Affine Dynamics
In this chapter, we obtain the equations of motion that govern the simplest, non-trivial, deformable dynamics of an ellipsoidal body. We will employ volume averaging (also known as the method of moments or the virial method) to this end.
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Affine Dynamics
3.1 Introduction As we saw in the previous chapter, the linear momentum balance equations (2.58) for the stresses are coupled, non-linear partial differential equations. Additional complication is introduced by the constitutive description that for geophysical materials is often not straightforward. Also, formulae for gravitational attraction are typically unavailable for shapes more complicated than those of a homogeneous ellipsoid. Finally, the body’s physical environment may itself be changing, leading to timevarying boundary conditions. Thus, solving for the dynamics of a geophysical system is very difficult. Analytical results are possible only in the simplest of geometries for idealized linear material responses. Computational approaches are also not straightforward, as they are often hampered by the presence of several widely separated time scales over which the dynamics proceeds. For example, in a landslide, the time scale over which the landslide descends is much longer than the time scale over which particles constituting the landslide interact. Finally, current geophysical continuum models tend to be rather crude. They are designed to capture only those aspects of the material’s behavior that are relevant to the problem at hand. Because of this, the hope that a large multi-physics, multi-scale, computational model may capture all aspects of a geophysical phenomenon may presently be unrealistic. Much progress may be made by following an alternative approximate approach. In the spirit of Saint-Venant’s semi-inverse method (see, e.g., Timoshenko and Goodier 1970, p. 293), we investigate the response of a continuous body to its environment by making systematic approximations to its actual time-dependent motion in terms of low-order kinematic fields; several examples of the latter are provided in Sect. 2.3. This process is similar to the Galerkin method, wherein an arbitrary field that varies in space and time is approximated by projecting it onto a weighted finite sum of independent, spatial basis functions, with the weights dependent on time; see, e.g., Boyd (2001, Chap. 3). Together the basis functions and the weights account for
© 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_3
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3 Affine Dynamics
variation of the original field in both space and time. For example, the Nth-order approximation of a function f (x, t) in terms of the basis functions Pi (x) and weights wi (t) will be N f (x, t) = wi (t)Pi (x). i=1
As Boyd (2001) shows, Galerkin approximations are often very good. When investigating the motion of a body, the basis functions play the role of low-order kinematic fields. Once the basis set is chosen, the knowledge of the timeevolution of the weights defines a body’s deformable dynamics at the corresponding level of approximation. Because the weights are finite in number, and depend only on time, only a finit
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