Full text of DOI:10.1557/mrs2001.44

www.mrs.org/publications/bulletin
reaches the value needed to make new
crack surface; that is, when the energy-
release rate is 2ꢂ, where ꢂ is the surface
energy (extending the crack makes two
surfaces). This holds for brittle solids. For
an ideal brittle solid, only the breaking of
bonds on the crack plane is involved. Real
materials are different: the linear elastic
response that is assumed in Equation 1
ceases to hold in regions sufficiently close
to the crack tip. Not only does the re-
sponse become nonlinear, but irreversible
deformation processes occur (which is what
makes putting Humpty together again so
difficult). This has two far-reaching conse-
quences: (1) the local stress state deviates
strongly from that in Equation 1, and
(2) crack growth requires more energy
than just that needed for creating new sur-
face. It was Irwin’s1 great insight that the
Griffith scenario still holds if one inter-
prets ꢂ as the surface energy plus the plas-
tic dissipation in the advancing crack-tip
region. This interpretation is valid pro-
vided that the region over which dissipa-
tive and nonlinear effects predominate is
small compared with the region over
which Equation 1 holds (small-scale yield-
ing). To illustrate that the additional term
can be large, the surface energy of many
materials with very different fracture re-
sistances is of the order of 1 J/m2; the
energy-release rate of structural metals (e.g.,
aluminum and steel alloys) is typically
one to two orders of magnitude greater.
The success of engineering fracture me-
chanics has been to relate measurements
of the effective ꢂ from tests on small labo-
ratory specimens to the initiation of crack
growth in structures and components. The
approach is a purely phenomenological
one: the fracture energy is not related to the
underlying physical processes. The focus
of this article is on the interaction between
the dominant dissipative mechanism in
crystalline metals, dislocation plasticity,
and material separation. First, however,
we point out some other challenging un-
resolved issues in fracture.
Micromechanics
of Fracture:
Connecting Physics
to Engineering
A. Needleman and E.Van der Giessen
mode of crack loading. Three modes are
possible: Mode I involves opening of the
crack; Mode II involves sliding of the crack
faces perpendicular to the crack front;
Mode III is also a sliding mode, but with
the direction of sliding parallel to the
crack front. The opening mode, Mode I,
is of the greatest significance in practical
applications and has received the most at-
tention in the engineering and materials
science fields. However, the sliding modes
are what cause earthquakes, and so they
are of great interest to geophysicists. In ad-
dition, in heterogeneous solids, such as
composites and multilayers, the mismatch
in mechanical properties leads to mixed-
mode behavior (i.e., opening and sliding)
in the vicinity of the crack tip, even if the
remote loading is purely Mode I. The
function fij depends on the mode of load-
ing, but is otherwise independent of the
forces that act to cause crack growth. All of
the details about the geometry and the
forces only affect the value of K. The physi-
cal significance of Equation 1 is that the
energy available to drive crack growth, the
energy-release rate , is proportional to K2.
Whether or not the driving force (for
Humpty, the height of the fall) is sufficient
to cause fracture depends upon the pre-
sumed presence of unavoidable flaws or
small cracks (in fracture, as in many other
subjects, growth is much better under-
stood than nucleation). Given a solid body
with an initial crack subject to some exter-
nal forces, will the crack grow, or will it re-
main stationary? To decide this, a fracture
criterion is needed. Such a criterion has to
connect the physics of the actual material-
separation process to the driving force.
Fracture mechanics dates from Griffith’s
work, where he postulated that a crack
will grow when the energy-release rate
Introduction
Humpty Dumpty sat on a wall
Humpty Dumpty had a great fall
All the king’s horses and all the king’s men
Couldn’t put Humpty together again.*
This nursery rhyme is one of the oldest,
most familiar, and shortest treatises on
fracture mechanics in the literature. Never-
theless, it makes several important points:
(1) fracture occurs in response to some
driving force (for Humpty, a “great fall”),
(2) fracture involves material separation,
and (3) fracture is irreversible.
Of these, the driving force for fracture
is probably the best-understood aspect.
Fracture generally occurs because of the
propagation of crack-like defects under the
influence of a surrounding stress field. If
the surrounding material behaves like a
linear elastic solid (i.e., it follows Hooke’s
law), the stress state near the crack tip is
described by
K
ꢀij
ꢀ
fij ,
(1)
2ꢁr
ꢀ
The single-crack picture implicit in the
considerations just described implies that
fracture splits an object into two pieces,
but this is not always the case. In a suf-
ficiently ductile material, the crack may
arrest before it severs the solid into sepa-
rate pieces. For brittle solids, once a crack
begins to run, it runs very fast. The crack
repeatedly bifurcates,2,3 with the solid
eventually shattering into many pieces.
