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Continuum models of ductile fracture : A review
Jacques Besson
To cite this version:
Jacques Besson. Continuum models of ductile fracture : A review. International Journal of Damage
Mechanics, SAGE Publications, 2010, 19, pp.3-52. �10.1177/1056789509103482�. �hal-00550957�
Continuum Models of Ductile
Fracture: A Review
J. B
ESSON
*
Centre des Mate
´
riaux, Mines ParisTech
UMR CNRS 7633, BP 87, 91003 Evry Cedex, France
ABSTRACT: The past 20 years have seen substantial work on the modeling of
ductile damage and fracture. Several factors explain this interest. (i) There is a
growing demand to provide tools which allow to increase the efficiency of structures
(reduce weight, increase service temperature or load, etc.) while keeping or increasing
safety. This goal is indeed first achieved by using better materials but also by
improving design tools. Better tools have been provided which consist (ii) of material
constitutive equations integrating a physically-based description of damage processes
and (iii) of better numerical tools which allow to use the improved constitutive
equations in structural computations which become more and more realistic. This
article reviews the material constitutive equations and computational tools, which
have been recently developed to simulate ductile rupture.
KEY WORDS: ductile rupture, models, numerical simulation.
INTRODUCTION
P
REDICTIVE NUMERICAL SIMULATIONS
of ductile fracture may be of
great interest in industrial situations for which full-scale experimental
approaches are either too costly or even impracticable. This is the case of
ductile tearing of gas pipelines over several hundred meters, of crack
propagation in large nuclear vessels or of ductile tearing of aircraft fuselage.
For such applications, simulations should predict crack paths, stability,
stress states, etc. Several techniques may be used to achieve these objectives.
The approach based on Rice (1968) J-integral is widely used for industrial
applications but suffers from various limitations: (i) It can only deal with
preexisting cracks and cannot be applied to model crack initiation and pro-
pagation from a notch. (ii) It is not a material intrinsic property as it strongly
depends on specimen geometry as experimentally shown in Sumpter and
International Journal of D
AMAGE
M
ECHANICS
, Vol. 19—January 2010
3
1056-7895/10/01 0003–50 $10.00/0 DOI: 10.1177/1056789509103482
ß The Author(s), 2010. Reprints and permissions:
http://www.sagepub.co.uk/journalsPermissions.nav
*Author to whom correspondence should be addressed. E-mail: jacques.besson@ensmp.fr
Forbes (1992) and Sumpter (1993) using single edge notch bending (SENB)
specimens and center crack panels (CCP). This as led to the introduction of
the J–Q two parameters approach (O’Dowd and Shih, 1991, 1992). (iii) It can
hardly be applied to complex geometries such as welds. In the case of welds, the
J–Q description of fracture proposed by O’Dowd and Shih, (1991) was extended
to account for plastic mismatch between materials (Zhang et al., 1996, 1997). It
was also proposed to compute local crack tip values for both J and Q (Kim
et al., 1997, 1999). Approaches using critical crack top opening displacement
(CTOD) or crack tip opening angle (CTOA) suffer from the same limitations
(Dawicke et al., 1997; James and Newman, 2003; Mahmoud and Lease, 2003).
The limitations of the previous approach (so called ‘Global Approach’)
have led to the development of more physically-based descriptions of
fracture which belong to the ‘Local Approach to Fracture’
1
(Pineau, 1980;
Berdin et al., 2004; Pineau, 2006). The approach is referred to as ‘local’ as a
detailed and physically-based description of damage phenomena is used to
represent the rupture process zone. Within this framework, damage and
rupture can be represented on a surface (cohesive zone model) or in the
volume (continuum damage mechanics). Both methods can be implemented
in the finite element (FE) method.
The first approach is mainly limited to predefined known crack paths
(Roychowdhury and Dodds Jr, 2002; Cornec et al., 2003) because they
exhibit strong mesh dependency and over-estimate cracked areas if inserted
between each volume element in a FE mesh (Scheider and Brocks, 2003).
The model can clearly not account for diffuse ductile damage which occurs
in metals before damage localization within a thin band. A nonpredefined
crack path could possibly be modeled using advanced numerical techniques
such as X-FEM (Sukumar et al., 2000) based on the partition of unity
method and coupled with a cohesive zone model (Moes and Belytschko,
2002). Applications of this methodology are still limited to elastic solids
(i.e., fatigue or brittle rupture) or small-scale yielding conditions.
