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Dispersion analysis of a strain-rate dependent ductile-to-brittle transition model

Harm Askes1, Juha Hartikainen2, Kari Kolari3, Reijo Kouhia2

1The University of Sheffield, 2TKK, 3VTT

The X Finnish Mechanics Days, December 3-4, 2009, Jyv ¨ askyl¨ a

Partially funded by the Academy of Finland (121778)

OUTLINE

• MOTIVATION
• THE MODEL
• DISPERSION ANALYSIS

– Basics – Viscous material – Elastic damaging material – Full transition model

• CONCLUDING REMARKS

Photograph: Kari Kolari, Helsinki, Feb 2007 2/14

MOTIVATION

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

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THE MODEL (1-D CASE)

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ σ = ρ∂ψ ∂ǫ = βEǫe = βE

• ǫ − ǫi

dǫi dt = ∂ϕ ∂σ =

• ϕd

(tps

vpη)nβσr

|σ| βσr np−1 + 1 tps

vpβ

|σ| βσr p sign dǫ dt

• dβ

dt = −∂ϕ ∂Y = −ϕtr tdβ Y Yr r ϕtr ≥ 0 ϕtr ≈ 0 when ˙ ǫi < η and ϕtr > 1 when ˙ ǫi > η ϕtr = 1 pn 1 tps

vpη

|σ| βσr pn ∼ ˙ ǫi η ϕd = 1 r + 1 Yr tdβ Y Yr r+1 , Y = 1 2E(ǫe)2 = 1 2E σ β 2

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DISPERSION ANALYSIS - Basics

Dispersion = waves of different wavelengths have different phase speeds Analysis = put the harmonic wave into the equation of motion u(x, t) = A exp [i(kx − ωt)] , ρd2u dt2 − dσ dx = 0 Dispersion relation ω = Ω(k) Phase velocity v = ω k, group velocity vR = ∂ω ∂k Nice illustration by Greg Egan

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DISPERSION ANALYSIS - Basics

Non-dimensional quantities: τ = t/te, te = L/ce, where ce =

• E/ρ

ξ = x/L, ¯ u = u/L, s = σ/σr Relative strain e = ǫ/ǫr where ǫr = σr/E Non-dimensional equation of motion d2¯ u dτ 2 − ǫr ds dξ = 0, simply ¨ ¯ u − ǫrs′ = 0 Non-dimensional constitutive equations ⎧ ⎪ ⎨ ⎪ ⎩ s = ǫr−1βǫe = ǫr−1β(ǫ − ǫi) ˙ ǫi = f(β, s) ˙ β = g(β, s)

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DISPERSION ANALYSIS - Viscous material

Constitutive equations: ˙ s = ǫr−1(˙ ǫ − ˙ ǫi), ˙ ǫi = (τ ps

vp)−1sp

Linearization at s∗: ˙ s′ = ǫr˙ ǫ − as′, where a = p τvp sp−1

∗

Equation of motion:

...

¯ u − ˙ ¯ u′′ + a¨ ¯ u = 0 Damped harmonic wave: ¯ u(ξ, τ) = ¯ A exp(−¯ αξ) exp[i(¯ krξ − ¯ ωτ)] Dispersion relation: i¯ ω(¯ ω2 − ¯ k2

r + ¯

α2) + 2¯ ω¯ α¯ kr − a¯ ω2 = 0 Solution ¯ ω = ¯ kr

• 1 + 1

4(a/¯

kr)2 , ¯ α = a √ 2

• 1 +
• 1 + (a/¯

ω)2

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DISPERSION ANALYSIS - Viscous material

cR = dω dkr = ce d¯ ω d¯ kr and c = ω kr = ce ¯ ω ¯ kr now cR > ce anomalous dispersion

p = 4 → τvp = 102, 104 ← p = 1, 8 ↑ τvp = 102, 104 ↓ (c, cR)/ce krL 10−6 10−4 10−2 1 102 104 0.25 0.50 0.75 1.00 αL ωte 10−6 10−4 10−2 1 102 104 10−6 10−4 10−2 10−2 1

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DISPERSION ANALYSIS - Elastic damaging material

Constitutive equations: ˙ s = ǫr−1(˙ ǫ − ˙ ǫi), ˙ β = −(τd)−1β−2r−1s2r Linearization at s∗, β∗ results in the equation of motion:

...

¯ u − h0 ˙ ¯ u′′−h1¨ ¯ u+h2¯ u′′ = 0 where h0 = β∗, h1 = τd−1β−2r−2

∗

s2r

∗ ,

h2 = (2r + 1)τd−1β−2r−2

∗

s2r

∗

Dispersion relation: ¯ kr

4 − a1¯

k2

r − a2 0¯

ω2 = 0, ¯ α = a0¯ ω/¯ kr a0 = (h2 − h0h1)¯ ω2 2(h2

0¯

ω2 + h2

2) ,

h2 − h0h1 = 2r τdβ∗ s∗ β∗ 2r > 0 a1 = h−1

0 (¯

ω2 − 2h2a0)

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DISPERSION ANALYSIS - Elastic damaging material

cR < ce normal dispersion

r = 4,

• ǫ0td = 10−2

r = 2, 4 ↑

• ǫ0td = 10−1, 10−2, 10−1 →

(c, cR)/ce krL 10−6 10−4 10−2 1 102 104 1 1.5 2.0 2.5 3.0 p = 2 p = 4

• ǫ0td = 10−1

c/ce krL 10−6 10−4 10−2 1 102 104 1 1.5 2.0 2.5 3.0

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DISPERSION ANALYSIS - Full transition model

Equation of motion:

....

¯ u − h0¨ ¯ u′′ − h1

...

¯ u + h2 ˙ ¯ u′′−h3¨ ¯ u = 0, h0 = β∗ h1 = gβ + (s∗/β∗)gs−β∗fs h2 = β∗gβ h3= β∗(gβfs − fβgs) Dispersion relation: ¯ kr

4 − a1¯

k2

r − a2 0¯

ω2 = 0, ¯ α = a0¯ ω/¯ kr a0 = (h2 − h0h1)¯ ω2+h2h3 2(h2

0¯

ω2 + h2

2)

a1 = h−1

0 (¯

ω2 − 2h2a0+h3)

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DISPERSION ANALYSIS - Full transition model - rate 10η

E = 40 GPa, σr = 20 MPa, tps

vp = 1000 s, td = 1 s, η = 10−3 s−1, p = r = n = 4

e s 1 2 2 3 4 0.25 0.50 0.75 1.00 1.25 e β 1 2 2 3 4 0.25 0.50 0.75 1.00 1.25 7 6 5 4 3 2 1 c/ce ωte 10−810−610−410−2 1 102 104 0.5 1.0 1.5 2.0 2.5 3.0 7 6 5 4 3 2 1 cR/ce ωte 10−810−610−410−2 1 102 104 0.5 1.0 1.5 2.0 2.5 3.0

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DISPERSION ANALYSIS - Evolution of the cut-off frequency

20η 10η 5η η e s 1 2 3 4 0.25 0.50 0.75 1.00 1.25 20η 10η 5η η e ωcte 1 2 3 4 0.002 0.004 0.006 0.008 0.010 0.012

Emerges near the peak stress The saturation value of ωc depend on the loading rate

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CONCLUDING REMARKS

• Both anomalous and normal dispersion depending on state and loading rate
• The model is not able to slow down the high frequency components
• Emerging cut-off frequency
• Length scale of the localization zone?
• Stability?

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