Mesh-independent modelling of diffuse cracking in cohesive grain-based materials

Mesh-independent modelling of diffuse cracking in cohesive grain-based materials

Public defense Zoltan Csati

Nantes 21st October 2019

Institut de Recherche en Génie Civil et Mécanique

G M

Topics

1 Motivation

2 About the title

3 Workflow

4 Physical model

5 Contributions

6 Kinematics

7 Governing equations

8 Discretization

9 Stable mixed method

10 Cracking

11 Examples

12 Conclusions

Motivation About the title Workflow Physical model Contributions Kinematics Governing equations Discretization Stable mixed method Cracking Examples Conclusions

Motivation of the thesis

https://tinyurl.com/ybuseuph

Zoltan Csati Mesh-independent modelling of diffuse cracking 1 / 44

Motivation About the title Workflow Physical model Contributions Kinematics Governing equations Discretization Stable mixed method Cracking Examples Conclusions

Motivation of the thesis

Fig. 2 from DOI: 10.1007/s12665-017-6696-4

Motivation About the title Workflow Physical model Contributions Kinematics Governing equations Discretization Stable mixed method Cracking Examples Conclusions

Motivation of the thesis

Experiments 3 reality

7 occasionally costly

Numerical simulations 3 rapid prototyping 3 parametric studies

7 representative microstructure 7 needs calibration

Zoltan Csati Mesh-independent modelling of diffuse cracking 3 / 44

Motivation About the title Workflow Physical model Contributions Kinematics Governing equations Discretization Stable mixed method Cracking Examples Conclusions

Motivation of the thesis

Experiments 3 reality

7 occasionally costly

Numerical simulations 3 rapid prototyping 3 parametric studies

7 representative microstructure

7 needs calibration

Motivation About the title Workflow Physical model Contributions Kinematics Governing equations Discretization Stable mixed method Cracking Examples Conclusions

Motivation of the thesis

Multiple cracking in cohesive grain-based materials

Offer a computational tool for microstructural simulations

Capture the relevant physics but be computationally efficient

No user intervention

Studied material: granit

Zoltan Csati Mesh-independent modelling of diffuse cracking 4 / 44

Motivation About the title Workflow Physical model Contributions Kinematics Governing equations Discretization Stable mixed method Cracking Examples Conclusions

Motivation of the thesis

Multiple cracking in cohesive grain-based materials

Offer a computational tool for microstructural simulations

Capture the relevant physics but be computationally efficient

No user intervention

Studied material: granit

Motivation About the title Workflow Physical model Contributions Kinematics Governing equations Discretization Stable mixed method Cracking Examples Conclusions

About the title

Cohesive grain-based materials

Zoltan Csati Mesh-independent modelling of diffuse cracking 5 / 44

Motivation About the title Workflow Physical model Contributions Kinematics Governing equations Discretization Stable mixed method Cracking Examples Conclusions

About the title

Diffuse cracking

Motivation About the title Workflow Physical model Contributions Kinematics Governing equations Discretization Stable mixed method Cracking Examples Conclusions

About the title

Mesh-independent

3 no meshing burden 3 simpler data structures

3 precompute the Jacobian matrix

Zoltan Csati Mesh-independent modelling of diffuse cracking 7 / 44

Motivation About the title Workflow Physical model Contributions Kinematics Governing equations Discretization Stable mixed method Cracking Examples Conclusions

About the title

Mesh-independent

3 no meshing burden 3 simpler data structures

3 precompute the Jacobian matrix

Motivation About the title Workflow Physical model Contributions Kinematics Governing equations Discretization Stable mixed method Cracking Examples Conclusions

Fig. 15 (d) from DOI: 10.1520/GTJ20170098

How can we get such simulation results?

Zoltan Csati Mesh-independent modelling of diffuse cracking 8 / 44

Motivation About the title Workflow Physical model Contributions Kinematics Governing equations Discretization Stable mixed method Cracking Examples Conclusions

Fig. 15 (d) from DOI: 10.1520/GTJ20170098

How can we get such simulation results?

Motivation About the title Workflow Physical model Contributions Kinematics Governing equations Discretization Stable mixed method Cracking Examples Conclusions

Workflow

1 Identify the most important phenomena we want to model

2 Make use of the a priori known crack paths

3 Construct a non-conforming discretization

4 Ensure the stability of the scheme

5 Verify the discretization

6 Couple it with crack propagation

7 Validate the model on various test cases

Zoltan Csati Mesh-independent modelling of diffuse cracking 9 / 44

Motivation About the title Workflow Physical model Contributions Kinematics Governing equations Discretization Stable mixed method Cracking Examples Conclusions

Physical model

Complexity due to the microstructure

Behaviour at the meso-scale

Motivation About the title Workflow Physical model Contributions Kinematics Governing equations Discretization Stable mixed method Cracking Examples Conclusions

Physical model

Complexity due to the microstructure Behaviour at the meso-scale

E

ν

E

ν

E

ν

No crack

Zoltan Csati Mesh-independent modelling of diffuse cracking 10 / 44

Motivation About the title Workflow Physical model Contributions Kinematics Governing equations Discretization Stable mixed method Cracking Examples Conclusions

Physical model

Complexity due to the microstructure Behaviour at the meso-scale

E

ν

E

ν

E

ν

J u K

Open cracks everywhere, floating

grains

Motivation About the title Workflow Physical model Contributions Kinematics Governing equations Discretization Stable mixed method Cracking Examples Conclusions

Physical model

Complexity due to the microstructure Behaviour at the meso-scale

E

ν

E

ν

E

ν

λ

λ

λ

λ

λ

λ

λ

λ

λ

Lagrange multipliers “glue” the grains together

Zoltan Csati Mesh-independent modelling of diffuse cracking 10 / 44

Motivation About the title Workflow Physical model Contributions Kinematics Governing equations Discretization Stable mixed method Cracking Examples Conclusions

Physical model

Complexity due to the microstructure Behaviour at the meso-scale

E

ν

E

ν

E

ν

λ

λ

λ

λ

To much “glue” causes

instability

Motivation About the title Workflow Physical model Contributions Kinematics Governing equations Discretization Stable mixed method Cracking Examples Conclusions

Physical model

Complexity due to the microstructure Behaviour at the meso-scale

E

ν

E

ν

λ

λ

E

ν

J u K (λ

) G

= G

− a t

|{z}

Allow gradual crack opening, cohesive zone

Zoltan Csati Mesh-independent modelling of diffuse cracking 10 / 44

Motivation About the title Workflow Physical model Contributions Kinematics Governing equations Discretization Stable mixed method Cracking Examples Conclusions

Contributions

CutFEM with Lagrange multipliers for Q1 elements with arbitrary discontinuities

Mixed-mode crack propagation Non-uniform fracture energy Damage-driven solution procedure

Robust discretization and crack propagation

Motivation About the title Workflow Physical model Contributions Kinematics Governing equations Discretization Stable mixed method Cracking Examples Conclusions

Kinematics

Discrete crack model

= ⇒

Discontinuity in the displacement field: J u K 6= 0 Suppress discontinuity for cracks not yet appeared

Zoltan Csati Mesh-independent modelling of diffuse cracking 12 / 44

Motivation About the title Workflow Physical model Contributions Kinematics Governing equations Discretization Stable mixed method Cracking Examples Conclusions

Kinematics

Discrete crack model

= ⇒

Discontinuity in the displacement field: J u K 6= 0

Suppress discontinuity for cracks not yet appeared

Motivation About the title Workflow Physical model Contributions Kinematics Governing equations Discretization Stable mixed method Cracking Examples Conclusions

Kinematics

Discrete crack model

= ⇒

Discontinuity in the displacement field: J u K 6= 0 Suppress discontinuity for cracks not yet appeared

Zoltan Csati Mesh-independent modelling of diffuse cracking 12 / 44

Motivation About the title Workflow Physical model Contributions Kinematics Governing equations Discretization Stable mixed method Cracking Examples Conclusions

Governing equations

Γ _D

Γ _N Ω

σ · ∇ = 0, x ∈ Ω σ = C : ε, x ∈ Ω ε = ∇ ^s u, x ∈ Ω u = u D , x ∈ Γ _D σ · n = t N , x ∈ Γ _N

Γ ³ _D

Γ ² _N Ω ¹

Ω ³ Ω ² Γ ¹ _D

Γ ¹ Γ ² Γ ³

σ ^m · ∇ = 0, x ∈ Ω ^m σ ^m = C ^m : ε ^m , x ∈ Ω ^m ε ^m = ∇ ^s u ^m , x ∈ Ω ^m u ⁱ = u ⁱ _D , x ∈ Γ ⁱ _D σ ⁱ · n ⁱ = t ⁱ _N , x ∈ Γ ⁱ _N JuK ⁱ = 0 , J σ K

i · n ⁱ = 0 , x ∈ Γ ⁱ

Motivation About the title Workflow Physical model Contributions Kinematics Governing equations Discretization Stable mixed method Cracking Examples Conclusions

Governing equations

Γ _D

Γ _N Ω

σ · ∇ = 0, x ∈ Ω σ = C : ε, x ∈ Ω ε = ∇ ^s u, x ∈ Ω u = u D , x ∈ Γ _D σ · n = t N , x ∈ Γ _N

Γ ³ _D

Γ ² _N Ω ¹

Ω ³ Ω ² Γ ¹ _D

Γ ¹ Γ ² Γ ³

σ ^m · ∇ = 0, x ∈ Ω ^m σ ^m = C ^m : ε ^m , x ∈ Ω ^m ε ^m = ∇ ^s u ^m , x ∈ Ω ^m u ⁱ = u ⁱ _D , x ∈ Γ ⁱ _D σ ⁱ · n ⁱ = t ⁱ _N , x ∈ Γ ⁱ _N JuK ⁱ = 0, J σ K

i · n ⁱ = 0, x ∈ Γ ⁱ

Zoltan Csati Mesh-independent modelling of diffuse cracking 13 / 44

Motivation About the title Workflow Physical model Contributions Kinematics Governing equations Discretization Stable mixed method Cracking Examples Conclusions

Governing equations

Weak form

Find u ∈ V and λ ∈ Λ such that

a(u, v) + b(λ, v) = f (v), ∀ v ∈ V

b(u, µ) = g(µ), ∀µ ∈ Λ

Motivation About the title Workflow Physical model Contributions Kinematics Governing equations Discretization Stable mixed method Cracking Examples Conclusions

Discretization

Discrete weak form

Find u _h ∈ V _h and λ _h ∈ Λ _h such that

a _h (u _h , v _h ) + b _h (λ _h , v _h ) = f _h (v _h ), ∀ v _h ∈ V _h b _h (u _h , µ _h ) = g _h (µ _h ), ∀µ _h ∈ Λ _h

Requirements FEM

Decouple mesh from domain Stability

Adapted method CutFEM

Lagrange multipliers No interfacial mesh

Zoltan Csati Mesh-independent modelling of diffuse cracking 15 / 44

Motivation About the title Workflow Physical model Contributions Kinematics Governing equations Discretization Stable mixed method Cracking Examples Conclusions

Discretization

Discrete weak form

Find u _h ∈ V _h and λ _h ∈ Λ _h such that

a _h (u _h , v _h ) + b _h (λ _h , v _h ) = f _h (v _h ), ∀ v _h ∈ V _h b _h (u _h , µ _h ) = g _h (µ _h ), ∀µ _h ∈ Λ _h Requirements

FEM

Decouple mesh from domain Stability

Adapted method CutFEM

Lagrange multipliers

No interfacial mesh

Motivation About the title Workflow Physical model Contributions Kinematics Governing equations Discretization Stable mixed method Cracking Examples Conclusions

Discretization

Discrete weak form

Find u _h ∈ V _h and λ _h ∈ Λ _h such that

a _h (u _h , v _h ) + b _h (λ _h , v _h ) = f _h (v _h ), ∀ v _h ∈ V _h b _h (u _h , µ _h ) = g _h (µ _h ), ∀µ _h ∈ Λ _h Requirements

FEM

Decouple mesh from domain Stability

Adapted method CutFEM

Lagrange multipliers No interfacial mesh

Zoltan Csati Mesh-independent modelling of diffuse cracking 15 / 44

Motivation About the title Workflow Physical model Contributions Kinematics Governing equations Discretization Stable mixed method Cracking Examples Conclusions

CutFEM

M ⁱ M ^j M Ω ⁱ Ω ^j

V _h ⁱ := span{ ψ ˆ ⁱ _j | j ∈ M ⁱ } V h = M

i∈I s

V _h ⁱ

Motivation About the title Workflow Physical model Contributions Kinematics Governing equations Discretization Stable mixed method Cracking Examples Conclusions

CutFEM

M ⁱ M ^j M Ω ⁱ Ω ^j

V _h ⁱ := span{ ψ ˆ ⁱ _j | j ∈ M ⁱ } V h = M

i∈I s

V _h ⁱ

Zoltan Csati Mesh-independent modelling of diffuse cracking 16 / 44

Motivation About the title Workflow Physical model Contributions Kinematics Governing equations Discretization Stable mixed method Cracking Examples Conclusions

CutFEM

M ⁱ M ^j M Ω ⁱ Ω ^j

V _h ⁱ := span{ ψ ˆ ⁱ _j | j ∈ M ⁱ }

V h = M

i∈I s

V _h ⁱ

Motivation About the title Workflow Physical model Contributions Kinematics Governing equations Discretization Stable mixed method Cracking Examples Conclusions

