An S-N curve (also known as a Wöhler curve) visualizes the relationship between alternating stress amplitude and the number of cycles to failure for a material under fatigue loading. Both axes typically use logarithmic scales, with stress on the y-axis and cycle count on the x-axis. This plot is fundamental for predicting material fatigue life and identifying key material properties such as ultimate strength, yield strength, and endurance limit.
""" anyplot.ai
sn-curve-basic: S-N Curve (Wöhler Curve)
Library: plotly 6.7.0 | Python 3.13.13
Quality: 87/100 | Updated: 2026-05-20
"""
import os
import numpy as np
import plotly.graph_objects as go
# Theme
THEME = os.getenv("ANYPLOT_THEME", "light")
PAGE_BG = "#FAF8F1" if THEME == "light" else "#1A1A17"
ELEVATED_BG = "#FFFDF6" if THEME == "light" else "#242420"
INK = "#1A1A17" if THEME == "light" else "#F0EFE8"
INK_SOFT = "#4A4A44" if THEME == "light" else "#B8B7B0"
GRID = "rgba(26,26,23,0.10)" if THEME == "light" else "rgba(240,239,232,0.10)"
# Okabe-Ito palette
BRAND = "#009E73" # position 1 — Basquin fit line
C2 = "#C475FD" # position 2 — test data markers
C3 = "#4467A3" # position 3 — ultimate strength
C4 = "#BD8233" # position 4 — yield strength
C5 = "#AE3030" # position 5 — endurance limit
# Data: Steel fatigue test data (Basquin model)
np.random.seed(42)
A = 1200 # Fatigue coefficient (MPa)
b = -0.12 # Fatigue strength exponent
stress_levels = np.array([600, 550, 500, 450, 400, 350, 320, 300, 280, 260, 250, 240])
cycles_base = (stress_levels / A) ** (1 / b)
cycles = []
stress = []
for s, n_base in zip(stress_levels, cycles_base, strict=False):
n_samples = np.random.randint(2, 5)
scatter = 10 ** (np.random.normal(0, 0.15, n_samples))
for factor in scatter:
cycles.append(n_base * factor)
stress.append(s)
cycles = np.array(cycles)
stress = np.array(stress)
# Material properties
ultimate_strength = 650 # MPa
yield_strength = 450 # MPa
endurance_limit = 230 # MPa
# Basquin fit line
fit_cycles = np.logspace(2, 7, 100)
fit_stress = A * fit_cycles**b
n_knee = (endurance_limit / A) ** (1 / b) # transition point to infinite life
# Plot
fig = go.Figure()
fig.add_trace(
go.Scatter(
x=fit_cycles,
y=fit_stress,
mode="lines",
name="Basquin Fit",
line={"color": BRAND, "width": 3},
hovertemplate="Cycles: %{x:.2e}<br>Stress: %{y:.0f} MPa<extra>Basquin Fit</extra>",
)
)
fig.add_trace(
go.Scatter(
x=cycles,
y=stress,
mode="markers",
name="Test Data",
marker={"color": C2, "size": 12, "line": {"color": PAGE_BG, "width": 1.5}},
hovertemplate="Cycles: %{x:.2e}<br>Stress: %{y:.0f} MPa<extra>Test Data</extra>",
)
)
x_range = [100, 1e7]
# Infinite-life zone fill — opacity raised so the region registers as a meaningful cue
fig.add_hrect(y0=200, y1=endurance_limit, opacity=0.11, fillcolor=BRAND, layer="below")
# Reference lines — excluded from legend; annotated directly at right edge instead
fig.add_trace(
go.Scatter(
x=x_range,
y=[ultimate_strength, ultimate_strength],
mode="lines",
name=f"Ultimate Strength ({ultimate_strength} MPa)",
line={"color": C3, "width": 2, "dash": "dash"},
showlegend=False,
