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Arkadi Berezovski, Mihhail Berezovski, Jüri Engelbrecht Numerical simulation of nonlinear elastic wave propagation - p. 1/28
Numerical simulation of nonlinear elasticwave propagation in piecewise homogeneous
media
Arkadi Berezovski, Mihhail Berezovski, Jüri EngelbrechtCentre for Nonlinear Studies, Institute of Cybernetics at Tallinn University of Technology,
Akadeemia tee 21, 12618 Tallinn, Estonia
● Outline
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
Outline
■ Motivation: experiments and theory
● Outline
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
Outline
■ Motivation: experiments and theory■ Formulation of the problem
● Outline
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
Outline
■ Motivation: experiments and theory■ Formulation of the problem■ Wave-propagation algorithm
● Outline
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
Outline
■ Motivation: experiments and theory■ Formulation of the problem■ Wave-propagation algorithm■ Comparison with experimental data
● Outline
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
Outline
■ Motivation: experiments and theory■ Formulation of the problem■ Wave-propagation algorithm■ Comparison with experimental data■ Conclusions
● Outline
Motivation
● Experiments by Zhuang et al.
(2003)
● Time history of shock stress
● Time history of shock stress
● Theory by Chen et al. (2004)
● 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. 3/28
Experiments by Zhuang et al. (2003)
(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.)
● Outline
Motivation
● Experiments by Zhuang et al.
(2003)
● Time history of shock stress
● Time history of shock stress
● Theory by Chen et al. (2004)
● 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. 4/28
Time history of shock stress
0
0.5
1
1.5
2
2.5
3
3.5
1 1.5 2 2.5 3 3.5 4
Str
ess
(GP
a)
Time (microseconds)
experiment
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.
● Outline
Motivation
● Experiments by Zhuang et al.
(2003)
● Time history of shock stress
● Time history of shock stress
● Theory by Chen et al. (2004)
● 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. 5/28
Time history of shock stress
0
0.5
1
1.5
2
2.5
3
3.5
1 1.5 2 2.5 3 3.5 4
Str
ess
(GP
a)
Time (microseconds)
experimentsimulation - linear
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.
● Outline
Motivation
● Experiments by Zhuang et al.
(2003)
● Time history of shock stress
● Time history of shock stress
● Theory by Chen et al. (2004)
● 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
Theory by Chen et al. (2004)
■ Analytical solution of one-dimensional linear wavepropagation in layered heterogeneous materials
● Outline
Motivation
● Experiments by Zhuang et al.
(2003)
● Time history of shock stress
● Time history of shock stress
● Theory by Chen et al. (2004)
● 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
Theory by Chen et al. (2004)
■ Analytical solution of one-dimensional linear wavepropagation in layered heterogeneous materials
■ Approximate solution for shock loading by invoking ofequation of state
● Outline
Motivation
● Experiments by Zhuang et al.
(2003)
● Time history of shock stress
● Time history of shock stress
● Theory by Chen et al. (2004)
● 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
Theory by Chen et al. (2004)
■ Analytical solution of one-dimensional linear wavepropagation in layered heterogeneous materials
■ Approximate solution for shock loading by invoking ofequation of state
■ Wave velocity, thickness and density for the laminatessubjected to shock loading, all depend on the particlevelocity
● Outline
Motivation
● Experiments by Zhuang et al.
(2003)
● Time history of shock stress
● Time history of shock stress
● Theory by Chen et al. (2004)
● 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
Theory by Chen et al. (2004)
■ Analytical solution of one-dimensional linear wavepropagation in layered heterogeneous materials
■ Approximate solution for shock loading by invoking ofequation of state
■ Wave velocity, thickness and density for the laminatessubjected to shock loading, all depend on the particlevelocity
■ Chen, X., Chandra, N., Rajendran, A.M., 2004. Analytical solution to the plate impactproblem of layered heterogeneous material systems Int. J. Solids Struct. 41,4635–4659.
● Outline
Motivation
● Experiments by Zhuang et al.
