Numerical implementation of damage and fracture models for ...
Experimental and Numerical Characterization of Damage and … · Experimental and Numerical...
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Experimental and Numerical Characterization ofDamage and Application to Incremental Forming
PhD thesis presentation
Carlos Felipe Guzman
Department ArGEnCoUniversity of Liege, Belgium
February 1st, 2016
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Simple geometries
Cooking pots
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Simple geometries
Manufactured by Deep Drawing
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More complex geometries
Planes and car prototypes
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More complex geometries
Implants
Manufactured by ???
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More complex geometries
Implants
Manufactured by ???
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Single point incremental formingSPIF
Hirt et al. [2015] Schafer and Dieter Schraft [2005]
A sheet metal is deformed by a small tool.
The tool could be guided by a CNC (milling machine, robot).
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Single point incremental formingSPIF
Video
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Single point incremental formingSPIF
Advantages
Dieless, with high sheetformability.
Easy shape generation.
For rapid prototypes, smallbatch productions, etc.
Challenges
Poor geometrical accuracy.
Process slowness.
Characterization of servicelife.
The increased formability.
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Single point incremental formingSPIF
Advantages
Dieless, with high sheetformability.
Easy shape generation.
For rapid prototypes, smallbatch productions, etc.
Challenges
Poor geometrical accuracy.
Process slowness.
Characterization of servicelife.
The increased formability.
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The high formability of SPIF
crack
70◦
crack
sine law:
tf = t0 sinα⇒ tf ≈ 0.35
ε� 1.0
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The high formability of SPIF
crack
Detail:
tf ≈0.25mm
t0 =1.0mm
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The high formability of SPIFWhy formability is so high?
Forming Limit Curves
Reddy et al. [2015]
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Methodology
Hypothesis
The crack is preceded by damage.
Damage is governed by microvoid nucleation, growth andcoalescence.
Damage is observed in SPIF [Lievers et al., 2004].
Tasks
1 Implementation of a damage model (Gurson) in the LAGAMINE FEcode.
2 Identification of the material parameters of the damage model.
3 Evaluate the model to understand the process mechanics leading tofracture.
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Methodology
Hypothesis
The crack is preceded by damage.
Damage is governed by microvoid nucleation, growth andcoalescence.
Damage is observed in SPIF [Lievers et al., 2004].
Tasks
1 Implementation of a damage model (Gurson) in the LAGAMINE FEcode.
2 Identification of the material parameters of the damage model.
3 Evaluate the model to understand the process mechanics leading tofracture.
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Thesis
Main question
Is the Gurson model with a shear extension able to predict failurein SPIF process?
Objectives
Efficient numerical model.
Limitations of the damage model (if any).
Reproduce the SPIF process mechanics.
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Thesis
Main question
Is the Gurson model with a shear extension able to predict failurein SPIF process?
Objectives
Efficient numerical model.
Limitations of the damage model (if any).
Reproduce the SPIF process mechanics.
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Presentation contents
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Contents
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Constitutive modeling
Elasticity
ε =1
2Gsσ − ν
E
1
3tr (σ) I
PlasticityHill [1948] yield locus
Fp =
√1
2(σ − X) : H : (σ − X)− σY
(εP)= 0
Isotropic hardening: Swift law
σY
(εP)= K
(εP + ε0
)nKinematic hardening: Armstrong and Fredrick [1966]
X = CX
(Xsatε
P − XεP)
Damage . . .
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Constitutive modeling
Elasticity
ε =1
2Gsσ − ν
E
1
3tr (σ) I
PlasticityHill [1948] yield locus
Fp =
√1
2(σ − X) : H : (σ − X)− σY
(εP)= 0
Isotropic hardening: Swift law
σY
(εP)= K
(εP + ε0
)nKinematic hardening: Armstrong and Fredrick [1966]
X = CX
(Xsatε
P − XεP)
Damage . . .
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Constitutive modeling
Elasticity
ε =1
2Gsσ − ν
E
1
3tr (σ) I
PlasticityHill [1948] yield locus
Fp =
√1
2(σ − X) : H : (σ − X)− σY
(εP)= 0
Isotropic hardening: Swift law
σY
(εP)= K
(εP + ε0
)nKinematic hardening: Armstrong and Fredrick [1966]
X = CX
(Xsatε
P − XεP)
Damage . . .
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The damage model
Basic hypothesis
Material deterioration that leads to material failure.
Associated with the evolution of micro voids.
Cross section (2000x)
Anne Mertens, ULg
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The damage modelVoid evolution
Base material
Nucleation Growth Coalescence
Lassance et al. [2007]
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The damage modelVoid evolution
Base material
Nucleation Growth Coalescence
Lassance et al. [2007]
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The Gurson [1977] model
Approach
Micromechanics based yield criterion.
Damage variable: void volume fraction (porosity).
