Page 169 - The Combined Finite-Discrete Element Method

152 DEFORMABILITY OF DISCRETE ELEMENTS

This strain tensor is called a right Green–St. Venant strain tensor. It is worth mentioning
that although strains are small, rotations in the combined finite-discrete element method
are always finite. With the right stretch tensor U a material is first stretched in the principal
directions. This is followed by rotation. Thus, the right small strain tensor corresponds to
the initial configuration in a sense that strain components expressed using a global triad
are correct when applied to the initial configuration.
An equivalent small strain tensor is obtained using the left stretch tensor V:

T
T
1
T
1
1
E = (FF − I) = [(VR)(VR) − I] = (VV − I) (4.104)
˜
2 2 2
This strain tensor is called the left Green–St. Venant strain tensor. The left stretch tensor
is defined in such a way that rotation occurs before stretching, i.e. stretching in three
principal directions occurs on the rotated configuration. As rotation in the combined
finite-discrete element method is always finite regardless of the strains, the strain tensor
defined by equation (4.104) is generally different from the strain tensor obtained using
equation (4.103).
The right Green–St. Venant strain tensor refers to the initial configuration. The left
Green–St. Venant strain tensor refers to the deformed (current) configuration. Since strains
are small, to obtain stresses from strains a small strain elasticity constitutive law can be
employed. For homogeneous isotropic material the stress-strain relationship is given by
Hooks law. Hooks law in terms of principal stresses and strains is given by
1 − ν ν ν
σ 1 = E ε 1 + ε 2 + ε 3 (4.105)
(1 + ν)(1 − 2ν) 1 − ν 1 − ν
1 − ν ν ν
σ 2 = E ε 1 + ε 2 + ε 3 (4.106)
(1 + ν)(1 − 2ν) 1 − ν 1 − ν
1 − ν ν ν
σ 3 = E ε 1 + ε 2 + ε 3 (4.107)
(1 + ν)(1 − 2ν) 1 − ν 1 − ν
Additional decomposition of the small strain tensor in the form

     
0 0 0 0 0 0
ε 1 ε 1 − ε s ε s
0 0 = 0 0 + 0 0 (4.108)
 ε 2   ε 2 − ε s   ε s 
0 0 ε 3 0 0 ε 3 − ε s 0 0 ε s
where
1
ε s = (ε 1 + ε 2 + ε 3 ) (4.109)
3
separates change of volume of the material element from the change in shape of the
material element. Hooks law expressed in terms of the strain components given by
equation (4.108) is as follows:

1 1
σ 1 = E(ε 1 − ε s ) + Eε s (4.110)
(1 + ν) (1 − 2ν)
1 1
σ 2 = E(ε 2 − ε s ) + Eε s (4.111)
(1 + ν) (1 − 2ν)