A constitutive model for strain-rate dependent ductile-to-brittle - PowerPoint PPT Presentation

A constitutive model for strain-rate dependent ductile-to-brittle transition Juha Hartikainen 1 , Kari Kolari 2 , Reijo Kouhia 1 1 Aalto University, 2 VTT NSCM-23, October 21-22, 2010, KTH Stockholm

OUTLINE • Motivation • The model – Thermodynamic formulation – Helmholtz free energy – Dissipation potential • Concluding remarks Photograph: Kari Kolari, Helsinki, Feb 2007 2/13

MOTIVATION E. M. Schulson: Brittle failure of ice, Engineering Fracture Mechanics 68 (2001) 1839–1887. 3/13

Brittle failure in compression M.S. Paterson, Experimental deformation and faulting in Wombeyan marble, Bull. Geol. Soc. Am. , 69 (1958) 465–476. Occurrence to splitting mode is sensitive to strain rate: J.F . Dorris (1985) 4/13

THERMODYNAMIC FORMULATION Helmholtz free energy ψ ( ǫ e , d ) where ǫ e = ǫ − ǫ i . γ = − ρ ˙ Clausius-Duhem Inequality ψ + σ : ˙ ǫ ≥ 0 γ = ∂ϕ ∂ σ : σ + ∂ϕ Dissipation potential ϕ ( σ , y ) such that ∂ y · y ≥ 0 � � � � � � σ − ρ∂ψ ǫ i − ∂ϕ d − ∂ϕ ˙ = ⇒ : ˙ ǫ e + ˙ : σ + · y = 0 ∂ ǫ e ∂ σ ∂ y σ = ρ∂ψ ǫ i = ∂ϕ d = ∂ϕ ˙ ˙ = ⇒ ∂ ǫ e ∂ σ ∂ y = ⇒ γ ≥ 0 satisfies CDI 5/13

Helmholtz free energy Integrity basis 2 tr ǫ 2 3 tr ǫ 3 I 6 = d · ǫ 2 I 2 = 1 I 3 = 1 I 1 = tr ǫ e , e , e , I 4 = � d � , I 5 = d · ǫ e · d , e · d � 1 2 λI 2 � ρψ = (1 − I 4 ) 1 + 2 µI 2 λµ + H ( σ ⊥ ) λ + 2 µ ( I 4 I 2 1 − 2 I 1 I 5 I − 1 + I 2 5 I − 3 4 ) + (1 − H ( σ ⊥ ))( 1 2 λI 4 I 2 1 + µI 2 5 I − 3 4 ) 4 5 I − 3 − 2 I 6 I − 1 2 I 4 I 2 + I 2 � � + µ 4 4 Heaviside function H ( σ ⊥ ) takes into account the “crack” opening/closure 6/13

Helmholtz free energy - alternative expression Integrity basis ˆ ˆ 2 tr ǫ 2 3 tr ǫ 3 I 5 = ˆ I 6 = ˆ d · ǫ 2 I 2 = 1 I 3 = 1 I 1 = tr ǫ e , e , e , I 4 = � d � , · d , d · ǫ e ˆ e ˆ · d where ˆ d = d / � d � = d /I 4 2 λI 2 � 1 � ρψ = (1 − I 4 ) 1 + 2 µI 2 λµ 1 − 2 I 1 ˆ I 5 + ˆ 1 + µ ˆ + H ( σ ⊥ ) λ + 2 µI 4 ( I 2 I 2 5 ) + (1 − H ( σ ⊥ )) I 4 ( 1 2 λI 2 I 2 5 ) � � 2 I 2 + ˆ I 2 + µI 4 5 − 2 I 6 Stresses are continuous when the “crack” closes 7/13

Dissipation potential Decomposed as ϕ ( σ , y ) = ϕ d ( y ) ϕ tr ( σ ) + ϕ vp ( σ ) where � r +1 � ( y + y 0 ) · M · ( y + y 0 ) 1 Y r ϕ d = τ d (1 − I 4 ) H ( ǫ 1 − ǫ tresh ) Y 2 2( r + 1) r � 1 � p � n ϕ tr = 1 � σ ¯ pn τ vp η (1 − I 4 ) σ r � p +1 1 σ r � σ ¯ ϕ vp = p + 1 τ vp (1 − I 4 ) σ r and M = n ⊗ n , y 0 = βY r n 8/13

MODEL CHARACTERISTICS • Elastic stiffness is reduced monotonously due to damage • Qualitatively predicts correct brittle failure mode in compression/tension Animation • The constraint for the damage � d � ∈ [0 , 1] is satisfied automatically • The transition function ϕ tr deals with the change in the mode of deformation through the damage evolution such that ϕ tr ≥ 0 and ϕ tr ≈ 0 when � ˙ ǫ i � < η and ϕ tr > 1 when � ˙ ǫ i � > η ; • CDI is satisfied a priori for any admissible isothermal process • The dissipation potential is a non-convex function with respect to the thermodynamic forces σ and y 9/13

Uniaxial stress-strain behaviour 1.5 ǫ 0 /η = 10 ˙ 1 σ/σ r ǫ 0 /η = 1 ˙ 0.5 ǫ 0 /η = 0 . 1 ˙ 0 0 1 2 3 4 ǫ/ǫ r 10/13

NUMERICAL EXAMPLE von-Mises solid ¯ σ = σ eff , E = 40 GPa, ν = 0 . 3 , σ r = 20 MPa, τ vp = 1000 s transition strain rate η = 10 − 3 s − 1 , p = r = n = 4 ✲ u prescribed y, v ✛ ✲ L ✻ F B ✲ ❄ z, w x, u 11/13

Load-displacement curves, 12 × 6 mesh τ d η = 10 − 3 ǫ 0 /η = 10 ˙ 1.5 2 1.5 1 F/BHσ r F/BHσ r τ d η = 10 0 1 τ d η = 10 − 1 τ d η = 10 − 2 τ d η = 10 − 3 ǫ 0 /η = 10 . ˙ 0.5 ǫ 0 /η = 1 . 0 ˙ 0.5 ǫ 0 /η = 0 . 1 ˙ 0 0 0 0.2 0.4 0.6 0.8 1 0 0.1 0.2 0.3 0.4 0.5 0.6 100 u/L 100 u/L 12/13

CONCLUSIONS AND FURTHER DEVELOPMENTS • Thermodynamically consistent formulation • Predicts correct failure modes in tension/compression • Easily extensible to more realistic creep and plasticity models • Length scale ?? • Alternative formulation using ψ ∗ ( σ , α ) and ϕ (˙ ǫ i , Z ) !! 13/13