Nonlinear Finite Element Analysis of Concrete Structures

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where J^ is defircu by
J 3 = 3 (S 1 + S 2 + S 3 } 1
•Z S . . S ., 3, .
3 i] jk ki
and s.. is the stress deviator tensor defined by
ID
s.. = a. . --r 6.. a.,
ID ID 3 ID kk
where the usual tensor notation is employed with indices running
from 1 to 3. The principal values of the stress deviator tensor
are termed s,, s_ and s.,.I., is the first invariant of the stress
tensor; J 2 and J^ are the second and third invariants of the
stress deviator tensor. The often applied octahedral normal
stress a and shear stress T are related to the preceding ino
2°
variants by a
o
= I,/3
1
and T o
=2 J->/3.
2
The invariants of eq. (2)
have a simple geometrical interpretation when eq. (1) is considered
as a surface in a Cartesian coordinate system with axes
a , ø 2 and o, - the Haigh-Westergaard coordinate system - and
the necessary symmetry properties of the failure surface appear
explicitly when use is made of these invariants.
For this purpose, any point, P(o,, o 2 > o^t in the stress space
is described by the coordinates (£, p, 6), in which C is the
projection on the unit vector e= (1, 1, 1)/ \/Ton the hydrostatic
axis, and )) are polar coordinates in the deviatoric
plane, which is orthogonal to (1, 1, 1) , cf. fig. 1. The length
P|ff 1 ,ff 2 ' ff 3'
•» ffi
Fig. 2.1.1: (a) Haigh-Westergaard coordinate system;
(b) Deviatoric plane
b)
of ON is