Stress and Strain

When external forces act on a solid, the body tends to deform. The internal restoring forces per unit area are described by stress, and the fractional deformation produced is described by strain. These two quantities together capture the mechanical response of solids.

Concept

Stress is defined as the internal restoring force per unit area acting on a deformable body:

$\sigma = \frac{F}{A}$

Here $F$ is the magnitude of the applied (or internally generated) force perpendicular or tangential to the area $A$. The SI unit of stress is the pascal (Pa) where $1\,\text{Pa} = 1\,\text{N/m}^2$. Stress is a tensor quantity, but at this level we treat it as a scalar magnitude with a specified direction.

There are three principal types of stress:

1. Tensile (longitudinal) stress — force pulls the body apart along the axis. $\sigma_t = F/A$.
2. Compressive stress — force pushes the body together along the axis.
3. Shear (tangential) stress — force acts parallel to the surface. $\sigma_s = F_{\parallel}/A$.
4. Hydraulic/Volumetric stress — uniform pressure from all sides, equal to the pressure $P$ applied to the body.

Strain is the dimensionless ratio of change in dimension to original dimension:

• Longitudinal strain: $\varepsilon_\ell = \Delta L / L$
• Shear strain: $\varepsilon_s = \tan\theta \approx \theta$ for small angles
• Volume strain: $\varepsilon_v = \Delta V / V$

Because strain is a ratio of like quantities, it carries no SI unit and no dimension.

Derivation

Consider a uniform cylindrical rod of original length $L$ and cross-sectional area $A$, pulled with a force $F$ at each end. The rod elongates by $\Delta L$.

Step 1 — Define longitudinal stress:

$\sigma = \frac{F}{A}$

Step 2 — Define longitudinal strain:

$\varepsilon = \frac{\Delta L}{L}$

Step 3 — For an isotropic elastic material in the proportional region, the ratio $\sigma / \varepsilon$ is a constant called Young's modulus $Y$:

$Y = \frac{\sigma}{\varepsilon} = \frac{F/A}{\Delta L / L} = \frac{F L}{A\,\Delta L}$

Step 4 — For shear, a tangential force $F$ produces angular displacement $\theta$:

$\sigma_s = \frac{F}{A}, \quad \varepsilon_s = \theta, \quad G = \frac{\sigma_s}{\varepsilon_s}$

Step 5 — For volumetric stress, pressure $P$ produces fractional volume change $\Delta V / V$:

$B = -\frac{P}{\Delta V / V}$

The negative sign indicates that an increase in pressure reduces volume.

Worked Example

A steel wire of length $2.0\,\text{m}$ and cross-sectional area $1.0\,\text{mm}^2 = 1.0\times 10^{-6}\,\text{m}^2$ is stretched by hanging a load of $20\,\text{kg}$. Take $g = 9.8\,\text{m/s}^2$. Find the stress and strain. (Given $Y_\text{steel} = 2.0\times 10^{11}\,\text{Pa}$.)

Force: $F = mg = 20 \times 9.8 = 196\,\text{N}$.

Stress: $\sigma = F/A = 196 / 10^{-6} = 1.96\times 10^{8}\,\text{Pa}$.

Strain: $\varepsilon = \sigma / Y = 1.96\times 10^{8} / 2.0\times 10^{11} = 9.8\times 10^{-4}$.

Elongation: $\Delta L = \varepsilon \cdot L = 9.8\times 10^{-4} \times 2.0 = 1.96\times 10^{-3}\,\text{m} \approx 2\,\text{mm}$.

Common Confusions

• Stress is not pressure, though they share units. Pressure is always normal and compressive; stress is more general and can be tensile, compressive, or shear.
• Strain has no units. Writing "0.001 m/m" is acceptable shorthand, but the ratio itself is dimensionless.
• The "area" $A$ in $\sigma = F/A$ is the original cross-sectional area at small strains (engineering stress). True stress uses instantaneous area.
• Shear strain $\theta$ must be in radians when small-angle approximation is used.

Key Takeaways

• Stress is internal restoring force per unit area, measured in pascals.
• Strain is the fractional deformation and is dimensionless.
• Three types: longitudinal, shear, and volumetric.
• Stress and strain are linearly proportional in the elastic (Hookean) regime.