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

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

Outline

q Outline s Motivation: experiments and theory
Motivation

Formulation of the problem

Wave-propagation algorithm

Comparison with experimental
data

Discussion


Outline

Motivation
s Formulation of the problem


data

Discussion


Outline

Motivation

s Wave-propagation algorithm
data

Discussion


Outline

Motivation

s Comparison with experimental data
data

Discussion


Outline

Motivation

s Comparison with experimental data
data
s Conclusions
Discussion


Experiments by Zhuang et al. (2003)

q Outline

Motivation
q Experiments by Zhuang et al.
(2003)
q Time history of shock stress
q Theory by Chen et al. (2004)



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.)


Time history of shock stress
3.5
experiment
q Outline
3
Motivation
(2003) 2.5
Stress (GPa)
q Theory by Chen et al. (2004) 2

1.5

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.


3.5
experiment
simulation - linear
q Outline
3
Motivation
(2003) 2.5
Stress (GPa)
q Theory by Chen et al. (2004) 2

1.5

Comparison with experimental 1
data

Discussion
0.5

0
1 1.5 2 2.5 3 3.5 4
Time (microseconds)
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.


Theory by Chen et al. (2004)

q Outline s Analytical solution of one-dimensional linear wave
Motivation
propagation in layered heterogeneous materials
(2003)



data

Discussion



Motivation
(2003)
s Approximate solution for shock loading by invoking of
equation of state



data

Discussion



Motivation
(2003)
equation of state
s Wave velocity, thickness and density for the laminates
subjected to shock loading, all depend on the particle
Wave-propagation algorithm velocity
data

Discussion



Motivation
(2003)
equation of state
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.



Motivation
(2003)
equation of state
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.



q Outline

Motivation
(2003)



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.


Geometry of the problem

q Outline

Motivation

q Geometry of the problem
q Formulation of the problem


data

Discussion


s Basic equations
Conservation of linear momentum and kinematical compatibility:

q Outline
∂v ∂σ ∂ε ∂v
Motivation
ρ = , =
∂t ∂x ∂t ∂x
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
v(x, t) the particle velocity.

data

Discussion


s Basic equations

q Outline
∂v ∂σ ∂ε ∂v
Motivation
ρ = , =
∂t ∂x ∂t ∂x

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 Basic equations

q Outline
∂v ∂σ ∂ε ∂v
Motivation
ρ = , =
∂t ∂x ∂t ∂x

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.



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.
q Wave-propagation algorithm

data

Discussion



Motivation
s Numerical ﬂuxes are determined by solving the Riemann
Comparison with experimental problem at each interface between discrete elements
data

Discussion



Motivation
data
s Reﬂection and transmission of waves at each interface are
Discussion
handled automatically



Motivation
data
Discussion
s Second-order corrections are included



Motivation
data
Discussion
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.


1.4
experiment
q Outline
1.2
Motivation

Formulation of the problem 1
Stress (GPa)
0.8
data
q Time history of shock stress 0.6
q Time history of particle
velocity
q Time history of particle 0.4
velocity
0.2
0
0 0.5 1 1.5 2 2.5
q Time history of shock stress Time (microseconds)
q Time history of shock stress Composite: 8 units of polycarbonate, each 0.74 mm thick, and 8 units of stainless steel,
each 0.37 mm thick.
Discussion Flyer velocity 561 m/s and ﬂyer thickness 2.87 mm.


1.4
experiment
simulation - nonlinear
q Outline
1.2
Motivation

Formulation of the problem 1
Stress (GPa)
0.8
data
velocity
velocity
0.2
0
0 0.5 1 1.5 2 2.5
each 0.37 mm thick.
Nonlinearity parameter A: 300 for polycarbonate and 50 for stainless steel.


Time history of particle velocity
0.35
experiment
q Outline
0.3
Motivation

Particle velocity (km/s)
Formulation of the problem 0.25

0.2
data
velocity
velocity
0.05
0
4 5 6 7 8 9
each 0.37 mm thick.


Time history of particle velocity
0.35
experiment
q Outline
0.3
Motivation

Particle velocity (km/s)

0.2
data
velocity
velocity
0.05
0
4 5 6 7 8 9
each 0.37 mm thick.
Nonlinearity parameter A: 300 for polycarbonate and 50 for stainless steel.


