Continuum Mechanics
In this chapter we cover the rudiments of solid mechanics as will be required in later development.
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Continuum Mechanics
2.1 Introduction In applications to be addressed later, we will be interested in an object’s motion, as well as, its internal force distribution. The internal force distribution is described through a stress tensor. At the same time, a description of the motion of a body, or its kinematics, involves knowledge of how each material point in its interior evolves with time. The kinematics is, in turn, intimately linked to the internal and external force distributions via appropriate constitutive and balance laws. To follow the evolution of an object, we relate its configuration at any time t to its known state at some previous time. If this previous time is a fixed reference time, say the time when the body was in its virginal state, we obtain the so-called Lagrangian description of the motion. On the other hand, knowing the configuration of the body at time t in terms of its arrangement an infinitesimal instant before, constitutes the Eulerian description of motion. We will have occasion to employ both these representations. Similarly, the stress state can be elaborated in terms of the object’s present, or its reference state, leading to the Cauchy stress tensor, or the first Piola–Kirchoff tensor, respectively. These two tensors will be related to each other by a knowledge of the body’s kinematics. We will preferentially employ the Cauchy stress tensor. Finally, the kinematical response of a body to applied forces, or, conversely, the stress distribution resulting from an imposed kinematics, are both governed by constitutive laws employed to model the mechanical behavior of the object’s constituent material. We will discuss one in detail which will be employed extensively later. In this chapter, we provide a quick summary of essential continuum mechanics. For more details we refer the reader to the excellent book by Spencer It is (1980). important to note that we will develop relevant equations in a frame O, eˆ i (t) with origin O and which rotates at angular velocity (t), to which we associate the tensor Ω(t). This is motivated by our ultimate aim to investigate Solar System objects, all of which rotate. In this, our formulation isa departure from standard ones that are done with respect to a fixed inertial frame I , ˆii . © 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_2
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2 Continuum Mechanics
2.2 Motion in a Rotating Coordinate System We will be employing several rotating coordinate systems while investigating the motion of Solar System bodies. For this, we will need to relate rates of change of scalar, vector and tensor quantities as seen by observers in these coordinate systems.
2.2.1 Vectors and Tensors Consider the coordinate systems O, eˆ i (t) and I , ˆii . Time differentiation in the rotating frame O will be indicated by ‘˙’, while ‘˚’ will identify time derivatives in the inertial frame I . The rotat
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