Numerical Modeling: An Efficient Tool for Analyzing the Behavior of Constructions - Multiscale Geomechanics: From Soil to Engineering Projects - page 261

266 Multiscale Geomechanics

Yield function:

f(σ p ,σ q ,α)=σ q −µσ p Fρ(α d )≤0

F=(1−bln σ p

σ c )

σ c = σ co exp(βα v )

α d
a+α d

ρ(α d )=r ela +

Flow rule:

α v = λ p Ψ v

Ψ v = ξ(ρ)(µ − σ q with

σ p )

α d = λ p Ψ d

∂f
∂σ d

with Ψ d =

ρ(α d )= λ p l ρ l ρ = (1−ρ) 2

a

with

Variables

Generic Isotrope Multimechanism Interface

stresses: σ p and σ q p p k and q k σ nn and τ σ nn and τ and q

ε v and k ε nn and γ [u n ] and [u t ] strains: p and q ε v and

hardening:
α v

ε v ε v p ε nn [u n ]

k [u t ]

p γ p

α q

ρ(α q ) r r k r k r

Parameters

sin φ pp tan φ pp tan φ pp µ M

M µ sin ψ tan ψ tan φ

σ c0 p c p c σ c0 σ c0

a = a 1 +(a 2 −a 1 )ξ(ρ)
ξ(δ p )=0 δ p ≤ hys if pseudo-elastic domain

0 <ξ(δ p ) < 1 hys ≤ δ p ≤ mob

if hysteretic domain

ξ(δ p )=1 δ p ≥ mob

if plastic domain

Elasticity:

σ p
σ ref

n e

E max = E ref

ν = cst

Table 9.1. Comparison of constitutional relations for different models of the ECP family