CONTINUUM MECHANICS for ENGINEERS

FIGURE 4.1
Position of typical particle in reference configuration X A and current configuration x i.
and we call the components X A the material coordinates, or sometimes the
referential coordinates, of the particle X. Upper-case letters which are used as
subscripts on material coordinates, or on any quantity expressed in terms of
material coordinates, obey all the rules of indicial notation. It is customary
to designate the material coordinates (that is, the position vector X) of each
particle as the name or label of that particle, so that in all subsequent configurations
every particle can be identified by the position X it occupied in the
reference configuration. As usual, we assume an inverse mapping
X = Φ –1 (X) (4.2-3)
so that upon substitution of Eq 4.2-3 into Eq 4.1-3 we obtain
x = κ [Φ –1 (X),t] = χ(X,t) (4.2-4)
which defines the motion of the body in physical space relative to the reference
configuration prescribed by the mapping function Φ.
Notice that Eq 4.2-4 maps the particle at X in the reference configuration
onto the point x in the current configuration at time t as indicated in
Figure 4.1. With respect to the usual Cartesian axes Ox 1x 2x 3 the current position
vector is
x= xieˆi (4.2-5)
where the components x i are called the spatial coordinates of the particle.
Although it is not necessary to superpose the material and spatial coordinate
axes as we have done in Figure 4.1, it is convenient to do so, and there are
no serious restrictions for this practice in the derivations which follow. We