Viscosity02

Viscous Damper

This material model does not respond to strain/displacement. To represent materials that respond to both displacement and velocity, see Maxwell and Kelvin.

See also damper elements Damper01 and Damper02.

Reference

1. 10.1016/j.ymssp.2021.108795

Theory

The quadrant damper is implemented.

The damping force is defined as a function of displacement and velocity (or strain and strain rate, depends on what the input is).

$$\sigma =\text{sign}(\dot{\epsilon })\eta (\epsilon ,\dot{\epsilon })|\dot{\epsilon }{|}^{\alpha }.$$

The damping coefficient is a function of strain and strain rate that can be expressed as follows, which shows different response in different quadrants.

$$\begin{array}{rl}\eta (\epsilon ,\dot{\epsilon })& ={\displaystyle \frac{{\eta }_1+{\eta }_2+{\eta }_3+{\eta }_4}{4}}+{\displaystyle \frac{{\eta }_1-{\eta }_2+{\eta }_3-{\eta }_4}{{\pi }^2}}\arctan \left(g_1\epsilon \right)\arctan \left(g_2\dot{\epsilon }\right)\\ & +{\displaystyle \frac{{\eta }_1-{\eta }_2-{\eta }_3+{\eta }_4}{2\pi }}\arctan \left(g_1\epsilon \right)+{\displaystyle \frac{{\eta }_1+{\eta }_2-{\eta }_3-{\eta }_4}{2\pi }}\arctan \left(g_2\dot{\epsilon }\right).\end{array}$$

Syntax

material Viscosity02 (1) (2) (3) [4] [5] [6] [7] [8]
# (1) int, unique tag
# (2) double, alpha
# (3) double, damping coefficient \eta_1
# [4] double, damping coefficient \eta_2, default: (3)
# [5] double, damping coefficient \eta_3, default: (3)
# [6] double, damping coefficient \eta_4, default: (3)
# [7] double, steepness factor g_1, default: 1E3
# [8] double, steepness factor g_2, default: 1E3