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muSpectre/doc/dev-docs/source/ConstitutiveLaws.rst
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.. toctree::
:maxdepth: 2
:caption: Contents:
.. _constitutive_laws:
Constitutive Laws
~~~~~~~~~~~~~~~~~
.. toctree::
:maxdepth: 2
MaterialLinearElasticGeneric
Testing Constitutive Laws
~~~~~~~~~~~~~~~~~~~~~~~~~
When writing new constitutive laws, the ability to evaluate the stress-strain
behaviour and the tangent moduli is convenient, but *µ*\Spectre's material model
makes it cumbersome to isolate and execute the :cpp:func:`evaluate_stress` and
:cpp:func:`evaluate_stress_tangent` methods than any daughter class of
:cpp:class:`MaterialMuSpectre<muSpectre::MaterialMuSpectre>` must implement
(e.g., :cpp:func:`evaluate_stress()
<muSpectre::MaterialLinearElastic1::evaluate_stress>`). As a helper object,
*µ*\Spectre offers the class
:cpp:class:`MaterialEvaluator<muSpectre::MaterialEvaluator>` to facilitate
precisely this:
A :cpp:class:`MaterialEvaluator<muSpectre::MaterialEvaluator>` object can be
constructed with a shared pointer to a
:cpp:class:`MaterialBase<muSpectre::MaterialBase>` and exposes functions to
evaluate just the stress, both the stress and tangent moduli, or a numerical
approximation to the tangent moduli. For materials with internal history
variables, :cpp:class:`MaterialEvaluator<muSpectre::MaterialEvaluator>` also
exposes the
:cpp:func:`MaterialBase::save_history_variables()<muSpectre::MaterialBase::save_history_variables>`
method. As a convenience function, all daughter classes of
:cpp:class:`MaterialMuSpectre<muSpectre::MaterialMuSpectre>` have the static
factory function :cpp:func:`make_evaluator()
<muSpectre::MaterialMuSpectre::make_evaluator>` to create a material and its
evaluator at once. See the :ref:`reference` for the full class description.
Python Usage Example
````````````````````
.. code-block:: python
import numpy as np
from muSpectre import material
from muSpectre import Formulation
# MaterialLinearElastic1 is standard linear elasticity while
# MaterialLinearElastic2 has a per pixel eigenstrain which needs to be set
LinMat1, LinMat2 = (material.MaterialLinearElastic1_2d,
material.MaterialLinearElastic2_2d)
young, poisson = 210e9, .33
# the factory returns a material and it's corresponding evaluator
material1, evaluator1 = LinMat1.make_evaluator(young, poisson)
# the material is empty (i.e., does not have any pixel/voxel), so a pixel
# needs to be added. The coordinates are irrelevant, there just needs to
# be one pixel.
material1.add_pixel([0,0])
# the stress and tangent can be evaluated for finite strain
F = np.array([[1., .01],[0, 1.0]])
P, K = evaluator1.evaluate_stress_tangent(F, Formulation.finite_strain)
# or small strain
eps = .5 * ((F-np.eye(2)) + (F-np.eye(2)).T)
sigma, C = evaluator1.evaluate_stress_tangent(eps, Formulation.small_strain)
# and the tangent can be checked against a numerical approximation
Delta_x = 1e-6
num_C = evaluator1.estimate_tangent(eps, Formulation.small_strain, Delta_x)
# Materials with per-pixel data behave similarly: the factory returns a
# material and it's corresponding evaluator like before
material2, evaluator2 = LinMat2.make_evaluator(young, poisson)
# when adding the pixel, we now need to specify also the per-pixel data:
eigenstrain = np.array([[.01, .002], [.002, 0.]])
material2.add_pixel([0,0], eigenstrain)
C++ Usage Example
`````````````````
.. code-block:: c++
#include"materials/material_linear_elastic2.hh"
#include "materials/material_evaluator.hh"
#include "common/T4_map_proxy.hh"
#include "Eigen/Dense"
using Mat_t = MaterialLinearElastic2<twoD, twoD>;
constexpr Real Young{210e9};
constexpr Real Poisson{.33};
auto mat_eval{Mat_t::make_evaluator(Young, Poisson)};
auto & mat{*std::get<0>(mat_eval)};
auto & evaluator{std::get<1>(mat_eval)};
using T2_t = Eigen::Matrix<Real, twoD, twoD>;
using T4_t = T4Mat<Real, twoD>;
const T2_t F{(T2_t::Random() - (T2_t::Ones() * .5)) * 1e-4 +
T2_t::Identity()};
T2_t eigen_strain{[](auto x) {
return 1e-4 * (x + x.transpose());
}(T2_t::Random() - T2_t::Ones() * .5)};
mat.add_pixel({}, eigen_strain);
T2_t P{};
T4_t K{};
std::tie(P, K) =
evaluator.evaluate_stress_tangent(F, Formulation::finite_strain);
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