薄膜力学8-断裂2.pdf

1、EM 397:Thin Film MechanicsVIII.Film Cracking under TensionNanshu LuThe University of Texas at AustinFall 2013Crack nucleation and growthSubstrateFilmA three-dimensional process:crack grows in both parallel and perpendicular directions,with a curved crack front.Cutting and ChannelingCutting:a crack g

2、rowing perpendicular to the interface;a plane strain problem,assuming long in the parallel direction.Channeling:crack(s)growing parallel to the interface,assuming a constant depth in the perpendicular direction.Elastic mismatchFor elastic films and substrates,the crack behavior depends on the elasti

3、c mismatch between them.Dundurs parameters:sfsffssffssfEEEE11111111fssffssfPlane strain:21EE43No mismatch:=0;Stiff film on compliant substrate:0;Compliant film on stiff substrate:0;If f=s=0.5,=0;If f=s=1/3,=/4;Both and change signs when the materials are switched.1-1-0.250.25Partially cracked filmsS

4、ubstrateFilmStress intensity factor at the crack tip:hhaFKI,a/hF01000When=0 or a/h 0:aKI12.1 Stiff film on compliant substrate:SIF increases monotonically;Compliant film on stiff substrate:SIF attains a maximum near the interfaceBeuth,IJSS 29,1657-1675(1992).Fully cracked filmsSubstrateFilma=h:the c

5、rack tip is at the interface,different crack-tip field due to elastic mismatch.sxxrhCr)0,(The stress singularity exponent,s,depends on the elastic mismatch(Zak and Williams,1963):01)1(12)cos(222ssWhen=0,s=0.5;For 0,s 0,s 0.5(more singular);For 1,s 1.s-11010.5Penetration or deflection?SubstrateFilmsp

6、GidGWhen both conditions are satisfied,the crack is likely to penetrate into the substrate if1idspGGHe and Hutchinson,IJSS 25,1053-1067,1989.Both the energy release rates depend on the elastic mismatch between the film and the substrate.Energy release ratea/h01000ffhEG2SubstrateFilmSubstrateFilmCrac

7、k channelingSubstrateFilm0hAssume the crack cuts through the film,with no substrate penetration,interface debonding,or spalling.The crack grows parallel to the interface,with a curved channel front.Three-dimensional analysis is required to determine the shape of the channel front(Nakamura and Kamath

8、,1992).The crack reaches a steady state when the length exceeds a few times the film thickness.Steady-state channeling cracksSubstrateFilm0hThe channel front maintains its shape as it advances,and the cross section(opening)profile behind the front attains the equilibrium shape of a plane-strain crac

9、k.Thus,the steady-state energy release rate of a channeling crack can be obtained from the plane strain problem,and the fracture resistance is the film toughness.Steady-state energy release rate0FilmSubstrateSubstrateh0Film(z)(b)zxAhead of the channel frontFar behind the channel frontEnergy release

10、rate:hSSdzzhG00)(21hSSdzaGhG0)(1orFrom dimensional analysis:fSSEhZG20),(Numerical results-1-0.500.510510152025303540Elastic mismatch,Dimensionless energy release rate,(GSSE1*)/(02h)Beuth 21XFEM=/4 Beuth,IJSS 29,1657-1675(1992);Huang et al.,Engineering Fracture Mech.70,2513-2526(2003).fSSEhZG20),(976

11、.1ZParallel channeling cracksfSSEhShZG20),(Assume simultaneous growth and homogeneous system.SHuang et al.,Engineering Fracture Mech.70,2513-2526(2003).Elastic shear-lag approximation0000(x)fhxxuEfuksuhEkxuffs220)0(u0)2/(SsffkhEl2lSElSuftanh)2/(20Average crack opening:Energy release rate:lSElGfSStan

12、h221200SXia and Hutchinson,JMPS 48,1107-1131(2000).Shear-lag lengthlSElGfSStanh220From the shear-lag model,sffkhEl2For thick substrates,compare the steady-state energy release rate with numerical solutions.For S :fffSSEhZElG2020),(2fhZl),(2For a thin substrate with the lower surface fixed:ssshkssffh

13、hEl2h aaaEG21Free-standing filmEffect of substrate constrainthEG22Film on elastic substratehahThe energy release rate increases if the crack penetrates into the substrate or if the film debond.Loss of constraintunderlayerCreeping or viscous underlayer:the constraint is gradually lost over time.Plast

14、ic underlayer:the substrate constraint is partially lost,and the driving force for channel cracks increases.Ratcheting underlayer:the constraint is gradually lost over thermal cycles,analogous to creep.Crack growth modulated by creepTensile FilmCreeping layerRigid substrateWhen t 0,4/121103.1)(ssfff

15、thhEtKFor a stationary long crack:225.0cssKHhEVSteady-state velocity:Huang,et al.,Acta Mater.50,4137(2002);Liang et al.,Experimental Mechanics 43,269-279(2003).Numerical simulation of crack growth00.511.522.5300.51K/Kc00.511.522.5300.511.5a/00.511.522.5300.20.40.6Vt0/t/t0Transient stateStationary cr

16、ack4/1)(DttKSteady state growth0tVStress contour as crack growsUniform mesh,200 by 20035 initial cracksSubcritical CrackingKcKthKVAfter 100 time steps1sinh220thKKVVKinetic law:On elastic substrate(no creep)Extended Finite Element Method(XFEM)Liang et al.,Int.J.Solids Struct.40,2343(2003).Measurement of film toughnessGenerate pre-cracks,e.g.,by a gentle scratching;Measure crack velocity under controlled stresses,using a bending fixture.Threshold toughness,GthToughness for fast fracture,GcSubcriti