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Ductile fracture of structural metals occurs mainly by the nucleation, growth and coalescence of voids. Here an overview of continuum models for this type of failure is given. The most widely used current framework is described and its limitations discussed. Much work has focused on extending void growth models to account for non-spherical initial void shapes and for shape changes during growth. This includes cases of very low stress triaxiality, where the voids can close up to micro-cracks during the failure process. The void growth models have also been extended to consider the effect of plastic anisotropy, or the influence of nonlocal effects that bring a material size scale into the models. Often the voids are not present in the material from the beginning, and realistic nucleation models are important. The final failure process by coalescence of neighboring voids is an issue that has been given much attention recently. At ductile fracture, localization of plastic flow is often important, leading to failure by a void-sheet mechanism. Various applications are presented to illustrate the models, including welded specimens, shear tests on butterfly specimens, and analyses of crack growth.
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Int J Fract
DOI 10.1007/s10704-016-0142-6
SPECIAL INVITED ARTICLE CELEBRATING IJF AT 50
Ductile failure modeling
Ahmed Amine Benzerga ·Jean-Baptiste Leblond ·
Alan Needleman ·Viggo Tvergaard
Received: 29 March 2016 / Accepted: 21 July 2016
© Springer Science+Business Media Dordrecht 2016
Abstract Ductile fracture of structural metals occurs
mainly by the nucleation, growth and coalescence of
voids. Here an overview of continuum models for this
type of failure is given. The most widely used current
framework is described and its limitations discussed.
Much work has focused on extending void growth mod-
els to account for non-spherical initial void shapes and
for shape changes during growth. This includes cases
of very low stress triaxiality, where the voids can close
up to micro-cracks during the failure process. The void
growth models have also been extended to consider the
effect of plastic anisotropy, or the influence of non-
local effects that bring a material size scale into the
models. Often the voids are not present in the material
from the beginning, and realistic nucleation models are
important. The final failure process by coalescence of
neighboring voids is an issue that has been given much
attention recently. At ductile fracture, localization of
A. A. Benzerga
Department of Aerospace Engineering, Texas A&M
University, College Station, TX, USA
J.-B. Leblond
Institut Jean le Rond d’Alembert, Sorbonne Universités,
Université Pierre et Marie Curie, Paris, France
A. Needleman (B)
Department of Materials Science and Engineering,
Texas A&M University, College Station, TX, USA
e-mail: needle@tamu.edu
V. Tvergaard
Department of Mechanical Engineering, The Technical
University of Denmark, Lyngby, Denmark
plastic flow is often important, leading to failure by
a void-sheet mechanism. Various applications are pre-
sented to illustrate the models, including welded spec-
imens, shear tests on butterfly specimens, and analyses
of crack growth.
Keywords Ductile failure ·Constitutive modeling ·
Micromechanics ·Porosity evolution
1 Introduction
The role played by void nucleation, growth and coa-
lescence in ductile fracture was identified in the 1940s
(Tipper 1949). However, it was not until the 1960’s
that the phenomenology of this process was well
documented (Rogers 1960;Beachem 1963;Puttick
1959;Gurland and Plateau 1963). In structural met-
als deformed at room temperature, the voids generally
nucleate by decohesion of second phase particles or by
particle fracture, and grow by plastic deformation of the
surrounding matrix. Void coalescence occurs either by
necking down of the matrix material between adjacent
voids or by localized shearing between well separated
voids, as has been described in a number of previous
review papers (Garrison and Moody 1987;Tvergaard
1990;Benzerga and Leblond 2010).
The first micromechanical studies of void growth
were carried out for a single void in an infinite elastic-
plastic solid, either a circular cylindrical void (McClin-
tock 1968) or a spherical void (Rice and Tracey 1969).
123
A. A. Benzerga et al.
A numerical study for a material containing a peri-
odic array of circular cylindrical voids (Needleman
1972) allowed for including the effect of the interaction
with neighboring voids, both in the early growth stages
and in the final stages approaching coalescence. This
numerical study considered a representative unit vol-
ume, containing a single void, with appropriate bound-
ary conditions to represent the full material. Such unit
cell analyses have become an important tool in the
analysis of several different aspects of ductile fracture.
It is also appreciated that unit cells containing many
voids have advantages over those with only one void, as
they can account for differences in void size or spacing
and also for localized plastic flow due to void clustering
or due to instabilities.
One approach to modeling void evolution in plastic
solids is to directly use isolated void growth analy-
ses as in the models of Beremin et al. (1981a) and
Johnson and Cook (1985). Such uncoupled models
have been recently reviewed by Pineau et al. (2016).
