Experimental and Numerical Investigation of Void Nucleation in an AlMgSi Alloy
Abstract: The nucleation of microvoids from second phase particles in an AlMgSi alloy has been investigated. The work has comprised detailed stress analyses of elastic particles situated in an elasticplastic matrix as well as damage mechanics analyses of the ductile fracture behaviour of smooth and notched axisymmetric tensile specimens. Based on the micromechanical analyses of isolated particles, general expressions relating local stresses in the particles to the stresses on a mesoscopic (continuum) level have been derived. These relationships have been incorporated into the classical Gurson model. Combined with a failure criterion based on a physical coalescence mechanism, a calibration with experimental results enabled the establishment of a set of micromechanical parameters which are transferable between different stress triaxialities.
Topics: Nucleation (52%), Damage mechanics (52%)
Summary (3 min read)
Jump to: [1 Introduction] – [Particle] – [2 Micromechanical stress analyses] – [2.1 Mesoscopic stresses and strains] – [2.2 Material descriptions] – [2.3 Stresses at the particlematrix interface] – [2.4 Stresses in the particle crosssection] – [3 Simulation of ductile fracture in axisymmetric tensile specimens] – [3.1.1 Material description] – [3.1.2 Mechanical behaviour] – [3.2 Numerical calculations] – [3.2.1 Nucleation model] – [3.2.2 Void growth] – [3.2.3 Void coalescence] – [3.2.4 Results] and [4 Conclusions]
1 Introduction
 A frequently observed problem with the Gurson model is the lack of transferability between different stress states.
 A set of micromechanical parameters determined for one particular stress state does not necessarily apply to other stress states.
 The void shape has previously been shown to have a significant effect on damage evolution [9, 101.
 This effect depends on stress triaxiality, and it was therefore expected t o also be of significance for the transferability problem.
 It was concluded that these effects cannot alone be responsible for the observed lack of transferability.
Particle
 FE model of material unit cell containing an isolated elastic particle.
 In current applications of the Gurson model, it is frequently assumed that the voids are either initially present or that they are nucleated upon the attainment of a certain nucleation strain.
 For larger particles, however, many investigations (e.g. [I, 31) have indicated that the stresses a t the particles are determined by the global stress state.
 The main aim of the present work is to investigate whether the adoption of a stress based nucleation model will improve the stress state transferability of micromechanical parameters for the Gurson model.
 In the first part of this paper, the main results of some micromechanical stress analyses of elastic particles surrounded by an elasticplastic matrix will be given.
2 Micromechanical stress analyses
 In order to evaluate the local stresses around an elastic particle situated in an elasticplastic matrix, an axisymmetric finite element (FE) model of a representative material element (unit cell) was established.
 The spring elements are introduced in order to control the stress triaxiality.
 In consistency with a common damage mechanics approach, it is convenient to distinguish between local and global quantities.
 In the present work, local quantities within the unit cell will be referred to as microquantities.
 Averaged, or global quantities will be denoted mesoquantities.
2.1 Mesoscopic stresses and strains
 The mesoscopic strains are defined as logarithmic strains in the following manner.
2.2 Material descriptions
 Where 8 and 5 are the uniaxial stress and.
 In the analyses reported here, Young's modulus was set to 70 GPa, the yield stress to 260 MPa and the hardening exponent to 0.1.
 The particle was specified as purely elastic with a Young's modulus of 130 GPa and a Poisson's ratio of 0.33.
2.3 Stresses at the particlematrix interface
 It was found that with an appropriate adjustment of the k,factor, the Argon model reasonably well predicts the maximum normal stress over the interface.
 For the spheroidal particles, the normalised stresses tended to vary more with the strain level than was the case for the spherical particle.
2.4 Stresses in the particle crosssection
 Experimental investigations of void nucleation from secondphase particles and inclusions often show an increased tendency for particle fracture with increasing particle aspect ratio (e.g. [2]).
 matrix interfacial stress can be evaluated Although the ratio increases with increasing particle aspect ratio for all stress triaxialities, the increase is too small t o be any major explanation to the abovementioned observations.
3 Simulation of ductile fracture in axisymmetric tensile specimens
 In order to investigate the effects of using a stress controlled nucleation model, detailed simulations of smooth and notched axisymmetric tensile specimens were performed.
 Similar specimens of an AlMgSi alloy have previously been tested experimentally, thus allowing for a determination of micromechanical parameters for this alloy.
 It has been shown [Ill that the nucleation phase in aluminium alloys may constitute the major part of the total ductility.
3.1.1 Material description
 The test material was taken from an extruded AlMgSi alloy.
 Chemical composition is shown in table 1. Metallographical investigations [7] revealed that the only constituent particles were AlFeSi.
