Isotropic Hyperelasticity: Gradient and Hessian w.r.t. the Metric (divided by two) returns the Kirchhoff stress

[adtzlr]

With tensortrax, it is possible to express the strain energy function of an isotropic hyperelastic solid in terms of the right or left Cauchy-Green deformation tensor. By applying the gradient on different variables, one obtains the appropriate stress measure. A transformation of one stress measure proofes the results.

import numpy as np
from tensortrax import hessian
from tensortrax.math import sqrt, trace, transpose
from tensortrax.math._linalg import det

def W(h, F):
    "An Exemplaric Isotropic Hyperelastic Strain Energy Function."
    
    # Metric Tensor
    g = 2 * h

    # Right Cauchy-Green Deformation Tensor and
    # Determinant of Deformation Gradient
    C = transpose(F) @ g @ F
    J = sqrt(det(C))
    
    return (J ** (-2/3) * trace(C) - 3) + 20 * (J - 1) ** 2 / 2

# Metric tensor and an exemplaric Deformation Gradient
g = np.eye(3)
F = g + np.arange(9).reshape(3, 3) / 10

# Kirchhoff stress
Jc, τ, ψ = hessian(W, full_output=True, wrt="h")(h=g/2, F=F)

# First Piola-Kirchhoff stress
A,  P, ψ = hessian(W, full_output=True, wrt="F")(h=g/2, F=F)

assert np.allclose(τ, P @ F.T)