Crack interaction with microstructure - CyberLeninka

Crack interaction with microstructure Designing microstructure for damage tolerance requires a detailed understanding of how an advancing crack interacts with the microstructure (and sometimes modifies it locally) at multiple length scales. Advances in experimental techniques, such as the availability of well-controlled straining stages for optical and electron microscopes, the focused ion beam, electron backscattered diffraction, and nanoindentation, enable probing at these length scales in real time and through interrupted tests. Simultaneously, increasing computational power coupled with new computational methods, such as finite element analysis (FEA) incorporating cohesive elements at the continuum level, discrete dislocation methodology at the mesoscopic level, and coupled atomistic/continuum methods that transitions atomic level information to the mesoscopic level, have made it possible to begin addressing these complex problems. By reviewing crack growth in a variety of multiphase alloys including steels, titanium aluminides, Mo alloys, and nanocrystalline metals, we demonstrate various aspects of crack interaction with microstructure, and how these problems are being addressed through experiments and computations. Sharvan Kumar* and William A. Curtin Division of Engineering, Brown University, Providence, RI 02912, USA *E-mail: [email protected]

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Developing materials to withstand stress, often combined with

their properties from a complex microstructure in which a

an extreme environment, is central to many current technologies

matrix phase is combined with a distribution of a second phase

such as aircraft structures, gas turbines, and lightweight engines

(particles, rods, plates) to achieve a desirable mechanical

for automotive and marine applications. These materials derive

response. Their properties can, in principle, be adjusted by tuning

SEPTEMBER 2007 | VOLUME 10 | NUMBER 9

ISSN:1369 7021 © Elsevier Ltd 2007

Crack interaction with microstructure

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the microstructure appropriately. Properties such as modulus,

chemistry modification. With the enhancements in the capabilities of

flow stress, ductility, fracture toughness, and fatigue resistance

in situ scanning and transmission electron microscopy (SEM and TEM,

can be optimized by engineering the grain size and texture; the

respectively) straining stages17-20, as well as the development of new

size, shape, and distribution of the second phase; and/or the

experimental techniques over the past two decades, including focused

chemistry, structure, and strength of the grain boundaries and

ion beam (FIB) for micromachining and characterization21, in situ Auger

interfaces between the matrix and second phase. A quantitative

spectroscopy22, electron backscattered diffraction (EBSD) coupled with

understanding of the relationship between these microstructural

selected area channeling patterns (SACP)23-25, Moire interferometry26,

variables and the performance of a material is critical to enhancing

synchrotron X-ray tomography27, and neutron diffraction28, substantial

its performance. Progress has been slow, however, particularly

progress is being made in furthering our understanding in this arena.

in the area of understanding microstructural effects on failure

Computer simulations that predict the response of a material to

resistance, which is determined by a complex interaction of

stress are an integral part of modern materials design and considerable

physical processes involving material features ranging in size from

progress has been made in developing computational methods to

1 nm to 100 µm or more.

address material behavior both at continuum length scales and at the

Classical research has involved careful examination of the surface

atomic scale. For example, continuum finite element simulations can

microstructure of polished specimens by periodically interrupting

accurately predict properties such as elastic modulus, yield stress, creep

mechanical tests and post-mortem examination of fractured surfaces.

