Uniaxial creep of long fibre reinforced metal matrix composites

Composites Engineering. Vol. 4, No. 12, pp. 1241-1255. 1994 Copyright 0 1995 Elsevier Science Ltd

Printed in Great Britain. All rights reserved 0961.9526/94 $7.00+ .oO

Pergamon

096%9526(94) EOOO73-5

UNIAXIAL CREEP OF LONG FIBRE REINFORCED METAL MATRIX COMPOSITES

J. F. Durodola,‘* C. Ruiz“ and B. Derbyb ‘Department of Engineering Science, *Department of Materials, University of Oxford,

Parks Road, Oxford 0X1, U.K.

(Received 21 March 1994; final version accepted 1 July 1994)

Abstract-A generalisation of McLean’s (Directionally Solidified Materials for High Temperature Service, 1983, The Metals Society, London) uniaxial creep model which accounts for the effects of inherent residual stresses, matrix primary creep and the treatment of the effect of possible instantaneous matrix yield upon loading is presented. The model and numerical micromechanics approach are used to analyse the uniaxial creep of Ti-6AI-4V/SiC metal matrix composites. Excellent agreement is obtained between the analytical and numerical models and predicted results are in reasonable agreement with published experimental results.

NOTATION

C

f” n s’

T V

6”

E’ u

V

00

elasticity constitutive tensor Young’s modulus hardening law (eqn (4)) creep law exponents deviatoric stress tensor time temperature volume fraction thermal coefficient of expansion Kronecker delta strain tensor in eqns (l)-(8); direct axial strain component in eqns (lo)-(30) deviatoric strain tensor stress tensor in eqns (l)-(8); direct axial stress component in eqns (lo)-(30) plastic multiplier Poisson’s ratio infinity

Subscripts

1,2 creep stress and time exponents, respectively c, f,m composite, fibre and matrix i, j, k, I = 1,2,3 material axes directions 0 initial strain R residual Y yield point

Left and right superscripts

equivalent c, e, p, 0 creep, elastic, plastic and thermal t time A increment

1. INTRODUCTION

The use of predictive methods for the estimation of creep of metal matrix composites such as Ti-6-WSiC based on the properties of the titanium alloy Ti-6Al-4V and the reinforcing silicon carbide, Sic, fibre is helpful for the design and lifting analysis of high

*Present address: Oxford Brookes University, School of Engineering, Gypsy Lane Campus, Headington, OX3 OBP.

1241

1242 J. F. Durodola et al.

temperature components such as turbine disks and bladed disks which can be made from these materials.

Although several analytical micro-mechanics models (Hashin, 1966, 1970; McDanels et al., 1967; De Silva, 1968; Soliman, 1969; Lou and Schapery, 1971; McLean, 1972; Kelly and Street, 1972; Garmong, 1974a,b; McLean, 1983; Pan and Weng, 1993) have been proposed for the creep of composites, two of these are widely known for predicting uniaxial creep of continuous fibre reinforced metal matrix composites (FRMMCs). These are the McDanels’ (1967) model and McLean’s (1983) model. McDanels considered a creeping fibre in a creeping matrix. In McLean’s model, the creep of the composite is considered to be due to matrix creep only, that is the fibre is taken to be only linearly elastic in response to the loading. In both models, iso-strain compatibility conditions are assumed; the rate of change of composite strain is equal to the rate of change of strain of the fibre and also equal to the rate of change of strain of the matrix. The average axial stress in the matrix and in the fibre are assumed to satisfy Voigt’s rule of mixtures.

The models do not incorporate residual stress and matrix primary creep effects. It should be indicated however that McLean (1986) gave a coupled system of differential equations governing the creep of an FRMMC in which residual stresses can be introduced as initial conditions during the solution of the system of equations. The models are nevertheless uniaxial and neglect lateral deformation effects. Ideally finite element method based micromechanics analysis (Crossman and Karlak, 1974, 1976; Foye, 1975; Schaffer and Adams, 1981; Min and Crossman, 1982) allow multiaxial stress, residual stress effects and time dependent matrix creep strain rate to be considered readily.

