2.1. Particle Mechanics Approach
Particle scale modeling of the thermally-assisted compaction process requires an extension of the discrete element method to account for the integration of heat conduction (e.g., [
25,
34,
35]). Starting from the well-known theory of Hertzian deformation [
20], heat conduction through the conforming contact of spherical particles [
2,
4] is adopted for the case of thermally-assisted compaction. Under steady state conditions, the total of the forces acting on individual particle
m from neighboring particles
and the total heat transferred to particle
m are zero, that is:
where
is the unit normal vector defined from
to
, i.e., from the center of particle
n to the center of particle
m.
Johnson studied the elastic deformation of locally spherical particles that are subject to a compression load by contact mechanics considerations [
36]. Small-strain deformation of conforming surfaces results in a flat circular contact area. The collinear, elastic, contact force between particles
m and
n is defined through reduced elastic modulus
, reduced particle radius
and overlap
between these particles. Specifically, the contact force is:
where:
Similar to previous studies in the literature [
35,
37], in the present study, a linear thermal expansion formulation is taken into consideration; that is
, where
is the thermal expansion coefficient,
is the reference temperature and
is the radius of the particle at the reference temperature. Due to the fact that the contact geometry depends highly on the heat conduction between the consecutive conforming particle pairs, it is expected to capture a distribution of the contact area formation throughout the compacted medium.
The major heat transfer mechanisms in compacted particle beds consist of conduction through solid particles, conduction through the contact area between two touching particles, conduction to/from interstitial fluid, heat transfer via convection, radiation between particle surfaces and radiation between neighboring voids [
34]. For a system of granular media where the thermal conductivity of the solid particles is much larger than that of the interstitial medium, the driving mechanisms for the heat transfer are the first two. Restricting attention to the problem of thermally-assisted compaction of spherical particles in a vacuum, we focus on thermal contact models that consider the conduction through solid particles and the contact areas between touching particles.
The analytical solution of the heat conduction through the solid phase of ordered spherical particles has been proposed by Chan and Tien [
2] and Kaganer [
3]. Moreover, the problem of heat transfer regarding the compaction of particles that are in or nearly in contact is deeply investigated by Batchelor and O’Brien [
4]. In an attempt to find the approximate effective thermal conductivity of ordered and randomly-packed granular beds, Batchelor and O’Brien discussed the heat flux across the flat, circular contact surface between smooth, conforming and elastic particles. In this study, we adopt Batchelor and O’Brien’s model for predicting the heat conductance, which is the ability of two touching surfaces to transmit heat through their contact interface. Heat flux across the contact area of two spherical, smooth particles is given by:
where
is the arithmetic mean of the thermal conductivities of two conforming particles and
is the Hertzian contact area. These are defined as:
The total heat flow to an individual particle, Equation (
2), is calculated by adding the heat flow, Equation (
7), across each contact surface shared with its neighboring particles. Thermal contact models introduced in the literature [
2,
4], Equation (
2), assume that the resistance to heat transfer inside the particle is significantly smaller than the resistance between the particles, i.e., a Biot number much less than one:
where
A is the cross-sectional area,
. This assumption was applied by several authors in earlier studies [
34,
38], which also enforces the condition of
, i.e., of small-strain deformation of elastic bodies in contact.
Referring to the previous experimental studies on regular and random packing of granular media, Walton points out that although the regular packing models are founded on strict assumptions, they are capable of capturing the vast majority of the characteristics of a real granular media [
33]. In the present study, we consider a simple cubic packing of identical elastic spheres, which are constrained between parallel planes of infinite extent. A compression load and a temperature gradient are applied along the major and finite direction. Stress and heat flux are defined to depend only on externally-applied thermal and mechanical loads, and the weight of the particles is neglected. For such regular packings, each layer of the arrangement is isothermal normal to the direction of applied load. Furthermore, since these transversely-oriented particles are, at most, at the contact point, for each particle there is only one pair of contact areas aligned with the direction of applied thermal and mechanical load. Due to the symmetry of the problem, it is sufficient to consider a single column of a square cross-section containing the longitudinally-compressed spheres together. The above-described set of concepts regarding regular packings is also encountered in the early work of Chan and Tien [
2] and Kaganer [
3]. Based on these assumptions, the specified granular media can be visualized as a chain of elastic particles compressed between two walls, which are maintained at different temperatures, as seen in
Figure 1. Details of the particle mechanics approach adopted in this study can also be found in detail in our earlier work [
39].
2.2. Conventional Continuum Mechanics Approach
There has been considerable research directed towards describing the macroscopic behavior of compacted granular materials by using various homogenization techniques and postulating continuum constitutive laws [
40]. Some of the previous studies on mathematical modeling of transport properties are aimed at estimating elastic-plastic mechanical properties, thermal and electrical conductivity of ordered and disordered arrangements. In addition to the particle-level approach, we also focus on a small-strain thermoelasticity model of continuum scale description that integrates the previously-proposed effective mechanical and thermal properties for granular beds under compaction. In this study, we refer to the particle mechanics approach and the conventional continuum mechanics approach as PMA and CMA, respectively.
The governing field equations of motion and energy of the analogous problem defined at the continuum scale are the following:
and
, where Cauchy’s stress,
, is formulated as a combination of classical linear elasticity theory and simple linear thermal expansion, that is:
where
is the identity matrix. The solution for the basic one-dimensional steady state thermoelastic, continuum problem, where body forces are neglected, depends linearly on elastic constants,
,
, thermal expansion and conduction coefficients,
and
k, respectively. Since
holds,
is referred as
, and it is defined positive for compression. The system of questions then reduces to:
with
positive for compression, and
.
Effective mechanical properties of granular beds are of great interest for numerous theoretical studies, some of which focuses on: (a) the principal elastic modulus for vertical compression of spherical particles without any lateral extension (Walton [
33]); (b) finite and incremental elasticity of random packing of identical particles using energy methods (Norris and Johnson [
41]); (c) enhancement of the derived formulas based on the pressure dependence of the elastic moduli of granular packings (Makse et al. [
28,
29]). The effective medium approach proposes the following elastic effective properties,
and
:
where
is the actual stiffness that depends on the bulk mechanical properties: Young’s modulus,
E, and Poisson’s ratio,
.
is the packing fraction, and
Z is the coordination number.
The effective thermal conductivity of a granular bed is substantially sensitive to the thermal and elastic properties of individual particles. In this study, we adopt Batchelor and O’Brien’s [
4] solution for effective thermal conductivity coefficient:
It has been shown that the above-mentioned thermal contact models provide accurate results in estimating steady and average temperature profiles for ordered granular packings [
42].
After implementing the effective mechanical and thermal properties in the resembling continuum description, the equation of stress becomes:
where
and
are the temperature at the constraining walls, and
is the compaction strain along the principle direction. The overall compaction force can simply be expressed as
, where
= max{.,0} (notice that since
and
are assumed to be positive for compressive stress and strain, the above equation is valid for positive values of the expression in the parentheses).