High temperature deformation and fracture of a fiber reinforced titanium matrix composite

Pergamon 0956-7151(95)00208-l

Acta mater. Vol. 44, No. 2, pp. 683-695, 1996 Elsevier Science Ltd Copyright 0 1996 Acta Metallurgica Inc. Printed in Great Britain. All rights reserved 1359-6454/96 $15.00 + 0.00

HIGH TEMPERATURE DEFORMATION AND FRACTURE OF A FIBER REINFORCED TITANIUM MATRIX COMPOSITE C. H. WEBER, Materials

Department,

University

Z.-Z. DU and F. W. ZOKY of California,

Santa

Barbara,

CA 93106, U.S.A

(Received 28 October 1994; in revised form 30 March 1995)

Abstract-The

longitudinal tensile properties of a SIC fiber-reinforced Ti alloy have been investigated over the temperature range of 20400°C. Experiments have been conducted at both constant strain rate and constant stress. The experimental results have been compared with predictions of models based on fiber fragmentation, matrix flow and creep, and load transfer via interfacial friction. The models predict trends that are similar to the experimental ones, though they generally overestimate the tensile strength and ductility and the creep rupture time. The possible origins of such discrepancies are briefly discussed.

1. INTRODUCTION

along a single plane [3]. The composite strength is then dominated by the weakest fibers: the full fiber bundle strength is never realized. Clearly, GLS conditions are desirable for high strength. Within the GLS regime, the fiber strength is dictated by a characteristic slip length, given by [l, 2,4]

Fiber reinforced titanium matrix composites (TMCs) exhibit good potential for structural applications requiring high specific stiffness and strength at low and moderate temperatures (to - 600°C). The largest benefits will be obtained by using unidirectionally reinforced materials in components that are subjected predominantly to axial loads (e.g. tie rods, actuators and fan blades). The design of such components will thus be dictated by the longitudinal properties of the composites. The present understanding of the longitudinal tensile strength of unidirectional TMCs is as follows. In order to achieve a high strength, the fiber/matrix interface must be relatively weak, characterized by a low debond energy, r, and a low sliding resistance, z. Under these conditions, fiber fracture is accommodated by debonding and sliding along the fiber/matrix interface, with minimal stress concentration around the fracture site. Consequently, the load carried by the broken fiber is shed to the surrounding matrix and fibers over a length that is large in comparison with the fiber spacing. The condition wherein the load is distributed uniformly over the entire section of the composite in the plane of a fracture site is one of “global load sharing” (GLS) [l, 21. In the GLS regime, fibers are able to break independently of one another. In contrast, when r and/or T are high, debonding and sliding do not occur extensively around the fracture sites, leading to large stress concentrations in the adjacent fibers: the so-called “local load sharing” (LLS) condition [l, 21. In the LLS regime, fracture generally occurs by a mechanism involving a co-operative spread of fiber breaks tTo whom

all correspondence

should

where R is the fiber radius, m is the Weibull modulus of the fibers, and S, is the reference strength corresponding to a length, L,. Provided the composite gauge length L is large compared with 26,, the fiber bundle strength and corresponding failure strain are independent of L and given by

with IJ* being a characteristic

strength

o* = (S~TL,/R)“(“+

‘I.

For L 6 26,, the bundle strength exceeds the value given in equation (2). Equations (2)-(4) predict an increase in uB and cB with increasing T. However, for sufficiently large values of T, a transition from GLS to LLS conditions occurs, with a concomitant reduction in tensile properties. Thus, an optimal value of T exists for maximum strength and ductility. The models based on the GLS assumption have recently been assessed by comparing the predictions of strength and failure strain with experimental results obtained on a SIC fiber reinforced Ti-6Al-4V

be addressed. 683

684

WEBER et al.:

DEFORMATION/FRACTURE

alloy at ambient temperature [5]. To conduct such comparisons, measurements were made of the key constituent properties, including the fiber strength distribution and the interface sliding areas. The predictions of both the ultimate tensile strength and the failure strain were in reasonable agreement with the experiments, with the predicted values exceeding the experimental ones by - 1O-20%. The experimental data also indicate that these properties are independent of gauge length over the range L = 12-200 mm, in accord with the predictions of the GLS model. Moreover, fiber pullout experiments by Marshall and co-workers [6, 71 indicate that the debond energy of C-coated fibers is negligibly small and that the sliding resistance can be well represented by a constant shear stress, Z. This work further justifies the use of models based on constant r for simulating composite behavior. The purpose of the present article is to extend the previous work on the TijSiC composite to the high temperature regime. Some general trends in the high temperature properties are anticipated, based on knowledge of the mechanisms operating in the composite at ambient temperature and in monolithic alloys at high temperatures. First, the load bearing capacity of the matrix is expected to be reduced, particularly at temperatures exceeding - 200°C (- 25% of the absolute melting point) wherein creep mechanisms are operative [S]. This effect will be manifested as a reduction in composite strength. Furthermore, the strength will be time-dependent (i.e. the composite will exhibit creep rupture). Second, the residual stress arising from thermal expansion mismatch will be diminished. In fiber reinforced TMCs, the matrix thermal expansion coefficient, c(, , exceeds that of the fibers, clr. Consequently, after cooling from the processing temperature, the interface experiences a normal compression. This compression is expected to control the interface sliding resistance through a friction coefficient that depends on the chemistry and morphology of the interface or coating. The magnitudes of the normal interface pressure and the sliding resistance are thus expected to decrease with increasing temperature. The reduction in z will, in turn, lead to a reduction in the fiber bundle strength, in accordance with equations (2) and (4). Third, chemical reactions may occur, degrading the fiber strength. This speculation is based on the reported reduction in fiber strength following composite consolidation [9]. The present study examines the contributions from each of these processes to the composite properties.