rates, and strain hardening rates for two-phase microstructures in

Significant knowledge of crack interaction with microstructure has been

terms of microstructural variables such as the crystallographic texture,

gained over the years; toughening and embrittlement mechanisms

particle volume fraction, and the elastic and plastic properties of the

understood1-16,

matrix and second phase. Atomistic and ab initio simulations can now

have been identified and

the former based on ideas

such as crack bridging by ductile ligaments, crack deflection by

compute a range of material parameters such as defect energies and

second-phase particles, microcrack formation, and stress-induced

traction-separation relations for ideal interfaces29-31, and can also be

phase transformations (Fig. 1), and the latter based on crack tip

used to model the atomic-scale processes directly in a small volume of material for short periods of time. Despite such notable successes, enormous challenges remain. In particular, efforts to simulate failure mechanisms, such as crack nucleation, fracture, and other localized phenomena not amenable to homogenization, remain rudimentary. Furthermore, capturing the influence of grain or particle size on constitutive response and failure resistance is fundamentally outside the scope of standard continuum and too large for direct atomistic simulation, leading to its introduction through ad hoc Hall-Petch-type scaling. With respect to modeling failure resistance of a material, simulations must capture in detail both the small-scale features and processes that trigger damage, as well as the larger scale surrounding regions that generate high local stresses. In particular, crack nucleation, crack growth, and flow strength dependence on microstructural scale, are all determined to a large extent by mesoscale phenomena (length scales of ~100 nm to ~1000 nm) involving the collective behavior of large numbers of dislocations and the inherently discrete nature of plastic flow. Furthermore, methodologies to relate atomic-, meso-, and continuumscale phenomena are needed. In this review, we focus our attention on progress being made on the experimental and computational fronts in understanding crack interaction with local microstructure in polycrystalline and multiphase structural materials. Thus, we consider crack interaction with grain boundaries, with second-phase particles and particle/matrix interfaces, and instances where crack-tip stresses and environment modify

Fig. 1 Schematic illustrations of ways in which an advancing crack interacts with microstructure or modifies it.

the microstructure in the vicinity of the crack tip, and thus affect subsequent crack growth. This review is by no means comprehensive,


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and only represents some of the recent activity in the field. We first

Often however, recognizing the sequence of deformation and

present experimental observations, from submillimeter length scale

cracking events enables a detailed understanding of the underlying

resolution to the submicron regime, where available in each of the

mechanisms and provides valuable insights for modeling the

categories identified above. Little has been done at the atomic scale,

phenomenon. In order to track the sequence of events ahead of a

primarily because of instrumentation limitations. We then present

growing crack and the process of damage evolution, in situ observation

advances in computational capabilities from the continuum through

of crack growth in an SEM or under an optical microscope can be

the mesoscopic to the atomic level. In this context, it is pertinent

informative at the millimeter to micron scale. However, since such

to recognize that whereas experimental studies are usually system

observations (whether in situ or ex situ) are made on the surface

specific, computational efforts tend to focus on technique development

of a specimen, care must be taken to use ‘tailored’ specimens and

and subsequent application to a specific material system.

specimen geometry, and assumptions must be made regarding the subsurface microstructure and crack path. For example, Vehoff et al.36

Experimental observations

used coarse-grained, thin specimens of Fe-2.7% Si to examine fatigue

Crack interaction with grain boundaries

crack nucleation at grain boundaries. They measured the orientation of

Several polycrystalline materials exhibit preferred crack paths within a

individual grains using EBSD and used the information as an input into

grain that could be a crystallographic cleavage plane or an interphase

finite element calculations to determine the effect of microstructure

interface. When such a crack arrives at a grain boundary, the change in

on stress evolution, and to predict crack nucleation and propagation

orientation of this preferred path in the adjacent grain serves to arrest

paths. Alternately, bicrystals with known orientations and slip bands

the crack and enhance cleavage-cracking resistance. This orientation

can be used. Single and bicrystals of NiAl have been deformed in situ

change can be quantified through a tilt and a twist misorientation

in a scanning force microscope (SFM), and the sequential progress

(Fig. 2). As an example, Argon and Qiao32-34 have examined how

of elastic deformation, dislocation emission, and discontinuous crack

cleavage cracks break through specific grain boundaries with known

growth tracked to understand crack interaction with grain boundaries37.

tilt and twist misorientations in Fe-3 wt.% Si bicrystals. They conclude

The study confirms that NiAl is intrinsically ductile but the presence

that twist misorientation has a more profound influence on fracture

of impurities is effective in pinning dislocations and promoting

resistance than the tilt misorientation. A similar conclusion has been

brittle crack growth. Furthermore, the grain boundary is shown to be

reached by Zhai et al.35 in a study on fatigue crack propagation in

inherently brittle, the extent of which depends on the Ni content.

the Al–Li alloy 8090. They argue that the area between the traces

Over the past two decades, there has been a significant amount

on the grain boundary plane of the crack planes across the boundary

of research on the mechanical behavior of intermetallic compounds

has to be fractured in order for the crack to propagate through the

with particular emphasis on the titanium aluminides. Various

boundary, which presents resistance to crack growth. These studies

research efforts have been devoted to understanding the mechanisms

were, however, conducted ex situ and the mechanisms and evolution

controlling crack nucleation and growth in these materials23-25,38-43.

of events were inferred from examination of the deformed/fractured

The microstructure of this alloy includes two phases, the major phase

specimens subsequently.

