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This is the first of three overviews on failure of metals. Here, brittle and ductile failure under monotonic loadings are addressed within the context of the local approach to fracture. In this approach, focus is on linking microstructure, physical mechanisms and overall fracture properties. The part on brittle fracture focuses on cleavage and also covers intergranular fracture of ferritic steels. The analysis of cleavage concerns both BCC metals and HCP metals with emphasis laid on the former. After a recollection of the Beremin model, particular attention is given to multiple barrier extensions and the crossing of grain boundaries. The part on ductile fracture encompasses the two modes of failure by void coalescence or plastic instability. Although a universal theory of ductile fracture is still lacking, this part contains a comprehensive coverage of the topic balancing phenomenology and mechanisms on one hand and microstructure-based modeling and simulation on the other hand, with application examples provided.
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By invitation only: overview article
Failure of metals I: Brittle and ductile fracture
A. Pineau
a
,
**
, A.A. Benzerga
b
,
c
,
d
,
*
, T. Pardoen
e
a
Centre des Mat
eriaux, Mines ParisTech, UMR CNRS 7633, B.P. 87, 91003 Evry, France
b
Department of Aerospace Engineering, Texas A&M University, College Station, TX 77843-3141, USA
c
Department of Materials Science and Engineering, Texas A&M University, College Station, TX 77843, USA
d
Center for Intelligent Multifunctional Materials and Structures (CiMMS), College Station, TX 77843, USA
e
Institute of Mechanics, Materials and Civil Engineering, Universit
e catholique de Louvain, 1348 Louvain-la-Neuve, Belgium
article info
Article history:
Received 12 September 2015
Received in revised form
18 December 2015
Accepted 22 December 2015
Available online 30 January 2016
Keywords:
Cleavage
Ductility
Fracture toughness
Voids
Fracture locus
abstract
This is the rst of three overviews on failure of metals. Here, brittle and ductile failure under monotonic
loadings are addressed within the context of the local approach to fracture. In this approach, focus is on
linking microstructure, physical mechanisms and overall fracture properties. The part on brittle fracture
focuses on cleavage and also covers intergranular fracture of ferritic steels. The analysis of cleavage
concerns both BCC metals and HCP metals with emphasis laid on the former. After a recollection of the
Beremin model, particular attention is given to multiple barrier extensions and the crossing of grain
boundaries. The part on ductile fracture encompasses the two modes of failure by void coalescence or
plastic instability. Although a universal theory of ductile fracture is still lacking, this part contains a
comprehensive coverage of the topic balancing phenomenology and mechanisms on one hand and
microstructure-based modeling and simulation on the other hand, with application examples provided.
©2016 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.
1. Introduction
Among the various damage mechanisms introduced in the
preface, we begin by studying those associated with brittle and
ductile fracture in metallic alloys. Brittle fracture includes both
cleavage and intergranular fracture. Ductile fracture encompasses
failure by cavitation or by plastic instability. The main objective of
this article is to overview the methodologies that are based on the
study of the mechanisms operating at a local, i.e. microscopic scale
and, through a multiscale approach, on the transfer of this local
information to the macroscale, that over which the performance of
structural components as well as materials characteristics or
propertiesare usually dened. Several reviews and books have
been published on this subject (see e.g. Refs. [1e3]) but very few of
them provide a comprehensive synthesis of the state of the art. In
particular, a special effort is made here to incorporate the most
recent developments in the theoretical and numerical modeling of
both brittle and ductile fracture.
The methodologies referred to above fall under what is now
called the local approach to fracture, which has been largely
developed for brittle fracture with the original Beremin model
introduced in the late 70's and early 80's [4,5]. Brittle fracture has
been reviewed recently by the authors [6]. In the present paper
emphasis is laid on the latest developments, in particular those
dealing with the multiple barrier models and the crossing of grain
boundaries by cleavage cracks. This aspect of brittle fracture has a
special importance when the materials are tested in the rising part
of the ductile-to-brittle transition (DBT) curve. The topic of ductile
fracture has also been independently reviewed by the authors in
two separate monographs [6,7]. Another review by Besson [8]
focused on modeling. While we defer to these reviews for many
details, the main mechanisms and concepts are overviewed for
completeness. In doing so, we lay emphasis on the latest de-
velopments adopting a narrative that seamlessly combines ductile
fractures in structural components and metalworking. In addition,
signicant advances have recently been made in developing more
robust models, which ultimately will reduce the many un-
certainties associated with currently used models.