In this work a generalisation of McLean’s (1983) creep model which accounts for the effects of residual stresses in the composite, primary creep of the matrix, as well as the treatment of possible instantaneous plastic flow of the matrix on loading (if the external load is sufficiently large) is presented. The magnitude of residual stresses are assumed to be due to a typical manufacturing temperature and cooling rate after Nimmer et al. (1991). Results of the model are compared with predictions based on micromechanic finite element analysis and with published experimental results for Ti-6-4/SiC composites under different loads and temperatures.

2. NUMERICAL APPROACH

The numerical micromechanics finite element approach (Foye, 1975; Schaffer and Adams, 1981) considers a uniform distribution of fibres (e.g. silicon carbide) in the matrix (e.g. Ti-6Al-4V) alloy. The uniform distribution was assumed to be maintained during the deformation experienced by the composite during fabrication and subsequent loading. Square array and hexagonal array periodic distributions (Fig. l(a) and (b)) were considered in this work in order to highlight possible effects of geometric differences in these models. Advantage is taken of the symmetry imposed by the constraints implicit in the uniform distribution so that only simple representative triangularly shaped parts of the unit cells, as shown in the figure, need to be analysed. Figure l(c) shows a typical finite element mesh used for the analysis of 35% fibre fraction hexagonally packed composite.

The manufacture of Ti-6-4/SiC composites by diffusion bonding or hipping takes place in the neighbourhood of 85O”C, and the service temperature conditions considered in this work are well below this temperature. At these temperatures the creep of Sic fibres is insignificant (Simon and Bunsell, 1984). The fibre was therefore assumed to be linearly elastic during manufacture and subsequent loading.

Creep is a thermally activated process and depending on the magnitude of applied stress, the instantaneous response of the matrix can consist of elastic and time independent plastic strains. Thus, at a given time t and assuming small strains, the total matrix strain can be related to elastic, plastic, creep and thermal strains by

‘& = l&e + t&p + f&C + tsO (1)

where the superscripts e, p, c and 8 are used to denote elastic, time independent plastic,

Uniaxial creep of metal matrix composites 1243

Hexagonal array Square array

(4 (W

000

olBP 000

Fig. 1. Transverse section of (a) hexagonal array and (b) square array distribution of fibres in the matrix considered; simplest models analysed are the triangular regions shown within the unit cells: (c) typical finite element mesh used for the analysis of 35% fibre volume fraction hexagonal array

pack composite.

creep and thermal strains, respectively. The constitutive law relating stress to strain using tensor notation is given by

‘~ij = ‘Cij,, (l&k, - ‘&fll - ‘&Ckr + ‘E~[) (2) where ‘C,, are the usual components of the elasticity tensor components (Snyder and Bathe, 1981). The incremental changes in the thermal, plastic and creep strains over a given time At are given by

A$, = ( t+ A‘o t+A‘T _ t, fT)ak, (3)

a(Y) A@, = d(‘A) f a( ok,)

(4)

A.$, = At[(l - p) ‘& + /3(“A’E;r)] (5) respectively and Tin eqn (3) is temperature in K. In eqn (4), dl is a scalar multiplier and fis a hardening law. The dot notation in eqn (5) is used to denote derivative of strain with respect to time and hence strain rate; 0 s /I s 1 is a scalar constant, the exact value of which depends on the explicit-implicit integration scheme used in the time incremental solution of the non-linear problem. The left superscript t and t + At notation in eqns (l)-(5) and in what follows is used to denote quantities at time t and t + At, respectively.

In the present analysis, the elastic constants in ‘C,,, and the thermal expansion coefficient (Y, as well as the yield stress used in the plastic flow definition, are taken as temperature dependent, Table 1. The creep strain rate for the Ti-6-4 matrix was approximated by

1244 J. F. Durodola et 01.

Table 1. Material properties used in the analysis at room and elevated temperatures

Temperature (K) &?a) v,

%I @fPa)

298 393 0.2 4.1 114 0.33 8.9 900 700 374 0.2 4.7 85 0.33 - 502 811 354 0.2 4.7 69 0.33 - 443

1173 354 0.2 4.7 21 0.33 10.5 35

using a strain hardening power law obtained from an empirical creep strain time relation- ship given by Nimmer et al. (1991). This is expressed as

‘i = a2[a0a’l t@“z-l) exp - (a3/T)J1’Q (6)

where ‘a and $I are equivalent stress and strain quantities given by

respectively, s$ and E& are deviatoric stress and strain tensor components, respectively; T is temperature in K, and a,, aI, a2 and a3 are constants given by

a,, = 3.6 x lo9 (MPa)-3.403 (h)-0.92s1; a, = 3.403;

a3 = 3.6 x 104(K). (9)

a2 = 0.9251 and

The ABAQUS finite element program (Hibbitt et al., 1992) with the creep law described by eqn (6) was used for the numerical micromechanics analysis. Twenty-node solid elements were used in the analysis; elastic perfectly plastic matrix behaviour following the Prandtl Reuss yield criterion and explicit time integration scheme with creep strain error tolerance of 10 - ’ were specified. A detailed description of typical finite element incremental solution methods for problems involving thermal, elastic, plastic and creep strains can be found in Snyder and Bathe (1981).

3. MODIFIED MCLEAN’S MODEL

McLean’s (1983) model for uniaxial creep of continuous fibre reinforced metal matrix composites is based on a steady state power law creep for the matrix, an initially (residual) stress free composite, and the load on the composite is assumed to produce only elastic deformation in the matrix and in the fibre. The model is thus not readily applicable to some problems.

The mismatch between the coefficients of thermal expansion (CTE) of Ti-6-4 and of Sic in Ti-6-4/SiC composites results in high tensile axial residual stress in the matrix (Durodola and Derby, 1994). Under creep loading conditions, a part of this inherent axial stress can induce further creep of the composite and can be significant. As with most alloys the yield stress of Ti-6-4 decreases with increasing temperature so that instantaneous matrix yield is possible under the applied stress and temperature conditions found during creep.

The model presented herein accounts for the inherent residual axial stress in the composite explicitly; treatment of the effect of possible instantaneous plastic flow of the matrix upon the loading of the composite and allows the ready use of a time dependent creep strain rate for the matrix.

The axial stress, of applied to the composite is assumed to produce uniform axial stresses (or) and (0,) in the fibre and in the matrix phases, respectively. Furthermore, as in the finite element micromechanics model, iso-strain rates along the axial direction are assumed for the composite and the constituent phases based on strain compatibility requirements. It follows therefore that

d, = km = Er (10)


where the subscripts c, f and m used here and in what follows denote composite, fibre and matrix terms and properties, respectively. Note also that the left superscript t notation used in eqns (l)-(9) to denote quantities at a given time t has been dropped as there is no further need to consider time increment effects. Also o and E henceforth refer to uniaxial stress and strain components, respectively.

The creep of the composite in this model is due to matrix creep. An ideal matrix creep constitutive law should be able to represent all possible creep stages (primary, steady and tertiary) and account for the effect of stress changes as stress in the matrix is relieved due to creep. Unfortunately no creep law which satisfies all these conditions has received wide acceptance. Assuming isothermal conditions, this work is based on the following creep strain rate law.

