Displacements and finite-strain fields in a hollow sphere subjected to large elastic deformations

Int. J . mech. 8~i. Pergamon Press. 1974. Vol. 16, pp. 777-788. Printed in Great Britain

DISPLACEMENTS A N D F I N I T E - S T R A I N F I E L D S IN A HOLLOW S P H E R E S U B J E C T E D TO LARGE ELASTIC D E F O R M A T I O N S T. L. CH~.~ and A. J. DUR~t,tx Civil and Mechanical Engineering Department, The Catholic University of America, washington, D.C., U.S.A.

(Received 25 September 1973, and in revised form 17 April 1974)

S m m n a r y - - T h l s paper presents the fields of displacements and strains in a hollow sphere diametrically loaded to seven levels of loads. Polyurethane rubber was used to make the hollow sphere a n d grids and moird gratings were printed on a meridian plane to measure displacements. The m a x i m u m applied vertical displacement reduced the outer diameter of the hollow sphere b y about 20 per cent. I t was found that the horizontal displacement of the ends of the horizontal diameter divided b y its deformed length is linear up to t h a t load. Strains were obtained b y photographic differentiation. Strains obtained for low levels of load were compared at two points to those obtained using a series b y Golecki for the inside boundary, a n d found appreciably lower. Results are presented in terms of dimensionless loads a n d Eulerian strains. The largest Eulerian strain recorded is about 30 per cent, the smallest is of the order of 0"001. Stress distributions will be given in another paper.

NOTATION

E K ~'r, n z

1V p P r, R

rc Ri, Re R~ R~ U, W

~u/~r, Ou/Oz,aw/~r, ~w/~z z,Z 8h 8~ 54

natural Young's modulus nondimensionalized load parameter moird fringe order for radial and vertical displacement components, respectively fringe order of spatial derivatives grating pitch vertical applied load radial co-ordinates in deformed and undeformed states, respectively radius of contact circle inside and outside radii of hollow sphere, respectively radius of the equatorial plane of hollow sphere in deformed state haft length of the vertical axis of hollow sphere in deformed state radial and vertical displacement components, respectively spatial derivatives of u and w displacements vertical co-ordinates in deformed and undeformed states, respectively Eulerian shear strains in tO, Oz and rz planes, respectively extension of radius of the equatorial plane of hollow sphere contraction of the vertical axis of hollow sphere 777

T. L. CHEN and A. J. DURELLI

778 trr, E

Eulerian radial, circumferential and vertical strains, respectively L Eulerian, Lagrangian and natural axial strains in uniaxial tensile test, respectively E E Eulerian normal and tangential strains along a curve e2~. g T boundary, respectively 0,® circumferential co-ordinates in deformed and undeformed states, respectively nondimensionalized deformation parameter natural Poisson's ratio Eulerian, Lagrangian and natural axial stresses in uniaxial tensile test, respectively angle of principal strain with horizontal direction angular co-ordinate in undeformed state 1. I N T R O D U C T I O N

THE STRESS and strain analysis of an elastomer subjected to large deformation usually involves two types of nonlinearity: the geometric one due to the change of configuration as a result of load application and the nonlinear constitutive relation of the material used. The latter can often be studied using the concept of the strain energy function as presented by Rivlin. 1 Alternatively, the problem can be simplified by using the linear natural stress and natural strain relation developed b y Parks and Durelli. n Besides the constitutive nonlinearity, the geometric nonlinearity remains to be coped with. T h a t this problem is difficult is verified by the fact t h a t relatively few theoretical solutions of large deformation problems can be found in the literature, particularly the three-dimensional ones. Experimental methods seem t hen to be a reasonable approach to the solution of this t ype of problem. I n a previous paper, n the embedded-moir~ technique has been described in its successful application to the determination of strain fields in the meridian plane of a solid sphere subjected to large diametral compressions. The strain fields were converted to stress fields in another paper following the two aforementioned approaches. 4 As a continuing effort to solve large deformation problems the objective of the present paper is to determine the strain field in the meridian plane of a hollow sphere subjected to large diametral compressions along its axis. Interest is centered upon the discontinuity of the structure in a form of a spherical cavity. I n a later paper the stress analysis will be conducted following two different approaches:the strain-energy function approach and the linearization approach using the concept of natural stress. In t h a t paper the results obtained using both approaches will be compared. 2. G E O M E T R Y AND LOADING The specimen is a hollow sphere, 6.85 in. o.d. and 3-480 in. i.d., of transparent polyurethane rubber compressed along its vertical diameter between two flat plates as shown in Fig. 1. The polar-cylindrical co-ordinate system is used for convenience. Seven levels of loads were applied. The maximum load reduced the outside diameter of the sphere by about 20 per cent.

Displacements and strains in sphere subjected to deformations

779

3. E X P E R I M E N T S The hollow sphere was made by cementing two hollow hemispheres cast in molds according to the following formula. Base resin: Hysol 2085, 100 pbw; hardener: Hysol 3462, 45 pbw. The castings were cured in an oven at 212°F for 48 h. A rubber sheet of 20 × 20 x 0.15 in. was also cast to calibrate the material. The meridian plane of the hollow sphere was printed with one-way 500 lines-per-in, gratings with ~ × ~ in. grids using the photosensitive emulsion described in ref. (5). The hollow sphere was immersed in white paraffin oil confined in a container with parallel glass walls to avoid the refraction of the beam of light. The entire setup was placed on a solid platform and dead weights were used for load application as seen in Fig. 1. The hollow sphere was loaded first with the grating lines in the horizontal direction. The camera was carefully adjusted to obtain one-to-one image size with the help of a master grating placed at the camera back. The grating was first photographed when the sphere was subjected to a very small initial load and then after each load of a series of seven loads of successively larger amounts. Next, the sphere was unloaded, rotated to have the grating lines vertical, loaded again, and the procedure was repeated. Two photographs were taken at each load level, one corresponding to the horizontal and one to the vertical displacements. A dial gauge was mounted to record the vertical displacem e n t applied to the sphere. To obtain the u and w isothetics for small deformation, the photograph of the grating obtained when the sphere was subjected to a v e r y small initial load, was used as master and superposed on the photograph of the grating obtained when the sphere was deformed. This procedure eliminated any correction for possible initial mismatch of the gratings. However, for large deformation, the configuration of the loaded sphere deviates appreciably from its original shape, and with t h a t procedure moir6 patterns cannot be obtained in some regions of the boundaries. To overcome this difficulty an independent master grating, big enough to cover the final configuration, was used to superimpose on each deformed grating. This procedure required, however, t h a t small initial mismatch be subtracted at each loading level. Typical moir6 patterns for small and large deformations are shown in Figs. 2 and 3, respectively. The contrast was enhanced b y performing optical spatial filtering e in order to obtain better moir6-of-moir6 patterns. E a c h of the u and w field isothetics was superposed on a photograph of itself and shifted in the r and z directions, successively, to produce four moir6-of-moir6 patterns of spatial derivatives of displacement components ~u/~r, ~u/~z, ~w/~r and ~w/~z. 7 Difficulties were encountered to obtain ~w/~r and ~w/~z by this procedure for the three highest load levels. I t was found more practical for these loads to use the alternative method of shifting the deformed grating over a photograph of itself. 7 A representative quadrant for each of these four fields is shown in Fig. 4. Details concerning the experiments which are not covered here can be found in a previous paper. 3 4. M A T E R I A L

PROPERTIES

The properties of the rubber used were determined in a uniaxial calibration test. The material behaves practically as an incompressible material up to as high as 200 per cent Lagrangian strain in uniaxial state of stress. The natural Poisson's ratio (~) is equal to 0.5 up to a natural strain of more t h a n 100 per cent. The stress-strain relation becomes linearized when natural stress and natural strain definitions arc used up to about 80 per cent natural strain as shown in Fig. 5. The slope of the line gives the natural modulus of elasticity E = 150 psi. 5. A N A L Y S I S

AND RESULTS

5.1. Representation o/deformation Using the co-ordinate system given in Fig. 1, the deformation for a hollow sphere subjected to diametral compression can be represented by

r = R+u(R,Z),

]

0 = ®,

)

z = Z + w(R, Z).

(5.1)

780

T. L. CHEN and A. J. DURELLI ~psi) %"0

3oo

• cE

28o

3

26o ~X

240 220

2oo 18o

¢.

Ey

16o

a~ ~- a--,~--150 psi 12(

Y

oL EL y'y

4~ 20

o;

3018

02 04 0.6 0.8 1.0 1.2 1.4 16

1:8 2:0 2:2 "~E|in/in|

FIG. 5. Uniaxial stress-strain relations for a polyurethane rubber in three different definitions. The displacement components u and w are independent of (9 and circumferential displacement component v vanishes everywhere in the sphere due to the rotational s y m m e t r y of the geometry and loading. 5.2. Displacements The governing equations for displacement components are U = pnr,

t

w = pnz,

I

(5.2)

where p is the grating pitch ( = 0.002 in.), n, the moir4 fringe order for displacements in the r direction and n, the moir4 fringe order for displacements in the z direction. The contraction of the vertical axis and the elongation of the horizontal axis of the sphere plotted vs the applied loa, ling are shown in Lagrangian and Eulerlan discriptions in Fig. 6. The radii of the contact circles at the loading area were also measured and are plotted in Fig. 6. I n addition, for each load level, the u displacement component along the equatorial plane, the w displacement component along the vertical axis and both the u and w displacement components along the inner boundary of the hollow sphere are shown in Figs. 7-9, respectively. The angular co-ordinate ~bindicating the undeformed position of points along the inner boundary was used. The expression of the loading is generalized b y using K = P/TrR~ E where P is the load applied, Ro is the outer diameter and E is Young's modulus of the material used. Alternatively, the deformation of the sphere at each stage of loading can be characterized by a dimensionless parameter defined as 8, contraction of vertical axis P = 2Ro = diameter of sphere To apply the results obtained to specimens of other sizes, the load has to be such t h a t the deformed shape is the same. The material should also be characterized as linear, using the same definitions as the ones used in this paper. The dimensionless parameters K and iz are related as shown in Fig. 6.

D i s p l a c e m e n t s a n d s t r a i n s in s p h e r e s u b j e c t e d t o d e f o r m a t i o n s

P x 103"

P fib)

180

781

~1~ m~+t ~m,, - ~

6j,

6v ,m-o

°

6v +m~

.

160

/ /Ro

140

///__o,,

/]//

120

//

lOO

;//

/

e

Dialgage

.'/

P r~:

v12

P ,3175 00"-'~'+:020:04 0:06 0:08 0:10 0:12 0:14 0:16 0:18 0:20 0:22:51~ 6 0104o:m 0:,2 0:,6 0'.~ 0:24 ~.~'80'.++ ~.++ 0'.40 o'.4~",:mo FIG. 6. R a d i i o f c o n t a c t circles a n d c h a n g e i n d i a m e t e r s of a hollow s p h e r e as a f l m e t i o n o f l o a d in L a g r a n g i a n a n d E u l e r i a n d e s c r i p t i o n s .

.p

5vl2

0

0.06

. ~

~t--5.12

Ro I Ri=1.97

0.05 0.04

i

~.-..-~o. og?2

I

0.0+; I

0.02

0.01

0.0163

I I

I I

o.olo9

I

I

o.o054

I

I i

.~,. , , ~BL 0'., 0.~ 016 0.7 0.8 0:9 t.O I.,

" ro

3177

FIG. 7. R a d i a l d i s p l a c e m e n t s o n t h e e q u a t o r i a l p l a n e of a h o l l o w s p h e r e s u b j e c t e d t o s e v e n levels of d i a m e t r a l c o m p r e s s i o n , i n E u l e r i a n d e s c r i p t i o n .

782

T . L . CHEN and A. J. DURELLI

••

r2

R "i

cL-.~R./ ~ \ ~,w \

K_-_L_ P 12

:,.o ~0.

-

0.0109

C 0.6

0t"4 o.3

3178 w't

i

,I

i

eo - . 18 -. J7 -. 16-. i5 - . i4 - . b - . 12-. i~ - . iO -.t~ -.(~ -.bz -.06 -.05 -.04 -.03 -.oz ~ 0

FIG. 8. Vertical displacements on the vertical axis of a hollow sphere subjected to seven levels of diametral compression, in Eulerian description.

u

u

'~R~) ~

3179

FIG. 9. Radial and vertical displacements along the inner boundary of a hollow sphere subjected to seven levels of diametral compression, in Lagrangian description.

Displacements and strains in sphere subjected to deformations

783

5.3. S t r a i n analysis The four spatial derivatives of the displacement components were computed with the following, and three similar equations (for the other three partial derivatives) ~u pitch N -~r = shift '

(5.3)

where "pitch" denotes the pitch of the undeformed grating in inches ( = 0.002 in.) and "shift" is the distance in inches which the deformed gratings were shifted over a copy of itself. N is the fringe order of spatial derivatives. For small deformation, the photographic differentiation of the u displacement field was sometimes difficult due to the low density of the moir4 fringes. Graphic differentiation was then performed whenever necessary. When the photographic differentiation is performed the images of the boundaries shifted in respect to each other will not coincide, and the true boundary should lie between the two images. Consequently, it is necessary to obtain the values of the spatial derivatives at the boundary b y extrapolation. Care must be exercised in doing that, particularly for ~w/Oz along the inner boundary where the gradient is high. T h e e x t r a p o l a t e d v a l u e s o f ~ w / ~ z a r e v e r i f i e d b y m e a s u r i n g t h e s p a c i n g of two neighboring isothetics along the vertical axis whenever possible. According to the definitions given in ref. (8), the strain-displacement relations for the direct (normal) and shear strains in Eulerian description are: e~, = 1 - 4[(1 -- ~u/~r) 2 + (~v/Or)2],

(5.4a)

e~oo -- u/r,

(5.4b)

e~ = 1 -- ~][(au/~z) ~ + (1 -- cOw/~z) 2]

(5.4c)

and

~o = ~= = o, = are sin

(5.5a)

(1 - Ou/~r) 8u/~z + ~w/~r(1 - ~w/~z) (1--e~) (1--e~)

(5.5b)

Thus by determining the spatial derivatives ~u/~r, ~u/~z, Ow/Sr, 8w/~z and radial displacement u at a point, the strain at the point can be completely determined. Along the vertical axis and at the equatorial plane the cross spatial derivatives vanish and the above equation further simplified, i.e. ~u e~ = ~ r ' U

e~o = - , r

(5.6a)

(5.6b)

~W

= ~z'

(5.6c)

while the shear strains vanish everywhere. Along the intersection of the meridian plane and the inner boundary, the principal directions are tangential, normal and binormal to the boundary and vary from point to point. Principal strains along this boundary can be computed b y first evaluating the strain matrix according to equations (5.4o,, b, c) and (5.5a, b, c), then finding its extrema and their directions using its tensorial form2 The angles can also be measured with a protractor. The strain distributions for points at the equatorial plane and along the vertical axis of the sphere for seven successive load levels are shown in Figs. 10-13. I t is to be noted that in Fig. 12 the curves near the loading area are dotted because of difficulties in reproducing the same b o u n d a r y conditions of the mating surfaces every time the sphere has to be reloaded. The strain distributions and angle of principal strain with the horizontal direction along the inner boundary are also shown in Fig. 14.

Fzo. 10. Radial and circumferential strains on the equatorial plane of a hollow sphere subjected to seven levels of diametral compression.

3170

EEzz( IN I IN /

RolRj=I.9/ 6 v.

LO 1.1

3J?J

FIO. 11. Vel~ical s~rain on the equatorial plane of a hollow sphere subjected to seven levels of diametral compression.

:-0.

~0.18

~0.17

-O.[~

-0.15

-0.14

-0.13

-o. t2

-0.11

-O. IG

-0.09

-0,08

-0,07

-0,0~

-0.05

-0.04

-0.03

-0.02

-0.01

~.5 0.6" 0.7 0.8 0.9

t~

oo

Displacements and strains in sphere subjected to deformations

o•

~8 IZ

.....

w .~Err

785

R,- - ~ \

c ~ . ~ % "~',

,<

.

.

.~.~ ~,~..9

.

0.3 . .

RolR/.=I.97

.

.

.

.

.

.

.

.

.

'2~

• 3172 '

FIG. 12. Radial and circumferential strains along the vertical axis of a hollow sphere subjected vertically to seven levels of diametral compression.

0.

