Formation
In this chapter we study the dynamics of a freely rotating rubble asteroid employing volume-averaged equations developed in Chap. 3 . Our mainly analytical approach provides an alternative to more computationally intensive paradigms, e.g., discrete eleme
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Formation
11.1 Introduction In this chapter, we explore the deformable dynamics and subsequent passage into equilibrium of a granular asteroid. Such an exploration is of interest for many reasons. These include, modeling the processes whereby granular asteroids may have formed beginning with a loose collective of grains (see, e.g., Weidenschilling 2011), or the re-aggregation of a rubble asteroid, following a disruptive flyby (Richardson et al. 1998) or collisions (Michel et al. 2001). We will specialize our investigation to the dynamics of rubble asteroids that are initially prolate. We will find that we succeed in recovering most of the results of Richardson et al. (2005), who obtained equilibrium shapes by computationally studying the passage into equilibrium of aggregates of discrete, interacting, frictionless, rigid spherical particles. The present mainly analytical approach aids in understanding and quantifying such numerical simulations, including the recent ones due to Sanchez and Scheeres (2011, 2012). Finally, this application, which follows the work of Sharma et al. (2009), will demonstrate how the volume-averaging method is extended beyond purely static approaches, such as those of Holsapple (2001), Holsapple (2004).
11.2 Governing Equations We have modeled granular aggregates as rigid-plastic materials in Sect. 2.10.1. When rigid, the dynamics of an asteroid is governed by the Euler’s equations (3.9): ˙ + × · = 0, ·
© Springer International Publishing Switzerland 2017 I. Sharma, Shapes and Dynamics of Granular Minor Planets, Advances in Geophysical and Environmental Mechanics and Mathematics, DOI 10.1007/978-3-319-40490-5_11
(11.1)
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11 Formation
where the right-hand side, representing the torque about the asteroid’s mass center, vanishes for freely-rotating asteroids, and is the angular velocity of the rigid asteroid. We recall that is Euler’s moment of inertia of the asteroid about its mass center and is related to its inertia tensor through (3.10). To complete the dynamical description of a rigid-plastic asteroid, it remains to obtain the equations of motion appropriate to the asteroid when it yields. The plastic flow of the material beyond yield is given by the flow rule (2.77). We retained a dilatation parameter ε in the flow rule (2.77) to control the extent of volume change post-yield. In a deviation from the preceding three chapters on stability, we will not assume an associative flow rule corresponding to ε = k 2 /9. Instead, we will take ε = 0, and, so, recover the flow rule for rigid-plastic materials that preserves volume during plastic flow. Thus, the volume is constant for all times, i.e., V˙ ≡ 0, so that, for an ellipsoid with V ∝ a1 a2 a3 , a˙ i = 0, and, further, tr = 0, ai
(11.2)
with the last equation following from (2.40). Setting ε to zero in (2.80) provides the average stress, post-yield, within the plastically flowing asteroid as
= −p + kp || .
(11.3)
The above, along with the kinematic constraint provided by the second of (11.2), completes the
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