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

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


Outline

■ Motivation: experiments and theory

● Outline

Motivation




data

Discussion


Outline

■ Motivation: experiments and theory■ Formulation of the problem

● Outline

Motivation




data

Discussion


Outline

■ Motivation: experiments and theory■ Formulation of the problem■ Wave-propagation algorithm

● Outline

Motivation




data

Discussion


Outline

■ Motivation: experiments and theory■ Formulation of the problem■ Wave-propagation algorithm■ Comparison with experimental data

● Outline

Motivation




data

Discussion


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


● Theory by Chen et al. (2004)





data

Discussion


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


(2003)








data

Discussion


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


(2003)








data

Discussion



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


Gage position: 3.41 mm from impact boundary.

● Outline

Motivation


(2003)








data

Discussion


Theory by Chen et al. (2004)

■ Analytical solution of one-dimensional linear wavepropagation in layered heterogeneous materials

● Outline

Motivation


(2003)








data

Discussion




■ Approximate solution for shock loading by invoking ofequation of state

● Outline

Motivation


(2003)








data

Discussion





■ Wave velocity, thickness and density for the laminatessubjected to shock loading, all depend on the particlevelocity

● Outline

Motivation


(2003)








data

Discussion






■ 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


(2003)








data

Discussion






■ 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


(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.

● Outline

Motivation


● Geometry of the problem

● Formulation of the problem



data

Discussion


Geometry of the problem

● Outline

Motivation






data

Discussion


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






data

Discussion




ρ∂v

∂t=

∂σ

∂x,

∂ε

∂t=

∂v

∂x



■ 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






data

Discussion




ρ∂v

∂t=

∂σ

∂x,

∂ε

∂t=

∂v

∂x



■ 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



● Wave-propagation algorithm


data

Discussion



■ Finite-volume numerical schemeLeVeque, R.J., 1997. Wave propagation algorithms for multidimensional hyperbolic

systems. J. Comp. Physics 131, 327–353.

● Outline

Motivation





data

Discussion





■ Numerical fluxes are determined by solving the Riemannproblem at each interface between discrete elements

● Outline

Motivation





data

Discussion






■ Reflection and transmission of waves at each interface arehandled automatically

● Outline

Motivation





data

Discussion







■ Second-order corrections are included

● Outline

Motivation





data

Discussion







■ 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




data



● Time history of particle

velocity


velocity













Discussion



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




data




velocity


velocity













Discussion



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


Nonlinearity parameter A: 300 for polycarbonate and 50 for stainless steel.

● Outline

Motivation




data




velocity


velocity













Discussion


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


● Outline

Motivation




data




velocity


velocity













Discussion


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)



Nonlinearity parameter A: 300 for polycarbonate and 50 for stainless steel.

● Outline

Motivation




data




velocity


velocity













Discussion



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




data




velocity


velocity













Discussion



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)



● Outline

Motivation




data




velocity


velocity













Discussion



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)


simulation - nonlinear


Nonlinearity parameter A: 180 for polycarbonate and zero for stainless steel.

● Outline

Motivation




data




velocity


velocity













Discussion



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


● Outline

Motivation




data




velocity


velocity













Discussion



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)



Nonlinearity parameter A: 230 for polycarbonate and zero for stainless steel.

● Outline

Motivation




data




velocity


velocity













Discussion



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


● Outline

Motivation




data




velocity


velocity













Discussion



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)



● Outline

Motivation




data




velocity


velocity













Discussion



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)


simulation - nonlinear


Nonlinearity parameter A: 90 for polycarbonate and zero for D-263 glass.

● Outline

Motivation




data




velocity


velocity













Discussion



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




data




velocity


velocity













Discussion



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)



Nonlinearity parameter A: 55 for polycarbonate and zero for float glass.

● Outline

Motivation




data




velocity


velocity













Discussion



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


● Outline

Motivation




data




velocity


velocity













Discussion



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)



Nonlinearity parameter A: 100 for polycarbonate and zero for float glass.

● Outline

Motivation




data

Discussion

● Nonlinear parameter

● Conclusions


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




data

Discussion


● Conclusions


Conclusions

■ Good agreement between computations and experimentscan be obtained by means of a non-linear model

● Outline

Motivation




data

Discussion


● Conclusions


Conclusions


■ 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




data

Discussion


● Conclusions


Conclusions



■ The influence of the nonlinearity is not necessary small

● Outline

Motivation




data

Discussion


● Conclusions


Conclusions



■ The influence of the nonlinearity is not necessary small■ Additional experimental information is needed to validate the

proposed nonlinear model

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

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