaaOGaaiykaiaad2 gadaWgaaWcbaGaamOAaaqabaGccaWGKbGaam4CaaWcbaGaeu4KdCea beqdcqGHRiI8aaaa@51E1@

This is the famous J integral.  It has the following properties:

1.      The crack tip energy integral is path independent, as long as the material enclosed by the contour is homogeneous.  There is no need then to shrink the contour down onto the crack tip MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa aaaaaaa8qacaWFtacaaa@37E6@  we get the same answer for any contour that encloses the crack tip.

2.      J=G  for an elastic solid - so the contour integral gives an elegant way to calculate the crack tip energy release rate.

 

Path independence of J: To show this, we first show that if the J integral is evaluated around any closed contour that does not enclose the crack tip, it is zero.  To see this, apply the divergence theorem

J= Γ (W δ j1 σ ij u i x 1 ) m j ds = A x j (W δ j1 σ ij u i x 1 )dA =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOsaiabg2 da9maapefabaGaaiikaiaadEfacqaH0oazdaWgaaWcbaGaamOAaiaa igdaaeqaaOGaeyOeI0Iaeq4Wdm3aaSbaaSqaaiaadMgacaWGQbaabe aakmaalaaabaGaeyOaIyRaamyDamaaBaaaleaacaWGPbaabeaaaOqa aiabgkGi2kaadIhadaWgaaWcbaGaaGymaaqabaaaaOGaaiykaiaad2 gadaWgaaWcbaGaamOAaaqabaGccaWGKbGaam4CaaWcbaGaeu4KdCea beqdcqGHRiI8aOGaeyypa0Zaa8quaeaadaWcaaqaaiabgkGi2cqaai abgkGi2kaadIhadaWgaaWcbaGaamOAaaqabaaaaOGaaiikaiaadEfa cqaH0oazdaWgaaWcbaGaamOAaiaaigdaaeqaaOGaeyOeI0Iaeq4Wdm 3aaSbaaSqaaiaadMgacaWGQbaabeaakmaalaaabaGaeyOaIyRaamyD amaaBaaaleaacaWGPbaabeaaaOqaaiabgkGi2kaadIhadaWgaaWcba GaaGymaaqabaaaaOGaaiykaiaadsgacaWGbbaaleaacaWGbbaabeqd cqGHRiI8aOGaeyypa0JaaGimaaaa@6FDD@

where A is the area enclosed by Γ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeu4KdCeaaa@375E@ .  To see that the area integral on the right hand side is zero, note that

W x j δ j1 = W ε kl ε kl x 1 = σ kl u k x l x 1 x j ( σ ij u i x 1 )= σ ij x j u i x 1 + σ ij u i x j x 1 = σ ij u i x j x 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaadaWcaa qaaiabgkGi2kaadEfaaeaacqGHciITcaWG4bWaaSbaaSqaaiaadQga aeqaaaaakiabes7aKnaaBaaaleaacaWGQbGaaGymaaqabaGccqGH9a qpdaWcaaqaaiabgkGi2kaadEfaaeaacqGHciITcqaH1oqzdaWgaaWc baGaam4AaiaadYgaaeqaaaaakmaalaaabaGaeyOaIyRaeqyTdu2aaS baaSqaaiaadUgacaWGSbaabeaaaOqaaiabgkGi2kaadIhadaWgaaWc baGaaGymaaqabaaaaOGaeyypa0Jaeq4Wdm3aaSbaaSqaaiaadUgaca WGSbaabeaakmaalaaabaGaeyOaIyRaamyDamaaBaaaleaacaWGRbaa beaaaOqaaiabgkGi2kaadIhadaWgaaWcbaGaamiBaaqabaGccqGHci ITcaWG4bWaaSbaaSqaaiaaigdaaeqaaaaaaOqaamaalaaabaGaeyOa IylabaGaeyOaIyRaamiEamaaBaaaleaacaWGQbaabeaaaaGcdaqada qaaiabeo8aZnaaBaaaleaacaWGPbGaamOAaaqabaGcdaWcaaqaaiab gkGi2kaadwhadaWgaaWcbaGaamyAaaqabaaakeaacqGHciITcaWG4b WaaSbaaSqaaiaaigdaaeqaaaaaaOGaayjkaiaawMcaaiabg2da9maa laaabaGaeyOaIyRaeq4Wdm3aaSbaaSqaaiaadMgacaWGQbaabeaaaO qaaiabgkGi2kaadIhadaWgaaWcbaGaamOAaaqabaaaaOWaaSaaaeaa cqGHciITcaWG1bWaaSbaaSqaaiaadMgaaeqaaaGcbaGaeyOaIyRaam iEamaaBaaaleaacaaIXaaabeaaaaGccqGHRaWkcqaHdpWCdaWgaaWc baGaamyAaiaadQgaaeqaaOWaaSaaaeaacqGHciITcaWG1bWaaSbaaS qaaiaadMgaaeqaaaGcbaGaeyOaIyRaamiEamaaBaaaleaacaWGQbaa beaakiabgkGi2kaadIhadaWgaaWcbaGaaGymaaqabaaaaOGaeyypa0 Jaeq4Wdm3aaSbaaSqaaiaadMgacaWGQbaabeaakmaalaaabaGaeyOa IyRaamyDamaaBaaaleaacaWGPbaabeaaaOqaaiabgkGi2kaadIhada WgaaWcbaGaamOAaaqabaGccqGHciITcaWG4bWaaSbaaSqaaiaaigda aeqaaaaaaaaa@A12C@

