1. Reference problem#
1.1. Geometry#
Analysis consists in determining the damage suffered by a part at a point at which the stress loading history is provided.
From a simple loading history defined by DEFI_FONCTION, the elementary cycles are extracted by the cycle counting method of RAINFLOW [R7.04.01], then the elementary damage associated with each cycle is calculated, by interpolation on the Wöhler curve of the material.
Various possibilities of introducing the Wöhler curve and the taking into account or not of an elasto-plastic concentration coefficient are tested:
The Wöhler curve is defined in the form:
\(\begin{array}{cc}{S}_{\mathrm{alt}}=\text{contrainte alternée}=1/2({E}_{C}/E)\Delta \sigma & X={\text{LOG}}_{10}({S}_{\mathrm{alt}})\\ N={10}^{\mathrm{a0}+\mathrm{a1X}+{\mathrm{a2X}}^{2}+{\mathrm{a3X}}^{3}}& D=\{\begin{array}{}1/N\text{si}{S}_{\mathrm{alt}}\ge {S}_{I}\\ 0.\text{sinon}\end{array}\end{array}\)
with:
\({E}_{C}\) = Young’s modulus associated with the fatigue curve of the material,
\(E\) = Young’s modulus used to determine the constraints,
the material constants \(\mathrm{a0}\), \(\mathrm{a1}\), \(\mathrm{a2}\), and \(\mathrm{a3}\),
and \({S}_{I}\) the endurance limit of the material.
The Wöhler curve is defined in the same form and in addition an elasto-plastic concentration coefficient defined by:
\(\mathrm{\{}\begin{array}{ccc}{K}_{e}\mathrm{=}1& \mathit{si}& \Delta \sigma <3{S}_{m}\\ {K}_{e}\mathrm{=}1+(1\mathrm{-}n)\mathrm{/}(\Delta \sigma \mathrm{/}3{S}_{m}\mathrm{-}1)\mathrm{/}(n(m\mathrm{-}1))& \mathit{si}& 3{S}_{m}<\Delta \sigma <3m{S}_{m}\\ {K}_{e}\mathrm{=}1\mathrm{/}n& \mathit{si}& 3m{S}_{m}<\Delta \sigma \end{array}\)
where
\({S}_{m}\) is the maximum allowable stress,
and \(n\) and \(m\) two constants that depend on the material.
The Wöhler curve is defined in Basquin’s analytic form: \(D=A{S}_{\mathrm{alt}}^{\beta }\)
Finally, we determine the total damage suffered by the room by adding up all the elementary damage by Miner’s linear rule.
1.2. Material properties#
Parameters for defining the Wöhler curve:
\(\mathrm{a0}\) |
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|
|
|
55.81 |
—43.06 |
11.91 |
—1.16 |
Parameters for defining the elasto-plastic concentration coefficient \(\mathrm{Ke}\):
\(\mathrm{Sm}\) |
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|
0.3 |
1.7 |
Parameters for defining the Wöhler curve in Basquin analytical form:
\(A\) |
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1.001730939 E-14 |
4.065 |
Charging history
\(t\) |
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\(\sigma (t)\) |