EJAS-13675.pdf

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European Journal of Applied Sciences – Vol. 10, No. 6

Publication Date: December 25, 2022

DOI:10.14738/aivp.106.13675. Partom, Y. (2022). Modeling Dynamic Compaction of Porous Materials. European Journal of Applied Sciences, 10(6). 577-586.

Services for Science and Education – United Kingdom

Modeling Dynamic Compaction of Porous Materials

Y. Partom

Retired, 18 HaBanim, Zikhron Ya,

akov 3094017, ISRAEL

ABSTRACT

To model dynamic compaction of a porous material we need: 1) an equation of state

(EOS) for the porous material in terms of the EOS of its matrix; and 2) a compaction

law. For an EOS people usually use Hermann's suggestion, as in his Pα [1] model. For

a compaction law people usually use the results of a spherical shell collapse analysis

(Carroll and Holt model [2]). In their original paper Carroll and Holt do both: the

quasi-static shell collapse and the dynamic shell collapse. In their dynamic analysis,

however, they ignore density changes of the matrix. In what follows we: 1) revisit

the spherical shell collapse problem but with density changes taken into account;

2) develop a dynamic compaction law based on our overstress principle; 3)

implement the different compaction laws mentioned above in a hydro-code; and 4)

run a planar impact problem and compare histories and profiles obtained with the

different compaction laws. We find that: 1) dynamic compaction laws give entirely

different results from quasi-static compaction laws; 2) taking density changes into

account do make a certain difference.

INTRODUCTION

To model dynamic compaction of a porous material we need: 1) an equation of state (EOS) for

the porous material in terms of the EOS of its matrix; and 2) a compaction law. Most compaction

models use Herrmann's EOS [1], and we're using that EOS too. As for a compaction law, we

distinguish the following:

• An assumed compaction law usually calibrated from tests

• A compaction law deduced from a model on the meso-scale

Examples for assumed compaction laws are:

• Instantaneous pore collapse

• Herrmann's Pα compaction law

An example for a compaction law deduced from a model on the meso-scale is the well-known

spherical shell model by Carroll and Holt [2]. Carrol and Holt developed quasi-static and

dynamic spherical shell models, both of them neglecting density changes.

Here we focus on dynamic compaction. We use and compare two approaches:

• Following Carrol and Holt, we revisit the spherical shell collapse model taking density

changes into account

• We use an Overstress approach model. According to this approach, the rate of pore collapse

(or porosity change) is proportional to the overstress from an assumed quasi-static

compaction law

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European Journal of Applied Sciences (EJAS) Vol. 10, Issue 6, December-2022

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Herrmann’s EOS

Herrmann [1] assumed that:

(1)

where E=internal energy, P=pressure, V=specific volume, r=density, α=distention ratio and

j=porosity. Herrmann's assumptions (first two equations in (1)) are still used today, although

we haven't seen them verified directly or indirectly.

To complete the equation of state one needs a α(P) or a j(P) relation, and this is what a

compaction law defines. Defining a j(P) relation, we employ the EOS in a hydro-code as follows:

The EOS is generally given by:

(2)

Using:

(3)

we get:

(4)

Together with conservation of energy:

(5)

where q is artificial viscosity, we finally get:

(6)

where the partial derivatives in Eq. (6) are given by:

( ) ( )

j = -

ö ç

a = a

r = a a = =

1 1

V E E P,

V P P ;

E P,V E P ,V ; m for matrix

m m m

( )

j ¶j

¶ =

= j

d E dV

E dP

E dE

E E P,V,

( )

d d

j j =

j = j

E dP

E d

E dE

+ ÷

ö ç

æ j

¶j

¶ =

dE = -(P + q)dV

E d

E P q

¶j

+ +

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Partom, Y. (2022). Modeling Dynamic Compaction of Porous Materials. European Journal of Applied Sciences, 10(6). 577-586.

URL: http://dx.doi.org/10.14738/aivp.106.13675

(7)

We see that we get two ordinary differential equations (ODEs) with two unknowns. These are

the second of Eqs. (3) and Eq. (6). These two ODEs apply separately to each computational cell

and during each time step. The independent variable (V) is known at the beginning and at the

end of the time step, and we assume (as usual) that it varies linearly during the time step. The

two ODEs can be integrated numerically by a standard ODE solver (e.g., 4th order Runge-Kutta).

Spherical shell model

We start with dynamic spherical shell collapse equations without density changes. The mass

conservation equation is:

(8)

where v is the radial velocity. Integrating with respect to r we get:

(9)

where a and b are the inner and outer boundary radii of the collapsing shell.

