Page 179 - The Combined Finite-Discrete Element Method

162 DEFORMABILITY OF DISCRETE ELEMENTS

The physical meaning of this transformation is best understood by considering the inverse
transformation
   
∂x c ∂x c ∂x c ∂x c


∂
x i ∂
y i ∂x i ∂y i i x j x
  
F =   =   (4.166)
∂y c ∂y c ∂y c ∂y c i y j y
   
∂
x i ∂
y i ∂x i ∂y i
The first line of F matrix is the vector

∂x c ∂x c
(4.167)
∂
x i ∂
y i
the components of which are given in global frame (i, j). It represents the gradient of the
function x c (x i ,y i ). Column

i x
i = (4.168)

i y
is the base vector of the initial frame

(i, j) (4.169)
expressed using the global base (i, j). Thus,

∂x c ∂x c
i x ∂x c
= (4.170)

∂x i ∂y i i y ∂
x i
Thesameappliesto

∂y c ∂y c
i x ∂y c
= (4.171)

∂x i ∂y i i y ∂
x i
Having understood the physical meaning behind the components of the deformation gra-
dient, it is evident that the same applies to the velocity gradient:

   
∂v xc ∂v xc ∂v xc ∂v xc
−1


∂x i ∂y i ∂
x i ∂
y i i x j x
  
L =   =   (4.172)
 ∂v yc ∂v yc   ∂v yc ∂v yc 
i y
j y
∂x i ∂y i ∂
x i ∂
y i
The deformation gradient is comprised of either rotation followed by stretch
F = VR (4.173)

or stretch followed by rotation
F = RU (4.174)