Page 186 - The Combined Finite-Discrete Element Method

CONSTANT STRAIN TETRAHEDRON FINITE ELEMENT 169

transformation of vector components from the deformed initial frame into the global frame
is given by
˘ ˘ ˘
 
  i x j x k x  
b x b ˘x
 
 b y  ˘ ˘ ˘  b ˘y  (4.197)
=  i y j y k y 
b z b ˘z
˘
˘
˘ i z j z k z
The transformation matrix
˘ ˘ ˘
 
i x j x k x
˘ ˘ ˘
 
 i y j y k y  (4.198)
˘ ˘ ˘
i z j z k z
is called the deformed initial transformation matrix.
Transformation of vector components from the global frame into the initial and deformed
initial frames is obtained using inverse initial transformation matrices:
−1

   −1     ˘ ˘ ˘  

i x j x k x
a
x i x j x k x a x b ˘x b x
˘
˘
˘


k y
 a
y  = 
i y
j y
  a y  and  b ˘y  =  i y j y k y   b y  (4.199)
a
z
i z
j z
k z a z b ˘z ˘ ˘ ˘ b z
i z j z k z
Deformation over the domain of a four noded tetrahedron is approximated using defor-
mation at four nodes of the tetrahedron. This approximation is therefore of the type
x c = α x x i + β x y i + γ x z i (4.200)
y c = α y x i + β y y i + γ y z i
z c = α z x i + β z y i + γ z z i
where x c ,y c and z c represent the current coordinates corresponding to the deformed
configuration, while x i ,y i and z i represent the initial coordinates corresponding to the
undeformed configuration. Both sets of coordinates refer to the global frame:
(i, j, k) (4.201)

It follows from (4.200) that the deformation gradient over the domain of the tetrahedron
finite element is constant. The easiest way to calculate this deformation gradient is to use
the deformed initial frame:
˘ ˘ ˘
(i, j, k) (4.202)
Using this frame, the following matrix of the deformation gradient is obtained:

 
∂x c ∂x c ∂x c
 ∂
x i ∂
y i ∂
z i 
 
 ∂y c ∂y c ∂y c 
F =   (4.203)
 
∂
x i ∂
y i
 ∂
z i 
∂z c ∂z c ∂z c
 
∂
x i ∂
y i ∂
z i