Page 173 - The Combined Finite-Discrete Element Method

156 DEFORMABILITY OF DISCRETE ELEMENTS

Using these logarithmic strain components, the constitutive law can be formulated for any
material, thus yielding the Cauchy stress components.

4.7 CONSTANT STRAIN TRIANGLE FINITE ELEMENT

In the combined finite-discrete element method, finite element discretisation of discrete
elements is also used to process contact interaction. To arrive at efficient contact interaction
algorithms, it is important to employ the simplest possible geometry of finite elements.
In 2D space this is a three noded triangle (Figure 4.12). The deformation of this element
is shown in Figure 4.13.
To describe this deformation, two frames are introduced:
• Initial frame: this frame corresponds to the initial (undeformed) configuration. The base
vectors of this frame are identical in magnitude and orientation with the two edges of
the finite element, as shown in Figure 4.14. Thus, base vectors are neither unit vectors
nor orthogonal to each other:
(i, j) (4.131)

• Deformed initial frame: this frame corresponds to the deformed (current) configuration.
The base vectors of this frame are identical in magnitude and orientation with two edges
of the deformed triangle, as shown in Figure 4.14. The base vectors are not unit vectors.
They are not orthogonal to each other either:

(i, j) (4.132)
˘ ˘

2

0
1

Figure 4.12 Geometry of the constant strain triangle finite element.

2
0
2

y
Initial
configuration 1
0
1
Deformed (current)
x configuration

Figure 4.13 Left: Initial configuration, right; deformed configuration.