- Email: [email protected]
On rubber disks under rotation or axisymmetric stretching
Int. J. Non-IAnear Mechanics. Vol. 8. pp. 539~550. Pergamon Press 1973. Printed in Great Britain
ON
RUBBER DISKS UNDER ROTATION AXISYMMETRIC STRETCHING WILLIAM W.
OR
FENG
Department of Mechanical Engineering, Carnegie-Mellon University, Pittsburgh, Pennsylvania 15213, U.S.A.
Abstract--The governing equations for a class of axisymmetric problems under large elastic deformation, concerned with a circular rubber disk with body force as well as non-uniform initial thickness, are formulated in terms of two coupled first-order ordinary differential equations with explicit derivatives. The following two problems subjected to different boundary conditions are solved: (a) Rotating disks with uniform initial thickness (13) Circular disks with non-uniform initial thickness under axisymmetric stretching at the outer boundary. In problem (b), the rubber disk whose initial thickness contour is ho = cr~ (where c and n are any constantsk or whose final thickness is a constant, is considered. Highly elastic materials with a Mooney strain-energy function are used for numerical calculations.
INTRODUCTION
Tim THEORYof large elastic deformations and non-linear continuum mechanics has been presented by Green and Adkins [1]. The strain distribution around a hole in a highly elastic sheet under axisymmetric deformation has been solved by Rivlin and Thomas [2]. The strain concentration for the same problem has been obtained by Yang [3] who introduced a phase plane integration method. In this paper, the body force of the disk, as well as the non-uniform initial thickness, is considered. The governing equations are formulated in terms of two coupled ordinary differential equations. The solutions can be obtained by any standard numerical integration method. Two problems are solved in this paper: (a) a disk rotating about its axis of symmetry with uniform initial thickness and an angular velocity co and (b) a circular disk with non-uniform initial thickness under axisymmetric stretching at the outer boundary. These problems are solved for the following cases: (i) a solid circular disk (ii) a circular disk with a concentric circular rigid inclusion and (iii) a circular disk with a concentric hole. In problem (b), an axisymmetric follower load is applied at the boundary of the disk. The follower load is one specified at a point of the disk and identified by its undeformed position. For purposes of illustration, numerical solutions subjected to the following conditions are presented in this paper. In problem (a)~ a uniform initial thickness disk is considered. The ratio of D~, the diameter of the hole or rigid inclusion to Do, the diameter of the disk, is 0"2. In problem (b), a disk with the thickness variation parameter n = - 1 and with a uniform final deformed thickness is considered. Furthermore, the ratio of D, the diameter of the hole or rigid inclusion to Do, the diameter of the disk, is 0.4. The strain energy function of the incompressible rubber disk is assumed to have the Mooney form. The instability phenomenon for rotating disks and the strain distribution for circular disks under axisymmetric stretching are discussed. Some possible techniques for obtaining approximate solutions are also mentioned. 539
540
WILLIAMW. FENG FORMULATION
For a circular thin disk where the thickness of the disk is small compared to the diameter of the disk, the diameter of the hole or the diameter of the rigid inclusion, the plane stress assumption will lead to a good approximation. The geometry of an axisymmetric disk can be described by polar coordinates. A point on the undeformed disk p(r, O, z) is deformed to p'(p, O, z) as shown in Fig. 1, where p is a function of r only; therefore the principal stretch
/
/
/
/
~
\\
]~
2:2::2;SL FIG. 1. Geometry of a disk under deformations.
ratios are 2x
dp dr
(1) 22 = pr
where the subscripts 1 and 2 in the above and subsequent equations denote the radial and tangential directions respectively. The third principal stretch ratio in the direction perpendicular to the disk can be obtained through the incompressible condition 2x2223 = 1, i.e. r
23 = -pp' -
(2)
where the prime in the above and subsequent equations denotes differentiation with respect to r. The strain energy function for an incompressible material is W = W(I1,I2)
(3)
where I1,12 are strain invariants, related to the principal strain ratios by 11 = 2 2 + 2 2 + 1
1
1 2 2 2122 (4)
The stress resultants measured per unit deformed length, in terms of the strain energy function, are
On rubber disks under rotation or axisymmetric stretching
T1 = 2ho23(2
-
+ 2 \WJtl/~W
T2 = 2 h o 2 3 ( 2 2 - 2 2 ) d W + 2 ~
541
va2OW) 0w)
,,,
where ho is the thickness of the disk before deformation and ho is a function of r. The equilibrium equations are satisfied automatically in the tangential and normal directions to the disk. The stress resultants must satisfy the equilibrium equation in the radial direction, i.e.
aT1
__+I(T dp p
1 _ T2)+ho23F=0
(6)
where F is the body force measured per unit deformed volume. In terms of the undeformed coordinate r, equation (6) reduces to d ~rr(PT1) = p'(T2 - ho2apF)
(7)
Introducing relations (1) and (2), equation (7) becomes d drr (r22 7"1) = 21 T2 - rhoF and the relation between
(8)
21, 22 and r is r = ~ (21 - 22)
(9)
Equations (8) and (9) are the governing equations for the axisymmetric plane stress problems. In general, they are two coupled first-order ordinary differential equations. T1 and T2 are functions of the principal stretch ratio 21 and 22. When the strain-energy function of the material and the boundary conditions are given, equations (8) and (9) can be integrated simultaneously by any standard numerical methods. If the disk is of an incompressible Mooney material, the strain-energy function is W(/1, 12) = C1(I1 - 3) + C2(I 2 - 3) = C1 [(11 - 3) + ot(I2 - 3)]
(i0)
where C1 and C2 are constants in the units of stress and a equals C2/C 1. Introducing (10) and the incompressibility conditions, equations (5) become 7"1 = 2Clh0 (~~
2:22a)(I + a222)
(11) T2= 2 C l h o ( ~
2:22a)(1 + a22)
Substituting equations (11) and (9) into equation (8), we obtain the governing equation for the axisymmetric problem for Mooney materials.
