2006 Numerical simulation of nonlinear elastic wave propagation in piecewise homogeneous media

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  • 1. Numerical simulation of nonlinear elastic wave propagation in piecewise homogeneous media Arkadi Berezovski, Mihhail Berezovski, Jüri Engelbrecht Centre for Nonlinear Studies, Institute of Cybernetics at Tallinn University of Technology, Akadeemia tee 21, 12618 Tallinn, Estonia Arkadi Berezovski, Mihhail Berezovski, Jüri Engelbrecht Numerical simulation of nonlinear elastic wave propagation - p. 1/28
  • 2. Outline q Outline s Motivation: experiments and theory Motivation Formulation of the problem Wave-propagation algorithm Comparison with experimental data Discussion Arkadi Berezovski, Mihhail Berezovski, Jüri Engelbrecht Numerical simulation of nonlinear elastic wave propagation - p. 2/28
  • 3. Outline q Outline s Motivation: experiments and theory Motivation s Formulation of the problem Formulation of the problem Wave-propagation algorithm Comparison with experimental data Discussion Arkadi Berezovski, Mihhail Berezovski, Jüri Engelbrecht Numerical simulation of nonlinear elastic wave propagation - p. 2/28
  • 4. Outline q Outline s Motivation: experiments and theory Motivation s Formulation of the problem Formulation of the problem Wave-propagation algorithm s Wave-propagation algorithm Comparison with experimental data Discussion Arkadi Berezovski, Mihhail Berezovski, Jüri Engelbrecht Numerical simulation of nonlinear elastic wave propagation - p. 2/28
  • 5. Outline q Outline s Motivation: experiments and theory Motivation s Formulation of the problem Formulation of the problem Wave-propagation algorithm s Wave-propagation algorithm Comparison with experimental s Comparison with experimental data data Discussion Arkadi Berezovski, Mihhail Berezovski, Jüri Engelbrecht Numerical simulation of nonlinear elastic wave propagation - p. 2/28
  • 6. Outline q Outline s Motivation: experiments and theory Motivation s Formulation of the problem Formulation of the problem Wave-propagation algorithm s Wave-propagation algorithm Comparison with experimental s Comparison with experimental data data s Conclusions Discussion Arkadi Berezovski, Mihhail Berezovski, Jüri Engelbrecht Numerical simulation of nonlinear elastic wave propagation - p. 2/28
  • 7. Experiments by Zhuang et al. (2003) q Outline Motivation q Experiments by Zhuang et al. (2003) q Time history of shock stress q Time history of shock stress q Theory by Chen et al. (2004) q Time history of shock stress Formulation of the problem Wave-propagation algorithm Comparison with experimental data Discussion (Original source: Zhuang, S., Ravichandran, G., Grady D., 2003. An experimental investigation of shock wave propagation in periodically layered composites. J. Mech. Phys. Solids 51, 245–265.) Arkadi Berezovski, Mihhail Berezovski, Jüri Engelbrecht Numerical simulation of nonlinear elastic wave propagation - p. 3/28
  • 8. Time history of shock stress 3.5 experiment q Outline 3 Motivation q Experiments by Zhuang et al. (2003) 2.5 q Time history of shock stress q Time history of shock stress Stress (GPa) q Theory by Chen et al. (2004) 2 q Time history of shock stress Formulation of the problem 1.5 Wave-propagation algorithm Comparison with experimental 1 data Discussion 0.5 0 1 1.5 2 2.5 3 3.5 4 Time (microseconds) Experiment 112301 (Zhuang, S., Ravichandran, G., Grady D., 2003.) Composite: 16 units of polycarbonate, each 0.37 mm thick, and 16 units of D-263 glass, each 0.20 mm thick. Flyer velocity 1079 m/s and flyer thickness 2.87 mm. Gage position: 3.41 mm from impact boundary. Arkadi Berezovski, Mihhail Berezovski, Jüri Engelbrecht Numerical simulation of nonlinear elastic wave propagation - p. 4/28
  • 9. Time history of shock stress 3.5 experiment simulation - linear q Outline 3 Motivation q Experiments by Zhuang et al. (2003) 2.5 q Time history of shock stress q Time history of shock stress Stress (GPa) q Theory by Chen et al. (2004) 2 q Time history of shock stress Formulation of the problem 1.5 Wave-propagation algorithm Comparison with experimental 1 data Discussion 0.5 0 1 1.5 2 2.5 3 3.5 4 Time (microseconds) Experiment 112301 (Zhuang, S., Ravichandran, G., Grady D., 2003.) Composite: 16 units of polycarbonate, each 0.37 mm thick, and 16 units of D-263 glass, each 0.20 mm thick. Flyer velocity 1079 m/s and flyer thickness 2.87 mm. Gage position: 3.41 mm from impact boundary. Arkadi Berezovski, Mihhail Berezovski, Jüri Engelbrecht Numerical simulation of nonlinear elastic wave propagation - p. 5/28