The inuence of crack tip constraint and stress triaxiality on
ductile and brittle fracture is of major importance for the assess-
ment of structural integrity of many industrial components. This
assessment is usually made by using linear and nonlinear fracture
*Corresponding author. Department of Aerospace Engineering, Texas A&M
University, College Station, TX 77843-3141, USA.
** Corresponding author.
E-mail addresses: andre.pineau@mines-paristech.fr (A. Pineau), benzerga@
tamu.edu (A.A. Benzerga).
Contents lists available at ScienceDirect
Acta Materialia
journal homepage: www.elsevier.com/locate/actamat
http://dx.doi.org/10.1016/j.actamat.2015.12.034
1359-6454/©2016 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.
Acta Materialia 107 (2016) 424e483
mechanics concepts. Compared with these concepts, micro-
mechanical models developed in the frame of the local approach to
fracture have the advantage that the corresponding material pa-
rameters for fracture can be transferred in a more general way
between various specimen geometries. In the early version of the
Gurson model for ductile fracture [9e11] , crack initiation and
propagation are a natural outcome of the local softening of the
material due to void coalescence, which starts when a critical void
volume fraction, f
c
, is reached over a characteristic distance, l
c
.In
principle, the parameters f
c
and l
c
can be determined from rather
simple tests such as tensile tests using smooth and notched round
bars in combination with numerical analyses of these tests, or from
micromechanical models. Similarly, the Weibull stress model
originally proposed by Beremin [5] provides a framework to
quantify the complex interactions among specimen size and ge-
ometry deformation level and material ow properties when
dealing with brittle (cleavage or intergranular) fracture. The Bere-
min model in its simplest form also uses two parameters only.
The identication and determination of the damage parameters
in the Gurson or in the Beremin model require a hybrid method-
ology of combined testing and numerical simulation. The full
description of this methodology is out of the scope of the present
paper. More details can be found elsewhere [6]. Here it is enough to
say that, contrary to the classical fracture mechanics methodology,
the local approach to fracture is not subject to any size requirement
for the specimens as long as the same fracture phenomena occur.
This article is organized according to failure modes: cleavage,
intergranular, and ductile fracture. In the part devoted to cleavage
the early theories for this mode of failure are briey presented rst.
Then more recent theoretical developments are presented and
applied to ferritic steels and other metals with either a BCC or HCP
structure. Intergranular fracture in ferritic steels is also briey
reviewed. Then, ductile fracture is presented in some detail.
2. Cleavage fracture
2.1. Preliminary remarks
Cleavage fracture preferentially occurs over dense atomic planes
(See Table 1). Three fracture surfaces observed on ferritic steels are
shown in Fig. 1a, b, c. These micrographs reveal that the orientation
of cleavage facets, change when they cross sub-boundaries, twin
boundaries or grain boundaries. Steps or ridges appear on the
fracture surface to compensate for the local misorientation, in
particular at grain boundaries. The crossing of grain boundaries by
cleavage cracks is analyzed in more detail in the following. For BCC
metals and in the case of mechanical twins, these steps look like
indentation marks which are named tongues(Fig. 1c). In order to
maintain the equilibrium of the crack front, the nearest steps gather
to form a single step of higher height leading to the formation of
riversas observed in Fig. 1a and b. These rivers align with the
direction of the local propagation of the cleavage cracks. On a
macroscopic scale the surfaces of the cleavage facets tend to be
normal to the maximum principal stress (mode I fracture).
