Page 332 - The Combined Finite-Discrete Element Method

DEFORMABILITY WITH FINITE ROTATIONS IN 3D 315

The global components of the base vectors of the deformed initial frame, as shown in
Figure 10.4.
 
˘ i x j x k x
˘
˘
 ˘ ˘ ˘  (10.2)
 i y j y k y 
˘
˘
˘ i z j z k z
are represented by a two-dimensional array FX[3][3].
At the end of Listing 10.38, the matrix of the tensor of velocity gradient
 
∂v xc ∂v xc ∂v xc
 ∂
x i ∂
y i ∂
z i   
v 1xc − v 0xc v 2xc − v 0xc v 3xc − v 0xc
 
 ∂v yc ∂v yc ∂v yc 
 =  v 1yc − v 0yc v 2yc − v 0yc v 3yc − v 0yc  (10.3)
L = 
 ∂
x i ∂
y i ∂
z i 
v 1zc − v 0zc v 2zc − v 0zc v 3zc − v 0zc
∂v zc ∂v zc ∂v zc
 
∂
x i ∂
y i ∂
z i
is also calculated and stored as the two-dimensional array L[3][3].
In Listing 10.38, the components of the global base vectors are also expressed in terms
of the base vectors of the initial frame. These components are as follows:
−1
  

i
x j
x k
x i x j x k x

i j k =  i
y j
y k
y  = 
i y
j y
 (10.4)
k y
i
z j
z k
z
i z
j z
k z
The matrix
 
i
x j
x k
x
(10.5)
i
y j
y k
y
 
i
z j
z k
z
is stored as the two-dimensional array FOinv[3][3]. In a similar way, the components of
the global base vectors are expressed in terms of the base vectors of the deformed initial
frame. These components are as follows:
  −1
˘ ˘ ˘
 
i ˘x j ˘x k ˘x i x j x k x

 ˘ ˘ ˘ 
ijk =  i ˘y j ˘y k ˘y  =  i y j y k y  (10.6)
i ˘z j ˘z k ˘z ˘ ˘ ˘
i z j z k z
The matrix
 
i ˘x j ˘x k ˘x
 i ˘y j ˘y k ˘y  (10.7)
i ˘z j ˘z k ˘z
is represented by the two-dimensional array FXinv[3][3].Both FOinv[3][3 and FXinv[3][3
are calculated using the MACRO YMATINV3, which is presented in the file frame.h.
It takes matrix FO as input and returns the inverse matrix FOinv, together with the
determinant voli of the matrix FO. In a similar way, this macro takes matrix FX and returns
its determinant volc, together with the inverse matrix FXinv. It is worth mentioning that
the ratio volc/voli represents volumetric stretch, i.e. the ratio between the volume of the
deformed tetrahedron and the volume of the initial (nondeformed) tetrahedron.