An analysis of the effect of hydrostatic pressure on the tensiledeformation of aluminum-matrix composites

C. Gonzalez, J. Llorca *

Department of Materials Science, Polytechnic University of Madrid, E.T.S. de Ingenieros de Caminos, 28040 Madrid, Spain

Received 17 December 2001; received in revised form 5 April 2002

Abstract

The effect of superposed hydrostatic pressure on the tensile deformation of particle-reinforced Al-matrix composites was analyzed

using a self-consistent approximation. The composite was represented in terms of an interpenetrating network of randomly

distributed spheres, which stand for the intact and damaged regions in the composite. Each sphere contained an intact or broken

ceramic particle at the center, and the model assumed that the fraction of damaged spheres increased during deformation. The load

partitioning between intact and damaged regions in the composite as well as the stress redistribution due to damage was computed

through a self-consistent scheme. It was shown that the tensile stresses in the ceramic particles, and thus the fraction of broken

particles, were reduced as the hydrostatic pressure increased. This led to a moderate improvement in the composite flow stress but

more significant gains were achieved in the strain at the onset of plastic instability. Both magnitudes increased with the hydrostatic

pressure until a saturation stress was reached. Particle fracture was completely inhibited at this point, and higher pressures did not

have any influence on the composite behavior, which was equivalent to that of the undamaged phase in the absence of hydrostatic

pressure. Using reasonable values for the matrix and reinforcement properties, the model predictions for the composite strength and

strain at the onset of plastic instability were in good agreement with the experimental data in the literature for high strength Al alloys

reinforced with SiC and Al2O3 particles.

# 2002 Elsevier Science B.V. All rights reserved.

Keywords: Hydrostatic pressure; Ductility; Mechanical behavior; Modeling; Damage; Composites

1. Introduction

Particle-reinforced Al-matrix composites exhibit very

poor tensile ductility (normally a few percent) and this

limitation has hindered their application in critical

structural components [1�/3]. However, various experi-

mental studies demonstrated that their fracture strain

increased dramatically when they were deformed under

superposed hydrostatic pressure [4�/8], and that the

composites were much more susceptible than the

unreinforced alloys to superposed hydrostatic stresses.

In particular, the fracture strain (determined from the

reduction of the area in the failure section) of the

unreinforced alloys increased two to four times as the

superposed hydrostatic pressure varied from 0.1 to 600

MPa; the composite fracture strain improved between

six and 20 times under the same conditions. As a result,

the fracture strain of the composites approached (and,

in some cases, exceeded) that of the unreinforced

counterparts when tested under superposed hydrostatic

pressure. This explains the good formability of particle-

reinforced Al-matrix composites by forging, extrusion

and rolling. These severe secondary processes enhance

the mechanical properties (especially the ductility) by

improving the spatial distribution of the reinforcement.

The analysis of the failure mechanisms showed that

voids were normally nucleated in the composite by

particle fracture. Liu and Lewandowski [6] tested up to

2% plastic strain tensile specimens of a 6061 Al alloy

reinforced with alumina particles. Afterwards they were

unloaded, sectioned, and the fraction of broken ceramic

particles was measured. It was reduced from 10 to 5% as

the hydrostatic pressure superposed on the axial stress

increased from 0.1 to 500 MPa. Similar results were

* Corresponding author. Tel.: �/34-913-365-375; fax: �/34-915-437-

845

E-mail address: jllorca@mater.upm.es (J. Llorca).

Materials Science and Engineering A341 (2002) 256�/263

www.elsevier.com/locate/msea

0921-5093/02/$ - see front matter # 2002 Elsevier Science B.V. All rights reserved.

PII: S 0 9 2 1 - 5 0 9 3 ( 0 2 ) 0 0 2 4 4 - 7

found by post mortem analyses of longitudinal sections

[7]. For instance, approximately 30�/40% of the ceramic

reinforcements were broken just below the fracture

surface in an underaged Al�/Cu�/Zn�/Mg alloy rein-forced with 15 vol. % SiC particles tested in tension at

atmospheric pressure. The fraction of broken reinforce-

ments was reduced by half when the material was tested

under 500 MPa of pressure.

