Chapter 9

 

Modeling Material Failure

 

 

9.6 Linear Elastic Fracture mechanics of interfaces

 

Many engineering applications require one material to be bonded to another.  Examples include adhesive joints; protective coatings; composite materials; and thin films used in the manufacture of microelectronic circuits.  In all these applications, techniques are required to predict the strength of the bond.

 

To this end, a great deal of work has been done over the past 20 years to extend linear elastic fracture mechanics to predict the behavior of cracks on, or near, the interface between two dissimilar brittle materials. 

 

 

9.6.1 Crack Tip Fields for a crack on an interface

 

The foundation for linear elastic interfacial fracture mechanics is based on an asymptotic analysis of the stress and strain fields near the tip of a crack.

 

The problem of interest is illustrated in the figure. A semi-infinite crack with a straight front that coincides with the x 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaBa aaleaacaaIZaaabeaaaaa@37DC@  axis lies on the interface between two linear elastic solids. The material above the crack has shear modulus and Poisson’s ratio μ 1 , ν 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiVd02aaS baaSqaaiaaigdaaeqaaOGaaiilaiabe27aUnaaBaaaleaacaaIXaaa beaaaaa@3CF3@ ; the material below the crack has shear modulus and Poisson’s ratio μ 2 , ν 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiVd02aaS baaSqaaiaaikdaaeqaaOGaaiilaiabe27aUnaaBaaaleaacaaIYaaa beaakiaaykW7aaa@3E8A@ .  In this section we give the complex variable solution that governs the variation of stress and displacement near the crack tip.   The solid is subjected to static remote loading, and is assumed to deform in plane strain. 

 

The complete stress and displacement fields for an interface crack are given in Section 5.3.6.  The solution is too long to type out here:  Instead, we summarize the key features.

 

Material parameters for an interface:  The solution is expressed in terms of several additional parameters

1.      Plane strain moduli E 1 =2 μ 1 /(1 ν 1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmyrayaafa WaaSbaaSqaaiaaigdaaeqaaOGaeyypa0JaaGOmaiabeY7aTnaaBaaa leaacaaIXaaabeaakiaac+cacaGGOaGaaGymaiabgkHiTiabe27aUn aaBaaaleaacaaIXaaabeaakiaacMcaaaa@438A@ , E 2 =2 μ 2 /(1 ν 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmyrayaafa WaaSbaaSqaaiaaikdaaeqaaOGaeyypa0JaaGOmaiabeY7aTnaaBaaa leaacaaIYaaabeaakiaac+cacaGGOaGaaGymaiabgkHiTiabe27aUn aaBaaaleaacaaIYaaabeaakiaacMcaaaa@438D@

2.      Bimaterial modulus 1 E * ={ 1 E 1 + 1 E 2 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca aIXaaabaGaamyramaaCaaaleqabaGaaiOkaaaaaaGccqGH9aqpdaGa daqaamaalaaabaGaaGymaaqaaiqadweagaqbamaaBaaaleaacaaIXa aabeaaaaGccqGHRaWkdaWcaaqaaiaaigdaaeaaceWGfbGbauaadaWg aaWcbaGaaGOmaaqabaaaaaGccaGL7bGaayzFaaaaaa@42B5@

3.      Dundur’s elastic constants

α= E 1 E 2 E 1 + E 2 β= ( 12 ν 2 )/ μ 2 ( 12 ν 1 )/ μ 1 2( 1 ν 2 )/ μ 2 +2( 1 ν 1 )/ μ 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySdeMaey ypa0ZaaSaaaeaaceWGfbGbauaadaWgaaWcbaGaaGymaaqabaGccqGH sislceWGfbGbauaadaWgaaWcbaGaaGOmaaqabaaakeaaceWGfbGbau aadaWgaaWcbaGaaGymaaqabaGccqGHRaWkceWGfbGbauaadaWgaaWc baGaaGOmaaqabaaaaOGaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7ca aMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaa ykW7caaMc8UaaGPaVlaaykW7cqaHYoGycqGH9aqpdaWcaaqaamaabm aabaGaaGymaiabgkHiTiaaikdacqaH9oGBdaWgaaWcbaGaaGOmaaqa baaakiaawIcacaGLPaaacaGGVaGaeqiVd02aaSbaaSqaaiaaikdaae qaaOGaeyOeI0YaaeWaaeaacaaIXaGaeyOeI0IaaGOmaiabe27aUnaa BaaaleaacaaIXaaabeaaaOGaayjkaiaawMcaaiaac+cacqaH8oqBda WgaaWcbaGaaGymaaqabaaakeaacaaIYaWaaeWaaeaacaaIXaGaeyOe I0IaeqyVd42aaSbaaSqaaiaaikdaaeqaaaGccaGLOaGaayzkaaGaai 4laiabeY7aTnaaBaaaleaacaaIYaaabeaakiabgUcaRiaaikdadaqa daqaaiaaigdacqGHsislcqaH9oGBdaWgaaWcbaGaaGymaaqabaaaki aawIcacaGLPaaacaGGVaGaeqiVd02aaSbaaSqaaiaaigdaaeqaaaaa aaa@891A@

