Page 168 - The Combined Finite-Discrete Element Method

CONSTITUTIVE LAW 151

The first case is the case of small strains, and the second case is the case of small
displacements. In the case of small strains and small displacements, a suitable strain
tensor is the so-called infinitesimal strain:

T
1
E = (∇u +∇u ) (4.96)
2
In the combined finite-discrete element method, the strain may be small in most problems
of practical importance. However, the displacements are almost never small, thus (4.96)
does not apply. Small strains only imply small stretches, while rotations and displacements
are large. In such a case of small strains and large rotations, the deformation gradient can
be decomposed into a stretch followed by rotation

F = RU (4.97)

Stretch U is a result of displacements u,i.e.

U = I +∇u (4.98)

Also,
T
C = F F = U 2 (4.99)
where

∂u ∂v ∂w
   ∂u ∂u ∂u 
1 + 1 +
∂x ∂x ∂x ∂x ∂y ∂z
   
   
∂u ∂v ∂w ∂v ∂v ∂v
2    
U =  1 +   1 +  (4.100)
 ∂y ∂y ∂y   ∂x ∂y ∂z 
   
∂u ∂v ∂w
   ∂w ∂w ∂w 
1 + 1 +
∂z ∂z ∂z ∂x ∂y ∂z
which after neglecting higher order terms yields
∂u ∂u ∂v ∂u ∂w
 
1 + 2 + +
∂x ∂y ∂x ∂z
 ∂x 
 
 ∂u ∂v ∂v ∂v 

2 ∂w 
+ 1 + 2 + (4.101)
 ∂y ∂x ∂y ∂z ∂y 

U = 
 
 ∂u ∂w ∂v ∂w ∂w 
+ + 1 + 2
∂z ∂x ∂z ∂y ∂z
This means that the small strain tensor can be approximated by
2
1
E = (U − I) (4.102)
2
In other words, if the strains are small, a small strain tensor (engineering strain) is obtained
using the formula
1
T
1
T
1
T
E = (F F − I) = [(RU) (RU) − I] = (U U − I) (4.103)
2 2 2