 These particles were oriented with their longitudinal axes in the direction of extrusion and had an average aspect ratio of about 2.7.
3.1.2 Mechanical behaviour
 Three specimens of each geometry were tested but the variation in mechanical response among the parallels was negligible.
 The smooth specimens were also provided with an extensiometer for more accurate recordings of displacements.
 These recordings were used for establishing a 'true' stressstrain curve, see Figure 4 .
3.2 Numerical calculations
 The Gurson model was implemented through the UMAT user subroutine in ABAQUS.
3.2.1 Nucleation model
 Using a purely stresscontrolled nucletion model may lead to some numerical difficulties.
 These are mainly attributable to the general softening effect of cavitation, i.e. a increment in porosity may give a negative increment in effective stress.
 The nucleation strain E N is computed separately for each point in the specimens according to the actual stress state.
 In order to include the lower 'half' of the normal distribution in equation ( 12), it is necessary to know the nucleation strain before the analysis actually reaches that strain.
 At the beginning of each increment, the matrix flow stress corresponding to a 'trial' strain equal to the current plastic strain plus 3SN is determined.
3.2.2 Void growth
 The nucleation process as well as the early stages of void growth are very complex.
 Large parts of the broken particles may remain bonded to the matrix and probably alter the growth pattern.
 Since these aspects are not known, it was decided not to include the influence of void shape in the present investigation.
3.2.3 Void coalescence
 By letting the void coalescence be a 'natural' consequence of the damage evolution, the calibration of micromechanical parameters can be based on the nucleation parameters rather than the coalescence parameters.
 In this way it is believed that more 'realistic' parameters can be established and thereby improve the chances of obtaining parameters that are stress state independent.
3.2.4 Results
 As could be expected, the incipient void nucleation starts a t a much higher strain level for the smooth specimen than for the three other specimen geometries.
 Looking a t Figure 5 , one may be led to the conclusion that the same results could have been obtained with a constant nucleation strain of about 5%.
 With a constant nucleation strain, the damage distribution would have been significantly altered.
 For specimen TSR08, for instance, final failure would have been predicted a t a much earlier stage because the most severely damaged region would then have been in the highly strained (but with relatively low stress triaxiality) region in front of the notch root.
4 Conclusions
 With an appropriate choice of micromechanical parameters, the Gurson model gives good predictions of ductility for a wide range of stress states.
 An appropriate nucleation model in combination with a micromichanical based coalescence criterion, shows good prospects with respect to the ability to determine physically realistic nucleation parameters.
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Experimental and Numerical Investigation of Void
Nucleation in an AlMgSi Alloy
O. Søvik
To cite this version:
O. Søvik. Experimental and Numerical Investigation of Void Nucleation in an AlMgSi Alloy. Journal
de Physique IV Proceedings, EDP Sciences, 1996, 06 (C6), pp.C6155C6164. �10.1051/jp4:1996615�.
�jpa00254443�
JOURNAL
DE
PHYSIQUE
IV
Colloque
C6,
supplCment au Journal de Physique
HI,
Volume
6,
octobre
1996
Experimental and Numerical Investigation of Void Nucleation in an AlMgSi
Alloy
O.P.
Sovik
Raufoss Hydro Automotive Research Centre,
P.O.
Box
41,
2831 Raufoss, Norway
Abstract:
The nucleation of microvoids from second phase particles in an AlMgSi alloy has been
investigated. The work has comprised detailed stress analyses of elastic particles situated in an
elasticplastic matrix
as
well
as
damage mechanics analyses of the ductile fracture behaviour of
smooth and notched axisymmetric tensile specimens.
Based on the micromechanical analyses of
isolated particles, general expressions relating local stresses in the particles to the stresses on a
mesoscopic (continuum) level have been derived. These relationships have been incorporated into
the classical Gurson model. Combined with
a
failure criterion based on a physical coalescence
mechanism, a calibration with experimental results enabled the establishment of a set of
microme
chanical parameters which are transferable between different stress triaxialities.
1
Introduction
During the last years, the constitutive model introduced by Gurson
[5]
has been used extensively
for simulating ductile damage. Despite the many simplifications adopted by the model, numerous
investigations have demonstrated its capability to predict ductility limits of engineering materials
and structures.
The successful application of the Gurson model relies upon an appropriate selection of the
micromechanical parameters included in the model.
A direct determination of these parameters
based on metallographical examinations is however difficult since there appears to be no unique
relation between the micromechanical parameters and the actual microstructural characteristics of
the material.
A
more commonly applied procedure for determining the micromechanical parameters
is based on a dual approach involving experimental testing as well as
FE
analyses of axisymmetric
tensile specimens (see
e.g.
[12]).
