Method of complex potentials in linear microstretch elasticity

International Journal of Engineering Science 44 (2006) 797–806 www.elsevier.com/locate/ijengsci

Method of complex potentials in linear microstretch elasticity D. Iesßan, L. Nappa

*

Department of Mathematics, ‘‘Al.I. Cuza’’ University, 700506 Iasßi, Romania Dipartimento di Costruzione e Metodi Matematici in Architettura, Universita` degli Studi di Napoli ‘‘Federico II’’, Via Monteoliveto 3, 80134 Napoli, Italy Received 29 June 2005; accepted 15 October 2005 Available online 1 August 2006

Abstract The paper is concerned with the plane strain of homogeneous and isotropic microstretch elastic bodies. We give a new representation of the solution in terms of complex potentials. The method is useful for the treatment of the constitutive equations established by Eringen in [A.C. Eringen, Microcontinuum Field Theories. I. Foundations and Solids, Springer-Verlag, New York, 1999], where the introduction of stress functions leads to difficulties. The complex variable technique is used to study Kirsch problem in the context of the theory presented in [A.C. Eringen, Microcontinuum Field Theories. I. Foundations and Solids, Springer-Verlag, New York, 1999]. Ó 2006 Elsevier Ltd. All rights reserved.

1. Introduction The theory of micromorphic continua was introduced by Eringen [1,2], in order to study materials whose microelements can deform independently from their centroidal motions. In the framework of micromorphic theory a material point is endowed with three deformable directors. When the directors are constrained to have only breathing-type microdeformations, then the body is a microstretch continuum [1,3]. The material points of the microstretch bodies can stretch and contract independently of their translations and rotations. The intended applications of the theory of microstretch continua are to composite fibrous materials, granular materials and porous bodies. The present article is concerned with the linear theory of microstretch elastic solids established in [1]. We study the plane strain problem in the equilibrium theory of homogeneous and isotropic bodies. The complex variable technique has been used in [4] to study the plane strain problem for a special class of microstretch elastic solids introduced in [3]. As in classical theory (see, e.g. [5–7]) the method of [4] rests on the representation of displacement, microrotation and microstretch function in terms of stress functions. In the present paper we give a different method which avoids the introduction of the stress functions. This method is useful in the treatment of the constitutive equations presented in [1], where the introduction of stress functions *

Corresponding author. Tel.: +39 081 2538039; fax: +39 081 5528838. E-mail address: [email protected] (L. Nappa).

0020-7225/$ - see front matter Ó 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijengsci.2005.10.010

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implies difficulties. In the theory of homogeneous and isotropic microstretch elastic solids established in [1] the stress tensor depends on the microrotation vector field and the microstretch function. The constitutive equations contain some terms which do not appear in [3]. The new terms have no contribution to the equations of Lame´ type and to the resultant stress vector and resultant moment. However these terms are important since they are involved in the boundary conditions. In Section 2 we present the field equations of the homogeneous and isotropic microstretch elastic solids in the case of equilibrium. Section 3 deals with the formulation of the basic boundary-value problems of the plane strain theory. In Section 4 we establish a representation of the displacements, microrotation and microstretch function in terms of a pair of complex analytic functions and two real functions which satisfy homogeneous Helmholtz equations. The boundary-value problems are reduced to the determination of these functions from prescribed values of certain combinations of these functions on the boundary of a plane region. The structure of potential functions is studied in Section 5 for several domains of interest. The method is applied in Section 6 to solve the Kirch problem. 2. Basic equations Throughout this section B is a bounded regular region of three-dimensional Euclidean space. We let B denote the closure of B, call oB the boundary of B, and designate by n the outward unit normal of oB. We assume that B is occupied by a linearly microstretch elastic body. The body is referred to a fixed system of rectangular Cartesian axes Oxi (i = 1, 2, 3). Throughout this paper Latin indices have the range 1–3, Greek indices have the range 1, 2 and the usual summation convention is employed. We use subscripts preceded by a comma for partial differentiation with respect to the corresponding coordinate. The linear strain measures eij and jij are defined by eij ¼ uj;i þ ejik uk ;

jij ¼ uj;i ;

ð2:1Þ

where u is the displacement field over B,u is the microrotation field over B, and eijk is the alternating symbol. The constitutive equations for a homogeneous and isotropic microstretch elastic solid are [1] tij ¼ kerr dij þ ðl þ jÞeij þ leji þ gwdij ; mij ¼ ajrr dij þ bjji þ cjij þ b0 esji w;s ;