CutFEM

M ⁱ M ^j M Ω ⁱ Ω ^j

V _h ⁱ := span{ ψ ˆ ⁱ _j | j ∈ M ⁱ } V h = M

i∈I s

V _h ⁱ

Zoltan Csati Mesh-independent modelling of diffuse cracking 16 / 44

Motivation About the title Workflow Physical model Contributions Kinematics Governing equations Discretization Stable mixed method Cracking Examples Conclusions

CutFEM

M ⁱ _Γ M

Γ ⁱ

Λ ⁱ _h = span

n ψ ˜ _j ⁱ | j ∈ M ˜ ⁱ _Γ o

ψ ˜ ⁱ _j = X

k∈M ⁱ _Γ

α _jk ψ _k | _Γ i

Λ _h = M

i∈I int ∪I D

Λ ⁱ _h

Motivation About the title Workflow Physical model Contributions Kinematics Governing equations Discretization Stable mixed method Cracking Examples Conclusions

Implementation

Zoltan Csati Mesh-independent modelling of diffuse cracking 18 / 44

Motivation About the title Workflow Physical model Contributions Kinematics Governing equations Discretization Stable mixed method Cracking Examples Conclusions

Implementation

Motivation About the title Workflow Physical model Contributions Kinematics Governing equations Discretization Stable mixed method Cracking Examples Conclusions

Implementation

Zoltan Csati Mesh-independent modelling of diffuse cracking 18 / 44

Motivation About the title Workflow Physical model Contributions Kinematics Governing equations Discretization Stable mixed method Cracking Examples Conclusions

Implementation

Motivation About the title Workflow Physical model Contributions Kinematics Governing equations Discretization Stable mixed method Cracking Examples Conclusions

Implementation

Zoltan Csati Mesh-independent modelling of diffuse cracking 18 / 44

Motivation About the title Workflow Physical model Contributions Kinematics Governing equations Discretization Stable mixed method Cracking Examples Conclusions

Implementation

Motivation About the title Workflow Physical model Contributions Kinematics Governing equations Discretization Stable mixed method Cracking Examples Conclusions

Implementation

Zoltan Csati Mesh-independent modelling of diffuse cracking 18 / 44

Motivation About the title Workflow Physical model Contributions Kinematics Governing equations Discretization Stable mixed method Cracking Examples Conclusions

Implementation

Motivation About the title Workflow Physical model Contributions Kinematics Governing equations Discretization Stable mixed method Cracking Examples Conclusions

Concepts

Stability Goal

ensure stability decrease dim Λ _h

easy-to-implement procedure

Zoltan Csati Mesh-independent modelling of diffuse cracking 19 / 44

Motivation About the title Workflow Physical model Contributions Kinematics Governing equations Discretization Stable mixed method Cracking Examples Conclusions

The reduction algorithm

λ 2

λ 3 λ 1

λ 4 λ 5

λ 6 λ 7

Motivation About the title Workflow Physical model Contributions Kinematics Governing equations Discretization Stable mixed method Cracking Examples Conclusions

The reduction algorithm

λ 2

λ 4

λ 6

Zoltan Csati Mesh-independent modelling of diffuse cracking 20 / 44

Motivation About the title Workflow Physical model Contributions Kinematics Governing equations Discretization Stable mixed method Cracking Examples Conclusions

The reduction algorithm

λ 2

λ 4

λ 6

Motivation About the title Workflow Physical model Contributions Kinematics Governing equations Discretization Stable mixed method Cracking Examples Conclusions

The reduction algorithm

λ 2

λ 4

λ 6

Zoltan Csati Mesh-independent modelling of diffuse cracking 20 / 44

Motivation About the title Workflow Physical model Contributions Kinematics Governing equations Discretization Stable mixed method Cracking Examples Conclusions

Assessing the stability

4 8 16 32 64 128 256 10 ⁻²

10 ⁻¹ 10 ⁰

1/h

p β h ;min

Motivation About the title Workflow Physical model Contributions Kinematics Governing equations Discretization Stable mixed method Cracking Examples Conclusions

Assessing the stability

4 8 16 32 64 128 256 10 ⁻²

10 ⁻¹ 10 ⁰

1/h

| α h ;min |

Zoltan Csati Mesh-independent modelling of diffuse cracking 21 / 44

Motivation About the title Workflow Physical model Contributions Kinematics Governing equations Discretization Stable mixed method Cracking Examples Conclusions

