,, ..

THE INTERFACIAI, SHEAR STRESS FAILURE

OF FLAWED FIBER COMPOS I'I'ES

by

Le Roy Allan Baldwin

Thesis submitted to the Graduate Faculty

of the Virginia Polytechnic Institute in

candidacy for the degree of

MASTER OF SCIENCE

in

Engineering Mechanics

Approved:

V. Maderspach

May, .1970.

Blacksburg, Virginia

ACKNOWLEDGEMENTS

The author wishes to thank his Graduate Pr?gram

Corrunittee chairman : . for his encouragement

and su9gestions which have helped to give expression to

the theory outlined in this thesis.··

Every member of the Graduate Program Committee is

sincerely thanked for their patience in reading the manu-

script and for their helpful su9gestions.

The author also wishes to thank , and

, colleagues at Xerox corporation,. whose

patient explanation of the rudiments of statistical theory·

and subsequent help with problem~ encountered in computer

programming helped greatly in the completion of this thesis.

Finally, many thanks are . owed to the author's wife,

, whose help with typi~g of earlier drafts and con- ..

, tinuing encouragement and understanding helped to make

completion of this thesis possible.

TITLE PAGE • • • • • ·• • • . . . . . . . . . .. ACKNOWLEDGEMENTS •.

.......... LIST OF FIGURES

LIST OF SYMBOLS . . . . . . . . ' . I.

II.

INTRODUCTION • . . . . . . THE INVESTIGATION . . .• . :• . . 1. The Strength of Flawed Fibers

1.1 Fiber Strerigth Tests · • •

. . . . . .

1.2 The Weibull Equation

1.3 The Kies Equation

. . .

i

ii

v

vii·

l

6

6

6

7.

8

1. 4 Derivation of the Weibull Equation 8

1.5 The Darn~ge Parameter m . . . . ~ . 2. Fiber Segment Length in a Composite

2 .• 1 Fiber Stress as a Function of Dis-

tance From the Nearest·Break

2.2 Link Failure Probability . . . Expected Fiber Segment Length . . . 2.3

2.4 'rhe Probability Function . . . 2.5 Characteristics of the Curve

3. Composite Strength . . . .- .. ~ . Derivation of the Equation

Calculation of Failure Stress .

11

12.

12

TABLE OF CONTENTS (Continued)•

III. DISCUSSION

1. Theory Applications ' . . 2. Theory Limitations

3. Future Directions . . . BIBLIOGRAPHY • ' . APPENDIX THE COMPUTER PROGRAM

VITA •• .• . -.

·.No.

1 A Fiber Segment in a Matrix Subject to

Tension . . . . . . - . . . . . . -. 2 · A Composite at the Instant of Failure •

3 Probable Break in a Moderately Stressed

Fiber Segment • . .. . . . . . .. . . . . . . 14 4 Probable Breaks in a High~y Stressed

Fiber Segment . . . . . . . . . .. . 15

5 P(.R.) versus 2., for several values of the

damage Para~neter 19.

6 P(t) versus JI, for several values of the

Interfacial Shear Stress . . . . . . . 20

7 P(t) versus t for several values of flawless

· fiber strength 21-

8 P(t) versus t for several values of fiber

diameter . . . . . . . . .. . . .. . . . 22

9 Mean fiber length as a function of the

damage parameter . . - . 10 Standard deviation of fiber le~gth as a

function of the damage parameter • • • • •

. 11 Mean fiber length as a function of inter-:-

facial shear stress • ·~ • . . 12 Standard deviation of fiber length as a

LIST OF FIGURES

13 Stress in a Fiber S~gment at Composite Failure

14 (J versus m c the damage parameter . 15 (J versus 1's the interf acial shear stress c 16 cr versus (J the flawless fiber strength . . • c 0

fiber/matrix interfacial shear

maximum.tensile stress "'

R; fiber segment lepgth

PU,} fiber segment length distribution

P fiber fp.ilure probability

er u C1

0

m

v

w

f (cr)

F (cr)

g(cr)

G (er)

x

D

N

Weibull equation fiber volume (or volume fraction)

fiber stress

minimum fiber strength

maximum fiber strength

Weibull dam~ge .coefficient

Kies damage coefficient .

small elemental volume of the total volume V

number of ylemental volumes inthe total fiber volume

"link" probability density function

"link" p.robc:tbility distribution function

"fiber" probability density function

"fiber" probability density function

example parameter used to demonstrate Poisson's approximation

fiber diameter

the number of unbroken fiber segments in a ite at the instant of composite.failure

:.. . ·~ ·.·

· .. · .......

. .. . . ~

. . ~- ' .:·

.:·.-

- .

··a.

F(x}

F(k).

F(k}

Jl

.. ___ SD

()"

a c

n

_p(R.)

M 1

.. · .. : .·.

-dist~~e -of a given. link in a ··_ the nearest · break _-_ -·

· maximum possib.le fiber s~gment ,le~gth .. ---· .

. _ curoula:tive probability. distribution functio~. for---. - a link at distance x from the neare{st break

· ntimber of links in distance from nearest. break (R, = _kD} . _ _ "

cumulative probability distribution function for the k th_- link of a fiber distance kD from 'the nearest break

probability that the kth link. will-~ fail . .

mean.fiber s~gment le*1gth

standard deviation of .fiber. s~gment length . . . . .

average_- stress in, a_ given·. f:i.ber s~gme~t.

.:.:.,:. ·- ...

, ... · ..

average fiber st.ress ·in a composite at the instant.· of.composite failure

number-. of links in · a fiber of ·maximum . le~gth a . . . .

.continuous function f;it to th.epoints calculated with tjle function J? (£}·

first moment of the function -I>(t)

. • .. ,_., . -,.' : ..

. . ';_·,.:,~ . -

· .. ··• .. ·- ;

._·,··. ·.-.·

: . .---.:.:.-::·:· ·:.

M second· momeiat of the ·function P (t) 2

.· .'·. ·: .

. : ··:.-·:-:. _ ...

:" ._:-··_;:. ~ ·.J: :

: >>·~:· ..

. , ... ··'-_·.· '. , . .,_,..

. . .. ~ . :" .

·.-_,·~:...: . ' ..

. · .. ~. :

-:~· , .

':· · ....

·:: ·,

: ··:' ·. --. . .· '· .

. ; '.:: .: .: ... · ..