ð2:2Þ

ri ¼ aw;i þ b0 eirs jsr ; h ¼ gerr þ bw

Here we have used the notations: tij is the stress tensor, mij is the couple stress tensor, ri is the microstretch stress vector, h is the microstress function, w is the microstretch function, k, l, j, g, a, b, c, a, b and b0 are constitutive constants. In the absence of body loads, the equilibrium equations of microstretch continua can be written as tji;j ¼ 0;

mji;j þ eirs trs ¼ 0;

ri;i  h ¼ 0:

ð2:3Þ

The surface force, the surface moment and the generalized surface force acting at a regular point of oB are defined by ti ¼ tji nj ;

mi ¼ mji nj ;

r ¼ ri ni ;

ð2:4Þ

respectively. In what follows we assume that the internal energy density is a positive definite quadratic form. This fact implies that [1] bð3k þ 2l þ jÞ  3g2 > 0; b > 0;

3a þ b þ c > 0;

2l þ j > 0; c þ b > 0;

j > 0;

c  b > 0:

a > 0;

ð2:5Þ

3. The plane strain problem We assume that the region B from here on refers to the interior of a right cylinder with the open crosssection R and the lateral boundary P. The rectangular Cartesian coordinate frame is supposed to be chosen

D. Iesßan, L. Nappa / International Journal of Engineering Science 44 (2006) 797–806

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in such a way that the x3-axis is parallel to generators of B. We denote by L the boundary of R. We consider the plane strain parallel to the x1, x2-plane. Thus, we have ua ¼ ua ðx1 ; x2 Þ;

u3 ¼ 0;

ua ¼ 0;

u3 ¼ uðx1 ; x2 Þ;

w ¼ wðx1 ; x2 Þ;

ðx1 ; x2 Þ 2 R:

ð3:1Þ

The above restrictions, in conjunction with the geometrical equations (2.1) and the constitutive relations (2.2), imply that eij, jij, tij, mij, ri and h are all independent of x3. It follows from (2.1) and (3.1) that the non-zero strain measures are given by eab ¼ ub;a þ eba3 u;

ja3 ¼ u;a :

ð3:2Þ

From (2.2) we conclude that the non-zero components of the stress tensor, couple stress tensor and internal hypertraction vector are tab, ma3, t33, m3a and ra. Further, tab ¼ keqq dab þ ðl þ jÞeab þ leba þ gwdab ; ma3 ¼ cja3 þ b0 e3ab w;b ; ra ¼ aw;a þ b0 e3ba u;b ;

ð3:3Þ

h ¼ geqq þ bw: The equations of equilibrium (2.3) become tba;b ¼ 0;

ma3;a þ e3ab tab ¼ 0;

ra;a  h ¼ 0

ð3:4Þ

on R. The non-zero surface tractions acting at a regular point of L are given by ta ¼ tba nb ;

m ¼ mq3 nq ;

r ¼ ra na

on L:

ð3:5Þ

To the field equations we must adjoin boundary conditions. In the case of the first boundary-value problem the boundary conditions are e on L; e; w ¼ w u¼u ð3:6Þ e are prescribed functions. The second boundary-value problem is characterized by the e and w where e ua; u boundary conditions ua ¼ e ua;

tba nb ¼ ~ta ;

~ ma3 na ¼ m;

ra na ¼ e r

on L;

ð3:7Þ

~ and r ~ are given. where ~ta , m From (3.2)–(3.4) we obtain the field equations in terms of displacement, microrotation and microstretch function ðl þ jÞDua þ ðk þ lÞuq;qa þ je3ub u;b þ gw;a ¼ 0; ð3:8Þ

cDu þ je3ab ub;a  2ju ¼ 0; aDw  guq;q  bw ¼ 0;

where D is the Laplacian. Existence and uniqueness results for the plane strain problems of the linear theory of microstretch elastic bodies have been presented in [8]. 4. The complex potentials In this section we establish a representation of the displacements, microrotation and microstretch function in terms of a pair of complex analytic functions and two real functions which satisfy homogeneous Helmholtz equations. In the case of constitutive Eq. (2.2) the use of stress functions leads to serious difficulties. In this paper we express the system of field Eq. (3.8) in complex coordinates and integrate this system directly. We introduce the complex coordinate z and z on R, and the complex displacement D by z ¼ x1 þ ix2 ;

z ¼ x1  ix2 ;