Cohesive zone model

Discrete crack model Cohesive zone model

fracture process zone

J u K ^eq

cohesive zone t eq

t eq,c

J u K ^eq,c mathematical

crack tip

physical crack tip

fully developed crack intact material

crack lip

Motivation About the title Workflow Physical model Contributions Kinematics Governing equations Discretization Stable mixed method Cracking Examples Conclusions

Cohesive zone model

Discrete crack model Cohesive zone model

fracture process zone

J u K ^eq

cohesive zone t eq

t eq,c

J u K ^eq,c mathematical

crack tip

physical crack tip

fully developed crack intact material

crack lip

Zoltan Csati Mesh-independent modelling of diffuse cracking 22 / 44

Motivation About the title Workflow Physical model Contributions Kinematics Governing equations Discretization Stable mixed method Cracking Examples Conclusions

Cohesive zone model

Discrete crack model Cohesive zone model

fracture process zone

J u K ^eq

cohesive zone t eq

t eq,c

J u K ^eq,c mathematical

crack tip

physical crack tip

fully developed crack intact material

crack lip

Motivation About the title Workflow Physical model Contributions Kinematics Governing equations Discretization Stable mixed method Cracking Examples Conclusions

Cohesive zone model

J u K ^eq t _eq

t eq,c

J u K ^eq,c

(a) Intrinsic CZM

J u K ^eq t _eq

t _eq,c

J u K ^eq,c

(b) Extrinsic CZM

Zoltan Csati Mesh-independent modelling of diffuse cracking 23 / 44

Motivation About the title Workflow Physical model Contributions Kinematics Governing equations Discretization Stable mixed method Cracking Examples Conclusions

Cohesive zone model

J u K ^eq t _eq

t _eq,c

J u K ^eq,c

(a) Intrinsic CZM

J u K ^eq t eq

t _eq,c

J u K ^eq,c

(b) Extrinsic CZM

return

Motivation About the title Workflow Physical model Contributions Kinematics Governing equations Discretization Stable mixed method Cracking Examples Conclusions

Failure criterion

c

f t

−f c

t t

t n

1 3 2

4

φ

ψ

Zoltan Csati Mesh-independent modelling of diffuse cracking 24 / 44

Motivation About the title Workflow Physical model Contributions Kinematics Governing equations Discretization Stable mixed method Cracking Examples Conclusions

Mixed method for the cohesive formulation

No crack

Z

Γ

µ · J u K dΓ = 0

Cohesive crack

Z

Γ

µ · ( JuK − JuK (λ)) d Γ = 0

Contact

Z

Γ

µ · ( J u K − χ(t n > 0) J u K (λ)) dΓ = 0

Final form

Z

Γ

η · ( JuK − χ(ζ _n > 0 ) JuK (ζ)) d Γ = 0

J u K = 0

t n , t t T 0

Motivation About the title Workflow Physical model Contributions Kinematics Governing equations Discretization Stable mixed method Cracking Examples Conclusions

Mixed method for the cohesive formulation

No crack

Z

Γ

µ · J u K dΓ = 0

Cohesive crack

Z

Γ

µ · ( JuK − JuK (λ)) d Γ = 0

Contact

Z

Γ

µ · ( J u K − χ(t n > 0) J u K (λ)) dΓ = 0

Final form

Z

Γ

η · ( JuK − χ(ζ _n > 0 ) JuK (ζ)) d Γ = 0

J u K = 0 t n , t t T 0

J u K t n

t n

t t

t t

Zoltan Csati Mesh-independent modelling of diffuse cracking 25 / 44

Motivation About the title Workflow Physical model Contributions Kinematics Governing equations Discretization Stable mixed method Cracking Examples Conclusions

Mixed method for the cohesive formulation

No crack

Z

Γ

µ · J u K dΓ = 0

Cohesive crack

Z

Γ

µ · ( JuK − JuK (λ)) dΓ = 0

Contact

Z

Γ

µ · ( J u K − χ(t n > 0) J u K (λ)) dΓ = 0

Final form

Z

Γ

η · ( JuK − χ(ζ _n > 0 ) JuK (ζ)) d Γ = 0

J u K = 0 t n , t t T 0

J u K t n

t n

t t

t t

t n

t n

t t

t t

J u n K = 0

J u K

Motivation About the title Workflow Physical model Contributions Kinematics Governing equations Discretization Stable mixed method Cracking Examples Conclusions

