CNRS Université Pierre et Marie Curie Paris From microstructural to macroscopic properties in failure of brittle heterogeneous materials s 0 s 0 Youngs modulus E eff ID: 1048552
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1. Laurent PonsonInstitut Jean le Rond d’AlembertCNRS – Université Pierre et Marie Curie, ParisFrom microstructural to macroscopic properties in failureof brittle heterogeneous materials
2. s0s0Young’s modulus:Eeff average (Elocal)Fracture energy:Gceff average (Gclocal)XPredicting the effective toughness of heterogeneous systems:A challenging multi-scale problem
3. s0s0Predicting the effective toughness of heterogeneous systems:A challenging multi-scale problems (r)rStress field diverges at the crack tip
4. s0s0Predicting the effective toughness of heterogeneous systems:A challenging multi-scale problems (r)rMacroscopic failure properties strongly dependent on material heterogeneitiesStress field diverges at the crack tip
5. s0s0Macroscopic failure properties strongly dependent on material heterogeneitiesOpens the door to microstructure design in order to achieve improved failure propertiesPredicting the effective toughness of heterogeneous systems:A challenging multi-scale problems (r)rStress field diverges at the crack tip
6. Application: Asymmetric adhesivesS. Xia, L. Ponson, G. Ravichandran and K. Bhattacharya, Phys. Rev. Lett. 2012 and International Patent 2011Hard directionEasy direction
7. Goal: Developing a theoretical framework that predicts the effective resistance of heterogeneous brittle systems Using it for designing systems with improved failure properties1- Theoretical approach: Equation of motion for a crack in an heterogeneous materialFailure as a depinning transition2- Confrontation with experiments in the case of materials with disordered microstructuresEffective fracture energy of disordered materials3- Application to material design in the context of thin film adhesivesEnhancement and asymmetry of peeling strengthApproach & outline:
8. What are the effects of heterogeneities on the propagation of a crack?zxGext1. Theory: deriving the equation of motion of a crack
9. What are the effects of heterogeneities on the propagation of a crack?Pinning of the crack front:zxGext1. Theory: deriving the equation of motion of a crack
10. Real material = GC(M) = <GC> + δGc(M)HomogeneousmaterialHypothesis:-Brittle material-Quasi-static crack propagationzxGexty1. Theory: deriving the equation of motion of a crackFracture energy fluctuations+
11. Real material = GC(M) = <GC> + δGc(M)Fracture energy fluctuationsHomogeneousmaterial+Hypothesis:-Brittle material-Quasi-static crack propagationzxGexty1. Theory: deriving the equation of motion of a crackRandom quenched noise with amplitude σGcFor disordered materials
12. Elasticity of the material Crack front as an elastic line: Real material = GC(M) = <GC> + δGc(M)Homogeneousmaterialzxf(z,t)MyGextJ. Rice (1985)1. Theory: deriving the equation of motion of a crackFracture energy fluctuations+
13. Elasticity of the material Crack front as an elastic line: Real material = GC(M) = <GC> + δGc(M)HomogeneousmaterialEquation of motion for a crack zxMyGextL. B. Freund (1990)J. Rice (1985)1. Theory: deriving the equation of motion of a crackf(z,t)Fracture energy fluctuations+
14. Elasticity of the material Crack front as an elastic line: Real material = GC(M) = <GC> + δGc(M)HomogeneousmaterialEquation of motion for a crack zxMyGextJ. Rice (1985)1. Theory: deriving the equation of motion of a crackJ. Schmittbuhl et al. 1995, D. Bonamy et al. 2008, L. Ponson et al. 2010f(z,t)Fracture energy fluctuations+
15. Elasticity of the material Crack front as an elastic line: Real material = GC(M) = <GC> + δGc(M)HomogeneousmaterialEquation of motion for a crack zxMyGextJ. Rice (1985)1. Theory: deriving the equation of motion of a crackCrack propagation as an elastic interface driven in a heterogeneous planef(z,t)J. Schmittbuhl et al. 1995, D. Bonamy et al. 2008, L. Ponson et al. 2010Fracture energy fluctuations+
16. Predictions on the dynamics of cracksGextGextVariations of the average crack velocity with the external driving forceVcrack1. Theory: deriving the equation of motion of a crackFor disordered materials