Final fracture of the composites was induced by the

growth and coalescence of voids in the matrix, irrespec-

tive of the hydrostatic pressure levels. The kinetics of

void growth was, however, modified by the externalpressure. It is already well established that composites

tested in tension at atmospheric pressure fail with very

little necking, e.g. specimen fracture occurred immedi-

ately after the onset of plastic instability, and in fact, the

tensile ductility of these composites is very well pre-

dicted by the Considere criterion [9�/12]. In contrast,

composites tested under superposed hydrostatic stress

presented extensive necking, indicating that the processof void growth and coalescence in the matrix was slowed

down by the external pressure [4�/8]. These experiments

also showed that the yield and the tensile strength of the

composites improved when tested under hydrostatic

pressure, but this behavior was not observed in the

unreinforced alloys, whose flow behavior remained

insensitive to the stress triaxiality. The enhanced yield

stress could be due to two sources, namely the pressure-induced generation of dislocations around the reinforce-

ments in the composite, thereby providing local hard-

ening of the matrix, and the suppression of damage by

particle fracture.

Although the influence of hydrostatic stresses on the

mechanical response of the composites is well documen-

ted from the experimental viewpoint, quantitative pre-

dictions through realistic micromechanical models havenot been attempted. The only previous work in this area,

to the authors’ knowledge, was done by Christman et al.

[13] within the framework of axisymmetric unit cell

calculations. Their analyses used the Gurson model to

study the process of void growth and coalescence in the

matrix in the presence of hydrostatic stresses up to the

specimen failure. On the contrary, this paper emphasizes

the effect of superposed hydrostatic pressure on thetensile stress�/strain curve before the onset of plastic

instability. The analysis is based on a self-consistent

model previously developed [14], which can take into

account the stress redistribution induced by particle

fracture during deformation and the effect of stress

triaxiality on the composite flow stress. Quantitative

predictions for the composite strength and ductility as a

function of the hydrostatic pressure levels were obtainedfor reasonable values of the matrix and reinforcement

properties, and the model results were compared with

experimental data in the literature for a 2014 Al alloy

reinforced with 20 vol. % SiC particles.

2. Model of the composite behavior

The composite was represented in terms of an

interpenetrating network of randomly distributedspheres, which stand for the intact and damaged regions

in the composite (Fig. 1). The intact spheres represent

the composite before any damage and are formed by a

spherical reinforcement surrounded by the metallic

matrix. The damaged spheres exhibit the same geometry

but the spherical reinforcement was broken by a penny-

shaped crack perpendicular to the z -axis. The reinforce-

ment volume fraction was the same in intact anddamaged regions. Damage in the composite was given

by the fraction of the damaged spheres, r . Both phases

were assumed to behave as isotropic, elasto-plastic

solids following the incremental (J2) theory of plasticity.

Within the framework of the mean-field approach, the

composite stress and strain tensors, s and e; can be

computed from the volume-averaged values of the

stresses and strains in each phase according to:

s�(1�r)si�rsd and e� (1�r)ei�red (1)

where the subindexes i and d stand for the behavior of

the intact and damaged phases, respectively. Assuming

that all the stresses and strains in Eq. (1) as well as r are

known at a given instant of the analysis, the composite

strain hardening rate can be computed as [14]:

ds

d e�

�@s

@e

�r

��@s

@r

�e

�@r

@e

�(2)

and the composite behavior is determined incrementally

by integrating along the loading path the effective stress

hardening rate given by Eq. (2). The first term in this

expression stands for the hardening contribution if thevolume fraction of intact and damaged regions remains

constant, while the second term introduces the stress

Fig. 1. Composite representation as an interpenetrating network of

intact and damaged regions. Each region is formed by a spherical

reinforcement (either intact or broken by a penny-shaped crack)

surrounded by the metallic matrix.

C. Gonzalez, J. Llorca / Materials Science and Engineering A341 (2002) 256�/263 257

redistribution due to the damage of dr material when

the prescribed boundary conditions are held constant.

