3.1.5.30. Jankowski Impact Material

This command is used to construct the uniaxial Jankowski Impact Material

uniaxialMaterial JankowskiImpact  $matTag $Kh $xi $Meff $gap <$n>

Argument	Type	Description
$matTag	integer	integer tag identifying material.
$Kh	float	nonlinear Hertz contact stiffness.
$xi	float	impact damping ratio.
$Meff	float	effective mass.
$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.

This material model follows the constitutive law

$$\begin{array}{r}{f}_c(t)=\{\begin{array}{}{k}_h(\delta (t)-g{)}^n+c_J(t)\dot{\delta }(t)& \dot{\delta }(t)>0\\ k_h(\delta (t)-g{)}^n& \dot{\delta (t)}\le 0\end{array}\end{array}$$

where t is time, $f_c(t)$ is the contact force, $k_h$ is the nonlinear Hertz contact stiffness ($Kh), $\delta (t)$ is the indentation, g is the initial gap ($gap), n is the indentation exponent ($n), and $\dot{\delta }(t)$ is the indentation velocity. Damping is only applied during the approach phase, when $\delta (t)>0$. The damping coefficient $c_J‘$ is computed as

$$c_h=2{\xi }_j\sqrt{m_{\text{eff}}{k}_h(\delta (t)-g{)}^{n-1}}$$

where $m_{\text{eff}}$ is the effective mass of the system ($Meff), computed using the masses of the coliding bodies $m_1$ and $m_2$:

$$m_{\text{eff}}=\frac{m_1{m}_2}{m_1+m_2}$$

The damping ratio ${\xi }_j$ ($xi) is usually related to the coefficient of restitution, represented by e. The recommended form of ${\xi }_j$ is

$$\xi =\frac{9\sqrt{5}}{2}(\frac{1-e^2}{e(e(9\pi -16)+16)})$$

Response of the JankowskiImpact material during impact:

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

[Jankowski2005]

Non-linear viscoelastic modelling of earthquake-induced structural pounding. Earthquake Engineering and Structural Dynamics 2005; 34(6): 595–611. DOI: 10.1002/eqe.434.

[Jankowski2006]

Analytical expression between the impact damping ratio and the coefficient of restitution in the non-linear viscoelastic model of structural pounding. Earthquake Engineering and Structural Dynamics 2006; 35(4): 517–524. DOI: 10.1002/eqe.537.

[Jankowski2007]

Jankowski R. Theoretical and experimental assessment of parameters for the non-linear viscoelastic model of structural pounding. Journal of Theoretical and Applied Mechanics (Poland) 2007.

[Hughes2020]

Hughes PJ, Mosqueda G. Evaluation of uniaxial contact models for moat wall pounding simulations. Earthquake Engineering and Structural Dynamics 2020(March): 12–14. DOI: 10.1002/eqe.3285.