.. DO NOT EDIT. .. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY. .. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE: .. "examples/heterogeneous/homogenization.py" .. LINE NUMBERS ARE GIVEN BELOW. .. only:: html .. note:: :class: sphx-glr-download-link-note :ref:`Go to the end ` to download the full example code. .. rst-class:: sphx-glr-example-title .. _sphx_glr_examples_heterogeneous_homogenization.py: 2 phase composite homogenization example ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ This tutorial studies the evolution of the mechanical properties of a composite material considering spherical and ellipsoidal reinforcements. .. GENERATED FROM PYTHON SOURCE LINES 7-14 .. code-block:: Python import numpy as np from simcoon import simmit as sim import matplotlib.pyplot as plt import os .. GENERATED FROM PYTHON SOURCE LINES 15-21 In this tutorial we will study mechanical properties of a 2-phase composite material considering spherical reinforcement. We will also explore the effect of inclusion aspect ratio on both the Eshelby tensor and the effective properties. First we define the number of state variables (if any) at the macroscopic level and the material properties. .. GENERATED FROM PYTHON SOURCE LINES 21-32 .. code-block:: Python nstatev = 0 # None here nphases = 2 # Number of phases num_file = 0 # Index of the file that contains the subphases int1 = 50 # Number of integration points along the long axis int2 = 50 # Number of integration points along the lat axis n_matrix = 0 # Phase number for the matrix props = np.array([nphases, num_file, int1, int2, n_matrix], dtype="float") .. GENERATED FROM PYTHON SOURCE LINES 33-42 There is a possibility to consider a misorientation between the test frame of reference and the composite material orientation, using Euler angles with the zā€“xā€“z convention. The stiffness tensor will be returned in the test frame of reference. We also specify which micromechanical scheme will be used: - `MIMTN` ā†’ Mori-Tanaka scheme - `MISCN` ā†’ Self-consistent scheme .. GENERATED FROM PYTHON SOURCE LINES 42-49 .. code-block:: Python psi_rve = 0.0 theta_rve = 0.0 phi_rve = 0.0 umat_name = "MIMTN" # Micromechanical scheme (Mori-Tanaka here) .. GENERATED FROM PYTHON SOURCE LINES 50-53 Run the simulation to obtain the effective isotropic elastic properties. Since both phases are isotropic and the reinforcements are spherical, the result is also isotropic. .. GENERATED FROM PYTHON SOURCE LINES 53-61 .. code-block:: Python L_eff = sim.L_eff(umat_name, props, nstatev, psi_rve, theta_rve, phi_rve) p = sim.L_iso_props(L_eff).flatten() np.set_printoptions(precision=3, suppress=True) print("L_eff = ", L_eff) print("Effective isotropic properties (E, nu):", p) .. rst-class:: sphx-glr-script-out .. code-block:: none L_eff = [[3622.978 859.796 859.796 0. 0. -0. ] [ 859.796 3622.978 859.796 -0. 0. 0. ] [ 859.796 859.796 3622.978 0. 0. -0. ] [ 0. -0. 0. 1381.591 0. 0. ] [ 0. 0. 0. 0. 1381.591 0. ] [ -0. 0. -0. 0. 0. 1381.591]] Effective isotropic properties (E, nu): [3293.16 0.192] .. GENERATED FROM PYTHON SOURCE LINES 62-65 Note: Even though both materials share the same Poisson ratio, the stiffness mismatch leads to a different *effective* Poisson ratio. .. GENERATED FROM PYTHON SOURCE LINES 67-73 Effect of aspect ratio on Eshelby tensor components --------------------------------------------------- The Eshelby tensor :math:`\mathbf{S}` relates the constrained strain inside an ellipsoidal inclusion to the eigenstrain. It depends on the inclusion shape and the matrix Poisson ratio. Let's visualize how its components vary with aspect ratio. .. GENERATED FROM PYTHON SOURCE LINES 73-104 .. code-block:: Python nu = 0.3 # Poisson ratio of the matrix aspect_ratios = np.logspace(-1, 1, 50) # From 0.1 to 10 S_11 = [] S_22 = [] S_33 = [] for ar in aspect_ratios: if ar > 1: S = sim.Eshelby_prolate(nu, ar) else: S = sim.Eshelby_oblate(nu, ar) S_11.append(S[0, 0]) S_22.append(S[1, 1]) S_33.append(S[2, 2]) fig, ax = plt.subplots(figsize=(10, 6)) ax.semilogx(aspect_ratios, S_11, "b-", label=r"$S_{1111}$", linewidth=2) ax.semilogx(aspect_ratios, S_22, "r--", label=r"$S_{2222}$", linewidth=2) ax.semilogx(aspect_ratios, S_33, "g-.", label=r"$S_{3333}$", linewidth=2) ax.axvline(x=1, color="k", linestyle=":", alpha=0.5, label="Sphere (ar=1)") ax.set_xlabel("Aspect ratio $a_1/a_3$", fontsize=12) ax.set_ylabel("Eshelby tensor components", fontsize=12) ax.set_title( f"Eshelby tensor diagonal components vs aspect ratio ($\\nu$={nu})", fontsize=14 ) ax.legend(fontsize=11) ax.grid(True, alpha=0.3) plt.tight_layout() plt.show() .. image-sg:: /examples/heterogeneous/images/sphx_glr_homogenization_001.png :alt: Eshelby tensor diagonal components vs aspect ratio ($\nu$=0.3) :srcset: /examples/heterogeneous/images/sphx_glr_homogenization_001.png :class: sphx-glr-single-img .. rst-class:: sphx-glr-timing **Total running time of the script:** (0 minutes 0.172 seconds) .. _sphx_glr_download_examples_heterogeneous_homogenization.py: .. only:: html .. container:: sphx-glr-footer sphx-glr-footer-example .. container:: sphx-glr-download sphx-glr-download-jupyter :download:`Download Jupyter notebook: homogenization.ipynb ` .. container:: sphx-glr-download sphx-glr-download-python :download:`Download Python source code: homogenization.py ` .. container:: sphx-glr-download sphx-glr-download-zip :download:`Download zipped: homogenization.zip ` .. only:: html .. rst-class:: sphx-glr-signature `Gallery generated by Sphinx-Gallery `_