Evidently α MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySdegaaa@3785@  is a measure of the relative stiffness of the two materials.  It must lie in the range 1<α<1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOeI0IaaG ymaiabgYda8iabeg7aHjabgYda8iaaigdaaaa@3BF0@  for all possible material combinations: α=1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySdeMaey ypa0JaaGymaaaa@3946@  indicates that material 1 is rigid, while α=1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySdeMaey ypa0JaeyOeI0IaaGymaaaa@3A33@  signifies that material 2 is rigid.  The second parameter does not have such a nice physical interpretation MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa aaaaaaa8qacaWFtacaaa@37E6@  it is a rough measure of the relative compressibilities of the two materials.  For Poisson’s ratios in the range 0<ν<1/2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGimaiabgY da8iabe27aUjabgYda8iaaigdacaGGVaGaaGOmaaaa@3C8A@ , one can show that that 1<α4β<1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOeI0IaaG ymaiabgYda8iabeg7aHjabgkHiTiaaisdacqaHYoGycqGH8aapcaaI Xaaaaa@3F3C@ .

4.      Crack tip singularity parameter

ε= 1 2π log( 1β 1+β ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyTduMaey ypa0ZaaSaaaeaacaaIXaaabaGaaGOmaiabec8aWbaaciGGSbGaai4B aiaacEgadaqadaqaamaalaaabaGaaGymaiabgkHiTiabek7aIbqaai aaigdacqGHRaWkcqaHYoGyaaaacaGLOaGaayzkaaaaaa@47DE@

For most material combinations the value of ε MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyTdugaaa@379D@  is very small MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa aaaaaaa8qacaWFtacaaa@37E6@  typically of order 0.01 or so.

 

Crack tip loading parameters The state of stress at the crack tip is characterized by three numbers: an arbitrary characteristic length L (a value of L=100μm MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGmbGaeyypa0JaaGymaiaaicdaca aIWaGaeqiVd0MaamyBaaaa@3974@  is often used); the phase angle of the loading ψ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHipqEaaa@3494@  and the magnitude of the stress intensity factor | K | MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaabdaqaaiaadUeaaiaawEa7caGLiW oaaaa@36B8@ .  Often, the energy release rate for the crack G is used in place of | K | MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaabdaqaaiaadUeaaiaawEa7caGLiW oaaaa@36B8@ . These are defined as follows

 Phase angle ψ= tan 1 ( σ 12 (r=L) σ 22 (r=L) ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHipqEcqGH9aqpciGG0bGaaiyyai aac6gadaahaaWcbeqaaiabgkHiTiaaigdaaaGcdaqadaqaamaalaaa baGaeq4Wdm3aaSbaaSqaaiaaigdacaaIYaaabeaakiaacIcacaWGYb Gaeyypa0JaamitaiaacMcaaeaacqaHdpWCdaWgaaWcbaGaaGOmaiaa ikdaaeqaaOGaaiikaiaadkhacqGH9aqpcaWGmbGaaiykaaaaaiaawI cacaGLPaaaaaa@4B12@

