Page 167 - The Combined Finite-Discrete Element Method

150 DEFORMABILITY OF DISCRETE ELEMENTS

~
i

~
j
~
j s ˆ~
zk
~
k

Deformed configuration
i

k Initial configuration

Figure 4.10 The second Piola-Kirchhoff stress components.

4.6 CONSTITUTIVE LAW

For an elastic body a constitutive law (physical equations) can be written as
ˆ
T(x) = T(F(p), p) (4.92)

where T is Cauchy stress tensor, x represents deformed configuration and p represents
initial configuration. A necessary and sufficient condition that the response is independent
of the observer is that
ˆ
T
ˆ
RT(F(p), p)R = T(RF(p), p) (4.93)
i.e. if rotation R is applied to the elastic body, the stress should not change. Actually, this
rotation could also be viewed as rotation of the global triad. Rotation of an elastic body
is equivalent to the rotation of the global coordinate system in the opposite direction by

R −1 = R T (4.94)

The body is called isotropic if

ˆ
ˆ
T(F(p), p) = T(F(p)R, p) (4.95)
Many materials cannot undergo large (finite) strains, and often it is the case that only
small strains are possible before fracture or failure occurs. In addition, in many problems
of practical engineering importance the deformation gradients are also small, resulting in
the deformed configuration being almost identical to the initial configuration.