TiAl and the minor phase Ti3Al, in a lamellar morphology within each grain called a ‘colony’. The microstructure is hierarchical in length scale; colonies are in the range 100–500 µm, and individual lamellae are 100–300 nm wide. An example of this microstructure is shown in Figs. 3a and 3b. In the early 1990s38-40, it was shown that uncracked ligaments between cracks play an important role in enhancing fracture toughness. The role of shear ligament toughening in these materials was also elucidated. Subsequently, Wang et al.41 conducted a systematic study of monotonic crack growth in precracked, coarse-grained, lamellar binary TiAl alloys using single-grain-thick, compact-tension specimens in situ in the SEM. They have demonstrated that in such a microstructure, cracks within a colony propagate along the Ti3Al/TiAl interface or within Ti3Al parallel to the interface with minimal resistance, but that upon arrival at certain colony boundaries (large twist misorientation of the lamellae), they experience significant

Fig. 2 Schematic describing the crack plane misorientation across a grain boundary in terms of the kink angle α (top), and the twist angle β (bottom).

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resistance to crossing the boundary. In agreement with an earlier study42, they found that a major component of the crack growth


(a)

(b)

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to a twist misorientation – is ineffective in arresting a crack at the boundary, because in this case the crack path in the two participating grains is coplanar. However, if two grains are stacked on top of each other and are in the way of an advancing crack, then a large kink misorientation between the top and bottom grains is effective at splitting the crack as opposed to a twist misorientation that produces a coplanar crack path (as illustrated in Figs. 5a and 5b). Thus both kink and twist misorientations combine to enhance participation of high-energy crack paths in polycrystalline materials exhibiting crystallographically preferred fracture paths.

Fig. 3 Microstructure of a binary lamellar Ti-46% Al alloy showing (a) ~500 µm size colonies and (b) 100–300 nm scale lamellae of the TiAl (γ) and Ti3Al (α2) phases. In this material, fracture occurs parallel to the lamellar interface. (Reproduced with permission from87. © 2001 Springer.)

More recently23-25, the intersection of deformation twins with grain boundaries and the consequential nucleation of microcracks at the boundaries has been studied in detail using SACP in conjunction with EBSD. Such microcracks are found to link up to form intergranular

resistance of lamellar TiAl arises from this colony boundary resistance,

cracks that then lead to the renucleation of the primary failure crack.

which forces the participation of translamellar high-energy fracture

Together, these studies enable an improved understanding of the

paths43. In addition, the in situ testing of these specially configured

mechanisms controlling crack nucleation and growth in this complex,

specimens has enabled observation of the sequence in which damage

hierarchical microstructure.

evolves in the vicinity of these colony boundaries and the mechanisms

The FIB is becoming an invaluable micromachining and

that contribute to the toughness enhancement. These include multiple

microstructural characterization tool. Uchic et al.21 have recently

cracking in the participating grains (Fig. 4a), plastic deformation of the

reviewed developments and applications of FIB tomography for

ligaments between the cracks, crack bridging by ligaments (Fig. 4b), and

three-dimensional materials characterization at the microscale and

the subsequent translamellar rupture of these ligaments (Fig. 4c). In

discussed future developments and needs. Holzapfel et al.44 have used

contrast, a kink misorientation across the grain boundary – as opposed

FIB tomography coupled with three-dimensional reconstruction to

(a)

(b)

(a)

(c)

(b)

Fig. 4 Effect of lamellar misorientation across a colony boundary on damage generated in the vicinity of the boundary: (a) multiple microcracking; (b) bridging ligament at the boundary caused by a large twist misorienation during the test as the crack crosses the boundary; and (c) ligament fracture later in the test, accompanied by substantial plastic deformation of the ligament. Arrows indicate direction of crack growth. (Part (a) reproduced with permission from87. © 2001 Springer. Parts (b) and (c) reproduced with permission from41. © 2003 Elsevier.)

Fig. 5 When the two grains are stacked on top of each other (grains 2 and 3), a twist misorientation (a) produces a coplanar crack in the two grains whereas a kink misorientation (b) results in crack splitting.