542
WILLIAM W. FENG ROTATING CIRCULAR DISK
A circular disk of uniform initial thickness rotates about its axis of symmetry. The angular velocity is co. Assume the body force is due to inertia force only; therefore F
=
)~pf.O2
where ~ is the density of the circular disk. The governing equation (8) reduces to
~=
rQ
1 . [0,2 _ 2 1223)(1 + ct22) _ 0o, _ 23~?~(1 + ~t22) q-24~22)(1 +~£2) 1,2f (12)
Introducing the dimensionless quantities r
R = - - and ro
fl-
Tro2O2 2C1
equations (9) and (12) reduce to 1 2~ = ~ ( 2 1
- 22)
and
;:1= R(1 + ~ ) ( 11
+ ~222)I(22 - 21~1223)(1 + ct22)- ( 2 1 - ~ ) ( 1
--flR2,~,2-
2()],1 - ,~,2)(~31~23 +
+ ct22)
0~21/~2t]
(13)
where the dots denote d~erential with respect to R. Equations (13) are the governing equations for rotating circular disks. Due to the presence of R, equations (13) are coupled differential equations. However, if we divide R by a scaling factor ~ apd multiply fl by ~2, equations (13) remain unchanged. This property makes the numerical calculation easier. The boundary value problem will be changed to an initial value problem by setting the parameter, that is not determined by the boundary condition, at a desired value, 2o, that is 2 1 ---- '~-2 ---- 20 at R = 0 for solid rotating disks, 21 = 20 at R = 0.2 for a rotating disk with a rigid inclusion and 2 2 ---- 20 at R = if2 for a rotating disk with a concentric hole. An arbitrary value of fl is assigned. Then, one solves equation (13) simultaneously for the successive points by increasing R until the outer boundary condition is satisfied. The final R(R:) is not likely to be equal to unity since fl is not likely to be the exact value. However, equations (13) remain unchanged when we divide R by ~ and multiply fl by ~2. We can choose ~ equal to R:. We obtain the exact fl value fl:( = fl~2) which will satisfy the governing equations and the boundary conditions at R = 0 (or 0.2) and R = 1. The physical properties can be obtained by replacing fl: for/3 and repeating the calculation.
On rubber disks under rotation or axisymmetric stretching
543
CIRCULAR DISK WITH NON-UNIFORM INITIAL THICKNESS
A circular disk of non-uniform initial thickness is subjected to outward stretching. The external load ~ measured per unit undeformed length is applied to the outer boundary of the disk. We assume that the external load is applied quasi-statically; therefore the body force may be considered to be zero. The governing equation (8) reduces to 2~=
(1 + ~ ) ( 1
2'2
+ ~222)
+~+~ 2122
+~+212
2122
h'(r)(21 +h-N
- 2t~11~)(i + 0t22)}
(14)
Equation (14) and the companion equation (9) are the governing equations for circular disks of Mooney materials. The governing equations are further simplified for the following two cases: (1) The initial thickness contour is h(r) = c f where c and n are constants, or
h'(r) h(r)
n r
=
-
(15)
Substituting equation (9) into (15), we have
h'(r) h(r)
n2'2
(16)
21 -- 2 2
Hence equation (14) becomes 2~ =
-22'
{1 + ~ 3
(1 + 24~)(1 + ~222)
2122
+ ~ (2~22a + ~ 1 + 2122 ) 2122
+ 2, - 2------S 1 -
(1 + d ~ )
(17)
Equations (17) and (9) are the governing equations for a circular disk with c#' as its initial thickness contour. However, it is noteworthy that there are two variables, 21 and 22, in equation (17). (2) The final deformed thickness is a constant; that is 2ah(r) = a constant or
h'(r)
- -
h(r)
-
2'2 + 2'1 22
(18)
21
Hence equation (12) becomes
21---{1( I
22
+ ~ + 2212 2122
(19)
Equations (19) and (9) are the governing equations for a circular disk with uniform final
WILLIAM W. F ~ O
deformed thickness. Again, it is worthy of note that there are two variables 2t and 22 in equation (19). Equations (17) and (9) or Equations (19) and (9) are two decoupled first-order ordinary differential equations. With the proper boundary conditions equation (17) or equation (19) can be integrated separately. The results can be shown on the 21 - 22 phase plane. However, equation (9) is used to find the principal stretch ratios distribution in the disk.