 Stress intensity magnitude   |K|= e iψ lim r0 ( σ 22 +i σ 12 ) (r/L) iε 2πr MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8sk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaGG8bGaam4saiaacYhacqGH9aqpca WGLbWaaWbaaSqabeaacqGHsislcaWGPbGaeqiYdKhaaOWaaCbeaeaa ciGGSbGaaiyAaiaac2gaaSqaaiaadkhacqGHsgIRcaaIWaaabeaaki aacIcacqaHdpWCdaWgaaWcbaGaaGOmaiaaikdaaeqaaOGaey4kaSIa amyAaiabeo8aZnaaBaaaleaacaaIXaGaaGOmaaqabaGccaGGPaGaai ikaiaadkhacaGGVaGaamitaiaacMcadaahaaWcbeqaaiabgkHiTiaa dMgacqaH1oqzaaGcdaGcaaqaaiaaikdacqaHapaCcaWGYbaaleqaaa aa@5749@

 Energy release rate G= | K | 2 2 E * cosh 2 πε MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4raiabg2 da9maalaaabaWaaqWaaeaacaWGlbaacaGLhWUaayjcSdWaaWbaaSqa beaacaaIYaaaaaGcbaGaaGOmaiaadweadaahaaWcbeqaaiaacQcaaa GcciGGJbGaai4BaiaacohacaGGObWaaWbaaSqabeaacaaIYaaaaOGa eqiWdaNaeqyTdugaaaaa@4845@

 Solutions to interface crack problems are also often expressed in terms of two stress-intensity factor like parameters K 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGlbWaaSbaaSqaaiaaigdaaeqaaa aa@347D@  and K 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGlbWaaSbaaSqaaiaaikdaaeqaaa aa@347E@ .  These are related to the crack tip parameters by

( K 1 +i K 2 )=| K | e iψ L iε | K |= K 1 2 + K 2 2 ψ= tan 1 { Im[ ( K 1 +i K 2 ) L iε ] Re[ ( K 1 +i K 2 ) L iε ] } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8 fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGa aiaabeqaaiWabaWaaaGceaqabeaadaqadaqaaiaadUeadaWgaaWcba GaaGymaaqabaGccqGHRaWkcaWGPbGaam4samaaBaaaleaacaaIYaaa beaaaOGaayjkaiaawMcaaiabg2da9maaemaabaGaam4saaGaay5bSl aawIa7aiaadwgadaahaaWcbeqaaiaadMgacqaHipqEaaGccaWGmbWa aWbaaSqabeaacqGHsislcaWGPbGaeqyTdugaaaGcbaWaaqWaaeaaca WGlbaacaGLhWUaayjcSdGaeyypa0ZaaOaaaeaacaWGlbWaa0baaSqa aiaaigdaaeaacaaIYaaaaOGaey4kaSIaam4samaaDaaaleaacaaIYa aabaGaaGOmaaaaaeqaaOGaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7 caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlabeI8a5jabg2da9iGacs hacaGGHbGaaiOBamaaCaaaleqabaGaeyOeI0IaaGymaaaakmaacmaa baWaaSaaaeaaciGGjbGaaiyBamaadmaabaWaaeWaaeaacaWGlbWaaS baaSqaaiaaigdaaeqaaOGaey4kaSIaamyAaiaadUeadaWgaaWcbaGa aGOmaaqabaaakiaawIcacaGLPaaacaWGmbWaaWbaaSqabeaacaWGPb GaeqyTdugaaaGccaGLBbGaayzxaaaabaGaciOuaiaacwgadaWadaqa amaabmaabaGaam4samaaBaaaleaacaaIXaaabeaakiabgUcaRiaadM gacaWGlbWaaSbaaSqaaiaaikdaaeqaaaGccaGLOaGaayzkaaGaamit amaaCaaaleqabaGaamyAaiabew7aLbaaaOGaay5waiaaw2faaaaaai aawUhacaGL9baaaaaa@872C@

 

 

 

Interpreting the crack tip fields

 The values of | K | MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaabdaqaaiaadUeaaiaawEa7caGLiW oaaaa@36B8@  and ψ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHipqEaaa@3494@  are is determined by the solid’s shape and how it is loaded (the value of   ψ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHipqEaaa@3494@  also depends on the choice of the characteristic length L). Once | K | MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaabdaqaaiaadUeaaiaawEa7caGLiW oaaaa@36B8@  and ψ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHipqEaaa@3494@  are known, however, the near tip fields always have the form given by the asymptotic solution.