where we have used the equilibrium equation σ ij /d x j =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8sk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqGHciITcqaHdpWCdaWgaaWcbaGaam yAaiaadQgaaeqaaOGaai4laiaadsgacaWG4bWaaSbaaSqaaiaadQga aeqaaOGaeyypa0JaaGimaaaa@3D90@ . 

 

Now, evaluate the integral around the closed contour shown on the right. Note that the integrand vanishes on C 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4qamaaBa aaleaacaaIYaaabeaaaaa@38AD@  and C 4 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4qamaaBa aaleaacaaI0aaabeaaaaa@38AF@  so that

C 1 (W δ j1 σ ij u i x 1 ) m j ds + C 3 (W δ j1 σ ij u i x 1 ) m j ds =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaapefabaGaai ikaiaadEfacqaH0oazdaWgaaWcbaGaamOAaiaaigdaaeqaaOGaeyOe I0Iaeq4Wdm3aaSbaaSqaaiaadMgacaWGQbaabeaakmaalaaabaGaey OaIyRaamyDamaaBaaaleaacaWGPbaabeaaaOqaaiabgkGi2kaadIha daWgaaWcbaGaaGymaaqabaaaaOGaaiykaiaad2gadaWgaaWcbaGaam OAaaqabaGccaWGKbGaam4CaaWcbaGaam4qamaaBaaameaacaaIXaaa beaaaSqab0Gaey4kIipakiabgUcaRmaapefabaGaaiikaiaadEfacq aH0oazdaWgaaWcbaGaamOAaiaaigdaaeqaaOGaeyOeI0Iaeq4Wdm3a aSbaaSqaaiaadMgacaWGQbaabeaakmaalaaabaGaeyOaIyRaamyDam aaBaaaleaacaWGPbaabeaaaOqaaiabgkGi2kaadIhadaWgaaWcbaGa aGymaaqabaaaaOGaaiykaiaad2gadaWgaaWcbaGaamOAaaqabaGcca WGKbGaam4CaaWcbaGaam4qamaaBaaameaacaaIZaaabeaaaSqab0Ga ey4kIipakiabg2da9iaaicdaaaa@6BB5@

Now reverse the direction of integration around C 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4qamaaBa aaleaacaaIZaaabeaaaaa@37A7@  (note that m = -n) to get

