Cold Compaction and Macro-Porosity Removal in Rubble-Pile Asteroids: 1. Theory

Abstract

Many asteroids are likely to be have been shattered by collisions into fragments and reaccumulated as gravitationally-bound rubble piles. These bodies may contain large porosities, although this picture may be complicated by compaction inside the asteroid body. Estimates of asteroid mass and volume imply a negative correlation between size and porosity for stony asteroids. Asteroids that are suspected to be metallic appear to contain larger porosities than stony asteroids of similar sizes. To understand these observations, we develop models for cold compaction of fragments of different materials. The initial boulder size distributions are assumed to be narrow. We focus on macro-porosity between the boulders and do not consider micro-porosity inside the boulders. In our model of silicate/chondritic boulders, compaction is assumed to occur through cataclastic fracturing, which creates small pieces that fill the pores between residual large boulders, leading to fractal-like distributions. This fracturing occurs when compression leads to stresses exceeding the tensile strength of the boulders. Combining this model with data on meteorite strength, we suggest that the compaction of chondritic boulders can be significant at pressures of several megapascals. In our model of metal boulders, we consider cold welding and boulder deformation (through ductile yielding or brittle-like fracturing, depending on the stress and intrinsic crack size). Given the properties of iron meteorites, we infer that compaction in metallic rubble piles, caused by ductile or brittle deformation, is small, and that cold welding may lead to large (≳50%) porosities if the boulders are of ∼1 m sizes.

Key Points

• Models are developed for cold compaction of rubble piles of different materials

• Silicate and chondritic assemblages may be substantially compacted through boulder fracturing at pressures of several megapascals

• Metal assemblages can preserve large porosities at least up to ∼10 MPa pressures

Plain Language Summary

The asteroid belt is a remnant of early planetary formation, and it has been continuously modified in the following ∼4.5 billion years through collisional evolution. Many asteroids are likely to have experienced disruptive collisions and exist as gravitationally-bound rubble piles. Voids are expected to exist between the rubble fragments, but they may be removed by the pressure inside the asteroid body. In this paper, we develop models for compaction of rubble piles of different compositions. At pressures inside asteroid bodies with diameters of a few hundred kilometers, assemblages of stony boulders can be substantially compacted through boulder fracturing, which creates smaller pieces that fill the pores between large ones. Assemblages of metal boulders likely preserve larger fractions of voids because of their greater strength and inter-boulder adhesion through cold welding. These models are used to discuss asteroid observations in a companion paper.

1 Introduction

The asteroid belt is a relic of planetary formation processes, and provides a window into the early evolution of the solar system. The features of the asteroid belt, as well as those of individual asteroids, have been continuously modified by collisions. After catastrophic disruptions of bodies larger than ∼1 km in diameter, the remnant bodies are mostly gravitationally-bound aggregates of fragments that are much smaller than the original bodies themselves (e.g., Benz & Asphaug, 1999; Melosh & Ryan, 1997; Michel et al., 2001). Models of asteroid collisional evolution suggest that many asteroids, probably including those that are several hundred kilometers in diameter, may have experienced disruptive collisions and exist as gravitationally-bound aggregates rather than monolithic objects (e.g., Campo Bagatin et al., 2001; Holsapple et al., 2002). Such aggregates of collision fragments are hereafter referred to as rubble piles. The theoretical prediction that rubble piles are a natural outcome of collisional evolution models needs to be tested against observations of asteroids. Density in particular is invoked to infer whether a body is a rubble pile (e.g., Britt et al., 2002; Carry, 2012). With the asteroid density inferred from mass and volume estimates and the material density inferred from spectral observations (together with comparison with analogous meteorites), the macro-porosity (i.e., the volume fraction of void between the blocks) can be estimated. Using the macro-porosity estimates, Britt et al. (2002) classified asteroids into three groups: essentially solid objects, heavily fractured (but coherent) bodies with macro-porosities around ∼20%, and collisionally disrupted rubble piles with macro-porosities over ∼30%. This simple picture can be complicated by at least two factors: (a) The size variation or polydispersity of rubble-pile boulders may reduce the macro-porosity to less than ∼20%, as suggested by recent studies on the small near-Earth rubble pile, (162173) Ryugu (Grott et al., 2020; Herbst et al., 2021). (b) In large asteroids, the pressure is probably large enough to cause porosity reduction through compaction, even in the absence of thermally activated creep. Carry (2012) noticed that there is a moderate positive correlation between size and density for stony and carbonaceous asteroids and hypothesized that it is caused by pressure-induced compaction. There has been a lack of quantitative analysis on the effect of cold compaction in asteroids. Many studies involving this effect cited the discussion in Britt et al. (2002), which used results for sand compression (Hagerty et al., 1993; Yamamuro et al., 1996) to speculate that compaction of rubble-pile blocks may initiate at ∼10 MPa but perhaps remain modest until ∼100 MPa. However, as pointed out by Britt et al. (2002) himself, this analogy may not apply because asteroid blocks likely have disparate properties from sand grains. Asteroids of different types consist of blocks of different materials (ordinary chondrite-like materials for S-type bodies, carbonaceous chondrite-like materials for C-type bodies, and perhaps iron meteorite-like materials for M-type bodies) and likely exhibit various resistance to compaction. It is yet unclear whether such variation can explain densities observed for asteroids of different types.

M-type asteroids are usually thought to be parent bodies of iron meteorites (e.g., Dollfus et al., 1979; Neeley et al., 2014; Shepard et al., 2015). However, recent observations suggest that the densities of two large M-type asteroids, (16) Psyche and (216) Kleopatra (∼3.8 ± 0.3 × 10³ kg/m³ and ∼3.4 ± 0.5 × 10³ kg/m³, respectively), are substantially lower than those of iron meteorites, ∼7.5 ± 0.5 × 10³ kg/m³ (Elkins-Tanton et al., 2020; Fry et al., 2018; Marchis et al., 2021). This implies that either the two large M-type asteroids are dominated by non-metal components or they contain large (≳50%) macro-porosities. M-type asteroids are of interest to planetary scientists because they may provide unique opportunities to investigate planetary cores. Whether Psyche is an exposed core of an ancient planetesimal is a topic of ongoing debate. The detection of silicates, especially hydrated phases, on the surface of Psyche has cast doubt on this view (e.g., Fornasier et al., 2010; Landsman et al., 2018), but recent hypervelocity experiments suggest that these features can be explained by impacts of silicate projectiles at typical asteroid collision velocities (Libourel et al., 2019). The most serious challenge for considering Psyche an exposed core is that its bulk density is much lower than the density of iron. Thus, a critical issue in this debate is whether high porosities (∼50%) can be preserved against compaction caused by the self-gravitation of a body of Psyche's size (e.g., Elkins-Tanton et al., 2020; Shepard et al., 2021). The densities of S-type (stony) asteroids around Psyche's size typically exhibit bulk densities close to their meteorite analogs, namely ordinary chondrites, indicating negligible porosities (e.g., Carry, 2012). It is still not a priori clear whether the high porosities (≳50%) required by interpreting Psyche as a metallic body can actually be preserved in that asteroid's interior. Moderate to high porosities (∼20%–40%) are also invoked in several models that interpret Psyche as a mixed metal and silicate body (e.g., Cantillo et al., 2021; Elkins-Tanton et al., 2020). To explore whether bodies of candidate materials (e.g., CB chondrites) can support void space under Psyche's internal pressure is important for testing these models.

In typical asteroid bodies (up to a few hundred kilometers in diameter), heating from disruptive collisions is almost negligible because strongly heated materials are likely to escape instead of being re-accreted (e.g., Keil et al., 1997; Love & Ahrens, 1996). If the parent body of a rubble pile had sufficiently cooled, and the heat-producing radio-nuclides had decayed by the time of the disruptive collision, the re-accreted rubble pile is likely to remain cold since its formation. In this case, the porosity is controlled by cold compaction inside the asteroid body. In this study, we develop models for cold compaction of rubble piles of different materials. In Section 2, we discuss the packing of boulders in uncompressed assemblages and develop a simple model to evaluate the effect of boulder size distribution. In Section 3, we develop a model for compaction of silicate and chondritic boulders, for which fracturing of boulders is considered the main mechanism for boulder deformation and assemblage compaction. In Section 4, we develop a model for compaction of metal boulders, considering the effects of cold welding and boulder deformation (through ductile yielding and/or brittle-like fracturing, depending on stress conditions and pre-existing crack sizes within the boulders). These models will be applied in a companion paper (Z. Zhang et al., 2022) to discuss observations regarding the densities and porosities of asteroid bodies.

2 Some Preliminary Considerations

2.1 Macro-Porosity and Micro-Porosity

The porosity of an asteroid can be separated into micro-porosity and macro-porosity. The former represents voids and cracks inside the boulders; the latter represents void space between the boulders (e.g., Britt et al., 2002). High micro-porosities (e.g., ∼40%–50%) have been observed in some carbonaceous chondrites (CCs) and the boulders of recently visited C-type asteroids, (162173) Ryugu and (101955) Bennu (e.g., Cambioni et al., 2021; Flynn et al., 2018; Grott et al., 2019; Okada et al., 2020). Such materials are from bodies that were formed from flocculent dust particles in the early solar system (e.g., Okada et al., 2020). To remain highly micro-porous, such materials must have avoided high temperatures that can cause dehydration and thermal annealing since their formation. In contrast, ordinary chondrites (OCs) are thought to have experienced temperatures that are high enough to cause thermal annealing; over ∼90% OC samples were metamorphosed at ≳1000 K (e.g., E. Scott & Krot, 2014); the <∼10% OC samples that were metamorphosed at ≲1000 K are thought to correspond to outermost layers of their parent bodies (e.g., McSween et al., 2002). In these bodies, micro-porosities are expected to be low because of annealing, which is consistent with the view that micro-porosities in OCs were mostly produced by impacts (e.g., Consolmagno et al., 2008). The samples returned from the S-type asteroid, (25143) Itokawa, also exhibit evidence for long-term thermal annealing (e.g., T. Nakamura et al., 2011) and low micro-porosities (e.g., Tsuchiyama et al., 2014). The formation of OC parent bodies appears to occur only in the first ∼2.5 Myr of the solar system, which is early enough for them to be substantially heated by the decay of ²⁶Al with a 0.7 Myr half-life (in contrast with the formation of CC parent bodies, which extended at least to ∼3.5 Myr; e.g., Fujiya et al., 2012; Doyle et al., 2015). The early formation of OC parent bodies (inner solar system planetesimals) may be a natural outcome of the proto-planetary disk evolution (e.g., B. Liu et al., 2022). According to these disk evolution models, late-forming micro-porous OC-like bodies are unlikely to exist. It is possible that some OC-like bodies were formed late and remained micro-porous but that such materials from these bodies have not been sampled as meteorites because they may be too friable to survive atmospheric entry. However, given the information from available meteorites, we assume that S-type asteroid boulders have similar densities as OC meteorite samples in order to interpret the densities of S-type asteroids. As to iron meteorites, their parent bodies must have formed following the segregation and solidification of metallic liquids (e.g., Goldstein et al., 2009). Such materials cannot preserve primitive micro-porosity.

The samples returned from the C-type rubble pile Ryugu exhibit higher micro-porosities than all carbonaceous chondrites (E. Nakamura et al., 2022; Yada et al., 2022). However, the identification of carbonaceous chondrites with relatively high micro-porosities (∼40%) suggests that at least a portion of meteorites with such micro-porosities can survive atmospheric entry and arrive more or less intact at the Earth's surface (the average micro-porosity of CI chondrite is ∼35%, and that of the ungrouped C chondrite, Tagish Lake, is ∼40%; e.g., Hildebrand et al., 2006; Consolmagno et al., 2008; Flynn et al., 2018). In contrast, the micro-porosities of ordinary chondrites and iron meteorites hardly exceed ∼20% (e.g., Consolmagno et al., 2008; Flynn et al., 2018). The lack of high micro-porosities in ordinary chondrites and iron meteorites implies that their parent bodies (S-type and M-type bodies) are unlikely to contain high micro-porosities as in the C-type bodies. In this study, we focus on macro-porosity. When we refer to “porosity,” we mean, by default, the void space between the boulders.

2.2 The Packing of Equally Sized Boulders

The packing of boulders is controlled by the forces that are applied to them. These forces include cohesive (e.g., van der Waals) forces between the boulders, the elastic forces between the boulders (the resistance to deformation), and the gravitational attraction from the rest of the asteroid. For small particles in environments of low gravity and low overburden, the cohesive and elastic forces dominate the force balance (Scheeres et al., 2010). In this case, the particles are gathered in loose, chain-like structures of high porosities. Hence, the porosities of regolith layers on small asteroids can be very high, perhaps greater than ∼0.8 (e.g., Kiuchi & Nakamura, 2014). With increasing gravity and particle sizes, the cohesive forces become less important. In an asteroid greater than one hundred kilometers in diameter (which is of primary interest in this study), the cohesive forces are negligibly small compared to the gravitational attraction within the asteroid if the boulders are a few decimeters in size (Scheeres et al., 2010). In the interior of such an asteroid, the pressure may be large enough to fracture boulders into smaller sizes. If the overburden is transmitted all the way to the smallest boulders, the elastic forces would far exceed the cohesive forces, although the gravitational forces in this case are not that large. In the absence of cohesive effects, the boulders must be packed in a jammed state with large enough numbers of contacts to keep the system in mechanical equilibrium (i.e., force balance). For equally sized spherical boulders, jammed random packing leads to porosities between ∼0.36 (dense random packing) and ∼0.45 (loose random packing), with the specific value depending on the state of shaking and the friction between boulders (e.g., G. D. Scott, 1960; Song et al., 2008; Silbert, 2010). Irregularly shaped boulders may have different porosities, but this effect is not large unless their shapes are extremely non-spherical (e.g., Herbst et al., 2021; Zou & Yu, 1996). The size polydispersity may substantially reduce the porosity. This effect is discussed next.

