Page 240 - Biaxial Multiaxial Fatigue and Fracture

224 C. GAIER AND H. DANNBAUER

The theory applied in FEMFAT differs in some aspects from classical approaches like the
nominal stress concept or the local one and can be characterized by the terms ,,influence
parameter method“, “concept of synthetic S/N-curves” or “local stress concept”. As basic input
data S/N-curves of unnotched specimens are used and modified locally at each node of the FE-
mesh to obtain computed component S/N-curve at critical spots. Several influence parameters
can be considered as e.g. stress-gradient to take into account notch effects, mean-stress
influence, surface roughness, surface treatments, temperature, technological size, etc. Also
mean-stress rearrangements caused by plastic deformations are taken into account. The user
can choose between several analysis methods for the quantification of the influence parameters,
e.g. methods which are fixed by German guiding rules from the Research Committee for
Mechanical Engineering FKM (“Forschungskuratorium Maschinenbau”) [ 71 and such ones
which have been developed by ECS (“Engineering Center Steyr”) [ 1-61.
For proportional loads, which scale the magnitude of local stress states, but which do not
change the orientation of the principal axes, classical damage hypotheses like the maximum
principal stress hypothesis for brittle materials or the distortion energy (von Mises) criterion for
ductile ones can be applied in the high cycle fatigue domain. These hypotheses deliver results
with sufficient accuracy for the technical practice.
The situation is different for complex load situations, when non-correlating stochastic
external loads are applied to components with complicated shapes. The principal axes of local
stress tensors will change their orientations with time. For very special load combinations these
axes can even rotate, while the magnitude of the maximum principal stress remains almost
constant. A widely accepted and applied method to deal with such general load situations is the
critical plane approach (e.g. [8-131). At the critical spot of the component it is assumed, that the
plane is responsible (“critical”) for fatigue failure, for which the accumulated damage exceeds a
critical limit at first (theoretically 1). A still open question remains, how to combine the critical
plane method with classical rainflow counting procedures in a general way, for the assessment
of triaxial stress states and for different material behaviours (brittle/ductile). A method is
developed using a similar nomenclature for basic stress quantities as in [ 121 and [ 131.

THEORETICAL BACKGROUND

In fatigue science closed hysteresis loops in the local stress-strain-path have been recognized to
be responsible for local damages of the material. This has been realized for uniaxial stress or
strain states with constant directions. A procedure for counting such closed hyteresis loops is
the well known rainflow counting method (Matsuishi and Endo [ 141, see Fig. 1).
In reality the situation is much more complicated: Stresses and strains are symmetric tensors
of second order in the 3D-space, where cycle counting methods cannot be applied directly. The
stress tensor must be reduced by two steps to obtain scalar quantities, for which a rainflow
counting of amplitudes and mean values can be applied (Fig. 2): The first step is the well
known transformation into a plane, which is specified by its unit normal vector n:

A time dependent stress vector Sn(t) is obtained, consisting of three independent stress
components &At), S,,&) and S,,&).