NUCL 520NUCL 520

Radiation Effects and Reactor Materials

Topic 9: Mechanical Behavior ofTopic 9: Mechanical Behavior of

Materials

Topic 9: Mechanical Behavior of Topic 9: Mechanical Behavior of Irradiated Materials: Hardening Irradiated Materials: Hardening

Lectures 20Lectures 20--2222

Prof. Miloshevsky

OutlineOutline

ElastoElasto--plasticityplasticity ofof metalsmetalsElastoElasto--plasticityplasticity ofof metalsmetals

HardeningHardening mechanismsmechanismsHardeningHardening mechanismsmechanisms

HardeningHardening modelsmodelsgg

DeformationDeformation mechanismmechanism mapsmapspp

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ElastoElasto--plasticity of metalsplasticity of metals irradiation hardening and deformation: introduction ff t f i di ti t l i i i ld h id effect of irradiation on metals: increase in yield strength over widetemperature range <0.3Tm yield strength (yield point): stress level at which material behaviory g (y p )changes from reversible elastic to permanent and non-reversibleplastic deformation effect of irradiation on stress strain behavior in austenitic (fcc) and effect of irradiation on stress-strain behavior in austenitic (fcc) andferritic (bcc) SS: 1) increasing yield strength; 2) reduction in ductility

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ElastoElasto--plasticity of metalsplasticity of metals irradiation hardening and deformation: introduction d ili f h h i ( l i ) i l ductility: measure of how much strain (elongation) material cantake before rupture much higher increase in yield strength by irradiation than that due tog y g yultimate tensile strength for both fcc and bcc metals total loss of ductility (brittle (breaking) material at zero elongation) di i i d d h d i i f i d f bl ki radiation-induced hardening: creation of various defects blockingdislocation motion (plastic deformation) and strengthening material(loss of ductility)( f y)

o irradiation-produced defects causing hardening:• voids and bubbles, precipitates, defect clusters

i it d f t l t l• impurity-defect cluster complexes• faulted or unfaulted, vacancy or interstitial dislocation loops• dislocation lines (dislocation loops that have unfaulted and

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d s oca o es (d s oca o oops a ave u au ed a djoined dislocation network of original microstructure)

ElastoElasto--plasticity of metalsplasticity of metals review of basic elements of elasticity and plasticity theory relationship between stress σ and strain ε (Hooke’s law):

E - modulus of elasticitystress: load (force) per unit area tending to deform materialstress: load (force) per unit area tending to deform material strain: relative change in size or shape of material

material extension along x-direction & contraction along transversey- and z-directions under applied tensile (pulling apart) force transverse strain: as constant fraction of longitudinal strain

0 33 P i ’ iν ~ 0.33 - Poisson’s ratio

3D stress-strain table:

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ElastoElasto--plasticity of metalsplasticity of metals review of basic elements of elasticity and plasticity theory superposition of straincomponents:

addition of strain components:

- hydrostaticor mean stress

- volume strain relation between shear stresses and shear strains:relation between shear stresses and shear strains:

μ - shear modulus

th l ti t t i t t i l ti f i t i lid three elastic constants in stress-strain relations for isotropic solid: 21 elastic constants for general anisotropiclinear elastic solid; considerable reduction for

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solids with high degree of symmetry

ElastoElasto--plasticity of metalsplasticity of metals review of basic elements of elasticity and plasticity theory no coupling between expressions for normal & shear stress andstrain at small elastic deformations (solve for stress in terms of strain)

eliminate σ and σ : eliminate σyy and σzz :

in tensor notation:

δij - Kroneckerd lnotation: delta

σij - three equations for normal stress and six equations for shear stress

longitudinal stress expressed via shear modulus and Lame constant

where - Lame constant

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ElastoElasto--plasticity of metalsplasticity of metals review of basic elements of elasticity and plasticity theory hydrostatic (pure tension and compression) and deviatoric (shearstress) components of stress and strain relationship between distortion and stress deviator: relationship between distortion and stress deviator:

relationship between hydrostatic stress and mean strain: relationship between hydrostatic stress and mean strain:

where

K - bulk modulus or volumetric modulus of elasticity: ratio ofhydrostatic pressure -p to dilatation β - compressibility of solid

h d t ti t t 1) ibl f l ti l ho hydrostatic stress tensor: 1) responsible for elastic volume change;and 2) doesn’t affect yield stress and plastic deformation of solido stress deviator: 1) involves shear stress; and 2) responsible for

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plastic deformation

ElastoElasto--plasticity of metalsplasticity of metals review of basic elements of elasticity and plasticity theory two special cases of engineering stresses: plane stress & plane strain

plane stress state: zero component of one ofprincipal stresses such as in thin plate loaded inplane of plate

plane strain state: zero components ofone of principal strains with one muchgreater dimension than other two

using > principal strains:

using =>

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ElastoElasto--plasticity of metalsplasticity of metals review of basic elements of elasticity and plasticity theory principal stresses and strains: act normal to principal planes; noshearing stresses acting on these planes strain energy U: energy expended by action of external forces instrain energy U: energy expended by action of external forces indeforming elastic body

elastic energy: product of force F anddistance δ : - area under elastic portion ofstress-strain curve (lecture 10-13, slide 24)

h i i f lid d il change in strain energy of solid due to tensilestress in x-direction

Adx - volume iincrement

strain energy per unit volume or strain energy density

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ElastoElasto--plasticity of metalsplasticity of metals review of basic elements of elasticity and plasticity theory for pure shear stress:

elastic strain energy for three-dimensional stress state:

substituting expressions for strains gives

any stress component: derivative of U with respect toy p pcorresponding strain component

