This spatial description is called Eulerian description. Another description is
material field description. In this case, function g(r, t) defines displacement of
point A at time t regardless of where point A is located in space. This is called
material or local description (Fig. 2.17).
The coordinates of a point in local reference coordinate system (configuration)
r are referred to as material coordinates.
The coordinates of a point in spatial coordinate system x, y, z are referred to as
spatial coordinates. Of course, we can link the material coordinate system and spatial
coordinate system using deformation mapping.
In computational solid mechanics, we prefer to use material coordinates because
it is more convenient. We will explain this now.
2.7.2.1 Definition of Material (Local) Coordinates
In Fig. 2.18, we can define a coordinate system that is shaped in the original shape of
the line AB. We place the origin of our local coordinate axis r at point A. At point A r
¼ 0 and at point B the value of local coordinate r ¼1. By normalizing the coordinate
between 0 and 1, we can define displacement of any point between A and B with
respect to initial coordinates very easily. Of course, we can define displacement of
point A using global Cartesian coordinate system x, y and time or we can use material
(local) coordinate system (which is usually called local coordinate system in finite
element method terminology). Local coordinate r is in the axis direction of the initial
1-D element shown in the figure. The origin of the local coordinate system is at point
A’
A
B
B’
Fig. 2.17 Spatial
description of the new
location of point A
A
B
A’
B’
Fig. 2.18 Material (local)
coordinate system
description
32
2 Stress and Strain in Continuum