Intergranular fracture corresponds to another brittle mode of
failure observed in polycrystalline metals. This mode of failure is
often observed when the segregation of impurities such as P, As, S,
etc at grain boundaries takes place (See e.g. Ref. [12]). The
transition between cleavage and intergranular fracture takes place
when the ratio R
CI
is lower than one [3]. This ratio is dened as
R
CI
¼1:20 g
b
2g
S
(1)
where g
b
is the free energy (per unit area) of the boundary and g
S
the free energy of a surface exposed by cleavage. Cottrell [13e15]
has shown that, in pure metals, g
b
depends mainly on the macro-
scopic shear modulus,
m
, whilst g
S
depends on the macroscopic
bulk modulus, K. This means that the ratio R
CI
can be written as
Table 1
Cleavage planes in various materials.
Structure Cleavage plane Some materials
BCC {100} Ferritic steels, Mo; Nb, W
FCC {111} Very rarely observed
HCP {0002} Be, Mg, Zn
Diamond {111} Diamond, Si, Ge
NaCl {100} NaCl, LiF, MgO, AgCl
ZnS {110} ZnS, BeO
CaF
2
{111} CaF
2
,UO
2
, ThO
2
Fig. 1. Cleavage fracture surfaces (a) low alloy steel [439] eRivers; (b) Polycrystalline
zinc rivers originating from a grain boundary; (c) SEM micrograph showing the
presence of tongues(indicated by arrows) formed at the intersection of the main
(001) fracture plane with mechanical twins.
A. Pineau et al. / Acta Materialia 107 (2016) 424e483 425
R
CI
¼1:20 am
K(2)
where
a
is a numerical constant close to one. Table 2 gives the
values of R
CI
for a number of metals. In a large number of pure
metals, including pure Fe, intergranular fracture should be the
preferential mode of failure. However, in many cases, the segre-
gation of benecial impurities, such as carbon, along the grain
boundaries in iron tends to suppress intergranular brittleness.
Cleavage cracks can be blunted by the emission of dislocations.
Rice and Thomson [16] have investigated the conditions under
which this blunting mechanism operates. These authors compared
the energy needed for the propagation of a cleavage crack over a
unit area, that is 2g
S
, to the energy U
L
needed to nucleate a dislo-
cation loop of Burgers vector band of radius rin a slip plane
intersecting the cleavage plane. They derived a criterion for
intrinsicbrittleness corresponding to pure cleavage without crack
blunting. Intrinsic brittleness occurs when the ratio between the
shear modulus,
m
, and the bulk modulus, K, is lower than a critical
value given by
ðm=KÞ
CD
¼10g
S
=bK:(3)
The propensity for blunted cleavage (called ductile fracture in
Ref. [16]) increases with the ratio
m
/K. A number of values for this
ratio are reported in Table 2 which indicates that, in almost all
metallic materials, blunted cleavage should occur.
The critical radius r
*
of the dislocation loop is only a few b, which
is of the same order of magnitude as the dislocation core. This casts
doubt on the validity of the preceding calculation. This explains
why subsequently, the above theoretical transition from blunted
cleavage to pure cleavage was reanalyzed by Rice [17] who used the
Peierls concept to analyze dislocation nucleation from a crack tip.
The shear stress on the slip plane is now a periodic function of
period b. The work done to displace the two half bodies on both
sides of the slip plane by Ref. b/2 is the unstable stacking fault energy
denoted g
us
. It was found that adding shear modes (II and III) has a
strong inuence on the value of the strain energy release rate for
the nucleation of a dislocation loop. With those calculations and
considering that blunting requires only the nucleation of partial
dislocations, it is nally found that in mode I and 10% shear modes,
the critical condition for blunting is g
S
=g
us
larger than 4.4 for FCC
and 2.4 for BCC metals.
The values of the ratio g
S
=g
us
reported in Table 3 were obtained
from Frenkel-Peierls-Nabarro potential or from embedded-atom-
models (EAM). These values are very approximate. This table
shows that in most FCC metals, except iridium, blunting is expected
to always take place. For BCC metals, lithium is expected not to
cleave. Conversely, pure cleavage should be observed before
dislocation nucleation for Cr, Mo and W and also Fe, Nb, V and Ta
which are closer to the border line.
It should be pointed out that these calculations apply for a
temperature of 0 K and that at higher temperatures thermal acti-
vation will favor crack blunting. Moreover these calculations apply
to pure metals and not to engineering materials in which cleavage
fracture is initiated from second-phase particles, as shown below.