D ¼ u1 þ iu2 :

ð4:1Þ

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Then, we have D¼4

o2 ; ozoz

uq;q ¼

oD oD þ ; oz oz

e3ab ub;a ¼ i

  oD oD  ; oz oz

ð4:2Þ

where a bar over a letter designates the complex conjugate. In complex coordinates the system of Eq. (3.8) can be expressed in the form   o2 D o oD oD ou ow þ ðk þ lÞ þ þg ¼ 0; 2ðl þ jÞ  ij ozoz oz oz oz oz oz   o2 u oD oD ð4:3Þ 4c  ij   2ju ¼ 0; ozoz oz oz   o2 w oD oD g þ 4a  bw ¼ 0: ozoz oz oz The first equation of (4.3) may be integrated to give the result   oD oD oD þ ðk þ lÞ þ  iju þ gw ¼ X0 ðzÞ; 2ðl þ jÞ oz oz oz

ð4:4Þ

where X is an arbitrary analytic complex function on z, and X 0 (z) = dX(z)/dz. The conjugate of this relations is   oD oD oD þ ðk þ lÞ þ ð4:5Þ þ iju þ gw ¼ X0 ðzÞ: 2ðl þ jÞ oz oz oz It follows from (4.4) and (4.5) that oD oD 1 þ ¼ ½X0 ðzÞ þ X0 ðzÞ  2gwðz; zÞ: oz oz 2ðk þ 2l þ jÞ We may now eliminate D and D from (4.6) and the third equation of (4.3) to give   o2 g 2 ½X0 ðzÞ þ X0 ðzÞ; p w¼ 4 2aðk þ 2l þ jÞ ozoz

ð4:6Þ

ð4:7Þ

where p2 ¼

1 ½bðk þ 2l þ jÞ  g2 : aðk þ 2l þ jÞ

ð4:8Þ

It follows from (2.5) that p2 > 0. If we substract (4.5) from (4.4) then we get oD oD 1  ¼ ½X0 ðzÞ  X0 ðzÞ þ 2ijuðz; zÞ: oz oz 2ðl þ jÞ In view of (4.9), the second equation of (4.3) reduces to   o2 ij ½X0 ðzÞ  X0 ðzÞ;  s2 u ¼ 4 2cðl þ jÞ ozoz

ð4:9Þ

ð4:10Þ

where s2 ¼

jð2l þ jÞ : cðl þ jÞ

From (4.7) we conclude that the function w can be expressed in the form g w¼W ½X0 ðzÞ þ X0 ðzÞ; 2 2ap ðk þ 2l þ jÞ

ð4:11Þ

ð4:12Þ

D. Iesßan, L. Nappa / International Journal of Engineering Science 44 (2006) 797–806

where W is a real function which satisfies the equation   o2  p2 W ¼ 0: 4 ozoz

801

ð4:13Þ

In view of (4.6) and (4.12) we get   oD oD g þ W; ¼ C½X0 ðzÞ þ X0 ðzÞ  oz oz k þ 2l þ j

ð4:14Þ

where C¼

g2 þ ap2 ðk þ 2l þ jÞ 2ap2 ðk þ 2l þ jÞ2

:

It follows from (4.10) that the function u can be expressed as u¼U

ij ½X0 ðzÞ  X0 ðzÞ; 2cs2 ðl þ jÞ

ð4:15Þ

where U is a real function which satisfies the equation   o2 2  s U ¼ 0: 4 ozoz

ð4:16Þ

By (4.5), (4.12)–(4.14) and (4.16) we obtain oD o2 U o2 W ¼ n1 X0 ðzÞ  n2 X0 ðzÞ þ 4iq1  4q2 ; oz ozoz ozoz

ð4:17Þ

where n1 ¼

k þ 3l þ 2j j g2 þ þ ; 4ðk þ 2l þ jÞðl þ jÞ 4ð2l þ jÞðl þ jÞ 4ap2 ðk þ 2l þ jÞ2

kþl j g2 þ  ; 4ðk þ 2l þ jÞðl þ jÞ 4ð2l þ jÞðl þ jÞ 4ap2 ðk þ 2l þ jÞ2 j g ; q2 ¼ 2 : q1 ¼ 2 2s ðl þ jÞ 2p ðk þ 2l þ jÞ

ð4:18Þ

n2 ¼

The Eq. (4.17) may be integrated to give D ¼ n1 XðzÞ  n2 zX0 ðzÞ  xðzÞ þ 4iq1

oU oW  4q2 ; oz oz

ð4:19Þ

where x is an arbitrary analytic complex function on z. The relations (4.12), (4.15) and (4.19) give a representation of w, u and D in terms of the complex analytic functions X, x and real functions U, W which satisfy homogeneous Helmoholtz equations. A simple calculation shows that the constitutive Eq. (3.3) may be written as t11 þ t22 ¼ ð2k þ 2l þ jÞuq;q þ 2gw; t11  t22 þ iðt12 þ t21 Þ ¼ ð2l þ jÞ½u1;1  u2;2 þ iðu1;2 þ u2;1 Þ ¼ 2ð2l þ jÞ

oD ; oz

t21  t12 ¼ jðu1;2  u2;1 þ 2uÞ; ou ow þ 2ib0 ; oz oz ow ou r1  ir2 ¼ 2a  2ib0 ; oz oz h ¼ guq;q þ bw: m13  im23 ¼ 2c

ð4:20Þ

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D. Iesßan, L. Nappa / International Journal of Engineering Science 44 (2006) 797–806

In view of (4.2), (4.12), (4.15) and (4.19) the relations (4.20) can be expressed in the form bð2k þ 2l þ jÞ  2g2 0 gð2l þ jÞ W; ½X ðzÞ þ X0 ðzÞ þ 2 k þ 2l þ j 2ap ðk þ 2l þ jÞ   o2 U o2 W t11  t22 þ iðt12 þ t21 Þ ¼ 2ð2l þ jÞ n2 zX00 ðzÞ þ x0 ðzÞ  4iq1 2 þ 4q2 2 ; oz oz t11 þ t22 ¼

t21  t12 ¼ cs2 U;   oU oW j b0 g þ 2ib0 i 2 þ 2 m13  im23 ¼ 2c X00 ðzÞ; oz oz s ðl þ jÞ ap ðk þ 2l þ jÞ   oW oU g b0 j  2ib0  2 þ 2 r1  ir2 ¼ 2a X00 ðzÞ; oz oz p ðk þ 2l þ jÞ cs ðl þ jÞ

ð4:21Þ

h ¼ ap2 W: We assume that the curve L is a piecewise smooth curve parameterized by its arc length s. If we use the relations     1 dz dz 1 dz dz  þ n1 ¼  i ; n2 ¼  ; 2 ds ds 2 ds ds then from (3.5) we obtain dz dz 2ðt1 þ it2 Þ ¼ ½t12  t21  iðt11 þ t22 Þ þ i½t11  t22 þ iðt12 þ t21 Þ ; ds ds   dz m ¼ Im ðm13  im23 Þ ; ds   dz ; r ¼ Im ðr1  ir2 Þ ds

ð4:22Þ

where Imfg denotes the imaginary part of the equality {}. We note that t11 þ t22 ¼ 2ð2l þ jÞfn2 ½X0 ðzÞ þ X0 ðzÞ þ q2 p2 Wg: With the help of (4.21)–(4.23) we find that   d oU oW n2 ½XðzÞ þ zX0 ðzÞ þ xðzÞ  4iq1 þ 4q2 t1 þ it2 ¼ ð2l þ jÞi ; ds oz oz    oU oW dz þ 2ib0  ik 1 X00 ðzÞ m ¼ Im 2c ; oz oz ds    oW oU dz  2ib0  k 2 X00 ðzÞ r ¼ Im 2a ; oz oz ds

ð4:23Þ

ð4:24Þ

where k1 ¼

j b0 g þ ; s2 ðl þ jÞ ap2 ðk þ 2l þ jÞ

k2 ¼

g b0 j þ : p2 ðk þ 2l þ jÞ cs2 ðl þ jÞ

ð4:25Þ

We denote by R1 and R2 the components of the resultant vector of external stresses applied to the contour L. It follows from (4.24) that  A Z oU oW 0 þ 4q2 R1 þ iR2 ¼ ðt1 þ it2 Þds ¼ ð2l þ jÞi n2 ½XðzÞ þ zX ðzÞ þ xðzÞ  4iq1 ; ð4:26Þ oz oz A L A

where fF gA denotes the change in value of the function F on passing once round the contour L in the conventional positive sense.

D. Iesßan, L. Nappa / International Journal of Engineering Science 44 (2006) 797–806

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The boundary conditions (3.6) can be expressed in the form n1 XðzÞ  n2 zX0 ðzÞ  xðzÞ þ 4iq1

oU oW  4q2 ¼ f ðsÞ; oz oz

ij ~ ðsÞ; ½X0 ðzÞ  X0 ðzÞ ¼ u 2cs2 ðl þ jÞ g ~ ½X0 ðzÞ þ X0 ðzÞ ¼ wðsÞ; Wðz; zÞ  z 2 L; 2ap2 ðk þ 2l þ jÞ

Uðz; zÞ 

ð4:27Þ

where f = u˜1 + iu˜2. In view of (4.24), the boundary conditions (3.7) become   d oU oW n2 ½XðzÞ þ zX0 ðzÞ þ xðzÞ  4iq1 þ 4q2 ð2l þ jÞ ¼ gðsÞ; ds oz oz    oU oW dz 00 e ðsÞ; þ 2ib0  ik 1 X ðzÞ ¼m Im 2c oz oz ds    oW oU dz ~ðsÞ; z 2 L;  2ib0  k 2 X00 ðzÞ ¼r Im 2a oz oz ds

ð4:28Þ

where gðsÞ ¼ ið~t1 þ i~t2 Þ. 5. The structure of potentials In this section we investigate the arbitrariness and the structure of potentials X, x, U and W for several domains of interest. First, we investigate what is difference in the forms of two sets of potentials (X, x, U, W) and (X*, x*, U*, W*) that correspond to the same functions tab, ma3, ra and h. The relations (4.21) demand that Re½X0 ðzÞ ¼ R e½X0 ðzÞ; W ¼ W ; U ¼ U ; n2 zX00 ðzÞ þ x0 ðzÞ ¼ n2 zX00 ðzÞ þ x0 ðzÞ: Thus we conclude that XðzÞ ¼ X ðzÞ þ iAz þ q1 ;

xðzÞ ¼ x ðzÞ þ q2 ;

U ¼ U ;

W ¼ W ;

ð5:1Þ

where A is a real constant and qa are complex constants. If the origin of coordinates is taken within R, the functions X and x will be determined uniquely if A, q1 and q2 are chosen so that Xð0Þ ¼ 0;

ImX0 ð0Þ ¼ 0;

xð0Þ ¼ 0:

ð5:2Þ

Consider now the situation in which the two sets of potentials correspond to the same functions ua, u and w. In this case the extent of arbitrariness in choosing the potentials cannot be greater than that indicated in (5.1). From (4.19), the equality of displacements requires that A = 0 and n1 q1 ¼ q2 . In this case we can choose q1 so that X(0) = 0. We note that in a bounded simply connected region, X and x are single-valued analytic functions. Let us consider the case when the domain R is multiply connected and bounded. We assume that the boundary L consists of m + 1 simple closed contours Lj such that the exterior contour Lm+1 contains within it the contours Lk(k = 1, 2, . . . , m). In what follows we assume that the functions ua, u and w, and the stress functions tab, ma3, ra and h are single-valued. First, from (4.21) we find that U and W and their derivatives up to the second order must be single valued, and that the complex potentials have the form m X ðzAk þ Bk Þ logðz  zk Þ þ X1 ðzÞ; XðzÞ ¼ k¼1

xðzÞ ¼

m X k¼1

ð5:3Þ C k logðz  zk Þ þ x1 ðzÞ:

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Here zk is point in the simply-connected region Rk bounded by Lk, Ak are real constants, Bk and Ck are complex constants, and X1 and x1 are analytic and single-valued functions on R. Then, from (5.3), (4.19), (4.15) and (4.12) we get ½DLk ¼ 2pi½ðn1 þ n2 ÞzAk þ n1 Bk þ C k ; ½uLk ¼ 2pðn1 þ n2 ÞAk ;

½wLk ¼ 0;

where ½ Lk denotes the change in value of the function inside on passing once round the contour Lk in the conventional positive sense. Since ua and u must be single-valued, we obtained the conditions Ak ¼ 0;

n1 Bk þ C k ¼ 0: ðkÞ

ð5:4Þ

ðkÞ

We denote by ðR1 ; R2 Þ the resultant of stress vector applied to the contour Lk. By (4.24), (4.26) and (5.2) we find ðkÞ

ðkÞ

R1 þ iR2 ¼ 2pð2l þ jÞðn2 Bk  C k Þ:

ð5:5Þ

It follows from (5.4) and (5.5) that 1  ðkÞ ðkÞ R1 þ iR2 ; C k ¼ n1 Bk : Bk ¼  2p

ð5:6Þ

Thus, from (5.3) we obtain m  1 X ðkÞ ðkÞ XðzÞ ¼  R1 þ iR2 logðz  zk Þ þ X1 ðzÞ; 2p k¼1 m  1 X ðkÞ ðkÞ n1 xðzÞ ¼ R1  iR2 logðz  zk Þ þ x1 ðzÞ: 2p k¼1

ð5:7Þ

We consider now that the domain R is unbounded with certain contours L1, L2, . . . , Lm as internal boundaries. We assume that the origin z = 0 is taken outside R, and suppose that the functions tab, ma3, ra and h are bounded in the neighborhood of the point at infinity. We can show that for sufficiently large jzj we have XðzÞ ¼ 

1 ðF 1 þ iF 2 Þ log z þ ðc1 þ ic2 Þz þ X0 ðzÞ; 2p

1 n1 ðF 1  iF 2 Þ log z þ ðd 1 þ id 2 Þz þ x0 ðzÞ; 2p 1 X Uðz; zÞ ¼ ðUn einh þ Un einh ÞK n ðsrÞ; xðzÞ ¼

ð5:8Þ

n¼0

Wðz; zÞ ¼

1 X

ðWn einh þ Wn einh ÞK n ðprÞ:

n¼0

Here, ca and da are real constants, Un and Wn are complex constants, X0 and x0 are single-valued analytic functions on R including the point at infinity, Kn are modified Bessel functions of order n and m X RðkÞ z ¼ reih : ð5:9Þ Fa ¼ a ; k¼1

For sufficiently large jzj the functions X0 and x0 can be represented in the form 1 1 X X U n zn ; x0 ðzÞ ¼ V n zn : X0 ðzÞ ¼ n¼0

ð5:10Þ

n¼0

Let tab be the limiting value of tab(P) as the point P tends to infinity. It follows from (4.21), (4.23) and (5.8) that t11 ¼ ð2l þ jÞð2n2 c1  d 1 Þ; t22 ¼ ð2l þ jÞð2n2 c1 þ d 1 Þ;

t12 ¼ t21 ¼ ð2l þ jÞd 2 :

ð5:11Þ

D. Iesßan, L. Nappa / International Journal of Engineering Science 44 (2006) 797–806

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The constant c2 is related to the rigid rotation e at infinity by c2 ¼ ð2l þ jÞe:

ð5:12Þ

6. The stresses around a hole In this section we use the results from Sections 5 and 6 to solve the plane strain problem for an unbounded region with a circular hole of radius c whose centre is at the origin of coordinates. This problem has been considered in [4] in the context of a special form of constitutive equations. In this case the domain R is defined by R ¼ fðx1 ; x2 Þ 2 R2 ; x21 þ x22 > c2 g. We suppose that the hole is free of loads and that the body is subjected to a uniform axial tension at infinity acting in the x1–direction. The conditions at infinity are t11 ¼ P ;

t22 ¼ t12 ¼ t21 ¼ 0;

ma3 ¼ 0;

ra ¼ 0;

h ¼ 0;

ð6:1Þ

where P is a given constant. If we take into account the arbitrariness in choosing X and x we find that the boundary conditions (4.28) on the boundary of the hole reduce to oU oW þ 4q2 ¼ 0; n2 ½XðzÞ þ zX0 ðzÞ þ xðzÞ  4iq1 oz  oz  oU oW dz Im 2c þ 2ib0  ik 1 X00 ðzÞ ¼ 0; oz oz ds    oW oU dz  2ib0  k 2 X00 ðzÞ ¼ 0; for jzj ¼ c: Im 2a oz oz ds

ð6:2Þ

In this case the solution has the form given by (5.8) and (5.10). We note that F1 = F2 = 0. In the analysis of stresses the constants c2,U0 and V0 can be set equal to zero. From (5.11) and (6.1) we find that c1 ¼

1 P; 4n2 ð2l þ jÞ

d1 ¼ 

1 P; 2ð2l þ jÞ

d 2 ¼ 0:

ð6:3Þ

By using (5.8) and (5.10) in (6.2) we obtain XðzÞ ¼

1 1 Pz þ U 1 ; 4n2 ð2l þ jÞ z

1 1 1 Pz þ V 1 þ 3 V 3 ; 2ð2l þ jÞ z z z z Uðz; zÞ ¼ iR  K 2 ðsrÞ; z z z z Wðz; zÞ ¼ H þ K 2 ðprÞ; r ¼ ðzzÞ1=2 ; z z xðzÞ ¼ 

ð6:4Þ

where U1 ¼

1 Pc2 ; 2ð2l þ jÞQ

V3 ¼

1 ½n þ 2q1 csTK 3 ðscÞ þ 2q2 cpSK 3 ðpcÞPc4 ; 2ð2l þ jÞQ 2

R ¼ TU 1 ;

H ¼ SU 1 ;

V1 ¼

1 Pc2 ; 2ð2l þ jÞ

Q ¼ n2 þ 2q1 scTK 1 ðscÞ þ 2q2 pcK 3 ðpcÞ;

4 f8aK 2 ðpcÞ þ 2pcb0 ½K 1 ðpcÞ þ K 3 ðpcÞg; 3c4 K 4 S ¼ 4 f8cK 2 ðscÞ  2scb0 ½K 1 ðscÞ þ K 3 ðscÞg; 3c K 16ac K ¼ 2 K 2 ðscÞK 2 ðpcÞ þ 4psb20 ½K 1 ðpcÞ þ K 3 ðpcÞ½K 1 ðscÞ þ K 3 ðscÞ: c

T ¼

ð6:5Þ

806

D. Iesßan, L. Nappa / International Journal of Engineering Science 44 (2006) 797–806

Let ur and uh be the components of the displacement vector field in polar coordinates. Then, ur þ iuh ¼ eih D: It follows from (4.12), (4.15), (4.19) and (6.4) that n1  n2 1 Pr  ðV 1  n1 U 1 Þ þ u cos 2h þ iv sin 2h; ur þ iuh ¼ r 4n2 ð2l þ jÞ   j U u ¼ 2 RK 2 ðsrÞ  1 sin 2h; 2cs2 ðl þ jÞr2   gP g þ 2 HK ðprÞ þ cos 2h; w¼ 2 4ap2 n2 ðk þ 2l þ jÞð2l þ jÞ 2ap2 ðk þ 2l þ jÞr2 where 1 Pr 1 2  3 V 3 þ q1 RK 2 ðsrÞ þ 2pq2 H ½K 1 ðprÞ þ K 3 ðprÞ; u ¼ n2 U 1 þ r 2ð2l þ jÞ r r 1 Pr 1 8  3 V 3 þ 2q1 R½K 1 ðsrÞ þ K 3 ðsrÞ þ q2 HK 2 ðprÞ: v ¼ n2 U 1  r 2ð2l þ jÞ r r Similarly, by using (4.21) and (6.4) we can determine the stresses. We note that the behaviour at infinity of displacements and stresses is the same as in classical theory of elasticity (see [5, Section 56a], and [6, p. 291]). The details of this argument can be found in [5, Section 36]. If b0 = 0 then we obtain the solution given in [4]. References [1] A.C. Eringen, Microcontinuum Field Theories. I. Foundations and Solids, Springer-Verlag, New York, 1999. [2] A.C. Eringen, Mechanics of Micromorphic Materials, in: Go¨rtler, H. (Ed.) Proc. 11th Int. Congress of Appl. Mech., Springer-Verlag, New York, 1964. [3] A.C. Eringen, Theory of thermo-microstretch elastic solids, Int. J. Engng. Sci. 28 (1990) 1291–1301. [4] D. Iesßan, A. Scalia, On complex potentials in equilibrium theory of microstretch elastic bodies, Int. J. Engng. Sci. 41 (2003) 1989–2003. [5] N.I. Muskhelishvili, Some Basic Problems of the Mathematical Theory of Elasticity, Nordhoff, Groningen 1954. [6] I.S. Sokolnikoff, Mathematical Theory of Elasticity, McGraw-Hill Book Company, New York, 1956. [7] A.E. Green, W. Zerna, Theoretical Elasticity, second ed., Clarendon Press, Oxford, 1968. [8] D. Iesßan, L. Nappa, On the plane strain of microstretch elastic solids, Int. J. Engng. Sci. 39 (2001) 1815–1835.