Predictive model for void nucleation and void growth controlled ductility in quasi-eutectic cast aluminium alloys

Acta Materialia 53 (2005) 2739–2749 www.actamat-journals.com

Predictive model for void nucleation and void growth controlled ductility in quasi-eutectic cast aluminium alloys G. Huber

a,b

, Y. Brechet b, T. Pardoen

c,*

a

Max-Planck Institut fu¨r Metallforschung, Heisenbergstr. 3, D-70569 Stuttgart, Germany LTPCM-ENSEEG, Domaine Universitaire de Grenoble BP75, F-38402, Saint Martin dÕHeres, France De´partement des Sciences des Mate´riaux et des Proce´de´s, Universite´ catholique de Louvain, IMAP, Place Sainte Barbe 2, B-1348 Louvain-la-Neuve, Belgium b

c

Received 18 November 2004; received in revised form 14 February 2005; accepted 15 February 2005 Available online 8 April 2005

Abstract A micromechanical model for the ductility of plastically deforming materials containing a homogeneous distribution of brittle inclusions is developed and applied to quasi-eutectic cast aluminium alloys. The model includes a micro–macro void nucleation condition, the initial penny shape nature of the voids, and the growth and coalescence regimes. The model is validated by comparing the predictions to experimental results obtained under different levels of stress triaxiality and for different heat treatments controlling the hardness of the matrix. The material parameters are directly identified on the microstructure. A parametric study on the ductility in uniaxial tension of materials with penny shape voids demonstrates the complex couplings existing between the void nucleation condition, the state of hardening, the strain hardening capacity and the particle volume fraction.
2005 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Casting; Aluminium alloys; Ductility; Micromechanical modelling

1. Introduction By contrast with wrought alloys, the optimisation of cast alloys is mainly driven by their processability, i.e., resistance to hot tearing, ability for mould filling. Comparatively, properties such as ductility or fracture toughness and their relationship to the microstructure have up until now received less attention [1–3]. Nevertheless, the limiting factors for the use of cast alloys are often their fatigue resistance and their ductility [4,5]. These properties depend on the microstructure resulting from the solidification conditions and further heat treatments, but the quantitative relationship is far less understood than is, for instance, the yield stress [3]. This lack of understanding delays the improvement of casts, and *

Corresponding author. Tel.: +32 10 475134; fax: +32 10 474028. E-mail address: [email protected] (T. Pardoen).

undermines their application for structural critical components [4]. The quality of castings (gas content, porosity) primarily controls the ductility and fatigue properties [4], but when this quality is ensured owing to the continuous improvement in casting techniques, the influence of microstructural features, such as the eutectic phase or the precipitation achieved with additional heat treatments, becomes the key issue. The present paper deals with a predictive model for the ductility of cast aluminium alloys, as a function of microstructural features resulting from solidification and further heat treatments, in order to guide the development of more ductile alloys. From a general point of view, this study is part of a recent effort pursued by several teams which aims at improving the micromechanics of ductile fracture based on new advances in the modelling of the nucleation, growth and coalescence of voids, see, e.g. [6–12].

1359-6454/$30.00
2005 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.actamat.2005.02.037

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Quasi-eutectic alloys of the 300 series provide, from the fundamental viewpoint, the following advantages:
It is a sort of natural ‘‘in situ’’ composite with perfectly elastic particles (eutectic silicon) which are, for the quasi-eutectic composition, relatively evenly distributed. Initial damage always occurs by particle fracture.
The plastic properties of the matrix in this composite can be varied independently from the features of the reinforcing particles with an appropriate heat treatment; here, two ageing conditions were chosen, namely T6 and A. Changing the hardening of the matrix has a very marked effect of void nucleation. Several new developments have been required in order to address the fracture mechanisms in this material, involving (i) a proper account of the initial penny shape of the damage and of its shape evolution, and (ii) the implementation of a physical void nucleation condition. These new developments, motivated by in situ tensile tests in the SEM, and added to other recent enhancements of the classical Gurson model [8,10,12,13] will be validated from the results of the experiments performed on both T6 and A alloys. The identification of the microstructural parameters entering the model will be directly based on metallographic analysis and in situ testing. The validation is essentially provided by the comparison of the model with the results of mechanical tests performed at different level of stress triaxiality for the two different heat treatments considering that the conditions for void nucleation remain identical. The paper is structured as follows. Section 2 introduces the materials and the experiments for the microstructural characterisation, the identification of elementary mechanisms, the fractographic analysis and the characterisation of the mechanical behaviour. Section 3 presents the damage model. After careful identification of the parameters of the model, the predictions for the ductility are compared to the experimental data for T6 and A alloys in Section 4 which ends with a systematic parametric study about the influence of the microstructure on ductility.