The
FE
analyses calculate the macroscopic response of the specimens
using the constitutive equations ofthe Gurson model. The choice of nucleation parameters (and the
Tvergaard parameters
ql
and
qz)
is either arbitrary or based on metallographical examinations. The
final step is to compare the experimentally observed response with the numerical calculations and
then adjust the coalescence parameters so that the fulfilment of the chosen coalescence criterion
coincides with failure in the tested specimens.
A frequently observed problem with the Gurson model is the lack of transferability between
different stress states. A set of micromechanical parameters determined for one particular stress state
does not necessarily apply to other stress states. In a recent investigation by the author
[8],
the effect
of void shape on the transferability of micromechanical parameters was studied. The void shape has
previously been shown to have a significant effect on damage evolution
[9,
101.
This effect depends
on stress triaxiality, and it was therefore expected to also be of significance for the transferability
problem. Although it was demonstrated that void shape effects may play a significant role, it was
concluded that these effects cannot alone be responsible for the observed lack of transferability.
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jp4:1996615
JOURNAL
DE
PHYSIQUE
IV
Particle
Figure
1:
Axisymmetric FE model of material unit cell containing an isolated elastic particle
In the present work, another possible source of the stress state dependence of
micromechan
ical parameters is investigated; namely the nucleation model applied. In current applications of
the Gurson model, it is frequently assumed that the voids are either initially present or that they
are nucleated upon the attainment of a certain nucleation strain. The latter may be a reasonable
assumption in the case of very small particles
(
<
1
pm
)
since the local stresses then are mainly
determined by the accomodation of plastic strains around the particles. For larger particles, however,
many investigations
(e.g. [I,
31)
have indicated that the stresses at the particles are determined by
the global stress state. In such cases,
a
nucleation model based on a constant nucleation strain may
be inappropriate. The main aim of the present work is to investigate whether the adoption of a stress
based nucleation model will improve the stress state transferability of micromechanical parameters
for the Gurson model.
In the first part of this paper, the main results of some micromechanical stress analyses of
elastic particles surrounded by an elasticplastic matrix will be given. In the second part, the results
from the micromechanical analyses will be used in connection with the Gurson model in an attempt
to establish micromechanical parameters that are transferable between different stress states.
2
Micromechanical stress analyses
In order to evaluate the local stresses around an elastic particle situated in an elasticplastic matrix,
an axisymmetric finite element (FE) model of a representative material element (unit cell) was
established.
The FE mesh is shown in Figure
1.
The spring elements are introduced in order to
control the stress triaxiality. The
FE
model is described in more detail in
[9,
101.
In consistency with a common damage mechanics approach, it is convenient to distinguish
between local and global quantities. In the present work, local quantities within the unit cell will
be referred to as microquantities. Averaged, or global quantities will be denoted mesoquantities.
With this notation, the term macroscopic should only be used to describe an assembly of unit cells
(e.g. a structure or a component).
2.1
Mesoscopic stresses and strains
The mesoscopic stress components are defined in terms of the net forces acting at the cell edges in
the following way
El1
=
Fl
C33
=
F3
4r(L0
+
ul)(Ho
+
us)
'
GO
+
~1)~
(1)
where
Fl
and
F3
are the concentrated loads applied in the radial and axial direction, respectively.
The former is assumed to be the integral of a force distributed along the circumference.
Due to the axisymmetry,
Ell
=
E22
and
C33
are principal stresses and the mesoscopic
effective and hydrostatic stresses can thus be expressed
The mesoscopic stress triaxiality factor is defined as
where
8
=
Cll/C33
is the stress proportionality factor.
The mesoscopic strains are defined as logarithmic strains in the following manner
The corresponding mesoscopic effective strain is conveniently defined
2
Ve
=
I
7733

7711
I
2.2
Material descriptions
The matrix material is assumed to obey a uniaxial stressstrain relation of the following form:
ET

00
;
EL
~(5)
=
E

00
where
8
and
5
are the uniaxial stress and.strain, respectively,
00
is the uniaxial yield stress,
E
is
Young's modulus and
n
is the hardening exponent.
In
the analyses reported here, Young's modulus
was set to 70
GPa, the yield stress to 260 MPa and the hardening exponent to 0.1.
The particle was specified as purely elastic with a Young's modulus of 130
GPa and
a
Poisson's ratio of 0.33.
2.3
Stresses at the particlematrix interface
The interactions between the matrix and particle were modelled using the special contact formulation
in ABAQUS
[6]. In this approach, the contact problem is defined in terms of two surfaces, of which
one
is
denoted the master and the other the slave. If the two surfaces are brought into contact,
ABAQUS will automatically introduce the constraints required for forcing the nodes on the slave
surface to remain on the master surface as long
as
the contact conditions prevail. In the present
work, the contact was specified
as
permanently bonded. The advantage of using this procedure is
that the stresses at the interface are automatically calculated and given with components normal to
and tangential to the interface, respectively.
