A material may be predicted to fail according to different failure theories; which condition(s) can be a valid failure criterion depending on the material and loading?

Introduction / Context:
Different materials fail under different controlling stress/strain states. Classical failure theories provide criteria to assess safety: maximum principal stress (Rankine), maximum principal strain (St. Venant), maximum shear stress (Tresca), and energy-based theories like total strain energy or distortion energy (Beltrami, von Mises). This question checks awareness that multiple criteria may be relevant depending on the material behavior (brittle versus ductile) and design code.

Given Data / Assumptions:

• General 3D stress state possible.
• Allowable limits σ0 (direct), shear limit, and strain/energy limits are known from material data or codes.
• We are not restricting ourselves to only one theory.

Concept / Approach:
Brittle materials (e.g., cast iron): Rankine (maximum normal stress) often governs.Ductile materials (e.g., mild steel): Shear- or energy-based criteria (Tresca or von Mises) are typically appropriate.High strain sensitivity: Maximum principal strain criterion may be used for certain materials/components.Energy viewpoint: Exceeding total allowable strain energy can indicate failure risk under complex states.

Step-by-Step Solution:
Map each option to a known failure theory.Recognize that design codes allow different criteria per material class and service condition.Therefore, all listed conditions can represent valid failure triggers in the proper context.

Verification / Alternative check:
Compare predicted safety margins using Rankine vs. Tresca vs. von Mises in a biaxial state; the controlling (lowest margin) criterion dictates design in that scenario.

Why Other Options Are Wrong:

• Any single choice alone ignores other established, code-recognized criteria.

Common Pitfalls:
Applying a ductile-material criterion to brittle materials (or vice versa) without justification; assuming one universal theory fits all cases.

Final Answer:
All the above.