The simulation of time dependent behaviour of cement bound materials - PowerPoint PPT Presentation

Elastic Composite Non-linear Inelastic matrix Micro-cracks Damage Conclusion The simulation of time dependent behaviour of cement bound materials with a micro-mechanical model Robert Davies and Dr Anthony Jefferson BRE Institute of Sustainable Engineering, Cardiff University SSCS2012 /Aix en Provence,France /May 29-June 1 2012

Elastic Composite Non-linear Inelastic matrix Micro-cracks Damage Conclusion Introduction Micromechanical models: Simple mechanisms considered at micro (meso) scales which are expected to capture the macroscopic behaviour Viable alternative to phenomenological models

Elastic Composite Non-linear Inelastic matrix Micro-cracks Damage Conclusion Introduction Micromechanical models: Simple mechanisms considered at micro (meso) scales which are expected to capture the macroscopic behaviour Viable alternative to phenomenological models Inelastic strain may derive from: shrinkage creep micro-cracking differential thermal expansion ageing Need to simulate inelastic behaviour in the matrix phase alone

Elastic Composite Non-linear Inelastic matrix Micro-cracks Damage Conclusion Content Outline Elastic two-phase composite 1 Non-linear behaviour within the composite 2 A two-phase composite with inelastic strain in the matrix only 3 Micro-cracks in the matrix 4 Damage predictions 5 Concluding remarks 6

Elastic Composite Non-linear Inelastic matrix Micro-cracks Damage Conclusion Content Outline Elastic two-phase composite 1 Non-linear behaviour within the composite 2 A two-phase composite with inelastic strain in the matrix only 3 Micro-cracks in the matrix 4 Damage predictions 5 Concluding remarks 6

Elastic Composite Non-linear Inelastic matrix Micro-cracks Damage Conclusion Elastic two-phase composite Two phase composite idealisation with Spherical inclusions(Eshelby solution with modified Mori-Tanaka averaging) Penny-shaped micro-cracks(Budiansky and O‘Connell) ¯ σ = f Ω · σ Ω + f M · σ M (1) ¯ ε = f Ω · ε Ω + f M · ε M + ε a (2)

Elastic Composite Non-linear Inelastic matrix Micro-cracks Damage Conclusion Elastic two-phase composite Two phase composite idealisation with Spherical inclusions(Eshelby solution with modified Mori-Tanaka averaging) Penny-shaped micro-cracks(Budiansky and O‘Connell) ¯ σ = f Ω · σ Ω + f M · σ M (1) ¯ ε = f Ω · ε Ω + f M · ε M + ε a (2) Average stress-strain relationship σ = D M Ω : ¯ ¯ ε e = D M Ω : (¯ ε − ε a ) (3) D M Ω = ( f Ω · D Ω : T Ω + f M · D M ) · ( f Ω · T Ω + f M ) − 1 , (4a) T Ω = I 2 s + S Ω : A Ω , (4b) A Ω = [( D Ω − D M ) : S Ω + D M ] − 1 : ( D Ω − D M ) (4c)

Elastic Composite Non-linear Inelastic matrix Micro-cracks Damage Conclusion Content Outline Elastic two-phase composite 1 Non-linear behaviour within the composite 2 A two-phase composite with inelastic strain in the matrix only 3 Micro-cracks in the matrix 4 Damage predictions 5 Concluding remarks 6

Elastic Composite Non-linear Inelastic matrix Micro-cracks Damage Conclusion Non-linear behaviour within the composite Inelastic strains in the inclusion D Ω : ( ε o + ε c + ε IN ) = D M : ( ε o + ε c + ε IN − ε τ ) (5a) ε c = S Ω : ( ε τ + ε IN ) (5b)

Elastic Composite Non-linear Inelastic matrix Micro-cracks Damage Conclusion Non-linear behaviour within the composite Inelastic strains in the inclusion D Ω : ( ε o + ε c + ε IN ) = D M : ( ε o + ε c + ε IN − ε τ ) (5a) ε c = S Ω : ( ε τ + ε IN ) (5b) Inelastic strains in the matrix Secant moduli method (Weng) D Ω : ( ε o + ε c ) = D Secant : ( ε o + ε c − ε τ ) (6a) ε c = S secant : ε τ (6b) Elastic constraint method (Weng) D Ω : ( ε o + ε c + ε IN ) = D M : ( ε o + ε c + ε IN − ε τ ) (7a) ε c = S Ω : ( ε τ − ε IN ) (7b) (Nemat Nasser and Hori( 1993 ), Mura ( 1987 ) and Weng ( 1988 ))

Elastic Composite Non-linear Inelastic matrix Micro-cracks Damage Conclusion Content Outline Elastic two-phase composite 1 Non-linear behaviour within the composite 2 A two-phase composite with inelastic strain in the matrix only 3 Micro-cracks in the matrix 4 Damage predictions 5 Concluding remarks 6

Elastic Composite Non-linear Inelastic matrix Micro-cracks Damage Conclusion Two material problem with matrix undergoing transformation strain ε IN . Ω remains elastic. Matrix phase Inclusion phase σ M = D M : ε M = D M : ( ε o + ε c − ε IN ) σ Ω = D Ω : ε Ω = D Ω : ( ε o + ε c ) (8) (9) Consistency equation D Ω : ( ε o + ε c ) = D M : ( ε o + ε c − ε τ ) (10) Constrained strain ε c = S Ω : ( ε τ − ε IN ) (11) Transformation eigenstrain ε τ = A Ω : ( ε o − S Ω : ε IN ) (12)

