experimental and numerical characterization of damage and … · experimental and numerical...

Experimental and Numerical Characterization ofDamage and Application to Incremental Forming

PhD thesis presentation

Carlos Felipe Guzman

Department ArGEnCoUniversity of Liege, Belgium

February 1st, 2016

Simple geometries

Cooking pots

2

Simple geometries

Manufactured by Deep Drawing

3

More complex geometries

Planes and car prototypes

4

More complex geometries

Implants

Manufactured by ???

5

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).

6


Video

7


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.

8

The high formability of SPIF

crack

70◦

crack

sine law:

tf = t0 sinα⇒ tf ≈ 0.35

ε� 1.0

9

The high formability of SPIF

crack

Detail:

tf ≈0.25mm

t0 =1.0mm

9

The high formability of SPIFWhy formability is so high?

Forming Limit Curves

Reddy et al. [2015]

10

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.

11

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.

12

Presentation contents

13

Contents

14

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 . . .

15

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

16

The damage modelVoid evolution

Base material

Nucleation Growth Coalescence

Lassance et al. [2007]

17

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

18


Approach




σY 2− 1︸︷︷︸

Von Mises

+2f cosh

(3

2

σmσY

)− f 2 = 0

18


Approach




σY 2− 1 + 2f cosh

(3

2

σmσY

)− f 2︸︷︷︸

Damage

= 0

18


Approach




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

18

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

19

GTN extensionTvergaard [1982]

Fp(σ, f ∗, σ) =σ2eq

σ2 − 1︸︷︷︸Von Mises

+ 2q1f∗ cosh

(−3

2q2σmσ

)− q3 (f ∗)2

︸︷︷︸Damage

= 0

20

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)

20

NucleationChu and Needleman [1980]


fnucleation = AεPM︸︷︷︸Strain

+B (σeq + cσM)︸︷︷︸Stress

21





A(εPM) =1√2π

fNSN

exp

[−1

2

(εPM − εN

SN

)2]

B(σ) = 0

21





A(εPM) =1√2π

fNSN

exp

[−1

2

(εPM − εN

SN

)2]

B(σ) = 0


fN , εN , SN

21

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


fcr , fF

(fu =

1

q1

)

22

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 →∞

23

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.

24

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.

25

Contents

26

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)

27

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)

27

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

28

Integration scheme

Equations set

Fp(σ,X,H) = 0

dεP = dλ∂Fp

∂σ

dH = h(dεP ,σ,H)


εn+1 = εn + ∆tεn+1

∆t = tn+1 − tn


Γ(Y) = 0

Yi ={

∆εp ,∆εq , n1, n2, n3, n4, n5, εPM , f

}

N-R iteration

Γi +9∑

j=1

∂Γi

∂YjdYj = 0

28

Integration scheme

Equations set

Fp(σ,X,H) = 0

dεP = dλ∂Fp

∂σ

dH = h(dεP ,σ,H)


εn+1 = εn + ∆tεn+1

∆t = tn+1 − tn


Γ(Y) = 0

Yi ={

∆εp ,∆εq , n1, n2, n3, n4, n5, εPM , f

}N-R iteration

Γi +9∑

j=1

∂Γi

∂YjdYj = 0

28

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


0

0.25

Void volumefraction [−]

GUR3DextNahshon2009

29

Numerical validationShear test Nahshon and Xue [2009]


0 2.0

Equivalent strain [−]

0

2.5

Equivalentstress ratio [−]

kω = 0

kω = 1

kω = 3

GUR3DextNahshon2009

0 2.0


0

0.4


kω = 1

kω = 3

GUR3DextNahshon2009

30

Contents

31

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

32

Experimental setup

Uniaxial Zwickmachine

Bi-axial machine

A B

C

Load capacity: ±100 kN

33

Digital Image CorrelationDIC

Contactless method for displacements and strains.

Pattern tracking.