2. Materials and experiments 2.1. Materials The alloy selected for this study was an Al–Si–Mg base alloy containing 11% Si and 0.3% Mg. A very small amount of Sr (134 ppm) was added as a modifier of the eutectic in order to favour a more rounded shape of the eutectic silicon particles. The Fe and Mn content were, respectively, 0.12 and 1.029 wt%. These elements lead to brittle intermetallic particles. After homogenisation

and quench, an ageing treatment results in a fine precipitation of Mg2Si in the aluminium matrix. A low temperature treatment leads to a peak aged state labelled T6 providing the maximum hardness of the aluminium matrix surrounding the silicon particles. Another heat treatment of equivalent duration but at higher temperature leads to a coarser precipitation microstructure resulting in an overaged state, labelled A, which has a significantly smaller yield stress than the T6 condition. More details about the processing conditions and microstructure are provided in Appendix A. The various heat treatments are summarised in Table 1. 2.2. Characterisation of the microstructure and fracture mechanisms The microstructural characterisation had three complementary aims: (i) to quantify the relevant features of the eutectic structure; (ii) to analyse the sequence of fracture events occurring under mechanical loading; (iii) to provide, through a qualitative analysis of the fracture surface, a comparison with the scenario proposed from the in situ SEM observations (see Section 3). (i) Analysis of the eutectic structure with SEM. In both alloys (A and T6) the mean dendrite arm spacing was equal to 42 lm. From the viewpoint of fracture mechanisms, the incidence of the state of fine precipitation is mainly through its influence on yield stress and work hardening of the matrix surrounding the eutectic particles. It will only be quantified through mechanical tests (see further). By contrast, since silicon particles in the eutectic phase are the nucleation site for damage, a careful characterisation of their size, aspect ratio and volume fraction was conducted using image analysis methods, see Table 2. (ii) In situ tensile testing and fractography. In situ tensile tests on flat polished specimens were carried out within an SEM in order to observe the sequence of Table 1 Heat treatment conditions for the T6 and the A heat treatments T6

A

10 h at 540 C (solutionizing) Quenching into cold water 6 h at 170 C Air cooling

10 h at 540 C (solutionizing) Quenching into cold water 6 h at 300 C Air cooling

Table 2 Quantitative characteristics of the eutectic particles

T6 A

Rp (lm)

Mp (lm)

mp (lm)

Wp

Lp (lm)

fp (%)

3.4 3.4

4.8 4.8

2.5 2.5

1.9 1.9

5 4.55

14 15

Rp is the average radius defined as the square root of the average area; Mp and mp are the average major and the minor axis, respectively; Wp is the average aspect ratio of the particle; Lp is the average interparticle distance; fp is the particle volume fraction.

G. Huber et al. / Acta Materialia 53 (2005) 2739–2749

mechanisms leading to fracture. The sequence of events in the damage accumulation process in both A and T6 materials are summarised in Fig. 1. The two materials, A and T6 present qualitatively a similar behaviour: the first step is cracking of the silicon particles, giving birth to a penny-shaped crack. With increasing deformation, one observes further particle cracking and crack opening. Progressively the cavities nucleated from the initial cracks grow, resulting in the blunting of the initial penny shape crack and finally coalesce. This progressive damage accumulation leads to final fracture. The only obvious difference between the two materials is in the nucleation step: in the T6 sample, particle cracking occurs simultaneously with the onset of plastic yielding, whereas it occurs much later in the softer material A

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in which significant surface relief and slip bands are observed before any nucleation events. The analysis of the fracture surfaces was performed with an SEM using, on the same observation area, both chemical X-ray analysis and secondary electrons for evidencing the chemical features, and backscattered electrons for the relief. The observations of the fracture surface (Fig. 2) show a typical ductile fracture pattern with equiaxed dimples inside which one detects the cracked Si particles. 2.3. Mechanical testing The room temperature uniaxial true stress–true strain curves of the A and T6 materials obtained using smooth

Fig. 1. Micrographs taken from in situ tensile tests describing microstructure and the different stages of the damage process.

