.. _HystereticPoly:

HystereticPoly Material
^^^^^^^^^^^^^^^^^^^^^^^

This command is used to construct the uniaxial HystereticPoly material producing smooth hysteretic loops and local maxima/minima. It is based on a polynomial formulation of its tangent stiffness.

.. function:: uniaxialMaterial HystereticPoly $matTag $ka $kb $alpha $beta1 $beta2 <$delta>

.. csv-table:: 
   :header: "Argument", "Type", "Description"
   :widths: 10, 10, 40

   $matTag, |integer|, integer tag identifying material.
   $ka, |float|,  Tangent stiffness of the initial "elastic" part of the loop.
   $kb, |float|, Tangent stiffness of the asymptotic part of the loop.
   $alpha, |float|, Parameter governing the amplitude of the loop.
   $beta1, |float|, Parameter governing the shape of the asymptotic region of the loop.
   $beta2, |float|, Parameter governing the shape of the asymptotic region of the loop.
   $delta, |float|, Asymptotic tolerance (optional. Default 1.0e-20).

.. note::

   Determination of constitutive parameters is quite intuitive and is reported below, although, their identification can be performed by the strategy formulated in [SessaEtAl2020]_ implemented in the freeware available `here <http://bit.ly/35F5x7Q>`_.
   
   
The original formulation of HystereticPoly is reported in [VaianaEtAl2019]_. Minor changes have been made in its implementation for OpenSees and make reference to the enhanced formulation reported in [SessaEtAl2022]_ and [Sessa2022]_.

The model may reproduce either force-displacement or stress-strain relationships. It is formulated by means of two asymptotic lines (blue) linked by transition curves (red):

.. figure:: figures/HystereticPoly/HystereticPoly01.gif
	:align: center
	:figclass: align-center

Parameter $alpha governs the transition between such curves so that:

:math:`u_0=\frac{1}{2}\left[\left(\frac{k_a-k_b}{\delta}\right)^{1/\alpha}-1\right]`

:math:`\bar{f}=\frac{k_a-k_b}{2}\left[\frac{\left(1+2u_0\right)^{1-\alpha}-1}{1-\alpha}\right]`

Where :math:`\bar{f}` is the value at which the asymptotic line crosses the vertical axis and :math:`2u_0` is the generalized displacement for which the transition curve reaches the asymptotic line.

In general, $alpha= :math:`\alpha` influences the amplitude of the loop:

.. figure:: figures/HystereticPoly/HystereticPoly02.gif
	:align: center
	:figclass: align-center

while parameters $beta1 and $beta2 modify the shape of the loop:

.. figure:: figures/HystereticPoly/HystereticPoly03.gif
	:align: center
	:figclass: align-center


.. admonition:: Example 

   The following constructs a HystereticPoly material with tag **1**, parameters corresponding to line (d) of the table above and tolerance $delta = :math:`10^{-20}`.

   1. **Tcl Code**

   .. code-block:: tcl

      uniaxialMaterial HystereticPoly 1  100.0 10.0 20.0 -10.0 10.0 1.0e-20 

   2. **Python Code**

   .. code-block:: python

      uniaxialMaterial('HystereticPoly', 1, 100.0, 10.0, 20.0, -10.0, 10.0, 1.0e-20)


Code Developed by: `Salvatore Sessa <https://www.docenti.unina.it/salvatore.sessa2/>`_, University of Naples Federico II, Italy 


.. [VaianaEtAl2019] Vaiana, N., Sessa, S., Marmo, F. and Rosati, L. (2019). "An accurate and computationally efficient uniaxial phenomenological model for steel and fiber reinforced elastomeric bearings." Composite Structures, 211: 196-212. `DOI: https://doi.org/10.1016/j.compstruct.2018.12.017 <https://doi.org/10.1016/j.compstruct.2018.12.017>`_

.. [SessaEtAl2020] Sessa, S., Vaiana, N., Paradiso, M. and Rosati, L. (2020). "An inverse identification strategy for the mechanical parameters of a phenomenological hysteretic constitutive model.", Mechanical Systems and Signal Processing, 139: 106622. `DOI: https://doi.org/10.1016/j.ymssp.2020.106622 <https://doi.org/10.1016/j.ymssp.2020.106622>`_

.. [SessaEtAl2022] Sessa, S., Vaiana, N., Paradiso, M. and Rosati, L. (2022). "Thermodynamic Compatibility of the HystereticPoly Uniaxial Material Implemented in OpenSees.", Advanced Structured Materials, 175: 565 - 580. `DOI: https://doi.org/10.1007/978-3-031-04548-6_27 <https://doi.org/10.1007/978-3-031-04548-6_27>`_

.. [Sessa2022] Sessa, S. (2022). "Thermodynamic compatibility conditions of a new class of hysteretic materials.", Continuum Mechanics and Thermodynamics, 34(1): 61 - 79. `DOI: https://doi.org/10.1007/s00161-021-01044-w <https://doi.org/10.1007/s00161-021-01044-w>`_