Archive of Applied Mechanics 70 (2000) 659±669 Ó Springer-Verlag 2000

Singular potentials and double-force solutions for anisotropic laminates with coupled bending and stretching D. D. Zakharov, W. Becker

Summary Exact solutions are obtained in the framework of the classical theory of laminates subjected to the action of normal moments, double forces, double moments or momentless double dipoles. Seven cases of such loads are considered and completed by considering the case of given transversal discontinuity of normal de¯ection. It is shown that, in contrast to the case of in®nite straight dislocations in a pure in-plane problem, the energy of this eighth solution depends on the discontinuity orientation. Some numerical examples are presented. Besides the formal value, the obtained double-force and double-moment solutions, as well as dimensionless double dipoles, can be used to construct kernels of additional boundary integral equations (BIE). Due to the coupling phenomena in the BIE system for the region with a corner point, additional variable such as corner forces appear and require the mentioned equation. Key words Composite, anisotropy, laminate, double force, dislocation

1 Introduction The problem of double force solutions naturally arises when solving boundary value problems for elastic solids, especially by means of complex potential techniques. It is connected with cases of concentrated loads, dislocations and with single-valued conditions for physically meaningful variables. A few results for classical formulations of such problems are presented in [1±4] for homogeneous, isotropic and anisotropic media; a detailed review is given in [5]. In our case, we consider a 2D formulation of the problem of thin anisotropic laminates under coupled bending and stretching, due to an asymmetrical layup. Using an averaged model of laminates, [6], and the complex potentials developed in [7, 8], we focus our attention on the analytical solutions for double forces with or without moments, and for double moments with respect to longitudinal axes, as well as on the given discontinuity of normal de¯ection. The main goal is to obtain for the cases of practical importance the complete system of analytical solutions and conditions of single valuedness, and the correspondence to the integrals of concentrated loads over faces, or to the integrals of loads over arbitrary curvilinear boundaries of ®nite regions, occupied by the laminate. Let us just mention two possible applications. The ®rst one concerns the boundary value problem for a multiply-connected region. Due to the static equilibrium, the load on the entire boundary should be totally self-equilibrated. But the load on each hole, for example, may be not

Received 22 June 1999; accepted for publication 6 March 2000 D. D. Zakharov Russian Academy of Sciences, Institute for Problems in Mechanics, 101 Vernadskogo Avenue, 117526 Moscow, Russia W. Becker (&) UniversitaÈt-GH Siegen, Institut fuÈr Mechanik und Regelungstechnik, FB-11, Paul-Bonatz-Strasse 9-11, D-57068 Siegen, Germany Fax: +49 271 740 2461 e-mail: [email protected] This work was supported by DFG in the framework of project No. BE 1090/5-1 which is gratefully acknowledged.

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self-equilibrated, and may cause non zero integral values of forces and moments. The standard way to proceed to the self-equilibrated holes and external boundary, both for the implementation of the complex potential method and the direct/indirect boundary element method, is to extract the multivalued partial solution responsible for such non zero integrals, [1, 2]. Respective fundamental solutions have been previously analyzed in [6, 7], and the present paper gives solutions related to a torque, which often appears in practice. These solutions depend on the orientation of constitutive couples, causing such torque. The second case concerns the implementation of the direct boundary element method to the region with corner points. Fundamental solutions permit us to obtain four boundary integral equations, [9], which are suf®cient for the region with a smooth boundary. But in each corner point, due to the coupling phenomena and similarly to the pure bending problem, [10], the additional variable ± corner force (or a jump of the tangent moment) may appear. It requests an additional independent equation, [10]. Using the presented doubleforce and double-moment solutions, one can obtain equations with respect to the strains and curvatures at the left-hand side, and on multiplying by the stiffness matrix ± the equations with respect to the force and moment tensors. Such equations were well-studied for the pure in-plane problem, [11], where they possess some dual properties with respect to the equations following from the use of usual fundamental solutions. Next, tangent moment is obtained as an evident linear combination of the stress couples, while its jump (i.e. the corner force) results as a difference of the limit values, when tending to the corner point on the boundary. This yields the desired additional equation. Despite the wide spread use of the ®nite element approach, the boundary element method often gives an alternative, permitting to deal with problems of smaller dimensions, and to analyze in details the behaviour of laminate near corner points. In addition, the obtained double-force and double-moment solutions may be used as asymptotic solutions in the case of similar loads, when the distances between concentrated forces or moments are much less than the distance where the laminate response is analyzed.

