Thermal stresses in fibre reinforced composites

.I. Mech.

Phys. Solids,1969. Vol. 17, pp. 387 ta 403.

THERMAT, STRESSES

Pergamon

Press.

P&led

IN FIBRE

in Great

Britain.

REINFORCEL)

COMPOSITES

,\ ~rm:onri’rrc~n analysis is nlade of the dimensional changes, stresses and plastic flow accompanying an allotropic transformation in one component of a fibre composite, taking into account the simultaneous stress relaxation which occurs. The theory predicts t,be transformation dimensional changes observed experimentally under various temperature cycling conditions and provides an estimate of the stresses. Extending the theory t.o include the effect of a variation in temperature, sevcrai categories of coeEcieuts of thermal expansion arc obtained, none of which correspond to t.he rule of mixtures. The predicted transition from one coe%cient of expansion to another during continuous cooling is confirmed experimentally and the resulting dimensional changes verified.

1.

INTKoBtICTroX

.\t.r. ~*n_\zc”rrc.~r~ fihrc composites would experience variations in temperature and, in addition, several of the components envisaged for USCin fibre composites undergo an allotropic transformation. The dimensional changes, stresses and plastic flow resulting from such variations are important in design especially against thermal fatigue and in estimating the microstructural stability of the composite. In order to analyse these effects the nature of stresses in a fibre composite and the characteristics of creep and relaxation in metals are taken to be as given by DE SILVA (I%%). It will be seen later that in this case it is also necessary to take into account lateral effects since there is a finite lateral contribution to the dimensional changes and the extent of plastic flow even when the difference in Poisson’s ratios (qn -- vr) + 0. It is simpler to begin by examining the effect of an allotropic transformation alow, where a volume change occws in one component. The theory can then be c~xtended to include the effect of a variation in temperature. Throughout the analysis second order effects have been assessed and neglected to present important aspects with greater clarity and to provide a practically usable solution,

NOMENCLATURE

Young’s modulus Poisson’s ratio volume fraction of fibres and matrix respectively 387

I)

l’hcrrnal stresses in fibrc reinforced composites

389

L

kFIG. I. The position of the matrix

in tlrc composite unit after the transformation material compatibility.

if there is no

During the transformation the stresses could rise to a point at which they cause the matrix to yield.* It is found that the normal yield criteria will be first satisfied at the fibre-matrix interface and that the yield zone will then spread radial11 outwards. The analysis can be extended to include this propagation but for simplicity we assume that the period for the plastic zone to spread is relatively small, and that the matrix does not work-harden. Then, once the matrix yields it cannot balance a further increase in (a,),, meaning that the fibres cannot extend further. That is, the longitudinal extension of the composite will cease. However, plastic flow can occur in a radial direction and will do SO to accommodate the remaining change in volume. Taking a subscript y to denote the values at yield, we have

and

(ii) Finite

weep

effects

Generally the transformation occurs as a nucleation and growth process over a range of temperature and requires a finite time. As the transformation begins in the matrix a compressive stress develops in it and a tensile stress in the fibres. These stresses cause creep at the temperatures concerned. As the transformation proceeds and the temperature decreases the stresses and extensions increase, continuously opposed by creep. *For simplicity

of presentation

we ignore the variation of yield stress with temperature

as the mean value in the temperature

range considered.

and regard the yield stress

:I!)0

:-,

1. /:

(“,;T)t/l

‘I’hrrnd

Similarly,

301

stresses in fibre reinforced composites

in the radial direction,

SlJ’ = ;

is.

1

S,r j- St’ -

S7]‘.

(2)

Hy analogy from (AS). .,l {s [X,], -

67) = II S,?/+- ;

C s,t/.

S1Ihhtitllting from (1) iLIlt (2),

L

.rI”[“~]~-fi?7}1~~~s~-S~--S~~-f.L’1, Simplifying

and integrating

The transformation l)ositc rxtension [hc]dl

=

from ti to tz,

is complete

when x _

1’ p+S~‘-S~’ 1 1

>

.

