Thermal stress analysis of a silicon carbide/aluminum composite

Thermal Stress Analysis of a Silicon Carbide/Aluminum Composite by E.E. Gdoutos, D. Karalekas and I.M. Daniel

ABSTRACT--Thermal deformations and stresses were studied in a silicon-carbide/aluminum filamentary composite at temperatures up to 370~ (700~ Longitudinal and transverse thermal strains were measured with strain gages and a dilatometer. An elastoplastic micromechanical analysis based on a one-dimensional rule-of-mixt_qres model and an axisymmetric two-material composite cylinder model was performed. It was established that beyond a critical temperature thermal strains become nonlinear with decreasing longitudinal and increasing transverse thermal-expansion coefficients. This behavior was attributed to the plastic stresses in the aluminum matrix above the critical temperature. An elastoplastic analysis of both micromechanical models was performed to determine the stress distributions and thermal deformation in the fiber and matrix of the composite. While only axial stresses can be determined by the rule-of-mixtures model, the complete triaxial state of stress is established by the composite cylinder model. Theoretical predictions for the two thermal-expansion coefficients were in satisfactory agreement with experimental results.

Introduction Operational conditions for advanced aerospace applications demand high performance materials capable of withstanding extremely high temperatures and operating loads. These materials must maintain dimensional stability and mechanical strength and toughness when subjected to wide temperature ranges. Various types of metallic (MMC) and ceramic-matrix fiber-reinforced composites have been proposed as candidate materials for such structural and machine applications. In many metal matrix composites, a high thermal-expansion mismatch between the matrix and fiber results in high thermal stresses. For example, high residual thermal stresses developed in the matrix during cooling from consolidation temperatures may result in premature yielding even before application of external loading. The study of the thermal-expansion behavior and the resulting thermal stresses is an important step in the characterization of the composite. A number of investigations have been reported on the problem of thermal-expansion behavior of MMC's. 1-6 They derived predictions of the linear thermal expansion in exact form or in terms of upper and lower bounds. However, a more limited number of investigations has dealt with nonlinear thermal expansion of metal matrix composites due to matrix yielding. 7-12

E.E. Gdouros (SEM Member) is Professor, Democritus University of Thrace, GR-671 O0 Xanthi, Greece. D. Karalekas is Manager, Research and Development, EUCAT, Athens, Greece. LM. Daniel (SEM Fellow) is Professor, Northwestern University, Evanston, IL 60208. Original manuscript submitted: June 14, 1990. Final manuscript received: March 21, 1991.

202 * September 1991

This paper presents the results of an experimental investigation of the thermal-expansion behavior of a SiC / A1 composite for temperatures up to 370 ~C (700 ~F). Theoretical predictions are made by two elastoplastic micromechanical models based on a one-dimensional rule-of-mixtures model and an axisymmetric composite cylinder model. The experimental results are in good agreement with the theoretical predictions. Experimental Procedure The material used in this investigation was SiC/6061-A1 composite which consisted of an aluminum matrix reinforced with silicon carbide fibers. The fibers were SCS-2 filaments of 140-t~m (5.6 x 10-3 in.) diameter (Textron Specialty Materials, Inc.). The material was obtained in the form of a 30.5 x 30.5-cm (12 x 12-in.) eight-ply unidirectional plate. It was produced by diffusion bonding. The fiber volume ratio was measured to be 0.44. The ply thickness was 0.178 mm (7 x 10-3 in.). Two types of experiments were performed to measure the coefficients of thermal expansion in the longitudinal and transverse to the fiber direction. In the first type a prismatic specimen of dimensions 15.2 x 1.27 cm (6.0 x 0.5 in.) was instrumented with strain gages (Micromeasurements gages tYPe EA-00-125AD-120) and a thermocouple. The specimen was thermally cycled between room temperature and 177~ (350 ~F). Longitudinal and transverse strains and temperature were recorded at 14~ (25~ intervals using a data logger. In the second type of experiment, longitudinal and transverse thermal strains were obtained at temperatures up to 370~ (700~ using a dilatometer. Special cylindrical specimens, 10-mm (0.39-in.) long and 3 mm (0.12 in.) in diameter, were used. For this purpose three layers of the composite plate were first glued together to produce sufficient thickness for preparation of the cylindrical specimens. The longitudinal and transverse thermal strain curves are shown in Fig. 1. The results from the two types of experiments are in agreement over the common temperature range, From Fig. 1 it is observed that the longitudinal thermal strain is linear with temperature up to a strain of approximately e = 0.002 which is close to the strain of the proportional limit of 6061-T4 aluminum at ambient temperature. Beyond that point the thermal strain increases at a lower rate indicating a decreasing coefficient of thermal expansion (CTE). It can be seen from Fig. 1 that the transverse thermal strain deviates from linearity at a strain of less than 0.001. Unlike the longitudinal strain the transverse thermal strain beyond the proportional limit increases at a faster rate, indicating a transverse CTE increasing with temperature. The variation of the coefficients of thermal expansion with temperature is shown

