RSC under combined loading

The result should match Figure 5 in Wang, Jin, O’Gara, John F., Tucker, Charles L: "An objective model for slow orientation kinetics in concentrated fiber suspensions: Theory and rheological evidence", Journal of Rheology, 52, 1179, 2008: DOI: 10.1122/1.2946437.

import matplotlib.pyplot as plt
import numpy as np
from scipy.integrate import solve_ivp

from fiberoripy.closures import IBOF_closure
from fiberoripy.orientation import integrate_ori_ode, rsc_ode

# geometric factor
xi = 1.0

# time steps
t0, tf = 0, 400
t = np.linspace(t0, tf, 500)

# initial fiber orientation state
A0 = 1.0 / 3.0 * np.eye(3)

fig, axes = plt.subplots(3, 1, figsize=(4, 6))

for ax, ratio in zip(axes, [0.1, 0.12, 0.2], strict=True):

    def L(t, ratio=ratio):
        """Velocity gradient."""
        return np.array([[-ratio, 0.0, 1.0], [0.0, ratio, 0.0], [0.0, 0.0, 0.0]])

    # computed solution
    sol = solve_ivp(
        integrate_ori_ode,
        (t0, tf),
        A0.ravel(),
        t_eval=t,
        args=(L, IBOF_closure, rsc_ode, {"xi": 1.0, "Ci": 0.01, "kappa": 0.1}),
    )

    ax.plot(sol.t, sol.y[0], label="A11")
    ax.plot(sol.t, sol.y[4], label="A22")
    ax.plot(sol.t, sol.y[2], label="A13")

    ax.set_xlabel("Time $t$ in s")
    ax.set_ylim([0, 1])
    ax.grid()
    ax.legend()
plt.tight_layout()
plt.show()