A note on the physical interpretation of the second Piola–Kirchhoff stress

01

Interpreting the stress measures

The second Piola–Kirchhoff stress tensor $\boldsymbol{S}$ is extensively used in finite-deformation mechanics, especially in Total Lagrangian formulations, but its physical interpretation requires care.

The Cauchy stress tensor $\boldsymbol{\sigma}$ has the most direct physical interpretation: it represents the actual traction on a surface of the current configuration. The first Piola–Kirchhoff stress tensor $\boldsymbol{P}$ relates forces in the current configuration to areas defined in the initial configuration.

Let $W$ denote the strain-energy density, i.e. the strain energy per unit volume of the initial configuration.

The Green–Lagrange strain $\boldsymbol{E}$ and the second Piola–Kirchhoff stress $\boldsymbol{S}$ form a conjugate strain–stress pair. In engineering notation, the variation of the strain-energy density may be written as:

Here, the engineering-component vectors are:

This relation expresses the conjugate character of the Green–Lagrange strain $\boldsymbol{E}$ and the second Piola–Kirchhoff stress $\boldsymbol{S}$.

For the linear constitutive law used on this site:

where $\boldsymbol{D}$ is the constitutive matrix of Hooke’s law.

In one dimension:

This association makes PK2 very convenient in the Total Lagrangian formulation. However, $\boldsymbol{S}$ should not be interpreted as the physical traction acting directly on a current surface.

02

One-dimensional example

Consider the data of Example 1 in Chapter 14, using N, mm, and MPa:

Let $\Delta$ denote the elongation. The stretch ratio is:

The Green–Lagrange strain is:

or:

The second Piola–Kirchhoff stress, denoted here by the scalar $S$, is:

03

From PK2 to the applied force

At finite deformation, it is not correct to identify the applied force directly with the initial area multiplied by PK2 stress:

The force referred to the initial area is instead related to the first Piola–Kirchhoff stress:

The two Piola–Kirchhoff stress measures satisfy:

Here, bold $\boldsymbol{F}$ denotes the deformation gradient, while the non-bold $F$ used in the numerical example denotes the applied force. Recall:

In one dimension, the deformation gradient reduces to the stretch ratio:

Therefore:

is the one-dimensional counterpart of $\boldsymbol{P}=\boldsymbol{F}\boldsymbol{S}$. Consequently:

and:

04

Numerical solution

For the specified load:

Let:

Then:

The correct equilibrium equation is:

Therefore:

or:

The positive physical root is approximately:

Hence:

This agrees with the finite-element result obtained with the program used in Chapter 12, as reported in Example 1 of Chapter 14.

05

The incorrect shortcut

If PK2 were incorrectly used directly in $F=A_0S$, the equation would be:

This would give approximately:

The difference between 30.8 mm and 28.8 mm is explained by the missing stretch factor:

which converts PK2 into PK1 in the one-dimensional relation:

06

Final remark

The second Piola–Kirchhoff stress has a precise and important mechanical role, but it is not the physical traction measure acting directly on a surface of the current configuration. Its usefulness in the Total Lagrangian formulation comes from its natural association with the Green–Lagrange strain tensor.

In one dimension, the conversion is: