Page 177 - The Combined Finite-Discrete Element Method

160 DEFORMABILITY OF DISCRETE ELEMENTS

Use of three noded finite element results in the deformation gradient being constant over
the domain of the finite element. This is because the deformation over the domain of
finite element is a linear function of type

x c = α x x i + β x y i
(4.153)
y c = α y x i + β y y i

where x c and y c represent current coordinates (deformed configuration), and x i and y i
represent the initial coordinates (undeformed configuration).
Deformation over the domain of the finite element is expressed in terms of the defor-
mation at nodes of the triangle. Deformation at nodes of a triangle is defined by the initial
nodal coordinates corresponding to the initial configuration, and current nodal coordinates
corresponding to the deformed configuration.
The easiest way to calculate the deformation gradient for a three noded triangle is to
use the deformed initial frame
˘ ˘
(i, j) (4.154)
Using this frame, the matrix of the deformation gradient tensor is

 
∂x c ∂x c
∂
x i ∂
y i
 
F =   (4.155)
∂y c ∂y c
 
∂
x i ∂
y i
where x c and y c are the current global coordinates given in the global frame

(i, j) (4.156)

while

x and
y (4.157)

are the current coordinates expressed using the initial frame. In other words,

x c = x c (
x i ,
y i )
(4.158)
y c = y c (
x i ,
y i )

and the physical meaning of operators

∂x c ∂x c
and (4.159)
∂
x i ∂
y i
is the change in global current coordinate when from point p in the initial configuration
one moves towards point q, which is in the direction of the initial base vectors

(i, j) (4.160)