Page 286 - Biaxial Multiaxial Fatigue and Fracture

270 A. BUCZYNSM AND G. GLINKA

density contributed by each pair of corresponding stress and strain components. It was confirmed
later by Singh et al. [4] that the accuracy of the additional energy equations was also good when
used in an incremental form. Because the ratios of strain energy density increments seem to be
less dependent on the geometry and constraint conditions at the notch tip than the ratios of
stresses or strains the analyst is not forced to make any arbitrary decisions about the constraint
while using these equations. However, the additional strain energy density equations [4, 71 have
a theoretical drawback indicated by Chu [8], namely the estimated elastic-plastic notch tip strains
and stresses may depend slightly on the selected system of coordinates. Fortunately, the
dependence is not very strong and with suitably chosen system of axis it could be sufficiently
accurate for a variety of engineering applications. It was also found that the set of equations
involving the strain, stress and the strain energy density increments can be singular at some
specific ratios of stress components, which is due to the conflict between the plasticity model
(normality rule) and strain energy density equations. Such a conflict can be avoided if the
principal idea of Neuber involving only shear stresses is implemented in the incremental form.
Namely, it should be noted that the original Neuber rule (5a) was derived for bodies in pure shear
stress state. It means that the Neuber equation states the equivalence of only distortional strain
energy density. Therefore, in order to formulate the set of necessary equations for a multiaxial
non-proportional analysis of elastic-plastic stresses and strains at the notch tip, the equality of
increments of the total distortional strain energy dens@ was used. As a consequence ~ all
-.
equations were written in terms of deviatoric stresses and strains.

DEVIATORIC STRESS-STRAIN RELATIONSHIPS

The notch tip deviatoric stresses of the hypothetical linear-elastic input are determined as:

(7)

The elastic deviatoric strains and strain increments can be calculated from the Hooke law.

det; = ~
2G

The actual near the notch tip deviatoric stress components in the notch tip can analogously be
defined as:

The incremental deviatoric elastic-plastic stress-strain relationships based on the associated
Prandtl-Reuss flow rule can be subsequently written as:

In engineering notation the deviatoric stress-strain relations are written as: