Upload
nguyenkiet
View
231
Download
4
Embed Size (px)
Citation preview
Cavitation Behavior of Heterogeneous Plastic Cavitation Behavior of Heterogeneous Plastic Solids : Fracture of Brittle BMGsSolids : Fracture of Brittle BMGsSo ds actu e o tt e GsSo ds actu e o tt e Gs
R.Narasimhan
Department of Mechanical Engineering
Indian Institute of Science, Bangalore
Collaborators
Prof U RamamurtyProf.U. Ramamurty
Indrasen Singh
IISc, Bangalore
P.Murali
Prof.Y.W.Zhang
T.F.GuoProf.H.Gao
11
T.F.Guo
IHPC, SingaporeBrown University, USAParag Tandaiya
IIT, Mumbai
Introduction• Upon slow cooling, most metals
– solidify in an ordered crystalline structure. • When quenched rapidly (~ 105 K/s), some binary alloys
– solidify in a disordered form => metallic glasses or amorphous metals– Ribbons/wires of thickness ~10 μm
• Bulk Metallic Glasses (BMGs)– Multi-component alloy systems; e.g., Vitreloy-1 => – Cooling rates ~ 1-100 K/s ; bulk form (thickness > 1 mm) specimens
• BMGs have superior properties – High yield strength ~ 2 GPa– High yield strain ~ 2%– Young’s modulus ~ 100 GPa – High corrosion resistance– Ability for net shape forming– Fracture toughness ??
2
Applications
Golf club, face material made of Zirconium-based BMG
Tungsten + Vitreloy composite KEP rods
Titanium-based BMG alloy tooth implant
• Other applications: MEMS devices, Electronic casings, hinge components in mobiles, Ophthalmic knives, etc.
Microscopic and macroscopic deformation behavior
• Crystalline metals versus metallic glasses Shear bands near spherical indent
STZLocal cluster of atoms that cooperatively rearrange toDi l ti ti
1. Atomic structure: Ordered lattice Lack long range order
2. Deformation Dislocation motion Operation of Shear
(Trichy et al. 2005)
cooperatively rearrange to accommodate shear strain
Dislocation motion
pmechanism: Transformation Zones (STZ)
3. Plastic Pressure insensitive Pressure/normal stress sensitive
yielding: Volume preserving Dilationaly g p g
Strain hardening Strain softening-perfectly plastic
Homogeneous deformation Inhomogeneous / localized deformations at
Low temp and high stresses => Shear banding
4
Low temp. and high stresses => Shear banding
Fracture behavior of BMGs
F t t h K f 2 MP 1/2 f b ittl MG t 80 MP 1/2 fFracture toughness KIc can vary from 2 MPam1/2 for brittle MGs to 80 MPa m1/2 for ductile MGs
Lewandowski et al. (2005)
Correlation between fracture energy and ν for various metallic glasses.
Note Gc increases by 3-4 orders of magnitude as νchanges from 0.3 to 0.4.
Higher ν => Lower shear modulus to bulk modulus ratio => Enhanced plasticity & toughness – Too simplistic !
Mechanistic reasons for brittle &
5
Mechanistic reasons for brittle & ductile behaviors not fully understood.
Fracture Behavior of Ductile BMGs• As‐Cast vit‐1 specimen • Tandaiya et.al., Acta Mat, 2013
B f k i i i i C k j fBefore crack initiation Crack trajectory post-fracture
• Note intense shear banding around notch root prior to fracture initiation.
• Incipient crack growth occurs inside a dominant shear band AB
• Crack initially follows curved shear band ABC and then propagates straight ahead of notch tip.