Within the local approach, the second description of damage is based on a
volume representation of degradation phenomena. This method is based on
constitutive equations coupling plasticity and damage at the material point
level so that the materials models are often referred to as ‘Continuum Damage
Mechanics’ (CDM). Such models are reviewed in the following after a brief
description of physical damage mechanisms leading to ductile rupture.
Micromechanical models or descriptions are then presented; such approaches
are used to derive semi-empirical constitutive equations which can be used in
FE softwares and which can account for the three main stages of ductile rupture.
A more phenomenological approach is then presented. Both approaches
1
The term Local Approach to Fracture was first proposed by Pineau (1980) to describe this methodology.
4
J. B
ESSON
are compared. As damage growth leads to a strong decrease of the load
bearing capacity (softening), FE simulations carried out with CDM models
are prone to strain and damage localization so that results are not robust and
strongly mesh size dependent. Some solutions to this problem are also
reviewed. Some information is given about numerical techniques needed to
perform simulation of ductile rupture. The conclusion focuses on the flat to
slant fracture transition in thin sheets as this phenomenon is only partly
understood and simulated and epitomizes most of the difficulties currently
encountered in the domain of modeling and simulation of ductile fracture.
PHYSICAL PROCESSES OF DUCTILE RUPTURE
Ductile fracture can be described as a three stages process (see review in
Garrison and Moody, (1987)). Voids are first initiated at material defects
(mostly inclusions). Note that voids may also preexist in the material. Due to
large plastic deformation, these voids then grow in particular in situations
where the stress triaxiality is large.
2
When voids are large enough they tend to
coalesce to form microcracks and eventually a macroscopic crack that leads
to macroscopic failure. Figure 1 gives several examples of ductile rupture for
one aluminum alloy (2000 series) and various steels. Void initiating particles
are coarse intermetallic particles containing Fe and Si (2024 aluminum alloy),
elongated (X52 steel) or spherical (A508 steel) manganese sulfides (MnS),
spherical CaS particles (X100 steel). These examples are ideal cases where
only one inclusion type (in general the coarser ones) is at the origin of fracture.
This situation prevails at high stress triaxialities. However, engineering alloys
always contain several inclusion populations corresponding to different
length scales. At low stress triaxialities, void nucleation in narrow bands of
secondary voids is often observed. This failure mechanism is often referred
to as ‘void sheeting’ (Garrison and Moody, 1987). Examples are given in
Figure 2 for the 2024 aluminum alloy and the X100 steel which were used to
illustrate primary cavity growth. In the first case (Figure 2(a)), dispersoids
containing Zr, Mn, or Cr are sites for secondary nucleation. In the second
case (Figure 2(b)), Fe
3
C carbides can initiate secondary voids.
Observation and quantification of these processes can help developing
relevant models and fitting model parameters required to perform structural
simulations. Observations have, for a long time, been limited to studies of
fracture surfaces and of polished cross sections of broken or damaged specimens.
The recent development of X-ray tomography (see e.g., Maire et al., 2001;
Morgeneyer et al., 2008) now allows the direct observation of bulk damage
processes. Using this technique, error on damage quantification induced by
2
The stress triaxiality ratio is defined as: ¼
1
3
kk
=
eq
where
kk
is the trace of the stress tensor and
eq
the
von Mises equivalent stress.
Continuum Models of Ductile Fracture
5
20 mm
20 mm
20 mm
(a) (b)
(c) (d)
1 mm
Figure 1. Examples of ductile failure by internal necking on voids initiated at primary
inclusions. (a) Al 2024 (Bron, 2004): voids are initiated on coarse intermetallic particles
containing Fe and Si. (b) A508 steel (Tanguy, 2001) formation of a macroscopic crack by void
coalescence in a notched bar. (c) X52 steel (Benzerga, 2000): voids are initiated on
manganese sulfides (MnS). The photograph of a cross section of the material shows the
coalescence of two voids by internal necking. (d) X100 steel (Luu, 2006): voids are initiated
on spherical CaS inclusions shown by arrows.
(a) (b)
20 μm 100 μm
Figure 2. Examples of ductile failure involving two populations of cavities. (a) Al 2024
(Bron, 2004) secondary voids are nucleated on dispersoid particles having a typical size
between 0.05 and 0.5mm. These particles contain Zr, Mn, or Cr. (b) X100 steel (Luu, 2006):
secondary voids are nucleated on Fe
3
C carbides.
6
J. B
ESSON