3.5
experiment
q Outline
3
Motivation

Stress (GPa)
2
data
velocity
q Time history of particle 1
velocity
0.5
0
1 1.5 2 2.5 3 3.5 4 4.5
q Time history of shock stress Composite: 16 units of polycarbonate, each 0.37 mm thick, and 16 units of stainless
steel, each 0.19 mm thick.


3.5
experiment
simulation - linear
q Outline
3
Motivation

Stress (GPa)
2
data
velocity
velocity
0.5
0
1 1.5 2 2.5 3 3.5 4 4.5


3.5
experiment
simulation - linear
q Outline
3 simulation - nonlinear
Motivation

Stress (GPa)
2
data
velocity
velocity
0.5
0
1 1.5 2 2.5 3 3.5 4 4.5
Nonlinearity parameter A: 180 for polycarbonate and zero for stainless steel.


4
experiment
q Outline 3.5
Motivation
3

Stress (GPa) 2.5
data 2
velocity
velocity 1
0
1 1.5 2 2.5 3 3.5 4 4.5 5


4
experiment
q Outline 3.5
Motivation
3

Stress (GPa) 2.5
data 2
velocity
velocity 1
0
1 1.5 2 2.5 3 3.5 4 4.5 5
Nonlinearity parameter A: 230 for polycarbonate and zero for stainless steel.


3.5
experiment
q Outline
3
Motivation

Stress (GPa)
2
data
velocity
velocity
0.5
0
1 1.5 2 2.5 3 3.5 4
q Time history of shock stress Composite: 16 units of polycarbonate, each 0.37 mm thick, and 16 units of D-263 glass,
each 0.20 mm thick.


3.5
experiment
simulation - linear
q Outline
3
Motivation

Stress (GPa)
2
data
velocity
velocity
0.5
0
1 1.5 2 2.5 3 3.5 4
each 0.20 mm thick.


3.5
experiment
simulation - linear
q Outline
3 simulation - nonlinear
Motivation

Stress (GPa)
2
data
velocity
velocity
0.5
0
1 1.5 2 2.5 3 3.5 4
each 0.20 mm thick.
Nonlinearity parameter A: 90 for polycarbonate and zero for D-263 glass.


1
experiment
q Outline

Motivation 0.8

Stress (GPa)
0.6
data
0.4
velocity
velocity
0
1 1.5 2 2.5 3 3.5 4 4.5
q Time history of shock stress Composite: 7 units of polycarbonate, each 0.74 mm thick, and 7 units of ﬂoat glass, each
0.55 mm thick.


1
experiment
q Outline

Motivation 0.8

Stress (GPa)
0.6
data
0.4
velocity
velocity
0
1 1.5 2 2.5 3 3.5 4 4.5
0.55 mm thick.
Nonlinearity parameter A: 55 for polycarbonate and zero for ﬂoat glass.


3
experiment
q Outline
2.5
Motivation


Stress (GPa) 2

data 1.5
velocity 1
velocity
0
1 1.5 2 2.5 3 3.5 4
0.55 mm thick.


3
experiment
q Outline
2.5
Motivation


Stress (GPa) 2

data 1.5
velocity 1
velocity
0
1 1.5 2 2.5 3 3.5 4
0.55 mm thick.
Nonlinearity parameter A: 100 for polycarbonate and zero for ﬂoat glass.


Nonlinear parameter

q Outline
Exp. Specimen Units Flyer Flyer Gage A A
Motivation
soft/hard velocity thickness position PC other
(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.


Conclusions

q Outline s Good agreement between computations and experiments
Motivation
can be obtained by means of a non-linear model


data

Discussion
q Nonlinear parameter
q Conclusions


Conclusions

Motivation
s The nonlinear behavior of the soft material is affected not
only by the energy of the impact but also by the scattering
data induced by internal interfaces
Discussion
q Conclusions


Conclusions

Motivation
Discussion s The inﬂuence of the nonlinearity is not necessary small
q Conclusions


Conclusions

Motivation
Discussion s The inﬂuence of the nonlinearity is not necessary small
q Conclusions s Additional experimental information is needed to validate the
proposed nonlinear model


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

Recommended

Recommended

More Related Content

What's hot

What's hot (9)

Viewers also liked

Viewers also liked (6)

Recently uploaded

Recently uploaded (20)

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