2. THEORETICAL

BACKGROUND

Some basic results pertaining to the longitudinal and transverse tensile response of fiber reinforced MMCs are presented below, with a view to providing insight into the roles of various constituent properties

OF A Ti MATRIX

COMPOSITE

in the composite response. In this treatment, the fibers are assumed to be elastic at all temperatures of interest, with a Young’s modulus, Ef. Moreover, their strengths are assumed to follow the two-parameter Weibull distribution, characterized by a modulus, m, and a reference stress, S,, for a reference length, L, . At low temperatures and high strain rates, the matrix is taken to be elastic-perfectly plastic, with a yield stress 02 that varies only moderately with strain rate. In contrast, at high temperatures and low strain rates, the matrix is assumed to follow power-law creep. For uniaxial loading, the creep law is given by i, = 6, I-&,,+ i, (a, /aO)

(5)

where Q, is the matrix stress, cm is the matrix strain, n is the creep exponent, E,,, is the matrix Young’s modulus, i, and Q,, are the reference strain rate and stress (1 MPa), respectively, and the dot denotes the derivative with respect to time. 2.1. Longitudinal behavior The longitudinal tensile response, a,(e), of a unidirectional MMC can be written as a,(t) =f&@) + (1 +f)&l(~)

(6)

where f is the fiber volume fraction, and &(e) and C?,,,(E)are the volume-average axial stresses in the fiber and the matrix, respectively. Upon initial loading, both phases are essentially elastic such that the composite response is linear, with a modulus, E,=fE,+

(1 -f)E,,,.

(7)

At low temperatures, yielding occurs at a strain, ez, at which the total matrix strain (thermal plus mechanical) reaches the unconstrained matrix yield strain, 6;. Upon further loading (c > E:), the slope of the curve (or tangent modulus) is dictated by the fiber properties. Provided the extent of fiber failure is small, the slope in this regime is x_fEf. Fiber bundle failure then occurs when the average fiber stress reaches us. This occurs at a strain, t,* , of cc*=tg-t;

(8)

where ERR is the residual axial strain in the fibers. The corresponding stress, ad, is 0: =f& + (1 -f)oL.

(9)

The matrix yield stress, a&, is related to the composite yield strain through u&=(~,y+&)E,,,

(10)

with t& being the residual axial strain in the matrix. At high temperatures, the matrix creeps in accordance with equation (5). If the composite is loaded at a constant strain rate, the matrix stress asymptotically approaches a steady state value, given by OJug = (i/i,)““.

(11)

WEBER et al.: DEFORMATION/FRACTURE OF A Ti MATRIX COMPOSITE Moreover, at higher stresses, fiber breakage occurs, resulting in partial shedding of the load from the broken fibers to the surrounding matrix and fibers. Provided z is sufficiently low and r x 0, the axial strain in the matrix remains essentially uniform. The distribution of axial strain and stress in the fiber can be evaluated through a shear lag analysis, leading to a constitutive law for the fragmenting fibers (originally derived by Curtin) [4] ri,=&,[l

-$;T+‘].

The response of the obtained by combining matrix [equation (1 1)] with the requirements

(12)

composite in this regime is the constitutive laws for the and the fibers [equation (12)] for mechanical equilibrium

685

As the matrix creeps, it sheds load onto the (elastic) fibers and thus the composite strain increases with time. As t + co, the matrix stress asymptotically approaches zero, whereupon the composite strain approaches a saturation level, L,, that is dominated by the elastic properties of the fibers and is given by E, = o/fEf.

(16)

It should be emphasized that equations (14)-( 16) are only valid at low stress levels where the degree of fiber fracture is minimal. At higher stress levels, some of the fibers break. In this regime, the appropriate constitutive law for the fibers is the one derived by Curtin [equation (12)]. Combining this result with equations (5) and (6) yields the result [1 1]

~~/~o-GfW,)[l-UI~NWQ.Y+~~~~ ““=(l

-f)“{l

+[fEr/(l

[equation (6)] and axial strain compatibility, the result a,(c) =fE,t[l

-fiE,][l

yielding

- (1/2)(tE,/a.)“+‘] + (1 -floo(i/co)““.

(13)

Fracture occurs at a strain equivalent to the fiber bundle failure strain, eB, [given by equation (3) or (S)], with a corresponding stress evaluated from equation (13) at t = tg. These results are the same as those corresponding to composites in which the matrix exhibits perfectly-plastic, rate-independent behavior. The longitudinal creep response under a constant applied stress, CT,is also evaluated by combining the appropriate constitutive laws with the requirements for equilibrium and compatibility. In this case, it is instructive to consider two regimes, governed by the extent of fiber fracture. (The purpose of this distinction will become apparent later in the paper.) At low stress levels, the extent of fiber fracture is minimal, and thus the fiber stress is essentially uniform along its length. In this regime, the creep response follows the relation:

(17)

+2)/2)(~E,/cr,)“+‘]}

-((m

This result is subsequently referred to as the Curtin-McLean model since it combines Curtin’s analysis of fiber fragmentation with McLean’s analysis of the evolution of matrix stress. The accumulated strain with time is obtained by integrating equation (17) numerically. Assuming that GLS conditions prevail over the entire loading history, composite fracture occurs at a critical strain, ec, at which i + cc. This critical strain, obtained from equation (17) is *_

1

_

l/b

+U -f)&

+ 1)

(18)

5%

ur

This result differs from the one corresponding to a fiber bundle embedded in a perfectly-plastic rate-independent matrix (Curtin’s result) by a factor of [(l + (1 -f)E,,,/fEr]li(mf’). The limiting case wherein (1 -f)E,/SE,= 0 is identical to Curtin’s result [equation (3)]. Trends in the failure strain with both the Weibull modulus and the elastic modulus ratio (1 -f )E,,,/fEf are shown in Fig. 1. For typical values

p’

R

(I-f)E,/fE,=1.5

(14) with t being the longitudinal strain. This result was originally derived by McLean [lo]. Equation (14) predicts an initial (elastic) strain, tl, at t = 0 of L,= u/E,.

