.. _HertzDamp:

Hertz Damp Material
^^^^^^^^^^^^^^^^^^

This command is used to construct the uniaxial Hertz Damp Material 

.. function:: uniaxialMaterial Hertzdamp $matTag $Kh $xiNorm $gap <$n>

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

   $matTag, |integer|, integer tag identifying material.
   $Kh, |float|,  nonlinear Hertz contact stiffness.
   $xiNorm, |float|, normalized impact damping ratio.
   $gap, |float|, initial gap.
   $n, |float|, indentation exponent (optional with default value of: 1.5).

.. note::

   This material is implemented as a compression-only gap material, so $gap should be input as a negative value.
   
.. Description::
This material model follows the constitutive law

  .. math:: f_c (t) = k_h (\delta(t) -g)^n + c_h \dot{\delta}(t)

where t is time, :math:`f_c (t)`  is the contact force, :math:`k_h` is the nonlinear Hertz contact stiffness ($Kh), :math:`\delta(t)` is the indentation, g is the initial gap ($gap), n is the indentation exponent ($n), and :math:`\dot{\delta}(t)` is the indentation velocity. The damping coefficient :math:`c_h` is computed as

   .. math:: c_h = \overline{\xi} \frac{k_h}{\dot{\delta}_0} (\delta(t) - g)^n

where :math:`\dot{\delta}_0` is the pre-impact indentation velocity. The normalized impact damping ratio :math:`\overline{\xi}` ($xiNorm) is usually related to the coefficient of restitution, represented by e. The recommended form of :math:`\overline{\xi}` is
   
   .. math:: \overline{\xi} = \frac{8}{5} \frac{1-e}{e}

Response of the Hertzdamp material during impact:

   .. figure:: figures/hertzdamp_response.png
      :align: center
      :figclass: align-center

Note that the flat displacement from 0 to roughly minus 0.01 inch displacement is caused by the gap parameter.

Code Developed by: Patrick J. Hughes, UC San Diego


.. [Lankarani1990] Lankarani HM, Nikravesh PE. A Contact Force Model with Hysteresis Damping for Impact Analysis of Multibody Systems. Journal of Mechanical Design 1990; 112(3): 369–376. DOI: 10.1115/1.2912617.

.. [Muthukumar2006] A Hertz contact model with non-linear damping for pounding simulation. Earthquake Engineering and Structural Dynamics 2006; 35(7): 811–828. DOI: 10.1002/eqe.557.

.. [YeK2009] Ye K, Li L, Zhu H. A note on the Hertz contact model with nonlinear damping for pounding simulation. Earthquake Engineering and Structural Dynamics 2009; 38: 1135–1142. DOI: 10.1002/eqe.

.. [Hughes2020]  Evaluation of uniaxial contact models for moat wall pounding simulations. Earthquake Engineering and Structural Dynamics 2020(March): 12–14. DOI: 10.1002/eqe.3285.