Page 264 - Marine Structural Design

240 Part I1 UItimate Strength

V
I-- V 0 0 0
1-v 1-v
V
1 - 0 0 0
1-v
1 0 0 0
E(1- v) 1-2v
[De] = 0 0
(1 + .)(I - 2v) 2(1- v)
1-2v 0
2(1- v)
1-2v
2(1- v)

1 -v -v 0 0
1 1 - v 0 0 O l
0
1 0 0
".l-fl 2(l+v) 2(1+v) 2(1+ :i
0
0
v)
In the elastic region, the relationship between stress increment and strain increment may be
written based on Ep(A.4) as follow.

(do} = [D.l(dE} or {A€} = [c"l(do} (A4
where, A is an increment.
12.5.3 Yield Criterion
The stress condition for the initiation of plastic deformation is called yield criterion and is
generally written as a yield function$

f(4, J, , J, ) = 0 ('4.9)
where, J, , J2, J, are the invariants and expressed as,
J, = ox + oY +or I
r:
J, =-( oxoy + oyez + c20x)+ + ri + r: (A.lO)
-
J, = o,oyc, o,r; - cyr: - u,ri + 2ryrz,rly
The geometrical surface for the yield criterion in a stress space is called yield surface. Because
the first approximation of the yield fimction has no relation with the hydrostatic pressure for
the metallic materials, the yield criterion can be expressed as,
&, J;)= 0, J; = 0 (A.11)
where, J', , J', , J', are called the invariants of deviatoric stress as shown below,