 Since ψ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHipqEaaa@3494@  quantifies the ratio of shear to opening stress ahead of the crack tip, it is qualitatively equivalent to the ratio tan 1 ( K II / K I ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaaciGG0bGaaiyyaiaac6gadaahaaWcbe qaaiabgkHiTiaaigdaaaGccaGGOaGaam4samaaBaaaleaacaWGjbGa amysaaqabaGccaGGVaGaam4samaaBaaaleaacaWGjbaabeaakiaacM caaaa@3DF8@ , where 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@  are the mode 1 and mode 2 stress intensity factors for a crack in a homogeneous solid.

 The opening and shear stresses along θ=0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8sk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaH4oqCcqGH9aqpcaaIWaaaaa@364C@  ahead of the crack tip can be calculated from

σ 22 +i σ 12 = | K | e iψ 2πr ( r L ) iε MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aaS baaSqaaiaaikdacaaIYaaabeaakiabgUcaRiaadMgacqaHdpWCdaWg aaWcbaGaaGymaiaaikdaaeqaaOGaeyypa0ZaaSaaaeaadaabdaqaai aadUeaaiaawEa7caGLiWoacaWGLbWaaWbaaSqabeaacaWGPbGaeqiY dKhaaaGcbaWaaOaaaeaacaaIYaGaeqiWdaNaamOCaaWcbeaaaaGcda qadaqaamaalaaabaGaamOCaaqaaiaadYeaaaaacaGLOaGaayzkaaWa aWbaaSqabeaacaWGPbGaeqyTdugaaaaa@524B@

 The crack opening displacements behind the crack tip can be calculated from

δ 2 +i δ 1 = 4| K | e iψ E * (1+2iε)cosh(πε) r 2π ( r L ) iε MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiTdq2aaS baaSqaaiaaikdaaeqaaOGaey4kaSIaamyAaiabes7aKnaaBaaaleaa caaIXaaabeaakiabg2da9maalaaabaGaaGinamaaemaabaGaam4saa Gaay5bSlaawIa7aiaadwgadaahaaWcbeqaaiaadMgacqaHipqEaaaa keaacaWGfbWaaWbaaSqabeaacaGGQaaaaOGaaiikaiaaigdacqGHRa WkcaaIYaGaamyAaiabew7aLjaacMcaciGGJbGaai4BaiaacohacaGG ObGaaiikaiabec8aWjabew7aLjaacMcaaaWaaOaaaeaadaWcaaqaai aadkhaaeaacaaIYaGaeqiWdahaaaWcbeaakiaaykW7caaMc8+aaeWa aeaadaWcaaqaaiaadkhaaeaacaWGmbaaaaGaayjkaiaawMcaamaaCa aaleqabaGaamyAaiabew7aLbaaaaa@64ED@

 The complex exponent appearing in these expressions is scary MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa aaaaaaa8qacaWFtacaaa@37E6@  to understand what it means note that

r iε =exp(iεlogr)=cos(εlogr)+isin(εlogr) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOCamaaCa aaleqabaGaamyAaiabew7aLbaakiabg2da9iGacwgacaGG4bGaaiiC aiaacIcacaWGPbGaeqyTduMaciiBaiaac+gacaGGNbGaamOCaiaacM cacqGH9aqpciGGJbGaai4BaiaacohacaGGOaGaeqyTduMaciiBaiaa c+gacaGGNbGaamOCaiaacMcacqGHRaWkcaWGPbGaci4CaiaacMgaca GGUbGaaiikaiabew7aLjGacYgacaGGVbGaai4zaiaadkhacaGGPaaa aa@5C65@

so this term indicates that the stresses oscillate near the crack tip.  We will discuss this in more detail below.

 

 

Oscillations in the stress and displacement fields

 

The asymptotic crack tip field for an interface crack is strikingly different to the corresponding solution for a homogeneous solid.  In fact, the results are somewhat disturbing, and have been the cause of much anguish in the fracture mechanics community.

 

We have already noted that the stress fields are oscillatory near the crack tip.  The stress distributions are plotted on the right as a function of log(εr/L) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiGacYgacaGGVbGaai4zaiaacIcacqaH1o qzcaWGYbGaai4laiaadYeacaGGPaaaaa@38B3@ .  Note that the results are shown for an unphysically large value of ε MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabew7aLbaa@320F@  - for practical values the oscillations are so slow it is hard to see them.