Mixed method for the cohesive formulation

No crack

Z

Γ

µ · J u K dΓ = 0

Cohesive crack

Z

Γ

µ · ( JuK − JuK (λ)) dΓ = 0

Contact

Z

Γ

µ · ( J u K − χ(t n > 0) J u K (λ)) dΓ = 0

Final form

Z

Γ

η · ( JuK − χ(ζ _n > 0 ) JuK (ζ)) d Γ = 0

J u K = 0 t n , t t T 0

J u K t n

t n

t t

t t

t n

t n

t t

t t

J u n K = 0 J u K

Zoltan Csati Mesh-independent modelling of diffuse cracking 25 / 44

Motivation About the title Workflow Physical model Contributions Kinematics Governing equations Discretization Stable mixed method Cracking Examples Conclusions

Internal variable

Damage-driven computation

Free energy

ϕ( J u K , d) = 1 2

1 d − 1

J u K · k · J u K Cohesive traction

t = ∂ϕ

∂ J u K

= 1

d − 1 k · J u K

Energy release rate y = − ∂ϕ

∂ d = 1

2d ² J u K · k · J u K Weak constraint

Z

Γ

η · JuK − χ(ζ _n > 0 ) dk ⁻¹ · ζ

d Γ = 0

Motivation About the title Workflow Physical model Contributions Kinematics Governing equations Discretization Stable mixed method Cracking Examples Conclusions

Internal variable

Damage-driven computation Free energy

ϕ( J u K , d) = 1 2

1 d − 1

J u K · k · J u K

Cohesive traction

t = ∂ϕ

∂ J u K

= 1

d − 1 k · J u K

Energy release rate y = − ∂ϕ

∂ d = 1

2d ² J u K · k · J u K Weak constraint

Z

Γ

η · JuK − χ(ζ _n > 0 ) dk ⁻¹ · ζ

d Γ = 0

Zoltan Csati Mesh-independent modelling of diffuse cracking 26 / 44

Motivation About the title Workflow Physical model Contributions Kinematics Governing equations Discretization Stable mixed method Cracking Examples Conclusions

Internal variable

Damage-driven computation Free energy

ϕ( J u K , d) = 1 2

1 d − 1

J u K · k · J u K Cohesive traction

t = ∂ϕ

∂ J u K

= 1

d − 1 k · J u K

cf. extrinsic CZM

Energy release rate y = − ∂ϕ

∂ d = 1

2d ² J u K · k · J u K Weak constraint

Z

Γ

η · JuK − χ(ζ _n > 0 ) dk ⁻¹ · ζ

d Γ = 0

Motivation About the title Workflow Physical model Contributions Kinematics Governing equations Discretization Stable mixed method Cracking Examples Conclusions

Internal variable

Damage-driven computation Free energy

ϕ( J u K , d) = 1 2

1 d − 1

J u K · k · J u K Cohesive traction

t = ∂ϕ

∂ J u K

= 1

d − 1 k · J u K

Energy release rate y = − ∂ϕ

∂ d = 1

2d ² J u K · k · J u K

Weak constraint Z

Γ

η · JuK − χ(ζ _n > 0 ) dk ⁻¹ · ζ

d Γ = 0

Zoltan Csati Mesh-independent modelling of diffuse cracking 26 / 44

Motivation About the title Workflow Physical model Contributions Kinematics Governing equations Discretization Stable mixed method Cracking Examples Conclusions

Internal variable

Damage-driven computation Free energy

ϕ( J u K , d) = 1 2

1 d − 1

J u K · k · J u K Cohesive traction

t = ∂ϕ

∂ J u K

= 1

d − 1 k · J u K

Energy release rate y = − ∂ϕ

∂ d = 1

2d ² J u K · k · J u K Weak constraint

Z

η · JuK − χ(ζ _n > 0 ) d k ⁻¹ · ζ

d Γ = 0

Motivation About the title Workflow Physical model Contributions Kinematics Governing equations Discretization Stable mixed method Cracking Examples Conclusions

Damage evolution

Introduce energy dissipation

G ˙ ≥ 0 y − y c (G) ≤ 0 (y − y c (G)) ˙ G = 0 Discretization → compute ∆G

Update ∆d

Zoltan Csati Mesh-independent modelling of diffuse cracking 27 / 44

Motivation About the title Workflow Physical model Contributions Kinematics Governing equations Discretization Stable mixed method Cracking Examples Conclusions

