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(1)

Computational & Multiscale

Mechanics of Materials

CM3

www.ltas-cm3.ulg.ac.be

Damage to crack transition for ductile materials using a

cohesive-band /discontinuous Galerkin framework

Julien Leclerc, Ling Wu, Van-Dung Nguyen, Ludovic Noels

The research has been funded by the Walloon Region under the agreement no.7581-MRIPF in the context of the 16thMECATECH call.

(2)

Goal: To capture the whole ductile failure process

– A diffuse stage

• Damage onset / nucleation, growth…

– Followed by a localized stage

• Damage coalescence

• Crack initiation and propagation • …

Introduction

[http://radome.ec-nantes.fr/]

𝐹

𝑢

𝐿

Diffuse damage stage

Localized damage

stage

Physical

process

(3)

State-of-the-art

– 2 approaches modeling material failure:

• Continuum Damage Models (CDM)

– Lemaitre-Chaboche, – Gurson,

– …

• Discontinuous: Fracture mechanics – Cohesive zone,

– XFEM – …

(4)

Material properties degradation modelled through internal variables evolution

– Form 𝛔 𝜺; 𝑍 𝑡

• With internal variables history 𝑍 𝑡′

– Lemaitre-Chaboche model,

– Gurson model,

• Porosity evolution 𝑓𝑉

Continuous Damage Model (CDM) implementation:

– Local form

• Mesh-dependency

– Non-local form needed

[Bažant 1988] • Introduction of a characteristic length 𝑙𝑐 • In terms of Weight functions: 𝑓𝑉 𝒙 = 𝑉

𝐶𝑊 𝒚; 𝒙, 𝑙𝑐 𝑓𝑉 𝐲 𝑑𝒚

– Implicit formulation [Peerlings et al. 1998]

• New non-local degrees of freedom 𝑓𝑉

• New Helmholtz type equation to be solved 𝑓𝑉− 𝑙𝑐2Δ 𝑓𝑉 = 𝑓𝑉

State of art: Continuous approaches

The numerical results change without convergence

𝜎

(5)

Continuous:

Continuous Damage

Model (CDM)

Discontinuous:

+

Capture the diffuse damage stage

+

Capture

stress

triaxiality

and

Lode

variable effects

-

Mesh

dependency

without

(implicit) non-local form

-

Numerical

problems

with

highly

damaged elements

-

Cannot

represent

cracks

without remeshing / element deletion at

𝐷 → 1

(loss

of

accuracy,

mesh

modification ...)

-

Crack initiation observed for lower damage

values

(6)

Based on fracture mechanics concepts

• Characterized by

• Strength sc &

Critical energy release rate GC

One of the most used methods:

– Cohesive Zone Model (CZM) modelling

the crack tip behavior

– Integrate a Traction Separation Law (TSL):

• At interface elements between two elements

• Using element enrichment (EFEM) [Armero et al. 2009] • Using mesh enrichment (xFEM) [Moes et al. 2002] • …

State of art: Discontinuous approaches

𝛿

𝒕

𝒕

d

t

d

c

G

C

s

c

(7)

Cohesive elements

– Inserted between volume elements

• Zero-thickness no triaxiality accounted for

• Intrinsic Cohesive Law (ICL)

• Cohesive elements inserted from the beginning • Efficient if a priori knowledge of the crack path

• Mesh dependency[Xu & Needelman, 1994]

• Initial slope modifies the effective elastic modulus

• This slope should tend to infinity [Klein et al. 2001]:

• Alteration of a wave propagation • Critical time step is reduced

• Extrinsic Cohesive Law (ECL)

• Cohesive elements inserted on the fly when the failure criterion is verified[Ortiz & Pandolfi 1999]

• Complex implementation in 3D (parallelization)

State of art: Discontinuous approaches

d

t

t

d

t

G

C

d

c

s

c

d

t

d

c

G

C

s

c

(8)

Hybrid framework

[Radovitzky et al. 2011]

