State the name associated with the shear strain energy (distortion energy) theory for elastic failure under multiaxial stress.

Introduction / Context:
Designing ductile materials under multiaxial stress often uses an energy-based failure criterion. The shear (distortion) strain energy theory—also called the Von Mises or Hencky–Von Mises criterion—is widely adopted in metal design because it correlates well with yielding in ductile metals.

Given Data / Assumptions:

• We focus on the distortional component of strain energy as the trigger for yield.
• Elastic behavior up to yield point is assumed.

Concept / Approach:
Von Mises theory states that yielding begins when the second deviatoric stress invariant (J2) reaches the same value as in uniaxial tension at yield. Physically, only the distortion energy causes yielding; hydrostatic (volumetric) energy does not.

Step-by-Step Solution:
1) Associate names with criteria: Rankine → maximum principal stress; St. Venant → maximum principal strain; Guest/Tresca → maximum shear stress; Haigh → total strain energy; Von Mises → distortion energy.2) Select Von Mises for shear strain (distortion) energy theory.

Verification / Alternative check:
Modern codes and FEA post-processing commonly report Von Mises equivalent stress for ductile materials, aligning with this theory.

Why Other Options Are Wrong:

• Rankine/St. Venant: Principal stress/strain criteria.
• Guest or Trecas: Maximum shear stress, not energy-based.
• Haig: Total strain energy, not distortional-only.

Common Pitfalls:

• Interchanging Tresca and Von Mises; Tresca gives a hexagonal yield surface, Von Mises a circular one in principal stress space.

Final Answer:
Von Mises.