This transition from crack growth to frag-
mentation is a topic of current research
interest and is far from completely under-
stood. There is not yet a complete answer,
for example, to the question of what con-
stitutes the limiting crack speed in a brittle
where r is the distance from the crack tip,
fij is a function of the angular position rela-
tive to the crack tip (which depends on
the material’s elastic constants), and K is the
stress-intensity factor that depends on the
* The real Humpty Dumpty is said to have been
a powerful cannon used during the English
Civil War (1642–1649). It was mounted on top
of St. Mary’s at the Wall Church in Colchester,
defending the city against siege in the summer
of 1648. The church tower was hit by the enemy
and the top of the tower was blown off, sending
Humpty tumbling to the ground.
MRS BULLETIN/MARCH 2001
211

Micromechanics of Fracture: Connecting Physics to Engineering
solid. What is clear, both theoretically and
experimentally, is that crack speeds can
exceed elastic wave speeds, and such in-
tersonic cracks are accompanied by shock
waves.4 Crack growth faster than an elas-
tic wave speed occurs more readily under
Mode II (sliding) conditions than under
Mode I (opening) conditions, and differ-
ences in elastic properties (as across an
interface) promote such fast crack growth.
The prediction of the crack path, the total
consumed energy, the fracture-surface
characteristics, and so on are current topics
of research. These aspects are crucial in
order to address issues of design of
fracture-resistant materials. In heteroge-
neous materials, cracks meander through
the microstructure, and microscale altera-
tions in the crack path can have a major
effect on macroscale fracture resistance.
Dissipative mechanisms in crystalline
metals include mesoscale dislocation plas-
ticity in the crack-tip region and larger-
scale (hundreds of microns to millimeters)
plasticity associated with the nucleation
and coalescence of voids at second-phase
particles. Chemical and electrical inter-
actions in the crack-tip region can affect
the energy required for fracture. For ex-
ample, the diffusion of impurities to the
crack-tip region may directly affect either
dissipative processes or cohesion there.
Figure 1. The various relevant scales that may determine the response of a crack in a
Another chemical effect of importance is
the creation of an oxide layer on newly
exposed free surface.
macroscopic component (a). Zooming in to the near-tip region, we observe (b) the plastic
zone governed by macroscopic continuum plastic flow, inside of which we notice that
(c) the material is polycrystalline, with the plastic deformation being different in different
grains. Even closer to the tip of the crack, we observe that (d) plastic deformation occurs
via localized slip on particular slip planes, caused by the motion of individual dislocations.
The final illustration (e) shows the atomic arrangement on either side of the crack.
The Scales in Fracture
The micromechanics of fracture attempts
to link the macroscopic fracture resistance
to the underlying physical mechanisms of
irreversible deformation and material sepa-
ration. The specific physical mechanisms
are material-dependent and, in general,
involve a broad range of length and time
scales. To make matters specific, we focus
on room-temperature crack growth in crys-
talline metals under circumstances where
dislocation glide is the dominant mecha-
nism of plastic dissipation. However, many
of the issues are generic and are pertinent
in a broader range of circumstances.
tion of surfaces may only contribute a
small fraction of the total energy-release
rate, it can still be controlling. This is be-
cause dissipative mechanisms can operate
only if fracture is delayed sufficiently to
allow them to come into play. Indeed, as
pointed out in Reference 5, the surface
energy can play a valve-like role because
small changes in atomic-scale fracture re-
sistance can lead to large changes in plas-
tic dissipation.
crack is modeled as being mathematically
sharp), but the singularity depends on the
strain-hardening characteristics of the ma-
terial; (2) the stress distribution near the
crack tip depends on the material proper-
ties, but is independent of the loading con-
ditions and the geometry; and (3) there is
a single parameter characterizing the am-
plitude of the singularity that embodies
the effects of geometry and loading.
ꢀ
At smaller scales, this description loses
Figure 1 is a highly simplified illustration
of the important length scales involved
for cleavage crack growth in a relatively
ductile polycrystalline metal. The relevant
length scales range from that of the
macroscale object to the atomic scale, in-
cluding the various microstructural length
scales in between that are associated with,
for example, particles, grains, and defect
structures.