E; = A,a”,LP2 (11)

where A,,,, n,, n2 are constants for a given temperature. Equation (11) is a time hardening creep rate based on a general stress, time, temperature empirical creep law (White and Goodman, 1994) and should not be confused with the constant stress Norton-Bailey power law. Creep rate laws of the form of eqn (11) are often criticised for the infinite creep rate which they predict at time t = 0 when n2 is less than zero. Although other creep rate laws which obviate the problem at initial time have been proposed (Frost and Ashby, 1982), it is still difficult to obtain published experimental creep data based on these other forms. The present analysis is based on the empirical creep law of Nimmer et al. (1991) (for Ti-6-4 alloy) which takes the form of eqn (11) under isothermal conditions. Typi- cally, the results of preliminary numerical analyses showed that predictions based on the time hardening law and its equivalent strain hardening law are not significantly different. Thus, the analytically more convenient time hardening form is used for the development of the model.

Since the fibre response is linearly elastic and plastic flow in the matrix is time independent, the effective strain rates in the fibre and matrix are given by

1 dor “f=zdt (12)

f

1 da Cm = E, dt

2 + A,rs~tn2 (13)

where Ef and E,,, are fibre and matrix Young’s moduli, respectively. By considering equilibrium along the axial direction assuming that the Voigt’s rule of mixture applies, then

oc = T/ma&) + V,of(t) (14)

where V, and Vf are matrix and fibre volume fractions, respectively. Expressing crm in terms of CT, and rrr and differentiating with respect to time, t and substituting into eqn (13) and considering eqn (10) gives

(15)

The integration of eqn (12) to obtain the stress in the fibre at a given time and substitution of the initial conditions (i.e. of = oRf at t = 0, where oRf is the axial residual stress component in the fibre) gives

0r = oaf + E,e, = (TRf + Ef c, . (16)

Note that ef is strain in the fibre due to applied external load only, i.e. independent of strain history. The governing differential equation is obtained by using eqns (16) and (12) to substitute for the stress in the fibre and its rate of change with time in eqn (15). This gives the first order differential equation

& = p(q - E,)” t”2 (17)

1246 J. F. Durodola et of.

where r, p and q are constants given by

4 6 r=-; &An 4’ 0,

,=,_aRff *c and ’ = (E,V, + E,l$);)l-l (18)

Equation (17) can be solved for the overall composite strain E, at a given time by integrat- ing between limits, time, t = 0 and t = t, i.e.

s

G(t) de pz+’

8,(O) Ph - KY’ = - n2 + 1 (1%

for n2 # - 1. The left hand side of eqn (19) is of two different forms depending on the possibilities n, = 1 and n, > 1; the solutions for both cases can be expressed in a similar form to McLean’s results and are given by

- (coo - fzO) exp - Prt nz+l

E, = E, ( > - n2 + 1

E, = E, - (E, - EO) [l + pr”’ (n1 - 1) (&, - &(Jn’- 1 t”2+ I/(n2 + l)]“’ -“I (21)

respectively, where so is used to denote the instantaneous elastic strain, s,(O), of the composite on loading and E, = q/r is the composite strain when the stress in the matrix is completely relieved. Note that in this case, e,,, includes strain due to the relief of residual stresses.

3.1. Effect of initial plastic deformation

In the event of plastic flow at the instance of loading, a part of the applied load is carried by the fibre only. This load is equal to the product of the fibre stiffness, volume fraction and the plastic strain, so that, oC can be expressed as

6, = of + VfE& (22)

where of is the part of the applied load which generates elastic strain in the composite. Thus, in cases involving plastic deformation, of should replace a, in the calculations and the plastic strain component et should be added to the overall strain given by eqns (20) and (21) in order to obtain the true overall strain of the composite at a time t. In the case of elastic perfectly plastic matrix behaviour the plastic strain, et and of are given by

J% aC--(crYm-o~m) J% I

(f = c +n - ORrn) (24)

m

respectively; aYm is the matrix yield stress and oRm is the axial component of the residual stress in the matrix, and E, = (Ef Vf + E,,, V,) is the composite stiffness.