C 1 (W δ j1 σ ij u i x 1 ) m j ds = C 3 (W δ j1 σ ij u i x 1 ) n j ds MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaapefabaGaai ikaiaadEfacqaH0oazdaWgaaWcbaGaamOAaiaaigdaaeqaaOGaeyOe I0Iaeq4Wdm3aaSbaaSqaaiaadMgacaWGQbaabeaakmaalaaabaGaey OaIyRaamyDamaaBaaaleaacaWGPbaabeaaaOqaaiabgkGi2kaadIha daWgaaWcbaGaaGymaaqabaaaaOGaaiykaiaad2gadaWgaaWcbaGaam OAaaqabaGccaWGKbGaam4CaaWcbaGaam4qamaaBaaameaacaaIXaaa beaaaSqab0Gaey4kIipakiabg2da9maapefabaGaaiikaiaadEfacq aH0oazdaWgaaWcbaGaamOAaiaaigdaaeqaaOGaeyOeI0Iaeq4Wdm3a aSbaaSqaaiaadMgacaWGQbaabeaakmaalaaabaGaeyOaIyRaamyDam aaBaaaleaacaWGPbaabeaaaOqaaiabgkGi2kaadIhadaWgaaWcbaGa aGymaaqabaaaaOGaaiykaiaad6gadaWgaaWcbaGaamOAaaqabaGcca WGKbGaam4CaaWcbaGaam4qamaaBaaameaacaaIZaaabeaaaSqab0Ga ey4kIipaaaa@6A10@

showing that the integral is equal for any two contours that start and end on the two crack faces.

 

 

 

9.4.8 Calculating energy release rates using the J integral

 

The J integral has many applications.  In some cases it can be used to compute energy release rates.  For example, consider the problem shown below.  A cracked linear elastic cracked sheet is clamped between rigid boundaries.  The bottom boundary is held fixed; the top is displaced vertically by a distance Δ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeuiLdqeaaa@375C@ .  Calculate the energy release rate for the crack.

 

For this case G=J, and we can easily evaluate the J integral around the contour shown. To do so, note that

1.      Far behind the crack tip ( x 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xq qrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8 fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGa aiaabeqaaiWabaWaaaGcbaGaamiEamaaBaaaleaacaaIXaaabeaaki abgkziUkabgkHiTiabg6HiLcaa@383B@  ) the solid is stress free.  The J integral vanishes on Γ 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xq qrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8 fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGa aiaabeqaaiWabaWaaaGcbaGaeu4KdC0aaSbaaSqaaiaaigdaaeqaaa aa@3451@  and Γ 5 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xq qrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8 fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGa aiaabeqaaiWabaWaaaGcbaGaeu4KdC0aaSbaaSqaaiaaiwdaaeqaaa aa@3455@

2.      The displacement field is constant on x 2 =±h/2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xq qrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8 fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGa aiaabeqaaiWabaWaaaGcbaGaamiEamaaBaaaleaacaaIYaaabeaaki abg2da9iabgglaXkaadIgacaGGVaGaaGOmaaaa@3941@  so that u i / x 1 =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xq qrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8 fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGa aiaabeqaaiWabaWaaaGcbaGaeyOaIyRaamyDamaaBaaaleaacaWGPb aabeaakiaac+cacqGHciITcaWG4bWaaSbaaSqaaiaaigdaaeqaaOGa eyypa0JaaGimaaaa@3B4D@  there.  In addition m 1 =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xq qrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8 fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGa aiaabeqaaiWabaWaaaGcbaGaamyBamaaBaaaleaacaaIXaaabeaaki abg2da9iaaicdaaaa@35A5@  on Γ 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xq qrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8 fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGa aiaabeqaaiWabaWaaaGcbaGaeu4KdC0aaSbaaSqaaiaaikdaaeqaaa aa@3452@  and Γ 4 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xq qrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8 fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGa aiaabeqaaiWabaWaaaGcbaGaeu4KdC0aaSbaaSqaaiaaisdaaeqaaa aa@3454@ .   The J integral vanishes on Γ 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xq qrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8 fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGa aiaabeqaaiWabaWaaaGcbaGaeu4KdC0aaSbaaSqaaiaaikdaaeqaaa aa@3452@  and Γ 4 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xq qrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8 fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGa aiaabeqaaiWabaWaaaGcbaGaeu4KdC0aaSbaaSqaaiaaisdaaeqaaa aa@3454@ , therefore.