Thus, this expression for the stress hardening rate

assumes that two different mechanisms operate con-secutively during each infinitesimal strain increment.

The material initially deforms without any variation in

the volume fraction of each phase. The changes in the

stresses and strains in the intact regions then lead to a

small increment dr in the volume of damaged material,

and the elastic stress redistribution associated with this

process is taken into account by the second term of the

equation.The details of the analysis to compute both terms in

Eq. (2) can be found in [14], and only a very brief outline

is given here. Basically, the first term in Eq. (2) can be

computed using the classical self-consistent model

developed by Hill [15�/17] to simulate the elasto-plastic

deformation of multiphase materials with random

microstructures. The only difference between the classi-

cal approach and our analyses [14] was the use of anisotropic form (instead of anisotropic) for the elasto-

plastic tangent stiffness tensor of both phases. As

indicated by other authors [18,19], the classical incre-

mental self-consistent model based on the anisotropic

tangent stiffness tensor greatly overpredicts the compo-

site response. Nonetheless, this problem disappeared

with the isotropic form of the tangent stiffness tensor

[14], although this approximation is rigorously validonly under proportional loading.

The stress redistribution due to damage (second term

in Eq. (2)) was computed under two hypotheses. Firstly,

it was assumed that damage occurs very rapidly (as

compared with the strain rate) and thus the prescribed

boundary conditions remain constant. Secondly, da-

mage reduces the effective elastic constants and this

leads to an elastic stress relaxation in the effectivematerial, ds which can be calculated by derivation of

its elastic constitutive equation given by:

s�Leleel (3)

where s and eel stand for the composite stress and elastic

strain prior to any phase change.

Thus:

ds

dr�Lel deel

dr�

dLel

dreel (4)

where Lel is the elastic stiffness tensor for the two-phase

material, which can be computed from the elastic

properties and the volume fraction of each phase using

the self-consistent method. Eq. (4) represents a set of

equations in which the terms of ds=dr and deel=dr

corresponding to prescribed boundary conditions arezero. This is the case of the derivative of the normal

stresses in the y and z directions, of all the tangential

stresses, and of the longitudinal strain in the z direction

when the material is strained along the z axis under

superposed hydrostatic pressure. The components of

dLel/dr necessary to solve the set of equations, can be

obtained by the procedure given in [14].

3. Constitutive equations of each phase

The self-consistent model assumes that the intact and

damaged regions in the composite material stand for

isotropic, elasto-plastic solids whose behavior is char-

acterized by two elastic constants and by the yield

surface, which depends on the plastic strain. They were

determined from numerical simulations of the uniaxial

deformation of an axisymmetric cylindrical cell contain-ing one ceramic sphere (either intact or broken) at the

center. This approach has been extensively used ([20�/

22]) to model the tensile deformation of a three-

dimensional array of hexagonal cells containing one

particle (either intact or broken) at the center, which can

be regarded as an approximation to the spherical

regions containing the particles depicted in Fig. 1.

The finite element discretizations of one half of theaxisymmetric unit cells are shown in Fig. 2, and more

details of the finite element analyses can be found

elsewhere [22], The ceramic spheres were assumed to

be linear elastic and isotropic, and the elastic constants

of SiC (450 GPa and 0.17) were used in the numerical

simulations. The matrix was modeled as an isotropically

hardening elasto-plastic solid following the incremental

(J2) theory of plasticity. The Al elastic constants (70GPa and 0.33) were selected, and the relation between

the matrix flow stress, sm; and the corresponding plastic

strain, em; was represented by the power-law equation:

sm�Aenm (5)

Values of A (�/600 MPa) and n (�/0.1) representa-

tive of high strength Al alloys were chosen for the

analyses presented in this paper. The unit cells depicted

in Fig. 2 were subjected to unaxial deformation along

the z axis, and the evolution of the flow stress as afunction of the plastic strain in the intact and damaged

axisymmetric cells is plotted in Fig. 3. These curves were

used as the constitutive equations for the intact and

damaged regions in the composite (independently of the

hydrostatic stress level), although it should be noted that

the damaged regions are not isotropic. This limits the

current application of the self-consistent model to

loading paths where damage occurs during uniaxialdeformation along the z axis, the ceramic spheres being

always broken by cracks perpendicular to the loading

axis.