Fp(σ, f , σY ) =σ2eq
σY 2− 1 + 2f cosh
(3
2
σmσY
)− f 2 = 0
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The Gurson [1977] model
Approach
Micromechanics based yield criterion.
Damage variable: void volume fraction (porosity).
Fp(σ, f , σY ) =σ2eq
σY 2− 1︸ ︷︷ ︸
Von Mises
+2f cosh
(3
2
σmσY
)− f 2 = 0
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The Gurson [1977] model
Approach
Micromechanics based yield criterion.
Damage variable: void volume fraction (porosity).
Fp(σ, f , σY ) =σ2eq
σY 2− 1 + 2f cosh
(3
2
σmσY
)− f 2︸ ︷︷ ︸
Damage
= 0
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The Gurson [1977] model
Approach
Micromechanics based yield criterion.
Damage variable: void volume fraction (porosity).
Fp(σ, f , σY ) =σ2eq
σY 2− 1︸ ︷︷ ︸
Von Mises
+ 2 f cosh
(3
2
σmσY
)− f 2︸ ︷︷ ︸
Damage
= 0
Matrix mass conservation:
f = (1− f ) tr εp
1 material parameter:
f0
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GTN extension
The Gurson-Tvergaard-Needleman (GTN) extension:
Nucleation [Chu and Needleman, 1980].
Void growth (classical volumetric assumption).
Coalescence [Tvergaard and Needleman, 1984].
f = fnucleation + fgrowth
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GTN extensionTvergaard [1982]
Fp(σ, f ∗, σ) =σ2eq
σ2 − 1︸ ︷︷ ︸Von Mises
+ 2q1f∗ cosh
(−3
2q2σmσ
)− q3 (f ∗)2
︸ ︷︷ ︸Damage
= 0
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GTN extensionTvergaard [1982]
Fp(σ, f ∗, εPM) =σ2eq
σ2Y
− 1︸ ︷︷ ︸Von Mises
+ 2 q1 f∗ cosh
(−3 q2 σm
2σY
)− q3 (f ∗)2
︸ ︷︷ ︸Damage
= 0
Matrix hardening:
σY = σY (εPM)
2 material parameters:
q1, q2 (q3 = q12)
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NucleationChu and Needleman [1980]
f = fnucleation + fgrowth
fnucleation = AεPM︸︷︷︸Strain
+B (σeq + cσM)︸ ︷︷ ︸Stress
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NucleationChu and Needleman [1980]
f = fnucleation + fgrowth
fnucleation = AεPM︸︷︷︸Strain
+B (σeq + cσM)︸ ︷︷ ︸Stress
A(εPM) =1√2π
fNSN
exp
[−1
2
(εPM − εN
SN
)2]
B(σ) = 0
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NucleationChu and Needleman [1980]
f = fnucleation + fgrowth
fnucleation = AεPM︸︷︷︸Strain
+B (σeq + cσM)︸ ︷︷ ︸Stress
A(εPM) =1√2π
fNSN
exp
[−1
2
(εPM − εN
SN
)2]
B(σ) = 0
3 material parameters:
fN , εN , SN
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CoalescenceTvergaard and Needleman [1984]
f ∗ =
{f if f < fcrfcr + Kf (f − fcr ) if f > fcr
fcr fFVoid volume fraction (f)
fcr
fu
Effectiveporosity (f ∗)
Kf
Kf =fu − fcrfF − fcr
2 material parameters:
fcr , fF
(fu =
1
q1
)
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CoalescenceTvergaard and Needleman [1984]
f ∗ =
{f if f < fcrfcr + Kf (f − fcr ) if f > fcr
fcr fFVoid volume fraction (f)
fcr
fu
Effectiveporosity (f ∗)
Kf
Kf =fu − fcrfF − fcr
2 material parameters:
fcr , fF
(fu =
1
q1
)
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Shear extensions
Coupling of stress and damage history.
Triaxiality: measure of the stress state.
[Pineau and Pardoen, 2007]
T (I1, J2) =σmσeq
T → 0 =⇒ εf →∞
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Shear extensions
Coupling of stress and damage history.
Triaxiality: measure of the stress state.
[Pineau and Pardoen, 2007]
T (I1, J2) =σmσeq
T → 0 =⇒ εf →∞
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Shear extensionsFailure modes
Cavity controlled (T = 1.10) Shear controlled(T = 0.47)
[Barsoum and Faleskog, 2007]
GTN model → No damage is predicted when T = 0.At low triaxiality, void shape evolution becomes important.
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Shear extensionsFailure modes
Cavity controlled (T = 1.10) Shear controlled(T = 0.47)
[Barsoum and Faleskog, 2007]
GTN model → No damage is predicted when T = 0.At low triaxiality, void shape evolution becomes important.
