Thí problem o f determ ining the elastic moduli for com posite filled w ith spherical particles is studiecly several authors [10,11]... Conclusions.[r]
Trang 1Analysis o f stress-strain relationship o f titanium dioxide epoxy composite tube under pressure and thermal load
Tran Quoc Quan"^
i ' \ ! i ' ị hĩỉ\ i'rKìt\ ()f Finnnccrir)^ a n d Ị echnolci'j^w Ỉ4 4 X]tcm Thuv, tỉarioi l-ỉcỉnaiyì
R e c e iv e d 21 iV lav2011
VNU Journal o f Science, Mathematics - Physics 27 (2011) 91-99
A b s tr a c t T o d a y , titan iu m d io x id e p article is w id e ly u sed in sh ip b u ild in g , p etro le u m in du stry an d
other field s T h e g o a l o f this p aper is to stu d y th e e ffe c t that titan iu m d io x id e p a rticles h a v e on the
stress, strain o f c o m p o s ite tu b es under p ressure and therm al load From the o b ta in ed result,
com m en ts and s u g g e s tio n s are g iv e n on the m anufacture and u se o f e n g in e e r in g tub es.
Key ^vords: c o m p o site , e n g in e e r in g tub e, titan iu m d io x id e particle, therm al, pressure
I Introduction
Additive and reinforced particles arc often used lo increase the abrasion resistant, crack resistant, fireprooi’ w aterproof ability and strength o f polym er m aterials [1] In V ietnam , in the past few years, com posite material w ith titanium dioxide particles is w idely used m shipbuilding and other indusưies [2,3] 'fhus, many researches concerning the physio-m echanical role o f titanium dioxide particle in com posite are carried out T itanium dioxide/polystyren com posite is studied experim entally in [4] In [5], the authors investigate the optical and mechanical properties o f com posite m em brane filled with titanium dioxide parliclcs In [6], the engineering m odi’li OS 3-phase com posite (glass fiber, titanium dioxide particle, polym er m atrix) are determ ined by experim ent and base on that, a bending analysis for com posite plates used in shipbuilding is done [7]
One o f the m ost com m on com posite structures is engineering tube such as w ater tubes, oil
tu b e s [1]* Recently, som e analysis for com posite tubes have been done; in [8], the authurs study the stress-strain relationship for com posite cylinder under unsteady, axisym m etric, plane tem perature field In [9], the inner/outer pressure is taken into account beside the heat ữ ansfer in an analysis for titanium dioxide/PV C com posite This p ap er’s goal IS to calculate the stress and sưain for engineering tubes made o f titanium dioxide/epoxy com posite under pressure and therm al load w ith constant tem perature increment
T he com posite m aterial IS considered to have periodic structure w ith small particle volum e ratio The particles have the sam e diam eter and the interaction betw een particle and m atrix is neglected B y using the equation o f theory o f therm oelasticity and com posite m aterial, the authur has solved the stress-strain relationship o f engineering tubes made o f titanium dioxide/epoxy com posite, thereby clarifying the role o f titanium dioxide particle in im proving the com p osite’s m echanical properties
Email: quantq@vnu.edu.vn
91
Trang 22 PnH em form ulation
2.7 PỉO)lem
Ccn;idering a com posite cylinder with inner radius a and outer rad iu s b filled w ith spherical particles The cilynder is put under inner pressure, outer pressure P 2 and a tem perature increm ent
A J = Ĩ ~ T ^ (T’o IS the initial tem perature) The m aterial IS assum ed to be elastic, isotropic and the
interatìim betw een particle and m atrix is igriored (all p artiles’ radius are equal) T he elastic moduli and thtrnal coefficient o f the m atrix are Aj, / i ,, a , , and those o f the p article are /1 , , Ơ-, The equavi a t properties o f the com posite cilynder are X \ Ị u \ a or K \ ịấ \ a .