Another approach is to incorporate porosity evolution
into the constitutive formulation and several porous
plastic constitutive frameworks have been developed
for analyzing ductile failure problems, (e.g. Gurson
1975,1977;Rousselier 1981,1987). Probably the most
widely known and most widely used porous ductile
material model is that developed by Gurson based on
micromechanical studies (Gurson 1975,1977), using
averaging techniques similar to those applied by Bishop
et al. (1945). Here, we focus on ductile failure modeling
based on the Gurson framework (Gurson 1975,1977)
and its modifications. Some improvements were added
to Gurson’s model early on (Chu and Needleman 1980;
Tvergaard 1981,1982b;Tvergaard and Needleman
1984), resulting in a modified Gurson model (the so-
called GTN model), which has since been used exten-
sively to analyze a variety of problems. In many appli-
cations the material does not contain voids initially, so
the representation of gradual void nucleation during
the deformation process is important. Porous ductile
material models were also developed early on by fitting
experiments for powder metallurgy materials (Shima
and Oyane 1976), and in fact the approximate yield sur-
faces obtained by these different methods are in rather
good agreement for a given void volume fraction.
The Gurson model is limited by a number of assump-
tions, e.g. the voids are embedded in a standard Mises
solid, and the voids are taken to remain spherical inde-
pendent of the stress state. At low stress triaxiality,
i.e. low mean tensile stress relative to the effective
Mises stress, voids tend to elongate, and this can have a
strong influence on predictions of ductile failure. Early
studies that extended the Gurson model to account for
void shape effects are given by Gologanu et al. (1993,
1994,1997). Other early work on the effect of shape
changes was presented by Ponte Castañeda and Zaid-
man (1994). Regarding the effect of anisotropy the
Gurson model was extended by Benzerga and Besson
(2001) to consider a spherical void embedded in an
elastic-plastic matrix that obeys Hill’s quadratic yield
condition (Hill 1948). This model was further extended
by Keralavarma and Benzerga (2008,2010)toalso
account for non-spherical voids embedded in the same
anisotropic solid.
The final failure in void containing materials typi-
cally occurs by void coalescence, where the ligament
between neighbor voids necks down to zero thick-
ness and leaves the characteristic fibrous fracture sur-
face. An important contribution to the modeling of this
mechanism has been given by Koplik and Needleman
(1988). However, quite often the final failure is associ-
ated with a shear band instability (Rice 1976;Needle-
man and Rice 1978), leading to a so-called void-sheet
failure, where voids grow to coalescence inside a nar-
row layer of material (Rogers 1960) and the fracture
surface shows that the voids have been smeared out
during coalescence. In materials containing two size
scales of voids or inclusions from which voids nucle-
ate, it is sometimes observed that plastic flow localiza-
tion develops between two larger voids and that final
failure involves void-sheet failure by the small scale
voids between the larger voids (Cox and Low 1974;
Van Stone et al. 1985). In a model of this phenomenon
the small scale voids have been represented in terms
of the Gurson model and localization leading to void-
sheet failure between larger voids has been predicted
(Tvergaard 1982a).
Recently there has been increasing interest in the
behavior of porous materials under low stress triaxial-
ity, such as simple shear, where the standard material
models do not predict void growth to coalescence. Full
three dimensional analyses for shear specimens con-
taining spherical voids have been carried out by Bar-
soum and Faleskog (2007a) in order to model experi-
ments on ductile fracture in double notched tube speci-
mens loaded in combined tension and torsion (Barsoum
and Faleskog 2007b). In plane strain cell model analy-
ses for a material containing a periodic array of cir-
123
Ductile failure modeling
cular cylindrical voids (Tvergaard 2008,2009,2012;
Dahl et al. 2012), it has been found that in stress states
similar to simple shear the voids flatten out to micro-
cracks, which rotate and elongate until interaction with
neighboring micro-cracks gives coalescence; stresses
pass through a maximum so that failure is predicted.
This mechanism has also been found in three dimen-
sions for initially spherical voids (Nielsen et al. 2012).
Thus, under high stress triaxiality the void volume frac-
tion increases until ductile fracture occurs, whereas the
void volume fraction disappears under low stress tri-
axiality, as the voids become micro-cracks. The signif-
icant void shape changes at low stress triaxiality are
accounted for in the models of Gologanu et al. (1993,
1994,1997) and Ponte Castañeda and Zaidman (1994)
mentioned above, but to deal with failure in simple
shear the models must be extended to describe void
closure into micro-cracks and the interaction between
these micro-cracks.
