of 16
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SGYVIFV
"m`kisgnum`rpgysifs.ik" 0
BD VQIFIQY \IVFHVIQYBD@VQIFIQY @KA SD@VQIFIQY
Qgb prhpbrty hl ` m`tbri`d ehay ey virtub hl wgifg it rbj`iks its hrijik`d fhklijur`tihk (i.b. sg`pb `ka sizb)
wgbk tgb bxtbrk`d ablhrmikj lhrfb is rbmhvba is f ddba bd stifity. Qgb prhpbrtyhl tgb m`tbri d ehay ey virtub
hl wgifg it ahbs kht rbj`ik its hrijik`d fhklijur`tihk wgbk tgb bxtbrk`d lhrfb is rbmhvba is f`ddba pd`stifity.
Ablhrmikj lhrfb ? @k bxtbrk`d lhrfb `ppdiba th ` ehay wgifg fg`kjbs its sizb hr sg`pb hr ehtg is f`ddba
ablhrmikj lhrfb.
Sbrlbftdy Bd`stif ehay ? @ ehay is s`ia th eb pbrlbftdy bd`stif il it fhmpdbtbdy rbj`iks its hrijik`d lhrm wgbk
tgb ablhrmikj lhrfb is rbmhvba. Vikfb kh m`tbri d f`k rbj ik fhmpdbtbdy its hrijik`d lhrm sh tgb fhkfbpt hl
pbrlbftdy bd`stif ehay is hkdy `k iab`d fhkfbpt. @ qu`rtz liebr is tgb kb`rbst `pprh`fg th tgb pbrlbftdy bd`stif
ehay.
Sbrlbftdy Sd`stif ehay ? @ ehay is s`ia th eb pbrlbftdy pd`stif il it ahbs kht rbj`ik its hrijik`d lhrm bvbk
sdijgtdy wgbk tgb ablhrmikj lhrfb is rbmhvba. Vikfb bvbry m`tbri`d p`rti`ddy rbj`ik its hrijik`d lhrm hk tgb
rbmhv`d hl ablhrmikj lhrfb, sh tgb fhkfbpt hl pbrlbftdy pd`stif ehay is`dshhkdy `k iab`d fhkfbpt. S`r`llik
w`x, wbt fd`y `rb tgb kb`rbst `pprh`fg th ` pbrlbftdy pd`stif ehaibs.
F`usb hl Bd`stifity ?Ik ` shdia, `thms `ka mhdbfudbs `rb `rr`kjba ik sufg ` w`y tg`t b`fg mhdbfudb is`ftba uphk ey tgb lhrfbs aub th tgb kbijgehrikj mhdbfudbs. Qgbsb lhrfbs `rb nkhwk `s iktbrmhdbfud`r
lhrfbs. Zgbk kh ablhrmikj lhrfb is `ppdiba hk tgb ehay, b`fg mhdbfudb hl tgb shdia (i.b. ehay) is ik its
bquidierium phsitihk `ka tgb iktbr mhdbfud r lhrfbs ebtwbbk tgb mhdbfudbs hl tgb shdia `rb mikimum.
Hk `ppdyikj tgb ablhrmikj lhrfb hk tgb ehay, tgb mhdbfudbs bitgbr fhmb fdhsbr hr jh l`r `p`rt lrhm b`fg
htgbr. @s ` rbsudt hl tgis, tgb mhdbfudbs `rb aispd`fba lrhm tgbir bquidierium phsitihk. Ik htgbr whras,
iktbrmhdbfud r lhrfbs jbt fg`kjba `ka rbsthrikj lhrfbs `rb abvbdhpba hk tgb mhdbfudbs. Zgbk tgb ablhrmikj
lhrfb is rbmhvba, tgbsb rbsthrikj lhrfbs erikj tgb mhdbfudbs hl tgb shdia th tgbir rbspbftivb bquidierium
phsitihks `ka gbkfb tgb shdia (hr tgb ehay) rbj`iks its hrijik`d lhrm.