··. _· .~-.:.~: .· · .

···---_ .• _.

· ... ;,;:

. '• .· .. ··. ·.· .·.,..'

' < :.:·

: . ,:_·.·:--· ... / .:·

. ·.· ~. ' . . ...

·· ... '

' ... -.. _: .... ·.-·

. ! . ·· .. '::'·.· .. : .:·

-··: .... ·> . .· .·.<:. ••• ••••••••••• ••

. ":··· "_::". . :~ -.. :_: ._.· :)

.,. - ~ . : ., . . _"' .. . ·.:._;·

:-·

I. 'INTRODUCTION . . . .· - . . . . . . ; .

. . -~ . ···'' -; ...

The. literature on th.e subject ~f filler composite • • • c. : • •

stre~gth abounds with dissirnila,.r failure theories. Some -~. "::.: __ .. ,. . ·. . - - . . .· .·. . ·. ··. . . ·, . . ..

postulate continuous flawless fibers others. include sta- .·

tistically fl~wed fibers.· Nume~ous ntdd~~ of fiber/matrix . . . . . . . .

interaction are post:ula·ted .. The'. comple~lty and diversity

o.f the literature leads one tO the conclusion that, •. for . ·. ,· .. ·· '· -· '. : .· . '

several types of composite, no clear und,erstandi~g of the · ·

failure mechanism exists at this time.

One of .the more ;Pr?mising failure models SlJ.<;rgested

in the. literature, was proposed by Barratt (Re·f. 1) • He

_SlJ.9gested that statistically flawed fibers (such as. glass . . ' .. - . .,, . ; . . ·. . . . . . .. . . ' . . . : . ~ . :· ,

filaments) would break ~gain. and ~gain in. a composite as :>

.. ·the stres:s increased thu$ ·becomi~1g s·horte·r ·and. shorter a.s: · ... ··

the stress became.<Jreaterand.greateruntil the matrix

could no lo~g~r transfer·addftional stress··f:rom broken

-.: :'.-

. ·,.·· .... ·

. fibers to other fibers,.. It was SlJ.$fges·ted that.at this.

instant the_ composite·failed because the broken fiber

segments pulled out of the matrix~'. . : .. ,_-., .. :. .. . . :

. . . . .

Kelly (Ref •. 2} a:rgues that stress concentrations·,•. ,

near the ertds of fibers c.ause yi~ldi~9, i~ ductile matrice·~.- •.. ·

This.yieldi~g yield •Stress ,.

2

acting on the fiber* causing the fiber tensile stress to

build up to some maximum stress cr , (See Fig. 1} from a max ··· · value of zero at the fiber end. The Parratt model thus

assumes that as the composite stress is increased the.

fibers break into ever shorter segments so that at com-

posite failure, the shear stress, Ts' acts along the whole

length of all fibers, producing the situation shown sche-

matically in Fig. 2.

At the instant of composite failure the fiber seg-

ment lengths ".Q," will be distributed according to a prob-

ability function P(.Q,). It will be the task of this thesis·

to derive the function P(.Q,) and apply it to a prediction

of composite failure.

Metcalfe and Schmitz (Ref. 4 & 5) have developed

an experimental technique for applying a Weibull (Ref. 6)

type function to the determination the expected failure

strength of a_ given length fiber in a tension test. Sev-

eral writers (Ref. 3, B, & 9) have attempted to use a

Weibull type function in the form of a length/strength

curve. The length/strength curve is usually mqdified by

* An assumption suggested by Kelly (Ref.2} and used by Riley and Reddaway· (Ref .3) will be adopted in this thesis i.e., that Ts. has a constant value over the region on which it acts. This results in a linear increase in fiber stress as shov".rn in Fig. 1. The assumption merely simplifies the development~ It is not essential to the validity of the theory.

l

···'··

: ·'

_<I

- •. •

.·.:.:· .··.,.

:._ ....

. t .,,_._ ..

- . , ...

·.··

··.'·'..

T ·i--------S

(J max

.. ~ .. · .... ,

··-_,. ·,.·

·~ ....

· ... "'.

k=-------

.........

SHEAR STRESS 11 "[-II AND FUNCTION OF ":X." 8 FOR A

·JECT TO TENSION

- .·. ' .

. '"·

. ~··

· __ ... :··

·-· ~· . ' ·. . .. :~ ..

. . .. -...... / .. · . <.·.

FIBER TENSILE ST.RESS ucr" AS A FI~ER SEGMENT IN A MATRIX SUB-

:: . · ... ·~ : .... .. ·

.. ·:.· ·· ... ·.

·.·,-.., -,·?·;·

..... :.· ... ·· .. , ..

. ·.:-.:

'.··, .... ... . ··

· .... , ·:; . : . .

..... :

. .-.....

4 •_:-"

·~,~-lA/ , ___ -_-_-_-_-_-__.-'"~i · "··==:?'.o=:Ja=============:

SCHEMATIC REPRESENTATION OF FIBER STRESS IN A ITE AT THE INSTANT OF .FAILURE

-.._,.

the method of Coleman (Ref. 10), to give fiber bundle

strength versus bundle le~gth. It will be shown that the

Weibull function is derived based on the assumption that

stress in the fiber goes to zero immediately when the

fiber fails a.t any point which is clearly not the case in

a fiber reinforced composite. It will be shown that, while

direct use -of the Weibull function is not a valid way to

describe fiber segment length in the composite, it is

proper to infer, from a: Weibull analysis of individual

fiber failure tests, a cumulative probability distribution

function for an arbitrarily small finite length of fiber.

The cumulative probability distribution function may then

be used to develop a fiber segment le~gth distribution

function. This distribution function will be used to com-·

pute the aver~ge fiber stress at composite failure based

on the Parratt failure model.

This thes.is is, therefore, concerned with the

development of an analytical technique which will permit

prediction of composite failure stress based on the Parratt

failure model. Thus it allows an experimental evaluation

of the v~lidity_of this failure model.

II. THE INVESTIGATION

1. The Strength of Flawed Fibers

In this section the strength of individual fibers

is discussed. Experimental techniques for the determin:-

a tion of the length/strength characteristics of fibers and

application of test results to a Weibull type analysis is

discussed. The Weibu.11 "expected failure stress" equation

is derived and some con.clusions are drawn as to how hand-

ling test results might be improved.

