SECOND-ORDER ELASTIC EFFECTS IN AN INFINITE PLANE

SECOND-ORDER ELASTIC EFFECTS IN AN INFINITE PLANE CONTAINING A RIGID CIRCULAR INCLUSION

A. P. S. SELVADURAI

Department of Civil Engineering, The University o f Aston in Birmingham, Birmingham (Great Britain)

(Received: 9 February , 1973)

S U M M A R Y

The problems treated in this paper deal with the second-order effects in an incompressible isotropic elastic infinite plane containing a loaded rigM circular inclusion. The analysis is based on the 'displacement function' method used for the solution of the second-order problems for incompressible elastic materials. The displacement function method is particularly suitable for the problems considered, since, in each case, the displacement boundary conditions are prescribed at the rigid inclusion- elastic medium interface. The solutions are presented in closed form, in terms of elementary functions, to a set of problems relevant to the study of stress eoncentrations, which occur at the fibres of reinforced rubber-like materials and in bonded rubber mountings.

l . INTRODUCTION

The theory of successive approximations has been widely applied to obtain approximate solutions to problems in finite elasticity. 1 In the method of successive approximations we assume that the stress, strain and displacement components may be expressed as power series in terms of a small real parameter, e, the choice of this parameter depending on the problem under consideration. The non-linear differential equations of finite elasticity for the incompr~essibility condition, equations of equilibrium and boundary conditions can be reduced to sets of linear equations by considering the corresponding powers of e. In second-order elasticity we take the theory as far as the second order in e.

The solution of the second-order problem for an incompressible isotropic elastic material by making use of a 'displacement function' approach has been discussed by Selvadurai and Spencer 2 for axially symmetric problems, and Selva- durai 3 for plane strain problems. By introducing a displacement function we make

299

Fibre Science and Technology (6) (1973)--O Applied Science Publishers Ltd, England, 1973--Printed in Great Britain

300 A . P . S . SELVADURAI

direct use of the incompressibility of the material and reduce the solution of the second-order problem to the solution of an equation of the form V4W = g(R, 19) w h e r e V 2 is Laplace's operator and g(R, 19) depends only on the first-order solution. This method of solution of the second-order problem is particularly convenient when displacement boundary conditions are prescribed.

This paper deals with the application of the displacement function technique to certain problems of an incompressible elastic infinite plane loaded by a rigid circular inclusion. A brief summary of the second-order theory is given in Section 2. hi Section 3 the problem of an infinite elastic plane containing a bonded rigid circular inclusion, subjected to a combined force and a couple at its centre is considered while, in Section 4, we consider the problem of a bonded rigid circular inclusion subjected to a force at its centre, and in addition the infinite elastic plane is subjected to a state of uniform biaxial stress at infinity. Section 5 deals with a similar problem; the force at the centre of the rigid circular inclusion is replaced by a couple. Finally, in Section 6, the problem of a cylindrical cavity in an infinite plane containing a smooth oversize circular rigid inclusion is considered. In this case the infinite plane is subjected to a state of uniform biaxial stress at infinity.

In this paper the rigid circular inclusion only is considered; applications of the 'displacement function' technique to problems where the bonded inclusion has elastic characteristics will be treated in a subsequent paper.

2. FUNDAMENTAL EQUATIONS

A detailed derivation of the general theory is given by Selvadurai, a and in this section only the relevant results will be briefly mentioned.

(a) Kinematics of deformation We consider the motion of a generic particle Po(R, ®) in the undeformed body

Bo which is in a state of zero stress or of uniform hydrostatic stress, whose polar coordinates with reference to a fixed origin 0 are R, 19. After deformation let the same particle assume a position P(r, O) in the deformed body B, where r, 0 are the polar coordinates of P with reference to 0.

In plane strain the deformation gradients in the R, 19 directions are given by the matrix:

dr 1 & ]

Rg-6 F = f (2.1)

[ t30 r 00 rOR R - ~

SECOND-ORDER ELASTIC EFFECTS IN AN INFINITE PLANE 301

We consider incompressible elastic materials for which:

r (O?R O0 Or 9 0 ) = 1 (2.2) det F = ~ DO DO

We make the assumption that the displacement components u and v in the R and 19 directions, of the displacement vector connecting Po and P can be ex- panded as a power series in terms of the small real dimensionless parameter e in the form:

u = eul(R, 19) + •2u2(R , 19) -~- 0(8 3) (2.3) 1) = /3Vl(R, O) -[- 82/32(R, O) -]- 0(/~ 3)

By considering the geometry of the deformation it can be shown (Selvadurai 3) that in the first-order the incompressibility condition (eqn. (2.2)) reduces to:

Oul ul 1 dvl _ 0 (2.4) +-f f + R DO

and, in the second-order, to:

0U 2 U 2 10/3 2 (OHI~ 2 0/31 ( 1 0 U l - - ~ ) (2.5)

are given by:

where:

B1 1 (1) = 2 Du----!" 8 R '

BI2 (1) = B2~ (1) = _ _

= 2 D U 2 B11(2) OR

1 Oul Dr1 /31 + R O® DR R

+ B,11 ; B22(2) = 2 0/32 __~ 0-O + 2 + B ' 2 2

B12(2) = B21(2)= 1 3u2 Or2 I) 2 R DO + OR R + B*I 2

and = [DUl~ 2 [L DUl~ 2 0/)1 /312

B ' 1 1 \ ' - ~ ] + \ R O®] + 2 v R DR R 2

B = FF r (2.6)

B = I + eB (a) + e2B (2) (2.7)

B22 (1) = - - _ 2 Or1 ut . R o o

-- H(R, o )

The first- and second-order components, B (1) and B (2) respectively of the Cauchy- Green strain matrix:

302 A. P. S. SELVADURA!

vi ., .12 B'22 \ O R ] + \OR - 2 R 2 0-0 + ~-5

UlU1 Vl (~Vl Vl ~Ul OUl ~/0uI -~) l OUl ('~ OVl R ) a * ~ 2 = R----- 5- + R E 06) R ~ R + ~-R \OR + -R - ~ ~ +

(b) Constitutive equations The general constitutive equation for an isotropic incompressible elastic material

under plane strain conditions can be expressed in the form :

S = - p I + #B (2.8) where:

,,01 &o Sod

is the symmetrical Cauchy stress matrix referred to the deformed body, p is an indeterminate scalar pressure, # is the linear elastic shear modulus and I is the unit matrix.

By making use of the power series expansions for S and p, the first- and second- order constitutive equations reduce to:

S (1) = - p l l +/~B (~) S (2) = - p 2 1 +/~B (2) (2.9)

The components of S ~1) are:

Srr (1) -~- - P l + 11 2 OR J

Soo (1) = - P l +/~ - ~ + 2

[ 1 0 u l &'l R 1 ] (2.10) S,o ~ ~ ~ = I~ ~ + OR

and the components of S (2) can be written as:

[2 °u2] Srr (2) = --P2 + I "z [ OR J + Trr

Soo (z) = - P 2 + la - ~ + 2 + Too

S~o (2) = I~ ~ + OR + Tro (2.11)

where: T,~ = /~B*I~; Too = #B*2z; T,o = laB**2 (2.12)


(c) Equations of equilibrium By expressing the differential operators O/Or and 0/00 in terms of power series

expansions, the differential equations of equilibrium in the deformed configuration can also be expressed in terms of power series in e. In the absence of body forces the first-order equations of equilibrium are

OS,, (1) 1 OSro (1) S e e ( 1 ) - Soo (1) + +

OR R 019 R = 0

OS, o (1) 10Soo (1) Sro (1) O---'-R--" + R 0----O-- + 2-- - -~ = 0 (2.13)

and the second-order equations of equilibrium are:

OSe, (2) 1 OSro (2) Srr (2) -- Soo (2) O-'---R + R 019 + R = Na(R, 19)

OSro (2) 10Soo (2) Sro (2) - - + + 2 - = N2(R, 19) (2.14)

OR R 019 R where:

NI(R ' 19) = Ou~ 0gre (1) "t- 1 (0U 1 OR OR -R \OR

1 Ou 1 OSoo (x) N2(R, 19) = +

R OR 019

~ ) OS,, (1) 1 Oul OS~o (1)

019 R OR 019

1 OU I OSeo (1) /21 (See (1) - Soo (1)) + + - - R 019 OR R R

1 OU 1 OSoo (1) Ou I OSeo (1) + R 019 OR OR OR

+ R \OR ~ + 2"-R 2 S'°(x) (2.15)

(d) Displacement functions We observe that the first-order incompressibility condition (eqn. (2.4)) can be

identically satisfied by the introduction of a displacement function, qJ 1, such that:

1 c~W1. 0qsl (2.16) Ul R 019 ' Vl OR

By making use of the first-order constitutive equations (eqns. (2.10)), the first-order equations of equilibrium (eqns. (2.13)) can be written as:

# 0 0Pl + [VZW1] = 0 0R Vg

1 0pl ~R R 019 P [VZqJl] = 0 (2.17)

304 A . P . S . SELVADURA[

where: 0 2 [ ¢3 1 O 2

V 2 -- + + - - ~ OR 2 R ~ R 2 302

By eliminating p~ and W~ in turn from eqns. (2.17), we obtain:

and V4W~ = 0 (2.18)

V2pl = 0 (2.19)

It has been shown (Selvadurai 3) that, by adopting a second-order displacement function, ~P2, such that:

1 0W 2 H 2 - - .~ Ht2

R 0 ®

where:

OW2 v 2 = - - - + v' 2 (2.20)

0R

U 1 Ou 1 U '2 - - 1 ut 2 1 vl 2 + _ _ _

2 R 2 R R O 0

0u 1 2 - u 1 (2.21)

JR

the second-order incompressibility condition (eqn. (2.5)) is identically satisfied, and the second-order equations of equilibrium (eqns. (2.14)) can be reduced to the

+ _ m

form:

where:

__[Orrr F I ( R ' ®) = [ OR

- 0--R + R [V2W2] = F , ( R , ®)

0p2 pR 0 - - 0-"O - - ~ [V2klI2] = Fz (R ' ®)

1 OT, o T . - Too +

R 06) R

(2.22)

OH + # ~-~ + V2u'2

F2(R, 6)) = - [ R OT'° OT°° [ dR + - ~ +2T, o

u'2 2 0vP2] _ N I ( R ' 6 ) ) ] R 2 R z 0 0 ] ]

OH v'2 2 Ou2] _ R N 2 ( R , ®)] + ~' ~ + nv2v'2 - -Y + -k o o !


By eliminating P2 and ~2 in turn from eqns. (2.22) we obtain:

1 [~F, OFa]

_[63F1 F1 1 ~ F 2 ] v2P2 = + - 2 +

(2.23)

(2.24)

A solution of eqn. (2.23) consists of a particular integral and solutions to the homogeneous equation V4W2 = 0. The complete second-order solution is obtained by selecting that solution which will satisfy the appropriate boundary conditions of the problem. To the second-order in e the displacement and stress components are given by: u = eu 1 +/~2u2; /) = e/) 1 --[- /32/)2; S = gS (1) -q- ~28(2).

3. INFINITE ELASTIC PLANE CONTAINING A RIGID CIRCULAR INCLUSION SUBJECTED TO

A COMBINED FORCE AND COUPLE

The second-order solution to the problem of an infinite elastic plane bounded internally by a bonded rigid circular inclusion and loaded at its centre by a concentrated force has been given by Selvadurai. 3 Here we examine the problem where the rigid circular inclusion is subjected simultaneously to a concentrated force (T) and a couple (Go) at its centre (Fig. 1). The force T causes a rigid body translation 6 of the rigid inclusion in the x-direction and the couple causes a rigid

¥,Y

I

t

= - X , ×

Fig. 1. Rigid circular inclusion subjected to a concentrated force and couple.

306 A. P. S. SELVADURAI

body rotat ion, p. Perfect bonding between the rigid inclusion and the surrounding elastic med ium is assumed, thereby prescribing displacement boundary condit ions at the inclusion e las t i c -medium interface R = a (a > 0). We note that, in classical elasticity, on account of the linearity of the field equat ions the principle of super- posit ion of solutions is applicable. It must be emphasised that, in general, the principle of superposi t ion is not applicable to problems of second-order elasticity theory. This is due to the presence of products , and other non-l inear terms of the first-order d isplacement and stress componen t s in the equat ions for the second- order d isplacement and stress components , incompressibil i ty condit ion and equat ions o f equil ibrium. A first-order displacement funct ion of the form :

= - + 2 a R l n sin O - 21a 2 In (3.1) zW~ a

which satisfies eqn. (2.18) gives the first-order displacement componen ts :

~u~ = - + 2 a l n cos ®

eVl = - - 2a 1 + In sin ® + 21 (3.2) a

where: ~ Go A1 --

4Xl~fa is a non-dimens ional constant .

The first-order componen t s are:

( S , ~1) 6 8 a _ 4 c o s ®

_ cos o z It a

_ ( ( 5 a 3 a 2 ) - 4 ~ - 5 sin ® - 221 ~-~ (3.3)

it a

We note that the first-order stress componen t s (eqns. (3.3)) satisfy the boundary condi t ions:

f ~ (S~r (~) O Sro (1) sin R d O T COS O)

i ~ (Sr~(~)sin O + S~0(l) cos O) R d O = 0 ~z

f '~ S~o (1) R 2 dO = (3.4) Go

and Sr~ (~), So® (~), St® (~) tend to zero as R ~ oo.


We choose the small real dimensionless parameter e as 6/a. On substituting the expressions for the first-order displacement and stress components (eqns. (3.2) and (3.3)) in eqns. (2.22), the inhomogeneous differential equation for the second-order displacement function (eqn. (2.23)) reduces to:

where: V4W2 = Fl l (R , (9) + 21F12(R, (9)

( a 6 a 2 a 2 R) F 1 ,(R, (9) -- - 96 ~-~ + 8 ~-~ + 32 ~ In sin 2(9

a5 a 3 ) F12(R,®) = --96~5 + 16~--~ cos(9

A particular integral of eqn. (3.5) can be written in the form:

t'll/2p ~--- LI'/2p(11) -F" 21kl/2p(12)

where: LFI2p(I1)- [ l a 6 ( R ) 2 ] - ~ ~-~ + a 2 In Ra + a2 In sin 20

(3.5)

(3.6)

(3.7)

2) ( 212a4 ) u2p(2 = 2 R-3

a 4 a 2 a 4 R\ v2p(11)= - 3 ~ + R - 2 R 3 1 n a ) s in20

I a $ 0 3 a 3 R) v2p(12) = 2 R 4 R 2 R2 In cos ® (3.10)

a3 u2p (l = - ~-~ + 2 ~ + ~ l n s inO

The displacement components derived from the particular solution (eqns. (3.7) and (2.20)) are:

1) 2) (22). 1) (3.9) ~/2p = U2p (1 + ~l/~/2p (1 -~ ~12U2p , V2p = /)2p (1 "q- ~lV2p (12)

where: [ a 4 a2 ( a 4 ~ ) R ] [ a 6 0 4 ~ ]

u2v(t~) -- -2~-~ + -~ - 2 ~-3 - 2 In cos20 + -~-3 + 2 R3

kI./2p(12) ( l a 5 a 3 R ) - 2 R 3 R In cos ® (3.8)

30X A. P. S. SELVADURAI

By considering the kinematics of deformation of the rigid circular inclusion it can be shown that the second-order displacement components must satisfy the boundary conditions:

U,(Q, 0) = 0: L’&z, 0) = 0 (3.11)

and the second-order stress components should tend to zero as R + co. To satisfy these boundary conditions we require five independent solutions ofthe homogeneous equation V4Y’, = 0.

These solutions are:

VIZH = ix, + 3) sin20 + (2 + a,R) cos 0 + cc,@ (3.12)

where c1r, a2.. ., are arbitrary constants. On satisfying the boundary conditions (eqns. (3.11)) we obtain the final expression

for the second-order displacement and stress components as follows:

s (2) rr

where :

lnRcos20 + a

u2 (12) =

(22) = -- _ !a i

1 a4 2

u2 2R3 2R

sin 0

v2 (11) = sin 20

la3 _--- - a3 In E 2 R4 2 R2 R2

cos 0 a

(3.13)

(3.14)

(3.15)

SECOND-ORDER ELASTIC EFFECTS IN AN INFINITE PLANE

a6 a 4 a 2 S~,11 (2) = 4~--~ - 8~-~ + 2~- 5

3O9

+ 4(-~4 - 2 ~2) l nR] cos 2 0

( a 6 a 4 a 2 a 4 R a) + 6~-~ - 10 ~--~ + 6 ~-Z + 8 ~-~ In

and

respectively, where, on the contour, c:

131 0 = 0 + ~ ( r o , 0) ,

to order e.

P~ = f (S,~ cos 0 - S,o sin O)r o dO c

PY = f (S,, sin 0 + S,o cos O)ro dO ¢

ro = R + eu~(ro, O)

(3.17)

(3.18)

Srrl2 (2) = -2~--~ + 4~- 3 In sin 0

a 4 a 2

Srr22(2) -- ( R4 R--2)

( a° a" Sooll (2~= -4~ -~ + 2 ~ - ~ - 4 ~ - ~ 1 n c o s 2 0

+ 2 -2 +2N-8 ln

Soo~2(2)= 8 ~-g - 2~-5 - 4~--g In s inO

Soo22 (2) : (3 a 4 a 2

[ a 6 a" a 2 (a 4 a 2 ) R ] Sr011(2)= 4 ~ - ~ - 6 ~ - - ~ + 2 ~ - 5 + 4 ~-~+~-- ~ In s in2®

S, o12 (2~ = 4 ~ - g - 6~- 5 + 4~-51n cosO

Sr022 (2) = 0 (3.16)

The stress components (eqn. (3.14)) tend to zero as R ~ oo. The resultant force exerted on the inclusion can be evaluated by considering the force resultants in the x and y directions (Fig. 1) transmitted across any contour r = ro in the deformed body. The force resultants in the x and y directions are:

310 A.P.s . SELVADURAI

We express Px and Py in power series of E (to order e2) as:

= 8 2 p (2) Px = ePx (t) + /32px (2), py gpy(l) + y where:

PY¢ " = f ~ (Sr /1)s in ® + Sro (I) COS O)r o dO

Pr(2) = f~,, [S~r(2) sin

(3.19)

(3.20)

(3.21)

® + Srot2) c o s ® - Ull---:-:--~ sin 0 + ---7-~ cos®

U I .ql_ (Srr (1) COS (~ -{- St0 (1) s i n O)

1 ~v1¢.~ (1)sin 0 + S r o ( l ) c o s O ) ] r o d O + J

f~[ [OS'/') OSr°(') sin O) ex(2) = rx Srr (2) COS ( ~ - - Sro (2) sin ® - ul [ ~ cos ® O ~

(3.22)

Vl R (Srr(1) sin O + Sro (1) cos O)

1 c~vl ] + ~ ~ (S,, (1) cos O - S~0 O) sin O) ro dO (3.23)

Similarly, the resultant couple (M) acting on the rigid inclusion can be written as:

M = aM (1) + g2M(2) (3.24) to order e 2, where:

M (1) = sroCl)ro 2 dO (3.25)

and

M ~z) = _~ S,o ( 2 ) - u ~ + R 0®] r ° 2 d O (3.26)

On evaluating Px (~), Px ~2), Py(J), . . ., M ~2) we have:

and

6 ePx (a) = 8npa6, EM (j) = G o -

a

px~2~ = py~2) = M(2) = 0

In the problems where the rigid circular inclusion is subjected to only a concentrated force at its centre, it has been observed 3 that, by the symmetry of the problem


py(1) py(2), M (1) and M (2) are all zero and on evaluating the integral (eqn. (3.23)), Px (2) was found to be equal to zero. I t may also be o f interest to note that, in the particular case where the rigid circular inclusion is subjected to a combined force and a couple at its centre, there is no second-order contr ibut ion to Px, Py and M. The applied force and couple at the centre o f the inclusion, to order e 2, are given by:

T = 8npa6, M = Go

4. INFINITE ELASTIC PLANE UNDER BIAXIAL STRESS CONTAINING A RIGID CIRCULAR INCLUSION SUBJECTED TO A CENTRAL FORCE

We consider the problem o f an infinite plane, conta ining a bonded rigid circular inclusion, which is subjected to a state o f uni form biaxial stress at infinity (Fig. 2). In addit ion, the rigid circular inclusion is subjected to a concentrated force at its centre which acts in the x-direction.

Y,y

TI

.7. T21

• ~ T I

2o

=~ X,x

Fig. 2. Infinite plane under biaxial stress. Rigid circular inclusion subjected to concentrated force.

312 A.P.S. SELVADURAI

The first-order displacement function for this problem is given by:

4p aa --a - a + -~--~2 sin 2 0

(a 3 R) ] + 22 ~ + 2aR In sin ® (4.1)

where T1 and T 2 are the uniform stresses acting at infinity.

T 1 -- T 2 4#6 a = - and 22 = (4.2)

TL + T2 a(Ta + T2)

In this, and subsequent problems treated in this paper, the small real dimensionless parameter e is chosen to be equal to (TI + T2)/4p.

The first-order displacement and stress components are:

and

ul = aa - 2 ~ + cos 2® + 22 + 2aln cos ®

(2a ) [a3 ( v l = a a - + - ~ sin 20 +22 ~ - 2 - 2 a 1 + l n sin®

I ( ( ° 3 ) l a 2 a 4 a _ 4 ~-3 cos ® St, (1} = p 2 + a 2 + 8 ~ - ~ - - 6 - ~ cos2® + 2 2 8 ~

(4.3)

S o 0 ( 1 ) = p [ 2 + a ( - 2 + 6 ~ 4 ) cos 2 0 + 22 (4 ~3 ) c o s ® ]

S,o (1~ = # a - 2 + 4 ~ - 2 - 6R-~ s in2® + 2 2 - 4 ~ - 5 s in® (4.4)

respectively. It may be verified that the first-order stress components (eqns. (4.4)) reduce to a

state of uniform biaxial stress at infinity and give force and couple resultants:

p ( l ) = T, py(1) = 0, M (1) = 0

on the rigid circular inclusion-elastic medium interface R = a. By substituting the first-order displacements and stress components (eqns. (4.3)

and (4.4)) into eqns. (2.22), we obtain the inhomogeneous differential equation for the second-order displacement function:

V 4 ~ l 2 = ,~22F11(R, O) + o'~,2[F13(R, O) + F3I(R , O)] -[- tr2Fa3(R, ®) (4.5)


where F~ I(R, 0) is given by eqns. (3.6) and:

( a a 3 a 7 a R) F ~ 3 ( R , 0) = 32~- 5 - 48~--g - 320 ~--g + 64~-51n sin3®

( a_3 a 5 -- a 5 R) F 3 I ( R , 0 ) = - 16 R ~ + 32 ~-~ 192 ~-~ In sin ®

(48 a 2 a 8 ) F33(R,®)= ~ - 7 - 9 6 ~ - 2 4 0 ~ sin 40 (4.6)

A particular integral of eqn. (3.5) can be written in the form:

~'I'/2p • 222t'IJ2p(11) "~ G,,~2(~I'/2p(13) "~- ~'IJ2p (31)) "l- 0"2tlJ2p(33) (4.7)

where Ut/2p(11) is given by eqns. (3.8) and

[ 2 R l a 7 ( _ ~ ) R] W2/13) - - - 2 R 5 + a R - In sin3®

[ a s ( ~ a 5 ) R] W2/31) = -~--g + ~-5 In sin ®

{R 2 a 2 1 a8~ ~F2/33) - \ 4 ~ ~ ~ ] sin 40 (4.8)

For this problem, the second-order displacement components should satisfy the conditions:

u2(R , ®) = /)2(R, ®) = 0 (4.9)

on R = a, and the second-order stress components should either reduce to zero or constitute a state of uniform biaxial stress at infinity.

These boundary conditions can be identically satisfied by making use of the solutions to the homogeneous equation V4tt I /2 = 0:

~2~ = ~-~ + sin rn® (4.10) m=l

where c~1, ~2 . . . . . ill, f12,..., are arbitrary constants.

The final expressions for the second-order displacement and stress components can be written in the form:

U2) lU2 (11)) lU2 (13) + U2(31) t 0" 2 lU2(33)t . . . . + - - (4.11)

v~l = ~ i v ~ ' ~ i + ~ - / v ~ ( ~ + v ~ b Iv~(~)l

314 A. P. S. SELVADURAI

and

S°°(2) = 222- S o o l l -.t- 022` S0013 (2) + S0031 (2) - + 0"2+ S0033 (2) -(4.12) . (2) tt

]'1 -t- 5"tO 3 1 I

respectively, where"

U2(I 1), v2(l 1), Srrl 1(2), So011(2) , Srol 1(2) a re given by eqns. (3.15) and (3.16) a n d :

(°3 a, 1 //2(I 3 ) = - - R--- ~ + ~ ~ + 3 ~-2 ~-~ In cos 30

U,,;(31) : 1 _ 5 a 3 a 5 5 a 7 ( a 3) R] [ 2 R 2 + 5 R 4 2 R 6 + a - ~ - ~ In cos 0

(_~ a+ a +) (~ a2 a + ,,6 3a +) U2 (33) = - 2 ~ +~-~ c o s 4 0 + R R 3 + 3 R 5 2 R -7

[2 la5 ( a3 aS) ~-] v2 (13~= ~--4 + ~-7 - 3~--~ In sin30

[ 3a3 a 5 l a 7 ( a3) R 1 v2 ( 3 u = a ~ - ~ +~-~ }~-~ a +~--~ In s inO

U 2 ( 3 3 ) = - - 2 ~-3 + ~-3 sin 40 (4.13)

(2) [ - 4 a a 3 a s a 7 S,,~3 = ~ + 10~- 3 - 16R--g + 10 R--- 5

[_~ a3 (2) _ 4 + 16 ~ 40 Srr 31 - -

+ - 4 R - 12R- 5 + 12~--g In

a 5 a 7 ~-~ + 20 R---- 5

cos 30

a3 + 4 - 4~-3 + 12~-g In cos®


( a4 a6 aS) Srr33 (2) = - 2 + 6~- 2 - 8 ~ + 6~-g cos40

Soo13(2) = [4 R -

315

a2 a 4 a 6 a 8 )

+ 2 - 4 ~ +20~- 2 - 36~-~ + 15~-~

a 3 a s a 7

6~- 3 + 4~--g - 10~- v

( R a3 aS) R] + 4 - 4 ~ - ~ - 12~-g In cos 30

[ R a7 ( R a3 aS) R ] Soo31~2)= 4 +4~- 5 + - 4 + 4 ~ - 3 - 12~--g In cos®

( a a a 6 a8\ ~2) 2 - 10 + 8 - 6 ) c o s 4 0

a2 a 4

+ - 2 + 4 ~ - ~ - 4 ~ - ~ + 4

a3 a s a 7 a _ 2 - 12~--g + 1 0 - - S~013{2) = 4 ~ ~-5 R 7

a 6 a s )

a3 a') + 4 - 4~-3 + 12~-~ In sin 3 0

[ a3 a7 ( R a3 aS) R] S, o3~2)= -4~--~+4~--q+ - 4 - 4 ~ - 3 + 1 2 ~ - g In sinO

a* a 6 a s ) S,o33 ~2) = 2 - 4 ~ 2 + 4 ~ - ~ - 8~-~ +6~-g sin4® (4.14)

By making use of the second-order stress components (eqn. (4.12)) and the expressions for the second-order force and couple resultants (eqns. (3.22), (3.23) and (3.26)) it can be shown that:

px(2) = py(2) = M(2) = 0

In addition, it may be verified that, in the special case when 3 = 0 and T 2 = 0,

the second-order displacement and stress components (eqns. (4.11) and (4.12)) reduce to those given by Adkins et al. 4 for the extension of an infinite elastic body containing a rigid circular inclusion.

3 1 6 A. I'. S. SELVADURAI

5. INFINITE ELASTIC PLANE UNDER BIAXIAL STRESS CONTAINING A RIGID CIRCULAR INCLUSION SUBJECTED TO A COUPLE AT THE CENTRE

In this problem the bonded rigid circular inclusion is subjected to a concentrated couple (Go) at its centre (Fig. 3), and the infinite plane is subjected to a state o f uni form stress at infinity.

The first-order d isplacement funct ion for this problem is given by:

/~k[/1 (T~ + T 2 ) [ (1R 2 l a 3 ) R ] = - a + sin 2 ® -- 23 a2 In (5 .1 )

where: T 1 -- T 2 Go

o" - T1 + T2 a n d 2 3 = rca2(T 1 + T2 ) (5 .2 )

The first-order displacement and stress c o m p o n e n t s are:

u I = a~ R + cos 2®

vl = a~ - + s i n 2 0 + 2 3 - ~ - (5.3)

Y,y

T~

Fig. 3.

XTx

I-- ~ 2 - a

Infinite plane under biaxial stress. Rigid circular inclusion subjected to couple.

and


[ ( a2 a') ] Srr (I) = ~ 2 + a 2 + 8~-- i - 6~--~ cos 20

Soo (1) =/~ [2 + a ( - 2 + 6~-~4) cos2® ]

[ ( a 4 ) a21 S,o (1) = /~ a - 2 + 4 ~ 2 - 6~- 2 s in20 - 223~-- ~ (5.4)

respectively.

From eqns. (5.3), (5.4) and (2.22) we obtain the inhomogeneous differential equation for the second-order displacement function (eqn. (2.23)) as:

V4I"I) 2 = o'2F33(R, O) -q- o'23F23(R, f~)) (5.5)

where F 3 3 ( R , O) is defined in eqns. (4.6) and ( a_~ a 6 )

F 2 3 ( R , O) = --16R~ -- 192~-~ cos2® (5.6)

A particular integral of eqn. (5.5) is given by:

ltt/Ep ----- 0"2~1/2p(33) 21- O'~31tIJ2p(23) (5.7)

where W2p (33) is given by eqns. (4.8) and

tlJ 2p(23) (--a2 In R la;4 ) = cos 20 (5.8) a 2

The displacement boundary conditions:

u 2 ( g , O) = v 2 ( R , O) = 0 (5.9)

at the rigid inclusion-elastic medium interface and the stress boundary conditions at infinity can be identically satisfied by making use of the homogeneous solution of eqn. (2.23)

UIJ2H-~- (~22 "3 t- r/~) s in40 + (r /3+ ~/~--~) cos 20 + r/sO (5.10)

where r/1, r/z . . . . . are arbitrary constants. On satisfying the boundary conditions (eqn. (5.9)) we obtain the final expression

for the second-order displacement and stress components, which could be written as:

u2} o.2]u2(33)! o.~3!u2(23)}_ .31_ ~32_ Iu2(22,} U2 i ~ = iU2(33) i + - ( 5 . 1 1 ) fp2 (23) } [/)2 (22) }

318

and

A. P. S. SELVADURAI

800(2) a 2- 80033 (2) " + O'2 3- Sooz3 (2) - + 232. So02z (2J (5.12) tz

where u2(33)~ u2(22)~ v2(33)~ u2 (22) (2) . . . . , 8roa3 , 8,022 (2) are given by eqns. (3.t5), (3.16), (4.13) and (4.14), and

04 a6 a2 /22(23) = ~ R - + 3 ~-3 - 2 ~ + 2 ~- In sin 2 0

v2 (23)-- -~ -5+~-~ cos 20

(2) = ~6 ~-~ ~-~ ~-~ ~',r23 - 18 + 4 - 8 In s i n 2 0

a2 a 4 a 6 ) Soo23 (2) = - 2 ~ - ~ + 2 ~ - ~ + 8 ~ - ~ s i n 2 0

( 02 a" a 6 a 2 R) S,023 (2) = -6~-~ + 2~--~ + 2~--~ + 4~- i l n cos 2 0

(5.13)

(5.14)

From the second-order stress components (eqn. (5.12)) and integrals (eqns. (3.22), (3.23) and (3.26)) we may verify that there is no second-order contribution to Px,

Py and M.

6. OVERSIZE RIGID CIRCULAR INCLUSION 1N AN INFINITE ELASTIC PLANE UNDER BIAXIAL STRESS

An infinite elastic plane contains a circular cavity of radius a. To this circular cavity is fitted a rigid circular inclusion of radius a(1 + ~) where ~ is a small quantity. The contact between the rigid circular inclusion and the infinite elastic plane is assumed to be smooth. The infinite elastic plane is also subjected to a state of uniform biaxial stress at infinity (Fig. 4). We assume that no separation takes place at the inclusion-elastic medium interface due to the action of the biaxial state of stress.


Yy

319

T I -.,,-----

Fig. 4.

I I .. L

,t_.~2a(l+ ( ) ,.q

I-I" i i I-I IT.--

T I

~- X,x

Infinite plane under biaxial stress. Cylindrical cavity containing an oversize rigid inclusion.

The first-order displacement function for this problem is:

( )[ (~.. 1.) ] etP1 = .T~ + T2. aa 4p a 2 a z sin 2 0 + 2 4 ®

where: T1 - Tz 4p~

Ti + T 2 T1 + T2

and the first-order displacement and stress components are:

and

ul = aa - cos 2 0 + 24-~

v l = a a ( - R ) sin 2®

[ ( a ' ) o'] S,, (1) = # 2 + a 2 + 4 ~ - 5 cos 2® - 224 ~-~

(6.1)

(6.2)

(6.3)

320 A.P .S . SELVADURAI

So® ( l~ = # 2 - 2a cos 2 0 + 2 , ~ 4

1 S,o (1) = # ~ - 2 + 2-R--~ s i n 2 0 (6.4)

respectively. Since the circular inclusion is rigid, the first-order displacement Ul (eqn. (6.3)),

satisfies the boundary condition:

ul(a, O) = a,~, 4

on the surface R = a. Owing to the smooth contact at the interface the shear stress component S,0 (1)

is zero on R = a. The first-order stress components combine to form a state of uniform biaxial stress at infinity. The first-order displacement components (eqns. (6.3)) indicate that, in general, the circular cavity will assume a certain ovality due to the biaxial state of stress. For a circular inclusion of small oversize it is possible that some clearance might appear at, say, points A for T~ > T2.

From the first-order stress components (eqns. (6.4)) it is therefore possible to obtain a relationship between T1, T2 and ~ when separation will just occur at the points A. For a given value of ~ the 'critical values' of T~ and T2 are related by the equation:

2T 1 - T 2 - - 2p~ = 0 (6.5)

In what follows we shall assume that there is no loss of contact at the rigid inclusion-elastic medium interface due to the effects of moderately large strains. The inhomogeneous differential equation (eqn. (2.23)) reduces to:

where: V4~IJ 2 = o-2F44(R, 0 ) + o'~,4F45(R, ®)

48( a 2) F 4 4 ( R , O ) = ~ - 5 1 - ~ sin 4 0

a2( a2) F45(R,®) = -48~-~ 1 - ~ sin 2 0

A particular integral of eqn. (6.5) is:

L]12p = O'2U~']Zp(44) "F O'~4Ul]2p(45) where:

( 1 ~2p( 45 )= - 3 a 2 - ~ - ~ l n R s i n 2 ® a

(6.6)

(6.7)

(6.8)

(6.9)


In order to satisfy the boundary conditions at the rigid inclusion-elastic medium interface:

u2(a, O) = 0 Sro(2)(a, O) = 0 (6.10)

and at infinity, we make use of the solutions of the homogeneous equation:

• 2 n = ~-5+~--~ s i n 4 0 + P3 +~-~ s i n 2 ® + p s ® (6.11)

where p ~, p2, • .. are arbitrary constants. Finally, the expressions for the second-order displacement and stress components

can be written in the form:

u2,55,1

V2i (6.12)

and

Soe( 2 )

8"°( 2 ) t ~ '

(2,1 (2, s . . (2,1 Srr44 I Srr45 1

= a2', S0044(2) ', _{_ 0.24 . S0045(2) . ..~ 2 4 . 2 S0055(2) .

! i b

I S,044 Sro,~5 (2~ S, o55 j

respectively, where:

[ l a 2 7a4 3 a 6 ) (R u2(**) = \2 R 8 R 3 + 8 ~ COS 4 0 +

I a 2 1 a 4 \

4n 4 ~ ) [ a 2 a 4 ( a 2 a*) R]

Uz (45)= 2-~ - - 2 ~ - 3 - - 6-~- + 2 ~ - 5 In cos 20

= (1 a 2 1 a 4) u2 (55~ \ ~ ~

(~ a 2 5 a* 3 a 6 v2(44) = R 16 R 3 + 8 R-g] sin 4 0

a 2 a 4 a 4 R \ v2 (45) = 2 ~ - - 2 ~ - 2 R -'31na-) s i n 2 0

(6.13)

1)2 (55) = 0 (6.14)

322 A . P . S . SELVADURAI

( 53 a 4 15 a 6 ) ( 5 a 2 a 4 ) Srr , , ~z' = --2 + ~ ~-~ ~ ~ COS40 + 2 ~ - ~ +4-R- 2

E a2 a4 ( a2 a4) S~,45 (2~ = - 2 6 ~ + 1 4 ~ + 12 2~-5 + ~ - ~ In cos 2 0

( a2 a4) Srr55 ~2) = --~-~ + 3 - ~

Soo4~Z,=(2 7a4 15a 6 ) ( 5 a 2 ) S R 4 +-4-R-g cos4® + - 2 + ~ - i

Soo45 ~z) = ~2~- 5 - 6~-£ - 12~-21n c o s 2 0

( a 2 ~ a 4 1 5 a 6 ) S~o44 ~2) = 2 - 2 ~ + ~ ~-~ s i n 4 0

[ 0 2 ( a 2 ) a2 ( a 2 ) R] Sr045 ~2) = - 1 2 - ~ 1 - ~-5 + 12~-5 1 + ~-5 In s in2®

Sro55 ~2) = 0 (6.15)

We further observe that, owing to the symmetry of the problem and the assumed frictionless conditions at the rigid inclusion-elastic medium interface, the resultant force and couple acting on the rigid inclusion are zero. In the second-order case, the biaxial stress components T 1 and T 2 which will just cause separation at point A are related by the equation:

a2(~ l e ) + t r ( 6 - 1224e)+ ( 2 - 2 2 , + 2J.42e) = 0

REFERENCES

1. A. E. GREEN and J. E. ADKINS, Large elastic deformations and non-linear continuum mechanics, 2nd ed., Oxford University Press, 1971.

2. A. P. S. SELVADURAI and A. J. M. SPENCER, Second-order elasticity with axial symmetry, Int. J. Engng Sci., 10 (1972) pp. 97-114.

3. A. P. S. SELVAOURAI, Plane strain problems in second-order elasticity theory (in press). 4. J. E. ADKINS, A. E. GREEN and R. T. SrtmLD, Finite plane strain, Phil. Trans. Roy. Soc. Ser. A,

246 (1953) p. 181.

Documents

SECOND-ORDER ELASTIC EFFECTS IN AN INFINITE PLANE