 

Both normal and shear stresses oscillate with increasing frequency as the crack tip is approached.  As a result, it is difficult to unambiguously separate the loading into normal and shear components MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa aaaaaaa8qacaWFtacaaa@37E6@  an opening stress induces just as much shear near the crack tip as does shear loading, and vice-versa.

 

Even more disturbingly, the crack opening displacements show the same oscillatory character. This means that the solution predicts that the crack faces overlap near the crack tip, which is clearly unphysical.

 

It is possible to find a solution that corrects for the overlapping crack faces (Comninou, J. Appl Mech. 44, 631 1977).  This solution predicts that the crack faces touch just behind the crack tip for all combinations of remote load.  There is a square root singularity in shear stress at the crack tip (so it’s strictly always loaded in mode II).  The zone of contact is extremely small, however MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa aaaaaaa8qacaWFtacaaa@37E6@  typically of the order of a few nanometers for most practical crack sizes and materials, and probably much smaller than the process zone.

 

The standard procedure in LEIFM (linear elastic interfacial fracture mechanics) is to ignore the overlap between crack faces, and accept the asymptotic field described in the beginning of this sub-section as characterizing the stress and strain fields for an interface crack.  The oscillatory singularity is, after all, no less physical than a square root singularity.  The asymptotic field is expected to represent actual stress and strain fields in an annular region, which is small compared with specimen geometry, and large compared with the process zone.

 

 

 

9.6.2 Phenomenological theory of interface fracture

 

Phenomenological fracture mechanics for interfaces is based on the same reasoning that is used in fracture mechanics of homogeneous solids.  We anticipate 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 field is not accurate;

2.      A K dominant zone, outside the process zone, but small compared with specimen dimensions, where the asymptotic linear elastic field accurately describes the deformation;

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

Material failure (crack growth or fatigue) is a consequence of the failure mechanisms in the process zone.  As usual, we do not attempt to model the failure process in detail, and instead assume that the fields in the process zone are controlled by the fields in the region of K dominance.  In interface fracture, we use the stress state at the reference length L ahead of the crack tip to characterize the loading experienced by the process zone.  As we have seen, this stress is characterized by the energy release rate G (or alternatively the stress intensity magnitude | K | MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaabdaqaaiaadUeaaiaawEa7caGLiW oaaaa@36B8@  ), together with the phase angle ψ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHipqEaaa@3494@

 

Fracture Criterion The critical condition for an interface crack to propagate is therefore given by a fracture criterion of the form

G= G i (ψ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4raiabg2 da9iaadEeadaWgaaWcbaGaamyAaaqabaGccaGGOaGaeqiYdKNaaiyk aaaa@3DE6@

where G i (ψ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGhbWaaSbaaSqaaiaadMgaaeqaaO GaaiikaiabeI8a5jaacMcaaaa@37DD@  is the fracture toughness of the interface and ψ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHipqEaaa@3494@  is the phase angle, defined using some appropriate choice of length L.    The fracture toughness is a function of phase angle, just as the fracture toughness of a homogeneous solid subjected to mixed mode loading is a function of K II / K I MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGlbWaaSbaaSqaaiaadMeacaWGjb aabeaakiaac+cacaWGlbWaaSbaaSqaaiaadMeaaeqaaaaa@37E5@

 

The fracture resistance of the interface G i (ψ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4ramaaBa aaleaacaWGPbaabeaakiaacIcacqaHipqEcaGGPaaaaa@3C14@  must be measured experimentally.  Several specimens are available for this purpose.  Examples include sandwich specimens (e.g. Leichti & Knauss, Exp. Mech. 22 383 1982; see also Suo & Hutchinson Mat. Sci & Eng A107 1989 135) and 4 point bend specimens (e.g. Charalambides et al Mechanics of Materials 8 269, 1990).  Experiments show that G i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4ramaaBa aaleaacaWGPbaabeaaaaa@37DC@  increases rapidly with phase angle: the typical variation of fracture toughness with phase angle is sketched in the figure. In fact, many experimental data seem to be fit by G i = G 0 /cos(ψ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4ramaaBa aaleaacaWGPbaabeaakiabg2da9iaadEeadaWgaaWcbaGaaGimaaqa baGccaGGVaGaci4yaiaac+gacaGGZbGaaiikaiabeI8a5jaacMcaaa a@425C@ .

