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