Acta metall, mater. Vol. 41, No. 1, pp. 175-189, 1993 0956-7151/93 $5.00 + 0.00 Printed in Great Britain. All rights reserved Copyright © 1992 Pergamon Press Ltd

THERMOMECHANICAL DEFORMATION MODELING OF A12xxx-T4/SiC.p COMPOSITES

M. KARAYAKA and HUSEYIN SEHITOGLUI" Department of Mechanical and Industrial Engineering, University of Illinois at Urbana-Champaign,

Urbana, IL 61801, U.S.A.

(Received 18 July 1991; in revised form 8 June 1992)

Abstraet--A constitutive model for metal matrix composites is developed and its capabilities for predicting cyclic isothermal and cyclic thermomechanical behavior are demonstrated. The silicon carbide particulate reinforced Al2xxx-T4 alloy was studied experimentally and theoretically with the model. Cyclic stress--strain behavior of 15 and 20% reinforced silicon carbide particulate reinforced A12xxx-T4 were successfully predicted at temperatures of 20, 200 and 300°C at strain rates between 3 x 10-5s -~ and 3 x 10-3s -t. The thermomechanical stress-strain behaviors (T~in = 100°C, T~x=200, 300°C) were studied experimentally and the results were closely predicted when temperature-strain phasing was in-phase and out-of-phase. This study clarifies the influence of mechanical property mismatch in the elastic and in the inelastic ranges vs the thermal property mismatch on composite and the matrix behaviors. The transverse and hydrostatic stresses in the matrix, developed during cyclic loading, are reported for both isothermal and thermomechanical loading conditions.

1. INTRODUCTION

Many models using constituent properties and their interactions have been proposed for the evaluation of macroscopic behavior of composite materials [1-29]. Almost all of these models were designed to simulate monotonic, isothermal, elastic and time-independent deformations. There is a need to develop time and temperature dependent constitutive models for metal matrix composites, capable of handling cyclic defor- mations, as these materials are candidates for high temperature applications where thermomechanical fatigue is of paramount concern. The current models are designed to evaluate overall effective properties, while in fatigue and fracture studies a measure of internal stresses and strains, albeit approximate, is needed. This study is concerned with the thermo- mechanical cyclic deformation behavior of metal matrix composite materials. A two-state variable visco-plastic relation is developed for the A12xxx-T4 (matrix) alloy. Using this model and Eshelby's equiv- alent inclusion method the composite (silicon carbide particulate reinforced Al2xxx-T4) behavior was pre- dicted. The model leads to a general description of deformation of metal matrix composites under ther- momechanical loading.

Earlier micromechanistic models concentrated on predicting elastoplastic behavior of polycrystalline materials. The self consistent method, first proposed by Hershey [1, 2] and developed by Kroner [3] and Budiansky and Wu [4], was a procedure for the

fPresent address: Mechanics and Materials Program, National Science Foundation, 1800 G Street, NW., Room 1108, Washington, DC 20550, U.S.A.

estimation of elastic-plastic behavior of polycrys- talline aggregates which utilized the single crystal properties. This method is based on a single crystal, with uniform plastic strain as its only transformation strain, embedded in a matrix with the unknown polycrystalline properties. The self consistent method for multiphase materials was established by Hill [5, 6]. Hill's procedure was based on the solution of an elastic inclusion problem by Eshelby [7, 8]. In his procedure instantaneous moduli were used to model the inelastic behavior of the polycrystalline materials. The self consistency approach has been applied to polycrystals undergoing elastic-plastic deformation [9, 10] and steady-state creep deformation [11-13].

While the self consistency method has been used to predict elastic-plastic behavior of multiphase ma- terials, the elastic behavior of the composite materials has been investigated by using Eshelby's equivalent inclusion method. Eshelby's equivalent inclusion method [7, 8] provides a solution to the strain field disturbance occurring in an elastic medium due to an arbitrarily shaped region (inhomogeneity) with differ- ent elastic constants from the remainder. This region undergoes a transformation which is restrained by the surrounding material, and constrained strains are generated in the inhomogeneity and the matrix. When the inhomogeneity has eUipsoidal shape, Eshelby showed that the strain within the inhom- ogeneity is uniform. Direct use of Eshelby's equival- ent inclusion method for predicting the elastic modulus of a composite material is applicable to the dilute volume fraction of reinforcement [14]. The interaction among the reinforcements is implicitly implemented in the self consistency model by assuming that the single fiber is embedded into an

175

176 KARAYAKA and SEHITOGLU: THERMOMECHANICAL DEFORMATION MODELING

equivalent medium. The self consistency method is applicable to a higher volume fraction of reinforce- ments than Eshelby's direct method.

The plastic deformation of the matrix with elastic inclusion has been treated within the context of Eshelby's equivalent inclusion method as follows. Tanaka and Mori [15] were the first to apply this approach to the case of an inclusion and matrix with different elastic moduli. Plastic deformation in the matrix was assumed to be uniform and restored to the reinforcement as transformation strain. This theory was used to determine the yield strength and the hardening rate of crystals with disc, needle and spherical inhomogeneities. With this approach, Tanaka e t al. [16] has demonstrated the anisotropy of deformation under uniaxially applied stress for ellip- soidal reinforcement geometries.

At a finite volume fraction of reinforcment the stress and strain in the matrix are disturbed from the remotely applied loading due to interaction of the particles. Eshelby's equivalent inclusion method was improved by means of volumetric average stress concept [17-19] to account for inclusion of matrix interaction at the finite volume fraction of reinforce- ment. In order to satisfy the equilibrium condition, the volumetric average of the perturbed stress and strain can be forced to vanish when integrated over the entire composite domain. Mori and Tanaka [17] considered plastically deforming inclusions in an elastic domain with the same elastic modulus as the matrix. The volumetric averaging method was used by Wakashima e t al. [20] for inclusions which have different elastic moduli than the matrix to study overall thermoelastic properties of the two-phase systems. The model was used to predict macroscopic thermal expansion behavior of 32 and 44% re- inforced unidirectional tungsten-copper composites. They noted that a uniform temperature change in the composite results in internal stresses due to the mismatch of thermal expansion coefficients.

A more important problem, one in which the elastic reinforcement is embedded in an elastic-plastic domain has recently been addressed by Arsenault and Taya [21] and Tandon and Weng [22] who utilized Esheiby's modified equivalent inclusion method. Ar- senault and Taya predicted the thermal residual stress and the yield strength of whisker reinforced com- posites. Their interest was on the effect of residual stress on the decreased composite yield strength; therefore cyclic deformation was not considered. Tandon and Weng accounted for the disturbance of the field variables by interaction of the inclusions, and predicted monotonic stress-strain behavior. Since their approach was not incremental, it can not be used directly to model cyclic loading with tempera- ture dependent material properties.

