Understanding Theories of Failure

When materials are subjected to complex stress conditions, it's important to predict when they will fail. Theories of failure help us determine the elastic limit of a material under combined stresses based on its properties measured in a simple tension test.

Total Strain Energy Theory Explained

The total strain energy theory is one of the important theories used to predict material failure under multiaxial stress states. This theory states that failure of a material at the elastic limit under any combination of stresses occurs when the total strain energy per unit volume at that point is equal to the total strain energy per unit volume at the elastic limit in a simple tension test.

In a simple uniaxial tension test, the stress at the elastic limit is the yield strength, $\sigma_y$. The strain energy per unit volume is given by $\frac{1}{2} \sigma_y \epsilon_y$, which for an elastic material (using Hooke's Law $\epsilon_y = \frac{\sigma_y}{E}$) can be written as $\frac{1}{2} \frac{\sigma_y^2}{E}$.

Under a general three-dimensional stress state with principal stresses $\sigma_1, \sigma_2, \sigma_3$, the total strain energy per unit volume (U) is given by:

$$U = \frac{1}{2E}(\sigma_1^2 + \sigma_2^2 + \sigma_3^2) - \frac{\nu}{E}(\sigma_1 \sigma_2 + \sigma_2 \sigma_3 + \sigma_3 \sigma_1)$$

According to the total strain energy theory, failure occurs when this U equals the strain energy density at yield in simple tension:

$$\frac{1}{2E}(\sigma_1^2 + \sigma_2^2 + \sigma_3^2) - \frac{\nu}{E}(\sigma_1 \sigma_2 + \sigma_2 \sigma_3 + \sigma_3 \sigma_1) = \frac{1}{2} \frac{\sigma_y^2}{E}$$

Multiplying by 2E:

$$\sigma_1^2 + \sigma_2^2 + \sigma_3^2 - 2\nu(\sigma_1 \sigma_2 + \sigma_2 \sigma_3 + \sigma_3 \sigma_1) = \sigma_y^2$$

This equation represents the criterion for failure according to the total strain energy theory.

Identifying the Total Strain Energy Theory

The total strain energy theory for the failure of a material at the elastic limit is also widely known by the name of its proponent.

• Guest’s or Tresca’s theory is based on Maximum Shear Stress.
• St. Venant’s theory is based on Maximum Principal Strain.
• Rankine’s theory is based on Maximum Principal Stress.
• Haig’s theory is the name given to the theory based on Total Strain Energy.

Therefore, the total strain energy theory is known as Haig's theory.

Revision Table: Common Theories of Failure

Additional Information on Haig’s Theory

Haig's theory, or the total strain energy theory, is generally suitable for ductile materials, although it provides a closer prediction for brittle materials than some other theories like Tresca or Von Mises. However, the theory that usually shows the best agreement with experiments for ductile materials is the Distortion Energy Theory (Von Mises theory), which is derived from the total strain energy theory by subtracting the strain energy associated with hydrostatic stress.

The assumption behind Haig's theory is that the total energy absorbed by the material per unit volume is the critical factor for failure, regardless of how that energy is stored (whether causing change in volume or distortion).