2.2 Bcsc equations
2.1 i D eterm ine the com posite's elastic moduli
Thí problem o f determ ining the elastic moduli for com posite filled w ith spherical particles is studiecly several authors [10,11] ỉn this paper, the result o f V anin-N guyen D inh D ue [12] IS used;
3/r + Ơ
92 T.Q Quan / VNU Jo u rn a l o f Science, M athem atics - Physics 2 7 (2 0 1 Ỉ) 91-99
here
^ 1 + 4 0 , 1 ( 3 ^ ) 'j^
' \ - A G L ( i K y i ' ' ’ l + ( 8 - I O v - , ) / / i
( 2)
where
K , + ^ G , 8 -IO k , 3
(3)
/»
H ex ệ = —— is the particle volum e ratio (TV is the total num ber o f the particles, V is the volume )f the i-th particle (i ” 1, 2, ,7V), V IS the com posite volum e) / / , /Vị are the shear modulisandAT* are the bulk m odulus o f com posite, m atn x and particle, respectively I ', , are th e fo is so n ’s ratio o f m atrix and particle The advantage o f these resu lts IS that the interaction between particle and m atrix is taken into account
2.22 The effective iherm al expansion coefficient o f cotnposite
The current paper uses the result from [13] for calculating the therm al ex p an sio n coefficient:
In viich a * is the effective therm al expansion coefficient Oi com posite; a jjO r.a re the therm al
expansoi coefficients o f m atnx and particle, respectively
Trang 3T.Q Q uan / VNU Jo u rn a l o f Science, M athem atics - Physics 27 (20 Ị Ỉ) 91-99 93
2.2.3 The H o o k e ’s la w f o r the therm oelastic problem
We study the th erm oclastic defon nation o f the material under m echanical and therm al loads (the
c o m p o s i t e IS b e i n g heated f r o m TJj t o T ).
The m echanical stress-strain relationship follows H ooke’s law
Thereby, the th e m io elastic relationship between sfrain and stress IS [14]
The inverse ex pression for stress IS
cr,.=Ầ'0ô,^ + 2 ụ e , - { ư + 2 ụ ' ) a \ ợ -T ,)5 ,J (7) This IS the stress-strain relationship in therm oelasticity theory
2.3 C alculating th e stress a n d strain
W hen the cy lin d er is lon g enough w e have the plane stress stage T he equations w ritten for the
cylindncal coordinate sy stem {r, 0, z) are [14]
From the sym m etric property, all points only have radial displacem ent The displacem ent filed has the follow ing form
Ur = « r ( ' ' ) , = U g = 0 ( 8 ) The C auchy strains are
" đ r ’ - ~ r ' d r r
The stress-strain relatio n sh ip is
(T^ = r ỡ + 2 /^ V ^ -(3 /l* + 2 / ) a * A T (10a)
ơgg = Ả '0 + 2 ụ Sgg -(3 /1 ’ + 2 / Ầ ) â ầ T (10b)
= ẵ '9 - (3 à ' + 2ju’) a A T , CT„ = ơ^g = ơ ^ g =0 (lOc)
The equibrilium eq u atio n is
d ơ ^ 1
Introduce (lO a) and (lO b) into (11) w e get the differential equation for
This Euler differential equ ation w ith the boundary condition:
^rr r=a ~ ~P\ ’ ^rr r=t> Pi
T he solution o f (12) is
Trang 494 T.Q Q uan / VNU Jo u rn a l o f Science, M athem atics - P hysics 2 7 (20 Ỉ Ỉ) 91-99
3
+ (3A* + 2 / ) a AT
{b - a )
£ < ^ + 3 /íV aT
i p , - p ^ ) c r h ' 1
{ p ^ - p , ) a h - 1
2 / i ‘ ( ố “ - Cl') r
r —
Th; lon-zero strains are
=
2 ( r + / )
1
+
/ • \
K ' +
3
+ Ì K ' a ' A T
h - a
+
^00 ~
2 { Ả ' + ụ )
1
2 ụ \ b - - c r ) r-
l ị - ì (h^ - a ' ) r '
i P 2 - ! h W b " 1
r + ^-3
f 4 ^ + 3 A 'V A T
-
a-2 ụ ' { h ^ - a - ) r { p ^ - p , ) a h ^ 1