2.3 The Role of Boulder Size Distribution

The sizes of boulders in rubble-pile asteroids are usually described by a power law distribution (e.g., Hartmann, 1969; Michikami & Hagermann, 2021). The boulders can be neither infinitely large nor infinitely small. Here, we assume that the power law distribution applies between a minimum size s_min and a maximum size s_max, such that

(1)

where n(s) is the distribution density (i.e., n(s)ds gives the number of boulders with size between s and s + ds; all the symbols are described in Table 1), β is the power law exponent, C is a constant of proportionality (which describes the size of the assemblage and is thus an extrinsic parameter), and f(x₁, x₂; x) is a boxcar function that satisfies f(x₁, x₂; x) = 1 for x ∈ (x₁, x₂) and f(x₁, x₂; x) = 0 for x ∉ (x₁, x₂). The integration of this distribution density leads to

(2)

for s ∈ (s_min, s_max), for s ∈ (0, s_min), and N(s) = 0 for s ∈ (s_max, +∞). Here, N(s) is the cumulative distribution function (the number of boulders greater than size s). If the size s is small compared with the maximum size s_max, the cumulative distribution function is approximately N(s) ≈ Cs^−β, which recovers the commonly used expression for a power law distribution. The constant of proportionality C describes an extrinsic property, and it does not affect intrinsic properties, such as porosity. Thus, the truncated power law distribution can be captured by the two intrinsic parameters, the power law exponent β and the ratio of the minimum and maximum boulder sizes,

(3)

Table 1. Symbols

a	Radius of an individual boulder
s	Size (diameter) of an individual boulder
smin	Minimum boulder size of the initial (uncompressed) assemblage
smax	Maximum boulder size of the initial (uncompressed) assemblage
si	Characteristic boulder size of the initial (uncompressed) assemblage
sf	Critical boulder size in a compressed assemblage (boulders of this size may be strong enough to withstand compression)
Π	Shape constant of boulders (π/6 for spheres)
N(s)	Cumulative number of boulders with sizes greater than s
n(s)	Boulder size distribution function, n(s) = −dN(s)/ds
Σ(s)	Cumulative volume of boulders with sizes greater than s
ΣSK	Total volume of boulders in the load-supporting skeleton
ΣTOT	Total volume of all boulders in the assemblage
α	Boulder size ratio, α = smin/smax
αc	The critical boulder size ratio, describes the largest size variation of boulders that can occur in a same skeleton
β	Power law exponent of the initial boulder size distribution
βc	Transition value of the power law exponent, βc ≈ 2.5
ɛ	Void ratio (void-to-boulder volume ratio) of the assemblage
ɛ0	Void ratio of equally sized non-adhesive boulders
ɛSK	Void ratio of the load-supporting skeleton
φ	Porosity of the assemblage, φ = ɛ/(1 + ɛ)
φ0	Porosity of equally sized non-adhesive boulders, φ0 = ɛ0/(1 + ɛ0) ≈ 0.4
φSK	Porosity of the load-supporting skeleton, φSK = ɛSK/(1 + ɛSK)
ϕ	Boulder volume fraction, ϕ = 1 − φ
P	Pressure applied to the boulder assemblage
ϵ	Force inhomogeneity factor in the boulder assemblage
E′	Effective Young's modulus
	Specific crack formation energy of a homogeneous brittle material, dominated by the creation of new surfaces (against surface tension)
	Specific crack formation energy of a inhomogeneous brittle material, dominated by the opening of crack band (e.g., through friction between sliding grains)
	Specific crack formation energy of a ductile material, dominated by the deformation in the plastic zone near the propagating crack tip
KB	Fracture toughness of a brittle material,
KQ	Fracture toughness of a quasi-brittle material,
KD	Fracture toughness of a ductile material,
c	Crack (flaw) size
ω	Power law exponent for the flaw size distribution
w	Crack band width
ℓ	Material inhomogeneity length scale
L	Boulder-to-crack size ratio
k	Geometry constant determined by the shape of cracks
W	Crack width to material inhomogeneity size ratio, W = w/ℓ
λ	A dimensionless coefficient, λ = WL/(2k)
σeff(s)	Effective tensile strength of boulder of size s
σ0	Effective tensile strength of chondritic boulders in the small size limit
σY	Ductile yield stress of metal
σH	Indentation hardness of metal
x	Radius of contact between boulders
Z	Average number of contacts per boulder
	Pull-off force of adhesive boulders
Θ	Reduced pressure in an assemblage of adhesive boulders,
ρs	Boulder density

In what follows, we discuss how these two parameters affect porosities of assemblages.

In general, boulder size polydispersity affects porosity through two mechanisms (e.g., Yu et al., 1996). (a) The mixing mechanism: boulders of comparable sizes form a skeleton that is slightly different from that of equally sized ones (Figure 1a). This mechanism applies when there is little variation in boulder size. With this mechanism only, the porosity reduction is small. (b) The unmixing mechanism: boulders of very different sizes behave like independent components. The unmixing mechanism is dominant when boulder size variation is large, and the porosity reduction is potentially large. This mechanism can be further divided into two scenarios. (a) When the number of small boulders is small, they simply fit into the pore space of the skeleton supported by the large boulders, without filling it (Figure 1b). This scenario is hereafter referred to as undersaturated configuration. (b) When the number of small boulders is large, they replace the large boulders to form the load-supporting skeleton. The large boulders become separated from each other and embedded in the skeleton of small boulders. This scenario is hereafter referred to as oversaturated configuration. (A note on nomenclature: Here, we use the concept of saturation in a somewhat different way from its original meaning. In thermodynamics, saturation refers to a limit state, beyond which the system would be out of equilibrium. In this paper, saturation means inserting small boulders to fill the pores until no more boulders of similar sizes can be inserted, unless the spacing between the original large boulders is changed. Correspondingly, undersaturation means to fit the pores with small boulders without filling them, and oversaturation means to insert small boulders into the pore space beyond the point that the original large boulders are separated. These terms are not connected to the equilibrium and stability of the system.)

Figure 1

Mechanisms of porosity reduction due to boulder size polydispersity. (a) Boulders of comparable sizes form a skeleton that is slightly different from that of equally sized ones. (b) Small boulders (gray) fit in the pores between large boulders but do not affect the skeleton structure. (c) Small boulders (gray) are inserted into pore space beyond the point that the large ones are separated, and they replace the large ones to form the load-supporting skeleton. Shown in the inserts are illustrations of the boulder size distributions of the simplified assemblages in the figures (solid curves) and those of the analogous truncated power law distributions (dashed curves). The mechanism illustrated by (a) is referred to as the mixing mechanism, and the mechanisms illustrated by (b and c) are in general referred to as unmixing mechanisms. The scenario of (b) is referred to as an undersaturated assemblage, and that of (c) is referred to as an oversaturated assemblage.

Now, we apply this simple picture to an assemblage of boulders following a truncated power law size distribution (Equation 1). The boulder size variation is described by the size ratio α = s_min/s_max. When α_c(β) < α < 1 (where α_c(β) is a critical size ratio i.e., considered for now as a function of the exponent β), only the mixing mechanism applies (Figure 1a). When α < α_c(β), the unmixing mechanism would also apply. As can be seen from the size distribution function (Equation 1), smaller values of β lead to smaller fractions of small boulders, and greater values of β lead to larger fractions of small boulders. Hence, the undersaturated configuration applies for small β (Figure 1b), and the oversaturated configuration applies for large β (Figure 1c), as indicated by the inserts in the figure.

To facilitate the analysis, it is useful to calculate the cumulative volume Σ(s) of boulders greater than size s. With the distribution density given by Equation 1, we write

(4)

for s ∈ (s_min, s_max) and Σ(s) = 0 for s ∈ (s_max, +∞). Here, v(x) = Πx³ is the volume of an individual boulder of size x, in which Π is a constant determined by the shape of the boulder (e.g., for spherical boulders, s is diameter and Π = π/6). For the undersaturated assemblage (Figure 1b), the small (gray) boulders do not affect the packing of the large, skeleton-forming (white) boulders. The small boulders fit the pores between the skeleton-forming ones without changing the total volume of the assemblage. In this case, the pore space Σ_P of the assemblage is calculated as

(5)

where Σ_P,SK(L) is the volume of void space between the large, skeleton-forming (white) boulders, and Σ_SM is the total volume of the small, interstitial (gray) boulders. The void ratio (void-to-boulder volume ratio) of the assemblage, ɛ = Σ_P/Σ_TOT, is thus calculated as

(6)

where and Σ_SK(L) is the total volume of the large, skeleton-forming (white) boulders, and Σ_TOT is the total volume of all the boulders (i.e., Σ_TOT = Σ_SK(L) + Σ_SM). The controlling skeleton is assumed to have an effective void ratio Σ_P,SK(L)/Σ_SK(L) = ɛ_SK(β), which is a function of the exponent β (and also the packing structure, see discussion below). Applying this effective void ratio, Equation 7 becomes

(7)

For boulders following the truncated power law size distribution with a sufficiently small β, we assume that boulders of size s ∈ (s_maxα_c(β), s_max) occur as skeleton-forming ones, and those with s ∈ (s_min, s_maxα_c(β)) occur as small interstitial ones. Under this assumption, the volume of the skeleton-forming boulders is Σ_SK(L) = Σ(s_maxα_c(β)), the volume of the interstitial boulders is Σ_SM = Σ(s_min) − Σ(s_maxα_c(β)). Note that by definition of Equation 4, the total volume of the boulders is Σ_TOT = Σ(s_min), and also, in writing the relation for Σ_SK(L), we have used Σ(s_max) = 0 (i.e., there are no boulders bigger than s_max). Therefore, we write Σ_SK(L)/Σ_TOT = Σ(s_maxα_c(β))/Σ(s_min) and Σ_SM/Σ_SK(L) = 1 − Σ(s_maxα_c(β))/Σ(s_min). Substituting Equation 4 into these relations leads to

(8)

and

(9)

Substituting these relations into Equation 6, we write

(10)

As the derivation explicitly involves the assumption of s_min < s_maxα_c(β), the above equation applies only when the boulder size variation is large enough so that α = s_min/s_max < α_c(β).

For the over-saturated assemblage (Figure 1c), the volume of pore space in the entire assemblage, Σ_P, is approximately equal to that of the pores between the small, skeleton-forming (gray) boulders, Σ_P,SK(S). The void ratio, ɛ = Σ_P/Σ_TOT, is calculated as

(11)

where Σ_SK(S) is the total volume of the small, skeleton-forming boulders, and Σ_TOT is the total volume of the boulders. Again, we assume that the controlling skeleton has an effective void ratio Σ_P,SK(S)/Σ_SK(S) = ɛ_SK(β), which is a function of the exponent β and the packing structure. Thus, Equation 11 becomes

(12)

For boulders following the truncated power law distribution with a sufficiently large β, we assume that boulders of sizes s ∈ (s_min, s_min/α_c(β)) occur as skeleton-forming ones (as described above, α_c(β) describes the greatest size variation of boulders that can occur within a given skeleton, beyond which the unmixing mechanism applies). In this case, the volume of the small, skeleton-forming boulders is Σ_SK(S) = Σ(s_min) − Σ(s_min/α_c(β)). The total volume of the boulders is still Σ_TOT = Σ(s_min). Substituting Equation 4 into these relations, we write

(13)

Substituting Equation 13 into Equation 12, we express the void ratio ɛ as

(14)

As the derivation explicitly involves the assumption of s_min/α_c(β) < s_max, the above equation applies only when the boulder size variation is large enough, α = s_min/s_max < α_c(β).

The undersaturated configuration (Figure 1b) applies for small β values, and the oversaturated configuration (Figure 1c) applies for large β values. The transition between the two scenarios can be expected to occur at some intermediate β value (which is hereafter referred to as β_c). We suggest that the transition distribution is represented by that of the random Apollonian packing, which is constructed by continuously filling the space of an initial packing by inserting boulders at randomly positioned centers in the pores with the largest possible size without redistributing the existing boulders (Figure A1; e.g., Anishchik & Medvedev, 1995; Varrato & Foffi, 2011). This procedure implies that the random Apollonian packing corresponds to a configuration that is marginally saturated, that is, neither undersaturated nor oversaturated (recall that, in this paper, saturation means inserting small boulders to fill the pores until no more boulders of similar sizes can be inserted, unless the spacing between the original large boulders is changed). The random Apollonian packing is usually described by a power law boulder size distribution with β = β_c ≈ 2.5 (e.g., Anishchik & Medvedev, 1995; Varrato & Foffi, 2011). At this transition distribution, Equations 10 and 14 should coincide with each other, which requires , or equivalently, . When this requirement is satisfied, the two equations both become ɛ ≈ α^1/2/(1 − α^1/2) at β = β_c ≈ 2.5, and the porosity is expressed as

(15)

Since both Equation 10 and Equation 14 apply only when α < α_c(β), the above expression also requires α < α_c(β_c). This expression is consistent with the scaling relation φ ∼ α^1/2 given by the numerical simulations of Apollonian packings (e.g., Anishchik & Medvedev, 1995; Varrato & Foffi, 2011, see Appendix A1 for details). It is interesting to note that, the effective skeleton void ratio ɛ_SK(β_c) is absent in Equation 15, which implies that the packing structure (loose vs. dense packing) of the skeleton does not affect the porosity of the assemblage. This is because the effect of the skeleton packing structure on the void fraction (indicated by the value of ɛ_SK(β_c)) is offset by the effect of the maximum size variation of boulders in this skeleton (indicated by the value of α_c(β_c)). This can be illustrated, for example, under the framework of the undersaturated assemblage (Figure 1b). A looser structure of packing implies larger sizes of pores between boulders in the skeleton, which tends to increase the porosity. Meanwhile, as a result of the increased sizes of the pores, more boulders can fill interstitial space, which tends to reduce the porosity. These two effects happen to cancel each other. Mathematically, the latter effect (that more boulders would occur as interstitial ones) is reflected by an elevated value of α_c(β) (i.e., the size ratio of the smallest and largest boulders that occur in a same skeleton), as required by the condition for Equations 10 and 14 to coincide at β = β_c, that is, .