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Shear extensions
Nahshon and Hutchinson [2008]
f = fg + fn + fshear
fshear = kωf ω(σ)σdev : εP
σeq
1 material parameter: kω.
Note: ω(σ) is a scalar functions of the stress.
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Shear extensions
Nahshon and Hutchinson [2008]
f = fg + fn + fshear
fshear = kωf ω(σ)σdev : εP
σeq
1 material parameter: kω.
Note: ω(σ) is a scalar functions of the stress.
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Contents
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Numerical implementation
Based on Ben Bettaieb et al. [2011b,a]
Complete GTN model:
Kinematic hardening (classical non-linear).Nucleation and coalescence (GTN model).Shear [Nahshon and Hutchinson, 2008].
Matrix anisotropy (Hill type) [Benzerga and Besson, 2001]:
q =
√1
2(σ − X) : H : (σ − X)
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Numerical implementation
Based on Ben Bettaieb et al. [2011b,a]
Complete GTN model:
Kinematic hardening (classical non-linear).Nucleation and coalescence (GTN model).Shear [Nahshon and Hutchinson, 2008].Matrix anisotropy (Hill type) [Benzerga and Besson, 2001]:
q =
√1
2(σ − X) : H : (σ − X)
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Integration scheme
Equations set
Fp(σ,X,H) = 0
dεP = dλ∂Fp
∂σ
dH = h(dεP ,σ,H)
Backward EulerAravas [1987]
εn+1 = εn + ∆tεn+1
∆t = tn+1 − tn
Ben Bettaieb et al. [2011b]
Γ(Y) = 0
Yi ={
∆εp ,∆εq , n1, n2, n3, n4, n5, εPM , f
}N-R iteration
Γi +9∑
j=1
∂Γi
∂YjdYj = 0
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Integration scheme
Equations set
Fp(σ,X,H) = 0
dεP = dλ∂Fp
∂σ
dH = h(dεP ,σ,H)
Backward EulerAravas [1987]
εn+1 = εn + ∆tεn+1
∆t = tn+1 − tn
Ben Bettaieb et al. [2011b]
Γ(Y) = 0
Yi ={
∆εp ,∆εq , n1, n2, n3, n4, n5, εPM , f
}N-R iteration
Γi +9∑
j=1
∂Γi
∂YjdYj = 0
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Integration scheme
Equations set
Fp(σ,X,H) = 0
dεP = dλ∂Fp
∂σ
dH = h(dεP ,σ,H)
Backward EulerAravas [1987]
εn+1 = εn + ∆tεn+1
∆t = tn+1 − tn
Ben Bettaieb et al. [2011b]
Γ(Y) = 0
Yi ={
∆εp ,∆εq , n1, n2, n3, n4, n5, εPM , f
}
N-R iteration
Γi +9∑
j=1
∂Γi
∂YjdYj = 0
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Integration scheme
Equations set
Fp(σ,X,H) = 0
dεP = dλ∂Fp
∂σ
dH = h(dεP ,σ,H)
Backward EulerAravas [1987]
εn+1 = εn + ∆tεn+1
∆t = tn+1 − tn
Ben Bettaieb et al. [2011b]
Γ(Y) = 0
Yi ={
∆εp ,∆εq , n1, n2, n3, n4, n5, εPM , f
}N-R iteration
Γi +9∑
j=1
∂Γi
∂YjdYj = 0
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Numerical validationHydrostatic test Nahshon and Xue [2009]
Gurson parametersq1 1.0 fN 0.04 f0 0.005q2 1.0 εN 0.30 fc 0.15q3 1.0 SN 0.10 ff 0.25
0 0.25
Volumetric strain [−]
0
4.5
Hydrostaticstress ratio [−]
GUR3DextNahshon2009
0 0.25
Volumetric strain [−]
0
0.25
Void volumefraction [−]
GUR3DextNahshon2009
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Numerical validationHydrostatic test Nahshon and Xue [2009]
Gurson parametersq1 1.0 fN 0.04 f0 0.005q2 1.0 εN 0.30 fc 0.15q3 1.0 SN 0.10 ff 0.25
0 0.25
Volumetric strain [−]
0
4.5
Hydrostaticstress ratio [−]
GUR3DextNahshon2009
0 0.25
Volumetric strain [−]
0
0.25
Void volumefraction [−]
GUR3DextNahshon2009
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Numerical validationShear test Nahshon and Xue [2009]
Gurson parametersq1 1.0 fN 0.04 f0 0.005q2 1.0 εN 0.30 fc 0.15q3 1.0 SN 0.10 ff 0.25
0 2.0
Equivalent strain [−]
0
2.5
Equivalentstress ratio [−]
kω = 0
kω = 1
kω = 3
GUR3DextNahshon2009
0 2.0
Equivalent strain [−]
0
0.4
Void volumefraction [−]
kω = 1
kω = 3
GUR3DextNahshon2009
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Numerical validationShear test Nahshon and Xue [2009]
Gurson parametersq1 1.0 fN 0.04 f0 0.005q2 1.0 εN 0.30 fc 0.15q3 1.0 SN 0.10 ff 0.25
0 2.0
Equivalent strain [−]
0
2.5
Equivalentstress ratio [−]
kω = 0
kω = 1
kω = 3
GUR3DextNahshon2009
0 2.0
Equivalent strain [−]
0
0.4
Void volumefraction [−]
kω = 1
kω = 3
GUR3DextNahshon2009
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Contents
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Material presentation
DC01 ferritic steel (EN 10330).
