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**Question** - 132Principles of Composite Material Mechanics

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FIGURE 4.2

Maximum stress, Maximum strain, and Tsai-Hill failure surfaces in o,, o, space.

Strength of a Continuous Fiber-l

FICURE 4.3

Comparison of predicted failure >r

epoxy. (From Burk, R.C. 1983. A.trc"

Reprinted with permission.)

comparison of theoretical faili

data for a unidirectional g

strengths along the principal

criteriory we would expect d

stress is uniaxial along tho-

in the Maximum Stress Critet

in biaxial stress siLuatio*.1

l"hmately typical tbr compo#

Experimental biaxial failurr

surfaces can be obtained br i

specimens. Biaxial stress fielc

ot-axis uniaxial loading tesb I

:n equations (2.31), the applir

l,rading test shown in figure {

along t
...Read More
he principal material:

T--

w:here the applied normal sl

mrportance of the sign of thr

4.2.1 Maximum Stress Criterion

The Maximum Stress Criterion for orthotropic laminae was ap-parently

iirrt ,ogg"tted in t920 by ]enkins l24l as an extension of the Maximum

Nor-ui5t "ss Theory 1or nanUne's Theory) for isotropic T1l:ttg"'which

is covered in elementary mechanics of materials courses [251. This crite-

iiotr pt"aigls failure *hut any prinl4ral rnaterial axis stress. component

exceeds the corresponding strengtt . Thus, in order to avoid failure accord-

;; r;-his;tit"tiol", tne f"ollowlig set of inequalities must be satisfied:

(4.2)

where the numerical values of sr(-) and sr(-) are assumed to be positive' It

is assumed that shear failure along the principal material axes is-indepen-

JJ "f the sign of the shear stresi trr' Thus, only the Tagni.tude of t" is

important, as"sho*n in the last of equations $'2)'.As :ll9!!,1!.!.!*.'"yy-:!

the'shear strength for off_axis loading may dep-end on^the sign of lhe"ihear stress.

The failure suiface for the Maximum Stress Criterion in ot - 62 space

is a rectangle, as shown in figure 4.2. Note that this failure surface is

na"p""a"it of the shear anditress r12, and that the criterion does not

u..ol^t for possible interaction between the stress components' That is'

the predictea n*itirrg value of a particular stress component is-the same

wh"'th"r or not otheistress components are present' Figure4.3 shows a

o.

{vfechanics

i

[parently

lvlaximum

fh, which

lUs iift-,

prponenF

ftacc@-

Sri"a, 'o

!

I

| (4.2)

t

I

Wagth of a Continuous Fiber-Reinforced Lamina

o2 kpsi

20

Maximum strain Test data (typical)

01 = or cos2 e

02 = o, sin2 0

Trz = -o' sin 0 cos 0

133

,"*("/

6"

/ /"/Tsai-Hillrsar-nrx"4g5,

///,rto2

EGURE 4.3

comparison of predicted failure surfaces with experimental failure data for graptite/cpory. (From Burk, R.c. 1983. Astronautics and Aeronautics,2r(6),58-62. Copyright AIAA.Reprinted with permission.)

comparison of theoretical failure surfaces with experimental biaxial failuredata for a unidirectional graphite/epoxy composite [26]. since thestrengths along the principal material directions provide the input to thecriterion, we would expect the agreement to be good when the applied

. stress is uniaxial along those directions. Due to lack of stress interactionin the Maximum stress cnjg19-a, howgver, the agreement is not so goodin biaxial stress situations-%* -r*.r*n-. ia r r;f^-

Experimental biaxial failure data for Comparison with predicted failuresurfaces can be obtained by applying biaxial loading directly to the testspecimens. Biaxial stress fields can also be generatea inairecity by usingoff-axis uniaxial loading tests [27] or off-ans ihear-loading tests. Accordin[to equations (2.37), the applied normal stress, o,, in the off-axis uniadalloading test shown in figure 4.4 produces the following biaxial stress statealong the principal material axes

ot kpsi

(4.3)

yhere the applied normal stress, ox, may be positive or negative. Theimportance of the sign of the applied strlss in the interpreta"tion of the

20 40 60 80 100 120-.,

=. '. -.::.

i or.tPu!"

[urface is

idoes not

I That is,

1*re same

Ishows a

chnnics

de of

value

as shown

r sffess

e prin-

L Stress

LL, the

lPa

node of

Strength of a Continuous Fiber-Reinforced Lamina 737

Again there is no shear stress along the principal material axes' since

Trz = t', (cos2 0 - sin2 0) = t*1sos' 45o - sin2 45"; = g

So transverse tension is now the goveming mode of failure' and the corre-

sponding value of the off-axis shear stress required to produce failure is

now onlY

t* = 44'8MPa

So simply changing the sign.of .the off-axis shear stress from positive to

negative produces Jto*pfJtay different mode of failure and a much lower

failure stress.

4.2.2 Maximum Strain Criterion

In 1967, Waddoups [29] proposed the lvlaximum Strain Criterion for

orthotropic laminae ;;; eitension of the Maximum Normal Strain

Theory (or Saint V""uiti Theory) for isotropic materials' which is also

discussed ir., "f"rr."r,iuiv mechanics of materials courses [25]. This crite-

,ior, pr"dicts failure wlien any- principal material axis strain component

exceeds the corresponding ultimate strain' In order to avoid failure

according to this ;il;; the following set of inequalities must be

satisfied:

-ef)

rvhere the numerical values of et() and er(-) are assumed to,be positirre

and the ultimate ,truit, ur" all eigineerinf strains as defined by equation

G.il. et with the Ma^im"m StrJss Criterion' it is assumed that shear

failure along the pttt.ip"f -aterial axes is independent of the sign of the

shear strain 1t. .. ,r A r

Due to the similarity of equation (4'5) and equation (4'2)' * 11t11f i z r -i,

surface for the Maximum Strain Criterion in et - t2 sPace ls a rectang'j n '11 '

"i*itu, to that of the Maximum Stress Criterion in o' . 02 Space' In 01 - o' u ,

;;;; how"rr"r, the Maximum Strain Criterion failure surface is a

* skewed parallelogtarn, as shown in figure 4'2 and figure f'3 The thape

;il;;;dl"l"du* cun be deduced-by combining the lamina stress-

strain relatiorr"tip,-i'l eqriation (2'24)' with the relationships given in

mode of

138 Principles of Composite Materisl Mechanics

equation (4.L). For example' the limiting strain associated with the positive

1di$-ction ls

Strength of a Continuous Fiber-Reinforced

interaction between the stress comPon

lf," "qr;utiotts for plane stress lead to el

i" ""i mechanics of materials book' t

criterion or von Mises Criterion (circa ea

ouadrattc interactjon criteria for predi<

dooic metal s [25].In 1948, Htll [30J sugg

.Ja U" modified to include the effecr

i oodutty itottopic metals dwng large 1

ttt""-ai-""sional state of stress along

frft"lzg axes) in such a matefiaT' the fai

the Hill Criterion in c.1,C,2, and cu space

A@i;r; - o r.\;2,iB(o, - ot)t + C(or - ozf

or-{)O) =-- --to

which is the equation of a straight hne having.intercept 1t:]l:9] and slope

1 /v.^ (hs.4.2\. Asimilar development usinjihe limiiing strain along the

#il;? direction Yieldd the equation:

sl*) Qr - \ltzozet=Ei='Er- h

oz=vzror+s$1)

(4.6)

(4.7)

(4.8)

whichistheequationforastraightlinehavingintercept(0'sr(*))andslope

v,,. These rines form,il ffi;;?J iop ,ia"r.?espectively, oj tlr" parallel-

ogru- shown n nf""-|V' and similar.consiieration of the limiting

strains in the negatrv" i"#iJi.ections yields equations for the remain-

ing two sides. It shourd Le noted, however, that iepending on the mag-

nitudes of the '"*;;;;";ii't3'1"a

stiffiresses' the intercepts of the

Maximum strain Criterion parallelogram may not be the same as those

oi the Maxim* St'"" ititerion rectingle in #ess space' For some mate-

rials, the lines defi#il;t ;11-?i'- of the Maximum Strain

Criterion parutt"togto:'intercept the hotizontal axis at stresses less than

the measured tensile '"i t"-pff:ve longitudinal strengths' which con-

tradicts experimentaf "*'la"*" [5'8]' According to Wu [5]' such conttadic-

tions develo P as a'"*f':of u"'ulnbigt'ous coirversion from strain space

to stress space unles';;;;;ih;riatical constraints on the properties

are satisfied. only ft';;;;;it -u*'iut' ut" ut" intercepts always the

same for the maxlmtr'm;;;";; and maximum strain criteria' As with the

Maximum Stress C'iltl;;' the Maximum Strain Criterion does not

account for possible tt"'"ttit" between stress comPonents' and the pre-

dicted failure,"'tut"'al"l"tJi tnt* g*a ugl"-""t withexperimental

i"rJa]",r"*'""'tii::*Jj*ruH;m.?ffi :H,fi'":""i

f;""1il?f|,J",,i#',uo because tn" '""'tii"g equations are relativelv

simple.

