Stress and Strain — Revision Notes

⚡ 30-Second Revision

Stress ($sigma$) — Internal restoring force per unit area. $\sigma = F/A$. Unit: $Pa$ or $N/m^2$.
Normal Stress — Force perpendicular to area. Tensile (pulling), Compressive (pushing).
Shear Stress — Force parallel to area. Causes shape change.
Volumetric Stress — Uniform pressure, causes volume change.
Strain ($epsilon$) — Dimensionless ratio of change in dimension to original dimension.
Longitudinal Strain — $\epsilon_L = \Delta L/L$. Change in length.
Shear Strain — $\epsilon_S = \Delta x/h = \tan \theta \approx \theta$. Angular deformation.
Volumetric Strain — $\epsilon_V = \Delta V/V$. Change in volume.
Elastic Limit — Max stress before permanent deformation.

2-Minute Revision

Stress is the internal resistance a material offers to deformation, quantified as restoring force per unit area ($sigma = F/A$). Its unit is Pascal ($N/m^2$). It can be normal (perpendicular force, causing elongation or compression) or shear (parallel force, causing shape change).

Strain, on the other hand, is the dimensionless measure of deformation itself, expressed as the fractional change in dimensions. Longitudinal strain ($Delta L/L$) relates to length change, shear strain ($Delta x/h$ or $heta$) to angular deformation, and volumetric strain ($Delta V/V$) to volume change.

Understanding these definitions is crucial. Remember that stress is the 'cause' (internal resistance) and strain is the 'effect' (deformation). Materials behave elastically up to a certain elastic limit, beyond which they deform permanently.

This foundational knowledge is key to understanding Hooke's Law and the various moduli of elasticity.

5-Minute Revision

Let's quickly recap Stress and Strain, the twin pillars of material elasticity. **Stress ($sigma$)** is the internal restoring force developed within a body per unit cross-sectional area, opposing an external deforming force. Its SI unit is the Pascal ($Pa$ or $N/m^2$). We categorize it into:

1
Normal Stress — Force acts perpendicular to the area. It's tensile if it stretches the material (e.g., a hanging wire) or compressive if it squeezes it (e.g., a pillar). Formula: $\sigma_N = F/A$.
2
Shear Stress — Force acts parallel or tangential to the area, causing a change in shape. Formula: $\sigma_S = F/A$.
3
Volumetric Stress — Uniform pressure applied from all sides, causing a change in volume. It's numerically equal to pressure.

**Strain ($epsilon$)** is the dimensionless measure of deformation, representing the fractional change in a body's dimensions. It has no units because it's a ratio of similar quantities. Its types are:

1
Longitudinal Strain — Ratio of change in length ($Delta L$) to original length ($L$). $\epsilon_L = \Delta L/L$.
2
Shear Strain — Ratio of relative displacement ($Delta x$) of layers to their perpendicular distance ($h$), or the angle of shear ($heta$) in radians. $\epsilon_S = \Delta x/h \approx \theta$.
3
Volumetric Strain — Ratio of change in volume ($Delta V$) to original volume ($V$). $\epsilon_V = \Delta V/V$.

Remember, stress is the internal resistance, and strain is the resulting deformation. A material's elastic limit is the maximum stress it can endure without permanent deformation. Beyond this, it enters the plastic region.

For NEET, practice numerical problems involving calculating stress and strain, paying close attention to unit conversions (e.g., mm to m, cm$^2$ to m$^2$) and correctly identifying the type of stress or strain required.

For instance, if a $5,\text{cm}$ side cube has its top face displaced by $0.1,\text{mm}$ due to a $200,\text{N}$ tangential force: Shear stress = $200,\text{N} / (0.05,\text{m})^2 = 8 \times 10^4,\text{Pa}$.

Shear strain = $0.1 \times 10^{-3},\text{m} / 0.05,\text{m} = 0.002$.

Prelims Revision Notes

Stress ($sigma$)

Definition — Internal restoring force per unit cross-sectional area.
Formula — $\sigma = F/A$.
SI Unit — Pascal ($Pa$) or $N/m^2$. Also $kg,m^{-1}s^{-2}$.
Types of Stress

* Normal Stress: Force perpendicular to area. * Tensile Stress: Causes elongation ($Delta L > 0$). * Compressive Stress: Causes shortening ($Delta L < 0$). * Shear (Tangential) Stress: Force parallel to area. Causes change in shape. * Volumetric (Hydraulic) Stress: Uniform force perpendicular to all surfaces. Causes change in volume. Numerically equal to pressure.

Key Point — Stress is an internal property, not just external force/area.

Strain ($epsilon$)

Definition — Ratio of change in dimension to original dimension.
Unit — Dimensionless (no unit).
Types of Strain

* Longitudinal Strain: Change in length per unit original length. $\epsilon_L = \Delta L/L$. * Shear Strain: Angular deformation. Ratio of relative displacement ($Delta x$) to perpendicular distance ($h$). $\epsilon_S = \Delta x/h = \tan \theta \approx \theta$ (for small $\theta$ in radians). * Volumetric Strain: Change in volume per unit original volume. $\epsilon_V = \Delta V/V$.

Key Point — Strain is a relative measure of deformation.

Important Concepts

Elasticity — Property of a material to regain original shape/size after deforming force removal.
Plasticity — Property of a material to undergo permanent deformation.
Elastic Limit — Maximum stress a material can withstand without permanent deformation.
Stress vs. Pressure — Stress is internal, can be normal/tangential. Pressure is external, always normal and inwards (scalar). Volumetric stress is equivalent to pressure.

Calculation Tips for NEET

Always convert all given values to SI units (meters, Newtons, Pascals) before calculation.
For area, remember $A = \pi r^2$ for circular cross-sections.
Be careful with powers of ten in calculations.
Identify the correct type of stress/strain based on the problem description.

Vyyuha Quick Recall

To remember the types of Stress and Strain: Nice Students Verify Long Stories Vigorously.

Normal Stress, Volumetric Stress
Longitudinal Strain, Volumetric Strain

(And don't forget Shear Stress and Shear Strain, which are the other main types!)