A stress-strain curve shows how a material responds as it is loaded. The horizontal axis is strain, and the
vertical axis is stress. From the curve, important material properties can be estimated.
1. Stress
Engineering stress is force divided by original cross-sectional area:
\[
\sigma=\frac{F}{A_0}.
\]
Stress has pressure units, such as \(\mathrm{Pa}\), \(\mathrm{MPa}\), or \(\mathrm{GPa}\).
2. Strain
Engineering strain is the fractional change in length:
\[
\varepsilon=\frac{\Delta L}{L_0}.
\]
Strain is dimensionless, but it is often shown as a decimal, percent, or microstrain.
3. Young’s modulus
In the initial linear elastic region, stress is approximately proportional to strain:
\[
\sigma=E\varepsilon.
\]
The slope of this early linear region is Young’s modulus:
\[
\boxed{
E=\frac{\Delta\sigma}{\Delta\varepsilon}
}.
\]
4. Elastic limit
The elastic limit is the approximate point where the material starts to depart from linear elastic behavior.
Below this region, unloading would ideally return the material close to its original shape.
A practical numerical estimate compares each data point with the fitted elastic line:
\[
\text{deviation}
=
\frac{|\sigma_{\mathrm{data}}-\sigma_{\mathrm{fit}}|}{|\sigma_{\mathrm{fit}}|}.
\]
5. Yield point and offset yield strength
Some materials do not have a sharp yield point. A common engineering estimate is the offset-yield method,
often using a \(0.2\%\) strain offset.
The offset line is drawn parallel to the elastic slope:
\[
\sigma_{\mathrm{offset}}
=
E(\varepsilon-\varepsilon_{\mathrm{offset}}).
\]
The yield strength is estimated where this offset line intersects the measured curve:
\[
\boxed{
\sigma_y=\sigma(\varepsilon_y)
}.
\]
6. Ultimate tensile strength
The ultimate tensile strength is the maximum engineering stress on the curve:
\[
\boxed{
\sigma_{\mathrm{UTS}}=\max(\sigma_i)
}.
\]
It is the highest stress reached before the material begins to weaken or neck significantly.
7. Breaking point
The breaking point is the final point supplied on the curve. It estimates the fracture stress and fracture strain:
\[
\sigma_f,\qquad \varepsilon_f.
\]
8. Ductility
Ductility can be estimated from fracture strain:
\[
\boxed{
\text{ductility}\approx \varepsilon_f\times100\%
}.
\]
A larger fracture strain means the material stretches more before breaking.
9. Resilience
The modulus of resilience is the energy density absorbed in the elastic region:
\[
\boxed{
U_{\mathrm{res}}\approx
\int_0^{\varepsilon_y}\sigma\,d\varepsilon
}.
\]
On the graph, this is the area under the curve up to the yield point.
10. Toughness
Toughness is the total energy density absorbed before fracture:
\[
\boxed{
U_{\mathrm{tough}}\approx
\int_0^{\varepsilon_f}\sigma\,d\varepsilon
}.
\]
On the graph, this is the full area under the stress-strain curve.
11. Trapezoidal integration
For discrete data points, the area under the curve is estimated using trapezoids:
\[
U\approx
\sum_i
\frac{\sigma_i+\sigma_{i+1}}{2}
(\varepsilon_{i+1}-\varepsilon_i).
\]
Because stress has units of \(\mathrm{Pa}\) and strain is dimensionless, this area has units
\(\mathrm{Pa}=\mathrm{J/m^3}\). It represents energy per unit volume.
12. Typical curve regions
| Region or point |
Graph meaning |
Material property |
| Linear elastic region |
Nearly straight initial part |
Young’s modulus \(E\) |
| Elastic limit |
First clear departure from linearity |
Approximate limit of elastic behavior |
| Yield point |
Start of important plastic deformation |
Yield strength \(\sigma_y\) |
| Maximum stress |
Highest point on the curve |
Ultimate tensile strength |
| Fracture point |
Last point on the curve |
Ductility and breaking stress |
| Area under curve |
Stress integrated over strain |
Energy density, resilience, toughness |
13. Worked example idea
Suppose the early part of an aluminum alloy curve has points close to a straight line. A linear fit may give
\[
E\approx70\ \mathrm{GPa}.
\]
If the \(0.2\%\) offset line intersects the curve near
\[
\varepsilon_y\approx0.004,
\qquad
\sigma_y\approx170\ \mathrm{MPa},
\]
then the material’s offset yield strength is approximately \(170\ \mathrm{MPa}\).
If the maximum stress is around \(330\ \mathrm{MPa}\), then
\[
\sigma_{\mathrm{UTS}}\approx330\ \mathrm{MPa}.
\]
Key idea: the slope gives stiffness, the yield point marks the start of significant plastic deformation, and the
area under the curve estimates energy absorbed per unit volume.
