Engineers want to make sure materials don’t break when they twist. To do this, they use something called Unified Failure Criteria. This is a method that looks at different ways materials can fail when they are twisted. It's really important to know how materials react to different kinds of pressure.

There are two main ideas to think about:

1. Maximum Shear Stress Criterion: This idea says that a material will fail when it can’t handle the maximum twist it experiences compared to its strength. For a round shaft that is twisting, we can think of it this way:

   $\tau_{max} = \frac{T}{J} \cdot \frac{r}{2}$

   Here, T is the twisting force (or torque), J is a number that tells us how the shape of the material resists twisting, and r is the distance from the center to the edge. Engineers look at the maximum twist stress and compare it to how much stress the material can handle before it fails.

2. Distortion Energy Criterion (von Mises): This idea checks how much energy is stored in the material when it is twisted. According to this theory, the material starts to bend or break when the energy it can handle reaches a certain point. This can be written as:

   $\sigma_1 - \sigma_2 = \sqrt{3} \cdot \tau_{max}$

By using these two ideas, engineers can really understand how strong materials are when they twist. They also consider other factors like the shape of the material, its surface condition, and how fast the loading happens. This helps make sure buildings, bridges, and other structures stay safe and reliable even when they are twisted.