VQUBVVZgbk ablhrmikj lhrfb is `ppdiba hk tgb ehay tgbk tgb bqu d rbsthrikj lhrfb ik hpphsitb airbftihk is abvbdhpba
iksiab tgb ehay. Qgb rbsthrikj lhrfbs pbr ukit `rb` hl tgb ehay is f`ddba strbss.
strbss =@
L
ehaytgbhl@rb`
lhrfbrbsthrikj
Qgb ukit hl strbss isK/m5. Qgbrb `rb tgrbb typbs hl strbss
0. Dhkjituaik`d hr Khrm`d strbss
Zgbk heobft is hkb aimbksihk`d tgbk lhrfb `ftikj pbr ukit `rb` is f`ddba dhkjituaik`d strbss.
It is hl twh typbs ? (`) fhmprbssivb strbss (e) tbksidb strbss
Bx` m p db s ? (i) Fhksiabr ` edhfn hl shdia `s sghwk ik lijurb. Dbt ` lhrfb L eb `ppdiba th tgb l`fb wgifg g`s
`rb` @. Ubshdvb L
ikth twh fhmphkbkts ?
Lk= L sik f`ddba khrm`d lhrfb `ka L
t= L fhs f`ddba t`kjbkti`d lhrfb.
Khrm`d (tbksidb) strbss =@
Lk=
@
sikL
5. Q`kjbkti`d hr sgb`r strbss
It is ablikba `s tgb rbsthrikj lhrfb `ftikj pbr ukit `rb` t`kjbkti d th tgb surl`fb hl tgb ehay. Ublbr th sghwk ik
lijurb `ehvb.
Q`kjbkti`d (sgb`r) strbss=@
Lt=
@
fhsL
Qgb bllbft hl strbss is th prhaufb aisthrtihk hr ` fg`kjb ik sizb, vhdumb `ka sg`pb
(i.b. fhklijur tihk hl tgb ehay).
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SGYVIFV
"m`kisgnum`rpgysifs.ik" 5
6. Eudn strbss ? Zgbk lhrfb is `ftikj `dd `dhkj tgb surl`fb khrm`d th tgb `rb , tgbk lhrfb
`ftikj pbr ukit `rb` is nkhwk `s prbssurb. Qgb bllbft hl prbssurb is th prhaufb vhdumb
fg`kjb. Qgb sg`pb hl tgb ehay m`y hr m`y khtfg`kjb abpbkaikj uphk tgb ghmhjbkbity
hl ehay.
Bx`mpdb 0.
Lika hut dhkjituaik`d strbss `ka t`kjbkti`d strbss hk ` lixba edhfn.
Vhd. Dhkjituaik`d hr khrm`d strbss d=
52
63sik033
= 2 K/m5
Q kjbkti`d strbss t=
52
63fhs033
= 5m/K62
Bx`mpdb 5.
Lika hut Eudn strbss hk tgb spgbrif`d heobft hl r`aius
03fm il
`rb` `ka m`ss hl pisthk is 23 fm5 `ka 23 nj rbspbftivbdy lhr `
fydikabr liddba witg j`s.
Vhdutihk ? pj`s
= `p@
mj = 10323
0323
+ 0 032 = 5 032 K/m5
Eudn strbss = pj`s
= 5 032 K/m5
VQU@IKQgb r`tih hl tgb fg`kjb ik fhklijur tihk (i.b. sg`pb, dbkjtg hr vhdumb) th tgb hrijik`d fhklijur tihk hl tgb ehay
is f`ddba str`ik,
i.b. Vtr`ik,=ihkfhklijur`thrijik`d
ihkfhklijur`tikfg`kjb
It g`s kh ukit
Qypbs hl str`ik ? Qgbrb `rb tgrbb typbs hl str`ik
(i) Dhkjituaik`d str ik ? Qgis typb hl str ik is prhaufba wgbk tgb ablhrmikj lhrfb f`usbs ` fg`kjb ik dbkjtg
hl tgb ehay. It is ablikba `s tgb r`tih hl tgb fg`kjb ik dbkjtg th tgb hrijik`d dbkjtg hl tgb ehay.
Fhksiabr ` wirb hl dbkjtg D ? Zgbk tgb wirb is strbtfgba ey` lhrfb L, tgbk dbt tgb fg`kjb ik dbkjtg hl tgb wirb
isD.
Dhkjituaik`d str`ik , =
dbkjtghrijik`ddbkjtgikfg`kjb hr Dhkjituaik d str ik =
DD
(ii) \hdumb str`ik ? Qgis typb hl str`ik is prhaufba wgbk tgb ablhrmikj lhrfb
prhaufbs ` fg`kjb ik vhdumb hl tgb ehay `s sghwk ik tgb lijurb. It is ablikba `s tgb
r`tih hl tgb fg`kjb ik vhdumb th tgb hrijik`d vhdumb hl tgb ehay.
Il \ = fg`kjb ik vhdumb \ = hrijik`d vhdumb
v= vhdumb str`ik =
\
\
(iii) Vgb`r Vtr`ik ?Qgis typb hl str`ik is prhaufba wgbk tgb ablhrmikj lhrfb f`usbs ` fg`kjb ik tgb sg`pb
hl tgb ehay. It is ablikba `s tgb `kjdb tgrhujg wgifg ` l`fb hrijik`ddy pbrpbkaifud r th tgb lixba l`fb is turkba
`s sghwk ik tgb lijurb.
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SGYVIFV
"m`kisgnum`rpgysifs.ik" 6
t`khr=x
GHHNBV D@Z @KA MHARDRV HL BD@VQIFIQY@ffhraikj th tgis d`w, witgik tgb bd`stif dimit, strbss is prhphrtihk`d th tgb str`ik.
i.b. strbssstr`ik
hr strbss = fhkst`kt str ik hrstr`ik
strbss= Mhaudus hl Bd`stifity..
Qgis fhkst`kt is f`ddba mhaudus hl bd`stifity.
Qgus, mhaudus hl bd`stifity is ablikba `s tgb r`tih hl tgb strbss th tgb str`ik.
Mhaudus hl bd stifity abpbkas hk tgb k turb hl tgb m`tbri d hl tgb ehay `ka is ikabpbkabkt hl its aimbksihks
(i.b. dbkjtg, vhdumb btf.).
Rkit ? Qgb Vd ukit hl mhaudus hl bd`stifity is Km5 hr S`sf`d (S`).
QYSBV HL MHARDRV HL BD@VQIFIQYFhrrbsphkaikj th tgb tgrbb typbs hl str`ik tgbrb `rb tgrbb typbs hl mhaudus hl bd`stifity.
0. Yhukj's mhaudus hl bd`stifity (Y) 5. Eudn mhaudus hl bd`stifity (N)
6. Mhaudus hl rijiaity ().
0. Yhukj'smhaudus hl bd stifity
It is ablikba `s tgb r`tih hl tgb khrm`d strbss th tgb dhkjituaik`d str`ik.
i.b. Yhukj's mhaudus (Y) =str`ik`dDhkjituaik
strbss`dDhkjituaik
Khrm`d strbss = L/@,Dhkjituaik`d str`ik =D/D
Y =D@
LD
D/D
@/L
Bx`mpdb 6.
Lika hut tgb sgilt ik phikt E, F `ka A
Vhd. DE
=D@E
=@Y
MjD
@Y
LD = 03: 032.503
0.30303
= 1 03
6 m = 1mm
DF
=DE
+DEF
= 1 036 + 03: 03103
5.3033
= 1 03
6 + 2 036 = 9mm
DA= DF +DFA = 9 036 + 03: 03003
02.3033
= 9 036 + 02 036 = 51 mm
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SGYVIFV
"m`kisgnum`rpgysifs.ik" 1
BDHKJ@QIHK HL UHA RKABU IQV VBDL ZBIJGQDbt rha is g`vikj sbdl wbijgt Z, `rb` hl frhss-sbftihk @ `ka dbkjtg D. Fhksiabrikj hk bdbmbkt `t `
aist`kfb x lrhm ehtthm.
tgbk xD
ZQ
bdhkj`tihk ik ax bdbmbkt =
@Y
Qax
Qht`d bdhkj`tihk @Y5
ZD
D@Y
axZx
@Y
Qaxs
D
3
D
3
Khtb ?Hkb f`k ah airbftdy ey fhksiabrikj tht`d wbijgt `t F.M. `ka usikj bllbftivb dbkjtg/5.
Bx`mpdb 1.
Jivbk Y = 5 0300 K/m5 = 031 nj/m6
Lika hut bdhkj tihk ik rha.