SE, so that the final longitudinal

com-

[hc]dl is given by f

((n

+

c)

6,

-t

(A

-

u)

17dt -

fi

td,

+

c

$

(tdi’

-

17dt’)

*

(3)

> It

is found that Af > 12 and B > (I, thus making the last term much smaller than

the others. extension

Therefore

it is seen that creep in the fibres increases the longitudinal

of the composite

while creep in the matrix

normal creep and the presence of enhanced c%scrpt in cases where VI is extremely large stresses. The transverse

displacement

decreases

it.

Due to more

creep, the matrix creep effect is larger,

small when the fihres have to sust,ain veq

of the composite

[&*I, at any instant is given by

where Z” = (K/L) x is the displacement due to the transformation alone, &” is the lateral creep displacement at r = R and is given by (see Appendix 2b) Cz 5;

= B

s

L

[(%?),,, -

‘j (%)#J at

where the stresses hare the values at 1‘ = If. icnced due to the maintenance

\vhrrc (I’,),

and (&),

and ((p/L) 22+ &’ composite is

C!, is the lateral displacement of material compatibility and

expcr-

are given by (A-)4 with 61 and St replaced by (z - 6% - 71~) Thus, the final lateral extension of the respectively.

vz’}

‘I’tlis will be greater than without

creep effects because

the longitudinal

extension

Thmnal

stresses in fibrr reinforced conq)ositcs

898

c

LENGTH FIG. 3. Schematic

in tcnqmature

length curve obtained while cooling through the transformation range.

from B to C

on

the theory of the transformation.

us with a basis for determining cycling

conditions.

expansion

It is interesting

are obtained,

Consider

the behaviour

of the composite

to note that several categories

none of which correspond

a fibre composite

in the fully

temperature

is decreased

the matrix

(in general),

the fibrcs preventing

it from

It also provides under thermal of coefficients

(unstressed)

to contract

doing

so.

tensile stress in the matrix and a balancing compressive taneously creep occurs under these stresses.

state.

This causes a longitudinal stress in the fibres.

i7 bs COMPOSITE

FIBRES

of the coqoncnts

If the

more than the fibres

MATRIX

I+o. 4. Virtual displacements

of

to the rule of mixtures.

annealed

attempts

temperature

due to a fall in temperature

62’.

Simul-

._

l’hermal stresses in fibre rcinforrrd

longitudinal

and lateral directions.

3!)5

ron~positrs

.Z Feinkeramic-Maschinen

meter was used in which the specimen

Electronic

is kept under a vacuunl

Dilate-

of N 5 x 10P5 mm

Hg. Its temperature and lcugth rclativc to a push rod are automatically and continuously recorded. An acscurncy of about O.1 p could bc obtained in the lengtll measurements

and it was possible

Specimens were annealed before to relic\-c any existing strcssrs.

To use the theory to y&Y

md

to automate

the required

thermal

after the passage through

experinirntal

treatment.

the transformation

results it is necessary to first compute

t,llc I-arious creep parameters. Normally, liowc\-cr, lack of creep data on the conlponents dots not make this possible. As mentioned in Section 2 (ii), thercforc, it is more convenient to measure experimentally [A,], and [A,*], a.nd proceed as in Appendix 2 to obtain an estimate of the stresses. Even so, 7% and tz” must be c\-aluatcd and often sullicient data arc not available to do this. Thcrcforc, wl1e11 such

theoretical

calculations

are made

in this investigation

(i.c.

stresses in Figs. 6 and 9), we make the simplifyin, (7 assumptions

to obtain

the

that creep in tlit

fibres is negligible and that ~1and y’ nlaintaiu their proportionality effects are prcscnt in the matrix.

cvcn

when

creep

Resides t,he possibilit,y, in principle, of predict.ing experimental results, t I w immediate usefulness of the theory is that it provides a theoretical basis for unclcrstanding

the effect of an allotropic

iI tiljrc composite

urlder various

transformation

thrrmal

and a teniperaturc

cbycling conditions.

