NASA Technical Reports Server (NTRS) 20030067948: Micromechanics Modeling of Fracture in Nanocrystalline Metals

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AIAA 2002-1315 L7
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MICROMECHANICS MODELING OF
FFLACTURE IN NANOCRYSTALLINE
METALS
E.IH. Glaessgen, R.S.Piascik,
1.S. Raju and C.E. Harris
NASA Langley Research Center
Hampton, VA 23681
4C3rd AIANASM WASC WAHSIASC
!Structures, Structural Dynamics,
and Materials Conference
April 22-25, 2002/Denver, CO
For permission to copy or republish, contact the American Institute of Aeronautics and Astronautics
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AIAA-2002-1315
MICROMECHANICS MODELING OF FRACTURE IN NANOCRYSTALLINE METALS
E.H. Glaessgen', R.S. Piascikt. 1.S. Raju' and C.E. Harris4
NASA Langley Research Center, Hampton, VA 23681, U.S.A.
.
Abstract Classical Approaches
Fracture of standard engineering metals containing
Nanocrystalline metals have very high theoretical
micro-sized structure is thought of in terms of brittle
strength, but suffer from a lack of ductility and
and ductile fracture. Brittle metals usually have either
toughness. Therefore, it is critical to understand the
body-centered cubic (BCC) or hexagonal close packed
mechanisms of deformation and fracture of these
(HCP) atomic structure whereas ductile metals have
materials before their full potential can be achieved.
face-centered cubic (FCC) atomic structure. In BCC
Because classical fracture mechanics is based on the
and HCP metals, there are few planes for dislocation
comparison of computed fracture parameters, such as movement allowing for little plastic deformation. In
stress intlmsity factors, to their empirically
FCC metals, there are many planes for dislocation
determined critical values, it does not adequately
movement and hence they can undergo significant
describe the fundamental physics of fracture required
plastic deformation. Figure 2 shows these three types
to predict the behavior of nanocrystalline metals.
of atomistic structures.
Thus, micromechanics-based techniques must be
considered to quanti@ the physical processes of
deformation and fracture within nanocrystalline
In general, high strength materials often exhibit
metals. This paper discusses hndamental physics-
brittle fracture and low toughness whereas low
based modeling strategies that may be useful for the
strength materials exhibit ductile fracture and high
prediction Iof deformation, crack formation and crack
toughness. Current fracture models employ classical
growth within nanocrystalline metals.
fracture mechanics. Classical fracture mechanics is
based on the premise that brittle fracture (in plane
Introduction strain) will occur when KI > KK-, i.e. when the
computed value of the stress intensity factor, KI, is
Fracture processes in materials such as greater than or equal to the fracture toughness, KK.
nanocrystalline metals (see Figure 1 layered metals In classical fracture mechanics, fracture toughness is
I),
and powder metallurgy-formed materials cannot be considered to be a property of the material, and the
modeled by traditional fracture mechanics-based plane strain fracture toughness, KIC, is the lowest
concepts. While these materials often exhibit high value of material toughness.
strength, they also tend to have low ductility and low
fracture toughness. Low ductility and toughness is a
result of the nano-scale structure; here, Hall-Petch
behavior is no longer valid when the structural size
approaches the size of dislocations, thus disabling the
mechanisms that produce ductility and toughness.'
Therefore, it is critical to understand the mechanisms
of deformation and fmcture of these materials before
their full potential can be achieved This requires a
.3
new understanding at the micromechanics level. This
paper discusses fundamental physics-based modeling
strategies that may be useful for the prediction of
deformation, crack formation and crack growth within
nanocrystalline metals.
Figure 1. A 3D View of Microstructure of Extruded
A I-'Ti-Cu bulk Nanoaystalline Metal.
(TEM Images)'
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:a) BCC (b) FCC c) HCP
Figure 2. Three Atomic Structures Found in Typical Metals.
