Page 181 - Mechatronic Systems Modelling and Simulation with HDLs

170 8 MICROMECHATRONICS

The local coordinates of arbitrary points (x, y) of the element are thus found
to be:

4 4

x = h i x i , y = h i y i (8.2)
i=1 i=1

where x i or y i denote the coordinates of the element nodes. In the isoparametric
formulation the element displacements are interpolated similarly to the coordinates,
but this time the node displacements are multiplied by the form functions.

4 4

u = h i u i , v = h i v i (8.3)
i=1 i=1

Here u and v are the local element displacements at an arbitrary point of the
element and u i and v i are the corresponding element displacements at the nodes.
The node displacements are furthermore summarised in the form of a vector:

T
ˆ u = u 1 v 1 u 2 v 2 u 3 v 3 u 4 v 4 (8.4)

Thus (8.3) can be drawn up in matrix form:

u(s, t) = Hˆu (8.5)

The next step is the transition from the element displacements to the element strain
ε, which is defined by the derivative of the displacements with respect to the local
coordinates:

∂u ∂v ∂u ∂v ∂w

T
ε = ε xx ε yy γ xy ε zz = + (8.6)
∂x ∂y ∂y ∂x ∂z
The last entry of ε corresponds with the hoop strain, which is triggered by a
displacement in the x-direction over the whole circumference according to the
rotational symmetry.
We can thus draw up the strain-displacement transformation matrix B, which
effects the transformation of the displacements to the strains.

 
∂h 1 ∂h 2 ∂h 3 ∂h 4
0 0 0 0
∂x ∂x ∂x ∂x
 
 
∂h 1 ∂h 2 ∂h 3
 0 0 0 0 ∂h 4 


∂y ∂y ∂y
 (8.7)
ε = B · ˆu, B =  ∂y 

 ∂h 1 ∂h 1 ∂h 2 ∂h 2 ∂h 3 ∂h 3 ∂h 4 ∂h 4 
 ∂y ∂x ∂y ∂x ∂y ∂x ∂y ∂x 


 
h 1 h 2 h 3 h 4
0 0 0 0
R R R R