Short Answer:

The theory used for brittle materials is the Maximum Principal Stress Theory, also known as the Rankine’s Theory. This theory states that failure in a brittle material occurs when the maximum principal stress in a complex stress system reaches the ultimate tensile stress obtained from a simple tension test.

In simple words, brittle materials like cast iron, glass, stone, and concrete fail when the maximum normal stress in the material exceeds their strength limit. This theory is suitable for brittle materials because they fail by fracture due to normal stress, not by shear or yielding as in ductile materials.

Detailed Explanation :

Theory Used for Brittle Materials

The Maximum Principal Stress Theory (Rankine’s Theory) is the most suitable and widely used theory for brittle materials. Brittle materials are those that fracture suddenly without significant plastic deformation when the applied stress exceeds their ultimate strength. Examples include cast iron, ceramics, glass, concrete, and stone.

This theory assumes that brittle materials fail when the maximum normal (principal) stress in the material reaches the same value as the ultimate stress obtained from a simple tensile test. Since brittle materials fail due to direct normal stress and not by shear or distortion, this theory accurately predicts their failure.

Definition

According to the Maximum Principal Stress Theory,

“Failure occurs when the maximum principal stress in a material reaches the limiting stress obtained from a simple tension or compression test.”

Mathematically, the condition of failure is expressed as:

where,

•  = maximum principal stress in the system,
•  = ultimate tensile or compressive stress of the material.

Thus, the material will fail when its maximum principal stress equals the yield or ultimate value obtained experimentally.

Mathematical Form of the Theory

Let the three principal stresses at a point be , where
= maximum principal stress,
= minimum principal stress.

According to the Rankine Theory,

• For tensile failure:

• For compressive failure:

where,

•  = ultimate tensile stress,
•  = ultimate compressive stress.

Thus, failure occurs when either the maximum tensile or compressive principal stress equals the material’s strength in that direction.

For Two-Dimensional Stress System

In most practical applications, stresses act in two perpendicular directions. The failure conditions then become:

or

The safe condition for the material is:

This ensures that the stresses acting in the material remain below its strength limits.

Reason for Suitability in Brittle Materials

Brittle materials, unlike ductile ones, have very little capacity for plastic deformation. When a brittle material is subjected to stress, it deforms elastically until it reaches the fracture point. Once the normal stress exceeds the ultimate tensile or compressive limit, the material cracks or breaks suddenly.

This failure occurs mainly due to normal tensile stress, not due to shear stress. Therefore, the Rankine Theory, which focuses only on maximum normal stress, is most appropriate for predicting failure in brittle materials.

For brittle materials:

• The tensile strength is usually much less than the compressive strength.
• Hence, failure generally occurs under tension when cracks develop along the plane of maximum tensile stress.

Graphical Representation

In a principal stress diagram (σ₁ vs σ₂), the Rankine’s theory is represented by a square or rectangle bounded by the tensile and compressive strength limits of the material.

• Points inside the rectangle represent safe conditions.
• Points on or outside the boundary represent failure.

The simple rectangular shape of this diagram makes the theory easy to visualize and apply in design calculations.

Assumptions of Rankine’s Theory

1. The material fails when the maximum normal stress equals the ultimate stress in a simple test.
2. Shear stress has no effect on failure.
3. The material is homogeneous and isotropic (same properties in all directions).
4. The failure criterion is independent of loading direction or stress combination.

These assumptions make the theory simple but limit its accuracy for ductile materials, where shear plays an important role.

Advantages

1. Simple to apply:
   The theory involves only maximum principal stresses, making it easy to use.
2. Suitable for brittle materials:
   Accurately predicts failure in materials that fail by cracking under normal stress.
3. Useful for tension and compression:
   Can be applied for both tensile and compressive failure conditions.
4. Practical in structural design:
   Widely used in designing brittle components such as concrete beams, stone pillars, and cast-iron structures.

Limitations

1. Not suitable for ductile materials:
   Ductile materials fail due to shear, not normal stress.
2. Neglects shear effect:
   The theory ignores shear stress, which can contribute to material yielding.
3. Not applicable for combined stresses:
   Fails to give accurate results when materials are subjected to complex multiaxial loading.
4. Experimental deviation:
   The theory predictions deviate from experimental results for materials under complex stress systems.

Applications

• Design of brittle materials like cast iron, concrete, glass, and ceramics.
• Used in structural engineering for analyzing stone columns, concrete beams, and masonry walls.
• Applied in machine design where components are brittle and subjected mainly to tensile or compressive loads.

Comparison with Other Theories

Type of Material	Suitable Theory	Basis of Failure
Brittle Material	Rankine’s Theory	Maximum principal stress
Ductile Material	Von Mises Theory or Tresca Theory	Shear or distortion energy

This comparison clearly shows that Rankine’s Theory is ideal for brittle materials, while Von Mises and Tresca are more accurate for ductile materials.

Conclusion

The Maximum Principal Stress Theory (Rankine’s Theory) is the most suitable and widely accepted theory for predicting the failure of brittle materials. It assumes that failure occurs when the maximum principal stress equals the ultimate tensile or compressive strength obtained from simple tests. Because brittle materials fail suddenly by cracking under normal stress, this theory gives accurate results for their behavior under complex loading. It remains one of the simplest and most useful theories in the field of strength of materials and structural design.