A, or it can be located at any point between A and B using a linear interpolation
function. As a result, if point A moves with respect to initial location due to loading,
origin A also moves. Therefore, coordinate r will define displacement of point A at
any time regardless of where point A is (Fig. 2.19).
We will use the terminology local coordinate rather than material coordinates. For
a 2-D case, let’s x-y define our global Cartesian coordinate system and r define local
coordinate system. Let dr define relative deformation of B with respect to
A (Fig. 2.20), where r is the material (local) coordinate axis. It is always in the
natural direction of the member, u is the displacement vector along spatial x axis and
v is the displacement vector along spatial y axis.
The local coordinate relative displacement dr in spatial Cartesian coordinate
system x, y can be given by
du
dr
¼
du
dx
dx
dr
þ
du
dy
dy
dr
dv
dr
¼
dv
dx
dx
dr
þ
dv
dy
dy
dr
ð3:13Þ
It is important to point out that the left-hand side of this equation represents strain
components with respect to the initial length. Equation (3.13) can be written in
matrix form:
A
= 1
B
= 0
Fig. 2.19 Material (local)
coordinate system
O
B
A
B’
Fig. 2.20 Local (material)
coordinate system
2.7 Deformation and Strain
33