Page 176 - The Combined Finite-Discrete Element Method

CONSTANT STRAIN TRIANGLE FINITE ELEMENT 159

The matrix

˘ ˘
i x j x
(4.146)
˘
˘ i y j y
is called the deformed initial transformation matrix.
Transformation of vector components from the global frame into the initial frame is
obtained using an inverse initial transformation matrix:

−1

a
x i x j x a x
= (4.147)
a
y
i y
j y a y
Transformation of vector components from the global frame into the deformed initial
frame is obtained using an inverse deformed initial frame matrix:

−1
˘ ˘
b ˘x i x j x b x
= (4.148)
˘
b ˘y ˘ i y j y b y
Transformation of vector components from the initial into the deformed initial frame
is obtained as follows:

a = a
x i + a
y j (4.149)

˘
= 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
= (4.150)
a ˘y
i ˘y
j ˘y a
y
Transformation of vector components from the deformed initial frame into the initial
frame is obtained as follows:
a = a ˘x i + a ˘y j ˘ (4.151)
˘

= 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
−1
˘ ˘
a
x i
x j
x a ˘x i ˘x j ˘x a ˘x
= = (4.152)
˘
a
y ˘ i
y j
y a ˘y
i ˘y
j ˘y a ˘y