A unified constitutive equation for the composite materials is developed in this paper. Temperature and strain rate dependency of the mechanical properties and the inelastic deformation of the matrix are

accounted for in the model. The model is used to characterize cyclic and monotonic stress-strain be- haviors of silicon carbide particulate reinforced Al2xxx-T4 under isothermal and thermomechanical fatigue loading conditions. The volumetric average stresses and strains of the matrix and the fiber are reported, and these provide the basis of the thermo- mechanical fatigue life modeling for metal matrix composites [23].

2. BACKGROUND

The transformation and inhomogeneity problem solved by Eshelby considers a domain ~p, which has different elastic constants than the rest of the medium, embedded in an infinite elastic body D [Fig. l(a)]. Since its deformation is constrained by the surrounding matrix, a perturbed strain field result. The problem, then, is to find how a remotely applied loading, E °, is disturbed by the existence of the inhomogeneity. The total strain on the inhomo- geneity is summation of the remotely applied strain and the constraint strain

r _ r r r 0 c n 17 ij - - C ijklE kl .~- C ijkl( E kl -~- E kl ) . (1)

In the "equivalent inclusion method" the domain ~p is replaced by an equivalent inclusion which has the same elastic constants as the matrix. The equivalent inclusion is assumed to undergo a stress free trans- formation under a fictitious eigenstrain (transform- ation strain), E*. The resulting total stress in the equivalent inclusion is

C,yk/(Ekl + E~ - E*). (2) 0 " ~ = m 0

Eshelby proved that the stress field in a single ellipsoidal inhomogeneity is constant and equations (1) and (2) are equivalent

r _ _ r r _ _ r 0 c n a,j - C#k~Ekt -- C,ykt (Ek~ + Ekt ) m O c n

"~- C ijkl ( £ kl "~ (" kl - - E ~ / ) . (3)

Superscripts r and m denote reinforcement (inhomo- geneity) and matrix, respectively. Isotropic or anisotropic stiffness tensors may be used in equation (3), depending on the constituent properties.

The relation between the eigenstrain, e*, and the constrained strain, E~7, is given by Eshelby's well known tensor, S~kt, as

c n * % - S~ktEkt. (4)

Sokt depends only on reinforcement geometry and Poisson's ratio of the matrix. S~kt for different re- inforcement geometries may be found in a book by Mura [24].

The procedure for determining stress in the in- clusion is described in Fig. 1, where the net strain, eigenstrain and stresses in the matrix and the re- inforcement are given for different stages of the problem. Figure l(a) describes the classical Eshelby problem given by equations (1)-(3). Figure l(b)

KARAYAKA and SEHITOGLU: THERMOMECHANICAL DEFORMATION MODELING 177

a) Eshelby's Equivalent Inclusion Model

Net m o Strain e/J =£/J

I Eigen pm strain E# = 0

b) Single Inclusion in Infinite Elastic-Plastic Matrix Under Thermomechanical Loading

i igen £pr= 0 strain

~r ~ r :_o cr~ Stress ~ u # ~ e ~ )

Transl. ** , strain E/j =E/j

£~e in £me

me in m eq +E~ +O#.~T

in Ill £~- +0j/AT

me e#

c) Eshelby's Modified Equivalent Inclusion Model

£c9"e , AT

D~o O @

®

C e ~m #+e/j

0 0

Stress O~a# =C~a~°~j m me m me m ce ~m

Net r o cn me cn in m me cn ee r Strain e#. = ~#+e~ e#. +e#+~. 0. +0~ ~T ~ +e~ e# +g#

r m in (O#-O 0 ) AT-~#

m in (0~/j-0#)AT-eq

c r _ me cn pr. r ce ~r

r 0~AT

c r . m e cn in .m _ pr~

e#+0~ AT £, in r E* in ,_m #- #-tO#-0"#)Ar

Fig. 1. Schematic illustration of Eshelby's model modified to handle thermal and mechanical mismatch under thermomechanical deformations• Net strain, eigenstrain and stress in the matrix and in the reinforcement are tabulated: (a) Eshelby's equivalent inclusion model, single inclusion in infinite matrix; (b) thermo-mechanical loading of a single inclusion in infinite elastic-plastic domain; (c) Eshelby's

equivalent inclusion model modified for finite volume fraction of reinforcement.

demonstrates the case of a single elastic inclusion in an infinite elastic-plastic matrix under thermomech- anical loading. In this case [Fig. l(b)], the transform- ation strains of the reinforcement and the matrix are modified, without disturbing the stress fields, and Eshelby's equivalent inclusion method is applicable. The matrix is subjected to a total strain composed of elastic, E~ °, and inelastic, in • % , strams and has tempera- ture dependent material properties, while the re- inforcement remains elastic. In this case the reinforcement has its own eigenstrain, 0,~ AT, due to thermal loading. Thermal strain in the matrix, 0,~ AT, and the inelastic strain in the matrix, Eu ~, are restored in the domain ~0 as eigenstrains. After these modifi- cations the total eigenstrain of the inhomogeneity, E~ ~, is

- - r Ill e ~ _ O # A T _ O i ~ A T _ in__ th i, % - % - %. (5)

The stress in the reinforcement is

a~. = C~kt(Ekm[ 4- E~,~ -- E~). (6)

This equivalent model can simulate the current prob- lem without disturbing the stress fields in domain D. The matrix is entirely elastic [Fig. l(b) r.h.s.] and the inhomogeneity is subjected to an eigenstrain of E~'. The domain D is subjected to, e,~ °, which is also entirely elastic.

The inhomogeneity with eigenstrain of E pr can be replaced by an equivalent inclusion with the same properties as the matrix with the addition of fictitious eigenstrain, e*. Then the transformation strain which

relates the stresses in the equivalent inclusion to the current problem is

* in th E** = E~ 4- E~ r = % - E,~ + % . (7)

This transformation strain is noted in Fig. l(b). The transformation strain is also related to constraint strain through Eshelby's tensor as

cn -- ** E ij - - S i j k l E k l • (8)

Eshelby's equivalency condition is written as

= C~k , {E f f + e~ + E~ -- e~ -- e~} (9)

In the current problem, domain D contains a finite volume fraction of reinforcement and is subjected to isothermal and thermomechanical loading conditions [Fig. l(c)]. For finite volume fraction of fibers the remote strain field, E~, is no longer equivalent to the undisturbed matrix strain field. It is replaced by the composite elastic strain field, E~. Average matrix elastic strain, E ~c, differs from remotely applied strain by the additional term g~

E~ . . . . . (I0)