When the power law exponent β is large (or small), the boulder volume is dominated by boulders of the lower (or upper) end of the size distribution (for a power law distribution described by Equation 1, the total volume of boulders within a logarithmic size bin is proportional to s^3−β). In this case, the boulder assemblage behaves similarly to a equally sized one, and the mode of packing (loose vs. dense packings) plays a dominant role. So far, we have considered both ɛ_SK and α_c as functions of the mode of packing and the exponent β. In what follows, as a crude approximation, we neglect their variation with β. In Figure 2, we show some results of calculations from Equation 10 for β < 2.5 and Equation 14 for β > 2.5, as well as the relation φ = ɛ/(1 + ɛ), under the assumption of ɛ_SK = 0.56 for dense random packing (φ_SK = ɛ_SK/(1 + ɛ_SK) = 0.36) and ɛ_SK = 0.82 for loose random packing (φ_SK = ɛ_SK/(1 + ɛ_SK) = 0.45) of equal-size spheres. The critical boulder size ratios are taken as α_c ≈ 0.13 and α_c ≈ 0.20, respectively, as required by the condition for Equations 10 and 14 to coincide at the transition exponent β = β_c ≈ 2.5, that is, . The results suggest that the porosity remains close to those in equally sized assemblages at large or small power law exponents (β ≲ 2 or β ≳ 4) regardless of the boulder size ratio α, while the porosity can be largely reduced at intermediate β values if the size ratio α is small. Also shown in Figure 2 for comparison are the predictions of the widely used linear-mixture model (Yu & Zou, 1998). At small or large β values (β ≲ 2 or β ≳ 4), our model matches the linear-mixture model. At intermediate β values (2 ≲ β ≲ 4), our model predicts lower porosities than the linear-mixture model. However, our model is more consistent with the results from simulations on Apollonian packings (see Figure A2 and discussion in Appendix A1) and the results of experimental and numerical investigations on the optimum packing that the minimum porosity occurs at β ≈ 2.5 (e.g., Fuller & Thompson, 1907; Huang, 1963; Oquendo-Patiño & Estrada, 2022; Suzuki et al., 1985), rather than at β ≈ 2.7 as predicted by the linear-mixture model. Therefore, the application of this model is appropriate in this context (also see discussion in Section 3.1), although we do not argue that it can in general replace the linear-mixture theory for problems such as the macro-porosities of small rubble-pile asteroids with low internal pressures (e.g., Grott et al., 2020; Herbst et al., 2021). A detailed comparison between the model described here and the linear-mixture model is provided in Supporting Information S1.

Figure 2

The variation of assemblage porosity with boulder size distribution. The boulder sizes are assumed to follow a truncated power law distribution (Equation 1); α is the ratio of the minimum and maximum boulders sizes, and β is the power law exponent. The calculation is performed using Equation 10 for β < 2.5 or Equation 14 for β > 2.5, together with the relation φ = ɛ/(1 + ɛ). In (a), the controlling skeleton is assumed to form a dense random packing structure with an effective void ratio ɛ_SK = 0.56 (effective porosity φ_SK = 0.36). In (b), the controlling skeleton is assumed to form a loose random packing structure with an effective void ratio ɛ_SK = 0.82 (effective porosity φ_SK = 0.45). The crosses indicate results predicted by the model of Yu and Zou (1998).

2.4 The Mono-Size Approximation

Laboratory experiments suggest that the sizes of impact fragments follow two-segment power law distributions, for s ∈ (s_i, s_max) and for s ∈ (s_min, s_i), in which s_i is the transition size, and β₁ and β₂ are exponents of the two segments (β₁ > β₂) (e.g., Davis & Ryan, 1990; Fujiwara et al., 1989). If the large-fragment segment is sufficiently steep (β₁ ≳ 4) and the small-fragment segment is sufficiently shallow (β₂ ≲ 2), the effect of size polydispersity on porosity is small (since the porosities of assemblages following truncated power law distributions with β ≳ 4 and β ≲ 2 are close to those of equally sized assemblages; Figure 2), and the population of boulders with greatest volume occurs around the transition size s_i (Figure 3b, solid curve). Recent studies on asteroid boulders suggest that their sizes follow the Weibull distribution, or similarly, the log-normal distribution (Grott et al., 2020; Herbst et al., 2021; Schröder et al., 2021). As an example, we consider a Weibull distribution, , where s_i is the characteristic size and ξ_w is a constant parameter (e.g., Grott et al., 2020; Schröder et al., 2021). If ξ_w is sufficiently large (e.g., ξ_w ≈ 0.6, as for crater ejecta boulders on Equation 4 Vesta; Schröder et al., 2021), most of the boulder volume occurs around the characteristic size s_i (as in two-segment power law distributions with β₁ ≈ 5 and β₂ ≈ 1.5; Figure 3b), and the effect of size polydispersity on porosity can be expected to be small. In these cases, the assemblage can be assumed to be comprised of boulders of a single size s_i, and the porosity is assumed to be φ₀ = 40% (i.e., an intermediate value between the dense and loose random packings of equally sized cohesionless spheres).

Figure 3

The cumulative size distribution and cumulative volume distributions for examples “narrow” (a and b) and “wide” (c and d) size distributions. N(s) is the cumulative number of boulders larger than size s, Σ(s) is the cumulative volume of boulders larger than size s, and Σ_TOT is the total volume of all boulders in the distribution. The solid curves are two-segment power law distributions, for s ∈ (s_i, s_max) and for s ∈ (s_min, s_i), and the dashed curves are Weibull distributions, for s ∈ (s_min, s_max). The cumulative volume distribution is calculated as , where n(s) = −dN/ds is the distribution function such that n(s)ds is the number of boulders with s ∈ (s, s + ds), v(s) = Πs³ is the volume of an individual boulder of size s (and Π is a constant), and x here is the variable of integration. For all examples, s_min = 10⁻²s_i and s_max = 10²s_i; for distributions in (a) and (b), β₁ = 5, β₂ = 1.5, and ξ_w = 0.6; for distributions in (c) and (d), β₁ = 3.5, β₂ = 2.5, and ξ_w = 0.1.

Observations of asteroids show that boulder size distributions vary among rubble piles. The sizes of boulders on the Ryugu can be described by a Weibull distribution with ξ_w ≈ 0.1 or a log-normal distribution that fits the observed boulder sizes (Grott et al., 2020; Herbst et al., 2021). For boulders of sizes between ∼5 and ∼100 m, the distribution can be fitted by a power law with β ≈ 2.7 (Michikami et al., 2019). With this distribution, the boulder volume is distributed over a wide range of sizes (Figure 3d), and the macro-porosity of Ryugu is estimated to be ∼15% (Grott et al., 2020; Herbst et al., 2021). Nevertheless, this example may not represent rubble piles in general. The sizes of boulders on the likely rubble-pile asteroid, (4179) Toutatis, are fitted by a power law with β ≈ 4.4 (for sizes larger than ∼20 m, below which boulders cannot be well identified in the images taken by the flyby spacecraft; Jiang et al., 2015). With the distribution of the Toutatis boulders, the size polydispersity hardly affects the assemblage porosity (Figure 2). The densities of S-type asteroids (243) Ida and (433) Eros indicate macro-porosities of ∼20%–30% (e.g., Carry, 2012, as discussed in Section 2.1, the micro-porosities of S-type asteroid boulders are likely similar to ordinary chondrites and thus unlikely to affect the macro-porosity estimates). These results imply that the boulder size distribution varies among rubble piles, which is expected because the fragment size distribution depends on parameters such as collision velocity and projectile and target sizes as well as mechanical properties of the target (e.g., Ryan & Melosh, 1998) and thus varies among collisions.

As demonstrated by the discussion above, a narrow distribution leads to a clustering of boulder volume around the characteristic size s_i and a porosity close to that of a mono-sized assemblage, while a wide distribution leads to a spread over a wide range of sizes and a porosity that is much smaller than the mono-sized value (Figures 2 and 3). Since the role of size polydispersity varies among rubble piles and the purpose of this study is to examine the effect of pressure, we consider assemblages with narrow distributions (e.g., for two-segment power law distributions with β₁ ≳ 4 and β₂ ≲ 2 or for more continuous distributions with the transition from d ln N/d ln s ≲ −4 to d ln N/d ln s ≳ −2 occurring within one order of magnitude in size), for which the assemblage can be assumed to consist of boulders of a single size s_i, and the porosity can be assume to be ∼40% (when cohesion effects are absent). The characteristic size s_i is hereafter referred to as the initial boulder size (here we use the word “initial” because we consider the fracturing of boulders upon compression, which leads to the creation of boulders of smaller sizes, in the following section).

Although we focus on assemblages with narrow size distributions, we discuss in this paragraph how those with wide size distributions can be treated, for the sake of completeness. If the distribution is shallow for small boulders and steep for large boulders (as is the case for a Weibull distribution), we can define a characteristic s_i for the skeleton-forming boulders, such that boulders smaller than this size scale fit into the pores without filling them (like the gray ones in Figure 1b) while boulders larger than this size scale are separated from each other (like the white ones in Figure 1b). This characteristic size s_i corresponds to the size of the “controlling component” in the linear-mixture model (e.g., Yu & Zou, 1998). As the transition power law distribution between the configurations of Figures 1b and 1c occurs as the Apollonian packing with β ≈ 2.5, we suggest that s_i corresponds to the size at which d ln N/d ln s ≈ −2.5. Since the large boulders (with s > s_i) contribute to the solid volume but not the pore volume, they are inactive during compaction (i.e., collapse of pore space). In the active part of the assemblages (i.e., for boulders with s ≲ s_i), the load is supported by those with sizes around s_i. In a brittle boulder assemblage, compaction occurs through boulder fracturing, which creates small fragments to fill the pores between residual large ones (see Section 3). The originally small boulders play a similar role as the newly created small fragments. In Section 3, we suggest that brittle compaction leads to a power law size distribution between a critical size s_f and the initial size s_i, and the pore space can be accordingly estimated for boulders with s ∈ (s_i, s_f). Then, by subtracting the volume of boulders with s < s_f from the estimated pore space and adding the volume of boulders with s > s_i to the boulder volume, the porosity can be finally calculated. From this description, it is clear that the role of a wide size distribution is always to reduce the porosity, even after compaction occurs. The predictions of our model for initially mono-sized assemblages represent upper bounds of the porosities that rubble-pile bodies can preserve.

2.5 Summary and Remarks

The porosity of a boulder assemblage can be separated into the contributions of micro-porosity (voids and cracks inside the boulders) and macro-porosity (void space between the boulders). Here, we focus on macro-porosity and do not attempt to evaluate the variation of micro-porosity. Throughout this paper, we use “porosity” to represent macro-porosity.

The packing of boulders is affected by several factors. The cohesive effects may lead to chain-like structures and very high porosities in fine-particle regolith (and perhaps also in assemblages of meter-sized metal boulders, see discussion in Section 4.3). For silicate and chondritic boulders of decimeter or larger sizes on hundred-kilometer-diameter asteroids, the role of cohesive forces is likely to be small (e.g., Scheeres et al., 2010). For cohesionless equally sized spherical boulders, the porosity lies between ∼0.36 for dense random packing and ∼0.45 for loose random packing, depending on the state of shaking and the friction between the boulders (e.g., G. D. Scott, 1960; Song et al., 2008). The role of boulder shape is likely to be small unless their shapes are extreme (e.g., Zou & Yu, 1996).

The boulder size polydispersity may substantially reduce the porosity of an assemblage. A simple model is developed for boulders following the truncated power law size distribution (Equation 1), which is usually assumed for rubble-pile boulders (e.g., Hartmann, 1969; Michikami & Hagermann, 2021). The results (Figure 2) suggest that the porosities are close to those in equally sized assemblages at large or small power law exponents (β ≲ 2 or β ≳ 4) regardless of the boulder size ratio (α = s_min/s_max), while the porosities could be largely reduced at intermediate exponents (especially at β ≈ 2.5) if the size ratio α is small. This model is also applied as a part of the brittle compaction model in the following section.

Although Itokawa and Ryugu exhibit wide boulder size distributions (with β ≈ 3.1 and β ≈ 2.7 in the power law fit), this may not be the case for all asteroids. Toutatis, for example, exhibits a boulder size distribution of β ≈ 4.4. With β ≳ 4, most of boulder volume occurs around the minimum size of the steep power law size distribution, and the effect of the size polydispersity on porosity is small (Figure 2). Two-segment distributions have been observed in laboratory experiments (e.g., Davis & Ryan, 1990; Fujiwara et al., 1989). If the power law exponent of the steep segment is large and that of the shallow segment is small (β₁ ≳ 4 and β₂ ≲ 2), most of boulder volume occurs around the transition size between the two segments, and the porosity is likely close to that of a mono-sized assemblage (Figure 2). In the following sections, we consider assemblages that are comprised of effectively equally sized boulders, with a porosity of ∼40%. The boulder size is characterized by a value s_i, which is referred to as the initial size in the following text.

3 Compaction of Silicate and Chondritic Boulders

3.1 The Compaction Caused by Fracturing of Brittle Boulders

When the elastic limits are exceeded, a portion of brittle boulders would be fractured into fragments that can fill the pores between the residual intact boulders, leading to reduction in porosity of the assemblage. Whether a boulder would be fractured depends on the configuration of packing and the number of contacting neighbors. When a boulder is surrounded by numerous smaller, load-supporting neighbors (like the white ones in Figure 1c), the stress in its interior tends to be homogeneous, and the boulder is unlikely to be crushed. This can be illustrated intuitively by considering the limit that a boulder is surrounded by an infinite number of infinitely small neighbors, in which the boulder is effectively embedded in a fluid phase and would not be fractured since it feels same stress from all different directions. Considering the effect of coordination number only, one may assume that a boulder can survive only if it is surrounded by enough neighbors of smaller sizes. Since every boulder of each size must be surrounded by progressively smaller sizes, a fractal (scale-independent) behavior is expected to emerge, and the boulder size distribution is expected to follow a power law. This mechanism has been invoked by Sammis et al. (1987) to explain the size distribution of grains in fault gouges. The scale-independent rule would result in a power law size distribution of the fragment sizes, for s_F ∈ (0, s_i), where s_i is the characteristic initial boulder size, N_F(s_F) is the number of fragments greater than s_F, and the power law exponent is β_F ≈ 2.5 (e.g., McDowell & Daniell, 2001; McDowell et al., 1996; Sammis et al., 1987). The subscript “F” distinguishes this case from the initial boulder size distribution that is discussed in Section 2.3. This size distribution (a power law in which β_F ≈ 2.5) describes the production of small fragments that are barely enough to protect the large boulders; thus it coincides with the transition between the oversaturated (Figure 1c) and undersaturated (Figure 1c) configurations; that is, the random Apollonian packing (e.g., Anishchik & Medvedev, 1995; Varrato & Foffi, 2011). Apart from the number of neighbors, whether fracture occurs also depends on the strength of the boulder. The strength of boulders typically increases with decreasing size because small boulders are unlikely to contain large cracks (see discussion in Section 3.2). A stable state would be reached for a boulder assemblage when large boulders are all separated and protected by smaller fragments and the smallest fragments are small and thus strong enough to withstand compressive loading. Therefore, the fracturing process is usually thought to lead to a power law boulder size distribution for s_F ∈ (s_f, s_i), in which s_i is the initial boulder size and s_f is the critical size that is small enough for boulders to withstand compression, and the power law exponent is β_F ≈ 2.5 (e.g., McDowell & Daniell, 2001; McDowell et al., 1996). The porosity can thus be estimated (using Equation 15) as

(16)

if the size ratio s_f/s_i is sufficiently small (≲0.1; recall that Equation 15 assumes α < α_c).

