> For the complete documentation index, see [llms.txt](https://tlcfem.gitbook.io/suanpan-manual/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://tlcfem.gitbook.io/suanpan-manual/material/material1d/viscosity/viscosity02.md).

# Viscosity02

Viscous Damper

This material model does not respond to strain/displacement. To represent materials that respond to both displacement and velocity, see [`Maxwell`](/suanpan-manual/material/material1d/viscosity/maxwell.md) and [`Kelvin`](/suanpan-manual/material/material1d/viscosity/kelvin.md).

See also damper elements [`Damper01`](/suanpan-manual/element/special/damper01.md) and [`Damper02`](/suanpan-manual/element/special/damper02.md).

## References

1. [10.1016/j.ymssp.2021.108795](https://doi.org/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).

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

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{align\*} \eta\left(\varepsilon,\dot\varepsilon\right)
&=\dfrac{\eta\_1+\eta\_2+\eta\_3+\eta\_4}{4}+\dfrac{\eta\_1-\eta\_2+\eta\_3-\eta\_4}{\pi^2}\arctan\left(g\_1\varepsilon\right)
\arctan\left(g\_2\dot\varepsilon\right)\\\[4mm]&+\dfrac{\eta\_1-\eta\_2-\eta\_3+\eta\_4}{2\pi}\arctan\left(
g\_1\varepsilon\right)+\dfrac{\eta\_1+\eta\_2-\eta\_3-\eta\_4}{2\pi}\arctan\left(g\_2\dot\varepsilon\right). \end{align\*}
$$

## 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
```


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