CMOS cameras, resolution 1280x800

AOI

refstep

34

Experimental test campaignSpecimens

Tensile

Shear

30

120

3

Plane strain

3058

120

R2

35

Plasticity tests

Tensile and shear test

0 0.5

Strain [−]

0

500

Stress [MPa]

45

RD,TD

Tensile

Shear

Bauschinger test

36

Plasticity tests


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

36

Plasticity tests


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

36

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

37

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%

38

Contents

39

GTN characterizationMethodology

Difference with plasticity

Microscopic scale,heterogeneous deformation.

Force vs. displacement insteadof stress vs. strain.

Coupling between variables.

40

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

41

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

42

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

43

Strain prediction

0 2.75 5.5

Displacement [mm]

0

0.30

0.6

Strain [−]

R=10

hole

Strain localization is not captured

44

Strain prediction

0 1.25 2.5

Displacement [mm]

0

0.20

0.4

Shear strain [−]

shear

44

DIC vs. FE predictionsAxial strain

notch R = 5

DIC Numerical

0.45

−0.05

notch R = 10

0.40

0.00

45

DIC vs. FE predictionsAxial strain

hole

DIC Numerical

0.60

0.00

shear

0.10

−0.25 46

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.

47

Contents

48

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.

49

Literature review summaryShape inaccuracies

Springback, bending.

Elastic strains.

50

Literature review summaryFormability

Forming Limit Curve (FLC):classic approach.

Through the thickness shear,Bending-under-tension, cycliceffects, etc.

51

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.

52

Finite element type

• ••••

•• ••• ⊗

⊗⊗

⊗

Shell

• •

•••

•• •

••⊗

⊗...

Solid-shell

• •

•••

•

• •

••

⊗

Brick

RESS solid-shell element [Alves de Sousa, 2006].

Numerical technique: Enhanced assumed strain (EAS)

ε = εcom + εEAS

53

Line testDescription

Most basic SPIF test.

Experimental data by Hans Vanhove (KULeuven).

54

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

55

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

55

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

56

Numerical predictionsExperimentally there is no crack!

Shape: bottom surface

0 45 90

X [mm]

-90

-45

0

Z [mm]

expnum

cross section

x

y

57

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

57

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

58

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◦

59

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◦

59

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

60

Contents

61

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.

62

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.

63

Experimental and Numerical Characterization ofDamage and Application to Incremental Forming

PhD thesis presentation

Carlos Felipe Guzman

Department ArGEnCoUniversity of Liege, Belgium

February 1st, 2016

Material presentationTexture measurements

Incomplete pole figures:

(110) (200) (211)

Philip Eyckens, KULeuven

65

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.

66

Integration schemeConsistent tangent matrix, algorithm approach

σ = σ(ε)

dσ = D : dε ; D :=∂σ

∂ε

εn εn+1

σn

σn+1

Dalgo

Dcont

67


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

67


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

67

Numerical validationHydrostatic test, Aravas [1987]

0 0.4


0

2.5

Hydrostaticstress ratio [−]

GUR3DextAravas1987

0 0.4


0

0.4


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 -

68

Tensile testAravas [1987]

0 1.0


0

1.8

Equivalentstress ratio [−]

GUR3DextAravas1987

0 1.0


0

0.1


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 -

69

Numerical validationShear test Xue [2008]


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

70

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

71

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

72

SPIF Two-slope pyramidMesh and boundary conditions

x

y

•O •P

(ux)O = − (ux)P(uy )O = − (uy )P(uz)O = (uz)P

73

SPIF Two-slope pyramidNumerical predictions

Porosity f

Eq. macro. strain εq

74

SPIF Two-slope pyramidNumerical predictions

Porosity f Eq. macro. strain εq

74

Cone testMesh and boundary conditions

x

y

•O•O

•P

75

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.

76

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.

77

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.

78

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.

79

experimental and numerical characterization of damage and … · experimental and numerical...

Documents