4.2.3 Quadratic lnteraction Criteria

The so-called quadratic interaction cnteria'also evolved from early failure

theories for isotropic#;ffi;;"ky differ from the maximum stress

and maxim"," 't'"i^ "iit"'1"-io

tnut ti-rey include terms to account for

b zeto equation (4'9) reduces to:

B+C=-

nhere Yr is the yield strength along L c

J""i,ft" 2 and"3 directions give the eq

a+c=S; A'

where Y,andY,atethe uniaxial yi"19 t

tt, t"tp"ctively' The yield strengths--

assumed io be the same' Solving equahol

il.ourty fot A, B, and C, we find that

11,^=

":*E11,u=

":*t11

"= x*E

egt#ffi ffiE Jait"re is Predicu

".il-l"f test along the 1 direction n'ith c

t40 Principles of Composite Material Mechanics

Similarly, for pure shear tests along the 2g' gI' and 12 planes' equation

(4.9) gives:

j'r.r:.: li tli,i f 611111 11 ;,

:r*-:lcrFal stress di,t

sq::-:l-. that drir-e :

[.r ::c l'l-mental er-ide

::r,-:oe thre slip an*J c

.mq" il.d the Hill G

ffiWdrustatic state of

;rr::-inq. horr-ever.

::nrn ;:'-.iuce shear s

[t :r::: -]nlear terrns. c

16;r*,n:€.-er" a_il of thes

mu-n,iaric interastio

,nr -i:1. Tsai and I

01il' i :ensrrr polrnor

rnewr. ;uqgested ear

,Swrr quadratic i-l

11!ruuil.edbytteE

mrdltttere *re contracti

irmpner:::nentallv det

rum:u. :wpectivelr-. n

'rs,, 15 rust tre < 1, i

fimr :-,c case of plar

rttirfl;]uuifi E Jn t f .1 5 ) becr

l-.-6:

mitilmer ffe linear tt

lbrrernnu:-e tlFre shea-r sl

unr, fc ;r,ggi of the :

,$ffimwft.i o. rennains. Fl

mm,U := = 6:: ilr€ ft

gfrmErg:h-< in tensior

mtlri ar;;r'ru-nt intera,

',ffiilr' .:[-, '-l the streng;

,mff il:tc -;niadal arrc

uruueu -,,-th the HiLl (

Itursts l-fi uniarial

tmro[ri{L{ illLrtion of t]

zo=i; zr=$; ,,=+ (4'13)

Yh Ysi

where Y12,Yzz,and Y., are the yieid strengths in shear associated with the

12,23, u"a gi Planes, resPectivelY'

The extension of the Hilt Criteiion to prediction of failure in an ortho-

tropic, transversely isotropic lamina was suggested'by Azzi a-nd{sai [31]

and Tsai [32]; the resulting equation is oftei"refe""d to as the Tsai-Hill

Criterion. If the 123 directions are assumed to be the principal material

axes of the transver,"if i*oopiclamina' with the L direction being along

the reinforcement dirJction, i? plane stress is assumed (ou = "= r" = 0)'

and if Hill's unlrottopit yieid stiengths are replaced by the'corresponding

effective lamina t";;tit;';;ii= "'

v'l Yt= s'' and Yrz = sm' and

equation (4.9), equati u:n'(+'tZ1'u'td "quitio" (+'f g) reduce to the equation

foi the Tsai-Hill failure surface:

o? oroz oi , 11, -',_-- - l- ,T 2 -'sL si si $t-t

(4.r4)

AswiththeHillequation'failureis'avoide'difthe-left-handsideod

equation (a.la) is 'r,tiiJr"irt"e is predictea ir tneleft-hand side is 2l'

The failure surface generated by thii equation is an ellipse' as shown in

figure 4.2' The "ifip'" 'flo*" i" fig"'" d'2.i' ty**etric about ti:::ilT

because of the assumpt-" of equaf strengths in tension and compresslon'