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Fig. 2. Fracture surfaces observed in chemical composition contrast revealing Si particles, and in backscattered electrons; (a,b) for the T6 materials; (c,d) for the A materials.

cylindrical specimens are shown in Fig. 3. The crosshead speed was equal to 0.5 mm/min. The yield stress of the T6 material at r0 = 234 MPa is almost three times larger (MPa) 400

300

T6

200 A 100

0 0

0.02

0.04

0.06

0.08

0.1

0.12

Fig. 3. Uniaxial true stress–true strain curves: (a) T6; (b) A.

than the yield stress of the A material, r0 = 87 MPa, while their strain hardening capacity is very similar (see Fig. 3 which shows only a small difference in the necking strain). Tensile tests on notched round bars with two sizes of the notch radius (see Fig. 4) were also performed. The reduction in diameter of the minimum cross-section was continuously measured by means of a radial extensometer. Both the initial and final (i.e., after fracture occurred) diameters of the minimum cross-section were measured using a travelling microscope. The measurement of the final diameter is useful for correcting possible errors in the measurement of the current diameter, essentially due to the initial positioning of the extensometer which is never perfectly located at the minimum crosssection. Two or three tensile tests were performed at room temperature for each geometry and material. The ductility ef is defined as the average longitudinal strain at fracture
 D0 ef ¼ 2 ln ; ð1Þ Df

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3. Microstructure-based modelling 3.1. Physical model

Fig. 4. Dimensions of the cylindrical notched round specimens.

where D0 and Df are the initial and final diameters of the minimum cross-section. The ductilities of the T6 and A samples are reported in Table 3. In order to classify and compare the specimens in terms of the stress state, the best indicator in the case of ductile fracture is the stress triaxiality ratio [14] defined as the ratio T ” Rh/Re, where Rh is the hydrostatic stress and Re the von Mises equivalent stress. The stress triaxiality history in the most damaged region of the smooth and notched specimens, which is always the centre of the minimum cross-section, was computed using the finite-element program ABAQUS 5.8 [15] by simulating the loading of the full specimen geometry. An average value of T is given in Table 3 for each alloy and geometry. In the case of uniaxial tension, T is exactly equal to 1/3 before necking and then steadily increases. In the T6 alloy, fracture occurs after a limited amount of necking and the mean stress triaxiality is thus not much larger than 1/3. In notched round specimens, the stress triaxiality is essentially constant with plastic deformation (at least for moderate plastic strains) in the centre of the minimum section of the specimen. As expected [14,16], the ductility significantly decreases with increasing triaxiality. The T6 material is more than three times less ductile than the A material whatever the stress triaxiality level.

Table 3 Average stress triaxiality T and ductility ef for the two alloys and the three types of tensile test specimens Uniaxial

T6 A

AE1.2

AE6

T

ef

T

ef

T

ef

0.35 0.45

0.18 0.62

0.73 0.68

0.11 0.37

1.19 1.21

0.048 0.215

Several assumptions or approximations can be motivated from the analysis of the sequence of events observed during the in situ tensile tests, see Section 2.2 and Fig. 1, in order to develop a model for the damage and fracture process: (i) The initial void configuration consists of flat, pennyshape microcracks resulting from the fracture of the Si particles. Neglecting the differences between the dendritic and interdendritic regions, which is reasonable in the quasi-eutectic composition, the initial microstructure is idealised by regularly distributed spheroidal particles with initial spacing Lpx = Lpy and Lpz and radii Rpx = Rpy and Rpz (see schematic picture in Fig. 1). The dimensionless geometrical parameters are vp = Rpx/Lpx, Wp = Rpz/Rpx, and kp = Lpz/Lpx. As no orientation effects are present in the material, the parameter kp is taken equal to 1. The particle volume fraction fp can be expressed in terms of vp, Wp and kp. Penny-shape voids of initial radii Rx0 = Ry0 and Rz0 nucleate when the stress in the particle is large enough for inducing particle fracture [17,18]. The initial void shape W0 = Rz0/ Rx0 is obviously extremely small as the critical crack opening in Si is on the order of a few nanometres and particle diameter is several micrometers. The initial void volume fraction f0 is related to the particle void volume fraction fp by f0 ¼ W 0 fp :

ð2Þ

The presence of a small amount of initial porosity in the matrix resulting from the casting is not taken into account (see the recent [19] which involves a simulation of the effect of pores on the stress transfer within the Si particles). (ii) A critical stress based nucleation criterion is first motivated by the observation that void nucleation in the T6 state (high yield stress) occurs early in the deformation process while it happens much later in the annealed state (low yield stress). Secondly, Si is a brittle material whose fracture resistance is thus essentially controlled by the size of the largest inner defect. (iii) The fracture process is also controlled by a stable void growth stage as can be deduced from the significant opening of the microcracks (see Fig. 1). The damage process is thus intimately related to both the void dilatation and the change of the void shape. (iv) The void coalescence process (or void linking process) leading to small cracks is stable and controlled by plastic deformation (Fig. 1). This also means that the tearing resistance of the alloy constituting the matrix between the Si particles is good. This fracture process is for instance very different than the process occurring in some AA5XXX Al–Mg alloys where the ductile fracture

