[IMPL] Implement Damage Accumulation via various versions of Palmgren Miner rule

[JanHPapuga]

ℹ️ General Information

Component Name: Damage Accumulation Rule according to Palmgren-Miner

Component Location: core/damage_cumulation

Suggested Python Names: damage_cumulation_elementary, damage_cumulation_basic, damage_cumulation_haibach

FABER WG Relation: 4.5

Brief Description: three functions to deal with the Palmgren-Miner linear damage accumulation depending on the choice of its version:

Priority: 10

Technical Complexity: 2

Estimated Effort: 1

Dependencies: -

Implementation Details
📋 Specification

There are at least these three functions to deal with the Palmgren-Miner linear damage accumulation depending on the choice of version:

1. Elementary version - the same slope k of the S-N curve below and above the fatigue limit
2. Basic version - the S-N curve gets horizontal at the fatigue limit, no damage for stresses beneath
3. Haibach version - the original slope_k is modified below fatigue limit to 2*slope_k-1

Inputs
(1) Parameters of the S-N curve in the power law sig^k*N=C: (a) slope k, (b) constant C
(2) Fatigue limit value sig_FL
(3) Pairs of load spectrum: stress sig, number of occurrences n

Outputs
Damage caused by each input pair

Required behavior
For the stresses at and above the fatigue limit, the damage is computed from the input law:
D=n/N=n*sig^k/C

For the response below the fatigue limit, the response differ based on the version to be implemented:
1) Palmgren-Miner elementary
The same rule as above the fatigue limit is applied to compute damage

2) Palmgren-Miner basic
D=0

3) Palmgren-Miner by Haibach
D=nsig^(2k-1)/(C*sig_FL^(k-1))

Resources for understanding if necessary

1. Graphical interpretation of the change in the S-N curve based on the chosen version can be found e.g. in this open-access paper: http://dx.doi.org/10.5545/sv-jme.2013.1348
2. The principles of the linear damage accumulation can be grasped from this post: https://community.sw.siemens.com/s/article/calculating-damage-with-miner-s-rule

Tests/validation information if necessary
For k=5, C=1e15, sig_FL=137.97
n=5000, sig=150: D=3.8e-3 for any variant
n=100000, sig=110:
Elementary: D=1.61e-2
Basic: D=0
Haibach: D=6.51e-3