C. Gonzalez, J. Llorca / Materials Science and Engineering A341 (2002) 256�/263258

4. Damage criterion

A damage criterion for particle fracture has to be

provided to determine the evolution of damage during

deformation. Experimental observations (see [22] and

references therein) have demonstrated that the particle

strength follows the Weibull statistics and thus the

volume fraction of broken particles, r , can be computed

as:

r�1�exp

���sp

s0

�m�(6)

where sp is the average tensile stress acting on the

ceramic particle in the z direction and m and s0 stand,

respectively, for the Weibull modulus and characteristic

strength of the Weibull distribution. They were esti-

mated by quantitative microscopy, and typical values

for SiC reinforcing particles are in the range 3B/m B/6

and 1 GPa5/s05/2 GPa [23�/25].The actual magnitude of the sp as a function of the

axial strain in the intact phase, ezi was also computed

through the axisymmetric model of the intact cell (Fig.

2(a)). The axisymmetric unit cell was initially subjected

to hydrostatic pressure and then deformed in tension

along the z axis. The results obtained for different levels

of superposed hydrostatic pressure, plotted in Fig. 4,

were used to compute the fraction of broken reinforce-ments at each step of the incremental analysis.

Thus, the incremental procedure to compute the

composite stress�/strain curve was the following. Start-

ing from a previous step, where the stresses and strains

in both phases as well as their respective volume

fractions are known, the new stresses and strains in

both phases after an increment Dez in axial deformation

were computed using the first term in Eq. (2). The axialstrain increased in the intact phase, and the new stresses

acting on the reinforcements were determined from the

curves in Fig. 4. Then, the corresponding increase in the

Fig. 2. Finite element discretization of one half of the axisymmetric unit cells used to compute the constitutive equations of the intact and damaged

regions in the composite, (a) intact cell, (b) damaged cell.

Fig. 3. Constitutive equations for the intact and damaged phases in

the composite. They were obtained through the finite element

simulation of unaxial deformation of the axisymmetric unit cells in

Fig. 2. The matrix properties are also plotted for comparison.

C. Gonzalez, J. Llorca / Materials Science and Engineering A341 (2002) 256�/263 259

volume fraction of the damaged phase, Dr , wasdetermined from Eq. (6), and was used to compute the

stress relaxation due to damage using the second term in

Eq. (2).

5. Results and discussion

The composite, containing 20 vol. % of intact ceramic

spheres, was initially subjected to a uniform hydrostatic

pressure, and then uniaxially strained along the z axis.

Tensile stresses developed in the ceramic particles during

uniaxial deformation, which led to particle fracture. Theself-consistent model*/described above*/was used to

compute the composite behavior, taking into account

the presence of intact and damaged regions as well as the

stress redistribution due to the progressive particle

fracture. Typical values of the Weibull parameters

were used for the particle strength distribution (m�/3

and s0�/1.2 GPa). The evolution of the composite flow

stress, s; with the equivalent plastic strain, e; is plottedin Fig. 5(a) for superposed hydrostatic pressures be-

tween 0 and 600 MPa1. The corresponding strain

hardening rates, ds=de; also plotted, show that the

composite hardening capacity increased with the hydro-

static pressure. This result cannot be attributed to the

behavior of the intact and damaged regions because the

Von Mises yield criterion is insensitive to the hydrostatic

stress component, and the differences in hardening ratewere due to the progress of damage during deformation.

This is shown in Fig. 5(b) where the fraction of broken

reinforcements, r , is plotted as a function of the

equivalent plastic strain for the materials depicted in

Fig. 5(a). The superposed hydrostatic pressure reduced

markedly the tensile stresses in the particles, and

following Eq. (6), damage by particle fracture practically

disappeared in the composites tested under 600 MPa of

hydrostatic pressure.

Once the composite stress�/strain curve has been

computed, it is possible to determine the ultimate tensile

strength, su; as the stress at the onset of plastic

instability, which can be obtained from the Considere

criterion according to:

Fig. 4. Average stress in the ceramic reinforcements,/sP; as a function

of the axial strain in the intact phase, ezi . The results are plotted for

different levels of superposed hydrostatic pressure, sh.