(2003)
● Time history of shock stress
● Time history of shock stress
● Theory by Chen et al. (2004)
● 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
Theory by Chen et al. (2004)
■ Analytical solution of one-dimensional linear wavepropagation in layered heterogeneous materials
■ Approximate solution for shock loading by invoking ofequation of state
■ Wave velocity, thickness and density for the laminatessubjected to shock loading, all depend on the particlevelocity
■ Chen, X., Chandra, N., Rajendran, A.M., 2004. Analytical solution to the plate impactproblem of layered heterogeneous material systems Int. J. Solids Struct. 41,4635–4659.
■ Chen, X., Chandra, N., 2004. The effect of heterogeneity on plane wave propagation
through layered composites. Comp. Sci. Technol. 64, 1477–1493.
● Outline
Motivation
● Experiments by Zhuang et al.
(2003)
● Time history of shock stress
● Time history of shock stress
● Theory by Chen et al. (2004)
● 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. 7/28
Time history of shock stress
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.
● Outline
Motivation
Formulation of the problem
● Geometry 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. 8/28
Geometry of the problem
● Outline
Motivation
Formulation of the problem
● Geometry 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. 9/28
Formulation of the problem■ Basic equations
Conservation of linear momentum and kinematical compatibility:
ρ∂v
∂t=
∂σ
∂x,
∂ε
∂t=
∂v
∂x
ρ(x, t) is the density, σ(x, t) is the one-dimensional stress, ε(x, t) is the strain, and
v(x, t) the particle velocity.
● Outline
Motivation
Formulation of the problem
● Geometry 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. 9/28
Formulation of the problem■ Basic equations
Conservation of linear momentum and kinematical compatibility:
ρ∂v
∂t=
∂σ
∂x,
∂ε
∂t=
∂v
∂x
ρ(x, t) is the density, σ(x, t) is the one-dimensional stress, ε(x, t) is the strain, and
v(x, t) the particle velocity.
■ Initial and boundary conditionsInitially, stress and strain are zero inside the flyer, the specimen, and the buffer, but the
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.
● Outline
Motivation
Formulation of the problem
● Geometry 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. 9/28
Formulation of the problem■ Basic equations
Conservation of linear momentum and kinematical compatibility:
ρ∂v
∂t=
∂σ
∂x,
∂ε
∂t=
∂v
∂x
ρ(x, t) is the density, σ(x, t) is the one-dimensional stress, ε(x, t) is the strain, and
v(x, t) the particle velocity.
■ Initial and boundary conditionsInitially, stress and strain are zero inside the flyer, the specimen, and the buffer, but the
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.
■ Stress-strain relation
σ = ρc2p ε(1 + Aε)
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.
● Outline
Motivation
Formulation of the problem
Wave-propagation algorithm
● Wave-propagation algorithm
Comparison with experimental
data
Discussion
Arkadi Berezovski, Mihhail Berezovski, Jüri Engelbrecht Numerical simulation of nonlinear elastic wave propagation - p. 10/28
Wave-propagation algorithm
■ Finite-volume numerical schemeLeVeque, R.J., 1997. Wave propagation algorithms for multidimensional hyperbolic
systems. J. Comp. Physics 131, 327–353.
● Outline
Motivation
Formulation of the problem
Wave-propagation algorithm
● Wave-propagation algorithm
Comparison with experimental
data
Discussion
Arkadi Berezovski, Mihhail Berezovski, Jüri Engelbrecht Numerical simulation of nonlinear elastic wave propagation - p. 10/28
Wave-propagation algorithm
■ Finite-volume numerical schemeLeVeque, R.J., 1997. Wave propagation algorithms for multidimensional hyperbolic
systems. J. Comp. Physics 131, 327–353.
