Rankine's theory, also known as the maximum distortion energy theory or the shear strain energy theory, is used to analyze the failure of brittle materials under complex stress conditions. It is one of the theories of failure that falls under the field of strength of materials in mechanical engineering.

Brittle Materials:
- Brittle materials are those that undergo very little or no plastic deformation before failure. Examples of brittle materials include ceramics, glass, and cast iron.
- Brittle materials fail by fracture, where cracks propagate through the material, leading to sudden and catastrophic failure.
- Due to their limited ability to deform, brittle materials are more susceptible to failure under complex stress conditions, such as combined tension and shear or multiaxial stress states.

Rankine's Theory:
- Rankine's theory is based on the assumption that failure occurs when the maximum distortion energy per unit volume exceeds the distortion energy per unit volume at the yield point in uniaxial tension.
- It considers the shear stress and normal stress components acting on a material and combines them to calculate the distortion energy.
- The theory assumes that failure occurs when the distortion energy exceeds a critical value, which is determined experimentally for different materials.
- Mathematically, the theory is expressed as:
Distortion Energy per unit volume = (1/2) * (σ₁² + σ₂² + σ₃²) - (σ₁σ₂ + σ₂σ₃ + σ₃σ₁)

Application to Brittle Materials:
- Rankine's theory is particularly applicable to brittle materials because it takes into account the shear stress component, which plays a significant role in their failure.
- Brittle materials are known to have low tensile strength but high compressive strength. The theory considers the maximum normal stress components, which are important for brittle materials.
- By calculating the distortion energy and comparing it with the critical value, the theory helps determine whether or not a brittle material will fail under a given set of stress conditions.
- It is worth noting that Rankine's theory is not applicable to ductile materials, which exhibit significant plastic deformation before failure. For ductile materials, the von Mises yield criterion or the maximum shear stress theory is commonly used.

In conclusion, Rankine's theory is used to analyze the failure of brittle materials under complex stress conditions. It considers both normal and shear stress components and calculates the distortion energy per unit volume. By comparing this energy with a critical value, the theory helps determine the failure potential of brittle materials.