  • 10. Theory by Chen et al. (2004) q Outline s Analytical solution of one-dimensional linear wave Motivation q Experiments by Zhuang et al. propagation in layered heterogeneous materials (2003) q Time history of shock stress q Time history of shock stress q Theory by Chen et al. (2004) q Time history of shock stress Formulation of the problem Wave-propagation algorithm Comparison with experimental data Discussion Arkadi Berezovski, Mihhail Berezovski, Jüri Engelbrecht Numerical simulation of nonlinear elastic wave propagation - p. 6/28
  • 11. Theory by Chen et al. (2004) q Outline s Analytical solution of one-dimensional linear wave Motivation q Experiments by Zhuang et al. propagation in layered heterogeneous materials (2003) q Time history of shock stress s Approximate solution for shock loading by invoking of q Time history of shock stress q Theory by Chen et al. (2004) equation of state q Time history of shock stress Formulation of the problem Wave-propagation algorithm Comparison with experimental data Discussion Arkadi Berezovski, Mihhail Berezovski, Jüri Engelbrecht Numerical simulation of nonlinear elastic wave propagation - p. 6/28
  • 12. Theory by Chen et al. (2004) q Outline s Analytical solution of one-dimensional linear wave Motivation q Experiments by Zhuang et al. propagation in layered heterogeneous materials (2003) q Time history of shock stress s Approximate solution for shock loading by invoking of q Time history of shock stress q Theory by Chen et al. (2004) equation of state q Time history of shock stress s Wave velocity, thickness and density for the laminates Formulation of the problem subjected to shock loading, all depend on the particle Wave-propagation algorithm velocity Comparison with experimental data Discussion Arkadi Berezovski, Mihhail Berezovski, Jüri Engelbrecht Numerical simulation of nonlinear elastic wave propagation - p. 6/28
  • 13. Theory by Chen et al. (2004) q Outline s Analytical solution of one-dimensional linear wave Motivation q Experiments by Zhuang et al. propagation in layered heterogeneous materials (2003) q Time history of shock stress s Approximate solution for shock loading by invoking of q Time history of shock stress q Theory by Chen et al. (2004) equation of state q Time history of shock stress s Wave velocity, thickness and density for the laminates Formulation of the problem subjected to shock loading, all depend on the particle Wave-propagation algorithm velocity Comparison with experimental s Chen, X., Chandra, N., Rajendran, A.M., 2004. Analytical solution to the plate impact data problem of layered heterogeneous material systems Int. J. Solids Struct. 41, Discussion 4635–4659. Arkadi Berezovski, Mihhail Berezovski, Jüri Engelbrecht Numerical simulation of nonlinear elastic wave propagation - p. 6/28
  • 14. Theory by Chen et al. (2004) q Outline s Analytical solution of one-dimensional linear wave Motivation q Experiments by Zhuang et al. propagation in layered heterogeneous materials (2003) q Time history of shock stress s Approximate solution for shock loading by invoking of q Time history of shock stress q Theory by Chen et al. (2004) equation of state q Time history of shock stress s Wave velocity, thickness and density for the laminates Formulation of the problem subjected to shock loading, all depend on the particle Wave-propagation algorithm velocity Comparison with experimental s Chen, X., Chandra, N., Rajendran, A.M., 2004. Analytical solution to the plate impact data problem of layered heterogeneous material systems Int. J. Solids Struct. 41, Discussion 4635–4659. s Chen, X., Chandra, N., 2004. The effect of heterogeneity on plane wave propagation through layered composites. Comp. Sci. Technol. 64, 1477–1493. Arkadi Berezovski, Mihhail Berezovski, Jüri Engelbrecht Numerical simulation of nonlinear elastic wave propagation - p. 6/28
  • 15. Time history of shock stress q Outline Motivation q Experiments by Zhuang et al. (2003) q Time history of shock stress q Time history of shock stress q Theory by Chen et al. (2004) q Time history of shock stress Formulation of the problem Wave-propagation algorithm Comparison with experimental data Discussion Reproduced from: Chen, X., Chandra, N., Rajendran, A.M., 2004. Analytical solution to the plate impact problem of layered heterogeneous material systems Int. J. Solids Struct. 41, 4635–4659. Arkadi Berezovski, Mihhail Berezovski, Jüri Engelbrecht Numerical simulation of nonlinear elastic wave propagation - p. 7/28