3.      Far ahead of the crack tip x 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xq qrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8 fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGa aiaabeqaaiWabaWaaaGcbaGaamiEamaaBaaaleaacaaIXaaabeaaki abgkziUkabg6HiLcaa@374E@ , the displacement, stress and strain energy density can easily be calculated as

u 2 = x 2 Δ/h, u 1 = u 3 =0 σ 22 =E(1ν)Δ/(1+ν)(12ν)h σ 11 = σ 33 =EνΔ/(1+ν)(12ν)h W=E(1ν) Δ 2 /2(1+ν)(12ν) h 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xq qrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8 fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGa aiaabeqaaiWabaWaaaGceaqabeaacaWG1bWaaSbaaSqaaiaaikdaae qaaOGaeyypa0JaamiEamaaBaaaleaacaaIYaaabeaakiabfs5aejaa c+cacaWGObGaaiilaiaaykW7caaMc8UaamyDamaaBaaaleaacaaIXa aabeaakiabg2da9iaadwhadaWgaaWcbaGaaG4maaqabaGccqGH9aqp caaIWaaabaGaeq4Wdm3aaSbaaSqaaiaaikdacaaIYaaabeaakiabg2 da9iaadweacaGGOaGaaGymaiabgkHiTiabe27aUjaacMcacqqHuoar caGGVaGaaiikaiaaigdacqGHRaWkcqaH9oGBcaGGPaGaaiikaiaaig dacqGHsislcaaIYaGaeqyVd4MaaiykaiaadIgacaaMc8UaaGPaVlaa ykW7caaMc8UaaGPaVlabeo8aZnaaBaaaleaacaaIXaGaaGymaaqaba GccqGH9aqpcqaHdpWCdaWgaaWcbaGaaG4maiaaiodaaeqaaOGaeyyp a0Jaamyraiabe27aUjabfs5aejaac+cacaGGOaGaaGymaiabgUcaRi abe27aUjaacMcacaGGOaGaaGymaiabgkHiTiaaikdacqaH9oGBcaGG PaGaamiAaaqaaiaadEfacqGH9aqpcaWGfbGaaiikaiaaigdacqGHsi slcqaH9oGBcaGGPaGaeuiLdq0aaWbaaSqabeaacaaIYaaaaOGaai4l aiaaikdacaGGOaGaaGymaiabgUcaRiabe27aUjaacMcacaGGOaGaaG ymaiabgkHiTiaaikdacqaH9oGBcaGGPaGaamiAamaaCaaaleqabaGa aGOmaaaaaaaa@92EE@

The contribution to the J integral from Γ 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeu4KdC0aaS baaSqaaiaaiodaaeqaaaaa@3847@  follows as

Γ 3 (W δ j1 σ ij u i x 1 ) n j ds = E(1ν) Δ 2 2(1+ν)(12ν)h MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xq qrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8 fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGa aiaabeqaaiWabaWaaaGcbaWaa8quaeaacaGGOaGaam4vaiabes7aKn aaBaaaleaacaWGQbGaaGymaaqabaGccqGHsislcqaHdpWCdaWgaaWc baGaamyAaiaadQgaaeqaaOWaaSaaaeaacqGHciITcaWG1bWaaSbaaS qaaiaadMgaaeqaaaGcbaGaeyOaIyRaamiEamaaBaaaleaacaaIXaaa beaaaaGccaGGPaGaamOBamaaBaaaleaacaWGQbaabeaakiaadsgaca WGZbaaleaacqqHtoWrdaWgaaadbaGaaG4maaqabaaaleqaniabgUIi YdGccqGH9aqpdaWcaaqaaiaadweacaGGOaGaaGymaiabgkHiTiabe2 7aUjaacMcacqqHuoardaahaaWcbeqaaiaaikdaaaaakeaacaaIYaGa aiikaiaaigdacqGHRaWkcqaH9oGBcaGGPaGaaiikaiaaigdacqGHsi slcaaIYaGaeqyVd4MaaiykaiaadIgaaaaaaa@60CF@

4.      The energy release rate is therefore

G= E(1ν) Δ 2 2(1+ν)(12ν)h MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4raiabg2 da9maalaaabaGaamyraiaacIcacaaIXaGaeyOeI0IaeqyVd4Maaiyk aiabfs5aenaaCaaaleqabaGaaGOmaaaaaOqaaiaaikdacaGGOaGaaG ymaiabgUcaRiabe27aUjaacMcacaGGOaGaaGymaiabgkHiTiaaikda cqaH9oGBcaGGPaGaamiAaaaaaaa@4B80@

Symmetry conditions show that the crack must be loaded in pure mode I, so the stress intensity factor can also be computed.