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UNIVERSITE DE LIMOGES Mechanics and Modeling of Materials and Structures of Civil Engineering University of Limoges, Boulevard Jacques Derche, Egletons, France DUBOIS Frédéric Modeling of crack growth initiation in wood timber: an approach by the G v integral MECHANICS AND MODELLING OF MATERIALS AND STRUCTURES OF CIVIL ENGINEERING University of Limoges, Civil Engineering Department, Egletons, France

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UNIVERSITE DE LIMOGES Mechanics and Modeling of Materials and Structures of Civil Engineering University of Limoges, Boulevard Jacques Derche, Egletons, France DUBOIS Frédéric Modeling of crack growth initiation in wood timber: an approach by the G v 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

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UNIVERSITE DE LIMOGES Mechanics and Modeling of Materials and Structures of Civil Engineering University of Limoges, Boulevard Jacques Derche, Egletons, France Thermodynamic approach Thermodynamic functions for viscoelasticity t 0 kl ijklij dtJt Elastic strain energy dVt tU V e F klij t 0 ijkl t 0 dd t2J tJ2 2 1 t F Energy balance vise WUW V e U vis W W S

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UNIVERSITE DE LIMOGES Mechanics and Modeling of Materials and Structures of Civil Engineering University of Limoges, Boulevard Jacques Derche, Egletons, France Thermodynamic approach Thermodynamic functions for viscoelasticity t 0 kl ijklij dtJt Energy balance V e U vis W W S e WUW Viscous dissipation work dVd tW V vis t - 0 D klij t 0 t 0 dd t2Jt D

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UNIVERSITE DE LIMOGES Mechanics and Modeling of Materials and Structures of Civil Engineering University of Limoges, Boulevard Jacques Derche, Egletons, France Thermodynamic approach Griffith fracture energy balance a a state (a) state (b) a W a W a U a W svise svisew GGGG

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UNIVERSITE DE LIMOGES Mechanics and Modeling of Materials and Structures of Civil Engineering University of Limoges, Boulevard Jacques Derche, Egletons, France Thermodynamic approach Viscoelastic energy release rate Ue Wvis V Ws W svisew GGGG Energy release rate a U G e v Viscoelastic energy release rate Ws Crack growth initiation criterion sv GG a G a G sv

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UNIVERSITE DE LIMOGES Mechanics and Modeling of Materials and Structures of Civil Engineering University of Limoges, Boulevard Jacques Derche, Egletons, France Modeling of the linear viscoelastic behavior Incremental formulation 1m M m ijkl o l,k ijklij Finite difference integration 1nnn t ~ tt M

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UNIVERSITE DE LIMOGES Mechanics and Modeling of Materials and Structures of Civil Engineering University of Limoges, Boulevard Jacques Derche, Egletons, France Modeling of the linear viscoelastic behavior Finite element algorithm 1 ~ n tF n t ext F n tu T K d T BMB T K 1 d n t T n tF 1 ~ 1 ~ MB

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UNIVERSITE DE LIMOGES Mechanics and Modeling of Materials and Structures of Civil Engineering University of Limoges, Boulevard Jacques Derche, Egletons, France Free energy partition dVt tU V e F klij t 0 ijkl t 0 dd t2J tJ2 2 1 t F + Kelvin Voigt properties Viscoelastic fracture algorithm

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UNIVERSITE DE LIMOGES Mechanics and Modeling of Materials and Structures of Civil Engineering University of Limoges, Boulevard Jacques Derche, Egletons, France Viscoelastic energy release rate partition a U G e v Viscoelastic fracture algorithm

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UNIVERSITE DE LIMOGES Mechanics and Modeling of Materials and Structures of Civil Engineering University of Limoges, Boulevard Jacques Derche, Egletons, France Viscoelastic energy release rate partition M m m v G o v G v G 1 dV V p kl p ij p ijkl k a p v G 2 1 Elastic energy release rate definition a p e U p v G dV t tU V p e p F p kl p ij p ijkl p k 2 1 F Viscoelastic fracture algorithm

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UNIVERSITE DE LIMOGES Mechanics and Modeling of Materials and Structures of Civil Engineering University of Limoges, Boulevard Jacques Derche, Egletons, France Path-independent Integral G Viscoelastic fracture algorithm klijijkl kW 2 1 a W G G dC C jkki u ijkk WG,,, C C p v G Generalization for ²

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UNIVERSITE DE LIMOGES Mechanics and Modeling of Materials and Structures of Civil Engineering University of Limoges, Boulevard Jacques Derche, Egletons, France Path-independent Integral G v Viscoelastic fracture algorithm dC C jkki u ijkk WG,,, klijijkl kW 2 1 p kl p ij p ijkl p k 2 1 F a W G a p e U p v G dC uG C j,k p k,i p ij k,k p p v F M m m v G o v G v G 1 p ij p ijkl k lk p ij, Elastic stress :

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UNIVERSITE DE LIMOGES Mechanics and Modeling of Materials and Structures of Civil Engineering University of Limoges, Boulevard Jacques Derche, Egletons, France Geometry and mesh definition Numerical validation 200mm 100mm tP tP 10MPa

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UNIVERSITE DE LIMOGES Mechanics and Modeling of Materials and Structures of Civil Engineering University of Limoges, Boulevard Jacques Derche, Egletons, France o JJ tt Viscoelastic properties Numerical validation 200/ exp15 80/ exp121 tt t LR G RR E E LR RR E LRLL E /100 0/1/ 0//1 o J MPa15000E LL MPa RR E600 MPa LR G LR Pine spruce Creep function

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UNIVERSITE DE LIMOGES Mechanics and Modeling of Materials and Structures of Civil Engineering University of Limoges, Boulevard Jacques Derche, Egletons, France Creep loading : analytic solution Numerical validation Correspondence principle tC K )t(G 2 t 87,19 t G Gv method 200/t /t 80/t /t v exp t G

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UNIVERSITE DE LIMOGES Mechanics and Modeling of Materials and Structures of Civil Engineering University of Limoges, Boulevard Jacques Derche, Egletons, France Creep loading : analytic solution Numerical validation G Gv

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UNIVERSITE DE LIMOGES Mechanics and Modeling of Materials and Structures of Civil Engineering University of Limoges, Boulevard Jacques Derche, Egletons, France Creep loading : numerical simulation by G v Numerical validation

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UNIVERSITE DE LIMOGES Mechanics and Modeling of Materials and Structures of Civil Engineering University of Limoges, Boulevard Jacques Derche, Egletons, France Creep loading : Path-independence integration for G v Numerical validation Crack lips C6 C5 C4 C3 C2 Crack lips C6 C5 C4 C3 C2

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UNIVERSITE DE LIMOGES Mechanics and Modeling of Materials and Structures of Civil Engineering University of Limoges, Boulevard Jacques Derche, Egletons, France 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 G v 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

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

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UNIVERSITE DE LIMOGES Mechanics and Modeling of Materials and Structures of Civil Engineering University of Limoges, Boulevard Jacques Derche, Egletons, France DUBOIS Frédéric Modeling of crack growth initiation in wood timber: an approach by the G v integral MECHANICS AND MODELLING OF MATERIALS AND STRUCTURES OF CIVIL ENGINEERING University of Limoges, Civil Engineering Department, Egletons, France

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