The normal stress or cleavage stress,
s
c
, theoretically needed to
fracture a crystal by cleavage can be determined provided that the
bonding energy, U, between the atoms located across the cleavage
plane is known. It can be shown that
s
c
can be expressed as [3].
s
c
¼ðEg
S
=bÞ
1=2
:(4)
With the typical values valid for iron, E¼200 GPa, b¼0.3 nm,
g
S
z0:1mbz1J=m
2
, Eq. (4) leads to
s
c
zE/10. Mechanisms of stress
amplication must therefore be invoked since the measured
cleavage stresses are much lower than this theoretical value.
2.2. Mechanisms of stress intensication
The rst mechanism implies the formation of slip bands and,
under given circumstances, of mechanical twins as sources of stress
concentration [18,19]. It is assumed that cleavage initiates when the
local normal stress due to dislocation pile-up reaches a critical
value
s
Ic
over a critical distance. This mechanism which is based on
a criterion of cleavage initiation is not fully satisfactory because it
predicts that the calculated cleavage stress is temperature depen-
dent which is not the general rule. The second mechanism due to
Cottrell [20] assumes that cleavage is growth controlled. In this
model the condition for propagating a microcrack initiated at the
intersection of two slip planes is calculated. This model predicts
that the cleavage stress can be written as
s
f
¼k
0
y
d
1=2
with k
0
y
¼2mg
S
pð1nÞk
y
(5)
where dis the grain size and k
y
is the coefcient of the HallePetch
equation predicting the variation of yield strength
s
y
with grain
Table 2
Transition parameters for fracture.
m
/Kis the ratio of the shear modulus to bulk modulus; R
CI
quanties the risk of intergranular fracture versus cleavage; (
m
/K)
CD
is the ratio
required for the transition between cleavage and ductile fracture.
Metal Au Ag Cu Pt Ni Rh Ir Nb Ta V Fe Mo W Cr
m
/K0.11 0.19 0.22 0.24 0.34 0.52 0.52 0.25 0.31 0.32 0.33 0.48 0.52 0.82
R
CI
1.09 1.02 0.99 0.97 0.87 0.71 0.70 0.97 0.91 0.89 0.88 0.75 0.71 0.42
(
m
/K)
CD
0.36 0.43 0.57 0.38 0.49 0.39 0.32 0.59 0.55 0.65 0.56 0.35 0.45 0.68
Table 3
Materials properties to evaluate the ratio g
u
=g
us
[17].
Material g
S
ðT¼0Þ
m
slip
bg
S
=g
uS
g
S
=g
uS
(J/m
2
) (GPa) (nm) (Frenkel) (EAM)
FCC metals
Ag 1.34 25.6 0.166 8.8 12.5
Al 1.20 25.1 0.165 8.1 11.5
Au 1.56 23.7 0.166 11.0 15.7
Cu 1.79 40.8 0.147 8.3 11.8
Ir 2.95 198 0.156 2.7 3.8
Ni 2.27 74.6 0.144 5.9 8.4
Pb 0.61 7.27 0.201 11.6 16.6
Pt 2.59 57.5 0.160 7.8 11.2
BCC metals
Cr 2.32 131 0.250 1.1 1.6
Fe 2.37 69.3 0.248 2.2 3.2
K 0.13 1.15 0.453 4.0 5.7
Li 0.53 3.90 0.302 7.3 10.4
Mo 2.28 131 0.273 1.0 1.5
Na 0.24 2.43 0.366 4.4 6.2
Nb 2.57 46.9 0.286 3.1 4.4
Ta 2.90 62.8 0.286 2.6 3.7
V 2.28 50.5 0.262 2.8 4.0
W 3.07 160 0.274 1.1 1.6
Diamond Cubic
C 5.79 509 0.145 2.2 3.1
Ga 1.2 49.2 0.231 2.9 4.2
Si 1.56 60.5 0.195 3.7 5.2
A. Pineau et al. / Acta Materialia 107 (2016) 424e483426
size:
s
y
¼s
i
þk
y
d
1=2
(6)
where
s
i
is the lattice friction stress which is a function of tem-
perature. The cleavage stress predicted from Eq. (5) is independent
of temperature.
Smith [21] modied the Cottrell theory to account for the fact
that in mild steels cleavage fracture initiates from very brittle
platelets of cementite particles located along the grain boundaries.
Initiation and propagation of cleavage cracks from particles have
been reanalyzed using the theory of the deformation of heteroge-
neous solids containing inclusions [22,23]. The inclusions were
assumed to have an oblate or prolate spheroidal shape (Fig. 2). One
example of such inclusions in steel is shown in Fig. 3. The (local)
maximum principal stress
s
I
applied to the particle is given by
Ref. [23]:
s
I
¼S
I
þkS
eq
s
0
(7)
where S
I
is the remote maximum principal stress, S
eq
is the remote
von Mises effective stress,
s
0
is the initial yield strength of the
matrix, and kis a factor that depends on particle shape and loading
orientation. In this equation, stress intensication corresponds to
the second term on the right hand side. The criterion for brittle
fracture of the particle thus writes:
s
I
¼
s
Ic
with
s
Ic
the critical stress
considered as a material parameter. Note that this criterion does
not distinguish between particle cracking and particle-matrix
decohesion so that
s
Ic
can also be viewed as the smallest of the two.
At failure of the particle, the energy stored in the matrix
delineated by the representative elementary volume (RVE) shown
in Fig. 2 is given by
E
V
¼S
eq
ε
eq
¼s
2
0
E
p
s
Ic
=s
0
þl
Tþ2=3þl1s
Ic
=s
0
þl
Tþ2=3þl(8)
where ε
eq
is the equivalent strain, E
p
is the tangent modulus of the
material, TS
m
/S
eq
is the stress triaxiality ratio with S
m
¼S
kk
/3
the mean normal stress,
l
¼w
2
/2
a
2/3 þ2/3
a
with w¼c/a, the
aspect ratio of the particle,
a
¼3/2 w
2
(2L1) and L¼ln((2 w)/w)
[12]. The surface energy of the crack in the particle can be written as
E
S
¼3=4g
S
ðf=hÞ, where f¼2/3c
2
h/R
3
is the volume fraction of
inclusions.
At failure, the ratio E
V
/E
S
is usually much greater than one, as
illustrated in Fig. 4. This indicates that most of the energy is spent in
plastic deformation of the matrix. This ratio is plotted in Fig. 4 as a
function of Tand for various values of
s
0
and w, with
s
Ic
¼2000 MPa, g
S
¼2J=m
2
,h¼10
6
m which are typical values
for a ferritic steel. It is thus found that most of the energy is spent in
the plastic deformation of the matrix. This is largely veried at low
T, low yield strength and for spherical particles where it is observed
that E
V
z10
3
E
S
. A similar conclusion would have been reached if
one had assumed that fracture took place from the surface sepa-
rating the particle from the matrix.
Fig. 2. A representative volume element containing an inclusion and bilinear constitutive equation for the matrix.
Fig. 3. Cleavage fracture of an oxide inclusion in a low alloy steel. Loading is horizontal.
A. Pineau et al. / Acta Materialia 107 (2016) 424e483 427
2.3. Multiple barriers models and crossing of grain boundaries
2.3.1. Basic model
Schematically, cleavage of ferritic steels most frequently results
from the succession of three elementary steps illustrated in Fig. 5
(See also [24e29]):
Slip induced cracking of brittle particle (most often carbide) in
ferritic steels;
Propagation of the microcrack across the particle or the particle/
matrix interface and then along a (100) cleavage plane of the
neighboring matrix;
Propagation of the grain-sized or, in bainitic steels, packet-sized
crack to neighboring grains across the grain boundary.
The rst event is governed by a critical stress
s
Ic
through Eq. (7).
This expression applies when the particle size is larger than
0.1e1
m
m[24]. Below this size a dislocation-based theory should be
used [30]. The values of
s
Ic
can be statistically distributed (see e.g.
Refs. [31,32]). Eq. (7) shows that for a given stress state the strain
necessary to initiate particle cracking increases with temperature
because of the variation of yield strength with temperature.
The various mechanisms in cleavage fracture can be described in
terms of local values of the fracture toughness, K
c=f
Ia
(carbide/ferrite)
and K
f=f
Ia
(ferrite/ferrite), or g
c=f
and g
f=f
in terms of free surface
energy, that must be reached by the crack to cross the rst barrier
(particle/matrix) and the second barrier (grain boundary), as
schematically shown in Fig. 6. The crossing of the rst barrier has
been analyzed previously. The grain boundary crossing will be
analyzed in detail below. A number of studies have shown that in
ferritic steels, the crack arresting boundaries were those which are
largely misorientated [33e35] and in particular those with a large
twist angle [12,36e40]. The particle and grainsize distribution
functions (f
c
,f
g
) have thus to be considered, as schematicallyshown
in Fig. 7 [35]. In this gure the critical values of the particle and
grain size, C
*
and D
*
are simply related to the local value of the
maximum principal stress,
s
I
, by a Grifth-like expression [41].
C
¼ dK
c=f
Ia
s
I
!
2
and D
¼ dK
f=f
Ia
s
I
!
2
:(9)
Very few experimental results have been published in the
literature to test the validity of this basic model. However a number
of results are gathered in Table 4 where the details concerning a
study on a bainitic steel [35] are given. It is worth noting that the
local values of the calculated fracture toughness K
c=f
Ia
and K
f=f
Ia
are
much lower than the macroscopic fracture toughness, K
Ic
. Several
reasons can be invoked to explain this difference. The rst one lies
in the calculations. The local values for the maximum principal
stress due to stress concentration related to crystallographic as-
pects can be much larger than the macroscopic stress used in the
calculations. The second reason is related to possible dynamic ef-
fects, not accounted for in these calculations, as discussed in the
next section. In Table 4, it is also worth noting that K
f=f
Ia
is larger
than K
c=f
Ia
. This conclusion combined with other observations ob-
tained from acoustic emission measurements [35] suggest that the
micromechanisms operating during fracture toughness de-
terminations at increasing temperature are not necessarily the
same. At very low temperature, cleavage can be controlled by the
initiation of microcracks from carbides, while at increasing tem-
perature cleavage is controlled by the propagation of microcracks
arrested at grain boundaries [42]. In such conditions the cleavage
stress is not necessarily constant with temperature. A number of
authors have shown that the energy g
c=f
is independent of tem-
perature and is of the order of 7 J/m
2
[26,28] while g
f=f
is much
larger and is strongly temperature dependent (See e.g.
Refs. [25e29]). The ratio g
f=f
=g
c=f
can reach values as large as
10e50.
2.3.2. Dynamic behavior
The dynamic behavior of a microcrack nucleated from a carbide
particle and propagating within the ferrite matrix was studied by
Kroon and Faleskog [43]. These authors carried out unit cell-type
dynamic FEM calculations (Fig. 8). The initiation of cleavage was
modeled explicitly by introducing a small pre-existing crack within
the carbide. This microcrack propagated through the carbide and
eventually into the surrounding ferrite. The carbide was assumed to
be purely elastic and the ferrite to be elastic-viscoplastic with a
yield strength at vanishing zero strain rate equal to R
e
. Macroscopic
constitutive equations allowing for different strain rate sensitivity
were adopted. Crack growth resistance was simulated using a
cohesive surface, where the traction was governed by an expo-
nential cohesive law. Crack growth rates as large as the Rayleigh
wave velocity and strain rates as large as 10
4
e10
6
s
1
were
Fig. 4. Variation of the ratio E
V
/E
S
(see text) with stress triaxiality S
m
/S
eq
for f¼0.10,
E
p
¼2000 MPa (tangent modulus),
s
1c
¼2000 MPa, gS¼2 J/m
2
,h¼1
m
m, and for
various values of (a) the particle aspect ratio kwith
s
0
¼500 MPa; (b)
s
0
with k¼1/2.
A. Pineau et al. / Acta Materialia 107 (2016) 424e483428