  • 16. Geometry of the problem q Outline Motivation Formulation of the problem q Geometry of the problem q Formulation of the problem Wave-propagation algorithm Comparison with experimental data Discussion Arkadi Berezovski, Mihhail Berezovski, Jüri Engelbrecht Numerical simulation of nonlinear elastic wave propagation - p. 8/28
  • 17. Formulation of the problem s Basic equations Conservation of linear momentum and kinematical compatibility: q Outline ∂v ∂σ ∂ε ∂v Motivation ρ = , = ∂t ∂x ∂t ∂x Formulation of the problem q Geometry of the problem ρ(x, t) is the density, σ(x, t) is the one-dimensional stress, ε(x, t) is the strain, and q Formulation of the problem v(x, t) the particle velocity. Wave-propagation algorithm Comparison with experimental data Discussion Arkadi Berezovski, Mihhail Berezovski, Jüri Engelbrecht Numerical simulation of nonlinear elastic wave propagation - p. 9/28
  • 18. Formulation of the problem s Basic equations Conservation of linear momentum and kinematical compatibility: q Outline ∂v ∂σ ∂ε ∂v Motivation ρ = , = ∂t ∂x ∂t ∂x Formulation of the problem q Geometry of the problem ρ(x, t) is the density, σ(x, t) is the one-dimensional stress, ε(x, t) is the strain, and q Formulation of the problem v(x, t) the particle velocity. Wave-propagation algorithm Comparison with experimental s Initial and boundary conditions data Initially, stress and strain are zero inside the flyer, the specimen, and the buffer, but the Discussion initial velocity of the flyer is nonzero: v(x, 0) = v0 , 0<x<f f is the size of the flyer. Both left and right boundaries are stress-free. Arkadi Berezovski, Mihhail Berezovski, Jüri Engelbrecht Numerical simulation of nonlinear elastic wave propagation - p. 9/28
  • 19. Formulation of the problem s Basic equations Conservation of linear momentum and kinematical compatibility: q Outline ∂v ∂σ ∂ε ∂v Motivation ρ = , = ∂t ∂x ∂t ∂x Formulation of the problem q Geometry of the problem ρ(x, t) is the density, σ(x, t) is the one-dimensional stress, ε(x, t) is the strain, and q Formulation of the problem v(x, t) the particle velocity. Wave-propagation algorithm Comparison with experimental s Initial and boundary conditions data Initially, stress and strain are zero inside the flyer, the specimen, and the buffer, but the Discussion initial velocity of the flyer is nonzero: v(x, 0) = v0 , 0<x<f f is the size of the flyer. Both left and right boundaries are stress-free. s Stress-strain relation σ = ρc2 ε(1 + Aε) p cp is the conventional longitudinal wave speed, A is a parameter of nonlinearity, values of which are supposed to be different for hard and soft materials. Arkadi Berezovski, Mihhail Berezovski, Jüri Engelbrecht Numerical simulation of nonlinear elastic wave propagation - p. 9/28
  • 20. Wave-propagation algorithm q Outline s Finite-volume numerical scheme Motivation LeVeque, R.J., 1997. Wave propagation algorithms for multidimensional hyperbolic Formulation of the problem systems. J. Comp. Physics 131, 327–353. Wave-propagation algorithm q Wave-propagation algorithm Comparison with experimental data Discussion Arkadi Berezovski, Mihhail Berezovski, Jüri Engelbrecht Numerical simulation of nonlinear elastic wave propagation - p. 10/28
  • 21. Wave-propagation algorithm q Outline s Finite-volume numerical scheme Motivation LeVeque, R.J., 1997. Wave propagation algorithms for multidimensional hyperbolic Formulation of the problem systems. J. Comp. Physics 131, 327–353. Wave-propagation algorithm q Wave-propagation algorithm s Numerical fluxes are determined by solving the Riemann Comparison with experimental problem at each interface between discrete elements data Discussion Arkadi Berezovski, Mihhail Berezovski, Jüri Engelbrecht Numerical simulation of nonlinear elastic wave propagation - p. 10/28
  • 22. Wave-propagation algorithm q Outline s Finite-volume numerical scheme Motivation LeVeque, R.J., 1997. Wave propagation algorithms for multidimensional hyperbolic Formulation of the problem systems. J. Comp. Physics 131, 327–353. Wave-propagation algorithm q Wave-propagation algorithm s Numerical fluxes are determined by solving the Riemann Comparison with experimental problem at each interface between discrete elements data s Reflection and transmission of waves at each interface are Discussion handled automatically Arkadi Berezovski, Mihhail Berezovski, Jüri Engelbrecht Numerical simulation of nonlinear elastic wave propagation - p. 10/28