– Discontinuous Galerkin (DG) framework

• Test and shape functions discontinuous

• Consistency, convergence rate, uniqueness

recovered though interface terms

• Interface terms integrated on interface elements

– Combination with extrinsic cohesive laws

• Interface elements already there

• Switch to traction separation law • Efficient for fragmentation simulations

State of art: Discontinuous approaches

𝒖

t

t

𝒖

t

d

c

G

C

s

c (a-1)+ (a)+ x (a+1) -u (a+1)+ (a) -(a-1)0 𝐏: 𝛁0𝜹𝒖𝑑Ω + 𝜕IΩ0 𝜹𝒖 ⋅ 〈𝐏〉 ⋅ 𝑵 −𝑑𝜕Ω+ 𝜕IΩ0 𝒖 ⋅ 〈𝑪 el: 𝛁 𝟎𝜹𝒖〉 ⋅ 𝑵−𝑑𝜕Ω+ 𝜕IΩ0 𝒖 ⊗ 𝑵 −: 𝛽𝑠𝑪el ℎ𝑠 : 𝜹𝒖 ⊗ 𝑵 −𝑑𝜕Ω=0

(9)

Continuous:

Continuous Damage

Model (CDM)

Discontinuous:

Extrinsic Cohesive

Zone Model (CZM)

+

Capture the diffuse damage stage

+

Capture

stress

triaxiality

and

Lode

variable effects

+

Multiple crack initiation and propagation

naturally managed

-

Mesh

dependency

without

(implicit) non-local form

-

Numerical

problems

with

highly

damaged elements

-

Cannot

represent

cracks

without remeshing / element deletion at

𝐷 → 1

(loss

of

accuracy,

mesh

modification ...)

-

Crack initiation observed for lower damage

values

-

Cannot capture diffuse damage

-

No triaxiality effect

-

Currently valid for brittle / small scale

yielding elasto-plastic materials

(10)

Goal:

– Simulation of the whole ductile failure process with accuracy

Main idea:

– Combination of 2 complementary methods in a single finite element framework:

• Continuous (non-local damage model)

+ transition to

• Discontinuous (Hybrid DG model)

– However triaxiality effects are important Cohesive zone model not adequate!!!

Goals of research

Damage to crack transition

𝐹

𝑢

𝐿

CDM

(diffuse damage)

Hybrid DG

(localized

damage)

Numerical

model

(11)

Hybrid DG model: use of a Cohesive Band Model (CBM)

– Hypothesis

• In the last stage of failure, all damaging process occurs in an uniform thin band

– Principles

• Substitute TSL of CZM by the behavior of a uniform band of thickness ℎb [Remmers et al. 2013]

– Methodology

[Leclerc et al. 2018]

1. Compute a band strain tensor 𝐅b = 𝐅 + 𝒖 ⊗𝑵

𝑏 + 1

2𝛁𝑇 𝒖

2. Compute a band stress tensor 𝛔b 𝐅b; Z 𝜏 using the same CDM as bulk elements 3. Recover a surface traction 𝒕( 𝒖 , 𝐅) = 𝛔b. 𝒏

– What is the effect of ℎ

b

(band thickness)

• A priori determined with underlying non-local damage model to ensure energy consistency

Damage to crack transition – Principles

𝐅, 𝛔

𝑵, 𝒏

Bulk

𝒖

𝒉

b

𝐅

b

, 𝛔

b

𝑵, 𝒏

Band

Bulk

𝐅, 𝛔

(12)

Elastic damage material model

– Constitutive equations

• Helmholtz energy: 𝜌𝜓 𝜺, 𝐷 = 1

2 1 − 𝐷 𝜺: 𝐻: 𝜺

• Non-local maximum principal strain: 𝑒 − 𝑙𝑐2Δ 𝑒 = 𝑒

• Damage evolution 𝐷 𝜅 = 1 − 𝐷 𝛽 𝜅 + 𝛼 𝜅𝑐−𝜅 𝜅 with 𝜅 = max𝑡′ 𝑒 𝑡 ′

– 1D non-local test

Damage to crack transition for elastic damage – Proof of concept

0

0.5

1

𝐷

Non-local only

(13)

Non-local elastic damage to Cohesive Band Model transition

– Influence of ℎ

b

(for a given 𝑙

c

) on response in a 1D elastic case

[Leclerc et al. 2018]

