The scope of this note is to assess the localized compliance of a double cantilever beam specimen determined by stress intensity factors and
Castigliano's theorem. It will be compared to the compliance obtained from simple beam theory and thus, the stress intensity factors will also
be from simple beam theory.
The local compliance is denoted λij which translates to the deflection of Pi at x1i due to a force couple, Pj, acting at x1j. Castigliano's theorem
states:
a

 


2  KI x 1i Pi  KI x 1j Pj
da
λji =  
Pi Pj
E 

0
Consider a single cantilever beam with a concentrated load, P, at distance l from the base
The deflection at distance x from base is:
y ( x P E I l) 
P x
2
6  E I
for x =< l
 ( 3l  x )
With I=(h3 )/12 for a beam of unit depth and the height h with δ=2y being the normal opening displacement of the double cantilever we get:

h

12
δ( x P E h l)  2 y  x P E 
3

l factor 

2
4  P x  ( 3  l  x )
E h
3
for x =< l
We now switch to a coordinate system where x1 originates at the free end of the beams and is positive toward the crack tip. Thus, we need to
replace x and l appropriately with respect to the new reference:
l was the distance from the load, Pi, to the crack tip:
And x the distance from the crack tip to the point where we wnt to know the opening displacement:
 
x  x 1j a  a  x 1j
l x 1i a  a  x 1i
The displacement at x1j due to a force couple, Pi, acting at X1i is then:




δji x 1i x 1j a P E h  δ a  x 1j P E h a  x 1i factor 

 

2
4  P a  x 1j  2  a  3  x 1i  x 1j
E h
3
for x1i =< x1j
And the the compliance at x1j for a force couple acting at x1i is thus:


λji x 1i x 1j a E h 

δji x 1i x 1j a P E h
From Bettis's theorem we get:
λij = λji
P



 

2
4  a  x 1j  2  a  3  x 1i  x 1j
E h
3
for x1i =< x1j
Compliance from stress intensity factors
To determine the local complaince using stress intensity factors we use:

12 P

KI x 1 a h P 
3
h

 a  x1

2
,where a-x1 is the distance from the force couple P to the crack tip.
For Pi and Pj respectively we get:

12 Pi

KI x 1i a h Pi 
3
h



 a  x 1i

KI x 1j a h Pj 
2
12 Pj
3
h


 a  x 1j
2
According to the Castigliano theorem, the localized compliance can be determined by:
a
2

4  a  2  a  3  a x 1i  3  a x 1j  6  x 1i x 1j
2  KI x 1i a h Pi  KI x 1j a h Pj


da factor 
λij x 1i x 1j a h E   
3
E 
Pi Pj
E h




 

0
but this is different from the local compliance obtained directly from simple beam theory. However, we get the same result if we change the limits
of the integral from 0 to a in to from x1j to a!:
a

2 
λij x 1i x 1j a h E   
E

x



 
KI x 1i a h Pi  KI x 1j a h Pj
Pi Pj
1j
 da factor


 

2
4  a  x 1j  2  a  3  x 1i  x 1j
E h
3