  • 23. Wave-propagation algorithm q Outline s Finite-volume numerical scheme Motivation LeVeque, R.J., 1997. Wave propagation algorithms for multidimensional hyperbolic Formulation of the problem systems. J. Comp. Physics 131, 327–353. Wave-propagation algorithm q Wave-propagation algorithm s Numerical fluxes are determined by solving the Riemann Comparison with experimental problem at each interface between discrete elements data s Reflection and transmission of waves at each interface are Discussion handled automatically s Second-order corrections are included Arkadi Berezovski, Mihhail Berezovski, Jüri Engelbrecht Numerical simulation of nonlinear elastic wave propagation - p. 10/28
  • 24. Wave-propagation algorithm q Outline s Finite-volume numerical scheme Motivation LeVeque, R.J., 1997. Wave propagation algorithms for multidimensional hyperbolic Formulation of the problem systems. J. Comp. Physics 131, 327–353. Wave-propagation algorithm q Wave-propagation algorithm s Numerical fluxes are determined by solving the Riemann Comparison with experimental problem at each interface between discrete elements data s Reflection and transmission of waves at each interface are Discussion handled automatically s Second-order corrections are included s Success in application to wave propagation in rapidly-varying heterogeneous media and to nonlinear elastic waves Fogarty, T.R., LeVeque, R.J., 1999. High-resolution finite volume methods for acoustic waves in periodic and random media. J. Acoust. Soc. Amer. 106, 17–28. LeVeque, R., Yong, D. H., 2003. Solitary waves in layered nonlinear media. SIAM J. Appl. Math. 63, 1539–1560. Arkadi Berezovski, Mihhail Berezovski, Jüri Engelbrecht Numerical simulation of nonlinear elastic wave propagation - p. 10/28
  • 25. Time history of shock stress 1.4 experiment q Outline 1.2 Motivation Formulation of the problem 1 Wave-propagation algorithm Stress (GPa) 0.8 Comparison with experimental data q Time history of shock stress q Time history of shock stress 0.6 q Time history of particle velocity q Time history of particle 0.4 velocity q Time history of shock stress q Time history of shock stress 0.2 q Time history of shock stress q Time history of shock stress q Time history of shock stress q Time history of shock stress 0 q Time history of shock stress 0 0.5 1 1.5 2 2.5 q Time history of shock stress Time (microseconds) q Time history of shock stress q Time history of shock stress Experiment 112501 (Zhuang, S., Ravichandran, G., Grady D., 2003.) q Time history of shock stress Composite: 8 units of polycarbonate, each 0.74 mm thick, and 8 units of stainless steel, q Time history of shock stress each 0.37 mm thick. Discussion Flyer velocity 561 m/s and flyer thickness 2.87 mm. Arkadi Berezovski, Mihhail Berezovski, Jüri Engelbrecht Numerical simulation of nonlinear elastic wave propagation - p. 11/28
  • 26. Time history of shock stress 1.4 experiment simulation - nonlinear q Outline 1.2 Motivation Formulation of the problem 1 Wave-propagation algorithm Stress (GPa) 0.8 Comparison with experimental data q Time history of shock stress q Time history of shock stress 0.6 q Time history of particle velocity q Time history of particle 0.4 velocity q Time history of shock stress q Time history of shock stress 0.2 q Time history of shock stress q Time history of shock stress q Time history of shock stress q Time history of shock stress 0 q Time history of shock stress 0 0.5 1 1.5 2 2.5 q Time history of shock stress Time (microseconds) q Time history of shock stress q Time history of shock stress Experiment 112501 (Zhuang, S., Ravichandran, G., Grady D., 2003.) q Time history of shock stress Composite: 8 units of polycarbonate, each 0.74 mm thick, and 8 units of stainless steel, q Time history of shock stress each 0.37 mm thick. Discussion Flyer velocity 561 m/s and flyer thickness 2.87 mm. Nonlinearity parameter A: 300 for polycarbonate and 50 for stainless steel. Arkadi Berezovski, Mihhail Berezovski, Jüri Engelbrecht Numerical simulation of nonlinear elastic wave propagation - p. 12/28
  • 27. Time history of particle velocity 0.35 experiment q Outline 0.3 Motivation Particle velocity (km/s) Formulation of the problem 0.25 Wave-propagation algorithm 0.2 Comparison with experimental data q Time history of shock stress q Time history of shock stress 0.15 q Time history of particle velocity q Time history of particle 0.1 velocity q Time history of shock stress q Time history of shock stress 0.05 q Time history of shock stress q Time history of shock stress q Time history of shock stress q Time history of shock stress 0 q Time history of shock stress 4 5 6 7 8 9 q Time history of shock stress Time (microseconds) q Time history of shock stress q Time history of shock stress Experiment 112501 (Zhuang, S., Ravichandran, G., Grady D., 2003.) q Time history of shock stress Composite: 8 units of polycarbonate, each 0.74 mm thick, and 8 units of stainless steel, q Time history of shock stress each 0.37 mm thick. Discussion Flyer velocity 561 m/s and flyer thickness 2.87 mm. Arkadi Berezovski, Mihhail Berezovski, Jüri Engelbrecht Numerical simulation of nonlinear elastic wave propagation - p. 13/28
