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For the isotropic material, the stresses are expressed in terms of the displacements as
[Chapter 1, Eq. (1.35)]
E
ν
2 G
ν
σ ij =
ν) δ ij kk +
2 G
ij =
ν δ ij u k,k +
G
(
u i, j +
u j,i )
(9.97)
(
1
+ ν)(
1
2
1
2
Substitution of Eq. (9.96) into (9.97) leads to
σ
=
+
D ki j p k dS
S ki j u k dS
D ki j p Vk dV
(9.98)
ij
S
S
V
where
1
r α { (
1
D ki j
=
1
2
ν)(δ
ki r ,j
+ δ
kj r ,i
δ
ij r ,k
) + β
r ,i r ,j r ,k
}
4
απ(
1
ν)
2 G
r β
β
r
S ki j =
n [
(
1
2
ν)δ ij r ,k + ν(δ ik r ,j + δ jk r ,i ) γ
r ,i r ,j r ,k ]
+ βν(
a i r ,j r ,k +
a j r ,i r ,k )
ij
1
+ (
1
2
ν)(β
a k r ,i r ,j
+
a j
δ
+
a i
δ
) (
1
4
ν)
a k
δ
ik
jk
4
πα(
1
ν)
in which
5 , 4 for three-dimensional problems and two-dimensional
plane strain problems, respectively, and r is the distance from the point where the stress
is computed to the boundary. The a j are the direction cosines of the outer normal n of
boundary S
α =
2 , 1 ,
β =
3 , 2, and
γ =
.
9.4 Computational Considerations: Interpolation Functions
and Element Matrices
In Sections 9.2 and 9.3, the integral equations and the methods of boundary discretization for
the boundary element method are developed. In this section, some computational aspects
will be considered.
The equations used for the construction of the element matrices are given in Eqs. (9.34)
and (9.92). Since the shape functions for the boundary element method can be the same as
those used for the finite element method, the shape functions given in Chapter 6 for one- and
two-dimensional problems (except those which use nodal derivatives as nodal variables)
can be used as the shape functions for the boundary elements of two- and three-dimensional
problems. Among these shape functions, the isoparametric type of shape function deserves
more attention here since they are widely used in the boundary element method.
The concept of isoparametric elements is that the same shape functions are used to
describe both the displacement inside the element and the geometry of the element. For two-
dimensional problems, the most widely used elements are linear and quadratic elements
(Fig. 9.6a) with the shape functions
Linear elements:
1
2 (
1
2 (
N 1
=
1
ξ)
N 2
=
1
+ ξ)
Quadratic elements:
1
2 ξ(ξ
1
2 ξ(
N 1
=
1
)
N 2
= (
1
ξ)(
1
+ ξ)
N 3
=
1
+ ξ)
(9.99)
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