 

To apply LEIFM, then, it is necessary (i) to measure the fracture resistance of the interface as a function of phase angle; (ii) calculate energy release rate and phase angle for the interface crack in the structure or component of interest, and (iii) apply the fracture criterion to assess the load bearing capacity of the component.

 

 

9.6.3 Stress intensity factors for some interface cracks

 

The solutions to interface crack problems are most conveniently expressed in terms of the stress intensity factors K 1 , K 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGlbWaaSbaaSqaaiaaigdaaeqaaO GaaiilaiaadUeadaWgaaWcbaGaaGOmaaqabaaaaa@36EF@ , because these are linear in stress and can therefore be superposed.  Stress intensity factors have been computed for many standard specimen geometries (usually using a numerical technique).  A few examples are shown below.

 

STRESS INTENSITY FACTORS FOR INTERFACE CRACKS

K 1 +i K 2 =( σ 22 +i σ 12 ) πa ( 1+2iε ) ( 2a ) iε MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4samaaBa aaleaacaaIXaaabeaakiabgUcaRiaadMgacaWGlbWaaSbaaSqaaiaa ikdaaeqaaOGaeyypa0ZaaeWaaeaacqaHdpWCdaWgaaWcbaGaaGOmai aaikdaaeqaaOGaey4kaSIaamyAaiabeo8aZnaaBaaaleaacaaIXaGa aGOmaaqabaaakiaawIcacaGLPaaadaGcaaqaaiabec8aWjaadggaaS qabaGcdaqadaqaaiaaigdacqGHRaWkcaaIYaGaamyAaiabew7aLbGa ayjkaiaawMcaamaabmaabaGaaGOmaiaadggaaiaawIcacaGLPaaada ahaaWcbeqaaiabgkHiTiaadMgacqaH1oqzaaaaaa@57A9@

 

K 1 +i K 2 = 2 π coshπε F 2 +i F 1 b 1/2+iε MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4samaaBa aaleaacaaIXaaabeaakiabgUcaRiaadMgacaWGlbWaaSbaaSqaaiaa ikdaaeqaaOGaeyypa0ZaaOaaaeaadaWcaaqaaiaaikdaaeaacqaHap aCaaaaleqaaOGaci4yaiaac+gacaGGZbGaaiiAaiabec8aWjabew7a LnaalaaabaGaamOramaaBaaaleaacaaIYaaabeaakiabgUcaRiaadM gacaWGgbWaaSbaaSqaaiaaigdaaeqaaaGcbaGaamOyamaaCaaaleqa baGaaGymaiaac+cacaaIYaGaey4kaSIaamyAaiabew7aLbaaaaaaaa@5335@

K 1 +i K 2 = 2 π coshπε F 2 +i F 1 (2a) 1/2+iε ( a+b ab ) 1/2+iε MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4samaaBa aaleaacaaIXaaabeaakiabgUcaRiaadMgacaWGlbWaaSbaaSqaaiaa ikdaaeqaaOGaeyypa0ZaaOaaaeaadaWcaaqaaiaaikdaaeaacqaHap aCaaaaleqaaOGaci4yaiaac+gacaGGZbGaaiiAaiabec8aWjabew7a LnaalaaabaGaamOramaaBaaaleaacaaIYaaabeaakiabgUcaRiaadM gacaWGgbWaaSbaaSqaaiaaigdaaeqaaaGcbaGaaiikaiaaikdacaWG HbGaaiykamaaCaaaleqabaGaaGymaiaac+cacaaIYaGaey4kaSIaam yAaiabew7aLbaaaaGcdaqadaqaamaalaaabaGaamyyaiabgUcaRiaa dkgaaeaacaWGHbGaeyOeI0IaamOyaaaaaiaawIcacaGLPaaadaahaa WcbeqaaiaaigdacaGGVaGaaGOmaiabgUcaRiaadMgacqaH1oqzaaaa aa@6223@

 

 

 

The point force solutions can be used to calculate stress intensity factors for interface cracks that are subjected to nonuniform stress fields, following the procedure given in Section 9.3.3.  The energy release rate and phase angle can be calculated using the formulas given in the preceding section.  For example, for the slit crack with length 2a subjected to uniform stress far from the crack, the energy release rate and phase angle are