17. Predictions on the dynamics of cracksGextGextVariations of the average crack velocity with the external driving forceStablePropagatingToughening effectEffective fracture energy:Vcrack1. Theory: deriving the equation of motion of a crack
18. Predictions on the dynamics of cracksGextGextVariations of the average crack velocity with the external driving forceStablePropagatingToughening effectEffective fracture energy:Power law variation of the crack velocityCrack velocity:Vcrack1. Theory: deriving the equation of motion of a crack
19. Predictions on the dynamics of cracksGextGextVariations of the average crack velocity with the external driving forceStablePropagatingToughening effectEffective fracture energy:Power law variation of the crack velocityCrack velocity:Fluctuations of velocityIntermittent dynamics of cracksPower law distributed fluctuations of velocityVcrack1. Theory: deriving the equation of motion of a crack
20. Variations of the average crack velocity with the external driving force2. Confrontation with experiments on disordered materialsConfrontation with experimental observationsFracture test of a disordered brittle rockL. Ponson, Phys. Rev. Lett. 2009
21. Variations of the average crack velocity with the external driving force2. Confrontation with experiments on disordered materialsConfrontation with experimental observationsFracture test of a disordered brittle rockL. Ponson, Phys. Rev. Lett. 2009Critical regime
22. Variations of the average crack velocity with the external driving forceL. Ponson, Phys. Rev. Lett. 20092. Confrontation with experiments on disordered materialsConfrontation with experimental observationsFracture test of a disordered brittle rockSubcritical regime(thermally activated)Critical regime
23. Variations of crack velocity as a function of timeFracture test of a disordered brittle rock2. Confrontation with experiments on disordered materialsConfrontation with experimental observationsDéfinition of the size S of a fluctuationFluctuations of velocityL. Ponson, Phys. Rev. Lett. 2009Subcritical regime(thermally activated)Critical regimeVariations of the average crack velocity with the external driving forceD. Bonamy, S. Santucci and L. Ponson, Phys. Rev. Lett. 2008
24. Variations of crack velocity as a function of timeFracture test of a disordered brittle rock2. Confrontation with experiments on disordered materialsConfrontation with experimental observationsDéfinition of the size S of a fluctuationFluctuations of velocityL. Ponson, Phys. Rev. Lett. 2009Subcritical regime(thermally activated)Critical regimeVariations of the average crack velocity with the external driving forceD. Bonamy, S. Santucci and L. Ponson, Phys. Rev. Lett. 2008
25. Variations of crack velocity as a function of timeFracture test of a disordered brittle rock2. Confrontation with experiments on disordered materialsConfrontation with experimental observationsDéfinition of the size S of a fluctuationFluctuations of velocityL. Ponson, Phys. Rev. Lett. 2009Subcritical regime(thermally activated)Critical regimeVariations of the average crack velocity with the external driving forceDistribution of fluctuation sizesExperimental results, Maloy, Santucci et al.Theoretical predictionsP(S) ~ S-with ~ 1.65D. Bonamy, S. Santucci and L. Ponson, Phys. Rev. Lett. 2008
26. Variations of crack velocity as a function of timeFracture test of a disordered brittle rock2. Confrontation with experiments on disordered materialsConfrontation with experimental observationsDéfinition of the size S of a fluctuationFluctuations of velocityL. Ponson, Phys. Rev. Lett. 2009Subcritical regime(thermally activated)Critical regimeVariations of the average crack velocity with the external driving forceD. Bonamy, S. Santucci and L. Ponson, Phys. Rev. Lett. 2008Distribution of fluctuation sizesExperimental results, Maloy, Santucci et al.Theoretical predictionsP(S) ~ S-with ~ 1.65Failure of disordered brittle solids as a depinning transition
27. Application: Effective fracture energy of disordered solidsPropagation direction2. Effective fracture energy of disordered materialsEquation of motion of the crackFracture energy randomly distributed with standard deviation σGc
28. Application: Effective fracture energy of disordered solidsPropagation direction2. Effective fracture energy of disordered materialsEquation of motion of the crackEffective fracture energy given by the depinning thresholdFracture energy randomly distributed with standard deviation σGcEffect of disorder strength σGc? Of its distribution (Gaussian, bivalued…)?z