The forces at contacts in boulder assemblages are highly inhomogeneous, even when the boulder sizes are uniform and the load is isotropic. As observed in photoelastic experiments, the load supported by the boulders concentrates into force chains so a portion of contacts carry larger forces than average while others carry less (e.g., C.-H. Liu et al., 1995). The fracturing of boulders is usually thought to be driven by the contrast in forces applied from different directions. In large boulders that are surrounded by numerous small neighbors, this force contrast is small because the force fluctuation occurs over length scales comparable to the distance between contacts (which is small compared to the size of the boulder) and do not penetrate to the boulder interior. In small boulders with few neighbors, force contrasts can be much larger. Although small boulders are stronger than large ones, natural and laboratory observations suggest that the number of neighbors plays a more important role; hence, small boulders are more likely to fracture than coexisting large ones, and the mode of fracture is likely similar to that caused by diametrical compression of spherical or irregular boulders (e.g., Sammis et al., 1987; Tsoungui et al., 1999). In the diametrical compression test, a boulder of size s breaks when the load reaches a critical value that is comparable to σ_eff(s)s² (e.g., Hiramatsu & Oka, 1966; Jaeger, 1967), in which σ_eff(s) is the effective tensile strength of the material and is a function of the size s (see Section 3.2 for details). Here, the fracturing is controlled by the tensile strength, as in the Brazilian test (in which a disk is diametrically loaded) of material tensile strength (e.g., Jaeger & Hoskins, 1966), because (a) the diametrical compression leads to extension on free surfaces of the boulder and thus tensile stress in the boulder interior, even though compressive stress is created near the contact surfaces, and (b) the tensile strength of the material is much smaller than its compressive strength (e.g., Hiramatsu & Oka, 1966; Jaeger, 1967). At a loading pressure P, the characteristic force applied to boulders of size s scales as . Thus, the critical condition for boulders of size s_f to survive compression can be expressed as

(17)

where ϵ is a constant depending on the force inhomogeneity (load concentration in force chains). In this study, we assume ϵ = 3 (see Appendix A2 for discussion). If the relation between the effective tensile strength σ_eff and the boulder size s is known, then the critical boulder size s_f could be solved as a function of the pressure applied to the assemblage through Equation 17, which would consequently lead to the relation between the pressure and the porosity of the assemblage through Equation 16. These two relations (i.e., Equations 16 and 17) form the basic framework for the model of compaction of brittle boulder assemblages.

3.2 Size Effect on Strength of Meteoritic Materials

In general, material samples of larger sizes exhibit lower strength than those of smaller sizes because they are likely to contain more and larger flaws. The effect of flaw size can be demonstrated by the simple example of a flat plate of uniform thickness b that contains a straight crack of length c (e.g., Griffith, 1921; Knott, 1973). In homogeneous materials, the crack can be considered to be narrow (i.e., of width much smaller than its length). If the surface of the crack is traction-free, then stress would be released in a region surrounding this crack with area proportional to the square of the crack length c, leading to a potential energy release of , in which σ is the ambient tensile stress, E′ is the effective Young's modulus, and k is a constant describing the shape of the stress relief region. Here, σ²/(2E′) is the elastic energy per unit volume; for plane stress, E′ = E, where E is the Young's modulus; for plane strain, E′ = E/(1 − υ²), where υ is the Poisson's ratio (e.g., Knott, 1973). For typical solids, Poisson's ratio is ∼0.3; the difference in E′ between plane stress and plane strain is small (roughly ∼10%) and not distinguished in the following text. The formation energy of the crack is proportional to its surface area, , in which is the specific crack formation energy per unit area for a homogeneous brittle material. In a homogeneous brittle material, the energy of crack formation is dominated by the creation of new surface, so is controlled by the surface tension of the material. The propagation of a crack requires that the release of elastic energy compensates the energy cost to increase crack surface area, that is, . To meet this requirement, the applied stress needs to exceed a critical value σ_eff that is inversely proportional to the square root of the crack length; given that and , we obtain

(18)

where σ_eff is the effective tensile strength of the material, and is the fracture toughness, which is usually used to describe the strength of a material because it is easier to measure through experiments (e.g., Knott, 1973). Although this derivation assumes the geometry of a flat plate, Equation 18 can also be shown to apply for bodies of different geometries (but with different values of k). In samples with multiple flaws, the strength is controlled by the largest one (i.e., , in which c_max is the length of the largest pre-existing flaw). The sizes of pre-existing flaws are usually assumed to follow

(19)

where N_FLAW(c) is the cumulative number of flaws with length greater than c per unit volume, is a constant, and ω is the exponent for the power law flaw size distribution (e.g., Bryson et al., 2018; Housen & Holsapple, 1999). In a boulder of size s (and volume v(s) = Πs³, in which Π is the shape constant), the total number of flaws greater than a certain length c is . In such a boulder, the size of the largest flaw, c_max, should lead to N_FLAW(c_max)v(s) = 1, and therefore be estimated as

(20)

Substituting the above relation into leads to a Weibull-type size effect law on strength (e.g., Trustrum & Jayatilaka, 1983; Weibull, 1939),

(21)

This size effect law is widely used to describe the strength of homogeneous brittle materials and invoked in brittle compaction models to explain compression experiments on assemblages of single crystals (such as sands and granular water ice discussed in Appendix A2). Housen and Holsapple (1999) combined observations of flaw sizes in terrestrial rocks, from millimeter-size cracks in laboratory samples to ten-km faults and fractures in the field, and suggested that ω ≈ 3 applies for rocks all these scales. A recent study on meteorite samples suggests similar distribution exponents could apply for most types of meteorites (Bryson et al., 2018). If the meteorite strength follows Equation 21 with ω ≈ 3, the size effect should be readily observed in laboratory tests: If the sample size is increased by a factor of four, the strength is expected to decrease by a factor of two. In the results of meteorite strength tests, the size effect appears to be much weaker or even absent (e.g., Pohl & Britt, 2020, also see Figure 4), but this is probably because such measurements are scarce and limited to millimeter- to centimeter-sized samples (see next paragraph for details).

Figure 4

The results of strength tests of meteorite samples. (a) The effective tensile strengths of meteorites of different types. (b) The relation between sample size and effective tensile strengths of meteorite Tsarev (a L5 ordinary chondrite). The triangles are individual test results, the red circles are average effective tensile strengths of samples of the same size, the solid black curve indicates the strength-size relation given by σ_eff(s) = σ₀(1 + s/(λℓ))^−1/2 with σ₀ = 63 MPa and λℓ = 0.04 m, and the dashed black curve indicates the strength-size relation given by σ_eff(s) = σ₀(s/λℓ)^−1/2 with the same values of σ₀ and λℓ. Shown in the lower left corner of (b) is a zoom-in of the laboratory test results of meteorite Tsarev. The data are collected from the review of Pohl and Britt (2020).

A similar discrepancy between the prediction of size effect and laboratory tests is also observed for concretes at small sample sizes, even when the samples are notched (so the crack lengths are explicitly prescribed to be proportional sample sizes). However, the size effect becomes observable for concrete samples that are sufficiently large (e.g., Bažant & Chen, 1997). To explain the results of concrete strength, Bažant (1984) proposed a blunt crack band theory to account for the effect of finite width of cracks (or crack bands) in inhomogeneous brittle materials. In inhomogeneous materials, cracks need to propagate along tortuous paths to pass around strong inclusions such as aggregates in concrete structures (e.g., Bažant & Oh, 1983). Therefore, the crack should actually be considered a crack band with an effective width that is proportional to the size of structural heterogeneities (such as the size of aggregates in concretes). In the theory of Bažant (1984), the release of elastic energy associated with a crack band depends on not only its length but also its width. In a simple illustrative example of a flat plate (of uniform thickness b) with a crack band of length c and width w, the area of stress relief can be considered to consist of a “stress diffusion” region with area proportional to c² and the crack band itself with area wc (Bažant, 1984). The release of elastic energy is . Hence, the rate of elastic energy release associated with the crack band prorogation is . The formation energy of the crack band is expressed as , where is the specific crack formation energy per unit area for an inhomogeneous brittle material (or quasi-brittle material following Bažant's terminology). In an inhomogeneous brittle material, in which the cracks are blunted, the energy flowing into the fracture front is consumed mainly by the opening of the crack band (e.g., through friction between the sliding grains/aggregates of the material) and is several orders of magnitude greater than that required to overcome the surface tension to create new surfaces (thus, the specific crack formation energy for a quasi-brittle material, , is several orders of magnitude greater than that for a homogeneous brittle material, ; e.g., Bažant & Kazemi, 1990). In this case, the crack formation energy is expressed as , in which is the formation energy of the crack band per unit volume that may be considered as a material property (e.g., Bažant, 1984, 1989). The rate of energy consumption associated with the crack band propagation is . To meet the requirement of , the applied stress needs to exceed

(22)

Although this relation is derived for flat plates, the general analysis of Bažant (1984) suggests that it also applies for other geometries (with different values of geometry constant k). If the crack length c is proportional to the sample size s (i.e., c = s/L where L is the characteristic ratio of sample and flaw sizes), and the crack band width w is proportional to the inhomogeneity size scale ℓ (i.e., w = Wℓ where W is the characteristic ratio of crack band width to inhomogeneity size), the strength of the structure varies with size s as

(23)

where is the effective tensile strength in the small sample limit, and λ = LW/(2k) is a dimensionless coefficient that may be experimentally determined. Introducing the fracture toughness for quasi-brittle materials (e.g., Bažant & Kazemi, 1990; Knott, 1973), we express the small-sample-limit tensile strength as σ₀ = K_Q(2/w)^1/2.

In the theory of Bažant (1984), the crack band length c is assumed to be proportional to the sample size s. In notched concretes, the cracks can be set explicitly to be proportional to the sample sizes. In reinforced concretes, the cracks are thought to grow stably until they reach a specific portion of the sample sizes (e.g., Bažant, 1984). It is thus not surprising that the relation Equation 23 matches experiments of notched and/or reinforced concretes (e.g., Bažant et al., 1994, and references therein), as well as other notched inhomogeneous materials such as granites (for which ℓ is taken as the typical grain-size; e.g., C. Zhang et al., 2018). Interestingly, Equation 23 also describes the strengths of unnotched and unreinforced concretes in several types of tests, such as double-punching tests on cylindrical samples (e.g., Bažant et al., 1994; Marti, 1989), probably because the maximum lengths of pre-existing cracks (or crack bands) in the unnotched plain concretes are also approximately proportional to the sample sizes, as would be the case for rocks with ω ≈ 3 (see Equation 20 and recall that ω ≈ 3 holds widely for terrestrial and meteoritic rocks; e.g., Bryson et al., 2018; Housen & Holsapple, 1999). If this argument applies, we expect that Equation 23 can also be used to scale the strength of rubble-pile boulders from those of analogous meteorites. Applying this size dependence, we suggest that meteorite strength measurements do not reveal a clear size-strength relationship because they are limited to small (up to centimeter-sized) samples and the measured strength represents the strength σ₀ in the small sample limit.

3.3 Compaction of Rubble-Pile Asteroids

In this section, we combine the framework described in Section 3.1 and the size effect law for boulder strength described in Section 3.2 to derive a relation between pressure and porosity for assemblages of silicate or chondritic rubble-pile boulders. Substituting the size effect law Equation 23 (with s = s_f) into Equation 17, we obtain the critical boulder size,

(24)

Apparently, Equation 24 applies for P < σ₀/ϵ only. According to Equation 23, the strength becomes size-independent and stays at a saturation level σ₀ in the limit of small samples; if the tensile stress externally applied to a boulder exceeds this saturation level, the boulder could not survive the compression no matter how small it is. When the pressure is greater than σ₀/ϵ, the resulting comminution would follow the Sammis et al. (1987) model, in which fracture probability depends on the number of nearest neighbors but not on the boulder size. In this case, the boulder sizes are expected to follow a power law distribution with s_f → 0 (Sammis et al., 1987), which would lead to essentially zero porosity if Equation 16 applies. This is not strictly true because Equation 23 applies for fracturing through the failure of the bonding of the constituent particles rather than fracture of the particles themselves. Hence, Equation 23 must become invalid for samples as small as individual constituent particles of the material (i.e., when s is comparable to ℓ). Taking into account the variation in fracture mechanism around s ≈ ℓ may avoid the embarrassing predictions of negative boulder sizes made by Equation 24 at pressures greater than σ₀/ϵ. However, this would not affect the prediction that fragment pieces as small as s ≈ ℓ are produced at P ≈ σ₀/ϵ. If the material inhomogeneity scale ℓ is much smaller than the initial boulder size s_i, the difference between s ≈ ℓ and s → 0 would be small. If we consider boulders with initial sizes around one m (s_i ≈ 1 m) and material inhomogeneities on scales of chondrules (ℓ ≈ 10⁻³ m), a critical boulder size of s_f ≈ ℓ would lead to a porosity of (Equation 16), which is close to zero. Here, we simply assume s_f ≈ 0 (and φ ≈ 0) for P ≳ σ₀/ϵ. This approximation becomes less valid with decreasing values of the initial boulder size s_i. Therefore, although we apply this approximation for all s_i values, caution must be taken when the discussion involves small initial boulder sizes.

