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- 1. Introduction Experimental part DIC Data Usage Hyperelastic Viscoelasticity Cohesive zone Comparison Conclusions 1/31 Investigation of dynamic viscoelastic fracture of elastomers Vasudevan Kamasamudram, Michel Coret, Nicolas Moës GeM Laboratory, Centrale Nantes, CNRS, France February 21, 2021
- 2. Introduction Experimental part DIC Data Usage Hyperelastic Viscoelasticity Cohesive zone Comparison Conclusions 2/31 Overview - Fracture of elastomers Treloar(1944) Knauss (1970) Mueller and Knauss (1971) Rivlin, Thomas(1953) Stevenson, Thomas(1972) Gent, Marteny(1982) Tsunoda(2000) Lake(2000) Petersan(2004) Zhang(2009) Morishita(2016) Corre(2019) Willis (1965) Geubelle (1998) Graham, Walton (1995) Schapery (1975) Freund(1990) Marder(2005) Chen(2011) Dynamic fracture is when the crack speed is more than about 30% of the Shear wave speed, cs
- 3. Introduction Experimental part DIC Data Usage Hyperelastic Viscoelasticity Cohesive zone Comparison Conclusions 3/31 A small background Dynamic Cracks by LEFM Upper bound from LEFM is cR in Mode-I and cs in Mode-III Extended to include Transonic cracks in Mode-II - cs < v < cd Freund(1979) Transonic Mode-II cracks in experiments Rosakis(2000) Dynamic Cracks in elastomers Rupture of ballons - Stevenson et al(1972), membranes - Gent et al (1982), Lake (2000) Experiments on latex Rubber - Petersan et al (2004) establish Transonic Mode-I cracks Further studies by Marder (2006), Chen et al (2011) through lattice models
- 4. Introduction Experimental part DIC Data Usage Hyperelastic Viscoelasticity Cohesive zone Comparison Conclusions 4/31 Some Characteristics Observations by Chen et al(2011) Mode-I crack speed can exceed cs Crack speed dependence on Energy Release Rate, G = ψ(λ)h0, when v < cs λ, when v > cs
- 5. Introduction Experimental part DIC Data Usage Hyperelastic Viscoelasticity Cohesive zone Comparison Conclusions 5/31 Current study Goals of this study Examine the phenomenon for Polyurethane elastomers Check the behaviour of transonic cracks for different heights Essential ingredients for FE simulations for Transonic cracks Role of Local Hyper-elastic stiffening Role of Viscoelasticity and calibration of model Rate dependence in fracture process
- 6. Introduction Experimental part DIC Data Usage Hyperelastic Viscoelasticity Cohesive zone Comparison Conclusions 6/31 More Experiments Corre (2019) performs experiments on 40 mm geometry Transonic cracks observed in that study Additional experiments on 20 mm and 60 mm
- 7. Introduction Experimental part DIC Data Usage Hyperelastic Viscoelasticity Cohesive zone Comparison Conclusions 7/31 Experimental protocol Corre (2019) Specimen first stretched at 20 mm/min a to b Crack initiated with a razor blade at target stretch level b Crack propagation monitored by HS camera c a b c Blade
- 8. Introduction Experimental part DIC Data Usage Hyperelastic Viscoelasticity Cohesive zone Comparison Conclusions 8/31 Results Corre (2019) and Current study 1 1.5 2 2.5 3 3.5 4 4.5 5 0 20 40 60 80 Stretch Crack speed (m/s) Crack Speed vs Applied Stretch 20 mm 40 mm 60 mm cs
- 9. Introduction Experimental part DIC Data Usage Hyperelastic Viscoelasticity Cohesive zone Comparison Conclusions 9/31 Observations Crack speed < cs for λ <2.5 Crack speed > cs for λ >2.5 Large scatter in 20 mm and 60 mm geometries in current study - hence could not compare with Chen (2011) Sticking to results from Corre (2019) for the rest of the study
- 10. Introduction Experimental part DIC Data Usage Hyperelastic Viscoelasticity Cohesive zone Comparison Conclusions 10/31 Can a hyperelastic model with stiffening behaviour predict the Transonic cracks?