  • 28. Time history of particle velocity 0.35 experiment simulation - nonlinear q Outline 0.3 Motivation Particle velocity (km/s) Formulation of the problem 0.25 Wave-propagation algorithm 0.2 Comparison with experimental data q Time history of shock stress q Time history of shock stress 0.15 q Time history of particle velocity q Time history of particle 0.1 velocity q Time history of shock stress q Time history of shock stress 0.05 q Time history of shock stress q Time history of shock stress q Time history of shock stress q Time history of shock stress 0 q Time history of shock stress 4 5 6 7 8 9 q Time history of shock stress Time (microseconds) q Time history of shock stress q Time history of shock stress Experiment 112501 (Zhuang, S., Ravichandran, G., Grady D., 2003.) q Time history of shock stress Composite: 8 units of polycarbonate, each 0.74 mm thick, and 8 units of stainless steel, q Time history of shock stress each 0.37 mm thick. Discussion Flyer velocity 561 m/s and flyer thickness 2.87 mm. Nonlinearity parameter A: 300 for polycarbonate and 50 for stainless steel. Arkadi Berezovski, Mihhail Berezovski, Jüri Engelbrecht Numerical simulation of nonlinear elastic wave propagation - p. 14/28
  • 29. Time history of shock stress 3.5 experiment q Outline 3 Motivation Formulation of the problem 2.5 Wave-propagation algorithm Stress (GPa) 2 Comparison with experimental data q Time history of shock stress q Time history of shock stress 1.5 q Time history of particle velocity q Time history of particle 1 velocity q Time history of shock stress q Time history of shock stress 0.5 q Time history of shock stress q Time history of shock stress q Time history of shock stress q Time history of shock stress 0 q Time history of shock stress 1 1.5 2 2.5 3 3.5 4 4.5 q Time history of shock stress Time (microseconds) q Time history of shock stress q Time history of shock stress Experiment 110501 (Zhuang, S., Ravichandran, G., Grady D., 2003.) q Time history of shock stress Composite: 16 units of polycarbonate, each 0.37 mm thick, and 16 units of stainless q Time history of shock stress steel, each 0.19 mm thick. Discussion Flyer velocity 1043 m/s and flyer thickness 2.87 mm. Arkadi Berezovski, Mihhail Berezovski, Jüri Engelbrecht Numerical simulation of nonlinear elastic wave propagation - p. 15/28
  • 30. Time history of shock stress 3.5 experiment simulation - linear q Outline 3 Motivation Formulation of the problem 2.5 Wave-propagation algorithm Stress (GPa) 2 Comparison with experimental data q Time history of shock stress q Time history of shock stress 1.5 q Time history of particle velocity q Time history of particle 1 velocity q Time history of shock stress q Time history of shock stress 0.5 q Time history of shock stress q Time history of shock stress q Time history of shock stress q Time history of shock stress 0 q Time history of shock stress 1 1.5 2 2.5 3 3.5 4 4.5 q Time history of shock stress Time (microseconds) q Time history of shock stress q Time history of shock stress Experiment 110501 (Zhuang, S., Ravichandran, G., Grady D., 2003.) q Time history of shock stress Composite: 16 units of polycarbonate, each 0.37 mm thick, and 16 units of stainless q Time history of shock stress steel, each 0.19 mm thick. Discussion Flyer velocity 1043 m/s and flyer thickness 2.87 mm. Arkadi Berezovski, Mihhail Berezovski, Jüri Engelbrecht Numerical simulation of nonlinear elastic wave propagation - p. 16/28
  • 31. Time history of shock stress 3.5 experiment simulation - linear q Outline 3 simulation - nonlinear Motivation Formulation of the problem 2.5 Wave-propagation algorithm Stress (GPa) 2 Comparison with experimental data q Time history of shock stress q Time history of shock stress 1.5 q Time history of particle velocity q Time history of particle 1 velocity q Time history of shock stress q Time history of shock stress 0.5 q Time history of shock stress q Time history of shock stress q Time history of shock stress q Time history of shock stress 0 q Time history of shock stress 1 1.5 2 2.5 3 3.5 4 4.5 q Time history of shock stress Time (microseconds) q Time history of shock stress q Time history of shock stress Experiment 110501 (Zhuang, S., Ravichandran, G., Grady D., 2003.) q Time history of shock stress Composite: 16 units of polycarbonate, each 0.37 mm thick, and 16 units of stainless q Time history of shock stress steel, each 0.19 mm thick. Discussion Flyer velocity 1043 m/s and flyer thickness 2.87 mm. Nonlinearity parameter A: 180 for polycarbonate and zero for stainless steel. Arkadi Berezovski, Mihhail Berezovski, Jüri Engelbrecht Numerical simulation of nonlinear elastic wave propagation - p. 17/28
