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European Journal of Applied Sciences – Vol. 11, No. 2

Publication Date: April 25, 2023

DOI:10.14738/aivp.112.14385.

Partom, Y. (2023). Modeling Dynamic Rate Dependent Pore Closure with a Range of Pore Sizes. European Journal of Applied

Sciences, Vol - 11(2). 491-497.

Services for Science and Education – United Kingdom

Modeling Dynamic Rate Dependent Pore Closure with a Range of

Pore Sizes

Y. Partom

18 Habanim, Zikhron Ya'akov 3094017, Israel

ABSTRACT

Previously we presented a model (which we call PORT) with rate dependent pore

closing/opening, for which we assumed that all pores close/open with the same

dynamics [1]. Here we upgrade PORT to take into account different dynamics as

function of pore size. We represent different pore sizes by their volume (v), and we

have k discrete pore sizes. We therefore call this model VK. For the ith pore size we

have ni, i=1,k pores per unit mass. We’re not aware of information on pore size

distributions of porous materials, and we assume arbitrarily that pore sizes are

initially distributed with a log-normal distribution. Similar to PORT, we define

quasistatic pore closure curves that depend on pore volume v, and we compute the

rate of pore closure with a linear overstress equation relative to these curves. From

the values of v and vdot (rate of change of v) we then compute (for each cell and for

each time) the overall porosity  and its rate of change dot. Finally, we compute

Pdot and Tdot (P=pressure and T=temperature) in the same way as in PORT, using

the equation of state of the porous material. To show how our VK model works, we

apply it to a simple 1D problem: a 20GPa sustained pressure pulse enters a porous

aluminum target. We show histories of pressure, temperature and porosity at

several locations into the target. We compare these curves to the ones obtained for

k=1 (which are as in PORT).

INTRODUCTION

Some time ago we developed a model for time dependent pore closing/opening in porous

materials [1, and see also references therein], which we call PORT (POR for porous and T to

indicate that we also calculate temperature changes). In PORT we assume that all pores

close/open with the same dynamics and can be therefore represented by a single variable

which is the porosity . But it makes more sense to assume that pores of different sizes have

different closing/opening dynamics. For instance, we tend to assume intuitively that it would

be harder to close small pores than to close large pores. Accordingly, we develop here a time

dependent pore closure model that treats pores of different sizes, and assumes for them

different dynamics.

PORE DESCRIPTION AND INITIAL CONDITIONS

We represent a pore size by its volume v. At this stage of model development, we ignore the

influence of pore shape. We consider a finite number k of pore volumes. For each pore volume

vi(i=1,k) there are ni pores of the same volume per unit mass. The total void volume per unit

mass (=void) and the porosity  are then given by:

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493

Partom, Y. (2023). Modeling Dynamic Rate Dependent Pore Closure with a Range of Pore Sizes. European Journal of Applied Sciences, Vol - 11(2).

491-497.

URL: http://dx.doi.org/10.14738/aivp.112.14385.

Figure 1. Initial pore volume distribution.

PORE CLOSURE EQUATIONS

As in PORT, here too the pore closure equations are rate equations (ODEs), that we integrate

separately for each computational cell and for each time step. The number of rate equations is

k+2, k for the different pore volumes and 2 for pressure and temperature. For each of the k pore

volumes (and for each cell and time step) we have a separate quasistatic closure curve, and we

assume that a quasistatic closure curve depends on pore volume by:

P P exp v v qsi c i c = −( ) (6)

where the parameters Pc,vc are common to all quasistatic closure curves. These parameters can

be calibrated from quasistatic tests. We determine the rate of pore closure by the overstress

equation:

i c qsi qsi i ( )

i

v A P P P P V 0 v 0

otherwise :v 0

= −   

=

(7)

Knowing the values of vi and vdoti, we proceed to calculate  and dot by the following

equations:

i i i i

i i

2

void v n voiddot v n

void voiddot void V

V V V

= =

 =  = −

 

(8)