3.2. Creep strain, strain rate and time to prescribed overall strain

The effective creep strain at a time t is obtained by subtracting so from E, in eqns (20) and (21) which gives

&Z = (E, -so)[l -exp- rs)] (25)

&E = (E, - EO)[l - (1 + prnl(nl - l)(s_ - eo)nl-1tn2+1/(n2 + 1)]“‘-“1] (26)

for the case n, = 1 and n, > 1, respectively, and E,C denotes creep strain only. The time taken for the overall composite strain to attain a given value (which is useful for design


analysis) is given by

(27)

p;“y;n; ;jl) I(&_ - E, + &,p)(‘-“‘) - (E, - &py I l/(n2+ 1) t= (28)

for n, = 1 and n, > 1, respectively, and the creep strain rate at time t is given by

ii = pr(e, - eO) tflZexp - Prt fl2+1

( > ___ n2 + 1

E,C = prnl(eE - eJ”l tn2 [l + prnl (nl - 1) (E, - &J-l tn2+‘/(n, + l)](l+nl)‘(‘-nl) (30)

also for the cases n, = 1 and n, > 1, respectively.

4. RESULTS AND DISCUSSION

Ti-6-4/SiC composite model systems used as test cases were assumed to be fabricated at 1 I73K and cooled down to room temperature at the rate of 0.68K/s. These conditions are typical for the manufacture of Ti-6-4/SiC composites (Nimmer et al., 1991). Results obtained using McLean’s (1983) model, and its modified form presented herein, are compared with results obtained using a more comprehensive multiaxial numerical finite element micromechanics model. Comparisons are also made with published experimental results. Results presented in the case of the numerical approach are those of a hexagonal array fibre distribution in the matrix unless otherwise stated.

4.1. Comparison of analytical and numerical models

Although McLean’s (1983) model was shown to agree with experimental results, the comparison of the predicted and experimental results of Schwenker et al. (1993) for Ti-6-4/SiC did not show good agreement. Two factors which may be responsible for the difference are the effects of residual stresses and the lack of representation of primary creep of the matrix in the creep constitutive law used in the model. It is helpful to assess the performance of the model for cases in which these factors are not significant.

The model’s predictions were compared with finite element numerical micromechanics predictions using the same steady state creep law and without the incorporation of the effect of residual stresses. The steady state creep law constants used were taken from Schwenker et al. (1993) and are given in Table 2. Both low and high volume fraction (20% and 60% fibre volume fraction) Ti-6-4/SiC composites were considered. Axial loading of 750 MPa along fibre alignment direction and a temperature of 427°C were considered in both cases. The results are compared in Fig. 2; it can be seen in the figure that close (within 2%) agreement is obtained using both methods, even at the high fibre volume fraction of 60%. This agreement at high volume fraction indicates that even though the variation of the effective creep strain is greater within the matrix of the high volume fraction composite, Fig. 3, the net axial creep is well approximated in comparison with the multiaxial numerical model prediction.

The generalized model given in this work allows a ready incorporation of the effect of residual stresses and both time dependent and time independent creep constitutive laws can be used (note that the time dependent creep power law in eqn (11) reduces to a

Table 2. Constants used with the creep law eqn (11) in calculations; constants for the McLean model were taken from Schwenker et 01. (1993), constants for the modified model were calculated

from the creep law given by Nimmer et al. (1991)

Temperature (“C)

427 538

nI

8 7

McLean

A m (MPa-“I s-‘)

2.58 x 1O-29 2.12 x 10-23

nI

3.403 3.403

Modified McLean

n2 A m

(MPa-“’ s-0.9251 )

- 0.075 7.90 x 10-I’ -0.075 9.00 x lo-l4


0 Numeriical

Time (Hr)

0.0075 - 0 Numerical

0 MCLH”

.r

2 u) 0.0050-

if

5

Time (Hr)

Fig. 2. Comparison of predictions using McLean’s model and numerical micromechanics finite element analysis for Ti-6-4/SiC (a) 20% fibre volume fraction (b) 60% fibre volume fraction

composites; applied load and temperature were 750 MPa and 427”C, respectively.

