Page 171 - The Combined Finite-Discrete Element Method

154 DEFORMABILITY OF DISCRETE ELEMENTS

The Green–St. Venant strain tensor due to the shape changing stretch has no volume
changing component in it. In a similar way, the Green–St. Venant strain tensor due to
the volume changing stretch has no shape changing component in it. In other words, by
using these two strain tensors, change in the shape of the material element is completely
separated from change in the volume of the material element. A constitutive law for
homogeneous isotropic material is derived by analogy with the constitutive law described
above in terms of small strains. The physical equations obtained are as follows:

E E
˜
˜
T = E d + E s (4.121)
(1 + ν) (1 − 2ν)
It was explained above that the strains are small because the material fails or fractures
as significant strains are reached. This is not the case with all materials; for instance,
rubber can undergo very large strains before failure. However, under hydrostatic pressure
many other materials can undergo significant change in volume without being damaged.
To cater for such materials that undergo small shape change but finite volume change,
the first part of the above constitutive law should be independent of the current volume
of the material element. As T is the Cauchy stress tensor, this is not the case, for if the
volume decreases the surface tractions on the surfaces of material element will decrease
as well. The change in surface area of each of the six surfaces of the material element
with change in volume is proportional to
(| det F|) 2/3 (4.122)

Thus, to take into account the change in the surface area of the surfaces of the material
element with a change in the volume of the material element, the following modification
of the above constitutive law is adopted:

E 1 E 1
˜
˜
T = E d + E s (4.123)
(1 + ν) (| det F|) 2/3 (1 − 2ν) (| det F|) 2/3
Practical implementation of the above concepts is described in detail in the follow-
ing sections.
It is worth noting that this constitutive law applies only to linear elastic homogeneous
and isotropic materials. However, in principle, it is possible to implement any constitutive
law. For instance, for rubber-like materials undergoing large (finite) strains, it is easier
to express the constitutive law in terms of logarithmic strains. However, in such a case,
base vectors coinciding with principal stretch directions must be employed. These base
vectors are orthogonal to each other. The matrix of the left stretch tensor is therefore
diagonal, i.e.
 
s 1 0 0
V =  0 s 2 0  (4.124)
0 0 s 3

The matrix of the left Cauchy–Green strain tensor is also diagonal:

 2 
s 1 0 0
T 2
B = VV =  0 s 2 0  (4.125)
0 0 s 3 2