1.0 mm thickness.
Microstructure:
Anne Mertens, ULg
Mn C Al Ni,Cu,Cr,P0.21 0.049 0.029 <0.025
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Experimental setup
Uniaxial Zwickmachine
Bi-axial machine
A B
C
Load capacity: ±100 kN
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Digital Image CorrelationDIC
Contactless method for displacements and strains.
Pattern tracking.
CMOS cameras, resolution 1280x800
AOI
refstep
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Experimental test campaignSpecimens
Tensile
Shear
30
120
3
Plane strain
3058
120
R2
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Plasticity tests
Tensile and shear test
0 0.5
Strain [−]
0
500
Stress [MPa]
45
RD,TD
Tensile
Shear
Bauschinger test
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Plasticity tests
Tensile and shear test
0 0.5
Strain [−]
0
500
Stress [MPa]
45
RD,TD
Tensile
Shear
Bauschinger test
-0.5 0 0.4
Strain [-]
-300
0
300
Stress [MPa]
10%20%
30%
Precharge degree
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Plasticity tests
Tensile and shear test
0 0.5
Strain [−]
0
500
Stress [MPa]
45
RD,TD
Tensile
Shear
Bauschinger test
-0.5 0 0.4
Strain [-]
-300
0
300
Stress [MPa]
30%
Stagnation
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Identification of material parameters
Hill [1948] parameters → Classicalsimulated annealing.
Hardening (K , n, ε0, Cx , Xsat) →Inverse optimization (OPTIM).
error norm =
√√√√ N∑i=1
(yFEi − y exp
i
)2
Initialparameters
FEMsimulations
Numerical-experimentalcomparison
New set
Acceptable?
Identifiedparameters
yes
no
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Identification of material parameters
Hill [1948] parameters → Classicalsimulated annealing.
Hardening (K , n, ε0, Cx , Xsat) →Inverse optimization (OPTIM).
error norm =
√√√√ N∑i=1
(yFEi − y exp
i
)2
Initialparameters
FEMsimulations
Numerical-experimentalcomparison
New set
Acceptable?
Identifiedparameters
yes
no
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Identification of material parameters
Tensile test
0 0.2
Strain [−]
100
400
Stress [MPa]
Expnum
Bauschinger test
-0.4 0 0.4
Strain [−]
-300
0
300
Stress [MPa]
Expnum 10%
30%
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Identification of material parameters
Tensile test
0 0.2
Strain [−]
100
400
Stress [MPa]
Expnum
Bauschinger test
-0.4 0 0.4
Strain [−]
-300
0
300
Stress [MPa]
Expnum 10%
30%
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Contents
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GTN characterizationMethodology
Difference with plasticity
Microscopic scale,heterogeneous deformation.
Force vs. displacement insteadof stress vs. strain.
Coupling between variables.
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GTN characterizationMethodology
Difference with plasticity
Microscopic scale,heterogeneous deformation.
Force vs. displacement insteadof stress vs. strain.
Coupling between variables.
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GTN parameters characterization
Automatic optimization (OPTIM) issues
CPU time, iterations, etc.
Sensitivity of nucleation, coalescence parameters.
Introduction of weights in the error norm.
Approach:
d1 d2Displacement
Force Plasticity Nucleation Coalescence
Lagamine Optim
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GTN parameters characterization
Automatic optimization (OPTIM) issues
CPU time, iterations, etc.
Sensitivity of nucleation, coalescence parameters.
Introduction of weights in the error norm.