The early constitutive models for porous ductile
solids did not incorporate an effect of the third stress
invariant J3, but recently there has been more focus on
this through the effect of the Lode parameter. It has been
found in some fracture tests under loads including shear
(Bao and Wierzbicki 2004;Barsoum and Faleskog
2011) that the effective plastic strain to failure does
not decrease monotonically with increasing stress tri-
axiality. This has been further investigated by Xue et al.
(2013), where tension-torsion fracture experiments are
modeled using an extension of the Gurson model by
Nahshon and Hutchinson (2008), which has been made
J3dependent by adding an extra damage term that
allows for failure prediction even at zero hydrostatic
tension. This extension of the Gurson model (Nahshon
and Hutchinson 2008) has been compared with cell
model studies for voids in shear fields (Tvergaard and
Nielsen 2010) and it has been found that the model can
capture quantitative aspects of softening and localiza-
tion in shear. Xue et al. (2013) modeled tension-torsion
fractures by finding the localization strain in a shear
stress state with varying amounts of superposed tensile
stress, and showed that the failure strain does not vary
monotonically with the stress triaxiality. On the other
hand, it has been found in other recent fracture tension-
torsion experiments (Haltom et al. 2013;Papasidero
et al. 2015) that the strain to failure decreases monoton-
ically with stress triaxiality. Improved ductile failure
modeling is therefore needed to gain insight into the
fundamental reasons for such conflicting reports.
Recent reviews of the field are available (Benz-
erga and Leblond 2010;Besson 2010). Nevertheless,
developments that have occurred in the past five years,
notably pertaining to the influence of void shape effects
at low triaxiality, the micromechanically-based model-
ing of void coalescence and new applications, justifies
a review paying special attention to these aspects.
The paper focuses on studies extending and using
the framework orginated by Gurson (1975,1977) and is
organized as follows. Section 2gives the current frame-
work, through an introduction to the Gurson and GTN
models. Section 3discusses void nucleation, while
Sect. 4presents a number of models for void growth.
Studies of void coalescence are discussed in Sect. 5.
Localization of plastic flow is introduced in Sect. 6,
and finally applications of the ductile failure models
are presented in Sect. 7.
2 Current framework
The Gurson flow potential (Gurson 1975,1977)was
obtained from the approximate homogenization and
limit-analysis of a hollow sphere made of a rigid,
ideal-plastic Mises solid. It was subsequently modified
by Tvergaard (1981) and Tvergaard and Needleman
(1984) to take the form
Φ(σ,f)=σ2
eq
¯σ2+2q1fcosh 3q2σm
2¯σ
1q3f2=0(1)
during plastic flow.
Here, σeq =(3
2σ:σ)1/2,σm=1
3tr σ(σis the
Cauchy stress tensor and σits deviator), ¯σis the matrix
flow strength, the qi,i=1,2,3 are Tvergaard’s (1981)
parameters (in practice q3=q2
1is generally used) and
fis given by
f=f,f<fc
fc+(1/q1fc)( ffc)/( fffc), ffc(2)
where fis the porosity (the void volume fraction)
and fcand ffare specified parameters. The constant
1/q1=fuis the value of fat zero stress carry-
ing capacity, i.e. at material failure. As fffand
ffuthe material loses all stress carrying capacity.
The rate of deformation tensor, D, the symmetric
part of ˙
F·F1(where F=¯
x/∂xwith xand ¯
xbeing
the positions of a material point in the reference and
current states, respectively), is written as the sum of an
123
A. A. Benzerga et al.
elastic part (most commonly in applications a hypoe-
lastic part), De, and a plastic part, Dpwith the plastic
flow rule given by
Dp=˙
Λ∂Φ
σ(σ,f)(3)
For a rate independent matrix material
˙
Λ=0ifΦ(σ,f)<0
0ifΦ(σ,f)=0(4)
while for a rate dependent matrix material, ˙
Λis a non-
negative function of state.
Assuming small elastic strains, the constitutive rela-
tion can be expressed in terms of the Cauchy stress σ
and the rate of deformation tensor Din the form
ˆ
σ=Lel ·(DDp). (5)
Here the Jaumann rate of Cauchy stress ˆ
σis given by
ˆ
σ=Dσ
Dt +σ·ΩΩ·σ(6)
where Dσ/Dt is the material time derivative of Cauchy
stress and Ωis the antisymmetric part of ˙
F·F1.
With the moduli Lel taken as constants, as generally
done in practice, the relation Eq. (5) is a hypoelastic
relation not a hyperelastic relation. However, if needed,
a hyperelastic relation can be expressed in the form Eq.
(5) but the elastic moduli are then stress dependent.
