Page 19 - Biaxial Multiaxial Fatigue and Fracture

4 K. DANG VAN, A. BIGNONNETAND 1L. FAYARD

fatigue cannot be reduced to plane cracks simply characterised by their length a, submitted to
simple in mode I loading and in a linear elastic regime. This makes the calculation of the
parameters which are supposed to govern the propagation extremely difficult. The transcription
of some parameters which are justified in Fracture Mechanics to fatigue is still poorly founded
and therefore ambiguous for a structure. It is the case for instance of the J parameter and the J
derived parameter usually invoked to correlate fatigue testing results on specimens. For all
these reasons the local approaches based on crack initiation are still preferred by engineers.
As for welded structures, the structural method developed for the offshore industry in the
70’ and the 80’s (AWS and API codes) and particularly following Radenkovic’s proposals [3],
is without any doubt the one which allows the most interesting industrial calculations. In this
method, the welded connection is characterised by a design stress S, more or less clearly
defined, which better describes the local stress at the <<Hot Spot),. The latter is obtained either
by the extrapolation of strain values measured by strain gauges close to the Hot Spot of the
structure or by a thin-shell finite element calculation. S is a ccgeometricalv stress which can be
deduced from the nominal stress using a Stress Concentration Factor obtained by experiment
and parametric formula. This stress is used to predict the fatigue tests on welded tubular nodes
submitted to various service loading by means of a unique S-N line.
It is this idea which is considered in the following, giving a clearer interpretation of the
design stress based on the Dang Van classical fatigue criterion.

DANG VAN CLASSICAL FATIGUE CRITERION

This model is presented in detail in [4]; at the same time, different structural applications are
given. In this model the material is considered as a heterogeneous structure submitted to
cyclic loading.
The investigation of the asymptotic behaviour of elasto-plastic structures submitted to cyclic
loading is a very important topic in solid mechanics; a great deal of research work is devoted to
this subject and direct analysis methods are derived theoretically: the techniques used for
estimating the limit cycle of stress are based on a cinematic and a static approach. Melan [5],
Koiter [6] theorems for elasto-perfectly plastic materials and their extensions by Mandel et al.
[7] for combined isotropic and cinematic hardening material and more recently by Q.S.
Nguyen [8] for a wide class of inelastic materials called generalised standard materials (which
contains the previous ones), are the main theoretical results supporting the proposed fatigue
model. These theorems give (sufficient or necessary) conditions for the existence of an elastic
shakedown regime for a given cyclic loading of an elasto-plastic structure. The key point of the
proof is the associative property of the plastic model, i.e. the fact that the plastic flow (or
plastic flow and generalised hardening parameters) is normal to the convex plasticity domain
(or to the convex domain defined by the function on the ccgeneralised force spacew for the
generalised standard material). The existence of an elastic shakedown state means that a
residual stress pattern builds up and reaches a stabilised state which does not change anymore
under further cycling: the stabilised stress cycle is then purely elastic; it also means that the
dissipated energy is bounded as well as plastic strain. These properties are elegantly
summarised in the following citation due to Professor W. Koiter [6]: ccif the total amount of
plastic work performed in the loading process is accepted as suitable criterion for assessing the
overall deformation, boundedness of the overall deformation may be proven if the structure has
a safety factor greater than one with respect to shakedown)>.
In Dang Van’s fatigue model, since the material is considered as a structure, the previous
results hold with some adaptation: its theoretical foundation is based on an elastic shakedown
hypothesis at all scales of material description near the fatigue limit which corresponds