Contents

Fatigue life depends on both stress amplitude and mean stress. S-N curves capture the amplitude–cycles relationship at a fixed stress ratio; the Goodman diagram extends this to arbitrary $R$-ratios by correcting for mean stress. For fiber composites, the S-N slope is steeper than for metals and the mean stress sensitivity can differ markedly between tension and compression.

S-N relationship

The Basquin (power-law) S-N equation relates stress amplitude $\sigma_a$ to reversals to failure $2N_f$ [2]:

\[\sigma_a = \sigma_f' \left(2N_f\right)^{b} \tag{1}\]

where $\sigma_f’$ is the fatigue strength coefficient and $b < 0$ is the fatigue strength exponent. Rearranging for cycles:

\[N_f = \frac{1}{2}\left(\frac{\sigma_a}{\sigma_f'}\right)^{1/b} \tag{2}\]

The stress ratio $R = \sigma_{\min}/\sigma_{\max}$ defines the loading asymmetry. In terms of mean stress $\sigma_m$ and amplitude $\sigma_a$:

\[R = \frac{\sigma_m - \sigma_a}{\sigma_m + \sigma_a} \qquad \Longleftrightarrow \qquad \sigma_m = \sigma_a\frac{1+R}{1-R} \tag{3}\]

Fully reversed cycling ($R = -1$) has $\sigma_m = 0$ and gives the reference S-N curve. For $R > -1$ the positive mean stress reduces fatigue life.

Modified Goodman criterion

The Goodman criterion [1] assumes a linear interaction between stress amplitude and mean stress:

\[\frac{\sigma_a}{\sigma_{a,0}} + \frac{\sigma_m}{\sigma_u} = 1 \tag{4}\]

where $\sigma_{a,0}$ is the $R = -1$ amplitude at the same $N_f$ and $\sigma_u$ is the ultimate tensile strength. Substituting eq. (3) into eq. (4) and solving for $\sigma_a$:

\[\sigma_a(N_f, R) = \frac{\sigma_{a,0}(N_f) \cdot \sigma_u \cdot (1-R)}{\sigma_u(1-R) + \sigma_{a,0}(N_f)(1+R)} \tag{5}\]

This is the S-N amplitude corrected for mean stress at any $R$-ratio. The denominator grows with $R$, reducing the allowable amplitude. For $R = -1$ the formula returns $\sigma_{a,0}$ exactly.

Constant-life diagram

At a fixed number of cycles $N$, eq. (5) traces a straight line in $(\sigma_m, \sigma_a)$ space — the classic Goodman line [4]. Multiple iso-$N$ lines form a fan:

\[\sigma_a = \sigma_{a,0}(N)\left(1 - \frac{\sigma_m}{\sigma_u}\right) \tag{6}\]

The $\sigma_m$-intercept is always $\sigma_u$ regardless of $N$; the $\sigma_a$-intercept decreases as $N$ increases. For composite laminates the fan is often piecewise linear with different slopes for compression-dominated loading [3].

Interactive charts

Chart 1 — S-N curves at four $R$-ratios using the Goodman correction (eq. 5). Chart 2 — Constant-life diagram (Goodman fan) at $N = 10^3$, $10^5$, $10^7$ cycles.

Python

import numpy as np

def basquin(N, sf_prime, b):
    """R = -1 stress amplitude from Basquin equation."""
    return sf_prime * (2 * N) ** b

def goodman_sn(N, R, sf_prime, b, sigma_u):
    """Goodman-corrected amplitude at stress ratio R (eq. 5)."""
    a0  = basquin(N, sf_prime, b)
    if R == -1:
        return a0
    num = a0 * sigma_u * (1 - R)
    den = sigma_u * (1 - R) + a0 * (1 + R)
    return np.where(den > 0, num / den, np.nan)

# Material: glass-epoxy composite (indicative values)
sigma_u  = 500.0   # MPa, ultimate tensile strength
sf_prime = 1000.0  # MPa, fatigue strength coefficient
b        = -0.08   # fatigue strength exponent

N = np.logspace(1, 8, 400)
for R in [-1, 0.1, 0.5, -2]:
    sa = goodman_sn(N, R, sf_prime, b, sigma_u)
    at_1e6 = float(goodman_sn(1e6, R, sf_prime, b, sigma_u))
    print(f"R = {R:5.1f}  σ_a at 10^6 cycles: {at_1e6:.1f} MPa")

# Constant-life diagram at N = 10^5
N_cld  = 1e5
a0     = basquin(N_cld, sf_prime, b)
sm     = np.linspace(-sigma_u, sigma_u, 300)
sa_cld = a0 * (1 - sm / sigma_u)           # Goodman line (eq. 6)

References

[1] Goodman, J. (1899). Mechanics Applied to Engineering. Longmans, Green, London.

[2] Basquin, O. H. (1910). The exponential law of endurance tests. Proceedings of ASTM, 10, 625–630.

[3] Vassilopoulos, A. P. & Keller, T. (2011). Fatigue of Fiber-Reinforced Composites. Springer.

[4] Sendeckyj, G. P. (1981). Fitting models to composite materials fatigue data. ASTM STP 734, 245–260.

For composite laminates the Goodman criterion is often non-conservative: the constant-life diagram is piecewise linear with different slopes for tension-dominated and compression-dominated loading, and the $R$-ratio dependence of the S-N slope $b$ must be measured separately for each fiber orientation.