The term g~ represents average elastic strain dis- turbance of the matrix due to finite volume fraction of reinforcement. The average matrix stress is dis- turbed from the composite stress by 6!y

m _ c -m (11)

178 KARAYAKA and SEHITOGLU: THERMOMECHANICAL DEFORMATION MODELING

The term ~ , which is to be determined, represents the average stress disturbances in the matrix. Simi- larly, the reinforcement stress and elastic strain are defined as follows [see Fig. l(c)]

e~ = e~ + g,~ (12)

~r,~ = tr,~+ 6,~.. (13)

In the model the particles are considered to be randomly distributed in the matrix in such a way that the composite is homogeneous at a macroscopic scale. For a homogeneous body in static equilibrium a volume integral of the disturbance of field variables should vanish over the entire body, D. In that case

f ~odV = 0 (14)

D gO dV = 0. (15)

Then

r c B vr (a 0 -- a~j) + Vm(a ~ -- fro) - 0 (16) c.¢ ce __ V v ( e ~ - - E o ) + V m ( E ~ e - - % ) - - O (17)

where vr and vIn are reinforcement and matrix volume fractions, respectively. The computation of effective properties of a heterogeneous media by using volu- metric averaging of field variables [equations (16) and (17)] is also known as the "direct" approach [25-29].

Following above modifications, Eshelby's equival- ence condition [equation (3)] is rewritten as

r _ c ~ r _ r c¢ ~ m c a i n t h a ij - - ~ ij "~- a ij - - C ijkl { e kl "~- e kl dr- e kl ~- e kl - - e kl }

. . . . . in th e~}. ( l g ) - - C ijkt { E kl "~- ~11 "31- E kl Jr- e kl - - E kl - -

From equations (10), (12), (17) and (18) the matrix and the reinforcement strain disturbances are calcu- lated as

~ m = cn th +E~) (19.1) Eli -- Vr(Ei j -- Eij

~ r _ _ c n t h in Eij- Vm(Eiy - e o +Eiy ). (19.2)

Note that as volume fraction of reinforcement goes to 0 the matrix strain disturbance, g~j, becomes 0 and reinforcement strain approaches the one given by the Eshelby modified inclusion method.

3. INCREMENTAL CONSTITUTIVE EQUATION FOR THE C O M P O S I T E

The composite was allowed to undergo an arbi- trary strain-temperature history. For a given loading history, strain increments of the matrix and the reinforcement should be written in terms of the incremental loading parameters. Using equations (8) and (19) the equivalency condition [equation (18)] reduced to the following equation which leads to solution of incremental transformation strain, ~ .

m .~ __ .ce

- - [ C ok t "~- I) m A C l j m n Smnkl]~k , - - A C i j k l E k l

m K +Vm AC,j.~(Sm~t-J.~t)(~k~ - ~ ) - ¢OktEk, (20)

where Ek~ is given by

Note that E o - '~- d E f f / d t where t is time. Then i~ is substituted into equations (10) and (!2)

to solve the matrix elastic strain, i~ ' , and the re- inforcement elastic strain, ~ , in terms of the loading

'~ and T. parameters E,.j The matrix elastic strain rate is

• m e 1 . c e 2 . t h . i n i n 3 £ij = O ijklEkl + O ijkl(Ekl - - e k l ) "~- ~ i j k l O kl (21)

and the reinforcement elastic strain rate is

• re 4 . c e 5 ' t h " i n m 6 = ekl) (22) ('ij O ijklEkl dr" O ijkl(Ekl - - + ~ i j k l O kl '

The inelastic strain rate of the matrix, E0, "~ is explicitly related to the composite strain through the constitutive equations of the matrix. The strain con- centration tensors in equations (21) and (22) are given as follows

[ C ijmn .~_ l - - m m Vm ACijtsStsmn]Omnk I - - [Cijkl ..~ ACijmnSmnkf]

(23.1)

m 2 - - m [Ci jmn -~ I) m A C i j t s S t s m n ] O m n k l - V r C i j m n ( ~ m a k l - Smnkl )

(23.2)

[C~jmn -~- l) m A C i j t s S t s m n ] O 3 m n = - - U r S i j k l E k K (23.3)

ra i) m A C i j t s S t s m n ] O rnnkl - - C ijk t [Co, ~+ 4 _ m (23.4)

[ C ijmnm .~_ Um A C i j t s S t s m n ] O m.k 1 5

__ In - Vm Com, (Sm,kl -- 6m,~t) (23.5)

[C~jm. + V m ACijtsStsmn]O 6 = VmSijkl EK (23.6)

where, ACuk ~ = C~kt- Co%. Equation (23.1) leads to the solution of O~kt, equation (23.2) leads to the solution of 2 OUkt, and so on.

The strain coefficient tensors given in equation (21) and (22) satisfy the following identities

1)In O lk t -~- V r O 4kl = 6 ijkl (24.1)

2 5 __ v ~ O ijkt + v , O okt -- 0 (24.2)

v m O a + v , O r = O . (24.3)

TO determine matrix, reinforcement and composite stresses, equations (21) and (22) should be coupled with the constitutive equations of the matrix and reinforcement.

When the matrix undergoes elastic-plastic defor- mation under thermal loading conditions the follow- ing equations are valid

• m _ m .m . tro -- Cokt(ekt -- E~ -- O~ I") + go ~ (25)

where

dC~kt m go = ~ (ekt -- ekt -- O~ AT).

For the elastic reinforcement

r . r #fj = Cukt(Ek,- 0~tT). (26)

KARAYAKA and SEHITOGLU: THERMOMECHANICAL DEFORMATION MODELING 179

By combining equations (16), (21), (22), (25) and (26) the composite stress increment was found in terms of composite elastic strain, matrix inelastic strain and temperature increments

• c _ _ 1 m 4 r . c e ¢7ij- [Vm ~) ijklC klmn + l)fO ijklC klmn]Emn 5 " in [VrCg~klACktmnAO,, m - v~ O okt A C ~ , E ~, +

+Vmgij]~'+V r " m 6 (27) ACuk~ Ckt,,, Om,"

In order to solve this equation, the matrix inelastic strain needs be defined in terms of the current state variables. At this point the formulation is general. Any constitutive model, which relates the inelastic strain rate to the current state variables of the matrix, can be used to solve for the reinforcement, matrix and composite stresses. The current model utilizes a unified constitutive equation developed for A12xxx- T4.

