Plasticity with Isotropic and Kinematic Hardening Example

import os
import numpy as np
import matplotlib.pyplot as plt
import simcoon as sim

plt.rcParams["figure.figsize"] = (18, 10)

This plasticity model combines both isotropic and kinematic hardening with a power-law isotropic hardening and linear kinematic hardening.

Seven parameters are required:

1. The Young modulus \(E\)

2. The Poisson ratio \(\nu\)

3. The coefficient of thermal expansion \(\alpha\)

4. The initial yield stress \(\sigma_Y\)

5. The isotropic hardening parameter \(k\)

6. The isotropic hardening exponent \(m\)

7. The kinematic hardening modulus \(k_X\)

The constitutive law is given by:

\[\begin{split}{\sigma}_{ij} & = L_{ijkl}\left({\varepsilon}^{\textrm{tot}}_{kl}-\alpha_{kl}\left(T-T^{\textrm{ref}}\right)-{\varepsilon}^{\textrm{p}}_{kl}\right) \\ \dot{\varepsilon}^{\textrm{p}}_{ij} & =\dot{p}\Lambda_{ij}, \quad \Lambda_{ij}=\frac{3}{2}\frac{\sigma'_{ij} - X_{ij}}{\overline{\sigma - X}} \\ \dot{X}_{ij} & = \frac{2}{3} k_X \dot{\varepsilon}^{\textrm{p}}_{ij} \\ \Phi & =\overline{\sigma - X}-\sigma_{Y}-kp^m\leq 0\end{split}\]

where \(X_{ij}\) is the kinematic hardening (back stress) tensor resulting from linear kinematic hardening, and the isotropic hardening follows a power-law evolution \(kp^m\).

umat_name = "EPKCP"  # 5 character code for combined isotropic-kinematic hardening
nstatev = 14  # Number of internal variables

# Material parameters
E = 67538.0  # Young's modulus (MPa)
nu = 0.349  # Poisson ratio
alpha = 1.0e-6  # Thermal expansion coefficient
sigma_Y = 300.0  # Initial yield stress (MPa)
k = 1500.0  # Isotropic hardening parameter
m = 0.3  # Isotropic hardening exponent
k_X = 2000.0  # Kinematic hardening modulus

psi_rve = 0.0
theta_rve = 0.0
phi_rve = 0.0
solver_type = 0
corate_type = 1

props = np.array([E, nu, alpha, sigma_Y, k, m, k_X])

path_data = "../data"
pathfile = "EPKCP_path.json"

The loading path is read in Python and the simulation runs in memory: no result file is written, and the histories come back as component-first arrays.

blocks, T_init, _ = sim.solver.load_path_json(os.path.join(path_data, pathfile))

res = sim.solver.solve(
    blocks,
    umat_name,
    props,
    nstatev,
    T_init=T_init,
    solver_type=solver_type,
    corate=corate_type,
    orientation=(psi_rve, theta_rve, phi_rve),
)

Plotting the results

We plot the stress-strain curve showing both isotropic and kinematic hardening.

e11, e22, e33, e12, e13, e23 = res["Strain"]
s11, s22, s33, s12, s13, s23 = res["Stress"]
time, T = res["Time"], res["Temp"]
Wm, Wm_r, Wm_ir, Wm_d = res["Wm"]

fig = plt.figure()

# First subplot: Stress vs Strain
ax1 = fig.add_subplot(1, 2, 1)
plt.grid(True)
plt.tick_params(axis="both", which="major", labelsize=15)
plt.xlabel(r"Strain $\varepsilon_{11}$", size=15)
plt.ylabel(r"Stress $\sigma_{11}$ (MPa)", size=15)
plt.plot(e11, s11, c="blue", label="EPKCP model")
plt.legend(loc="best")

# Second subplot: Work terms vs Time
ax2 = fig.add_subplot(1, 2, 2)
plt.grid(True)
plt.tick_params(axis="both", which="major", labelsize=15)
plt.xlabel("time (s)", size=15)
plt.ylabel(r"$W_m$", size=15)
plt.plot(time, Wm, c="black", label=r"$W_m$")
plt.plot(time, Wm_r, c="green", label=r"$W_m^r$")
plt.plot(time, Wm_ir, c="blue", label=r"$W_m^{ir}$")
plt.plot(time, Wm_d, c="red", label=r"$W_m^d$")
plt.legend(loc="best")

plt.show()