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study the spreading of microcracks in a Ni-base superalloy and the

shown that a variety of options are available (Fig. 1). These include

interaction of microcracks with carbide particles at grain boundaries.

the direct transmission of pile-up dislocations across grain boundaries,

The advantage of FIB tomography over classical X-ray tomography

absorption into the boundary, emission of matrix dislocations into

and synchrotron sources lies in the scale of resolution, which for FIB

an adjacent grain at a different location, transformation into grain

tomography is from ~10 nm to 100 µm. This is ideal for examining

boundary dislocations, and even ejection back into the original grain.

crack interaction with local microstructure. The study recognizes the

Under certain circumstances, grain boundary cracking has been

importance of subsurface orientation of grain boundaries in interpreting

reported as a means of accommodating stress build-up at a grain

measured surface crack growth rate. The study also confirms that the

boundary. More recently, Couzinié et al.47,48 have used weak-beam

crack bypasses the grain boundary carbides by forming steps, which

microscopy to study the interactions of matrix dislocations with

is more important than the reorientation of the crack plane across

specific well-characterized grain boundaries in detail. They have

the boundary. Toda

et al.27

have reported three-dimensional in situ

interact with a Σ = 3 grain boundary, it has to recombine into a perfect

conditions in a high-strength 2024-type Al alloy using high-resolution

dislocation before converting into grain boundary dislocations. These

(0.7 µm) synchrotron X-ray tomography. They observed the crack tip

details determine the ease with which relaxation can occur to relieve

geometry, crack bifurcation, and crack surface topology, as well as crack

crack-tip stresses. The role of grain boundary chemistry (S segregation

interactions with high-angle grain boundaries delineated by a special

in Ni) on dislocation transmission/pile-up at grain boundaries and

technique (Ga wetting of grain boundaries). This in situ study, coupled

subsequent cracking has also been studied, and provides useful insights

with post-deformation reconstruction of images, illustrates that the

into the underlying embrittlement mechanisms49.

simplified models that have been used to explain the underlying

When multiple phases are present, the deformation response ahead

mechanisms responsible for crack closure (plasticity- versus roughness-

of the crack tip will likely be different in the two phases. Tang et al.50

induced closure) are over-simplified, and that the process is primarily

have demonstrated this by examining the development of dislocation

attributable to the generation of Mode III displacements associated

structures in the plastic zone ahead of a crack tip in a duplex stainless

with the local crack surface topology.

steel deformed in the TEM. They confirm that dislocation configuration

The examination of dislocation activity (in the vicinity of the

and crack propagation are vastly different in the ferrite and coexisting

crack tip and otherwise) and the interaction of dislocations with grain

austenite phases. In the ferrite phase, the crack advances in a fairly

boundaries, second-phase particles, and other interphase interfaces has,

straight manner and the plastic zone is broad because of the ability

until now, largely remained in the domain of TEM studies. While in situ

of the dislocations emitted from the crack tip to cross slip. In the

straining in the TEM provides significant insight into microstructural

austenite phase, however, dislocations are confined to narrow strips

events at the crack tip on the mesoscopic scale, caution must be

and form inverse pile-ups, and the crack propagates in a zigzag manner.

exercised in extending these findings to bulk specimens. The electron-

The factors controlling the activation of slip systems in the two phases

transparent regions in specimens used in TEM studies are necessarily

have also been delineated.

thin (usually ~100 nm), and therefore lack the constraint in the third

A topic of significant recent research has been the deformation

dimension that is present at the crack tip in bulk specimens. Constraint

response of nanocrystalline metals and alloys, which has led to a

affects crack growth behavior, but the local stress state around a

number of in situ TEM straining studies involving the examination of

microstructural feature can differ significantly from the global stress

crack-tip activities, the role of grain boundaries in acting as dislocation

state, and the individual dislocation/microstructure interactions that

sources and sinks, and grain boundary sliding51-55. Furthermore, novel

drive fracture also respond to the local stress state. In this context,

probing and measurement techniques including nanoindentation within

in situ TEM studies are relevant for fracture. For example, Zhong

the TEM, the use of microlithography, and microelectromechanical

et al.45

(MEMs)-based techniques to impart deformation and measure

have used an in situ straining stage in the TEM to examine

the interaction of a growing crack with high- and low-angle grain

displacements have been developed19,20,56-59. In particular, attention

boundaries in a high-strength pipeline steel with an ultrafine acicular

is drawn to the examination of the deformation response and

ferrite structure. Furthermore, they examined the effect of a film of

simultaneous crack growth studies of free-standing, nanoscale Al and

martensite + retained austenite (M/A) at a low-angle grain boundary

Au thin film specimens in situ in the TEM57 (Fig. 6).