2 Basic relations of a 2D model for anisotropic laminates Consider a laminate in 3D space with cartesian coordinates x ˆ ia xa , x3 ˆ z (a ˆ 1; 2), which occupies a region X in its plane. The total thickness is 2h ˆ z‡ z , and the laminate consists of N elastic plies zj  z  zj‡1 (j ˆ 1; 2; . . . ; N) with general 3D anisotropy, perfectly joined at their interfaces. In addition, some loads may be distributed over the faces under some boundary conditions at the edges oX  ‰z ; z‡ Š (z ˆ z1 , z‡ ˆ zN‡1 ). Under usual assumptions for suf®ciently thin laminates, [8], the internal stress and strain state is described by the classical 2D laminate theory, and transverse shear effects can be neglected. Namely, for the longitudinal displacements v ˆ ia va and for the normal de¯ection w, we follow Kirchhoff's kinematical relations v ˆ u…x†

zrw;

w ˆ w…x†; r ˆ ia oa ; j

with corresponding expressions for the longitudinal stresses rab , stress resultants Qab and couples Mab j

rab ˆ vab …Cj †v; 

Qab Mab



1 Cj ˆ kcmn kj ; cmn ˆ Gm n G0

  mab ˆD ; jab



D1 Dˆ D2

mab ˆ 12…ob ua ‡ oa ub †…1 ‡ d12 ab †;

…m; n ˆ 1; 6; 2† zj‡1

 D2 ; D3

Dk ˆ

jab ˆ

o2ab w…1 ‡ d12 ab † …ab ˆ 11; 12; 22† ;

XZ j

Cj zk

1

dz ;

zj

where mab , jab are the components of strains and curvatures. Here, and in what follows, Gj ˆ kgmn kj ˆ const (m; n ˆ 1; 2; . . . ; 6) is the stiffness matrix of the 3D Hooke's law for the jth ply, whose lines and columns correspond to the stresses and strains with indices 11; 22; 33; 23; 13; 12, respectively. The matrix Cj contains reduced stiffnesses of the jth ply, each of them is expressed via the determinant G0 of …3; 4; 5†  …3; 4; 5†-minor from Gj , and via the determinant Gm n of the minor, bordering G0 by mth line and nth column. Matrices Dk are the

matrices of the membrane (k ˆ 1), membrane-bending (k ˆ 2) and bending (k ˆ 3) integral stiffnesses, d12 ab is a Kronecker delta. Operators vab …Cj † acquire the form (1 $ 2)

v11 …C† ˆ i1 …c11 o1 ‡ c16 o2 † ‡ i2 …c16 o1 ‡ c12 o2 †; v12 …C† ˆ i1 …c16 o1 ‡ c66 o2 † ‡ i2 …c66 o1 ‡ c26 o2 † : Transversal forces Qaz are obtained by differentiation (to sum with repeating Greek indices)

Qaz ˆ ob Mab : Equations of equilibrium are written in the form

ob Qab ˆ

oa Qaz  o2ab Mab ˆ

Ta ;

Tz

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…a; b ˆ 1; 2† ;

…1†

where

Ta ˆ r‡ az

Tz ˆ Tz ‡ oa …z‡ r‡ az

raz ;

z raz †;

Tz ˆ r‡ zz

rzz ;

r az ˆ raz jzˆz ;

…2†

are effective loads on the faces. The boundary conditions may be given in the form of known displacements and slopes or in the form of known stress resultants and couples on the total boundary oX, as well as on its parts.