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ElastoElasto--plasticity of metalsplasticity of metals plasticity: introduction dependence of plastic deformation on loading path by which final

state achieved (only initial and final states for elastic deformation) power-law hardening relationship: dependence of stress required to power-law hardening relationship: dependence of stress required to

cause metal to flow plastically on strain (flow curve)εp - plastic strain K - stress at εp = 1.0 n - strain hardening exponentp p p

flow curves for elastic n = 1 (left) perfectly plastic n = 0 (middle)and plastic behavior with intermediate value of n (right):

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ElastoElasto--plasticity of metalsplasticity of metals plasticity: models goal: development of empirical mathematical relations forgoal: development of empirical mathematical relations forpredicting plastic yielding under stresses yielding: second invariant of stress deviator exceeded some critical

l ( Mi )value (von Mises) applying this expression to uniaxial tension test with σ1 =σy, σ2 =σ3= 0, (σ - yield stress) to determine value of k0, (σy yield stress) to determine value of k

substituting this expression into that for k gives familiar form ofvon Mises yield criterion

in presence of shear stresses in presence of shear stresses

yielding: differences in stresses on right side of von Mises equation

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y g g qexceed yield stress in uniaxial tension σy

ElastoElasto--plasticity of metalsplasticity of metals plasticity: models relation between shear stress σ and principal stresses for pure shear relation between shear stress σs and principal stresses for pure shearstress state (as in torsion test) =>

where k - yield stress in pure shear

von Mises criterion: less yield stress intorsion than that in uniaxial tension by th T ( i h t ) it i i ldi h another Tresca (or maximum shear stress) criterion: yielding whenmaximum shear stress reached in uniaxial tension test

σ1 - algebraically largest principal stress σ3 - algebraically smallest principal stress

for uniaxial tension σ1 = σy, σ2 = σ3 = 0 and shearing yield stressσ = σ /2 :σsy = σy/2 : => for pure shear stress state σ1 = −σ3 = k and σ2 = 0:

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ElastoElasto--plasticity of metalsplasticity of metals tension test: introduction subjection of sample to increasing uniaxial tensile force with subjection of sample to increasing uniaxial tensile force withsimultaneous observations of elongation of sample stress-strain diagram of data from load-elongation measurementsresulting in engineering stress-engineering strain curve parameters for describing stress-strain curve: 1) yield, tensile andfracture strengths; 2) uniform fracture strains; and 3) reduction in areafracture strengths; 2) uniform, fracture strains; and 3) reduction in area engineering stress-strain curves & parameters for fcc and bcc metals

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ElastoElasto--plasticity of metalsplasticity of metals tension test: definitions average longitudinal stress S: load P divided by original area A average longitudinal stress S: load P divided by original area A0

average linear strain e: ratio of change in length δ to original length L0 average linear strain e: ratio of change in length δ to original length L0

i i t i i t i t t d f ti engineering stress-engineering strain curve: not true deformationcharacteristics of material; based on original dimensions of samplechanging continuously during tension test true stress σ and strain ε: based on instantaneous values of crosssectional area and length given by

close true strain and engineering strain at small values of strain

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g g(<0.2); significant divergence at large values of strain

ElastoElasto--plasticity of metalsplasticity of metals tension test: definitions relationship between true and engineering stress using conservation relationship between true and engineering stress using conservationof volume:

recovering of solid to its original dimensions upon load removal up recovering of solid to its original dimensions upon load removal upto certain limiting load elastic limit: no elastic behavior of material beyond this load limit

o permanent deformation of solid upon removal of load exceedingelastic limit

yield stress σy or YS: stress at which material plasticity beginsy y p y g offset yield strength: stress corresponding to intersection of stress-strain curve and line parallel to elastic part of curve and offset bystrain of 0.2%

tensile strength or ultimate tensile strength (UTS): maximum load

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divided by original cross sectional area of sample:

ElastoElasto--plasticity of metalsplasticity of metals tension test: definitions true stress at maximum load: true tensile strength given by maximum true stress at maximum load: true tensile strength given by maximumload divided by sample cross sectional area at maximum load

o eliminating Pmax gives fracture stress: stress at point of failure given by fracture stress: stress at point of failure given by

o initially uniform accumulation of strain in gage section of sampleo initially uniform accumulation of strain in gage section of sampleup to UTS with further necking or localized deformations true uniform strain εu : strain at maximum load:u

true fracture strain εf : - true strain based on original area and areaafter fracture Af

with- reduction i t

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with in area at fracture

ElastoElasto--plasticity of metalsplasticity of metals tension test: definitions local necking strain ε : strain required to deform sample from local necking strain εn: strain required to deform sample frommaximum load to fracture:

onset of plastic instabilit : increase in stress d e to decreasing cross onset of plastic instability: increase in stress due to decreasing crosssectional area becomes greater than load-carrying ability of metalo necking at point of maximum load defined by condition dP = 0

o from conservation of volume:

o point of tensile instability:strain hardening exponent:

- from true stress-true straincurve: point where rate of

i h d i lstrain hardening exponent: strain hardening equals stress

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simple expression for the true uniform strain:

ElastoElasto--plasticity of metalsplasticity of metals yield strength: introduction yield strength: onset of plasticity key parameter in determining yield strength: onset of plasticity - key parameter in determiningmechanical behavior of metals understanding of yielding: through examining behavior ofdislocations in metal under stress pile-up of dislocations formed by Frank-Read sources on slipplanes at grain boundaries precipitates or sessile dislocationsplanes at grain boundaries, precipitates or sessile dislocations

o high stress onleading dislocation in

o back stress on dislocationsfurther from obstacle opposing

pile-up due to 1)applied shear stress;and 2) interaction with

pp gtheir motion on slip plane

and 2) interaction withother dislocations onslip planeo initiation of yielding or nucleation