Elastic Composite Non-linear Inelastic matrix Micro-cracks Damage Conclusion Individual phase and composite equations σ M = D M : ε Mel = D M : ( ε M − ε IN ) (13) σ Ω = D Ω : T Ω : ( ε M − S Ω : ε IN ) (14) Substitution and rearranged total strain equation ε M = ( f Ω · T Ω + f M ) − 1 : (¯ ε + f Ω · T Ω · S Ω : ε IN ) (15) Constitutive equation σ = D M Ω : (¯ ¯ ε − ε INEQ ) (16) where ε INEQ = D M Ω − 1 · ( f Ω · D Ω · T Ω : S Ω + f M · D M − f Ω · D M Ω · T Ω : S Ω ) : ε IN (18)

Elastic Composite Non-linear Inelastic matrix Micro-cracks Damage Conclusion Individual phase and composite equations σ M = D M : ε Mel = D M : ( ε M − ε IN ) (13) σ Ω = D Ω : T Ω : ( ε M − S Ω : ε IN ) (14) Substitution and rearranged total strain equation ε M = ( f Ω · T Ω + f M ) − 1 : (¯ ε + f Ω · T Ω · S Ω : ε IN ) (15) Constitutive equation σ = D M Ω : (¯ ¯ ε − ε INEQ − ε a ) (17) where ε INEQ = D M Ω − 1 · ( f Ω · D Ω · T Ω : S Ω + f M · D M − f Ω · D M Ω · T Ω : S Ω ) : ε IN (18)

Elastic Composite Non-linear Inelastic matrix Micro-cracks Damage Conclusion Content Outline Elastic two-phase composite 1 Non-linear behaviour within the composite 2 A two-phase composite with inelastic strain in the matrix only 3 Micro-cracks in the matrix 4 Damage predictions 5 Concluding remarks 6

Elastic Composite Non-linear Inelastic matrix Micro-cracks Damage Conclusion Penny-shaped micro-cracks in the matrix σ M = D M : ( ε M − ε IN − ε F RM ) (19) σ Ω = D Ω : ε Ω = D Ω : ( ε o + ε c ) (20) Consistency equation D Ω : ( ε o + ε c ) = (1 − ω ) D M : ( ε o + ε c − ε τ ) (21) Constrained strain ε c = S Ω : ( ε τ − ε IN ) (23)

Elastic Composite Non-linear Inelastic matrix Micro-cracks Damage Conclusion Penny-shaped micro-cracks in the matrix σ M = D M : ( ε M − ε IN − ε F RM ) (19) σ Ω = D Ω : ε Ω = D Ω : ( ε o + ε c ) (20) Consistency equation D Ω : ( ε o + ε c ) = D M : ( ε o + ε c − ε τ − ε FRM Ω ) (22) Constrained strain ε c = S Ω : ( ε τ − ε IN ) (23)

Elastic Composite Non-linear Inelastic matrix Micro-cracks Damage Conclusion Penny-shaped micro-cracks in the matrix σ M = D M : ( ε M − ε IN − ε F RM ) (19) σ Ω = D Ω : ε Ω = D Ω : ( ε o + ε c ) (20) Consistency equation D Ω : ( ε o + ε c ) = D M : ( ε o + ε c − ε τ − ε FRM Ω ) (22) Constrained strain ε c = S Ω : ( ε τ − ε IN ) (23) Transformation eigenstrain ε τ = A Ω : ( ε o − S Ω : ε IN ) − B Ω : ε FRM Ω (24) where B Ω = [( D Ω − D M : S Ω + D M ] − 1 · D M (25)

Elastic Composite Non-linear Inelastic matrix Micro-cracks Damage Conclusion Local model of fracture strain e FRM = C LM : s M (26a) s M = N · σ M (26b) e FRM = C LM : N · σ M (26c) � N T · e FRM · ds ε FRM = (26d) ε F RM = ( I 2 s + C add ) − 1 · C add : D M : ( ε M − ε IN ) (27a) ε F RM Ω = ( I 2 s + C add : D M : V Ω ) − 1 · C add : D M : U Ω : ( ε M − S Ω : ε IN ) (27b) where C add = 1 ω ( θ, ψ ) � � N ε : C LM : N · 1 − ω ( θ, ψ ) sin ( ψ )d ψ d θ (28a) 2 π π 2 π 2 U Ω = I 2 s + ( S Ω − I 2 s ) : A Ω , (28b) V Ω = I 2 s + ( S Ω − I 2 s ) : B Ω (28c)

Elastic Composite Non-linear Inelastic matrix Micro-cracks Damage Conclusion Inelastic strain and micro-cracks in the matrix Constitutive equation σ = D M Ω FR : (¯ ¯ ε − ε INFREQ ) (29) where D M Ω F R = F : H (30a) − 1 : G ) − J ] : ε IN ε INF REQ = [( D M Ω F R (30b) F = f Ω · D M U Ω − f Ω · D M : V Ω · ( I 2 s + C add : D M : V Ω ) − 1 · C add : D M : U Ω + f M · D M − f M · D M C add · ( I 2 s + C add : D M ) − 1 · D M (30c) H = [ f Ω · D − 1 : D M : U Ω + f M · I 2 s Ω : D M : V Ω · ( I 2 s + C add : D M : V Ω ) − 1 · C add : D M : U Ω ] − 1 − f Ω · D − 1 (30d) Ω G = f Ω · D M U Ω : S Ω − f Ω · D M : V Ω · ( I 2 s + C add : D M : V Ω ) − 1 · C add : D M : U Ω : S Ω + f M · D M − f M · D M C add · ( I 2 s + C add : D M ) − 1 · D M (30e) J = [ f Ω · D − 1 : D M : U Ω − f Ω · D − 1 : Ω Ω D M : V Ω · ( I 2 s + C add : D M : V Ω ) − 1 · C add : D M : U Ω ] : S Ω (30f) (30g)