Withollt

\.ariation in rccoursc

to

nuy creep data, the theory also permits us to ~al~~ulat~~ the magnitude of tile variolls changes that occur if no creep effects were prcscnt. 111 addition. it shows whether tllcsc \-alues give an upper or lower bound

to those obtained

whcii crcrp effects

lm\~:~il.

1)iiring unintcrruptccl

cooling

of a Cu spccimcn

contiiuiol~sly

reinforc4

with

40 per cent by volume of W wires the theory predicts that the composite will first clisplay the temperature coefficient x.cl gi\.en by (5) and then the coefficirnt xrS2 gi\-en bv (7). would

Since c(cl > ccP and xc2 ~1 xP it was expected

first record

This was observed shown in Fig. 5.

a decrease expcrimrntally

To obtain

in length

longitudinally,

and the curve recorded

that the dilatomc+cr and then

an increase.

by the dilntomcter

the actual length changes of the specimen

is

the contrac-

tion of the machine push rod must he subtracted. IGgure 6 shows the actual longitudinal and lateral length changes ol&Lincd c~spcrimcntally when a Cu-W composite 2.745 cnl long and 0.56 cm in diamctel was cooled from 49O’C.

In Table 1 the dimensional

with the theoretical bounds obtained by calculating The agreement is good except for the temperature

changes observed

arc compared

the case without creep effect.s. coeflicient after matrix yield.

Here the obser\-ed value of (2.8) is less than the theoretical lower bound, which is obtained as follows. The tcmpcrat~urc: c~ocfticient after yield in this system is (Lx~ ! (I /I,) (hj/r17’)] front (‘7). ancl hinc*c tlrj/O - 0. wc: 1lai.c the iurq\ialit!

:$!I(;

.\. IL

‘I‘.

,,1:

SIl.V.\

:r1rtl

c:. .\.

(‘ll.\l,\\,,‘,i

3’37

Thermal stresses in fibre reinforced conkposites

TEMPERATURE

500

TEMPERAT%E

“C

400

“C

I

c&=2*8 > [4*6]

-J

E’x. (i. The actual length clranges while cooling a Cu-IV composite with the theoretically calculated values (without creep effects) in brackets. The 3 sign indicates the theoretical value to bc a lower or upper bound respectively (see Table 1). Also shown are the theorctirally calculatwl stresses (with crerp cffect,s).

Figure 7 shows the length changes rccordcd longitudinallyon a E’c-Fe& specimen 3.68 cm long and 143 cm in diameter. As predicted from (3) the transformation length change is much smaller (10.7 TVduring the cooling pure Fe (122.9 p). In addition, it is seen that the transformation while cooling is greater than that obtained be given later. transformation

during heating.

cycle) than for length change

The reason for this will

It is first necessary to see the reason for the variation in longitudinal length change with cooling or heating UZP found experimentally

(Fig. 8). The stresses existing just prior to the transformation, say on cooling, depend on the temperature range through which cooling has taken place and the rate of cooling. For a fixed temperature range a faster rate of cooling gives less relaxation (i.c. a smaller 6 factor) and increases the stresses. If the rising stresses cause the 1nntri.u to >,icld (in tension) and it does not strain-harden, the stresses t.hcln remain

3!)S

stressesin film

‘I’her~nal

imtil the matrix extension

yields

after

c~itses. Therefore

which

rcinf’orrecl

they remain

constant

a slower rate of cooling

mn

composites

and the longitudinal

not only decreases

the trrtns-

foriuation length change by increasing the 6 factor (see ccluation 3) hut nlso does so by &creasing the initial tensile stress which reduces tlic stress range the IllidXG pass through before yielding in compression. \\‘v now examine the reason for the longitudinal transformation length changc~ on ~wctli~~g being greater than that on heating (see Figs. 7 aud 8). For the Fe matris twuld

the total

crq

rate during

the c( -+ y change

is much greater than that during

the y -+ x change for two reasons (CLIKARTI and SIIEIWY, 1964).