■ Numerical fluxes are determined by solving the Riemannproblem at each interface between discrete elements
● Outline
Motivation
Formulation of the problem
Wave-propagation algorithm
● Wave-propagation algorithm
Comparison with experimental
data
Discussion
Arkadi Berezovski, Mihhail Berezovski, Jüri Engelbrecht Numerical simulation of nonlinear elastic wave propagation - p. 10/28
Wave-propagation algorithm
■ Finite-volume numerical schemeLeVeque, R.J., 1997. Wave propagation algorithms for multidimensional hyperbolic
systems. J. Comp. Physics 131, 327–353.
■ Numerical fluxes are determined by solving the Riemannproblem at each interface between discrete elements
■ Reflection and transmission of waves at each interface arehandled automatically
● Outline
Motivation
Formulation of the problem
Wave-propagation algorithm
● Wave-propagation algorithm
Comparison with experimental
data
Discussion
Arkadi Berezovski, Mihhail Berezovski, Jüri Engelbrecht Numerical simulation of nonlinear elastic wave propagation - p. 10/28
Wave-propagation algorithm
■ Finite-volume numerical schemeLeVeque, R.J., 1997. Wave propagation algorithms for multidimensional hyperbolic
systems. J. Comp. Physics 131, 327–353.
■ Numerical fluxes are determined by solving the Riemannproblem at each interface between discrete elements
■ Reflection and transmission of waves at each interface arehandled automatically
■ Second-order corrections are included
● Outline
Motivation
Formulation of the problem
Wave-propagation algorithm
● Wave-propagation algorithm
Comparison with experimental
data
Discussion
Arkadi Berezovski, Mihhail Berezovski, Jüri Engelbrecht Numerical simulation of nonlinear elastic wave propagation - p. 10/28
Wave-propagation algorithm
■ Finite-volume numerical schemeLeVeque, R.J., 1997. Wave propagation algorithms for multidimensional hyperbolic
systems. J. Comp. Physics 131, 327–353.
■ Numerical fluxes are determined by solving the Riemannproblem at each interface between discrete elements
■ Reflection and transmission of waves at each interface arehandled automatically
■ Second-order corrections are included■ 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 acousticwaves 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.
● Outline
Motivation
Formulation of the problem
Wave-propagation algorithm
Comparison with experimental
data
● Time history of shock stress
● Time history of shock stress
● Time history of particle
velocity
● Time history of particle
velocity
● Time history of shock stress
● Time history of shock stress
● Time history of shock stress
● Time history of shock stress
● Time history of shock stress
● Time history of shock stress
● Time history of shock stress
● Time history of shock stress
● Time history of shock stress
● Time history of shock stress
● Time history of shock stress
● Time history of shock stress
Discussion
Arkadi Berezovski, Mihhail Berezovski, Jüri Engelbrecht Numerical simulation of nonlinear elastic wave propagation - p. 11/28
Time history of shock stress
0
0.2
0.4
0.6
0.8
1
1.2
1.4
0 0.5 1 1.5 2 2.5
Str
ess
(GP
a)
Time (microseconds)
experiment
Experiment 112501 (Zhuang, S., Ravichandran, G., Grady D., 2003.)Composite: 8 units of polycarbonate, each 0.74 mm thick, and 8 units of stainless steel,each 0.37 mm thick.Flyer velocity 561 m/s and flyer thickness 2.87 mm.
● Outline
Motivation
Formulation of the problem
Wave-propagation algorithm
Comparison with experimental
data
● Time history of shock stress
● Time history of shock stress
● Time history of particle
velocity
● Time history of particle
velocity
● Time history of shock stress
● Time history of shock stress
● Time history of shock stress
● Time history of shock stress
● Time history of shock stress
● Time history of shock stress
● Time history of shock stress
● Time history of shock stress
● Time history of shock stress
● Time history of shock stress
● Time history of shock stress
● Time history of shock stress
Discussion
Arkadi Berezovski, Mihhail Berezovski, Jüri Engelbrecht Numerical simulation of nonlinear elastic wave propagation - p. 12/28
Time history of shock stress
0
0.2
0.4
0.6
0.8
1
1.2
1.4
0 0.5 1 1.5 2 2.5
Str
ess
(GP
a)
Time (microseconds)
experimentsimulation - nonlinear
Experiment 112501 (Zhuang, S., Ravichandran, G., Grady D., 2003.)Composite: 8 units of polycarbonate, each 0.74 mm thick, and 8 units of stainless steel,each 0.37 mm thick.Flyer velocity 561 m/s and flyer thickness 2.87 mm.
