> 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/hysteresis/boucwen.md).

# BoucWen

Bouc-Wen Model

The `BoucWen` model is a phenomenological model. Compared to the original formulation, the following modifications are applied.

1. $$A=1$$.
2. $$\gamma+\beta=1$$.

## Theory

The evolution of internal displacement $$z(t)$$ is governed by the differential equation,

$$
\Delta{}z=\dfrac{\Delta{}u}{u\_y}\left(1-\left(\gamma+\text{sign}\left(z\cdot\Delta{}u\right)\beta\right)
\Big|z\Big|^n\right).
$$

Then,

$$
F=aF\_y\dfrac{u}{u\_y}+\left(1-a\right)F\_yz.
$$

For state determination, $$z$$ is solved by using the Newton method.

## Syntax

```
material BoucWen (1) (2) (3) (4) (5) (6)
# (1) int, unique material tag
# (2) double, elastic modulus
# (3) double, yield stress
# (4) double, hardening ratio
# (5) double, beta, a positive parameter
# (6) double, n, a positive exponent
```

## Caveat

Since it is a phenomenological model, the non-observable internal "displacement" $$z$$ has no physical meaning. For small loops, it violates plasticity postulates.

It is recommended to use a value greater than unity for $$n$$.

## Example

```
material BoucWen 1 2E5 400 .01 1E-1 2
materialTest1D 1 1E-3 10 20 30 40 50 60 30
```

![example one](https://1124082496-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2FEquxr3H9WpPg4iPvT2mB%2Fuploads%2Fgit-blob-b694179830d6b0d1758ba4ecf7979cfaff230c88%2FBoucWen.EX1.svg?alt=media)

```
material BoucWen 1 2E5 400 .01 2. .6
materialTest1D 1 1E-3 10 20 30 40 50 60 30
```

![example two](https://1124082496-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2FEquxr3H9WpPg4iPvT2mB%2Fuploads%2Fgit-blob-59fdf4591fb376a60b8971ff8c240cbad48580fe%2FBoucWen.EX2.svg?alt=media)


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