An engineering stress-strain curve visualizes the relationship between applied stress (MPa) and resulting strain (dimensionless) in a material under uniaxial tensile loading. The curve reveals distinct mechanical behavior regions — elastic deformation, yielding, strain hardening, and necking — culminating in fracture. It is the foundational plot for characterizing material mechanical properties and comparing material performance.
""" pyplots.ai
line-stress-strain: Engineering Stress-Strain Curve
Library: letsplot 4.9.0 | Python 3.14.3
Quality: 92/100 | Created: 2026-03-20
"""
import numpy as np
import pandas as pd
from lets_plot import *
from lets_plot.export import ggsave as export_ggsave
LetsPlot.setup_html()
# Data - Mild steel tensile test simulation
np.random.seed(42)
# Material properties for mild steel
youngs_modulus = 210000 # MPa
yield_strength = 250 # MPa
uts = 400 # MPa (ultimate tensile strength)
fracture_strain = 0.35
uts_strain = 0.22
yield_strain = yield_strength / youngs_modulus # ~0.00119
# Elastic region (0 to yield)
n_elastic = 60
strain_elastic = np.linspace(0, yield_strain, n_elastic)
stress_elastic = youngs_modulus * strain_elastic
# Yield plateau (mild steel has a distinct yield point)
n_plateau = 20
strain_plateau = np.linspace(yield_strain, 0.015, n_plateau)
stress_plateau = yield_strength + np.random.normal(0, 1.5, n_plateau)
# Strain hardening region (from end of plateau to UTS)
n_hardening = 120
strain_hardening = np.linspace(0.015, uts_strain, n_hardening)
stress_hardening = yield_strength + (uts - yield_strength) * (
1 - np.exp(-8 * (strain_hardening - 0.015) / (uts_strain - 0.015))
)
stress_hardening += np.random.normal(0, 1.0, n_hardening)
# Necking region (UTS to fracture)
n_necking = 60
strain_necking = np.linspace(uts_strain, fracture_strain, n_necking)
stress_necking = uts - (uts - 280) * ((strain_necking - uts_strain) / (fracture_strain - uts_strain)) ** 1.5
stress_necking += np.random.normal(0, 1.5, n_necking)
# Combine all regions
strain = np.concatenate([strain_elastic, strain_plateau, strain_hardening, strain_necking])
stress = np.concatenate([stress_elastic, stress_plateau, stress_hardening, stress_necking])
df = pd.DataFrame({"strain": strain, "stress": stress})
# 0.2% offset line for yield point determination
offset_val = 0.002
offset_line_strain = np.linspace(offset_val, offset_val + yield_strength / youngs_modulus + 0.003, 50)
offset_line_stress = youngs_modulus * (offset_line_strain - offset_val)
offset_line_stress = np.clip(offset_line_stress, 0, yield_strength + 30)
df_offset = pd.DataFrame({"strain": offset_line_strain, "stress": offset_line_stress})
# Key points
yield_point_strain = offset_val + yield_strength / youngs_modulus
yield_point_stress = yield_strength
fracture_stress = stress_necking[-1]
df_points = pd.DataFrame(
{
"strain": [yield_point_strain, uts_strain, fracture_strain],
"stress": [yield_point_stress, uts, fracture_stress],
"label": [f"Yield Point ({yield_strength} MPa)", f"UTS ({uts} MPa)", f"Fracture ({fracture_stress:.0f} MPa)"],
"type": ["Yield", "UTS", "Fracture"],
}
)
# Consolidated annotations DataFrame
df_annotations = pd.DataFrame(
{
"x": [yield_point_strain + 0.012, uts_strain + 0.015, fracture_strain - 0.045, 0.008, 0.007, 0.005, 0.11, 0.29],
"y": [yield_point_stress + 15, uts + 10, fracture_stress - 30, 130, 60, 350, 350, 350],
"label": [
f"Yield Point\n({yield_strength} MPa)",
f"UTS ({uts} MPa)",
"Fracture",
f"E = {youngs_modulus // 1000} GPa",
"0.2% offset",
"Elastic",
"Strain Hardening",
"Necking",
],
"group": ["yield", "uts", "fracture", "modulus", "offset", "region", "region", "region"],
}
)
# Colorblind-safe palette: blue, purple, gray (avoids orange/green pair)
color_yield = "#9467BD" # purple
color_uts = "#D62728" # red
color_fracture = "#7F7F7F" # gray
color_main = "#306998" # Python blue