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y gof crack at obstacle due to high stress

ElastoElasto--plasticity of metalsplasticity of metals yield strength: analysis estimation of number of dislocations in pile up of length L under estimation of number of dislocations in pile-up of length L undershear stress σs on slip plane by summing x-direction forces betweeneach dislocation under condition of mechanical equilibrium

at large distances from pile-up: array of n dislocations acting as at large distances from pile-up: array of n dislocations acting assingle dislocation with Burgers vector nb and force nbσs

analysis of stress at head of pile-up by Stroh: tensile stress normal toline OP in neighboring grain

maximum value of σ at θ = 70.50 yielding

h t ti i l OP

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shear stress acting in plane OP :

ElastoElasto--plasticity of metalsplasticity of metals yield strength: analysis id b t l i b d ith di t f h d f consider obstacle as grain boundary: with distance r from head ofpile-up in grain 1 to nearest dislocation source in grain 2 and length ofpile-up L taken to be equal to grain diameter d yielding: shear stress in pile-up σs reaches shear stress causingyielding σsy or σs = σsy : - stress to nucleate

li i i 2slip in grain 2σsi - friction stress or stress opposing dislocation motion in slip plane yielding equation in terms of normal stress with σ = σ /m yielding equation in terms of normal stress with σs = σ /m

S h idt f t d fi d ti

- Hall-Petch equationdescribing grain size

m - Schmidt factor defined as ratioof resolved shear to axial stress

g gdependence of yield stress

increase in yield strength with decreasing grain size: describes well

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few to hundreds of micrometer grains, but fails for nanometer grains

Hardening mechanisms Hardening mechanisms irradiation hardening: introduction two mechanisms of metal strengthening by irradiation: source two mechanisms of metal strengthening by irradiation: sourcehardening and friction hardening source hardening: increase in unpinning or unlocking stressrequired to release and start dislocation moving on its glide plane friction hardening: resistance to dislocation motion caused bynatural or radiation produced obstacles lying close to or in slip planenatural or radiation-produced obstacles lying close to or in slip plane both source and friction concepts used to describe total hardeningresulting from radiation-induced defects unclear distinction between source and friction hardening:deformations produced by friction hardening found similar to those ofsource hardening

g

source hardening loss of distinction: distance between defect clusters less thansource length producing observed critical shear stress - source cannot

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operate without interference from lattice clusters

Hardening mechanisms Hardening mechanisms source hardening: observations for fcc and bcc metals h d i f d i i di t d f d b th i di t d d source hardening: found in irradiated fcc and both unirradiated andirradiated bcc metals in unirradiated bcc metals, source hardening thought to be causedg gby pinning or locking of dislocation lines by impurity atoms

o required unpinning of dislocation line from impurities beforeFrank Read source can operate under applied stressFrank-Read source can operate under applied stresso required larger stress than that to move dislocation causing dropin yield stress continuing at constant flow stress (Luders strain

in irradiated fcc metals, increase in stress due to irradiation-produced defect clusters in vicinity of Frank Read sources thus

region) until onset of work hardening (plot for bcc in slide 15)

o at stress levels sufficient to release source, moving dislocations can

produced defect clusters in vicinity of Frank-Read sources, thusexpanding loops and permitting source multiplication

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destroy small clusters reducing stress needed to continue deformation

Hardening mechanisms Hardening mechanisms source hardening: Frank-Read source of dislocations i i di t d f t l i i t f F k R d i in unirradiated fcc metals, unpinning stress of Frank-Read sources inmetal required to initiate dislocation motion

μ - shear modulus l(= 2R) - distance betweenμ shear modulusb - Burgers vector

l( 2R) distance between pinning points

stress inversely proportional to distance between pinning points

Frank-Read sourcefor production ofdi l idislocations

l t 10 13see lectures 10-13, slides 29-30

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Hardening mechanisms Hardening mechanisms source hardening: properties gradual onset of yielding characteristic in fcc metals: explained by gradual onset of yielding characteristic in fcc metals: explained bydistribution of stresses required to operate sources at low applied stress, generation of dislocations by sources easiestto operate (with large separation between pinning points) pile-up of dislocations exerting back stress on dislocation sourceceasing its operation and hence plastic strainceasing its operation and hence plastic strain activation of more dislocation sources with increasing appliedstress and increase in dislocation multiplication requirements for source hardening: bowing out of dislocation linesegment between pinning points requiring strong pinning release of dislocation at lower values of applied stress if dislocation release of dislocation at lower values of applied stress if dislocationsegment is able to unlock itself before bowing occurs

o realization of unlocking dislocation line segment: pinning points

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consisted of small dislocation loops or defect clusters

Hardening mechanisms Hardening mechanisms source hardening: edge dislocation-loop interactions hardening by faulted loops: interaction of stress fields between loop hardening by faulted loops: interaction of stress fields between loopand edge dislocation moving on its slip plane, located parallel to anddisplaced a distance y from plane of loop

loops arranged in row with Burgersvector bl, radius Rl, spacing l and att d ff di t f dstand-off distance y from edge

dislocation of Burgers vector be

l i l only σyy term exerting stress on loopacting to expand or contract it

force on loop due to σyy componentof stress from edge dislocation:

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Hardening mechanisms Hardening mechanisms source hardening: edge dislocation-loop interactions work to expand loop:

substituting for stress σyy and differentiating with respect to x givesf b t l d d t i di tiforce between loop and edge segment in x-direction

maximum force at angle of 400 between distance vector and glideplane of dislocation written as function of r/y for ν = 1/3 and b = be

given F = σ bl => (Singh et al ) given F = σsbl => (Singh et al.)

expression for shear stress in terms of loop spacing at y = 1.5rconsistent with observed microstructure

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consistent with observed microstructure

Hardening mechanisms Hardening mechanisms friction hardening: introduction t i d t t i l ti d f ti ft t d fl stress required to sustain plastic deformation often termed as flowstress or friction stress friction forces responsible for resisting dislocation motion throughf f p g gcrystal lattice due to dislocation network and obstacles such as defectclusters, loops, precipitates, voids, etc. long range and short range sources of friction hardening long-range and short-range sources of friction hardening long-range friction stresses: caused by dislocation-dislocationinteraction through their stress fieldsg short-range friction stresses: direct interaction between movingdislocations and discrete obstacles in slip plane t t l f i ti t t b th l (LR) total friction stress σF necessary to overcome both long-range (LR)and short-range (SR) forces in order to move dislocation

where

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with contributions due to precipitates, voids and loops

Hardening mechanisms Hardening mechanisms friction hardening: long-range stresses i i f l f l i i i b i origin of long-range forces: repulsive interaction between moving

dislocation and components of dislocation network of solid dislocations on glide planes exerting forces on each other due to dislocations on glide planes exerting forces on each other due to

their stress fields representing long-range stress fields maximum force between edge dislocations occurring at angle θ = 00

using ν = 1/3 and where ρd - dislocation density =>

where α ∼ 0.44

stress to overcome this force giving stress to overcome this force giving yield strength in terms of grain size-dependent unpinning stress

dislocation density varying with grain

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y y g gsize d as ρd = 1/d- see slide 22

Hardening mechanisms Hardening mechanisms friction hardening: long-range stresses l bl A Li l {Li (b ) 32 GP 0 3 } example problem: A Li sample {Li (bcc): µ = 32 GPa; a = 0.3 nm}

exhibited an yield strength of 10 MPa. a) Estimate the dislocationdensity assuming that the strengthening is all due to dislocationsy g g g(strain hardening); b) Estimate the strengthening if the dislocationdensity increases by a factor of 100 following cold working (orexposure to radiation)exposure to radiation)

a) for bcc: m

f i li it 1 d 32 GP 32 000 MPfor simplisity, assume α ~ 1 and µ = 32 GPa = 32 000 MPa

b)

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Hardening mechanisms Hardening mechanisms friction hardening: short-range stresses origin of short-range forces: direct contact interaction between

moving dislocation and obstacle that lies in its slip plane

classified into athermal and thermall acti ated interactions: classified into athermal and thermally activated interactions:

o athermal stress interaction: independent of temperature resultingin dislocation bowing around obstaclein dislocation bowing around obstacle

o thermally activated stress interaction: dislocation overcomingobstacle either by cutting through or climbing over itobstacle either by cutting through or climbing over it

o for both processes, addition of energy required

dependence of friction stress due to dispersion of barriers onaverage separation distance between obstacles in slip plane ofmoving dislocation: (see next slide)

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moving dislocation: (see next slide)

Hardening mechanisms Hardening mechanisms friction hardening: short-range stresses unit area of slip plane intersected by spherical objects of diameter d

randomly distributed throughout solid at concentration of N cm-3

o intersection of slip plane byo intersection of slip plane byany sphere with its centerwithin slab centered on slipplaneNd - number of obstacles inthis volume element as wellthis volume element as wellas number of intersectionsper unit area on slip plane

o product of number of intersections per unit area Nd and square ofdistance between obstacles l2 gives unity yielding distance betweenobstacles as

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obstacles as

Hardening mechanisms Hardening mechanisms friction hardening: precipitates dislocation-precipitate interaction: short-range physical contact of

dislocation with precipitate obstacle bowing of dislocations between precipitate obstacles similar to that bowing of dislocations between precipitate obstacles similar to that

as in Frank-Read source

obstacles left surrounded by obstacles left surrounded bydislocation loops presentingstronger obstacles to nextdi l tidislocations

determination of short-range stress due to array of obstacles ofd it N d i ddensity N and size d line tension of edge dislocation:

di l i di

- neglected dislocation core energy

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R - grain radiusrc - dislocation core radius core energy

Hardening mechanisms Hardening mechanisms friction hardening: precipitates

relation between line tension and shear stress: =>

h R l/2 d l b l iwhere R = l/2 and l - obstacle spacing

using gives: withusing gives: with

relation between yield stress σy & shear stress σs by Taylor factor M

=>

increment in yield strength due to obstacles of size d number increment in yield strength due to obstacles of size d, numberdensity N and strength α

- dispersed barrier hardening

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dispersed barrier hardening

Hardening mechanisms Hardening mechanisms friction hardening: precipitates

dislocation cutting byprecipitate obstacle

shearing of obstacle by dislocation with top and bottom halvesg y pdisplaced along glide plane by amount equal to magnitude of Burgersvector of dislocation

separation of two parts resulting in two smaller obstacles due tosuccessive shearing of obstacle on the same plane

molecular dynamics (MD) simulations of dislocation-obstaclecutting: visualization of complicated micro-structural processes withqualitative interpretation