At low pressures under which boulders do not fracture, the porosity is approximately equal to that of the uncompressed assemblage, φ ≈ φ₀ ≈ 0.4. At pressures that are relatively large (under which boulder fracture is significant) but still lower than σ₀/ϵ, the porosity can be estimated using Equation 16 with the critical boulder size characterized by Equation 24. Therefore, the relation between pressure and porosity is expressed as

(25)

where H(x) is the Heaviside function that satisfies H(x) = 1 if x is positive and H(x) = 0 if x is negative. It should be noted that the compaction occurs through boulder fracturing and is an irreversible process; once an assemblage is compacted, the porosity would not increase back to the initial value even if the load is removed. Therefore, the pressure here represents the maximum value that the assemblage has ever experienced in its history. We also note that the porosity reduction is caused by the production of small fragment pieces that fill the pores between the residual large boulders (in other words, by widening the boulder size distribution); if the boulder size distribution is initially “wide” (as is the case for Ryugu), the effect of brittle compaction would be small (also see discussion in Section 2.4).

Tensile strength tests have been performed on several types of meteorites (Pohl & Britt, 2020, and references therein) and the results are summarized in Figure 4a. These measurements were performed on small samples (millimeters to centimeters), so the reported values can be considered as the strength of the small sample limit, σ₀. Carbonaceous (C, CI, and CM) chondrites are relatively weak, with σ₀ between ∼1 and ∼10 MPa; ordinary (H, L, and LL) chondrites exhibit greater strength but also with large variations, from a few to tens of megapascals. The low strength values of carbonaceous chondrites are probably caused by the porous structure of the carbonaceous materials (i.e., large micro-porosities that occur as micro-cracks and voids between mineral grains inside the boulders).

The results of size scale effect, especially that on tensile strength, are scarce. To our knowledge, only the measurements on Tsarev (a L5 ordinary chondrite) performed by Zotkin et al. (1987) cover a considerable range of size (Pohl & Britt, 2020). The results exhibit a weak size dependence but may fit Equation 23 with σ₀ = 60 MPa and λℓ = 0.04 m (Figure 4b). This value of λℓ could be explained if we take the scale of material inhomogeneity as the typical size of chondrules, ℓ ≈ 10⁻³ m, and the dimensionless coefficient λ as a typical value that is reported from experiments on concretes, λ ≈ 40 (e.g., Bažant et al., 1994; Marti, 1989). The ratio of the crack band width and the material inhomogeneity size is typically W = w/ℓ ≈ 4 (Bažant, 1984); for penny-shaped cracks, the geometry factor is k = 2/π (Sack, 1946; Sneddon, 1946); this implies a characteristic boulder-to-crack size ratio of L = 2kλ/W ≈ 10. With the expression of σ₀ = K_Q(2/w)^1/2 (as derived in the discussion following Equation 23) and the assumption of w ≈ 0.004 m, a small-sample-limit strength σ₀ ≈ 60 MPa implies a fracture toughness of , which is close to the upper limit of terrestrial igneous rocks (e.g., Balme et al., 2004). This comparison implies that the material strength of strong ordinary chondrites is comparable to that of strong terrestrial rocks (since Tsarev is among the strongest chondritic meteorites). In the following text, we assume that σ₀ is in the range of ∼1–50 MPa and λℓ is approximately ∼0.04 m.

A few examples of calculations using Equation 25 are shown in Figure 5. These results suggest that, depending on the choice of parameters, compaction is likely to be significant under pressures of several to 10 MPa, which can typically be reached in relatively large asteroids (i.e., with diameters greater than ∼100 km). We do note that, due to the lack of experimental data, the estimates of strength of large boulders would be associated with large uncertainties. However, the extrapolated low strength of large boulders is consistent with the low strength of meteoroids inferred from their breakup during atmospheric entry (Popova et al., 2011). Moreover, we can at least argue that the critical boulder size s_f is as small as the laboratory test-pieces (millimeter to centimeter sizes) when the pressure reaches P ≈ σ₀/ϵ. If the initial boulder size is much greater than the laboratory-tested meteorite samples, then the conclusion that compaction would be almost “complete” at the pressure of P ≈ σ₀/ϵ is likely to hold, regardless of the specific assumptions of the size scale effect.

Figure 5

The compaction of assemblages of chondritic boulders. Shown in different panels are results with different small-sample-limit tensile strength σ₀. In each panel, the curves of different colors are results of different initial boulder sizes s_i (black for s_i = 1 m, blue for s_i = 10 m, and red for s_i = 10² m). The calculations are performed using Equation 25 (assuming λℓ = 0.04 m).

3.4 Summary and Remarks

Silicate and chondritic materials are brittle at present asteroid temperatures. In a chondritic rubble pile, compaction occurs through boulder fracturing, which creates small pieces to fill the pores between the residual large ones. Boulder fracturing caused by external compression is usually considered to result in a truncated power law size distribution between the initial boulder size s_i and the critical boulder size s_f (which is smaller than s_i), with an exponent close to the transition value β ≈ β_c ≈ 2.5 (e.g., McDowell & Daniell, 2001; Sammis et al., 1987). The porosity can be expressed using Equation 16, as discussed in Section 2.3. In general, smaller boulders are statistically stronger than larger ones because they are less likely to contain large cracks. An assemblage is stable when the skeleton-forming boulders are small enough to withstand compressive loading, so the critical size s_f can be estimated using Equation 17 (e.g., McDowell, 2005; Tsoungui et al., 1999). Equations 16 and 17 form the basic framework of the brittle compaction model, and some experimental observations that may validate this framework are discussed in Appendix A2.

A critical part of this model is the size dependence of boulder strength. The widely used Weibull-type size effect law is not supported by strength measurements of meteorite samples (e.g., Pohl & Britt, 2020). This is probably because the cracks in inhomogeneous materials (such as chondrites) are blunted and should be described by the model of Bažant (1984), which was proposed to explain similar discrepancies between the Weibull theory and experimental results of concretes. The size scale effect is therefore assumed to follow Equation 23, which explains the weak size dependence at laboratory-scale samples but predicts notable size dependence at larger scales. Although this extrapolation is yet to be tested against experimental data, it is perhaps consistent with the low strength of meteoroids, which is inferred from their breakup during atmospheric entry (e.g., Popova et al., 2011). Moreover, even though the estimates on strength of large boulders are associated with large uncertainties, it seems clear that the critical boulder size s_f would be as small as the laboratory test-pieces (millimeter to centimeter sizes) when the pressure reaches P ≈ σ₀/ϵ (in which σ₀ is the tensile strength of small meteorite samples and ϵ is the constant of force inhomogeneity), regardless of the assumptions of the size scale effect. Therefore, we assume with some confidence that, if the initial boulder size is much greater than the laboratory-tested meteorite samples, the compaction is likely to be almost “complete” at the pressure of P ≈ σ₀/ϵ. Assuming the Bažant-type size dependence (Equation 23), the porosity is calculated using Equation 25. With reasonable values of involved parameters (e.g., ϵ = 3, λℓ = 0.04 m, σ₀ on the order of a few to tens of megapascals, depending on the considered material, and s_i of one to one hundred meters), the results (Figure 5) suggest that compaction is likely to be significant, or perhaps even “complete,” when the pressure reaches a few megapascals, the typical pressure in the interior of hundred-kilometer-diameter asteroids. (Again, we note that “porosity” here refers to macro-porosity excluding micro-porosity. Although the equations here may imply that the compaction is “complete,” our model does not suggest that the voids are completely removed because micro-pores may be present inside the boulders.)

4 Compaction of Metal Boulders

Metals are different from silicate or chondritic materials in the following aspects: (a) meteoritic iron exhibits ductile rheology under low strain rate conditions, perhaps even at temperatures as low as ∼100 K (e.g., Gordon, 1970). When the deviatoric stress exceeds a critical value, the material undergoes plastic deformation. (b) Metal surfaces, if not covered by oxide layers, can be cold-welded together (e.g., Anderson, 1960; Merstallinger et al., 2009). If clean metal surfaces are pushed together, they could be adhered together because metallic bonds can be formed between atoms on different sides of the interfaces. In what follows, we discuss how these properties affect the compaction of metal boulder assemblages.

4.1 The Role of Ductile Yielding

The densification of metal particles has been extensively investigated, both theoretically and experimentally, by the community of powder metallurgy (e.g., Fischmeister, 1982; Hewitt et al., 1974). Under low temperature conditions, particle deformation occurs mainly through ductile yielding. Unlike brittle fracture, the criteria of ductile yielding (in terms of stress distribution) is independent of size scale effects. Hence, when discussing the role of ductile yielding, the results for fine-particle powders may also be applicable for piles of large metal boulders. Theoretical and experimental results suggest that, for metal powders, cold densification is rather small when pressure is small compared to the yield stress of the material, even though yielding may already occur around the contacts between particles (e.g., Fischmeister, 1982; Hewitt et al., 1974). In general, the interior pressure in a M-type asteroid is much smaller than the yield stress of typical meteoritic iron (see below for details), so the role of ductile yielding is likely to be minor. In what follows, we also demonstrate this through a simple but quantitative model.

As an example, we consider the deformation of a spherical boulder with radius a and volume v = (4/3)πa³. When the boulder is pushed at a contact by a normal force toward its center, it undergoes deformation that leads to a displacement y such that the initial point contact spreads into a contact surface with a finite radius x. When the contact radius x is much smaller than the boulder radius a, the stress distribution around the contact is similar to that around a normal load to an infinite half space. The theory of indentation suggests that a ductile half space is indented if the normal stress surpasses an indentation hardness

(26)

where σ_H is the indentation hardness, σ_Y is the material yield stress, and χ is a constant that is typically taken as χ = 3 (e.g., Tabor, 1996). The indentation hardness is greater than the yield stress because an indentation does not entail one-dimensional compression. When a force of magnitude is applied to a boulder, indentation occurs to ensure that the contact stress does not exceed the indentation hardness. Hence, the contact area is

(27)

where x is the contact radius. When the contact displacement y is small compared to the boulder radius a and the elastic distortion of the surface around the contact is neglected, we approximate the Pythagorean relation x² + (a − y)² = a² to first order in y as

(28)

In an isotropically compressed boulder assemblage, the applied pressure P and the average normal contact force are related to each other via

(29)

where ϕ = 1 − φ is the boulder volume faction, and Z is the average number of contacts per boulder (e.g., Arzt, 1982; Bagi, 1996). This formulation can be intuitively understood as follow: The product of the average number of contacts, Z, and the contact force, , gives the total magnitude of forces applied to a boulder; the ratio of the total force and the boulder surface area 4πa² gives the average pressure within the boulder, ; the volume average of the pressure in the boulders and the pressure in the void gives the pressure of the assemblage, . Equations 26-29 (with χ = 3) lead to

(30)

If all contacts undergo a common dimensionless displacement y/a, the domains assigned to all individual boulders retain their original shapes, with their dimensions reduced by a factor of (1 − y/a). The assemblage also retains its original shape with the dimensions also reduced by a factor of (1 − y/a), and the volume is reduced by a factor of (1 − y/a)³. The boulder materials are conserved, so the packing fraction is ϕ = ϕ₀(1 − y/a)⁻³, in which ϕ₀ is the packing fraction of rigid spheres. When y/a is small, this relation is approximated as

(31)

Substituting Equation 26 (with χ = 3) and Equation 30 with (ϕ = ϕ₀) into Equation 31, we evaluate the compaction caused by ductile yielding (in the small deformation limit) as

(32)

The temperature of the largest M-type asteroid, Psyche, is around ∼150 K (Bierson et al., 2022). For a metallic body around the size of Psyche, a disruptive collision may lead to a global temperature increase of ∼150 K (Keil et al., 1997). At temperatures of ∼300 K, the yield stresses of iron meteorites (such as Canyon Diablo, Gibeon, Odessa, and Sikhote Alin) are typically are typically ∼150–400 MPa (e.g., Slyuta, 2013). In assemblages of non-adhesive spheres, Z ∈ (4, 6) (e.g., Silbert, 2010; Song et al., 2008). In the largest M-type asteroid, Psyche, the central pressure is ∼20 MPa, and half of the mass is situated in regions of pressures less than ∼10 MPa. In this case, the porosity reduction caused by the ductile yielding of the iron boulders is at most a few percent (e.g., with Z = 4, σ_Y = 200 MPa, and at the pressure of P = 10 MPa, the porosity reduction caused by ductile yielding is (2/Z) (P/σ_Y) ≈ 0.03; since the calculation suggests that the densification is small, the assumptions of small deformation used in the derivation are validated a posteriori). Hence, we suggest that the densification caused by ductile yielding is likely to be small, at least in asteroids up to the size of Psyche (which is the largest M-type asteroid in the asteroid belt).

4.2 The Role of Boulder Fracturing

Metals (ductile materials) may also develop brittle-like fractures. This phenomenon became well known after catastrophic failures, leading to, for example, the loss of over two hundred Liberty ships during World War II (e.g., Irwin, 1957; Marder & Fineberg, 1996). The failure of metals depends on their sizes. Large ship-plates may fracture before general yielding, with fracture surfaces of brittle cleavage facets, but small test-pieces cut from the fractured ship-plates only break with fibrous appearances after yielding. This effect can be explained as follows. In a ductile material, a crack needs to open by a critical amount through plastic deformation before it can extend, and this opening needs to be accommodated by a plastic zone. If a sample is small and the plastic zone traverses through it before the critical opening is reached, it breaks ductilely after general yielding. If a sample is much larger than the plastic zone and the critical opening can be easily reached, it breaks in a brittle manner without yielding outside the plastic zone (e.g., Irwin, 1957; Knott, 1973; Orowan, 1954). The particles that are of interest to powder metallurgy (up to millimeter sizes) are smaller than the size of the plastic zone around propagating crack tips (see below for details), so they behave in purely ductile manners. However, because boulders in rubble-pile asteroids are presumably much larger than powder particles, brittle-like failure is probably present.

