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A constitutive model for strain-rate dependent ductile-to-brittle transition

Juha Hartikainen1, Kari Kolari2, Reijo Kouhia1

1Aalto University, 2VTT

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.

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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(ǫe, d) where ǫe = ǫ − ǫi. Clausius-Duhem Inequality γ = −ρ ˙ ψ + σ : ˙ ǫ ≥ 0 Dissipation potential ϕ(σ, y) such that γ = ∂ϕ ∂σ : σ + ∂ϕ ∂y · y ≥ 0 = ⇒

• σ − ρ∂ψ

∂ǫe

• : ˙

ǫe +

• ˙

ǫi − ∂ϕ ∂σ

• : σ +
• ˙

d − ∂ϕ ∂y

• · y = 0

= ⇒ σ = ρ∂ψ ∂ǫe ˙ ǫi = ∂ϕ ∂σ ˙ d = ∂ϕ ∂y = ⇒ γ ≥ 0 satisfies CDI

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Helmholtz free energy

Integrity basis I1 = tr ǫe, I2 = 1

2tr ǫ2 e,

I3 = 1

3tr ǫ3 e,

I4 = d, I5 = d·ǫe·d, I6 = d·ǫ2

e·d

ρψ = (1 − I4) 1

2λI2 1 + 2µI2

• + H(σ⊥)

λµ λ + 2µ(I4I2

1 − 2I1I5I−1 4

+ I2

5I−3 4 ) + (1 − H(σ⊥))(1 2λI4I2 1 + µI2 5I−3 4 )

+ µ

• 2I4I2 + I2

5I−3 4

− 2I6I−1

4

• Heaviside function H(σ⊥) takes into account the “crack” opening/closure

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Helmholtz free energy - alternative expression

Integrity basis I1 = tr ǫe, I2 = 1

2tr ǫ2 e,

I3 = 1

3tr ǫ3 e,

I4 = d, ˆ I5 = ˆ d·ǫeˆ ·d, ˆ I6 = ˆ d·ǫ2

eˆ

·d where ˆ d = d/d = d/I4 ρψ = (1 − I4) 1

2λI2 1 + 2µI2

• + H(σ⊥)

λµ λ + 2µI4(I2

1 − 2I1ˆ

I5 + ˆ I2

5) + (1 − H(σ⊥))I4(1 2λI2 1 + µˆ

I2

5)

+ µI4

• 2I2 + ˆ

I2

5 − 2I6

• Stresses are continuous when the “crack” closes

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Dissipation potential

Decomposed as ϕ(σ, y) = ϕd(y)ϕtr(σ) + ϕvp(σ) where ϕd = 1 2(r + 1) Yr τd(1 − I4)H(ǫ1 − ǫtresh) (y + y0)·M·(y + y0) Y 2

r

r+1 ϕtr = 1 pn 1 τvpη

• ¯

σ (1 − I4)σr pn ϕvp = 1 p + 1 σr τvp

• ¯

σ (1 − I4)σr p+1 and M = n ⊗ n, y0 = βYrn

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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 ϕtr deals with the change in the mode of deformation through the

damage evolution such that ϕtr ≥ 0 and ϕtr ≈ 0 when ˙ ǫi < η and ϕtr > 1 when ˙ ǫi > 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

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Uniaxial stress-strain behaviour

˙ ǫ0/η = 10 ˙ ǫ0/η = 1 ˙ ǫ0/η = 0.1 ǫ/ǫr σ/σr 4 3 2 1 1.5 1 0.5 10/13

NUMERICAL EXAMPLE

von-Mises solid ¯ σ = σeff, E = 40 GPa, ν = 0.3, σr = 20 MPa, τvp = 1000 s transition strain rate η = 10−3 s−1, p = r = n = 4 z, w x, u y, v

✛ ✲

L

❄ ✻

B

✲

F

✲ uprescribed

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Load-displacement curves, 12×6 mesh

˙ ǫ0/η = 0.1 ˙ ǫ0/η = 1.0 ˙ ǫ0/η = 10. τdη = 10−3 100 u/L F/BHσr 1 0.8 0.6 0.4 0.2 1.5 1 0.5 τdη = 10−3 τdη = 10−2 τdη = 10−1 τdη = 100 ˙ ǫ0/η = 10 100 u/L F/BHσr 0.6 0.5 0.4 0.3 0.2 0.1 2 1.5 1 0.5 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(˙

ǫi, Z) !!

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