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qualitative interpretation

Hardening mechanisms Hardening mechanisms friction hardening: precipitates mechanisms summarized by Dieter for strengthening and hardeningof materials due to obstacle cutting

additional work: for shearing precipitate by dislocation due toincreasing surface area and creation of step of width b

extra energy: for creation of new interface within precipitate due toshearing of ordered precipitate structures

hardening: due to difference between elastic moduli of matrix andprecipitate affecting line tension of dislocation and requiringadditional stress to cut precipitateadditional stress to cut precipitate

strengthening: due to difference in Peierls stress (force to movedi l ti ithi l f t ) b t i it t d t i

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dislocation within plane of atoms) between precipitate and matrix

Hardening mechanisms Hardening mechanisms friction hardening: precipitates Russell & Brown model for describing hardening due to differencein moduli between precipitate and matrix shear yield stress: function of obstacle spacing l in slip plane and shear yield stress: function of obstacle spacing l in slip plane andcritical angle ϕ at which the dislocation can cut obstacle

equilibrium for dislocation crossing interface:E1 & E2 - energies per unit length before and after crossing1 2 g p g gθ1 & θ2 - angles between dislocation and normal to interface when E1 < E2 => =>

where E∞ - energy per unit length of

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where E energy per unit length ofdislocation in infinite media

Hardening mechanisms Hardening mechanisms friction hardening: voids dislocation cutting through voids: dislocation segments always meetvoid surface at right angles and leave no dislocation ring after passagethrough void or bubbleg force to cut through void

UV - elastic strain energyR - radius of void l - void spacing elastic void-dislocation energy

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yield stress and

Hardening mechanisms Hardening mechanisms friction hardening: loops dislocation-loop interaction stress of order:

shear stress:

yield strength increment due to loops:

observation of interactions between dislocations and loops:

through stress fields: 1) drag of SIA perfect loop in direction ofg ) g p pmoving edge dislocation; 2) rotation and glide of SIA Frank loop

through direct contact: 1) screw dislocation shearing Frank loopwith absorption into screw dislocation core; 2) absorption and re-emission of loop away from original absorption point with screwdislocation cross-slipped onto different glide plane

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dislocation cross slipped onto different glide plane

Hardening mechanisms Hardening mechanisms friction hardening: effect of temperature mean velocity of dislocation segment:

ΔG(σs) - activation Gibbs free energy for cutting or bypassing obstacle β di i l t b it d f B tβ - dimensionless parameter b - magnitude of Burgers vectorν - frequency σs - shear stress activation Gibbs free energy: activation Gibbs free energy:ΔF - total free energy - stress at 0 K

- for random array of obstaclesfor random array of obstacleswhere 0 ≤ p ≤ 1 and 1 ≤ q ≤ 2

where with p = 2/3 and q = 3/2with p 2/3 and q 3/2

strain rate equation for discrete obstacle controlled plasticity- stress & temperature dependence

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p pof dislocation-obstacle interaction

Hardening mechanisms Hardening mechanisms summary of hardening due to long-range and short-range obstacles

values for α based on experimental work varying by significantamount depending on obstacle type

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p g yp

Hardening models Hardening models superposition of hardening mechanisms dependence of irradiated metal microstructure on dose &temperature: 1) defect clusters & loops (low dose); 2) dislocations(increased dose); 3) voids, bubbles and precipitates (high temperature)(increased dose); 3) voids, bubbles and precipitates (high temperature) hardening of true irradiated microstructure: different types, sizesand number densities of obstacles for moving dislocations long-range stresses and short-range obstacles long-range internal stresses: groups of dislocations of the same signof Burgers vectorof Burgers vector effective stress for pushing dislocation over short-range obstacles:difference between applied stress σa and stress σLR necessary formoving dislocations through long-range stress field

total stress: composed of stress due to two types of

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total stress: composed of stress due to two types ofhardening as if each acted independently

short-range obstaclesHardening models Hardening models

dependence of hardening on strengths and relative concentrationsof short-range obstacles in irradiated microstructure two obstacles with high strengthstwo obstacles with high strengths no distinction in dislocation-obstacle interaction effective obstacle distance: expressed through the sum of areadensities of two obstacles in glide plane

and giving

σ1 and σ2 - critical (short-range) stresses of obstacles of type 1 or 2 two obstacles with different strengthsg obstacles with extremely weak and strong forces and distances

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bowing between two strong obstacles & cutting through weak ones

two obstacles with different strengths f di l i h h k & b l

Hardening models Hardening models

movement of dislocation through many weak & two strong obstacles

increase in angle betweeni hb i b h fneighboring branches of

dislocation at weak obstacles

smaller (at given stress)force with which dislocationpressed against weakp essed aga s weaobstacles

smaller angle between neighboring branches of dislocation at smaller angle between neighboring branches of dislocation atstrong obstacles and increase in force acting on them stress needed to push dislocation through this obstacle configuration

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two obstacles with different strengthsHardening models Hardening models

h ll i d i f f b l ( thermally activated surmounting of two types of obstacles (noextreme condition F1 << F2)

waiting time t until dislocation gets enough thermal energy to waiting time ts until dislocation gets enough thermal energy toovercome obstacle ~ N1t1 +N2t2

if N t >> N t effective flow stress due to obstacles of type 1; if N1t1 >> N2t2, effective flow stress due to obstacles of type 1;type 2 is “transparent” for dislocation under applied stress for type 1

&

- for small and high concentration of type 2 obstacles

f b t l f i il t th - root-sum-square for obstacles of similar strengths: root sum square superposition law

for obstacles of dissimilar strengths: - linear i i l

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for obstacles of dissimilar strengths: superposition law

