Transversely Isotropic Elasticity Example

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

In transversely isotropic elastic materials, there is a single axis of symmetry. The material behaves isotropically in the plane perpendicular to this axis (the transverse plane). Eight parameters are required:

1. The axis of symmetry (1, 2, or 3)

2. The longitudinal Young modulus \(E_L\)

3. The transverse Young modulus \(E_T\)

4. The Poisson ratio in the transverse-longitudinal plane \(\nu_{TL}\)

5. The Poisson ratio in the transverse-transverse plane \(\nu_{TT}\)

6. The shear modulus in the longitudinal-transverse plane \(G_{LT}\)

7. The coefficient of thermal expansion in the longitudinal direction \(\alpha_L\)

8. The coefficient of thermal expansion in the transverse direction \(\alpha_T\)

The elastic stiffness tensor for a transversely isotropic material with axis 1 as the symmetry axis is written in the Voigt notation formalism as:

\[\begin{split}\mathbf{L} = \begin{pmatrix} L_{11} & L_{12} & L_{12} & 0 & 0 & 0 \\ L_{12} & L_{22} & L_{23} & 0 & 0 & 0 \\ L_{12} & L_{23} & L_{22} & 0 & 0 & 0 \\ 0 & 0 & 0 & G_{TT} & 0 & 0 \\ 0 & 0 & 0 & 0 & G_{LT} & 0 \\ 0 & 0 & 0 & 0 & 0 & G_{LT} \end{pmatrix}\end{split}\]

where \(G_{TT} = E_T / (2(1+\nu_{TT}))\) is the shear modulus in the transverse plane.

The thermal expansion tensor is:

\[\begin{split}\boldsymbol{\alpha} = \begin{pmatrix} \alpha_L & 0 & 0 \\ 0 & \alpha_T & 0 \\ 0 & 0 & \alpha_T \end{pmatrix}\end{split}\]

umat_name = "ELIST"  # 5 character code for transversely isotropic elastic subroutine
nstatev = 1  # Number of internal variables

# Material parameters
axis = 1  # Symmetry axis
E_L = 4500.0  # Longitudinal Young's modulus (MPa)
E_T = 2300.0  # Transverse Young's modulus (MPa)
nu_TL = 0.05  # Poisson ratio (transverse-longitudinal)
nu_TT = 0.3  # Poisson ratio (transverse-transverse)
G_LT = 2700.0  # Shear modulus (longitudinal-transverse)
alpha_L = 1.0e-5  # Thermal expansion (longitudinal)
alpha_T = 2.5e-5  # Thermal expansion (transverse)

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

props = np.array([axis, E_L, E_T, nu_TL, nu_TT, G_LT, alpha_L, alpha_T])

path_data = "../data"
pathfile = "ELIST_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 (6, N) 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 in the loading direction (direction 1).

e11, e22, e33, e12, e13, e23 = res["Strain"]
s11, s22, s33, s12, s13, s23 = res["Stress"]

fig = plt.figure()

plt.grid(True)
plt.xlabel(r"Strain $\varepsilon_{11}$")
plt.ylabel(r"Stress $\sigma_{11}$ (MPa)")
plt.plot(e11, s11, c="blue", label="Loading direction 1")
plt.legend(loc="best")

plt.show()