[IMPL] Implement S-N curve Kohout-Věchet model

[MartinNesladek]

ℹ️ General Information

Component Name: Kohout-Věchet curve

Component Location: material_laws/SN/

Suggested Python Name: wohler_kohout_vechet

FABER WG Relation: 2.1

Brief Description: Stress to life and life to stress calculation via the Kohout-Věchet curve

Priority: 9

Technical Complexity: 3

Estimated Effort: 4

Dependencies: -

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Implementation Details

📋 Specification

A function implementing the Kohout-Věchet representation of an S–N curve, allowing the computation of the number of cycles ($N$) from a given stress amplitude ($\sigma_a$), and vice versa. The parameters ($A, B, C$) and ($\beta$) are regression constants (coefficients and exponent) obtained from fatigue testing.

Mathematical Formulation

Stress amplitude from life:

$$ \displaystyle \sigma_a = A\left( C\frac{N+B}{N+C} \right)^{\beta} $$

$$  \displaystyle \sigma_a = A\left( C\frac{N+B}{N+C} \right)^{\beta} $$

Use the Newton iterative scheme to get life ($N$) from stress amplitude:

$$ \displaystyle N^{i+1}=N^{i}-\frac{f(N^i)}{f'(N^i)}, $$

where

$$ \displaystyle f(N) =A\left( C\frac{N+B}{N+C} \right)^{\beta} - \sigma_a $$

and

$$ \displaystyle f'(N) =A\beta C^{\beta}\frac{(N+B)^{\beta-1}(C-B)}{(N+C)^{\beta+1}} $$

$$  \displaystyle N^{i+1}=N^{i}-\frac{f(N^i)}{f'(N^i)}, $$

$$  \displaystyle f(N) =A\left( C\frac{N+B}{N+C} \right)^{\beta} - \sigma_a $$

$$  \displaystyle f'(N) =A\beta C^{\beta}\frac{(N+B)^{\beta-1}(C-B)}{(N+C)^{\beta+1}} $$

Inputs

1. Kohout-Věchet model regression parameters

Parameter	Symbol	Type	Description	Units	Constraints
KV_A	$A$	array of floats	K-V coefficient	-	$>0$
KV_B	$B$	array of floats	K-V coefficient	-	$>0$
KV_C	$C$	array of floats	K-V coefficient	-	$>0$
KV_beta	$\beta$	array of floats	K-V exponent	-	$<0$

2. Stress / Strain values or life

Parameter	Symbol	Type	Description	Units	Range
stress_amp	$\sigma_a$	array of floats	Stress amplitude	MPa	$(0; \infty)$
life	$N$	array of floats	Number of cycles	-	$(0; \infty)$

Outputs

Parameter	Type	Description	Units	Range
$N$	array of floats	Number of cycles	-	$(0; \infty)$
$\sigma_{a}$	array of floats	Stress amplitude	-	$(0; \infty)$

Expected Behavior

🔧 Implementation Guidelines

Function Signature

# Suggested function signature

Code Structure

Error Handling

✅ Validation & Testing

Test Cases

Test Case	Inputs	Expected Outputs	Notes
Example 1	$\sigma_{a} = 400 MPa; A = 16,651.6; B = 7,214; C = 960,478; \beta = -0.351$	$N = 35,360$

Validation Criteria

• Mathematical accuracy verified against literature
• Edge cases handled appropriately
• Output format matches specification

📚 References & Resources

Kohout, J., Věchet, S., 2001. A new function for fatigue curves characterization and its multiple merits. Int. J. Fatigue 23, 175–183. https://doi.org/10.1016/S0142-1123(00)00082-7

📝 Technical Notes

Performance Considerations

Edge Cases to Handle

Special Requirements