2. Benchmark solution#

2.1. Calculation method used for the reference solution#

Analytical solution

  • calculation of stresses and deformations. For a loading under simple tension, a uniaxial stress state is obtained that is homogeneous at all points:

\(\sigma =\left[\begin{array}{ccc}\sigma & 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]\) and \(\varepsilon =\left[\begin{array}{ccc}\varepsilon & 0& 0\\ 0& \gamma & 0\\ 0& 0& \gamma \end{array}\right]\)

So the equivalent quantities are \(\{\begin{array}{}{\sigma }_{\mathrm{VMIS}}=\mid \sigma \mid ={\sigma }_{\mathrm{TRESCA}}\\ {\sigma }_{\mathrm{VMIS}-\mathrm{SG}}=\sigma \end{array}\)

and \(\{\begin{array}{}{\varepsilon }_{\mathrm{INVA}-2}=\frac{2}{3}\mid \varepsilon -\gamma \mid \\ {\varepsilon }_{\mathrm{INVA}-\mathrm{2SG}}=\frac{2}{3}\mid \varepsilon -\gamma \mid \ast \mathrm{sign}\left[\frac{\varepsilon +2\gamma }{3}\right]\end{array}\)

  • then manual calculation of the cycles using the RAINFLOW method, as well as the loading amplitudes (\(\frac{\Delta \sigma }{2}\) or \(\frac{\Delta \varepsilon }{2}\)).

cycles

\(\Delta \sigma /2\)

\(\Delta {\varepsilon }_{\mathrm{INVA}-2}/2\)

1 2 3 4

1 ».

0.5

3.5 »

0.8667 0.433315 0.8667 3.03335

  • finally transfer these values to the Wöhler or Manson-Coffin curves to estimate the unit damage at each cycle \(i\), i.e. \({\mathrm{Du}}_{i}=\frac{1}{{N}_{i}}\) (\({N}_{i}\) being the number of cycles to break for a given amplitude), as well as the cumulative damage \(D=\underset{i}{\Sigma }D{u}_{i}\) (linear accumulation rule of MINER).

Note:

We will use as equivalent stress \({\sigma }_{\mathrm{VMIS}-\mathrm{SG}}\) and as equivalent deformation \({\varepsilon }_{\mathit{INVA}\mathrm{-}\mathrm{2SG}}\mathrm{=}\frac{2}{3}\mathrm{\mid }\varepsilon \mathrm{-}\gamma \mathrm{\mid }\mathrm{\times }\mathit{sign}\left[\frac{\varepsilon +2\gamma }{3}\right]\) .

2.2. Benchmark results#

  • Given the values of the load parameters used, we simply obtain at the end of loading (increment 8) \(\sigma =-3.\) \(\varepsilon =-3.\) \(\gamma =0.9\) \({\varepsilon }_{\mathit{INVA}\mathrm{-}2}\mathrm{=}2.6\).

  • For the calculation of the damage, we obtain:

\({D}_{\mathit{Wöhler}}\mathrm{=}\mathrm{4,8133}\mathrm{.}{10}^{\mathrm{-}3}\mathrm{=}\underset{i\mathrm{=}1}{\overset{4}{\Sigma }}D{u}_{i}\)

\({D}_{\mathit{Manson}}\mathrm{=}\mathrm{4,67}\mathrm{.}{10}^{\mathrm{-}3}\mathrm{=}\underset{i\mathrm{=}1}{\overset{4}{\Sigma }}D{u}_{i}\)

2.3. Bibliographical references#

  1. DOWNING and SOCIE, 1982. « Simple Rainflow Counting Algorithms. » Int. J. Fatigue, January 1982 (p. 31).