The resistance of a ductile material against fracturing can be described by its fracture toughness, , where is the specific crack formation energy of the ductile material (which is dominated by the plastic flow associated with crack propagation, instead of the creation of new surfaces, and thus much greater than those of brittle materials; e.g., Knott, 1973). Measurements on fracture toughness of iron meteorites are rare. The Young's moduli of iron meteorites are reported to be ∼150–200 GPa (e.g., Dyaur et al., 2020); the specific crack formation energy of iron meteorites at ∼300 K can be inferred from impact strength tests to be a few 100 kJ/m² (e.g., A. A. Johnson et al., 1979); these results overlap with the typical values low-carbon steels and imply fracture toughness values of ∼100- (e.g., Ashby, 1989). Given the properties of a ductile material, including the fracture toughness K_D and the yield stress σ_Y, an intrinsic length scale from the dimensional analysis is . The analysis of stress around a crack suggests that the size of the crack tip plastic zone is (e.g., Knott, 1973). With K_D in the range of ∼100- (e.g., Ashby, 1989) and σ_Y in the range of ∼150–400 MPa (e.g., Slyuta, 2013), the plastic zone size c_PZ is likely to be on the order of ∼0.1 m. When the crack length c is larger than the plastic zone size c_PZ, the effective tensile strength can be estimated using the framework of Griffith's theory (e.g., Irwin, 1957; Knott, 1973; Orowan, 1954) as

(33)

where k is a geometry constant depending on the shape of the crack. Assuming that the crack size c is proportional to the boulder size s (i.e., s/c = L, where L is the characteristic boulder-to-crack ratio), we express the effective tensile strength of a boulder of size s as

(34)

Substituting the size dependence of Equation 34 (with s = s_f) into the framework of the model of brittle boulder compaction (Equations 16 and 17), we estimate the porosity as

(35)

As discussed in Section 3.3, this estimate is valid only if fracturing is significant such that the calculated porosity is smaller than the porosity of equally sized boulders. Assuming penny-shaped cracks, we take a geometry constant of k = 2/π (Sack, 1946; Sneddon, 1946). Assuming a fracture toughness of (Ashby, 1989), a force inhomogeneity factor of ϵ = 3 (Appendix A2) and a boulder-to-crack size ratio of L = 10 (see below for discussion), we find that Equation 35 predicts φ ≳ 0.5 at the pressure of ∼10 MPa if the initial boulder size is s_i ≲ 1 km. This calculation indicates that role of brittle-like fracturing is likely minor in metallic rubble piles unless the boulders are kilometer-sized. We also note that the assumption on the boulder-to-crack size ratio (L = 10) is crude. Although this value is inferred from the strength of ordinary chondrites (see Section 3.3 for details), it is probably not appropriate for metals. As metals may undergo stress release through plastic deformation, the formation of large cracks may be prohibited. Moreover, existing cracks may also be partly annealed, even at low temperatures, through cold welding. Hence, the above calculation may to some extent overestimate the role of brittle-like fracturing, although it already implies that this failure mechanism is likely to only be of minor importance.

4.3 The Potential Role of Cold Welding

If clean metal surfaces are pushed together, they can be welded together even under low temperature environments, leading to interfaces potentially as strong as the bulk material (e.g., Anderson, 1960; Merstallinger et al., 2009). If the metal boulders are cold-welded into chain-like structures of low coordination numbers, the porosity can probably exceed ∼50% (as in cohesive particle regolith discussed in Section 2.2). In cohesive assemblages, the packing structure is controlled by the competition between adhesive and externally applied forces. When external forces are mainly from isotropic compression, the porosity is controlled by a dimensionless parameter that is usually referred to as the reduced pressure,

(36)

where P is the assemblage pressure, s is the boulder diameter, and is the pull-off force of a pair of adhered boulders (e.g., Lammali et al., 2021; Than et al., 2017). This dimensionless parameter characterizes the importance of compression-induced forces relative to adhesive forces between the boulders. It can also be regarded as the ratio of the pressure, P, and the adhesion strength, . When the reduced pressure is much smaller than one, the adhesive forces are much larger than the pressure-induced forces, and the packing could be very loose. When the reduced pressure is much greater than one, the adhesion effects are negligible, and the packing reduces to that of cohesionless particles. The transition between the extreme scenarios occurs at intermediate values of the reduced pressure, around Θ ∼ 1. The quantitative relation between Θ and φ can be inferred from the compression of adhesive particles. Güttler et al. (2009) suggested that the relation between pressure and porosity of adhesive particles can be described as a function φ = φ₀ + (φ₁ − φ₀)/(1 + (P/p)^η), where φ₀ is the porosity of non-adhesive particles, φ₁ is the unconsolidated porosity of adhesive particles, p is the characteristic pressure, and η is a constant. Here, we assume that p is proportional to the strength of adhesion, . Adopting the formulation of Güttler et al. (2009), we suggest that the relation between φ and Θ can be described by

(37)

where Ξ is a constant. Here, the porosity of non-adhesive particles is taken as φ₀ = 0.4 (as in previous sections); the unconsolidated porosity of adhesive particles is taken as φ₀ = 0.7, as for particles feeling adhesive forces that are ∼10³ times stronger that their gravitational forces (e.g., Kiuchi & Nakamura, 2014, also see next paragraph for discussion); with this choice of φ₀ and φ₁, the compression experiments on adhesive particles reported by Güttler et al. (2009), Omura and Nakamura (2017), and Than et al. (2020) can be fitted by Ξ = 0.4 and η = 1 (see Appendix A3 for detailed discussion). With these parameters, we obtain the following relation between the porosity and the reduced pressure,

(38)

This compaction occurs through the rearrangement of the boulders and is an irreversible process; once an assemblage is compacted, it would not return to the unconsolidated structure even if the load is removed. Thus, the pressure here corresponds to the maximum value that the assemblage has ever experienced. Specifically, in the deep part of a rubble pile, where the static overburden is large, the maximum pressure corresponds to the hydrostatic pressure; in the shallow part, where the static overburden is small, the maximum pressure likely corresponds to the pressure caused by the kinetic energy the fragments during their landing onto the body in the re-accretion process (see Z. Zhang et al., 2022, for details).

The operation of cold welding requires a contact of finite area between metal surfaces. Such contact can be created between two boulders when they collide with each other during the re-accretion of the rubble-pile body. If a collision between two spherical boulders of the same diameter s at a relatively velocity u is purely elastic, the maximum normal stress at the contact during the collision is (see Appendix B2). When the collision velocity reaches a threshold value of ∼0.1 m/s, the contact stress reaches the indentation hardness σ_H. At higher collision velocities, plastic deformation occurs such that the contact stress does not surpass the ductile indentation hardness σ_H and only a portion of the kinetic energy is stored as elastic energy (with the rest dissipated by plastic flow). Under the assumption that Hertz's elastic theory approximately applies even when the contact stress reaches the indentation hardness σ_H, the storage of elastic energy adjacent to a plastically created contact of radius x is estimated as (see Appendix B1). In the absence of cold welding at contact (e.g., due to the oxidation and contamination of metal surfaces in the Earth's atmosphere), such elastic energy can be used to estimate the rebound velocity, and the results are verified by experiments on the coefficient of restitution (e.g., K. L. Johnson, 1987). With the operation of cold welding at contact during collision, rebounding occurs only if the stored elastic energy is greater than the energy required to break the cold welded contact, and sticking occurs when the elastic energy is smaller than the energy of contact breaking. The energy required for contact failure can be estimated as , in which is the specific crack formation energy (see Section 4.2 for description of this concept). The critical condition for sticking between boulders can be estimated by equating the storage of elastic energy, , and the energy of contact breaking, , which leads to a critical contact radius, , where is the fracture toughness of the metal (also see Section 4.2 for discussion of this quantity). Substituting Equation 26 with χ = 3 (e.g., Tabor, 1996) into this formulation of the critical contact radius, we obtain

(39)

Assuming that the collision is fully plastic, the radius of the collision-created contact can be estimated as (Appendix B2). Equating this collision-created contact radius and the critical contact radius given by Equation 39, we obtain a maximum collision velocity for sticking between boulders. Assuming a fracture toughness of , a yield stress of σ_Y = 200 MPa, and a boulder diameter of s = 2 m, we obtain a sticking velocity of u_A ≈ 0.3 m/s, which is much smaller than the escape velocities of hundred-kilometer-sized asteroids (e.g., ∼160 m/s for Psyche). The implies that sticking does not occur upon the initial landing of the boulders to the accreting rubble pile (at velocities around the body's escape velocity); but rather, it occurs after the kinetic energy is mostly dissipated (after a few cycles of rebounding).

The critical diameter (twice the critical radius) of contacts between sticking boulders, , is smaller than the characteristic size of the crack tip plastic zone, . This implies that the failure of such contacts, if occurs, would be in a ductile manner. The ductile failure of a contact embedded in a large surface resembles the inverse process of ductile indentation into a infinite half space, and the pulling strength of such contact should be identical to the indentation hardness σ_H (e.g., Orowan, 1949). Hence, we estimate the pull-off force of a pair of cold welded boulders as . Substituting the expressions of the critical contact radius (Equation 39) and the indentation hardness (Equation 26 with χ = 3) into this relation, we obtain the pull-off force,

(40)

This force is also the maximum contact force resulting from collisions that lead to sticking between boulders. With a fracture toughness of and a yield stress of σ_Y = 200 MPa, Equation 40 yields a pull of force of , which is roughly three orders of magnitude greater than the gravitational force of a boulder, , for a boulder with radius s = 2 m, but becomes comparable the gravitational force for a boulder with radius s = 10 m (here, the gravitational acceleration is assumed to be around that on the surface of Psyche, g ≈ 0.1 m/s², and the boulder density is assumed to be around the average value of iron meteorites, ρ_s ≈ 7.5 × 10³ kg/m³). The pressure-induced force, which scales as Ps², becomes comparable to the gravitational force, , when the pressure reaches the level of ∼ρ_sgs, or at a depth that is on the order of the boulder diameter s. The boulder size is typically much smaller than the rubble-pile radius, and the pressure-induced forces of an asteroid are much greater than the gravitational forces of individual boulders essentially everywhere in the asteroid interior. For this reason, we do not consider the role of gravitational forces of individual boulders in the following text. Substituting the pull-off force given by Equation 40 into Equation 36, we estimate the reduced pressure Θ as

(41)

where s_i is the characteristic size of metal boulders of a rubble-pile body. Substituting the above expression of the reduced pressure (Equation 41) into the relation between reduced pressure and porosity of cohesive assemblages (Equation 38) and taking a numerical approximation of the constant coefficient (note that χ = 3 and Ξ = 0.3 themselves are numerical approximations), we express the porosity of cold welded metal boulders as

(42)

Calculations are performed using Equation 42, assuming fracture toughness K_D values between 100 and 200 MPa ⋅ m^1/2 (e.g., Ashby, 1989), yield stress σ_Y values between 200 and 300 MPa (e.g., Slyuta, 2013), and characteristic boulder sizes between 1 and 5 m (Figure 6). The results suggest that (a) the cold welding between meter-sized metal boulders may substantially increase the porosity, especially at relatively low pressures (e.g., several megapascals) and (b) the role of cold welding decreases with increasing boulder size and is likely to be minor if the boulders are greater than ∼5 m in diameter.

Figure 6

The compaction of assemblages of metal boulders. Shown in different panels are results assuming different values of fracture toughness K_D and yield stress σ_Y of the metal. In each panel, solid curves of different colors are results of different initial boulder sizes s_i (black for s_i = 1 m, blue for s_i = 2 m, and red for s_i = 5 m); these calculations are performed using Equation 42. The dashed green curves are results assuming boulder sizes predicted by the Glenn and Chudnovsky (1986) model (i.e., Equation 43); these compaction calculations are performed using Equation 44.

These calculation results suggest that the effect of cold welding is sensitive to the boulder size, and is important only if the boulders are no more than a few meters in diameter (Figure 6). The sizes of boulders of metallic rubble piles (especially their interiors) are not constrained by direct observations. Iron meteorites found on the Earth are at most one to a few meters in diameter. This is probably a lower bound because large meteoroids may be fragmented during atmospheric entry. Another guide can be provided by applying the dynamic fragmentation theory to asteroid collision processes. Studies on collisional fragmentation of stony asteroids (e.g., Melosh et al., 1992; Ryan & Melosh, 1998) usually apply the theory of Grady and Kipp (1980), which assume that the fragmentation is controlled by the propagation of pre-existing cracks at relatively low stress levels (due to the flaw-weakening effects). In metallic bodies, stress-reducing cracks are probably rare (see discussion in Section 4.2), and fracturing may be prohibited at stresses below the yield value. In this case, the model of Glenn and Chudnovsky (1986), which assumes that fragmentation occurs only if a critical stress for fracture-initiation is reached, appears to be more appropriate (see Grady, 2017, for discussion on the comparison of different fragmentation models and the situations where they apply). According to the Glenn and Chudnovsky (1986) model, at strain rates relevant to large asteroid collisions (which can be estimated as the ratio of the asteroid collision velocity, typically ∼5 km/s, and the target diameter, e.g., ∼100 km, to be on the order of ∼0.05 s⁻¹ and regarded to be in the low strain rate regime), the characteristic fragment size can be estimated as

(43)

where s_GC is the characteristic boulder size predicted by the Glenn and Chudnovsky (1986) model. Its value is, not surprisingly, proportional to the critical crack size that is required by the Griffith-type theory to result in a strength comparable to the ductile yield stress, since the energy balance in low strain rate regime is primarily that between elastic energy and fracture formation energy (Glenn et al., 1986). Assuming material properties used in the previous calculations (i.e., K_D of ∼100–200 MPa ⋅ m^1/2 and σ_Y of ∼200–300 MPa), we find that s_GC is on the order of ∼1 m. Substituting s_i = s_GC into Equation 42, we obtain

(44)

The results of calculations assuming the Glenn-Chudnovsky size (the dashed green curves in Figure 6) suggest that the cold welding between metal boulders may notably increase the porosities of their assemblages under pressures that can be reached in metallic rubble piles.

4.4 Summary and Remarks

Metals exhibit different properties from silicates: (a) meteoritic irons are ductile (more accurately, undergo elastic deformation at relatively low stress levels and plastic deformation when the deviatoric stress reaches the yield value) even at current temperatures of asteroids and (b) clean metal surfaces can be cold-welded together when they are in contact. In asteroid bodies that are up to the size of Psyche (the largest M-type body), the pressure is small compared with the yield stress of meteoritic iron. Under these pressures, ductile yielding occurs only in regions around the contacts between the boulders, and it plays very minor role in affecting the porosity. Meter-sized metal boulders may stick together through cold contact welding after low-velocity collisions (of a few meters per second). The cold-welded interfaces are as strong as the bulk material, and the assemblages of cold-welded boulders are considered here as cohesive assemblages. This effect may notably increase the porosity of the assemblage (from ∼40% for non-welded cohesionless boulders to ≳60% for cold welded ones), especially at pressures up to a few megapascals (Figure 6).