(LAGB) on resistance to crack growth. Whereas a LAGB by itself is

In spite of all these advances, however, it is worth noting that

ineffective in impeding crack growth, the presence of the M/A film

atomic-level understanding of crack-tip activities, including dislocation

provides an effective barrier to dislocation motion and resistance to

emission, their absorption at grain boundaries, and the consequential

crack growth. Lee

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determined that when a pair of Shockley partial dislocations in Cu

observation of fatigue crack closure under steady-state plane-strain

et al.46

have used in situ deformation in the TEM

changes in the atomic structure at the affected grain boundaries, all

to understand dislocation interactions with grain boundaries in 310

remain to be studied experimentally. Although adequate resolution is

stainless steel, α-titanium, and the intermetallic compound Ni3Al, and

available today with modern electron microscopes to examine post-



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their in situ fracture study in the TEM note that SiC whiskers in a Mg alloy matrix (AZ91) are effective in blocking dislocations emitted from an advancing crack tip. Eventually the particle/matrix interface cracks, producing microcracks that link with each other and are connected to the main crack, which leads to catastrophic failure. They also note that twins present ahead of the crack tip in the Mg alloy matrix are instrumental in discouraging crack propagation. When the size of the second-phase particles are on the nanoscale, as is usually the case in high-strength precipitation-hardened alloys, dislocations emitted from the crack tip can interact with these particles by either bowing around them or shearing them. Vivas et al.68 have used in situ straining in the TEM to examine dislocation interaction with precipitates in a high-strength Al alloy. From this study, they were able to determine the strength of the precipitate phase (i.e. its shear Fig. 6 Quantitative, in situ TEM test results for 100-nm-thick freestanding Al films. Three TEM test results are shown along with an SEM test result to demonstrate experimental consistency. Insets: microstructures corresponding to the different stress-strain states for experiment 1 (triangle data markers). The rightmost insets show crack growth through the grains and between the grains. The top right inset shows the film after fracture. (Reproduced with permission from59. © 2004 National Academy of Sciences, USA.)

resistance). Pettinari et al.69 have provided several case studies aimed at extracting quantitative information from in situ TEM straining results for dislocation interactions with various obstacles in several metallic systems. Appel and coworkers70,71 have used in situ straining in the TEM to understand crack-tip interaction with lamellar interfaces and crack propagation across such interfaces in titanium aluminides. They found

deformed specimens, these instruments do not lend themselves readily

cracks (i) cleaving along the {111} planes in individual TiAl (γ phase)

to dynamic studies.

lamellae; (ii) deflection of the crack at the interface between two γ lamellar variants; and (iii) formation of mechanical twins ahead of

Crack interaction with second phase and other interfaces

the crack tip as a result of crack tip stress activation of the numerous

We now briefly examine work relating to cracks interacting with

on fracture, suggesting that crack-tip shielding rather than crack-tip

second-phase particles and other interfaces including interphase

blunting is the underlying mechanism. Based on these findings, they

interfaces.

have proposed that solute additions and second-phase particles that

The interaction of cracks with coarse second-phase particles under monotonic and cyclic loading has been extensively studied in the

available sources. They delineated the beneficial role of twinning

promote twinning might be beneficial to enhancing the toughness of these alloys.

metal- and ceramic-matrix composites arena4-8,11,60-62. In contrast, the presence of micron-sized particles in ductile metals is known to lead

Crack-tip stress-induced microstructure modification

to void nucleation, growth, and coalescence, either by particle fracture

As a last example of crack interaction with microstructure, the

or by particle/matrix interface debonding (Fig. 1), leading to ductile

stress ahead of the crack tip can, under certain circumstances,

fracture by dimpled rupture1,63-65. The stress field ahead of a growing

produce substantial microstructural modification that can aid or

crack is usually adequate to produce such interface debonding/particle

deter subsequent crack growth. Thus at high temperatures, during

fracture. However, fine details at the mesoscopic level, which lead to

monotonic and cyclic loading, local recrystallization and cavitation

such decohesion/cracking including dislocation emission/absorption at

ahead of the crack tip (Fig. 7) have recently been reported in a two-

the crack tip and at the particle/matrix interface, have to the best of

phase Mo–Si–B alloy, which is detrimental to crack growth72. In other

our knowledge not been fully resolved experimentally, particularly in

instances, phase transformations occur ahead of the crack tip. This has

multiphase metallic alloys. Jiao et al.66 have examined crack/particle

been observed in situ in the TEM for martensite formation in shape-

interactions in an Al2O3/SiC nanocomposite, where the typical particle

memory alloys73,74 and stress-assisted hydride formation at crack tips,

size is of the order of 200 nm, using in situ deformation in the TEM.

which causes embrittlement in certain Ti alloys but not others75,76.

