DUBOIS Frédéric

1. Modeling of crack growth initiation in wood timber: an approach by the Gqv integral DUBOIS Frédéric MECHANICS AND MODELLING OF MATERIALS AND STRUCTURES OF CIVIL ENGINEERING University of Limoges, Civil Engineering Department, Egletons, France

2. Modeling of crack growth initiation in wood timber: an approach by the Gqv integral Thermodynamic approach Modeling of the linear viscoelastic behavior Viscoelastic fracture algorithm Numerical validation DUBOIS Frédéric MECHANICS AND MODELLING OF MATERIALS AND STRUCTURES OF CIVIL ENGINEERING University of Limoges, Civil Engineering Department, Egletons, France

3. Thermodynamic approach t ( ) ¶ s x ( ) ( ) ò kl e = - x × x t J t d ij ijkl ¶ x - 0 Energy balance Elastic strain energy W ( ) ( ) ò S t dV = U t F e V U W e vis t t ( ) × - x é ù 2 J t V 1 ( ) ( ) ( ) ò ò ijkl s x s b t d d = × ê ú F ij kl ( ) - × - x - b 2 J 2 t ë û = + - - W U W 0 0 e vis Thermodynamic functions for viscoelasticity

4. Thermodynamic approach t ( ) ¶ s x ( ) ( ) ò kl e = - x × x t J t d ij ijkl ¶ x - 0 t ( ) ( ) ò ò t = t d dV W t D vis V - U 0 W e vis = + W U W e vis Thermodynamic functions for viscoelasticity Energy balance Viscous dissipation work W S V t t ( ) ( ) ( ) ( ) ò ò & s x s b = × - x - b d d t J 2 t D ij kl - - 0 0

5. Thermodynamic approach state (b) state (a) D a a - = + G G G G w e vis s D D D D U W W W e vis s = + + D D D D a a a a Griffith fracture energy balance

6. Thermodynamic approach D W V D D Ue Wvis D Ws Crack growth initiation criterion - = + G G G G w e vis s ¶ U D Ws e = - G < G G v v s ¶ a ¶ ¶ G G v s Viscoelastic energy release rate = ¶ ¶ a a Viscoelastic energy release rate Energy release rate

7. Modeling of the linear viscoelastic behavior M å å e = P o m P = P + P ( ) ( ) ( ) ~ Finite difference integration ij ijkl D e = × D s + e M t t t ijkl ijkl ijkl - n n n 1 k , l = Incremental formulation m 1

8. Modeling of the linear viscoelastic behavior { } ~ { } ( ) { } ( ) ( ) × D = D + K u t F t F t - T n ext n n 1 - T 1 ò = × × W K B M B d T W { } ~ ( ) { } ( ) T ò ~ = × × e W B M F t t d - - n 1 n 1 W Finite element algorithm

9. Viscoelastic fracture algorithm ( ) ( ) ò t dV = U t F e V t t ( ) × - x é ù 2 J t 1 ( ) ( ) ( ) + Kelvin Voigt properties ò ò ijkl s x s b t d d = × ê ú F ij kl ( ) - × - x - b 2 J 2 t ë û - - 0 0 Free energy partition

10. Viscoelastic fracture algorithm ¶ U e - = G v ¶ a Viscoelastic energy release rate partition

11. Viscoelastic fracture algorithm M ¶ 1 p p p p ò å = - × × e × e G k dV o m v = + ij G G G ijkl kl ¶ v v v a 2 V = m 1 Elastic energy release rate definition p ¶ 1 U p p p p e p ( ) ( ) F ò = × × e × e p p k = - G = t dV F U t v ijkl ij kl 2 ¶ a e V Viscoelastic energy release rate partition

12. ² Viscoelastic fracture algorithm q q = = 1 1 1 1 ¶ W e q q = = 0 0 = = - 2 2 G Gq ¶ a [ ] ò C C = - × q + s × × q G W u dC q e k , k ij i , k k , j C q q = = 0 0 1 1 q q = = 0 0 2 2 1 = × × e × e W k e ijkl ij kl 2 Generalization for p G v Path-independent Integral Gq

13. Viscoelastic fracture algorithm [ ] ¶ ò W e = - × q + s × × q G W u dC q e = - k , k ij i , k k , j Gq C ¶ a å p p p s = × e k ij ij ijkl 1 k , l = × × e × e W k e ijkl ij kl 2 M p ¶ å 1 U o m é ù p p p p e p ò p p p q = q + q p F = × × e × e G G G F k q = - × q + s × × q = - dC G u G Elastic stress : v v v v ê ú v k , k k , j ijkl ij kl 2 ¶ ë û a ij i , k C = m 1 Path-independent Integral Gqv

14. Numerical validation t P ( ( ) ) P P t t Mesh details around the crack tip 100mm 200mm Crack 200mm Geometry and mesh definition ( ) 10MPa

15. Numerical validation = E 15000 MPa LL = E 600 MPa RR Pine spruce = G 700 MPa LR ( ) ( ) - - æ ö æ ö = b × ( ) t / 80 t / 200 J J t t b = + × - + × - t ç ÷ ç ÷ 1 2 1 exp 5 1 exp o è ø è ø n = - n 0 . 4 é ù 1 / E / E 0 LR LL LR RR ê ú = - n J / E 1 / E 0 o LR RR RR ê ú ê ú 0 0 1 / G ë û LR Viscoelastic properties Creep function

16. Numerical validation ( ) ( ) = × b t t G 19 , 87 ) ( - - é ù t / 80 t / 80 2 - × × - 8 2 exp 2 exp K ( ) = ê ú = × t G ( t ) G 0 . 136 Gv method ( ) v ( ) - - C t t / 200 t / 200 ê ú - × × - 5 exp 2 exp ë û Creep loading : analytic solution Correspondence principle

17. Numerical validation Creep loading : analytic solution G Gv

18. Numerical validation Creep loading : numerical simulation by Gqv

19. Numerical validation C6 C6 C5 C5 Crack lips Crack lips C4 C4 C3 C3 C2 C2 Creep loading : Path-independence integration for Gqv

20. Conclusion and Perspectives Conclusion Separation of the free energy and the viscous dissipation for the crack growth initiation process Development of the new path-independence integral Gqv Development of the energy partition by using a generalized Kelvin Voigt model for the mechanical behavior Adaptation of the approach with an orthotropic viscoelastic behavior algorithm using the finite element method

21. Conclusion and Perspectives 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

22. Modeling of crack growth initiation in wood timber: an approach by the Gqv integral DUBOIS Frédéric MECHANICS AND MODELLING OF MATERIALS AND STRUCTURES OF CIVIL ENGINEERING University of Limoges, Civil Engineering Department, Egletons, France