The momentum equation is:

(10)

where is the radial stress component and is the tangential stress component. For in

Eq. (10) we have from Eq. (9):

(11)

We represent the difference in Eq. (10) by its average over the shell thickness:

( )

( ) m m 2

E V

V P

P E

E E

E 1

¶ - ¶

- j = ¶j

¶j

¶ = ¶j

¶ = - j ¶

¶ = ¶

- j = ¶

¶ = ¶

v 0

÷ =

ö ç

æ +

r = \

÷ =

ö ç

æ +

r = r

( ) ( )

( )

4 2

2 2 2

2 2

a a

d vr 0 ; vr a a

vr 0

vr 0 ;

r r

= =

= ¶

¶ = ¶

( ) 0

v r s r 0 + s - s = ¶

¶s

r ! +

sr ss v!

( ) 4

4 2

2 2 1

2 r

a a

a a 2aa

v ! ! !! ! ¶

¶ = + +

¶ =

sr - ss

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European Journal of Applied Sciences (EJAS) Vol. 10, Issue 6, December-2022

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(12)

where G is the shear modulus, the subscript 0 refers to the initial configuration, and where we

check that: .

Finally, substituting into the momentum equation and integrating from a to b we get:

(13)

Assuming Pb is given, we get from Eqs. (13) two simultaneous first order ODEs:

(14)

which can be integrated numerically.

In Fig. 1 we show an example of results obtained from these equations. The material is a

stainless-steel shell with 20% porosity and Pb=20GPa.

Figure 1. Inner and outer boundary histories for a dynamic spherical shell collapse with density

changes not considered

We see from Fig. 1 that towards closure, the inner boundary velocity becomes extremely fast,

and will probably get unstable.

We performed similar computations with different values of the boundary pressure Pb.

( ) ÷

ö ç

+ - -

- s - s @ s - s = e - e = b a

b a

b a 2G 2G 0 0 0 0

r s r s r s

sr - ss £ Y

( ) ( )

( ) 0 0 b

r s 4 4

4 2

2 2 b 1

or schematically : a F a ,b ,a,b,a,P

a a

a a 2aa

!! !

!! ! ! "

s - s

÷ +

ö ç

æ + - r

÷ =

ö ç

æ + -

3 3

0 0 b

b a b a

a z ; z F(a ,b ,a,b, z,P )

= + -

! = ! =

0.2

0.4

0.6

0.8

1.2

0 0.05 0.1 0.15 0.2 0.25

Shell radius (mm)

Time (microsec)

Phi=20%

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Partom, Y. (2022). Modeling Dynamic Compaction of Porous Materials. European Journal of Applied Sciences, 10(6). 577-586.

URL: http://dx.doi.org/10.14738/aivp.106.13675

In Fig. 2 we show closure times for different values of Pb obtained from these computations. We

see from Fig. 2 that for Pb<1GPa closure time becomes long, and the cavity might not close

completely.

Figure 2. Hole closure times for a dynamic spherical shell collapse with density changes not

considered.

Next, we upgrade the analysis to take density changes into account.

Mass conservation equation is now:

(15)

where is the average density across the shell thickness, and is the average density rate of

change across the shell thickness. We're only able to take average density changes into account.

We use a constant (average) bulk modulus K so that:

(16)

The momentum equation is:

(17)

0.5

1.5

2.5

0 5 10 15 20 25

Closure time (microsec)

Outer boundary pressure (GPa)

phi=20%

r ÷ = - ø

ö ç

æ +

¶ ÷ =

ö ç

æ +

r + r ! ! v

v 0 ;

r r!

0 b

K P

; K

P 1

; P P

P 1

+ = r

r @ +

@ ÷

r = r +

! !

( ) 0

v r s r + s - s = ¶

¶s

r! +

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and the over whole mass conservation is:

(18)

We see that for a constant Pb value, the mass conservation equation is the same as before, and

we may also proceed as before.

In Fig. 3 we show the porosity history for the same problem as before, and compare it to the

what we obtained from the constant density run.

Figure 3. Porosity closure history. Red (lower curve): no density changes. Blue (upper curve)

considering density changes.

We see from Fig. 3 thatthe influence of density changes is small and that it slows the compaction

process.

If Pb doesn’t change over time, the system of equations gets extremely complicated. We

therefore choose not to work with these equations. Instead, we approximate Pb(t) by a staircase

curve so that for each time step we have a constant Pb. The error introduced can be checked by

changing the integration time step.

The implementation of this scheme into a hydro-code is similar to that of Eqs. (3,6). The only

difference is that we use rates:

( ) 3

3 3 0 b a b - a

r - =

0.05

0.1

0.15

0.2

0.25

0 0.05 0.1 0.15 0.2 0.25 0.3

No density change

With density change

Porosity

Time (microsec)

Phi=20%

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Partom, Y. (2022). Modeling Dynamic Compaction of Porous Materials. European Journal of Applied Sciences, 10(6). 577-586.

URL: http://dx.doi.org/10.14738/aivp.106.13675

(19)

Identifying the matrix pressure and density with the average shell pressure and density, we

finally have:

(20)

In Figs. 4 and 5 we compare results of a planar shock run with the dynamic shell model to a

similar run with an equivalent quasi-static shell model (which we haven't discussed here).