  • 32. Time history of shock stress 4 experiment q Outline 3.5 Motivation 3 Formulation of the problem Wave-propagation algorithm Stress (GPa) 2.5 Comparison with experimental data 2 q Time history of shock stress q Time history of shock stress q Time history of particle 1.5 velocity q Time history of particle velocity 1 q Time history of shock stress q Time history of shock stress q Time history of shock stress 0.5 q Time history of shock stress q Time history of shock stress q Time history of shock stress 0 q Time history of shock stress 1 1.5 2 2.5 3 3.5 4 4.5 5 q Time history of shock stress Time (microseconds) q Time history of shock stress q Time history of shock stress Experiment 110502 (Zhuang, S., Ravichandran, G., Grady D., 2003.) q Time history of shock stress Composite: 16 units of polycarbonate, each 0.37 mm thick, and 16 units of stainless q Time history of shock stress steel, each 0.19 mm thick. Discussion Flyer velocity 1043 m/s and flyer thickness 5.63 mm. Arkadi Berezovski, Mihhail Berezovski, Jüri Engelbrecht Numerical simulation of nonlinear elastic wave propagation - p. 18/28
  • 33. Time history of shock stress 4 experiment simulation - nonlinear q Outline 3.5 Motivation 3 Formulation of the problem Wave-propagation algorithm Stress (GPa) 2.5 Comparison with experimental data 2 q Time history of shock stress q Time history of shock stress q Time history of particle 1.5 velocity q Time history of particle velocity 1 q Time history of shock stress q Time history of shock stress q Time history of shock stress 0.5 q Time history of shock stress q Time history of shock stress q Time history of shock stress 0 q Time history of shock stress 1 1.5 2 2.5 3 3.5 4 4.5 5 q Time history of shock stress Time (microseconds) q Time history of shock stress q Time history of shock stress Experiment 110502 (Zhuang, S., Ravichandran, G., Grady D., 2003.) q Time history of shock stress Composite: 16 units of polycarbonate, each 0.37 mm thick, and 16 units of stainless q Time history of shock stress steel, each 0.19 mm thick. Discussion Flyer velocity 1043 m/s and flyer thickness 5.63 mm. Nonlinearity parameter A: 230 for polycarbonate and zero for stainless steel. Arkadi Berezovski, Mihhail Berezovski, Jüri Engelbrecht Numerical simulation of nonlinear elastic wave propagation - p. 19/28
  • 34. Time history of shock stress 3.5 experiment q Outline 3 Motivation Formulation of the problem 2.5 Wave-propagation algorithm Stress (GPa) 2 Comparison with experimental data q Time history of shock stress q Time history of shock stress 1.5 q Time history of particle velocity q Time history of particle 1 velocity q Time history of shock stress q Time history of shock stress 0.5 q Time history of shock stress q Time history of shock stress q Time history of shock stress q Time history of shock stress 0 q Time history of shock stress 1 1.5 2 2.5 3 3.5 4 q Time history of shock stress Time (microseconds) q Time history of shock stress q Time history of shock stress Experiment 112301 (Zhuang, S., Ravichandran, G., Grady D., 2003.) q Time history of shock stress Composite: 16 units of polycarbonate, each 0.37 mm thick, and 16 units of D-263 glass, q Time history of shock stress each 0.20 mm thick. Discussion Flyer velocity 1079 m/s and flyer thickness 2.87 mm. Arkadi Berezovski, Mihhail Berezovski, Jüri Engelbrecht Numerical simulation of nonlinear elastic wave propagation - p. 20/28
  • 35. Time history of shock stress 3.5 experiment simulation - linear q Outline 3 Motivation Formulation of the problem 2.5 Wave-propagation algorithm Stress (GPa) 2 Comparison with experimental data q Time history of shock stress q Time history of shock stress 1.5 q Time history of particle velocity q Time history of particle 1 velocity q Time history of shock stress q Time history of shock stress 0.5 q Time history of shock stress q Time history of shock stress q Time history of shock stress q Time history of shock stress 0 q Time history of shock stress 1 1.5 2 2.5 3 3.5 4 q Time history of shock stress Time (microseconds) q Time history of shock stress q Time history of shock stress Experiment 112301 (Zhuang, S., Ravichandran, G., Grady D., 2003.) q Time history of shock stress Composite: 16 units of polycarbonate, each 0.37 mm thick, and 16 units of D-263 glass, q Time history of shock stress each 0.20 mm thick. Discussion Flyer velocity 1079 m/s and flyer thickness 2.87 mm. Arkadi Berezovski, Mihhail Berezovski, Jüri Engelbrecht Numerical simulation of nonlinear elastic wave propagation - p. 21/28
