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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 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.

©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

“properties”are usually deﬁned. 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,

signiﬁcant advances have recently been made in developing more

robust models, which ultimately will reduce the many un-

certainties associated with currently used models.

The inﬂuence 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 ﬂow properties when

dealing with brittle (cleavage or intergranular) fracture. The Bere-

min model in its simplest form also uses two parameters only.

The identiﬁcation 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 brieﬂy presented ﬁrst.

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 brieﬂy

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

“rivers”as 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 deﬁned 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 beneﬁcial 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

“intrinsic”brittleness 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 inﬂuence 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 ﬁnally 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

ampliﬁcation must therefore be invoked since the measured

cleavage stresses are much lower than this theoretical value.

2.2. Mechanisms of stress intensiﬁcation

The ﬁrst 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 coefﬁcient 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

quantiﬁes 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] modiﬁed 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 intensiﬁcation 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=3þl1s

Ic

=s

0

þl

Tþ2=3þl(8)

where ε

eq

is the equivalent strain, E

p

is the tangent modulus of the

material, T≡S

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 veriﬁed 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 ﬁrst 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 ﬁrst barrier

(particle/matrix) and the second barrier (grain boundary), as

schematically shown in Fig. 6. The crossing of the ﬁrst 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 “grain”size distribution

functions (f

c

,f

g

) have thus to be considered, as schematicallyshown

in Fig. 7 [35]. In this ﬁgure 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 Grifﬁth-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 ﬁrst 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