The relationship between the characteristic behavior tip opening angle (CTOA) are often used and tend to
of standard engineering materials and the types of be associated with empirically determined length
fracture criteria that are applicable for these materials scales resulting in two-parameter criteria for crack
is shown in Figure 3.4 In this figure, LEFM denotes growth. Additionally, energy-based methods such as
linear elastic fracture mechanics. The CTOD and the J-integral are unable to separate the relative
CTOA are critical crack tip opening displacement and contributions of the energy driving plastic
angle, respectively, and K-R and J-R curves are crack deformation and the energy spent on the creation of
growth resistance curves based on K and the J- new crack surfaces to the perceived crack growth
integral, respectively. The critical J-integral, Jlc., is resistance in ductile materials.
yet another fracture criterion. In Figure 3, fracture
behavior is captured in terms of empirical parameters
Computational Strategies
(K,(., J-R curves, CTOA, etc.) that are devoid of
physics-based understanding. For example, plane
strain fracture toughness (KK,) is an empirical Because classical fracture mechanics is based on the
quantity that is only applicable to linear elastic comparison of computed fracture parameters (such as
fracture in thick materials. the stress intensity factor) to their empirically
determined critical values, the concept does not
adequately describe the fundamental physics of
Such empirical concepts used in classical fracture
fracture required to predict the behavior of
mechanics are not physics-based and do not capture
nanocrystalline metals. In a large part, classically
the fundamental mechanisms associated with crack
formulated fracture mechanics (and classical
growth. For example, in many medium and low
continuum mechanics as a whole) is unsuitable for
strength materials under plane stress, methods such
the prediction of deformation and hcture of
as crack tip opening displacement (CTOD) or crack
nanocrystalline metals because it does not account for
LEFM holds
Brittle KI Stress Intensity Factor
KIC Critical value at fracture
(e g 7075 AI, Glass, -
JIC
KIC
LEFM holds
JIC
Medium Strength
(e.g. 2024 AI) -
J-R or K-R curves
LEFM invalid
CTOD or CTOA
/
\
LEFM holds*- KIC
Low Stength JIC
’
(e.g. Carbon
Where plane strain conditions exist, Steel)
e.g., for very thick sections J-R or K-R curves
LEFM invalid- CTOD or CTOA
Figure 3. Relationship Between Material Behavior and Suitable Fracture Mechanics Analysis.
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Table I. Characteristic Scaling Lengths and Associated Simulation Methods
the length-scale dependencies that tend to dominate intractable as the number of atoms considered
the behavilor of these materials. That is, the details increases, therefore, approximations to these
of microstructural and nanostructural features are not interactions have been developed in the form of
considered. Table 1, adapted from reference 5, empirical and semi-empirical potentials describing
presents bounds on the domain of application for the potential energy of the interactions among the
typical simulation methods over a broad range of atoms.6 Among the best known of the relationships
length scales. for non-bonding potentials is the Lennard-Jones
potential, Qi,
This paper discusses three of the physics-based
analysis approaches shown in Table 1 to predict
deformation, crack formation and crack growth in
metallic materials. These approaches are molecular
dynamics analysis6, cohesive zone models' and L J
variants of plasticity theory'. To varying degrees,
these approaches have the ability to model aspects of where E is the depth of the energy well, o is the van
der Waals radius and r,/ is the separation distance
the material architecture and their effects on fracture
behavior while classically formulated fixture between the i'" and jIh atoms in a pair. The r,;' term
represents the attractive contribution to the van der
mechanics cannot model these aspects. These three
computational methods will be discussed next Waals forces between neutral molecules. The other
component of the van der Waals interactions mimics
followed by a brief discussion of their effects on the
the Coloumb interaction of the nuclei and the Pauli
emerging field of computational materials.
repulsion between overlapping electron clouds and is
modeled by the short ranged ri'* term.6
Molecular Dynamics Analysis
The Lennard-Jones potential was originally developed
Gaining accurate understanding of the mechanisms of and is most accurate for interactions among atoms of
fracture at a crack tip requires modeling at the noble gasses.8 However, it has been used for a broad
atomistic level. Quantum mechanical solutions for variety of gasses, liquids and solids. Great care is
the interaction among atoms rapidly become needed in the selection of these potentials and hrther
A
(a) Before Deformation (b) After 10% Deformation
Figure 4. Nanocrystalline Copper Sample Containing 100,000A toms.'