Nonlinearity parameter A: 300 for polycarbonate and 50 for stainless steel.
● Outline
Motivation
Formulation of the problem
Wave-propagation algorithm
Comparison with experimental
data
● Time history of shock stress
● Time history of shock stress
● Time history of particle
velocity
● Time history of particle
velocity
● Time history of shock stress
● Time history of shock stress
● Time history of shock stress
● Time history of shock stress
● Time history of shock stress
● Time history of shock stress
● Time history of shock stress
● Time history of shock stress
● Time history of shock stress
● Time history of shock stress
● Time history of shock stress
● Time history of shock stress
Discussion
Arkadi Berezovski, Mihhail Berezovski, Jüri Engelbrecht Numerical simulation of nonlinear elastic wave propagation - p. 13/28
Time history of particle velocity
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
4 5 6 7 8 9
Par
ticle
vel
ocity
(km
/s)
Time (microseconds)
experiment
Experiment 112501 (Zhuang, S., Ravichandran, G., Grady D., 2003.)Composite: 8 units of polycarbonate, each 0.74 mm thick, and 8 units of stainless steel,each 0.37 mm thick.Flyer velocity 561 m/s and flyer thickness 2.87 mm.
● Outline
Motivation
Formulation of the problem
Wave-propagation algorithm
Comparison with experimental
data
● Time history of shock stress
● Time history of shock stress
● Time history of particle
velocity
● Time history of particle
velocity
● Time history of shock stress
● Time history of shock stress
● Time history of shock stress
● Time history of shock stress
● Time history of shock stress
● Time history of shock stress
● Time history of shock stress
● Time history of shock stress
● Time history of shock stress
● Time history of shock stress
● Time history of shock stress
● Time history of shock stress
Discussion
Arkadi Berezovski, Mihhail Berezovski, Jüri Engelbrecht Numerical simulation of nonlinear elastic wave propagation - p. 14/28
Time history of particle velocity
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
4 5 6 7 8 9
Par
ticle
vel
ocity
(km
/s)
Time (microseconds)
experimentsimulation - nonlinear
Experiment 112501 (Zhuang, S., Ravichandran, G., Grady D., 2003.)Composite: 8 units of polycarbonate, each 0.74 mm thick, and 8 units of stainless steel,each 0.37 mm thick.Flyer velocity 561 m/s and flyer thickness 2.87 mm.
Nonlinearity parameter A: 300 for polycarbonate and 50 for stainless steel.
● Outline
Motivation
Formulation of the problem
Wave-propagation algorithm
Comparison with experimental
data
● Time history of shock stress
● Time history of shock stress
● Time history of particle
velocity
● Time history of particle
velocity
● Time history of shock stress
● Time history of shock stress
● Time history of shock stress
● Time history of shock stress
● Time history of shock stress
● Time history of shock stress
● Time history of shock stress
● Time history of shock stress
● Time history of shock stress
● Time history of shock stress
● Time history of shock stress
● Time history of shock stress
Discussion
Arkadi Berezovski, Mihhail Berezovski, Jüri Engelbrecht Numerical simulation of nonlinear elastic wave propagation - p. 15/28
Time history of shock stress
0
0.5
1
1.5
2
2.5
3
3.5
1 1.5 2 2.5 3 3.5 4 4.5
Str
ess
(GP
a)
Time (microseconds)
experiment
Experiment 110501 (Zhuang, S., Ravichandran, G., Grady D., 2003.)Composite: 16 units of polycarbonate, each 0.37 mm thick, and 16 units of stainlesssteel, each 0.19 mm thick.Flyer velocity 1043 m/s and flyer thickness 2.87 mm.
