S-N curves

In order to set a suitable design criteria, I am looking to compare two classes of S-N curves for a fatigue design, viz., E and F2, and I cannot not find a handy plot to refer to, and it is frustrating when standards fail to include. So, I channel it to write some code to roll my own:

S-N curves
S-N curves

The basic S-N curve equation is as follows, which one may know is from Paris-Erdogan law (fracture mechanics):

$$ N = k_1 \cdot S^{-m} $$

The standard does describe it in its logarithmic form, which is as follows:

$$ \log N = \log k_1 - m \cdot \log S $$

and then it goes on to furnish its two sets of key components that form parts of the equation — highlighted below.1

For graphing purposes, the above can also be written as:

$$ S = \left(\frac{N}{k_1}\right)^{-\frac{1}{m}} $$

For example, and I am writing this for myself since I struggle with logarithms, if

$$ log_{10}k_1 = 12.18 $$


$$ k_1 = (10)^{12.18} $$

From ISO 19902:2007

Code: sncurves.py for plotting hotspot stresses versus number of cycles:

#!/usr/bin/env python
# encoding: utf-8
sncurves.py -- 2016 ckunte
May 7: Initial commit.
Apr 29, 2020: Code simplified
import numpy as np
import matplotlib.pyplot as plt

# a = log_10(k1)
a = [
12.18, 16.13, # TJ
14.61, 17.01, # B
13.23, 16.47, # C
11.78, 15.63, # D
11.62, 15.37, # E
11.40, 15.00, # F
11.23, 14.71, # F2
11.00, 14.33, # G
10.57, 13.62  # W1
m = [3.0, 3.5, 4.0, 5.0] # Slope
r = [1.8E6, 1.0E5, 4.68E5, 1.0E6] # Range limit for curves

def style():
    plt.rcParams['grid.linestyle'] = ':'
    plt.rcParams['grid.linewidth'] = 0.5

def sncurve(curve, r_start, r_mid, r_end, a1, m1, a2, m2, \
    # For slope 1 (m1)
    n = np.arange(r_start, r_mid, 1.0E4)
    s = (n / 10**a1)**(-1 / m1)
    plt.loglog(n, s, basex=10, color=graphcolor, linewidth=2.0, \
    # For slope 2 (m2)
    n = np.arange(r_mid, r_end, 1.0E9)
    s = (n / 10**a2)**(-1 / m2)
    plt.loglog(n, s, basex=10, color=graphcolor, linewidth=2.0)

if __name__ == '__main__':
    # Plot all
    sncurve('TJ curve', 1, r[0], 1.0E12, a[0], m[0], a[1], \
        m[3], 'black')
    sncurve(' B curve', 1, r[1], 1.0E12, a[2], m[2], a[3], \
        m[3], 'magenta')
    sncurve(' C curve', 1, r[2], 1.0E12, a[4], m[1], a[5], \
        m[3], 'blue')
    sncurve(' D curve', 1, r[3], 1.0E12, a[6], m[0], a[7], \
        m[3], 'orange')
    sncurve(' E curve', 1, r[3], 1.0E12, a[8], m[0], a[9], \
        m[3], 'green')
    sncurve(' F curve', 1, r[3], 1.0E12, a[10], m[0], a[11], \
        m[3], 'olive')
    sncurve('F2 curve', 1, r[3], 1.0E12, a[12], m[0], a[13], \
        m[3], 'brown')
    sncurve(' G curve', 1, r[3], 1.0E12, a[14], m[0], a[15], \
        m[3], 'deeppink')
    sncurve('W1 curve', 1, r[3], 1.0E12, a[16], m[0], a[17], \
        m[3], 'olivedrab')
    plt.xlabel('Number of cycles, N')
    plt.ylabel('Hotspot stress $\sigma$, MPa')

  1. Playing with logarithms is fraught with error, as they are not the same as plain algebra — Leonhard Euler’s gift to the world, which reminds me I should jog my memory from high-school. There is a very nice podcast recording (mp3) on Euler’s e that also discusses the history of logarithms, which was in ancient days allowed to transform “complicated” multiplication into simple addition, which makes perfect sense in a world that had no computers, no calculators, and no slide rules.