Ro I Ri= 1.97

O.6 O.tTra

0.156 0.092

0.056

p=0.037

• I I f i t f I f I I i I I i I f - . 3 4 -;32 - . 3 0 - . 2 8 - . 2 6 - . 2 4 -.22 -.20 - . 1 8 -.16 - . 1 4 -.12 -.10 -.08 -.06 -.04 -.02

0.4

o

.~2 .~ .~

,:zEz( INI IN ) FIG. 13. Vertical strains along the vertical axis of a hollow sphere subjected vertically to seven levels of diametral compression.

The results for small deformation were compared at two points of interest to the series solution obtained b y Golecki z° for the inside boundary. The experimental values are found to be lower than the theoretical. This can be explained by the fact t h a t the solution assumes point loading which is physically impossible even for small strain. Points at the intersection of the inner boundary and the vertical axis (where the strain is the largest) and the inner boundary and the horizontal axis are of special interest. Strains at these places are plotted as a function of loads in Fig. 15.

786

T.L. C~

a n d A. J. DURELLI

~-

~-

9EA

IJ=0.037 O

0.066 O.O92 0.115 ..---..--- _

%

,

E E (T E EN 4e

E~ E~

90'

EXPERIMENTAL RESULTS

GOLECKI'S

SOLUTION

COMPARABLE TO p=0.0T/

/ij;= 0,176

¢ 70"

~~-~,~

50"

So'

~=o.o~ ] 0.o66 1

l

'~

- ~ ~

O.llS IMOIREDATA

e %7

C

/o..5 ~0660.066 ~ 0 . 0 3 7

^ ...... o ~

o"

0.136

/

0.136|

l

io"

I

t

2o"

/ i NFi N i TI:S1 M A C ~ I ~

~

I

~"

i,

i

~"

l

i

~"

i

l

6o"

I

i

70"

!

I

l

8O"

A

90" 3174

FIG. 14. Principal strains e~0, e~ a n d e~, a n d their o r i e n t a t i o n on the inner b o u n d a r y of a hollow sphere subjected vertically to seven levels of diametral compression, in L a g r a n g i a n description.

Displacements and strains in sphere subjected to deformations

787

EE(INIIN)

,28 •26

C,~o z,

~ )C

.24 .2? •18

,1~

"i

4,,

.° ~

.04;

EErr)h

~

~

~o

J~o

i~o

i~o

I

i

2jo P IIb

0

FIG. 15. Strains at the intersection of the axes and the inner boundary.

Acknowledgements--The research program reported in this paper was supported fiaancially b y the Office of Naval Research under contract N0014-67-A-0377-0011, b y the National Science F o u n d a t i o n under grant No. GK-16907 and by the Army Research Office (Durham) under Grant No. 31-124-72-G71. The authors would like to express their appreciation to N. Perrone of ONR, to C. J. Astill of NSF and to J. Murray of ARO for their understanding of the work and their encouragement. Credit should be given to V. J. Parks for m a n y valuable suggestions, to H. Miller for help with the experimental work, to J. Sala for the preparation of the manuscript and to J. Vossoughi for the drawing of the figures. REFERENCES

1. R. S. RIVI,IN, in Rheology (edited b y F. R. EIRICH), Vo]. 1, Chapter 10. Academic Press, New York (1956). 2. V. J. P ~ K S and A. J. DU~LLI, Int. J. Non-linear Mech. 4, 7 (1969). 3. A. J. DU~EL~ and T. L. CHEN, Int. J. Non-linear Mech. 8, 17 (1973). 4. T. L. CH~.N and A. J. D~ELLI, Int. J. Solid Struct. 9, 1035 (1973). 5. A. J. D~ELLX and V. J. PARKS, Moirg Analysis of Strain, p. 261. Prentice-Hall, ~ e w Jersey (1970). 6. F. CHIANG,Expl Mech. 9, 523 (1969). 7. A. J. D ~ E L ~ and V. J. PARKS, Op. cit, p. 183. 8. A. J. D~ELT.I and V. J. P~-RKS, 01O.cit, p. 96. 9. A. J. D ~ E ~ and V. J. P ~ x s , o19. cit, p. 122. 10. J. GO~CKI, Archs Mech. Stos. 9, 301 (1957)•