(15)

0.71 2

0

1 4

-

1 6

-

I 8

.

I 10

I

I 12

I

I 14

Weibull Modulus, m

Fig. 1. Trends in the fracture strain with the fiber Weibull modulus, m, and the elastic modulus ratio (1 -j$!Z,/‘Y,.

686

WEBER et al.: DEFORMATION/FRACTURE

OF A Ti MATRIX COMPOSITE

matrix yields and the subsequent response is no longer elastic. In this regime, the failure strain reduces to E, = eB [equation (3)]. Consequently, the two critical strains (c, and tg) are expected to bound the failure strain for a matrix which exhibits rate-dependent creep at low strain rates and either elastic behavior or rate-independent yielding at high strain rates.

;

Strain, E

/

2.2. Transverse behavior In the transverse orientation, the composite erties are essentially dominated by those matrix. The two are related in the following Under constant stress conditions (c?,,,= 0), the creep law [equation (5)] reduces to a power-law form

propof the way. matrix of the

i, = &(a/$>“. The steady state composite predicted to follow

(19)

creep

rate

i, is then

i, = io(o/cQ Strain, E

Fig. 2. Schematics showing: (a) the evolution of strain rate C with accumulated strain, 6; and (b) the partitioning of the stress between the fibers and the matrix under conditions of constant applied stress. Note that the instability point (6 = co) occurs at a strain, 4, greater than the fiber bundle failure

strain,

cB.

of the Weibull modulus (m 24) and the elastic modulus ratio [( 1 -f )E,,,/'Ef 6 0.51, the failure strain differs from the Curtin result by < 10%. The physical origin of the difference in the failure strains for the rate-dependent and rate-independent matrix materials can be understood with the aid of the schematic in Fig. 2. Upon initial loading, the fibers and the matrix respond elastically, with the initial stresses in the two phases being oE,/E, and oE,,,/E,, respectively. As the matrix creeps and sheds load onto the fibers, the fibers begin to break, causing an acceleration in the creep rate, i. At sufficiently high values of t! (near the maximum point in the stressstrain curve of the fibers), the matrix stress rises rapidly. Indeed, in the limit wherein i + co, the matrix response is completely elastic. The rapid “hardening” of the matrix at strains comparable to cg delays the onset of instability, to a strain &c> cB. Since the matrix is elastic at the point of instability, the predicted failure strain [equation (18)] depends on the matrix modulus but not on the matrix creep characteristics (i.e. i,, go and n). It should be noted that the dependence of the failure strain on the matrix modulus is, to some extent, an artefact of the constitutive law used to characterize the matrix response. Notably, the law in equation (5) predicts elastic behavior at high strain rates, with no provision for rate-independent yielding. Clearly, at sufficiently high stress levels, the

where G0 is the reference is related to u0 through

(20)

stress for the composite

G/a0 =g(f,

and

n).

(21)

A comprehensive set of results for g (A n) has been obtained through finite element analyses of unit cell models [12]. When the fiber/matrix interface is well bonded and for modest values of f (50.35), g is insensitive to both the creep exponent and the spatial arrangement of fibers, and is given approximately by g x 2/$. The main role of the fibers is to constrain the matrix from contracting along the fiber direction, leading to plane strain conditions. When the fibers separate from the matrix, they contribute negligibly to the flow or creep resistance of the composite [13-151. Indeed, the composite behavior is then similar to that of a ductile metal containing an array of cylindrical holes. Finite element calculations have been performed to evaluate the effects of the fiber content, A the creep exponent, n, and the spatial arrangement of fibers on g [14]. The results of 1.21

0

I

,

0.1

I

(

0.2

I

, ,

0.3

I

,

I

0.4

,

0.5

I

(

0.6

Fiber Volume Fraction, f

Fig. 3. Trends in the reference stress ratio, g, with fiber content, A for weakly-bonded fiber composites (CPD refers to the close-packed direction).

WEBER et al.:

DEFORMATION/FRACTURE

OF A Ti MATRIX COMPOSITE

681

Fig. 4. Cross section of the TijSiC composite showing a uniform fiber distribution. these calculations are shown in Fig. 3. For n = co, the results can be described empirically by g(f, n = co) = 1.1547 - Sf

(22)

where 6 and K are numerical coefficients that depend on the fiber arrangement and the loading direction. For a hexagonal arrangement of fibers, the coefficients are: (i) 6 = 1.381, K = 0.72 for load applied normal to the close-packed direction; and (ii) 6 = 1.130, K = 0.38 for load applied parallel to the close-packed direction. For finite values of n, g increases only slightly with decreasing n. The results in Fig. 3 can be reinterpreted to describe the low temperature flow response of the composite (in the absence of creep). This is accomplished by replacing &/a, with the ratio of the flow stress of the composite to that of the matrix, and replacing n with l/N, where N is the exponent in the power-hardening law.

