The correct answer is option D, Mises-Hencky.

The maximum distortion energy theory, also known as the von Mises yield criterion or the Mises-Hencky theory, is a widely used criterion in the field of mechanics to predict the failure of ductile materials under complex stress states.

Here is a detailed explanation of the answer:

1. Background:
- In the field of mechanics, various theories have been postulated to predict the failure of materials under different stress conditions.
- The maximum distortion energy theory was developed to predict the yielding and failure of ductile materials subjected to multiaxial stress states.
- This theory is particularly applicable to materials that exhibit plastic deformation and do not have a clear yield point.

2. The Mises-Hencky Theory:
- The maximum distortion energy theory was independently proposed by Richard von Mises and Ludwig Hencky in the early 20th century.
- Richard von Mises, an Austrian mathematician, and Ludwig Hencky, a Czech engineer, both contributed to the development of this theory.
- Hence, the theory is often referred to as the von Mises yield criterion or the Mises-Hencky theory.

3. Principle of the Theory:
- The maximum distortion energy theory states that failure occurs when the distortion energy per unit volume exceeds a critical value.
- Distortion energy, also known as the von Mises stress or equivalent stress, is a measure of the deformation energy stored in a material due to plastic deformation.
- It is calculated using the principal stresses and is given by the following equation:
Distortion Energy = sqrt(0.5 * (σ₁ - σ₂)² + (σ₂ - σ₃)² + (σ₃ - σ₁)²)
- According to the theory, failure occurs when the distortion energy exceeds a material-specific limit, known as the yield strength or the yield stress.
- This criterion is often used to predict the yielding and failure of ductile materials, such as metals and alloys, under complex stress states.

4. Application and Significance:
- The maximum distortion energy theory is widely used in the field of mechanical engineering and materials science.
- It enables engineers to predict the failure of materials subjected to complex stress states, which is important for the design and analysis of structural components.
- The theory provides a practical and reliable approach to assess the structural integrity and safety of components under various loading conditions.
- The use of this theory has contributed to the development of robust and efficient designs in industries such as aerospace, automotive, and civil engineering.

In conclusion, the maximum distortion energy theory, also known as the von Mises yield criterion or the Mises-Hencky theory, was postulated by Richard von Mises and Ludwig Hencky. This theory is widely used to predict the failure of ductile materials under complex stress states and has significant applications in the field of mechanical engineering.