The evolution of the void volume fraction has two
terms; a growth term from approximate matrix incom-
pressibility (elastic volume change neglected) and a
term arising from void nucleation, so that
˙
f=(1f)tr Dp+˙
fnuc (7)
It is worth noting that the term (1f)tr Dpthat
accounts for void growth is one of the rare rigor-
ous equalities (presuming no elastic volume change)
derived from homogenization.
Another key relation of the framework is
˙
Wpσ˙
¯=(1f)σ:Dp(8)
where ˙
Wpσ˙
¯is the plastic dissipation rate in the
matrix material. There are two parts to this relation: (i)
the statement that the macro or overall plastic dissipa-
tion rate is equal to the plastic dissipation rate in the
matrix material; and (ii) that the plastic dissipation rate
in the matrix material can be represented by the product
of “average” values (or representative values) of flow
strength and matrix plastic strain rate.
Note that Eq. (8) implies that during plastic flow
˙
Λ=¯σ˙
¯
(1f)σ:(∂Φ/∂ σ)(9)
Although the analysis leading to Eq. (1) was based
on the assumption of a perfectly plastic, rate indepen-
dent matrix material, this framework is widely used
to model porous materials with strain and strain rate
hardening matrix materials.
Equations (1)to(8) together with a constitutive char-
acterization of the matrix material provide a constitu-
tive framework for analyzing porosity evolution in duc-
tile solids. This framework, now generally referred to
as the GTN relation, differs from a purely phenomeno-
logical damage mechanics framework in that, to some
extent at least, it is based on micromechanical analyses.
However, it is important to remember that the micro-
mechanical analyses leading to the form of the flow
potential, Eq. (1) and extensions that will be discussed
subsequently, are obtained from analyses predicting the
yield surface of a non-hardening, rate independent solid
containing voids. The extensions to use Eq. (1)for
other matrix constitutive characterizations, for exam-
ple strain hardening and strain rate hardening matrix
materials, are phenomenological as are the extensions
to include the qiparameters and void coalescence via
Eq. (2). Thus, this framework as used is a cross between
a micromechanically based model and a phenomeno-
logical damage mechanics theory.
The micromechanical basis of this material model
can be regarded as pertaining to isothermal porosity
evolution in lightly (strain and/or strain rate) hardening
solids, mainly structural metals at room temperature.
Even in this context there are potentially significant
effects that will not be discussed in the following: (i)
for sufficiently small voids the increased hardening due
to the size dependence of plastic flow in metals delays
void growth (e.g. Hussein et al. 2008;Segurado and
Llorca 2009); (ii) the voids in structural metals typically
nucleate from second phase particles and the constraint
provided by the particle can enhance void growth at low
values of the stress triaxiality (e.g. Fleck et al. 1989);
123
Ductile failure modeling
and (iii) at sufficiently high loading rates as can occur
near a crack tip in a dynamically loaded component or
structure material inertia can affect porosity evolution
(e.g. Ortiz and Molinari 1992;Jacques et al. 2012).
A goal of recent analyses is to develop flow poten-
tials that extend the micromechanical basis of this
framework. For example, the form of Eq. (1) is based
on analysis of spherical voids. The effects of void
shape and void shape changes, which are particularly
important at low values of the stress triaxiality, are not
accounted for. An ultimate goal is to develop a unified,
micromechanically based flow potential that incorpo-
rates void nucleation, void growth and void coales-
cence. Progress is being made on incorporating void
coalescence but incorporating void nucleation is essen-
tially unexplored.
The aim of the GTN relation is to provide a basis for
predicting ductile failure/fracture or at least the reduc-
tion in strength due to porosity evolution. Modeling the
localization of deformation and creation of new free
surface requires the incorporation of a length scale into
the formulation as will be discussed in several contexts
subsequently.
3 Void nucleation
In ductile fracture formulations based on a single dam-
age variable, namely the void volume fraction f,void
nucleation is represented through a rate equation of the
form of Eq. (7). This formulation goes back to Gurson
(1975) and was further developed by Chu and Needle-
man (1980) who accounted for two possible contribu-
tions that we write as
˙
fnuc =D˙
¯ε+B(c1˙σeq +c2˙σm), (10)
with the first term representing strain-controlled nucle-
ation (Goods and Brown 1979), and the second term
representing stress-controlled nucleation (Argon et al.
1975;Beremin et al. 1981b), with the requirement that
c1˙σeq +c2˙σm>0. Generally the factor c1is taken as 1
or 0 and the factor c2is introduced here based on find-
ings by Needleman (1987) using cell model analyses.
