An iterative approach to mechanical properties of MMCs at the onset of plastic deformation

MATERIALS SCIENCE & ENGINEERING ELSEVIER

Materials Science and Engineering A229 (1997) 203-218

A

An iterative approach to mechanical properties of MMCs at the onset of plastic deformation A. Roatta, P.A. Turner, M.A. Bertinetti, R.E. Bolmaro * Instituto de Fisica Rosario, Facultad de Ciencias Exactas, Ingenieria y Agrimensura, CONICET-UNR, Universidad Nacional de Rosario, By. 27 de febrero 210 bis, Rosario 2000, Argentina Received 9 June t996

Abstract The current work presents a generalized Eshelby model allowing interaction among reinforcing particles under a Mori-Tanaka like scheme. Different aspect ratios and geometries are studied in the elastic and incipient elasto-plastic regime for a model SiC-A1 composite. The fibers are taken as purely elastic and the matrix is regarded elastic perfectly plastic responding to a Von Mises yield criterion. The phenomenon of plastic localization in the vicinities of the inclusions is carefully described for different reinforcement volume fractions and thermo-mechanicai loading. Equivalent stress, hydrostatic pressure and elastic and plastic strains are depicted as contour levels around a representative inclusion. Effective coefficients of thermal expansion of composites are calculated both under purely elastic composite response and at the onset of plastic localized deformation. The influence of plastic strain over those effective coefficients is shown to be detectable. The simulated stress-strain curves show the influence of interaction stresses over macroscopic yield stress by isolating this phenomenon from matrix hardening. Accumulated elastic energies and plastic work are calculated to show the different nature of purely thermal and mechanical loads. © 1997 Elsevier Science S.A. Keywords: Eshelby model; Von Mises yield criterion; Deformation

1. Introduction The presence of thermal residual stresses (TRS) in discontinuous metal matrix composites (DMMCs) is due to the difference in coefficients of thermal expansion (CTE) between reinforcements and matrix and the fact that the composite is cooled from the processing or annealing temperatures. Sometimes, an intermediate temperature is important due to the occurrence of some phase transformation that may create second phase inclusions or modify their size and/or shape. A portion of stresses generated by the CTE mismatch might relax by plastic deformation and the accumulation of dislocations and diminishing subgrain size can become extra mechanisms for composite strengthening [1-4]. Fully plastic D M M C s stress-strain curves are always above the corresponding pure matrix curves. Although, there is an early bulk plastic yielding detectable at stresses lower than the pure matrix yield stress, either in trac* Corresponding author. 092t-5093/97/$17.00 © 1997 Elsevier Science S.A. All rights reserved.

tion or compression, shown as deviation from the linear response. Moreover the difference between traction and compression test curves is a two-phase simile Bauschinger effect. The early non linear behavior is due to a microscopic local yielding effect happening in the reinforcement neighborhood owing to the elastic inhomogeneity represented by the reinforcement embedded in the matrix (elastic constants mismatch). Plastic early relaxation is detected as a slope change in the stressstrain curves in the elasto-plastic regime and it is shown as a decrement of the stiffness. It has been shown that plastic localization can also provide extra strengthening mechanisms by raising matrix work hardening and preventing failure by debonding or cracks [5]. The strengthening mechanisms presented by these composites show some subtleties and anomalies not well understood until now [6,7]. In the particular case of SiC/At composite, the ratio between composite and aluminum matrix yield stresses has been explained by the additive effect of at least three mechanisms. Dislocation density strengthening has been approached by

204

A. Roatta et at./Materials Science and Engineering A229 (t997) 203-218

Arsenault and Shi [3] by punching of prismatic dislocations due to TRS. Taya and Mori [8] and Taya et al. [9] have also attempted to explain strengthening by that geometrically necessary dislocation accumulation but Arsenault [10] showed it is also necessary to introduce the concept of diminishing subgrain size strengthening. Both mechanisms together with the non relaxing TRS due to CTE mismatch and elastic inhomogeneity can account for the dramatic increment, sometimes referred to as synergic, in composite yield stresses under some thermo-mechanical treatment conditions [9]. A great deal of discussion has emerged about experimental and theoretical interpretation of those different mechanisms [7,11,12]. Dislocation punching and subgrain size decrement effects can be partially precluded to operate by some heat treatment and TRS mechanism modelization can be tested in a SiC-A1 composite when this sole mechanism can account for yield stress raising. The onset of plastic deformation is not clear from experiments and it has been rarely approached by theoretical means. It is expected a great influence of the amount of localized plastic relaxation either in the values of effective CTE's or TRS accumulation: The elastoplastic transition of composite materials is accepted to happen in two stages: (i) the matrix yields and a slope change appears in the stress-strain curve and (ii) the reinforcement yields until failure happens by debonding, void nucleation and growth or crack propagation. The process of plastic yielding during the first stage is very complicated. The matrix plastic yielding is not massive but localized in the reinforcement vicinities and the stress-strain curve slope changes gradually as fast as new plastic matrix regions are incorporated. Plastic yielding in the reinforcements can be completely absent and the second stage is reduced to failure and debonding as soon as enough stresses are accumulated mainly in the interfaces. The purpose of the current paper is the study of the first elastoplastic transition by continuum mechanics. We will also study the plastic relaxation due to CTE mismatch and cooling. The study will be conducted for different inclusion shapes and varying volume fractions. The plastic work due to CTE mismatch and cooling can give some hints about dislocation density increment and consequent strengthening. The enduring TRS will show the effective influence of cooling over the composite strengthening. Hardening due to prismatic dislocation punching will not be addressed in the current paper but in a coming one [13]. Section 2 will show a brief of an Eshetby type model extended to include local plastic relaxation. Detailed explanation of the model has been given elsewhere to which the reader is referred for further details [5]. Section 3 will show the main features of the interaction model developed to include finite volume fraction of particles effects as well as plastic localized strain. The

interaction scheme will allow to calculate effective CTE, both with and without plastic relaxation, for different reinforcement concentrations and inclusion aspect ratios. The Eshelby scheme has been previously used, under the assumption of homogeneous plastic relaxation acting as a transformation strain in a finite volume fraction composite [14,15]. We will show that some experimental results are better explained by plastic localization rather than by homogeneous plastic strain. The present innovation allows the plastic relaxation to be calculated as a field variable respectful of inclusion geometry. The results will be shown in Section 4. The influence of finite volume fraction in the local behavior and the expected stress-strain curves will be presented. In Section 5 we present conclusions and we discuss some experimental results previously published in the literature.