3 Complex potentials Like the classical approach with potentials of Kolosov±Muskhelishwili±Lekhnitskii, [1, 3], or the approach of Stroh, [3, 12], functions of complex variable permit us to obtain ef®ciently some basic analytical solutions, [7, 8], to the homogeneous problem. To this end we introduce complex potentials as follows: X X wˆ 2Re wk …fk †; ua ˆ 2Refbak w0k …fk †g ‡ u0a …x†; fk ˆ x1 ‡ lk x2 ; …3† k

b1k ˆ

k

 p13 p22

p12 p23 p0

 k

;

b2k ˆ

 p11 p23

p13 p12



p0

k

;

p0k ˆ fp11 p22

p…l† ˆ det P…l†; p…lk † ˆ 0; Im lk > 0 …k ˆ 1; 2; 3; 4† 2 3 2 p12 p13 p11 1 l 0 0 P ˆ 4


p22 p23 5 ˆ DDDT ; D ˆ 4 0 1 l 0 0 0 0 1 Symm:


p33

0 0 2l

p212 gk ;

3 0 0 5 : l2

Here, lk are ordinary roots of the eighth order characteristic polynomial p…l†; they are complex by virtue of the ellipticity of the operator of system (1), [7]. Additional term u0a …x† is a linear function with respect to the coordinates xa . Four potentials wk …fk † satisfy the equations of equilibrium (1) under Ta ˆ 0, Tz ˆ 0, and are sought as a solution to the problem of complex analysis: to ®nd complex functions inside the region by their real parts, known from the boundary conditions at oX, [8]. By substitution of expressions (3) into formulas for the stress resultants and couples, we obtain

fQab ; Mab g ˆ

X k

qa1k ‡ lk qa2k ˆ 0;

2Refqab ; mab gk w00k …fk †;

Qaz ˆ

X k

2Refqaz w000 …f†gk

q1zk ‡ lk q2zk ˆ 0 ;

with respective complex coef®cients qabk , mabk , qazk hq i ab ˆ DDTk ; qazk ˆ ma1k ‡ lk ma2k …ab ˆ 11; 12; 22† : mab k

…a; b ˆ 1; 2†

Since functions of complex variables may be multivalued, we should also satisfy some additional conditions of single-valuedness for the meaningful functions, and conditions of one-toone correspondence to the integral load on the faces and edges.

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4 Conditions of single-valuedness and of correspondence to the integral load Let us consider an in®nitely extended region, and single out an arbitrary subregion X from the region, occupied by a laminate, and denote as Fa , Fz and Mb , Mz the following integrals: Z Z Fa ˆ Ta dX ˆ Qa2 dx1 Qa1 dx2 …a ˆ 1; 2† ; …4† X

Z Fz ˆ

oX

Z

Tz

dX ˆ

X

Ma ˆ … 1†a‡1

Z

Q2z dx1

Q1z dx2 ;

…5†

oX

…xb Tz

z‡ s‡ b ‡ z sb †dX

X a‡1

Z …xb Q1z

ˆ … 1†

oX

Z Mz ˆ

Mb1 †dx2

…x1 T2

…xb Q2z

Mb2 †dx1

…ab ˆ 12; 21† ;

…6†

Z x2 T1 †dX ˆ

X

…x2 Q11

x1 Q12 †dx1 ‡ …x1 Q22

x2 Q12 †dx2 :

…7†

oX

Formulas (4), (5) express integral values of the longitudinal and transversal forces, while formulas (6), (7) correspond to the integral values of moments. They are obtained using equilibrium equations (1) and Green's formulas, to proceed from the integral over X to the integral over its boundary. Introduce a function

Dk ˆ ‰wk ŠjoX ; which is a jump of the potential in the current point x at getting around the loop oX, and let the prime denote its derivative. Then, D00k should satisfy eight linear equations

X k

X k

X

2Reflka‡b 2 D00k g ˆ 0; 2Refq23k D00k g

k

2Reflak 1 bbk D00k g ˆ 0 ;

Fz ˆ 0 …a; b ˆ 1; 2† :