In this paper the model is used to simulate uniaxial cyclic loading conditions. Under axial loading con- ditions fourth order tensors are reduced to 3 x 3 matrices, and second order tensors are reduced to 3 x 1 vectors. Three parameters need to be specified to solve for reinforcement, matrix and composite stress and strain rates. Under uniaxial loading, trans- verse strain increments of the composite can not be specified under elastic-plastic loading. Transverse composite stress rates are z e r o ( # ~ 2 = 0 " ~ 3 = 0 ) , and the elastic strain rate in the loading direction, ~ , is specified. Temperature is varied linearly with the longitudinal net strain of the composite during ther- momechanical fatigue loading such that prespecified minimum and maximum temperatures coincide with maximum and minimum mechanical strain levels. With these boundary conditions equation (27) can be solved for 6"~1, E2~, and E~3. Then the corresponding matrix and reinforcement strain increments can be calculated using equations (21) and (22).

4. UNIFIED CONSTITUTIVE EQUATION FOR THE AI2xxx-T4

different deformation mechanisms (plasticity vs power law creep) that lead to different strain rate sensitivity regimes. The normalized equivalent inelas- tic strain rate vs effective stress to drag stress ratio, 6/K, for unreinforced A12xxx-T4, and simulations for 15, 20, 30% SiC particulate reinforced Al2xxx-T4 are given in Fig. 2. The flow rule of the unreinforced A12xxx-T4 is obtained from yield strength measure- ments at different strain rates and temperatures [32]. The simulations for 15, 20 and 30% SiC were ob- tained from the model outlined in Section 3. The flow rule, which relates the internal stress state of the matrix to the inelastic strain rate is

f exp[(61K) ~°1- 1 ] ( ~ )

EI~/A~L(~/K)46 ( ~ )

(if~K) >I 1

(if/K) < 1.

(28)

This relation accounts for both power law creep, (O/K) < 1, and the plastic deformation, (6/K)>t 1, mechanisms. The higher slope (10.1) in the plasticity mechanism regime represents rate-insensitive ma- terial behavior. The power law creep exponent was found to be 4.6, which is consistent with the models based on dislocation climb. These models predict a stress exponent near 4. Figure 2 could be used to obtain yield strength of the 15, 20 and 30% SiCp reinforced composite for a given strain rate and temperature.

In equation (28), A = 9.8 x 10 l~ exp[-AH/R(T + 273)] l/s, AH/R = 18,722 K and K is the drag stress of the matrix which typically evolves with defor- mation. The variation of initial values of K with temperature is given as, K = h3 + ha T MPa. For T i> 323 K, h3 = 420 MPa, ha = -0 .30 MPa/K and for T < 323 K, h 3 = 620 MPa and h 4 = - - 1.66 MPa/ K. sij and ~ j represent deviatoric stress and deviatoric back stress respectively and 6 is the effective stress defined as [(3/2(s~j- ~ ) ( s u - ~j)]l/2.

A unified constitutive model suitable for elevated temperature isothermal, creep and thermomechanical loading has been developed for the unreinforced material (A12xxx-T4). Incremental inelastic strain given by this unified model is used in equation (27) to calculate the volumetric average stresses. The proposed constitutive equation is capable of predict- ing stress hardening, cyclic hardening or softening, strain rate sensitivity, and recovery effects for a wide range of temperature-strain histories. The functions were derived directly from the experiments as out- lined in Refs [30-32]. The details of the material properties and the microstructure is given in Refs [33, 34]. The material is produced by ALCOA and is designated as Alcoa MB85.

Since cyclic deformation at different strain rates and temperatures are considered in this work, the flow rule for the matrix should account for the

[ . . . . , . . . . , , , t , 1 , [ . . . . J .... i , i I , i

0 % A 1 2 x x x - T 4 F l o w R u l e

a l ¢ 4.s | I l l S i m u l a t i o n s

v A = ~ , lO.l ] [ l [

, o , sio,

. . . . i . . . . i i [ i i i i [ . . . . i . . . . i i i i i i

2 3 4 5 6 7 8 9 2 3 4 5 6 7 8 9 0.1 1 0 ~/K

Fig. 2. Flow rule relating the inelastic strain rate to effective stress for A12xxx-T4 and simulations for 15, 20 and 30%

SiCp A12xxx-T4.

180 KARAYAKA and SEHITOGLU: THERMOMECHANICAL DEFORMATION MODELING

- - RT, e = 1.10 -4 I/see 400 • ...... " ....................

/

~ C , E = 3.10"Sl/see

200 " ! ~ -. 3 0 0 ° C , " . - 3 , 1 0 " S l / s e e AI 2xxx-T4, 0% SiCp i /~.. .~- '"" . . . . . Monotonic Behavior

100[ '~'~'"'" _ - - Experiment ~ . ................. 300 ° C, e = 3.10"Sl/see ..... Simulation

0 0 . 0 ~ ~ ~ ( ; ; . . . . 0 : ( ~ ; ' " 0 0 1 0 ' 0 0 1 2 ' 0014

Strain

Fig. 3. Monotonic stress-strain behavior of A12xxx-T4, experiments and simulation.

Two state variables were used to represent com- bined isotropic and kinematic hardening. We note that since the composite behavior over a broad range of temperatures and strain rates is of interest, a unified creep-plasticity model is chosen. The drag stress state variable, K, is related to the stress surface size. The deviatoric back stress variable, ct~j, rep- resents the stress surface center in the deviatoric stress space. The internal stress state evolves throughout the deformation history in a recovery-hardening format as

• , .in , (29) otij = 2 /3h~Ei j - - r ~ U

[ ( = hk - - rk + Ok 7" (30)

where a~j is the deviatoric back stress rate, ii~ is the inelastic strain rate, and/~ is the drag stress rate. The back stress hardening term, h~, represents the evol- ution of back stress during high strain rate exper- iments in which recovery is small. The recovery term, r~, represents a decrease in back stress to zero as time or temperature increases. The term hk represents the hardening of drag stress. Softening effects in the material during cyclic loading are represented by rk. The term 0k accounts for changes in drag stress with

temperature. Determination of the flow rule and functions hk, h~, r~, r k and 0 k from experimental results can be found in Ref. [32].

5. S I M U L A T I O N S ON AI2xxx-T4, AI2xxx-T4 15% SiCp AND AI2xxx-T4 20% SiCp

Cyclic and monotonic stress-strain behaviors of unreinforced and silicon carbide particulate re- inforced A12xxx-T4 under isothermal and thermo- mechanical loading conditions were simulated. The size and shape distribution of the silicon carbide distribution was found to be fairly uniform [33, 34] such that they are characterized by spherical re- inforcements. The reinforcement was assumed to be elastic (Er= 450GPa) for the loading conditions considered. The Poisson's ratio of the matrix and the particulate were 0.33 and 0.17 respectively and inde- pendent of temperature.

In the simulations net strain and the temperature of the composite varies with time. The net strain increment is related to thermal strain and mechanical strain increments as follows

(E ij )net __ Emj __ m " .rn • m - - " - - o i j r ' q - ( g i j ) m e c h (31)

5 0 0 , , , i , , l l l , a , , l l . , , l l l , , i , 4 1 , i , i q , l , , ~ , l , , , , i , , , , i , . , , i , , , , i , , , , i , , , , l ~ , , ,

400 ............... T=200 *C, ~ = 3.10 "s 1/see

o°OO oO"°°'°

~ ° " 'I'=200 °C, ~ = 3.10 "5 1/see

200 = 3.10 "s 1/see

.. A12xxx-T4, 15% SiCp 100 F".f ..- ........ ~" n, .... ~ ,. ~ Monotonic behavior

~.--~'- T=300 °(3, e = 3.10 1/see i Experiment "" ..... Simulation

Strain

Fig. 4. Monotonic stress-strain behavior of Al2xxx-T4 15% SiCp, experiments and simulation.

KARAYAKA and SEHITOGLU: THERMOMECHANICAL DEFORMATION MODELING 181

500 .,,. l,.., U,,,,u,,,.l,,,. l..,,l,,,, l.,,,l*.,.u,.,.l,,,, i,, *,i,.,. l I,,,i,,,

o.- ............ T=200 °C, + = 3.10 .3 I/see

300 =200 °C, e = 3.10 -5 1/sec

,~ 200 ~ ° C , e = 3.10 -3 1/sec

m f../" _ ....... A12xxx-T4, 20% SiCp T=300 °C, e = 3.10 -5 1/see Monotonic behavior

100 -- Experiment ..... Simulation

o m r , , , I , , , ~ 1 ~ , , , I ~ , , , I , , , , I , , , , I , , , , I , , , , 1 , , , ~ I , , , ~ I ~ , ~ I , , , , I , , , , I , , , , I , ,

0.000 0.002 0.004 0.006 0.008 0.010 0.012 0.014

Strain

Fig. 5. Monotonic stress-strain behavior of Al2xxx-T4 20% SiCp, experiments and simulation.

for the matrix and similar expressions may be written for the composite and the reinforcement.

The thermal expansion coefficients of the unrein- forced A12xxx-T4 and the SiC were 30 x 10 6 1/K and 5 × 10 -6 1/K respectively. The mechanical strain components were used in the stress-strain plots for the matrix, reinforcement and the composite. In the isothermal experiments the net strain component is equivalent to the mechanical strain component. In the in-phase thermomechanical fatigue (TMF IP) loading case the maximum temperature of the cycle coincided with the maximum mechanical strain. In the out-of-phase thermomechanical fatigue (TMF OP) loading case the maximum temperature co- incided with the minimum mechanical strain of the cycle.

The stress-strain behavior of the A12xxx-T4 were obtained by setting v r = 0 in the unified constitutive equation of the composite material model. Simulation of the monotonic stress-strain behavior of the unreinforced Al2xxx-T4 is summar- ized in Fig. 3. The stress-strain behavior was simulated at room temperature, 200 and 300°C, at

several strain rates. The strain rate and temperature sensitivity of the material behavior were successfully simulated with the constitutive equation. Simulations of the monotonic stress-strain behavior of the 15 and 20% reinforced materials are given in Figs 4 and 5, respectively. The elastic modulus and yield strength were predicted successfully. Predicted stress levels during inelastic deformation correlated well with the experiments for the strain levels con- sidered.

Simulation of the isothermal cyclic stress-strain behavior of A12xxx-T4 is presented in Fig. 6. At low strain rates, the matrix alloy undergoes cyclic soften- ing which is closely predicted with the model. In this case, K level decreases due to the r k term. A cyclic stress-strain behavior simulation of the reinforced material, A12xxx-T4 15% SiCp, is given in Fig. 7(a). The broken lines are the simulations and the solid line shows the experimental results. The simulation indi- cating the composite behavior, matrix behavior and reinforcement behavior in the loading direction is illustrated in Fig. 7(b). It is noted that the matrix strain range is higher than the composite strain range,

I I

400 "_Al 2xxx-T4, 0% SiCp First Cycle - IF, T = 200 °C /

.(Ae11)m~ h = 0.0152, ~ = 3.10 ~ 1/see

'~ ~ S ~::" - -- Experiment

200 Simulation ~ •

~ o

a -200

-400

i i i i I i i i i i i i i |

-0.010 -0.005 0.000 0.005 0.010 Longitudinal Mechanical Strain, (e11)mech

Fig. 6. Isothermal fatigue, cyclic stress-strain behavior of Al2xxx-T4, experiments and simulation, longitudinal stress-strain behavior.

182 KARAYAKA and SEHITOGLU: THERMOMECHANICAL DEFORMATION MODELING

o

D

r ~

3

400

200

(a) I

AIg-,x~-T4,15% SiCp IF, T=300 °C (AellC)m.ch = 0.01, ~= 3.10 ~ Usec

-200

400 i i

-0.010

/ F i r s t Cycle

~~'~Stable Cycle

, I , i , I

-0.005 0.000 0.005

Longitudinal Mechanical Strain, (£1iC)m~h

0.010

D

3

200

-200

(b)

I

- AI~-T4 ,15% SiCp IF, Tffi300 °C (A£11C)mech ffi 0.01, ~ = 3.10 "5 1/sec

400 i

-0.010

I

inforcement Composite

/ M a t r i x

I I -0.OOS 0.000 0.005

Longitudinal Mechanical Strain, (£n)mech 0 . 0 1 0

0

-100

(c) 200 . . . . I . . . . l . . . . I . . . .

.Am-~,-T4,15% SiCp IF, T=300 °(3 (AsnC)mech= 0.01, ~ = 3.10 ~ 1/sec / P a r t i c u l a t e

~ _ _ ~ # , f o m p o s i t e

: ) 08 " i i I I I | I I l I I I , l , , , ,

010 -0.005 0.000 0.005 ).010

Transverse Mechanical Strain, ( £ 2 2 ) m e c h

Fig. 7. Isothermal cyclic stress-strain behavior of Al2xxx-T4 15% SiCp, Experiments and Simulation: (a) longitudinal stress-strain behavior; Co) longitudinal stress-strain behavior of composite, matrix and

particulate; (c) transvers¢ stress-strain behavior of composite, matrix and particulate.

KARAYAKA and SEHITOGLU: THERMOMECHANICAL DEFORMATION MODELING 183

o

)

2.0

1.5

1.0

0.5

ii! .... - .