2. The Eshelby model extended to include plastic strain

Let D be an infinitely extended material, with the elastic moduli Cijkl containing an inhomogeneous inclusion t-l, with the elastic moduli C*kv The material is subjected to a uniform applied stress Z and a non-uniform, non-elastic strain e n~ and the inclusion has an arbitrary shape. Hooke's' law is written as: +

= C%(Ekj +

--

Vxsa

(1) (2)

where E is the total strain at infinity and g is the strain disturbance. The non-elastic strain is uniform in the inclusion where plastic deformation is not permitted and has two terms in the matrix. One is originated in thermal expansion, uniform but different from the inclusion strain, and the other is plastic. That is, e "~ = Ez1 V x ~

e n~= e TM + ~PI(X)

(3) Vx~D

-

f~

Actually, the plastic deformation is extended only to the vicinity of the particle. At infinity the matrix remains elastic. Eq. (1) and Eq. (2) must satisfy the equilibrium equation in the absence of body forces, ~ijj = 0, and continuity of displacement and traction at the inclusion-matrix interface. Following Johnson and Lee [t6], Johnson [17] and Roatta and Bolmaro [5], by using the formalism of the elastic Green's functions, we arrive to the following integral equation for the strain disturbance f:

A. Roatta et al./Materials Science and Engineering A229 (I997) 203-218

205

constants of the matrix and the inclusion, respectively, and A S = $ * - $. The coefficients A and B are defined by

=~ ACijk~Eiif (G~,~ q-Gm,km)dV'

A, ij(X, Xo)

+ l ACijklfa (Gml,k,.+ Gnl,knOgijdV'

! dV' (Gm~,k~(X-X)¢ + G,,,km(X -- X))

~--1 AGkX ~

2

- -2 "ijklCij

1

BI j(X, Xo)

(Gm ,k + G~l,km)d V'

+E 2

Jg2 (~ml,kn -{'- 6nl,knO d r '

Cijkl fM (~mi,kn-{- ~ . l , k t 0 ~ 1 dV'

= 1 CijkI ~

2

(4)

where the identity aX'kOX'~Gml(X--X) = axk3x~Gml ( X - X ' ) has been used and 6Im and a ( X - X ' ) are the K_ronecker and Dirac delta functions, AC = C* - C, M denotes the plastic zone of the matrix and the comma denotes differentiation respect to the X' components. In the elastic case (e,~l = 0), Eq. (4) reproduces the Eshelby solution for strain disturbance and in the elasto-plastic case it is solved numerically by using a selfconsistent and iterative method. Even though the plastic zone limits are not fixed a priori we can consider that, for each iteration step, the limit is known and we performed the derivation taking M as constant. The iterative technique requires initial estimates for #ij and e~i, which were taken as the elastic solution for the fit'st one and zero for the second one. Furthermore, we must discretize the integration domain and select a mesh for the matrix and particle, which depends on the geometry of the inclusion. The strain is considered constant on each mesh element and the accuracy of the results depends on the mesh size. It is necessary to select a yield criterion and a plastic law to determine the size and shape of the plastic zone and the plastic strain of the matrix. They are unknowns of the problem, together with the disturbance strain ~. In our case we adopt Von Mises yield criterion, the associated Prandtt-Reuss flow rule and ideal plastic behavior. The integrals of Eq. (4) are approximated by summation of integrals over the mesh elements. From here on the inclusion is assumed elliptical. In order to work with Eq. (4), we reorder it to obtain firstly a linear system for points inside the inclusion

gm.(X)--

X,~

A. j(X, Xo) ij(Xo)

= - - ASrrmijEij --k Smnij~ij * TI

-

(Gm~kn(X -- X')

+ GnI,~(X- X')) d V'

(6)

where Vo is the volume of the mesh element Xo. Some combinations of A and B coefficients will need to be calculated under special assumptions of symmetry for different inclusion geometries. Secondly, for points in the matrix (X~D - f2), Eq. (4) is reduced to:

~m~(X)

=

-

ADm~ij(X)E0 @

+

D mnij (X)%TI

- Dr j(X)%TM

Am j(X,XO
-- E Bmnij(X, Xc)G~l(x¢)

(7)

XceM

where D(X) and D*(X) are the Eshelby tensors for points outside the inclusion [18] calculated using C~j~ and Cij~a, * respectively; and AD(X) = D*(X) - D(X)

(8)

In each iteration the strain in the precipitate is obtained through the resolution of a non homogeneous linear system (Eq. (5)) whose coefficients are integrals, over the mesh elements, of tensorial products of second derivatives of elastic Green's functions and elastic constants of inclusion and matrix. The independent terms depend on the amount of plastic strains assigned to each element. With the strain already calculated inside the inclusion we calculate the strains and equivalent stresses in the matrix, being the last ones compared with the yield stress of the matrix. Plastic strain is assigned to each mesh element by: o'<~(X) - Cry pl Ae~q(X) = p E

(9)

where: pl Ae~q(X). difference between plastic equivalent strains in the (n + 1)-th and n-th iteration for the mesh dement

x,

-- SnmijE'ijTM

Z B j(X, Xo)P,;(Xo)

Jv~

(5)

XcEM

where Xo is the center of the mesh element, $ and $* are the Eshelby tensors [18] calculated using the elastic

O-~q)(X): equivalent stress in the n-th iteration, O-y: yield stress of the matrix, E: Young's modulus of the matrix and p: proportionality factor that takes values dose to refit. For calculating the plastic strain components we employ the Prandtl-Reuss relation:

A. Roatta et at./Materials Science and Engineering A229 (1997) 203-218

206

_o~_ 3 d%z~ , delj - - - _ aij 20eq

(10)

50 6

where oij is the deviatoric stress tensor. Eq. (10) can be incrementally written in the following manner: pl

e~t('+ :>(X) = e~l(")(X)+ 3 Ae~P~(X)zlj(.)(X )

(11)

O

2~ .=_ g 4

/A--A--A--A--A--A--&--A--A--A

*/'~/0--0--0--0--0--0 --0--0--0--0 /A~/~-×-×-×-×-×-×-×-×-×

(,9

0

"1" e,

-50

o

o

where (n) and (n + 1) identify the (n)-th and (n + t)-th steps respectively and X identifies the position of each calculation cell of the discretized space. In some elements the stresses are well beyond the yield surface, particularly in the initial steps for high thermal or mechanical loads. Notwithstanding, when a criterion expressed by Eq. (11) is applied, it provides fast and smooth convergence. No limitations imposed by the high ratio between elastic constants used in the present work have been found, other than the number of iterations. The convergence is reached when the equivalent stress reproduces the assumed ideal plastic law for all X e M . The elements will be included among the ones plastically relaxing whenever the equivalent stress exceeds the yield stress, and they will be taken out of the plastic region just when the plastic equivalent strain is negligible or slightly negative for the current iteration step. Convergence (measUred as relative difference between Von Mises equivalent stresses and yield stress) is achieved after a number of iterations going from 1-20. The higher the initial stress compared with the yield stress the greater the number of iterations. The error, calculated as the difference between stresses for two successive steps, can be reduced below almost any arbitrarily imposed limit, but the actual limitation must be judged by comparison with analytic results and inspection of the interface behavior where we expect stronger demands over the model. Typical errors of 1-3% were tolerated and the iterative scheme was terminated when we reached those values. Analytical solutions can be obtained for stress, displacements, strains and energy associated with a misfitting spherical precipitate, inclusion, inhomogeneity or second hard phase, provided that the load, either thermal or mechanical, also keeps radial symmetry. Comparison between the results of the current Eshelby extended model and the analytical one due to Lee et al. [19] shows perfect agreement in the elastic region. The plastic area is determined within an error of half mesh Table 1 Elastic constants, thermalcoefficientsand yieldstresses Material

AI

SiC

Ref.

v p, (OPa) (°C-1" 106) O-y (MPa)

0.33 25.68 23.6 97.0

0.17 182.48 4.3 --

[9]

g

Sphere

K E

2

2

1

-100

--A--20%

"--- " ~ %

E "0 -150 ~

111 0 1.0

"~--~-~-i~-g--t~-~-N-N-~ 1.5

2.0

2.5

-200

da

Fig. 1. Von Mises equivalent plastic strain and hydrostatic pressure as a function of radial distance from particle center in units of a (particle radius), for 0, 10 and 20% volume fraction of reinforcements. Spherical particles subject to 200°K cooling.

size and the amount of plastic strain is reasonably well calculated and the stresses, once they have been relaxed by local plastic yielding, superpose almost perfectly with the analytical model [5].

3. Interaction m o d e l

In the present section general concepts about internal stresses and interaction among inclusions in a composite material, originally developed by Mori and Tanaka [20], will be reviewed. Let D be the domain occupied by a heterogeneous material. The average of the internal stresses in D always satisfies, by definition and independently of the origin of the internal stress field, the condition -~ l f D 5ij d D = <~ij)D = 0

(12)

where V is the volume of domain D and the internal stresses were defined as 5ij = o'ij- D= o'ij- Xij

(13)

where #ij is the local stress deviation from their average. In Eq. (12) the notation for the spatial averages was also introduced. When D is an infinite domain and the internal stresses arise from the presence of an inhomogeneity with domain f2 included in :l), the internal stress field is given by the solution of the Eshetby problem for the inhomogeneous inclusion, in the case of a pure elastic behavior, or by the extended Eshelby scheme (Section 2), when local plastic deformation occurs in the matrix. This solution will be called c?~ whether plastic strain is happening or not. The integral in Eq. (12) is written as:

A. Rostra et al./ Materials Science and Engineering A229 (I997) 203-218

207

Table 2 Interaction stresses for different volume fractions and inclusion geometries Inclusion shape (2a: minor or unique axis)

AT-- --200°K Misfit strain 0.39%

10% 20%

E ~m (MPa) Eint (MPa)

E = 0.90 ay

10% 20%

E int (MPa)

E int (MPa)

Sphere

Oblate

~11

0"22

13.8 33.4

13.8 33.4

1.49 2,97

-~ l f D a~]°dD= l f n a { d D + - p l f D _ a : ~ d D = 0

G33

13.7 33.1

1.49 2.97

(14)

where the integral over the entire domain D has been split in two parts: the integral over the inhomogeneity domain and the integral over the matrix domain (that is the complementary domain). When domain D is infinite the relative spatial dimension of the inhomogeneity to the spatial dimension of the matrix is null. The meaning of Eq. (14) is that the presence of the inhomogeneity produces internal stresses that are not null in its own domain and in a volume of the matrix that is an infinitesimal of the whole domain D, but the contribution of these stresses to the average is null. When the domain D is finite, Mori and Tanaka [20] showed that the integral over the matrix domain is null for the case of a matrix that has a shape and an orientation equal to the ones of the inhomogeneity. In such a case the average of the internal stresses over D is written

::o

p

~

dD =-~

3 2 dD

(15)

but now the Eshelby solution does not satisfy the condition given by Eq. (12). The solution for the finite domain problem can always be written as the superposition of the solution for the infinite domain problem plus the so called image stress field. This last term accounts for the presence of a surface close to the inhomogeneity. The internal stresses are: Gij = e ~ + o'i~'

(16)

and its average is, replacing Eq. (16) in Eq. (12) and, applying the Mori and Tanaka theorem (Eq. (t5)): -~ 1 fD eij dD = l f n # g d D + p1 fD O'ij imdD=0

(17,

which allows the average of image stresses over the domain D to be calculated as: gl ,m\/ D = -- "g ( # g ) n O'ij

(18)

The above stated solution can be extended to the case of a material having a finite concentration of inhomogeneous inclusions. The presence of the inclusions in the domain D will produce a deviation of the stress in

-4.55 --9.1

(4x4x I)

~11

~22

28.1 52.8

28.1 52.8

4.90 8.65

4.90 8.65

Prolate (1 x 1 x 4) ~33

5.3 10.5 -- 1.16 -2.0

~II

~22

10.7 21.5

10,7 21.5

0.52 0.82

0.52 0.82

G33

38.4 61,9 -16,0 -27.2

the matrix. Mori and Tanaka [20] showed that the local internal stress in the matrix is equal to the average of the stresses plus local fluctuations due to the closest inclusions. The average in the matrix domain of these local fluctuations is null. In such a case the average of internal stresses in the matrix domain, M, is: (~ij)ra = - f (~?2)a

(19)

where f is the volume fraction of inhomogeneities. This average can be thought as the interaction stresses among inhomogeneities, that is, the effect produced over one inhomogeneity by other inhomogeneities. In such a way, the interaction is taken into account via a mean field approach. It is necessary, in this approach, to know the stress inside the inhomogeneity that corresponds to the solution for the infinite domain problem for determining the interaction stress (image stress, see Eq. (18)). To solve this problem it is possible to introduce a new inhomogeneous inclusion in D, assuming that the number of inclusions is so high that the presence of a new one will not appreciably modify the volume fraction f This inclusion will be subjected to the external boundary condition plus the effect of the other inclusions represented by an interaction strain that is given in Eq. (19). The problem to solve is Xij -1- E!.nt --:J -I- G~~ = C~kl(Ekl + ffkl int -]- ffkI ~*e -- e T ) V X ~ ' ~

Xij + - . + ¢i~ = Cij~(Zkl+ ~ yj.~t = Cijkl(Ekl + Xij "1- __.

T ~kJ:

e~ t) in FD (boundary condition) (20)

When plastic relaxation is allowed in the matrix, Eq. (20) has to be modified to include an extra plastic strain term, e ~ st, as it was written in Eq. (2). For both cases the interaction stress is defined by 5"~int

0 = -f(gi~)a

(21)

The system represented by Eq. (20) and Eq. (21) can be solved by iterative methods, among others. The iteratire method assumes an initial state without interaction, •int= 0, and no plastic cells. Once the due cells have been allowed to yield, an iteration step is performed and --g '}-~fintis calculated. A new loop checks again every cell looking for over shots in applied stresses and plastic

A. Roatta et al./ Materials Science and Engineering A229 (I997) 203-218

208

strain is corrected whenever necessary. Once the interaction stress remains constant the iteration process is terminated. Iteration (n + 1) is related with iteration (n) by the following equation: el~*("+ ') = - f {Ci]-k~@~'(~))n -- £>a -- ei~t(~)}

(22)

J I

¢q LO ¢0 II L_

4. Results and discussion

/

11

-¢0

I

I !

I

b,-

4.t. Local plastic yielding by thermal load

¢0 0

By simple cooting, the matrix develops local plastic yielding in the vicinity of the inclusion. The isolated !.0 %

8 max. 16.5x104 7

at r=3.52a e=0.17

~

O0

"l l l l

6

I l l

C'q

5

U

O

e/z

3

Fig. 3. Von Mises equivalent plastic strain contour levels induced by 200°K cooling around one representative oblate inclusion with an aspect ratio given by 4 x 4 x t, for 20% volume fraction, in units of 10 -3. r and 0, dotted line and _a are defined as in Fig. 2.

