DUBOIS Frédéric

UNIVERSITE

DE LIMOGES

Mechanics and Modeling of Materials and Structures of Civil Engineering University of Limoges, Boulevard Jacques Derche, 19300 Egletons, France

Dubois@egletons.unilim.fr

DUBOIS Frédéric

Modeling of crack growth initiation in wood timber:

an approach by the Gv integral

MECHANICS AND MODELLING OF MATERIALS AND STRUCTURES OF CIVIL ENGINEERING

University of Limoges, Civil Engineering Department, Egletons, France

UNIVERSITE

DE LIMOGES

Mechanics and Modeling of Materials and Structures of Civil Engineering University of Limoges, Boulevard Jacques Derche, 19300 Egletons, France

Dubois@egletons.unilim.fr

DUBOIS Frédéric

Modeling of crack growth initiation in wood timber:

an approach by the Gv integral

MECHANICS AND MODELLING OF MATERIALS AND STRUCTURES OF CIVIL ENGINEERING

University of Limoges, Civil Engineering Department, Egletons, France

Thermodynamic approach

Modeling of the linear viscoelastic behavior

Viscoelastic fracture algorithm

Numerical validation

UNIVERSITE

DE LIMOGES

Mechanics and Modeling of Materials and Structures of Civil Engineering University of Limoges, Boulevard Jacques Derche, 19300 Egletons, France

Dubois@egletons.unilim.fr

Thermodynamic approach

Thermodynamic functions for viscoelasticity

t

0

klijklij dtJt

Elastic strain energy

dVttUV

e F

klij

t

0

ijklt

0

ddt2J

tJ2

2

1tF

Energy balance

vise WUW

V eU visW

WS

UNIVERSITE

DE LIMOGES

Mechanics and Modeling of Materials and Structures of Civil Engineering University of Limoges, Boulevard Jacques Derche, 19300 Egletons, France

Dubois@egletons.unilim.fr

Thermodynamic approach

Thermodynamic functions for viscoelasticity

t

0

klijklij dtJt

Energy balance

V eU visW

WS

vise WUW

Viscous dissipation work

dVdtWV

vis

t

-0

D

klij

t

0

t

0

ddt2Jt D

UNIVERSITE

DE LIMOGES

Mechanics and Modeling of Materials and Structures of Civil Engineering University of Limoges, Boulevard Jacques Derche, 19300 Egletons, France

Dubois@egletons.unilim.fr

Thermodynamic approach

Griffith fracture energy balance

Ue Wvis

V

Ws

Wa a

state (a)

state (b)

a

W

a

W

a

U

a

W svise

svisew GGGG

UNIVERSITE

DE LIMOGES

Mechanics and Modeling of Materials and Structures of Civil Engineering University of Limoges, Boulevard Jacques Derche, 19300 Egletons, France

Dubois@egletons.unilim.fr

Thermodynamic approach

Viscoelastic energy release rate

Ue Wvis

V

Ws

W

svisew GGGG

Energy release rate

a

UG e

v

Viscoelastic energy release rate

Ws

Crack growth initiation criterion

sv GG

a

G

a

G sv

UNIVERSITE

DE LIMOGES

Mechanics and Modeling of Materials and Structures of Civil Engineering University of Limoges, Boulevard Jacques Derche, 19300 Egletons, France

Dubois@egletons.unilim.fr

Modeling of the linear viscoelastic behavior

Incremental formulation

oijkl

1ijkl

mijkl

Mijkl

kl kl

ijkl

oijklk

1ijklk m

ijklk M

ijklk

1ijkl m

ijkl M

ijkl

oijkl

1ijkl

mijkl

Mijkl

kl kl

ijkl

oijklk

1ijklk m

ijklk M

ijklk

1ijkl m

ijkl M

ijkl

1m

M

mijkl

oijklijkl

l,kijklij

Finite difference integration 1nnn t~tt M

UNIVERSITE

DE LIMOGES

Mechanics and Modeling of Materials and Structures of Civil Engineering University of Limoges, Boulevard Jacques Derche, 19300 Egletons, France