Hardening models Hardening models two obstacles with different strengths bi i f d l d l f id combination of root-square-sum and linear sum models for widerange of obstacle strengths

with

- weighting parameter

with

weighting parameter

S = 1 & S = 0 for linear sum & root sum square lawq

summary of superpositionrules for hardeningrules for hardening

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Hardening models Hardening models two obstacles with different strengths d l d f l i ib i h d i f i model predictions of relative contributions to hardening of variousmicrostructure obstacles as function of dose for three temperatures

near 300 0C, maximum hardening dominated by contribution ofblack dots, small loops and He bubblesblack dots, small loops and He bubbles above about 400 0C, contribution of voids and bubbles to hardening at lower doses, hardening due to voids and loops; at higher doses,

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hardening due to dislocation microstructure and voids

Hardening models Hardening models hardening in polycrystals ff f i b d i h d i i l lli l effect of grain boundaries on hardening in polycrystalline metals:increase in flow stress due to grain orientations and grain boundaries Hall-Petch relation for dependence of tensile yield stress on grain Hall Petch relation for dependence of tensile yield stress on grainsize d: σi - friction stress and ky - unpinning stress for small grain sizes, increase in friction stress σi with little effecton ky due to effect of irradiation

for larger grain sizes, greaterincrease in yield stress withincrease in yield stress withreduction of ky to almost zero linear increase in dislocationdensity with strain ε in solidundergoing plastic deformation

i h

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with β - constant

Hardening models Hardening models hardening in polycrystals i i ld using , yield stress:

- equivalent to Hall-Petch equation withβ - measure of work hardenability four stages of developmentf di ti h d i i Fof radiation hardening in Fe-

Mn-C steel alloy

stage A: 1) very low doses stage A: 1) very low doses(1015 to 1016 n/cm2); 2)increase in ky with negligibleyincrease in σi; 3) no change inn and K; 4) increase in upperyield point and Luder’s strain

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y e d po t a d ude s st a

Hardening models Hardening models hardening in polycrystals f f d l f di i h d i i F M C ll four stages of development of radiation hardening in Fe-Mn-C alloy

stage B: 1) around 1018

n/cm2; 2) increase in σi butli l h i k 3) d

stage C: 1) about 3×1018 n/cm2; 2)increase in σi and decrease in ky; 3)decrease of strain hardening exponentlittle change in ky; 3) decrease

of both n and K; 4) reducedslope of stress-strain curve; 5)

decrease of strain hardening exponentn but slight increase of K; 4) smallchange in slope of stress-strain curve;

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p )increase in Luder’s strain 5) small decrease in Luder’s strain

Hardening models Hardening models hardening in polycrystals f f d l f di i h d i i F M C ll four stages of development of radiation hardening in Fe-Mn-C alloy

stage D: 1) >5×1018 n/cm2;2) further increase of σi andfall of ky nearly to zero; 3)decrease of both n and K; 4); )further decrease in slope ofstress-strain curve; 5) neardisappearance of Luder’sdisappearance of Luder sstrain

experimentally observed that ky either increase or decreasesuggesting that grain size effect is not well-established assuggested theoretically

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gg y

Hardening models Hardening models saturation of irradiation hardening increase in yield strength Δσy as N1/2 with N proportional to total

fluence giving irradiation hardening proportional toti i i b f b t l ith fl- continuous increase in number of obstacles with fluence

contradiction with observations of dislocation loop density andsize saturation by several dpay p

effect of irradiation dose onmeasured tensile yield strengthfor several 300 SS irradiatedand tested at temperature ofabout 300 0C good fit of hardening withϕt1/2 through about 5 dpa

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overestimated hardening at higher dose

Hardening models Hardening models saturation of irradiation hardening account for saturation of hardening at higher doses (Makin & Minter) postulation: no new cluster or zone formed in neighborhood of

existing cluster or zone in displacement cascade time rate of change of density of zones

Σs - macroscopic scattering cross section

existing cluster or zone in displacement cascade

ζ - number of zones created per neutron collision (∼1)

s p gϕ - fast neutron flux V - zone volume

1-VN - fraction of solid volume available for creation of new zones

integration gives

increment in yield strength

where and- solid curve in figure

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where and in previous slide

Hardening models Hardening models comparison of measured and predicted hardening application of hardening models for predicting irradiation hardeningof austenitic SS and irradiated microstructures dominated by loops correlation between measured and calculated yield strength for a setcorrelation between measured and calculated yield strength for a setof alloys (Was & Busby (2003)) irradiated with 3.2 MeV

0protons at 360 0C to dose of5.5 dpa hardening model equation: hardening model equation:

hardening due to bubble-precipitate pair in contact (Kelly) fitting

parameter α

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Hardening models Hardening models comparison of measured and predicted hardening effects of solutes such as Cu and Ni, temperature and irradiationflux on hardening in ferritic SS used in reactor pressure vessels (RPV) components of microstructure contributing to hardening of ferritic

formation of extremelyll C h

components of microstructure contributing to hardening of ferriticRPV steel (Odette et al. 1993)

small Cu rich precipitates(CRP), unstable matrixfeatures (UMF) & stablef ( )matrix features (SMF)o SMFs: defect cluster-solute complexeso UMFs: recovering smallvacancy and interstitial

solute complexes

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yclusters

Hardening models Hardening models comparison of measured and predicted hardening model for increase in yield strength in RPV steels due to irradiation

