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7/30/2019 Hart Maier

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Max Planck Institute for Metals Research, Stuttgart

Alexander Hartmaier

Crack-Tip Plasticity and

Fracture Toughness

International Max Planck Research School for

Advanced Materials

1st Summer School in Stuttgart


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Short phenomenology of fracture

Modeling plastic zones with discrete dislocations Dislocation nucleation at crack tips

Identifying dominant deformation mechanisms

Theoretical description of crack-tip plasticity

Overview


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Overview


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Standard tensile tests Fracture tests

tensile test with homogeneous

specimen

homogeneous plasticity

necking (slip localization)

failure by tearing

global behavior

3-pt-bending with pre-notchedspecimen

confined process zone (yielding)

stress concentration at crack tip

failure by cleavage or general yielding

local behavior, sensitivity to flaws

Mechanical testing


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Process zone astougheningmechanism forceramics

Needle-likemicrostructure inSi3N4

Crack has to doadditional work onits path

Fracture and process zones

(Aldinger, 1999)


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Toughening ofbrittle Cr by Cuinclusions

Crack has todeform Cu particleand to re-nucleateafterwards


(Flaig, 1994)


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(Abraham, Walkup, Gao, Duchaineau, Diaz De La Rubia, Seager; 2002)

Large-Scale molecular dynamics simulation for copper


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Overview


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elasticityinteraction of straightdislocations in infinite mediumwith semi-infinite crack(Lin & Thomson, 1986)

materials science dislocation mobility

nucleation criterion

failure criterion

numericsdynamical evolution ofdislocation population

Discrete Dislocation model


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materials science

dislocation mobility: thermallyactivated, viscous motion(tungsten: Schadler, 1964)

nucleation criterion:homogenous nucleation atfixed source position(refinements: Roberts, 1996)

failure criterion: dislocationshielding of sharp crack tip(Lin & Thomson, 1986)


m(T) =T

+

fdis(rsrc) > 0

v(i

)dis = v0f(i)

dis0 b

m(T)exp

Q

kT

ktip > kcrit = 2MPa

m

ktip = Kb

1

i

fr1/2i ,i


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numericsdynamical evolution ofdislocation population

constant temperature T,constant loading rate K

introduction of super-dislocations to savecomputing time


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Overview


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homogeneous nucleationoverestimates ductility atlow temperatures

dislocation nucleation inbrittle materials occurs atdiscrete sites

(Roberts, Booth, Hirsch,1994; Hsia, Gao, Xin, 2001;

Zhou, Thomson, 1991; Xu,Argon, Ortiz, 1997)

Dislocation nucleation

(Gumbsch, Riedle, Hartmaier, Fischmeister; 1998)


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dislocation nucleation atjogs produces inefficientdislocations for shielding

cross-slip mechanisms cantransform jogging intoblunting dislocations(Hartmaier, 2000; Narita,Takahara, Higashida; 2002)


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Shielding of complete cracktip only after coalescence ofhalf loops

Translation into 2D model:1. nucleate dislocation lines

at source position r

2. shielding taken into

account after motionover additional(incubation) distance

= ()

(Roberts, 1996)


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fracture toughness at lowtemperatures is nucleationlimited

results from refinednucleation model

better agreement withexperiments in lowtemperature regime


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Predeformation experiments

experimental work:

predeformation to 5% plastic strain prior tocrack initiation

facilitates dislocation nucleation

obstructs dislocation motion

(Gumbsch, Riedle, Hartmaier,

Fischmeister; 1998)


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low temperatures: deformation is nucleation limited (exceptpredeformed material)

intermediate temperatures: deformation is mobilitycontrolled (saturation in nucleation sites)

high temperatures: transition to ductility not only due to

shielding (crack-tip blunting must be taken into account,dislocation multiplication)

Deformation mechanisms


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Overview


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Force balance at crack tip

G = gt +Nd

j=1

gd(j)

G =K2 (1 )

Egt =

ktip2 (1 )

E

Ndj=1

gd(j)= C

ktip

kc

sNq

Total force on defects =force on crack tip + force on dislocations

Identification with energy release

rate(Weertman, Lin, Thomson, 1983)

Result of numerical

simulations;C, s, q only dependenton elastic constants andBurgers vector


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Force balance at crack tip

Kc =k2c+ CNq

Fracture toughness is only a function of

number of dislocations


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1. number of

dislocations is only afunction of velocity ofleading dislocation

2. Arrhenius relation

between loading rateand temperature forall points of constantfracture toughness

3. Scaling relation forpoints of constantfracture toughness

Scaling relation

T2 =

k

QlnK1

K2+

1

T1

1

P(Kc) = AK

expQkT

N 1

K

KcK=0

vdis(1)

dK


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Scaling relation is verified

for simulation results(left, with constant m)

and for experimental data

(bottom)

Scaling relation


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Modeling:

Discrete Dislocation Dynamics needs phenomenological input,but yields information on deformation mechanisms.

DDD paves way to predictive descriptions of crack-tip plasticityand fracture toughness.

Fracture:

Irreversible processes at stress concentrations determine

toughness of a material Dislocation nucleation is necessary condition for plastic

relaxation, but in general not rate limiting

Crack-tip plasticity can be described as thermally activatedprocess with same characteristics as dislocation mobility

Conclusions


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