On the basis of earlier studies (e.g., Goods and Brown
1979;Argon et al. 1975), it was suggested by Chu and
Needleman (1980) that Dand Bare functions of ¯εand
c1σeq +c2σm, respectively, and that they follow a nor-
mal distribution. For the strain controlled term,
D(¯ε) =fN
sN2πexp1
2¯εN
sN2(11)
where fNrepresents the volume fraction of void-
nucleating particles, Nis some average nucleation
strain and sNis a standard deviation.
For the stress controlled term, with σNthe average
nucleation stress,
B=fN
sN2πexp1
2c1σeq +c2σmσN
sN2(12)
if (c1σeq +c2σm)is at its maximum over the deforma-
tion history. Otherwise B=0.
At a more fundamental level, an energy criterion
is necessary for void nucleation (Goods and Brown
1979). When this criterion is satisfied, a sufficient con-
dition may be formulated in terms of stresses. On the
other hand, attainment of a critical strain is neither
necessary nor sufficient for void nucleation. In addi-
tion, a strain-controlled criterion does not capture the
dependence of void nucleation upon stress triaxiality,
a fact that is inferred from both experiments (Beremin
et al. 1981b) and analysis (Needleman 1987). It would
also predict an increasing amount of void nucleation
with decreasing stress triaxiality, simply because of
the larger amounts of accumulated plastic strain at
low triaxialities. In practice, however, use of a strain-
controlled nucleation may be a convenient way of rep-
resenting the outcome of a more basic stress-based cri-
terion. An example in this regard was discussed by
Needleman (1987).
Analyses of localization carried out within the
framework of Rice (1976) indicated that strain-
controlled and stress-controlled nucleation can lead
to quite different predictions of macroscopic ductility,
interpreted as the onset of a bifurcation in the set of
governing partial differential equations (see Sect. 6for
details). Of particular significance is that the hydro-
static stress dependence of ˙
fnuc in Eq. (10) gives rise
to non-symmetry of the tangent matrix, which favors
early flow localization.
In ductile fracture formulations that employ addi-
tional damage variables, such as the void shape and
orientation, Eqs. (710) may still be used. This model
of nucleation does not refer to a specific nucleation
mechanism and thus does not distinguish between par-
ticle debonding and particle cracking for instance. In
some material systems, void nucleation is deformation
123
A. A. Benzerga et al.
induced and may occur at twins (Kondori and Benz-
erga 2014;Rodriguez et al. 2016)orbycleavagein
brittle phases (Joly et al. 1990). In such cases as well
as for particle cracking, damage initiates in the form
of penny-shape cracks. It is not clear what role, if
any, the shape of incipient voids plays on subsequent
fracture events but enhanced void nucleation formu-
lations are needed to elucidate such effects. Also, the
formulation in Eq. (10) rests on empirical experimen-
tal evidence, some basic analyses in the 1970’s and
early 1980’s and further corroborated by the microme-
chanical simulations of Needleman (1987). In the lat-
ter, constitutive relations are specified independently
for the matrix, the particle and the interface. However,
the analyses were limited to axisymmetric loadings,
spherical particles in a plastically isotropic matrix, with
debonding as the only nucleation mechanism. There
is a great deal of interest in recent years in ductile
fracture at low triaxiality of stress states, and in par-
ticular under shear dominated loadings. Under such
circumstances, additional complexity arises due to so-
called void-locking effects (see Pineau et al. 2016 for
an overview). There is a need to extend the analysis
basis to account for particle shape effects, nucleation
by cracking or at sites other than particles. Some of that
has been accomplished (e.g. Xu and Needleman 1993;
Hu and Ghosh 2008) but has not yet been translated
into useful expressions for modeling void nucleation
in a Gurson-type constitutive framework. It is worth
mentioning in this context the work of Horstemeyer
and co-workers (Horstemeyer and Gokhale 1999), who
included the effects of the third stress invariant in a phe-
nomenological void nucleation criterion, as well as the
work of Lee and Mear (1999) who in the spirit of earlier
work (Wilner 1988) conducted a large series of analy-
ses providing a micromechanical basis to formulate a
nucleation criterion that distinguishes particle debond-
ing from particle cracking (see Benzerga and Leblond
2010).
There have been a limited number of microme-
chanical studies of the effects of particle size and dis-
tribution on void nucleation. A highly idealized two
dimensional study of void clustering effects on void
nucleation by inclusion debonding was carried out by
Shabrov and Needleman (2002). The particle distrib-
utions in Fig. 1were analyzed for applied overall in-
plane stress states of σyy =σand σxx =ρσ so that
the stress triaxiality increases with increasing values
of ρ.
