In the study of material science and engineering, theories of failure are crucial for predicting when a material will yield or fracture under various loading conditions. These theories provide criteria based on different stress and strain states to determine the point of failure for both ductile and brittle materials. Understanding these theories is fundamental for safe and efficient engineering design.

Theories of Failure Matching

Let's delve into each theory and its corresponding proponent to understand their principles and applications:

Normal Stress Theory (Rankine's Theory)

• The Max Normal Stress Theory, often referred to as Rankine's Theory, is primarily utilized for brittle materials.
• According to this theory, failure occurs when the maximum principal stress ($\sigma_1$) in a component under complex loading reaches the material's yield stress ($\sigma_y$) for ductile materials or its ultimate tensile strength ($\sigma_{ut}$) for brittle materials, as determined from a simple tensile test.
• Mathematically, the criterion for failure is expressed as $\sigma_1 = \sigma_y$ (for ductile materials) or $\sigma_1 = \sigma_{ut}$ (for brittle materials).

Shear Stress Theory (Guest and Tresca Theory)

• The Max Shear Stress Theory, widely recognized as Guest and Tresca Theory, is a commonly applied theory for ductile materials.
• This theory posits that yielding of a material initiates when the maximum shear stress ($\tau_{max}$) in the component reaches the maximum shear stress at the yield point in a simple uniaxial tensile test.
• In a uniaxial tensile test, the maximum shear stress is $\sigma_y/2$. Therefore, according to the Tresca criterion, failure occurs when $\tau_{max} = \sigma_y/2$. In terms of principal stresses ($\sigma_1$, $\sigma_2$, $\sigma_3$), this is expressed as $\frac{(\sigma_{max} - \sigma_{min})}{2} = \frac{\sigma_y}{2}$, which simplifies to $|\sigma_1 - \sigma_3| = \sigma_y$.

Strain Energy Theory (Haigh's Theory)

• The Max Strain Energy Theory, also known as Haigh's Theory or the Beltrami-Haigh theory, proposes that failure occurs when the total strain energy per unit volume stored in a stressed component equals the strain energy per unit volume at the yield point in a simple tensile test.
• This theory considers the total energy absorbed by the material up to the point of failure.
• The total strain energy per unit volume ($U$) for a three-dimensional stress state is given by $U = \frac{1}{2E}(\sigma_1^2 + \sigma_2^2 + \sigma_3^2) - \frac{\nu}{E}(\sigma_1\sigma_2 + \sigma_2\sigma_3 + \sigma_3\sigma_1)$. Failure occurs when $U = U_y$, where $U_y = \frac{\sigma_y^2}{2E}$ is the strain energy density at yield in simple tension.

Distortion Energy Theory (Von Mises Theory)

• The Max Distortion Energy Theory, famously known as Von Mises Theory or the Maximum Octahedral Shear Stress Theory, is one of the most accurate and widely used theories for ductile materials.
• This theory suggests that yielding occurs when the distortion energy per unit volume in a stressed material equals the distortion energy per unit volume at the yield point in a simple tensile test. Distortion energy is the portion of the total strain energy that causes a change in shape (distortion) rather than a change in volume (hydrostatic strain).
• The Von Mises yield criterion is mathematically expressed as: $\sigma_v = \sqrt{\frac{(\sigma_1 - \sigma_2)^2 + (\sigma_2 - \sigma_3)^2 + (\sigma_3 - \sigma_1)^2}{2}} = \sigma_y$, where $\sigma_v$ is the Von Mises equivalent stress.

Failure Theories Summary

Based on the analysis of these fundamental theories of failure, the correct matching is:

• A. Max Normal Stress Theory – 4. Rankine's Theory
• B. Max Shear Stress Theory – 3. Guest and Tresca Theory
• C. Max Strain Energy Theory – 2. Haigh's Theory
• D. Max Distortion Energy Theory – 1. Von Mises Theory