1.1 Fiber Strength Tests

Several investigators have successfully .tested· lo~g

fibers with conventional tension testing techniques but

fibers less than about 2.5 cm. in le~gth presented major

problems in determining true_ gage length. While the long

fibers appeared to fit a Weibull type cu;r-ve well enough,

data obtained by the method of Sinclair (Ref. 11) seemed

to indicate a substantially different strength, for very·

short fibers, than would be predicted by the extrapolation

of the Weibull curve developed from tests on lo~ger_ gage

le~gths. Metcalfe and Schmitz (Ref •. 4 & 5) developed a

technique by means of which very short fibers could be

failure tested in simple tension. Their work

least two families .o:E: flaws exist.;

drawn. glass fibers. On~ family of .flaws consists·

of severe flaws widely spaced, the other family consists·

of mild flaws very closely spaced. (Some preliminary work

by Bartenev and Sidorov (Ref.· 12) indicates that there may

be as many as four distinct families of flaws in virgin

glass fiber.)

1.2 The Weibull E9uatio~

Glass fiber failure data is usually presented by

attempti~g to fit it·to the equation

a-a·· m P = 1 - exp [ -V ( u ) ]

cr ~ L2.l

0

where P is the probability that a fiber of volume V will.

fail at or below stress a .•

This equation was first proposed by W. Weibull in

"A Statistical Theory of the Strength of Materials" (Ref.

6) in 1939. When applied to glass it is.usually found

that the data are well fitted if .a is set equal to zero. u

The parameter V in equation 1.2.1 may be loosely inter-

preted as the fiber· volume and the para!Ileter "rn",. ,known ·

as the damage coefficient, must be determined from experi..-

rnental data. There are two perplexing problems with this.··

~. ·~ ·. . ... ~ . .

.· :.·.·.·

~. ·. >·

<:.·'_ :~i· ... · ..... · .... ·. . . .. , . . . , .··._, . -.,: .: :.· .. -:• _. -. ,· ~ .. --:~:. < ' ·. : . ;, :· : ... ' .: ·. . .

';.·

·' ~- :. '· . . ... ~ : . ; :·: ·_: ... : :

:·. ,·-

, '. . .... . . ·'·

eous * not an un~o~on. ;Johlemw~t~ eiu~irical · ~q\lations · ... ·. · .. ·. .:.·: . . ,: ...

but somewhat d.isquieti~g. anyw.ay •. Th~ s~cond problem is the .... ·< '.: .. · ..

· fact that : the I?robability of fail~re P only becomes. 1 as· .. · ·· .. · ·. ··

the fiber stres·s cr approaches· infinity~ . . ·-~ --:

.. 1. 3 · The Kies Equation· ..

. . . . .

· In an effort tq . overcome the second pl:oblem 'Kies • · . ..

(Ref. 7) postulated;a slightly modified equation· ... : ~

,···

<J··- <J ·. a. P = 1 - . exp[ .:..v ( .. u ) 1 1.3.1 o ... cr

0 : . -~·:·.

where a. is a di.ffeient dam~ge parameter.. Kies points ou't . . ,.. .· . . .

that where fiber stre~gth a is small in .. comparison to o 0

the Weibull'.equati!?n is more convenient and equally accur-.

ate. The Kies equation can thus be best applied to the .

._: ,,.

'· ''.:.

_;;

·1. 4 Derivation of the .Weibull Equation< ·-··-.· ..

. The followi~g is based largely on the derivation• .. · .. . . . . . · .. published by Broutman and Krock. {Re~.~ 13J. ·

.. ·'.

We may conceptually partition a flawed fiber 0£, .·

volume, v, into e la'.t"g~ number, w, of eleme.ntal volumes,:

. v. Where IJl = V /v ~ We may . th.ink of. the fiber then· as a > •

...... ·.· .. ·-.....

·,'·,_.·_· ..

.··.·:_.;.

Chain COnSi~ti-!19.<0f W individual links I the failure Of ap.y· ··. ·,. ., ' . . :

: .··.:_:_· .. :·

·. .. . ~ : :·

. . . . ~ ' ..

* If the paramete:i; V ih .equation 1.2 .• l .is. considered ... to be a fiber. .volume .'.:fraction rather than f.iber volume,-:

-~.' · dimensional hom~_geneity is restored.

9

one of which would constitute. failure of the whole fiber •.

We may express the failure probability of a. given "link"

by a link probability density function f (o) where f (cr)do·

is the probability that a given link will fail at a stress

between a and cr + der. Next a link probability distribution

function F (er) may be defined

F (cr) = r f (cr) dcr 1.4.1·

0

Thus F(a) is the probability that a. given link will fail

at a stress between zero and cr ~ ··

Now a fiber probability density function g(cr} is

defined such that. g (a) do is the probability that a .fiber

will break at a stress in the range between er and cr + dcr.

The probability density function is defined as follows:

. g ( cr} da = w [ 1 1.4.2

where 1 - F (cr) is the probability that a given link will · w-1 not break in the range zero to cr and [l - F(cr)] is the

probability that w - 1 links will not break in this range.

f (cr} dcr is the probabili.ty that one of the links will break

in the rang~ from cr to cr + dcr. ·The factor w is necessi-

tated because there are w different ways that the fiber

failure can take place. For example, link #1 could fail

and the rest not, _or link #2 could fail and the rest not

or link #3 could fail and the rest not, etc.

:' ·: .,

··-:: · .....

~: .. :· ....

G (cr) is defined•.· as follows:· ·· "' . . .. <, . . ·

· .. ·.,.· r r .· . ·.... :::;:"·'"

G (cr) g ( cr) dcr w[l . . w-1 · f (cr) do = = - F (cr) ] .. 1.4.3. ~.:

0 0 ; .:·.">·::. . .•.

which gives. .. ... ; ...

' . . . . .

G(cr) = i - (1 - F(cr)]~- 1.4.4

Thus G(cr) is the probability that a fiber composed of w: .· elemental volumes will break cl:t a str~ss level in.the ra~ge·

·· ...

from zero to cr. Weibu1i found -thfl :E&ilure data well fitted ··-,1.

·if he set ·

cr'"'"cr rn. F (cr). = ( · u ) · .. ··.' ..

a ·o

-: . ·.. ·: .. : ..

· ... , i

or for the special case of glass fiber, .

F (cr) = .·(~)m . a . 0 ·. . .