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majority of materials. One exception involves certain
classes of nanocrystalline materials wherein the
characteristic length scales are very small. Typical
grain diameters in nanocrystalline metals range from
5 to 50 nanometers.' To resolve the issue of length
scale in most other metals and larger domain
problems in nanocrystalline materials, a multiscale
modeling strategy has been proposed. The multiscale
modeling strategy uses molecular dynamics models
to provide constitutive input into cohesive zone
relationships that can be embedded along regions of
separation within finite element models of the grain
structures.
Figure 5. Fracture Along the Primary Cleavage
{ 1 IO} Planes of NiAI."'
Cohesive Zone Models
work is needed in developing accurate potential
energy functions for specific types of atomic Cohesive zone models assume cohesive interactions
interactions. Once developed, the interatomic among the grains of a material and permit the
potentials can be used in molecular dynamics (MD) appearance of fracture surfaces in the continuum.6
simulations of deformation and fracture. One of the popular cohesive zone models is attributed
to Tvergaard and Hutchinson," where the normal and
shear components of the traction and displacement are
Molecular dynamics analysis mimics elementary combined into single measures, f and A, respectively,
atom istic path-dependent processes by solving the so that the responses are coupled.12 The coupled
equations of motion for all atoms in the domain.6 cohesive zone model (CCZM) given in reference 12
MD simulations have yielded results such as the defines a traction potential, @
prediction of the deformation of nanocrystalline
copper as shown in Figure 4. In this figure, stacking @(dn,dl)= s:,$r(A')dA' (2)
faults left behind by partial dislocations that have run
A
through the grains during the deformation processes where A is a nondimensional measure of the relative
are clearly seen.9 As another example, fracture along normal (&) and tangential (SI) displacements as
the primary cleavage { 1 10) planes of NiAl is shown defined by
in Figure 5."
112
[
($
(
Atomistic simulations containing several hundred (3)
million atoms are possible, however this number of A = + ;)2]
atoms corresponds to a cube of less than IO00 atoms
on each side. One thousand atoms of titanium span
where S,, and 8,a re the critical values for the normal
approximately 400 nm. Thus, it is readily apparent
and tangential modes, respectively, and the initiation
that an atomistic approach is not practical for
of a crack is assumed to occur when A reaches a value
determining deformation and fracture in the vast
4
Displacement, h I
I1
Figure 6. Traction-DisplacementR elationship in CCZM.
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t t t t t
.c
4
(a)G rain Structure and Far-Field Loading. (b) Local Decohesion at Location A.
Figure 7. Decohesion Among Gmin Boundaries of a Crystalline Metal.”
of unity. A graphical representation of the CCZM is heterogeneous character of crystalline micro~tructure.~
shown in Figure 6. For example, polycrystalline plasticity models have
been used to examine strain localization at triple
Grain boundaries in metals exist because of points and grain b0~ndaries.I~ Compared with
incohereno: in the atomic lattice. Decohesion may standard continuum plasticity models, the predictions
occur between the atoms within an individual grain of the polycrystalline plasticity-based analyses show
(intragranular) or along the boundaries between the a more heterogeneous and physically realistic strain
grains (intergranular). Currently, most analyses that field. Recently, crystalline plasticity has been
implement cohesive zone models consider only the extended to a more general theory based on classical
intergranular failure problem. Figure 7 shows results crystalline kinematics; classical macroforces;
that are typical of a finite element calculation that microforces for each slip system consistent with a
uses cohesive zone models to predict separation microforce balance; a mechanical version of the
between the grains.I2 Figure 7(a) is a representation Second Law of Thermodynamics that includes, via
of the grain structure, and Figure 7(b) is a detail of the microforces, work performed during slip; and a
the decohesion among the grains at location A. rate-independent constitutive theory that includes
While the results in reference 12 use heuristically dependencies on plastic strain gradient^.'^
derived relationships to define the coupled cohesive
zone modl:l, the CCZM can also be developed from Based on the general framework of Cosserat couple
results of molecular dynamics analyses allowing the stress theory, strain gradient plasticity allows for the
physical insight of the MD analysis to be embedded inclusion of a single material length scale, I, and
in the more computationally efficient finite element reduces to the conventional Jz plasticity theory when
models. geometric length scales are large compared with 1.”.