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resistance is controlled by void nucleation. In these alloys, void nucleation is rapidly followed by an unstable void linking process limiting the amount of plastic flow [8]. The purpose of the next sub-section is to present the micromechanical model that has been developed in order to meet all these requirements. 3.2. A model for the nucleation, growth and coalescence of voids 3.2.1. Before void nucleation The material is assumed to be initially perfectly dense and homogeneous. The response of the material before voids start to nucleate is modelled using the isotropic J2 flow theory. This is a reasonable assumption considering that no crystallographic or morphological texture is expected in these cast alloys which have an equiaxed solidification structure. 3.2.2. Void nucleation The nucleation of voids by cleavage of the Si particles is assumed to occur when the maximum principal stress in the particle reaches a critical value, denoted as rc max rparticle ¼ rc : princ

ð3Þ

A critical maximum principal stress corresponds physically to the critical stress in the particle required to reach the critical stress intensity factor for the propagation of the sub-micron defects present within the particles [17,18]. Following the Eshelby theory [20] and the ‘‘secant modulus’’ extension to plastically deforming matrix proposed by Berveiller and Zaoui [21], the maximum max principal stress in an elastic inclusion rparticle can be princ related to the overall stress state (described with the Cauchy stress tensor r) by the following expression: max rparticle ¼ rmax princ þ kðre  r0 Þ; princ

ð4Þ

where rmax princ is the maximum overall principal stress, re is the overall effective von Mises stress, r0 is the yield stress of the matrix, and k is a parameter of order unity which is a function of the inclusion shape and loading direction. For the fracture of spherical inclusions, k ranges between 1 and 2. This void nucleation criterion was initially proposed by the Beremin group [22]. Considering that the fracture stress of brittle particles decreases when the particle size increases due to a statistically larger maximum defect size, the rate of increase of the void volume fraction associated to the nucleation of new voids will be a function of the plastic strain rate: p f_ nucl ¼ gð
ep Þ
e_ ; p

ð5Þ

where
e_ is the effective plastic strain rate of the matrix material. A polynomial form for the function gð
ep Þ is chosen

6

4

2

gð
ep Þ ¼ a1
ep þ a2
ep þ a3
ep þ a4 :

ð6Þ

When criterion (3) is fulfilled, with the corresponding effective plastic strain noted
ep ¼
epc , the nucleation starts and takes place during a range of strain D
ep ; rc is thus the critical fracture stress of the largest particles. Nucleation continues on the smaller particles when increasing the matrix hardening in a range of strains D
ep . For numerical convenience, the parameters ai are chosen in such a way as to avoid discontinuities in the porosity evolution: both gð
epc Þ and gð
epc þ D
ep Þ as well as their first derivative are taken equal to 0. 3.2.3. Void growth As soon as voids have started to nucleate, the accumulation of plastic deformation causes the enlargement of the voids and an increase of the void volume fraction which, by stating volume conservation, writes f_ growth ¼ ð1  f Þ_epii ;

ð7Þ

e_ pij

where are the ij components of the overall plastic strain rate tensor. The full evolution law for the void volume fraction is written as f_ ¼ f_ growth þ f_ nucl :

ð8Þ

The initially flat voids progressively open with plastic flow leading to a change of the void shape index parameter S = ln W = ln (Rz/Rx) while assuming that the void remain spheroidal with current void radii Rx = Ry and Rz. The evolution law for S is
p  _S ¼ 3 ð1 þ h1 Þ e_ p  e_ kk d : P þ h2 e_ p : ð9Þ kk 2 3 This evolution law has been derived by Gologanu et al. [7] from a micromechanical analysis of the growth of a spheroidal void in a J2 perfectly plastic material. This analysis leads to analytical expressions for h1, h2 as a function of S and f (for the full expressions see [6,7,10]). P is a projector tensor, defined by ez  ez and ez is a unit vector parallel to the main cavity axis; d is the Kronecker tensor. For the sake of simplicity, we will assume here that the main void axis ez does not rotate. The hardening behaviour of the matrix material is related to the overall stress and plastic strain rate through the energy balance initially proposed by Gurson [13]: p

ry
e_ ð1  f Þ ¼ r_ep ;

