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

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

example two