in Fig. 2. Up to 65~ (150~ the measured CTEs are constant, i.e., a, = 9.1 /zE/~ (5.03 # e / ~ and a~ = 17.8 ~ e / ~ (9.90 # d ~ For the micromechanical prediction of thermal deformation, the stress-strain behavior and the thermal strain behavior of the aluminum used to manufacture the composite at various temperatures are needed. During consolidation of the composite the aluminum experienced a heat-treatment process which changed its temper. It was suggested by the composite supplier (Textron) that the aluminum in the composite system has properties close to those of the T4 temper. Aluminum of type 6061-T4 was thus obtained and was tested in tension at various temperatures. Rectangular specimens of dimensions 20.32 x 1.27 x 0.16 cm (8.0 x 0.5 x 0.062 in.) were prepared with steel tabs bonded with a high strength adhesive (FM-36). The aluminum specimens were loaded in tension in an Instron testing machine with grips capable of withstanding temperatures up to 540~ (1000~ inside a thermal chamber. A water-cooling system was installed to cool the grips for tests at elevated temperatures. Special hightemperature strain gages were used to record axial and transverse strains. Axial strains were also monitored with a special high temperature extensometer (MTS Corp.). A data-acquisition system (Metrabyte Corp.) was used to acquire, process and plot the data. Stress-strain curves

for aluminum at various temperatures from 24 ~ to 288 ~ C (75 ~ to 550~ are shown in Fig. 3. Only the portions of the curves up to 0.01 strain are shown. It is seen that the proportional limit decreases with increasing temperature. Values of the proportional limit a , , of aluminum at various temperatures are shown in Table 1. For the nonlinear part of the curves the stresses corresponding to the same strain decrease with increasing temperatures. The thermal deformation of aluminum was determined using a dilatometer and a cylindrical' specimen 10-ram long and 3 mm in diameter. The thermal strain curve of aluminum up to 370~ (700~ is shown in Fig. 4. The curve is linear up to 232~ (450~ with a coefficient of thermal expansion c~,, = 24.8 ~ e / ~ (13.8 a e / ~ Above this temperature the strain-temperature curve becomes nonlinear and the CTE increases. Values of c~, at various temperatures are shown in Table 1.

Rule-of.mixtures Model A one-dimensional rule-of-mixtures (RAM) model was used for the prediction of the thermal-expansion behavior of the composite based on the constituent properties. Emphasis was placed on the prediction of the nonlinear part of the longitudinal and transverse strain-temperature curves. The silicon carbide filter is assumed to be isotropic and linear elastic up to failure. Its stress-strain-temperature behavior is given by oi

8' 7. oooo---;~'~

~~5

~

*****ccM

**o****o*****

2~ o ?

16o

~do

sdo

460

sdo

6do

Temperature, T, (eF)

~oo

Fig. 1--Longitudinal, e~, and transverse, E2, strain versus temperature curves as obtained by experiment (continuous lines) and predicted by the rule of mixtures model (circles) and the composite cylinder model (asterisks)

I

- -

15l~

o o

6. = ~ a,,

e. -

1

E,.

+ a, AT

+

(2a)

a= a,.,

where e= and am are the strain and of elasticity, o.,y the proportional and n material parameters and thermal expansion. The quantities functions of temperature.