Figure 2a) shows the variation of normal stresses along the interface between a spherical
C6158 JOURNAL
DE
PHYSIQUE IV
0.4
0 10 20 30 40 50 60 70 80 90
Angular position (degrees)
a>
I
1
1.5
2
2.5
3
3.5
4
Pallicle aspect ratio,
W
b)
Figure
2:
a)
Variation of normal stresses at the interface of a spherical particle and matrix.
The effective mesoscopic strain is close to
5
%
b)
estimated values of ki for different
particle shapes
particle and the matrix. The angular position is referred to the radial direction,
i.e. an angular
position of
90
"
corresponds to the pole of the particle. The normal stresses are normalised with
respect to Argon's
[I]
estimate of the maximum interfacial normal stress
where
ki is
a
constant depending on the particle shape.
The maximum normalised stress is close to unity for all the stress states studied and it
remains so over a relatively wide range of deformation. The calculations therefore lend credibility to
the Argon model.
Similar calculations as those depicted in Figure
2
a) were performed for particles with aspect
ratios
W
(=
p3/p1) ranging from one to four. It was found that with an appropriate adjustment
of the k,factor, the Argon model reasonably well predicts the maximum normal stress over the
interface. However, for the spheroidal particles, the normalised stresses tended to vary more with
the strain level than was the case for the spherical particle. Figure
2
b) shows the estimated value
of the
kifactor as a function of particle shape.
2.4
Stresses in the particle crosssection
Figure
3
shows the distribution of normal stresses (~33) across the crosssection of particles with
aspect ratios ranging from
1
to
4.
The stresses are relatively constant over the crosssection, but the
stress level increases markedly with increasing aspect ratio. The stress raising effect of a superimposed
hydrostatic stress is also apparent.
The crosssectional normal stress in the centre of the particle was found to
be
conveniently
approximated by the following expression
where
kc is a constant. For all the calculations reported here, the kcfactor was found to be approx
imately
1.25
times ki as given in Figure
2b).
Figure 3b) shows the variation of the crosssectional normal stress with effective strain. The
stresses are normalised with those predicted by equation
(8).
As can be seen, the normalised stresses
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[...]
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Additional excerpts
...Nous trouvons aussi certaines incohérences, certains auteurs choisissent N C f f [81, 199, 203, 204], ce qui est incorrect d'un point de vue phénoménologique étant donné que le phénomène de coalescence ne peut précéder la nucléation des cavités....
[...]
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Abstract: The effects of void nucleation occurring during the deformation history on forming limit curves are considered for both inplane and punch stretching employing a constitutive model of a porous plastic solid. Both plastic strain controlled and stress controlled nucleation processes are simulated by a two parameter void nucleation criterion. For inplane stretching, plastic strain controlled nucleation can have, in certain circumstances, a significantly destabilizing effect on the forming limit curve. However, within the framework of plane stress theory which neglects the enhance ment of the hydrostatic stress due to necking, a stress controlled nucleation process is not found to be significantly destabilizing. In punch stretching a ductile rupture criterion, which limits the maximum volume fraction of voids, as well as the ap pearance of a well defined thickness trough, is adopted as a localized necking criterion. Only plastic strain controlled void nucleation is considered here in outofplane stretching. The resulting forming limit curves have the same shape as those obtained previously with void nucleation neglected.
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Abstract: The previously proposed conditions for cavity formation from equiaxed inclusions in ductile fracture have been examined. Critical local elastic energy conditions are found to be necessary but not sufficient for cavity formation. The interfacial strength must also be reached on part of the boundary. For inclusions larger than about 100A the energy condition is always satisfied when the interfacial strength is reached and cavities form by a critical interfacial stress condition. For smaller cavities the stored elastic energy is insufficient to open up interfacial cavities spontaneously. Approximate continuum analyses for extreme idealizations of matrix behavior furnish relatively close limits for the interfacial stress concentration for strain hardening matrices flowing around rigid nonyielding equiaxed inclusions. Such analyses give that in pure shear loading the maximum interfacial stress is very nearly equal to the equivalent flow stress in tension for the given state of plastic strain. Previously proposed models based on a local dissipation of deformation incompatibilities by the punching of dislocation loops lead to rather similar results for interfacial stress concentration when local plastic relaxation is allowed inside the loops. At very small volume fractions of second phase the inclusions do not interact for very substantial amounts of plastic strain. In this regime the interfacial stress is independent of inclusion size. At larger volume fractions of second phase, inclusions begin to interact after moderate amounts of plastic strain, and the interfacial stress concentration becomes dependent on second phase volume fraction. Some of the many reported instances of inclusion size effect in cavity formation can thus be satisfactorily explained by variations of volume fraction of second phase from point to point.
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