- due to CRPs - due to SMFscomposition dependence- composition dependence

of hardening due to SMFs for Cu (<0.1%) steels (Odette et al. 2005)

t d d f very strong dependence ofhardening on Cu content in ferriticSS alloys (Kasada et al. 2001)

o increase in hardening with Cuconcentration for both low andhigh dose rate

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Hardening models Hardening models comparison of measured and predicted hardening effect of dose rate on yield strength increment through term Δσyp

- plateau value of hardening ff ti fl ϕ f fl- effective fluence ϕr - reference flux

term X: F and β - fitting parameters h d i f SS ith 0 4% C d 1 25% Ni i di t d t 290 0C hardening of SS with 0.4% Cu and 1.25% Ni irradiated at 290 0Cdue to CRPs vs. fluence and vs. effective fluence (Odette et al. 2005)

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Hardening models Hardening models radiation anneal hardening (RAH) ddi i l h d i h i i li f b additional hardening mechanism occurring upon annealing of bccmetals following irradiation radiation anneal hardening in Nb irradiated to 2×1018 n/cm2 and

o hardening: begins at ~120

gannealed for 2 h (Ohr et al. 1970)

0C, increases to maximum at~180 0C, then decreases

2 d k i h d io 2nd peak in hardening: at~300 0C with further yieldstrength drop due tog precoveryo hardness peaks: first due to O andsecond due to C interstitial impurities o annealing-enabled migration

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second due to C interstitial impurities g gof interstitials to defect clusters

Hardening models Hardening models correlation between hardness and yield strength indentation or shear punch measurements of irradiated samples diamond pyramid tip used in Vickers microhardness measurementswith tip impression into sample (left) and flow pattern during Vickerswith tip impression into sample (left) and flow pattern during Vickersindentation of metal (right) (McClintock & Argon (1966))

o determination of value of hardness measure of resistance of solido determination of value of hardness, measure of resistance of solidto deformation, from shape of indent and magnitude of applied loado yield strength and maximum loads taken from plot of punch load

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y g p pversus punch displacement

Hardening models Hardening models correlation between hardness and yield strength indentation measurements: plastic flow of metal around indenter tipimplying plastic rather than elastic properties of metal (Tabor, 1956) measurement of yield stress of metal: Huber Mises criterion for measurement of yield stress of metal: Huber-Mises criterion forplastic deformation during indentation: maximum shear stress reachescritical value k : σy - yield stress pressure normal to surface of indenter tip:

combining two equations:

for Vickers hardness:

0 927 - ratio of base area of pyramid (projected area) to side area of0.927 - ratio of base area of pyramid (projected area) to side area ofpyramid (contact area) combining equations gives:

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Hardening models Hardening models correlation between hardness and yield strength yield stress determined from Vickers hardness measurements:

C = 0 364 for σ and H in units of kg/mm2with

C 0.364 for σy and Hv in units of kg/mmC = 3.55 for σy in MPa and Hv in kg/mm2

empirical correlation between hardness and yield strength foraustenitic and ferritic SS (Busby et al. 2005):

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Hardening models Hardening models deformation in irradiated metals l f d ili d k h d i i i di d l loss of ductility and work hardening in irradiated metals uniform elongation as function of square root of dose for 300 seriesaustenitic SS irradiated and tested at ∼300 0C (Odette & Lucas, 1991)austenitic SS irradiated and tested at 300 C (Odette & Lucas, 1991) drop in ductility from20-30% to less than 1%b 4 dby ∼4 dpa decrease in workhardening due tohardening due todecrease in differencebetween σUTS and σy

ith increasing dosewith increasing dose variations in trueuniform elongation εu

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g uis equal to n

Hardening models Hardening models deformation in irradiated metals variation of uniform ductility in austenitic SS with dose andtemperature (left) and temperature (right) (Lucas, 1993)

significant decrease in εu with minimum at dose that decreases withtemperature; minimum at 300 0C: major concern for light water reactors

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temperature; minimum at 300 0C: major concern for light water reactors

Hardening models Hardening models deformation in irradiated metals loss of uniform ductility and work hardening: due to interactionbetween dislocations and irradiated microstructure irradiation hardening: 1) pinning of dislocations by obstacles; 2) irradiation hardening: 1) pinning of dislocations by obstacles; 2)unfaulting of Frank loops and incorporation into dislocation network loop unfaulting: first mechanism (Strudel & Washburn, 1964)

o intersection of mobile dislocationwith small Frank loop formingSchockley partial on loop planeSchockley partial on loop plane

o generation of helical segment oni i l di l ti d toriginal dislocation due to

interaction of Shockley partial withfaulted loop eliminating loop

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Hardening models Hardening models deformation in irradiated metals loop unfaulting: second mechanism (Gelles, 1981)

o interaction of Frank loopl i i l f filying in plane of figurewith perfect dislocationmoving on other planeg p

o annihilation of Frank loop and creation of remnant coil becoming apart of dislocation network loop unfaulting: third mechanism (Foreman & Sharp, 1969)

o intersection of mobile dislocationwith loop in which loop glides onitself and becomes part of glidedislocation

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dislocation

Hardening models Hardening models deformation in irradiated metals

loop unfaulting: fourth mechanism (Tanigawa et al., 1996)

f l i i d bo unfaulting triggered byformation of Shockley partialloop inside Frank loopp p

result of unfaulting reactions: removal of faulted dislocation loopsfrom microstructure and growth of dislocation network density as