Approach:
d1 d2Displacement
Force Plasticity Nucleation Coalescence
Lagamine Optim
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Macroscopic test campaign
R = 5
T ≈0.6-0.7
ω ≈0.25-0.4
R = 10
T ≈0.5-0.7
ω ≈0.2-0.4
hole
T ≈0.35-0.6
ω ≈0.0
shear
T ≈0.0
ω ≈1.0
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Force predictions
Nucleation Coalescence ShearSet name f0 fN εN SN fc fF kω
set1 0.0055 0.135 0.25set2 0.0008 0.0025 0.175 0.42 0.0045 0.145 0.25set3 0.0025 0.170 0.075
0 2.5 5.0
Displacement [mm]
0
3000
6000
Force [N]
R=5R=10
hole
0 1.25 2.5
Displacement [mm]
0
500
1000
Force [N]
shear
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Force predictions
Nucleation Coalescence ShearSet name f0 fN εN SN fc fF kω
set1 0.0055 0.135 0.25set2 0.0008 0.0025 0.175 0.42 0.0045 0.145 0.25set3 0.0025 0.170 0.075
0 2.5 5.0
Displacement [mm]
0
3000
6000
Force [N]
R=5R=10
hole
0 1.25 2.5
Displacement [mm]
0
500
1000
Force [N]
shear
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Force predictions
Nucleation Coalescence ShearSet name f0 fN εN SN fc fF kω
set1 0.0055 0.135 0.25set2 0.0008 0.0025 0.175 0.42 0.0045 0.145 0.25set3 0.0025 0.170 0.075
0 2.5 5.0
Displacement [mm]
0
3000
6000
Force [N]
R=5R=10
hole
0 1.25 2.5
Displacement [mm]
0
500
1000
Force [N]
shear
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Strain prediction
0 2.75 5.5
Displacement [mm]
0
0.30
0.6
Strain [−]
R=10
hole
Strain localization is not captured
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Strain prediction
0 1.25 2.5
Displacement [mm]
0
0.20
0.4
Shear strain [−]
shear
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DIC vs. FE predictionsAxial strain
notch R = 5
DIC Numerical
0.45
−0.05
notch R = 10
0.40
0.00
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DIC vs. FE predictionsAxial strain
hole
DIC Numerical
0.60
0.00
shear
0.10
−0.25 46
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Discussion
Results
Loss on load carrying capacity is captured.
Strain localization is not captured.
Limitations of the GTN model.
Source of errors
Parameters q1 and q2 were not calibrated.
Hardening stagnation.
Mesh sensitivity.
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Contents
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Literature review summary
Simulate SPIF is not easy
Small contact zone with a very long path.
High strains.
Incremental deformation, simulation time.
Sensitivity of force prediction to FE choice, constitutive law.
Boundary conditions, grip modeling.
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Literature review summaryShape inaccuracies
Springback, bending.
Elastic strains.
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Literature review summaryFormability
Forming Limit Curve (FLC):classic approach.
Through the thickness shear,Bending-under-tension, cycliceffects, etc.
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Literature review summaryDamage
Definition
Mechanism of degradation leading to fracture (Damage 6= formability)
Malhotra et al. [2012].
Shear itself cannot explainhigher fomability:
Early localization: Noodletheory.
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Literature review summaryDamage
Definition
Mechanism of degradation leading to fracture (Damage 6= formability)
Malhotra et al. [2012].
Shear itself cannot explainhigher fomability:
Early localization: Noodletheory.
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Finite element type
• ••••
•• ••• ⊗
⊗⊗
⊗
Shell
• •
•••
•• •
••⊗
⊗...
Solid-shell
• •
•••
•
• •
••
⊗
Brick
RESS solid-shell element [Alves de Sousa, 2006].
Numerical technique: Enhanced assumed strain (EAS)
ε = εcom + εEAS
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Finite element type
• ••••
•• ••• ⊗
⊗⊗
⊗
Shell
• •
•••
•• •
••⊗
⊗...
Solid-shell
• •
•••
•
• •
••
⊗
Brick
RESS solid-shell element [Alves de Sousa, 2006].
Numerical technique: Enhanced assumed strain (EAS)
ε = εcom + εEAS
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Line testDescription
Most basic SPIF test.
Experimental data by Hans Vanhove (KULeuven).
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Numerical-Experimental validation
Shape: top and bottom surface
-91 0 91
X [mm]
-6
0
1
Z [mm]
Exp
GTN+shear
cross section
x
y
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Numerical-Experimental validation
Shape: top and bottom surface
-91 0 91
X [mm]
-6
0
1
Z [mm]
Exp
GTN+shear
Force
0 0.2 0.8 1.0 1.8
Ref. Time [s]
0
1000
2000
Force [N]
Exp
GTN+shear
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Two-slope pyramidDescription
For shape accuracy assessment.
Experimental DIC shape by Amar Behera (KULeuven).
60◦
30◦
y
z
1 35 52 6 52 35 1
60
90
182
182
x
y
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Numerical predictionsExperimentally there is no crack!
Shape: bottom surface
0 45 90
X [mm]
-90
-45
0
Z [mm]
expnum
cross section
x
y
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Numerical predictionsExperimentally there is no crack!
Shape: bottom surface
0 45 90
X [mm]
-90
-45
0
Z [mm]
expnum
Forces: no experiments available
0 225 450
Ref. Time [s]
0
3000
6000
Force [N]
no coalescence
with coalescence
With coalescence, the model predictsfracture. . . prematurely
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Cone testDescription
Benchmark for failure angles.