ð10Þ

where the mean yield stress of the matrix material ry is a function of
ep : ry  hð
ep Þ. In order to calculate the evolution of the three variables f, S and ry as well as the stress state r, use is made of an associated flow rule:

G. Huber et al. / Acta Materialia 53 (2005) 2739–2749

e_ pij ¼ c_

oU ; orij

ð11Þ

where U is the flow potential proposed by Gologanu et al. [7] for a porous materials involving spheroidal voids. This flow potential is written as
g 2 C r U  2 r0 þ grgh X  þ 2qðg þ 1Þðg þ f Þ cosh j h ry ry 2

2

 ðg þ 1Þ  q2 ðg þ f Þ ¼ 0;

ð12Þ

where r 0 is the deviatoric part of the Cauchy stress tensor; rgh is a generalised hydrostatic stress defined by rgh ¼ r : J ; X is a tensor associated to the void axis and defined by 2/3 ez  ez  1/3 ex  ex  1/3 ey  ey; J is a tensor associated to the void axis and defined by (1  2a2)ez  ez + a2ex  ex + a2ey  ey; i i is the von Mises norm; C, g, g, j, a2 are analytical functions of the state variables S and f, q is a parameter that has been calibrated has a function of f0, W0, and n (see [7] for complete expressions). The stresses are calculated from r_ ij ¼ Lijkl ðe_ kl  e_ pkl Þ;

ð13Þ

where the Lijkl are the elastic moduli. This void growth model has been worked out and validated in [7,10] with proper extension to strain hardening.

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3.3. Computational aspects The constitutive model described in the previous section has been implemented in the general purpose finite element code ‘‘ABAQUS Standard’’ through a User defined MATerial subroutine (UMAT) [15] with a fully implicit integration scheme based on the return mapping algorithm [24] for each steps of the deformation and damage process (i.e., for the J2 flow theory before nucleation, for the extended Gurson model and for the coalescence response). When the void nucleation criterion is fulfilled in one integration station, the parameters ai of the void nucleation function (6) are calculated. The numerical procedures for the rest of the model integration have been described elsewhere [10]. Note that the uniaxial response ry  hð
ep Þ is introduced in the model point by point to allow an exact match with the experimental curve. Very refined meshes are used for all the simulation with typically 50 fully integrated four-noded axisymmetric elements on half the cross-section.

4. Assessment of the model and discussion 4.1. Identification of the parameters

3.2.4. Coalescence The process of void growth by more or less homogenous plastic deformation of the matrix surrounding the voids is, at some point, interrupted by the localisation of the plastic flow in the ligament between the voids. This point corresponds to the onset of coalescence. Thomason [23] proposed the following condition for the onset of coalescence: sffiffiffi# "
2 rz 2 1v 1 2 ¼ ð1  v Þ a þb ; ð14Þ v expðSÞ v ry 3 where the two parameters a and b have been fitted in as a function of the average value of the strain hardening exponent n: a(n) = 0.1 + 0.22n + 4.8n2 (0 6 n 6 0.3) and b = 1.24, see [8]; v is the relative void spacing equal to Lx/Rx. The relative void spacing is a new variable entering the problem which of course is essential in controlling the localisation of plasticity in the intervoid ligament. For the sake of simplicity, we have assumed in Eq. (14) that coalescence occurs in a band normal the main void axis (which is assumed to remain parallel here to the main loading axis z). A detailed modelling of the coalescence process (e.g. [8,12]) after it has been initiated is with little consequence for the present problem because the fracture strain almost corresponds to the onset of coalescence in the most damaged region (i.e., the extra overall deformation required after the onset of coalescence to break the specimen is small).

The parameters required for predicting the ductility with the model developed in the previous section are (i) the flow curve, (ii) the initial state of damage characterised by f0, S0 and v0, and (iii) the void nucleation parameters rc and D
ep . (i) The flow curve of the matrix ry  hð
ep Þ is approximated by the overall flow curve shown in Fig. 3 thus neglecting the effect of the silicon particle on the material strength. (ii) The initial porosity f0 is related to the particle volume fraction fp through Eq. (2). Physically the initial void being almost flat, one should use f0 ! 0 and W0 ! 0 (or S0 ! 1). As shown in Fig. 5, a preliminary set of calculations for pure uniaxial tension conditions (i.e., without necking) with various initial void aspect ratio W0 at constant volume fraction of particles fp demonstrates that, to a very good approximation (and obviously within the limit of validity of the model), the ductility can be considered as independent of the initial void aspect ratio W0 (when keeping f0 proportional to W0) when W0 is typically lower than 0.02. Hence, in order to avoid numerical problems appearing at very low f0 and W0 the initial shape W0 has been taken equal to 0.01 and f0 is thus equal to 0.01fp for all the calculations performed in this study. The particle volume fraction fp is equal to 15% and 14% for materials A and T6, respectively, leading to f0 = 1.5 · 103 and 1.4 · 103 for materials A and T6, respectively. Finally, the distribution of particles being isotropic, v0 = (3f0/4pW0)1/3.

