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1
2
2
J           (     2 ),
2 x y y z z x xy yz zx
4
1 1
2
2
J      (        2 )     .
3 x y z x yz y xz z xy xy yz zx
4 4

3.7 Generalized Hooke's law

Using the principle of the independence of action of the forces
(superposition), volumetric stress state, given by the principal
stresses, represented as the sum of three linear stress states A , B
and C (fig. 3.11).

 2  2
 3  3

 1  1 =  1  1 + +

 3  3
 2  2
А В С

Figure 3.11

Using the formulas (2.10) and (2.17), we write the expression of
deformation for all three cases:
 
A
A
A
for A :   1 ,       1 ;
1 2 3
E E
 
B
B
B
for B :       2 ,   2 ;
1 3 2
E E
 
C
C
C
for C :       3 ,   3 .
1 2 3
E E
In simultaneous action of three principal stresses the full
relative deformation

B
C
A
B
C
B
A
A
C
       ;        ;        ,
1 1 1 1 2 2 2 2 3 3 3 3
or
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