ψ= tan 1 { Im[ ( σ 22 +i σ 12 )(1+2iε) (L/a) iε ] Re[ ( σ 22 +i σ 12 )(1+2iε) (L/a) iε ] } G= ( σ 22 2 + σ 12 2 )(1+4 ε 2 )πa 2 E * cosh 2 (πε) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakqaabeqaaiabeI8a5jabg2da9iGacshaca GGHbGaaiOBamaaCaaaleqabaGaeyOeI0IaaGymaaaakmaacmaabaWa aSaaaeaaciGGjbGaaiyBamaadmaabaGaaiikaiabeo8aZnaaBaaale aacaaIYaGaaGOmaaqabaGccqGHRaWkcaWGPbGaeq4Wdm3aaSbaaSqa aiaaigdacaaIYaaabeaakiaacMcacaGGOaGaaGymaiabgUcaRiaaik dacaWGPbGaeqyTduMaaiykaiaacIcacaWGmbGaai4laiaadggacaGG PaWaaWbaaSqabeaacaWGPbGaeqyTdugaaaGccaGLBbGaayzxaaaaba GaciOuaiaacwgadaWadaqaaiaacIcacqaHdpWCdaWgaaWcbaGaaGOm aiaaikdaaeqaaOGaey4kaSIaamyAaiabeo8aZnaaBaaaleaacaaIXa GaaGOmaaqabaGccaGGPaGaaiikaiaaigdacqGHRaWkcaaIYaGaamyA aiabew7aLjaacMcacaGGOaGaamitaiaac+cacaWGHbGaaiykamaaCa aaleqabaGaamyAaiabew7aLbaaaOGaay5waiaaw2faaaaaaiaawUha caGL9baaaeaacaWGhbGaeyypa0ZaaSaaaeaacaGGOaGaeq4Wdm3aa0 baaSqaaiaaikdacaaIYaaabaGaaGOmaaaakiabgUcaRiabeo8aZnaa DaaaleaacaaIXaGaaGOmaaqaaiaaikdaaaGccaGGPaGaaiikaiaaig dacqGHRaWkcaaI0aGaeqyTdu2aaWbaaSqabeaacaaIYaaaaOGaaiyk aiabec8aWjaadggaaeaacaaIYaGaamyramaaCaaaleqabaGaaiOkaa aakiGacogacaGGVbGaai4CaiaacIgadaahaaWcbeqaaiaaikdaaaGc caGGOaGaeqiWdaNaeqyTduMaaiykaaaaaaaa@92F9@

 

Note that, for a fixed value of L, the phase angle depends on the size of the crack.   This is a general feature of interface cracks: the mode mixity depends on specimen size.  However, the value of ε MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaH1oqzaaa@346D@  for most material pairs is very small, so the variation with specimen size is very weak.

 

 

9.6.4 Crack Path Selection

 

A final issue that is of great interest in interfacial fracture is the question of crack path selection.  An interface crack can either propagate along the interface, or deflect into one of the two materials adjacent to the interface.  In addition, a crack approaching transverse to an interface may be deflected along it MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa aaaaaaa8qacaWFtacaaa@37E6@  this is a mechanism for trapping cracks in composite materials.

 

A rather involved stress analysis is required to answer these questions, but the results are simple. A crack approaching perpendicular to an interface (as shown on the left above) will deflect along the interface as long as

G i (π/2)< G cm /4 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4ramaaBa aaleaacaWGPbaabeaakiaacIcacqaHapaCcaGGVaGaaGOmaiaacMca cqGH8aapcaWGhbWaaSbaaSqaaiaadogacaWGTbaabeaakiaac+caca aI0aaaaa@42C3@

where G i (π/2) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4ramaaBa aaleaacaWGPbaabeaakiaacIcacqaHapaCcaGGVaGaaGOmaiaacMca aaa@3D72@  is the interface toughness for a phase angle of 90 degrees, and G cm MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4ramaaBa aaleaacaWGJbGaamyBaaqabaaaaa@39CF@  is the fracture toughness of the material on the far side of the interface.  If this condition is satisfied, the crack remains trapped in the interface and will not kink out of it.