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G. Huber et al. / Acta Materialia 53 (2005) 2739–2749

0.3

0.2

ε

f

f = 14%

0.1

p

f =W f 0

0 p

0 0

0.5

1

W0

Fig. 5. The ductility ef is independent of the initial void shape W0 for low W0 at constant particle volume fraction fp. The example is given here for flow properties typical of the studied aluminium alloys (i.e., n = 0.1 and r0/E = 0.002).

(iii) Considering the relatively small dispersion in particle size in the present material, a small value of the parameter D
ep equal to 2% was arbitrarily chosen. The only ‘‘free’’ parameter to be calibrated is thus the critical stress for void nucleation rc. (This calibration will also account for the uncertainty in the determination of D
ep ). The critical stress rc has been identified as equal to 550 MPa based on the response of the material A in uniaxial tension, i.e., by finding the value that reproduces the experimental ductility. The effect of the factor k in the range 1–2 has been tested and no major effect has been observed: rc changes by no more than 100 MPa when increasing k from 1 to 2. The value k = 2 was chosen for this study. It is worth mentioning that modelling the ‘‘A’’ Al alloy without accounting for delayed void nucleation, i.e., using rc = 0, underestimates the ductility by more than a factor of two. For the sake of qualitatively justifying the critical stress estimated above, one can refer to two very simple arguments. First of all by performing the energy balance between the cracked and uncracked particle, the fracture condition can be written
 r2 4pR3 ð15Þ > 2Gc ðpR2 Þ; 2E 3 where Gc is the critical energy release rate. For Si, E = 110 GPa, Gc  8 J/m2 while an approximation of R can be taken from Table 2 as 4 lm, which gives rc = 1100 MPa. That is factor of two larger than the calibrated value. Note also that this p simple approach gives a fracture stress which scales as 1/ R: the larger the particles, the earlier the crack will nucleate. (This relationship combined with the experimental dispersion of particle sizes, the void nucleation condition (4) and the flow curve ry  hð
ep Þ offers a means to estimate D
ep . The value D
ep will also depend on the stress state because the void nucleation condition (4) not only depends

on the effective stress but also on the principal stress value. In the present problem, this extra complexity is definitely not necessary but it might be more important in more heterogeneous and less ductile materials.) Second, by taking the solution for a mode I crack running parallel to the diameter and the axis of plain cylinder [25] and applying it for a mode I crack in a plain sphere, one can estimate the crack length corresponding to an imposed rc = 550 MPa with Gc  8 J/m2 as equal to about 100 nm which does not seem unrealistic for a typical defect in the Si particles. Except for a small difference between the initial porosities, the microstructure and nucleation parameters are essentially similar for both conditions T6 and A. These two materials only differ by their flow properties. 4.2. Assessment of the model The tensile tests on the smooth and notched specimens for each condition (A and T6) were simulated using the parameters identified in the previous section. Fig. 6 presents for both the A and T6 materials the comparison between the experimental and predicted ductility as a function of the mean stress triaxiality in the most damaged region. Good agreement is obtained considering the experimental scatter and the fact that only one parameter, rc, has been adjusted. For the case of the T6 treatment, rc = 550 MPa leads to very early nucleation as also observed experimentally while nucleation occurs much later in the A material (at a strain equal to more than half the fracture strain). In other words, depending on the state of hardening in the matrix

0.8

experimental model

0.6

A f

0.4

0.2

T6 0 0. 4

0. 8

1.2

T Fig. 6. Variation of the ductility as a function of the mean stress triaxiality in the most damaged region, comparison of experiments and modelling for: (i) the A heat treatment using fp = 15%, W0 = 0.01, rc/r0 = 6.3 and the proper flow properties in uniaxial tension; (ii) the T6 heat treatment using fp = 14%, W0 = 0.01, rc/r0 = 2.3 and the proper flow properties in uniaxial tension.