29. 2. Effective fracture energy of disordered materialsTheory: A simplified linear model inspired by LarkinPropagation directionA. Larkin and Y. Ovchinnikov (1979)
30. 2. Effective fracture energy of disordered materialsTheory: A simplified linear model inspired by LarkinValidity range: so that Propagation directionA. Larkin and Y. Ovchinnikov (1979)Front geometryLarkin lengthzz
31. 2. Effective fracture energy of disordered materialsTheory: A simplified linear model inspired by LarkinValidity range: so that Typical resistance felt by a domain of size L if L<ξ if L<ξPropagation directionA. Larkin and Y. Ovchinnikov (1979)zzFront geometryLarkin length
32. 2. Effective fracture energy of disordered materialsTheory: A simplified linear model inspired by LarkinValidity range: so that Typical resistance felt by a domain of size L if L<ξ if L<ξLarkin argument: critical depinning force set by the Larkin domains Effective fracture energy ifIndividual pinningCollective pinningPropagation direction ifA. Larkin and Y. Ovchinnikov (1979)zzFront geometryLarkin length
33. 2. Effective fracture energy of disordered materialsSimulations: collective vs individual pinningV. Démery, A. Rosso and L. Ponson (2013)Collective pinningIndividual pinningFollows theoretical predictionsDepends on σGc onlyDepends on more parameters (strongest impurities)
34. 2. Effective fracture energy of disordered materialsSimulations: collective vs individual pinningV. Démery, A. Rosso and L. Ponson (2013)Collective pinningIndividual pinningFollows theoretical predictionsDepends on σGc onlyDepends on more parameters (strongest impurities)Disordered induced toughening relevant for the design of stronger solids
35. Peeling of heterogeneous adhesivesMf(z)zxEquation of motion of the peeling front3. Toughening and asymmetry in peeling of heterogeneous adhesivesFpVan Karman plate theoryLocal driving force:L. Ponson et al. (2013)
36. Peeling of heterogeneous adhesivesMf(z)zxEquation of motion of the peeling frontFpVan Karman plate theoryLocal driving force:External driving force:Displacement controlledHypothesis Quasi-static propagation Weakly heterogeneous Brittle system3. Toughening and asymmetry in peeling of heterogeneous adhesivesL. Ponson et al. (2013)
37. Peeling of heterogeneous adhesivesMf(z)zxEquation of motion of the peeling frontFpVan Karman plate theoryLocal driving force:External driving force:Displacement controlledLocal field of resistance:if M belongs to a pinning siteelsewhere3. Toughening and asymmetry in peeling of heterogeneous adhesivesL. Ponson et al. (2013)
38. Peeling of heterogeneous adhesivesMf(z)zxEquation of motion of the peeling frontFpVan Karman plate theoryLocal driving force:External driving force:Displacement controlledLocal field of resistance:if M belongs to a pinning siteelsewhereSimilar to crack fronts in 3D elastic solidsGextJ. Rice (1985)3. Toughening and asymmetry in peeling of heterogeneous adhesivesL. Ponson et al. (2013)
39. zzxxδf/dDeformation of the frontTheoretical predictions ContrastΔGc/Gc0 1.251.000.750.500.25δfExperiments on single defects: test of the approach3. Toughening and asymmetry in peeling of heterogeneous adhesives
40. zzxxδfExperiments on single defects: test of the approach3. Toughening and asymmetry in peeling of heterogeneous adhesivesComparison with experiments Δf/dδf (μm)z (mm)
41. From the local field of fracture energy …1. Theory: deriving an equation of motion for a peeling front
42. … to the effective adhesion properties1. Theory: deriving an equation of motion for a peeling frontMf(z)zxPeeling force G per unit length (N/m)Average position of the peeling front (mm)Peeling force G per unit length (N/m)Average position of the peeling front (mm)Effective peeling strength Gmax
43. … to the effective adhesion properties1. Theory: deriving an equation of motion for a peeling frontMf(z)zxPeeling force G per unit length (N/m)Average position of the peeling front (mm)Peeling force G per unit length (N/m)Average position of the peeling front (mm)GmaxGmaxeasyGmaxhardAverage position of the peeling front (mm)Peeling force G per unit length (N/m)Mf(z)zxEasydirectionHard directionStrength asymmetry