- length of unfaulted dislocation loop of radius Rl

Nl - number of dislocation loops per unit volume

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Deformation mechanism maps Deformation mechanism maps characterization of plastic deformation: shear stress, strain or strain

t d t trate and temperature five groups of deformation mechanisms (Frost and Ashby, 1982) flow above ideal shear strength flow above ideal shear strength low-temperature plasticity by dislocation glide low-temperature plasticity by twinningp p y y g power-law creep by dislocation glide or climb and glide diffusional creep subdivision of each of mechanisms into additional mechanismsdepending on choice of independent variables: stress & temperature ortemperature & strain ratetemperature & strain rate deformation mechanism map: representation of deformationmechanism in σ-T space with σ represented by normalized stress σs/μ

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and T represented by temperature ratio T/Tm

Deformation mechanism maps Deformation mechanism maps deformation mechanism map for 316 SS with grain size of 50 μmand deformed at strain rate of 10-8 s-1 for nirradiated (left) andand deformed at strain rate of 10-8 s-1 for unirradiated (left) andirradiated (right) metal to 1 dpa at 10-6 dpa/s (Zinkle & Lucas, 2003)

relationship between two independent variables σ and T and relationship between two independent variables σxy and T anddependent variable : strain rate contours above ideal shear strength with

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strain rate approaching infinity:α ~ 0.05 - 0.1

Deformation mechanism maps Deformation mechanism maps strain rate in discrete obstacle controlled plasticity regime

Q - activation energy required to overcome obstacle without aid

- athermal component of flow stress

Q gy qfrom external stress twinning stress σt in polycrystalline metals defined by critical stressfor infinite separation of partialsfor infinite separation of partials

strain rate equation for twinningq g

Qt - activation free energy to nucleate twin without aid of external stressσ stress required to nucleate twinning in absence of thermal activation

- constant

σt - stress required to nucleate twinning in absence of thermal activation effect of irradiation: 1) reduced dislocation glide at <0.5T/Tm; 2)irradiation-enhanced softening at >0.5T/Tm with expansion of

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mdislocation glide regime; 3) twinning at low T/Tm and high stresses

Deformation mechanism maps Deformation mechanism maps localized deformation l i d li i ti f d f t l t d h t i it t clearing and elimination of defect clusters and coherent precipitatesin dislocation glide plane by multiple shearing: dislocation channeling result of channeling: nearly zero work hardening and macroscopicg y g puniform strain with highly localized deformation within channels TEM images of dislocation channels in Fe-18Cr-12Ni irradiated to5 5 dpa at 360 0C with 3 MeV protons and strained to 7% at 288 0C5.5 dpa at 360 0C with 3 MeV protons and strained to 7% at 288 0C

localized hightstress: common

feature betweenchannel deformationin irradiated andunirradiated metals

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Deformation mechanism maps Deformation mechanism maps localized deformation di l ti h l h t i d b idth 0 1 i 1 3 dislocation channels: characterized by width ~0.1 μm, spacing ~1-3μm and amount of strain in channel channels initiation, propagation and termination at grain boundaries, p p g g propagation of channels of surface grains to surface producing stepon surface SEM image of surface ofirradiated SS samples to 5.5 dpawith 3 2 MeV protons at 360 0Cwith 3.2 MeV protons at 360 Cand strained to 7% plastic strain in288 0C argon

o Atomic Force Microscopyused to characterize magnitudeof surface step

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of surface step

Deformation mechanism maps Deformation mechanism maps localized deformation i t ti f di l ti h l ith f ti t

channel strain γ as function of height h

intersection of dislocation channels with surface creating step s onsurface defined by height h and width w (Was, 2006) channel strain γ as function of height hand width w

number of dislocations in channel nrelated to step height

low number of residual dislocations inh l 50channels ~50o slip transfer to neighboring grainso reaction of dislocations with grain boundaryo eact o o d s ocat o s w t g a bou da y

at intersection of dislocation channel and grain boundary: transferof dislocations either to adjoining grain or pile up at grain boundary

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Deformation mechanism maps Deformation mechanism maps localized deformation dditi ll b d d f ti t i i h i l t i i additionally observed deformation twinning or mechanical twinning:localized deformation mechanism caused by partial dislocations formation of deformation twins in fcc metals with low stacking formation of deformation twins in fcc metals with low stackingfault energy (SFE) by glide of Shockley partial dislocation of samesign on successive {111} planes clearing of defects by glide of partial dislocations formed fromdissociation of ordinary dislocations separation of partials or width of stacking fault separation of partials or width of stacking faultγSFE - stacking fault energy

o large separation of partials in low stacking fault energy metalsg p p g gy localized deformation: strong function of dose with localization ofdeformation increasing with dose, and preferential twinning at low

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temperatures

Deformation mechanism maps Deformation mechanism maps localized deformation stress based (left) and strain based (right) deformation mode maps stress-based (left) and strain-based (right) deformation mode mapsfor irradiated austenitic SS alloys describing deformation mode asfunction of dose (Farrell et al., 2004)

for stress-based deformation, increase in yield strength and

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increase in elastic deformation with higher dose

Mechanical Behavior of Irradiated Materials: Hardening Mechanical Behavior of Irradiated Materials: Hardening Summary increase of hardness and yeild strength of materials (irradiation

hardening) due to various irradiation-produced defects d i d ili ( b i l i l) decrease in ductility (more brittle material)

two mechanisms of irradiation hardening: source hardening and

occurrence of radiation hardening: at temperatures less than <0.4Tmand at radiation damage of >0.1 dpa

two mechanisms of irradiation hardening: source hardening andfriction hardening source hardening: increase in applied stress required to unpin

and move dislocation on its slip plane friction hardening: resistance to dislocation motion on its glide

plane caused by long range and short range defect sources

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plane caused by long-range and short-range defect sources