G. Huber et al. / Acta Materialia 53 (2005) 2739–2749

the ductility of these materials can be controlled by both the void nucleation and void growth regimes. None of these two stages can be neglected. The number of parameters to be calibrated is kept to a minimum also by the use of a physical void coalescence condition (14) which avoids the use of a critical porosity parameter whose status is very much debated in the literature (see, e.g. [7,26,27]). It is important to mention that if exactly the same analysis is attempted with a regular version of the Gurson model for spherical voids growing spherical, one will need to calibrate an effective initial porosity f0. Two tests would, thus, be necessary to tune both rc and f0. Furthermore, as there is a clear coupling between the stress triaxiality and the void shape (see the terms entering the hyperbolic cosines in Eq. (12)), nothing guarantees that a model for spherical voids would allow the encompassing of the range of stress triaxiality addressed here.

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2 void nucleation strain

n = 0.1 W = 0.01 0

1.5

ε

f

σc/σ0 = 6

1

σc/σ0 = 4.5 0.5

σc/σ0 = 0

σc/σ0 = 3

0 0

0.1

0.2

(a)

where E/r0 is taken equal to 1000 and n is taken equal to 0.1 or 0.25 to cover relatively low and high strain hardening capacity of typical Al matrix.
The critical stress for void nucleation rc/r0 is varied between 0 and 6.
The particle volume fraction fp is varied between 0.01 and 0.5, while the initial voids have a penny shape (W0 = 0.01) resulting from the fracture of the particle. The initial void volume fraction is given by Eq. (2). Fig. 7(a) shows the evolution of the ductility as a function of the particle volume fraction for a relatively low strain hardening material (n = 0.1) and different void nucleation stress. The strain corresponding to nucleation is also indicated. Let us define the ‘‘void growth strain’’ as the strain increment required to bring freshly nucleated void to the coalescence condition. For large particle volume fraction, the void growth strain is always small and the ductility is mostly controlled by the

0.5

void nucleation strain

W = 0.01 0

1.5


The flow properties of the matrix are represented by
n r ep ¼ 1þE ; ð16Þ r0 r0

0.4

2

n = 0.1 σ /σ = 6

4.3. Parametric study The model has been used to derive general trends about the ductility of materials showing a sequence of damage mechanism similar to the cast Al alloys studied in this work. The goal of this section is to provide a better understanding of the couplings between the resistance to void nucleation, the state of hardening, the strain hardening capacity, and the volume fraction of second phase particles. Uniaxial tensile tests (involving necking) have been simulated using the model presented in Section 3 with the following mechanical and microstructural characteristics:

0.3

fp

c

ε

f

0

1 n = 0.1 σ /σ = 0 c

0

n = 0.25 σc / σ0 = 0

0.5

n = 0.25 σ /σ = 6 c

0

0 0

(b)

0.1

0.2

0.3

0.4

0.5

fp

Fig. 7. Parametric study about the variation of the ductility of materials with initial penny-shape voids (W0 = 0.01) as a function of the particle volume fraction fp for various void nucleation stress rc/r0: (a) for a strain hardening exponent n = 0.1; (b) comparison for two different strain hardening exponents (n = 0.1 and 0.25).

nucleation step: nucleation is rapidly followed by void coalescence because the relative spacing between the voids is quite small. For lower particle volume fraction, the ductility is controlled by void growth when the void nucleation stress is not too high. For large void nucleation stress, the void growth strain is much lower than for low void nucleation stress leading again to a void nucleation controlled ductility. This last intriguing effect has two origins. The first reason is that voids nucleated late in the deformation process are much closer to each other (because of the lateral contraction) than if they were nucleated earlier, and thus much closer to the coalescence condition. The second reason is an artefact of the uniaxial tensile test and of the necking process. A void nucleated early undergoes first a low stress triaxiality, equal to 1/3 before the onset of necking, and elongates significantly without any lateral growth before being subjected to an increasing stress triaxiality associated to the development of the neck. The elongation

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of the voids decreases the stress concentration in the ligament between the voids, delaying the onset of void coalescence (see condition (14)). Voids nucleated much later, when the necking is already well developed, are directly subjected to a moderate stress triaxiality and moderate tensile stress while still very flat which is favourable to the onset of void coalescence (see again Eq. (14))). This artefact originating from the complex stress state history of uniaxial tension with necking shows again that, as advocated from a long time by several groups [14,26], notched specimens with constant stress triaxiality should be preferably chosen for ductility assessment. Fig. 7(b) compares the evolution of the ductility as a function of the particle volume fraction for materials presenting two different strain hardening capacity (n = 0.1 and n = 0.25), for two different nucleation stress (0 and 6). Increasing the strain hardening has two opposite effects: (i) by raising the stress more rapidly, a larger strain hardening capacity tends to accelerate nucleation; (ii) by delaying the onset of necking and by giving rise to shallower necks, it decreases the mean stress triaxiality and thus slows down the void growth rate. For immediate nucleation (low nucleation stress), a large strain hardening exponent leads to higher ductility while for large nucleation stress, a large strain hardening brings about lower ductility. Note that it is possible with Al alloys to change the strain hardening by changing the temperature without affecting the yield stress, hence the ratio rc/r0.

