Different Stress Measures

In large deformation analysis, there are multiple ways to define stress. Different FEA software packages output different measures by default. Understanding these measures is essential for correct result interpretation.

The Three Main Stress Measures

1. Cauchy Stress $\boldsymbol{\sigma}$ (True Stress)

Force per unit current area. This is the "true" stress that acts on material in its deformed state.

$$\mathbf{t} = \boldsymbol{\sigma} \cdot \mathbf{n}$$

where $\mathbf{n}$ is the current normal vector.

2. First Piola-Kirchhoff Stress $\mathbf{P}$

Force per unit reference area. This is the nominal or engineering stress.

$$\mathbf{P} = J \boldsymbol{\sigma} \cdot \mathbf{F}^{-T}$$

3. Second Piola-Kirchhoff Stress $\mathbf{S}$

Completely in reference configuration. Work-conjugate to Green-Lagrange strain.

$$\mathbf{S} = \mathbf{F}^{-1} \cdot \mathbf{P} = J \mathbf{F}^{-1} \cdot \boldsymbol{\sigma} \cdot \mathbf{F}^{-T}$$

Physical Interpretation

Consider a rubber specimen being stretched:

Conversion Formulas

If you have one stress measure, you can compute the others:

$$\mathbf{P} = J \boldsymbol{\sigma} \cdot \mathbf{F}^{-T}$$

$$\mathbf{S} = J \mathbf{F}^{-1} \cdot \boldsymbol{\sigma} \cdot \mathbf{F}^{-T}$$

$$\boldsymbol{\sigma} = \frac{1}{J} \mathbf{F} \cdot \mathbf{S} \cdot \mathbf{F}^T$$

Python Code

import numpy as np

def convert_cauchy_to_pk1(sigma, F):
    """P = J * sigma * F^{-T}"""
    J = np.linalg.det(F)
    F_inv_T = np.linalg.inv(F).T
    P = J * sigma @ F_inv_T
    return P

def convert_cauchy_to_pk2(sigma, F):
    """S = J * F^{-1} * sigma * F^{-T}"""
    J = np.linalg.det(F)
    F_inv = np.linalg.inv(F)
    S = J * F_inv @ sigma @ F_inv.T
    return S

# Example
F = np.array([[1.5, 0.2], [0.1, 0.8]])  # Deformation gradient
sigma = np.array([[100, 20], [20, 50]])  # Cauchy stress (MPa)

P = convert_cauchy_to_pk1(sigma, F)
S = convert_cauchy_to_pk2(sigma, F)

print(f"Cauchy σ:\n{sigma}")
print(f"\n1st P-K P:\n{P}")
print(f"\n2nd P-K S:\n{S}")

When to Use Each Measure

Automotive Application: Engine Mount Rubber

An engine mount under preload experiences large deformation. The FEA software reports stress, but which measure?

• LS-DYNA d3plot: Cauchy stress by default
• Abaqus UMAT: Typically works with 2nd P-K internally
• Engineering reports: Often convert to "engineering stress" (1st P-K divided by reference area)

Understanding these differences prevents factor-of-2 errors in fatigue life predictions!

Key Takeaways

• Cauchy σ: True stress, current configuration, symmetric
• 1st P-K P: Nominal stress, force/reference area, NOT symmetric
• 2nd P-K S: Reference configuration, symmetric, for material laws
• For small strains, all three are essentially equal
• Know your FEA software's default output!

What's Next

In the next lesson, we'll review FEM & Shape Functions — how continuous fields are approximated with nodal values and interpolation.