Normal stress and shear stress are very important when figuring out how materials fail, especially in the field of materials mechanics. Knowing how these types of stress work with material properties helps us predict when materials might break and how to design safer structures.

What is Normal Stress?

Normal stress is the force acting on a material's surface. It pushes or pulls perpendicularly (straight on) to that surface. We can calculate normal stress with this formula:

$$\sigma = \frac{F}{A}$$

Here, $\sigma$ is the normal stress, $F$ is the force applied, and $A$ is the area where the force is acting.

Normal stress can be divided into two types:

1. Tensile Stress: This pulls on the material and makes it stretch.
2. Compressive Stress: This pushes on the material and makes it shrink.

When a material feels tensile stress, it can get longer, but with compressive stress, it might get shorter or even buckle.

What is Shear Stress?

Shear stress is a bit different. It acts parallel to the surface of the material. We can calculate shear stress like this:

$$\tau = \frac{V}{A}$$

In this formula, $\tau$ is the shear stress, $V$ is the force pushing sideways, and $A$ is the area the force affects.

Shear stress makes materials change shape by sliding. This can happen without the material getting bigger or smaller in volume.

Why Are Stress Types Important?

The way normal and shear stress combine with material properties can lead to different ways materials can fail.

How Do We Know When Materials Will Fail?

To predict when a material will fail under combined normal and shear stress, we use several criteria:

1. Maximal Normal Stress Criterion (Rankine’s Criterion): This says failure happens if the maximum normal stress in a material reaches its breaking point (ultimate strength). Mathematically, this means:

   $$\sigma_{max} \geq \sigma_{ult}$$

   or

   $$\sigma_{min} \leq -\sigma_{c}$$

   Here, $\sigma_{ult}$ is the ultimate tensile strength, and $\sigma_{c}$ is the compressive strength.

2. Maximal Shear Stress Criterion (Tresca Criterion): Failure is predicted when the maximum shear stress reaches half the material's yield strength:

   $$\tau_{max} \geq \frac{\sigma_{y}}{2}$$

   where $\sigma_{y}$ is the yield strength.

3. Von Mises Criterion (Distortion Energy Criterion): This one is good for ductile materials. It considers the total energy from deformation. Failure happens when the equivalent stress reaches the yield strength:

   $$\sigma_{eq} = \sqrt{\sigma_1^2 + \sigma_2^2 - \sigma_1\sigma_2 + 3\tau^2} \geq \sigma_{y}$$

   Here, $\sigma_1$ and $\sigma_2$ are the main stresses.

What is Strain?

Understanding strain is important when studying material failure. Strain tells us how much a material has changed shape.

• Normal Strain ($\epsilon$) measures how much a material stretches or shrinks:

$$\epsilon = \frac{\delta L}{L_0}$$

Here, $\delta L$ is the change in length, and $L_0$ is the original length.

• Shear Strain ($\gamma$) measures the change in angle between two lines that were originally at a right angle:

$$\gamma = \frac{\delta x}{L}$$

Here, $\delta x$ is the sideways movement, and $L$ is the original length.

We also connect stress and strain through material properties, like:

• Young's Modulus ($E$) for normal stress.
• Shear Modulus ($G$) for shear stress.

This is shown in Hooke’s Law:

• For normal stress and strain:

$$\sigma = E \cdot \epsilon$$

• For shear stress and shear strain:

$$\tau = G \cdot \gamma$$

Real-World Stress Situations

In real life, materials face complex situations with both normal and shear stress at the same time. For example, when twisting a material, it feels shear stress, while pushing on it feels normal stress. The mix of these stresses can make predicting failure more complicated.

We can also use something called Mohr’s Circle to see how normal and shear stress relate. It helps visualize stress states on a 2D plane.

Ductile vs. Brittle Materials

Materials are classified as ductile or brittle, which influences how they react to stress and strain:

• Ductile Materials: These stretch a lot before breaking, absorbing energy. They usually fail after significant deformation, and we often use shear and Von Mises criteria to predict their failure.

• Brittle Materials: These break suddenly without much stretching. Their failure risk is assessed more using normal stress criteria, focusing on their maximum tensile strength.

What About Fatigue?

Fatigue is another reason materials fail. It happens when a material goes through cycles of stress. Even if the stress is less than the material's breaking point, repeated cycles can start cracks.

We use the S-N curve (Wöhler curve) to show the relationship between stress levels over time and how long the material lasts before it fails. It’s vital to consider both normal and shear stresses to ensure structures can handle repeated stress.

Final Thoughts

In conclusion, understanding the relationship between normal stress and shear stress is very important for knowing how materials fail. Different criteria can help predict when materials will break under different loads.

Knowing about ductility, brittleness, and fatigue is crucial when using these ideas in real life. By applying these principles, engineers can create materials and structures that are safe and perform well, avoiding failure through careful material selection and stress analysis. Understanding these concepts is key for safe engineering practices in materials mechanics.