Fitting experimental data to a Neo-Hookean solid.

neo_hookean_fit.py

# -*- coding: utf-8 -*-
"""
Compute the material parameters for a neo-hookean solid from
data obtained from a uniaxial stress test. The used expressions
are from _[1]

@author: Nicolas Guarin-Zapata

References
----------
..[1] Neo-Hookean solid. (2016, April 9). In Wikipedia, The Free
Encyclopedia. Retrieved 22:28, June 9, 2016, from
https://en.wikipedia.org/w/index.php?title=Neo-Hookean_solid&oldid=714459225

"""
from __future__ import division
import numpy as np
import matplotlib.pyplot as plt
from scipy.optimize import fsolve, minimize


def jaco(J, C1, D1, lam):
    return D1*J**5 - D1*J**2 + C1/(3*lam)*J - C1*lam**2/3


def fun(C1, D1, lams):
    J_vec = np.ones_like((lams))
    for k, lam in enumerate(lams):
        aux = fsolve(jaco, 1.0, args=(C1, D1, lam))
        J_vec[k] = aux
    return 4*C1/(3*J_vec**(5/3))*(lams - J_vec/lams) +\
           2*D1*(J_vec - 1)


def error_fun(x, lams, sigmas):
    C1, D1 = x
    S = fun(C1, D1, lams)
    return np.linalg.norm(S - sigmas)**2


C1, D1 = [3/4, 1/4]
lams = np.linspace(1, 5, 50)
sigmas = fun(C1, D1, lams)
sol = minimize(error_fun, (2, 2), args=(lams, sigmas), method='Nelder-Mead')
sigmas_fit = fun(sol.x[0], sol.x[1], lams)
plt.plot(lams, sigmas, 'ko')
plt.plot(lams, sigmas_fit, 'r')
plt.show()