Dubois@egletons.unilim.fr

Modeling of the linear viscoelastic behavior

Finite element algorithm

1~

ntFntextFntuTK

dT BMBTK 1 dntT

ntF 1~

1~ MB

UNIVERSITE

DE LIMOGES

Mechanics and Modeling of Materials and Structures of Civil Engineering University of Limoges, Boulevard Jacques Derche, 19300 Egletons, France

Dubois@egletons.unilim.fr

Free energy partition

dVttUV

e F

klij

t

0

ijklt

0

ddt2J

tJ2

2

1tF

dVk2

1

dVk2

1

dVk

2

1

k2

1

U

V

mkl

mij

mijkl

M

1m

V

okl

oij

oijkl

V

M

1m

mkl

mij

mijkl

okl

oij

oijkl

e

dVk2

1

dVk2

1

dVk

2

1

k2

1

U

V

mkl

mij

mijkl

M

1m

V

okl

oij

oijkl

V

M

1m

mkl

mij

mijkl

okl

oij

oijkl

e

+ Kelvin Voigt properties

Viscoelastic fracture algorithm

UNIVERSITE

DE LIMOGES

Mechanics and Modeling of Materials and Structures of Civil Engineering University of Limoges, Boulevard Jacques Derche, 19300 Egletons, France

Dubois@egletons.unilim.fr

Viscoelastic energy release rate partition

a

UG e

v

dV

V

mkl

mij

mijklk

a

M

m

dV

V

okl

oij

oijklk

a

dV

V

M

m

mkl

mij

mijklk

okl

oij

oijklk

avG

2

1

1

2

11

2

1

2

1

dV

V

mkl

mij

mijklk

a

M

m

dV

V

okl

oij

oijklk

a

dV

V

M

m

mkl

mij

mijklk

okl

oij

oijklk

avG

2

1

1

2

11

2

1

2

1

Viscoelastic fracture algorithm

UNIVERSITE

DE LIMOGES

Mechanics and Modeling of Materials and Structures of Civil Engineering University of Limoges, Boulevard Jacques Derche, 19300 Egletons, France

Dubois@egletons.unilim.fr

Viscoelastic energy release rate partition

M

m

mvGo

vGvG

1dV

V

pkl

pij

pijklk

apvG

2

1

Elastic energy release rate definition

a

peUp

vG

dVttUV

p

e pF p

klpij

pijkl

p k2

1 F

Viscoelastic fracture algorithm

UNIVERSITE

DE LIMOGES

Mechanics and Modeling of Materials and Structures of Civil Engineering University of Limoges, Boulevard Jacques Derche, 19300 Egletons, France

Dubois@egletons.unilim.fr

Path-independent Integral G

Viscoelastic fracture algorithm

klijijklkW 2

1

a

WG

G

1x

2x

Failure zone

1x

2x

Failure zone

dCC jkkiuijkkWG ,,,

02

11

02

01

C

02

11

02

01

C

pvGGeneralization for

²

UNIVERSITE

DE LIMOGES

Mechanics and Modeling of Materials and Structures of Civil Engineering University of Limoges, Boulevard Jacques Derche, 19300 Egletons, France

Dubois@egletons.unilim.fr

Path-independent Integral Gv

Viscoelastic fracture algorithm

dCC jkkiuijkkWG ,,,

klijijklkW 2

1pkl

pij

pijkl

p k2

1 F

a

W

Ga

peUp

vG

dCuGC

j,kp

k,i

p

ijk,kpp

v

F

M

m

mvGo

vGvG

1

pij

pijklk

lk

pij

,

Elastic stress :

UNIVERSITE

DE LIMOGES

Mechanics and Modeling of Materials and Structures of Civil Engineering University of Limoges, Boulevard Jacques Derche, 19300 Egletons, France

Dubois@egletons.unilim.fr

Geometry and mesh definition

Numerical validation

200mm

200mm

100mm

tP

tP

Mesh details around the crack tip

Crack

P t 10MPa

UNIVERSITE

DE LIMOGES

Mechanics and Modeling of Materials and Structures of Civil Engineering University of Limoges, Boulevard Jacques Derche, 19300 Egletons, France