44. 2. Confrontation with experiments on a model heterogeneous adhesiveAdhesive:PDMS thin film produced by spin coatingSubstrate:Transparent sheet printed with a standard printerAdhesion energy:PDMS-inkPDMS-transparent sheetGc1 = 12 J.m-2Gc2 = 4 J.m-2Contrast:Gc1/Gc0 ≈ 3Thickness between100µm and 3mmA model system for heterogeneous adhesionLocal field Gc(M) of local adhesion energy perfectly controled and known
45. 2. Confrontation with experiments on a model heterogeneous adhesiveAsymmetric adhesivesS. Xia, L. Ponson, G. Ravichandran and K. Bhattacharya, Phys. Rev. Lett. 2012 and international patent 2011Hard directionEasy direction
46. Asymmetric adhesivesHard directionEasy directionOptimization of the asymmetry by changing shape and contrast of pinning sites2. Confrontation with experiments on a model heterogeneous adhesiveS. Xia, L. Ponson, G. Ravichandran and K. Bhattacharya, Phys. Rev. Lett. 2012 and international patent 2011
47. 3. Optimization and design of adhesivesAlgorithm predicting Gmax from the local field Gc(x,z) Genetic algorithmA. Rosso and W. Krauth, PRE 2001BIANCA algorithm, Vincenti et al., J. Glob. Opt. 2010 Elementary cell Lz x LxDefect Front in the difficult direction Front in the easy directionOptimization procedure
48. Optimization resultThin defects with U shapeGc1 as upper bound of GhardGc0 as lower bound of Geasy3. Optimization and design of adhesivesAsymmetry ≤ Gc1/Gc0
49. Parametric study3. Optimization and design of adhesivesDefect shapez/df/dGc1Gc0Equilibrium shape of a front crossing a stripe with larger adhesion energy M. Vasoya, J.B. Leblond and L. Ponson IJSS 2012y/d
50. Parametric study3. Optimization and design of adhesivesDefect shapef/dGc1Gc0Equilibrium shape of a front crossing a stripe with larger adhesion energy M. Vasoya, J.B. Leblond and L. Ponson IJSS 2013Contrast C = Gc1/Gc0Defect width d/Ly normalized by the cell widthAsymmetryy/d
51. Beyond asymmetry:how achieving enhanced peel strength
52. Beyond asymmetry:how achieving enhanced peel strengthHeterogeneities of adhesion energyEffective peeling strength bonded by the max of the local GcHeterogeneities of elastic stiffness
53. Adhesives with elastic heterogeneities: experimental study
54. Adhesives with elastic heterogeneities: experimental study
55. Dramatic increase of the effective peeling forceAdhesives with elastic heterogeneities: experimental study
56. bPeeling mechanism: homogeneous tapeBending stiffness: D = EI with moment of inertia: I = bh3/12
57. Peeling mechanism: homogeneous tapebBending stiffness: D = EI with moment of inertia: I = bh3/12
58. Peeling mechanism: homogeneous tapebBending stiffness: D = EI with moment of inertia: I = bh3/12
59. WF0 = ∆Es + ∆EelDuring the peeling process, for a propagation over Δc:Peeling mechanism: homogeneous tapebBending stiffness: D = EI with moment of inertia: I = bh3/12
60. F0 Δc (1-cosθ0) Gc bΔcXPeeling mechanism: homogeneous tapebWF0 = ∆Es + ∆EelDuring the peeling process, for a propagation over Δc:Bending stiffness: D = EI with moment of inertia: I = bh3/12
61. F0 Δc (1-cosθ0) Gc bΔcXPeeling mechanism: homogeneous tapebWF0 = ∆Es + ∆EelDuring the peeling process, for a propagation over Δc:Fc = bGc/(1-cosθ0)Peeling forceR. S. Rivlin 1944Bending stiffness: D = EI with moment of inertia: I = bh3/12
62. Simplest system that could give rise to tougheningOne interface + inextensible tapeToughening mechanism: heterogeneous tape
63. Simplest system that could give rise to tougheningOne interface + inextensible tapeToughening mechanism: heterogeneous tape
64. Simplest system that could give rise to tougheningOne interface + inextensible tapeToughening mechanism: heterogeneous tape
65. WF0 = ∆Es + ∆EelDuring the peeling process, for a propagation over Δc:Simplest system that could give rise to tougheningOne interface + inextensible tapeToughening mechanism: heterogeneous tape
66. Variation of the bending energyEuler-Bernoulli beam theorySimplest system that could give rise to tougheningOne interface + inextensible tapeToughening mechanism: heterogeneous tapeWF0 = ∆Es + ∆EelDuring the peeling process, for a propagation over Δc:
67. Simplest system that could give rise to tougheningOne interface + inextensible tapeToughening mechanism: heterogeneous tapeWF0 = ∆Es + ∆EelDuring the peeling process, for a propagation over Δc:Euler-Bernoulli beam theory
