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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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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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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
Fracture and process zones
(Flaig, 1994)
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Fracture and process zones
(Abraham, Walkup, Gao, Duchaineau, Diaz De La Rubia, Seager; 2002)
Large-Scale molecular dynamics simulation for copper
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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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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)
Discrete Dislocation model
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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Discrete Dislocation model
numericsdynamical evolution ofdislocation population
constant temperature T,constant loading rate K
introduction of super-dislocations to savecomputing time
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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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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
dislocation nucleation atjogs produces inefficientdislocations for shielding
cross-slip mechanisms cantransform jogging intoblunting dislocations(Hartmaier, 2000; Narita,Takahara, Higashida; 2002)
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Dislocation nucleation
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Dislocation nucleation
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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Dislocation nucleation
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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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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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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