i‘ (%)

A ,699 B ,696 C ,694 D ,691 E ,688 F .a5 G ,683 H .680 I ,677 J ,675 K ,672 L ai

(4 (b)

i’ (%)

A ,176 B ,171 C ,178 D ,178 E ,179 F ,180 G .I81 H .1n2 I ,183 J ,184 K ,185 L .I86

Fig. 3. Variation of effective creep strain (eqn (7)) after 500 h within the matrix of Ti-6-4/SiC (a) 20% fibre volume fraction (b) 60% fibre volume fraction composites; applied load and

temperature were 750 MPa and 427°C respectively.


Table 3. Estimated axial residual stresses in 35% fibre volume fraction Ti-6-4LSiC composite at different temperatures assuming no prior heat treatment; elastic-viscoplastic model as in

Durodola and Derby (1994), composite cylinder model as in Durodola and Ruiz (1993)

Temperature (“C)

421 538

Finite element (elastic-viscoplastic model)

-

URf WW aR,(MPa)

- 287 155 -164 88

Composite cylinder (elastic model)

URf (MW uRm W'4

- 219 150 -213 115

steady state law if n2 = 0). This generalized model is compared with the numerical finite element model using the same loading conditions as in the case for McLean’s model above, but a time dependent creep strain rate law, eqn (II), was used. The matrix time dependent creep law constants used were obtained from the creep law given by Nimmer ef al. (1991) and are given in Table 2. The plot of the predictions using both models are given in Fig. 4; close agreement (within 2%) with the numerical model is obtained for the composites considered. Thus, the generalized model for uniaxial creep has been verified to be in close agreement with the multiaxial finite element numerical model.

4.2. Comparison of predictions with experimental results

Schwenker et al. (1993) presented results of experimental uniaxial creep analysis of 35% fibre volume fraction composites under different axial loading stresses and two different temperatures, namely 427°C and 538°C. They compared their results with analytical predictions using McLean’s (1983) uniaxial creep model in which the effect of residual stresses and matrix primary creep effects cannot be incorporated. The creep constitutive law constants used are given in Table 2.

The modified McLean’s model presented herein and finite element analysis were applied to a number of the experimental cases reported by Schwenker et al. (1993). The strain hardening creep constitutive law eqn (6) was used in the finite element numerical analysis and the corresponding isothermal time hardening law eqn (11) was used in the modified model. The values of the constants A,, n, and n2 used at different temperatures are those given in Table 2. Residual stresses were estimated using an elastic-viscoplastic micromechanics finite element analysis (Durodola and Derby, 1994) and an elastic composite cylinder model approach (Durodola and Ruiz, 1993). The values of stresses based on the elastic-viscoplastic model which are more realistic were used in the calculations.

The predictions using the models are compared with three experimental results in Fig. 5. It can be seen from the plots that the results of the modified model and the finite element method are in reasonable agreement with the experimental results, particularly at low loading stresses and temperatures. At 427°C the models under-predict creep, while at 538°C composite creep is overestimated by the models. In all cases, closer agreement is obtained at long creep times than at short times. This seems to suggest that the creep law used is more representative of the steady state creep of the matrix than the primary creep stage. The lack of incorporation of the effect of residual stresses is largely responsible for the wider difference between the predictions using the McLean’s (1983) model and the experimental results.

