Stress Transformation General Method (Cauchy’s Equation)

What is Cauchy’s Equation in Stress Tensor?

Cauchy’s Equation is a key concept in mechanics that links the internal stress in a material to the orientation of a specific plane within it. The equation is:

T = σ * n

where:

• T is the stress vector acting on a specific plane in the material.
• σ is the stress tensor (a 3×3 matrix representing stresses in different directions).
• n is the unit normal vector perpendicular to the plane.

Cauchy’s Equation allows us to determine the stress vector T on any plane with a known orientation inside a stressed body.

How to Calculate Normal and Shear Stress

1. Calculate the Stress Vector T: Use Cauchy’s Equation to find T by multiplying the stress tensor σ by the unit normal vector n.
2. Normal Stress (σₙ): The normal stress is the component of T that acts perpendicular to the plane. It can be calculated as the dot product of T and n: σₙ = T • n
3. Shear Stress (σₛ): The shear stress is the component of T that acts parallel to the plane. It can be found by removing the normal component from T and is calculated as: σₛ = √(|T|² – σₙ²)

where |T| is the magnitude of the stress vector T.