The Tsai_Hiil equatriori-:;-d" used i,hen tensile and compressive

strengths ure diffe'ent"Ui tt-Ott u3ing the appropriate value of sl and s1

for each quadrant ot strlss ,pui". For-examplL, if o, is positive and o' is

negative, the vatues.oi';-;;JJ;;' I would bf used in equation (4'1a)' The

resulting failure suriace Is no longer symmetric about the origin' as shown

for the case of gr"ph;7;;;"t; figure4'3' Although such a procedure

is inconsistent with rr.," "rJ"^'prions used in formulating the original von

Mises and HilI Criteria, it has been successfully used for some composites

125,321.As shown - fig"'" 4'3' the procedure seems to work reasonablv

well for the graphite-izp"y material :T.":pt for.the fourth qrradrant of

stress space. O,," -uy-tto uito""t for different strengths in tension and

compression is to inciude terms that are linear in the normal stresses 01'

02, ;d 03, as suggested by Hoffman [33]'

In addition to tf," ir""iorrty meniioned limitations of the quadratic

interaction criteria uaid on the von Mises model, there is another problem'

Since the von Mises and Hill Criteria are phenomenolgsical--tfreories for

the prediction of yi"iJi"j in ductile metais' the equations are based on

141

Incs

[on

Strength of a Continuous Fiber_Reinforced Lamina

J3)

ihe

principal stress differences and the corresponding shear stresses and

;SffJHLi:ffi"'# and dislocauo', *o'"*ent in metauic crystals.

carrse rhe ctin .-r ,-.^,"^1l,glq"rts that a hydrostatic state of stress does notroes nol

ffi jl";:irdn%H::*'i:n:l"n:li'-f:l;;iF;,:d*,hvie,d-

firol + F22ol + Fooo|+ fio1 + go2 *2F1261o2 = |(4.1.6)

[ro-

311

fiIl

rial

ng

0),

Fgnd

brrt

t4)

EH.

ln

IrIL

re

9r

ts

te

|Ir

re

TI

cs

ty

k

d

ilt

coupling, however, a hydrostati" rt# ;i;#, #'"; ;r,l;HTi:ff"tffi:ican produce shear strains and failure. Hofiman s equation [ad], Uy virtueof its linear terms, could predi"t f"ii;t; f;; the hydrostatic srate of stress.However' all of these theories trt" ""a a u" speciil cases of a *oru g"r"rutquadratic interaction cnterion, *ru"f, *ifiU" discussed next.rn7971, Tsai and wu.[34] proporuJut i-p.orr"a and simplified versionof a tensor polynomiar ruiiitr"'*,uory ro, lr.,irot.opic materials that had

ffl,:Hg,:"'r.:1,:-Ti::lj Got,denblat and Kopnoi, f3tt. ilil rsai_wu

where the contracted notation i, j = 1,2, ...,6is used, and { and Fl, areexperimentany determined strength i"1ro* of the second and fourthrank, respectively. * o,t9"j t" uuoiZ ruti,rr"l ,rr" t"ft_h";;;id" Jiuqrutio'(4'15) musr be < i, and failure ir;;J#;'*h"., the left-hand side is >1.

:fiiff]L.r$iS;J:"* *itr' i' =-",,10, on = ',,Ji,;;;= xst=',

'.ffi"Ti'"Xtfi H:,ffi:ffiffi ;;ld*r;ii#';;4".:''i:#;:H|;

F;oi+Fi,o,c,=1(4.1s)

where the linear terms in the shear stress <16 = T, have been droppedbecause the shear strength alongj*ffi;al material axes is not affectedby the sign of the shea*t "rr."Th.rr',;;t " quadratic term in the shearstress 06 remains. However, the tinear t"rXr-in tf," .or_ui;o""""" o, = o'and o, = o22 are retained n""u"r"1i"y'iutu t ra account the differentstrengths in tension and comprerrio.,. ii Jdtior,, t1,",".''.#,ro,o, ot",inro account interaction u"t*""r tn" ".rry1 il;r;;. r,iriii. inf ""ceptionof Fo' arl the shength tensors in eq.,utionla.16) can be expressed in termsof the uniaxiar and shear strengtis ;;id the same approach that wasused wirh the Hill Crirerion. r"r 8*r-pi", ior the tur,riorii.Jli-prurrio'tests with uniaxial stresses o, = s.(+) irrj o, =-sL(_), respectively, simulta_neous solution of the two "q,ru'orir;"i;g from equafi on (4_.16)yields:

-14' = 7i- and E =+-1si'rt' -' rl*) sl-)(4.17) ...Read Less

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