  • 36. Time history of shock stress 3.5 experiment simulation - linear q Outline 3 simulation - nonlinear Motivation Formulation of the problem 2.5 Wave-propagation algorithm Stress (GPa) 2 Comparison with experimental data q Time history of shock stress q Time history of shock stress 1.5 q Time history of particle velocity q Time history of particle 1 velocity q Time history of shock stress q Time history of shock stress 0.5 q Time history of shock stress q Time history of shock stress q Time history of shock stress q Time history of shock stress 0 q Time history of shock stress 1 1.5 2 2.5 3 3.5 4 q Time history of shock stress Time (microseconds) q Time history of shock stress q Time history of shock stress Experiment 112301 (Zhuang, S., Ravichandran, G., Grady D., 2003.) q Time history of shock stress Composite: 16 units of polycarbonate, each 0.37 mm thick, and 16 units of D-263 glass, q Time history of shock stress each 0.20 mm thick. Discussion Flyer velocity 1079 m/s and flyer thickness 2.87 mm. Nonlinearity parameter A: 90 for polycarbonate and zero for D-263 glass. Arkadi Berezovski, Mihhail Berezovski, Jüri Engelbrecht Numerical simulation of nonlinear elastic wave propagation - p. 22/28
  • 37. Time history of shock stress 1 experiment q Outline Motivation 0.8 Formulation of the problem Wave-propagation algorithm Stress (GPa) 0.6 Comparison with experimental data q Time history of shock stress q Time history of shock stress 0.4 q Time history of particle velocity q Time history of particle velocity q Time history of shock stress 0.2 q Time history of shock stress q Time history of shock stress q Time history of shock stress q Time history of shock stress q Time history of shock stress 0 q Time history of shock stress 1 1.5 2 2.5 3 3.5 4 4.5 q Time history of shock stress Time (microseconds) q Time history of shock stress q Time history of shock stress Experiment 120201 (Zhuang, S., Ravichandran, G., Grady D., 2003.) q Time history of shock stress Composite: 7 units of polycarbonate, each 0.74 mm thick, and 7 units of float glass, each q Time history of shock stress 0.55 mm thick. Discussion Flyer velocity 563 m/s and flyer thickness 2.87 mm. Arkadi Berezovski, Mihhail Berezovski, Jüri Engelbrecht Numerical simulation of nonlinear elastic wave propagation - p. 23/28
  • 38. Time history of shock stress 1 experiment simulation - nonlinear q Outline Motivation 0.8 Formulation of the problem Wave-propagation algorithm Stress (GPa) 0.6 Comparison with experimental data q Time history of shock stress q Time history of shock stress 0.4 q Time history of particle velocity q Time history of particle velocity q Time history of shock stress 0.2 q Time history of shock stress q Time history of shock stress q Time history of shock stress q Time history of shock stress q Time history of shock stress 0 q Time history of shock stress 1 1.5 2 2.5 3 3.5 4 4.5 q Time history of shock stress Time (microseconds) q Time history of shock stress q Time history of shock stress Experiment 120201 (Zhuang, S., Ravichandran, G., Grady D., 2003.) q Time history of shock stress Composite: 7 units of polycarbonate, each 0.74 mm thick, and 7 units of float glass, each q Time history of shock stress 0.55 mm thick. Discussion Flyer velocity 563 m/s and flyer thickness 2.87 mm. Nonlinearity parameter A: 55 for polycarbonate and zero for float glass. Arkadi Berezovski, Mihhail Berezovski, Jüri Engelbrecht Numerical simulation of nonlinear elastic wave propagation - p. 24/28
  • 39. Time history of shock stress 3 experiment q Outline 2.5 Motivation Formulation of the problem Stress (GPa) 2 Wave-propagation algorithm Comparison with experimental data 1.5 q Time history of shock stress q Time history of shock stress q Time history of particle velocity 1 q Time history of particle velocity q Time history of shock stress q Time history of shock stress 0.5 q Time history of shock stress q Time history of shock stress q Time history of shock stress q Time history of shock stress 0 q Time history of shock stress 1 1.5 2 2.5 3 3.5 4 q Time history of shock stress Time (microseconds) q Time history of shock stress q Time history of shock stress Experiment 120202 (Zhuang, S., Ravichandran, G., Grady D., 2003.) q Time history of shock stress Composite: 7 units of polycarbonate, each 0.74 mm thick, and 7 units of float glass, each q Time history of shock stress 0.55 mm thick. Discussion Flyer velocity 1056 m/s and flyer thickness 2.87 mm. Arkadi Berezovski, Mihhail Berezovski, Jüri Engelbrecht Numerical simulation of nonlinear elastic wave propagation - p. 25/28