l6
The technique has been used to show strain, curvature
Variants of Plasticity Theory and strength dependencies of materials containing
rigid particles on the prescribed radius of the
Because cd the complexities involved in developing particles. For materials with a strain hardening index
meaningfill representations of atomistic or grain between 1 and 10, strength was shown to increase by
configurations and the computational power required values of between approximately 6 and 8 when the
to solve the resulting systems of equations, ratio of material length scale to particle size increased
approaches allowing for homogenization of the from 0 to 1.
domains of interest may facilitate study of the
deformation and fracture problems. One such top- Another variant on P ,lasticity theory is discrete
down approach is to use models based on variants of dislocation plasticity. In discrete dislocation
plasticity theory. Among these approaches are plasticity, dislocations are treated as line singularities
polycrystalline plasticity, strain gradient plasticity in an elastic solid. The formulation consists of two
and discrete dislocation plasticity. parts: a many body interaction problem involving the
discrete dislocations and a fairly conventional solid
Polycrystalline plasticity replaces the microstructure mechanics boundary value problem. The long-range
of the material by an appropriate constitutive interactions between dislocations are accounted for
description and can, to some extent, account for the using the continuum elasticity fields, while the short-
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range interactions are incorporated through a set of systems exceeding a few hundred million atoms. In
constitutive rules. The technique has been used to contrast, the cohesive zone models are suited for
predict the local deformations inchdin5 localization larger domains including modeling decohesion along
of plastic flow within small (48p.m ) blocks of individual grain boundaries, but do so with a loss of
material under tension and bending and may be a detail near the crack tip. Plasticity-based models
useful intermediary between atomistic and continuum tend to be even more homogenized, but lend
form u~aiot ns. ” themselves to modeling larger domains.
Computational Materials The real power of these techniques is not seen when
the methods are implemented individually, but
Over the centuries, the materials processing rather, when the methods are combined. For
community has taken an Edisonian approach to example, molecular dynamics solutions can be used
materials processing. This Edisonian paradigm is as the basis for discrete dislocation plasticity models
archaic and needs to be replaced. Simultaneously, of deformation and as the foundation for cohesive
the mechanics of materials community has developed zone relationships used in polycrystalline models of
analytical approaches based on empirically crack initiation.
determined parameters such as modulus of elasticity,
strength and toughness. These empirical parameters Through these efforts, the mechanics of materials and
offer limited insight into the internal mechanisms materials processing communities have embarked on
from which they arise. Approaches such as the ones a bold paradigm shift that will lead a transition from
outlined in this paper have the potential to the traditional separate, empirical and heuristic
revolutionize mechanics of materials and couple it to qualitative methods that have existed for centuries to
materials processing allowing material integrated physics-based quantitative methods that
microstructures to be designed and processing will allow numerical experiments to drive materials
parameters to be determined before physical processing. Many early steps have been taken to
processing. develop useful techniques for modeling materials
over a broad range of length scales. Some of them
The work being undertaken to address the closely have been outlined in this paper.
interwoven goals of developing techniques to predict
the effects of processing parameters on microstructure References
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