Fig. 5. Effect of superposed hydrostatic pressure on (a) composite flow

stress and strain hardening rate, (b) fraction of broken reinforcements.

The Weibull parameters for Eq. (6) are given in the figure.

1 Evidently, the composite flow stress under axisymmetric loading is

given by sz�/sx , where sz is the longitudinal stress, and sx �/sy is the

hydrostatic stress.

C. Gonzalez, J. Llorca / Materials Science and Engineering A341 (2002) 256�/263260

s�ds

de(7)

if the dilatational strain associated with the fracture of

the reinforcements is negligible. Recent analyses [22]

have demonstrated that the composite tensile strength,

su; and the strain at the onset of plastic instability. eu;are proportional, respectively, to the matrix tensile

strength and strain hardening exponent. In addition,

su and eu depended mainly on two parameters, s0/A and

m , which controlled the development of damage by

particle fracture during deformation. Following these

previous findings, the model predictions for the compo-

site strength su (normalized by the matrix strength,

smu �/Ann) are plotted in Fig. 6 as a function of thesuperposed hydrostatic pressure for reasonable values of

s0/A and m . Similarly, the composite strain at the onset

of plastic instability, eu; (normalized by the matrix strain

hardening exponent, n ) is plotted in Fig. 7 for the same

values of s0/A and m . This figure shows that su and eu

increased with the superposed hydrostatic pressure until

a saturation stress was reached. Particle fracture was

completely inhibited at this point, and higher pressuresdid not have any influence on the composite behavior,

which was equivalent to that of the undamaged material

in the absence of hydrostatic pressure. The figures, also

show that the influence of hydrostatic stresses on the

composite properties increased as the particle/matrix

strength ratio decreased. Hydrostatic stresses modified

the ductility more than the flow stress, and this is in

agreement with the influence of particle fracture on

these properties. Composite hardening is maximum at

low strains, precisely the regime where damage is

minimum, and thus the composite flow stress is only

mildly affected by damage. On the contrary, the

ductility*/according to the Considere criterion*/is

dictated by the composite hardening rate at medium

and high strains, which is highly dependent on the

particle fracture rate.

The trends in Fig. 6 for the effect of superposed

hydrostatic pressure on the composite flow stress are in

good agreement with the experimental results compiled

by Lewandowski and Lowhaphandu [8]. Basically, they

found that hydrostatic pressure increased the yield and

tensile strength of the Al�/SiC and A1�/Al2O3 compo-

sites by an amount similar to that in Fig. 6. Moreover,

the improvement in flow stress was mainly observed for

low levels of hydrostatic pressure, and saturation was

reached for pressures in excess of 700 MPa. The model

results can be compared with the experimental data of

Fig. 6. Effect of hydrostatic pressure on the composite tensile strength,

su (normalized by the matrix strength, smu ) for different values of s0/

A and m .

Fig. 7. Effect of hydrostatic pressure on the composite strain at the

onset of plastic instability, eu (normalized by the matrix strain

hardening exponent, n ) for different values of s0/A and m .

C. Gonzalez, J. Llorca / Materials Science and Engineering A341 (2002) 256�/263 261

Vasudevan et al. [4]. who measured the stress�/strain

curve up to the onset of plastic instability at different

levels of superposed hydrostatic pressure in a 2014 Al

alloy reinforced with 20 vol. % SiC particles. As the

matrix properties were not detailed in the paper, the

experimental values of the composite strength and

ductility were normalized by those obtained in the

absence of hydrostatic stresses (Fig. 8). The eu predic-

tions for m�/4 and s0/A�/2.0�/2.5 are in good agree-

ment with the experimental data in the whole range of

hydrostatic stresses (Fig. 8(b)), whereas the strength

predictions followed the experimental data up to 400

MPa of hydrostatic pressure (Fig. 8(a)). Beyond thispressure, the strength predictions underestimated the