0

1

2

y/a Fig. 2. Von Mises equivalent plastic strain contour levels induced by 20WK cooling around one representative prolate inclusion with an aspect ratio given by 1 x i x 4, for 20% volume fraction, in units of 10-3. r and 0 are spherical coordinates and the zenithal angle 0 is measured from the z=axis. The outer dotted Iine shows the limit of the meshed region and a represents the shortest semiaxis.

inclusion in an infinite matrix has already been analyzed in two previous papers [5]. We will address here the problem of the interaction among plastically rigid particles distributed in a finite elasto-plastic matrix. Three inclusion geometries will be addressed in the calculation: (a) Prolate ellipsoids (fiber shape) with an aspect ratio given by 1 x 1 x 4 aligned with the largest axis in the z direction; (b) Spherical particles and (c) Oblate ellipsoids (disk shape) with an aspect ratio given by 4 x 4 x 1 with the shortest axis in the z direction. The interaction is taken into account via a mean field theory by using the models developed in Section 2 and Section 3. Varying volume fractions for different inclusion geometries will be compared in order to study the

A. Roatta et at./Materials Science and Engineer#zg A229 (1997) 203-218

effect of both phenomena, plastic relaxation and interaction, over the composite mechanical properties. Inclusion volume fractions higher than 20% can be found in the literature but its theoretical study by using a mean field theory approach, does not make sense because beyond 20% volume fraction of inclusions the plastic regions would superimpose. Equally randomly distributed particles would be at a mean distance of about 3.4 a for 20% volume fraction and 4.3 a for t0% volume fraction, where a is the minor semiaxis of the particle. Since the plastic phenomena is non-linear, it is unlikely that a mean field theory can capture the main characteristics of the mechanical properties of high volume fraction composites. In fact there is experimental evidence that, even in linear and relatively well understood properties, like elastic behavior, the usual self-consistent or Mori-Tanaka models can not explain the behavior of high second phase volume fraction composites [21]. Clustering is unavoidable for high volume fraction of inclusions and the average behavior is a rather complex function of the distribution. Further discussion about this topic will be conducted in Section 5. Both materials were taken as thermal and elastically isotropic and the matrix is assumed to follow an ideal elasto-plastic law. Material constants are show in Table 1. Fig. 1 shows the Von Mises equivalent plastic strain as a function of the radial distance from the center of the particle, in units of a, for 0, 10 and 20% volume fraction of spherical particles subject to 200°K cooling and no applied stress. A 0% volume fraction of reinforcements means no interaction, and it is calculated by placing an isolated particle in an infinite matrix. The results are shown here for the sake of comparison. It is observed that interaction among particles, as a mean field, has no influence over the plastic region. This is due to the symmetry of the problem. Spherical particles under a scalar thermal interaction field can produce just hydrostatic interaction stresses and they are unable to relax by plastic yielding. Table 2 shows the mean interaction stress field for 10 and 20% volume fractions. The effect of the superposition is observed in the change of character of the hydrostatic stress close to the particle interface. We can also see in Fig. 1 that the hydrostatic compressive region surrounding the inclusion has switched to a hydrostatic dilatational state beyond r = 1.55 a, for 10% volume fraction, and beyond r = 1.35 a_, for 20% volume fraction, where _ais the particle radius. Differences of less than t% were observed in the calculated interaction stress components. The discrete nature of the calculation method can introduce indeterminations in the plastic region and plastic work by as much as 5%. This is mainly due to the incorporation or withdrawal of a complete arc of elements into the domain of plastically relaxing elements, at some distance from the interface.

25

I

I

209

I

I

i

Elastic Case 20

~'15

q,

o

o v

wlO I0

\\\

. . . . . law of mixture i_O_sphere --/x-- prelate %3

)o

,, + "~

%

- - V - - prolate C~22

--0-- oblate %3 - - + - - oblate 0:22 i

0.0

t

i

0.2

i

0,4

r

0.6

1

0.8

i

t ,0

inclusion volume fraction 25 Plastic Case O.

~xT=-200°K

20

k,

15

0

b v

uJ 10 h0

"';Lq., "',h;,,

. . . . . law of mixture --13-- sphere 0:33 - - 0 - - sphere 0:22 - - A - - prelate %3

"%

- - V - - prolate 0:22 ---0-- oblate 0(33 - - + - - oblate 0:22 0.0

0.2

0.4

0.6

0.8

1.0

inclusion volume fraction Fig. 4. Effective Coefficients of thermal expansion (CTE) vs. volume fraction of inclusions for (a) AT = - 10°K when plastic relaxation is absent and (b) A T = - 2 0 0 ° K when plastic relaxation becomes important.

A. Roatta et at./Materials Science and Engineering A229 (i997) 203-218

210

(e}D = (1 --f)(e int + eAT) + f C * - l ( a ' : } n + f a * AT (25)

24

2ai

•

O~ oblate

0

22-'

f=20%

<>--O-~a

~22Prolate

21-

) 2o2 V V - V - % 19' []~sphera D / D ~

~

O

- - V ~

~

.

O

V.....

--rq--

Finally the CTEs in the longitudinal (z-axis) and transversal (x-y plane) directions can be calculated by:

V

[]

(e:j)D a~=

--•

x.

t.U 18 tO 17

O~22oblate/+~ +

+-+~

.~./k ~

A ~

A~

16

ZX-

15

Z~~/k" "~33pr°late

14 0

5

I

I

I

100

150

200

,~T [°K] Fig. 5. Effective Coefficients of thermal expansion (CTE) vs. cooling interval AT for longitudinal (z-axis) and transversal (x-y plane)

directions for 20% inclusionvolumefraction and differentincIusion geometries. The same cooling interval applied to protate inclusion composites, for 20% volume fraction, affects the plastic region as shown in Fig. 2. The maximum plastic strain has diminished by about 5% with respect to an isolated particle, with no other important modification over the contour levels. For oblate particles the results are shown in Fig. 3 for 20% volume fraction. The maximum plastic strain diminished by 10% from 0% to 20% volume fraction together with an increment of the plastic region in the z-axis direction until 0 = 70°. The plastic relaxation seems to be happening in a wider region and in a more homogeneous fashion than the case of an isolated oblate inclusion in an infinite matrix. In order to evaluate average results let us make the following considerations. The total average strain in the whole composite can be written as: (e}i) = (1 --f)(E}M + f (e}n = (1

- f ) ( E + e int) + f C * - : ( o ' ~ } a + f ( ~ T ) a

(23) The thermal properties of the composite, and its interaction with the elastic and plastic properties, can be studied by stating Eq. (20) in a slightly different manner: ~int' ~ ij +Gi~=

"* ~.i.t

~

~int ij 4-- G~ 2 = ~kl(ffi~l t "J- g~l -- E~Jast)

VX•D

- - ~-~

E!nt

- .Ant u = /,
and the total average deformation comes out as:

AT

(26)

These results are applied to different volume fractions, inclusion geometries and cooling intervals provoking elastic or elasto-plastic deformations. The effective CTEs are shown in Fig. 4a) for A T = - 1 0 ° K (when plastic relaxation is absent) and b) for A T = - 2 0 0 ° K (when plastic relaxation becomes important). If it were not for the plastic relaxation both temperature intervals, under the assumption of constant CTE's for both phases, would render the same values due to the linear nature of the phenomenon. Fig. 4a) shows that effective CTEs are isotropic in the case of spherical particles while prolate and oblate inclusions induce an anisotropic behavior. These results are in good agreement with other simulations already presented in the literature [14-17,21,22]. The curves are interrupted at a volume fraction of 39% for prolate inclusions and at 80% for oblate inclusions. At those volume fractions the iterative scheme does not converge and a more general method should be used. A discussion about agreement with experimental results will be done in Section 5. Fig. 4b) shows the influence of plastic relaxation. The CTEs become more isotropic and closer to the rule of mixtures calculation. In fact the spherical particle case is hardly distinguished from that simple rule. This tendency is more easily observed in the Fig. 5 where the longitudinal (z-axis) and transversal (x-y plane) CTEs are shown in function of AT for 20% inclusion volume fraction and different geometries. The limit AT intervals for which strains are kept in the elastic regime are - 4 4 ° K , - 3 6 ° K and - 3 7 ° K for spherical, prolate and oblate particles respectively. The difference between oblate and prolate cases is probably of numerical origin. Spherical particles concentrate lower stresses and they reach higher AT before plastic relaxation proceeds. The plastic strain induces softening along the longitudinal axis (z-axis) for protate and spherical inclusions and around the oblate inclusion equator and a consequent increment in longitudinal and transversal CTEs respectively. The plastic strain being just of shear nature one could suspect that there would be no effect other than the developed anisotropy of CTEs. The volumetric CTEs would not change appreciably and just a reorientation of thermoelastic effects should appear. Table 3 shows the volumetric CTEs for 10 and 20% volume fraction of inclusions, with and without plastic relaxation. They show that the non-linear nature of the plastic behavior, contrary to expected results, leads to volumetric CTEs consistently