They have determined that the intragranular particles are not effective

Furthermore, in situ TEM straining and observation of crack-tip activity

in deflecting the crack and microcracking around particles is not

has led to an improved understanding of environmental effects on

frequent. However, intergranular particles are effective in deflecting

dislocation mobility77-79, and these observations have also been

grain boundary cracks back into the grains, and this was thought to

relevant to isolating embrittlement mechanisms in various classes of

be an important toughening mechanism. In contrast, Zheng et al.67 in

alloys.


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(a)

(a)

(b)

(c)

(b)

(c)

Fig. 7 (a) Microstructure ahead of a growing crack tip in a Mo–Si–B alloy, obtained by interrupting a 1400°C three-point bend test at a loading rate of 10–5 mm/s. Isolated patches of recrystallized regions are observed; a higher magnification image from within one such patch confirms creep cavities at triple junctions. (b) An SEM image showing the crack path and fine recrystallized grains (3–5 µm) on either sides of the crack; the white arrow illustrates grain boundary relief from sliding. (c) Even finer grains (1–2 µm) are observed ahead of the main crack and evidence is seen for linking of creep cavities to generate several micron-long microcracks. (Reproduced with permission from72. © 2007 Elsevier.)

Fig. 8 Finite element analysis incorporating cohesive elements capture (a) plastic deformation of bridging ligaments across grain boundaries and (b) multiple cracking at grain boundaries. (c) Schematic of the geometry of a two-colony lamellar structure showing low-toughness planes distributed in the angled colony. (d) The stress field and crack configuration just after microcrack nucleation indicating heterogeneous toughnesses. (Parts (a) and (b) reproduced with permission from87. © 2001 Springer. Parts (c) and (d) reproduced with permission from88. © 2003 Elsevier.)

Computational advances

of bridging ligaments and multiple cracking at grain boundaries

The problem of crack interaction with grain boundaries can and

(Figs. 8a and 8b). Bi-colony models with stochastic lamellar toughness

should be approached computationally at the continuum, mesoscopic,

representing thickness variations in real microstructures88 show how

and atomistic levels. FEA coupled with cohesive elements that obey

low-toughness lamellae away from the main crack could cause the

prescribed traction-displacement laws have been used to address crack

preferential microcracking (Figs. 8c and 8d) at scales (5–10 µm) seen

propagation problems at the continuum level80-86. Crystal plasticity

experimentally. More recently, Catoor and Kumar89 have extended

can be incorporated to accommodate anisotropic crystallographic

this approach to three dimensions to examine the twist crack problem

characteristics. These problems have frequently been addressed in two

in oriented bicrystals of Zn (where atomic-scale fracture anisotropy

dimensions, as a full three-dimensional analysis is computationally

is present), and have also incorporated crystal plasticity to address

intensive.

the anisotropic response of Zn. Wei et al.90-92 have used a full

Arata et al.87 have modeled crack growth response across colony

three-dimensional FEA with cohesive surfaces and crystal plasticity to

boundaries in lamellar TiAl using FEA and cohesive surfaces, and

address grain boundary sliding, cavitation, and intergranular separation

examined crack interaction with colony boundaries. The results of

in nanocrystalline Ni and Mg.

the computations are in qualitative agreement with experimental observations of microstructural features such as plastic deformation

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(d)


The problem of void nucleation, growth, and coalescence in ductile fracture has been studied extensively. The spacing and distribution


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of void-nucleating inclusions is the key microstructural feature for setting crack-growth resistance. Dimensionality is important in these

(a)

problems, and Tvergaard and Needleman93,94 have demonstrated that a three-dimensional distribution of spherical inclusions exhibits increased crack-growth resistance compared with a two-dimensional distribution of cylindrical inclusions. While there is extensive work on void coalescence by plastic deformation95-98, void nucleation has only recently been studied by Shabrov and Needleman99. Void nucleation has been postulated to occur by decohesion of the interface between the inclusion and the surrounding matrix. A computational analysis using a cohesive surface framework for the inclusion interface has shown that inclusion clustering in three dimensions and inclusion

(b)

size both influence void nucleation. As a case study, experiments on a Ti-modified 4330 steel have revealed the main mechanism of void nucleation to be particle cracking. Interrupted tensile tests on samples designed to vary the stress triaxiality have identified locations of cracked TiN inclusions (Fig. 9a)100. Calculations show that TiN cracking occurs over a narrow stress range, with larger inclusions cracking at lower stresses101. A ‘nucleation stress’ has been identified as the weighted sum of the hydrostatic tension and the effective stress. Calculations of void nucleation by particle cracking (Fig. 9b) show that the effective nucleation stresses are nearly independent of triaxiality

(c)

(Fig. 9c). Many failure mechanisms in advanced metallic materials are controlled by mesoscale interactions between defects at scales ranging from 100 nm to 1000 nm. For example, crack nucleation at the second phase/matrix interface or in second phases results from elevated stresses caused by the formation of dislocation structures. Similarly, fatigue cracks propagate as a result of a complex interaction between dislocation structures near the crack tip and the atomic-scale process of material separation at the crack tip itself. These processes are especially dominant in nanostructured materials where meso- and atomic-scale phenomena, such as dislocation emission and absorption from grain boundaries, often control the deformation. Many of these processes are fundamentally outside the scope of continuum models – stresses in continuum elastic-plastic materials simply cannot attain the high levels needed to drive failure. Recent advances in the discrete dislocation (DD) method102-105 provide new opportunities to address

Fig. 9 (a) Contours of the stress driving void nucleation in a notched tensile bar and experimentally determined cracked particles (filled squares) and uncracked particles (open squares). (b) Three-dimensional stress around a debonded void. (c) Void nucleation stress versus triaxiality. (Part (a) reproduced with permission from101. © 2004 Springer. Parts (b) and (c) courtesy of M. N. Shabrov and A. Needleman.)

these issues. The DD method models plastic flow by explicitly tracking

being driven by the interactions among the dislocations, defects,

the motion of individual dislocations. Three-dimensional models

and applied loading in any particular problem. Significantly, the DD

are the most physically realistic104,105, but two-dimensional models

method can show trends that are not predicted in continuum models,

permit the study of complex boundary value problems involving, for

such as fatigue crack growth, the scaling of the fatigue threshold with

example,

cracks102,103.

Two-dimensional models use physically based

elastic modulus, fracture toughness versus film thickness in thin film

constitutive relations to account for dislocation nucleation, interaction,

metals106, and deformation in micro- and nanoscale structures107.

and annihilation, and include crack nucleation and propagation through

A representative example is illustrated in Fig. 10a, which shows the

cohesive zones that approximate the atomic-scale mechanisms of

tensile stress field and dislocation structure formed around a fatigue

material separation. Material size-scale effects, fatigue crack growth,

crack growing from an initially cracked inclusion particle. Fig. 10b

the large fluctuations in stress associated with dislocation structures,

shows the predicted crack growth rates, exhibiting threshold and

and crack nucleation can all emerge naturally from such computations,

power-law regimes, for several values of elastic mismatch. Despite such


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(a)

(a)

(b)

(b)

Fig. 10 (a) Tensile stress field and dislocation distribution as calculated using the DD method for growth of a fatigue crack from an initially cracked inclusion. (b) Fatigue crack growth rate around initial cracked inclusion for several inclusion moduli, as predicted by the DD model. (Courtesy of S. Groh, W. A. Curtin, V. Deshpande, A. Needleman, and E. Van der Giessen, unpublished research.)

Fig. 11 (a) Dislocation pile-up at a twin grain boundary in Al, as predicted using the CADD model. (b) Equilibrium atomistic configuration and reference lattice mesh for a five-dislocation pile-up on a tilt boundary in Al after shear loading to 331 MPa followed by tensile loading to 10.3 GPa, showing a large number of micro-voids near the boundary. (Part (b) reproduced with permission from120. © 2007 Institute of Physics Publishing Ltd.)

successes, improved methods are needed to include three-dimensional

increasing applications and successes, and such studies can show

effects108,

important crack/grain-boundary and crack/interface interactions111-115.

pursue full three-dimensional analysis more

efficiently109,

and/or couple to continuum plasticity models to describe deformation

However, even with high-performance computing, these models

away from local high-stress regions, such as near a crack tip or

are restricted to small volumes of material loaded at very high

boundary. The latter requires appropriate treatment of the discrete/

strain rates for very short periods of time. To overcome the size and

continuum boundary, including ‘passing’ of dislocations to a continuum

rate bottlenecks, a number of multiscale models are emerging to

field, and of the long-range dislocation fields. DD simulations will also

couple atomistic and continuum levels so that explicit atomic-scale

play an important role in calibrating constitutive relations used in the

resolution is used only where needed, such as at a crack tip or grain

continuum computations that include strain-gradient terms to account

boundary, with the remainder of the solid modeled as a continuum.

for highly localized

deformation110.

Both continuum and meso-scale simulations rely on

42

The quasicontinuum method116, now a decade old, has been applied to the study of crack/grain-boundary interactions, demonstrating

phenomenological descriptions of atomic-scale phenomena, such as

quite complex phenomena such as boundary migration and dislocation

dislocation nucleation and interactions within the bulk, on interfaces,

nucleation from the grain boundary as a crack approaches117. A

and at crack tips or voids. A critical issue, therefore, is to determine

coupled atomistic/discrete-dislocation (CADD) technique118,119 pushes

accurate constitutive parameters for the work of separation and peak

the multiscale concept further by permitting the continuum region to

strength of interfaces in cohesive zone models, kinetic parameters

deform plastically using discrete dislocations, with dislocations able to

governing interface sliding and mobility, and atomistic-scale

pass freely between the atomistic and continuum regions, as dictated

dislocation/interface interactions. Understanding these phenomena at

by the physics of the problem (Fig. 11a). This method provides a

the atomic scale could also influence material design. Conventional

seamless connection between mesoscale and atomistic simulations,

atomistic and molecular dynamic simulations are finding ever

while accounting for important phenomena occurring at both scales



REVIEW

simultaneously. Relevant to our discussion here is the important

Newer techniques like FIB offer considerable promise for the three-

potential role of dislocation structures in reducing the strength of an

dimensional examination of this problem, but damage from the ion

interface as a result of elevated stress at, or defects induced in, the

beam must be considered. Three-dimensional X-ray tomography does

interface by dislocation impingement. To this end, the CADD method

not have the resolution needed for mesoscopic-level examination

has been used to study multiple dislocations from a single source

and neither does neutron microscopy124,125, which is currently in its

impinging on an initially planar grain boundary under tension and shear

infancy. However, both of these techniques, which could enable the

loadings, as shown schematically in Fig. 11b120,121. Computations show

viewing of thicker specimens than currently possible with a TEM, hold

that both interface structure and dislocation type influence dislocation

promise and, with further development, should have better resolution

absorption, transmission, reflection, and damage nucleation. The

in principle.

general phenomena have been related to dislocation/grain-boundary interaction

rules122,123,

and include all the types of observations

On the computational front, new methodologies coupled with powerful computing capabilities are evolving at an unprecedented

noted above in in situ studies of Ni3Al46-48. In addition, the results

pace, particularly at the atomic and mesoscopic levels. We anticipate

are suggested to provide effective constitutive parameters for use in

that they will provide a much-needed insight that is unavailable from,

mesoscale DD simulations as outlined.

and thus complements, experimental work. Despite these advances, reliable material input parameters remain necessary for computations

Outlook

to be realistic, and such material inputs are not readily available for

A snapshot is provided here of the experimental and computational

enough complex materials systems, particularly for the description of

approaches being used to further our understanding of crack interaction

interface properties. Transitioning from predominantly two- to three-

with microstructure and its consequences for damage tolerance in

dimensional analyses comes at a substantial computational penalty

multiphase structural metals and alloys.

but is eventually necessary to produce quantitative comparisons with

From our review of recent experimental work, it is clear that significant research and new findings relating to crack interaction with microstructure in materials with complex multiphase microstructures

experiments and, subsequently, reliable design of microstructures and predictions of performance. Direct contact between computational models and experiments

has been either deduced from failed specimens and/or based on in situ

remains a challenge, as simulations are often performed on idealized

SEM and TEM studies. While failed specimens provide valuable insights

systems while experiments can include more complexity and

into understanding the underlying mechanisms, often the sequence

unknown factors (e.g. segregation of embrittling chemical species

of events, as well as mesoscopic- and atomic-level details are absent.

to boundaries). Nonetheless, as experimental methods steadily

In situ SEM observations rely on information gathered from the

move toward three-dimensional in situ nanoscale resolution and

specimen surface and thus assumptions must be made regarding

as computational models increase in size, complexity, and realism,

subsurface microstructure and events. To offset this disadvantage,

the confluence of these tools will lead to important scientific and

specimens are specially designed to have the surface microstructure

technological advances in the creation of advanced structural materials

traverse the thickness of the specimen. In situ TEM studies have

in the coming years.

been invaluable in providing information at the mesoscopic level, particularly the interaction of dislocations with interfaces and the second phase, but there will always remain questions regarding the applicability of the findings to bulk specimens. TEM studies, however, have been of relevance in the validation of computations that, until recently, have been predominantly two dimensional.

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