general theory of small elastic deformations superposed on finite elastic deformations

General Theory of Small Elastic Deformations Superposed on Finite Elastic DeformationsAuthor(s): A. E. Green, R. S. Rivlin and R. T. ShieldSource: Proceedings of the Royal Society of London. Series A, Mathematical and PhysicalSciences, Vol. 211, No. 1104 (Feb. 7, 1952), pp. 128-154Published by: The Royal SocietyStable URL: http://www.jstor.org/stable/98846 .

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128 R. Berman and D. K. C. MacDonald

greater effect on electrical than on thermal transport. At present, however, in the absence of any adequate theoretical explanation of the phenomenon, further

speculation seems hardly profitable.

REFERENCES

Berman, R. & MacDonald, D. K. C. 1951 Proc. Roy. Soc. A, 209, 368. Blackman, M. 1935a Proc. Roy. Soc. A, 148, 365, 384. Blackman, M. I935b Proc. Roy. Soc. A, 149, 117, 128. Bremmer, H. & de Haas, W. J. 1936 Physica, 3, 672. Cornish, F. N. H. & MacDonald, D. K. C. 1951 Phil. Mag. 42, 1406. Griineisen, E. & Goens, E. I927 Z. Phys. 44, 615. de Haas, W. J., de Boer, J. & van den Berg, G. J. I933 Physica, 1, 1115. Lees, C. H. I908 Phil. Trans. 208, 381. Makinson, R. E. B. I938 Proc. Camb. Phil. Soc. 34, 474. Mendoza, E. W. & Thomas, J. G. I951 Phil. Mag. 42, 291. Mott, N. F. & Jones, H. I936 Theory of properties of metals and alloys. Oxford University

Press. Sondheimer, E. I. 1950 Proc. Roy. Soc. A, 203, 75.

General theory of small elastic deformations

superposed on finite elastic deformations

BY A. E. GREEN, King's College, Newcastle upon Tyne, R. S. RIVLIN, British Rubber Producers' Research Association

AND R. T. SHIELD, King's College, Newcastle upon Tyne

(Communicated by G. R. Goldsbrough, F.R.S.-Received 8 September 1951)

Using tensor notations a general theory is developed for small elastic deformations, of either a compressible or incompressible isotropic elastic body, superposed on a known finite deformation, without assuming special forms for the strain-energy function. The theory is specialized to the case when the finite deformation is pure homogeneous. When two of the principal extension ratios are equal the changes in displacement and stress due to the small superposed deformation are expressed in terms of two potential functions in a manner which is analogous to that used in the infinitesimal deformation of hexagonally aeolotropic materials. The potential functions are used to solve the problem of the infinitesimally small indentation, by a spherical punch, of the plane surface of a semi-infinite body of incompressible isotropic elastic material which is first subjected to a finite pure homogeneous deformation symmetrical about the normal to the force-free plane surface.

The general theory is also applied to the infinitesimal deformation of a thin sheet of incompressible isotropic elastic material which is first subjected to a finite pure homogeneous deformation by forces in its plane. A differential equation is obtained for the small deflexion of the sheet due to small forces acting normally to its face. This equation is solved completely in the case of a clamped circular sheet subjected to a pure homogeneous deformation having equal extension ratios in the plane of the sheet, the small bending force being uniformly distributed over a face of the sheet. Finally, equations are obtained for the homogeneously deformed sheet subjected to infinitesimal generalized plane stress, and a method of solution by complex variable technique is indicated.

128 R. Berman and D. K. C. MacDonald

greater effect on electrical than on thermal transport. At present, however, in the absence of any adequate theoretical explanation of the phenomenon, further

speculation seems hardly profitable.

REFERENCES

Berman, R. & MacDonald, D. K. C. 1951 Proc. Roy. Soc. A, 209, 368. Blackman, M. 1935a Proc. Roy. Soc. A, 148, 365, 384. Blackman, M. I935b Proc. Roy. Soc. A, 149, 117, 128. Bremmer, H. & de Haas, W. J. 1936 Physica, 3, 672. Cornish, F. N. H. & MacDonald, D. K. C. 1951 Phil. Mag. 42, 1406. Griineisen, E. & Goens, E. I927 Z. Phys. 44, 615. de Haas, W. J., de Boer, J. & van den Berg, G. J. I933 Physica, 1, 1115. Lees, C. H. I908 Phil. Trans. 208, 381. Makinson, R. E. B. I938 Proc. Camb. Phil. Soc. 34, 474. Mendoza, E. W. & Thomas, J. G. I951 Phil. Mag. 42, 291. Mott, N. F. & Jones, H. I936 Theory of properties of metals and alloys. Oxford University

Press. Sondheimer, E. I. 1950 Proc. Roy. Soc. A, 203, 75.

General theory of small elastic deformations

superposed on finite elastic deformations

BY A. E. GREEN, King's College, Newcastle upon Tyne, R. S. RIVLIN, British Rubber Producers' Research Association

AND R. T. SHIELD, King's College, Newcastle upon Tyne

(Communicated by G. R. Goldsbrough, F.R.S.-Received 8 September 1951)

Using tensor notations a general theory is developed for small elastic deformations, of either a compressible or incompressible isotropic elastic body, superposed on a known finite deformation, without assuming special forms for the strain-energy function. The theory is specialized to the case when the finite deformation is pure homogeneous. When two of the principal extension ratios are equal the changes in displacement and stress due to the small superposed deformation are expressed in terms of two potential functions in a manner which is analogous to that used in the infinitesimal deformation of hexagonally aeolotropic materials. The potential functions are used to solve the problem of the infinitesimally small indentation, by a spherical punch, of the plane surface of a semi-infinite body of incompressible isotropic elastic material which is first subjected to a finite pure homogeneous deformation symmetrical about the normal to the force-free plane surface.

The general theory is also applied to the infinitesimal deformation of a thin sheet of incompressible isotropic elastic material which is first subjected to a finite pure homogeneous deformation by forces in its plane. A differential equation is obtained for the small deflexion of the sheet due to small forces acting normally to its face. This equation is solved completely in the case of a clamped circular sheet subjected to a pure homogeneous deformation having equal extension ratios in the plane of the sheet, the small bending force being uniformly distributed over a face of the sheet. Finally, equations are obtained for the homogeneously deformed sheet subjected to infinitesimal generalized plane stress, and a method of solution by complex variable technique is indicated.



Small deformations superposed on finite deformations 129

1. INTRODUCTION

In a recent series of papers Rivlin (I 948 a-I 949 c) has discussed a variety of problems concerned with the finite deformation of an elastic body. Most of these problems are solved exactly for an incompressible isotropic body using a completely general form for the strain-energy function. Contributions have also been made by Green & Shield (I950) to the list of exact solutions for an incompressible body, and these writers (I95I) have also solved two special problems of small deformations superposed on a known finite deformation, the solutions being valid for a compressible isotropic body withno restrictions on the strain-energy function. In the present paper a general theory is developed (in ? 3) for small elastic deformations of an isotropic elastic body superposed on a known finite deformation, the theory being available for both compressible and incompressible bodies, again without assuming special forms for the strain-energy function.