3. EXPERIMENTS

were conducted at temperatures up to 600°C. The longitudinal behavior of this material at ambient temperature has been described elsewhere [S]. During the tests, the specimens were surrounded by a tubular stainless steel susceptor which, in turn, was heated by an induction furnace. The susceptor/induction coil combination was designed to give a hot zone of either 25 or 100mm. The heating rates were typically lOO”C/min., with temperature fluctuations of 5 3°C at 600°C. Axial strains were measured using a 12.5 mm contacting extensometer. Two types of tests were performed. In the first, tve specimens were loaded at a fixed strain rate: either 1.7 x 10-7, 1.7 x 10m5 or 8.3 x lo-’ SK’. At the highest strain rate, fracture occurred in 52 min. These tests were conducted at 20, 300 and 600°C. In the second, the specimens were subjected to a constant load, with the applied stresses ranging between 750 and 850 MPa, at a temperature of 600°C. The time taken to apply the load was typically 6 10 s. The creep response was monitored for periods of up to 60 h, or until fracture.

3.1. Composite properties

3.2. Constituent properties

The material used in this study was a Ti-6Al-4V alloy reinforced with 32% of continuous, aligned SiCt fibers [5] (Fig. 4). The composite panel was comprised of six plies with a total thickness of 1.0 mm. Tensile tests were conducted on straight specimens using a servohydraulic testing machine. The specimens were 6 mm wide and N 150 mm long and loaded using hydraulic wedge grips. The tests

In order to assess the creep models presented in Section 2, a number of constituent properties are required. These include: (i) the fiber strength distribution; (ii) the sliding resistance of the fiber/matrix interface; and (iii) the creep characteristics of the matrix. The fiber strength distribution was obtained from a series of tensile tests conducted on individual fibers extracted from the as-consolidated composite. The extraction was accomplished by dissolving the matrix

tSM- 1240, produced by British Petroleum.

688

WEBER

et al.:

DEFORMATION/FRACTURE

OF A Ti MATRIX

COMPOSITE

temperature was measured by a thermocouple in contact with the specimen. The temperature fluctuations during the test were 55°C. Tests were conducted at 20, 300 and 600°C. In calculating the sliding resistance, the shear stress was assumed to be uniform along the interface and given by r = PI2aR(t - u)

Strain, E (%) Fig. 5. Effects of fiber-matrix separation on the transverse stress-strain response and the rate of acoustic emission at ambient temperature.

HF solution. To determine whether in a concentrated the strength characteristics change during high temperature exposure, strips of the composite were first heat-treated in air for periods of either 100 or 500 h at 600°C. The tests were conducted in a dedicated fiber tensile testing machine with a fiber gauge length of 25 mm. For each condition, at least 50 fibers were tested. The results were interpreted using the two-parameter Weibull function. In some cases, the axial strains were measured with a laser extensometer during testing, allowing the Young’s modulus of the fibers to be determined (Er = 360 GPa). The sliding resistance of the fiber/matrix interface was measured using the fiber pushout test [16,17]. Specimens for pushout were prepared by cutting thin sections, -SOOpm, transverse to the fibers. The specimen were subsequently polished to a final thickness of -4OOpm (approx. eight times the fiber radius). Calculations by Liang and Hutchinson [18] indicate that this thickness is sufficient to minimize the effects of the free surfaces on the pushout results. Consequently, the measured sliding stress is expected to be independent of specimen thickness. The fibers were pushed out using cylindrical SIC indentors, - 300 pm tall, that had been machined from the Sic fibers themselves. To prevent the indentor from coming into contact with the matrix during pushout, the matrix on top of the polished specimen was etched to a depth of - 30 pm, leaving the fibers protruding above the matrix surface. The pushout apparatus consisted of a flat support base with a strain gauged cantilever beam. For the tests performed at ambient temperature, a 200pm rod was placed into a 220 pm hole in the middle of the base. The rod was put into contact with both the bottom side of the fiber to be pushed and the cantilever beam, allowing measurement of fiber displacement during pushout. For tests performed at elevated temperature, displacements were measured remotely using an LVDT. High temperature tests were performed by placing a thin resistive heating plate between the specimen and the base of the pushout apparatus [16]. The

(23)

where P is the applied load, t is the section thickness and u is the sliding displacement. The matrix flow and creep properties were inferred from tensile tests conducted on the composite transverse to the fibers. Preliminary tests on the composite indicated that the interfaces undergo progressive debonding during creep testing. The debonding effectively reduces the reference stress 6,, resulting in some ambiguity in the interpretation of the results. To circumvent this problem, the specimens in subsequent tests were first loaded at ambient temperature to a stress sufficient to completely separate the interfaces. The degree of separation was determined using acoustic emission. Figure 5 shows both the stress-strain curve and the rate of acoustic emission during this portion of the test. The non-linearity in the stress-strain curve (0 z 250 MPa) is associated with the onset of debonding, as manifest in the high rate of acoustic emission. Upon further loading, the rate of acoustic emission decreases and reaches a negligible value at a strain of -0.3%. This strain was subsequently taken to be the value required for complete interfacial separation. It is of interest to note that there is no permanent strain upon unloading, implying that the matrix remains elastic during the debonding process. Similar behavior has been observed in other TijSiC composites containing weak interfaces [13]. Transverse creep tests were conducted at 600°C at stress levels of 35, 50 and 80 MPa. Once a steady state creep rate was attained, the stress was either increased or decreased to a new level and the strain monitored until a new steady state creep rate was attained. For comparison, tests were also conducted at a fixed strain rate of 8.3 x 1O-5 SK’at temperatures of 20, 300 and 600°C.

Fig.

6. Schematic showing fibers following

the extension of the exposed matrix dissolution.