DC01, 1.0 mm ⇒ α =67◦
α
x
z
182mm
30mm
DC01 steel, 1.0 mm⇒ failure angle: 67◦
φ182mm
x
y
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Numerical predictions
Force prediction (no experiments)
0 300 601
Ref. Time [s]
0
1250
2500
Force [N]
45
47
48
50
Fz s(48◦)
The crack is predicted at α =48◦
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Numerical predictions
Force prediction (no experiments)
0 300 601
Ref. Time [s]
0
1250
2500
Force [N]
45
47
48
50
Fz s(48◦)
Aerens et al. [2009] formula:
Fz s = 0.0716Rmt1.57dt
0.41∆h0.09
. . . (α− dα) cos (α− dα)
α =47◦ 1219.70Nα =48◦ 1222.49Nα =67◦ 1158.01N
The crack is predicted at α =48◦
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Analysis of fracture prediction
1 Predicted force overestimation.
2 Bad modeling of the deformation.
3 Limitations of the GTN model.
Porosity for the 47◦ cone
no fracture
Porosity for the 48◦ cone
εf ≈0.8
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Analysis of fracture prediction
1 Predicted force overestimation.
2 Bad modeling of the deformation.
3 Limitations of the GTN model.
Porosity for the 47◦ cone
no fracture
Porosity for the 48◦ cone
εf ≈0.8
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Contents
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Conclusions
Contributions
Fully implicit implementation of the GTN+shear model.
Extensive experimental data and material identification.
Good shape prediction in SPIF (FE element type).
Issues
The chosen damage model is capable to predict failure in theSPIF process but not accurately.
GTN model uncouples the hardening and damage.
Force prediction in SPIF.
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Conclusions
Contributions
Fully implicit implementation of the GTN+shear model.
Extensive experimental data and material identification.
Good shape prediction in SPIF (FE element type).
Issues
The chosen damage model is capable to predict failure in theSPIF process but not accurately.
GTN model uncouples the hardening and damage.
Force prediction in SPIF.
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Perspectives
Modification of the hardening in the GTN model [Leblond et al.,1995].
Implement different type of damage model [Lemaitre, 1985; Xue,2007].
Effect of hardening stagnation on damage.
SPIF
Remeshing + Damage in LAGAMINE.
Different EAS modes solid-shell.
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Perspectives
Modification of the hardening in the GTN model [Leblond et al.,1995].
Implement different type of damage model [Lemaitre, 1985; Xue,2007].
Effect of hardening stagnation on damage.
SPIF
Remeshing + Damage in LAGAMINE.
Different EAS modes solid-shell.
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Experimental and Numerical Characterization ofDamage and Application to Incremental Forming
PhD thesis presentation
Carlos Felipe Guzman
Department ArGEnCoUniversity of Liege, Belgium
February 1st, 2016
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Material presentationTexture measurements
Incomplete pole figures:
(110) (200) (211)
Philip Eyckens, KULeuven
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Shear extensions
Xue [2008]
f ∗ → D
D = Kf
(q1 f + Dshear
)Dshear = kg f
1/3gθ(σ)εeq εeq
Nahshon and Hutchinson [2008]
f = fg + fn + fshear
fshear = kωf ω(σ)σdev : εP
σeq
1 material parameter: kg or kω.
Note: gθ(σ) and ω(σ) are scalar functions of the stress.
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Shear extensions
Xue [2008]
f ∗ → D
D = Kf
(q1 f + Dshear
)Dshear = kg f
1/3gθ(σ)εeq εeq
Nahshon and Hutchinson [2008]
f = fg + fn + fshear
fshear = kωf ω(σ)σdev : εP
σeq
1 material parameter: kg or kω.
Note: gθ(σ) and ω(σ) are scalar functions of the stress.
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Shear extensions
Xue [2008]
f ∗ → D
D = Kf
(q1 f + Dshear
)Dshear = kg f
1/3gθ(σ)εeq εeq
Nahshon and Hutchinson [2008]
f = fg + fn + fshear
fshear = kωf ω(σ)σdev : εP
σeq
1 material parameter: kg or kω.
Note: gθ(σ) and ω(σ) are scalar functions of the stress.