The challenge lies in the fact that all scales
are connected, and all may contribute to
the total fracture energy. It is worth noting
that although the atomistics of the separa-
Even though the example in Figure 1 is
highly idealized, it is useful to consider
the various scales and their connections in
some more detail:
accuracy because the anisotropic nature
of the individual grains becomes notice-
able, which gives rise to stress and strain
fluctuations with wavelengths equal to
the grain size (Figure 1c). These can be
described by means of crystal plasticity
models, which are continuum-based but
which recognize the discrete nature of the
slip systems along which plastic flow
occurs. If the plastic zone is in a single
crystal, the near-tip fields are completely
different from the HRR fields. When hard-
ening at the crystal level can be ignored,
the stress field is piecewise uniform in
ꢀ
When the size of the plastic zone spans
sufficiently many grains (Figure 1b), one
may describe the plastic deformation by
means of a macroscopic, and often iso-
tropic, continuum model. This gives rise
to stress and strain fields (referred to as
the HRR fields)6,7 that differ from Equa-
tion 1, but which share several key charac-
teristics with it: (1) the stresses are singular
(the stress is infinite at r ꢀ 0, when the
212
MRS BULLETIN/MARCH 2001

Micromechanics of Fracture: Connecting Physics to Engineering
various sectors around the tip,8,9 with
Short-range dislocation interactions are
and to numerically resolve relevant length
scales, the calculations are carried out
using a cohesive strength that is lower by
a factor of 3–4 than expected from atom-
istics. Nevertheless, the results are instruc-
tive regarding the role of dislocation
structures in transmitting high stresses to
the crack tip. A typical result of such simu-
lations is shown in Figure 2. The opening
stress reaches sufficiently high values
ahead of the crack tip to open the cohesive
surface and thus for the crack to propa-
gate. The peak stress level, according to
conventional continuum plasticity, is about
a factor of 4 lower, and crack growth would
not be predicted by such an analysis.
stresses of the order of the critical resolved
shear stress for slip.
incorporated by a set of constitutive rules
that in principle can be derived from
atomistic modeling.
ꢀ
Zooming in to the tip of the crack, the
individual dislocations that mediate plas-
tic flow become noticeable (Figure 1d). At
this scale, energy dissipation is associated
with the motion of large numbers of dis-
locations moving through the lattice. Each
dislocation induces a localized stress con-
centration that gives rise to large local
stress fluctuations. This landscape continu-
ously changes as dislocations move or are
generated from the crack tip itself or from
nearby sources.
Conventional continuum plasticity the-
ory (Figures 1b or 1c) predicts that stress
levels at a blunted crack tip are limited to
being of the order of a few times the flow
strength (roughly 10ꢃ3 times Young’s modu-
lus). However, the strength of atomic
bonds, as predicted (for instance) from
first-principles computations, is roughly
an order of magnitude larger. Thus, a key
issue is how stresses that are high enough
to cause atomic separation are transmitted
to the crack tip. Our recent dislocation dy-
namics study12 indicates that this gap may
be bridged by the stress concentrations
associated with discrete dislocations (see
also References 13 and 14).
ꢀ
The actual separation process takes
These calculations12 indicate that dis-
locations play a dual role in the fracture
process. On the one hand, dislocation ac-
tivity gives rise to values of the crack driv-
ing force at initiation that are significantly
higher than for an elastic solid with the
same cohesive properties. On the other
hand, it is the local stress concentration as-
sociated with discrete dislocations in the
place at the atomic scale (Figure 1e) and
occurs inside the dislocation plasticity
zone (for a recent perspective on atomistic
analyses of fracture phenomena see the
May 2000 issue of MRS Bulletin10). The
stress fields from the next higher-up scale
cause local bonds between atoms to be
stretched and broken when the stresses at
this scale reach values of the order of the
bond strength. This process may be as-
sisted by other atoms that have diffused to
the crack tip from the environment.
The calculations in Reference 12 have
significant limitations: they are two-
dimensional; the short-range dislocation-
interaction rules are plausible choices
rather than directly derived from atomistics;
Many details are left out in this discus-
sion, but it emphasizes that fracture—that
is, the creation of new surface—is a highly
localized effect at the atomic scale that is
driven by the applied load via the stress
fields on smaller and smaller length scales.
It is the precise communication down
these scales that determines whether or
not crack growth occurs and how much
energy is dissipated along the way (e.g.,
see Reference 11). The success of predic-
tions of macroscopic fracture properties
on the basis of atomic properties relies
entirely on the accuracy with which the
intermediate scales can be bridged. In
transferring information between models
at different size scales, there are subtle
issues of capturing the behavior at the
smaller size scale in some appropriate
average sense, particularly when defect
structures are involved.