l ụ { b - - a )
Th; lon-zero stress are
_ p ỵ - p , b - ( p, - p , ) a ^ h ^ 1
h ^ - a ^ b ^ - a ^ _ p , a - p ỵ { p ^ - p , ) a b ^ 1
l 2 2 i 2 2 1
< 7
-(13)
(14a)
(14b)
(15a)
(15b)
p y ~ - P i b ^ _ (3Ẩ* + 2 ụ ' ) n '
b ' - a ^ Ả '+ I.i' A‘ + / /
Frcrr equation (1 3 )^ (15c), we can see that the displacem ent, sữ ain , stress depend on n o t only pressuE elastic m oduli, the thickness o f the tube but also tem perature
(15c)
3 N urerical results
C oisder a com posite cylinder having the follow ing properties
Epoxy ratrix [4]: E, = 2.75 (G Pa), v^, = 0 3 5 , a , = 54 X1 0 ^ /" c
Titian (loxide particle [4]: £ , - 1 4 7 (G Pa), V = 0 2 , «2 - 4 X1 O'" / “ c
(16) (17) Asiune that the tube has inner radius ( 3 = 1 3 (cm), outer radius Ố = 14 (cm ) It is put under inner pressuR /?] =50 (M Pa) and outer pressure V , initial tem perature Tg - 1 5 ° c , c u ư e n t tem perature
T = 1 2 i 'c
Trang 53.1 The dependence o f K ' , JU , a on p a rtic le volum e ratio ệ
3.1.1 A nalytical expressio n
The relationship b etw een the elastic m odule for com ponent m aterials [14]
K , ụ
-3(1- 2 v ) 2(1+ k)
Thus, bu lk m od ulus and P o isso n ’s ratio for m atrix and particle are
=^3.0556 (G Pa), = 81 6667 (GPa)
=1.0185 (G Pa), /i, = 61 25 (GPa) The effective m oduli fo r com posite cylinder can be calculted using (2) and (4)
^ ; 3 0 5 5 6 ( H - 0.4222^)
(1 -0 4 2 2 2 ^ )
1 0185 ( 1 1 1025 ^)
1 - 0 9 4 5 ^
0 0 4 1 2 -0 0 3 7 8 ^
“ ~ 762.2154 + 320.7288^
5.7.2 G raphs
T.Q Q uan / VN U Jo u rn a l o f Science, M athem atics - Physics 27 (2011) 91-99 95
(18)
(19)
(20)
From Figure 1, it can b e seen that increasing the particle volum e ratio leads to decrease o f the therm al expansion c o efficien t and increase o f the elastic m oduli, w hich m eans im provem ent o f the
m echanical and th erm al p ro p erties o f the com posite cylinder
3.2 The dependence o f stress, strain, displacem ent on p a rticle volum e ratio ệ a n d radius r
3.2.1 A n a lytica l expression
T he sfress co m p o n en ts are calculated as in (15a), (15b) and (15c)
= - 0 1 9 5 2 + ( GPa)
cr^ = 0 1 9 5 2
-r
24.5362
(GPa)
(21a)
(21b)
Trang 696 T.Q Q uan / VNU Journal o f Science M athem atics - P h ysics 2 7 (2 0 Ì I) 91-99
- 100.
0.235[9.18(1 + 0.4222^X1 - 0.945^) - 2.04(1 + 1.1025^)0 - 0.4222^)]
~ 9.18(l + 0 4 2 2 2 ^ ) ( l- 0 9 4 5 i) + 1.02(l + l 1 0 2 5 ^ X 1 -0 4 2 2 2 ^ ) 9.3,06.1,0185(1 + 0.4222^X1 + 1.1025^X 0.0412 - 0 0 3 7 8 ^ ) (762.2154 - 3 2 0 7 2 8 8 ^)[9 18(1 + 0.4222^)(1 - 0 945^) +1.02(1 + 1.1025^)(1 - 0.4222^)]
Strain com ponents are calculated from (14a) and (14b)
-0 2 9 2 5 ( 1 - 0 4 2 2 2 ^ X 1 - 0 9 4 5 ^ )
(21c)
9.18(1 + 0.4222^X1 - 0 945^) + 1.02(1 + 1 10 25 ^)0 - 0 4 2 2 2 ^ )
1377(1 + 0 4 2 2 2 ^ )0 - 0 09 45^)(0.0412 - 0 0 3 7 8 ^ ) +
9.18(1 + 0.4222^X1 - 0.945^) +1.02(1 + 1.1025^)(1 - 0 4 2 2 2 ^ )(7 6 2 2 154 + 320.7288^)
1 2 0 4 5 3 (1 -0 9 4 5 ^ ) 1 (1 + 1.1025^) -0.2925(1 - 0.4222^X1 - 0.945^)
(22a)
9.18(1 + 0.4222^X1 - 0.945^) + 1.02(1 + 1.1025^)(1 - 0.4222^)
1377(1 + 0.42^)(1 - 0.0945^)(Q.0412 - 0.0378^) 9.18(1 + 0.4222^)0 - 0.945^) +1.02(1 + 1 1025^)(1 - 0.4222^)(762.2154 + 320.7288^)
1 2 0 4 5 3 (1 -0 9 4 5 ^ ) 1 (1 + 1.1025^)