WEBER et al.: DEFORMATION/FRACTURE OF A Ti MATRIX COMPOSITE

689

imental data yields y, x -0.6. By comparison, the corresponding value for commercially pure Ti is [S] Ym-

0

0.2

0.6

0.4

Strain

0.8

Kh/Eo,> d(T,Tm)

(26)

1.2

-

with Ei being the modulus of Ti at ambient temperature. The ratio of these parameters is 7,/y,,, = 0.5. If the temperature dependence of the composite modulus is indeed dominated by that of the matrix, the parameters yc and y, are expected to be related through

I .o

Y,/v, = [(l

(%)

Fig. 7. Longitudinal tensile stress-strain curves, showing effect of temperature at a fixed (rapid) strain rate and effects of strain rate at 600°C. The residual axial strains were measured using a selective etching procedure. The specimen used for these measurements was - 15 mm wide (transverse to the fibers) and - 75 mm long (parallel to the fibers). Two narrow slits, - 2 mm apart and 50 mm long, were machined along the length of the specimen, using electrodischarge machining. The region outside of the narrow tongue between the slits was masked with silicone rubber. The entire specimen was immersed in an HNO, solution to dissolve the matrix between the slits, allowing the fiber to relax. The positions of the tops of the fibers were then measured using confocal microscopy. The precision of the individual measurements is - 1 pm. The extension of the fibers, A, was evaluated from the differences in positions of the exposed and embedded fibers, as shown schematically in Fig. 6. The axial residual fiber strain, t f‘, was calculated using the relation tr”= -A/%

=

-fEi/E:l.

(27)

This result assumes that the temperature sensitivity of the matrix alloy (Ti-6Al-4V) is the same as that of pure Ti. The ratio predicted by equation (27) is y,/y, = 0.39, slightly lower than the experimental value. The difference may be attributed to creep deformation occurring in the composites even at low stress levels, resulting in anomalously low measured moduli at elevated temperatures. At larger strains, the curves exhibit non-linearity, a result of matrix yielding or creep. The strain at the onset of non-linearity decreases with increasing temperature, consistent with the reduction in the flow and creep resistance of the matrix. Beyond this point, the slopes of all the curves approach a constant value, given by fEf = 115 GPa, independent of temperature. In this regime, the matrix stress is essentially constant. Finally, both the strain and the stress at fracture decrease with increasing temperature (Fig. 8). Similar trends are observed at lower strain 2000 k = 8.3 x lo.5

s-1

1800

(24)

where e is the length of exposed fiber. Additional details of the procedure are presented elsewhere [ 191.

4. EXPERIMENTAL

RESULTS 8oo0100

4.1. Composite

700

behavior

Temperature (C)

The stress-strain curves measured under fixed strain rate conditions are presented in Fig. 7. At the high strain rate, the initial elastic modulus decreases with increasing temperature, from - 201 GPa at ambient to - 164 GPa at 600°C. This trend is due mainly to the reduction in the matrix modulus. To provide a comparison, the temperature dependence of the composite modulus can be characterized by a dimensionless parameter (25)

I

I

I

I

I

I

(b)

1.4

constantEP

0

100

200

300

400

500

600

700

Temperature (“C)

where Ef is the composite modulus at ambient temperature and T,,, is the absolute melting point of Ti (1933 K). Linear regression analvsis of the exoerI

Fig. 8. Trends in: (a) fracture stress; and (b) fracture strain with temperature for the constant strain rate tests. Also shown for comparison are the model predictions.

690

WEBER

et al.:

DEFORMATION/FRACTURE

0.80,

I

(a) 0.75 -

OF A Ti MATRIX 600

COMPOSITE

I

I

I

;=8.3xlO”s-’

500

0.70 0.65 -

0.60 / 0.55 -

/ I d&

0.50 - s-

,:* /'

14

1O“O

10'

103

Time, 1 o-4

105

Transverse

t(s)

I

I

I

I

I

I

0.55

0.60

0.65

0.70

0.75

0.80

Total Strain, E (%) Fig. 9. Longitudinal creep response under constant stress conditions: (a) evolution of strain with time; and (b) reduction in strain rate with accumulated strain.

rates at 6OO”C, with the exception that the initial modulus coincides withf&. Evidently, the strain rate is sufficiently low to relax the matrix stress to a negligibly small value. The creep curves (strain vs time) measured under constant stress at 600°C are shown in Fig. 9(a) and the corresponding trends in creep rate i with accumulated strain c in Fig. 9(b). Also shown in Fig. 9(a) are the strain levels t, and t, predicted from equations (15) and (16), assuming that no fibers fail. For the stress levels examined here, the creep strain eventually exceeds the predicted saturation level, cS. Moreover,

z E I g cn

1000 900 800 -1.29 GPa

700

I

600 10-4

I lo-’

I

I

I

Time-to-Failure

I 102

100

0.8

Strain

1(b)

’ 0.50

0.4

0

(h)

Fig. 10. Failure times for tests conducted at 6OO”C, either at constant stress (no arrows) or constant strain rate (arrows). The solid lines are predictions of the Curtin/McLean model, whereas the dashed lines are obtained from the approximate solution [equation (30)].

1.2

1.6

2

(%)

stress-strain curves for constant rate tests at 600°C.

strain

the creep rate decreases monotonically with strain over the entire life of the composite. The response is transient in nature, with no evidence of either a steady state creep or an acceleration in creep rate prior to fracture. Failure occurs at a critical strain of -0.73-0.76%, essentially independent of applied stress. These values are similar to those measured under fixed strain rate conditions: -0.7-0.8%. Trends in the failure times for the constant stress tests are shown in Fig. 10. Evidently the short specimens (L = 25 mm) exhibit a longer creep life: roughly an order of magnitude greater than that of the long specimens (L = 100 mm). Also shown for comparison are the failure times for the tests conducted at a constant strain rate. The failure times for the latter tests provide an upper bound to those expected under constant stress conditions, since the majority of the time is spent at lower levels of stress. Consequently, these datum points are accompanied by arrows that show qualitatively the expected trends for constant stress tests. 4.2. Constituent properties The transverse stress-strain curves measured under constant strain rate conditions (i = 8.3 x 10-j SK’) are shown in Fig. 11. At ambient temperature, the limiting stress 0: is -430 MPa. By comparison, the yield stress of the monolithic Ti alloy measured using indentation is o&z 900 MPa. The strength ratio, u,“/cT:, 0 0.48, is consistent with the range of values predicted by equation (22): g = 0.42-0.55. At elevated temperatures, a limit stress is also obtained, though the stress levels are diminished. The temperature dependence of the limit stress can be described empirically by a:/&

