7.2 Deformation in Solids

When an external force is applied to a solid, its shape or size changes. This change is called deformation. The study of how solids deform under applied forces is essential in engineering and material science, as it helps determine a material's strength, stiffness, and elasticity. For more on the types of solids that undergo these deformations, see Classification Of Solids→.

1. Tensile and Compressive Deformation

Deformation in one dimension occurs when a force is applied along a single axis of a solid:

• Tensile Deformation: A pulling (stretching) force is applied along the length of the solid, causing it to elongate. The length increases while the cross-sectional area decreases. Example: stretching a rubber band or a metal wire.
• Compressive Deformation: A pushing (squeezing) force is applied along the length, causing the solid to shorten. The length decreases while the cross-sectional area increases. Example: compressing a spring or a column under load.

Both types involve a change in length along the axis of the applied force and are described quantitatively using stress and strain.

2. Stress

Stress is the internal restoring force per unit area that a material develops in response to an external force. It measures the internal forces that particles of a material exert on each other.

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

Unit: The SI unit of stress is the Pascal (Pa), which equals one Newton per square metre ($N/m^2$).

There are three primary types of stress:

3. Strain

Strain is the measure of deformation of a material. It is the fractional change in a physical dimension (length, volume, or angle) in response to stress. Strain is a dimensionless quantity because it is the ratio of two similar quantities.

There are three corresponding types of strain:

4. Modulus of Elasticity

The modulus of elasticity measures a material's stiffness or its resistance to elastic deformation. Within the elastic limit, it is the constant ratio of stress to strain. A higher modulus indicates a stiffer material. This relationship is governed by Hooke's Law.

There are three types of elastic moduli, corresponding to the three types of stress and strain.

Young's Modulus (Y)

Also known as the tensile modulus, Young's modulus measures the resistance to a change in length. $Y=\frac{\text{Tensile Stress}}{\text{Tensile Strain}}=\frac{F/A}{\Delta L/L}$

Shear Modulus (S or G)

Also known as the modulus of rigidity, it measures the resistance to shearing or twisting deformation. $S=\frac{\text{Shear Stress}}{\text{Shear Strain}}=\frac{F/A}{\Delta x/h}$

Bulk Modulus (B)

Bulk modulus measures the resistance to a change in volume (compression). $B=\frac{\text{Volume Stress}}{\text{Volume Strain}}=\frac{-F/A}{\Delta V/V}$

The negative sign indicates that an increase in pressure (stress) leads to a decrease in volume.

5. Difference Between Stress and Pressure

Q: What is the difference between stress and pressure?

A: Stress is a measure of the internal forces within a material, while pressure is an external force applied per unit area. Although both have the same units (Pascals), pressure is a scalar quantity, whereas stress is a tensor (it has both magnitude and direction associated with the surface it acts on).

6. Significance of Young's Modulus

Q: What does it mean if a material has a high Young's Modulus?

A: A high Young's Modulus means the material is very stiff. It requires a large amount of stress to produce a small amount of strain (deformation). Steel, for example, has a high Young's Modulus.

7. Elasticity and Plasticity

Materials behave differently depending on the amount of stress applied:

• Elastic Deformation: When stress is within the elastic limit, the material returns to its original shape and size after the force is removed. The deformation is reversible.
• Elastic Limit: The maximum stress that can be applied to a material such that it still returns to its original dimensions upon removal of the force. Beyond this point, permanent deformation occurs.
• Plastic Deformation: When stress exceeds the elastic limit, the material does not return to its original shape. The deformation is permanent and irreversible.

8. Elastic Potential Energy (Strain Energy)

When work is done to deform a material elastically, energy is stored within it as elastic potential energy (also called strain energy). This energy is released when the deforming force is removed.

$U=\frac{1}{2}F\Delta L$

where $F$ is the applied force and $\Delta L$ is the extension. This equals the area under the force-extension graph.

The strain energy density (elastic potential energy per unit volume) is: $u=\frac{1}{2}\times \text{Stress}\times \text{Strain}$