Issue 
A&A
Volume 567, July 2014



Article Number  A120  
Number of page(s)  15  
Section  Planets and planetary systems  
DOI  https://doi.org/10.1051/00046361/201423648  
Published online  24 July 2014 
Online material
Appendix A: Geometric model for the computation of the effective stress
The geometric approach to the deformation theory of hot pressing and the conclusions for the plastic flow have been investigated by Kakar & Chaklader (1967) and Rao & Chaklader (1972). However, the detailed derivation of the formulae was not provided. Thus, we present the theory in detail and improve some aspects that were treated in a simplified manner in the studies mentioned above.
In the study by Kakar & Chaklader (1967) four basic systems of packing are considered, defined by the geometric arrangement of the grains and by the coordination number (this is the number Z of the contact points, an average value for equally sized spheres). In the simple cubic packing (SCP) every grain has four neighbours in a horizontal layer and two additional neighbours in the layers above and below. Orthorhombic (ORP) packing corresponds to six neighbours in a layer (plus two above and below). The rhombohedral packing (RHP) has coordination number 12 and the bodycentred cubic packing (BCCP) 8. Although in a unidirectional force field the bodycentred cubic packing is an unstable arrangement, we consider this case for the sake of completeness. Furthermore, although stability varies with the packing type, we assume that the symmetry is sustained throughout the whole compaction process. Assuming homogeneous and isotropic composition of the grains and symmetrical spread upon application of stress, for different packings the spheres will deform to polyhedra upon complete compaction, e.g. cubes for SCP, hexagonal prisms for ORP, rhombic dodecahedra for RHP, and tetrakaidecahedra for BCCP. During the deformation, necks develop at the contact points. The form of the grains will change, whereas the volume will remain constant.
For each kind of packing a specific unit cell of a certain form which is occupied by a dust particle is considered. The form of a unit cell is defined by the polyhedra mentioned above. From Figs. A.2–A.4, the volume V of such a unit cell can be computed. For SCP, this is the volume of a cube, and, since , we obtain (A.1)For ORP, the volume of a hexagonal prism is (A.2)For RHP we compute (A.3)and for BCCP (A.4)Thus, the volume of the unit cell can be expressed as the product of a constant β_{1} (the prefactor of the bracket) and the power .
Fig. A.1
General deformation geometry at any contact point between two grains. 

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The deformation of the BCC packing proceeds in two stages. For this reason we describe the theory for the other cases and the first stage of the BCCP deformation first, and then consider the second deformation stage of the BCC packing separately.
Prior to the hot pressing, the grain touches the surface of the unit cell at Z points where the neighbouring grains touch each other (Z_{1} for BCCP). During compaction, the grains deform plastically at the contact points and develop necks. The crosssection of such a neck between two grains has a circular form and is called the face (the contact points have evolved to contact crosssections). Still, the volume of the particle needs to be kept constant. Thus, the current radius of the grain is assumed to increase and the volume of those parts of the grain now lying outside of the unit cell (the caps) need to be subtracted from the current sphere volume (we note that the initial sphere volume is normalised to 1): (A.5)where r is the current grain radius and (A.6)is the volume of a cap. Using the relation h = r − y defined by the geometry of the packings (Fig. A.1), we obtain the normalised volume of the particle for the first three packings as (A.7)This defines the geometric constants α and β from Table 1 as α = Zπ/ 3 and β = (2Z − 4)π/ 3. From Eq. (A.7) the current radius of the particle is computed:
Fig. A.2
Unit cells and their intersections with an inscribed grain for the SCP and the ORP (top row from left to right) along with the top views of the unit cells and of the grain at its initial and critical (final) stages of deformation (bottom row). 

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(A.8)Given the ratio a/r, the current radius r of the sphere during deformation can be found (in arbitrary units corresponding to unit volume of the sphere). The ratio a/r is computed from the expression for the bulk density D, defined as the ratio of the volume of the particle to the volume of the unit cell. This last value is β_{1}(r^{2} − a^{2})^{3/2} (see Table 1). From the normalised grain volume 1 and the volume of the unit cell we obtain (A.9)Prior to compaction, i.e. for a = 0, the radius equals the initial radius (r = r_{0}), hence (A.10)holds, thus, (A.11)This is solved for (a/r)^{2}: (A.12)For the computation of the effective stress, we also need the area of the crosssection of the unit cell. In terms of the geometries shown in Figs. A.2–A.4, the crosssection of a unit cell is the area of its projection onto the horizontal plane. For SCP this is a square with the edge length 2y, for ORP a regular hexagon with the edge length , for RHP a square with the edge length 2y, and for BCCP an irregular octagon with four edges of the length , four edges of the length and the internal angle of 135 degrees. The areas of those polygons can be written as γy^{2} (or as for BCCP; see Table 1 for the values of γ).
Fig. A.3
Unit cell of the RHP and its intersections with an inscribed grain (left panel), the top right view of the unit cell and of the grain at its initial and critical (final) stage of deformation (central panel), and a single face of the unit cell at the critical stage of the deformation (right panel). 

