4. Crossland and Dang Van Papadopoulos criteria#
The criteria [bib9] and [bib13] allow metal structures subject to stresses imposed as a result of a large number of cycles to distinguish between damaging loads and others.
The criteria can be classified into two categories according to the nature of their approach:
macroscopic approach: Crossland criterion,
microscopic approach: Dang Van Papadopoulos criterion.
The Crossland and Dang Van Papadopoulos criteria apply to periodic uniaxial or multiaxial loads.
The purpose of these criteria is not to determine a damage value, but a \({R}_{\text{crit}}\) criterion value such as:
\(\{\begin{array}{c}{R}_{\text{crit}}\le 0\text{pas}\text{de}\text{dommage}\\ {R}_{\text{crit}}>0\text{dommage}\text{possible}(\text{fatigue})\text{.}\end{array}\)
4.1. Crossland criterion#
The Crossland criterion is empirical in nature and is written only on the basis of macroscopic variables.
In fact, from test campaigns, it was possible to note that the amplitude of cission as well as the hydrostatic pressure played a fundamental role in the fatigue mechanisms of structures.
That is why, Crossland postulated the criterion:
\({R}_{\text{crit}}={\mathrm{\tau }}_{a}+a{P}_{\text{max}}-b\)
where
\({\mathrm{\tau }}_{a}=\frac{1}{2}\underset{0\le {t}_{0}\le T}{\text{Max}}\underset{0\le {t}_{1}\le T}{\text{Max}}\parallel {\mathrm{\sigma }}_{({t}_{1})}^{D}-{\mathrm{\sigma }}_{({t}_{0})}^{D}\parallel =\) Cission amplitude
with \({\mathrm{\sigma }}^{D}\) stress tensor deviator.
\({P}_{\text{max}}=\underset{0\le t\le T}{\text{Max}}(\frac{1}{3}\text{trace}\mathrm{\sigma })=\) maximum hydrostatic pressure.
\(a=({\mathrm{\tau }}_{0}-\frac{{d}_{0}}{\sqrt{3}})/(\frac{{d}_{0}}{\sqrt{3}})\text{et}b={\mathrm{\tau }}_{0}\)
with:
\({\mathrm{\tau }}_{0}=\) endurance limit in pure alternating shear,
\({d}_{0}=\) endurance limit in pure alternating traction-compression.
4.2. Dang Van Papadopoulos criterion#
It appeared that the initiation of fatigue cracks is a microscopic phenomenon occurring on a grain scale. This is why fatigue criteria based on local microscopic variables have been postulated.
The criterion implemented [bib8], [bib9], and [bib10] in*Code_Aster* is the Dang Van Papadopoulos criterion, which is written in the form:
\({R}_{\mathrm{crit}}={k}^{\ast }+a{P}_{\mathrm{max}}-b\)
where:
\({k}^{\ast }=\frac{R}{\sqrt{2}}\) |
yes |
\(R=\underset{0\le t\le T}{\text{Max}}\sqrt{({\mathrm{\sigma }}^{D}(t)-{C}^{\ast }):({\mathrm{\sigma }}^{D}(t)-{C}^{\ast })}\) |
\({k}^{\ast }=R\) |
yes |
\(R=\underset{0\le t\le T}{\text{Max}}\sqrt{{J}_{2}(t)}=\underset{0\le t\le T}{\text{Max}}\sqrt{\frac{1}{2}({\mathrm{\sigma }}^{D}(t)-{C}^{\ast }):({\mathrm{\sigma }}^{D}(t)-{C}^{\ast })}\) |
with:
\(R\), the radius of the smallest sphere circumscribed to the loading path in the space of the constraint deviators;
\({J}_{2}(t)\), the second invariant of constraint deviators;
\({C}^{\ast }=\underset{C}{\text{Min}}\underset{t}{\text{Max}}\sqrt{({\mathrm{\sigma }}^{D}(t)-C):({\mathrm{\sigma }}^{D}(t)-C)}\), the center of the hypersphere.
Note:
This is the definition of \(R\) that uses \({J}_{2}(t)\) that is programmed.
\({P}_{\mathrm{max}}=\) maximum hydrostatic pressure \(=\underset{0\le t\le T}{\text{Max}}(\frac{1}{3}\text{trace}\mathrm{\sigma })\)
\(a=({\mathrm{\tau }}_{0}-\frac{{d}_{0}}{\sqrt{3}})/(\frac{{d}_{0}}{3})\) and \(b={\mathrm{\tau }}_{0}\)
with:
\({\tau }_{0}=\) endurance limit in pure alternating shear,
\({d}_{0}=\) endurance limit in pure alternating traction-compression.
Papadopoulos’s basic idea is to write that the grain obeys a VonMises plasticity criterion instead of the Tresca type plasticity criterion used by Dang Van.