Vhd.
m`ss hl sg`aba phrtihk
)x(m
m
^wgbrb m = tht d m`ss =@ d]
Q = m5
x
5
x
mQ (d x)5
5
xQ =
5
m 5(5 x5)
tgis tbksihk widd eb m`ximum `t @
5
m 5`ka mikimum `t E (zbrh), bdhkj`tihk ik bdbmbkt hl wiatg ax=
@Y
Qax
Qht`d bdhkj`tihk
ax@Y5
)x(m
@Y
Qax
3
555
=
3
65
5
6
xx
@Y5
m
=
@Y6
@
@Y6
m
6@Y5
5m 555565
= 00
6165
0356
)2.0()133(03
Y6
= 9 036 m = 9mm
Bx`mpdb 2.
Lika hut tgb bdhkj`tihk ik edhfn. Il m`ss, `rb` hl frhss-sbftihk `ka yhukj
mhaudus hl edhfn rb m, @ `ka Y rbspbftivbdy.
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SGYVIFV
"m`kisgnum`rpgysifs.ik" 2
Vhd.
@ffbdbr`tihk, ` =m
Ltgbk Q = m` wgbrb m =
mx
Q =
mx
m
L=
xL
Bdhkj`tihk ik bdbmbkt ax =@Y
Qax
tht`d bdhkj`tihk, =
h@Y
Qaxa =
h
Y@
Lxax=
@Y5
L
Khtb ? -Qry tgis prhedbm, il lriftihk is jivbk ebtwbbk edhfn ka surl fb ( = lriftihk fhbllifibkt), ka
F`sb ? () L 4 mj () L ; mj
@ks. Ik ehtg f`sbs `kswbr widd eb@Y5L
5. Eudn m haudus ?
It is ablikba `s tgb r`tih hl tgb khrm`d strbss th tgb vhdumb str`ik
i.b. E=str`ik\hdumb
bssurbSr
Qgb strbss ebikj tgb khrm`d lhrfb `ppdiba pbr ukit `rb` `ka is bqu`d th tgb prbssurb `ppdiba (p).
E=\
p\
\\
p
Kbj tivb sijk sghws tg`t ikfrb`sb ik prbssurb (p) f`usbs abfrb`sb ik vhdumb (\).Fhmprbssieidity ? Qgb rbfiprhf`d hl eudn mhaudus hl bd`stifity is f`ddba fhmprbssieidity. Rkit hl fhmprbssieidity
ik Vd is K-0 m5 hr p`sf`d-0 (S`-0).
Eudn mhaudus hl shdias is `ehut lilty timbs tg`t hl diquias, `ka lhr j`sbs it is 037 timbs hl shdias.
Eshdias
; Ediquias
; Ej`sbs
Ishtgbrm`deudnmhaudus hl bd stifity hl j`sE= S (prbssurb hl j`s)
@ai`e`tifeudnmhaudus hl bd`stifity hl j`sE = S wgbrb =v
p
F
F.
Bx`mpdb
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SGYVIFV
"m`kisgnum`rpgysifs.ik" j = 03 ms5) hr g = 03 nm
It m`y eb khtba tg`t tgb gbijgt hl Mhukt Bvbrbst is kb`rdy 9 nm.
QHUVIHK FHKVQ@KQ HL @ ZIUBF =
5
r1Zgbrbis mhaudus hl rijiaity r `kais r`aius `ka dbkjtg hl wirb rbspbftivbdy..
(`) Qhqub rbquirba lhr twistikj ey `k`kjdb = F
(e) Zhrn ahkb ik twistikj ey`k`kjdb, Z =5
0F5.
\IVFHVIQYZgbk ` shdia ehay sdiabs hvbr `khtgbr shdia ehay, ` lriftihk`d-lhrfb ebjiks th `ft ebtwbbk tgbm. Qgis lhrfb
hpphsbs tgb rbd`tivb mhtihk hl tgb ehaibs. Vimid`rdy, wgbk d`ybr hl ` diquia sdiabs hvbr `khtgbr d`ybr hl tgb
s`mb diquia, ` lriftihk d-lhrfb `fts ebtwbbk tgbm wgifg hpphsbs tgb rbd tivb mhtihk ebtwbbk tgb d`ybrs. Qgis
lhrfb is f`ddba 'iktbrk`d lriftihk d-lhrfb'.
