Bouc-Wen-Baber-Noori Model
The BWBN model is an extension of the BoucWen model with stiffness degradation, strength degradation and pinching effect.
Syntax
Copy material BWBN (1) [2...18]
# (1) int, unique material tag
# [2] double, elastic modulus, default: 2E5
# [3] double, yield stress, default: 4E2
# [4] double, hardening ratio, default: 1E-2
# [5] double, \beta (>0), default: 0.5
# [6] double, exponent n (>0, normally >=1), default: 1.0
# [7] double, initial \nu (>0), default: 1.0
# [8] double, slope of \nu (>0), default: 0.0
# [9] double, initial \eta (>0), default: 1.0
# [10] double, slope of \eta (>0), default: 0.0
# [11] double, initial \phi (>0), default: 1.0
# [12] double, slope of \phi (>0), default: 0.0
# [13] double, \zeta (1>\zeta>0), default: 0.0
# [14] double, slope of A (>0), default: 0.0
# [15] double, p (>0), default: 0.0
# [16] double, q (>0), default: 0.0
# [17] double, \lambda (>0), default: 1.0
# [18] double, density, default: 0.0 History Variable Layout
Theory
The evolution of internal displacement z ( t ) z(t) z ( t )
η Δ z = h Δ u u y ( A − ν ( γ + sign ( z ⋅ Δ u ) β ) ∣ z ∣ n ) . \eta\Delta{}z=h\dfrac{\Delta{}u}{u_y}\left(A-\nu\left(\gamma+\text{sign}\left(z\cdot\Delta{}u\right)\beta\right) \Big|z\Big|^n\right). η Δ z = h u y Δ u ( A − ν ( γ + sign ( z ⋅ Δ u ) β ) z n ) . Then,
F = a F y u u y + ( 1 − a ) F y z . F=aF_y\dfrac{u}{u_y}+\left(1-a\right)F_yz. F = a F y u y u + ( 1 − a ) F y z . For state determination, z z z e e e normalized quantity.
e = ( 1 − a ) ∫ z d u . e=\left(1-a\right)\int{}z~\mathrm{d}u. e = ( 1 − a ) ∫ z d u . The trapezoidal rule is used so that
Δ e = ( 1 − a ) 2 z + Δ z 2 Δ u . \Delta{}e=\left(1-a\right)\dfrac{2z+\Delta{}z}{2}\Delta{}u. Δ e = ( 1 − a ) 2 2 z + Δ z Δ u . The evolutions are
ν = ν 0 + δ ν e , η = η 0 + δ η e , A = 1 − δ A e , h = 1 − ζ 1 exp ( − ( z ⋅ sign ( Δ u ) − q z u ζ 2 ) 2 ) , ζ 1 = ζ ( 1 − exp ( − p e ) ) , ζ 2 = ( ϕ 0 + δ ϕ e ) ( λ + ζ 1 ) , z u = 1 ν n = ν − 1 / n . \begin{align*} \nu&=\nu_0+\delta_\nu{}e,\\[3mm] \eta&=\eta_0+\delta_\eta{}e,\\[3mm] A&=1-\delta_Ae,\\[3mm] h&=1-\zeta_1\exp\left(-\left(\dfrac{z\cdot\text{sign}\left(\Delta{}u\right)-qz_u}{\zeta_2}\right)^2\right),\\[3mm] \zeta_1&=\zeta\left(1-\exp\left(-pe\right)\right),\\[3mm] \zeta_2&=\left(\phi_0+\delta_\phi{}e\right)\left(\lambda+\zeta_1\right),\\[3mm] z_u&=\sqrt[n]{\dfrac{1}{\nu}}=\nu^{-1/n}. \end{align*} ν η A h ζ 1 ζ 2 z u = ν 0 + δ ν e , = η 0 + δ η e , = 1 − δ A e , = 1 − ζ 1 exp ( − ( ζ 2 z ⋅ sign ( Δ u ) − q z u ) 2 ) , = ζ ( 1 − exp ( − p e ) ) , = ( ϕ 0 + δ ϕ e ) ( λ + ζ 1 ) , = n ν 1 = ν − 1/ n . Parameters
Strength degradation is controlled by u u u δ ν = 0 \delta_\nu=0 δ ν = 0
Stiffness degradation is controlled by η \eta η δ η = 0 \delta_\eta=0 δ η = 0
Pinching is controlled by h h h ζ = 0 \zeta=0 ζ = 0 p = 0 p=0 p = 0
Examples
Vanilla Model
The default behavior is similar to a bilinear hardening material.
Copy material BWBN 1
materialtest1d 1 4E-4 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 80 Strength degradation
A positive δ ν \delta_\nu δ ν
Copy material BWBN 1 2E5 4E2 0 .5 1. 1. 1E0 1. 0. 1. 0. 0. 0. 0. 0. 1. 0.
materialtest1d 1 4E-4 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 80 A positive δ A \delta_A δ A
Copy material BWBN 1 2E5 4E2 0 .5 1. 1. 0. 1. 0. 1. 0. 0. 1E0 0. 0. 1. 0.
materialtest1d 1 4E-4 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 80 Stiffness degradation
A positive δ η \delta_\eta δ η
Copy material BWBN 1 2E5 4E2 0 .5 1 1 0 1 1E1 1 0 1 0 1 1 0 0
materialtest1d 1 4E-4 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 80 Pinching Effect
The pinching effect is governed by ϕ 0 \phi_0 ϕ 0 δ ϕ \delta_\phi δ ϕ ζ \zeta ζ p p p q q q λ \lambda λ
Copy material BWBN 1 2E5 4E2 0 .5 1. 1. 0. 1. 0. 1. 1E1 1. 0. 1E1 1E0 1. 0.
materialtest1d 1 1E-3 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 80 
