Error in the Cauchy stress formula?

[r03ert0]

Dear Taichi devs,

In engine/mpm_solver.py, the formula for the Cauchy stress is

stress = 2 mu (F-R) F^T + I lambda J (J-1).

Isn't the formula missing a J^(-1) factor?

The Piola-Kirchhoff tensor for the fixed corrotated model is

P = 2 mu (F-R) + lambda (J-1) J F^(-T),

The Cauchy stress is defined as stress = (1/J) P F^(T), which should give

stress = 2 mu (F-R)/J + lambda (J-1) I.

However, the code in the mpm_solver.py (lines 452, 550) is:

stress = 2 * mu * (
    self.F[p] - U @ V.transpose()) @ self.F[p].transpose(
    ) + ti.Matrix.identity(ti.f32, self.dim) * la * J * (J - 1)