The general theory is then specialized in ? 4 to the case when the finite deformation is pure homogeneous. When two of the principal extension ratios are equal, the

expressions for the changes in the stress components due to the small superposed deformation are formally somewhat similar to the stress-strain relations for the infinitesimal deformation of a hexagonally aelotropic material (?? 5, 6). Accordingly, potential functions may be used for the solution of certain problems in a manner

largely analogous with that employed in the latter case. Their use is illustrated in

? 7 by the problem of the infinitesimally small indentation, by a spherical punch, of the plane surface of a semi-infinite body of incompressible isotropic elastic material which is first subjected to a finite pure homogeneous deformation

symmetrical about the normal to the force-free plane surface. In the last two sections of the paper, the theory is applied to the infinitesimal

deformation of a thin sheet of isotropic elastic material which is first subjected to a finite pure homogeneous deformation by forces in its plane. In order to simplify the analysis, the discussion is limited to incompressible materials and to two types of superposed infinitesimal deformation.

In the first case (?8), the homogeneously deformed sheet is considered to be

clamped on its periphery and to be bent by small forces acting normally to its major surface. A differential equation (8-23) is obtained for the deflexion of the sheet. As an example, this is solved completely for the case when the sheet is circular and is subjected to a pure homogeneous deformation having equal extension ratios in the plane of the sheet and the bending forces are uniformly distributed over its

major surface. In the second case (?9), the homogeneously deformed sheet is considered to be

subjected to infinitesimal generalized plane stress, and it is shown how complex variable techniques can be used to solve problems of this type.

2. NOTATION AND FORMULAE FOR FINITE DEFORMATIONS

We shall adopt the notations which have been used by Green & Zerna (I950) and Green & Shield (1950, I95I), and we briefly summarize the relevant formulae here.

Vol. 2II. A. 9



130 A. E. Green, R. S. Rivlin and R. T. Shield

The points PO of an unstrained and unstressed elastic body Bo at rest at time t = 0 are defined by a system of rectangular Cartesian co-ordinates xi or by a general curvilinear system of co-ordinates 0/, and the line element ds0 of Bo is given by

ds8 = dxi dxi = dOi dok. (2-1)

The usual summation convention is used and Latin indices take the values 1, 2, 3. An index which is repeated more than twice is not summed.

The body Bo is now strained or deformed, so that at time t the points P0 have moved to new positions P to form a strained body B. The curvilinear co-ordinates

Oi in Bo, which move with the body as it is deformed, form a curvilinear system in B so that the line element ds in B, for a given time t, is given by

ds2= Gikdi dok. (2*2)

The position vector of P relative to P0 is the displacement vector v(01, 02, 08, t), and Ei, Ei are the covariant and contravariant base vectors at P of the curvilinear co-ordinate system 0i. If r(01, 02, 03) is the position vector of P0, referred to the origin of the xi co-ordinate system, then

Ei = r, v,, Gik = E.' Ek, (2-3)

where a comma denotes partial differentiation with respect to 0i, and if Gik is the contravariant metric tensor of B,

GirGrk = , Ei = GikEk. (2.4)

One form of the stress equations of motion is

Tikll-+pFk = pfk, (2.5)

where Tik is the contravariant stress tensor, Fk andfk are the contravariant body force and acceleration tensors referred to the base vectors Ek and p is the density of the strained body B. The double line denotes covariant differentiation with

respect to the medium B, that is with respect to Oi and the metric tensor components Gik, Gik. For covariant differentiation we need the Christoffel symbols of B which are given by (

ikg = lGr'si, k + Gsk,i s) (2-6)

When the body B0 is homogeneous and isotropic the strain-energy function W, measured per unit volume of the unstrained body B0, is a function of the strain invariants, so that = W( 2 3), (2 7)

where 11 = grsr,,s = gr S, = G/g, G i G ik, g = gik . (2-8)

In this case the stress-strain relations take the form

Tik = (gik + TBik + pGik, (2.9)

2 3W 2 3W W 2 where ( vI i3

T 4D' p = 28 I I', (2-10)

Bik = I gik _9irgskG( =r eirmeksnrs Gn, (2-11) = - ~~ rs -ekngr (' 1



Small deformations superposed on finite deformations

with erst, as usual, equal to + 1 or -1 according as r, s, t is an even or odd

permutation of 1, 2, 3, and equal to 0 otherwise. When the material is incompressible, the incompressibility condition G = g, or

I3 = 1, holds at all points of the body, and the strain-energy W is a function of

I1 and I2 only. The stress-strain relation (2.9) is still valid, but in this case

= 2 a = 2 a

and p is a scalar invariant function of the co-ordinates Oi for each value of the time t. When surface forces are prescribed at a boundary,

Tikni = pk (2.12)

where P = PkEk, n =nEi, (2.13)

P being the surface force vector and n the unit outward normal vector to the surface.

3. GENERAL THEORY

We now consider a deformation of the body Bo which is such that the state of strain and stress at any time differs only slightly from the state in a known finite deformation of the body B0. We suppose that the points PO of the unstrained body B0 are displaced to P' and form, at time t, a strained body B' which may be obtained

by an infinitesimal displacement of the points P of a strained body B, where the deformation of the body Bo to the body B is assumed to be completely determined.

The state of the body B is described as in ? 2, so that the displacement vector PoP'

may be written v (01, 2, ,03, t) = v(01, 0, 03, t) + w(0l, 02, 03, t), (3-1)

where v is the displacement vector PoP and e is a constant which is small enough to allow us to neglect the second and higher powers of e compared with e.

The covariant base vectors of the co-ordinate system 0i at points P' of the body B' are denoted by Ei + seE = r,i + v,i + ew,i, so that

E* = w,i. (3-2)

The displacement vector w may be expressed in terms of components in various

ways, the most convenient for our purpose being

W - wmEm = wmEm, (3-3)

so that wm, wm are the components of w referred to base vectors at points P of the

body B. Hence E = WmJiEm =- wmiEm, (3-4)

where, as before, the double line denotes covariant differentiation with respect to the medium B.

The covariant metric tensor of the body B', evaluated at a given time t, is

ik + eGtk = (Ei + CE'). (Ek + eEk),

so that, to the first order in e,

Gk = Ei. E;+Ek E' = wi1 k+Wk li (3I5) 9-2

131




if we use (3 4). The contravariant metric tensor of B' is Gik + eG'ik, and since we have (Gik + eG%k) (Gkm + eG'km) =

m t

it follows that kGkm + ik G'km = 0

Alternatively, if we solve for G'ik we obtain

G'ik = GirGskG'. (3.6)

The contravariant base vectors at P' in the body B' are

Ei + E'i = (Gik + eG'ik) (Ek + eE1),

and therefore E'i = GikE + GikEk. (3-7)

If the determinant of the metric tensor components Gik + eGk is denoted by G + eG', then

| (k(Gik+eGk)| = | S+eGkrG = | Gkr I(G+eG') = 1 + ,

and therefore, to the first order in e,

G' = GGikG'. (3-8)

The strain invariants associated with the body B' are I + 6eI, I2 + eI, 3 + eIj, where, from (2.8), + s(Gs+

I1 + el = grs(GrS + s eG,S) (3 (39) 12+612 e rs(Grs Grs)(3+e), (3.9)

I + e3= (G + eG')/g,

and, to the first order in e, this gives - grsG!j,

2 + grs(GtrsI3 + Grs), (3.10) I3 G'lg I3Gik't.

For homogeneous isotropic materials the strain-energy function W for the body B has the form (2-7), and for the strained body B' the strain-energy becomes

W(Ih + eI1, I2 + eI2, I3 +I,13).

The scalar invariants d , T, p which are defined by (2.10) and which, for the body B, are functions of I,, I2, 13 become functions of I1 + el e, I2+ eI', 13+ e1' for the strained body B' and may be denoted by qD + e', T + eTp', p+ ep'. By Taylor's expansion, to the first order in e, we obtain

= a() a(i ,D D - +I 2 +I3a

1ai 2a 3 3t-

p' = + I' + ' -aI1 jI2 38J




and, using (2-10), these may be written

' = AI + FI2 + E3I-2I I3,

W 2 32 W ~21 3

hr= FIA +BI=+DI-' =I3S, (3.11)

2 a2W 2 a2W 2 a2W where A= B - C

43 ai2 a3 3 yI3aI1 vi3 ai1a2} 2 W 2 2W 2 2W

(312)

The strain-energy function W which appears in (3-12) is given by (2.7) and depends only on I1, I2, I3. The quantities A, B, ..., F are therefore invariants which depend only on 1l, I2, 13.

For an incompressible material, the strain-energy is a function of 1 and 12 only and also I = 0. The invariants A, B and F can still be found from equations (3.12) by writing I3 = 1, and from (3*11) we have

(' = AI+ FI, T' = FI +BIB, (3.13)

but p' cannot be found from the strain-energy function and is a scalar invariant function of the co-ordinates for each value of the time t.

The tensor Bik in (2.11) becomes Bi+ eB'ik for the strained body B', where

Blik = (ikggrs - girgsk) Gs = eirm eksng rsGC g (3-14)

Similarly, from (2 6), the Christoffel symbols for B' are

r( '

= 2 (rs+e k + sk, i{ - + Gik,s + C(si, k +Gsk, i ik,s)}

and therefore, to the first order in e,

) = \ Grs(Gsi, k + Gsk, i- Gik,s) + 20" (Gsi, k Gk, i Gk,s) (3-15)

The symmetrical stress tensor for the strained body B' is rik + er'ik, referred to the curvilinear co-ordinates Oi in B', where, from (2-9),

r'ik = ('gik + tI'Bik + ITB'ik +p'Gik +pG'ik. (3-16)

If the contravariant components of the body force and acceleration vectors for B' are respectively Fk + eFk,fk + ef'k referred to the base vectors Ek + 6EE, and if

p + ep' is the density of B', then the equations of motion for B', which correspond to the equations (2-5) for B, are

Irik . + e7,10 + k({~} + { }') (Tr+ + 6?'ir) + ( + { (Tik + eT'ik) Tk, i +i , + r) +irir i)(r+e')+(Fr) (fIr)

+ (p + ep') (Fk + eF'k) = (p + ep') (fk + efk).

133




Hence, using equations (2-5), we have to the first order in e,

r 'ik i+ ir} T+ rTik+pF'k +p'Fk = pf,k +p'fk. (3-17)

The double line still represents covariant differentiation with respect to the medium B. Also, as mass elements are conserved, we have

(p + ep') /(G+eG') = p 4G, GI

so that P' =-P 2G (3.18)

When surface forces are applied at the boundary surface of B', the boundary condition (2-12) becomes

(Tik + T'ik) (n + en') = Pk + ePk, (3.19)

where ni + en* are the covariant components of the unit normal to the surface of B', referred to the base vectors Ei + eE'i, and where pk + P'k are the contravariant

components of the surface force vector. The surface condition (3-19) may be reduced to the first order in e, but it is generally more convenient to leave the condition in the form (3-19).

4. SMALL DEFORMATION SUPERPOSED ON FINITE PURE HOMOGENEOUS

DEFORMATION: RECTANGULAR CARTESIAN CO-ORDINATES

We now assume that the body Bo is deformed into the body B by uniform finite extensions along three perpendicular directions, and that B is maintained in

equilibrium without the aid of body forces. The strained body B' is then obtained from the body B by superposing a small deformation of the type described in ? 3. We take our moving co-ordinates Oi to coincide with a fixed rectangular Cartesian

system of co-ordinates (x, y, z) in the strained body B, the axes being directed along the three perpendicular directions. Thus we have

01 = X, 02 , 03=. (4-1)

If the rectangular axes xi which define the points P0 of the unstrained body Bo are taken to coincide with the axes (x, y, z), then

x y z x i=y z 2 X' 3 -z ' (4.2)

where A1, A2, A3 are the constant extension ratios. It follows from ? 2 that

0 0\

gik= 0| A ? = 0 As

?) -

A2A2A2a (4-3 and Ik= itc= ik,G 1,Em= E }= (43)

ad20, 0, 2 3

and Gik= Gik = 8ik, G=1 EM = E = 0. (4-4)




Also, from (2-8), 7 ;t2_t_ 2 \2 2 122 22 IA- +A +A , z~=+ A+i2+ , 2t - A= A A, (4.5)

and from (2.10) we see that D, TI and p are independent of x,y,z and, from (2 11),

BI _=A _(A2+A2 , tB22=A (A + ), = A3(A1 + A2),} B12 = B23 = B31 = 0.

(

The stress components rk for the body B are given by (2.9) and are 11

Af + T*2A2A2 +A) +p

T22 = (D/A2 + Ap A2(A2+At) +p (4 T33 EA2 + TA2(A2 A2)+1 (+ 2

712 - 7T23 -- 31 = 0,

so that the stress equations of equilibrium are satisfied when the body forces are zero.

From (4-4) and (3.3) we see that wm = wm, and from (3-5), (3-6) and (4-4),

Glik aW a.Wk (4.8)

Equations (3-10) now reduce to

/ au av aw\ 1 = 2 (A2-+A2+A2 a ax 2ay 3 az

am av w 2 = 2A(A + A3) 2A,32 + 2A) 2A(A + A,2) a, (4.9)

!~ 2 2?3 a2 ay az I3 =

2A!AiA|^+^+

where we have written w, = , w2 v, W3 W. (4*10)

The coefficients A, B, ..., F in (3.12) are constants, and the tensor B'ik, which is

given by (3.14), has components

3ll 2) a A2v aw\ / aW au) B'33 2A (Au a+ ava .,11 = A+2)1 B22 2+A , 3 -- 3'.3 ax 2ay 2='

= 12 (t2 2t ) 3 =-u2 : 2 (~~a ) :t ~,U 2' 2 +ava B131_A2A t2

am\ 1 - -Ta- aX 2 3 'a ) 3 1 ax +a}I

(4.11) and substitution in (3-16) from (3-11), (4-3) to (4.11). shows that the components of the tensor r'ik are given by

aum av aw ' T Cll =

X + C12 y + C13

22 a av aw (412) '33 =C 21 - - + C22FY

+ (4a 12) au av aw:

T C31 aX + C32 y + 33 a~ ,

135




23 - C44 +

T'31_c ax )

Ttlg. __ C66 + N, &)t ,v

(4.13)

where c= -r11 + 2AA4 + 2BA4(A2 + A2)2 + 2CA4 A4 A4

+ 4DA41A2A(A2 + A2) + 4EA4A2|A + 4FA (A2 + A2),

c 2 = - A2 + TA2(A2 - A ) +p + 2A A2A2 + 2BA A2(A2 + A) (A + A2) (4-14)

+ 2CA4A4A + 2DA2A2 AA(A2 A2 + A2 A2+ 2A2A|2) 2 3 - 1 /2 z3 2 33 1 2

2

+ 2EA2 A2A2(A2 + A2) + 2FA2A2(A2l + A2 + 2A),

c22, 33 being obtained from cl by cyclic permutation of A1, A2, A3 and 11, T22, T33; and c23, C31 being obtained from c12 by cyclic permutation of A1, A2, A3. Also

C21- C12 = T11 - T22

C32 - 23 22 - 33, (4-15)

C13- 31 =T33-- 11,

Cs = -TA2 A - p,

c =- TA1A2- p.

When the material is incompressible a 1 and 1' = 0, so that we have

A2A23 au av aw A2a|= , ++- = o, ax ay 8z

(4.16)

(4.17)

and the stress-tensor components T'ik are obtained, from (3-16), (3.13) and (4-3) t+n /4.-1 1 in ,the form

a u v aw T'11 = p' + a11- + a12 -y ala

T'22 = au av aw +' = P+'al2 +a22 y+a23Y i

T3-p ax av aw ,33 _ p u Sv aw

r'a = p' a13 - + 2 y+ as33 ax ay az,

(4-18)

= - 2 + 2A1{A + B(A2 + A2)2 + 2F(A2 + A2)},

a22 = - 2p + 2A{A + B(A + A2)2 + 2F(A + A2)},

a33 = -2p + 2A4{A + B(A2 + A)2 (A + 2A) + ) },

a12 = 2TA2 A + 2A A2{A + ( + ( A) (A2 + A2) + F(A2 + A2 + 2A)}, a23 = 2A A2 + 2A 2A2{A +B(A + A2) (A2 + A2) +F(A2 + A + 2A)},

a13 = 2TA A2 + 2A A{A + B(A2 + A2) (A2 + A2) + (A + A2 + 2A)}

The remaining stress components are still given by (4-13) and (4-16).

where

(4.19)

v% \ r LI)1,, VI%V J VJ" .L




From equation (3-15) we have

ik 2(Gri',k+Grk,'i -

Gik,r) =aoa (42o0)

and since we have already assumed that Fk = 0, fk = 0, the equations of motion

(3-17) reduce to ar' + a ar T__, Tfir

? + T7ik - wr= pf 'k, - aoiaorik aoiaor

if we also take F'k = 0. This equation may be written in the form

a lik q

Tir aWk awj aio(rTk wrk+rk2E = pf'k (4.21)

3[ + r0 aO~! - Pf,k,

if we interchange the dummy suffices in the last term on the left of the equation and remember that the components rik are constants.

For some purposes it is more convenient to have the stress components in the strained body B' referred to rectangular Cartesian co-ordinates yi which coincide with the axes (x, y, z). Thus = + , (422)

.Vi = 0i+ewe, (4.22) and we shall denote the components of the stress tensor referred to the yi-axes by trs +et'r, these being related to the stress components 7ik +T'ik by the tensor transformation

trs + et'rs = ayr ay ( + 6T'mn)

= (rm +0') ((s ? n)'ws (Tmn 6T'mn) (4*23)

Thus, to the first order in e,

trs = Trs '

t'rs =- T'rs +ms +Trm s (424) a--m -Tm

and we see that the equations of motion take the alternative form

ft'rs _o = pf'r, (4.25)

where, sincefr = 0, ef'r are also the acceleration components along the yi-axes. This result may also be obtained by observing that the equations of motion of the body B', referred to the yi-axes, are

Y (trs + etrs) = epf 'r

and, using (4-22) and remembering that the stresses trs are constant, this reduces to (4-25) when we retain terms up to the order e.

We now examine conditions which must be satisfied at a boundary surface of B' at which surface forces are prescribed. Suppose such a surface is given in the

parametric form - K I , _ ,, W _ ta .oa9 1 k-'1, 2 2, (73)

--= , y ZJ- =tJ k% ZJU



A. E. Green, R. S. Rivlin and R. T. Shield

and let n be the unit normal to this surface so that

n = (%. + enr) (Er + eE'r).

Referred to the yi-axes, the unit normal to the surface (4-26) has direction cosines

proportional to F/layi. A simple tensor transformation shows that the covariant

components nr + enr of n, referred to the base vectors Er + eE'r, are

/ 8F ay. a F nr en

=ayi k

Or yr,

where k is a scalar. Since n is a unit vector it follows that

k{ (Grs +G'rs) =l,

or, to the first order in e, aEF aF asWr aw Ai

-1 I .- e I 6 8 W r _aI aF\ oj aFos a J

(4.27)

(4.28)

(4-29)

The contravariant components of the external force at the surface (4.26) of B', referred to Ek + eEl, are therefore

pk + ep'k = (zik + 6T'ik) (ni + en)

(4-30) aF = k(7ik q_ie7ki) ii.

Alternatively, we can refer the components of surface traction to the yi-axes. If these components are denoted by Qi + eQ'i, then

Qi + eQ'i = (pk + ep,k) ay aok

and therefore

= k(Trk ?rC'rk ) (3 + awt OF

Q -T fS / a[ J '

aF aW(aFaW, aWs\ ~Q, _ a( r .-k awi ?--Qi ar0 aO86s 8aO aOr Q-

{ax aT k2 az aWa F W

a- F aF aFw a,S ' mam r+ ao-

The theory has now been taken to a point from which applications may be made to special problems. For example, we may now examine the state of stress in a cylinder of constant cross-section which is extended finitely by a uniform force

parallel to its length and which then receives a small twist about an axis parallel to its length. This problem has, however, been solved previously (Green & Shield I95I)

(4.31)

(4.32)

138




without the use of the present general theory and will not be discussed again here. In the next sections we shall consider other special cases of the previous theory.

5. SOLUTION IN TERMS OF POTENTIAL FUNCTIONS: COMPRESSIBLE CASE

We now consider a special case of the previous section in which the finite extensions are maintained by equal forces in the x1 and x2 directions, so that

A==A2==A, A3 = u, (5-1)

and, from (4-7), 7rl = T22 = ()A2 + A(A2+2 + p2) p, (52) T33 = ()D2 + 2TA212 +qp.

From (4-12) to (4.16) and (5-1) we have

au av aw' T 1- Cjllw+Cl2ay+C'l3TZ

T22 = au av w5y+13) r'll: cX + clla +C13 (Z

T133 =C + \+ C w

a, av aw T+22 = 012 X- ?11 --13 (5'3)

T, =(11 C12)(ay1 aV) '

T'j.tt _ /? i _ a[ _

lawx az

~,2 = c44 + , ay (5.4)

7'13 = C44 q8-Z)

where the coefficients cl, c12, ... now take the values

cin = - 11 + 2A4{A + B(A2 + a2)2 + CA4C4 + 2DA2g2(A2 + 2) + 2EA212

+ 2F(A2 +2)},

+ 2A4{A + B(A2 + a2)2 + CA4aut + 2DA21a2(A2 + C2) + 2EA2a12

+ 2F(A2 +U2)},

C13 = - DA2 + TA2(i2 - A2) +p

+ 2A242{A 2B (2 + 2A2)(A2 + ) + EA(A + 32)+ A2

+ F(3A2 +2)},

C33 = -T33 + 2#t4{A + 4BA4 + CA8 + 4DA6 + 2EA4 + 4FA2},

C31 - C13 = (A2 -#2) (0 + IA2) = T1 _ T33,

C1- C12 = - 2(TA4 + p),

C44 = - (FA21t2 + p).

Also, from (4-24), tll 711, t22 711, t33 = 733, t12 t23 = t31 = 0, (5.6)




t= 7.+11 27. 1 '12 = T12 +Tl - 11 -r r'1 + 2+r1 - , T + ax, x

av aw av tt22 = Tr22 + 2r11- t'23 = 723 + + 33 (7) ay' ay az'

t33 = T733 + 2T33 tW _ r'31 + 1 +T 33

and, with the help of (5-3) and (5.4), equations (5-7) become

au av aw ' t'll = d1 + d1 + d3 ,

3uc -v 3w t'22= d2 + dll +l8 (5+8)

t 12 = (d a- y + a

t3 d u + d33v\ ~z'

aw

3v aw t'23 = d44d +d55y, (5.9)

au aw t'31 = d44 a +d55

where dll = cl + 2T11, d33 = C33 + 2T33,

d12=c12, d131 =3, d31= C31,

dil - d12 = c, - c, 2Tl1 = (- + 2A2 + 2), (10)

d44 = c44 + T33 = -2(D + TA2),

d55 44 + Tll = A2((D+ - 2),

d55- d44 = 3 - C13 = d3 - d3 = 711 -_ 33.

The equations (5-3), (5.4) and (5.8), (5.9) are in some respects similar to the

corresponding equations in infinitesimal elasticity theory when the material is

hexagonally aeolotropic. Stress systems in hexagonal aeolotropic materials have been studied by Elliott (I948, I949) and Shield (I95I), and a similar method of solution may be followed here except for certain differences which arise in the

boundary conditions. We assume that the displacements may be expressed in the form

a=- avy=- w =kaz (5.11) 9x 3y v z ' w=, where k is a constant. The stresses t'rs may then be found from (5-8) and (5.9) and, if the strained body is in equilibrium, the stress equations (4-25) with f'r = 0 are satisfied if

sd11 +a + 2 + Kd44 + k(d13 + d55)} = -

___+5 __a _ +d3 a24 =?(5.12) (d31 +d44+ kd55) (2+

+ kd33- =0.

140




A suitable three-dimensio: al non-zero solution of these equations can be found if the equations are identical, and this occurs if

k(dl3 + d55) + d44 kd33 = V. (5-13) dl, =cd55+ + dl - d44

This gives a quadratic equation

d11d55 v2 + {(d13 + d5)2 - l d33 - 44d55} + d33d44 = (5.14)

for v with roots v1 and v2. The possible functions 0 are then solutions of the equations

a2 32

2 + v 2 0= (CC=1,2), (5.15)

where V2y = a 2- (5-16)

and the corresponding values of k are denoted by k1 and k2. In its present form the theory is valid for a body which is homogeneous and

isotropic in its unstrained state, and we may proceed to write down the appropriate values of the stresses and displacements in terms of the functions <0,. We shall not, however, continue this general discussion here but will consider the special case of an incompressible body in more detail in the next section.

6. SOLUTION IN TERMS OF POTENTIAL FUNCTIONS: INCOMPRESSIBLE CASE

When the material is incompressible and in the case of the problem of ?5, we have, from (4.17),

au 3v aw A2= 1, +-; +=z =' (6.1)

and the scalar invariant p is found in terms of the given stresses r11, T33 from (5.2). Also, from (4.18), (4.19) and (6-1), we find that

T'2 , au av ( T PL= +a +P

ax 8y au 3v L =T P +>22 =Pt,

x e

(6'2)

aw T'33 =

p'+7, where

a = -2A2,a2 - 2p + 2A2(A2 - 2) {A + BA2(A2 + j/2) + F(2A2 +2)1, = 2A2(A2 -t2) {t + A + BA2(A2 +f2) + F(2A2 + /2)}, (6.3)

y = - 2TA2A2 - 2p + 2/t2(/2 - A2) {A + 2BA4 + 3FA2}.

The remaining components r'12, T'23, T'31 are still given by (5.4), and we note that

11i--C12 -fl.

141

(6-4)




From (5.7) and (6.2) we obtain a3u a3v '

t'll = p/ +a+b av-, ax ay'

t'22 = p' + bx +au a , (6.5)

az where a= + 2T11, b =, c = y + 2r33, (6'6)

the components t'12, t'23, t'31 being given by (5-9) and

d11-d2 = a-b, d44- =/2(( + TA2), d55-d44 - T11-33. (6-7)

The method of solution is now similar to that used for the general compressible material. Assuming the displacements to be of the form (5.11), we have, because of the incompressibility condition (6-1),

V2l0+ka 0= 0. (6.8)

The stress components t'r may be found from (5-9) and (6.5), and it is found that the equations of equilibrium are satisfied if

{p' +(d(44 +d55 -a) a 0,

{' -) = 0.

These equations have been derived with the help of (6.8), and a suitable three- dimensional non-zero solution can be found if

d + kd55 - ka = (c - d44-kd55) k,

or k2d55 + k(d44+ d55 - a - c) + d44 = . (6.9)

If kl, k2 are the roots of (6-9) then possible functions b are solutions of the equations

(v~ + L,/ 0 = 0 (c - 1,2). (6.10)

The displacements (5.11) are

a .a v +-2 wa = k + + a, (6. 1) U x ax ay ay W az az ( 1 and also we have

p' = (kla-k d55 -d44) a + (k2a-k 255-d4)

___2 ax2 \ (6-12) -l-kl(k d55-c + d44) a + 2(2d55 c + d44)

a22 J aZ + akd Z2 d4 ~




apart from a constant which represents a hydrostatic pressure. The stress com-

ponents T'rs can now be found in terms of the functions 01, 02 by substituting in

(5-4) and (6-2) from (6-10), (6-11) and (6-12), and we have

a'll = (d551 + V d4 + v k- 2 202 21 2(b-a) T'22 = (d55 + ) + (d ) 102-211 (a + )(b-a) a2 a2

7/22

d44

V2 + dU V2 1

2 ~2

T'33 kl(kld55+d44-- 2T33) a2+ k2 (kid55 + d44-2T3) a202

r'12 = ( - f (2x l a+ ) ax ay + ax

r23 = C44 (1 +2 kl +I (1 +c) a2

r'31 a0(l+ ^+2+l ) T = C44 ( + kl) az ax + (1 + 2) az ax (6-13)

When the strain-energy function has the particular form

W = C1(I -3) + C2(2-3), (6.14)

where C1, C2 are constants, suggested by Mooney (1940) for rubber-like materials, we have we have a = 4A2(C1+C2 A2), c = 41U2(C1 +C2),

(6.15) d44 = 2a2(C1 C2A2), d5 = 2A2(C1 +2A2),

and the quadratic equation (6-9) for k has the roots

k,= 1, k2 = 6 =u 3. (6-16)

7. PUNCH PROBLEMS

The theory of the preceding section will now be restricted to cases for which T33 is zero. From (5.2), since the material is incompressible, this implies that the scalar function p is given by 2A2}. (7

p = -#2{D + 2TFA2}. (7.1)

We suppose first that the strained body B, which is obtained from the unstrained

body Bo by the finite deformation of ? 5, occupies the semi-infinite region z > 0, and the body B' is obtained from B by the small indentation of the plane surface by a rigid punch. The plane boundary is the surface

F(x, y, z) _ z = 0,

and we have aF/3aO = 8m. It follows from (4-32) that on this surface

Qi = Ti =

0 }

\Qi_z3t ! T(7.2) Qti =_ T3i




If we assume that the surfaces of the punch and body B' are ideally smooth so that the frictional forces are zero, and if the remaining part of the boundary z = 0 of B' is free from applied force, then along the whole boundary z = 0

T'23 T'31 = 0,

and we see from (7-2) that this implies Q1 = Q'2 = 0. With (6.13), these conditions will be satisfied if

(l+ kl) '- +(1 +k2) = 0 ( = 0).

This condition will be satisfied if we put

v= lx(xvYl /z \V2 l?zk\(xr,Vk) 01 I+k,X Z Y?i02 ? Y (7.3) 51I = 1l+ ]k x, , T' ^)

2

=

l

+

X' x,

where V2X(x, , yz)= O.

Similar expressions have been used by Shield (195I) to solve problems involving hexagonal aeolotropic materials.

If we restrict our attention to rigid punches, which have an axis of symmetry in the direction of the z-axis and which are moved parallel to this direction, then the

remaining boundary conditions on z = 0 require

aX w = or (x, y, 0) = f(r) (r2 = 2 + y2 < r2)

a 2 3 \ ( 7 - 4 ) 3 K (X,y, 0) = 0 (r2 = x2+y2>r2),

kl k2 where cr = - - where r=1 l+kl l+k2 (75)

K (kld55 + d44) kl _ (k2d55 + d44) V2 1 + ki 1 + k2

and where r0 is the radius of the circle of contact between the punch and the surface z - 0 in the strained body B'. The known function ef(r) defines the shape of the

punch. The solution of the problem may now be completed by using Hankel transforms as has been done by Harding & Sneddon (I945) and Elliott (i949) for similar problems in classical elasticity for isotropic and hexagonal aeolotropic bodies respectively. Alternatively, we can use the method of Green (i949) which was based on a simple solution of the electrified disk problem given by Copson ( 947). We will not examine the problem in detail here but will restrict our attention to the distribution of stress under the punch. For this we need the value of a2X/az2 at z = 0 which is given by (see Green 1949)

a2X 4d rTo ts(t)dt L aZ2, y, r)=; drJ r (7 r 6)

1 d rfo tf(t) ( S(r) - 2idr cf(r-t dt

iff(r) is continuously differentiable in O < r < ro.

144




As an example we consider the small indentation by a rigid sphere of radius R so that f(r) = b + 1-r2/r2, (7-7)

where b is a constant to be determined. The actual displacement of the surface under the punch is = = eb ? 6e(l-r2/r2), (7-8)

and hence, approximately, r2 = 2Re. (7-9) a2X From (7-6) and (7-7) a simple calculation gives the value of - (x, y, 0) for r < rO, and,

if we remember that the stress must be finite at r = rt, we have b = 1 and

a2x 8(r2 -r2) r >Z2a (x, y,) = r2) (710)

The total pressure N exerted by the punch is

16eKr _ 8K r_ 8K

3od 3o' R 3cr where d = 2e = r2/R is the maximum depth of penetration by the punch.

The substitution of (6.15), (6.16) and (7.5) in this expression shows that for the

special case of the Mooney material the total pressure is

N 16(C + CA2) (A9 + A6 + 3A3-1)R (7 12) 3A4(A3 + 1)

If we put A = 1 in this expression and remember that 6(C1 + C2) is then the value of Young's modulus E for this material for small strains, we find that

N = EdWRB, (7.13) 9

which agrees with the classical value for an incompressible material in which Poisson's ratio is 0.5 (see Harding & Sneddon I945).

The value of N given by (7.12) is zero when A is approximately 3 and is negative for values of A less than 3. This result seems to indicate that, when a body is bounded

by a plane surface and is acted on by an all-round compressive force in planes parallel to the bounding surface, the equilibrium becomes unstable at certain critical values of the compressive force.

8. SMALL BENDING OF STRETCHED PLATES

In this and the next section we develop approximate theories for the small

bending and stretching of a plate which has first been extended finitely by uniform stresses .11,T 22 in its plane. Although the theory could be developed for a com-

pressible body we shall restrict our attention to incompressible bodies in order to avoid undue complications in the analysis. Using the notation of ?4, we imagine that the unstrained body Bo is bounded by the parallel planes x3 = ? h, and that it is first extended finitely by uniform forces in the xl,x2 directions only. So, putting T33 = 0 in (4-7) and eliminating p,

11 = (A2-Aj2)((+-A22), T22 (A2-A) (( +Ai),)

p =-A {( + (A2 + A2)}.

145

Vol. 2ii. A. IO




We repeat here the incompressibility conditions (4.17),

au av aw 1A2 A3=, ?-X ++ =- 0. (8.2)

The stress components T'ik are given by (4-13), (4-16), (4-18) and (4-19), the value of p in (4.16) and (4.19) being given by (8-1), and if we eliminate p' and use (8-2) we find that

r'll-T'33 = 2{2Aj(31+TA2)+ (A|2-Aj)2 (A + 2FA2+BA4)}- ax

+, 2{(A + A2 A2 + (A21- A3) (A2 - A2) [A + F(A2 + A+ A+ BA 12]} a '22_'33 = 2{)A32+NA2A+ (Ai-A2) (A2-A2) [A ?+(A1 +Ak2) +BA2A]} -

+ 2{2A2(0 + TA2) + (2 - A32)2 (A + 2FA1 + BA4)} ,

(8.3)

T'12 = - (TA12 2 p) (a +

=T23 - A2( TA+2)(a + ay) (8.4)

T31= A2(DT + a. 7'r A 21)ax a( z

Remembering that 11, T22 are the onlynon-zero components of 7rs, we find from (4.24)

tll 11, t22 = T22, t'S = 0 otherwise,

t,ll = T11 + 2T11, t13 T13 + T118W ax ax' V aw (8'5)

t22 = T22 +2T22 a t'23 = '23 + T22 (

av au axy ay' t'll2 = t+T~,, t'33 = T'r3,

au av so that t'1 -l'33 = + cc2l

au av 22 --t'33 = 3C- + 22 (8.6)

t2 = C66 (2 'a + I a,

where cn = 2(A + A) (D + TA) + 2(A2-A2)2 (A + 2FA2 + BA4) c = 2((D+A32)(T + (2) + 2( - A A) F(A + 2A2+A2)+BAA}, C12 + ( 12-1 3 2 3 2 )+ 2

c22 = 2(A2 + A2) (( + TA2) + 2(A-2 - (A + 2FA2 + BA4),

C66= +TA3 A2,




these coefficients being different from those defined in previous sections. It is convenient to repeat here the equations

at'll at'12 at'13 aX + = o , ax ay az

at'12 at'22 at'23 a+ - a + a = (8.8) ax y +z

at'13 at'23 at'33 + - + -a = , ax 8y az which must be satisfied if the final strained body B' is in equilibrium.

We now imagine that after the finite extension of the plate by stresses Tl1, r22, the

plate is bent by small transverse forces which are antisymmetrical about the plane z = O. The faces of the deformed plate are the surfaces z = + h, where

h = A3ho = ho/AlA2,

and if we impose the boundary conditions

13 = T'23 = 0, T33 = t'33 = + q (z = h),

where q is a function of x and y, we see, from (4.32), that

Q = Q2=Q3 = Q = Q'2 0, Q'3 = Tq (z = +h). (8-9)

Stress resultants and stress couples may be defined in a number of ways. We consider the stress couple about the y direction defined by

G1 + eG1 (tll + et'1) yddy 3 (8-10)

where the integration is through the thickness of the deformed plate, taken with

respect to y3 keeping y, and Y2 constant. Since t11 is constant,

ty3 dy3 = Tll[y2],

and, from (4.22), Y = z + ew(x, y, z),


S t1ldy33 T= -1[zw(x, y, Z)]z-th.

Also, to the first order in e, the term

6f t'lly3dy3

can be evaluated by replacing y3 by z and integrating with respect to z from -h to h. Thus, finally, h

a, = 0, OG = mll[zw]+ t'llzdz, (8-11) J-h

where [zw] = hw(x, y, h) + hw(x, y, - h). IO-2

147




Similarly, we may define other stress couples G2 + eG', H + eH', and we find that

2 = 0, GI =T 22[zw]+' t'22zdz,

G Oclv1, G 2 = +

hT *iv 2 S~P+ CL,(8.12) H-O 0, H' = f t'12zdz.

J-h

We define stress resultants N1 + eNd, N2 + eN; by

N + eN~ = f(t13 + et'13) dy3,

N2 + eN2 = f(t23 + et'23) dy3,


h N,0h N' fh/(8.13)

N2=0, N = t'23dz. J-h

Other stress resultants will be defined in the next section when we consider the

stretching of the plate instead of bending. The first two equations of equilibrium in (8.8) are now multiplied by z and

integrated with respect to z from - h to h, and the last equation in (8.8) is integrated directly with respect to z from -h to h. Then, using (8-5), (8-11), (8-12), (8-13) and the boundary conditions (8-9), we find that

ax ay

a+ -2_N =0, (8-14)

aN + aN2 + -q = 0. ax ay These equations are of the same form as the classical equations for the transverse

bending of a plate, but the stress couples have different forms. If we eliminate

N', N' we have 2 , 2,' 2 1, IY 2 Wti 1i~rVC3~a2GI a2H' a2'G

- +2 -+- = 2 =. (8.15) ax2 axay ay2

From (8 6), (8.11) and (8.12) we have

G= t'33z dz + - ac + el + [zw],

h 2h3 a 0? =

j t'33zdz + c +c2 + T22z] (8- 16) 2

66 'aX,1 ?22 -a 22lzwiI 2h3 / l; W i\ 3 66 2 ay 1 Ax '

=3 rh 3 rh where we have put u = h-J uzdz, i = 2h3 vzdz. (8-17)

J-h A2 -h

148




The numerical factor -h3 in the definitions (8.17) has been chosen so that if u, v are

proportional to z then iu, v are equal to the constant of proportionality. We also write

-^ ( -2) wdz. (8-18)

This definition is such that if w is independent of z then w is equal to w. It follows from (8.4) and the definitions (8-17) and (8-18) that we have

3 (1 T'2 dz - A2(D + A2) (819)

where we have used the rule of integration by parts. We also have, from (8.18),

_ 3r / z3\ - +h 3 rh / Z3 aw 2hw-= z \-- - -- 3h2z a dz

= [Z]2 h Zw 3h2) + A z

and therefore

[zw] = W 2hw- Z 3h2) - )x dza

The integral is of the order A3 (;+a-), so that we may evaluate it approximately,

as far as terms of order h3 are concerned, by assuming that u and v are proportional to z. This gives 43 ,au M

[zw] = 2h-- +5 t ) (8.20)

If we substitute this value of [zw] into the expression for Gj in (8.16) and use

equations (8-19) to eliminate ui and v we obtain

2h33 aii362 i3v4- h3

+ft33zdz + Aj + (D + TF2 ) dzx

G~ = 2~n~- -~^- ^+c j cD + T2 aX-r

f~ ^d

2

fh (S11_.6T11)~T213 a(C_6T),r2/ Z)

As in the corresponding classical theory we now neglect the last two terms in this

expression for GC and obtain approximately

2M a2;li a2i6 4h11 2 G; = 2hr11' - 2h3( ,, w+4 V2 (8.21) 3 11 ax2+C12 5

In the same way we have approximately

= 2hr2 --- Cl22 + c22a +- Vw,

xH ' __ _2c + w(8-22) 9h3 a26w H'- ~l +)2 a+ j) x y




With these values (8-21), (8-22) of G', GQ and H', equation (8.15) can be written

4w 4W 84 a4 a2 a2w8 a -4-2b + + T- (8.23)

aax4 + bax 2y2 ay4 lax2+Tay2- (823)

where we have put 2h3 2h3

a =- T ((C11--6Tll) C -- (C22 6T22), 3 3

b 2h3 (8.24) b= - {C12 +- 66(A+ A2) -- 5(Tll +22)},

T1= 2hrl, T2 = 2hT22.

When the two finite extensions A1, A2 are equal (and therefore, by (8-2), both equal to A3-), then (8-23) takes the simple form

aV41w-TV2w +q = 0, (8-25)

where a =- (cl- T, =T- (A2-) (D+?TA2),

cl1 = 2(A+ 4( +A)+ 2(A-- (A) +2F B) (8.26)

T = T1 T = 2h -11,

and 1V 82 82 +

As an example we solve (8-25) for the case of a circular sheet, stretched finitely by a uniform all-round tension T in its plane so that it becomes a sheet of radius r0. The sheet is then clamped at its edge and is subject to a constant small transverse

pressure uniformly distributed over a face of the sheet. If the resultant transverse force is denoted by W then W

q - r' (8.27)

and, changing the sign of w so that it is measured positively in the direction of W,

W aV4-- TVw -W 2 (8-28)

7Trr

together with the boundary conditions

= 0, =0 (r=ro), (8-29)

where r= -/(x2+y2). The solution of (8.28) subject to the boundary conditions

(8*29) may.be found by elementary analysis. If we suppose that T>0 and put 2 = Tr2/a, then wT

= "(1 _p2) 2(n (8-30) w - ?(-p) 2nlJ(n)

where p = r/ro and I0(x), 1(x) are modified Bessel functions of the first kind. We notice that when T = 0, n = 0, and the limiting form of (8.30) gives

_ _W _ (8.31 ) 1= 2nE(r -r2)o , (8-31) 512,Trr2E




where E is Young's modulus for small strains, and this is the classical result for an

incompressible material. On the other hand, if we consider the limiting form of

(8-30) as n -> oo (which corresponds with h - 0) we have

(I_T p-1- 1(p2), (8-32)

and this is the result for the small transverse displacement of a uniformly loaded membrane which is stretched to tension T.

If T < 0 the corresponding result to (8-30) is easily expressed in terms of Bessel functions Jo(x), J,(x) of the first kind.

9. GENERALIZED PLANE STRESS SUPERPOSED ON FINITE STRETCHING

We now consider the deformation produced by small forces in the plane of the

plate superposed upon the finite extension of the plate described in ? 8, the faces of the plate being free from surface traction.

We define stress resultants by

T, + el = f(tl+ et'l) dy3,

T2 +eTa = (t22 + t'22)dy3, (91) 2 eT^ dY3+e^d, (9. 1)

S + eS' = f (t12 + et'12) dy3,

where the integration is through the thickness of the deformed plate keeping y, and Y2 constant. To our order of approximation we have

rh T + eT - tl[z + ew]z-h + t'll dz

J-h

h = 2hT11 + er1[w] + ef t11 dz,

-h

where [w] = w(x, y, h) - w(x, y, -h),

so that T1 = 2hTll, Tl = t'11dz+Tl[w]. (9-2) -h

In the same way we obtain

T2= 2hT22, T = f th2dz+T22[W]

S = 0, ' =n S t'12 dz.

? -h

Since the faces of the plate are free from stress,

7'13 = T23 = 0 (Z = + h),

151




and if we integrate the first two equations of equilibrium in (8-8) with respect to z from -h to h and use this condition we obtain

-x --+ - = O)

1 ax1 ___a2

4 (9-4) as' a a-7 + - == o. ax ay

The form of these equations is the same as the form of the classical generalized plane-stress equations.

h rh

We now write U = udz, V = vdz, (9.5) h -h

and if we integrate the incompressibility condition (8-2) with respect to z through the plate we get au av

[wI = ---- x (9-6) ax ay The stress component t'33 = T'33 is zero on the faces of the plate, and as in the

classical generalized plane stress theory we shall neglect t'33 dz. h-

With this assumption, equations (8-6), (9-2), (9-3) and (9-6) give

T1 -- (a 1- -T11 + (2 - T )y

au av ax + (22 ay'

= (::2-T :;:::+:( z 7-T) (9.7)

S C66 (A2+ay + ) ax

These equations can be rearranged to give

aU - T+ s T' x -11 1 12S1 2'

av a = s21T +s22T,> (9-8) ay

aU av A2aU+A2av =

2 ITX = S66S

C22 - T 22 C - TC where s8n = A 2 S

11 C1 2 22

912 A- '[21 (9 9) 512

= A '1"821= ( A

66 -, A (11 - T11) (C22 -

22) - (12 - T) (12 -T22) c66

The stress equations of equilibrium (9-4) will be satisfied if we put

aO- a, a azs= 0 T'= T2 S'= - (9.10) a - 2 2 -

ax ay-




where the stress function 0 is a function of x and y. If we also write

= x+iy, = x-iy,

then equations (9 10) are equivalent to

T +-T = 4 - T,-T2+2iS' = -4 . (9.11)

The elimination of U, V from the relations (9.8) gives us

a2 a2 a2S' A22 y2ll 1 +x2) + 812 2 X2 (s2l T s22 2)-s66 a-

= ,

that is, if we use the expressions (9-10) for T', T' and S',

A8a4 + A2 + As12 +so) + A211_ = 0. (9.12) 1S22 aX4 1+(A821 +A2 812+ 66) az2 ay2 2 = .y4

This equation is analagous to the corresponding equation for generalized plane stress in an orthotropic material, and we may employ complex variable techniques to solve a variety of problems. Here, however, we shall only indicate briefly the general method of solution, and in order to simplify the algebra we confine attention to a neo-Hookean material defined by Rivlin (I948a). The strain-energy function for this material has the simple form

W- E(I- 3),

where E is the Young's modulus of the material for small strains, so that we have

( =-E, T =A=B=F= 0.

We shall also assume that the stress component r22 is zero so that

A2 =A3 = A, (9-13)

and it is found that

_E( = (A-A) (-E -)_ 1

S = AA+ ' ES22 = 2A(+A), E6 = 3, (914) E12- A-2 3A2) 22 =

~,12 12 + hQ 6

tsl2 = 2A22(A2 + A2) EX21- A2 q-A2

The equation (9-12) satisfied by the stress function 0 becomes

2 a2\ / a2 a2 \

2(A ~-+A2 ) (1+ (A2- +3A2)a 24 ~-) = A2 AI? ay 2 y2) A1 2 ?2 aYA

? 2

-0,

and since q( is a real function we can write the solution in the form

0 = f(e) +f(i}) + g(2) +g(z), (9.15)

where {~ = x+ioay, =_ x + i2y, (9-16)




the real quantities a,, a2 being given by o A _A A (A2 3A ) A + 3)

= ~ A 2 = 2 2 (9.17) 2 2

The stress resultants corresponding to the value (9-15) of 0 are, from (9-10),

T ; - --a -- T = - a{f'(l) +f(L)})- a2g (~2)+ g (2)),

2=- f"() +f"(f,) +q"(2) ( (9.18) s = - ial,{f(1-"() () -)- i{"( -g"(C).

The displacement resultants U, V can now be determined in terms of the functions f(,),g(g2) from (9-8) and (9-18), and we have, after simplification,

~-E4U = - -f [f (+13-) {f (G2) + ( +2)} (9) ?-E = i [+ yF-^

-EV =- [+ {f '( ;f)-f'( ,)}+ 2

)- ) (1)

REFERENCES

Copson, E. T. I947 Proc. Edinb. Math. Soc. 8, 14. Elliott, H. A. 1948 Proc. Camb. Phil. Soc. 44, 522. Elliott, H. A. i949 Proc. Camb. Phil. Soc. 45, 621. Green, A. E. 1949 Proc. Camb. Phil. Soc. 45, 251. Green, A. E. & Shield, R. T. 1950 Proc. Roy. Soc. A, 202, 407. Green, A. E. & Shield, R. T. I951 Phil. Trans. A, 244, 47. Green, A. E. & Zerna, W. 1950 Phil. Mag. 41, 313. Harding, J. W. & Sneddon, I. . 1945 Proc. Camb. Phil. Soc. 41, 16. Mooney, M. 1940 J. Appl. Phys. 11, 582. Rivlin, R. S. 1948a Phil. Trans. A, 240, 459. Rivlin, R. S. x948b Phil. Trans. A, 240, 491. Rivlin, R. S. x948c Phil. Trans. A, 240, 509. Rivlin, R. S. i948d Phil. Trans. A, 241, 379. Rivlin, B. S. I949a Proc. Camb. Phil. Soc. 45, 485. Rivlin, R. S. 1949b Proc. Roy. Soc. A, 195, 463. Rivlin, . . S 1949C Phil. Trans. A, 242, 173. Shield, R. T. 1951 Proc. Camb. Phil. Soc. 47, 401.



general theory of small elastic deformations superposed on finite elastic deformations

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