Page 99 - Introduction to Continuum Mechanics

84 Kinematics of a Continuum

Such a description is known as the material description. Other names for it are: Lagrangian
description and reference description.
II. Observing the changes at fixed locations, i.e., we express, ©,v,T etc. as functions of fixed
position and time. Thus,

Such a description is known as a spatial description or Eulerian description. The triple
(jEjjj^s) locates the fixed position of points in the physical space and is known as the spatial
coordinates. The spatial coordinates jq of a particle at any time t are related to the material
coordinates^, of the particle by Eq. (3.1.2). We note that in this description, what is described
(or measured) is the change of quantities at a fixed location as a function of time. Spatial
positions are occupied by different particles at different times. Therefore, the spatial descrip-
tion does not provide direct information regarding changes in particle properties as they move
about. The material and spatial descriptions are, of course, related by the motion. That is, if
the motion is known then, one description can be obtained from the other as illustrated by the
following example.

Example 3.2.1
Given the motion of a continuum to be

If the temperature field is given by the spatial description

(a)find the material description of temperature and (b)obtain the velocity and rate of change
of temperature for particular material particles and express the answer in both a material and
a spatial description.
Solution. (a)Substituting (i) into (ii), we obtain

(b) Since a particular material particle is designated by a specific X, its velocity will be given
by

t Note: the superposed ^ and the superposed ~ are used to distinguish different functions for the same
dependent variable.