1. Reference problem#
1.1. Modeling A#
The analysis consists in determining the criterion of CROSSLAND and the criterion of DANG VAN - PAPADOPOULOS at a point in a structure subjected to periodic radial multiaxial loading.
Criterion of CROSSLAND:
\({\tau }_{a}+a{P}_{\mathit{max}}\mathrm{-}b\mathrm{\le }0\) where:
\(\begin{array}{cc}{\tau }_{a}\mathrm{=}\frac{1}{2}\underset{0\mathrm{\le }{t}_{0}\mathrm{\le }T}{\mathit{Max}}\underset{0\mathrm{\le }{t}_{1}\mathrm{\le }T}{\mathit{Max}}∥\tilde{s}({t}_{1})\mathrm{-}\tilde{s}({t}_{0})∥\mathrm{=}& \frac{1}{2}\underset{0\mathrm{\le }{t}_{0}\mathrm{\le }T}{\mathit{Max}}\underset{0\mathrm{\le }{t}_{1}\mathrm{\le }T}{\mathit{Max}}\sqrt{\frac{1}{2}({\tilde{s}}_{11}^{2}+{\tilde{s}}_{22}^{2}+{\tilde{s}}_{33}^{2}+2{\tilde{s}}_{12}^{2}+2{\tilde{s}}_{13}^{2}+2{\tilde{s}}_{23}^{2})}\\ \text{amplitude de scission}& \text{avec}\tilde{s}\text{déviateur du tenseur des contraintes}\sigma \end{array}\)
\({P}_{\mathit{max}}\mathrm{=}\underset{0\mathrm{\le }t\mathrm{\le }T}{\mathit{Max}}(\frac{1}{3}\mathit{trace}(\sigma ))\mathrm{=}\) maximum hydrostatic pressure
\(a\mathrm{=}({\tau }_{0}\mathrm{-}\frac{{d}_{0}}{\sqrt{3}})\mathrm{/}\frac{{d}_{0}}{3}\) and \(b\mathrm{=}{\tau }_{0}\)
with \({\tau }_{0}\) = endurance limit in pure alternating shear
\({d}_{0}\) = endurance limit in pure alternating traction-compression
The criterion is: \(\mathrm{Rcrit}={\tau }_{a}+a{P}_{\mathrm{max}}-b\)
If \(\mathrm{Rcrit}\) is negative or zero, there is no damage. If \(\mathrm{Rcrit}\) is positive, there is likely to be damage.
Criteria of DANG VAN - PAPADOPOULOS:
\({K}^{\text{*}}+a{P}_{\mathit{max}}\mathrm{-}b\mathrm{\le }0\)
where \({K}^{\text{*}}\mathrm{=}R\mathrm{/}\sqrt{2}\) where \(R\) radius of the smallest sphere circumscribed to the loading path in the space of the stress deviators \(\tilde{s}\).
\(R\mathrm{=}\underset{0\mathrm{\le }t\mathrm{\le }T}{\mathit{Max}}\sqrt{(\tilde{s}(t)\mathrm{-}{C}^{\text{*}})\mathrm{:}(\tilde{s}(t)\mathrm{-}{C}^{\text{*}})}\) where \({C}^{\text{*}}\) is the center of the hypersphere
\({C}^{\text{*}}\mathrm{=}\underset{C\epsilon K}{\mathit{Min}}\underset{0\mathrm{\le }t\mathrm{\le }T}{\mathit{Max}}\sqrt{(\tilde{s}(t)\mathrm{-}C)\mathrm{:}(\tilde{s}(t)\mathrm{-}C)}\)
\({P}_{\mathit{max}}\mathrm{=}\underset{0\mathrm{\le }t\mathrm{\le }T}{\mathit{Max}}(\frac{1}{3}\mathit{trace}(\sigma ))\mathrm{=}\) maximum hydrostatic pressure
\(a\mathrm{=}({\tau }_{0}\mathrm{-}\frac{{d}_{0}}{\sqrt{3}})\mathrm{/}(\frac{{d}_{0}}{3})\) and \(b\mathrm{=}{\tau }_{0}\)
with \({\tau }_{0}\) = endurance limit in pure alternating shear
\({d}_{0}\) = endurance limit in pure alternating traction-compression
The criterion is: \({R}_{\mathit{crit}}\mathrm{=}{K}^{\text{*}}+a{P}_{\mathit{max}}\mathrm{-}b\)
If \(\mathit{Rcrit}\) is negative or zero, there is no damage. If \(\mathrm{Rcrit}\) is positive, there is likely to be damage.
1.1.1. Material properties#
\({\tau }_{0}\) = endurance limit in pure alternating shear = \(352.\mathit{MPa}\)
\({d}_{0}\) = endurance limit in pure alternating traction-compression = \(540.97\mathit{MPa}\)
1.1.2. Loading history#
\(t\) |
|||
\({\sigma }_{\mathrm{xx}}(t)\) |
—411. |
||
\({\sigma }_{\mathrm{xy}}(t)\) |
—205. |
||
\({\sigma }_{\mathit{yy}}(t)\mathrm{=}{\sigma }_{\mathit{zz}}(t)\mathrm{=}{\sigma }_{\mathit{xz}}(t)\mathrm{=}{\sigma }_{\mathit{yz}}(t)\) |
Charging is considered to be periodic.
1.2. B modeling#
The material properties and the loading history are identical and obtained from test cases SSLV135A.
1.3. C modeling#
The material properties and the loading history are identical and obtained from test cases SSLV135E.
1.4. D modeling#
The material properties and the loading history are identical and obtained from test case SSLV135F.
1.5. Modeling E#
The material properties and the loading history are identical and obtained from test cases SSLV135B.
1.6. F modeling#
The material properties and the loading history are identical and obtained from test cases SSLV135G
1.7. G modeling#
Material properties and loading history are identical and obtained from test cases SSLV135H and SSLV135A