Understanding Theories of Failure in Materials

When designing structures or machine parts, it's crucial to predict when a material will fail under different loading conditions. Theories of failure help engineers do this by establishing criteria based on stresses or strains within the material.

The question asks about a specific theory of failure where a ductile material begins to yield when the maximum principal strain reaches a certain value. Let's look at the prominent theories of failure and their criteria:

St. Venant's Theory: Maximum Principal Strain Theory

St. Venant's theory, also known as the Maximum Principal Strain Theory, postulates that failure (either yielding for ductile materials or fracture for brittle materials) occurs when the maximum principal strain in the complex stress system reaches the maximum strain at the elastic limit in a simple tension test.

For a 2D stress system with principal stresses $\sigma_1$ and $\sigma_2$, the principal strains $\epsilon_1$ and $\epsilon_2$ are given by:

• $\epsilon_1 = \frac{1}{E}(\sigma_1 - \nu\sigma_2)$
• $\epsilon_2 = \frac{1}{E}(\sigma_2 - \nu\sigma_1)$

where $E$ is Young's modulus and $\nu$ is Poisson's ratio.

According to St. Venant's theory, failure occurs when the maximum absolute principal strain reaches the strain at yield (or ultimate strength, depending on the material and failure definition) in a uniaxial tension test. The strain at yield in uniaxial tension is $\epsilon_y = \frac{\sigma_y}{E}$, where $\sigma_y$ is the yield strength.

So, the failure criterion is:

$\text{max}(|\epsilon_1|, |\epsilon_2|) = \epsilon_y = \frac{\sigma_y}{E}$

Substituting the strain expressions:

$\text{max}\left(\left|\frac{1}{E}(\sigma_1 - \nu\sigma_2)\right|, \left|\frac{1}{E}(\sigma_2 - \nu\sigma_1)\right|\right) = \frac{\sigma_y}{E}$

This simplifies to:

$\text{max}(|\sigma_1 - \nu\sigma_2|, |\sigma_2 - \nu\sigma_1|) = \sigma_y$

While often applied to brittle materials, the question specifically asks about a ductile material yielding based on the maximum principal strain reaching the yield strain. This description precisely matches St. Venant's theory.

Other Theories of Failure

Let's briefly examine the other options to see why they don't fit the description based on maximum principal strain:

• Rankine's Theory (Maximum Principal Stress Theory): This theory states that failure occurs when the maximum principal stress in the complex stress system reaches the ultimate tensile stress (for brittle materials) or yield strength (for ductile materials) in a simple tension test. The criterion is $\text{max}(|\sigma_1|, |\sigma_2|, |\sigma_3|) = \sigma_{\text{failure}}$. This theory is based on stress, not strain.
• Haigh's Theory (Maximum Strain Energy Theory): This theory proposes that failure occurs when the total strain energy per unit volume in the complex stress system equals the strain energy per unit volume at the elastic limit in a simple tension test. The criterion involves the sum of squared principal stresses and Poisson's ratio: $\sigma_1^2 + \sigma_2^2 + \sigma_3^2 - 2\nu(\sigma_1\sigma_2 + \sigma_2\sigma_3 + \sigma_3\sigma_1) = \sigma_y^2$. This theory is based on strain energy, not maximum principal strain directly.
• Guest Theory (Tresca's Theory or Maximum Shear Stress Theory): This theory suggests that failure occurs when the maximum shear stress in the complex stress system reaches the maximum shear stress at the elastic limit in a simple tension test. The maximum shear stress is $\tau_{\text{max}} = \frac{|\sigma_{\text{max}} - \sigma_{\text{min}}|}{2}$. In a simple tension test, the maximum shear stress at yield is $\sigma_y/2$. So the criterion is $\tau_{\text{max}} = \sigma_y/2$ or $|\sigma_{\text{max}} - \sigma_{\text{min}}| = \sigma_y$. This theory is based on shear stress, not principal strain.

Based on the descriptions, only St. Venant's theory is based on the criterion of maximum principal strain reaching a limit strain value, aligning perfectly with the question.

Conclusion on Theories of Failure

The theory that describes a ductile material beginning to yield when the maximum principal strain reaches the yield strain is St. Venant's theory, the Maximum Principal Strain Theory.

Revision Table: Key Theories of Failure

Additional Information on Material Behavior and Failure Theories

Ductile materials like mild steel typically exhibit significant plastic deformation before fracture. Their yielding behavior is often better predicted by theories based on shear stress or distortion energy (like Tresca or Von Mises theories) because yielding involves plastic slip along shear planes. However, the question specifically asks for the theory based on the maximum principal strain criterion, which is St. Venant's theory.

Brittle materials, such as cast iron or glass, fracture with little or no plastic deformation. Their failure is often governed by the maximum normal stress, making Rankine's theory more suitable for predicting their fracture.

Choosing the appropriate theory of failure depends on the material type (ductile or brittle) and the specific failure mode being considered (yielding or fracture).