(2b) stress, E . the modulus limit of aluminum, 3 a,. the coefficient of a . , , 3, n and tx.. are

Experiment ROM CCM

ooooo

-30 *****

~. l o .

where ej and as are the axial strain and stress, El the modulus of elasticity, al the CTE and A T the temperature change. It is assumed that o~tis constant. The aluminum matrix has a yield point much lower than the fracture stress of the fiber and exhibits a pronounced plastic deformation prior to fracture. Its thermomechanical behavior is described by the Ramberg-Osgood equation

o

o

o

a2

TABLE 1--VALUES OF THE QUANTITIES e~,y, c~=, 3 AND n OF ALUMINUM AI 6061-T4 AT VARIOUS TEMPERATURES

o

17o 2~o 37o ('e) ~6o 260 36o 4~o s6o 66o 700 Temperature,

T,

~

~

MPa

Ksi

24 121 171 232 288

75 250 340 450 550

86.2 79.3 72.4 60.3 41.4

12.5 11.5 10.5 8.7 6.0

/~e/~

#e/~

24.8 24.9 24.9 26.3 27.6

13.8 13.8 13.8 14.6 15.3

x 10-4

--

6.60 14.45 17.28 27.87 8.42

12.30 10.55 9.54 7.72 6.91

(OF)

Fig. 2--Variation of the two CTEs ~1 and ~z versus temperature according t o experiment and theoretical predictions by the ROM and CCM models

Experimenta/ M e c h a n i c s a 203

From the isostrain hypothesis and the equilibrium equation along the fiber direction the longitudinal stresses in the fiber, a's, and the matrix ~ ' . , for linear-elastic behavior are given by

a;

=

( o~,,,- cxt)EjE= V . A T Em V. + Es Vs

(or,,, - oq)EsE,, lisa T a~. = EmV. + E,Vs

(3b)

(4)

For higher temperature changes the matrix yields, and the stress a. in the matrix is determined from the following equation (Ra t" m .v '-")a"m + [1 + (-~s ) ( ~ ) ]

a,. +

[(c~m- ~ I ) E , A T . - {3a=,] = 0

(5)

The stress in the fiber is given by os

V, - a,. (--~/)

=

(6)

Having determined the stresses in the matrix and the fiber the longitudinal strain of the composite is determined from either eq (2a) or eq (2b). The transverse strain of the composite is calculated by ~,

=

c,, V s +

E,. V,

P!

(8a)

- ~ as + asA T r~s

e,= = - ~- o,.+ c~=AT

(8b)

where vj and v., are Poisson's ratios of the fiber and matrix, respectively. While vs is constant, v,. increases in the nonlinear range from its elastic value up to the limiting value of 0.5 for an incompressible material. In the transition region v. is determined as follows" E, v,. = 0.5 - (0.5 - v=') -~-

I

R1 K~ 0 2v~K~R~

- R1 -/(2 /(2 2 v 2 K 2 ( R ~ - R~)

In the two-material composite-cylinder model (CCM), the representative volume element for the micromechanical analysis of low fiber volume composites consists of an inner solid cylinder, simulating the fiber, and an outer hollow cylinder, representing the matrix. A fiber-reinforced composite is arbitrarily characterized as low fiber volume if the fiber-volume ratio is less than 65 percent. It is assumed that the two components are perfectly bonded at the interface. The CCM has been proposed by Hill, TM Hashin and Rosen" and Whitney and Riley? ~ The model has been used for the study of the elastoplastic behavior of two- and three-material composite cylinders by Hecker et al. ",18 In the present study the CCM is used for the study of the thermal-expansion behavior of a SiC/A1 composite. The cases of elastic and elastoplastic deformation of the matrix cylinder are considered separately. The inner cylinder always remains linearly elastic. Elastic Behavior

When both components of the composite cylinder are linearly elastic, the following equations for the radial displacement u, the radial and circumferential stresses or. and or0 and the axial stress a. are obtained from the thermoelasticity solution of the problem." u =Ar

B

+--

(10a)

r

a, = K [ A - ( 1 - 2 v )

B + ve,-(l+v)uAT]

(10b)

a8 = K [ A + ( I - 2 v )

B + w.-(l+v)otAT]

(lOe)

(9)

a, = K [2vA + (1 - v) e, - (1 + v)o~AT]

0

- R? l K1 vl - / ( 2 v2 (1 - 2 v2)K2R~ K, v~ - ( 1 - 2v~)K2R; 2 K ~ ( 1 - v,)R~ + K 2 ( 1 - v2)(R~ - R~) 0

[K,(1

o

(10d)

In these equations r is the radial distance from the center of the cylinder, e. is the axial strain, A and B are constants to be determined from the boundary conditions of the problem and K is the bulk modulus. The boundary conditions of the problem imply coritinuity of the radial displacement u and the radial stress or, along the boundary of the two materials. Furthermore, they should be finite at r = 0. Finally, on the outer surface the radial stress o. is assumed to be zero. For a temperature change A T, the constants A~, B~ (j = 1,2 refer to the inner and outer cylinder respectively) and the axial strain e. is determined from the following matrix equation.

KI(1 + v,)cq -/(2(1 + v2)cx2 (1 + v2)c~2K2 + vl)alR~ + K2(1 + v2)a2(R~- R~)

204 9 September 1991

Model

(7)

where the transverse strains in the fiber, c,t, and the matrix, e,,, are given by e,s =

Composite.cylinder Introduction

(3a)

where Vj and V= are the volume fractions of the fiber and matrix, respectively. The temperature at which the matrix starts to deform plastically is determined from the relation

~ ' . = ~.,

where v~ is the elastic value of the Poisson's ratio and E and Es are the elastic and the secant moduli, respectively.

]

][A] A2 B~ e,

=

(11) AT

w i t h Ba = 0. R, and R2 denote the radii of the inner and outer cylinders, respectively. After determination of Aj, B~ and e., the stresses and radial displacement in the two materials are calculated by eqs (10). The elasticity solution of the problem is used in conjunction with the von Mises yield criterion for the determination of the critical temperature at which the most stressed elements of the matrix along the fiber-matrix interface enter into the plastic domain of deformation. This temperature is determined from the equation a.lj = a..,

1

[ ( a , - o , p + ( o , - Oo) ~ + ( o o - a,)~] ~"

(13) The stress tr,,, the proportional limit of aluminum, is a function of temperature. Elastic=plastic

Behavior

When the temperature is increased beyond a critical

value plastic zones in the form of concentric cylindrical layers starting from the fiber-matrix interface are developed in the aluminum cylinder: The deformation in the aluminum becomes inhomogeneous and an elastic-plastic analysis is required for the determination of the stress components and the extent of the elastic-plastic boundary. The deformation theory of plasticity is used in conjunction with the von Mises yield criterion and the isotropic hardening rule for the solution of the elasticplastic problem. It has been shown that the results obtained by this theory coincide with those of the flow theory for proportional loading. However, for nonproportional monotonic loading without unloading, predictions based on deformation theory are quite reasonable. The fundamental assumption made is that the effective stress-strain curve a, lt = f ( d e , j l ) for an element in a triaxial state of stress coincides with the stress-strain curve in uniaxial tension. The effective strain is defined by de,ll --

[(de, - de,) ~ + (de, - deo) 2 +

,rE (1 + v,)

(14)

(deo - de,)2] '/2

30-

1

de, = -~, [ d o . - v ( d o , + doo)] 1

de, = - ~ . I d o , -

v(doo + do.)]

(15)

1

deo = if, [ d o o - v(da, + do,)l

(12)

where a, jt is given by ~.11 = - ~ -

where the value of Poisson's ratio v, is determined from eq (9). The stress-strain relations take the form

where de., de, and deo are the total, elastic plus plastic, strain increments and do,, da, and do0 are the stress increments. E, is the tangent modulus of the uniaxial stress-strain curve of the material in tension after the linearity point. In the elastic region E, is equal to the modulus of elasticity. For a work-hardening material beyond the proportional limit, E, decreases gradually as plastic deformation advances. Equations (15) along with the equations of equilibrium and compatibility are used for the solution of the elasticplastic problem. It is thus evident that the deformation theory of plasticity is actually a nonlinear elasticity theory with changing values of modulus of elasticity and Poisson's ratio depending on the amount of plastic deformation. This observation led to the following solution of the elastic-plastic problem of the composite cylinder. The aluminum ring was divided into N concentric layers with each layer having different elastic modulus and Poisson's ratio (Fig. 5). An elasticity analysis of an N + 1 material composite cylinder was then performed following an analogous procedure as in the case of the two-material cylinder. This solution served as a subroutine to a computer program written for the elastoplastic solution of the problem. Having determined the critical temperature at which the first layer at the fiber-matrix interface yields, the temperature is increased in small steps. For each step, the tangent modulus and Poisson's ratio of each layer are determined from the value of the equivalent strain in conjunction with the uniaxial stress-strain curve of the material in tension. For each layer, stresses and strains are determined at a representative point at the middle of its thickness. An iterative analysis was performed for each temperature step until convergence was achieved. For each temperature, the values of tangent modulus and Poisson's ratio for each layer were updated according to the value of the equivalent strain. In this way the complete history of stress and strain along the

~ r

ZOQ

250 350 451) 550

2520-

% .g L

s2 0.'2

'

0.'4 ' 0.'6 S t r o i n , r (Yr

'

O.~

'

1.0

Fig. 3--Stress-strain curves of 6061-T4 aluminum at different temperatures up to one percent

o

1~o 26o 36o ~6o s6o e6o 70o Temperature,

T, (OF)

Fig. 4--Strain-temperature curve of 6061-T4 aluminum up to 700 ~ F (370 ~C)

Experimental Mechanics

9 205

radius of the composite cylinder was determined as the temperature was increased incrementally.

Results and Discussion The various material parameters of silicon carbide and aluminum entering into the equations discussed before are needed for the analysis. The SiC remains linearly elastic up to failure and its thermomechanical constants are independent of temperature. On the other hand, aluminum exhibits pronounced plastic deformation and its thermomechanical parameters are temperature dependent and are obtained expeYimentally. Values of the elastic properties of SiC and A1 are shown in Table 2, while values of o,y, ~, n and ~,, are given in Table 1. E= is equal to 10 Msi for all temperatures. The post yield stress-strain curve of aluminum was represented by a polynomial of the form 5 o =

~,

C.e"

(16)

n=0

where the coefficients C, are temperature dependent. They were determined at various temperatures and then a least-squares regression analysis was performed to obtain the functions C, = C,(T). Both micromechanical models discussed before were used. For the two-material composite cylinder model, the outer radius Ra was taken to equal 1.51 R, which corresponds to a fiber volume ratio of 0.44. For the elastoplastic analysis, the matrix cylinder was divided into eight layers, each of thickness equal to 0.064 R~ (Fig. 5). The critical temperature at which the aluminum matrix starts to deform plastically was first determined by the two micromechanical models [eqs (4) and (12)]. It was found that ( A T ) = , = 74~ and 66~ (165~ and 150~ for the R a M and CCM models, respectively. In the CCM an incremental stress-strain analysis based on the deformation theory of plasticity as described previously was performed. The temperature was increased in steps of 13.89~ (25~ and the complete three-dimen-

sional stress distribution at the m i d p o i n t of the thickness of each layer of the aluminum ring was determined. Predictions of the longitudinal and transverse thermal strain of the composite by the R a M and the CCM are shown in Fig. 1 by circles and asterisks, respectively, together with the experimental results. The variation of the two CTEs, a~ and ct2, with temperature as it was determined experimentally and predicted by the two models is shown in Fig. 2. From Figs. 1 and 2 it is observed that the theoretical predictions are in good agreement with the experimental results for both the longitudinal and transverse strains. Furthermore, it can be seen that the predictions based on the CCM are closer to experimental results than those of the R a M . This result is attributed to the fact that in the CCM, the complete three-dimensional stress distribution is considered, while in the R a M , the effect of transverse stresses is omitted and only the longitudinal stresses are accounted for. As mentioned before in the elastoplastic analysis of the CCM by the deformation theory of plasticity, each layer in the aluminum ring is considered to be an elastic material with varying modulus of elasticity, E , , and Poisson's ratio, ~,, depending on the amount of plastic deformation. The variation of E , and v,, along the thickness of the aluminum ring ( 1 . 0 0 < r / R ~ < 1.51) for various temperatures is shown in Figs. 6 and 7. As plastic

11"

o=

~, 9"-60

~~

~8" UlE "

Material

r

200 (93) 225 007)

~5.

250 (121) 275

4"

x

~

1" 01.0

1.=2

1.'1 1.3 Normalized Radius,

1.~t r/R1

3OO 35OO'n') 500(250) 1.5

Fig. 6--variation of tangent modulus

E. along the thickness of the

oy

aluminum matrix ring according to

CCM at different temperatures /~E/~

AI 6061-T4 SCS-2

v

175 (79)

7-"45

--

TABLE 2--PROPERTIES OF ALUMINUM AI 6061-T4 AND SILICON CARBIDE SCS-2 FIBER AT ROOM TEMPERATURE E

~ (% 75 (24)

-75 10" "~

24.8

4.9

#el~

13.8 2.7

GPa

Msi

--

MPa

Ksi

69 365

10 53

0.33 0.22

76

11 0.50

~

50 077) 300 (149)

0.'$5

250 (121) m "O ~ 0.40'

225 (107)

=oo (95)

o Fig. 5--Composite cylinder

model

~

SCS~;~ -2 AL6~-TM~ 4

175(79)

x 0.35"

7s (24) 0.30

1.o

1:1

1.~

Normalized

1.'a

1),

~.~

Radius, r/R1

Fig. 7 - - V a r i a t i o n of P o i s s o n ' s ratio ~., along the thickness of the aluminum matrix ring according to CCM at different temperatures

206 9 September 1991

deformation advances from the fiber-matrix interface and spreads through the aluminum cylinder with increasing temperature, E,, decreases from its elastic value of 10 Msi for temperatures up to 66~ (155~ to the value of 0.2 Msi for T = 370~ (700~ On the other hand, u,, increases from its elastic value of 0.33 to its limiting value of 0.50. Figure 8 shows the variation of the axial, ~,, radial, ~,, and circumferential, ~, stresses along half the radius of the C C M for A T = 79~ (175~ All three stress components are constant in the fiber, while they vary along the thickness of the ring. The axial stress is tensile in the fiber and compressive in the matrix with increasing magnitude from the fiber/matrix interface to the outer radius of the composite cylinder. The radial stress is tensile in both the fil~er and matrix, while the circumferential stress is tensile in the fiber and compressive in the matrix. In the 1~iber it is equal to the radial stress,while in the matrix it takes its m a x i m u m value at the fiber/matrix interface. Note that high transverse stresses of the same order of magnitude as the axial stress are developed in the matrix and the fiber. These stresses are ignored in the R O M model. The variation of the o,, oo and ~, stresses

along the thickness of the aluminum matrix for various temperatures is shown in Figs. 9-11. In Fig. 11 the variation of a, stress is shown in a piecewise form, as it was determined in the eight layers of the matrix along which it is constant. Observe from Figs. 9-11 that all stresses increase with temperature but with a decreasing rate. The variation with temperature of the stress components ~. in the fiber and the matrix (o,t and tr,,,) according to the CCM and the ROM models and the stress o , / = try! in the fiber according to the CCM is shown in Fig. 12. Note that the stresses approach a plateau as the temperature approaches 370 ~C (700 ~F). The axial stress predicted by the two models differs by approximately 15 percent at high temperatures. Finally, Fig. 13 shows the variation of the effective strain ~o~ along the thickness of the matrix for different temperatures. This strain decreases from the fiber/matrix interface toward the outer radius of the composite cylinder and increases with temperature. Conclusions

A combined experimental/theoretical analysis was undertaken of the thermal-expansion behavior of a SiC/A1 filamentary composite from room temperature up to 370 ~C (700 ~F). The longitudinal and transverse thermalexpansion coefficients were determined experimentally.

