Page 185 - The Combined Finite-Discrete Element Method

168 DEFORMABILITY OF DISCRETE ELEMENTS

In a similar way, components of the base vectors of the initial frame written in the global
frame are given by

i = i x i + i y j + i z k (4.191)

j = j x i + j y j + j z k

k = k x i + k y j + k z k

Components of the base vectors of the deformed initial frame written in the global
frame are
˘
˘
˘
i = i x i + i y j + i z k (4.192)
˘
˘ j = j x i + j y j + j z k
˘
˘
˘
˘
˘
˘
k = k x i + k y j + k z k
˘
Any vector a can be written using either frame
a = a x i + a y j + a z k (4.193)

= a
x i + a
y j + a
z k

= a
x (i x i + i y j + i z k)

+ a
y (j x i + j y j + j z k)

+ a
z (k x i + k y j + k z k)

= (a
x i x + a
y j x + a
z k x )i

+ (a
x i y + a
y j y + a
z k y )j

+ (a
x i z + a
y j z + a
z k z )k

The transformation of vector components from one frame to another frame is given by

     
a x i x j x k x a
x
a y i y j y k y a
y
  = 

   (4.194)
a z
i z
j z
k z a
z
The transformation matrix
 

i x
j x
k x


(4.195)
i y j y

k y

i z j z k z
is called the initial transformation matrix. In a similar way, for any vector b
b = b x i + b y j + b z k
(4.196)
˘
= b ˘x i + b ˘y j + b ˘z k
˘