Dispersion analysis of a strain-rate dependent ductile-to-brittle - PowerPoint PPT Presentation

Dispersion analysis of a strain-rate dependent ductile-to-brittle transition model Harm Askes 1 , Juha Hartikainen 2 , Kari Kolari 3 , Reijo Kouhia 2 1 The University of Sheffield, 2 TKK, 3 VTT 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. 3/14

THE MODEL (1-D CASE) ⎧ = ρ∂ψ ∂ǫ = βEǫ e = βE ǫ − ǫ i � � σ ⎪ ⎪ ⎪ ⎪ � | σ | � | σ | ⎪ � � np − 1 � p � ⎪ d ǫ i � d ǫ � = ∂ϕ ϕ d 1 ⎪ ⎨ ∂σ = + sign ( t ps t ps vp η ) n βσ r d t βσ r vp β βσ r d t ⎪ � r ⎪ d β = − ∂ϕ ∂Y = − ϕ tr � Y ⎪ ⎪ ⎪ ⎪ ⎪ d t t d β Y r ⎩ ǫ i � < η ǫ i � > η ϕ tr ≥ 0 ϕ tr ≈ 0 when � ˙ and ϕ tr > 1 when � ˙ � 1 � | σ | � p � n ǫ i � ϕ tr = 1 ∼ � ˙ t ps pn vp η βσ r η � r +1 � 2 � Y � σ 1 Y r Y = 1 2 E ( ǫ e ) 2 = 1 ϕ d = , r + 1 t d β Y r 2 E β 4/14

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 )] , ρ d 2 u d t 2 − d σ d x = 0 Dispersion relation ω = Ω( k ) v = ω v R = ∂ω Phase velocity k , group velocity ∂k Nice illustration by Greg Egan 5/14

DISPERSION ANALYSIS - Basics Non-dimensional quantities: � τ = t/t e , t e = L/c e , where c e = E/ρ ξ = x/L, u = u/L, ¯ s = σ/σ r Relative strain e = ǫ/ǫ r where ǫ r = σ r /E Non-dimensional equation of motion d 2 ¯ u d s u − ǫ r s ′ = 0 ¨ d τ 2 − ǫ r d ξ = 0 , simply ¯ Non-dimensional constitutive equations = ǫ r − 1 βǫ e = ǫ r − 1 β ( ǫ − ǫ i ) ⎧ s ⎪ ⎨ ǫ i ˙ = f ( β, s ) ˙ ⎪ β = g ( β, s ) ⎩ 6/14

DISPERSION ANALYSIS - Viscous material ǫ i = ( τ ps s = ǫ r − 1 (˙ ǫ i ) , vp ) − 1 s p Constitutive equations: ˙ ǫ − ˙ ˙ a = p s ′ = ǫ r ˙ s p − 1 ǫ − as ′ , Linearization at s ∗ : ˙ where ∗ τ vp ... u ′′ + a ¨ u − ˙ Equation of motion: ¯ ¯ u = 0 ¯ αξ ) exp[ i (¯ u ( ξ, τ ) = ¯ Damped harmonic wave: ¯ A exp( − ¯ k r ξ − ¯ ωτ )] ω 2 − ¯ ω 2 = 0 α ¯ k 2 α 2 ) + 2¯ Dispersion relation: i ¯ ω (¯ r + ¯ ω ¯ k r − a ¯ Solution ¯ k r a ω = ¯ , α = ¯ √ � � 4 ( a/ ¯ 1 + 1 � k r ) 2 ω ) 2 2 1 + 1 + ( a/ ¯ 7/14

DISPERSION ANALYSIS - Viscous material c R = d ω d¯ ω c = ω ω ¯ = c e and = c e now c R > c e anomalous dispersion d¯ ¯ d k r k r k r k r τ vp = 10 2 , 10 4 ← τ vp = 10 2 , 10 4 ↓ p = 4 → p = 1 , 8 ↑ ( c, c R ) /c e αL 1 1 . 00 10 − 2 10 − 2 0 . 75 10 − 4 0 . 50 0 . 25 10 − 6 10 − 6 10 − 4 10 − 2 10 − 6 10 − 4 10 − 2 10 2 10 4 10 2 10 4 1 1 k r L ωt e 8/14