Metals may also develop brittle-like fractures. When brittle-like failure occurs, plastic deformation is concentrated in small zones near the tips of extending cracks, and the effective strength can be lower than the material yield stress. This type of failure is possible only in samples that are much larger than the plastic deformation zone, and the role of such fracturing is likely to be small in asteroids unless the fragment boulders are kilometer-sized.

5 Conclusions

The asteroid belt is a remnant of planetary formation processes, and it has been modified continuously in the following ∼4.5 billion years through collisional evolution. Many asteroids are likely to be have been shattered into fragments and exist as gravitationally-bound rubble piles. Rubble-pile asteroids may be expected to exhibit large porosities (∼40%), but this may be complicated by compaction under self-gravitation of the asteroids. A compilation of mass and volume estimates of asteroids implies a negative correlation between size and porosity for stony and carbonaceous asteroids (Carry, 2012). The density estimates of large M-type asteroids, such as Psyche and Kleopatra, indicate that these bodies contain large porosities (e.g., ∼50% if they are purely metallic) compared with asteroids of other types but similar sizes (e.g., Elkins-Tanton et al., 2020; Marchis et al., 2021). To understand these observations, models for cold compaction of asteroid boulders (with different materials for asteroids of different types) need to be developed.

We focus on the estimate of macro-porosity (i.e., void space between the boulders, rather than micro-cracks and voids inside the boulders). In the absence of compressive loading, the porosities of assemblages of equally sized non-adhesive boulders are typically ∼40%. A simple model is developed in Section 2.3 to estimate the effect of boulder size polydispersity, under the assumption that the boulders follow a truncated power law distribution. Although the porosity can be greatly reduced at intermediate power law exponents, it remains around the equally sized value at small and large power law exponents (see Figure 2). For simplicity, we focus on effectively equally sized boulder assemblages to investigate the effect of pressure.

Silicate and chondritic materials are brittle at present asteroid temperatures. In assemblages of this type of boulders, compaction occurs through boulder fracturing, which creates small boulders to fill the pores between the residual large ones. A general framework for brittle boulder compaction is developed (Equations 16 and 17), and the size scale dependence of the boulder strength is described using the model of Bažant (1984) (Equation 23). An expression for compaction of chondritic boulders is derived from these relations and is given in Equation 25. The results suggest that compaction can be significant at pressures of a few megapascals, which can be reached in hundred-kilometer-diameter asteroids (Figure 5).

Metals exhibit different properties from silicates: (a) meteoritic irons are ductile even at current temperatures of asteroids and (b) clean metal surfaces can be cold-welded together when they are in contact. In asteroid bodies up to the size of Psyche (the largest M-type body), the pressure is much smaller than the yield stress of meteoritic iron, and ductile yielding plays a very minor role in affecting the porosity. Meter-sized metal boulders may be cold-welded and stick with each other through contacts created by low-velocity collisions. This adhesion mechanism may significantly increase the porosity of an assemblage of metal boulders. Metals may also develop brittle-like fractures, if the boulders are sufficiently large to contain large cracks. Models are developed to examine the effects of cold welding and brittle-like fracturing. The model results suggest that (a) the porosity may be increased (to ≳60%) by cold welding of the metal boulders at relatively low pressures, and (b) brittle-like fracturing is unlikely to cause significant compaction of metal boulders at typical pressures in asteroid Psyche, unless the boulders are of kilometer sizes.

Acknowledgments

The authors thank Wladimir Neumann, Francis Nimmo, William Herbst, Laurent Montési (the editor), an anonymous reviewer, and an anonymous associate editor for their thoughtful comments. This work was supported by the NASA Discovery Mission grant NNM16AA09C, “Psyche: Journey to a Metal World” awarded to Arizona State University.

Appendix A: Comparison With Previous Results

A1 The Random Apollonian Packings

The random Apollonian packing of has been studied numerically by several groups (e.g., Anishchik & Medvedev, 1995; Varrato & Foffi, 2011), and their results are generally consistent with each other. Here, we focus on the simulations of Varrato and Foffi (2011). In each numerical experiment, an initial population N_init of spheres of size s_max are first inserted into a given volume, and new spheres osculatory to existing ones are progressively inserted into the unoccupied space. In these simulations, the size distributions of inserted spheres are not a priori given, but they can usually be approximately expressed as

(A1)

for s ∈ (s_min, s_max) and N_VF(s) = 0 for s ∈ (s_max, + ∞), in which s_min is the minimum boulder size that are involved in the numerical experiments, N_VF(s) is the total number of boulders that are greater than size s, β_c ≈ 2.5 is the power law exponent for the random Apollonian packing, and C is a constant given by

(A2)

where ϕ_init is the fraction of space occupied by the N_init boulders of size s_max, and φ_init = 1 − ϕ_init is the porosity when only the N_init boulders are inserted. According to the simulation results, the porosity φ_VF of the assemblage of this size distribution approximately follows

(A3)

where α = s_min/s_max is the size ratio. It is consistent with Equation 15 in terms of the exponent, but it has a different proportionality factor (φ_init). In what follows, we demonstrate that this difference is caused by the N_init boulders of size s_max and that Equation A3 can be recovered when the N_init boulders are considered into the simple analysis of Section 2.3.

Taking the derivative of Equation A2, we express the distribution density as

(A4)

where δ(x) is the Dirac delta function, and f(x₁, x₂; x) is a boxcar function that satisfies f(x₁, x₂; x) = 1 if x ∈ (x₁, x₂) and f(x₁, x₂; x) = 0 if x ∉ (x₁, x₂). Therefore,

(A5)

for s ∈ (s_min, s_max) and Σ_VF(s) = 0 for s ∈ (s_max, +∞), where Σ(s) is the total volume of boulders greater than size s, v(s) = Πs³ is the volume of a boulder with size s (Π is a shape constant; for spheres, s is diameter, and Π = π/6), and is the total volume of the N_init boulders of size s_max. The distribution Varrato and Foffi (2011) that is described by Equations A1, A4, and A5 is referred to as “VF distribution.”

The Apollonian packing represents a transition between the undersaturated configuration (Figure 1b) and oversaturated configuration (Figure 1c), and it can be described under both frameworks. Here, we consider it under the framework of the oversaturated assemblage (in which Equation 11 of the main text applies). As in Section 2.3, we assume that the effective void ratio of the controlling skeleton is Σ_P,SK(S)/Σ_SK(S) = ɛ_SK(β_c) and that the skeleton-forming boulders are in the size range (s_min, s_min/α_c(β_c)). The volume of boulders with s ∈ (s_min, s_min/α_c(β_c)) is Σ_SK(S) = Σ_VF(s_min) − Σ_VF(s_min/α_c(β_c)), while total boulder volume is Σ_TOT = Σ_VF(s_min). These relations, together with Equation A5, lead to

(A6)

Substituting Equation A6 and Σ_P,SK(S)/Σ_SK(S) = ɛ_SK(β_c) into Equation 11, we write

(A7)

The analysis in Section 2.3 suggests that at the transition value β ≈ 2.5. Substituting this relation into Equation A7 leads to

(A8)

and thus the porosity , which is consistent with Equation A3, that is, the results suggested by Varrato and Foffi (2011). In Anishchik and Medvedev (1995), the results are somewhat similar: The size distribution density is approximately a truncated power law plus a Gaussian-like peak centered around the upper truncation of the power law distribution (instead of the delta function in the Varrato and Foffi (2011) simulations), and the porosity is similar to that given by Equation A3. Since our simple model (with the above modification) recovers the results of Varrato and Foffi (2011), it should also explain the simulations of Anishchik and Medvedev (1995) when similar modifications are made.

To further illustrate that the Varrato and Foffi (2011) results are actually consistent with our simple analysis provided in Section 2.2, we consider a distribution

(A9)

where the constant C follows Equation A2, is an effective maximum size given by

(A10)

With this distribution density, we write the cumulative distribution function

(A11)

for and for , and the cumulative boulder volume

(A12)

for and for , as illustrated by the dashed curves in Figure A2. Here, a prime is added as superscript to distinguish from the VF distribution (which is illustrated by the solid curves in Figure A2). In what follows, the distribution described by Equations A8-A11 is referred to as “primed VF distribution.”

Figure A1

An illustration of the random Apollonian packing. An initial population of particles are first inserted. The pores are filled by largest possible particles that can be placed in them without redistributing the existing ones. In each step, the newly added particles are shown in gray. After repeating this process for many times, the space is filled with high particle fractions.

Figure A2

The comparison of the Varrato and Foffi (2011) simulations (blue data points) and the model described in A1 (black dashed curves). (a) The cumulative size distribution from the Varrato and Foffi (2011) simulations and the effective truncated power law (also referred to as the “primed VF distribution” in the text) described by Equation A11. (b) The cumulative particle volume distribution from the Varrato and Foffi (2011) simulations and that of the effective truncated power law described by Equation A12. (c) The porosity obtained in the Varrato and Foffi (2011) simulations at different values of minimum-to-maximum particle size ratio, α = s_min/s_max, and the prediction of our model given in Equation A13. Indicated by gray circles are the predictions of the linear-mixture model (Yu & Zou, 1998) given the particle size distribution of the Varrato and Foffi (2011) simulations. The gray circles given the linear-mixture predictions when the mono-sized assemblage porosity (an input parameter i.e., referred to as the “initial porosity” in the linear-mixture model papers) is 0.4, and the error bars indicate the range of linear-mixture model predictions with initial porosities in the range of 0.36–0.44.

Following the conclusion of Section 2.3 (Equation 15), we express the porosity of the assemblage of boulders of this primed size distribution as

(A13)

Substituting Equation A10 into Equation A13, one would find that the value given by Equation A13 is identical to the φ_VF value given by Equation A3. Hence, the primed VF distribution (Equations A8-A11; the dashed curves in Figure A2) can be considered as an effective truncated power law distribution for the VF distribution (Equations A1, A4, and A5; the solid curves in Figure A2). Substituting Equation A10 into Equation A12, one would find that the Σ_VF(s) value given by Equation A5 for s ∈ (s_min, s_max) is identical to the value given by Equation A12 (i.e., the solid and dashed curves coincide with each other for s ∈ (s_min, s_max) in Figure A2b). This reveals the reason why the primed VF distribution leads to the same porosity as the less idealized VF distribution. The two distributions lead to the same value of Σ_SK(S)/Σ_TOT (which determines the value of porosity, at least for the oversaturated assemblage, in which Equation 11 of the main text applies). To introduce this effective truncated power law distribution is also helpful in the following discussion on the results of brittle compaction experiments.

A2 The Sand Compaction Experiments

The brittle compaction model is based on two assumptions: (a) Under compression, the boulder sizes evolve into a truncated power law size distribution (with β ≈ β_c ≈ 2.5) between a characteristic initial boulder size s_i and the critical boulder size s_f; (b) the boulders of the critical size s_f have an average effective strength that is proportional to the external stress. These two assumptions lead to the basic framework of the brittle compaction model, Equations 16 and 17 (i.e., and σ_eff(s_f) = Pϵ).

To examine the validity of these two assumptions, we refer to results of compression experiments on sand assemblages (Altuhafi & Coop, 2011; McDowell, 2002; Nakata et al., 2001; Xiao et al., 2020). Most sand particles are single crystal grains and fracture through Griffith-type cracks (rather than Bažant-type blunt crack bands as assumed in our model for meteorite-like materials, see Section 3.2). Therefore, the size effect for sand particles should be described by a Weibull-like relation (Equation 21), or in a more specific form,

(A14)

where σ_ref is the reference strength of particles of the reference size s_ref, and ω′ = 3/(2ω) is the exponent for the Weibull-like size scale effect (and ω is the exponent for the flaw size distribution, as invoked in the Weibull-like size effect, Equation 21). As the Weibull size effect is statistical, the normal strength σ_eff represents the average of scattered individual values (although the size dependence on particle strength is systematic, the results of strength tests performed on particles of similar sizes may vary by up to one order of magnitude). Many studies report results of one-dimensional compression (rather than isostatic compression) experiments. In this case, Equation 17 should be modified as

(A15)

where σ_V is the vertical stress, and ϵ_V is the proportionality factor for one-dimensional compression (which is likely to be comparable to but different from that for isostatic compression, ϵ). Substituting Equation A14 (with s = s_f) into Equation A15 leads to

(A16)

Substituting Equation A16 into Equation 16 leads to

(A17)

Then, we examine how well this model prediction (Equation A17) explains the results of sand compaction experiments. A similar examination has been performed by de Bono and McDowell (2018). The previous study focused on explaining the slope of the log  φ-log  σ_V plot. Here, we also examine the effect of initial particle size s_i and the post-compression particle size distribution, and compare our results to several experimental studies (Figure A3). The parameters for the size effect on particle strength (ω′ and σ_ref) are obtained by fitting experimental data for individual particle strength (Figure A3, left column).

Figure A3

The results of compaction experiments performed on silicate sands. (a–c) The results from Nakata et al. (2001); (d–f) the results from McDowell (2002) and Altuhafi and Coop (2011) on the same type of sand; (g–h) the results from Xiao et al. (2020). The left column shows the relation between average effective strength and size of individual particles. The dashed curves are calculated using Equation A14 with ω′ given in the figure. The middle column shows the relation between vertical pressure and porosity in the compression experiments. The dashed curves are calculated using Equation A18, with given in the legend, and under the assumption that φ = 0.4 at low stress values. The right column shows the boulder size distribution after each compression experiment (inserts show zoom-ins of the small size ranges). The dashed curves are calculated from truncated power law distributions with β = 2.5, and the dotted curves in (i) are those with β ≈ 3.

