¼
S XX S XY S XZ
S YX S YY S YZ
S ZX S ZY S ZZ
It is important to point out that the second invariant of the deviatoric stress tensor
is a representation of shear stress; however, physical interpretation of the third
invariant is more complex.
2.7 Deformation and Strain
Let’s assume we have a deformable solid object defined by corners A to H. Under
external loading, the corner points displace to new locations given by points A
0 to H
0
(Fig. 2.12).
Using Cartesian coordinates X, Y, Z, we can easily measure the relative displacement of corner points (or any point in the object) with respect to their original
location. However, just knowing the relative displacement dr of each point in the
deformable body does not help us to understand the strain imposed on the body due
to new configuration. Therefore, we must establish a strain definition. There are
many definitions of strain. We can choose any definition of strain and calculate the
strain in the deformable body. The point we are trying to emphasize is that strain is a
definition (a human construct) calculated using measured displacement quantities.
Therefore, strain cannot be measured directly.
Strain formulation without any constraints on the magnitude of the strain gives
more accurate representation of the deformation. However, mathematical description
21
j
H
G
F
B
E
A
C
D
Z
Y
X
O
i
k
H’
G’
F’
E’
B’
A’
C’
D’
=
Fig. 2.12 Definition of strain deformation and rigid body motion
28
2 Stress and Strain in Continuum