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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
Services for Science and Education – United Kingdom
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:
( ) ( )
a
j = -
÷
ø
ö ç
è
æ
a = a
r
r = a a = =
=
1 1
V E E P,
V
V P P ;
E P,V E P ,V ; m for matrix
m
m
m
m
m m m
( )
j ¶j
¶
+
¶
¶
+
¶
¶ =
= j
d E dV
V
E dP
P
E dE
E E P,V,
( )
dP
dP
d d
P
j j =
j = j
dV
V
E dP
dP
E d
P
E dE
¶
¶
+ ÷
÷
ø
ö ç
ç
è
æ j
¶j
¶
+
¶
¶ =
dE = -(P + q)dV
dP
E d
P
E
V
E P q
dV
dP
j
¶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
m
m
m
m
m
m
m
m
m
m
V
E V
P
E
1
V P
V
P E
P
E E
V
E 1
V
V
V
E
V
E
P
E
1
1
P
P
P
E
P
E
¶
¶ - ¶
¶
- j = ¶j
¶
¶
¶
+
¶j
¶
¶
¶ = ¶j
¶
¶
¶ = - j ¶
¶
¶
¶ = ¶
¶
¶
¶
- j = ¶
¶
¶
¶ = ¶
¶
v 0
r
2
r
v 0
v 0
r
2
r
v
÷ =
ø
ö ç
è
æ +
¶
¶
r = \
÷ =
ø
ö ç
è
æ +
¶
¶
r = r
!
!
( ) ( )
( )
4
4 2
2
r
a
2 2 2
2 2
2
r
a a
v
d vr 0 ; vr a a
vr 0
r
vr 0 ;
r r
1
!
!
=
= =
= ¶
¶ = ¶
¶
ò
( ) 0
r
2
r
v r s r 0 + s - s = ¶
¶s
r ! +
sr ss v!
( ) 4
4 2
2
2 2 1
2 r
a a
r
a a 2aa
r
1
r
v
v
t
v
v ! ! !! ! ¶
¶ = + +
¶
¶
+
¶
¶ =
sr - ss