We do not fuss over smoothness assumptions:

(‡) Functions and the boundaries of regions are presumed to have continuity and differentiability properties sufficient to make meaningful the underlying analysis.

We work within a Euclidean space ε so that the phrase “region of space” connotes a region contained in ε.

We generally omit smoothness assumptions in sections not dealing specifically with fields that suffer jump discontinuities, which is the only loss of continuity that we discuss.

The kinematics discussed below, while general and applicable to both solids and fluids, is particularly important in the study of fluids.

Vorticity

The vorticity ω is defined by

Let w denote the axial vector of the spin W so that, by (2.36), W = w×. Then, choosing an arbitrary vector a and bearing in mind that the tensor a× is skew, it follows from (2.61)2, (2.64), and (3.19)2 that

Thus, ω = 2w and, since W = w×, we find that the spin and vorticity in an arbitrary motion are related by

This shows that the vorticity serves as a vectorial counterpart to the spin. Recall from (11.4) that the spin axis denotes the subspace ℒ of vectors a such that

EXERCISE

1. Use (17.2) to show that

Transport Relations for Spin and Vorticity

Kinematical relations central to a discussion of fluids consist of

(i) the relation

between the acceleration, the spatial time derivative v′, and the spatial gradient L = grad v of the velocity (§9.2);

(ii) the decomposition

of the velocity gradient L into a symmetric stretching tensor D and a skew spin tensor W;

(iii) the decomposition

of the stretching D into a deviatoric part D0 and a spherical part ⅓(trD)1.

We now establish several important relations for the transport of spin and vorticity. Since

it follows that

and (9.8) yields the first of

while the second follows from the first using (17.1) and (17.2).

Consider a motion χ. Since the mapping x = χ(X, t) is invertible in X for fixed t, it has an inverse X = χ−1(x, t), the reference map defined in (5.5) and (5.6). χ−1 associates with each time t and spatial point x in Bt, a material point X = χ−1(x, t) in B.

Using the reference map, we can describe the velocity as a function v(x, t) of the spatial point x and t:

The field v represents the spatial description of the velocity; v(x, t) is the velocity of the material point that at time t occupies the spatial point x.

More generally, let ϕ denote a scalar, vector, or tensor field defined on the body for all time. We generally consider ϕ to be a function ϕ(X, t) of the material point X and the time t; this is called the material description of ϕ. But, as with the velocity, we may also consider ϕ to be a function φ(x, t) of the spatial point x and t; this is called the spatial description and is related to the material description through

Similarly, a field φ(x, t) described spatially may be considered as a function ϕ(X, t) of the material point X and t; this is called the material description and is given by

• When there is no danger of confusion we use the same symbol for both the material and spatial descriptions.

This chapter discusses the classical theory of elastic solids, neglecting thermal effects. This theory is important not only in its own right but also because it represents a simple context in which to discuss the central steps in the construction of sound constitutive equations for solids.