Dubois@egletons.unilim.fr

oJJ tt

Viscoelastic properties

Numerical validation

200/exp1580/exp121 ttt

LRGRRERRELR

RRELRLLE

/100

0/1/

0//1

oJMPa15000ELL

MPaRRE 600

MPaLRG 700

4.0LR

Pine spruce

Creep function

UNIVERSITE

DE LIMOGES

Mechanics and Modeling of Materials and Structures of Civil Engineering University of Limoges, Boulevard Jacques Derche, 19300 Egletons, France

Dubois@egletons.unilim.fr

Creep loading : analytic solution

Numerical validation

Correspondence principletC

K)t(G

2 t87,19tG

Gv method

200/t200/t

80/t80/t

vexp2exp5

exp2exp28136.0tG

UNIVERSITE

DE LIMOGES

Mechanics and Modeling of Materials and Structures of Civil Engineering University of Limoges, Boulevard Jacques Derche, 19300 Egletons, France

Dubois@egletons.unilim.fr

Creep loading : analytic solution

Numerical validation

0

20

40

60

80

100

120

140

160

180

0 100 200 300 400 500 600 700 800 900 1000

Time (min)

Ene

rgy

rele

ase

rate

(N

/mm

-1)

Correspondence principleAnalytic solution

GGv

UNIVERSITE

DE LIMOGES

Mechanics and Modeling of Materials and Structures of Civil Engineering University of Limoges, Boulevard Jacques Derche, 19300 Egletons, France

Dubois@egletons.unilim.fr

Creep loading : numerical simulation by Gv

Numerical validation

0

20

40

60

80

100

120

140

160

180

0 100 200 300 400 500 600 700 800 900 1000

Time (min)

Vis

coel

asti

c en

ergy

rel

ease

rat

e (N

mm

-1)

Analytic solutionNumeric simulation

UNIVERSITE

DE LIMOGES

Mechanics and Modeling of Materials and Structures of Civil Engineering University of Limoges, Boulevard Jacques Derche, 19300 Egletons, France

Dubois@egletons.unilim.fr

Creep loading : Path-independence integration for Gv

Numerical validation

Crack lips

C6C5C4C3C2

Crack lips

C6C5C4C3C2

Crown C1 C3 C5 C7simulation error 11,15% 0,73% 0,64% 0,58%

Crown C9 C11 C13 C15simulation error 0,54% 0,51% 0,49% 0,47%

Crown C17 C19 C21 C23simulation error 0,45% 0,44% 0,42% 0,41%

UNIVERSITE

DE LIMOGES

Mechanics and Modeling of Materials and Structures of Civil Engineering University of Limoges, Boulevard Jacques Derche, 19300 Egletons, France

Dubois@egletons.unilim.fr

Conclusion

Conclusion and Perspectives

Separation of the free energy and the viscous dissipation

for the crack growth initiation process

Development of the new path-independence integral Gv

Adaptation of the approach with an orthotropic viscoelastic behavior algorithm using the finite element method

Development of the energy partition by using a generalized Kelvin Voigt model for the mechanical behavior

UNIVERSITE

DE LIMOGES

Mechanics and Modeling of Materials and Structures of Civil Engineering University of Limoges, Boulevard Jacques Derche, 19300 Egletons, France

Dubois@egletons.unilim.fr

Perspectives

Conclusion and Perspectives

Generalization for mixed mode fracture by using a M integral type

Adaptation for the crack growth process

Generalization for aging behavior for the mechano-sorptive effects

UNIVERSITE

DE LIMOGES

Mechanics and Modeling of Materials and Structures of Civil Engineering University of Limoges, Boulevard Jacques Derche, 19300 Egletons, France

Dubois@egletons.unilim.fr

DUBOIS Frédéric

Modeling of crack growth initiation in wood timber:

an approach by the Gv integral

MECHANICS AND MODELLING OF MATERIALS AND STRUCTURES OF CIVIL ENGINEERING

University of Limoges, Civil Engineering Department, Egletons, France