Nonlinear Finite Element Analysis of Concrete Structures

- 50 -
as the constitutive behaviour is fully described by fig. 3-1 b).
In the present section we proceed with the total formulation of
the plasticity theory as applied to membrane reinforcement.
The von Mises yield criterion is employed and considering in the
first place the incremental formulation of isotropic hardening
this means that the loading surfaces are given by
f - (I s i: Hif " a e leP »
(3 ' 1)
where s.. as usual is the deviatoric stress tensor and a is the
in
e
equivalent stress. In accordance with the assumption of isotropic
hardening a depends on the equivalent plastic strain e . The
increment of equivalent plastic strain e is defined by
where z. . denotes the plastic strain tensor. For uniaxial loading,
eq. (1) infers that f = a,, = a . Assuming the usual normality
rule following for instance from Drucker's postulates (1951) we
have
de?. = dX |£
where dX is a positive function. Use of eq. (1) yields
de ?j • dA rir 1 < 3 - 3 '
e
From the latter equation follows plastic incompressibility which
in turn implies that for uniaxial loading eq. (2) results in
de p = dz\ v
If eq. (3) is multiplied with itself we obtain
ax - (2 de?, deP j ) %
i.e.
dX » d£ p