● Outline
Motivation
Formulation of the problem
Wave-propagation algorithm
Comparison with experimental
data
● Time history of shock stress
● Time history of shock stress
● Time history of particle
velocity
● Time history of particle
velocity
● Time history of shock stress
● Time history of shock stress
● Time history of shock stress
● Time history of shock stress
● Time history of shock stress
● Time history of shock stress
● Time history of shock stress
● Time history of shock stress
● Time history of shock stress
● Time history of shock stress
● Time history of shock stress
● Time history of shock stress
Discussion
Arkadi Berezovski, Mihhail Berezovski, Jüri Engelbrecht Numerical simulation of nonlinear elastic wave propagation - p. 16/28
Time history of shock stress
0
0.5
1
1.5
2
2.5
3
3.5
1 1.5 2 2.5 3 3.5 4 4.5
Str
ess
(GP
a)
Time (microseconds)
experimentsimulation - linear
Experiment 110501 (Zhuang, S., Ravichandran, G., Grady D., 2003.)Composite: 16 units of polycarbonate, each 0.37 mm thick, and 16 units of stainlesssteel, each 0.19 mm thick.Flyer velocity 1043 m/s and flyer thickness 2.87 mm.
● Outline
Motivation
Formulation of the problem
Wave-propagation algorithm
Comparison with experimental
data
● Time history of shock stress
● Time history of shock stress
● Time history of particle
velocity
● Time history of particle
velocity
● Time history of shock stress
● Time history of shock stress
● Time history of shock stress
● Time history of shock stress
● Time history of shock stress
● Time history of shock stress
● Time history of shock stress
● Time history of shock stress
● Time history of shock stress
● Time history of shock stress
● Time history of shock stress
● Time history of shock stress
Discussion
Arkadi Berezovski, Mihhail Berezovski, Jüri Engelbrecht Numerical simulation of nonlinear elastic wave propagation - p. 17/28
Time history of shock stress
0
0.5
1
1.5
2
2.5
3
3.5
1 1.5 2 2.5 3 3.5 4 4.5
Str
ess
(GP
a)
Time (microseconds)
experimentsimulation - linear
simulation - nonlinear
Experiment 110501 (Zhuang, S., Ravichandran, G., Grady D., 2003.)Composite: 16 units of polycarbonate, each 0.37 mm thick, and 16 units of stainlesssteel, each 0.19 mm thick.Flyer velocity 1043 m/s and flyer thickness 2.87 mm.
Nonlinearity parameter A: 180 for polycarbonate and zero for stainless steel.
● Outline
Motivation
Formulation of the problem
Wave-propagation algorithm
Comparison with experimental
data
● Time history of shock stress
● Time history of shock stress
● Time history of particle
velocity
● Time history of particle
velocity
● Time history of shock stress
● Time history of shock stress
● Time history of shock stress
● Time history of shock stress
● Time history of shock stress
● Time history of shock stress
● Time history of shock stress
● Time history of shock stress
● Time history of shock stress
● Time history of shock stress
● Time history of shock stress
● Time history of shock stress
Discussion
Arkadi Berezovski, Mihhail Berezovski, Jüri Engelbrecht Numerical simulation of nonlinear elastic wave propagation - p. 18/28
Time history of shock stress
0
0.5
1
1.5
2
2.5
3
3.5
4
1 1.5 2 2.5 3 3.5 4 4.5 5
Str
ess
(GP
a)
Time (microseconds)
experiment
Experiment 110502 (Zhuang, S., Ravichandran, G., Grady D., 2003.)Composite: 16 units of polycarbonate, each 0.37 mm thick, and 16 units of stainlesssteel, each 0.19 mm thick.Flyer velocity 1043 m/s and flyer thickness 5.63 mm.