Thus 1. 4. 4 beb~mes-.. ....... -.:·:

G(cr) = 1 [l - <a<J )~]w. ::. 0

. . -.. . ... ·' '. :·.

Which may be simplified ,to the form of the _well known .· . '

·Weibull equation by means. of Poisson's· approximation · • ..m w - -wxm · ·· ·· · ·

(1 - [l x ] · - 1 - e where X << 1) ",}:

G(cr) = l .·. [· .. ( d)m]·-. .... ··· exp ..;.w ___;,. ·· -··· · .. · ·. a ... o: .. ·

; ~ ..

.· . .;: .. ::·, .. ·

1.4.5 -

1. 4. 6 .. ··.

. . ... . .. . .

1.4.7 ·. . ·. ,_., - ....

'For small diameter glass filaments it is convenient

an elemental volume, v, or "link", to be bounded by

fiber.cross section and a length measured along the fiber

axis, equal to the fiber diameter D. Thus the number of

"links", w = J1./D, where J1. is the total length of the fiber

and 1.4.8 becomes:

G (cr) = l - exp[- t (_q__)m] D a o

1. 4. 9

It is important to point out tJl.at equation 1.4.9 is dimen-

sionally homogeneous. .It is also interesting to note that

the second problem (G(cr) only approaching 1 when cr·ap-

proaches infinity) is introduced by the use of Poissons

approximation and not by the form of the function F (cr) •.

We would probably find much. greater conformity of failure

data to Equation L 4. 7 than to L 4. 8 in the region where

cr+cr • 0

In this age of the high speed digital computer,

use of. equation 1. 4. 7 should not present the i'iumerical

difficulties that it did when it was developed. Also if

t/D is used in place of the fiber volume V or length L

dimensional homogeneity is restored and Equation 1.4.9

rests on a more theoretical, iess empirical base.

1. 5 The Damage Parameter·"~"·

Metcalfe and Schmitz found that

in the range 5 to 10 for long gage lengths

12.

of the family of severe widely spaced flaws) and m c:

for short. gage lengths (characteristic of the family

mild closely spaced flaws). While the parameter m will

change when the quantity 9-/D is used in place of the quan-

tity V in Equatio:ri 1.2.l, for the purpose of illustrating

the analytical model proposed below, values in the region

m = 5 to m = 100 will be used.

2. Fiber Segment Length in ~Composite

The goal of this section will be to derive a re-.

lation for the expected lengths of unbroken fiber segments

in a composite at the instant of composite tensile failure.

In other words we wish to derive a probability function

P(9-) such that if there are N unbroken fiber segrrients in

a composite at the instant of co~posite failure, there

will be NP(9-)d9- fiber s~gments having lengths between 9,

and 9., + d9.. •.

2.1 Fiber Stress as a Function of Distance From

Nearest Break

Since fiber stress is assumed to.increase linearly·

with distance from a fiber break the maximum stress that

any link will "see" is· directly proportional to its dis.,-·

tance from the nearest break. It w.i.11 be assumed that

. ~. . . ·.·.

:;.:

. . , :.··~ . ' .··. . . =. .: ~ ,·

.. · ..

·, ... ·"-': .·,.·., .. · .. ... -· .· ·:. ~:

. ., - ·. ·.

stress that causes. the·._ le!1gth deter~ini!lg* breaks is ..

directly proportional to the dist~nce from the nearest ·. · .. ·> .'" .·. · ..

'· - : .. ·

··.break.·

In effect it is assumed that breaks alway~· occur

.·at the point in the fiber where the fiber stress· just · ·

reaches its maximum. · That is that breaks may occur at ..... ··.·.

points 1 and 5 in Fig. :fl:mt·n()t, for ~xample, at point 2. ·. ~,·~ .

·Of course all points .1 thro~gh. 5 are equally likely to.

break but as the stress con_tinues to increase more and

more points al0!1g the fiber will have reached their_ great-

est possible stress. Poirit·s such as 6 and 7 in Fig. 3

can never be subjected 'l:..o. a greater stres::; than that which

now acts on them so if · they·. ht;tve not broken. now, they are . ·- . . .

not goi!lg to. As. the stress increase~'· greater and_ great-.

er percent?-ges ·.of the. fibe;t" have . reached their maximum.

·stress and there is very'little space left for arbitrary

variation of fiber s~gment le!lgth. For example, if a.

break occurs at point B, c, or D in Fig. 4 I the fiber s~g- .·· ; .. ·

. men t lengths formed will be very close to. the s~gment .· L. ··.

lengths formed if the breaks had occurred at p9int A or ·

·'point E. ·Thus the function~ P (R.) developed in this thesis ·

. _.,

* Length determining breS.ks will be considered to··· be breaks tha't occur. sho:tt'iy before composite failure· .··· ... and establishby their distance from earlier breaks-the lengths the fiber segments will have at composite failure · ·

···'" - . ,:···.

f , .

·' .·

. :<

PROBABLE POINTS ATELY STRESSED

2

! A

I .T

8 D

, t t A c

. -·.· -,Q, E··-'--'---

1--------..__,1-i --....;...,...-~. D

16

should be a.good approximation if duril'.lg states of fairly

low stress the fibers break into moderate length segments

which remain unbroken until fairly high stresses are

reached at which point they break again. This segment

length formation model would seem to be supported by the

nature of the two families of flaws found in glass fibers

by Metcalfe and Schmitz.

We may therefore define fiber stress cr a distance.

x from the nearest break as follows

ITDT X cr = - s -

ITD 2 /4

'

2.2 Link Failur~ Probability

2 .1.1

Based on the derivation of the Weibull equation we

know that the failure probability of a given fiber link is

given by 1. 4. 6. Combining 1. 4. 6 with 2.1.1 we get

F (x) 2.2.1

the maximum length "a" that a fiber can attain is given

by

2.2.2

2.2.3

putting 2.23

2.3 Expected Fiber Segment Length.

Again recognizing a fiber to be a chain composed

of w links of length D such that w = J/,/D we may

failure probability of the kth link as

2.3.1

Similarily we may write the probability that link k will

not fail as:

;(k) = l - F(k) = l .... (kD}m ··' . a. 2.3.2

The probability P(.Q.) thata fiber segment in a

composite at composite.failure will have a length between

.Q. = kD and .Q. = (k + l)Dmaybe described as the probability

that link k breaks and that all the other links (1 to k-1)

will not break or

or

P(.Q.)D = F(l) • F{2) • F(3)

p (Ji)D

• • • F (k""."l)

(3D) m] . a

2. 4 The Probability Function

The function P(Ji) may be plotted by

the damage parameter m. F~gure 6 shows a family·

giving P(t) vs. t for several possible values of the

interfacial shear stress 'rs. Figures 7 and 8 show the

function P(t) vs. t for several different values of a and 0

D respectively.