…8† …9†

Seven equations (8) express the conditions of a single value for three curvatures, three strains, and the angle of rotation in the proper plane. Equation (9) represents the condition of one-toone correspondence to the integral transversal force, and is obtained by transforming (5) due to relations (1) and Green's formula. Similarly, for D0k we obtain

X k

2Refqa2k D0k g ‡ … 1†a Fa ˆ 0 …ab ˆ 12; 21† ;

  maak 0 2Re b 1 Dk ‡ … 1†a‡1 xa Fz ‡ Mb ˆ 0 ; lk k X X 2Refbak D0k g ˆ 0; 2Reflak 1 D0k g ˆ 0 : X

k

…10† …11† …12†

k

Four equations (12) are necessary to ensure the single values of longitudinal displacements and of slopes. In principle, in Eqs. (12) we also may include the intensities of dislocations: jumps of

the longitudinal displacements and of the slopes, but it is not necessary for our purpose to analyze the solution caused by singular potentials. By virtue of linear relations between strains/curvatures and forces/moments, the latter also acquire single values. Conditions of correspondence to the torque Mz , and to the single value of normal de¯ection obviously yield relations for Dk , [6, 7],

X k

X

  q11k 2Re Dk ‡ x1 F2 l2k

x2 F1 ‡ Mz ˆ 0 ;

2Re Dk ˆ 0 :

…13† …14†

k

As one can see, the system of equations (8)±(12) permits us to ®nd fundamental solutions ± a response of the in®nite laminate to the concentrated force Fz ˆ Fz d…x† where d…x† ˆ d…x1 †d…x2 † is a Dirac function (with Tz ˆ Fz , Ta ˆ 0), or to the concentrated longitudinal forces Fa ˆ Fa d…x† (with Ta ˆ Fa , Tz ˆ 0), and moments Ma ˆ … 1†a Ma ob d…x† (with ab ˆ 12; 21 and Ta ˆ 0, Tz ˆ Ma ) which are shown in Fig. 1. To this end, we may set the multivalued potentials in the form

wk …fk † ˆ 12Ak f2k ln fk




3 2

‡ Bk fk …ln fk

1† ‡ Ck ln fk ;

…15†

with the respective increments Dk and their derivatives

D00k ˆ 2piAk ;

D0k ˆ 2pi…fk Ak ‡ Bk †;

Dk ˆ 2pi…12f2k Ak ‡ Bk fk Bk ‡ Ck † :

Eight coef®cients Re Ak , Im Ak are determined by the linear system (8) and (9). Coef®cients Re Bk , Im Bk are found from Eqs. (10)±(12). In particular, the response to the concentrated moments Ma may be found by the formal differentiation of the previous A-solution, when setting Ma ˆ … 1†a ob Fz with respective substitution of the intensity Ma instead of Fz . It is also possible to obtain the analytical expressions for Ak and Bk using Fourier transformation of Eq. (1), [9]. Here, we give only the respective system of eighth order of linear algebraic equations for the coef®cients. Since these equations are exact we may call the potential in (15) an exact solution, even when the respective system is solved numerically. To ®nd out the coef®cients Re Ck , Im Ck we need some additional requirements, since two Eqs. (13), (14) are not suf®cient. In addition, the concentrated moment Mz can be obtained in different ways, using couples of longitudinal forces of different orientation. Below, we consider the solutions with different values of Ck and conditions for the singular (logarithmic) potentials. These solutions (including previous A- and B-terms) represent a complete system of multivalued components for the case of not self-equilibrated loads, concentrated at the origin. Like in the in-plane problem, [11], they may be used as kernels to obtain the boundary integral equations of the second kind with respect to the stress resultants and couples. Another possible application of these solutions is the opportunity to extract the multivalued part of potentials for the case of the boundary value problem for a multiply connected region X. For the region with a set of holes, the loads on each hole's boundary may be not self-equilibrated, this requirement for static problem is necessary only for the total boundary of the region. Thus, knowing the integral values of such boundary loads, P one can extract the multivalued part of solution with the help of logarithmic potentials k wk …fk flk † from (15), placing the point xl (flk ˆ xl1 ‡ lk xl2 ) inside the lth hole. The rest of the solution will correspond to the self-equilibrated load, which makes the boundary value problem much simpler.