Fig. 1 Inclusion distributions analyzed by Shabrov and Needle-
man (2002). Each distribution has the same volume fraction. The
particle size decreases as the distributions go from row a to row
bto row c
Fig. 2 The dependence of the void nucleation strain on the
imposed stress ratio ρ. The distribution labels correspond to those
in Fig. 1.FromShabrov and Needleman (2002)
Figure 2shows the effects of particle size and distri-
bution on the strain for void nucleation. The nucleation
strain is more sensitive to particle size and distribution
at smaller values of ρ(lower stress triaxiality values)
and has relatively little sensitivity when the stress triax-
iality is higher. Furthermore, Shabrov and Needleman
123
Ductile failure modeling
(2002) found that the value of cin Eq. (10)differed
significantly from c=1, the most commonly used
value in applications, and varied somewhat with parti-
cle size and distribution. The dependence on cis signif-
icant since, as will be seen in Sect. 6, the localization
of deformation can depend on the value of c. There
is a need for more, and more realistic, micromechan-
ical studies of void nucleation and a need to use such
studies to develop physically based void nucleation cri-
teria.
4Voidgrowth
4.1 Generalities
There are two basic methods to derive “homogenized”
models for porous plastic materials depicting the sec-
ond phase—void growth—of ductile fracture:
– The first was initiated by Gurson (1975,1977),
followed by many others (Gologanu et al. 1993,
1994,1997;Benzerga and Besson 2001;Monchiet
et al. 2006,2008;Keralavarma and Benzerga 2008,
2010;Madou and Leblond 2012a,b,2013;Madou
et al. 2013). Its principle consisted in combining
the theory of limit-analysis (equivalent to plastic-
ity theory in the absence of elasticity and strain
hardening) with homogenization of some “elemen-
tary cell” in some plastic porous material. The
shape of this cell was “adapted” to that of the
enclosed void: spherical/cylindrical for a spheri-
cal/cylindrical void, spheroidal and confocal with
the void if spheroidal, ellipsoidal and again confo-
cal with the void if ellipsoidal. Conditions of homo-
geneous boundary strain rate, as proposed by Man-
del (1964) and Hill (1967), were used. The matrix
was assumed to obey the Mises (isotropic) yield
criterion or the (Hill 1948) (orthotropic) criterion.
The second originated from homogenization meth-
ods extending the linear Hashin–Shtrikman bounds
to nonlinear composites (Ponte Castaneda 1991;
Willis 1991;Michel and Suquet 1992), and used
a technique of “comparison” with some reference
linear material. The early model of Ponte Castañeda
and Zaidman (1994), in spite of its accuracy for
deviatoric loadings, suffered from a notable over-
estimation of the overall yield limit under hydro-
static loading. This drawback was remedied in the
more recent model of Danas and Ponte Castañeda
(2009a,b) based on the “second-order homoge-
nization method” (Ponte Castaneda 2002).The
Ponte Castañeda and Zaidman (1994) yield surface
was also very recently improved by Agoras and
Ponte Castañeda (2013,2014) using an “iterative”
approach devised by Ponte Castaneda (2012).
Both approaches are presented in the sequel but
with major emphasis on the first one, which has been
followed by most authors and used more widely for
practical applications. As will be seen, a remark-
able, though still incomplete degree of convergence
between these approaches is apparent in very recent
works.
Starting with the work of Gurson (1975,1977)
flow potentials for modeling room temperature duc-
tile failure have mainly been based on analyses of iso-
lated voids or idealized distributions of voids in a rate
independent non-hardening solid. The analyses have
focused on developing expressions for the onset of plas-
tic yielding, i.e. yield surface. Presuming plastic nor-
mality at the microscale, the yield surfaces so computed
serve as plastic potentials. It is as plastic potentials as in
Eq. (3) that the derived expressions are typically used
in applications.
4.2 Gurson’s model
4.2.1 Original form
Gurson’s (1975,1977) model was derived from the
approximate homogenization and limit-analysis of a
hollow sphere made of some rigid, ideal-plastic mater-
ial obeying the Mises yield criterion and the associated
plastic flow rule, and subjected to conditions of homo-
geneous boundary strain rate (Mandel 1964;Hill 1967).
The overall criterion thus obtained is given by Eq. (1)
with qi=1fori=1,2,3 and ff.
Gurson (1975,1977) also showed that as a conse-
quence of homogenization combined with a classical
result of limit-analysis, the normality property obeyed
at the microscopic scale is preserved at the macroscopic
scale; thus the overall flow rule is a direct consequence
of the overall criterion, the overall plastic strain Dp
being given by Eq. (3) with Eq. (4).