hovertemplate=f"Ultimate Strength: {ultimate_strength} MPa<extra></extra>",
)
)
fig.add_trace(
go.Scatter(
x=x_range,
y=[yield_strength, yield_strength],
mode="lines",
name=f"Yield Strength ({yield_strength} MPa)",
line={"color": C4, "width": 2, "dash": "dash"},
showlegend=False,
hovertemplate=f"Yield Strength: {yield_strength} MPa<extra></extra>",
)
)
fig.add_trace(
go.Scatter(
x=x_range,
y=[endurance_limit, endurance_limit],
mode="lines",
name=f"Endurance Limit ({endurance_limit} MPa)",
line={"color": C5, "width": 2, "dash": "dash"},
showlegend=False,
hovertemplate=f"Endurance Limit: {endurance_limit} MPa<extra></extra>",
)
)
# Direct line labels via add_annotation — plotly-native, cleaner than legend entries
for y_val, label, color, anchor in [
(ultimate_strength, f"Ult. Strength<br>{ultimate_strength} MPa", C3, "top"),
(yield_strength, f"Yield Strength<br>{yield_strength} MPa", C4, "bottom"),
(endurance_limit, f"End. Limit<br>{endurance_limit} MPa", C5, "top"),
]:
fig.add_annotation(
x=8e6,
y=y_val,
text=label,
xanchor="right",
yanchor=anchor,
showarrow=False,
font={"color": color, "size": 12},
bgcolor=ELEVATED_BG,
bordercolor=color,
borderwidth=1,
borderpad=3,
)
# Fatigue knee annotation — where Basquin fit intersects endurance limit
fig.add_annotation(
x=n_knee,
y=endurance_limit,
text="Fatigue Knee<br>(~10⁶ cycles)",
showarrow=True,
arrowhead=2,
arrowcolor=INK_SOFT,
arrowsize=1.2,
ax=-70,
ay=-70,
font={"size": 10, "color": INK_SOFT},
bgcolor=ELEVATED_BG,
bordercolor=INK_SOFT,
borderwidth=1,
borderpad=3,
xanchor="center",
)
# Infinite-life zone label inside the green fill region
fig.add_annotation(
x=5e5,
y=218,
text="Infinite Life Zone",
showarrow=False,
font={"size": 10, "color": BRAND},
xanchor="center",
yanchor="middle",
)
# Style
fig.update_layout(
autosize=False,
paper_bgcolor=PAGE_BG,
plot_bgcolor=PAGE_BG,
hovermode="closest",
title={
"text": "sn-curve-basic · python · plotly · anyplot.ai",
"font": {"size": 16, "color": INK},
"x": 0.5,
"xanchor": "center",
},
xaxis={
"title": {"text": "Cycles to Failure (N)", "font": {"size": 12, "color": INK}},
"tickfont": {"size": 10, "color": INK_SOFT},
"type": "log",
"showgrid": True,
"gridwidth": 1,
"gridcolor": GRID,
"showline": True,
"linewidth": 1,
"linecolor": INK_SOFT,
"mirror": False,
"range": [2, 7],
"zerolinecolor": GRID,
},
yaxis={
"title": {"text": "Stress Amplitude (MPa)", "font": {"size": 12, "color": INK}},
"tickfont": {"size": 10, "color": INK_SOFT},
"type": "log",
"showgrid": True,
"gridwidth": 1,
"gridcolor": GRID,
"showline": True,
"linewidth": 1,
"linecolor": INK_SOFT,
"mirror": False,
"range": [2.3, 2.9],
"zerolinecolor": GRID,
},
legend={
"font": {"size": 10, "color": INK_SOFT},
"x": 0.05,
"y": 0.05,
"xanchor": "left",
"yanchor": "bottom",
"bgcolor": ELEVATED_BG,
"bordercolor": INK_SOFT,
"borderwidth": 1,
},
margin={"l": 80, "r": 60, "t": 80, "b": 60},
)
# Save
fig.write_image(f"plot-{THEME}.png", width=800, height=450, scale=4)
fig.write_html(f"plot-{THEME}.html", include_plotlyjs="cdn")
Part of S-N Curve (Wöhler Curve) on anyplot.ai.