Difference between Cauchy stress and nominal stress (1st Piola-Kirchhoff)

📚 Understanding Stress: Cauchy vs. Nominal (1st Piola-Kirchhoff)

In continuum mechanics, stress is a measure of the internal forces acting within a deformable body. Two important measures of stress are the Cauchy stress and the nominal stress (also known as the 1st Piola-Kirchhoff stress). They differ in how they relate forces to areas, especially when dealing with large deformations.

🎯 Definition of Cauchy Stress (True Stress)

The Cauchy stress, often referred to as true stress, represents the force acting on the current area of the deformed body. It provides a measure of the actual stress state within the material at its deformed configuration.

• 📏Formula: The Cauchy stress tensor, denoted by $\boldsymbol{\sigma}$, is defined as: $\boldsymbol{\sigma} = \frac{d\mathbf{f}}{dA}$, where $d\mathbf{f}$ is the force acting on the current area $dA$.
• ⚙️Reference Area: Current (deformed) area.
• 💡Use Case: Useful for understanding material behavior under large deformations, as it reflects the true stress state.

🧭 Definition of Nominal Stress (1st Piola-Kirchhoff Stress)

The nominal stress, or 1st Piola-Kirchhoff stress, relates the force acting on the current area to the original area of the undeformed body. It's a two-point tensor that maps vectors from the reference configuration to the current configuration.

• 📐Formula: The 1st Piola-Kirchhoff stress tensor, denoted by $\mathbf{P}$, is defined as: $\mathbf{P} = \frac{d\mathbf{f}}{dA_0}$, where $d\mathbf{f}$ is the force acting on the current area and $dA_0$ is the original (undeformed) area.
• 🏗️Reference Area: Original (undeformed) area.
• 🧭Use Case: Useful in computational mechanics for relating forces in the deformed configuration to the original geometry.

📝 Comparison Table

🔑 Key Takeaways

• 📍Area Consideration: The key difference lies in the area used for calculation. Cauchy stress uses the current area, while nominal stress uses the original area.
• 🧮Applications: Cauchy stress is crucial for understanding material response, while nominal stress is valuable in computational simulations.
• 💡Deformation Impact: For small deformations, the Cauchy stress and nominal stress are approximately equal. However, for large deformations, they diverge significantly.