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Integration schemeConsistent tangent matrix, algorithm approach
σ = σ(ε)
dσ = D : dε ; D :=∂σ
∂ε
εn εn+1
σn
σn+1
Dalgo
Dcont
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Integration schemeConsistent tangent matrix, algorithm approach
σ = σ(ε)
dσ = D : dε ; D :=∂σ
∂ε
εn εn+1
σn
σn+1
Dalgo
Dcont
σn+1 = C :(εn+1 − εPn+1
)
dσ = C : dε− C : d∆εP
Linearization
Relate dε with d∆εP
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Integration schemeConsistent tangent matrix, algorithm approach
σ = σ(ε)
dσ = D : dε ; D :=∂σ
∂ε
εn εn+1
σn
σn+1
Dalgo
Dcont
K : ∂∆εP = L : ∂σ Kim and Gao [2005]approach
D = C− C(K + LC)−1LC
∂Fp
∂σ,∂Fp
∂∆εP,∂Fp
∂Hβ,∂2Fp
∂σ2,
∂2Fp
∂σ∂∆εP, . . . Extension to
Kinematic hardening
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Numerical validationHydrostatic test, Aravas [1987]
0 0.4
Volumetric strain [−]
0
2.5
Hydrostaticstress ratio [−]
GUR3DextAravas1987
0 0.4
Volumetric strain [−]
0
0.4
Void volumefraction [−]
GUR3DextAravas1987
Elasto-plastic parameters Gurson parameters
E 210 GPa K 1200 MPa q1 1.5 fN 0.04 f0 0
ν 0.3 ε0 3.17 × 10−3 q2 1.0 εN 0.30 fc -n 0.1 q3 2.25 SN 0.10 ff -
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Tensile testAravas [1987]
0 1.0
Equivalent strain [−]
0
1.8
Equivalentstress ratio [−]
GUR3DextAravas1987
0 1.0
Equivalent strain [−]
0
0.1
Void volumefraction [−]
GUR3DextAravas1987
Elasto-plastic parameters Gurson parameters
E 210 GPa K 1200 MPa q1 1.5 fN 0.04 f0 0
ν 0.3 ε0 3.17 × 10−3 q2 1.0 εN 0.30 fc -n 0.1 q3 2.25 SN 0.10 ff -
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Numerical validationShear test Xue [2008]
Gurson parametersq1 1.5 fN 0.04 f0 0.00q2 1.0 εN 0.20 fc 0.05q3 2.25 SN 0.10 ff 0.25
0 4.0
Matrix plastic strain [−]
0
500
Equivalentstress [MPa]
kg = 0
kg = 0.25
kg = 3GUR3DextXue2008
0 4.0
Matrix plastic strain [−]
0
1.2
Damage [−]
kg = 3
kg = 0.25
kg = 0
GUR3DextXue2008
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Numerical validationShear test Xue [2008]
Gurson parametersq1 1.5 fN 0.04 f0 0.00q2 1.0 εN 0.20 fc 0.05q3 2.25 SN 0.10 ff 0.25
0 4.0
Matrix plastic strain [−]
0
500
Equivalentstress [MPa]
kg = 0
kg = 0.25
kg = 3GUR3DextXue2008
0 4.0
Matrix plastic strain [−]
0
1.2
Damage [−]
kg = 3
kg = 0.25
kg = 0
GUR3DextXue2008
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State variables analysis
0 0.2 0.8 1.0 1.8
Ref. Time [s]
0
f0
0.002
0.004
Porosity [−]
elem=118
elem=404
elem=690
indent 1
indent 2
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State variables analysis
0 0.2 0.8 1.0 1.8
Ref. Time [s]
0
f0
0.002
0.004
Porosity [−]
elem=118
elem=404
elem=690
indent 1
indent 2
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SPIF line testState variables analysis
0 0.2 0.8 1.0 1.8
Ref. Time [s]
-1.0
0
1.5
Triaxiality [−]
elem=118
elem=404
elem=690
0 0.2 0.8 1.0 1.8
Ref. Time [s]
0
6 · 10−4
1.2 · 10−3
Hyd. strain [−]
elem=118
elem=404
elem=690
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SPIF line testState variables analysis
0 0.2 0.8 1.0 1.8
Ref. Time [s]
-1.0
0
1.5
Triaxiality [−]
elem=118
elem=404
elem=690
0 0.2 0.8 1.0 1.8
Ref. Time [s]
0
6 · 10−4
1.2 · 10−3
Hyd. strain [−]
elem=118
elem=404
elem=690
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SPIF Two-slope pyramidMesh and boundary conditions
x
y
•O •P
(ux)O = − (ux)P(uy )O = − (uy )P(uz)O = (uz)P
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SPIF Two-slope pyramidNumerical predictions
Porosity f
Eq. macro. strain εq
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SPIF Two-slope pyramidNumerical predictions
Porosity f Eq. macro. strain εq
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Cone testMesh and boundary conditions
x
y
•O•O
•P
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References I
Aerens, R., Eyckens, P., van Bael, A., Duflou, J., 2009. Force prediction for single pointincremental forming deduced from experimental and FEM observations. The InternationalJournal of Advanced Manufacturing Technology 46 (9-12), 969–982.
Alves de Sousa, R. J., 2006. Development of a General Purpose Nonlinear Solid- Shell Element andits Application to Anisotropic Sheet Forming Simulation. Phd thesis, Universidade de Aveiro.
Aravas, N., 1987. On the numerical integration of a class of pressure-dependent plasticity models.International Journal for Numerical Methods in Engineering 24 (7), 1395–1416.