BOUNDARY CONDITIONS AND RESULTS The governing equations for a circular disk under radial stretching and during the rotation about its axis of symmetry are given in (13), (17) or (19) with the companion equation (9). In order to solve these differential equations, the appropriate boundary conditions should be described. Three cases of circular disks are studied in this paper" (a) a solid circular disk (b) a circular disk with a rigid inclusion (c) a circular disk with a concentric circular hole. The boundary conditions are, respectively: (i) Due to the symmetry property of the solid circular disk, the principal stretch ratios are equal at the origin, i.e. 21 = 2 2
r=O
(20)
and the stress resultant at the boundary is TI = 2hoC1 (,~1 - 2t~12~/(1 + ~222)
r=r
o
(21)
where ~1 is the stress resultant in the radial direction per unit of undeformed length. ~1 is a given value. A ~ of zero is assumed for the circular disk rotating about its axis of symmetry. (ii) A circular disk with a rigid inclusion. The ratio of the diameter of the rigid inclusion to the diameter of the disk is a; therefore 22 = 1
r =
aro
(22)
For the purpose of illustration, the values of a are assumed to be 1}2 for the rotating disk and 0.4 for the disk under axisymmetric stretch, respectively. The second boundary condition is the same as in equation (21). (iii) A circular disk with a surface-traction-free hole. The boundary condition at the hole is (23) The values of a are again taken to be 0.2 and @4 for the rotating disk and the disk under axisymmetric stretching respectively. The second boundary condition is the same as in equation (21). Some values of ~ are assigned for a circular disk under radial surface traction. The results plotted on the 21 - 22 phase plane are shown in Figs. 2 and 3. Since 21 and 22 are positive, only the first quadrant of the 21 - 22 phase plane is considered here. There are two lines worthy of special mention in Figs. 2 and 3. The stress resultant in the radial direction per unit of deformed length, T1, equals zero; therefore 2222 = 1. The stress resultant in the tangential direction per unit of deformed length, T2, equals zero; therefore 2122 = 1. These lines are shown in Figs. 2 and 3 and are the limits of the values of 21 and 22 in the first quadrant. Beyond these lines either T~ or T2 is negative; therefore the wrinkled zone appears in the disk. For a solid disk with a uniform deformed thickness the principal stretch ratios are constant values. The dark dots on the intersection of the 21 = 22 line and ~1 = 0
r = aro
On rubber disks under rotation or axisymmetric stretching
7.0
k k~_ L
545
kj= )'2 /
4o[/t ;,Oo:' t 0
t.O
20
30
5"0
40
6"0
7.0
)'2 FIG. 2. 21 -- 2 2 Phase plane for rotating disks of Mooney material (= = 0.1).
4.0
30 ~q 20
I0
/
I
I 0
I
2.0
I
:5"0
I
4.0
I
5"0
FIG. 3. '~l -- 22 Phase plane for disks of Mooney material (,, = 0.I) under axisymmetric stretching.
the constant ~ lines indicate the values of the principal stretch ratios. The principal stretch ratios vs r/r o are shown on Figs. 4-9. When the angular velocity of the rotating disk increases, the radius of the deformed disk increases. Due to the enlarging of the radius of the disk, the rate of the increasing inertia force of the disk increases. Due to the thinning and weakening of the disL the rate of the increasing internal restoring force decreases. When
546
WILLIAMW. FI~G
°r \\"-1 , o ~ 0
02
04
0.6
•
0-8
I'0
If o
FiG. 4. 2's vs r/r o for rotating disks of Mooney material (at = 0"1).
7.0
6.0
Xz 50
4.62
40 X _
50
~ , \~~-
- ~ . .
\
5'52
-_..
2.0
i~2~--~ \~~'-
1.0
I 0
0-2
[ 04
I
I
0-6
0.8
I.o
f/fo
FiG. 5.2's vs r/r o for rotating disks of Mooney material (ct = 0-1) with a concentric rigid inclusion.
On rubber disks under rotation or axisymmetric stretching 7-0
6.0 \
2.-rz
\
50
2.73\
-
Xz \\
\
4-0
3"0
2.0
I-0
-
\ \
\
\ _
\
\ \
x
\
2"31 \
"x
\
--
-
-
2 7 3 L ~ 0-2 0.4 r/r
I
I
0-6
0.8
I-0
o
FIG. 6. ZS VS r/ro for rotating disks of Mooney material (= = 0-1) with a concentric hole.
4.0
© ® 3'0 -
.11"
@
9-0
7-0 Xi
5"0
2-0
/_..i." j ~ ~~ J 1 1
-.~"
I
-.-"
3"0
I.O
0
i
I
I
[
0-2
0.4
0"6
0"8
I-O
r /r o
FiG. 7. ~-1 vs r/ro for disks of Mooney material (= = 0-1) and ho = c/r under axisymmetric stretching.
547
WILLIAM W. FENG 50
(3
\ \
4..0
,
\
® @
\ \
\
\
\
\
)'z
\
3'0 \
\
"~
~" ~ ~....... ~
2.0 ~-~. 1.0
~ ~ _J
9.0 ' 7"0
I"~"
J5"O
~ 1 3 ' 0 0
0.2
0-4
0.6
0.8
1.0
r/r o
I~G. 8. 22 VSr/r o for disks of Mooney material (a = 0-1) and h0 =
c/r
under axisymmetric stretching.
50
ii ®
4.0 --X
--
Xi,), 2 Q
30
2"0 .
I-0 0
m
I/-
-
I 0-2
0.4
~___}
[
I
0.6
0"8
3-0
I'0
r /t" o
FIG. 9. 2's
VSr/r o for disks of Mooney material (~ = ffl) and of constant deformed thickness under axisymmetric stretching.
the internal restoring force is smaller than the external inertia force, the instability phenom e n o n occurs. This p h e n o m e n o n is shown in Fig. 10. The curves of the # vs 20 graph decrease; therefore the rate of the deformation of the circular disk is increased when the angular velocity rises.
On rubber disks under rotation or axisymmetric stretching
7"0 t . . . . 6.0 ~
(~) // / (~ / / /
~---
(~11
I / ,o1-
549
>.o, //
:
o.o
, oo
-o
[ 1.0
I
2.0
I
I
3-0
4.0
I
5"0
6.0
X, FIG. 10. fl vs 2o for rotating disks of Mooney material (~t = 0-1) and neo-Hookean material (~, = frO).
