3.1.5.28. 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
\[\begin{split}f_c(t) = \left\{ \begin{array}{ }k_h (\delta(t)-g)^n + c_J(t) \dot{\delta}(t) & \quad \dot{\delta}(t) > 0 \\ k_h (\delta(t)-g)^n & \quad {\dot{\delta(t)} \leq 0} \end{array}\right.\end{split}\]
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
\[c_h = 2 \xi_j \sqrt{ m_{\textrm{eff}} k_h (\delta(t) -g)^{n-1}}\]
where \(m_{\textrm{eff}}\) is the effective mass of the system ($Meff), computed using the masses of the coliding bodies \(m_1\) and \(m_2\):
\[m_{\textrm{eff}} = \frac{m_1 m_2}{m_1 + m_2}\]
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.