= 1 - q(T/T,

- 1)

(28a)

where ~2 is the limit stress measured at ambient temperature To, and q is a numerical coefficient that describes the temperature sensitivity of the limit stress. Linear regression analysis of the data corresponding to a strain E = 0.8% yields a temperature sensitivity, q = 0.35. Recognizing that the composite limit stress is dictated by the matrix allows equation

WEBER et al.:

IO4

I

I

0.5

0

DEFORMATION/FRACTURE

1.0 Strain,

I

I

1.5

2.0

0,/o:,

691

-4

2.5

-5

E (?&)

Fig. 12. Transverse creep behavior for tests conducted under constant stress, with occasional changes in stress following attainment of a steady state response.

28(a) to be re-written c,:

OF A Ti MATRIX COMPOSITE

0

0.5

1

1.5

In (Strength/GPa)

Fig. 14. Fiber strength distributions for materials in the as-consolidated condition and following extended thermal exposure.

in terms of the matrix strength, = 1 - q(T/To - 1)

(28b)

with a; being the matrix strength at ambient temperature. The results of the constant stress transverse tests conducted at 600°C are shown in Fig. 12. The transient response following initial loading or following a load change persists for only a narrow strain range: typically, _ O.l-0.2%. Beyond the transient, a steady state creep rate is obtained. Moreover, a good correlation is obtained between steady state creep rates at a given stress following different loading histories (within a factor of -2). The steady state creep results are plotted in Fig. 13. Linear regression analysis of the results at low stress levels yields a n = 3, and a reference strain creep exponent, stress, rate, i, = 3.3 x lo-‘2s-’ , for a reference rr,, = 1 MPa. At high stresses, the power law apparently breaks down, resulting in a strain-rate insensitive yield strength of N 140 MPa. Following equation (22), the corresponding values of the reference strain rate and yield stress for the matrix at this temperature are L, = 4.12 x lo-l3 s-’ and a2 = 280 MPa.

The fiber strength distribution for the as-processed and heat treated conditions is presented in the usual logarithmic coordinates in Fig. 14. In this form, the data for the as-consolidated material appear linear, indicating that the strengths are consistent with the two parameter Weibull distribution. The relevant parameters characterizing this distribution are m = 5 and S, = 1.47 GPa for L, = 1 m. Following heat treatment for 100 h at 6OO”C, the fiber strength is degraded (S, x 1.26 GPa), though the Weibull modulus remains the same. This trend suggests that the entire population of strength-limiting flaws is more or less uniformly increased in size. Following heat treatment for 500 h at the same temperature, the strengths of the strongest fibers appear to remain unchanged relative to the 100 h treatment. However, the strengths of the weaker fibers are apparently enhanced. In the subsequent analysis, limits on the fiber strength distribution are established by using the values of S, corresponding to the as-consolidated state and the 100 h exposure, with m = 5. This exposure time is longer than the duration of any of the creep tests. The origins of the strength degradation following the initial 100 h exposure and the apparent

I

150

,

,

,

,

,

,

,

,

1

-0

a

0

s,=o

0

6,=2pm

-

600

7M)

10ar 20

0

Composite

Stress,

0

d (MPa)

Fig. 13. Steady state creep under transverse loading at 600°C. Note the power law creep behavior at low strain rates and the breakdown to yielding at higher strain rates.

~0111.111111 100 200

300

Temperature

Fig.

15.

400

500

(OC)

Influence of temperature on the interfacial sliding stress.

692

WEBER et al.: DEFORMATION/FRACTURE OF A Ti MATRIX COMPOSITE

1

: PO I 0

I 1

I 2

I 3

I 4

I 5

I 6

I 7

Position, x (mm) Fig. 16. Trends in the relative displacements, A, of the fibers exposed following matrix dissolution.

strengthening following the additional 400 h exposure are presently not understood. Figure 15 shows the variation in the interfacial sliding stress z with temperature. The data correspond to the values measured upon initially sliding (6, = 0) and following a sliding distance of 6, = 2 pm. Little change in sliding stress occurs over this range of sliding distance. The trends indicate that the sliding stress is sensitive to temperature, decreasing from N 120 MPa at ambient to -20 MPa at 600°C. This reduction is likely due to the relaxation in thermal residual stress at elevated temperatures. A further discussion on the effects of temperature on the interfacial sliding stress is presented elsewhere [16]. For the present purposes, the results can be fit by an empirical relation of the form z/z0 = 1 - j.?(T/T, - 1)

(29)

where z,, is the sliding stress measured at ambient temperature, T,,, and /I is a numerical coefficient that describes the temperature sensitivity of the sliding stress. Linear regression analysis of the data in Fig. 15 yields a value j? = 0.42. In principle, j? can be related to the thermal expansion mismatch between the fibers and the matrix and the friction coefficient for the fiber/matrix interface, though no effort is made to do so here. Measurements for residual strain determination are presented in Fig. 16. Each of the datum points correspond to a measurement on a single fiber. The data obtained on the fibers still embedded within the composite have been averaged to produce a zero reference point for the displacement measurements. The average displacement of the exposed fibers was 6 = 79 f 3 pm. Combining this result with equation (24) yields a residual fiber strain, of”= -(0.16 f O.Ol)%. 5. ANALYSIS