2O Fiber

Matrix

15".101

10-

10. -50

b"

500 400 3O0 250 209 175

76-

-10

260 20'$ 149 121 93 79

-52 x

-Io.o o.'3

2~ 1-

o.'s 0.'8 i.o 1.3 1.5 Radius, r

Fig. 8 - - V a r i a t i o n of the axial, a,s,,., radial cr,/,., and circumferential crej,,, stresses in the fiber and matrix along half the f i b e r / m a t r i x cross section for a t e m p e r a t u r e difference A T = 79 ~ C (175 ~ F)

Normalized Radius, r/R1.

Fig. l O - - V a r i a t i o n of the radial stress ~,,, along the thickness of the matrix at different temperatures as predicted by the CCM

20.0 v17"5~

~ r

~

22 ~o

7oo 43~1)

20 "130

soo (290) 400 (204)

oF r

I 15.0

700 4371)

sac 429o)

~ 12.5"

400 4204)

~

29o 4121)

300 (149)

10.0 "

200(93) P

7,5-

175 4 79)

~

300 (149)

.

250 4121)

~ "___r-~ 9~_ -90 ~ 12-

' 2oo (93) 175 (79)

5"01.0

1.'1 1.2 1.3 1.4 Norrnolized Rodius, r/R1

1.5

Fig. 9 - - V a r i a t i o n of the circumferential stress - cro,. along the thickness of the matrix at different t e m p e r a t u r e s as predicted by the CCM

'~

1.'~

,.'~

1.~

1.~

1.s

NorrnoEized Radius, r/R~

Fig. 1 1 - - V a r i a t i o n of the axial stress - a.,. along the thickness of the matrix at different t e m p e r a t u r e s as predicted by the CCM

E x p e r i m e n t a l M e c h a n i c s * 207

30 200 _ _

16'

CCM

150

v

~

*

-t~

lo.

~

15. lOO lO

o-n=o-w 5o

'r pc)

c

700 (371)

>=

~5o (~+~) 900 015) ss9 (2aa) 5oo (26o)

'-

"~ 050

Igo

2go

3go

450

sso

950 c)750

Temperature, T, (~

Fig. 12--Variation of the axial stress in the fiber, c,.~, and the matrix, c,,m, predicted by the CCM and the ROM models and the radial, ~,s, and circumferential, o0j, stresses in the fiber versus temperature. ~,,, for the CCM is the axial stress at the midpoint of the thickness of the aluminum phase

Elastoplastic micromechanical analyses based on the rule of mixtures and a composite cylinder model were performed. The main results of the present investigation may be summarized as follows. (1) Above a critical temperature of 66~ (150~ the longitudinal and transverse thermal strains become nonlinear resulting in decreasing longitudinal and increasing transverse CTEs with temperature. (2) The beginning of nonlinearity of the strain-temperature curves coincides with the development of plastic deformation in the aluminum matrix. (3) Elastic-plastic micromechanical analyses based on the rule of mixtures and the composite cylinder model were developed. In the analysis the changing material properties of the aluminum matrix, including the stressstrain curve and the CTE, were taken into consideration. (4) The complete three-dimensional stress distribution in both the fiber and the matrix was determined from the micromechanical analysis. (5) High triaxial stresses above the critical temperature resulting in plastic deformation of the matrix were developed. Plastic stresses are higher at the fiber/matrix interface and decrease away from it. (6) The transverse radial and circumferential stresses in the fiber and the matrix predicted by the CCM are of the same order of magnitude as the axial stresses and should not be ignored. These stresses are responsible for the development of excessive plastic yielding in the matrix. The ROM model does not take into account the transverse stresses. (7) The axial and transverse stresses developed in the fiber and matrix increase with temperature and approach limiting values. (8) Theoretical predictions by both models and the experimental results for the longitudinal CTE were in satisfactory agreement. However, for the transverse CTE the predictions of the CCM are much closer to the experimental values than those of the ROM model. Deviations between the predictions of the ROM model and experimental results are large at higher temperatures.

Acknowledgment The work described here was sponsored by the NASALewis Research Center, Cleveland, OH. We are grateful

208 9 September 1991

450 (232)

9,oo (2o4) 359 (177) 30o (149) 250 /121} 200 9,5 175 79

x "~ 1.0

1,2 13 1.4. Normalized Radius, r/R1

I.=1

1.5

Fig. 13--Variation of the matrix effective strain ~e//along the thickness of the matrix at different temperatures as predicted by the CCM

to Dr. C.C. Chamis of NASA for his encouragement and cooperation and to Mrs. Yolande Mallian for typing the manuscript.

