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**Question** - 132Principles of Composite Material Mechanics
-s[-)<or<s{])
-sF)<oz<sf)
Ittrl < srt
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) <e, < ell)
l+)-e)' <t2 1€1
lYt l<en
(4.5)
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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