Papadopoulos conducted a campaign of comparisons between the results provided by his criterion and experimental results, which shows that the predictions of the Papadopoulos criterion are excellent for affine loads; they are slightly less accurate for non-affine routes.
In his thesis [bib10] Papadopoulos shows that the Crossland criterion and the Dang Van Papadopoulos criterion give the same results for radial loads.
The algorithm used to calculate the radius of the smallest sphere circumscribed to the loading path in the space of the constraint deviators is the one proposed in [bib11]. It is a recurring algorithm that is based on the second invariant of constraint deviators.
Let’s note \({S}_{i}\) the value of the stress deviator at time \({t}_{i}\), \({O}_{n}\) the center of the hypersphere at time, the center of the hypersphere at iteration \(n,{R}_{n}\) the radius of the hypersphere at iteration \(n\) and \(x\) the « isotropic work hardening parameter » of the algorithm.
Algorithm initialization phase:
\(\begin{array}{}{O}_{1}=\frac{1}{N}\sum _{i=1}^{N}{S}_{i}\\ {R}_{1}=0\text{.}\end{array}\)
Iteration from step \(n\) to stage \(n+1\):
Assume \({O}_{n}\) and \({R}_{n}\) are known. We then calculate:
\(\begin{array}{}D=\parallel {S}_{i+1}-{O}_{n}\parallel \\ P=D-{R}_{n}\end{array}\)
If \(P>0\)
\(\begin{array}{}{R}_{n+1}={R}_{n}+x\text{.}P\\ {O}_{n+1}={S}_{i+1}+{R}_{n+1}\frac{{O}_{n}-{S}_{i}}{\parallel {O}_{n}-{S}_{i+1}\parallel }\end{array}\)
If \(P<0\)
\(\begin{array}{}{R}_{n+1}={R}_{n}\\ {O}_{n+1}={O}_{n}\end{array}\)
The algorithm ends when all \({S}_{i}\) points are in the hypersphere with center \({O}_{n}\) and radius \({R}_{n}\).
4.3. Calculation of damage value#
These two criteria applicable to periodic multiaxial loads make it possible to say whether there is damage or not:
\(\{\begin{array}{c}{R}_{\text{crit}}\le 0\text{pas}\text{de}\text{dommage}\\ {R}_{\text{crit}}>0\text{dommage}\text{possible}(\text{fatigue})\text{.}\end{array}\)
These criteria do not provide damage value. However, it may be interesting to calculate a damage value using the material’s Wöhler curves. To do this, it is necessary to define an equivalent constraint \({\mathrm{\sigma }}^{\ast }\), a value to be interpolated on the Wöhler curve.
Wöhler curves can be constructed from shear tests in which case the endurance limit is \({\mathrm{\tau }}_{0}\), but are more generally constructed from tensile compression tests where the endurance limit is \({d}_{0}({d}_{0}<{\mathrm{\tau }}_{0})\).
For there to be coherence between the criterion and the Wöhler curve, it is necessary that:
\(\begin{array}{}\left\{\begin{array}{c}{\mathrm{\sigma }}^{\ast }\le {\mathrm{\tau }}_{0}\text{pas}\text{de}\text{dommage}\\ {\mathrm{\sigma }}^{\ast }>{\mathrm{\tau }}_{0}\text{dommage}\end{array}\right\}\text{pour une courbe de Wöhler définie en cisaillement,}\\ \left\{\begin{array}{c}{\mathrm{\sigma }}^{\ast }\le {d}_{0}\text{pas}\text{de}\text{dommage}\\ {\mathrm{\sigma }}^{\ast }>{d}_{0}\text{dommage}\end{array}\right\}\text{pour une courbe de Wöhler définie en traction-compression}\text{.}\end{array}\)
It therefore seems possible to us to take:
\(\begin{array}{}{\mathrm{\sigma }}^{\ast }={R}_{\text{crit}}+{\mathrm{\tau }}_{0}\text{pour une courbe de Wöhler en cisaillement}(\text{ce qui est assez rare}),\\ {\mathrm{\sigma }}^{\ast }=({R}_{\text{crit}}+{\mathrm{\tau }}_{0})\ast ({d}_{0}/{\mathrm{\tau }}_{0})\text{pour une courbe de Wöhler en traction-compression}\text{.}\end{array}\)
In general, the user can take \({\mathrm{\sigma }}^{\ast }=({R}_{\text{crit}}+{\mathrm{\tau }}_{0})\ast \text{corr}\) where \(\text{corr}\) is a correction coefficient introduced by the user.
By default, this coefficient \(\mathrm{corr}\) is taken to be equal to \(({d}_{0}/{\mathrm{\tau }}_{0})\) (case of the Wöhler curve introduced in traction-compression).
Note:
In the literature, there is no presentation of an approach to using a criterion to calculate a damage value. However, we know that some manufacturers use such an approach, but without knowing the form adopted.
The approach implemented in Code_Aster is proposed by the AMA department.