The strain distribution in non-linear plane stress problems is a function of external loads as well as the initial geometrical configuration. It is found in this paper that the principal stretch ratios as well as the stress resultant are constants in the solid disk with uniform fmal deformed thickness; however the strain distribution changes drastically for the uniform final deformed thickness disk with a concentric hole or a rigid inclusion. The Newton-Ralphson's successive approximation method may be applied to this type of problem. The zero-order approximation should be guessed. The closer the zero-order approximation, the faster it converges to the exact solution. The governing equations for the disk of neo-Hookean material (~t = 0) under large principal stretch ratios, will reduce to linear differential equations. The solutions may be obtained easily. These solutions will be very close to the exact solution for the disk of neo-Hookean material and may be used as a zero order approximation for the disk of Mooney material. The second approximation method is to reduce the decoupled differential equations to an integral equation. Picard's successive approximation method may then be employed; however, the second-order approximation for both approximation methods must be integrated numerically. REFERENCES [1] A. E. G m ~ and J. E. APronS, Large Elastic Deformations, Clarendon Press, Oxford (1960). [2] R. S. P-aVLn~ and A. G. THOMAS, Large elastic deformations of isotropic material VIII. Strain distribution around a hole in a sheet, Phil. Trans. R. Soc. A, 243, 289 (1951). [3] W. H. YANO, Stress concentration in a rubber sheet under axially symmetrical stretching, J. Appl. Mech. 34; Trans. A S M E 89, E, 943-947 (1967). (Received 16 March 1972)
550
WILLIAM W . F~NO
R~um~--On formule, en termes de deux 6quations diff6rentielles ordinaires coupl6es du premier ordre avec des d6riv6es explicites, les 6quations fondamentales d'tm ensemble de probl6mes ~ sym6trie axial pour de grandes d6formations 61astiques d'un disque en caoutchouc avee des forces internes. On r6sout les deux probl6mes suivants ayants des conditions aux limits diff6rentes: (a) des diques en rotation avec une 6paisseur initiale tmiforme (b) des disques circulaires avec une 6paisseur initiale non uniforme soumis sur le bord externe ~ une extension/t sym6trie axiale. Dans le probl6me (b) on consid~re le disque en caoutchouc dont l'6paisseur initiale est h o = er~ (odc et n sont des constantes arbitraires) ou dont l'epaisseur finale est constante. On utilise pour les caleuls num6riques des mat6riaux fortement 61astiques avec une fonction d6formation-6nergie de Mooney. ZusammeafL~mg--Die bestimmenden Gleichung einer Klasse achsensymmetrischer Probleme mit grossen elastischen Verformungen, werden, ffir den Fall einer Kreisscheibe aus Gummi mit Massenkr~iften und ungleichf6rmiger Ausgangsdicke, mit Hilfe zweier gekoppelter gew6hnlicher Differentialgleichungenerster Ordnung mit expliziten Ableitungen formuliert. Zwei Aufgaben mit verschiedenen Randbedingungen werden gelSst: (a) Rotierende Scheiben mit gleichf'6rmiger Ausgangsdicke, (b) Kreisscheibenmit ungleichf6rmigerAusgangsdicke unter achsensymmetrischer Dehnung an der ausseren Begrenzung. Unter (b) werden Gummischeiben mit Ausgangsdickenproffl h o = cr~ (c und n sind beliebige Konstanten) oder mit konstanter Enddicke behandelt. Hochelastische Materialien mit einer Funktion die Dehnungsenergie nach Mooney werden f'tir die numerischen Berechnungen benutzt.
AUHoTaRHa--BbIBeheHI,I ypaaHenn~, onncuBammrle r~Jtacc ocec~IMMeTpHqHI,IX aa~aq o •onequux y n p y r n x aeqbopMaRnax Hpyr~oro peaHnoBoro aHci~a no~ ae~iCTBneM Maccoaux c r ~ , a TaKme npn Haqa;abHOft HepaaHOMepnoi~ ToJmtnHe. B CHCTeMy ypaBnenn~i BXO~AT ~Ba cortpflmeHuuX 05umtOa e n n u x and~dpepeHtma:ibHux ypaBHenHn nepBoro nop~J~Ha, coaepmamHe unHue npoaaBOaHue. PemeHu c~eayIOlane ABe aa~aqn c paa~nqnUMn KpaeB],tsn ycJIOBnrZun : (a) BpamalOli~ril~cn anCr C paanoMepHofi naqaJn, HOfi TO:ImnHOfi, (5) I~pyrJIu~t aHCI< C rtepaBHOMepHofi HaqaJibrtoia TO~I~HHOII, pacT~rnBaeMufi oceCHMMeTpnqHO ycn:mnMn, npn~omeHHUMn K BHemHeuy KoHTypy. B 3aaaqe (6) paccMaTpMnamTcfl aHCHH Ha peanHU, KoT0pux Haqa:n, Hue TOJ~mHm,t aaJIaHu qbopsty~o~ ho = cr ~, r a e C H n ;:IBSIItIOTC~:I npoH3BOJlbHbIMl/I HOHCTaHTaMH~ nJIH }Ee pe3HHOBYale aHCKtl, I~oHeqHa~l TOJIII~HHa
r~OTOpUX paBHOSepna. IIpu qncJ~eHHOM cqeTe. Hcno~baOBaH BUCOHoynpyrnfi MaTepna~ c qbyurR~e~ aHeprnr~ aeqbopMaImH Tuna MyH~.
ON
RUBBER DISKS UNDER ROTATION AXISYMMETRIC STRETCHING WILLIAM W.
OR
FENG
Department of Mechanical Engineering, Carnegie-Mellon University, Pittsburgh, Pennsylvania 15213, U.S.A.
Abstract--The governing equations for a class of axisymmetric problems under large elastic deformation, concerned with a circular rubber disk with body force as well as non-uniform initial thickness, are formulated in terms of two coupled first-order ordinary differential equations with explicit derivatives. The following two problems subjected to different boundary conditions are solved: (a) Rotating disks with uniform initial thickness (13) Circular disks with non-uniform initial thickness under axisymmetric stretching at the outer boundary. In problem (b), the rubber disk whose initial thickness contour is ho = cr~ (where c and n are any constantsk or whose final thickness is a constant, is considered. Highly elastic materials with a Mooney strain-energy function are used for numerical calculations.
INTRODUCTION
Tim THEORYof large elastic deformations and non-linear continuum mechanics has been presented by Green and Adkins [1]. The strain distribution around a hole in a highly elastic sheet under axisymmetric deformation has been solved by Rivlin and Thomas [2]. The strain concentration for the same problem has been obtained by Yang [3] who introduced a phase plane integration method. In this paper, the body force of the disk, as well as the non-uniform initial thickness, is considered. The governing equations are formulated in terms of two coupled ordinary differential equations. The solutions can be obtained by any standard numerical integration method. Two problems are solved in this paper: (a) a disk rotating about its axis of symmetry with uniform initial thickness and an angular velocity co and (b) a circular disk with non-uniform initial thickness under axisymmetric stretching at the outer boundary. These problems are solved for the following cases: (i) a solid circular disk (ii) a circular disk with a concentric circular rigid inclusion and (iii) a circular disk with a concentric hole. In problem (b), an axisymmetric follower load is applied at the boundary of the disk. The follower load is one specified at a point of the disk and identified by its undeformed position. For purposes of illustration, numerical solutions subjected to the following conditions are presented in this paper. In problem (a)~ a uniform initial thickness disk is considered. The ratio of D~, the diameter of the hole or rigid inclusion to Do, the diameter of the disk, is 0"2. In problem (b), a disk with the thickness variation parameter n = - 1 and with a uniform final deformed thickness is considered. Furthermore, the ratio of D, the diameter of the hole or rigid inclusion to Do, the diameter of the disk, is 0.4. The strain energy function of the incompressible rubber disk is assumed to have the Mooney form. The instability phenomenon for rotating disks and the strain distribution for circular disks under axisymmetric stretching are discussed. Some possible techniques for obtaining approximate solutions are also mentioned. 539
540
WILLIAMW. FENG FORMULATION
For a circular thin disk where the thickness of the disk is small compared to the diameter of the disk, the diameter of the hole or the diameter of the rigid inclusion, the plane stress assumption will lead to a good approximation. The geometry of an axisymmetric disk can be described by polar coordinates. A point on the undeformed disk p(r, O, z) is deformed to p'(p, O, z) as shown in Fig. 1, where p is a function of r only; therefore the principal stretch
/
/
/
/
~
\\
]~
2:2::2;SL FIG. 1. Geometry of a disk under deformations.
ratios are 2x
dp dr
(1) 22 = pr
where the subscripts 1 and 2 in the above and subsequent equations denote the radial and tangential directions respectively. The third principal stretch ratio in the direction perpendicular to the disk can be obtained through the incompressible condition 2x2223 = 1, i.e. r
23 = -pp' -
(2)
where the prime in the above and subsequent equations denotes differentiation with respect to r. The strain energy function for an incompressible material is W = W(I1,I2)
(3)
where I1,12 are strain invariants, related to the principal strain ratios by 11 = 2 2 + 2 2 + 1
1
1 2 2 2122 (4)
The stress resultants measured per unit deformed length, in terms of the strain energy function, are
On rubber disks under rotation or axisymmetric stretching
T1 = 2ho23(2
-
+ 2 \WJtl/~W
T2 = 2 h o 2 3 ( 2 2 - 2 2 ) d W + 2 ~
541
va2OW) 0w)
,,,
where ho is the thickness of the disk before deformation and ho is a function of r. The equilibrium equations are satisfied automatically in the tangential and normal directions to the disk. The stress resultants must satisfy the equilibrium equation in the radial direction, i.e.
aT1
__+I(T dp p
1 _ T2)+ho23F=0
(6)
where F is the body force measured per unit deformed volume. In terms of the undeformed coordinate r, equation (6) reduces to d ~rr(PT1) = p'(T2 - ho2apF)
(7)
Introducing relations (1) and (2), equation (7) becomes d drr (r22 7"1) = 21 T2 - rhoF and the relation between
(8)
21, 22 and r is r = ~ (21 - 22)
(9)
Equations (8) and (9) are the governing equations for the axisymmetric plane stress problems. In general, they are two coupled first-order ordinary differential equations. T1 and T2 are functions of the principal stretch ratio 21 and 22. When the strain-energy function of the material and the boundary conditions are given, equations (8) and (9) can be integrated simultaneously by any standard numerical methods. If the disk is of an incompressible Mooney material, the strain-energy function is W(/1, 12) = C1(I1 - 3) + C2(I 2 - 3) = C1 [(11 - 3) + ot(I2 - 3)]
(i0)
where C1 and C2 are constants in the units of stress and a equals C2/C 1. Introducing (10) and the incompressibility conditions, equations (5) become 7"1 = 2Clh0 (~~
2:22a)(I + a222)
(11) T2= 2 C l h o ( ~
2:22a)(1 + a22)
Substituting equations (11) and (9) into equation (8), we obtain the governing equation for the axisymmetric problem for Mooney materials.