Dislocation Dynamics
Near CrackTips
We now focus on the dislocation scale
illustrated in Figure 1d, because the stress
concentrations at this scale play a key role
in determining whether crack growth will
take place in a ductile or brittle manner. At
this scale, the dislocations can be modeled
as line singularities in an elastic solid; that
is, the elastic effect of interatomic inter-
actions is averaged out, but the dislocations
as carriers of plastic deformations are rep-
resented discretely. The long-range inter-
actions between dislocations are accounted
for through their continuum elasticity fields.
Figure 2. Distribution of the opening stress ꢀ22 (i.e., in the vertical direction) near the tip of a
stationary crack in a crystal with three slip systems (slip planes oriented at 0 and ꢄ60ꢅ with
respect to the horizontal crack plane; all lengths in micrometers). Beyond a distance of
0.5–1 ꢆm, the average stress fields agree rather well with continuum slip solutions where
the stress is uniform in sectors around the tip and is of the order of the flow strength.
Inside the very near-tip region of 0.5–1 ꢆm, however, the opening stress attains much
higher values.
MRS BULLETIN/MARCH 2001
213

Micromechanics of Fracture: Connecting Physics to Engineering
vicinity of the crack tip that leads to stress
levels of the magnitude of the cohesive
strength, causing the crack to propagate.
The source of dislocations in the crack-
tip region is material-dependent. In some
materials, such as silicon, the sole source
of dislocations is the crack tip itself. Then,
the key question for brittle or ductile re-
sponse is whether dislocation nucleation
precedes decohesion or vice versa (e.g.,
see Reference 15). Other materials have an
abundance of sources in the crack-tip re-
gion (this is the situation modeled in Ref-
erence 12). In either case, two conditions
need to be met for the material to behave
in a ductile manner: (1) the dislocations
must be nucleated, and (2) they need to be
mobile (see also Reference 14).
If very few dislocations are nucleated,
the stresses near the crack tip are essen-
tially those given by Equation 1, with local
deviations caused by the presence of iso-
lated dislocations. In such source-limited
circumstances, crack growth takes place,
with the energy-release rate differing only
slightly from the surface energy. On the
other hand, if ample dislocations nucleate
and can glide, unobstructed, away from
the crack-tip region, dislocation motion re-
laxes the stresses near the crack tip, lead-
ing to continued blunting without crack
propagation. In fact, fracture in crystalline
metals rarely occurs by cleavage when
there is large-scale plasticity. Continued
crack-tip blunting leads to large plastic
strains over a large enough region to ac-
tivate mechanisms of void nucleation,
growth, and coalescence. This microvoid
fracture mechanism, where the governing
processes take place over a size scale of
microns to hundreds of microns, is what
gives rise to the very high energy-release
rates of ductile structural metals. Interme-
diate between these two limits are the cir-
cumstances in Figure 2, where both plastic
flow and cleavage separation take place.
Discrete dislocation considerations are
of particular importance for fatigue crack
growth and for crack growth at metal–
ceramic interfaces. Atomistic calculations
of MgO/Ag adhesion16 give cohesive
strengths of 2–10 GPa and values of
the work of separation of the order of
0.1–1.0 J/m2, with the lower values ac-
counting for effects of small concentrations
of impurities and segregants. Macroscopi-
cally measured values of the work of
separation are typically 2–5 J/m2 for seg-
regated interfaces with sharp cracks and
greater than 200 J/m2 for clean interfaces
with blunt cracks.17 The macroscopically
inferred values of the cohesive strength
are several times the flow strength of
the metal.17 Thus, there is a significant
discrepancy between “top-down” and
“bottom-up” estimates of the fracture prop-
erties. This gap may be bridged through
consideration of the dislocation structures
that form near an interface crack.
computations on smaller size scales require
smaller time steps. More often than not,
time-step, rather than spatial-integration,
considerations are the computationally
limiting factor.
Multiscale modeling of fracture is in an
early stage of development, but there is
great potential for improving not only the
fracture resistance under mechanical load-
ing, but also the fracture resistance when
environmental effects come into play, for
example, through chemical interactions
and/or the presence of electrical and mag-
netic fields. Traditionally, the component
or structure of interest has been of a macro-
scopic size (centimeters to meters or larger),
but the reliability of micromechanical and
microelectronic structures and components
is likely to be of increasing importance.
