DAMAGE MODELS FOR CYCLIC AND DYNAMIC LOADING Thermodynamics and induced anisotropy Part IV- Strain rate effect - Regularization aspects
Rodrigue Desmorat LMT Cachan ENS Cachan, 61 av. du Pt Wilson 94235 Cachan Cedex
[email protected]
CISM course, Udine, may 2009
Numerical Modeling of Concrete Cracking
Outline I- Thermodynamics of damage (isotropic vs anisotropic damage) 1. 2. 3. 4. 5. 6.
Standard framework Non standard Mazars damage model Micro-defects closure effect : isotropic damage case Second order anisotropic damage Micro-defects closure effect: anisotropic damage case Positivity of intrinsic dissipation (second order damage case)
II- Cyclic and hysteretic isotropic damage models 1. 2. 3.
Laborderie model Damage model with internal sliding and friction Dissymmetry due to micro-cracks closure
III- An anisotropic damage model for concrete 1. 2. 3. 4.
Anisotropic damage model for monotonic applications Numerical scheme Effective damage – Extension to cyclic loading Multifiber modeling for (PSeudo-)Dynamics testing
IV- Strain rate effect - Regularization aspects 1. 2. 3. 4.
Nonlocal anisotropic damage model Visco-damage and strain rate effect Nonlocal versus visco-damage Internal time for nonlocal theories
III- STRAIN RATE EFFECT REGULARIZATION ASPECTS
IV- Strain rate effect - Regularization aspects 1. 2. 3. 4.
Nonlocal anisotropic damage model Visco-damage and strain rate effect Nonlocal versus visco-damage Internal time for nonlocal theories
1- NON LOCAL ANISOTROPIC DAMAGE MODEL Effective stress and elasticity law !
σ ˜ = (1 − D)−1/2 σ D (1 − D)−1/2
σ ˜ =E :"
"D
# $ 1 "tr σ#+ + + "tr σ#− 1 3 1 − ηDH
Nonlocal Mazars damage criterion (Pijaudier-Cabot Bazant 1987) f = !ˆ − κ(tr D) nl
1 !ˆnl (x) = Vr
!
Ω
α∞ (x − s) !ˆ(s) ds
! !ˆ = !!"+ : !!"+ Vr (x) =
!
Ω
α∞ (x − s)ds
Damage evolution law ˙ 2 ˙ = λ!"" D +
!
"κ # trD 0 κ(tr D) = a · tan + arctan aA a
Total of 5 material parameters: E, ν + damage threshold κ0 + damage parameters A, a
$
Nonlocal anisotropic damage model for concrete Effective stress and elasticity law !
σ ˜ = (1 − D)−1/2 σ D (1 − D)−1/2
σ ˜ =E :"
"D
# $ 1 "tr σ#+ + + "tr σ#− 1 3 1 − ηDH
Nonlocal Mazars damage criterion (Aifantis 1987, Peerlings 1996) f = !ˆ − κ(tr D) nl
!ˆnl − "2 ∇2 !ˆnl = !ˆ
! !ˆ = !!"+ : !!"+
Damage evolution law ˙ 2 ˙ = λ!"" D +
!
"κ # trD 0 κ(tr D) = a · tan + arctan aA a
Total of 5 material parameters: E, ν + damage threshold κ0 + damage parameters A, a
$
FE: Nooru-Mohamed test Nooru-Mohamed (1992) Desmorat, Gatuingt, Ragueneau (2004-2006)
classical mesh dependency
Local FE
D22 field
Non local FE
D22 field
F
FE: Reinforced concrete beam MECA Benchmark (Ghavamian & Delaplace 2003)
Non local FE
lc=150 mm
lc=250 mm
2- VISCO-DAMAGE AND STRAIN RATE EFFECT Strain rate effect in tension
(a) strain rate effect in tension
t σR
σ
!˙ Strength increase in tension (experiments)
! Strain-rate effect (model)
(a) strain effect in tension Strain raterate effect in compression
(b) strain rate effect in compression
Visco-plasticity : viscous stress
σ σv = KN p˙
R+X
σy
1/N
viscous stress function of accumulated plastic strain rate (Norton-Perzyna power law) p˙ =
σy
E εp
E
!
2 p p !˙ : !˙ 3
ε
εe
In plasticity
In visco-plasticity
f = |σ − X| − R − σy = 0 → f = |σ − X| − R − σy = σv f = (σ − X)eq − R − σy = σv
Visco-damage for isotropic damage (Ladevèze 1991, Dubé 1994) Visco-damage
Strain rate independent model
f = g(Y ) − D = 0
→
f = g(Y ) − D = Dv
D = g(Y ) − Dv < Dquasi-static = g(Y )
Strength increase when increasing strain rate
Strain energy release rate density ∂ψ ∂ψ ! 1 Y = −ρ =ρ = $e : E : $e ∂D ∂D 2
Visco-damage for isotropic damage (Ladevèze 1991, Dubé 1994) Visco-damage
Strain rate independent model
f = g(Y ) − D = 0
→
f = g(Y ) − D = Dv
D = g(Y ) − Dv < Dquasi-static = g(Y )
Strength increase when increasing strain rate
Power law
Dv = B D˙ 1/b
Law with damage rate saturation ! " 1 D˙ ∞ − D˙ Dv = − ln b D˙ ∞ Allix-Deü delay-damage law
Visco-damage for isotropic damage (Ladevèze 1991, Dubé 1994) Visco-damage
Strain rate independent model
f = g(ˆ !) − D = 0
→
f = g(ˆ !) − D = Dv
D = g(ˆ !) − Dv < Dquasi-static = g(ˆ !)