An assessment of the model predictions is made by comparing the composite properties measured exper-

imentally with those obtained from the models, incorporating the relevant constituent properties. Figure 8 shows such comparisons for the longitudinal strength and ductility. The predicted strength ad is evaluated from equation (9) incorporating the measured temperature-dependence of the sliding stress [equation (29), with /l = 0.421 into the fiber bundle strength [equation (2)], as well as the temperature-dependence of the matrix flow stress [equation (28b), with q = 0.35 and a& = 900 MPa]. Here the fiber strength characteristics are assumed to be the same as that of the as-consolidated material. The predicted trend (shown by the solid line) appears to be broadly consistent with the data. Also shown for comparison are curves that neglect either the temperature-dependence of the sliding stress (p = 0) or the temperaturedependence of the matrix yield stress (rl = 0). In both of the latter cases, the predicted strengths at elevated temperatures exceed the measured values. Moreover, the curves indicate that the strength reduction at elevated temperature is due primarily to the reduction in matrix yield strength. Nevertheless, the strength reduction associated with the fiber bundle (through the reduction in Z) is still substantial, being N 70% of that due to the matrix at 600°C. Prediction of the failure strain requires knowledge of the variation in the residual fiber strain cf” with temperature [equation (S)]. In principle, this variation can be calculated. However, for the present composite, the parameters governing the misfit strain, including the thermal expansion coefficients, the relevant temperature change, the contribution from phase transformations and the relaxation due to plastic flow and creep, are not well established. In light of these difficulties, no attempt has been made here to rigorously evaluate the variation in tf” with temperature. Instead, to provide a rudimentary assessment of the composite failure strain, extreme bounds on ERR were used. In one case, cf”was assumed to be constant over the temperature range of interest (20-6OO”Q at the level measured at ambient temperature: ERR = -0.16%. In the other case, the residual strain was assumed to vary linearly with temperature, reaching zero at the maximum temperature (600°C). The composite failure strain was then predicted from equation (8) incorporating the temperature-dependence of the sliding stress [equation (29)] into the prediction of the fiber bundle failure strain [equation (3)] and using the appropriate estimate of E:. The trends are shown in Fig. 8(b). For the case in which the residual strain is assumed constant, the predicted failure strains exceed the measured values by N 0.2%. For the case in which the residual strain is assumed to decrease linearly with temperature, the predictions lie close to the experimental results at 6OO”C,but overestimate the ones at 20 and 300°C. Simulations of the creep curves for the constant stress tests were performed in two ways. In the first, the fibers were assumed to remain intact, as in

WEBER et al.:

693

DEFORMATION/FRACTURE OF A Ti MATRIX COMPOSITE IO.5

(4

10-g ...... Mcl.ea#l Model IO-'0 10'

Time,

cunin/McleanMode Id

103

t(s)

Id

Time,

IO'

105

108

Time, t(s)

Id

IO'

105

Time, t(s)

t (s)

.... 10% 0=85OMPa 0

Expelirnnl

10-7

...... t?deean Mock, -

i

Cu*inh!cLeaan Model .

Fracl"n Iti

Id

IO5

Time, t(s)

Fig. 17. Comparisons of measured creep curves with model predictions.

McLean’s model [equation (14)]. This model predicts a limiting strain, E,, as t + co. In the second, the fibers were allowed to break in accordance with the Weibull distribution, with the tensile response of the fibers given by equation (12). The evolution of strain with time was then calculated by numerically integrating equation (17). For the latter simulations, the fiber strength distribution was assumed to follow either that of the pristine material (m = 5, go = 1.47 GPa) or that following the 100 h exposure at 600°C (m = 5, a,, = 1.29 GPa). The parameters characterizing the matrix creep response [equation (5)] were obtained from the transverse creep tests, as detailed in Section 4. The simulations are presented in terms of both accumulated strain and strain rate in Fig. 17. For

short times, the predicted strains underestimate the measured ones by -0.04-0.08%. This discrepancy is believed to be a result of transient creep in the matrix, a feature not incorporated in the creep law. However, the predicted strain rates are similar to the experimental ones. Moreover, the three sets of predicted curves for short times are similar to one another, suggesting that the effects of fiber failure on creep rate are initially minimal. At longer times, the predicted curves diverge from one another. In the cases where the fiber strength distribution is assumed to be the same as that of the thermally exposed material (with S,, = 1.29 GPa), the creep rates exhibit a minimum followed by a rapid acceleration to i -+ cc. In contrast, when the fiber strength distribution is taken to be the same as that of the as-consolidated material

694

WEBER

et al.:

DEFORMATION/FRACTURE

(S, = 1.47 GPa), the creep rates are predicted to decrease monotonically with time for applied stresses of 750 and 800 MPa; for 850 MPa, the curves again exhibit a minimum followed by a rapid acceleration. Because the creep rates diminish to extremely low values at high strains, the predicted failure times are sensitive to the fiber strength. For example, at a stress 0 = 800 MPa, the model predicts an infinite life for S, = 1.47 GPa and a creep rupture time of < 1 h for S, = 1.29 GPa (a reduction in S, of only - 12%). In all cases, the McLean model predicts a monotonically decreasing strain rate since all the fibers are assumed to remain intact. The predicted failure times based on the Curtin/McLean model are shown as solid lines in Fig. 10. Again, limits on the fiber strength distribution are obtained by using the two extreme values of S,, . Also shown by the dashed lines are predictions based on a simpler model derived from a combination of McLean’s creep model and Curtin’s result for fiber bundle failure. In this case, the effects of fiber failure on the composite creep rate are neglected, such that the strain-time response follows equation (14). The failure condition is then taken to be one in which the average fiber stress (taken simply as eE,) reaches the fiber bundle strength, cB. This approach has previously been considered by Du and McMeeking [1 11. The corresponding creep rupture time 1, is obtained from equation (14) by setting E = a,/&, whereupon ” =f(n

