Page 174 - The Combined Finite-Discrete Element Method

CONSTANT STRAIN TRIANGLE FINITE ELEMENT 157

j
2
0
j 2

y
1
0 i
1
x i

Figure 4.14 Left: initial frame, right; deformed initial frame.

The base vectors of the initial frame can be expressed using the base vectors of the
deformed initial frame:

i = i ˘x i + i ˘x j

˘

˘
(4.133)

˘

j = j ˘x i + j ˘y j

˘
Also, the base vectors of the deformed frame can be expressed using the base vectors of
the initial frame:
i = i
x i + i
x j
˘
˘
˘
(4.134)
˘
˘
˘ j = j
x i + j
y j
Any vectors a or b can be written using either base, as shown in Figure 4.15. Vector a
is, for instance, given by

a = 0.3i + 0.9j (4.135)
and vector b by
˘
b = 0.4i + 0.3j ˘ (4.136)
In implementation of the combined finite-discrete element method, both vectors are
defined using the global base:
(i, j) (4.137)

j
2
0
j
2
b
y a
1
i
0
1
i
x
Figure 4.15 Left: initial frame, right; deformed initial frame.