Page 175 - The Combined Finite-Discrete Element Method

158 DEFORMABILITY OF DISCRETE ELEMENTS

Components of vectors a and b in the global frame are as follows:

a = a x i + a y j
(4.138)
b = b x i + b y j

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

(4.139)

j = j x i + j y j

The components of the base vectors of the deformed initial frame written in the global
frame are
˘ i = i x i + i y j
˘
˘
(4.140)
˘
˘
˘
j = j x i + j y j
It follows that vector a can be written as
a = a x i + a y j (4.141)
= a
x i + a
y j

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

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

Thus

a x i x j x a
x
= (4.142)
a y
i y
j y a
y
The matrix

i x
j x
(4.143)

i y j y
is called the initial transformation matrix.
Vector b can be written as
b = b x i + b y j (4.144)
= b ˘x i + b ˘y j ˘
˘
˘
˘
˘
= b ˘x (i x i + i y j) + b ˘y (j x i + j y j)
˘
˘
˘
˘
˘
= (b ˘x i x + b ˘y j x )i + (b ˘x i y + b ˘y j y )j
Thus

˘ ˘
b x i x j x b ˘x
= (4.145)
˘
b y ˘ i y j y b ˘y