2.5 Characteristics of the Curve

It is interesti~g to note how the curve describing

· P (t) becomes progressively m6i'e "peaked" and shifts toward

longer lengths.for increasingvalues·of m. These char-

acteristics are quantitatively described by the qualities

of mean length t and standard deviation s0 which are.

plotted as functions of m .and. Ts in Figures 9, 10, 11,

and 12.

On examination of Figures.9, 10, and 12 certain

anomalies are observed. There is no reason to expect the

average fiber segment length to begin decreasing as m

increases beyond m = 70 but as shown in Figure 9, it does.

Similarly one would expect the standard deviation to

tinuously decrease for increasi~g values of mbut as

shown in F'igure 10 the standard deviation begins to in-

crease for increasing values of m above 40. The curve

Figure 12 also.exhibits erratic behavior for values of

. greater than about 3500 psi. The erratic behavior

to be associated with close proximity of the curve

. 19

T 10:00 psi 150 ·s

~ ·.

ao = 1 x 10 6 p$i m .. 45"~ 0"

140 D = 4 ·x lo-'+ inches

130 ,· ......

.-..... . ~ ......,, 120 P-f

i:: 0 ·ri 110 .µ u r·

i:: ;:I

rr.. 100 .·i ,· ·~

i:: 0 I •ri .µ 90 m = 25.0 ;:I

,.0 I ·ri I H .µ 80 I.· 00 ·ri I A

..i:::: 7 .µ .. bO i:: Q) . ·'

i-:1 .µ i:: Q)

s 5 00 ·~. : ..

' Q) ti)

. '·- ·.

H Q) 4 ,.n ·ri rr..

3

2

0.01 0.02 ·0.03 0.04 0.0~ 0.06 0.-07

FIBJ;:~ SEGMENT -~J;NGTH (~NCHES) .. ,,

FIGURE 5

, ...

·.· .. -···

80 ·:.;

75 . ,

70

65 ,.....,. ~ ._,, p., 60

s:::: 0

•rl .µ

55 CJ s::::

. ;j µ.,

s:::: 50 0

•rl .µ ;j

45 ,0 •rl ~-.µ. co 40 •rl A

..c: .µ 00 35 ~ (])

...:I .µ 30 s:::: (])

a 00 25 -(])

~

}-I (])

20 .0 •r-1 µ.,

15

10. . ' ~ .,.

5

·.·:.

- ::·

m = :15

CJ 0. = l. x .. D 4 x

. ' •. i.-.

·- ··~: ..

.. ~--,, .. ·

.20.

-_ ...

10 6 psi . io-r+ inches

,.~

.. :.:·

.i·· >:.:,,.·

..(

T =1000 psi s .. '.·

,, T =800p.s. s . . .

· .. ,

: - • !: ~ ...

·· .. \

·.:

·'

~I ·.

.•.:-;.

./ ...

·: -:-' .

/ ~, ... . :

'.;_ ,·

·~oL 02 .• 03 .04 .05 .06 '• ~ ....

SEGMENT

·.,

i:: 0 ·rl .µ

80

70

CJ 6 0 i:: ;::l

µ.,

i:: 0

·rl .µ ;::l 50 ..a ·rl H .µ (/J

·rl A

..c:: 40 .µ bO, i:: <!)

~

30

20

10 .

.01

/ , ... .. m. = 15

. ..

T = 1000 psi s

D = 4 X· lo-If inches \

- r .02 .03 .04 .07 .08

FIBER SEGMENT LENGTH (I~CHES)

I I I

ao= l.2xl0 6

psi

,....,, c< ~

P-1

Q 0

•rl .1-J (.) Q ;::l

r:r... Q 0

·rl .1-J ;::l

..a •rl

""' .1-J rJl

•rl A

. .i:::: .1-J bO Q Q)

.H

130 "" =· l x 10 6 ·psi

120 -D inches

110 -

100

90 -

80

7'0

60

,....,. '

Cf.)

<ll

0 .. 100

0.090

~ 0.085 i:::

H '-"

..c: '.µ

b.O i::: <ll

,_::i

.µ 'i:::

<ll s

0.080

0.075

0.070

b.O '<ll 0.065 . Cll

!-I <ll

..c 0.060 ·rl r:r.. Q)

~ 0.055 !-I <ll ' > ~

0.050

0.045

0.040

.0.035

·: ..

. 20 r

'30

T· s

ao

D

=

=

=

1000 ·psi

1 x 10 6 psi

4 x 10-'+ inches

I

60 70

1. 6

1. 5

1. 4

,....., N 1. 3 0 rl x

= 1. 2 ~

= 4-1 o.L! l":l 0

•rl .µ 1. 0 t\!

•rl > Q) '

A 0. g '"O

""' ctl '"O .

~ 0 ~- 8 .µ Cl)

0. 7

cr o· = 1 ;<

D"' 4 x .10- 4

0. 14·

0.13

0. 1~

,,....., Cll 0.11 (J)

..c: CJ i::::

·rl ."-.,/ 0.10

I~

..c: 0.09 .µ bO i:::: (J)

,_;r 0.08 .µ i:::: (J)

s 0.07 00. (J) ti)

1-1 0.06 (J) ,.a •rl ~

(J) 0.05 ti{) .

C1l 1-1 (J)

> 0.04 <t:

0.03

0.02

0.01

-

.

-a = 1 x 10 6 .. 0

D 4 x 10- 4

1500 2000 2500 ·3000 3500

STRESS "T " s

psi

inches

-·~

4-1 0

1. 4 .

1. 3

1. 2 -

0. 9

0.8

Q . 0. 7 0

•r-1 .µ cd

•r-1 :> Q. 6 Q)

Q

"Cl 1-l 0. 5 . cd

"Cl Q ~ cd .µ rJ) '0.4

0.3.