D isplacem ent is calculated from (13)
-0.2925(1-0.4222^X 1-0.945^)
(22b)
u =
+
9.18(1 + 0A222ệ)ạ - 0.945^') +1.02(1 + 1.1025ậ)(\ -0.4222^)
1377(1 + 0.4222^X1 - 0.0945^X0.0412 - 0.0378^) +
9.18(1 + 0A222ậ)ặ - 0.945^ ) +1.02(1 + 1.1025^)0 - 0.4222^ ')(762.2154 + 320.7288^ )
_ 12.0453(1-0.945^) J_
(1 + 1.1025^) r 3.2.2 G raphs
Sfress com ponents are presented in Fig 2 and Fig 3
r
-(23)
^ 13.4 13.6A 13.8
-0.1 -0.15
—
—•♦"“ Radial stress (G Pa)
*0.2 -0.25 -0.3
•—Circum ferential stress (G Pa) -0.35 - a t—-— • —
-0.4
Fig 2 Sfress components Ơ" andcr
Trang 7T.Q- Q uan / V N U J o u r n a l o f Science, M athem atics - Physics 27 (20 Ỉ I) 9 Ỉ-9 9 97
Fig 3 Stress component • From (21a), (21b), (21c), it can b e seen that only depends on ệ w hile , ơgg only depend
on r On Figure 2, Ơ and ơgỹ v ary slow ly in different directions (w hen r increases, decreases but
ƠỌQ increased), b u t th ey b o th hav e negative values O n the other hand, has the sam e value for any point in the cylinder O n F ig u re 3, increases w hen r increases and it alw ays have negative values Thus, the particle h elp to red uce the sừess concentration inside the tube
Strain com ponents , Eg o are presented in Fig 4 and Fig 5
Fig 4 Sưain component at r=a và r=b
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D isplacem ents u at r = <3 and r — b are presented in Fig 6.
F rom F ig ure 4,5 and 6, under m echanical and therm al loads, the sữ ain com ponents £ , E^g and
displacem ent at the sam e radius vary slow ly in different directions w hen particles are added W hen
ệ increases, the radial strain decreases (and has positive values) w hile radial displacem ent and
ang ular sừ ain increase (and have positive values)
Thus, u n d er m echanical and therm al loads, increasing the particle volum e ratio can help im prove the m echanical and therm al properties o f the com posite cylinder
4 Conclusions
In the cu rrent paper, the author has studied the follow ing points
1 T he dependence o f the com p osite’s elastic m oduli on the titanium dioxide particle volume ratio
Trang 92 Base on the th erm oelasticity theory and the m echanics o f c o m p o site structure, th e p r o b le m o f calculating stress, strain for a co m p o site cylinder under pressure an d therm al loads w a s set up and solved
3 The num erical results show that adding the titanium dioxide particle can help to im prove the
m echanical and ih e n n a l propertie s o f c o m p o site material T he elastic m oduli and t h e n n a l e x p a n sio n coefficient are e xperesscd explicitly from the c o m p o n e n t m ate ria ls ’ propertie s and v o lu m e ratios This can be the scientific base for the problem o f optim ization in design and m anufacture C om posite
m aterials with desired propertie s can be m anufactured by c hanging the m atrix and the particle w ith
p ro p er volum e ratios
A ck no w led gem ent I ’hc author w ould like lo express science thank to Professor N guyen D inh Due for offering help and m a n y v a luable suggestions
R eferences
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T.Q Qiiơn / VNU Jo u rn a l o f Science, M athem atics - Physics 27 (201!) 9 Ỉ-9 9 99