.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "examples/ex03_plasticity.py"
.. LINE NUMBERS ARE GIVEN BELOW.

.. only:: html

    .. note::
        :class: sphx-glr-download-link-note

        :ref:`Go to the end <sphx_glr_download_examples_ex03_plasticity.py>`
        to download the full example code.

.. rst-class:: sphx-glr-example-title

.. _sphx_glr_examples_ex03_plasticity.py:


Plasticity with Isotropic Hardening
-----------------------------------

.. topic:: Small-strain Plasticity

   * linear-elastic plastic material formulation with isotropic hardening

   * define a body force vector

   * extract state variables

The normal plastic strain due to a body force applied on a solid with a linear-elastic
plastic material formulation with isotropic hardening with young's modulus
:math:`E=210000`, poisson ratio :math:`\nu=0.3`, isotropic hardening modulus 
:math:`K=1000`, yield stress :math:`\sigma_y=355`, length :math:`L=3` and cross section
area :math:`A=1` is to be evaluated.

.. image:: ../../examples/ex03_plasticity_sketch.png

First, let's create a meshed cube out of hexahedron cells with ``n=(16, 6, 6)`` points
per axis. A three-dimensional vector-valued displacement field is initiated on the
numeric region.

.. GENERATED FROM PYTHON SOURCE LINES 25-34

.. code-block:: Python

    import numpy as np

    import felupe as fem

    mesh = fem.Cube(b=(3, 1, 1), n=(16, 6, 6))
    region = fem.RegionHexahedron(mesh)
    displacement = fem.Field(region, dim=3)
    field = fem.FieldContainer([displacement])








.. GENERATED FROM PYTHON SOURCE LINES 35-36

A fixed boundary condition is applied at :math:`x=0`.

.. GENERATED FROM PYTHON SOURCE LINES 36-38

.. code-block:: Python

    boundaries = {"fixed": fem.dof.Boundary(displacement, fx=0)}








.. GENERATED FROM PYTHON SOURCE LINES 39-41

The material behaviour is defined through a built-in isotropic linear-elastic plastic
material formulation with isotropic hardening.

.. GENERATED FROM PYTHON SOURCE LINES 41-44

.. code-block:: Python

    umat = fem.LinearElasticPlasticIsotropicHardening(E=2.1e5, nu=0.3, sy=355, K=1e3)
    solid = fem.SolidBody(umat, field)








.. GENERATED FROM PYTHON SOURCE LINES 45-50

The body force is created on the field of the solid body.

.. math::

   \delta W_{ext} = \int_v \delta \boldsymbol{u} \cdot \boldsymbol{b} ~ dv

.. GENERATED FROM PYTHON SOURCE LINES 50-53

.. code-block:: Python

    bodyforce = fem.SolidBodyGravity(field)
    b = fem.math.linsteps([0, 200], num=10, axis=0, axes=3)








.. GENERATED FROM PYTHON SOURCE LINES 54-57

Inside a Newton-Rhapson procedure, the vectors and matrices are assembled for the
given *items* and the boundary conditions are incorporated into the equilibrium
equations.

.. GENERATED FROM PYTHON SOURCE LINES 57-61

.. code-block:: Python

    step = fem.Step(items=[solid, bodyforce], ramp={bodyforce: b}, boundaries=boundaries)
    job = fem.Job(steps=[step]).evaluate()









.. GENERATED FROM PYTHON SOURCE LINES 62-67

A view on the field-container shows the deformed mesh and the normal plastic strain in
direction :math:`x` due to the applied body force. The vector of all state variables,
stored as a result in the solid body object, are splitted into separate variables. The
plastic strain is stored as the second state variable. The mean-per-cell value of the
plastic strain in direction :math:`x` is exported to the view.

.. GENERATED FROM PYTHON SOURCE LINES 67-76

.. code-block:: Python

    def plot(solid, field):
        "Visualize the final Normal Plastic Strain in direction X."

        plastic_strain = np.split(solid.results.statevars, umat.statevars_offsets)[1]
        view = fem.View(field, cell_data={"Plastic Strain": plastic_strain.mean(-2).T})
        return view.plot("Plastic Strain", component=0, show_undeformed=False)


    plot(solid, field).show()







.. tab-set::



   .. tab-item:: Static Scene



            
     .. image-sg:: /examples/images/sphx_glr_ex03_plasticity_001.png
        :alt: ex03 plasticity
        :srcset: /examples/images/sphx_glr_ex03_plasticity_001.png
        :class: sphx-glr-single-img
     


   .. tab-item:: Interactive Scene



       .. offlineviewer:: /home/docs/checkouts/readthedocs.org/user_builds/felupe/checkouts/latest/docs/examples/images/sphx_glr_ex03_plasticity_001.vtksz







.. rst-class:: sphx-glr-timing

   **Total running time of the script:** (0 minutes 2.611 seconds)


.. _sphx_glr_download_examples_ex03_plasticity.py:

.. only:: html

  .. container:: sphx-glr-footer sphx-glr-footer-example

    .. container:: sphx-glr-download sphx-glr-download-jupyter

      :download:`Download Jupyter notebook: ex03_plasticity.ipynb <ex03_plasticity.ipynb>`

    .. container:: sphx-glr-download sphx-glr-download-python

      :download:`Download Python source code: ex03_plasticity.py <ex03_plasticity.py>`


.. only:: html

 .. rst-class:: sphx-glr-signature

    `Gallery generated by Sphinx-Gallery <https://sphinx-gallery.github.io>`_