Solution procedure

Algorithm Quasi-static simulation

1: Precompute matrices and vectors

2: Initialize the damage field → d ⁽⁰⁾

3: for n from 1 to N step do . damage stepping loop

4: Update the “mass” matrix

5: Solve the mechanical problem → u ⁽ⁿ⁾ , ζ ⁽ⁿ⁾

6: Determine the energy dissipation → G ⁽ⁿ⁺¹⁾

7: Update the damage field → d ⁽ⁿ⁺¹⁾

8: end for

Motivation About the title Workflow Physical model Contributions Kinematics Governing equations Discretization Stable mixed method Cracking Examples Conclusions

Test examples

Verification Validation

Examples

Three-point bending test Brazilian test

Brazilian test with heterogeneity Compression test

Zoltan Csati Mesh-independent modelling of diffuse cracking 29 / 44

Motivation About the title Workflow Physical model Contributions Kinematics Governing equations Discretization Stable mixed method Cracking Examples Conclusions

Test examples

Verification Validation Examples

Three-point bending test Brazilian test

Brazilian test with heterogeneity

Compression test

Motivation About the title Workflow Physical model Contributions Kinematics Governing equations Discretization Stable mixed method Cracking Examples Conclusions

Three-point bending test

L v

c

Zoltan Csati Mesh-independent modelling of diffuse cracking 30 / 44

Motivation About the title Workflow Physical model Contributions Kinematics Governing equations Discretization Stable mixed method Cracking Examples Conclusions

Three-point bending test

0 0.05 0.10 0.15 0.20 0.25

0 5 10 15 20 25 30 35

CMOD (mm)

applied force ( kN )

experiment (Hoover et al. 2014)

∆d max = 0.1

∆d max = 0.01

∆d max = 0.001

0 0.05 0.10 0.15 0.20 0.25

0 5 10 15 20 25 30 35

CMOD ( mm )

applied force ( kN )

experiment (Hoover et al. 2014)

∆d _max = 0.1

∆d _max = 0.01

∆d _max = 0.001

Zoltan Csati Mesh-independent modelling of diffuse cracking 31 / 44

Motivation About the title Workflow Physical model Contributions Kinematics Governing equations Discretization Stable mixed method Cracking Examples Conclusions

Brazilian test

Proposed in 1943

Determine the tensile strength Widespread in rock testing Closed-form solution exists

https://tinyurl.com/yxd49ulo

Zoltan Csati Mesh-independent modelling of diffuse cracking 32 / 44

Motivation About the title Workflow Physical model Contributions Kinematics Governing equations Discretization Stable mixed method Cracking Examples Conclusions

Brazilian test

0 1 2 3 4 5

0 1 2 3 4 5

(a) Numerical model

0 20 40 60 80 100 10 ⁻³

10 ⁻² 10 ⁻¹ 10 ⁰

Number of elements

| P crit n um − P crit exact | | P ^{crit exact} |

(b) Relative error in the critical load

Motivation About the title Workflow Physical model Contributions Kinematics Governing equations Discretization Stable mixed method Cracking Examples Conclusions

Brazilian test

0 1 2 3 4 5

0 1 2 3 4 5

(a) Numerical model

0 20 40 60 80 100 10 ⁻³

10 ⁻² 10 ⁻¹ 10 ⁰

Number of elements

| P crit n um − P crit exact | | P ^{crit exact} |

(b) Relative error in the critical load

Zoltan Csati Mesh-independent modelling of diffuse cracking 33 / 44

Motivation About the title Workflow Physical model Contributions Kinematics Governing equations Discretization Stable mixed method Cracking Examples Conclusions

Brazilian test

0 1 2 3 4 5

−300

−200

−100 0

D (mm)

t n (MP a) 20

40 60 80 100 analytical

0 1 2 3 4 5

−300

−200

−100 0

D (mm)

t n (MP a) 20

40 60 80

100

analytical

0 1 2 3 4 5

−300

−200

−100 0

D (mm)

t n (MP a) 20

40 60 80

100

analytical

Zoltan Csati Mesh-independent modelling of diffuse cracking 34 / 44

Motivation About the title Workflow Physical model Contributions Kinematics Governing equations Discretization Stable mixed method Cracking Examples Conclusions

Brazilian test

0 0.2 0.4 0.6 0.8 1 0

1 2 3 4 5

d (a) ∆ d max = 0 . 1

0 0.2 0.4 0.6 0.8 1 0

1 2 3 4 5

d (b) ∆ d max = 0 . 01

0 0.2 0.4 0.6 0.8 1 0

1 2 3 4 5

d (c) ∆ d max = 0 . 001

Zoltan Csati Mesh-independent modelling of diffuse cracking 35 / 44

Motivation About the title Workflow Physical model Contributions Kinematics Governing equations Discretization Stable mixed method Cracking Examples Conclusions