Vupphsb ` diquia is ldhwikj ik strb`mdikba mhtihk hk ` lixba ghrizhkt`d surl`fb @E (Lij.). Qgb d`ybr hl tgb
diquia wgifg is ik fhkt`ft witg tgb surl fb is `t rbst, wgidb tgb vbdhfity hl htgbr d`ybrs ikfrb`sbs witg aist kfb
lrhm tgb lixba surl`fb. Ik tgb Lij., tgb dbkjtgs hl tgb `rrhws rbprbsbkt tgb ikfrb`sikj vbdhfity hl tgb d`ybrs.
Qgus tgbrb is ` rbd tivb mhtihk ebtwbbk `ao`fbkt d`ybrs hl tgb diquia. Dbt us fhksiabr tgrbb p`r ddbd d`ybrs `,
e `ka f. Qgbir vbdhfitibs `rb ik tgb ikfrb`sikj hrabr. Qgb d`ybr ` tbkas th rbt`ra tgb d`ybr e, wgidb e tbkas th
rbt`ra f. Qgus b`fg d`ybr tbkas th abfrb`sb tgb vbdhfity hl tgb d`ybr `ehvb it. Vimid`rdy, b`fg d`ybr tbkas th
ikfrb`sb tgb vbdhfity hl tgb d`ybr ebdhw it. Qgis mb`ks tg`t ik ebtwbbk `ky twh d`ybrs hl tgb diquia, iktbrk`d
t`kjbkti`d lhrfbs `ft wgifg try th abstrhy tgb rbd`tivb mhtihk ebtwbbk tgb d`ybrs. Qgbsb lhrfbs `rb f`ddba
'visfhus lhrfbs'. Il tgb ldhw hl tgb diquia is th eb m`ikt ikba, `k bxtbrk`d lhrfb must eb `ppdiba th hvbrfhmb tgbar`jjikj visfhus lhrfbs. Ik tgb `esbkfb hl tgb bxtbrk`d lhrfb, tgb visfhus lhrfbs whuda shhk erikj tgb diquia
th rbst. Qgb prhpbrty hl tgb diquia ey virtub hl wgifg it hpphsbs tgb rbd`tivb mhtihk ebtwbbk its
`ao`fbkt d`ybrs is nkhwk `s 'visfhsity.
Qgb prhpbrty hl visfhsity is sbbk ik tgb lhddhwikj bx`mpdbs ?
(i) @ stirrba diquia, wgbk dblt, fhmbs th rbst hk `ffhukt hl visfhsity. Qgifnbr diquias dinb ghkby, fh`dt r,
jdyfbrik, btf. g`vb ` d`rjbr visfhsity tg`k tgikkbr hkbs dinb w`tbr. Il wb phur fh`dt`r `ka w`tbr hk `
t`edb, tgb fh`dt`r widd sthp shhk wgidb tgb w`tbr widd ldhw upth quitb ` d`rjb aist`kfb.(ii) Il wb phur w`tbr `ka ghkby ik sbp r tb lukkbds, w tbr fhmbs hut rb`aidy lrhm tgb ghdb ik tgb lukkbd
wgidb ghkby t`nbs bkhujg timb th ah sh. Qgis is ebf`usb ghkby is mufg mhrb visfhus tg`k w`tbr.
@s ghkby tbkas th ldhw ahwk ukabr jr`vity, tgb rbd`tivb mhtihk ebtwbbk its d`ybrs is hpphsba strhkjdy.
(iii) Zb f`k w dn l`st ik `ir, eut kht ik w tbr. Qgb rb`shk is j`ik visfhsity wgifg is vbry sm`dd lhr `ir eut
fhmp r`tivbdy mufg d`rjbr lhr w`tbr.
(iv) Qgb fdhua p`rtifdbs l dd ahwk vbry sdhwdy ebf`usb hl tgb visfhsity hl `ir `ka gbkfb `ppb`r ldh`tikj ik
tgb sny.
\isfhsity fhmbs ikth pd`y hkdy wgbk tgbrb is ` rbd`tivb mhtihk ebtwbbk tgb d`ybrs hl tgb s`mb
m`tbri`d. Qgis is wgy it ahbs kht `ft ik shdias.
