Page 290 - Biaxial Multiaxial Fatigue and Fracture

274 A. BUCZWSH AND G. GLINKA

The deviatoric and the actual stress components Sir and doa at the end of each load increment
are determined from eq.(14-15).

where: n denotes the number of the load increments.
The actual strain increments, can finally be determined from the constitutive equation (3).

CYCLIC PLASTICITY MODEL

In order to predict the notch tip stress-strain response of a notched component subjected to
multiaxial cyclic loading, the incremental equations discussed above have to be linked with the
cyclic plasticity model. Several plasticity models are available in the literature. The most popular
is the model of Ref. [9] proposed by Mroz. According to Mroz [9] the uniaxial stress-strain
material curve can be represented in a multiaxial stress space by a set of work-hardening
surfaces.

In the case of a two-dimensional stress state (01 la = 012' = q3'= 0), such as that one at a notch
tip, the work-hardening surfaces can be represented by ellipses in the coordinate plane for which
the axes are defined by the directions of principal stress components (Fig. 5). The equation of
each work-hardening ellipse in the two-dimensional principal stress space is:

The essential elements of the plasticity model can be presented in such a case graphically in a
two-dimensional principal stress space.
The load path dependency effects are modeled by prescribing a rule for the translation of
ellipses in the d2a-03a plane. The translation of these ellipses is assumed to be caused by the
sought stress increment, represented in the principal stress space as a vector. The ellipses can be
translated with respect to each other over distances dependent on the magnitude of the stress/load
increment. The ellipses move within the boundaries of each other, but they do not intersect. If an
ellipse comes in contact with another, they move together as one rigid body.
However, it has been found that the ellipses in the original Mroz model may sometimes
intersect each other, which is not permitted. Therefore, Garud proposed [lo] an improved
translation rule that prevents any intersections of plasticity surfaces. The principle idea of the
Garud translation rule is illustrated in Fig. 6.

a) The line of action of the stress increment, Ad', is extended to intersect the next larger non-
active surface, fi, at point B2.