Material Modelling and Fundamental Solutions - The Boundary Element Method with Programming - page 85

The fundamental solutions for the tractions are obtained by first computing the
fundamental solutions for the strains and then applying Hooke's law. The fundamental
solutions for strains are obtained by taking the derivative of the displacement solution.
The tractions at point Q due to a unit load at P in x-direction are given by

C

2

2

TPQ , C 2 r cos T

xx 3 , x

r
C

ª º

2

ª º

TPQ , 2 rr cos T Cnr nr

(4.59)

¬ ¼

xy

¬

, x , y 3 , y , y , x , x

¼

r

1

C 1/ 4 SQ 1 , C 1 2 Q , cos T rn <

2 3

r

where Tis defined in Figure 4.10.
For a unit load in the y-direction we have

C

2 2

TPQ , C 2 r cos T

yy 3 , y

r
C

(4.60) ª º

2

TPQ , 2 rr cos T Cnr nr ª º

¬ ¼

x

¬

, xy , 3 , yx , , xy ,

¼

r

The combined expression is

C

(4.61)

2

ª º

TPQ (, ) C G 2 rr cos T C (1 G ( nr nr )

¬ ¼

ij 3 ij , i , j 3 ij j , i i , j

2

r

We note that the first part of the solution is symmetrical (i.e., the first part of T xy
equals T xy ) but the second part is not.
For the three-dimensional problem, the fundamental solution is obtained for point
loads in x ,y and z directions.

Figure 4.12 Notation for three-dimensional Kelvin solution (point load in x direction)