Isotropic elasticity (thermomechanical)

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

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

In thermoelastic isotropic materials three mechanical parameters and two thermal parameters are required:

1. The density \(\rho\)

2. The specific heat \(c_p\)

3. The Young modulus \(E\)

4. The Poisson ratio \(\nu\)

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

The elastic stiffness tensor is written in the Voigt notation formalism as

\[\begin{split}\mathbf{L} = \begin{pmatrix} L_{1111} & L_{1122} & L_{1122} & 0 & 0 & 0 \\ L_{1122} & L_{1111} & L_{1122} & 0 & 0 & 0 \\ L_{1122} & L_{1122} & L_{1111} & 0 & 0 & 0 \\ 0 & 0 & 0 & L_{1212} & 0 & 0 \\ 0 & 0 & 0 & 0 & L_{1212} & 0 \\ 0 & 0 & 0 & 0 & 0 & L_{1212} \end{pmatrix}\end{split}\]

with

\[L_{1111} = \frac{E(1-\nu)}{(1+\nu)(1-2\nu)}, \quad L_{1122} = \frac{E\nu}{(1+\nu)(1-2\nu)}, \quad L_{1212} = \frac{E}{2(1+\nu)}.\]

The increment of the elastic strain is given by

\[\Delta\varepsilon^{\mathrm{el}}_{ij} = \Delta\varepsilon^{\mathrm{tot}}_{ij} - \alpha \Delta T \delta_{ij}\]

In the thermomechanical framework, the thermal work terms \(W_t\), \(W_t^r\) and \(W_t^{ir}\) are also computed alongside the mechanical work terms.

umat_name = "ELISO"  # 5 character code for the elastic-isotropic subroutine
nstatev = 1  # Number of internal variables

# Material parameters
rho = 4.4  # Density
c_p = 0.656  # Specific heat capacity
E = 70000.0  # Young's modulus (MPa)
nu = 0.2  # Poisson ratio
alpha = 1.0e-5  # Thermal expansion coefficient

psi_rve = 0.0
theta_rve = 0.0
phi_rve = 0.0
solver_type = 0
corate_type = 2

props = np.array([rho, c_p, E, nu, alpha])

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

The loading path is read in Python and the simulation runs in memory: no result file is written. A thermomechanical run also carries the heat flux, the heat source and the thermal work terms.

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, the temperature evolution, the mechanical work terms and the thermal work terms.

fig = plt.figure()

# Get the data
e11, e22, e33, e12, e13, e23 = res["Strain"]
s11, s22, s33, s12, s13, s23 = res["Stress"]
time, T, Q, r = res["Time"], res["Temp"], res["Q"], res["r"]
Wm, Wm_r, Wm_ir, Wm_d = res["Wm"]
Wt, Wt_r, Wt_ir = res["Wt"]

# Stress vs Strain
ax = fig.add_subplot(2, 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="black", label="direction 1")
plt.legend(loc="best")

# Temperature vs Time
ax = fig.add_subplot(2, 2, 2)
plt.grid(True)
plt.tick_params(axis="both", which="major", labelsize=15)
plt.xlabel("time (s)", size=15)
plt.ylabel(r"Temperature $\theta$ (K)", size=15)
plt.plot(time, T, c="black", label="temperature")
plt.legend(loc="best")

# Mechanical work vs Time
ax = fig.add_subplot(2, 2, 3)
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")

# Thermal work vs Time
ax = fig.add_subplot(2, 2, 4)
plt.grid(True)
plt.tick_params(axis="both", which="major", labelsize=15)
plt.xlabel("time (s)", size=15)
plt.ylabel(r"$W_t$", size=15)
plt.plot(time, Wt, c="black", label=r"$W_t$")
plt.plot(time, Wt_r, c="green", label=r"$W_t^r$")
plt.plot(time, Wt_ir, c="blue", label=r"$W_t^{ir}$")
plt.legend(loc="best")

plt.show()