● Outline
Motivation
Formulation of the problem
Wave-propagation algorithm
Comparison with experimental
data
● Time history of shock stress
● Time history of shock stress
● Time history of particle
velocity
● Time history of particle
velocity
● Time history of shock stress
● Time history of shock stress
● Time history of shock stress
● Time history of shock stress
● Time history of shock stress
● Time history of shock stress
● Time history of shock stress
● Time history of shock stress
● Time history of shock stress
● Time history of shock stress
● Time history of shock stress
● Time history of shock stress
Discussion
Arkadi Berezovski, Mihhail Berezovski, Jüri Engelbrecht Numerical simulation of nonlinear elastic wave propagation - p. 19/28
Time history of shock stress
0
0.5
1
1.5
2
2.5
3
3.5
4
1 1.5 2 2.5 3 3.5 4 4.5 5
Str
ess
(GP
a)
Time (microseconds)
experimentsimulation - nonlinear
Experiment 110502 (Zhuang, S., Ravichandran, G., Grady D., 2003.)Composite: 16 units of polycarbonate, each 0.37 mm thick, and 16 units of stainlesssteel, each 0.19 mm thick.Flyer velocity 1043 m/s and flyer thickness 5.63 mm.
Nonlinearity parameter A: 230 for polycarbonate and zero for stainless steel.
● Outline
Motivation
Formulation of the problem
Wave-propagation algorithm
Comparison with experimental
data
● Time history of shock stress
● Time history of shock stress
● Time history of particle
velocity
● Time history of particle
velocity
● Time history of shock stress
● Time history of shock stress
● Time history of shock stress
● Time history of shock stress
● Time history of shock stress
● Time history of shock stress
● Time history of shock stress
● Time history of shock stress
● Time history of shock stress
● Time history of shock stress
● Time history of shock stress
● Time history of shock stress
Discussion
Arkadi Berezovski, Mihhail Berezovski, Jüri Engelbrecht Numerical simulation of nonlinear elastic wave propagation - p. 20/28
Time history of shock stress
0
0.5
1
1.5
2
2.5
3
3.5
1 1.5 2 2.5 3 3.5 4
Str
ess
(GP
a)
Time (microseconds)
experiment
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.
● Outline
Motivation
Formulation of the problem
Wave-propagation algorithm
Comparison with experimental
data
● Time history of shock stress
● Time history of shock stress
● Time history of particle
velocity
● Time history of particle
velocity
● Time history of shock stress
● Time history of shock stress
● Time history of shock stress
● Time history of shock stress
● Time history of shock stress
● Time history of shock stress
● Time history of shock stress
● Time history of shock stress
● Time history of shock stress
● Time history of shock stress
● Time history of shock stress
● Time history of shock stress
Discussion
Arkadi Berezovski, Mihhail Berezovski, Jüri Engelbrecht Numerical simulation of nonlinear elastic wave propagation - p. 21/28
Time history of shock stress
0
0.5
1
1.5
2
2.5
3
3.5
1 1.5 2 2.5 3 3.5 4
Str
ess
(GP
a)
Time (microseconds)
experimentsimulation - linear
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.
● Outline
Motivation
Formulation of the problem
Wave-propagation algorithm
Comparison with experimental
data
● Time history of shock stress
● Time history of shock stress
● Time history of particle
velocity
● Time history of particle
velocity
● Time history of shock stress
● Time history of shock stress
● Time history of shock stress
● Time history of shock stress
● Time history of shock stress
● Time history of shock stress
● Time history of shock stress
● Time history of shock stress
● Time history of shock stress
● Time history of shock stress
● Time history of shock stress
● Time history of shock stress
Discussion
Arkadi Berezovski, Mihhail Berezovski, Jüri Engelbrecht Numerical simulation of nonlinear elastic wave propagation - p. 22/28
Time history of shock stress
0
0.5
1
1.5
2
2.5
3
3.5
1 1.5 2 2.5 3 3.5 4
Str
ess
(GP
a)
Time (microseconds)
experimentsimulation - linear
simulation - nonlinear
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.