color_offset = "#E377C2" # pink
# Segment connector lines from key points to annotations (distinctive lets-plot feature)
df_segments = pd.DataFrame(
{
"x": [yield_point_strain, uts_strain, fracture_strain],
"y": [yield_point_stress, uts, fracture_stress],
"xend": [yield_point_strain + 0.011, uts_strain + 0.014, fracture_strain - 0.035],
"yend": [yield_point_stress + 12, uts + 8, fracture_stress - 22],
}
)
# Plot
plot = (
ggplot()
# Region background bands using geom_rect (distinctive lets-plot feature)
+ geom_rect(
aes(xmin="xmin", xmax="xmax", ymin="ymin", ymax="ymax", fill="region"),
data=pd.DataFrame(
{
"xmin": [0, 0.015, uts_strain],
"xmax": [0.015, uts_strain, fracture_strain],
"ymin": [0, 0, 0],
"ymax": [460, 460, 460],
"region": ["Elastic", "Strain Hardening", "Necking"],
}
),
alpha=0.35,
)
+ scale_fill_manual(
values={
"Elastic": "#DAE8FC",
"Strain Hardening": "#FFF2CC",
"Necking": "#F8D7DA",
"Yield": color_yield,
"UTS": color_uts,
"Fracture": color_fracture,
}
)
# Main stress-strain curve with tooltips (distinctive lets-plot feature)
+ geom_line(
aes(x="strain", y="stress"),
data=df,
color=color_main,
size=2.0,
tooltips=layer_tooltips()
.format("strain", ".4f")
.format("stress", ".1f")
.line("Strain: @strain")
.line("Stress: @stress MPa"),
)
# 0.2% offset line
+ geom_line(aes(x="strain", y="stress"), data=df_offset, color=color_offset, size=1.2, linetype="dashed")
# Segment connectors from points to labels (geom_segment - distinctive feature)
+ geom_segment(
aes(x="x", y="y", xend="xend", yend="yend"), data=df_segments, color="#999999", size=0.6, linetype="dotted"
)
# Key points with tooltips (distinctive lets-plot feature)
+ geom_point(
aes(x="strain", y="stress", fill="type"),
data=df_points,
color="white",
size=7,
shape=21,
stroke=1.2,
tooltips=layer_tooltips().line("@type").line("Strain: @strain").line("Stress: @stress MPa"),
)
+ guides(fill="none")
# Annotations - key points
+ geom_text(
aes(x="x", y="y", label="label"),
data=df_annotations.query("group == 'yield'"),
size=11,
color=color_yield,
hjust=0,
)
+ geom_text(
aes(x="x", y="y", label="label"), data=df_annotations.query("group == 'uts'"), size=11, color=color_uts, hjust=0
)
+ geom_text(
aes(x="x", y="y", label="label"),
data=df_annotations.query("group == 'fracture'"),
size=11,
color=color_fracture,
hjust=0.5,
)
# Elastic modulus annotation
+ geom_text(
aes(x="x", y="y", label="label"),
data=df_annotations.query("group == 'modulus'"),
size=10,
color=color_main,
hjust=0,
fontface="italic",
)
# Offset label
+ geom_text(
aes(x="x", y="y", label="label"),
data=df_annotations.query("group == 'offset'"),
size=9,
color=color_offset,
hjust=0,
fontface="italic",
)
# Region labels
+ geom_text(
aes(x="x", y="y", label="label"),
data=df_annotations.query("group == 'region'"),
size=12,
color="#666666",
fontface="italic",
)
# Styling
+ labs(
x="Engineering Strain",
y="Engineering Stress (MPa)",
title="line-stress-strain \u00b7 letsplot \u00b7 pyplots.ai",
)
+ scale_x_continuous(breaks=[0, 0.05, 0.10, 0.15, 0.20, 0.25, 0.30, 0.35])
+ scale_y_continuous(breaks=[0, 50, 100, 150, 200, 250, 300, 350, 400, 450])
+ ggsize(1600, 900)
+ theme_minimal()
+ theme(
axis_text=element_text(size=16, color="#555555"),
axis_title=element_text(size=20, color="#333333"),
plot_title=element_text(size=24, color="#222222", face="bold"),
panel_grid_major_x=element_blank(),
panel_grid_major_y=element_line(color="#E0E0E0", size=0.3),
panel_grid_minor=element_blank(),
plot_background=element_rect(fill="#FAFAFA", color="#FAFAFA"),
panel_background=element_rect(fill="transparent", color="transparent"),
axis_ticks=element_blank(),
axis_ticks_length=0,
plot_margin=[30, 40, 20, 20],
)
)
# Save
export_ggsave(plot, filename="plot.png", path=".", scale=3)
export_ggsave(plot, filename="plot.html", path=".")
Part of Engineering Stress-Strain Curve on anyplot.ai.