Page 282 - Biaxial Multiaxial Fatigue and Fracture

266 A. BUCZYNSKI AND G. GLINKA

loading history may contain from a few thousands to a few millions of cycles. Therefore
incremental elastic-plastic finite element analysis of such a history would required prohibitively
long computing time. For this reason more efficient methods of elastic-plastic stress analysis are
necessary in the case of fatigue life estimations of notched bodies subjected to lengthy cyclic
stress histories. One of such methods, suitable for calculating multiaxial elastic-plastic stresses
and strains in notched bodies subjected to proportional and non-proportional loading histories, is
discussed in the paper.

LOADING HISTONES
The notch tip stresses and strains are dependent on the notch geometry, material properties and
the loading history applied to the body. If all components of a stress tensor change
proportionally, the loading is called proportional. When the applied load causes the direction
and/or ratio of principal stresses to change in any load increment, the loading is termed non-
proportional. If plastic yielding takes place at the notch tip then the stress path in the notch tip
region is almost always non-proportional regardless whether the remote loading is proportional
or not. The non-proportional loadingktress paths are usually defined by successive increments of
load/stress parameters and all calculations need to be carried out incrementally. In addition the
material stress-strain response to non-proportional cyclic loading paths has to be appropriately
simulated, including the stress-strain non-Iinearity and the material memory effects.

THE STRESS STATE AT THE NOTCH TIP

For the case of general multiaxial laading applied to a notched body (Fig. l), the state of stress
near the notch tip is tri-axial in nature. There are six unknown stress and strain components.
Therefore there are twelve unknowns all together and a set of twelve independent equations is
required for the determination of all stress and strain components at a point near the notch tip.

The material constitutive relationships provide six equations, leaving six additional equations to
be established.

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THE MATERIAL CONSTITUTIVE MODEL
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In the case of proportional or nearly proportional notch tip stress paths the Hencky total
deformation equations of plasticity can be used in the analysis.