Figure 4. Pressure history for a planar shock into a 20% porous stainless-steel sample. Red:

quasi-static shell model. Blue: dynamic shell model.

( )

V V V

E P

E V

E P

P q E V P

E P q V

E V

E P

E E

E E P,V,

new old

- =

j ¶ ¶

¶ ¶j - ¶ ¶

+ + ¶ ¶

\ = -

= - +

j ¶j

¶ =

= j

! ! !

! !

! ! ! !

( )

( ) ( )

( ) ( ) m b m m b

3 3 0

0 0 b

P P ; ; P 1 P 1 P

; 3 1

a b a b a ;

a z ; z F a ,b , ,a, z,P

= r = r = - j = - j

- j = j = j - j

r = +

= = r

! !

4 5 6 7 8 9 10 11

Quasi-static shell model

Dynamic shell model

Pressure (GPa)

Time (microsec)

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European Journal of Applied Sciences (EJAS) Vol. 10, Issue 6, December-2022

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Figure 5. Pressure-volume history for a planar shock into a 20% porous stainless-steel sample.

Red: quasi-static shell model Blue: dynamic shell model.

We see that there's a big difference between the quasi-static and the dynamic models:

• For the quasi-static model, the pores close at a relatively low pressure, while for the

dynamic model they close along the whole pressure range.

• For the dynamic model there is no elastic precursor.

• For the quasi-static model, the shock is sharp, while for the dynamic model the shock is

smeared, like for a visco-plastic material.

Overstress model

The overstress approach is a general concept that can be used whenever a dynamic system has

a quasi-static equilibrium state. According to the overstress approach, the dynamic state always

tends towards the quasi-static equilibrium state at a rate that depends on the deviation (the

overstress) from the quasi-static state. For the problem of pore closure under hand, let the

quasi-static state be:

(21)

Then, at any stage during the pore closure process we have by the overstress approach:

(22)

The simplest form for Eq. (22) is a linear relation:

(23)

where the coefficient A may be calibrated from tests. Combining this with Eqs. (19), we get a

system of two first order ODEs to integrate for each cell during each time step as before.

0.11 0.12 0.13 0.14 0.15 0.16

Quasi-static shell model

Dynamic shell model

Pressure (GPa)

Specific volume (cc/g)

(P) jqs = jqs

j = ( - (j)) F P Pqs !

j = - ( - (j)) > (j) qs P Pqs ! A P P ; for

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Partom, Y. (2022). Modeling Dynamic Compaction of Porous Materials. European Journal of Applied Sciences, 10(6). 577-586.

URL: http://dx.doi.org/10.14738/aivp.106.13675

In Figs. 6 and 7 we show examples of such planar impact simulations with two values of A: 0.01

and 0.02 (GPa μs)-1. The quasi-static compaction law assumed in these examples is:

(24)

where Ym is the matrix flow stress.

Figure 6. Overstress model. Pressure history for a planar shock into a 20% porous Stainless- steel sample. (Red and blue lines are for different values of the coefficient A).

We see from Figs. 6 and 7 that the rate coefficient A determines the precursor level and the rise

time. The P(V) curve is similar to that obtained from the shell model, except for the level of the

precursor wave.

3 m 2

qs ; small number like 10 1 P Y n - d = ÷

ö ç

j + d = !

3 4 5 6 7 8 9 10 11

0.01 1/(GPa microsec)

0.02 1/(GPa microsec)

Pressure (GPa)

Time (microsec)

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European Journal of Applied Sciences (EJAS) Vol. 10, Issue 6, December-2022

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Figure 7. Overstress model. Pressure-volume histories for a planar shock entering a 20%

porous Stainless-steel sample. (Red and blue lines are for different values of the coefficient A).

SUMMARY

We focus on dynamic compaction of porous materials. We use Herrmann'EOS and two dynamic

compaction laws. The first compaction law is derived from Carrol and Holt’s spherical shell

model. We start from mass and momentum conservation equations and integrate them across

the shell radius. We obtain ODEs for Pressure (P) and porosity (j) as a function of time, to be

integrated numerically for each computational cell during each time step. We're able to account

for average density changes over the shell.

The second compaction law that we use is derived from the overstress approach. It requires a

quasi-static compaction law and a rate of approach to it that can be calibrated from tests.

For both compaction laws we show examples of a planar shock of 20GPa entering a 20% porous

Stainless-steel target.

References

1. W. Herrmann, Constitutive equation for the dynamic compaction of ductile porous materials, J. Appl. Phys.,

vol. 40, pp. 2490-2499, 1969.

2. M.M. Carroll and A.C. Holt, Static and dynamic pore collapse relations for ductile porous materials, J. Appl.

Phys. Vol. 43, pp. 1626-1636, 1972.

0.11 0.12 0.13 0.14 0.15 0.16

0.01 1/(GPa microsec)

0.02 1/(GPa microsec)

Pressure (GPa)

Specific volume (cc/g)