The poorer agreement between predicted and analytical results at higher stresses (e.g. 1172 MPa) and temperatures using the modified model and the finite element predictions is probably due to increased damage (e.g. fibre breakage, debonding, etc.) under the higher loading and more severe conditions. These result in reduction of the effective stiffness of the reinforcement and hence the greater load transferred to the matrix causes further creep of the composite. As a direct illustration of the effect of reduction in stiffness associated with the reinforcement, two analyses were carried out using two different fibre moduli, namely 418 and 374 GPa (see Fig. 6); creep load was taken as 965 MPa


0.10 0 Numerical ?? Modified McLean

0 100 200 300 400 f

Time (Hr)

lb)

?? ModlLd McLean

O.Ory 0 500

Time (Hr)

Fig. 4. Comparison of predictions using modified McLean’s model and numerical micromechanics finite element analysis for Ti-6-4/SiC (a) 20% fibre volume fraction (b) 60% fibre volume fraction composites; applied load and temperature were 750MPa and 427”C, residual

stresses not incorporated.

at 427”C, and 35% fibre volume fraction composite was considered. At 500 h the composite creep strain was estimated to increase by about 10% due to about 10% reduction in fibre stiffness.

4.3. Effect of residual stresses

The effect of inherent residual stresses in the composite have been highlighted in this work. Further consideration is necesary in order to give a better quantitative estimate of this effect under typical loading conditions. In general, residual stresses in metal matrix composites occur due to the mismatch in the thermal coefficient of expansion between the fibre and the matrix alloy phases and the fact that the composites are cooled down to room temperature from their high manufacturing temperature (residual stress free temperature). When the composite is heated from room temperature to a service temperature, which although high is nevertheless much lower than the manufacturing temperature, the residual stresses, although reduced, still exist in the composite. Table 3 gives typical estimated axial residual stress values in the composite at different temperatures. The stress is generally tensile in the matrix and compressive in the fibre.

When an external load is applied to the composite, the residual tensile stress in the matrix is increased. Thus, during the creep of the composite, the sum of the residual stress and the applied stress in the matrix constitutes the creep load, which is relieved in the matrix and unloaded onto the fibre if the creep time is sufficiently long. The composite creep strain at a given time is thus higher than if the residual axial stress in the matrix were not initially present. Figure 7 gives results from the numerical model with and without


(4 0 50, I I 1 , I

- - Experiment

0 N”tTWlCal

0 ModlRed McLean

Time (Hr)

(b) 0401I I 1

v I- - - Experiment

0 Modtfied McLean

0.30 0 NumerIcal

g + McLean

Time (Hr)

(c)o~40~-~

0.30- ?? Numenca

Fg + McLean

.E

; o.zo- ___--- ___---

% H = o+’

o.od I I I I

0 100 200 300 400 500

Time (Hr)

Fig. 5. Comparison of analytical and numerical predictions with experimental results of Schwenker et al. (1993) for 35% fibre volume fraction Ti-6-4/SiC composite under (a) 655 MPa and 538°C (b) 965 MPa and 427°C (c) 1172 MPa and 427’C; residual stress effects considered as

indicated in the text.

residual stresses in the composite; the creep load considered is 965 MPa at 427°C. It can be seen from the results that the effect of residual stresses account for about an 18% increase in creep strain after 500 h. In comparison with this temperature condition, the corresponding increase at lower temperatures will be higher than 18%) while a lesser value is expected at higher temperatures because residual stresses decrease with increasing service temperature.


0 fibre modulus = 374 GPa

?? hbre modulus = 418 GPa

Fig. 6. Predicted effect (numerical micromechanics approach) of effective fibre Young’s modulus on the creep of 35% fibre volume fraction Ti-6-4/SiC composite under 965 MPa and 427”C,

effect of residual stresses included.

with residual stresses

I I

200 300

Time (Hr)

Fig. 7. Predicted effect (numerical micromechanics approach) residual stresses on the creep of 35% fibre volume fraction Ti-6-4/SiC composite under 965 MPa and 427°C.

4.4. Effect of fibre distribution

Under a typical loading condition such as 965 MPa and 427”C, the effect of fibre distribution (i.e. hexagonal array or square array distribution) in the numerical model was found to be insignificant at low fibre volume fractions such as 20”/0. At higher volume fractions, such as 60%, there is a divergence in the predictions from the two geometric models as shown in Fig. 8. The hexagonal array predicts a higher creep strain by about 5% in comparison with the square array model after 500 h. Figure 9 shows the matrix effective creep strain distribution after 500 h in a 60% volume fraction composite, the square array shows significant variation in matrix strain when compared with the more homogeneous matrix flow in the hexagonal array. It appears, therefore, that the more anisotropic behaviour of the square array model composite offers greater resistance to axial creep than the more isotropic hexagonal array composite.