Armstrong, P., Fredrick, C., 1966. A Mathematical Representation of the Multiaxial BauschingerEffect. Technical report, Central Electricity Generating Board.
Barsoum, I., Faleskog, J., 2007. Rupture mechanisms in combined tension and shear-Experiments.International Journal of Solids and Structures 44 (6), 1768–1786.
Ben Bettaieb, M., Lemoine, X., Bouaziz, O., Habraken, A. M., Duchene, L., 2011a. Numericalmodeling of damage evolution of DP steels on the basis of X-ray tomography measurements.Mechanics of Materials 43 (3), 139–156.
Ben Bettaieb, M., Lemoine, X., Duchene, L., Habraken, A. M., 2011b. On the numericalintegration of an advanced Gurson model. International Journal for Numerical Methods inEngineering 85 (8), 1049–1072.
Benzerga, A. A., Besson, J., 2001. Plastic potentials for anisotropic porous solids. EuropeanJournal of Mechanics - A/Solids 20 (3), 397–434.
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References II
Chu, C. C., Needleman, A., 1980. Void Nucleation Effects in Biaxially Stretched Sheets. Journal ofEngineering Materials and Technology 102 (3), 249.
Gurson, A. L., 1977. Continuum theory of ductile rupture by void nucleation and growth: PartI-Yield criteria and flow rules for porous ductile media. Journal of Engineering Materials andTechnology 99 (1), 2–15.
Hill, R., may 1948. A Theory of the Yielding and Plastic Flow of Anisotropic Metals. Proceedingsof the Royal Society A: Mathematical, Physical and Engineering Sciences 193 (1033), 281–297.
Hirt, G., Bambach, M., Bleck, W., Prahl, U., Stollenwerk, J., 2015. The Development ofIncremental Sheet Forming from Flexible Forming to Fully Integrated Production of SheetMetal Parts. In: Brecher, C. (Ed.), Advances in Production Technology. Lecture Notes inProduction Engineering. Springer International Publishing, Cham, Ch. 9, pp. 117–129.
Kim, J., Gao, X., 2005. A generalized approach to formulate the consistent tangent stiffness inplasticity with application to the GLD porous material model. International Journal of Solidsand Structures 42 (1), 103–122.
Lassance, D., Fabregue, D., Delannay, F., Pardoen, T., 2007. Micromechanics of room and hightemperature fracture in 6xxx Al alloys. Progress in Materials Science 52 (1), 62–129.
Leblond, J.-B., Perrin, G., Devaux, J., 1995. An improved Gurson-type model for hardenableductile metals. European journal of mechanics. A. Solids 14 (4), 499–527.
Lemaitre, J., 1985. A Continuous Damage Mechanics Model for Ductile Fracture. Journal ofEngineering Materials and Technology 107 (1), 83.
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References III
Lievers, W., Pilkey, A., Lloyd, D., 2004. Using incremental forming to calibrate a void nucleationmodel for automotive aluminum sheet alloys. Acta Materialia 52 (10), 3001–3007.
Malhotra, R., Xue, L., Belytschko, T., Cao, J., 2012. Mechanics of fracture in single pointincremental forming. Journal of Materials Processing Technology 212 (7), 1573–1590.
Nahshon, K., Hutchinson, J. W., 2008. Modification of the Gurson Model for shear failure.European Journal of Mechanics - A/Solids 27 (1), 1–17.
Nahshon, K., Xue, Z., 2009. A modified Gurson model and its application to punch-outexperiments. Engineering Fracture Mechanics 76 (8), 997–1009.
Pineau, A., Pardoen, T., 2007. Failure mechanisms of metals. Comprehensive structural integrityencyclopedia 2.
Reddy, N. V., Lingam, R., Cao, J., 2015. Incremental Metal Forming Processes in Manufacturing.In: Nee, A. Y. C. (Ed.), Handbook of Manufacturing Engineering and Technology. SpringerLondon, London, Ch. 9, pp. 411–452.
Schafer, T., Dieter Schraft, R., dec 2005. Incremental sheet metal forming by industrial robots.Rapid Prototyping Journal 11 (5), 278–286.
Tvergaard, V., 1982. On localization in ductile materials containing spherical voids. InternationalJournal of Fracture 18 (4), 237–252.
Tvergaard, V., Needleman, A., 1984. Analysis of the cup-cone fracture in a round tensile bar. ActaMetallurgica 32 (1), 157–169.
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References IV
Xue, L., 2007. Damage accumulation and fracture initiation in uncracked ductile solids subject totriaxial loading. International Journal of Solids and Structures 44 (16), 5163–5181.
Xue, L., 2008. Constitutive modeling of void shearing effect in ductile fracture of porous materials.Engineering Fracture Mechanics 75 (11), 3343–3366.
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