Fracture Morphology in Ductile BMGs
A C t it 1 i• Tandaiya et.al., Acta Mat, 2013
As‐Cast vit‐1 specimen
• Fracture surface shows 3 distinct morphologies :
• Notch Blunting zone (smooth, featureless) ~ 10 - 12 μm
(CTOD at crack initiation measured from in-situ optical microscopy coupled with CCD camera ~ 22 μm)
• Taylor’s FMI region – relatively smooth region with ridge patterns running predominantly normal to the original notch front ~ 50 μm
D i k ti i• Dynamic crack propagation region : Coarse features involving ridges and deep valleys.Crack growth direction
Process zone (Notch blunting + FMI zone) size ~ 60 μm
Detailed fractography near notch front
As‐Cast vit‐1 specimen • Tandaiya et al Acta Mat 2013
Taylor’s meniscus instability region
p
NotchFront
Tandaiya et.al., Acta Mat, 2013
Crack growth directionCrack growth direction
Ridgepatterns
Crack growth directionNotchFront• Ridges run almost perpendicular to notch front.
• Ridges bridge the two fracture surfaces and rupture as the crack front advances.
• Local crack front meanders in different directions – uniform hydrostatic stress thro’ predominant specimen thickness.
Modified FMI model• Shear bands form ahead of blunting crack front Viscosity inside shear band is lowered
• Tandaiya et.al., Acta Mat, 2013
front. Viscosity inside shear band is lowered due to temperature rise and free vol evolution.
• Shear band subjected to normal stress σnand shear stress τ.
• Fingers with spacing λc develop as fluid flows inside shear band (nano-channel) and meniscus breaks down under suction gradient.breaks down under suction gradient.
• Fingers grow ahead as flattened cylinders under blunting crack front flow field.
• Shear band contains collinear elliptical c/s cavities (2a × 2b).
• Under action of σn , shear band widens and cavities grow plastically (creeping mechanism).
• Cavities assume shapes of notches and• Cavities assume shapes of notches and ligament bridging them ruptures – give rise to ridges on fracture surface.
Fracture Morphology in Brittle BMGs• Brittle Mg based glasses (low toughness KIc ~ 2 MPam1/2)
• Under low magnification : Mist, Transition, Mirror morphology .
•Wang et.al., J.App.Phys., 2009
p gy
• River lines seen in misty region.
• HRSEM images show :
• Cellular (spider web) structure in misty• Cellular (spider-web) structure in misty region.
• Nano-corrugations or nano-stripes in mirror region.
• Xia and Wang, Small, 2012g
HRSEM images
Mist Region Transition Region Mirror Region
100nm100nm
What is the cellular (spider-web) structure?
Xi d W S ll 2012• Xia and Wang, Small, 2012
• Mg-based glass; Very brittle (low toughness KIc ~ 2 MPam1/2)
AFM i f id b iAFM image of spider-web region
• Each cell : Hierarchical architecture with fine cavity at center surrounded by 4 conescones.
• AFM images show nano-voids of size ~ 100 nm; depth ~ 5-10 nm
100 nm
AFM images show nano voids of size 100 nm; depth 5 10 nm.
• Large number of nano-voids form near crack tip & ligaments connecting them neck down.
What are the nano-corrugations ?
W t l PRL 2007• Wang et.al., PRL, 2007
• Mg-based glass; Very brittle (low toughness KIc ~ 2 MPam1/2)
AFM images of matching surfaces
• Crack propagation directions C ac p opaga o d ec o sindicated by white & black arrows.
• Using red triangles as reference points, the 2 surfaces are p ,matched.
• Important Note : Peak-to-peak matching is observed between the g2 surfaces ⇒
• Striations are nano-voids
Line scans along green dotted lines in (a), (b)
Molecular Dynamics Simulations of Fracture in MGs• Murali et.al., PRL, 2011
• MD fracture simulations in 2 MGs: Fe80P20 & Cu50Zr50
• Simulation box size : 400 nm× 270 nm× 2 nm; Crack length : 68 nm; Periodic BCs applied //lto crack front : plane strain conditions.
• FeP glass shows:
• Crack extension with few shear bandsbands.
• Little blunting.
• Nano-voids form & coalesce with crackcoalesce with crack tip.
• CuZr glass shows:
• Extensive shear banding.
• Considerable blunting.
• No cavitation.• Summary : Cavitation occurs ahead of the tip for Fe-P glass whereas there is no cavitation for Cu-Zr.
Is cavitation possible ahead of tip ?• Cavitation in solids : Nucleation of voids in an initially void-free solid due to synergistic interplay of elastic & plastic deformation.
• Require : Hydrostatic stress σh = critical value σc (Hou & Abeyaratne)
• Continuum elastic-plastic computation of stress variation ahead of crack tip & cavitation stress levels :
• Note peak hydrostatic stress attained ahead of the tip falls well short of cavitation stress levels for both glasses.
• If peak σh/σy < σc/σy how can cavitation occur ahead of the tip for FeP glass as predicted by MD?MD?
Can low Poisson’s ratio promote cavitation ahead of tip leading to ductile-brittle transition?
• Continuum elastic-plastic analysisContinuum elastic-plastic analysis
• Peak hydrostatic stress ahead of tip also ↓ with ↓ in ν.
• Note cavitation stress decreases with ↓ in ν.
• Conclusion : Decreasing Poisson’s ratio does not help (σ /σ < σ /σ ) !• Conclusion : Decreasing Poisson s ratio does not help (σh/σy < σc/σy) !
• Also incorporating pressure sensitivity (Mohr-Coulomb) does not help !
• Cannot predict cavitation ahead of tip from continuum theories !!!
MD Simulations of Cavitation• Murali et.al., PRL, 2011, ,
FeP CuZr
• Plane strain equi-biaxial dilatation (plane strain) : Box size - 20 nm× 20 nm× 2 nm
• Multiple small voids nucleate, grow & coalesce.
• Single void nucleates and grows in simulation box.& coalesce.
• Contiguous crack path following nano-void coalescence.
• Void maintains circular shape under biaxial loading.
Hydrostatic stress versus Dilatation – From MD
• For FeP glass the peak hydrostatic stress attained for fo = 0 (i.e., cavitation stress) is about 1.4σo where σo is the yield strength.
• Not much sensitivity of peak hydrostatic stress to initial porosity fo (i.e., presence of an initial big void).
• The big void does not grow. Instead the tiny voids g g ygrow substantially (as if the big void was not present).
• For CuZr glass the peak hydrostatic stress for fo = 0 (cavitation stress) is about 3.5σo where σo is the yield strength.
• Peak hydrostatic stress drops significantly as initial porosity fo ↑.
• No additional voids nucleate in the presence of the pbig void – only the big void grows .
•Classical cavitation response .
Observation from MD simulations of cavitation• Murali et.al., PRL, 2011
• FeP shows:ρ/ρmean
FeP shows:
• Large local atomic number density ρ/ρmean fl ifluctuation.
• Voids nucleate in regions of low atomic density !atomic density !
• CuZr shows less atomic density fluctuation
• Summary : Intrinsic atomic density fluctuation seems to be key to understanding brittle cavitation behavior of FeP glass !
Characterization of atomic density fluctuations• Line scans of atomic density along the diagonals of the simulation boxLine scans of atomic density along the diagonals of the simulation box
• Local number density ρ~ Number of atoms
i hi h fpresent within sphere of radius equal to inter-atomic potential cut-off distance.
• ρ/ ρmean fluctuates from 0.8 to 1.2 for FeP and from 0.95 to 1.05 for CuZr
• Periodogram (Spectral Intensity) of density fluctuationsCuZr
• FeP shows many modes of fluctuations with λ 2 8 nmwith λ ~ 2–8 nm.
• CuZr shows one dominant mode with λ~ 3-4 nm.
• Higher intensity of fluctuations for FeP
Fluctuations in local cavitation stress
• Hydrostatic stress simulations conducted with smaller box sizes (12 configurations per box size)configurations per box size)
• Note FeP shows ~ 40% scatter in local cavitation stress as sample size ↓
• CuZr shows ~ 10% scatter in local cavitation stress as sample size ↓
• Hydrostatic (Virial) stress σh distribution ahead ofσh distribution ahead of crack tip from MD;
• Note peak σh falls within band of σc values for FeP &
t f C Znot for CuZr.
Continuum (FE) analysis of cavitation with distributed weak zones
• Doubly periodic distribution of weak zones subjected to biaxial straining.
W k A d i l ith
Indrasen et.al., JMPS 2013
Σ1, E1
•Weak zones : Assumed as circular with volume fraction : fow
• Volume fraction of small void : fo~ 5×10-5
(t t i it ti )
Weak zone
Lo(to trigger cavitation)
• Yield strength of weak zone : σow
• Yield strength of background : σo
LoSmall void (σo /E = 0.03; ν = 0.36; N = 0.05)
•2D plane strain FE analysis of 1/4th of a unit cell (by symmetry)
1h h dV
VσΣ = ∫
• Macroscopic hydrostatic stress
• Σha : Macroscopic hydrostatic stress in heterogeneous aggregateh h
VV ∫ aggregate
• Σhw : Macroscopic hydrostatic stress in weak zone
Macroscopic hydrostatic stress & void volume fraction versus dilatationversus dilatation
• Initially Σha ≈ Σhw and ↑ linearly with Θ
• Void starts growing unstably when Σhwreaches peak value (cavitation !)
•Σha attains a local peak at cavitation (Σc) & minimum & then ↑with Θ; reaches a global peak at a much larger Θ.
S i ti i t l f l ti &• Synergistic interplay of elastic energy & plastic dissipation in surrounding material dictates global peak in Σha
• Σc ≈ Peak value of Σhw ≈ LocalΣc Peak value of Σhw Local cavitation stress of weak zone Σcw
• Yield properties of weak zone governs Σc .
• Σc = 0.63 σo << Cavitation stress in absence of weak zones (2.2 σo)
Effect of yield strength ratio σow / σo
C ( f )• Cavitation (unstable growth of vanishingly small void) always occurs at peak Σhw
• Σc ≈ Peak value of Σhw ≈ Local cavitation stress of weak zone Σcw
• Σc /σo ↑ with σow / σo
• Difference between Σc and global peak in Σha ↓ with σow/σo ↑
• At high σow/σo : Σc = global peak in Σha
Cavitation stress versus yield strength ratio & volume fraction of weak zone
• For any σow / σo : cavitation stress Σ does not depend onstress Σc does not depend on fow
• Σc /σo ↑ with σow / σo
• Σc → 2.2 σo as σow / σo → 1
• Local yield properties of weak zone governs Σc .
Sensitivity of cavitation response to pre-existing void
Sq are region (2L × 2L ) s bjected to bia ial (plane strain)
• One quarter of square domain is modeled with a big void of radius ro at
• Square region (2Lo × 2Lo) subjected to biaxial (plane strain) stretching (E1, E2). Macroscopic (aggregate) stresses are Σ1 & Σ2 .
modeled with a big void of radius ro at center. Symmetry conditions imposed on X1 = 0 & X2=0.
• Four weak zones (vol fraction fow ) introduced along the diagonals of the square region.
• Small void in weak zone (vol fraction ~ 5 ×10-5) to trigger cavitation
• Yield strength of weak zones is σow ;that of the background material is σo(σo/E = 0.03).
• Analysis conducted for different sizes ro of centre void
FE analysis of cavitation behavior of BMGs
• Initial porosity fo = 0.005
• Volume fraction of weak zone : fow = 0.28
• Yield strength ratio σow/ σo = 0.6
• Note small void in weak zone growsweak zone grows much faster than the large centre void.