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Fig. A.4
Unit cells and their intersections with an inscribed grain for the two deformation stages of the BCCP (top row, first and second panels); the faces of the unit cell at the second critical deformation stage (top row, third and fourth panels); topview of the unit cell before the first critical stage (bottom row, left panel) and after the first but before the second critical stage (bottom row, central panel); and the arrangement of the spheres at the first critical deformation stage as seen in the plane defined by the centres of the three spheres (bottom row, right panel). 

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If a load m acts on a face then the stress on the face is equal m/ (γy^{2}). The total stress on the sphere is Zm/ (γy^{2}). In a planetesimal, the applied stress is the lithostatic pressure σ_{0} and is equal to the stress on the face: (A.13)From this the load acting on the crosssection of the unit cell follows as m = σ_{0}γy^{2} and can be divided into two components. It is equal to the sum of the load acting on the sphere through N nonvertical flat faces πa^{2}cos(θ)σ_{1} lying above the equator of the unit cell and of the load acting on the void space cσ_{2}, (A.14)where πa^{2} is the area of the neck, θ is the inclination angle of the faces, c is the crosssection of the porous part of the unit cell, γ is a geometric constant which depends on the packing (see Table 1), σ_{1} is the stress acting along the zaxis through the neck, and σ_{2} = 0 is the stress on the void space. The last equation is solved for σ_{1} to obtain (A.15)Above we have shown . From this we compute (A.16)Thus, (A.17)To obtain the particle volume in the second deformation stage of the BCCP, one needs to subtract Z_{1} = 8 caps of the volume and additionally Z_{2} = 6 caps of the volume : (A.18)This defines the geometric constants , and for BCCP. Similarly to the other cases, Eq. (A.18) implies (A.19)The power (a_{1}/r)^{2} is computed analogously to Eq. (A.12): (A.20)For (a_{2}/r)^{2} we consider the first critical stage c_{1} when the cap of the second kind with the radius a_{2} starts to form and express the relative density in the terms of y_{2},
Additional parameter values for the derivation of the effective stress.
(A.21)because . Then (A.22)and (A.23)follow. The relation between a_{1} and a_{2} is obtained from Eqs. (A.20) and (A.23) as (A.24)The load on the octagonal crosssection of the unit cell of the BCCP is (A.25)where σ_{1} is the effective stress on the faces of the first kind, σ_{2} on the second kind, and . Assuming equal load per face, we obtain and from this (A.26)where N = 5. Thus, (A.27)The geometric relations described above are valid from the beginning of the deformation until the faces start to touch each other. In the beginning, for the normalised grain volume 1 the initial grain radius r_{0} is equal to (3/4π)^{1 / 3}. The stage when the equations lose their validity corresponds to a certain particle radius r_{c}, the critical radius. This can be computed as
follows. For the first four cases, the critical face radius a can be expressed in terms of the critical particle radius as a_{c} = δr_{c} with the constants δ (equal to , 1/2, 1/2, and 1/2, respectively). Substituting this into Eq. (A.8) we obtain (A.28)From Eq. (A.10) we obtain the relative initial densities D_{0} (see Table 1). The critical density D_{c} is obtained by substituting r_{c} and a_{c} = δr_{c} into Eq. (A.9): (A.29)At the transition from the first to the second deformation stage of the BCC packing, contact points between the particle and the six rectangular faces of the unit cell form. Using δ_{1} = 1/2, the particle radius at this moment of deformation r_{c1} and the first critical density D_{c1} follow from Eqs. (A.8) and (A.9), or using δ_{1} = 0.5 and a_{2} = 0, from Eqs. (A.19) and (A.21), respectively. At the end of the second stage, all fourteen faces come in contact. Here, with and δ_{2} = 1/3, the second critical radius r_{c2} and the second critical density D_{c2} follow from Eqs. (A.19) and (A.21).
After the respective critical density is reached, the geometric approach to the computation of the effective stress σ_{1} can no longer be applied (i.e. for D ≥ D_{c} or ). In this case we use a continuous linear relationship between and .
© ESO, 2014
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