Failure theories at elastic limit (ductile materials): Which named theory corresponds to the “maximum shear stress” criterion for failure (also called the Coulomb–Tresca or Guest–Tresca criterion)?

Introduction / Context:
Failure theories map multiaxial stress states to an equivalent uniaxial condition to predict yielding. For ductile materials like mild steel, shear-based criteria are widely used because yielding initiates when shear reaches a critical value related to the material’s uniaxial yield strength.

Given Data / Assumptions:

• Elastic behavior up to yield; isotropic material.
• Multiaxial stress state with principal stresses σ1 ≥ σ2 ≥ σ3.
• No strain-rate or temperature effects considered.

Concept / Approach:

The maximum shear stress criterion (Guest–Tresca) states yielding begins when τ_max = (σ1 − σ3)/2 equals the uniaxial shear yield value (σ_y/2). It constructs a hexagonal yield locus in σ1–σ2 space and is conservative compared with the smooth elliptical Von Mises surface.

Step-by-Step Solution:

Verification / Alternative check:

For uniaxial tension (σ1 = σ_y, σ2 = σ3 = 0), τ_max = σ_y/2, matching the Tresca yield onset condition, validating the criterion.

Why Other Options Are Wrong:

• St. Venant (maximum principal strain) is not a standard modern failure criterion for yielding.
• Rankine uses maximum principal stress, more suited to brittle materials.
• Haigh relates to total strain energy and is seldom used for ductile yield prediction.
• Von Mises uses distortion energy and predicts yielding when sqrt( ( (σ1−σ2)^2 + (σ2−σ3)^2 + (σ3−σ1)^2 ) / 2 ) = σ_y, not maximum shear.

Common Pitfalls:

• Applying Von Mises and Tresca interchangeably without noting Tresca is slightly more conservative.
• Using these criteria for brittle materials where Rankine or Mohr–Coulomb may be more appropriate.

Final Answer:

Guest’s or Tresca’s theory (maximum shear stress).