FEA versus MD observationIndrasen et.al., JMPS 2013
• From FEA – After peak stress stage
•From MD simulation –After peak stress stage
• Large plastic strain accumulates near small voids ; Much less plastic strain near big voidvoid.
• Small voids have grown substantially whereas big void has not grown much –corroborates with MD observation.
Porosity fo Peak (σh /σo) % Drop
Hydrostatic stress versus Dilatation – From FEA
5x10-5 2.19 -0.005 1.92 12.30.015 1.71 21.90.02 1.63 25.6
• High cavitation stress σh/σo .
P k / iti t fWithout weak zone
• Peak σh/σo very sensitive to fo• Behavior like CuZr glass.
Porosity fo Peak (σh /σo) % Drop
5x10-5 1.50 -0.005 1.47 2.00.015 1.39 7.30.02 1.33 11.3
With k • Low cavitation stress σh/σo .
• Peak σh/σo less sensitive to fo• Behavior like FeP glass.
With weak zone
Cavitation bifurcations in composite cylinder with weak core• Analytical solution for sudden expansion of infinitesimal cylindrical void (f →0) at theAnalytical solution for sudden expansion of infinitesimal cylindrical void (fo →0) at the center of a composite elastic-plastic cylinder subjected to radial traction on boundary
• Weak inner core with vol fraction fow and yield strength ratio σow/σo
• Bifurcation at point ‘c’ from solid cylinder to one with infinitesimal cavity at center.
• Turning points : ‘p’ and ‘q’ : Nature of equilibrium solution changes : stable ⇔ unstable.
• For small fow : snap-cavitation possible for Σq < Σha < Σc :
• Void of finite size (f ~ 0.03) can suddenly appear at the center of the weak core
• Grows stably till turning point ‘p’ & then becomes unstable.
• As fow ↑ : only cavitation bifurcation to left is possible (like homogeneous plastic solids).
Effect of volume fraction of weak inner core on bifurcation
• As fow → 0 : Cavitation bifurcation to left at Σh ~ 2.45 σo
• As fow → 1 : Cavitation bifurcation to left at Σh ~ 0.7 σo
• Turning points ‘p’ and ‘q’ appear as fow ↑ f 0 d f f (2)from 0 and merge as fow → fow
(2)
• Σp < Σc for fow > fow(1)
Range of fow Global peak in Σha Cavitation stress No. of turning pointsg ow p ha g p
0 Σp Σp 0
(0, fow(1)) Σp Σc 2 (p, q)
[ fow(1), fow
(2)) Σc Σc 2 (p, q)
[ fow(2), 1] Σc Σc 0
Effect of σow / σo on transition weak zone volume fractions
III
II
I
C f• Region I : Two turning points ; Cavitation precedes attainment of global hydrostatic stress peak
• Region II : Two turning points ; Cavitation at global hydrostatic stress peak
R i III N i i Bif i i il h l i lid• Region III : No turning points; Bifurcation similar to homogeneous plastic solids
Experimental evidence for fluctuation in local properties of MGs
• Measured local indentation modulus M using Atomic Force Acoustic Microscopy (AFAM) for amorphous PdCuSi & crystallized PdCuSiMicroscopy (AFAM) for amorphous PdCuSi & crystallized PdCuSi.
• M exhibits a wide variation (about 30%) on a scale below 10nm in amorphous PdCuSi.
• The fluctuation is 10 to 30 times smaller in crystallized PdCuSi.
• Fluctuation in M attributed to significant spatial variation in local potential energy of a cluster of atoms.
BMG Fracture Models
Brittle BMG
N
crack path Ductile BMG
Nano-voids
Crack blunts significantly
Islands of weak material having
Original crack Shear Bands
significantly
Islands of weak material havingyield strength σyw < σy of background material
• Crack growth due to cavitation
• No cavitation because of more homogeneous initial local strength distribution;
• Crack growth due to cavitation ahead of tip in nearby weak island followed by coalescence. Low Kc
• Failure by crack growth in shear bands; High Kc