Fig. 1. Forces and moments

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5 Solution for double forces and double moments Consider the superposition of basic forces and moments (Fig. 1), which leads to the eight limit cases, shown in Fig. 2. Pictures on the left-hand side represent the cases …Fa …xa ; xb ‡ e† Fa …xa ; xb e†† (a; b ˆ 1; 2), with intensities Fa proportional to …2e† 1 . Thus, we obtain a natural transition from the distribution d…xa †fd…xb ‡ e† d…xb e†g=…2e† to the ®nal distribution ob d…x†. On the right-hand side in Fig. 2, similar cases are shown for moments. Let us sum up the results. Four cases for forces lead to the following four effective loads (2) in the equations of equilibrium (1): 664

(a) Ta ˆ Sa oa d…x†, Tz ˆ 0 (not to sum over a ˆ 1; 2) with Sa ˆ const. Such case of load is obtained from oa Fa and naturally called the center of extension in the direction xa with intensity Sa (or double force without moment, [5]); (b) Ta ˆ … 1†b Mza ob d…x†, Tz ˆ 0 (not to sum over ab ˆ 12; 21) with Mza ˆ const. This case of load is obtained as ob Fa and called couple of forces in the direction xa with torque Mza with respect to the transversal axis z, [5]. Combination of loads T1 and T2 with Mza ˆ Mz corresponds to the value of concentrated moment Mz in (13). When setting Mz1 ˆ Mz2 ˆ D, this combination corresponds to the dimensionless double dipole, [5], of intensity D. In contrast to the forces, four cases for moments result only in three types of effective loads in Eqs. (1) and (2): (c) Tz ˆ Ro212 d…x†, Ta ˆ 0 (ab ˆ 12; 21) with R ˆ const. Formally, it can be obtained from two limit cases as o2 M2 ˆ o1 M1 when setting Ma ˆ R. We call this case the center of rotation with intensity R (or double moment of the ®rst kind);

Fig. 2. Double forces and double moments

(d) Tz ˆ Ba o2b d…x†, Ta ˆ 0 (ab ˆ 12; 21) with Ba ˆ const. It is obtained as ob Ma with respective replacement of intensity. Let us call this case of load the center of bending with respect to xa with intensity Ba (or double moment of the second kind). Thus, the seven independent cases of concentrated loads (a)±(d) can easily be obtained by differentiation of the basic B-solution with simultaneous replacement of the intensities of forces Fa by Sa or … 1†a‡1 Mza , and the intensities of moments Ma by R or … 1†a‡1 Ba . Numerically, it means that coef®cients Ck should be equal to Bk , or to lk Bk , in agreement with the considered type of load. The mentioned solutions for double forces and for double moments were calculated for the unsymmetric cross-ply laminate [0 =90 ], made of two orthotropic plies with thickness h ˆ 0:5 mm, and elastic stiffnesses for the bottom ply in principal axes x1 ,x2 : c11 ˆ 135700 MPa, c12 ˆ 2710 MPa, c22 ˆ 10100 MPa, c66 ˆ 2920 MPa. In the system of polar coordinates x1 ˆ r cos u, x2 ˆ r sin u, the distributions of normal and tangent forces Qrr , Qru , normal and tangent moments Mrr , Mru , transversal force Qrz and displacements ur , uu , slopes hr , hu , and normal de¯ection w have been calculated. The distance from the origin is r ˆ 1 mm, intensities Sa ˆ Mza ˆ 1 N/mm, and Ba ˆ R ˆ 1 N. Before presenting the results let's make a remark: the value of h is given here only to obtain the integral stiffnesses; only these are essential to determine ua ,w,Qab ,Mab . From the viewpoint of stiffnesses we have no contradiction with the 2D theory from Sec. 2, which is valid as an asymptotical theory of thin laminates, [9]. The results are plotted in Figs. 3±6. As one can see, they are periodic with period p and symmetric or antisymmetric with respect to the phase shift 1=2p. This is explained by the orientation of basic loads. Due to the layup, which permits us to exchange the axes x1 $ x2 , z $ z, the numerical results also satisfy such changing of the signs and of the phase values. Thus, the plots for the case S2 ˆ 1 can be obtained from the case S1 ˆ 1 (Fig. 3), as well as the case Mz2 ˆ 1 is obtained from Mz1 ˆ 1 (Fig. 4) and B2 ˆ 1 is obtained from B1 ˆ 1 (Fig. 5) due to the respective symmetry of the variables' behaviour. Now, let us brie¯y describe the use of obtained solutions as kernels in the boundary integral equation with respect to the corner force, mentioned in the introduction. It is based on the reciprocity relations for statics, [9],

