Material Modelling and Fundamental Solutions - The Boundary Element Method with Programming - page 86

The solutions for the displacements in x,y and z directions due to a unit load in x-
direction can be written as

C

2

UPQ , Cr

xx 1, x

r

1

UPQCrr
r

,

xy , x , y

1

(4.62)

UPQCrr
r

,

xz , x , z

with
C 1/ 16 S G 1 Q , C 3 4 Q

1

Now the solution approaches zero, as the distance between source point P and field
point Q tends to infinity. However, this solution also approaches an infinite value, as r
tends to zero. This fact will pose some problems with integrating the fundamental
solutions which we will address later. The solution for load in x,y,z directions can be
written as a combined Equation

UPQCC (, ) ( G rr )

(4.63)

ij , 1 ij , i j ,

The solutions for stresses acting on a boundary surface with an outward normal
direction of n (see figure 4.8) are presented next .
The fundamental solutions for the tractions due to a unit load at P, in x-direction, are

C

2 2

TPQ , C 3 r cos T

xx 3 , x

2

r
C

ª º

2

ª º

TPQ , 3 rr cos T Cnr nr

(4.64)

¬ ¼

xy

¬

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

¼

2

r
C

ª º

2

ª º

TPQ , 3 rr cos T Cnr nr

¬ ¼

xz

¬

, x , z 3 , x , z , z , x

¼

2

r

with

(4.65)

C 1/ 8 SQ 1 , C 1 2 Q

2 3

The general solution for the tractions can be written as

C

2

ª º

TPQ (, ) C G 3 rr cos T C (1 G ( nr nr )

(4.66)

ij

¬

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

¼

2

r