68. Simplest system that could give rise to tougheningOne interface + inextensible tapeToughening mechanism: heterogeneous tapeWF0 = ∆Es + ∆EelDuring the peeling process, for a propagation over Δc:Euler-Bernoulli beam theoryh1/h2 = 2Fhet /Fhom ≈ 8
69. Adhesives with stripes of alternated stiffnessAdhesive described as a beam with alternating stiffness/bending rigidityWork of the peel force used to bend the stiffer domainsDriven away from the peel frontS. Xia, L. Ponson, G. Ravichandran and K. Bhattacharya, J. Mech. Phys. Solids (2013)For d>λbd
70. ConclusionsApplication: design of adhesives with new and improved propertiesEffectives adhesion properties of thin filmsSpatial distribution of heterogeneities at the microscale
71. AcknowledgementsTo my collaborators and studentTo my sources of fundingMarie Curie fellowship (FP7 of European Union)Integration grant (FP7 of European Union)Shuman Xia, Guruswami Ravichandran, Kaushik Bhattacharya (Caltech)Alberto Rosso (ENS, Paris)Vincent Demery (Post-doc)
72. …and perspectivesToughening in 3D brittle solidsExtension of this theoretical framework to quasi-brittle solidsInduced by collective pinning for materials with a disordered microstructureEffective resistance as a function of δGcInduced by elastic heterogeneities, inspired by the adhesion enhancement mechanisms
73. Direction ofpropagationK. Måløy et al. PRL 2006, D. Bonamy, S. Santucci and L. Ponson PRL 2008Intermittent crack dynamicsInteraction between a crack front and material heterogeneties: dramatatic effects at all scalesLarge scale roughness on fracture surfacesSandstoneL. Ponson et al. PRE 07Silica glass(courtesy of M. Ciccotti et al.).Scale of heterogeneities in sandstone
74. The crack tip as a magnifying glass of the material heterogeneitiess0s0
75. s (r)rThe crack tip as a magnifying glass of the material heterogeneitiesStress fied diverges at the crack tips0s0Macroscopic response depends strongly on the material properties at the microstructure scale
76. s (r)rStress fied diverges at the crack tips0s0Macroscopic response depends strongly on the material properties at the microstructure scaleThe crack tip as a magnifying glass of the material heterogeneitiesOpens the door to microstructure design in order to achieve improved failure properties
77. What are the effects of heterogeneities on the propagation of a crack?zxGext1. Theory: deriving an equation of motion for a crack or a peeling front
78. What are the effects of heterogeneities on the propagation of a crack?Pinning of the crack front:zxGext1. Theory: deriving an equation of motion for a crack or a peeling front
79. Resistance of a real material GC(M) = <GC> + δGc(M)Fluctuating partAverage resistance+Hypothesis Brittle material Quasi-static propagation Weakly heterogeneouszxGextMywith <δGc>=01. Theory: deriving an equation of motion for a crack or a peeling front=.
80. Elasticity of the material Crack front as an elastic line: zxf(z,t)MyGextJ. R. Rice (1985)Resistance of a real materialFluctuating partAverage resistance+= GC(M) = <GC> + δGc(M)with <δGc>=0.1. Theory: deriving an equation of motion for a crack or a peeling front
81. Elasticity of the material Crack front as an elastic line: Equation of motion for a crack zxMyGextL. B. Freund (1990)J. R. Rice (1985) GC(M) = <GC> + δGc(M)Resistance of a real materialFluctuating partAverage resistance+=f(z,t)with <δGc>=0.1. Theory: deriving an equation of motion for a crack or a peeling front
82. Elasticity of the material Crack front as an elastic line: Equation of motion for a crack zxMyGextJ. R. Rice (1985)J. Schmittbuhl et al. PRL 1995, L. Ponson PRL 2009, L. Ponson et al., IJF 2010 GC(M) = <GC> + δGc(M)Resistance of a real materialFluctuating partAverage resistance+=f(z,t)with <δGc>=0.1. Theory: deriving an equation of motion for a crack or a peeling front
83. Peeling of an PDMS thin film from a printed heterogeneous substrate2. Experiments: peeling of thin films with controlled heterogeneitiesExperimental setupPerturbation of the front: comparison theory/experimenth=1.2mmGcPDMS-ink = 1.4 J.m-2GcPDMS-transparent = 6 J.m-2
84. Experimental study of the effect of elastic heterogeneities3. Application to the design of adhesives with improved properties
85. Experimental study of the effect of elastic heterogeneities3. Application to the design of adhesives with improved properties