• Has effect on the totally dissipated energy Φ

• Could be chosen to conserve energy dissipation (physically based)

Damage to crack transition for elastic damage – Proof of concept

0

0.5

1

𝐷

Non-local only

Non-local – CBM

(14)

Non-local elastic damage to Cohesive Band Model transition

– Influence of ℎ

b

(for a given 𝑙

c

) on response in a 1D elastic case

[Leclerc et al. 2018]

• Has effect on the totally dissipated energy Φ

• Could be chosen to conserve energy dissipation (physically based) • For elastic damage: ℎb ≃ 5.4 𝑙𝑐

(15)

Study of triaxiality effect on a slit-plate

– Biaxial loading

• Loading at constant 𝐹𝑥/ 𝐹𝑦 ratio • Plane-strain state

– Comparison between:

• Pure non-local

• Non-local + cohesive zone (CZM) • Non-local + cohesive band (CBM)

Damage to crack transition for elastic damage – Proof of concept

𝐹

𝑦

𝑊

𝐹

𝑥

0

0.5

1

𝐷

Non-local - CZM

Non-local - CBM

Non-local only

(16)

Study of triaxiality effect on a slit-plate

– Reference dissipated energy Φ

ref

for non-local with

𝐹

𝑥

/

𝐹

𝑦

=0

Damage to crack transition for elastic damage – Proof of concept

Non-Local only Non-Local - CZM Non-Local - CBM

Error on total

diss. energy

-

CZM:

~30%

-

CBM:

~3%

𝐹

𝑥

=

𝐹

𝑦

1

2

𝐹

𝑥

=

𝐹

𝑦

1

2

(17)

Hyperelastic-based formulation

– Multiplicative decomposition

– Stress tensor definition

• Elastic potential 𝜓 𝐂e

• First Piola-Kirchhoff stress tensor

• Kirchhoff stress tensors – In current configuration 𝜿 = 𝐏 ⋅ 𝐅𝑇 = 2𝐅e ⋅𝜕𝜓 𝐂e 𝜕𝐂e ⋅ 𝐅 e𝑇 – In co-rotational space 𝝉 = 𝐂e ⋅ 𝐅e−1⋅𝜿 ⋅ 𝐅e−𝑇 = 2𝐂e ⋅𝜕𝜓 𝐂e 𝜕𝐂e

Logarithmic deformation

– Elastic potential 𝜓:

– Stress tensor in co-rotational space

Non-local porous plasticity model

𝐅 = 𝐅e ⋅ 𝐅p, 𝐂e = 𝐅e𝑇⋅ 𝐅e, 𝐽𝒆 = det 𝐅e 𝐏 = 2𝐅e ⋅𝜕𝜓 𝐂 e 𝜕𝐂e ⋅ 𝐅p −𝑇 𝜓 𝐂e = 𝐾 2ln 2 𝐽𝑒 +𝐺 4 ln 𝐂 e dev: ln 𝐂e dev 𝝉 = 𝐾ln 𝐽𝑒 𝑝 𝐈 + 𝐺 ln 𝐂e dev x Ω 𝐛 = 𝐅 ⋅ 𝐅𝑇 𝝈 = 𝜿 𝐽−1 X Ω0 𝐂 = 𝐅𝑇 ⋅ 𝐅 𝐒 = 𝐅−1⋅ 𝐏 𝝃 Ω𝜉 𝝃𝒄 Ω𝑐 𝐂e = 𝐅e𝑇⋅ 𝐅e 𝝉 = 𝑭e𝑇⋅ 𝜿 ⋅ 𝐅e−𝑇

𝐅

𝐅

p

𝐅

e

𝐑

e

𝐔

e

(18)

Porous plasticity (or Gurson) approach

– Assuming a J2-(visco-)plastic matrix

Damage to crack transition in porous elasto-plasticity

(19)

Porous plasticity (or Gurson) approach

– Assuming a J2-(visco-)plastic matrix

– Including effects of void/defect or porosity on plastic behavior

• Apparent macroscopic yield surface 𝑓(𝜏eq, 𝑝) ≤ 0 due to microstructural state:

Damage to crack transition in porous elasto-plasticity

(20)

Porous plasticity (or Gurson) approach

– Assuming a J2-(visco-)plastic matrix

– Including effects of void/defect or porosity on plastic behavior

• Apparent macroscopic yield surface 𝑓(𝜏eq, 𝑝) ≤ 0 due to microstructural state:

» Diffuse plastic flow spreads in the matrix » Gurson model

(21)

Porous plasticity (or Gurson) approach

– Assuming a J2-(visco-)plastic matrix

– Including effects of void/defect or porosity on plastic behavior

• Apparent macroscopic yield surface 𝑓(𝜏eq, 𝑝) ≤ 0 due to microstructural state: – Competition between two deformation modes:

» Diffuse plastic flow spreads in the matrix » Gurson model

» Before failure: coalescence or localized plastic flow between voids » GTN or Thomason models

(22)

Porous plasticity (or Gurson) approach

– Assuming a J2-(visco-)plastic matrix

– Including effects of void/defect or porosity on plastic behavior

• Apparent macroscopic yield surface 𝑓(𝜏eq, 𝑝) ≤ 0 due to microstructural state: – Competition between two deformation modes:

» Diffuse plastic flow spreads in the matrix » Gurson model

» Before failure: coalescence or localized plastic flow between voids » GTN or Thomason models

– Including evolution of microstructure during failure process

• Void growth by diffuse plastic flow

• Apparent growth by shearing

• Nucleation / appearance of new voids • Void coalescence until failure

(23)

Yield surface is considered in the co-rotational space

– Non-local form 𝑓 𝜏

eq

, 𝑝; 𝜏

Y

, 𝒁 𝑡

,

𝑓

𝑉

𝑡

≤ 0

• 𝜏eq is the von Mises equivalent Kirchhoff stress and 𝑝 the pressure

• 𝜏Y = 𝜏Y 𝑝, 𝑝 is the viscoplastic yield stress in terms of equivalent matrix plastic strain 𝑝 • 𝑓V is the porosity and 𝑓V, its non-local counterpart with 𝑓𝑉 − 𝑙𝑐2Δ 𝑓𝑉 = 𝑓𝑉

• 𝒁 is the vector of internal variables • 𝑙c is the non-local length

– Normal plastic flow 𝐃

p

– Microstructure evolution (spherical voids):

• Equivalent matrix plastic strain rate

• Porosity:

• Ligament ratio:

Non-local porous plasticity model

Microstructure parameters

𝜆𝐿

𝜒𝐿

𝐿

𝑓

V

𝑝 =

𝛕:𝐃p 1−𝑓𝑉0 𝜏Y

𝑓

𝑉

= 1 − 𝑓

𝑉

tr(𝐃

p

) +

𝑓

nucl

+

𝑓

shear

𝜒 =

𝜒 𝜒,

𝑓

𝑉

, 𝜅, 𝜆, 𝒁

𝐃

p

= 𝐅

p

⋅ 𝐅

p−𝟏

= 𝛾

𝜕𝑓

𝜕𝝉

=

𝑑

𝜕𝜏

eq

𝜕𝝉

+

𝑞

𝜕𝑝

𝜕𝝉

(24)

Plane strain specimen

[Besson et al. 2003]

– Only half specimen is modelled

– Three ≠ mesh sizes

Non-local porous plasticity – Comparison with literature results

Medium mesh (~8100 elements,𝑙m ≅ 0.75 𝑙c ) Coarse mesh (~4600 elements, 𝑙m ≅ 1.12 𝑙c ) Fine mesh (~15500 elements,𝑙m ≅ 0.5 𝑙c )

𝑒

0

= 5𝑚𝑚

𝑒

0

𝐹

𝐹

(25)

Gurson model

[Reush et al. 2003]

– Particularized yield surface

Verification of non-local model

Non-local porous plasticity – void growth

𝑓

V

→ 1

> 0.1

𝑓

V0

𝑓

V

0.01

𝑓

G

=

𝜏eq 2 𝜏Y2

+ 2𝑞

1

𝑓

𝑉

cosh

𝑞2𝑝 2𝜏Y

− 1 − 𝑞

3 2

𝑓

𝑉2

≤ 0

(26)