542
WILLIAM W. FENG ROTATING CIRCULAR DISK
A circular disk of uniform initial thickness rotates about its axis of symmetry. The angular velocity is co. Assume the body force is due to inertia force only; therefore F
=
)~pf.O2
where ~ is the density of the circular disk. The governing equation (8) reduces to
~=
rQ
1 . [0,2 _ 2 1223)(1 + ct22) _ 0o, _ 23~?~(1 + ~t22) q-24~22)(1 +~£2) 1,2f (12)
Introducing the dimensionless quantities r
R = - - and ro
fl-
Tro2O2 2C1
equations (9) and (12) reduce to 1 2~ = ~ ( 2 1
- 22)
and
;:1= R(1 + ~ ) ( 11
+ ~222)I(22 - 21~1223)(1 + ct22)- ( 2 1 - ~ ) ( 1
--flR2,~,2-
2()],1 - ,~,2)(~31~23 +
+ ct22)
0~21/~2t]
(13)
where the dots denote d~erential with respect to R. Equations (13) are the governing equations for rotating circular disks. Due to the presence of R, equations (13) are coupled differential equations. However, if we divide R by a scaling factor ~ apd multiply fl by ~2, equations (13) remain unchanged. This property makes the numerical calculation easier. The boundary value problem will be changed to an initial value problem by setting the parameter, that is not determined by the boundary condition, at a desired value, 2o, that is 2 1 ---- '~-2 ---- 20 at R = 0 for solid rotating disks, 21 = 20 at R = 0.2 for a rotating disk with a rigid inclusion and 2 2 ---- 20 at R = if2 for a rotating disk with a concentric hole. An arbitrary value of fl is assigned. Then, one solves equation (13) simultaneously for the successive points by increasing R until the outer boundary condition is satisfied. The final R(R:) is not likely to be equal to unity since fl is not likely to be the exact value. However, equations (13) remain unchanged when we divide R by ~ and multiply fl by ~2. We can choose ~ equal to R:. We obtain the exact fl value fl:( = fl~2) which will satisfy the governing equations and the boundary conditions at R = 0 (or 0.2) and R = 1. The physical properties can be obtained by replacing fl: for/3 and repeating the calculation.
On rubber disks under rotation or axisymmetric stretching
543
CIRCULAR DISK WITH NON-UNIFORM INITIAL THICKNESS
A circular disk of non-uniform initial thickness is subjected to outward stretching. The external load ~ measured per unit undeformed length is applied to the outer boundary of the disk. We assume that the external load is applied quasi-statically; therefore the body force may be considered to be zero. The governing equation (8) reduces to 2~=
(1 + ~ ) ( 1
2'2
+ ~222)
+~+~ 2122
+~+212
2122
h'(r)(21 +h-N
- 2t~11~)(i + 0t22)}
(14)
Equation (14) and the companion equation (9) are the governing equations for circular disks of Mooney materials. The governing equations are further simplified for the following two cases: (1) The initial thickness contour is h(r) = c f where c and n are constants, or
h'(r) h(r)
n r
=
-
(15)
Substituting equation (9) into (15), we have
h'(r) h(r)
n2'2
(16)
21 -- 2 2
Hence equation (14) becomes 2~ =
-22'
{1 + ~ 3
(1 + 24~)(1 + ~222)
2122
+ ~ (2~22a + ~ 1 + 2122 ) 2122
+ 2, - 2------S 1 -
(1 + d ~ )
(17)
Equations (17) and (9) are the governing equations for a circular disk with c#' as its initial thickness contour. However, it is noteworthy that there are two variables, 21 and 22, in equation (17). (2) The final deformed thickness is a constant; that is 2ah(r) = a constant or
h'(r)
- -
h(r)
-
2'2 + 2'1 22
(18)
21
Hence equation (12) becomes
21---{1( I
22
+ ~ + 2212 2122
(19)
Equations (19) and (9) are the governing equations for a circular disk with uniform final
WILLIAM W. F ~ O
deformed thickness. Again, it is worthy of note that there are two variables 2t and 22 in equation (19). Equations (17) and (9) or Equations (19) and (9) are two decoupled first-order ordinary differential equations. With the proper boundary conditions equation (17) or equation (19) can be integrated separately. The results can be shown on the 21 - 22 phase plane. However, equation (9) is used to find the principal stretch ratios distribution in the disk.