Strength increase when increasing strain rate
Mazars strain (concrete models) ! !ˆ = !!"+ : !!"+
Visco-damage for isotropic damage (Ladevèze 1991, Dubé 1994) Visco-damage
Strain rate independent model
f = g(ˆ !) − D = 0
→
f = g(ˆ !) − D = Dv
D = g(ˆ !) − Dv < Dquasi-static = g(ˆ !)
Strength increase when increasing strain rate
Power law
Dv = B D˙ 1/b
Law with damage rate saturation ! " 1 D˙ ∞ − D˙ Dv = − ln b D˙ ∞ Allix-Deü delay-damage law
Delay damage law extended to induced anisotropy Damage criterion in classical non viscous case
fini = !ˆnl − κ(tr D)
rewritten
f = g(ˆ !nl ) − tr D
Damage criterion in viscous case (delay-damage)
˙ 1 D˙ ∞ − tr D Dv = − ln b D˙ ∞
˙ f = g(ˆ !nl ) − tr D = Dv (tr D) !
˙ = D˙ ∞ 1 − e(−b(g(ˆ! tr D
Damage governed by the extensions
˙ = λ˙ !""2 D +
!
nl
)−tr D))
"
"
t σR
σ
!˙ Strength increase in tension (experiments)
! Strain-rate effect (model)
Allix-Deü delay damage law extended to induced anisotropy
!
˙ = D˙ ∞ 1 − e(−b(g(ˆ! tr D
nl
)−tr D))
"
Implicit numerical scheme... explicited
Replace
by
Delay damage law extended to induced anisotropy Damage criterion in classical non viscous case
fini = !ˆnl − κ(tr D)
rewritten
f = g(ˆ !nl ) − tr D
Damage criterion in viscous case (delay-damage) !
˙ f = g(ˆ !nl ) − tr D = Dv (H(tr !) tr D) Material strain rate effect in tension only
!
˙ = D˙ ∞ 1 − e(−b(g(ˆ! H(tr !) tr D
Damage governed by the extensions
˙ = λ˙ !""2 D +
˙ 1 D˙ ∞ − H(tr !) tr D Dv = − ln b D˙ ∞
nl
)−tr D))
"
"
Non local anisotropic model with delay-damage Effective stress and elasticity law σ ˜ =E :"
!
σ ˜ = (1 − D)−1/2 σ D (1 − D)−1/2
"D
# $ 1 "tr σ#+ + + "tr σ#− 1 3 1 − ηDH
Nonlocal Mazars damage criterion (Pijaudier-Cabot Bazant 1987) 1 !ˆ = Vr nl
f = !ˆnl − κ(tr D)
!
Ω
α∞ (x − s) !ˆ(s) ds
Delay-damage law ˙ = D˙ ∞ H(tr !) tr D
!
nl 1 − e(−b(g(ˆ! )−tr D))
Damage evolution law ˙ 2 ˙ = λ!"" D +
"
κ(tr D) = a · tan
g = κ−1 !
trD + arctan aA
Total of 5 (+1) +2 material parameters: E, ν + damage threshold κ0 + damage parameters A, a D˙ ∞ , b delay damage parameters
" κ #$ 0
a
Non local model with delay-damage (and effective damage) Effective stress and elasticity law σ ˜ =E :"
!
σ ˜ = (1 − D)−1/2 σ D (1 − D)−1/2
"D
# $ 1 "tr σ#+ + + "tr σ#− 1 3 1 − ηDH
Nonlocal Mazars damage criterion (Pijaudier-Cabot Bazant 1987) f = !ˆnl − κ(dε )
D : !!"+ dε = max !I
1 !ˆ = Vr nl
!
Ω
α∞ (x − s) !ˆ(s) ds
Delay-damage law ! " −b(g(ˆ "nl )−dε )) ( ˙ ˙ H(tr !) dε = D∞ 1 − e
Damage evolution law ˙ + ˙ = λ!"" D
κ(tr D) = a · tan
g = κ−1 !
trD + arctan aA
Total of 5 (+1) +2 material parameters: E, ν + damage threshold κ0 + damage parameters A, a D˙ ∞ , b delay damage parameters
" κ #$ 0
a
Strain rate effect in compression due to inertial effects
(PhD Chambart)
Figure 2: Inertial effects in compression
Impact on concrete bar
Local model - Delay-damage
Perzyna-type viscous damage law (saturation of damage rate, Allix & Deü 1997, Gatuingt & Desmorat 2006)
9(2:;(2:2>:*1/4*+),-2