Firstly,

normal

creep during the y + 01change is much kss than that during the x --f y change as the y pliasc is inherently about %OOtimes more creep resistant than the CLphase, and during

the y ir cc change

the Fe contaius

it

l)rt~~lot~liliarltly y matrix

with

regions of M. During the a --f y change there is an x matrix and the reverse is true. Sc~oi~dl~. cnhnnccd creep during the y -t x change is much less than that during

Ik:. 9. ‘l’he actuaf length charlges per ce~~ti?~tTewhile cooling a Fe-Fe2B s~%&iien (see also Table 2). The theoreticallp calculated stresses (vvith creep effects) are also shown.

the rx -+ y change as more vacancies created on heatiag

than interstitials

(the nucleating on cooling

phase b&g

since the activation

more dense) are energy for the

formation of the latter is greater. Therefore [&&,, > [&],,+, so that from (:3) the composite length change while cooling is much greater than that while heating. As the cooling or heating rate is increased the length change approaches a constant value corresponding to the matrix covering the full stress range from tensile yield to compressive yield. The yield stress of a Fe being considerably smaller than that of y Fe, this asymptotic value too is larger when cooling. Figure 9 shows the actual longitndinnl

and lateral transformation

length changrs

I ,ongitudinal length chances tluring transfornmtiotl (y/cl:1 Icngth) I,nteral length change during transforrnatiou (p/c111 dia.)

>-!I 19.3

‘l’hermal stresses in fibre reinforced composites

Ml

\vhWV E B = (1 + P) (1 ‘l’hc (lisplacemrnts

7J in a radial plane

2Y).

from the positions Q

U=Pr+

in

Fig. 1 are given by

,

(A2)

T

whcrc I’ and Q are constants.

Then

Q

dU

~,=-_=p--

f2 ’

dr

(A8) u po_-zl)+Q r Taking (t~~)~ as negative (i.e. compressive conditions are obtained. (i).

(~r)~ = 0

at

(ii).

(7;), = 0

+++~) at

i.e.

/

strain) and (eL)f positive,

the following

boundary

r = R,

(l-“_)(P_-S?)

i.e.

I.2 *

--vtn(&),=O.

7 = 0, Qf = 0.

(iii).

(~r)~, = (or),

at

(iv).

l(7Ji’)f] + ](U’),]

7 = p,

= 81 where & is the lateral misfit,

,,-(Pmp+$j =&.

i.r.

L [(ez), + (e&l = h,

(v). (iv).

where & is the longitudinal

misfit.

The compressive

force in the matrix is balanced by the tensile force in the fibres, i.e.

T’m [&II (e&

2/L %I (7% -

-

rm W,}]

=

r'f[Ef

(df

+

2Pfvf

{pf

+

vf (df)lg

Solving these equations and substituting into (Al) to (A3) it is possible to obtain expressions for ~77the stresses and strains in the fibre and matrix after the allotropic transformation is complete. Of particular interest is the longitudinal extension of the composite X, = 1; (e& where (e,)f is given by A (ez), = 23 $ _t C 3 in which

c

=

-

[2# (~nz ,% ~,,z + vf /3f pf) (1 -

2vm) ,%p2 -t 275 rs, vr (1 -t # Br R2)],

(A4)

/ \Ci)

Thermal stresses in fibre reinforced composites

403

i.e. = km [beh

-

3 {Mn

-

(4Jl.

(4,,}]

&.

Therrforr, 86’ = P km [(q&,

-

3 ((4m

Sinrilnr expressions are available for 67 and 67’. compute the various creep displacements.

-

If values of k nre known it is possible to