A…v; Tk † ˆ A…vk ; T† ‡ X fxc g

Z

…ukn Qnn ‡ uks Qns ‡ hkn Mnn ‡ wk Pn †dl

oX c wk DMns jxˆxc

Z

k …un Qknn ‡ us Qkns ‡ hn Mnn ‡ wPkn †dl ;

…16†

oX

Fig. 3. Forces (in N/mm) and moments (in N) for given S1 and respective displacements (in mm)/slopes (in radians) for the cross ply laminate. Displacement ®eld is increased 106 times

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666

Fig. 4. Forces (in N/mm) and moments (in N) for given Mz1 and respective displacements (in mm)/slopes (in radians) for the cross ply laminate. Displacement ®eld is increased 105 times

Fig. 5. Forces (in N/mm) and moments (in N) for given B1 and respective displacements (in mm)/slopes (in radians) for the cross ply laminate. Displacement ®eld is increased 105 times

Fig. 6. Forces (in N/mm) and moments (in N) for given R and respective displacements (in mm)/slopes (in radians) for the cross ply laminate. Displacement ®eld is increased 105 times

where k

Z

A…v; T † ˆ

…ua Tak ‡ wTzk †dX;

v ˆ …u; w†k ;

X c for each two states …v; T† and …v; T†k of laminate, occupying a region X in its plane. Here, DMns are jumps of tangent moment in the points xc , where they are not continuous, i.e. represent corner forces in corner points. When we set the facing loads in the form

Tak ˆ

ob d…x

x0 †;

Tbk ˆ

oa d…x

x0 †;

667

Tzk ˆ 0 …a; b ˆ 1; 2† ;

or in the form

Tak ˆ 0;

2 Tzk ˆ …1 ‡ d12 ab †oab d…x

x0 † ;

with corresponding displacement ®eld vk , then at the left-hand side of Eq. (16) we obtain

A…v; Tk † ˆ mab

or A…v; Tk † ˆ jab ;

respectively. Having at the left-hand side all possible strains and curvatures, and multiplying with the stiffness matrix D, we arrive at equations with respect to the stress resultants and couples. Moment Mns is just a linear combination of equations for moments Mab , which yields the equation with respect to the corner force. Of course, such an equation requires some additional simpli®cations which we do not discuss here.

6 Discontinuity of normal deflection To obtain the basis in the space of C-solutions, we need eight sets of numbers Re Ck , Im Ck (k ˆ 1; 2; 3; 4). Double-force and double-moment solutions give us only seven. It should be noted here that by virtue of Eqs. (12) and relations Ck ˆ Bk or Ck ˆ lk Bk all these solutions satisfy Eq. (14). In general, this is an independent requirement, which de®nes the discontinuity of normal de¯ection. Let us rewrite it in the form X 2RefDk g ‡ w ˆ 0 : …17† k