experimental data, and the difference may be attributed

to the nucleation of dislocations around the reinforce-

ments for hydrostatic pressures over 500 MPa, which

increased the matrix strength. Transmission electron

microscopy studies have confirmed that this can occur

provided that the pressurization generates shear stresses

large enough near the reinforcements [26,27].Finally, it should be noted that the improvements in

eu with hydrostatic pressure shown in Fig. 7 (by a factor

of two or three at most) are well below many experi-

mental data in the literature [5�/8], which reported

increments of between six and 20 times. The discrepancy

is due to the definition of ductility, which is taken as the

fracture strain in tension (as determined from the

reduction in the area at the failure section) instead ofthe plastic strain at the onset of plastic instability. The

difference between the two parameters is minimum in

the absence of superposed hydrostatic stresses because

the process of void growth and coalescence in the

matrix, which leads to the specimen rupture, is very

fast in these composites once the deformation has begun

to localize in a given section. As shown by numerical

simulations [28], the constraint imposed by the stiffceramic particles on the matrix plastic deformation leads

to the development of tensile hydrostatic stresses in the

matrix at the local level. They accelerate the void growth

and coalescence in the absence of a neck, which

generates the triaxial stresses required to promote void

growth and coalescence in the unreinforced alloys. The

scenario changes under superposed hydrostatic pressure,

which counterbalances the local hydrostatic stresses inthe matrix, and reduces the driving force for void

growth and coalcescence. Large plastic strains have to

be applied beyond the onset of plastic instability to grow

the voids in the matrix, leading to the formation of a

deep neck in the composite specimen.

The analysis of the kinetics of void growth and

coalescence in the matrix in the presence of hydrostatic

stresses is beyond the scope of this paper, but it wasstudied by Christman et al. [13]. As in the analyses

depicted in Fig. 2(a), the composite was made up of a

three-dimensional array of hexagonal cells containing

one intact reinforcement at the center, and the tensile

response of the composite was simulated in the context

of unit cell calculations. It was assumed that the void

nucleation in the matrix was controlled by the plastic

strain, and the subsequent growth and coalescence wasdictated by the Gurson plastic potential. Their simula-

tions showed that superposed hydrostatic pressure

delayed the process of ductile rupture in the matrix,

and thus modified significantly the composite strain to

Fig. 8. Influence of hydrostatic pressure on the tensile properties of a

2014 Al alloy reinforced with 20 vol. % SiC particles. Experimental

results from [4]. (a) Tensile strength, suh, normalized by the tensile

strength at atmospheric pressure, suatm; (b) Strain at the onset of plastic

instability, euh, normalized by the corresponding value at atmospheric

pressure, euatm.

C. Gonzalez, J. Llorca / Materials Science and Engineering A341 (2002) 256�/263262

failure but not the flow stress, which was fairly

insensitive to hydrostatic stresses.

6. Conclusions

The effect of superposed hydrostatic pressure on the

tensile deformation of particle-reinforced metal-matrix

composites was analyzed. The composite was repre-

sented in terms of an interpenetrating network of

randomly distributed spheres, which stand for the intact

and damaged regions in the composite. The modelassumed that damage during deformation occurred by

particle fracture, the particle strength following the

Weibull statistics. The load partitioning between intact

and damaged regions in the composite as well as the

stress redistribution due to damage was computed

through a self-consistent scheme. It was shown that

the tensile stresses in the ceramic particles, and thus the

fraction of broken particles, were reduced as thehydrostatic pressure increased. This led to a moderate

improvement in the composite flow stress but more

significant gains were achieved in the strain at the onset

of plastic instability, as determined by the Considere

criterion. Both magnitudes increased with the hydro-

static pressure until a saturation stress was reached.

Particle fracture was completely inhibited at this point,

and higher pressures did not have any influence on thecomposite behavior, which was equivalent to that of the

undamaged phase in the absence of hydrostatic pressure.

Using reasonable values for the matrix and reinforce-

ment properties, the model predictions for the compo-

site strength and strain at the onset of plastic instability

were in good agreement with the experimental data in

the literature for high strength Al alloys reinforced with

SiC and Al2O3 particles.

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