A Damage to Crack Transition Framework for Ductile Materials Accounting for Stress Triaxiality

Computational & Multiscale

Mechanics of Materials

CM3

www.ltas-cm3.ulg.ac.be

A Damage to Crack Transition Framework for Ductile Materials

Accounting for Stress Triaxiality

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 16th_{MECATECH call.}

•

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

•

State-of-the-art

– 2 approaches modeling material failure:

• Continuum Damage Models (CDM)

– Lemaitre-Chaboche, – Gurson,

– …

• Discontinuous: Fracture mechanics – Cohesive zone,

– XFEM – …

•

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 𝑙_𝑐 • In terms of Weight functions: 𝑓_𝑉 𝒙 = _𝑉

𝐶𝑊 𝒚; 𝒙, 𝑙𝑐 𝑓𝑉 𝐲 𝑑𝒚

– Implicit formulation

[Peerlings et al. 1998]

• New non-local degrees of freedom 𝑓

State of art: Continuous approaches

The numerical results change without convergence

𝜎

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

•

Based on fracture mechanics concepts

• Characterized by

• Strength s_c &

• Critical energy release rate G_C

•

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

•

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

•

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

State of art: Discontinuous approaches

𝒖

t

t

t

G

_C

s

_c (a-1)+ _(a)+ x (a+1) -u (a+1)+ (a) -(a-1) -Ω₀ 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

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

•

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

•

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 𝐅_b = 𝐅 + 𝒖 ⊗𝑵

ℎ_𝑏 + 1

2𝛁𝑇 𝒖

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

•

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

𝜅_𝑐−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

•

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 𝜅_𝑐−𝜅 𝜅 with 𝜅 = max_𝑡′ 𝑒 𝑡 ′

– 1D non-local test

Damage to crack transition for elastic damage – Proof of concept

0

0.5

1

𝐷

Non-local only

•

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

•

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

•

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

•

Study of triaxiality effect on a slit-plate

– Biaxial loading

• Loading at constant 𝐹_𝑥/ 𝐹_𝑦 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

•

Study of triaxiality effect on a slit-plate

– Biaxial loading

• Loading at constant 𝐹_𝑥/ 𝐹_𝑦 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

0.5

1

𝐷

Non-local - CZM

Non-local - CBM

Non-local only

•

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

•

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:

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 𝐂}e 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

•

Porous plasticity (or Gurson) approach

– Assuming a J2-(visco-)plastic matrix

Damage to crack transition in porous elasto-plasticity

•

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(𝜏_eq, 𝑝) ≤ 0 due to microstructural state:

Damage to crack transition in porous elasto-plasticity

•

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(𝜏_eq, 𝑝) ≤ 0 due to microstructural state:

» Diffuse plastic flow spreads in the matrix » Gurson model

•

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(𝜏_eq, 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

•

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(𝜏_eq, 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

•

Yield surface is considered in the co-rotational space

– Non-local form 𝑓 𝜏

_eq

, 𝑝; 𝜏

_Y

, 𝒁 𝑡

′

,

𝑓

_𝑉

𝑡

′

≤ 0

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

• 𝜏_Y = 𝜏_Y 𝑝, 𝑝 is the viscoplastic yield stress in terms of equivalent matrix plastic strain 𝑝 • 𝑓_V is the porosity and 𝑓_V, its non-local counterpart with 𝑓_𝑉 − 𝑙_𝑐2Δ 𝑓_𝑉 = 𝑓_𝑉

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

– Normal plastic flow 𝐃

p

– Microstructure evolution (spherical voids):

• Equivalent matrix plastic strain rate

• Porosity:

Non-local porous plasticity model

𝜆𝐿

𝑓

𝑝 =

𝛕:𝐃p 1−𝑓_𝑉0 𝜏_Y

𝑓

_𝑉

= 1 − 𝑓

_𝑉

tr(𝐃

p

) +

𝑓

_nucl

+

𝑓

_shear

𝐃

p

= 𝐅

p

⋅ 𝐅

p−𝟏

= 𝛾

𝜕𝑓

𝜕𝝉

=

𝑑

𝜕𝜏

_eq

𝜕𝝉

+

𝑞

𝜕𝑝

𝜕𝝉

•

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,𝑙_m ≅ 0.75 𝑙_c ) Coarse mesh (~4600 elements, 𝑙_m ≅ 1.12 𝑙_c ) Fine mesh (~15500 elements,𝑙_m ≅ 0.5 𝑙_c )