' ' ' ' I ' ~ ' ' I ' ' ' ' I ' ' ' ' I ' ' ' ° I ' ' ' ' i , , , , I

(t~n')~.¢h

A1 2xxx-T4 / SiCp ~ 0.030 IF, ~ =1.104 1/see Tffi300°C

o o

~ / / o.o12

0.006 0.003

, , , , | , , I I l I ~ I I I . . . . I , , , , I , , , , I

10 15 20 25 30 3,5

Vr, Percent Volume Fraction of SiCp

Fig. 8. Ratio of hydrostatic stress to effective stress in the matrix vs volume fraction of reinforcement isothermal fatigue simulations, T = 300°C.

while the composite stress range is higher than the matrix stress range. The matrix and the reinforcement stress-strain behaviors in the transverse direction are given in Fig. 7(c). The matrix stress and the reinforce- ment stress in the transverse directions are self equi- librating such that equation (14) is satisfied. The transverse stress in the matrix is approximately one fifth of the longitudinal stress component for the conditions considered.

To illustrate the multiaxiality of internal stress results further, the ratio of hydrostatic stress in the matrix to effective stress in the matrix, a~l# m, is plotted vs reinforcement volume fraction, vr, in Fig. 8. The hydrostatic stress, a~, is defined as (tr~ + trY2 + a~)/3 where trY2 and tr~ are transverse stresses and are equal in the case of particulate reinforcement. The results are presented at the maxi- mum strain of the cycle, whose ranges are given in the figure. When Vr = 0, a'~/~ ~ ratio is 0.33 under uniaxial loading. It is interesting to note that for the strain ranges of interest in fatigue research (0.003-0.012), appreciable hydrostatic stresses developed in the

matrix. When reinforcement volume fraction is 10%, the hydrostatic stresses are small except at large strains. For 20% volume fraction reinforcement the a~/e m ratio approached to 1. Beyond their influence on fatigue behavior, these results have implications in explaining early matrix fracture in metal matrix composites and lowered creep strain rates at elevated temperatures.

In Fig. 9, we illustrate the ratios of matrix strain range to composite strain range and the ratio of matrix stress range to composite stress range in the longitudinal direction. The simulations indicate clearly that the strain concentration in the matrix increases with increasing volume fraction of re- inforcement. The behavior at 20 and 300°C were examined; the mismatch in mechanical properties of matrix and the reinforcement is greater at 300°C compared to 20°C, and this is reflected in the results. The stress ratio, Acr~/Atr~t~ vs or results indicate that the stress matrix carries decreases with increasing v,, and the rate of decrease of Atr~ is higher at elevated temperatures.

A e l l m . . . . . . . . . . . .

A£1 t c

! ~ 1.0

~ A~II m Al2xxx-T4 / 8 iCp ............

0.5 IF, e =1.104 1/see, Aeu c ................................................. Aall c rO = o.oi

Simulation ~ T = 20°C ..... T = 300 °C

0.0 . . . . ' . . . . ! . . . . I . . . . I . . . . I , , , , I , , , , l i i i , | , , , , i

0 10 20 30 40

Vr, Percent Volume Fraction of SiCp

Fig. 9. Ratio of matrix strain (strcss) to composite strain (stress) isothermal fatigue simulations 7" = 20°C, T = 300°C.

AM 41/I--M

184 KARAYAKA and SEHITOGLU: THERMOMECHANICAL DEFORMATION MODELING

e~

D

"d

,-o

e~

400

200

-200

-400

i -0.010

I A12xxx-T4, 0% SiCp

TMF O P , (Ae n )mech m 0.015

T ~ n = 100 °C T ~ x = 200 °C

= 3.10 -4 1/see Experiment Simulation /

, , , I , , , , i I

-0.005 0.000

Fig. 10. TMF OP cyclic stress-strain behavior

I

S

0.005 0.010

Longitudinal Mechanical Strain, (en)mech

of AI2xxx-T4, experiments and simulation.

Simulations of the thermomechanical fatigue be- havior are presented in Figs 10 and 11 for the Al2xxx-T4 under Train= 100°C and Tmax=200°C conditions. A good correlation between the exper- iments and simulations was observed. In Fig. 10 the thermomechanical cycling is out-of-phase, denoted as TMF OP, where maximum mechanical strain co- incides with minimum temperature. In Fig. 11, ther- momechanical cycling is in-phase type, denoted as TMF IP, where maximum mechanical strain and maximum temperature coincides. The results of TMF OP simulations for A12xxx-T4 15%SiCp for Tmin -~- 100°C and Tma x = 300°C are presented in Fig. 12(a). We note that upon heating beyond 200°C, the yield strength decreased with increasing tempera- ture. The prediction of composite behavior was satis- factory. The transverse stress-strain simulation for the matrix is given in Fig. 12(b).

To understand the influence of mechanical prop- erty mismatch vs thermal property mismatch in the metal matrix composites under TMF IP and TMF OP conditions, it is worth studying Fig. 13(a) and (b) respectively. In these figures, the thermal expansion coefficient of matrix was 30 × 10 -6 K -l, while three

coefficients of thermal expansion, 30x 10-6K -1, 15 x 10-6K -l, and 5 x 10-6K -l were used for the reinforcement. The arrows indicate ranges of thermal and mechanical property mismatch for 0 r = 5 x 10 -6 K -~ case. When the thermal expansion coefficients of the particulate and matrix are equal (0 r= 0 m = 30 x 10 -6 K-l), the transverse stresses de- velop due to mechanical property mismatch only. As the thermal mismatch is increased, the transverse stresses in the TMF IP case gradually change sign, and the transverse stresses of opposite sign to the longitudinal stress component develop. In the TMF OP case, the behavior of transverse stresses differed. The transverse stresses due to mechanical property mismatch and thermal property mismatch add, and this resulted in transverse stresses that have the same sign as the longitudinal stress component.

The o'~/6 m ratio for the TMF IP case reflected the competition between thermal property vs mechanical property mismatch. When thermal property mis- match is dominant such as when Ae~l is small, the a~/# m ratio is near zero, indicating pure shear type of internal stresses. As the applied mechanical strain, (AE ~), is increased from 0.003 to 0.03, the mechanical

I I

400 - A12xxx-T4, 0% SiCp

N Train = 100 °C Tm~ = 200 °C 200 - e = 3.10-4 1/see /

- - Exper iment ...... Simulation

0 •

-200

-400 I I '

-0.010 -0.005 0.000 0.005 0.010

Longitudinal Mechanical Strain, (ezz)m~ h

Fig. 11. TMF IP cyclic stress-strain behavior of A12xxx-T4, experiments and simulation.