Brazilian test

0 50 100 150

0

50

100

150

Motivation About the title Workflow Physical model Contributions Kinematics Governing equations Discretization Stable mixed method Cracking Examples Conclusions

Brazilian test

0 0.2 0.4 0.6

0 0.1 0.2 0.3

vertical displacement (mm)

applied load (k N )

f t = 9 MPa f t = 6 MPa f t = 3 MPa

Zoltan Csati Mesh-independent modelling of diffuse cracking 37 / 44

Motivation About the title Workflow Physical model Contributions Kinematics Governing equations Discretization Stable mixed method Cracking Examples Conclusions

Brazilian test

(a) ∆ ˜ G max = 1 × 10 ^{− 1} (b) ∆ ˜ G max = 1 × 10 ^{− 2} (c) ∆ ˜ G max = 1 × 10 ^{− 3}

Motivation About the title Workflow Physical model Contributions Kinematics Governing equations Discretization Stable mixed method Cracking Examples Conclusions

Brazilian test

Zoltan Csati Mesh-independent modelling of diffuse cracking 39 / 44

Motivation About the title Workflow Physical model Contributions Kinematics Governing equations Discretization Stable mixed method Cracking Examples Conclusions

Compression test

Determine the compressive strength Widespread in rock testing

DOI: 10.1061/(ASCE)0899-1561(2009)21%3A9(502)

Motivation About the title Workflow Physical model Contributions Kinematics Governing equations Discretization Stable mixed method Cracking Examples Conclusions

Compression test

F = −F e y

e y

e x

u = 0 E

100E 100E

108 mm 10 . 8 mm 10 . 8 mm

54 mm

Zoltan Csati Mesh-independent modelling of diffuse cracking 41 / 44

Motivation About the title Workflow Physical model Contributions Kinematics Governing equations Discretization Stable mixed method Cracking Examples Conclusions

Compression test

Motivation About the title Workflow Physical model Contributions Kinematics Governing equations Discretization Stable mixed method Cracking Examples Conclusions

Compression test

Zoltan Csati Mesh-independent modelling of diffuse cracking 42 / 44

Motivation About the title Workflow Physical model Contributions Kinematics Governing equations Discretization Stable mixed method Cracking Examples Conclusions

Compression test

(a) Snapshot A (b) Snapshot B (c) Snapshot C (d) Final crack

Motivation About the title Workflow Physical model Contributions Kinematics Governing equations Discretization Stable mixed method Cracking Examples Conclusions

Conclusions

Two-field variational formulation

All DOF defined at the nodes of a Cartesian mesh Reduction algorithm on Lagrange multipliers Arbitrary discontinuities in Q1 elements Composite failure criterion

Non-uniform fracture energy Damage-driven solution procedure Verification and parameter studies

Qualitatively correct results on coarse meshes and with large damage steps

Zoltan Csati Mesh-independent modelling of diffuse cracking 44 / 44

Thank you for your attention!

The reduction algorithm in details Assembling the coupling matrix

The reduction algorithm in details

1 2 3

4 6

7 8

9

A B

C 5

D

The reduction algorithm in details Assembling the coupling matrix

The reduction algorithm in details

The reduction algorithm in details Assembling the coupling matrix

The reduction algorithm in details

The reduction algorithm in details Assembling the coupling matrix

The reduction algorithm in details

The reduction algorithm in details Assembling the coupling matrix

The reduction algorithm in details

The reduction algorithm in details Assembling the coupling matrix

The reduction algorithm in details

The reduction algorithm in details Assembling the coupling matrix

The reduction algorithm in details

The reduction algorithm in details Assembling the coupling matrix

The reduction algorithm in details

A B C D

0

1 ψ ˜ ₁

ψ ˜ ₂

The reduction algorithm in details Assembling the coupling matrix

Assembling the coupling matrix

λ ₁ λ ₁ λ ₁

λ 1

λ ₁ λ ₂

λ 2 λ ₂

The reduction algorithm in details Assembling the coupling matrix

Assembling the coupling matrix

λ ₁ λ ₁ λ ₁

λ 1

λ ₁ λ ₂

λ 2 λ ₂

The reduction algorithm in details Assembling the coupling matrix

Assembling the coupling matrix

λ ₁ λ ₁ λ ₁

λ 1

λ ₁ λ ₂

λ 2 λ ₂

The reduction algorithm in details Assembling the coupling matrix

Assembling the coupling matrix

λ ₁ λ ₁ λ ₁

λ 1

λ ₁ λ ₂

λ 2 λ ₂