𝑒

₀

= 5𝑚𝑚

𝑒

₀

𝐹

𝐹

•

Gurson model

[Reush et al. 2003]

– Particularized yield surface

•

Verification of non-local model

Non-local porous plasticity – void growth

𝑓

_V

→ 1

𝑓

_G

=

𝜏eq 2 𝜏_Y2

+ 2𝑞

1

𝑓

𝑉

cosh

𝑞₂𝑝 2𝜏_Y

− 1 − 𝑞

3 2

_𝑓

𝑉2

≤ 0

•

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 𝜏_Y2

+ 2𝑞

1

𝑓

𝑉

cosh

𝑞₂𝑝 2𝜏_Y

− 1 − 𝑞

3 2

_𝑓

𝑉2

≤ 0

•

Gurson model

[Reush et al. 2003]

– Phenomenological coalescence model:

• Replace 𝑓_V by an effective value 𝑓_V∗:

• 𝑓_𝐶 from concentration factor 𝐶_T𝑓 𝜒 [Benzerga2014]

Non-local porous plasticity – void growth and coalescence

𝑓

_𝑉∗

=

𝑓

𝑉

if

𝑓

𝑉

≤ 𝑓

𝐶

𝑓

_𝐶

+ 𝑅

𝑓

_𝑉

− 𝑓

_𝐶

if

𝑓

_𝑉

> 𝑓

_𝐶

𝜆𝐿

𝜒𝐿

𝐿

𝑓

_V

𝜒 =

𝜒 𝜒,

𝑓

_𝑉

, 𝜅, 𝜆, 𝒁

𝑢

max eig 𝝉 − 𝐶

_T𝑓

𝜒 𝜏

_Y

= 0

•

Gurson model

[Reush et al. 2003]

– Phenomenological coalescence model:

• Replace 𝑓_V by an effective value 𝑓_V∗:

• 𝑓_𝐶 from concentration factor 𝐶_T𝑓 𝜒 [Benzerga2014]

Non-local porous plasticity – void growth and coalescence

> 0.1

𝑓

_V0

𝑓

_V

0.01

𝑓

_𝑉∗

=

𝑓

𝑉

if

𝑓

𝑉

≤ 𝑓

𝐶

𝑓

_𝐶

+ 𝑅

𝑓

_𝑉

− 𝑓

_𝐶

if

𝑓

_𝑉

> 𝑓

_𝐶

𝜆𝐿

𝜒𝐿

𝐿

𝑓

_V

𝜒 =

𝜒 𝜒,

𝑓

_𝑉

, 𝜅, 𝜆, 𝒁

𝑢

max eig 𝝉 − 𝐶

_T𝑓

𝜒 𝜏

_Y

= 0

•

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

•

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

•

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 𝜏_Y2

+ 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

•

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 𝜏_Y2

+ 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

•

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:

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 𝜏_Y2

+ 2𝑞

1

𝑓

𝑉

cosh

𝑞₂𝑝 2𝜏_Y

− 1 − 𝑞

3 2

_𝑓

𝑉2

≤ 0

𝒖

N

N

𝐅, 𝛔

𝑵, 𝒏

Bulk

𝒖

_𝒉

b

𝐅

_b

, 𝛔

_b

𝑵, 𝒏

Band

Bulk

𝐅, 𝛔

•

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

– Crack path in cup-cone shape

[Besson et al. 2001]

Damage to crack transition for porous plasticity

Thomason coalescence

𝑓

_V0

𝑓

_V

> 0.1

0.01

Phenomenological coalescence

CDM-CBM

CDM only

•

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

– Crack path in cup-cone shape

[Besson et al. 2001]

Damage to crack transition for porous plasticity

Thomason coalescence

𝑓

_V

> 0.1

0.01

Phenomenological coalescence

CDM-CBM

CDM only

•

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

Computational & Multiscale

Mechanics of Materials

CM3

www.ltas-cm3.ulg.ac.be

Thank you for your attention

Computational & Multiscale Mechanics of Materials – CM3

•

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

•

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 tes t

•

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]

•

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]

•

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

•

Evolution of local porosity

– Shearing affect voids nucleation: 𝑓

_shear

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

2 𝜏_eq3

• where 𝑘_w 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

•

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−𝑓_𝑉0 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