BOUNDARY CONDITIONS AND RESULTS The governing equations for a circular disk under radial stretching and during the rotation about its axis of symmetry are given in (13), (17) or (19) with the companion equation (9). In order to solve these differential equations, the appropriate boundary conditions should be described. Three cases of circular disks are studied in this paper" (a) a solid circular disk (b) a circular disk with a rigid inclusion (c) a circular disk with a concentric circular hole. The boundary conditions are, respectively: (i) Due to the symmetry property of the solid circular disk, the principal stretch ratios are equal at the origin, i.e. 21 = 2 2
r=O
(20)
and the stress resultant at the boundary is TI = 2hoC1 (,~1 - 2t~12~/(1 + ~222)
r=r
o
(21)
where ~1 is the stress resultant in the radial direction per unit of undeformed length. ~1 is a given value. A ~ of zero is assumed for the circular disk rotating about its axis of symmetry. (ii) A circular disk with a rigid inclusion. The ratio of the diameter of the rigid inclusion to the diameter of the disk is a; therefore 22 = 1
r =
aro
(22)
For the purpose of illustration, the values of a are assumed to be 1}2 for the rotating disk and 0.4 for the disk under axisymmetric stretch, respectively. The second boundary condition is the same as in equation (21). (iii) A circular disk with a surface-traction-free hole. The boundary condition at the hole is (23) The values of a are again taken to be 0.2 and @4 for the rotating disk and the disk under axisymmetric stretching respectively. The second boundary condition is the same as in equation (21). Some values of ~ are assigned for a circular disk under radial surface traction. The results plotted on the 21 - 22 phase plane are shown in Figs. 2 and 3. Since 21 and 22 are positive, only the first quadrant of the 21 - 22 phase plane is considered here. There are two lines worthy of special mention in Figs. 2 and 3. The stress resultant in the radial direction per unit of deformed length, T1, equals zero; therefore 2222 = 1. The stress resultant in the tangential direction per unit of deformed length, T2, equals zero; therefore 2122 = 1. These lines are shown in Figs. 2 and 3 and are the limits of the values of 21 and 22 in the first quadrant. Beyond these lines either T~ or T2 is negative; therefore the wrinkled zone appears in the disk. For a solid disk with a uniform deformed thickness the principal stretch ratios are constant values. The dark dots on the intersection of the 21 = 22 line and ~1 = 0
r = aro
On rubber disks under rotation or axisymmetric stretching
7.0
k k~_ L
545
kj= )'2 /
4o[/t ;,Oo:' t 0
t.O
20
30
5"0
40
6"0
7.0
)'2 FIG. 2. 21 -- 2 2 Phase plane for rotating disks of Mooney material (= = 0.1).
4.0
30 ~q 20
I0
/
I
I 0
I
2.0
I
:5"0
I
4.0
I
5"0
FIG. 3. '~l -- 22 Phase plane for disks of Mooney material (,, = 0.I) under axisymmetric stretching.
the constant ~ lines indicate the values of the principal stretch ratios. The principal stretch ratios vs r/r o are shown on Figs. 4-9. When the angular velocity of the rotating disk increases, the radius of the deformed disk increases. Due to the enlarging of the radius of the disk, the rate of the increasing inertia force of the disk increases. Due to the thinning and weakening of the disL the rate of the increasing internal restoring force decreases. When
546
WILLIAMW. FI~G
°r \\"-1 , o ~ 0
02
04
0.6
•
0-8
I'0
If o
FiG. 4. 2's vs r/r o for rotating disks of Mooney material (at = 0"1).
7.0
6.0
Xz 50
4.62
40 X _
50
~ , \~~-
- ~ . .
\
5'52
-_..
2.0
i~2~--~ \~~'-
1.0
I 0
0-2
[ 04
I
I
0-6
0.8
I.o
f/fo
FiG. 5.2's vs r/r o for rotating disks of Mooney material (ct = 0-1) with a concentric rigid inclusion.
On rubber disks under rotation or axisymmetric stretching 7-0
6.0 \
2.-rz
\
50
2.73\
-
Xz \\
\
4-0
3"0
2.0
I-0
-
\ \
\
\ _
\
\ \
x
\
2"31 \
"x
\
--
-
-
2 7 3 L ~ 0-2 0.4 r/r
I
I
0-6
0.8
I-0
o
FIG. 6. ZS VS r/ro for rotating disks of Mooney material (= = 0-1) with a concentric hole.
4.0
© ® 3'0 -
.11"
@
9-0
7-0 Xi
5"0
2-0
/_..i." j ~ ~~ J 1 1
-.~"
I
-.-"
3"0
I.O
0
i
I
I
[
0-2
0.4
0"6
0"8
I-O
r /r o
FiG. 7. ~-1 vs r/ro for disks of Mooney material (= = 0-1) and ho = c/r under axisymmetric stretching.
547
WILLIAM W. FENG 50
(3
\ \
4..0
,
\
® @
\ \
\
\
\
\
)'z
\
3'0 \
\
"~
~" ~ ~....... ~
2.0 ~-~. 1.0
~ ~ _J
9.0 ' 7"0
I"~"
J5"O
~ 1 3 ' 0 0
0.2
0-4
0.6
0.8
1.0
r/r o
I~G. 8. 22 VSr/r o for disks of Mooney material (a = 0-1) and h0 =
c/r
under axisymmetric stretching.
50
ii ®
4.0 --X
--
Xi,), 2 Q
30
2"0 .
I-0 0
m
I/-
-
I 0-2
0.4
~___}
[
I
0.6
0"8
3-0
I'0
r /t" o
FIG. 9. 2's
VSr/r o for disks of Mooney material (~ = ffl) and of constant deformed thickness under axisymmetric stretching.
the internal restoring force is smaller than the external inertia force, the instability phenom e n o n occurs. This p h e n o m e n o n is shown in Fig. 10. The curves of the # vs 20 graph decrease; therefore the rate of the deformation of the circular disk is increased when the angular velocity rises.
On rubber disks under rotation or axisymmetric stretching
7"0 t . . . . 6.0 ~
(~) // / (~ / / /
~---
(~11
I / ,o1-
549
>.o, //
:
o.o
, oo
-o
[ 1.0
I
2.0
I
I
3-0
4.0
I
5"0
6.0
X, FIG. 10. fl vs 2o for rotating disks of Mooney material (~t = 0-1) and neo-Hookean material (~, = frO).