References 1. Levin, V.M., "On the Coefficients of Thermal Expansion of Heterogeneous Materials" (in Russian), Mekhanika Tverdogo Tela, 2, 88-94 (1967). 2. Van Fo F#, G.A., "'Elastic Constants and Thermal Expansion of Certain Bodies with Inhomogeneous Regular Structure," Soviet Physics, Doklady, 11, 176 (1966). 3. Van Fo Fy, G.A., "'Basic Relations o f the Theory of Oriented Glass-reinforced Plastics with Hollow Fibers" (in Russian), Mekhanika Polymerov, 2,, 763 (1966). 4. Schapery, R.A., "'Thermal Expansion Coefficients of Composite Materials Based on Energy Principles, '" J. Camp. Mat., 2, 380-404 (1968). 5. Rosen, B.W. and Hushin, Z., "'Effective Thermal Expansion Coefficients and Specific Heats of Composite Materials, "" Int. J. Eng. Sci.: 8, 157-173 (1970). 6. Dvorak, G.J. and Chen, T., "'Thermal Expansion of Three-phase Composite Materials, "' J. AppL Mech., 56, 418-422 (1989). Z Hoffman, C.A., "'Effects of Thermal Loading on Fiber-reinforced Composites with Constituents of Differing Thermal Expansivities, '" J. Eng. Mat. Tech., 95, 55-62 (1973). 8. Dvorak, G.J., Rao, M.S.M. and Tam, J.Q., "'Yielding in Unidirectional Composites Under External Load and Temperature Changes," J. Camp. Mat., 7, 194-216 (1973), 9. Flora, Y. and Arsenault, R.J., "'Deformation of Al-SiC Composites due to Thermal Stresses, "' Mat. Sci. Eng., 75, 151-167 (1985). 10. Kural, M.H. and Min, B.K., "'The Effects of Matrix Plasticity on the Thermal Deformation of Continuous Fiber Graphite~Metal Composites," J. Camp. Mat., 18, 519-535 (1984). 11. Min, B.K. and Crossman, F.W., "'History-dependent Thermomechanical Properties of Graphite~Aluminum Unidirectional Composites, "" Composite Materials: Testing and Design (Sixth Conference), ASTM STP 787, ed. LM. Daniel, Amer. Sac. for Test. and Mat., Philadelphia, 371-392 (1982). 12. Tompkins, S.S., "'Techniques for Measurement of the Thermal Expansion of Advanced Composite Materials, "' Metal Matrix Composites: Testing, Analysis and Failure Modes, ASTM STP 1032, ed. W.S. Johnson, Amer. Sac. for Test. and Mat., Philadelphia, 54-67 (1989). 13. Nadai, A., Theory of Flow and Fracture of Solids, McGraw-Hill, 2nd Ed., 387 (1950). 14. Hill, R., "'Theory of Mechanical Properties of Fibre-strengthened Materials: LElastic Behavior, II.lnelastic Behavior," J. Mech. Phys. Solids, 12, 199-218 (1974). 15. Hashin, Z. and Rosen, B.W., "'The Elastic Moduli of Fiberreinforced Materials, "' J. Appl. Mech., 31, 223-232 (1964). 16. Whitney, J.M. and Riley, M.B., "'Elastic Properties of Fiber Reinforced Composite Materials, ""AIAA J., 4, 1537-1542 (1966), 1Z Hecker, S.S., Hamilton, C.H. and Ebert, L.J,, "Elastoplastic Analysis of Residual Stresses and Axial Loading in Composite Cylinders, "" J. Mat., 5, 868-900 (1970). 18. Hamilton, C.H., Hecker, S.S. and Ebert, L.J., "Mechanical Behavior of UniaxiaUy Loaded Multilayered Cylindrical Composites,'" J. Basic Eng., Trans. ASME, Ser. D, 93, 661-670 (1971).