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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
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