KARAYAKA and SEHITOGLU: THERMOMECHANICAL DEFORMATION MODELING

400

200

"d 0

"~ -200

-400

o) I I

. . . . . . r

TMF OP, (A£nC)mech =0.015 Tmi" = 1 0 0 °C Tin, x = 300 °C

= 3.10 "4 1/see Experiment Simulation

I I I I I | I I I I I I I I I I I I

-0.010 -0.005 0.000 0.005 0.010

Longitudinal Mechanical Strain, (Ene)mech

gl D

r . n

300

200

100

0

-100

-200

-300

(b) i i i , i i , i , | , , , , l . i . i i , i , , i i i i , i , , i , i i , | ,

Al2xxx-T4, 15% SiCp / TMF OP, (Aerie)mech = 0.015 //

T.~. = i00 °C Tma x = 300 °C // ..__._..Reinforcement

i I , , i , , , , I , , , , I , , , , I , , , , i , i i I I i I i i I i i i

°0.004 -0.002 0.000 0.002

Transverse Mechanical Strain, (era)mech

0.004

Fig. 12. TMF OP cyclic stress-strain behavior of A12xxx-T4 15% SiCp: (a) experiments and simulation; (b) simulation--transverse stress-strain behavior of composite, matrix and particulate.

185

mismatch resulted in a decrease in hydrostatic stress component and upon further increase in mechanical strain, the hydrostatic stresses increased in the tensile direction.

The hydrostatic stress to effective stress ratio of the matrix at the maximum temperature of a thermo- mechanical fatigue cycle is given in Fig. 14(a) and for TMF OP and TMF IP cases, respectively. In the TMF OP case, the hydrostatic stresses are in com- pression at Tin,x, and the magnitude of hydrostatic stresses increases with increasing Vr- Note that these stress ratios are higher than observed in the TMF IP case.

6. DISCUSSION OF RESULTS

Before discussing the TMF stress-strain behavior, it is important to compare our model with the proposed models on the strengthening of the com- posite relative to the matrix in isothermal loading. When Mori-Tanaka's method is used in the context of direct approach, as adopted in this study, the

effective property predictions coincide with Hashin-Shtrikman's lower bound for spherical re- inforcements [27-29]. However, this method is not necessarily a lower bound approach for all reinforce- ment geometries. For disk-like reinforcements, pre- dictions of the proposed method coincide with Hashin-Sthrikman's upper bound [27]. For both continuum mechanics [25, 26] and finite element ap- proaches [35] the difference between the upper and lower bounds increase with increasing mismatch be- tween the mechanical properties of the constituents. Therefore, the results of the current approach could agree closely with other continuum mechanics models or finite element models depending on reinforcement volume fraction, reinforcement geometry and mech- anical property mismatch. Our current model pre- dicted an increase in (0.2% offset) yield strength of 25% for the 20% volume fraction material. The model derived by Duva [36], predicts much less strengthening than observed experimentally. Duva's model predicts that, for a matrix material with a strain hardening exponent of 0, yield strength would

186 KARAYAKA and SEHITOGLU: THERMOMECHANICAL DEFORMATION MODELING

(a)

60

~4o E

20

0

-2o

~ -40

~I -60

I I I I I

T.~ = 300°C 8" = 30.10 -e Thermal

or = 15 10.a~"'~~ ~ Property • " ' ' M i s m a t c h

¢= S. 10"s...~e~~,J~ "~ ~ I~..Mechanical T Property

Al 2xxx-T4, 20% SiCp T~, =i00°C Mismatch

TMF IP, (Aen ~ )~h = 0.01 era= 30.10 ~ I/°C = 1.104 1/see

I I I I I -0.004 -0.002 0.000 0.002 0.004

Matrix Transverse Mechanical Strain, (e99rn)mech

~4 D

14

m

(b) r

60

~ermal ,I Property - ~ |

40 Mismatch |

20 Mechanical ~/ Property "-,~A

0 Mismatch

-20 A1 2xxx-T4, 20% SiCp

TMF OP, (Aenc )m~h = 0.01 -40 0m= 30.10 "~ 1PC

= 1.10 "4 l/sec -60

-0.004 -0.002

r I - r I

T.~=300°C •

t t

er= 30.104

~ = 15.10 "s

~ = 5.10-s

L------

T=~ =100°C

0.000 0.002 0.004

Matrix Transverse Mechanical Strain, (e~rn)mech

Fig. 13. (a) TMF IP, transverse stress-strain behavior in the matrix for different thermal expansion coefficient mismatch conditions of matrix and particulate, simulation. (b)TMF OP, transverse stress-strain behavior in the matrix for different thermal expansion coefficient mismatch conditions of

matrix and particulate, simulation.

increase by 8%, and for a matrix material with a strain hardening exponent of 0.1 the strengthening would be 12%. The matrix alloy considered in this study has a strain hardening exponent between 0 and 0.1; therefore, Duva's results underpredict the strengthening. Bao et al's [37] finite element work also predicts a small degree of strengthening, similar to Duva's results, for volume fractions less than 0.2. The reason for this discrepancy between our results and of others is not clear. The use of rigid particles in Refs [36, 37] may result in early yielding around the particles and a lower apparent yield strength than observed experimentally, however the results con- verge to correct limit stresses. The elastic modulus of the composite and the stress-strain behavior at low strains is not predicted accurately when rigid particles are used. The analysis of Papazian and Levy [38] using the finite element method, with E = 485 GPa for the SiC in aluminum alloy, predicted the elastic modulus and strength consistent with the model

presented in this paper (i.e. 25% increase in strength for volume fraction of 20%). The model in this study, at strain levels near few percent strain, predicts stress levels higher than the experiments. Nevertheless, in view of the low ductility of the metal matrix com- posites as well as the occurrence of fatigue damage at low strains, the model offers considerable insight into the behavior of metal matrix composites.

It is difficult to compare the predicted internal stresses due to thermal and mechanical property mismatch with experiments, since experimental re- sults on internal stresses are not available for particle reinforced composites. Residual stresses due to cool- ing were evaluated in a whisker reinforced aluminum composite in Refs [21, 39]. The sign and magnitude of stresses reported in this study are in general agree- ment with those reported.

The results indicate that the transverse stresses generated due to coefficient of thermal expansion mismatch exceed those due to mechanical property

KARAYAKA and SEHITOGLU: THERMOMECHANICAL DEFORMATION MODELING 187

0.5

0.0

-0.5

-1.0

-1.5

-2.{] 0

to) . . . . I . . . . I ' ' ' ' I . . . . I ' ' ' ' I ' ' ' ' I ' ' ' ' I

A1 2xxx-T4 / SiCp TMF OP, e ~l.10 "4 1]sec T~nffi IO0°C, Tm~= 300°C Simulation

(Aeu')=~h

~ ~ ~ o.oo3

o . o 1 2

~ 0.018

0.030 , , , , I , ! I t I , , o , I , ~ i I , , , . | , o o , I . . . o |

5 10 15 20 25 30 35

vr, Percent Volume Fraction of SiCp

~ D

1.5

1.0

0.5

..~ 0.0

-0.%

(b) 2 . 0 . . . . I . . . . I . . . . I ' ' ' ' I ' ' ' ' I ' ' ' ~ I ' ' ' ' I

A1 2xxx-T4 / SiCp TMF IP, ~ =1.104 1/sec T~inffi 100°C, Tm~= 300°C Simulation ( A £ 1 1 e ) m e © h

/ 0.030

--.----- 0.018

0.012 0.006 0.003

. . . . , . . . . . . . . . . . . . . , . . . . , . . . . . . . . .