Thomason model

[Benzerga 2014, Besson 2009]

– Particularized yield surface

– Higher porosity to trigger coalescence

– No lateral contraction due to plasticity

Verification of non-local model

– For 𝜅 = 0.5; 𝜆 = 0.5; 𝑙

𝑐

= 50 𝜇m

Non-local porous plasticity – void coalescence

𝜒 → 1

𝑓

T

=

2 3

𝜏

eq

+ 𝑝 − 𝐶

T 𝑓

𝜒 𝜏

Y

≤ 0

1

𝜒

0

𝜒

(27)

Coupled non-local Gurson-Thomason

– Competition between 𝑓

G

and 𝑓

T

For 𝜅 = 0.5; 𝜆 = 0.5; 𝑙

𝑐

= 50 𝜇m

Non-local porous plasticity – void growth and coalescence

𝑓

G

=

𝜏eq 2 𝜏Y2

+ 2𝑞

1

𝑓

𝑉

cosh

𝑞2𝑝 2𝜏Y

− 1 − 𝑞

3 2

𝑓

𝑉2

≤ 0

𝑓

T

=

2 3

𝜏

eq

+ 𝑝 − 𝐶

T 𝑓

𝜒 𝜏

Y

≤ 0

> 0.1

𝑓

V0

𝑓

V

0.01

𝑢

(28)

Non-local Gurson model – CBM (arbitrary crack paths)

– Gurson material model

– Crack insertion at Thomasson criterion

– At crack insertion: Cohesive Band Model

– Comparison of two coalescence models

• Phenomenological approach:

• Thomason model:

Damage to crack transition for porous plasticity

𝑵 ⋅ 𝝉 ⋅ 𝑵 − 𝐶

T𝑓

𝜒 𝜏

Y

= 0

𝑓

𝑉

=

𝑓

𝑉

if

𝑓

𝑉

≤ 𝑓

𝐶

𝑓

𝐶

+ 𝑅

𝑓

𝑉

− 𝑓

𝐶

if

𝑓

𝑉

> 𝑓

𝐶

𝑓

G

=

𝜏eq 2 𝜏Y2

+ 2𝑞

1

𝑓

𝑉

cosh

𝑞2𝑝 2𝜏Y

− 1 − 𝑞

3 2

𝑓

𝑉2

≤ 0

𝑓

T

=

2 3

𝜏

eq

+ 𝑝 − 𝐶

T 𝑓

𝜒 𝜏

Y

≤ 0

𝒖

N

N

𝐅, 𝛔

𝑵, 𝒏

Bulk

𝒖

𝒉

b

𝐅

b

, 𝛔

b

𝑵, 𝒏

Band

Bulk

𝐅, 𝛔

(29)

Non-local Gurson model – CBM

– CBM insertion at Thomason criterion

– CBM with coalescence model

• Comparison of 2 coalescence models • For 𝜅 = 0.5; 𝜆 = 0.5; 𝑙𝑐 = 50 𝜇m

Damage to crack transition for porous plasticity

Thomason coalescence

𝑓

V0

𝑓

V

> 0.1

0.01

Phenomenological coalescence

CDM-CBM

CDM only

(30)

Round-bar test

– CBM insertion at Thomason

criterion

– CBM with Thomasson coalescence

model

– Nucleation not identified yet

– Crack path in cup-cone shape

[Besson et al. 2001]

Damage to crack transition for porous plasticity

Thomason coalescence

(31)

Objective:

– Simulation of material degradation and crack initiation / propagation

Methodology

– Combination of non-local Continuum Damage Model (CDM)

– Cohesive Band Model (CBM)

– Integrated in a Hybrid Discontinuous Galerkin framework

Proof of concept

– Elastic damage material model

– Cohesive band thickness controls the failure energy dissipation

Ductile materials

– Implementation of hyperelastic non-local porous-plastic model

• Coupled Gurson-Thomason model

– First results of the CDM-CBM transition

– Upcoming tasks:

• Enrichment of nucleation model and coalescence model • Calibration of the band thickness

• Validation/Calibration with literature/experimental tests

Conclusions

(32)

Computational & Multiscale

Mechanics of Materials

CM3

www.ltas-cm3.ulg.ac.be

Thank you for your attention

Computational & Multiscale Mechanics of Materials – CM3

http://www.ltas-cm3.ulg.ac.be/

B52 - Quartier Polytech 1

Allée de la découverte 9, B4000 Liège Julien.Leclerc@ulg.ac.be

(33)

Comparison with phase field

– Single edge notched specimen

[Miehe et al. 2010]

• Calibration of damage and CBM parameters with 1D case [Leclerc et al. 2018]

Damage to crack transition for elastic damage – Proof of concept

Non-local model T ens ion tes t Shearing tes t

(34)

Validation with Compact Tension Specimen

[Geers 1997]

– Better agreement with the cohesive band model than the cohesive zone model or the

non-local model alone

[Leclerc et al. 2018]

(35)

Evolution of local porosity

– Voids nucleation 𝑓

nucl

modifies porosity growth rate

• Linear strain-controlled growth

• Gaussian strain-controlled growth

• where 𝐴N, 𝑓N, 𝜖N, 𝑠N are material parameters

Porous plasticity – Voids nucleation

𝑓

𝑉

= 1 − 𝑓

𝑉

tr(𝐃

p

) +

𝑓

nucl

+

𝑓

shear

𝑓

nucl

= 𝐴

N

𝑝 with

𝐴

N

≠ 0

if 𝑓

𝑉

> 𝑓

N

𝐴

N

= 0

if 𝑓

𝑉

≤ 𝑓

N

𝑓

nucl

=

𝑓

N

√ 2𝜋𝑠

N2

exp −

𝑝 − 𝜖

N 2

2𝑠

N2

𝑝

(36)

Evolution of local porosity

– Shearing affect voids nucleation: 𝑓

shear

• Includes Lode variable effect 𝜁 𝛕 = −27 det 𝝉dev

2 𝜏eq3

• where 𝑘w is a material parameter

Porous plasticity – Voids nucleation

𝑓

𝑉

= 1 − 𝑓

𝑉

tr(𝐃

p

) +

𝑓

nucl

+

𝑓

shear

𝑓

shear

= 𝑓

𝑉

𝑘

w

1 − 𝜁

2

𝝉

𝝉

dev

: 𝐃

p

(37)

Predictor-corrector procedure

– Elastic predictor

– Plastic corrector (radial return-like algorithm)

• 3 equations

– Consistency equation: – Plastic flow rule:

– Matrix plastic strain evolution: • 3 Unknowns Δ 𝑑, Δ 𝑞, Δ 𝑝

• 3 linearized equations

– Consistency equation:

– Plastic flow rule:

– Matrix plastic strain evolution:

Integration algorithm

𝑓 𝜏

eq

Δ

𝑑 , 𝑝 Δ

𝑞 ; 𝜏

Y

Δ 𝑝 , 𝒁 Δ

𝑑, Δ

𝑞, Δ 𝑝 ,

𝑓

𝑉

= 0

𝐅

e𝐩𝐫

= 𝐅 ⋅ 𝐅

𝒏p−1

𝑝 =

𝛕:𝐃p 1−𝑓𝑉0 𝜏Y

𝐃

p

= 𝐅

p

⋅ 𝐅

p−𝟏

= 𝛾

𝜕𝑓

𝜕𝝉

=

𝑑

𝜕𝜏

eq

𝜕𝝉

+

𝑞

𝜕𝑝

𝜕𝝉

𝑓 𝜏

eq

, 𝑝; 𝜏

Y

, 𝒁 𝑡

,

𝑓

𝑉

𝑡

= 0

1 − 𝑓

𝑉0

𝜏

Y

Δ 𝑝 = 𝜏

eq

Δ

𝑑 + 𝑝Δ

𝑞

Δ

𝑑

𝜕𝑓

𝜕𝑝

− Δ

𝑞

𝜕𝑓

𝜕𝜏

eq

= 0

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