4.5. Time to prescribed strain

One of the advantages of simple models, such as the modified McLean’s model, is the ease and the relatively short time required to compute prescribed conditions to attain a given composite strain. The plot of time to attain total strain of 0.50% and 0.75% is com- puted for the unreinforced matrix and for the reinforced composite using the isothermal time hardening creep rate eqn (11). The temperature considered was 427°C and 35% fibre volume fraction composite was assumed. Residual stress as indicated in Table 3 was taken as inherent in the composite. The failure strength of the fibre was taken as 3.5 GPa and

Uniaxial creep of metal matrix composites

::;I

1253

Time (Hr)

Fig. 8. Predicted effect (numerical micromechanics approach) fibre array packing on the creep of 60% fibre volume fraction Ti-6-4/SiC composite under 965 MPa and 427°C; effect of residual

stresses included.

? (%)

A 1.30 B 1.24 C 1.17 C 0.79

D 1.11 D 0.76

E 1.04 E 0.74

r: 0.98 F 0.72

G 0.9 1 G 0.70

H as5 H 0.67

I 0.78 I 0.65

J 0.72 J 0.62

K 0.66 K 0.60

L 0.60 L 0.57

Fig. 9. Variation of effective creep strain (eqn (7)) after 500 h within the matrix of 60% fibre volume fraction Ti-6-4/SiC composite (a) square array pack (b) hexagonal array pack; applied load and temperature were 960 MPa and 427°C respectively; effect of residual stresses included.

calculations ensured that this stress was not exceeded. Results obtained are given in Fig. 10. The figure shows the significant improvement in creep resistance obtained through the reinforcement of the matrix using fibres.

5. CONCLUSIONS

It has been shown that McLean’s (1983) uniaxial creep model is in close (within 2%) agreement with numerical finite element model predictions for uniaxial creep of FRMMCs at low and high fibre volume fractions, assuming that there are no residual stresses and a steady state creep rate law is obtained.

A modified McLean’s model which allows a ready use of time dependent and time independent matrix creep rate laws and incorporates the effect of residual stresses has been given. This model has been verified against a numerical finite element micromechanics model and close agreement (within 2%) is obtained between the models at low and high fibre volume fractions.


tdal strain

10 ‘0 ?? 0.75%

0 050%

108

TiX-4 Ill&X I

.

102 Th4iSiC (35% V , Compmiie)

OL ’ I I I I III I 300 400 600 600 600 1000

Stress (MPa) Fig. 10. Predicted time for overall axial matrix and composite strain to attain 0.50% and 0.75% under different loads at 427°C using the modified McLean’s model; overall strain includes elastic,

time independent plastic and creep strains.

The predictions of the modified model and numerical finite element micromechanics model agree reasonably well with experimental results, particularly at low stresses and low temperatures. At higher loads and temperatures poorer agreement is obtained between predicted and experimental results. This is believed to be due to the fact that reinforcement damage is more important at higher loads and temperatures. Another important possibility is the fact that under long term exposure the accuracy of the matrix creep constitutive laws used in the models can be stress and temperature dependent.

The effect of fibre distribution (i.e. hexagonal array or square array distribution) on the numerical model results was found to be insignificant at low volume fractions, but at 60% fibre volume fraction a 5% creep strain difference (after 500 h under 965 MPa and 427°C creep load) was found between the two geometries.

Acknowledgement-The authors would like to thank the Science and Engineering Research Council, U.K. for providing funding for this project under grant GR/E/87660.

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Uniaxial creep of long fibre reinforced metal matrix composites

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