Crack growth under cyclic loading con-
ditions (fatigue-crack growth) is undoubt-
edly the most important mode of fracture
in practical applications. The essence of
fatigue is that crack growth occurs even
when the driving force for crack growth
is much smaller than what is needed for
the same crack to grow under monotonic
loading conditions. This is what makes
fatigue fracture so dangerous in practice
and so difficult to understand fundamen-
tally. Although much is known about the
phenomenology of fatigue, an understand-
ing is lacking of why fracture occurs at
lower values of the crack driving force
under cyclic loading conditions.
For crystalline metals, the irreversibility
of dislocation motion plays an essential
role in fatigue-crack growth. Under cyclic
loading conditions, the internal stress
state generated by the discrete disloca-
tions provides the bias for the evolution
of the dislocation structure that increases
the near-crack-tip opening stress and thus
provides the driving force toward fracture.
Acknowledgment
We are grateful for support from the
Materials Research Science and Engineering
Center program on Micro- and Nano-
Mechanics of Electronic and Structural Mate-
rials at Brown University (NSF grant
No. DMR-0079964).
References
1. G.R. Irwin, in Encyclopedia of Physics, Vol. VI,
edited by S. Flugge (Springer-Verlag, New
York, 1958) p. 551.
2. F.F. Abraham, D. Schneider, B. Land, D. Lifka,
J. Skovira, J. Gerner, and M. Rosenkrantz,
J. Mech. Phys. Solids 45 (1997) p. 1641.
3. X.-P. Xu and A. Needleman, J. Mech. Phys.
Solids 42 (1994) p. 1397.
4. A.J. Rosakis, O. Samudrala, and D. Coker,
Science 284 (1999)p. 1337.
5. J.R. Rice and J.-S. Wang, Mater. Sci. Eng., A
107 (1989) p. 23.
6. J.R. Rice and G. Rosengren, J. Mech. Phys.
Solids 16 (1968) p. 1.
7. J.W. Hutchinson, J. Mech. Phys. Solids 16
(1968) p. 13.
8. J.R. Rice, Mech. Mater. 6 (1987) p. 317.
9. W. Drugan, J. Mech. Phys. Solids (2001) in
press.
10. R.L.B Selinger and D. Farkas, guest editors,
“Atomistic Theory and Simulation of Fracture,”
MRS Bull. 25 (5) (2000) pp. 11–50.
11. J.W. Hutchinson and A.G. Evans, Acta
Mater. 48 (2000) p. 125.
12. H.H.M. Cleveringa, E. Van der Giessen, and
A. Needleman, J. Mech. Phys. Solids 48 (2000)
p. 1133.
Challenges and Outlook
Fracture is an archetypical multiscale
problem: length scales from the electronic
structure for chemical effects to the macro-
scopic can all come into play in determin-
ing the fracture resistance of a structure or
component. Engineering fracture mechan-
ics, which involves identifying fracture
parameters and transferring results from
small-scale tests to larger-scale structures,
is a well-developed subject, even though
a fundamental understanding of some is-
sues is still lacking. The main promise for
multiscale fracture modeling lies in appli-
cations to small-scale structures and to de-
signing materials with improved fracture
resistance.
Multiscale modeling of fracture proc-
esses can proceed in two ways: (1) by means
of data compression and transmission
across scales through passing effective
properties from one scale to a higher scale,
or (2) by the direct coupling of regions
modeled at one scale to another (e.g., an
inner region modeled atomistically, to an
intermediate region modeled in terms of
discrete dislocations, to an outer contin-
uum region). In this latter case, a key issue
that remains to be resolved is how to pass
defects from one size scale to the next. It
is also worth noting that as a general rule,
13. P.B. Hirsch and S.G. Roberts, Scripta Metall.
23 (1989) p. 925.
14. P. Gumbsch, J. Riedle, A. Hartmaier, and
H.F. Fischmeister, Science 282 (1998) p. 1293.
15. J.R. Rice and G.E. Beltz, J. Mech. Phys. Solids
42 (1994) p. 333.
16. T. Hong, J.R. Smith, and D.J. Srolovitz, Acta
Metall. Mater. 43 (1995) p. 2721.
17. A.G. Evans, J.W. Hutchinson, and Y. Wei,
Acta Mater. 47 (1999) p. 4093.
ꢀ
Materials Research Society online catalog for Proceedings is available at
www.mrs.org/publications/
www.mrs.org/publications/bulletin
214
MRS BULLETIN/MARCH 2001