0.2

0.1

cr 0 1 x 10 6 psi

D 4 x 10·- 4 ·inches

..._~-~---__;,~-.-~~-.-~~-..~~--r~~~r-~~.-~~,--~---' ' 500 lobo 1500 .. 2000 2soo 3000

either .end of the fiber s'.=gment length axis.

that a significant number of the length determining . . .

breaks are occurring in the nth link (where a = nD) when

this happens the round. off error (introduced when length ·

11 a 11 was defined to have integer value nD in the computer

program) begins to have a pronounced effect.

3. Composit~ Strength

In this section an equation is developed by means

of which the composite strength may be predicted. This

equation_ gives the average fiber stress at composite fail-

ure based on Parratts failure model. With the average

fiber stress known, composite strength is found directly.

by the well known "law of mixtures".

3.1 Derivation of ~Equation

Where a cross-section contains a large number of.

fibers, the percentage of stress in the composite cross

section that has value cr. is identical to the percentage l.

of fiber volume in the whole composite on which stress·

cJ. acts. This percentage is described by the following l.

ratio:.::

RATIO =

If we wish to

N P(i·)i· - l. 1 n Z: N P(ii)ii·

i=l

acting over

28

whole .composite. cross sect.ion, .we may sum all stresses in

the cross section times the corresponding percent~ge (or

ratio) o~ fiber volume on which.they act:

n I

i=l (J. ·. . .. . N P (9.. • ) 9...

l. l. l.

n I N p {9.,i) 9.,i.

i::::l

where n = a/D

=

n l

i=l n l

i=l

N p (Ji.} 9.,. l. .l.

N P(g,.)9., . ... J. l.

(J. l.

3.1.2

The stress in an individual fiber is assumed to be

as shown in F~g. 13. The aver~ge stress in the fiber will.

be

Since the stress cr ·is reached at the point max

x = .i/2

'fe have from 3 .1. 4 and 2 .1.1 that

(J max 4't' = s 1)2,

and from 3.1. 3 we get

a = _('t's/D) 9.,

or

21' s D

3 .1. 3 .

3.1.4

3.1.S

. 3.1.6

(J max

(J

'·i

29

-"-----------

£/2 -~ ' l £ )

STRESS IN A FIBER SEGMENT AT COMPOSITE FAILURE

FIGURE 13

r.

x

:i.

30.

Canceli?g N and D and substituti!lg

3 • 1. 2 we. get:

<J = c

n 1" . l (R, .. } 2 p (1 . ) ..

s i~l. i ·~ .. · i.

n D l

i=l 1. PU.}

l. l.

3.1.8

If a continuous function p(1} is fit to the values

determined by 3.1.8, we may write the aver?-ge fiber stress

at composite failure as follows. a

1" J 1 2 p(1} d.ll, s

0 <J = c

f D t p(R.) dt

3.1.9'

0

3.2 Calculation of Failure Stress

A computer program was written to solve Equation

3 ~ L 8 (See Appendix) for various values of the parameters ·

m, 1" , <J , and D. s 0 . The values reported f()r m by Metcalfe

and Schmitz ranged from approximately 5 to approximately

100. Figure 14 shows a plot of <J versus m for nominal c values of 1" , a , and D~ Rosen (Ref. 8) and others have s 0

su9"gested values for 5 in the region of 500 to 5000 psi.

Figure 15 shows a plot ·of cr. versus T for nominal values· .. c s of <Jo, m,. and D. <Jo, interpreted as the unflawed

fiber, is estimated to.be in the r~gion

230

220 -

T = 1000 psi 210 s

00 = 1 x 10 6 psi = (.)

lo-'+ ;~

b 200 D - 4 x inches = -·rl Cl) 190 p..

"" I 0 rl 180 x ~

Cl) Cl) 170 (!) H .µ ti)

H 160 (!)

,..c •rl µ..

150 (!)

00 Cl! H (!) 140 -:> <

130

120

. 110 ~·.

10 20 30 40 50

228 . m = 15

226 (J 0 = 1 x 10 6 psi

n, = 4 224

x ·10-'-+ inches

222

........,. ·rl 220 CfJ p;

"' I 218 0 -,.-j x

"-./

:: 216 -(.)

b ::

:-<._-·:_¢' CfJ 214 ... CfJ I Q) ~ +J Cf.)

212 1--1 Q)

..0 •rl µ:., 210 -Q) 0.0 Cl! 1--1 Q) 208 :>

<d ..

206 ..

-··: ·-·-

204 -

202

200

.·· ..

: ~- .

. ··;:-.

. : .'

· .. ·.·

..

_ ..

, ... ,; •rl !/l 0..

('t)

I 0 r-i x ..._, :

.CJ t>

!/l !/l C]J ~ .µ

. ti)

~ C]J

.0 ·o• --· -•ri µ.,

C]J ·- bO

Cl! ~ C]J

C·,·. > <

···1:

·-..:•'

•,·.,. .. · ,._,.·

. : .'·

_::·

.. ·. ~·· '~:--·

~· ..

.280 .: .. ·

.. ·-· .

. 2.60

240

··.I'·

220

200

180

. ·~ ~· .... _. ...

160

1·40~

. ";·" -.• ~ ..

.:'····

120

,, ·,.-

. ~ .·

T s .=

=

15

1000 psi

4 x 10.- 4

; •'

.·. ' .. ~'.:'. : .... · .. .-· ',1_··

·.···.;

. - . . '···'"

: •. : : ~ .. ··:

.. ·1··· .. ·-

· .. '·'

inc he '.-'··'

.".:;, ~ ..

.. ··'.·

--~~""""'~~~-.-~--~--~~--~~~--------•r--·--~~.~~~~1.--~~--~,--~.....: 9·· 7 8 9 10 11· 12 13 14

FIBER STRENGTIJ

34

to 1500 ksi. Figure 16 shows. cr for nominal values of m, - . 0

Ts'· and D. The fiber diameter D is the only parameter which

has directly measured, fully reliable, values. Fiber diam-

eter is a controllable function of the fiber drawi!lg con-

ditions. However, the values of cr and mare also directly 0

effepted by the drawing conditions. Therefore completeness

would demand that a plot be made of cr versus D for the c nominal values of m, T , and cr • However it was found that s 0

the value of crc was constant over the range of values of D

for which it was computed (D = 2 x 10- 4 inches to D = 8 x 10- 4 inches). Therefore no plot is included.