A. Roatta et a l . / Materials Science and Engineering A229 (1997) 203-218

211

Table 3 Volumetric CTEs for different inclusion geometries, cooling intervaIs and volume fractions Inclusion shape Cooling interval

Vol. fraction (%)

AT = -- 10°K Elastic A T = --200°K Plastic

10 20 10 20

Sphere ( x 10 -6 ° K - i )

Oblate 4 x 4 x 1 ( x 10 . 6 °K-1)

Prolate 1 x I x 4 ( x 10 -6 °K -1)

63.47 56.55 64.37 57.85

62.94 55.83 64.06 57.62

63.07 56.06 64.09 57.78

higher than when no ptastic relaxation is present. Clearly the main differences are due to the interaction among particles. Orientationally randomly distributed particles would produce a decrement of the volumetric CTE mainly because of the interaction and not because of plastic strain, which would slightly raise those values. The present calculations would confirm the conjecture of Lee et al. [23] in the sense that plastic relaxation can modify volumetric CTEs. The modification is the opposite to that informed by Lee et al., increasing the values rather than decreasing them. It is not clear whether their experimental values were measured under cooling or heating. Under cooling the plastic region developed around inclusions would prevent the matrix, having higher CTE, to compress the particles and smaller composite effective CTE's are expected.Similar analysis can be applied to heating measurements. The current results can be explained only as a result of interaction among particles. Greater modifications can be expected for higher cooling intervals and the differences can be as high as 10%. Table 4 shows accumulated elastic energies, both distortive and hydrostatic, together with ~plastic volumes and plastic work relative to inclusion volumes, for the three different aspect ratios and two inclusion volume fractions. The plastic works were summed over the plastified cells and divided by the inclusion volumes. The elastic energies were summed over the elementary representative volumes. 10% volume fraction is obtained by summing until r = 2.1 _a and 20% volume fraction until r = 1.7 _a, and the results normalized by the inclusion volume. There is no modification of the relative plastic work due to inclusion volume fraction increase when the composite is subjected to a pure temperature decrement. This result is expected for spherical inclusions because the superimposed interaction stress is purely hydrostatic and increments in hydrostatic stresses do not raise plastic relaxation. Notwithstanding the same result is approximately valid for oblate and prolate inclusions when interaction stresses have deviatoric components. The accumulated elastic energies are higher for 20% than for 10% volume fraction of inclusions, provided that we take care of multiplying the higher volume fraction results by two.

4.2. Local plastic yielding by mechanical loading The influence of interaction among particles over the plastic yielding induced by the presence of inclusions under traction is much more important than the one observed in the case of plastic relaxation due to CTE's mismatch. Fig. 6a-c show the Von Mises equivalent plastic strain contour levels for 0, 10 and 20% volume fraction of spherical inclusions when the composite is subject to E = 0.90 O'yield (traction). A decrement of all maxima can be observed due to the presence of a compressive interaction stress along z-axis and a traction in x-y plane (See Table 2). For 10% volume fraction the maximum has displaced over the interface being now close to 0 = 36 °, By increasing the volume fraction gradually the maximum absolute value of plastic strain is switched from that region to a place located along the z-axis. This effect is due to what is usually called development of triaxiallity because of the interaction among particles. Another effect is the general reduction of the plastic region per particle. For 20% volume fraction it has been reduced to about 1/3 of the plastic region for 0% volume fraction (Table 4). Once multiplied by the respective volume fractions, the plastic volume is actually greater by a factor of 1.17 for 20% volume fraction than for 10% volume fraction. Fig. 7a and b show contour levels for hydrostatic pressure at Z =0.90 O'yield and 0% and 20% volume fractions. At 20% volume fraction hydrostatic pressure has slightly decreased, comparing with 0% volume fraction, in the z-axis direction and it has increased by almost 36% in the x-y plane in the vicinity of the interface. This is also due to the same phenomenon of triaxiallity development. Interacting prolate inclusions, present in as much as 0, 10 and 20% volume fi'action, and subject to I; = 0.90 0"yield produce the results shown in Fig. 8a-c. The maximum value of the equivalent plastic strain has been reduced to 1/2 when we increased the inclusion volume fraction from 0 to 20%. The volume involved in plastic relaxation, per particle, has reduced approximately to 1/4 (for 10% volume fraction) and 1/t0 (for 20% volume fraction) of the values pertaining to 0% volume fraction (Table 4). Considering the volume fraction of inclusions the volume involved in plastic

212

A. Roatta et a l . / Materials Science and Engineering A229 (I997) 203-218

Table 4 Plastic work and elastic energies for different volume fractions and inclusion geometries

Inclusion shape-2a: minor or unique axis (Inclusion Vol. VI in units of [VI]=

a s)

Sphere(4/3 a)

Oblate(64/3 ~) Prolate (16/3 a)

Thermal mismatch

10% Plastic Vol./VI Plastic work Wp/VI in units of MJ m -3 Distortive elastic energy/V1in units of MJ m -3 Hydrostatic elastic energy/V1in units of MJ m -3

3.91 0.63 0.36 0.04

3.59 0.92 0.31 0.08

3.70 0.90 0.34 0.05

A T = -200°K

20% Plastic Vol./VI Plastic work Wp/VI in units of MJ m-3 Distortive elastic energy/Vi in units of MJ m -3 Hydrostatic elastic energy/VIin units of MJ m -3

3.22 0.60 0.24 0.025

3.50 0.91 0.20 0.06

3.89 0.93 0.21 0.035

Traction

0% PlasticVol./VI

1.31

0.30

1.62

10% Plastic ¥ol./V~ Plastic work Wp/V~in units of MJ m -3 Distortive elastic energy/VI in units of MJ m -a Hydrostatic elastic energy/V~in units of MJ m -3

0.75 0.024 0.36 0.056

0.07 0.0 0.36 0.067

0.39 0.02 0.29 0.038

20% Plastic Vol./V~ Plastic work Wp/Vr in units of MJ m-3 Distortive elastic energy/V1in units of MJ m -a Hydrostatic elastic energy/V1in units of MJ m -3

0.44 0.01 0.15 0.027

0.0 0.0 0.155 0.037

0.15 0.01 0.10 0.013

£ = 0.90 #y

strain for 20% volume fraction of inclusions is actually 4/5 of the corresponding to 10% volume fraction. The effect is the opposite to the one observed for spherical particles. The influence of this effect in the stress-strain curve will be discussed in Section 5. The plastic work, considering the doubling volume fraction, has slightly decreased for spherical particles go!rig from 10% to 20% of inclusions. For prolate inclusions the plastic work is kept constant with volume fraction increment. Comparing with the plastic volume involved in plastic relaxation, the different pace of plastic yield is evident. The elastic energies per inclusion have decreased with inclusion content even multiplying the results by two due to inclusion volume fraction doubling. These rather obvious results are generalized by saying that we can expect, with increasing inclusion contents, an increment of the accumulated elastic energies when the external action over the composite is one of thermal origin and a decrement when there is an external traction applied. In the first case there should be a maximum for the accumulated elastic energy for some particular inclusion volume fraction. In the second case there should be a steadily decrement of the total elastic energy until a minimum when inclusions reach 100% of the composite volume. The plastic work and plastic volume have reduced to zero for oblate inclusions going from 10 to 20% volume fraction (Table 4). For oblate inclusions the contour levels for elastic equivalent strain for 0, 10 and 20% volume fractions are shown in Fig. 9a-c. The plastic region has disappeared for 20% volume fraction due to the superimposed interaction stresses. Another effect

has been the gradual reduction of the elastic strain with the increment of inclusion volume fraction. We have also calculated the concentration of different field magnitudes in the vicinity of inclusions. It was found that diminishing gradients are obtained with increasing volume fractions. However, higher volume fraction of inclusions prevents plastic relaxation and stresses, hydrostatic pressures and elastic energies are higher than in the non-interaction case, that is to say in the vicinity of an isolated particle in an infinite matrix