5 10 15 20 25 3O Vr, Percent Volume Fraction of SiCD

35

Fig. 14. (a) Ratio of hydrostatic stress to effective stress in the matrix vs volume fraction of reinforcement at 300°C end ofa TMF OP cycle, simulation. (b) Ratio of hydrostatic stress to effective stress in the matrix

vs volume fraction of reinforcement at 300°C end of a TMF IP cycle, simulation.

mismatch. Furthermore, the thermal mismatch and mechanical property mismatch are additive in the TMF OP case which resulted in transverse stresses that were in-phase with the longitudinal component. It is noted that if the thermal mismatch strains increased, either by increasing the thermal expansion coefficient mismatch or increasing the temperature range, the transverse stresses, hence hydrostatic stresses, will increase considerably in the TMF OP case. In the TMF IP case, the transverse stresses due to mechanical mismatch are similar to the TMF OP case; however, with increasing thermal mismatch these stresses decreased. In TMF IP case [Fig. 13(a)], the increase in thermal mismatch resulted in a~ --* 0, then, upon further increase in mismatch a~ was negative at Tm~ end of the cycle. This resulted in compressive transverse stresses at the Tmax end of the cycle while the longitudinal stresses were tensile. Also, in the TMF IP cases, at Tm~ of the cycle, the hydrostatic stresses are of opposite sign of the effec- tive stress component for low to moderate strain

levels. As strain levels approach 0.03, the hydrostatic stresses are of the same sign as the effective stress component.

The progressive increase in compressive hydro- static stress in TMF OP with increasing volume fraction, v,, shown in Fig. 14(a), is beneficial in suppressing the creep damage. Under cyclic loading conditions of monolithic alloys, the benefits of com- pressive stresses under unsymmetric cycling have been documented [40-41]. In the composites, it is expected that these benefits will be accentuated due to the hydrostatic nature of the stress fields acting on the voids. The improvement in fatigue lives under TMF OP for the 20% SiCp material, reported in Ref. [23, 33], is consistent with the increased compressive hydrostatic stresses. The reinforced A12xxx-T4 had a TMF OP life X3 higher than the reinforced material life. The results shed light into the apparent improve- ment in the elongation, or superplastic behavior of A1-SiCp, noted in the literature, when samples were crept under thermal cycling conditions [42]. Upon

188 KARAYAKA and SEHITOGLU: THERMOMECHANICAL DEFORMATION MODELING

heating, due to thermal mismatch, the matrix devel- ops a compressive hydrostatic stress which suppresses cavitation. Although not the immediate motivation of our work, the analysis provided in this paper can be utilized to select temperature-stress histories con- ducive to enhanced superplastic flow of the metal matrix composites.

One important result in this study is that the strain range in the matrix relative to strain range in the composite is greater by approximately a factor of 1.20 for 20% SiCp reinforced composite. This ratio in- creases to 1.50 when the volume fraction of particu- late increased to 40% (Fig. 9). The dependence of the strain ratios, or strain concentrations, is similar at 20°C and at 300°C. The stress range in the matrix is appreciably smaller than in the composite. This ratio depends on the temperature, and for 20% volume fraction reinforcement at 300°C this ratio is approxi- mately 0.7. These results explain the shorter room temperature fatigue lifetimes obtained, based on strain range, for the 15% SiCp material compared to the unreinforced material reported in Ref. [43] on the same (Alcoa MB85) alloy. Similarly, the isother- mal fatigue experiments at 200°C (3 × 10 -3 I/s), reported in Ref. 33, exhibited lower lives for the reinforced material compared to the unreinforced A12xxx-T4.

Transverse stresses in the matrix increased as the strain amplitude and the volume fraction of particu- late were increased. When the strain amplitude was increased the effective stresses increased concurrently, however, the rate at which the hydrostatic stresses increased was greater. This results in tr~/t~ m ratios between 0.5 and 1.0 for applied strain ranges less than 0.01, while in the monolithic material this ratio would be 0.33 under tension.

At higher strain ranges, the hydrostatic stresses increased in the isothermal case (Fig. 8) and the increased buildup of hydrostatic stresses is primarily responsible for increase in composite strengthening. However, these hydrostatic stresses may result in fracture of the matrix ligaments and limit the tensile elongation of the material. The hydrostatic stress levels were comparable to those reported in finite element studies by Christman et al. [44] under monotonic loading conditions at similar strain levels.

The advantage of the model is its simplicity, and its capabilities for predicting the stress-strain behavior in the small strain range regime. A comparison of stress-strain simulations with experimental results indicate that the elastic moduli of the composite and yielding behavior were predicted closely for over 60 experiments. The initial slopes of the stress strain curves in the TMF experiments, which are influenced by elastic and thermal mismatch, were also predicted closely. Different reinforcement shapes can be ana- lyzed in the model by changing Eshelby's tensor although this has not been considered in this study.

7. CONCLUSIONS

1. The strengthening of the 15%SiCp and 20% SiCp composites relative to the matrix was pre- dicted closely over a broad range of temperature and strain rates.

2. Under isothermal cyclic loading conditions maximum hydrostatic stress to effective stress ratio in the matrix ranged from 0.4 to 0.8 for strain ranges of interest in fatigue research. These ratios are higher than the 0.33 value for the monolithic alloy.

3. The matrix strain range to composite strain range ratio was in the range 1.0 to 1.5, and the matrix stress range to composite stress range ratio varied from 1.0 to 0.5 depending on the volume fraction and temperature.

4. The shape of the TMF loops and the stress levels were predicted closely lending further credi- bility to the model proposed in this study.

5. Transverse stresses developed due to mechanical property and thermal property mismatches. Contri- bution of thermal property mismatch to the trans- verse stresses in the matrix would be greater than contribution due to mechanical property mismatch depending on Tm~ and T~n, and the applied mechan- ical strain. In the TMF OP case the transverse stresses generated due to thermal and mechanical property mismatch added. In the TMF IP case, transverse stresses due to the thermal and mechanical property mismatches are subtracted and their difference dic- tates the resultant transverse stresses.

Acknowledgments--This work is supported by a grant from the Ford Foundation. Collaboration with Dr John Allison, Material Science Department, Ford Motor Company is acknowledged.

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