5. Conclusions The present model for ductility of a quasi-eutectic cast aluminium alloy is to be seen as a step in a general attempt to account for the relationships between solidification and heat treatment parameters, microstructures, and damage mechanisms. It has evidenced the complex role of the nucleation growth and coalescence of voids in the control of the ductility. The importance of the ‘‘penny shape’’ crack geometry which initiates the damage process has been emphasised. The key parameters controlling the ductility have been identified as (i) the elastoplastic behaviour of the matrix material, mainly the yield stress and the work hardening behaviour and (ii) the void nucleation stress. A number of improvements of the model can be readily envisaged:
Improvements in the stochastic aspect of the nucleation step, which would allow consideration of the size effect and the size distribution of the eutectic particles (see [28,29] for more detailed approaches of the nucleation stage in cast Al alloys).
Improvements on the constitutive law for the matrix, with a better description of the local work hardening behaviour.


Incorporation of the effect of the surrounding particle on the void growth rate law.
Analysis of the cavity nucleation and growth kinetics in a non-homogeneous distribution of inclusion, or non-homogeneous plastic behaviour of the dendrite interior. This situation is reminiscent of the fracture of alloys with precipitate free zones studied previously [30]. These improvements are the next step to address hypoeutectic alloys, effects of modifiers as Sr or P, effects of solidification processes setting the scale of the microstructure. From a more general viewpoint, accounting for the initial penny shape of the freshly nucleated voids allows for linking the initial void volume fraction to the particle volume fraction, avoiding the introduction of an effective porosity parameter not directly related to a microstructural quantity. Many other materials involving brittle second phases will present a similar mechanism of particle fracture leading to initial flat voids, e.g., dual phase steels, WC–Co alloys, AlSiC composites. Note that the approach remains valid for partial interface decohesion as observed in AlSiC composites for instance. The present model, thus, offers a means for predicting the effect of large particle volume fraction on the damage mechanisms.

Acknowledgements The authors thank Dr. J.C. Ehrstrom and Dr. G. Laslaz for enlightening discussions. This project was partly supported by PECHINEY and a Franco Belgian Tournesol collaborative Program as well as the German Academic Exchange Service, DAAD.

Appendix A. Materials and heat treatments The ductility of cast aluminium alloys is known to depend strongly on the characteristics of the eutectic phase, on the iron content, and on the porosity. Accordingly, the alloy composition and solidification procedure were carefully controlled:
The silicon content is 10.77 wt%. Regarding the alloying, Mg was added in the form of pure Mg and Sr in the form of AlSr10. Mg is added in order to provide the structural hardening (0.29 wt%) and Sr was added to the alloy (134 ppm) as a modifier of the eutectic in order to promote more rounded shapes of the eutectic silicon. The Fe and Mn content are, respectively, 0.12 and 1.029 wt%. They provide brittle intermetallics, but their role is negligible compared to the influence of the major damage nucleation sites which are the silicon eutectic particles.

G. Huber et al. / Acta Materialia 53 (2005) 2739–2749


The furnace was held at a temperature of 750 C and degassing was performed with 5 l Ar/min for 10 min immediately before casting. Hydrogen in liquid aluminium sometimes causes porosity in the cast samples since the solubility of hydrogen in solid Al is much smaller (decreasing with about one order of the magnitude) than in liquid aluminium according to Engh et al. [31]. The expected porosity was revised by a density test. The enormous effect of a decreased porosity after degassing was observed already macroscopically. The initial porosity of all samples was estimated to be less than 2%.
The mould was preheated using a gas flame and by additionally casting one ‘‘dummy’’ just in advance of the sample used for the spectrochemical analysis. The materials used for heat treatment were cut from a region where the amount of porosity was very low. The solution heat treatment has been carried out in an air circulation furnace. The solid solution heat treatment temperature was above the solvus temperature and time was sufficient to ensure complete dissolution of Mg-containing particles. A consequence of the solid solution treatment is the spheroidisation and coarsening of the eutectic particles. Following solutionizing, the samples were quenched in a water bath at room temperature. After quenching, the T6treated samples were aged at 170 C for 6 h whereas the A-treatment comprised an aging at 300 C for the same duration.

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