- 11. Introduction Experimental part DIC Data Usage Hyperelastic Viscoelasticity Cohesive zone Comparison Conclusions 11/31 Calibrate Hyperelastic model Hyperelastic model W = ∑ n i,j=0 Cij (I1 − 3)i (I2 − 3)j 1 2 3 4 5 6 0 2 4 6 8 ·106 Stretch Nominal Stress (P a) Uniaxial Case Experiment Polynomial Model 1 2 3 4 0 2 4 6 ·106 Stretch Nominal Stress (P a) Pure Shear Case Experiment Polynomial Model Result of moving closer to Uniaxial case - large lateral displacements (Experimental dispalcements in black) 1 2 3 4 5 6 0 2 4 6 8 ·106 Stretch Nominal Stress (P a) Uniaxial Case Experiment Ogden Model 1 2 3 4 0 2 4 6 ·106 Stretch Nominal Stress (P a) Pure Shear Case Experiment Ogden Model
- 12. Introduction Experimental part DIC Data Usage Hyperelastic Viscoelasticity Cohesive zone Comparison Conclusions 12/31 FE Simulations with data from DIC Making use of data during crack propagation Use the Hyperelastic model to perform FE simulations Extract displacement along crack faces to impose as bc in FE (Crack speed is an input) DIC fails along the edge - Displacements extracted on green line (1.5 mm from red line) Experiments FE Model
- 13. Introduction Experimental part DIC Data Usage Hyperelastic Viscoelasticity Cohesive zone Comparison Conclusions 13/31 Methodology First stretched to target level a to b Integrated in time with bcs from DIC c Mixed u/p method (C3D8H) used with HHT-α for time integration a b c
- 14. Introduction Experimental part DIC Data Usage Hyperelastic Viscoelasticity Cohesive zone Comparison Conclusions 14/31 Comparison with experiments Once for subsonic case and once for Transonic case Comparison of FE results with experiments
- 15. Introduction Experimental part DIC Data Usage Hyperelastic Viscoelasticity Cohesive zone Comparison Conclusions 15/31 Hyperelastic, λ = 1.7 Velocity Fields (m/s) - Experimental 0 2.69 5.38 8.07 10.76 13.4 Velocity Fields (m/s) - FE 0 2.23 4.47 6.7 8.93 11.17 13.4
- 16. Introduction Experimental part DIC Data Usage Hyperelastic Viscoelasticity Cohesive zone Comparison Conclusions 16/31 Hyperelastic, λ = 1.7 Horizontal Displacement (m) - Experimental −6.68 −5.11 −3.54 −1.97 −0.4 1.17 ·10−3 Horizontal Displacement (m) - FE −6.68 −5.11 −3.54 −1.97 −0.4 1.17 ·10−3 *
- 17. Introduction Experimental part DIC Data Usage Hyperelastic Viscoelasticity Cohesive zone Comparison Conclusions 17/31 Hyperelastic, λ = 3.5 Velocity Fields (m/s) - Experimental 0 20.5 41 61.5 82 102.5 123 Velocity Fields (m/s) - FE 0 20 40 60 80 100 120 140 160 180 0
- 18. Introduction Experimental part DIC Data Usage Hyperelastic Viscoelasticity Cohesive zone Comparison Conclusions 18/31 Hyperelastic, λ = 3.5 Horizontal Displacement (m) - Experimental −1.33 −1.15 −0.97 −0.8 −0.62 −0.44 −0.26 0.1 0.28 0.46 ·10−2 Horizontal Displacement (m) - FE −1.33 −1.15 −0.97 −0.8 −0.62 −0.44 −0.26 0.1 0.28 0.46 ·10−2
- 19. Introduction Experimental part DIC Data Usage Hyperelastic Viscoelasticity Cohesive zone Comparison Conclusions 19/31 Hyperelastic, observations For λ = 1.7, crack speed is 15.7 m/s < cs There is a good qualitative match between FE and Experiments For λ = 3.5, crack speed is 55 m/s > cs Difference between FE and Experiments Even with a stiffening Hyperelastic curve, ‘Shock front’ like pattern seen in FE - not seen in the experiments
- 20. Introduction Experimental part DIC Data Usage Hyperelastic Viscoelasticity Cohesive zone Comparison Conclusions 20/31 Including Viscoelasticity Hyperelasticity alone seems insufficient Begin with Finite Linear Viscoelasticity Simo 1987 S = −JpC−1 + J− 2 3 DEV{∫ g(t − s) ∂ ∂s DEV{ ∂ψ̄0 ∂C̄ }} g(t) = g∞ + N ∑ i=1 gie − t τi g∞ and gi s are stiffness ratios and τi s are relaxation times