It is worth noting that Eq. (7) shows that the evolu-
tion of the internal parameter fis dictated by the flow
rule and thus, by what precedes, by the flow potential.
Therefore specifying this potential can, if void nucle-
123
A. A. Benzerga et al.
ation is neglected, quite remarkably, completely define
the model.
The Gurson model, in its original form, possesses
the following nice properties, which may serve for an
alternative, less rigorous but more intuitive derivation:
the criterion reduces to that of von Mises in the limit
of a zero porosity f;
for a purely deviatoric loading (σm=0), it pre-
dicts an overall yield stress equal to (1f)¯σ(with
q3=q2
1in Tvergaard’s modification this becomes
(1q1f)¯σ), in agreement with the elementary but
rigorous inequality σeq (1f)¯σresulting from
the Cauchy-Schwartz inequality;
for a purely hydrostatic loading (σeq =0), it predicts
an overall yield stress equal to 2
3¯σln f(in Tver-
gaard’s modification this becomes 2
3¯σln(q2f)), in
agreement with the exact result for a hollow sphere
resulting from an elementary calculation;
– for a low porosity fand a high triaxiality T=
σmeq , combination of Eqs. (1,3,7) essentially
yields (up to some multiplicative factor) the famous
exponential void growth law of Rice and Tracey
(1969), derived from the approximate limit-analysis
of a single void embedded in an infinite matrix;
– it formally looks like (without being completely
identical to) that for a hollow cylinder subjected to
some axisymmetric loading under conditions of gen-
eralized plane strain, the exact form of which is also
known from the work of Gurson (1975,1977).
The original reasoning of Gurson (1975,1977),
which involved a somewhat dubious expansion in pow-
ers of a parameter which was not really small, was
very recently reexamined and clarified by Leblond and
Morin (2014) using more rigorous mathematics (see
also Benallal et al. 2014). The main conclusions of this
work were twofold:
– Gurson’s criterion Eq. (1) provides a rigorous
“upper bound” for the exact overall yield locus of
the hollow sphere envisaged with the boundary con-
ditions considered, and also as a consequence for
that of a Hashin assembly of hollow spheres having
identical porosities (the same conclusion was also
reached by Benzerga and Leblond (2010), using a
different argument);
for the overall criterion, Gurson’s expansion pro-
cedure converges very quickly, and his first-order
criterion is almost identical to the final “converged”
one; but this is less true for the overall flow rule,
Gurson’s first-order truncation of the series involv-
ing a 25 % maximum error on the porosity rate (fur-
ther comments on this point are provided below).
4.2.2 Extended forms
In a sense, the first extension of Gurson’s model defined
by Eqs. (1,3,7) is due to Gurson himself, and pertains
to strain hardening. He assumed hardening to be of
isotropic type at the local scale, the yield stress in pure
tension of the material being now, instead of a mere con-
stant ¯σ, a given function σ() of the Mises equivalent
accumulated strain . Instead of extending his approxi-
mate homogenization of a hollow sphere made of ideal-
plastic material to the hardenable case, he adopted a
purely heuristic approach which consisted of assuming
that his overall yield criterion Eq. (1) remained applica-
ble to such a case, the parameter ¯σdenoting now some
“average value” of the local yield stress σ().More
precisely, he defined ¯σas the value of σcorresponding
to some “average value”, ¯,of, for which he proposed
Eq. (8) as an evolution law.
The meaning of Eq. (8) is that the plastic dissipation
(1f)σ:Dpin the real, inhomogeneously strained
material is heuristically identified to that, ¯σ˙
¯, in a ficti-
tious, homogeneously strained material with equivalent
accumulated strain ¯and yield stress ¯σ=σ(¯).One
remarkable feature of Eq. (8) is that it does not only
account for the hardening arising from the deviatoric
part of the overall plastic strain rate Dp, but also, in an
approximate way, for that arising from its hydrostatic
part, that is in fact from void growth.
The extended model thus defined however suffers
from the fact that the same parameter ¯σenters both
the “square” and “cosh” terms of the yield criterion,
which means that the effect of strain hardening is
implicitly assumed to be the same on the overall yield
stresses under purely deviatoric and purely hydrosta-
tic loadings. Leblond et al. (1995), using an extension
of the approximate homogenization analysis of Gur-
son (1975,1977) to the hardenable case, have shown
that this is only an approximation which may lead to
significant errors on the value of porosity rate. They
have proposed a variant of Gurson’s criterion Eq. (1)in
which different macroscopic parameters σ1,σ2, instead
of the single ¯σ, enter the “square” and “cosh” terms;
they have evidenced the improvement thus brought to
the prediction of the porosity rate through comparison
of the model predictions with the results of some micro-
123
Ductile failure modeling
mechanical numerical simulations of a spherical cell,
analogous to those of Koplik and Needleman (1988)
for a cylindrical one.
