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r
r + s + t , ξ 2 =
s
r + s + t ,and
Figure 10.3. Barycentric coordinates of a triangle, where ξ 1 =
t
r + s + t . This representation can be extended for tetrahedral elements. Barycentric
coordinates can be described from areas (left) or from distances (right).
ξ 3 =
obtained by linear interpolations of the tetrahedron nodal displacements. We can
write this as
ˆ
u = ξ 1 u 1 + ξ 2 u 2 + ξ 3 u 3 + ξ 4 u 4 , (10.2)
where ξ i are the shape functions and u i are the displacements of the four nodes of
the tetrahedron. We can rewrite this using the components of each displacement as
u 1 x
u 1 y
u 1 z
u 2 x
u 2 y
u 2 z
u 3 x
u 3 y
u 3 z
u 4 x
u 4 y
u 4 z
ξ 1
00 ξ 2
00 ξ 3
00 ξ 4
00
u =
ξ 1
00 ξ 2
00 ξ 3
00 ξ 4
0
0
00 ξ 1
00 ξ 2
00 ξ 3
00 ξ 4
or simply
u = H e u,
(10.3)
where H e is a 3
12 matrix containing the shape functions.
Let us now describe how to relate the displacements to the strains. From the
definition of the linear strain, we can explicitly write Equation (10.1) as
×
2 ∂u x
∂x
2 ∂u x
∂x
2 ∂u x
∂x
+ ∂u y
1
+ ∂u x
1
1
+ ∂u z
2 ∂u y
∂x
∂y 2 ∂u y
∂y
∂y 2 ∂u y
∂z
∂y
+ ∂u y
1
+ ∂u x
+ ∂u z
ε
=
,
∂x
2 ∂u z
∂y
∂z 2 ∂u z
∂z
2 ∂u z
∂z
∂z
+ ∂u y
1
+ ∂u x
1
+ ∂u z
∂x
∂y
∂z

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