- 21. Introduction Experimental part DIC Data Usage Hyperelastic Viscoelasticity Cohesive zone Comparison Conclusions 21/31 Calibration Calibrate the model using Dynamic Mechanical Analysis (DMA) and crack experiments (data from DIC) - Rich data with complex strains and strain rates Considered the reaction behind crack while calibrating Relaxation times obtained are 10−1 sec,10−2 sec, 10−3 sec, 10−4 sec, 10−5 sec, 10−6 sec and 10−7 sec
- 22. Introduction Experimental part DIC Data Usage Hyperelastic Viscoelasticity Cohesive zone Comparison Conclusions 22/31 Repeat the simulations from previous section with Viscoelasticity
- 23. Introduction Experimental part DIC Data Usage Hyperelastic Viscoelasticity Cohesive zone Comparison Conclusions 23/31 Viscoelastic, λ = 3.5 Horizontal Displacement (m) - Experimental −1.33 −1.15 −0.97 −0.8 −0.62 −0.44 −0.26 0.1 0.28 0.46 ·10−2 Horizontal Displacement (m) - FE −1.33 −1.15 −0.97 −0.8 −0.62 −0.44 −0.26 0.1 0.28 0.46 ·10−2 *
- 24. Introduction Experimental part DIC Data Usage Hyperelastic Viscoelasticity Cohesive zone Comparison Conclusions 24/31 Viscoelastic, λ = 3.5 Velocity Fields (m/s) - Experimental 0 20.5 41 61.5 82 102.5 123 Velocity Fields (m/s) - FE 0 20.5 41 61.5 82 102.5 123 *
- 25. Introduction Experimental part DIC Data Usage Hyperelastic Viscoelasticity Cohesive zone Comparison Conclusions 25/31 Comparison Hyperelastic/Viscoelastic, λ = 3.5 σyy (Pa) - Hyperelastic −0.3 8.33 · 10−2 0.47 0.85 1.23 1.62 2 ·107 * σyy (Pa) - viscoelastic −0.3 8.33 · 10−2 0.47 0.85 1.23 1.62 2 ·107 *
- 26. Introduction Experimental part DIC Data Usage Hyperelastic Viscoelasticity Cohesive zone Comparison Conclusions 26/31 Cohesive zone Previous sections used crack speed as input This section uses Hyper and Viscoelastic models from previous sections Use Cohesive Zone to predict crack speeds - path is known
- 27. Introduction Experimental part DIC Data Usage Hyperelastic Viscoelasticity Cohesive zone Comparison Conclusions 27/31 Cohesive zone Rate independent case - crack speeds were higher Inclusion of rate dependence - create a link between bulk and cohesive models Knauss and Losi (1993), Rahulkumar et al (2000) t = (1 − d)[∫ t −∞ g(t − s) ∂K[u] ∂s ds] Parameters t0 and δc determined from initiation tests Plane stress condition has been assumed (CPS4 Element)
- 28. Introduction Experimental part DIC Data Usage Hyperelastic Viscoelasticity Cohesive zone Comparison Conclusions 28/31 Comparison between experiments and simulations 1.4 2 2.5 3 3.5 4 4.2 0 20 40 60 80 Stretch Crack speed (m/s) Crack Speed vs Applied Stretch Experimental Rate independent
- 29. Introduction Experimental part DIC Data Usage Hyperelastic Viscoelasticity Cohesive zone Comparison Conclusions 29/31 Comparison between experiments and simulations 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3 3.2 3.4 3.6 3.8 4 4.2 0 10 20 30 40 50 60 Stretch Crack speed (m/s) Crack Speed vs Applied Stretch 40 mm Numerical 40 mm Experimental
- 30. Introduction Experimental part DIC Data Usage Hyperelastic Viscoelasticity Cohesive zone Comparison Conclusions 30/31 Conclusions Viscoelastic model calibrated using DIC data from crack experiments Able to go transonic with a Viscoelastic model Able to match experimental trend for 40 mm geometry Rate dependent cohesive zone linked to bulk Viscoelastic model