  • 40. Time history of shock stress 3 experiment simulation - nonlinear q Outline 2.5 Motivation Formulation of the problem Stress (GPa) 2 Wave-propagation algorithm Comparison with experimental data 1.5 q Time history of shock stress q Time history of shock stress q Time history of particle velocity 1 q Time history of particle velocity q Time history of shock stress q Time history of shock stress 0.5 q Time history of shock stress q Time history of shock stress q Time history of shock stress q Time history of shock stress 0 q Time history of shock stress 1 1.5 2 2.5 3 3.5 4 q Time history of shock stress Time (microseconds) q Time history of shock stress q Time history of shock stress Experiment 120202 (Zhuang, S., Ravichandran, G., Grady D., 2003.) q Time history of shock stress Composite: 7 units of polycarbonate, each 0.74 mm thick, and 7 units of float glass, each q Time history of shock stress 0.55 mm thick. Discussion Flyer velocity 1056 m/s and flyer thickness 2.87 mm. Nonlinearity parameter A: 100 for polycarbonate and zero for float glass. Arkadi Berezovski, Mihhail Berezovski, Jüri Engelbrecht Numerical simulation of nonlinear elastic wave propagation - p. 26/28
  • 41. Nonlinear parameter q Outline Exp. Specimen Units Flyer Flyer Gage A A Motivation soft/hard velocity thickness position PC other Formulation of the problem (m/s) (mm) (mm) Wave-propagation algorithm 112501 PC74/SS37 8 561 2.87 (PC) 0.76 300 50 Comparison with experimental 110501 PC37/SS19 16 1043 2.87 (PC) 3.44 180 0 data 110502 PC37/SS19 16 1045 5.63 (PC) 3.44 230 0 Discussion q Nonlinear parameter 112301 PC37/GS20 16 1079 2.87 (PC) 3.41 90 0 q Conclusions 120201 PC74/GS55 7 563 2.87 (PC) 3.37 55 0 120202 PC74/GS55 7 1056 2.87 (PC) 3.35 100 0 PC denotes polycarbonate, GS - glass, SS - 304 stainless steel; the number following the abbreviation of component material represents the layer thickness in hundredths of a millimeter. Arkadi Berezovski, Mihhail Berezovski, Jüri Engelbrecht Numerical simulation of nonlinear elastic wave propagation - p. 27/28
  • 42. Conclusions q Outline s Good agreement between computations and experiments Motivation can be obtained by means of a non-linear model Formulation of the problem Wave-propagation algorithm Comparison with experimental data Discussion q Nonlinear parameter q Conclusions Arkadi Berezovski, Mihhail Berezovski, Jüri Engelbrecht Numerical simulation of nonlinear elastic wave propagation - p. 28/28
  • 43. Conclusions q Outline s Good agreement between computations and experiments Motivation can be obtained by means of a non-linear model Formulation of the problem s The nonlinear behavior of the soft material is affected not Wave-propagation algorithm only by the energy of the impact but also by the scattering Comparison with experimental data induced by internal interfaces Discussion q Nonlinear parameter q Conclusions Arkadi Berezovski, Mihhail Berezovski, Jüri Engelbrecht Numerical simulation of nonlinear elastic wave propagation - p. 28/28
  • 44. Conclusions q Outline s Good agreement between computations and experiments Motivation can be obtained by means of a non-linear model Formulation of the problem s The nonlinear behavior of the soft material is affected not Wave-propagation algorithm only by the energy of the impact but also by the scattering Comparison with experimental data induced by internal interfaces Discussion s The influence of the nonlinearity is not necessary small q Nonlinear parameter q Conclusions Arkadi Berezovski, Mihhail Berezovski, Jüri Engelbrecht Numerical simulation of nonlinear elastic wave propagation - p. 28/28
  • 45. Conclusions q Outline s Good agreement between computations and experiments Motivation can be obtained by means of a non-linear model Formulation of the problem s The nonlinear behavior of the soft material is affected not Wave-propagation algorithm only by the energy of the impact but also by the scattering Comparison with experimental data induced by internal interfaces Discussion s The influence of the nonlinearity is not necessary small q Nonlinear parameter q Conclusions s Additional experimental information is needed to validate the proposed nonlinear model Arkadi Berezovski, Mihhail Berezovski, Jüri Engelbrecht Numerical simulation of nonlinear elastic wave propagation - p. 28/28