DISPERSION ANALYSIS - Elastic damaging material ˙ s = ǫ r − 1 (˙ ǫ i ) , β = − ( τ d ) − 1 β − 2 r − 1 s 2 r Constitutive equations: ˙ ǫ − ˙ Linearization at s ∗ , β ∗ results in the equation of motion: ... u ′′ = 0 u − h 0 ˙ u ′′ − h 1 ¨ ¯ ¯ u + h 2 ¯ ¯ h 1 = τ d − 1 β − 2 r − 2 s 2 r h 2 = (2 r + 1) τ d − 1 β − 2 r − 2 s 2 r where h 0 = β ∗ , ∗ , ∗ ∗ ∗ Dispersion relation: 4 − a 1 ¯ ω 2 = 0 , ¯ ω/ ¯ r − a 2 k 2 k r 0 ¯ α = a 0 ¯ ¯ k r � 2 r ω 2 a 0 = ( h 2 − h 0 h 1 )¯ h 2 − h 0 h 1 = 2 r � s ∗ 2 ) , > 0 ω 2 + h 2 2( h 2 0 ¯ τ d β ∗ β ∗ ω 2 − 2 h 2 a 0 ) a 1 = h − 1 0 (¯ 9/14

DISPERSION ANALYSIS - Elastic damaging material c R < c e normal dispersion ◦ ◦ ǫ 0 t d = 10 − 1 , 10 − 2 , 10 − 1 → ǫ 0 t d = 10 − 2 r = 4 , r = 2 , 4 ↑ ( c, c R ) /c e c/c e 3 . 0 3 . 0 ◦ 2 . 5 2 . 5 ǫ 0 t d = 10 − 1 p = 4 2 . 0 2 . 0 p = 2 1 . 5 1 . 5 1 1 10 − 6 10 − 4 10 − 2 10 − 6 10 − 4 10 − 2 10 2 10 4 10 2 10 4 1 1 k r L k r L 10/14

DISPERSION ANALYSIS - Full transition model .... ... u ′′ − h 1 u − h 0 ¨ u + h 2 ˙ u ′′ − h 3 ¨ Equation of motion: ¯ ¯ ¯ ¯ u = 0 , ¯ h 0 = β ∗ h 1 = g β + ( s ∗ /β ∗ ) g s − β ∗ f s h 2 = β ∗ g β h 3 = β ∗ ( g β f s − f β g s ) 4 − a 1 ¯ ω 2 = 0 , ¯ ω/ ¯ k 2 r − a 2 Dispersion relation: k r 0 ¯ α = a 0 ¯ ¯ k r ω 2 + h 2 h 3 a 0 = ( h 2 − h 0 h 1 )¯ ω 2 + h 2 2( h 2 0 ¯ 2 ) ω 2 − 2 h 2 a 0 + h 3 ) a 1 = h − 1 0 (¯ 11/14

DISPERSION ANALYSIS - Full transition model - rate 10 η vp = 1000 s , t d = 1 s , η = 10 − 3 s − 1 , p = r = n = 4 E = 40 GPa , σ r = 20 MPa , t ps 1 . 25 1 . 25 1 . 00 1 . 00 0 . 75 0 . 75 s β 0 . 50 0 . 50 0 . 25 0 . 25 0 1 2 2 3 4 0 1 2 2 3 4 e e c/c e c R /c e 3 . 0 3 . 0 1 1 2 2 2 . 5 2 . 5 3 3 4 4 2 . 0 5 2 . 0 5 6 6 7 7 1 . 5 1 . 5 1 . 0 1 . 0 0 . 5 0 . 5 10 − 8 10 − 6 10 − 4 10 − 2 10 2 10 4 10 − 8 10 − 6 10 − 4 10 − 2 10 2 10 4 1 1 ωt e ωt e 12/14

DISPERSION ANALYSIS - Evolution of the cut-off frequency 0 . 012 1 . 25 η 0 . 010 5 η 10 η 1 . 00 20 η 0 . 008 ω c t e 0 . 75 0 . 006 s η 0 . 50 0 . 004 5 η 10 η 20 η 0 . 25 0 . 002 1 2 3 4 1 2 3 4 e e Emerges near the peak stress The saturation value of ω c depend on the loading rate 13/14

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? 14/14