The theory developed in this thesis predicts, a

priori, that composite stre!lgths are less than cr 0/4.

Approximate values of CJ are fairly well established in 0

the literature (Ref.. 4 & 5). Therefore, it can be shown

that the majority of tested composites have strengths less

than CJ /4. Several theories of composite behavior (Ref. 8 0

& 9) predict composite strength to be substantially. great-

er than o /4. 0

The theory developed above gives values of com-

posite stre!lgth that are of the same order of m~gnitude as

experimentally determined strengths. It remains for an

extensive laboratory testing and evaluation program to

show direct correlation.

. ,.:.

... · ·:·:' .· . . .

., ... - .. : ·. :,-.

::.:. ~ .:· . ,.·, -. ,· ..... · ' . . :, ·~ ."

·: _... '. < ..... •

.. ......

. . . . ... < ..... _:(<·::>"" . .

•'.'··.' ".' .·:. ' .. ·

III• DISCUSSION .·--.-. ·,::',

· · i. Theory Applications _

.·It remains for an extensive laboratory testi~g al).,d ·

evaluation pr~gram to determine the ra~ge of application

· of the above theory. Once .the ra~ge of application is : . . ..

established it should be possible to shed new l~ght on the

nature of interfacial shear stress in fiber reinforced

composites •.. · The .Weibull damf!:ge parameter can be experi-

mentally determined by the method of Metcalfe and Schmitz

as cap.. the valu.e of cr • 0

Diameter D is easily measured and

composite strength can be found by standard tests. Equa-.

tion 3.1.8 can thenbe solved fcir 'ts.

More importantly, this theory applied to an exten- ·

sive testing pr~gram could provide a us'efui step toward

developi~g a consistent strength model for fiber rein-

forced materials.

2. Theory Limitations

In the course of the discussion of Fig. 3, it was

pointed out that the curve P(!) does not precisely describe

.all the fiber s~gment le~gthsbut that over half of the

fiber segments will be somewhat_ greater in length than that

predicted. . It is in fact possible for length 2a to· exist ··

·, 35.

. ;'' .. ".: . ~ ·* .-

··:-::f·.··

'I ::

·'.

·. "· -· '

. --,-:, , .. ··

.·· .. :

. ·,·. ···' ,:.·· .. ' . . ·. ·:. ~.

':"·.· .' '··

··.:·::

·in the composite at :failure.·.· ·It may be necessary, there-"-: .: ' .

fore, to determine a we~ghti~g factor with a value be.:.

tween 1 and 2 which would correct this discrepancy.·

This failure theory based upon Parratt' s model· n~g-:- .•• '

lects any tendency toward crack propagation in the matrix

surroundi~g a point where a fiber has b:r:-oken. Be~ause of . . .

this,· the theory takes no account of ·the influence on he~gh-::- . . ..

bori~g fibers of a break in anygiven·fiber. Instead it

treats the occurrence of brea~s in fibers as independent·

events as evidenced by eqµation (2. 3. 4). · ..... ·

3 •. E'uture Directions

In addition to the experimental approach outlined

above certain further· theo.retical ·investigations ·should

be undertaken ...

First an attempt should be made to establish the . .

we~ghti~g factor mentioned above. One approach to this

problem in~ght be the so c(llled Monte Carlo technique to •

be done on a d~gital computer using. a·bui.lt-in random

number. generator. A. theoretical fiber/matrix system would··

be defined in the computer a'moderate stress applied and

by the definition of "link-11 failure probabil,ity outlined

. ,• .

.-. ~-' '-;· . . . ... ' . '.·

' .. ·.

·~. : :.

above the number of breaks decided• The location of these · · ·

breaks would be picked by the random number_ generator •..

Next the r~gion of max shear.stress defined and the

· ...... · ··,-... -· ....

. ·' .. , ..

_1.:·

fiber stress calcul-ated. -___ The stress level is then -raised, -.. . .

the number of new breaks calculated, and their location

assigned_ by the rand_om number_ generator, new max shear

stress r~gions defined_ and new fiber stress calculated,. :.

This process is repeated until max shear stress acts over

the whole fiber thus defini~g composite failure. The

location of the breaks is defined and the fiber segment

le~gths tabulated. - Starti~g with an unstressed unbroken·

·fiber composite the whole series-is repeated again and

~gain until a sufficient population of s~gment lengths is

accumulated_to define.a distribution function which i.s

then compared with the __ function P .( t) and a we~ghti~g factor

established.

, Secondly it would be useful to -fit a curve to the --. . - .

function P (.t) so that ·equation 3 .) .• 9 could he solved in • ·- ·

closed form._ This would' probably .require a non-linear_

r~gression analysis. The ideal function p(l!.) would be one . ' . . .

- that incorporated the parameters m, - t , -<J :, and D as con~_ s _-0 -. . -

stants ··such that. a change in one of thes.e constants would

produce a cha~g~ in p CJi) that close.ly corresponds to the

cha~ge in P (.II.) for a similar constant cha~ge.

. --. --

_; . ·-

. .: : ·. :·~. _. .. :

. ·-... , .. ,. .... -.....

1. Parratt, N~J., Defects in Glass Fibers and Their Effect on the Streng:Eh{)f Plastic Moulding$7"Rubber and· Plastics Age 11 ,·March 1960, pp. 263-266.

2. Kelly, A., The Strengthening of Metals by Dispersed Particle8; Proc. Roy Soc.of London:-282 (1964) 63 Series A.

3. Riley, V.R. and Reddaway, J.L., Tensile Strength and Failure Mechanics of Fibre Composites," Journal of Materials Science #3 (1968), 41-46. ·

4. Metcalfe, A.G., and Schmitz, G.K., Effect of Length on the Stre~th of Silass Fibers, Proceedings ASTM 1964 pp. 1075-1093.

5. Metcalfe,.A;.G.· and Schmitz, G.K., Testing of Fibers, Materials Research arid Standards, Aprir-1967, Vol-ume 7, Number 4, pp. 146-152.

6. Weibull, A Statistical Theory of the Strength of ' Mater'fals, Handlingar RoyalSwedish Academy of Engineering Sciences,NR 151 (1939).