-K - l)Emi;if

-(1 -c&r,

f (l-f)

(30) Though this approach is less rigorous than the one based on the Curtin/McLean model, it provides a simple analytical result which is quite accurate for long rupture times. At shorter times, the simplified model yields conservative estimates of failure time. It is of interest to note that the failure times measured experimentally for the short gauge length fall between the predictions of the McLean-Curtin model using the two extreme values of S, (1.29 and the predicted curve for 1.47 GPa). Moreover, S,, = 1.29 GPa appears to overestimate only slightly the failure time for the longer gauge length. Upon initial inspection, this correlation would suggest that the model provides a reasonable description of the failure time, subject to the uncertainties in the fiber strength distribution. Despite this correlation, there is an important discrepancy between the experiments and the theory that suggests that the correlation may be somewhat fortuitous. Notably, none of the experimental curves exhibit a local minimum in creep rate preceding fracture (in contrast to the model prediction) indicating that the failure condition is not one in

OF A Ti MATRIX

COMPOSITE

which i + co. Consequently, it is likely that the model provides an upper bound estimate of the failure time.

6. CONCLUDING

REMARKS

The longitudinal stress-strain behavior and creep response of the TijSiC composite are broadly consistent with existing models based on fiber fragmentation, matrix flow and creep, and load transfer via interfacial friction. However, the predicted values of strength, ductility and rupture times are somewhat higher than the corresponding experimental results. Moreover, the creep experiments indicate that failure occurs with no prior acceleration in creep rate. In contrast, the GLS model for creep rupture suggests that failure should occur when i + co. The origin of this discrepancy and the true conditions for creep rupture are presently not understood. At high temperatures, the rupture time increases with decreasing gauge length (from 100 to 25 mm). In contrast, the tensile properties at ambient temperature have been shown to be independent of gauge length, over the range, G = 12-200 mm. To address this discrepancy, it should be noted that the model for fiber bundle failure is based on the assumption that the gauge length is larger than the characteristic transfer length (i.e. L > 26,). For shorter lengths, the strength is expected to increase with decreasing L; equivalently, at high temperatures, the rupture time is expected to increase with decreasing L. This transfer length varies roughly inversely with the sliding stress [equation (l)]. Using the relevant values of the material properties at 20°C yields 26, z 4 mm: considerably smaller than L. In contrast, at 600°C 26, z 20 mm, comparable to the length of the shorter specimens used for creep testing (L = 25 mm). It is surmised that the effective sliding stress under creep loading conditions may be lower than that measured in the pushout tests, a result of creep within the matrix parallel to the fiber/matrix interface. This effect would further increase 6, at high temperatures, making L < 26,. In this regime, the creep rupture times for the shorter specimens would be expected to exceed those measured on the longer specimens. Numerical studies by Du and McMeeking [11] confirm that such effects can occur over sufficiently long time periods. Studies in progress are aimed at addressing this issue further. Acknowledgements-Funding for this work was supplied by the DARPA University Research Initiative Program at UCSB under ONR contract N0014-92-J-1808. The authors gratefully acknowledge fruitful discussions with R. M. McMeeking.

REFERENCES 1. M. Y. He, A. G. Evans and W. A. Curtin, Acta metall. mater. 41, 871 (1993). 2. Z.-Z. Du and R. M. McMeeking, J. Computer-Aided mater. Design 1, 243 (1993).

WEBER

et al.:

DEFORMATION/FRACTURE

3. H. Cao, J. Yang and A. G. Evans, Acta metall. mater. 40, 2301 (1992). 4. W. A. Curtin, J. Am. Ceram. Sot. 74, 2837 (1991). 5. C. Weber, X. Chen, S. J. Connell and F. W. Zok, Actn metall. mater. 42, 3443 (1994). 6. D. B. Marshall, M. C. Shaw, W. L. Morris and J. Graves, Proc. Titanium Aluminide Composites Workshop (edited by P. R. Smith, S. J. Balsone and T. Nicholas). WL-TR-91-4020, Orlando, Fa (1990). 7. D. B. Marshall, M. C. Shaw and W. L. Morris, Acta metall. mater. 40, 443 (1992). 8. H. J. Frost and M. F. Ashby, Deformation Mechanism Maps, pp. 43-52. Pergamon Press, Oxford. 9. F. W. Zok. X. Chen and C. H. Weber. J. Am. Ceram. Sot. Submitted. 10. M. McLean, Comp. Sci. Tech. 23, (1985).

OF A Ti MATRIX

COMPOSITE

695

11. Z.-Z. Du and R. M. McMeeking, J. Mech. Phys. Solid?. 43, 701 (1995). 12. D. H. Zahl, S. Schmauder and R. M. McMeeking, Acta. metall. mater. 42, 2983 (1994). 13. S. Jansson, H. Deve and A. G. Evans, Metall. Trans. 22A, (1991). 14. Z.-Z. Du, C. H. Weber and F. W. Zok, to be published. 15. S. R. Gunawardena, S. Jannson and F. A. Leckie, Acta metall. mater. 41, 3147 (1993). 16. S. J. Connel and F. W. Zok, to be published. 17. P. D. Warren, T. J. Mackin and A. G. Evans, Acta metall. mater. 40, 1243 (1992). 18. C. Liang and J. W. Hutchinson, Mech. mater. 14, 207 (1993). 19. U. Ramamurty, F. Dary and F. W. Zok, Acfa metall. muter. submitted.