Int. J. Non-Linear Mechanics. Vol. 8. pp. 551-563. Pergamon Press 1973. Printed in Great Britain

P L A N E S T R A I N P R O B L E M S I N S E C O N D - O R D E R

E L A S T I C I T Y T H E O R Y

A. P. S. SELVADURM Department of Civil Engineering, The University of Aston in Birmingham, England

Abstract--The equations of second-order elasticity are developed in polar coordinates /7, 0 for plane strain deformations of incompressible isotropic elastic materials. By considering a 'displacement function' the second- order problem is reduced to the solution of an equation of the form V4~ ' = g(R, O) where V 2 is Laplace's differential operator and g(R, O) depends only on the first-order solution. The displacement function technique is then applied to obtain a second-order solution to the problem of an elastic body contained between two concentric rigid circular boundaries, when the outer boundary is held fixed and the inner is subjected to a rigid body trans- lation.

1. INTRODUCTION

T i m APPROXIMATE solution of problems in finite elasticity by the method of successive approximations is well established. A comprehensive exposition of this method is given in Green and Adkins Eli, Truesdell and Noll [2] and in a recent article by Spencer [3]. The fundamental assumption of this method is that the displacement and stress fields can be expanded as power series in terms of a small non-dimensional parameter E. The basic non-linear equations of finite elasticity can then be reduced to sets of linear equations from which the successive terms in the power series expansions can be determined.

The successive approximation procedure has been applied to several problems of plane finite elasticity by Adkins, Green and Shield [4], Adkins, Green and Nicholas [5], Carlson and Shield [6] and others who make efficient use of the complex variable method. An alternative formulation of the plane problem employs the Airy stress function approach used in classical elasticity.

This paper deals with the application of a "displacement function" approach to plane strain problems in second order elasticity, for an incompressible elastic material. In the context of an incompressible elastic material the use of a displacement function in contrast to an Airy stress function has received only limited attention. The formulation of the linear elastic problem in terms of the displacement function closely resembles the formulation of plane flows of a Newtonian viscous fluid in terms of Stokes' stream function. The use of a displacement function is particularly suitable for problems where displacement boundary conditions are specified but it can also be applied to problems where traction boundary conditions are prescribed.

As an illustration of this method we consider the problem of a cylindrical elastic annulus contained between two rigid boundaries, when the outer boundary is held fixed and the inner boundary is subjected to a rigid body translation. The displacement function method for incompressible elastic materials has been used by Selvadurai [7, 8] to obtain second- order solutions to wedge, dislocation, infinite plane and torsion problems, and by Selvadurai and Spencer [9] to axially symmetric three dimensional problems.

551

552 A . P . S . SELVADURAI

2. FINITE PLANE STRAIN- -PLANE POLAR COORDINATES

The position of a generic particle in the undeformed configuration is referred to a set of rectangular Cartesian coordinates XA(A = 1, 2, 3). After a deformation, the current position of the same particle is given by x~(X4) (i = 1, 2, 3). The cylindrical polar coordinates, (R, O, Z) of particles in the unreformed body and (r, 0, z) of particles in the deformed body are given by

X 1 = R cos O, X 2 ---- R sin O, X 3 = Z , (2.1)

X 1 = /. COS 0, X 2 = r sin 0, X 3 = Z.

For plane strain deformations

r = r(R, O), 0 = O(R, O), z = Z. (2.2)

The matrix F of deformation gradients in the R, O directions is

R ~ o

F = rdO I '

l_dR

We restrict our attention to incompressible materials for which

(2.3)

Or 90 Or 90) r ~ 9 0 ~-~ = 1. (2.4) det F = ~ ~ 0

The Cauchy--Green strain matrix referred to the (r, 0)'coordinates is

B FF r . (2.5)

| 90 dT .J.. /" d/. 90 / . 2 ( 0 0 ~ 2 -.L /.2 ( 9 0 ~ 2 /

, ]

The matrix S oi the physical components of the symmetrical Cauchy stress tensor (given by resolving the force per unit area in the deformed body in the directions of the axes in the deformed body~ referred to the current configuration is

F S" S'a] (2.6) S = l s , o Soo "

The constitutive equation for an incompressible isotropic elastic material expressed in plane polar coordinates can be stated in the form (e.g. Carlson and Shield [6])

S = - p l + laB, (2.7)

where p is an intermediate scalar pressure and # is the linear elastic shear modulus.

Plane strain problems in second-order elasticity theory '553

In the absence of body forces, the equat ions of equi l ibr ium in the deformed body ate

dS,, + - 1 dS,0 + S,, - S0o = O, dr r 80 r

dS,o 1 dSoo 2S,o + - - - + = 0 .

dr r 80 r

(2.8)

The componen t s of surface t ract ion in the r and 0 direct ions on a surface f ( r , 0) = 0 in

F o = n S,o + n o S . , (2.9)

the deformed body, are

F = n Srr + noS,o,

where

2 2 1, n d f /dr n r + n O = - - = . n o Of/rdO" (2.10)

3. SECOND-ORDER ELASTICITY

We now assume that the po la r coordina tes (r, 0) of a particle in the current configurat ion, whose equivalent coordinates in the undefo rmed conf igurat ion a te R, O, can be expressed as a power series in te rms of a small dimensionless p a r a m e t e r E, as follows

r = R + ~ ~r , (R, e ) , 0 = e + ~ ~'O,(R, e ) . (3.1) n = l I1=1

We also assume that the d i sp lacemem c o m p o n e n t s u and v in the R and O directions can be expressed as power series in the form

u = ~, d ' u ( R , O ) , v = ~ e~v(R,O) . (3.2) n = l n = |

By considering the de format ion of a generic particle it can then be shown that, to the second order in e., (3.1) can be expressed as

( r = R + E u 1 + E ~- u 2 - 2 R ] , (3.3)

R 2 ,]"

By subst i tut ing (3.3) into (2.4) we obta in the f irst-order incompressibi l i ty condi t ion as

du x u 1 1 Ov 1 - - - -- -- - - = 0, (3.4)

dR ' R ' R dO

and the second-order incompressibi l i ty condi t ion as

du 2 u 2 1 do z = H(R, 0) , (3.5)

554

where

A. P. S. SBLVAI)URm

fSul~=-t-Svl( lOu l ~ ) (3.6, H(R, e) = \sR} ' ~- 8e "

Also to the second-order in E, the Cauchy--Green strain matrix B can be expressed in the form

B = I + EB (1) + e2B (=), (3.7)

where the components of B (1) and B (2) are

8ul . 2 8v 1 u 1 1 0u 1 0v 1 v 1 B(~ ) = 2 8R B(212) R 8 0 + 2 ~- " no) = _ _ _ + , = - - , - 1 2 R 8 0 8 R R '

8u2 . 2 8% u= B (2) = 2 - ~ - + B* 1 , B(22) = R 0--O + 2 ~- + B~2, (3.8)

where

B(=)= 1 8u= By= v= 12 R 8 0 + 0R R + B~'2,

B*I \OR`] ÷ . 00`] ÷ 2 R OR R 2'

B*2 = O0 + + \OR , 2 - ~ O0 + v-L R 2,

B*= = R2 R OR + aR \OR _ + R ---~ 8---6 + -R 0 0 _ _ - ~ + _

If we further assume that the Cauchy stress matrix $ and the scalar pressure p can be expanded as power series in the forms

S = ~ d'S("', p = ~ ep(.), (3.9) n=O n=O

then from (2.7), we obtain the first and second-order constitutive equations as

$(1) = _p(t) I + izB(a), $(2) = _p(=)l +/zB(2). (3.10)

The components of S(1) are

(3.11) (10u, 0, ,) and for conciseness the components of $~2) can be written in the form

( 0< S(, 2) = -Pc=) + /* 2 OR,] + T ,

~2) = -P(2) + # 8 0 + + T~, (3.12)

Plane strain problems in second-order elasticity theory 555

1 0u 2 0v 2 _~) s ~ = , ~ oo + OR + T,,,

where

T = t~B* 1 ; Too = t~B~2; To = #B*2. (3.13)

By expressing the differential operators 0/dr and O/~ in terms of power series expansions of 6 the equations of equilibrium in the deformed configuration (2.8) can be reduced to

+ - - - - + =0 , OR R O0 R

05~ ) + -- _ _ OR R 00 + 2 - - - ~ - = 0 ,

in the first order, and in the second order to

S(r2) _ ~(2)

0S(2)0Rrr A_ --R1 __0S(~ )00 + R ",o = NI(R, 0),

O )~_2 , - F - - - -

OR

where

1 tOS(~--~) + 2 S(2)~ =- N2(R, 0), R 00

(3.14)

(3.15)

1) 1) 1) O" 1 0~r~'_t. 1:01) 1 DI~0~ (I, 1 OU 1 0~(r~ I, 1 Ou I 0~l'.l_Ul:.~rr =~(o..~ NI(R, O)-- 0R 0R ' ~ k ~ - f i - R ] 0 o R 0R 0O '~ R 0O 0R ' R \ R /'

(3.16) 0~[1 0S(I'A__~] ~ 0"1 0~(#~ ) 0"1 0~(1) L 1:0Vl Vl /0~( l ' j_o "1 .~(1)

N2(R'O)= ~'IR OR -~ 1( OO OR FOR ~ ' - R \ ~ - R - - R ] - ~ ' - - ~ - " "

4. D I S P L A C E M E N T F U N C T I O N S

We observe that the first-order incompressibility condition (3.4) can be identically satisfied by the introduction of a displacement function TI such that

1 0~' I 0~V 1 (4.1) U l = R 0 0 ' v l = 0R"

By making use of the first-order constitutive equations (3.11~ the first-order equations of equilibrium (3.14) can then be expressed in the form

- 0P<I)0R +R/Z 0-O0 [V2~,1 ] = 0, (4.2)

OPt1) - . ~ [ v ~ % ] = o, R 00

where 02 1 0 1 (9 2

V 2 ___ _ _ _t_ + R2 (4.3) 0R 2 ' R 0R 002,

is the Laplacian in plane polar coordinates referred to the undeformed body.

556 A . P . S . Sm~V.~U, UR~

By eliminating Pol and ~'1 in turn from (4.2) we have

V4 ~ t = 0, (4.4) and

V2p(1) = 0. (4.5)

We adopt a similar procedure in the case of the second-order equations. It may be verified that the second-order incompressibility condition (3.5) is identically satisfied if we express the displacement components u s and v s in terms of a second-order displacement function Ws by the relations

1 0 ~ ' 2 ,

u2 = R 8 0 + u2, (4.6)

0T2 v2 = - OR + rE'

where

2 vl Ou 1 1 u 2 1 v 1 ~ ( 4 . 7 ) #

Us= 2R 2 R ' R 00 '

0 u 1 #

v2 = - vl OR

We note that it is possible to obtain alternative expressions for u' 2 and v~, but for the present purposes it is sufficient to use the values of u' 2 and v 2 as defined by (4.7).

If we make use of (4.6) and the second-order constitutive equations (3.12), the second- order equations of equilibrium can be reduced to the form

# d _ 0p~s~ + _ _ _ [v,~,s] = F l ( r , o),

~R R 80 (4.8)

OP(2) - 80 - #Ra-~ rV2~'s] = Fs(g' O),

where

r o~. lOT. T - T. {OH ' 2 Ov~ ] FI(R,O)= - L T ~ - + ~ 0 o + ~ + ~ ~__~+VSu 7 r 2u" r 2 O@J -N~(R,O) ,

(4.9)

raT. 1DT,~, 2T,,# (1 0H._l_ ' 2 0u~'l, ] (R,O)= - L oR +XOO + r + # l t X ~ ' V2v'2 - R -~ v2 + ~ O o j - N 2 ( R ' O ) "

By eliminating P(2) and ~P2 in turn from (4.8) we obtain

1 rOF1 OF21 v'%

V~P~' = - L oR + --~ + ~ 8o f

(4.10)

(4.11)

Plane strain problems in second-order elasticity theory 557

The procedure for determining a second-order solution is as follows: From a knowledge of the first-order solutions for the displacement and stress components, the right hand side of (4.10) can be found. A solution of (4.I0) consists of a particular integral and solutions of the homogeneous equation V4~'2 - 0 . The correct homogeneous solution is that which satisfies the relevant boundary conditions of the problem. The particular solution together with the correct homogeneous solution then forms a complete solution of the second-order problem for an incompressible isotropic elastic material.

To the second-order in e the displacement and stress components are given by

U = 8U 1 + ~2U2; D = 8V 1 + ,~2D2; S = F~ (1) + ~2S(2). (4.12)

5. A CYLINDRICAL ANNULUS PROBLEM

A rigid cylinder of radius a(a > 0) lies inside and concentric with a rigid cylindrical region of radius b and the space between contains an incompressible elastic annulus which com- pletely adheres to the rigid surfaces (Fig,. 1). The outer boundary is held fixed and the inner boundary is subjected to a rigid body displacement 6 in the Xcdirection. We assume perfect bonding at the elastic material rigid medium interfaces thereby prescribing displacement boundary conditions at these surfaces.

x2" 2

FIG. 1. Geometry of the cylindrical annulus

A first-order displacement function of the form

8~P 1 = ~IR In R* - ~2R 3 + c~3R + -~ sin O,

provides the first-order displacement components which are

'E. 8ul = a l l n R * - ~ 2 R 2 + 0 c 3 + cosO,

(5.1)

558 A. P. S. Sm..VADURAI

where

-- *t I In R* + 3~2 R2 + ct s + sin O,

= a3(b2 0; a a a b 2 R

= - - ; R * = - % =- 2-~; ~* 2t/ b '

% = - 3~+ I

= (a 2 + b 2) In b - ae + b2' r/

and from the first-order incdmpressibility condition we note that ~ + % + The first-order stress components are

e = a L R + 4 ° t 2 R - 4 R 3 cosO,

e ~°° = 120t2R + 4~-~ cosO. #

~r0 = _ 4at2 R _ 4 sinO. /~ a

The small real dimensionless parameter e is chosen to be equal to 6 /a .

We note that owing to the symmetry of the problem the inner cylinder e: resultant moment about the origin. On evaluating the resultant force on any c] in the elastic material it may be shown that the force (N)that has to be applie~

. of the inner cylinder in the Xl-direction to maintain the rigid body translatio

41t/zt~(a 2 + b 2) N = (a 2 + b 2) In (a /b ) - a 2 + b 2"

If we let the outer boundary R = b recede to infinity we obtain the first-order s problem of an infinite plane containing a bonded rigid circular insert loade( by a concentrated force. The corresponding first-order displacement and stre~ can be obtained by setting

a 3

a t = a; *% 2' a 5 - a,

a 2 0t a = 0, R* R a

in equations (5.2) and (5.4). If the expressions (5.2) and (5.4) for the first-order displacement and stre~

are substituted in Equations (4.8), the inhomogeneous differential equation fi order displacement function, (4.10), reduces to

r 16°t,a2 l {sa~ lnR*-2a~ + 4ot:t a -4ata , } - 96~a]sin2, V'*~,~ =L R~' +k-~

Plane strain problems in second-order elasticity theory 559

A particular integral of (5.7) is

- 4 R 4] sin 20. (5.8)

The displacement components derived from the particular solution (5.8) and the Equations (4.6) are

U2p + R(20~20~ 3 + ~2~x 5 -- CXl~ 2 In R*) +

1 1 2 1 2 R1 ] + ~- {~ m R* + 3 ~ , - ~ + ~ , + ~ 3 ~ , ) + (~,~, - ~1~, ~ R*) cos 20

+ - ~ R3 + ~1~2 R + ~ ( - 2 ~ 2 ~ 4 - ~3~s - 4~3 - ~ 5 ) + R-~- Rs_], (5.9)

v2p = [3~R3 + R(-X2~1% + %% + ~1~2 In R*) +

+ ~ ( I ~ + ~ , ) + ~ ( - ~ 1 ~ , + ~,~, - ~1~, In R*) sin 2O.

By considering the kinematics of deformation of the inner and outer elastic material--- rigid medium interfaces it can be shown that the second-order displacement components must satisfy the boundary conditions

u2(R, O ) = v2(R, O) = 0, (5.10)

onR = a a n d R = b. On applying the boundary conditions (5.10) to the displacement components (5.9) we obtain

u2~(a, O) = Pl cos 20, v2p(a, O) = P3 sin 20, (5.11)

u2p(b, O) = P2 cos 20, V2p(b, O) = P4 sin 20,

where the constants Pl, P2 , " - , P4 are implied from (5.9). To satisfy the boundary conditions (5.10~ we therefore require four solutions of the

homogeneous equation W ~ 2 = 0. These solutions are

~ 2 r i = ( , l R 2 + ~R---~22 + ,3 + , , R ' ) s i n 2 0 , (5.12)

where ~1,/~2 . . . . . /~, are arbitrary constants. By making use of the boundary conditions (5.10), and (5.11) we obtain

~1 = -A~Pl ( a3b2 + a 5 + ab') + -~3 ( - a ' b - - bS) + ~A2 (a3 + 2ab2) + ~A2(b3 + 2a2b)'

]~2 =~aPl 5-4o -~P2a 'b ' + ~a3b'~+~-~2a'b 3,

560 A.P.S. S~LV~DURM

Pl ( _ 4 a S b 2 _ a3b 4 __ ab e) + ff.~A3 (a6b + a,b ~ + 4a2b s) #3 = -~

+ P3 (_ab 4 2a2b2) + P4 (_a4b _ 2a2b2), 22--- ~ - ~-~

f14 = aq)p~ + brpp2 ap3 bP4 (5.13) 223 223 222 222,

where ¢~} = (a 2 -+- b2).

The final expressions for the second-order displacement and stress components are

u 2 = [Ra{X2ct2 + 2fl ,} + Rt20t2~t 3 + ct2~t 5 - ~llX2 In R* + 2#x )

1 2 1 1 2 1 2 + ~ (2~1 In R* + 3~x20t 4 + ~Xa~X 5 -- ~0t 3 + ~0t 5 + 2fl3 )

1 ] + ~-~(u,% - ul=4inR* + 2fl2 ) c o s 2 0

I 1 oq~, ~24- I XCXl) + R3 Rs.j, + _Oc22R3 + ~xl~x2 R + ~(_2u2o % _ t 2

L..

o 2 = IRa(3m 2 - 4fl,) + R(-2X~lat2 + ~2~5 + ~lot2 In R* - 2fll )

1 1 2 1 ] + ~ (~ul + 4%~4) + ~-~ (--!2~1~4 + ~4~5 - ~1~4 In R* + 2fl2 ) sin 20,

(5.14)

and

(2) I S-~ -- 4~2R 2 + 4ot2ot 3 - 2ot2~x 5 2¢XlOt 2 4fl 1 + In R* +

1 (_2ot2 lnR* - 4at2o % - 2u3at s + ~ot 2 - ~ / 3 5 + o~ 2 - 8 # 3 )

+ ~-i(2ete, InR* - 2e4e 5 - 12fl2 ) + 4 cos20

[ 1 + - 2 ~ R 2 - 2 ~ + ~ 2 ~ + N (4~2~, + ~ )

1 + ~--Z (--5~tlot4 + 4~t3o~,, + 4tzl~41n R*) + 6-~21 R6_] '

Plane strain problems in second-order elasticity theory 561

S(2) [ - R2(22a 2 - 24# 4) + 6a2a s - 2a la 2 In R* - 4fll

1 1 2 1 a~__~_426 ] + R-- ~ (24¢x2¢x 4 + ]o~x) + ~-~ (4cxaot 4 + 6n4u 5 - 2~I~X,I " In R* + 12fl2 ) - 4 R _] cos 2 0

+ [ - 6 a 2 R 2 ÷ 20tl0t 2 + 40t2a 5 + 8a2a 3 + 8~1a 2 l n R *

1 . 1 + ~ ( - 4 a 2 a , + -~cx 2) . ~-~ (3ala 4 + 4o%a s - 4al(x,, In R*) + 2 (~2] (5.15)

R63 '

S(2) [ ro = R2(9~ 2 -- 12fl,) + 0~1~ 2 ÷ 2~2~ 5 -- 4~X2~ 3 -- 2~1~ 2 In R* - 4fll /t

1 i 2 + R-~ (~2 In R* - 6~x2~4. + ~1¢z 3 -- ~0f I + ~X3 2. + ~ 2 __ 4fl3)

+ 1 ( ~ 1 ~ , -- 2u,t~ s + 2 a l a , l n R * - 12a2)+ 4 ~2] s i n 2 0 , _IX

where al, o~ 2 . . . . . °~5, fix, r2 . . . . . f14, are defined by (5.3), (5.11) and (5.13). A relation between the applied force N and the rigid body t ranslat ion 6 for the second-order problem can be obtained by considering the resultant force in the Xl-d i rec t ion t ransmit ted across any con tour r = r o in the deformed body. This force resultant is

Pxl = Sc (S,, cos 0 - S,o sin O) r o dO (5.16)

where, on the contour C

r o = R + eul(ro, 0 ) , 0 = 0 + Evl(ro, 0 ) , (5.17)

to order e. r° Since the stress components ~ ) , ~ ) . . . . . etc., are in terms of polar coordinates of the

undeformed configuration, it is convenient to express (5.16) also in terms of R and O. The equat ion (5.16) can then be writ ten as

e~x = ~xi) + : ~ 2 , , (5.18)

to order E 2, where

and

//1) = i ( ~ ) cos O - ~ ) sin O) r o de ,

{aS('l) ~) o) cos O - 5 ̀(2) sin O - u I ~ - - ~ - cos O -- - - sin

1 ~v I ¢8(1 ) cos O - ~1) sin O) ] r o dO. - V ~ ( S ( 1 ) s i n O + S(~)c°s O) + R ~O" ~ R x Pr

(5.19)

(5.20)

562 A.P .S . SELVADURAI

On evaluating the expressions fo r /~ ) and P~) we have

t~ [ 4rclta(a2 + b 2) ] ,p~x] = a (a 2 + b2--~(--~ - - a 2 + b i ' pt2~ = 0. (5.21)

It is of interest to note that there is no second-order contribution to Pxl- This confirms the fact that from symmetry considerations, for large deformations, N must be an odd function of 6. Therefore the applied force N, given by (5.5), is valid to order c 3. The second-order solution to the problem of the infinite elastic plane loaded by a bonded rigid circular inclusion can now be obtained by using the values of ~x, ~2--- ~ts and R* given by (5.6) in equations (5.8) to (5.15).

For this particular case, the second-order displacement and stress components (5.14) and (5.15) reduce to

( , a, 1 ( 1 1 a" 1 a2) u 2 = ~ ~ -g+~ ~- l n R ' c o s 2 0 + 4 R 5 + 2 R 3 4 R- '

(5.22) ( l a'* l a 2 1 a " ) v 2 = - ~ R--- ~ + lnR' 4 R 2 R s sin 20,

~2)= a 6 _ 2 ~ + ~ - ~ + ~-Z 2~-~ lnR'

1/_3 a6 5a" 3 a 2 a" ) ' ~,2R 6 2R,*+~R---~+2-~ lnR' ,

cos 20 +

(5.23) ( aO ) la" a" ) /z = - R - Z + 2 R 2 ~- i lnR' c o s 2 0 + \ ~ 2R4+~R--~-2-~lnR' ,

St2)- a~ 3 a 4 1 a 2 ( a4 a2) 1 /~ ~ - ~ + ~ - ~ + ~-~+~-~ l n g ' s in20.

To the second-order in ~, the applied force N (per unit length) is given by 4x/~6. These results are in agreement with the solution, given by Carlson and Shield [6], to the problem where a stationary rigid circular inclusion is subjected to a resultant force at infinity.

Acknowledgements The work described in this paper was carried out at the Department of Theoretical Mechanies, University of Nottingham, supported by a Research Grant awarded by the Science Research Council. The author wishes to thank Professor A. J. M~ Spencer for many helpful discussions and generous assistance.

REFERENCES [1] A. E. GREEN and J. E. ADKINS, Large Elastic Deformations and Non-linear Continuum Mechanics. Oxford

University Press. 1st Edn. (1960), 2nd Edn. (1971). [2] C. TRUESDELL and W. NOEL, The Non-linear FieM Theories of Mechanics. Encyclopaedia of Physics vol. 3/3

(S. Flfigge, Ed.). Springer-Verlag Berlin (1965). [31 A. J. M. SPEI~CER, The static theory of finite elasticity, J. Inst. math. Applics. 6, 164 (1970). [4] J. E. ADKINS, A. E. GRImN and R. T. SHIELD, Finite plane strain, Phil. Trans. R. Soc. A, 246, 181 (1953). [5] J. E. ADrdNS, A. E. GREEN and G. C. NICHOLAS, Two-dimensional theory for f'mite deformations, Phil. Trans.

R. Soc. A, 247, 279 (1954). [6] D. E. CAm.sow and R. T. SHIELD, Second and higher order effects in a class of problems in plane finite

elasticity, Archs. ration. Mech. Analysis, 19, 189 (1965). [7] A. P. S. SELVADURAI, Problems in second-order elasticity theory. Ph.D. Thesis, University of Nottingham

(1970). [8] A. P. S. SELVAOURAt, Second-order effects in the torsion of a spherical annular region, Int. J. Engn 0 Sci. 11,

in press (1973).

Plane strain problems in second.order elasticity theory 563

[9] A. P. S. SV.~VADUt~ and A. J. M. S ~ c ~ , Second-order elasticity with axial symmetry; I--General theory, Int. J. Enong Sci. 10, 97 (1972).

(Received 6 March 1973)

Rtsnmt- -On d6veloppe les 6quations d'61asticit6 du second ordre dans les coordonn6es polaires R, 0 pour les dfformations planes de mat6riaux 61astiques isotropes incompressibles. En consid6rant une "fonction d6place- ment" le probl6me du second ordre est ramen6/t la r6solution d'une 6quation de la forme V4W = g(R, O) o6 V 2 est le Laplacien et g(R, 0 d6pend seulement de la solution du premier ordre. On applique ensuite la technique de la fonction d6placement pour obtenir une solution du second ordre au probl6me d'un corps 61astique contenu entre deux limites rigides circulaires concentriques lorsque la limite ext6rieure est malntenue fixe et que la limite interne est soumise/t une translation rigide.

Zusammeafasseng--Die Gleichungen der Elastizit~t zweiter Ordnung werden fdr Verformungen mit ebener Dehnung eines inkompressiblen isotropischen elastischen Stoffes in Polarkoordinaten R, 0 hergeleitet. Unter Verwendung einer "Versetfingsfunktion" wird das Problem zweiter Ordnung auf die L6sung einer Gleichung der Form V4eJ = g(R, 0) zurfickgeffihrt, wobei V 2 den Laplaceschen Differentialoperator darstellt und g(R, 0) nut yon der L6sung erster Ordnung abh~ngt. Das Verfahren wird dann angewendet, um die L6sung zweiter Ordnung fur den Fall eines elastischen K6rpers, der zwischen zwei konzentrischen starren krcisf'6rmigen Begrenzungen eingeschlossen ist, zu bestimmen, wobei die ~iussere Begrenzung festgehalten wird w~hrend die innere Begrenzung eine starre Translation erfiihrt.

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