7. Kies, J.A., The Strength of Glass, NRL Report #5098. April 3, --r9"58~

8. Rosen, B.W., "Mechanics of Composite. Strengthening" Fiber composite Matenals, Am. Soc. for Metals, Metals Park, Ohio· (1965). ·

9. Rosen, B.W., Tensile Failure of Fibrous Composites AIAA Journal, Vol. 2, No.ll, Nov. 1964, pp. 1985-1991. . .

10. Coleman, B.D., On the Strength of Classical Fibres · and Fibre BundleS, ·Journal of the Mechan1cs and

Ph'Ys.ics of Solids.· 7(1958) 60-70 November 1958.

11. Sinclair, D., A Bending Method.for Measurement Tensile Stre;ngth and.Young's Modulus of Glass Fibers, Journal of Applied Physics, May 1950, Vol;,.· 21, No. 5, pp~ 380·-386.

39

12. Bartenev, M. and Sidorov, A.B. Strength and Length of Glass Filaments, Glass Industry, February (1966)-.-

13. Broutman, L.J. and Krock, R.H., Modern Composite Materials, Addison-Wesley (1967).

14. Cameron, N.M., An Introduction to the Factors Influ-encing the strength of Gla-ssFibers, T & A.M. Report No. 186. Department of Theoretical and Applied Mechanics. University of Illinois, March 1961.

~ .' .

.:··.::"·

.·:'

... ·.·'"·-··

. . .

. - ·, .:: ... ~~: .

. (.· .·:·.-·.

'' ·.~· .. . ~ - ... ..... ,.

-·« .. ··

. .

·•· APPENDIX THE COMPUTER PROGRAM

·The computer pr9gram below is set up to compute a

value for the function P(t) as defined by equation 2.3.3

for every increment D of fiber s~gment le~gth·from t= 0

tot= .a (the maximum fiber s~gment le~gth)~

The computer pr9gram a1so calculat~s.the values of

the first and second moments of the function P (t) where

the jth moment M.. is def.ined as follows .• J. a

, M. = ·J . ' J . Al

0

Next moments M and M are used to define cr , t, 1 2. .· . c

SD as follows

C1 . c ..... · T sM2

.D M 1

which follows from equation 3 .1. 9

R, = M 1

by definition

S ~ (M - M2)1/2 D · 2' 1

by definition

A2

A3

A4

The relation of the symbols. used in the test of.this

thesis to those used in the computer pr9gram is. given

·· below. •.' -

·· .... ',,· ..

,: -- '. ....... . i

_, ... _.

40 . ·" ... · ...... .

... · .. . . : " . ·. --· .:·. . . .. ~ :• . .

... " .... ·· . . ·,.

.·. "·: . . ~ .... -.

--.".'·· .·

··• = ..• ":.-,,

-.. · -:

:,.······

PROB (K} = P (£) where JI. =

EL(K) = .R. = KD

D = D the f iher diameter

M = m the Weibull da.rnage parameter·

SIGMA = cr the maximum fiber stre~gth ' 0

TAU = T the interf acial shear stress . s ·.

A = a the maximum possible fiber length . ·

N = n the number of .·"links" in le~gth 11 a"

FIRST = M the first moment of P (.R.) 1

SECOND = M the second moment of P (JI.) . 2.

'·~· '•

SIGMAC = crc the average fiber stress at composite failure.

STAND = SD the standard deviation of JI,.

The followi~g computer pr~grarn was used to define P (9,) for

values of T ranging from T =. 500 psi to T = 1500 psi. s . '•. s . . s It can be used with only slight modification to define

P (R.) for various. values of the other parameters .•

200

300

DIMENSION PROB(3000};EL(3000) PRINT 200 FORMAT(1Hl,25X, 'FIRST' ,12X, 'SECOND' ,12X, 'SIGMAC-'J PRINT 300 FORMAT(lH+,sx, 'TAU') D=4.0E:._04 M=15 SIGMA=l.OE06 DO 3 J=5. I 15 TAU=FLOAT(JJ*l.OE02. A=SIGMA*D/{4. O*TAU). N=A/D . XN=A/D .. · . . IF(XN.;..FLOAT(N}-0.5}20,20,10 N=N+l

20 EL(l) =D PSUB=l.O . PROB (1) = (D/A) .**M DO 1 K=2,N EL(K)=FLOAT(K)*D PSUB=P.SUB* (1. o- (EL (K-1)/A) **M·

1 PROB (K) =PSUB * ((EL (K) /A) **M)/D FIRST-EL (1) *PROB (1) *D/2. 0 SECOND= (EL (1) **2) *PROB (1) *D/2. 0 DO 2K=l,N FIRST=FIRST+EL (K) *(PROB (K-1) +PROB (K)) *D/2. 0 .

2 SECOND=SECOND+ (EL(K) **2) *(PROB (K) +PR.OB (K~l) *D/2. 0 SIGMAC=TAU*SECOND/(D*FIRST) STAND=(SECOND-FIRST**2}**0.5 PRINT 100, FIRST,SECOND, STAND, SIGMAC

100 FORMAT (1H,20X,El3.5,4X,El3.5,4X,El3.5,4X,El3.5} 3 PRINT 400, TAU

400 FORMAT (lH+ I 4X,Ell. 5) STOP END

The vita has been removed from the scanned document

·>

·· ... \

' ... ·.··

... ·. '·.'·

THE INTERFACIAL SHEAR STRESS FAILURE

OF FLAWED FIBER COMPOSITES . · .·.:

by

Le Roy. Allan Baldwin ..

ABSTRACT •...

This thesis derives a Probability distribution func;.;. .··.·

tion which expresses the expect,ed le~gth of disqontinuou:s ·

fiber s~gments in a uni axially·. all~gned fiber reinforced· · . . .

composite. The stre~gth of elemental volumes of fiber,

used in this derivation are inferred from 1e~gth versus.··

stre~gth tests.on individual.fibers. The fiber.s are

assumed to have statistically distributed flaws that a.Ilow ·.

characterization of fiber stre~gth by a .Weibull type

equation •. ·

The probability distribution function is then used

.to compute the ayer~ge fiber stress at composite failure •.

. ' : .. · .· .... ·

This allows ·calculation of composite stre~gth by the well .. . :

;known law of mixtures •.. · .. ·.:,:

composite stre~gth. ·determined by this method show . . . . . .

·good correlation.with published experimental values for .. '

glass fiber composites.; , .

. . . . '""<>>· .. :,.··.

: · . .': ··. ;. ·:·· . . ~:.

:~ ..

._'···.: