Schmid’s law (Derivation of the Schmid factor)

Introduction

This tutorial explains Schmid's law, which describes the relationship between shear stress and normal stress in materials. Understanding Schmid's law is essential for fields such as materials science and engineering, particularly when analyzing how materials deform under stress. This guide will break down the derivation of the Schmid factor and its practical applications.

Step 1: Understand Forces in a Slip System

• A slip system consists of a slip plane and a slip direction.
• Forces applied to materials induce both normal and shear stresses.
• Recognize that the orientation of the applied force relative to the slip system is crucial for understanding how materials yield.

Step 2: Analyze Shear Force in a Slip Plane

• The shear force ((F_s)) acting on the slip plane can be calculated based on the applied normal force ((F_n)).
• The relationship can be expressed as:
  • (F_s = F_n \cdot \sin(\theta))
• Here, (\theta) is the angle between the normal force and the slip direction.

Step 3: Calculate Shear Stress

• Shear stress ((\tau)) is defined as the force per unit area acting parallel to the area:
  • (\tau = \frac{F_s}{A})
• Substitute the expression for shear force:
  • (\tau = \frac{F_n \cdot \sin(\theta)}{A})
• Ensure you use the correct area for the calculation, which is the area of the slip plane.

Step 4: Determine Critical Resolved Shear Stress (CRSS)

• The critical resolved shear stress is the minimum shear stress required to initiate slip in the material.
• It is a material property and varies with different materials.
• Use the relationship:
  • (\tau_{CRSS} = \frac{\sigma}{2})
• Here, (\sigma) represents the normal stress applied to the material.

Step 5: Apply Schmid's Law

• Schmid's law states that maximum resolved shear stress occurs at an angle of 45°:
  • At this angle, the shear stress is half the normal stress.
• The mathematical expression is:
  • (\tau_{max} = \frac{\sigma}{2})
• This relationship is fundamental in predicting when and how a material will yield.

Step 6: Examine an Example

• Consider a material subjected to a known normal stress.
• Calculate the shear stress using:
  • (\tau = \frac{F_n \cdot \sin(45°)}{A})
• Verify that the calculated shear stress meets or exceeds the CRSS to determine if yielding will occur.

Step 7: Understand Maximum Resolved Shear Stress

• The concept of maximum resolved shear stress helps in predicting failure in materials.
• Use the derived expressions to analyze different angles and forces acting on various slip systems.
• This understanding aids in designing materials that can withstand specific loads without failing.

Conclusion

Schmid's law provides a crucial framework for understanding shear and normal stresses in materials under load. By following these steps, you can analyze the behavior of materials, predict yielding, and apply this knowledge in practical scenarios such as material selection and structural design. For further exploration, consider studying different materials and their respective CRSS values to deepen your understanding of material mechanics.