The Boundary Element Method - Mechanics of Structures: Variational and Computational Methods - page 561

Then Eq. (9.70) can be written as

1 1

u ji = [ − r ,i j ] =

r r ,i r ,j

(9.72a)

16 π G ( 1 − ν) 16 π G ( 1 − ν)

for i = j and

1

u ji =

[2 ( 1 − ν) r ,kk − r ,i j ]

16 π G ( 1 − ν)

4

1 1

r −

1
r (

= ( 1 − ν) 1 − r ,i r ,i )

16 π G ( 1 − ν)

1

=

r [

( 3 − 4 ν) + r ,i r ,i ] (9.72b)

16 π G ( 1 − ν)

for i = j . Combine Eqs. (9.72a) and (9.72b) to obtain

1

u ji =

Gr [

( 3 − 4 ν)δ + r ,i r ,j ] (9.73)

ij

16 π( 1 − ν)

For two-dimensional plane strain problems, a similar procedure leads to

ln 1
r

r ,i r ,j

1

u ji = ( 3 − 4 ν) δ + (9.74)

ij

8 π( 1 − ν) G

The tractions p ji can be obtained by substituting the displacements into the strain-
displacement relations and the constitutive equations of Chapter 1, Eqs. (1.19) and (1.34),
to obtain

σ ij = S ki j a k

(9.75)

with

1
r α

S ki j =−

( − ν)(δ + δ − δ ) + β

[ 1 2

ki r ,j kj r ,i ij r ,k

r ,i r ,j r ,k ]

where 3 , 2 for three-dimensional problems and two-dimensional plane

strain problems, respectively. The tractions are [Chapter 1, Eq. (1.58)]

α = 2 , 1 , and β =

p i = σ ij a j on the boundary (9.76)

where here (and for the following two equations) a j is the direction cosine of the outer
normal to the boundary. Then from

p i = p ji a j (9.77)

it follows that

[

− 1

r ,i r ,j ] ∂

r

p ji = ( 1 − 2 ν)δ + β

n − (

1 − 2 ν)( r ,i a j − r ,j a i ) (9.78)

ij

4 απ( 1 − ν) r α ∂

where ∂ r /∂ n is the derivative of r with respect to the outer normal of the boundary.