FrictionSpringDamper Material
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

This command is used to construct a FrictionSpringDamper uniaxial material object. The material models the force-deformation response of a self-centering friction spring damper. The model is based on the behavior described by Wang et al. (2019) in series with an elastic stiffness.

.. function:: uniaxialMaterial FrictionSpringDamper $matTag $Ke $K $Kp $preload <-initialStrain $initStrain>

.. list-table::
	:header-rows: 1
	:widths: 15 15 70

	* * Parameter
	  * Type
	  * Description
	* * `$matTag`
	  * integer
	  * Integer tag identifying the material.
	* * `$Ke`
	  * float
	  * Elastic stiffness, as shown in the figure above.
	* * `$K`
	  * float
	  * Loading stiffness, as shown in the figure above.
	* * `$Kp`
	  * float
	  * Unloading stiffness, as shown in the figure above.
	* * `$preload`
	  * float
	  * Preload, as shown in the figure above.
	* * `$initStrain`
	  * float
	  * Optional initial strain. The default value is 0.0.

.. figure:: figures/FrictionSpringDamper.png
	:align: center
	:figclass: align-center
	:width: 70%

	Force-deformation response of the FrictionSpringDamper material.

The effective secondary tangent stiffness during loading is

.. math::

	K_{eq1} = \left(\frac{1}{K_e}+\frac{1}{K}\right)^{-1}

and the effective tangent stiffness during unloading is

.. math::

	K_{eq2} = \left(\frac{1}{K_e}+\frac{1}{K_p}\right)^{-1}

The force associated with the transition during unloading and reloading is

.. math::

	F_2 = F_{preload}\frac{K_p}{K}

where :math:`F_{preload}` is the specified `$preload`.

The optional `-initialStrain` argument shifts the strain supplied to the constitutive model such that

.. math::

	\epsilon = \epsilon_{applied} + \epsilon_{initial}

Thus, a nonzero initial strain can result in a nonzero initial force.

.. admonition:: Example 

	The following example creates a FrictionSpringDamper material with a tag of 1, elastic stiffness of 1000, loading stiffness of 200, unloading stiffness of 100, and preload of 50.

	**Tcl Code**

	.. code-block:: tcl

		uniaxialMaterial FrictionSpringDamper 1 1000.0 200.0 100.0 50.0

	**Python Code**

	.. code-block:: python

		ops.uniaxialMaterial('FrictionSpringDamper', 1, 1000.0, 200.0, 100.0, 50.0)

	An initial strain can optionally be specified.

	**Tcl Code**

	.. code-block:: tcl

		uniaxialMaterial FrictionSpringDamper 1 1000.0 200.0 100.0 50.0 -initialStrain 0.01

	**Python Code**

	.. code-block:: python

		ops.uniaxialMaterial('FrictionSpringDamper', 1, 1000.0, 200.0, 100.0, 50.0, '-initialStrain', 0.01)

References:

Wang, W., Fang, C., Zhao, Y., Sause, R., Hu, S., and Ricles, J. (2019).
"Self-centering friction spring dampers for seismic resilience."
*Earthquake Engineering & Structural Dynamics*, 48(9), 1045-1065. 
https://doi.org/10.1002/eqe.3174

Code Developed by: Mark D. Denavit, University of Tennessee, Knoxville