When comparing the experimental log  φ-log  σ_V curve with the predictions of Equation A17, we find that the experimental obtained slopes match reasonably well with the values of −0.5/ω′, as already demonstrated by the analysis of de Bono and McDowell (2018). However, directly putting the initial particle sizes into Equation A17 leads to a poor match with experimental data. The data can, instead, be better modeled by

(A18)

in which is an effective initial particle size that differs from the initial particle size s_i (see the legends in the middle column of Figure A3). The reason why the porosity results are better explained using values rather than the initial particle sizes (s_i) can be revealed by post-compression particle size distributions. The particle size distributions can be partly, but not entirely, approximated by truncated power law distributions (see Figures A3c and A3f dashed curves). The difference between the observed post-compression distributions (Figures A3c and A3f circles) and the truncated power law distributions (Figures A3c and A3f dashed curves) is similar to that between the VF distribution (Figure A2b solid curve) and the primed VF distribution (Figure A2b dashed curve) that we discuss in Appendix A1. The post-compression distributions are analogous to the VF distribution, and the truncated power law distributions that we invoke to fit the experimental data are analogous to the VF distribution. Hence, the initial particle size s_i and the effective initial particle size can be understood as s_max in Equation A13 and in Equation A3, respectively. Here, we distinguish between s_i and because the variations in s_i of the experiments are rather small (perhaps comparable to the difference between s_i and ). However, the difference between (the intersect of the dashed curve with the x-axis in Figures A3c and A3f) and s_i (the intersect of the experimental curve with the x-axis in Figures A3c and A3f) are generally no more than a factor of two. This difference is minor compared with the uncertainty in the initial boulder size s_i involved in the discussion of the main text. Thus, we do not introduce the concept of effective initial boulder size in the discussion on asteroids in the main text.

A challenge for the model is that the size distributions of the Xiao et al. (2020) experiments (Figure A3i circles) do not follow truncated power laws with β ≈ 2.5 (Figure A3i dashed curves) as assumed in the compaction model, but rather follow those with β ≈ 3 (Figure A3i dotted curves). A possible explanation for this discrepancy is that the distributions are modified by shear deformation associated with the one-dimensional (rather than isotropic) compression. Although the assumption that large particles must be surrounded by smaller ones typically leads to power law distributions with β ≈ 2.5 (Sammis et al., 1987), the increase in shear strain may make the distribution evolve toward a power law with β ≈ 3 (Sammis & King, 2007). In an experiment of Nakata et al. (2001), for example, the porosity changes from ∼0.4 to ∼0.2 (Figure A3b, blue circles); the assemblage volume changes from ∼Σ_TOT/0.6 to ∼Σ_TOT/0.8 (where Σ_TOT is the volume of boulders), with a vertical strain of ∼0.25 (in one-dimensional compression, volumetric change occurs entirely through vertical strain). In an experiment of Xiao et al. (2020), for example, the porosity changes from ∼0.5 to less than ∼0.1 (i.e., boulder fraction changes from ∼0.5 to over ∼0.9; Figure A3h, blue circles); the total volume of the assemblage changes from ∼Σ_TOT/0.5 to less than ∼Σ_TOT/0.9, with a vertical strain of ∼0.45. According to the simulations of Abe and Mair (2005), a shear strain of ∼0.4 could be sufficient to increase the exponent to β ≈ 3. As the experiments of Xiao et al. (2020) involved greater vertical (and thus shear) strains, they presumably also involved greater modification of the distribution exponent β (Sammis & King, 2007). If the shear deformation modifies the particle size distributions but does not induce further compaction (i.e., does not change the porosity), then we can also explain why the slope of the log  φ-log  σ_V curve is still well approximated by −0.5/ω′ (which is a prediction i.e., made under the assumption of β ≈ 2.5). In isotropic compression (without shear strain), the exponent likely stays around β ≈ 2.5, as suggested by theoretical models that do not include shear deformation (e.g., McDowell et al., 1996; Sammis et al., 1987).

Finally, we consider the constant of force inhomogeneity, ϵ_V. Considering both the compression curve and the particle size distribution, we find that the results of Nakata et al. (2001), McDowell (2002), and Altuhafi and Coop (2011) can be explained by ϵ_V = 8.5 (Figures A3b and A3e), while the results of Xiao et al. (2020) can be explained by ϵ_V = 3.5 (Figure A3h). These results are for one-dimensional compression, and the value for isotropic compression, ϵ, may be slightly smaller than ϵ_V (in the case of isotropic compression, the contrast in forces applied to the same particle from different directions to is likely to be smaller). In this study, as a conservative estimate, we assume ϵ = 3 for isotropic compression.

Cold compression experiments have also been performed for granular water ice assemblages at low temperatures (under which water ice exhibits brittle rheology) by Durham et al. (2005). Measurements of individual particle strength and post-compression grain size distribution were not performed, so an examination of microstructure as provided above is not possible. Their data suggest a power law relation between porosity φ and pressure P, or equivalently a linear relation between log  φ and log  P as demonstrated by the data re-analysis performed by Yasui and Arakawa (2009), which is consistent with our model prediction if the strength of single crystal ice particles follows the Weibull-type size effect.

A3 The Compression of Adhesive Particles

In this Appendix, we describe how we use experimental results on compression of adhesive particles to estimate the relation between reduced pressure Θ and porosity φ. Here, we collect the results on compression experiments performed by Yasui and Arakawa (2009), Güttler et al. (2009), Omura and Nakamura (2017), and Than et al. (2020). The experimental results are typically reported as effectively continuous pressure-porosity (P-φ) curves. Here, for the sake of convenience and feasibility, we collect around 10 points from each curve to characterize their results. The pressure data are converted to the reduced pressure using Equation 36 (i.e., , in which s is the diameter and a is the radius of the particles, and is the pull-off force). The particle diameters are available from the aforementioned papers, and the pull-off forces are estimated as follows.

Studies on compression of cohesive powders usually use assemblages of micrometer-sized dry particles (e.g., Güttler et al., 2009; Omura & Nakamura, 2017; Yasui & Arakawa, 2009). The adhesion between dry particles is often estimated using the Johnson-Kendall-Roberts model (K. L. Johnson et al., 1971) as

(A19)

where is the pull-off force of a pair of particles, γ is the surface energy of the material, and a is the particle radius. More generally, the pull-off force is usually expressed as , where a_eff is the effective radius of the pair of adhesive particles (denoted as particle i and particle j) given by 1/a_eff = 1/a_i + 1/a_j, where a_i and a_j are the radii of the two particles (e.g., K. L. Johnson, 1987). Therefore, the adhesion between a sphere of radius a and a flat surface (i.e., a_i = a and a_j → ∞) is given by , and the adhesion between two spheres of the same radius a (i.e., a_i = a_j = a) is given by Equation A19. Several compression experiments have been performed on micrometer-sized silica beads (Güttler et al., 2009; Omura & Nakamura, 2017; Yasui & Arakawa, 2009). The surface energy of silica has been measured using different methods, and a range of values have been reported. Among these results, perhaps the most relevant one is that provided by Heim et al. (1999), who measured the adhesion between micrometer-sized silica spheres and fitted the results using the Johnson-Kendall-Roberts model to provide a surface energy of γ = 0.019 J/m². Omura and Nakamura (2017) also performed experiments using glass beads, fly ash, and irregular alumina powders. The surface energies of fly ash and glass beads were not reported. Here, they are assumed to be effectively ∼10% the value of silica beads, as assumed by Kiuchi and Nakamura (2014) to explain the observed relation between regolith particle size and porosity. As to the surface energy of alumina, we use the results of Burnham et al. (1990), who measured the adhesion between micrometer-sized alumina particles and assumed the Derjaguin-Muller-Toporov model to suggest a value of γ = 0.041 J/m², which would correspond to γ = 0.055 J/m² if the Johnson-Kendall-Roberts model is assumed.

Than et al. (2020) studied the compaction of assemblages of wet (adhesive) glass beads. The adhesion between wet particles is estimated using the capillary model as

(A20)

where γ_L is the surface energy of the liquid (strictly, the liquid-air interface), and θ is the wetting angle of the liquid and the particle material (e.g., Herminghaus, 2005). Than et al. (2020) performed compression experiment to water-wetted glass beads. The surface energy of water is γ_L = 0.073 J/m². The wetting angle between water and the glass beads was not reported by Than et al. (2020), but the cos θ value is perhaps not very different from unity. Here, we assume perfect wetting (cos θ = 1). With these parameters, the capillary force (Equation A20) is ∼50 times the Johnson-Kendall-Roberts force (Equation A19) if the surface energy of glass beads is effectively ∼10% the value of silica beads.

The results are summarized in Figure A4. The data of dry silica beads from Güttler et al. (2009) and Omura and Nakamura (2017) and these of wet glass beads from Than et al. (2020) exhibit striking similarity around the transition values of , when it is between ∼0.3 and ∼30. This is quite remarkable, since the diameters of silica beads used by Güttler et al. (2009) and Omura and Nakamura (2017) are respectively 1.5 and 1.8 μm, while that of the glass beads used by Than et al. (2020) is 100 μm; accounting for the difference between dry and wet particles (between Equations A19 and A20), the difference in adhesion strength is still greater than one order of magnitude. These data can be fitted by Equation 38 (the black curve in Figure A4). The results of other experiments are somewhat scattered. The alumina particles exhibit greater resistance to compaction, probably because their irregular shapes lead to extra difficultly for particle rearrangement, which is required for compaction (Omura & Nakamura, 2017). The porosities of glass beads exhibit a rather low sensitivity to the variation in pressure, probably due to the low initial porosity caused by the relative large particle size (18 μm) and low effective surface energy of the glass beads (which leads to a relatively low ratio of the adhesive force and the gravitational force). The silica beads in the experiments of Yasui and Arakawa (2009) undergo porosity reduction at larger pressures than those in the experiments of Güttler et al. (2009) and Omura and Nakamura (2017), in terms of both the absolute value and the reduced value, but with a similar slope between φ and ln Θ. The reason of this discrepancy is not clear. A possible origin is the difference in surface “cleanliness” of the particles they used, which can lead a difference effective in the surface energy and thus the adhesion strength. Kiuchi and Nakamura (2014) invoked a difference by a factor of ∼2 in the effective surface energy (i.e., the adhesion force over particle size ratio) between silica particles used by Blum and Schräpler (2004) and Teiser et al. (2011) to discuss the relation between particle size and porosity. A difference in the effective surface energy (pull-off force over particle size ratio) by a factor of ∼5 would suffice to explain the discrepancy between the results of Yasui and Arakawa (2009) and those of Güttler et al. (2009) and Omura and Nakamura (2017).

Figure A4

The relation between reduced pressure Θ and porosity φ of adhesive particles. The data include results of Yasui and Arakawa (2009) for 1 μm diameter silica beads, Güttler et al. (2009) for 1.5 μm diameter silica beads, Omura and Nakamura (2017) for 1.8 μm diameter silica beads, 4.8 μm diameter fly ash particles, 18 μm diameter glass beads, and 6.5–23 μm diameter irregular alumina particles, and Than et al. (2020) for 100 μm wet glass beads. The black curve shows the relation given by Equation 38 (i.e., φ = 0.4 + 0.3/(1 + 0.4Θ)).

Appendix B: Contacts Between Colliding Metal Boulders

B1 Hertzian Elastic Contact

When two spherical boulders of same material and same radius a are pushed toward each other by a normal force , their deformation results in a displacement of magnitude y on each side of the contact and the initial point-contact spreads into a contact surface with a finite radius x. The average normal stress at the contact is . This normal stress is comparable to the shear stress around the contact, which scales as E(y/x), where E is the Young's modulus and y/x is a characteristic strain. Therefore,

(B1)

When the displacement y is small compared to the boulder radius a, it can be scaled as y ∼ x²/a (similar to that for plastic indentation described by Equation 28, but not exactly the same because the surfaces around the contact are distorted by the elastic stress). This geometrical relation, together with the scaling of Equation B1, leads to

(B2)

The rigorous calculation using Hertz's contact theory (e.g., K. L. Johnson, 1987) gives results that are consistent with the above scaling, which are expressed explicitly as

(B3)

where E′ = E/(1 − υ²) is the effective Young's modulus, in which E is the Young's modulus and υ is the Poisson's ratio. The solution of the displacement indicates that to result in a displacement of magnitude y requires a normal force of magnitude . Using this relation, we can calculate the total elastic energy as

(B4)

The factor of two is introduced to account for elastic energy stored on both sides of the contact. Substituting into the solutions of Equation B3, we obtain

(B5)

Substituting the relations given by Equation B5 into the expression of given by Equation B4, we can express the total elastic energy using a and x as

(B6)

or we can express the total elastic energy using σ_N and x as

(B7)

It is interesting to notice that the above expression (Equation B7) does not include the boulder radius a. When the contact formation involves plastic indentation, the contact stress is kept at the indentation hardness, that is, σ_N = σ_H. As the plastic indentation results in permanent deformation of materials around the contact, when the load is removed (in the absence of the cold welding effect), the surface would have a curvature that is probably much greater than the boulder radius a. In this case, during the unloading process, the elastic stress is approximately identical to that from a contact of the same radius but embedded in boulders of radius (e.g., K. L. Johnson, 1987). Since Equation B7 does not include the boulder radius a, it still applies even if a is replace by . Therefore, the elastic energy stored around a contact created by plastic indentation can be estimated as

(B8)

B2 Elastic and Plastic Collisions

Here, we consider the collision between two identical spheres of radius a at a collision velocity u. When the contact reaches the maximum radius x, the total work done by the contact force equals the kinetic energy of collision, , where M = (4/3)πρ_sa³ is the mass of a boulder. If the collision is purely elastic, kinetic energy is entirely stored as elastic energy . Equating the kinetic energy of collision, , and the elastic energy, , we obtain

(B9)

Substituting this Equation B9 into Equation B5, we obtain the maximum contact stress

(B10)

If the collision is deeply into the inelastic regime, most of the kinetic energy of collision is dissipated by the plastic flow. Here, we assume that the contact stress is approximately the indentation hardness σ_H. When the contact radius is x′, the contact force is . Neglecting the elastic deformation at the edge of the contact, we express the displacement on each side of the contact using the geometrical relation of Equation 28 as y′ = x′²/(2a). Taking differentiation of this relation, we obtain dy′ = (x′/a)dx′. When the contact radius x′ reaches the maximum value x, the total work done by the contact force is

(B11)

Again, the factor of two is introduced to account for the work done on both sides of the contact. Equating this work to the kinetic energy of collision, , we express the maximum contact radius x as

(B12)

Open Research

Data Availability Statement

The scripts that we used to perform calculations are available in Z. Zhang (2022).