[5]. Eq. (25) can be substantially simplified if we consider constant temperature. A macroscopic stress applied in that condition allows the calculation of effective z-axis Young's modulus by using the equation: ~"~33

Ez - - -

(27)

and the Poisson ratio by:

(~=>D

(28)

The direct application of the above stated equations would have rendered plastic relaxation in the whole meshed region for some percentage of matrix yield stress. Interaction among particles usually extends the stress range for which some rim cells are still purely elastic. Although, by starting the computation at those higher stresses the results would be incorrect because no cells would have remained elastic in the rim of the meshed region. The calculation is computationally performed by introducing a new loop in the program. The external stress is gradually increased step by step and

213

A. Roatta et at,/Materials Science and Engineering A229 (1997) 203-218

I

25 t

" " - ,

2.025] - - - ' ' " " - - . . . .

2.01

",

m a x . 8 . 7 x 1 0 "4 at

r=l.05a 0=0.63 "~ % %

1.5 ~

t.5

"~ 05

05

~--._6 \ \ 1.0 ~ 8 ~ ,

max.12x10-4at r=105 . a e=079 .

t

1.0

I,

\\x',N%-,.-10 0.5

=Sid: ,

-

0.0

'

i

0.5

1

1.0

12 '

1

'

1.5

I

2.0

'

0.0

2.5

0.5

1.0

1.5

2.0

2.5

y/a

y/a 2.5 - - . . . . " - ' " - - . .

2.0-

%

max.5.2x104at r=1.a5a e=O 1

% % %

•%

1.5

%

1.0 1,

0.5

0.0

5

0.5

115

1.0

210

2.5

yla Fig. 6. Von Mises equivalentplastic strain contour levels,in units of 10- 4, around a representativespherical inclusionfor (a) 0% volumefraction, (b) 10% volume fraction and (c) 20% volume fraction when the composite is subject to Z = 0.90 O'yield(tensile load along z-axis), r and 0 are defined as in Fig. 2. The outer dotted line shows the limit of the meshed region and a represents the radius of the particle. convergence is assured. For each stress increment the solution initially taken as valid is the one obtained with the previous smaller stress. By this mechanism we can reach stresses higher than the pure matrix yield stress before the whole meshed region deforms plastically. Fig. 10a and b show the stress-strain curves for all three different aspect ratios and volume fractions of inclusions equal to 10 and 20%. The rule of mixtures, as an upper bound, is shown to be always above the other curves. The maximum stress for each curve is the one at which the last cell of the meshed region, usually in the traction direction, has become plastic. As expected, the

maximum reinforcement ability is found for prolate fiber like inclusions with an increment in the Young's modulus of 29% and 62%, for 10% and 20% volume fraction, respectively. For spherical particles the increment is of 15% and 34%, for both volume fractions with respect to the matrix Young's modulus. Oblate inclusions behave similarly with an increment of 14% and 30%. The separation from the linear behavior is due to the plastic microregions developed around particles. In fact oblate inclusions do not induce any plastic yield, even for stresses higher than the matrix yield stress, as

A. Roatta et al./ Materials Science and Engineering A229 (1997) 203-218

214

demonstrated by the linearity of the curves. A 20% volume fraction of prolate inclusions shows the biggest separation from the linear behavior and the highest applied stress with and increment of approximately 25% over the matrix yield stress. We have been able to catch the onset of plastic strain. This elasto-plastic transition could be extended to higher stresses but the plastic regions would start to superimpose and interaction among particles can not be assumed as well described b y a mean field theory. Furthermore, from the microscopic crystallographic point of view, annihilation

2.5-

30~6

"''"-.%

2.0.

%% % %

1,5-

1.0-

0.5-

0.0

0,5

1.0

1.5

2.0

2.5

y/a

2.5

~

2.0

%%%%%%%%%%%

1.5 0~

1.0 0.5 0.0

0.5

1.0

1.5

2.0

2.5

y/a Fig. 7. Hydrostatic pressure contour levels in MPa around a representative spherical inclusion for Z = 0.90 ~yi~:a(traction along z-axis). (a) 0% volume fraction (b) 20% volume fraction. Dotted line and a defined as in Fig. 6.

and strong interaction among dislocations can be expected.

5. Discussion and conclusions

We have calculated effective CTEs of MMCs reinforced with different volume fractions of oblate, prolate and spherical inclusions. The calculation takes into account interaction among particles through a mean stress field and allows the matrix to plastically relax in the immediate vicinity of particles. The model does not consider matrix hardening by geometrically necessary or statistically distributed dislocations. If hardening is considered the volume fraction of matrix that would relax stresses by plastic yielding is expected to be smaller. Many of the available experimental values have been obtained in materials presenting preferential orientation distribution of particles, distribution of inclusion aspect ratios, complex thermomechanical treatment or some arbitrary combination of these and other factors. Particularly the synergyc effect of dislocation generation and interaction should be included in the model. No serious attempt to explain experimental results can be made without including that effect. Despite the limits imposed by the lack of matrix hardening the calculations are consistent with the results found in the literature. In the case of residual stresses almost the same considerations are valid. Internal stresses are not only due to CTEs mismatch and consequent TRS accumulation but also dependent on the punching of geometrically necessary dislocations, subgrain size decrement, particles orientation, aspect ratio distribution and, maybe particularly, on the relative arrangement of inclusions. The model is not able to simulate the interaction among particles taking into account the actual geometric arrangement of inclusions. Many authors [24-31] have approached the problem, mainly by FEM techniques, and have found this factor to be of great influence on the macroscopic properties. Although, Brockenbrough et al. [32] as well as Zahl et al. [30] found that, for continuous fiber reinforcement and loading perpendicular to the fiber axes, the average properties are rather insensitive to fiber distribution for volume fractions lower than 20%. Under this limit the agreement among calculated and experimental residual stresses is rather good [33]. Calculated TRS for oblate inclusions show the largest differences with experimental TRS and further study is due to the topic. Effective plastic response of two-phase composites has also been studied, by using FEM techniques, by Shen et al. [34] for different volume fractions of a variety of particle shapes. They regard, as well as we found, the hydrostatic stress accumulated in the matrix to be of paramount importance in the interpretation of the results.

A. Roatta et al, / Materials Science and Engineering A229 (1997) 203-218

% %

8

%

%

8

%

max. 4.7xl 0 -3 at r=4.2a e=O

7

7

215

% % I

max. 2 . 8 6 x l 0 -a

max. 2.38x103 I

at r : 4 . 2 a e : 0

at r=4.2a e=0

7

\

l l l 1 l l

6

6 1 l

5

5

4

-~ 4

3

3

3

2

2

2

5

63 U

63

4

N

0

1

2

0.5

0

y/a

1

y/a

2

0

1

2

y/a

Fig. 8. Von Mises equivalent plastic strain contour levels, in units of 10 -3, around a representative prolate inclusion with an aspect ratio given by 1 x 1 x 4, for (a) 0% volume fraction (b) 10% volume fraction and (c) 20% volume fraction when the composite is subject to Z = 0.90 0"yield (traction along z-axis), r and O, dotted line and _a are defined as in Fig. 2.