86. Experimental study of the effect of elastic heterogeneitiesDramatic increase of the effective peeling force3. Application to the design of adhesives with improved properties
87. Bending stiffness: EI Moment of inertia: I = bh3/12Peeling mechanisms: homogeneous tapeb3. Application to the design of adhesives with improved properties
88. Bending stiffness: EI Moment of inertia: I = bh3/12Peeling mechanisms: homogeneous tape3. Application to the design of adhesives with improved properties
89. Bending stiffness: EI Moment of inertia: I = bh3/12Peeling mechanisms: homogeneous tape3. Application to the design of adhesives with improved properties
90. During the peeling process, for a propagation over Δc:Bending stiffness: EI Moment of inertia: I = bh3/12R. S. Rivlin 1944Peeling mechanisms: homogeneous tapeWF0 = ∆Es + ∆Eel3. Application to the design of adhesives with improved properties
91. WF0 = ∆Es + ∆EelF0 Δc (1-cosθ0) Gc bΔcXBending stiffness: EI Moment of inertia: I = bh3/12Peeling mechanisms: homogeneous tapeDuring the peeling process, for a propagation over Δc:3. Application to the design of adhesives with improved properties
92. WF0 = ∆Es + ∆EelF0 Δc (1-cosθ0) Gc bΔcXBending stiffness: EI Moment of inertia: I = bh3/12R. S. Rivlin 1944Fc = bGc/(1-cosθ0)Peeling forcePeeling mechanisms: homogeneous tapeDuring the peeling process, for a propagation over Δc:3. Application to the design of adhesives with improved properties
93. Simplest system susceptible to give rise to tougheningOne interface + inextensiblePeeling mechanisms: heterogeneous tape3. Application to the design of adhesives with improved properties
94. Simplest system susceptible to give rise to tougheningOne interface + inextensiblePeeling mechanisms: heterogeneous tape3. Application to the design of adhesives with improved properties
95. Simplest system susceptible to give rise to tougheningOne interface + inextensiblePeeling mechanisms: heterogeneous tape3. Application to the design of adhesives with improved properties
96. Simplest system susceptible to give rise to tougheningOne interface + inextensibleWF0 = ∆Es + ∆EelDuring the peeling process, for a propagation over Δc:Peeling mechanisms: heterogeneous tape3. Application to the design of adhesives with improved properties
97. Simplest system susceptible to give rise to tougheningOne interface + inextensibleVariation of the bending energyEuler-Bernoulli beam theoryWF0 = ∆Es + ∆EelDuring the peeling process, for a propagation over Δc:Peeling mechanisms: heterogeneous tape3. Application to the design of adhesives with improved properties
98. Simplest system susceptible to give rise to tougheningOne interface + inextensibleVariation of the bending energyEuler-Bernoulli beam theoryWF0 = ∆Es + ∆EelDuring the peeling process, for a propagation over Δc:Peeling mechanisms: heterogeneous tape3. Application to the design of adhesives with improved properties
99. Simplest system susceptible to give rise to tougheningOne interface + inextensibleVariation of the bending energyEuler-Bernoulli beam theoryWF0 = ∆Es + ∆EelDuring the peeling process, for a propagation over Δc:h1/h2 = 2Fhet /Fhom = 8Peeling mechanisms: heterogeneous tapeS. Xia, L. Ponson, G. Ravichandran and K. Bhattacharya (Submitted)3. Application to the design of adhesives with improved properties
100. Toughening mechanism: numerical investigationS. Xia, L. Ponson, G. Ravichandran and K. BhattacharyaUS and international patent application 61/290, 133 (2010)Finite element simulations with cohesive zone model3. Application to the design of adhesives with improved properties
101. Application: Effective fracture energy of disordered solidsPropagation direction2. Effective fracture energy of disordered materialsNormalized equation of motion of the crackEffective fracture energy given by the depinning thresholdFracture energy fluctuationsdistributed in P(gc)Effect of disorder strength σ? Of its distribution P (Gaussian, bivalued…)?z
102. 2. Effective fracture energy of disordered materialsA simplified (linear) modelFront geometryValidity range: so that Larkin lengthTypical resistance felt by a domain of size L given by if L<ξ if L<ξLarkin argument: critical depinning force set by the Larkin domains Effective fracture energy if LLarkin<ξ if LLarkin<ξIndividual pinningCollective pinningPropagation direction