Nonlinear Problems - The Boundary Element Method with Programming - page 420

where c is the cohesion and M the angle of friction. If

slip occurs, that is,
the compatibility condition is no longer applied to that point in the direction tangential to
the interface.
The corresponding yield function is written as:

t c t tan I

s n

F t , t c t tan M

(15.67)

2 s n s n

The consequence is that when either F 1 or F 2 is zero the assembly is changed: Instead
of adding all stiffness coefficients we assemble the corresponding stiffness coefficients
for region I and II into different locations in K .
Consider, for example, the problem in Figure 15.13. The equations for compatibility
at node 1 are (since only one node is involved we have left out the subscript denoting the
local (region) node number):

I
n

II
n

I
s

II
s

u u u and u u u

(15.68)

n 1 s 1

With the vector of interface unknown only involving the ones at node 1

u

- ½

® ¾

n 1

u

(15.69)

c

¯ ¿

u

s 1

For the example with only one interface node we may write for the region stiffness
matrices (see 11.2.2.)

ª

I I

º ª

II II

º

t t t t

I nn ns II nn ns

K « » and K « »

(15.70)

I I II II

« t t » « t t »

¬ ¼ ¬ ¼

sn ss sn ss

and the following assembled interface stiffness matrix is obtained

I
nn

II
nn

I
ns

II
ns

ª º

t t t t

(15.71)

K «

¬

»
¼

«

I
sn

II
sn

I
ss

II
ss

»

t t t t

22

If F 1 =0 then the normal displacement and - as a consequence - also the shear
displacement of region I are independent of region II. The vector of interface unknown is
expanded to

I
n

- ½

u

° °

°

I
s

°

u

u

® ¾

(15.72) c

II
n

° u °

° °

II
s

° °

u

¯ ¿