Short Answer:

Distortion energy theory (von Mises theory) is one of the most accurate and widely used theories of failure for ductile materials. It states that yielding begins when the distortion energy per unit volume in a material under complex loading becomes equal to the distortion energy per unit volume at yield in a simple tension test.

In simple words, the von Mises theory predicts failure when the material’s shape-changing (distortional) energy due to applied stresses reaches the same value as that at the yield point in a tensile test. It is commonly used for metals such as steel and aluminum.

Detailed Explanation :

Distortion Energy Theory (von Mises Theory)

The distortion energy theory, also known as the von Mises yield criterion, was proposed by Richard von Mises in 1913. This theory is based on the idea that yielding in ductile materials occurs not because of changes in volume (like hydrostatic pressure) but due to distortion of shape caused by shear stresses.

When a material is subjected to external loads, two types of strain energy are developed:

1. Volumetric strain energy – associated with the change in volume of the material.
2. Distortion strain energy – associated with the change in shape of the material.

According to this theory, failure or yielding begins when the distortion strain energy per unit volume under complex loading equals that at the yield point under simple tension. Hence, this theory focuses on the part of the total energy that causes distortion, not volume change.

Basic Concept

When a material is stressed, the total strain energy stored per unit volume () is given by:

 

where,

• = volumetric strain energy per unit volume,
• = distortion strain energy per unit volume.

In hydrostatic stress, all principal stresses are equal (), and there is only a change in volume but no change in shape, hence no yielding occurs.
In contrast, when stresses are unequal, distortion occurs, leading to shape change, and yielding may begin.

Thus, von Mises assumed that distortion energy () is responsible for material yielding.

Mathematical Derivation

Let the three principal stresses be .
The total strain energy per unit volume is:

 

The volumetric strain energy is given by:

 

Hence, the distortion energy per unit volume is obtained by subtracting the volumetric part from the total:

 

Substituting and simplifying:

 

According to the von Mises criterion, yielding begins when this distortion energy equals that at yield in a simple tensile test, where the yield stress is .
Thus, at yielding:

 

This is known as the von Mises yield criterion.

Equivalent (von Mises) Stress

For practical design purposes, the above expression is often written in terms of an equivalent stress , which is given by:

 

The failure condition is then:

 

Hence, the von Mises stress represents a single value that can be compared directly with the material’s yield strength obtained from a simple tensile test.

Special Cases

1. Uniaxial Stress:
   If only one stress acts, say , and :

 

Hence, yielding occurs when .

1. Pure Shear Stress:
   For pure shear, , , and :

 

Yielding begins when:

 

This shows that yielding in pure shear occurs at 57.7% of the yield stress in tension.

Graphical Representation

In two-dimensional stress space, von Mises criterion forms an ellipse, while the maximum shear stress theory (Tresca) forms a hexagon.
The von Mises ellipse fits more accurately with experimental results for ductile materials, hence providing a more reliable prediction of yielding.

Advantages of Distortion Energy Theory

1. Accurate for Ductile Materials:
   It matches well with experimental results for materials like steel, copper, and aluminum.
2. Considers Combined Stresses:
   Can be applied to complex loading conditions involving normal and shear stresses.
3. Useful in Design:
   Commonly used in finite element analysis (FEA) and mechanical design to predict yielding.
4. Applicable to Elastic Behavior:
   It works well under linear elastic conditions, where Hooke’s law applies.
5. Energy-Based Concept:
   It relates directly to the internal energy changes responsible for deformation.

Limitations

1. Not suitable for brittle materials because they fail due to normal stress, not distortion.
2. Assumes homogeneous and isotropic materials.
3. Not applicable when the material exhibits plastic or non-linear behavior beyond yield point.
4. Ignores temperature effects and time-dependent deformations like creep.

Applications

• Design of machine components such as shafts, gears, and connecting rods.
• Used in Finite Element Analysis (FEA) for predicting yielding regions.
• Applied in pressure vessel design to calculate equivalent stresses.
• Used in aerospace, automotive, and structural engineering to ensure ductile material safety.
• Important in stress analysis for combined loading conditions.

Comparison with Other Theories

Theory	Failure Criterion	Best For
Maximum Stress Theory (Rankine)	Based on maximum normal stress	Brittle materials
Maximum Strain Theory (Saint-Venant)	Based on maximum strain	Brittle materials
Maximum Shear Stress Theory (Tresca)	Based on maximum shear stress	Ductile materials (less accurate)
Distortion Energy Theory (von Mises)	Based on distortion energy	Ductile materials (most accurate)

Conclusion

The distortion energy theory (von Mises theory) states that yielding in ductile materials starts when the distortion energy per unit volume under complex stress conditions equals that at yielding in a simple tensile test. The theory accurately predicts the onset of yielding and is represented by the von Mises stress equation:

 

It is the most reliable failure theory for ductile materials and is widely used in design and analysis for safe and efficient mechanical structures.