The results presented in the former paragraph, about the relative proportion of plastic volumes with respect to inclusion volume fractions, deserve further discussion. Spherical inclusions seem to induce higher plastic relaxation volumes for higher volume fractions of inclusions. This softening mechanism would predict that stress-strain curves should show a lower slope for higher volume fractions at the very onset of plastic strain. This is consistent with the results obtained by Lloyd [35] for 20 and 10% volume fractions of SiC in A1 composites. He found that strain hardening, for inclusions with an aspect ratio of 2, is higher for 10% volume fraction than for 20% volume fraction. The main difficult in judging these results is that the plastic localization starts at different total strains and while the

lower inclusion content composite is already in the elastoplastic transition the higher one is still in the elastic regime with a much higher slope. In fact Corbin and Wilkinson [36] obtained opposite results for almost the same composite, except for the thermomechanical treatment. The current model predicts lower plastic work for higher volume fractions of spherical particles. This fact can suggest an increment of the slope of stress-strains curves in the elastoplastic transition for higher volume fractions. Moreover we have to keep in mind that, as soon as the matrix yields, strain hardening can also modify the behavior and compensate the trend to develop larger plastic volumes. Further complications come from the results obtained for prolate inclusions for which the higher the inclusion volume

216

A. Roatta et a l . / Materials Science and Engineering A229

(1997) 203-218

max. 12.6x10 4 at r=4.6 e=r#2 2

min. 8.5x10 -4 at r=1.05 e=O "t

Q'k ~ l ' Q,,, I , ! ,,m ,,~

-10.5 12.5

""'11.5" ~ """

N

0

1

2

3

5

4

6

7

%

%

8

y/a max. 12.6x10 4 at r=4.2 e==/2 2

min. 8.1x104 at r=1.05 e=O 10.1 9.1

cu 1

12.1 ~

N

0

1

2

3

4

5

".

6

7

8

y/a max. 12.0x10 4 at r=1.2 e=~/2 min. 7.8x10 -4 at r=1.05 e=0

2

7.8 N

-

%

0

1

2

3

4

5

6

7

8

y/a Fig. 9. Elastic equivalent strain contour levels, in units of 10-4, around a representative oblate inclusion with an aspect ratio given by 4 x 4 x l, for (a) 0% volume fraction (b) 10% volume fraction and (c) 20% volume fraction when the composite is subject to Y~= 0.90 ~Ti~la (traction along z-axis), r and 0, dotted line and _a are defined as in Fig. 2.

fraction the smaller the plastic region while the relative plastic work is kept constant. The currently calculated behavior for linearity ending is the opposite to the one calculated by Shi et al. [37] who claimed that the linearity ending stress is higher for lower volume fraction of inclusions. If confirmed it could be related to particle arrangement and plasticzone-interconnection. This effect should appear at

higher strains and it is a different kind of localization phenomenon that cannot be simulated by the current model. Prangnell et al. [38] found, experimentally, that the stress for linearity ending decreases when the volume fraction and/or particle size increase, what is consistent with our predictions. Clearly again, it is necessary to introduce hardening mechanisms in the simulation to address this issue. The general trend

A. Roatta et al./ Materials Science and Engineering A229 (I997) 203-218

seems to be: the higher the volume fraction the higher the linearity ending stress. This rule seems to be valid for any aspect ratio of inclusions and the effect is induced by the average interaction stress. No clear result can be advanced for the apparent strain hardening rate without considering microscopic hardening mechanisms. Analytical calculations for elastoplastic behavior of MMCs are still limited to spherical symmetry. Recently, Bullough and Davis [39] were able to consider also an average stress field originated by the presence of many inclusions. Corbin and Wilkinson [40] also treated the clustering phenomena in a self-consistent manner without detailed consideration of topology of distributions. Attemps to introduce topology in relatively simple ways [41-43] were successful in simulating large deformation stress-strain curves. Although, the lack of proper treatment of plastic yield in the vicinity of inclusions render unreliable stress-strain behavior in the elastoplastic transition regime. The novelty of the present approach is the possibility of treating the plastic onset assigning plastic yielding to

12o-

f=lO~° o sphere prolate

100: O_ CO CO

80eo-

+

/"

oblate

./-'/.~

I

I....... law of mixtures ,, /, . ~ / ' J ,/*

.Q

CO

,,,.,,," ""]

//"

I

217

the right location in the matrix, as well as FEM techniques do. The main advantage over FEM simulations comes from the ability to calculate the average mechanical properties without imposing them a priori, as it is done by FEM when just a single particle is considered for simulation. Moreover, despite big computation power is currently available, it is still time consuming to compute stress-strain curves by 3D-FEM when interaction among many particles has to be taken into account. Watt et al. [44] have recently approached the problem by FEM by using a brick shape unit cell more appropriated for 2-D simulations than truly 3-D tests. The main disadvantage of the current model, that of not being able to incorporate distribution topology of particles, can be turned into an advantage. The interaction among closest inclusions cannot be separated from the average interaction field in a many particles FEM model. The resultant stress and strain fields are certainly dependent on the distribution topology. Although, some of the macroscopic characteristics should be independent of the actual distribution but only related to average stress and strain fields. Clearly more work is due to the topic in order to understand which properties are only dependent on the average fields and not on the details of the distribution. The current model is also unable to treat large deformations but, at the onset of plastic strain, can isolate the local behavior from the complications of a pre-established particle distribution.

/

,./."/

40-

References

o

200 0.0

'

4

0.2

'

I

0.4

'

I

0.6

'

I

0.8

'

I

1.0

'

I

1.2

strain [xlO "3] 120 I00

"~ Q.

f=20% -

//'"

0 sphere ZX prolate

/,/ /."'

x ~

.~--.~- / ~ f . ~ / /

-

8o-

co 60 CO

m

/.//

"N 40 /,//

20 0 0.0

012

0:4

0:6 018 strain [xlO "3]

1:0

112

Fig. 10. Stress-strain curves for all three different aspect ratios of inclusions; (a) 10% volume fraction (b) 20% volume fraction.

[1] R.J. Arsenault, The strengthening of 6061 aluminum by fiber and platelet silicon carbide, Mater. Sci. Eng., 64 (1984) 171-181. [2] R.J. Arsenault and R.M. Fischer, Microstructure of fiber and particulate sic in 6061 A1 composites, Scripta Metalt, I7 (1983) 67-71. [3] R.J. Arsenault and N. Shi, Dislocation generation due to differences between the coefficients of thermal expansion, Mater. Sci. Eng., 8I (1986) 175-187. [4] R.J. Arsenault, L. Wang and C.R. Feng, Strengthening of composites due to microstructural changes in the matrix, Aeta MetaIL, 39 (1991) 47. [5] A. Roatta and R.E. Bolmaro, An Eshelby inclusion based model for the study of strain and stress localization in metal matrix composites. I: general formulation and its application to round particles and II: fiber reinforcement and tamellar inclusions, Mat. Sci. Eng., in press (H-4241 and H-4296). [6] M. Taya and R.J. Arsenault, A comparison between a Shear Lag type model and an Eshelby type model in predicting the mechanical properties of a short fiber composite, Scripta metall., 21 (1987) 349-354. [7] K. Satya Prasad and Y.R. Mahajan. Effect of isothermal heat treatment on the mechanical properties of A1/20 v/o SiC composite, Scripta Metalt. Mater., 30 (1994) 1049-1054. [8] M. Taya and T. Moil, Dislocations punched-out around a short fiber metal matrix composite subjected to uniform temperature change, Aeta MetatL, 35 (1987) 155.

