Understanding Theories of Failure for Materials

When designing structures and components, engineers use theories of failure to predict when a material will fail under different types of stress conditions. Different theories are applicable to different types of materials, primarily categorized as ductile or brittle.

Suitable Theories for Ductile Materials

Ductile materials like steel, aluminum, and copper tend to yield and deform significantly before fracturing. Their failure is often related to shear stress.

Let's consider the theories mentioned in the question:

1. Maximum Principal Stress Theory (Rankine's Theory): This theory states that failure occurs when the maximum principal stress in the material reaches the yield strength (or ultimate strength for brittle materials) in a simple tension test. It does not account for the influence of other principal stresses, particularly in states of multi-axial stress. This theory is generally considered more suitable for brittle materials.
2. Maximum Strain Theory (St. Venant's Theory): This theory postulates that failure happens when the maximum principal strain reaches the strain corresponding to the yield strength (or ultimate strength) in a simple tension test. Like the maximum principal stress theory, it doesn't always accurately predict failure for ductile materials under various stress states.
3. Maximum Shear Stress Theory (Tresca's Theory): This theory suggests that yielding occurs when the maximum shear stress in the material reaches half of the yield strength obtained from a simple tension test (where the maximum shear stress is $\frac{\sigma_y}{2}$). Ductile materials fail in shear, making this theory a reasonably good predictor for their yielding behavior. It provides a conservative estimate compared to the Maximum Distortion Energy Theory.
4. Maximum Distortion Energy Theory (von Mises-Hencky Theory): This theory states that failure occurs when the distortion energy per unit volume in the material reaches the distortion energy per unit volume required to cause yielding in a simple tension test. Distortion energy is the energy associated with the change in shape of the material, excluding the energy associated with volume change. This theory is based on the shear stresses and is considered the most accurate theory for predicting the yielding of ductile materials under various stress states.

Based on experimental evidence and theoretical considerations, the theories most suitable for ductile materials are the Maximum Shear Stress Theory (Tresca) and the Maximum Distortion Energy Theory (von Mises).

Therefore, theories 3 and 4 are the most suitable for ductile materials.