The strain distribution in non-linear plane stress problems is a function of external loads as well as the initial geometrical configuration. It is found in this paper that the principal stretch ratios as well as the stress resultant are constants in the solid disk with uniform fmal deformed thickness; however the strain distribution changes drastically for the uniform final deformed thickness disk with a concentric hole or a rigid inclusion. The Newton-Ralphson's successive approximation method may be applied to this type of problem. The zero-order approximation should be guessed. The closer the zero-order approximation, the faster it converges to the exact solution. The governing equations for the disk of neo-Hookean material (~t = 0) under large principal stretch ratios, will reduce to linear differential equations. The solutions may be obtained easily. These solutions will be very close to the exact solution for the disk of neo-Hookean material and may be used as a zero order approximation for the disk of Mooney material. The second approximation method is to reduce the decoupled differential equations to an integral equation. Picard's successive approximation method may then be employed; however, the second-order approximation for both approximation methods must be integrated numerically. REFERENCES [1] A. E. G m ~ and J. E. APronS, Large Elastic Deformations, Clarendon Press, Oxford (1960). [2] R. S. P-aVLn~ and A. G. THOMAS, Large elastic deformations of isotropic material VIII. Strain distribution around a hole in a sheet, Phil. Trans. R. Soc. A, 243, 289 (1951). [3] W. H. YANO, Stress concentration in a rubber sheet under axially symmetrical stretching, J. Appl. Mech. 34; Trans. A S M E 89, E, 943-947 (1967). (Received 16 March 1972)
550
WILLIAM W . F~NO
R~um~--On formule, en termes de deux 6quations diff6rentielles ordinaires coupl6es du premier ordre avec des d6riv6es explicites, les 6quations fondamentales d'tm ensemble de probl6mes ~ sym6trie axial pour de grandes d6formations 61astiques d'un disque en caoutchouc avee des forces internes. On r6sout les deux probl6mes suivants ayants des conditions aux limits diff6rentes: (a) des diques en rotation avec une 6paisseur initiale tmiforme (b) des disques circulaires avec une 6paisseur initiale non uniforme soumis sur le bord externe ~ une extension/t sym6trie axiale. Dans le probl6me (b) on consid~re le disque en caoutchouc dont l'6paisseur initiale est h o = er~ (odc et n sont des constantes arbitraires) ou dont l'epaisseur finale est constante. On utilise pour les caleuls num6riques des mat6riaux fortement 61astiques avec une fonction d6formation-6nergie de Mooney. ZusammeafL~mg--Die bestimmenden Gleichung einer Klasse achsensymmetrischer Probleme mit grossen elastischen Verformungen, werden, ffir den Fall einer Kreisscheibe aus Gummi mit Massenkr~iften und ungleichf6rmiger Ausgangsdicke, mit Hilfe zweier gekoppelter gew6hnlicher Differentialgleichungenerster Ordnung mit expliziten Ableitungen formuliert. Zwei Aufgaben mit verschiedenen Randbedingungen werden gelSst: (a) Rotierende Scheiben mit gleichf'6rmiger Ausgangsdicke, (b) Kreisscheibenmit ungleichf6rmigerAusgangsdicke unter achsensymmetrischer Dehnung an der ausseren Begrenzung. Unter (b) werden Gummischeiben mit Ausgangsdickenproffl h o = cr~ (c und n sind beliebige Konstanten) oder mit konstanter Enddicke behandelt. Hochelastische Materialien mit einer Funktion die Dehnungsenergie nach Mooney werden f'tir die numerischen Berechnungen benutzt.
AUHoTaRHa--BbIBeheHI,I ypaaHenn~, onncuBammrle r~Jtacc ocec~IMMeTpHqHI,IX aa~aq o •onequux y n p y r n x aeqbopMaRnax Hpyr~oro peaHnoBoro aHci~a no~ ae~iCTBneM Maccoaux c r ~ , a TaKme npn Haqa;abHOft HepaaHOMepnoi~ ToJmtnHe. B CHCTeMy ypaBnenn~i BXO~AT ~Ba cortpflmeHuuX 05umtOa e n n u x and~dpepeHtma:ibHux ypaBHenHn nepBoro nop~J~Ha, coaepmamHe unHue npoaaBOaHue. PemeHu c~eayIOlane ABe aa~aqn c paa~nqnUMn KpaeB],tsn ycJIOBnrZun : (a) BpamalOli~ril~cn anCr C paanoMepHofi naqaJn, HOfi TO:ImnHOfi, (5) I~pyrJIu~t aHCI< C rtepaBHOMepHofi HaqaJibrtoia TO~I~HHOII, pacT~rnBaeMufi oceCHMMeTpnqHO ycn:mnMn, npn~omeHHUMn K BHemHeuy KoHTypy. B 3aaaqe (6) paccMaTpMnamTcfl aHCHH Ha peanHU, KoT0pux Haqa:n, Hue TOJ~mHm,t aaJIaHu qbopsty~o~ ho = cr ~, r a e C H n ;:IBSIItIOTC~:I npoH3BOJlbHbIMl/I HOHCTaHTaMH~ nJIH }Ee pe3HHOBYale aHCKtl, I~oHeqHa~l TOJIII~HHa
r~OTOpUX paBHOSepna. IIpu qncJ~eHHOM cqeTe. Hcno~baOBaH BUCOHoynpyrnfi MaTepna~ c qbyurR~e~ aHeprnr~ aeqbopMaImH Tuna MyH~.