Both the strain hardening and the strain rate harden-
ing extensions of the Gurson (1975,1977) framework
are only expected to be reasonable approximations for
lightly hardening solids since the form of Eq. (1)pre-
sumes a non-hardening, rate independent matrix mate-
rial. On the other hand, the validity of the expressions
governing the evolution of porosity, Eq. (7), and the
equivalence of matrix and macro plastic dissipation,
Eq. (8), are independent of the matrix material charac-
terization.
Another heuristic modification involving the char-
acterization of the matrix material is to represent the
matrix material as a rate dependent viscoplastic solid,
(Pan et al. 1983). As for strain hardening, the modi-
fication involves the relation between the matrix flow
strength ¯σand the matrix plastic strain rate ˙
¯. Specif-
ically, the matrix plastic strain rate can be written as
˙
¯0f(¯σ,g)(13)
where gis a measure of plastic flow resistance of the
matrix.
Extensions of the original model of Gurson (1975,
1977) to matrix materials exhibiting kinematic harden-
ing have also been proposed by Mear and Hutchinson
(1985), see also Becker and Needleman (1986), Tver-
gaard (1987), Leblond et al. (1995). But these exten-
sions are somewhat hampered by the ambiguities and
difficulties arising, already at the local scale, in the def-
inition of a “good” kinematic hardening rule in the con-
text of large strain plasticity.
Another extension pertains to the adaptation of the
model of Gurson (1975,1977) to more realistic, non-
spherical cell shapes. In order to bring the model pre-
dictions to better agreement with the results of some
micromechanical simulations, Tvergaard (1981)pro-
posed to modify Gurson’s original flow potential by
including the heuristic parameters qiin Eq. (1). Most
authors have adopted values of q2and q3equal to 1
and q2
1, respectively; Tvergaard’s proposed modifica-
tion then simply amounts to multiplying the porosity
fby the heuristic factor q1. Values of this parameter
of the order of 1.5 have been proposed both by Tver-
gaard (1981) as just mentioned, from comparisons with
micromechanical simulations, and Perrin and Leblond
(1990), from theoretical arguments.
The physical interpretation of the parameter q1is
however multi-faceted. For instance:
The study of void growth in an infinite medium
(zero porosity) by Huang (1991) led to the conclu-
sion that the prefactor in the Rice and Tracey (1969)
exponential void growth law was notably under-
estimated. Gologanu (1997) noted that correcting
this underestimation within the model of Gurson
(1975,1977) required introducing a q1-parameter
of the order of 1.6. The role of this parameter is
then to correct inaccuracies occurring in the model
in the limit of vanishingly small porosities, and has
nothing to do with the shape of the elementary cell.
The study of Leblond and Morin (2014) has shown
that introduction of a q1-parameter depending on
the triaxiality T, and of the order of 1.25 for small
T, is necessary to correct the inherent error on the
porosity rate made by Gurson’s (1977) model, as a
result of his truncation of a series at the first order.
In this context the introduction of q1is necessary
even for a spherical elementary cell and a finite,
nonzero porosity.
The introduction of the “qi”-parameters by Tver-
gaard (1981) was completed (i) by Chu and Needleman
(1980) by introducing an extra term connected to void
nucleation in the evolution law Eq. (7) of the porosity,
see Sect. 3above; and (ii) by Tvergaard and Needleman
(1984) through a heuristic modification of the porosity
in the yield criterion Eq. (1) and the associated flow rule
Eq. (3), aimed at phenomenologically accounting for
coalescence of voids, see Sect. 5below. The resulting
GTN model has been widely used for numerical, finite-
element based simulations of ductile rupture of actual,
full-size specimens and structures; a few examples will
be provided in Sect. 7below.
An important modification of the evolution law of
Gurson (1975,1977), Eq. (7) of the porosity, was
recently proposed by Nahshon and Hutchinson (2008).
The aim of this modification was to account in a heuris-
tic way for the development of damage evidenced in
micromechanical numerical simulations performed by
Tvergaard (2008,2009,2012), Dahl et al. (2012),
Nielsen et al. (2012) under conditions of low or vanish-
ing triaxiality T.TheNahshon and Hutchinson (2008)
modification can be written as
˙
¯
f=˙
f+kωfω(σ)
σ:Dp
¯σ(14)
123