218

A. Roatta et al./ Materials Science and Engineering A229 (1997) 203-218

[9] M. Taya, K.E. Lullay and D.J. Lloyd, Strengthening of a particulate metal matrix composite by quenching, Acta Metall. Mater., 39 (1) (1991) 73-87. [10] R.J. Arsenault, Composite Structures, I.H. Marshall (Ed.), Elsevier, 1987, p. 70. [1i] W.S. Miller and F.J. Humphreys, Strengthening mechanisms in particulate metal matrix composites, Scripta Metall. Mater., 25 (1991) 33-38. [12] R.J. Arsenault, Strengthening mechanisms in particulate MMC: remarks on a paper by Miller and Humphreys, 25 (1991) 26172621. [13] A. Roatta, P.A. Turner, M.A. Bertinetti and R.E. Bolmaro, MMCs strengthening by dislocation density increase. An interactive Eshelby approach, (Unpublished research). [14] P.J. Withers, W.M. Stobbs and O.B. Pedersen, The application of the Eshelby method of internal stress determination to short fiber metal matrix composites, Acta Metatl., 37 (11) (1989) 3061-3084. [15] S.C. Lin, C.C. Yang and T. Mura, Average elastic-plastic behavior of composite materials, Int. or. Solids Structures, 29 (14/15) (1992) 1859-1872. [16] W.C. Johnson and J.K. Lee, An integral equation approach to the inclusion problem of elastoplasticity, or. Appt. Mech., 49 (1982) 312. [17] W.C. Johnson, The Elasto-plastic Behavior of Inclusions, Ph.D. Thesis, Michigan Technological University (i980). [18] T. Mura, Micromechanics of Defects in Solids, Martinus Nijhoff, Dordrecht, 1987. [19] J.K. Lee, Y.Y. Earmme, H.I. Aaronson and K.C. Russell, Plastic Relaxation of the Transformation Strain Energy of a Spherical Precipitate: Ideal Plastic Behavior, Met. Trans., 1IA (1980) 1837-1847. [20] T. Mori and K. Tanaka, Average stress in matrix and average elastic energy of materials with misfitting inclusions, Acta Metall., 21 (1972) 571-574. [21] P. Kwon and C.K.H. Dharan, Effective moduli of high volume fraction particulate composites, Aeta Metatl. Mater., 43 (3) (1995) 1141-1147. [22] Y. Takao, in J.R. Vinson and M. Taya (Eds.), Recent Advances in Composites #z the United States and Japan, A S T M STP 864, American Society for Testing Materials, Philadelphia, 1985, pp. 685-699. [23] Y.S. Lee, M.N. Gungor, T.J. Batt and P.K. Liaw, Semi-empirical investigation of thermal expansion behavior of components in a two-phase particle-reinforced metal matrix composite, Mater. Sci. Eng., AI45 (199I) 37-46. [24] G. Bao, J.W. Hutchinson and R.M. McMeeking, Particle reinforcement of ductile matrices against plastic flow and creep, Acta Metatl. Mater, 39 (8) (1991) 1871-1882. [25] J.M. Papazian and P.N. Adler, Tensile properties of short fiberreinforced SIC/AL composites: part I. Effects of matrix precipitates, Metal. Trans., 21A (I990) 401-410. [26] D.B. Zahl and R.M. McMeeking, The influence of residual stress on the yielding of metal matrix composites, Acta Metall. Mater., 39 (6) (1991) 1117-1122.

[27] A. Levy and J.M. Papazian, Elastoplastic finite element analysis of short-fiber-reinforced SIC/A1 composites: effects of thermal treatment, Aeta Metall. Mater., 39 (I0) (1991) 2255-2266. [28] T.L. Dragone and W.D. Nix, Geometric factors affecting the internal stress distribution and high temperature creep rate of discontinuous fiber reinforced metals, Acta Metall. Mater., 38 (10) (1990) 1941-i953. [29] S. Ho and A. Saigal, Three-dimensional modeling of thermal residual stresses and mechanical behavior of cast SiC/A1 particulate composites, Acta Metall. Mater., 42 (10) (1994) 3253-3262. [30] D.B. Zahl, S. Schmauder and R.M. McMeeking, Transverse strength of metal matrix composites reinforced with strongly bonded continuous fibers in regular arrangements, Acta Metall. Mater., 42 (9) (1994) 2983-2997. [31] S. Kumar and R.N. Singh, Three-dimensional finite element modeling of residual thermal stresses in graphite/aluminum composites, Acta Metall. Mater., 43 (6) (1995) 2417-2428. [32] J.R. Brockenbrough, S. Suresh and H.A. Wienecke, Deformation of metal-matrix composites with continuous fibers: geometrical effects of fiber distribution and shape, Acta Metatl. Mater., 39 (5) (1991) 735-752. [33] L.F. Smith, A.D. Krawitz, P. Clarke, S. Saimoto, N. Shi and R.J. Arsenault, Residual stresses in discontinuous metal matrix composites, Mater. Sci. Eng., At59 (1992) L13-L15. [34] Y.-L. Shen, M. Finot, A. Needleman and S. Suresh, Effective plastic response of two-phase composites, Acta Metall. Mater., 43 (4) (1995) 170t-1722. [35] D.J. Lloyd, Aspects of fracture in particulate reinforced metal matrix composites, Aeta Metall. Mater., 39 (1) (1991) 59-71. [36] S.F. Corbin and D.S. Wilkinson, Low strain plasticity in a particulate metal matrix composite, Aeta Metatt. Mater., 42 (4) (1994) 13t9-1327. [37] N. Shi, B. Wither and R.J. Arsenault, An FEM study of the plastic deformation process of whisker reinforced SIC/AL composites, Acta Metatl. Mater., 40 (11) (1992) 2841-2854. [38] P.B. Prangnell, T. Dowries, W.M. Stobbs and P.J. Withers, The deformation of discontinuously reinforced MMCs-I: the initial yield behavior, Acta Metall. Mater. 42, 10 (1994) 3425-3436. [39] R. Bullough and L.C. Davis, The residual deformation fields in particle reinforced metal-matrix composites, Acta Metall. Mater. 43 (7) (1995) 2737-2742. [40] S.F. Corbin and D.S. Wilkinson, The influence of particle distribution on the mechanical response of a particulate metal matrix composite, Acta Metalt. Mater. 42 (4) (1994) 1311-t318. [41] K.S. Ravichandran, A simple model of deformation behavior of two phase composites, Acta Metall. Mater. 42 (4) (1994) 11131123. [42] Z. Fan and A.P. Miodownik, "lhe deformation behaviour of alloys comprising two ductile phases-I. Deformation theory, Aeta Metalt. Mater. 41 (8) (1993) 2403-2413. [43] Z. Fan and A.P. Miodownik, The deformation behaviour of alloys comprising two ductile phases-II, applications of the theory, Acta MetaU. Mater. 4I 8 (1993) 2415-2423. [44] D.F. Watt, X.Q. Xu and D.J. Lloyd, Effects &particle morphology and spacing on the strain fields in a plastically deforming matrix, Aeta Mater., 44 (2) (1996) 789-799.