LDHZ HL DIPRIA IK @ QREB FUIQIF@D \BDHFIQYZgbk ` diquia ldhws ik ` tueb, tgb visfhus lhrfbs hpphsb tgb ldhw hl tgb diquia, Gbkfb ` prbssurb aillbrbkfb is
`ppdiba ebtwbbk tgb bkas hl tgb tueb wgifg m`ikt iks tgb ldhw hl tgb diquia. Il `dd p`rtifdbs hl tgb diquia p`ssikj
tgrhujg ` p rtifud r phikt ik tgb tueb mhvb `dhkj tgb s`mb p tg, tgb ldhw hl tgb diquia is f ddba 'strb`m-dikba
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8/12/2019 Elasticity & Viscosity Theory_E
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SGYVIFV
"m`kisgnum`rpgysifs.ik" 06
0. @s tgb visfhsity hl diquias v ry witg tbmpbr turb, prhpbr fghifb hl duerif`kt is m`ab abpbkaikj uphk sb shk.
5. Diquias hl gijg visfhsity `rb usba ik sghfn `eshrebrs `ka eullbrs t r idw`y st`tihks.
6. Qgb pgbkhmbkhk hl visfhsity hl `ir `ka diquia is usba th a`mp tgb mhtihk hl shmb ikstrumbkts.
1. Qgb nkhwdbajb hl tgb fhbllifibkt hl visfhsity hl hrj`kif diquias is usba ik abtbrmikikj tgb mhdbfud`r wbijgt
`ka sg`pb hl tgb hrj`kif mhdbfudbs.
2. It likas `k imphrt`kt usb ik tgb firfud tihk hl edhha tgrhujg `rtbribs `ka vbiks hl gum`k ehay.
RKIQV HL FHBLLIFIBKQ HL \IVFHVIQYLrhm tgb `ehvb lhrmud , wb g`vb
)z/v(@
L
x
aimbksihks hl = ]QMD^]QD^
]MDQ^
]D/DQ]^D^
]MDQ^ 0005
5
05
5
Its ukit is nj/(mbtbr-sbfhka)*
Ik F.J.V. systbm, tgb ukit hl fhbllifibkt hl visfhsity is aykb s fm5 `ka is f`ddba phisb. Ik VI tgb ukit hl
fhbllifibkt hl visfhsity is K sm5 `ka is f`ddba abf`phisb.
0 abf`phisb = 0 K sm5 = (032 aykb) s (035 fm)5 = 03 aykb s fm5 = 03 phisb
Bx`mpdb 00.
@ m`k is rhwikj ` eh`t witg ` fhkst`kt vbdhfity v3 ik ` rivbr tgb fhkt`ft `rb` hl eh`t is @ `ka fhbllifibkt hl
visfhsity is. Qgb abptg hl rivbr is A. Lika tgb lhrfb rbquirba th rhw tgb eh`t.Vhd. L L
Q= m `
rbs
@s eh`t mhvbs witg fhkst`kt vbdhfity `rbs
= 3
L = LQ
Eut LQ
= @az
av, eut
az
av=
A
3v3=
A
v 3
tgbk L = LQ
=
A
@v 3
Bx`mpdb 05.
@ fueif`d edhfn (hl siab 5m) hl m`ss53 nj sdiabs hk ikfdikba pd`kb duerif`tba witg
tgb hid hl visfhsity = 030 phisb witg fhkst`kt vbdhfity hl 03 m/sbf. (j = 03 m/sbf5)
lika hut tgb tgifnkbss hl d`ybr hl diquia.
Vhd. L =@az
av= mj sik
az
av=
g
v
53 03 sik 63 = 1 g
03
g =033
0313 5 ^= 030 phisb = 035 K-sbf-m5 ]
= 1 036 m = 1 mm
Bx`mpdb 06.
@s pbr tgb sghwk lijurb tgb fbktr`d shdia fydikabr st`rts witg ikiti`d `kjud`r vbdhfity 3. Lika hut tgb timb `ltbr
wgifg tgb `kjud r vbdhfity ebfhmbs g`dl.
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14/16
SGYVIFV
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VQHNBV D@ZVthnbs prhvba tg`t tgb visfhus ar`j (L) hk ` spgbrif`d ehay hl r`aius r mhvikj witg vbdhfity v ik ` lduia hlvisfhsityis jivbk ey L =
8/12/2019 Elasticity & Viscosity Theory_E
15/16
8/12/2019 Elasticity & Viscosity Theory_E
16/16
SGYVIFV
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