Nonlinearity parameter A: 90 for polycarbonate and zero for D-263 glass.
● Outline
Motivation
Formulation of the problem
Wave-propagation algorithm
Comparison with experimental
data
● Time history of shock stress
● Time history of shock stress
● Time history of particle
velocity
● Time history of particle
velocity
● Time history of shock stress
● Time history of shock stress
● Time history of shock stress
● Time history of shock stress
● Time history of shock stress
● Time history of shock stress
● Time history of shock stress
● Time history of shock stress
● Time history of shock stress
● Time history of shock stress
● Time history of shock stress
● Time history of shock stress
Discussion
Arkadi Berezovski, Mihhail Berezovski, Jüri Engelbrecht Numerical simulation of nonlinear elastic wave propagation - p. 23/28
Time history of shock stress
0
0.2
0.4
0.6
0.8
1
1 1.5 2 2.5 3 3.5 4 4.5
Str
ess
(GP
a)
Time (microseconds)
experiment
Experiment 120201 (Zhuang, S., Ravichandran, G., Grady D., 2003.)Composite: 7 units of polycarbonate, each 0.74 mm thick, and 7 units of float glass, each0.55 mm thick.Flyer velocity 563 m/s and flyer thickness 2.87 mm.
● Outline
Motivation
Formulation of the problem
Wave-propagation algorithm
Comparison with experimental
data
● Time history of shock stress
● Time history of shock stress
● Time history of particle
velocity
● Time history of particle
velocity
● Time history of shock stress
● Time history of shock stress
● Time history of shock stress
● Time history of shock stress
● Time history of shock stress
● Time history of shock stress
● Time history of shock stress
● Time history of shock stress
● Time history of shock stress
● Time history of shock stress
● Time history of shock stress
● Time history of shock stress
Discussion
Arkadi Berezovski, Mihhail Berezovski, Jüri Engelbrecht Numerical simulation of nonlinear elastic wave propagation - p. 24/28
Time history of shock stress
0
0.2
0.4
0.6
0.8
1
1 1.5 2 2.5 3 3.5 4 4.5
Str
ess
(GP
a)
Time (microseconds)
experimentsimulation - nonlinear
Experiment 120201 (Zhuang, S., Ravichandran, G., Grady D., 2003.)Composite: 7 units of polycarbonate, each 0.74 mm thick, and 7 units of float glass, each0.55 mm thick.Flyer velocity 563 m/s and flyer thickness 2.87 mm.
Nonlinearity parameter A: 55 for polycarbonate and zero for float glass.
● Outline
Motivation
Formulation of the problem
Wave-propagation algorithm
Comparison with experimental
data
● Time history of shock stress
● Time history of shock stress
● Time history of particle
velocity
● Time history of particle
velocity
● Time history of shock stress
● Time history of shock stress
● Time history of shock stress
● Time history of shock stress
● Time history of shock stress
● Time history of shock stress
● Time history of shock stress
● Time history of shock stress
● Time history of shock stress
● Time history of shock stress
● Time history of shock stress
● Time history of shock stress
Discussion
Arkadi Berezovski, Mihhail Berezovski, Jüri Engelbrecht Numerical simulation of nonlinear elastic wave propagation - p. 25/28
Time history of shock stress
0
0.5
1
1.5
2
2.5
3
1 1.5 2 2.5 3 3.5 4
Str
ess
(GP
a)
Time (microseconds)
experiment
Experiment 120202 (Zhuang, S., Ravichandran, G., Grady D., 2003.)Composite: 7 units of polycarbonate, each 0.74 mm thick, and 7 units of float glass, each0.55 mm thick.Flyer velocity 1056 m/s and flyer thickness 2.87 mm.
● Outline
Motivation
Formulation of the problem
Wave-propagation algorithm
Comparison with experimental
data
● Time history of shock stress
● Time history of shock stress
● Time history of particle
velocity
● Time history of particle
velocity
● Time history of shock stress
● Time history of shock stress
● Time history of shock stress
● Time history of shock stress
● Time history of shock stress
● Time history of shock stress
● Time history of shock stress
● Time history of shock stress
● Time history of shock stress
● Time history of shock stress
● Time history of shock stress
● Time history of shock stress
Discussion
Arkadi Berezovski, Mihhail Berezovski, Jüri Engelbrecht Numerical simulation of nonlinear elastic wave propagation - p. 26/28
Time history of shock stress
0
0.5
1
1.5
2
2.5
3
1 1.5 2 2.5 3 3.5 4
Str
ess
(GP
a)
Time (microseconds)
experimentsimulation - nonlinear
Experiment 120202 (Zhuang, S., Ravichandran, G., Grady D., 2003.)Composite: 7 units of polycarbonate, each 0.74 mm thick, and 7 units of float glass, each0.55 mm thick.Flyer velocity 1056 m/s and flyer thickness 2.87 mm.
Nonlinearity parameter A: 100 for polycarbonate and zero for float glass.
● Outline
Motivation
Formulation of the problem
Wave-propagation algorithm
Comparison with experimental
data
Discussion
● Nonlinear parameter
● Conclusions
Arkadi Berezovski, Mihhail Berezovski, Jüri Engelbrecht Numerical simulation of nonlinear elastic wave propagation - p. 27/28
Nonlinear parameter
Exp. Specimen Units Flyer Flyer Gage A A
soft/hard velocity thickness position PC other
(m/s) (mm) (mm)
112501 PC74/SS37 8 561 2.87 (PC) 0.76 300 50
110501 PC37/SS19 16 1043 2.87 (PC) 3.44 180 0
110502 PC37/SS19 16 1045 5.63 (PC) 3.44 230 0
112301 PC37/GS20 16 1079 2.87 (PC) 3.41 90 0
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.
● Outline
Motivation
Formulation of the problem
Wave-propagation algorithm
Comparison with experimental
data
Discussion
● Nonlinear parameter
● Conclusions
Arkadi Berezovski, Mihhail Berezovski, Jüri Engelbrecht Numerical simulation of nonlinear elastic wave propagation - p. 28/28
Conclusions
■ Good agreement between computations and experimentscan be obtained by means of a non-linear model
● Outline
Motivation
Formulation of the problem
Wave-propagation algorithm
Comparison with experimental
data
Discussion
● Nonlinear parameter
● Conclusions
Arkadi Berezovski, Mihhail Berezovski, Jüri Engelbrecht Numerical simulation of nonlinear elastic wave propagation - p. 28/28
Conclusions
■ Good agreement between computations and experimentscan be obtained by means of a non-linear model
■ The nonlinear behavior of the soft material is affected notonly by the energy of the impact but also by the scatteringinduced by internal interfaces
● Outline
Motivation
Formulation of the problem
Wave-propagation algorithm
Comparison with experimental
data
Discussion
● Nonlinear parameter
● Conclusions
Arkadi Berezovski, Mihhail Berezovski, Jüri Engelbrecht Numerical simulation of nonlinear elastic wave propagation - p. 28/28
Conclusions
■ Good agreement between computations and experimentscan be obtained by means of a non-linear model
■ The nonlinear behavior of the soft material is affected notonly by the energy of the impact but also by the scatteringinduced by internal interfaces
■ The influence of the nonlinearity is not necessary small
● Outline
Motivation
Formulation of the problem
Wave-propagation algorithm
Comparison with experimental
data
Discussion
● Nonlinear parameter
● Conclusions
Arkadi Berezovski, Mihhail Berezovski, Jüri Engelbrecht Numerical simulation of nonlinear elastic wave propagation - p. 28/28
Conclusions
■ Good agreement between computations and experimentscan be obtained by means of a non-linear model
■ The nonlinear behavior of the soft material is affected notonly by the energy of the impact but also by the scatteringinduced by internal interfaces
■ The influence of the nonlinearity is not necessary small■ Additional experimental information is needed to validate the
proposed nonlinear model