List the different theories of failure.

Short Answer:

The theories of failure are scientific methods used to predict when a material will fail under complex loading conditions. They help engineers determine the safe limits of stress and strain in materials to avoid failure. Each theory is based on a specific failure criterion, such as stress, strain, or energy.

There are mainly five important theories of failure, namely:

  1. Maximum Principal Stress Theory
  2. Maximum Principal Strain Theory
  3. Maximum Shear Stress Theory
  4. Maximum Strain Energy Theory
  5. Maximum Shear Strain Energy Theory.
    These theories help in designing both ductile and brittle materials safely under different loading conditions.
Detailed Explanation :

Theories of Failure

When a material or a structural component is subjected to external loads, it experiences stresses in different directions. Failure occurs when these stresses exceed the material’s strength. However, in real engineering applications, the loading conditions are often complex and not limited to simple tension or compression.

To predict failure under such combined stress conditions, several failure theories have been developed. These theories help to determine whether the material will yield, fracture, or deform permanently when subjected to multiple stresses simultaneously.

Each theory provides a criterion that compares the complex stress condition to a simple test condition (like a uniaxial tensile test). If the equivalent stress calculated from the theory exceeds the yield or ultimate strength of the material, the material is said to fail.

Purpose of Theories of Failure

The main purpose of failure theories is to ensure safety and reliability in engineering design. Theories of failure help in:

  • Predicting when a material will start yielding or breaking.
  • Determining the safe working stress for design.
  • Comparing the behavior of materials under complex loading conditions.
  • Selecting suitable materials for specific applications (ductile or brittle).
  • Preventing unexpected failure and improving the factor of safety in design.

Different Theories of Failure

The following are the most widely used five theories of failure in strength of materials and mechanical design.

  1. Maximum Principal Stress Theory (Rankine’s Theory)
  • Statement:
    Failure occurs when the maximum principal stress in a material reaches the yield or ultimate stress from a simple tensile test.
  • Mathematical form:
  • Applicable for:
    Brittle materials such as cast iron, concrete, and glass.
  • Reason:
    Brittle materials fail by fracture due to normal stress and not by shear.

This theory ignores shear effects and is simple but not suitable for ductile materials.

  1. Maximum Principal Strain Theory (Saint-Venant’s Theory)
  • Statement:
    Failure occurs when the maximum principal strain in a complex stress system equals the strain at yield in a simple tensile test.
  • Mathematical form:
  • Applicable for:
    Ductile materials under elastic conditions, but not widely used due to low accuracy.

This theory is based on the concept of strain and is suitable when deformation, rather than stress, is the design concern.

  1. Maximum Shear Stress Theory (Tresca’s Theory or Guest’s Theory)
  • Statement:
    Failure occurs when the maximum shear stress in the material becomes equal to the shear stress at yield in a simple tension test.
  • Mathematical form:
  • Applicable for:
    Ductile materials like mild steel, copper, and aluminum.
  • Reason:
    Ductile materials fail due to shear yielding, not direct tension or compression.

This theory gives a conservative (safe) estimate and is widely used in the design of shafts, pressure vessels, and beams.

  1. Maximum Strain Energy Theory
  • Statement:
    Failure occurs when the total strain energy per unit volume in a complex stress system equals the strain energy at yield in a simple tension test.
  • Mathematical form:
  • Applicable for:
    Ductile materials, but not as accurate as energy-based shear theories.
  • Reason:
    It considers total energy, not only the part responsible for distortion or shape change.

This theory is more theoretical and less practical for engineering design.

  1. Maximum Shear Strain Energy Theory (Von Mises Theory or Distortion Energy Theory)
  • Statement:
    Failure occurs when the distortion (shear) strain energy per unit volume equals that at yield in a simple tensile test.
  • Mathematical form:
  • Applicable for:
    Ductile materials such as steel, aluminum, and copper.
  • Reason:
    It accurately predicts yielding caused by distortion energy and agrees closely with experimental results.

This theory is considered the most accurate and is widely used in modern engineering design, including finite element analysis (FEA).

Classification of Theories Based on Material Type

  • For Brittle Materials:
    1. Maximum Principal Stress Theory (Rankine)
    2. Maximum Principal Strain Theory (Saint-Venant)
  • For Ductile Materials:
    1. Maximum Shear Stress Theory (Tresca)
    2. Maximum Strain Energy Theory
    3. Maximum Shear Strain Energy Theory (Von Mises)

Comparison of Theories

  • The Rankine theory is simple and best for brittle materials.
  • The Tresca theory gives conservative but safe results for ductile materials.
  • The Von Mises theory gives results that closely match experimental data.
  • The Saint-Venant theory and Maximum Strain Energy theory are of limited practical use.

Thus, in modern engineering practice:

  • For brittle materials → Rankine’s theory is preferred.
  • For ductile materials → Von Mises or Tresca theory is preferred.

Importance of Theories of Failure

  1. Helps predict when materials will start yielding or breaking.
  2. Ensures safe and reliable design of mechanical components.
  3. Reduces the risk of unexpected failure in service.
  4. Assists in material selection and design optimization.
  5. Forms the foundation for advanced computational design methods.
Conclusion

The theories of failure provide a scientific basis for predicting the conditions under which materials fail under combined stresses. The five main theories — Maximum Principal Stress, Maximum Principal Strain, Maximum Shear Stress, Maximum Strain Energy, and Maximum Shear Strain Energy (Von Mises) — each apply to specific material types and stress conditions.
Among these, Rankine’s theory is used for brittle materials, while Tresca’s and Von Mises theories are most accurate for ductile materials. Understanding these theories ensures safe, efficient, and reliable design of engineering components.