When w ˆ 0, the normal de¯ection is continuous. The opposite case is similar to the so-called in®nite straight dislocation (screw) in the pure out-of-plane problem, [12, 13], but the dislocation intensity w 6ˆ 0, w ˆ ‰wŠjoX is given in the transversal direction, Fig. 7. This case also possesses some differences, since we use a 2D model of laminate, averaged just in the transversal direction. So, the eighth C-solution may be obtained by the following procedure: denote as el ˆ …e1 ; e2 ; . . . ; e8 †l the set of constants …Re C1 ; Im C1 ; . . . ; Re C4 ; Im C4 †l , where index l ˆ 1; 2; . . . ; 7 corresponds to the cases Sa , Mza , Ra , B (a ˆ 1; 2). We need to ®nd out a set of constants (a vector) e8 which can not be represented as a linear combination of vectors el (l  7); with real coef®cients. The one-dimensional subspace of vectors, orthogonal in the usual sense to each of el , possesses such a property, i.e. they should satisfy seven equations

X n

enl en8 ˆ 0 …l ˆ 1; 2; . . . ; 7; n ˆ 1; 2; . . . ; 8† :

Fig. 7. Normal de¯ection discontinuity

…18†

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Equation (17) permits us to ®x such a vector and to clarify its physical meaning. To illustrate the stress and strain state, the distributions of the forces and moments, as well as of the displacements/slopes, are shown in Figs. 8 and 9 for the case of unsymmetric cross-ply [0 =90 ], and angle-ply [ 45 =45 ] laminates, respectively. The parameters of the constitutive single-ply behaviour are the same. The cut-off is the set along the axis u ˆ p in the polar coordinate system, and intensity w ˆ 1 [mm]. The results also possess some symmetry with respect to the phase shifting, equal 3=4p or 1=4p, but normal de¯ection w on the boundary remains the same for both variants. Interesting that coef®cients Ck obviously do not depend on the cut-off orientation, since Eqs. (17), (18) do not contain such orientation angle. However, the energy of dislocation depends on the orientation. In fact, consider a ring r2 > r1 in polar coordinates r,u. Set an intensity w and a cut-off orientation as u ˆ u0 , i.e. u0  u < u0 ‡ 2p. Denote this region X. As we have seen, u0 does not appear in Eqs. (15), (16). Due to the classical 2D theory of laminates, [8], the following energy relations hold:

2P ˆ A ‡ E ; Z P ˆ 12 …Qab mab ‡ Mab jab †dX …ab ˆ 11; 12; 22† ; X

Z Aˆ X

…Ta ua ‡ Tz w†dX;

Z Eˆ

…Qa ua ‡ Ha ha ‡ Qnz w†dl ; oX

Fig. 8. Forces (in N/mm) and moments (in N) for given discontinuity w and respective displacements (in mm)/slopes (in radians) for the cross ply laminate. Forces and moments are multiplied by 10 5

Fig. 9. Forces (in N/mm) and moments (in N) for given discontinuity w and respective displacements (in mm)/slopes (in radians) for the angle ply laminate. Forces and moments are multiplied by 10 5

where Qa , Ha are the resultants and couples of stress components in the direction xa on the segment dl 2 oX; n is a unit outer normal to oX. Since Ta ˆ 0, Tz ˆ 0, total energy 2P equals E. Obviously, E ˆ I2 I1 ‡ Ic , where uZ0 ‡2p

Ia ˆ

Zr2 er dujrˆra ;

Ic ˆ

u0

u

‰eŠju00 ‡2p dr ; r1

e ˆ Qa ua ‡ Ha ha ‡ Qnz w ˆ

X

2Ref…qa2k ua ‡ …ma2k

k

wk …fk † ˆ Ck ln fk ;

fk ˆ rnk ;

qa2k †ha †nk w00k ‡ qazk nk w000 k wg ;

nk ˆ cos u ‡ lk sin u :

In the pure in-plane problem, all the dislocation energy is reduced to the integral Ic over the cut-off, which is independent of u0 , and depends on the radii as ln…r2 =r1 †, [11]. In our case we obtain

Ic ˆ 2w





1 r12

   1 X q2zk Ck ; Re r22 k n2k

and this result depends on u0 . Thus, the problem of an w-discontinuity under a coupled bending and stretching principally differs from the familiar pure in-plane problem on the in®nite straight dislocation.

7 Conclusion As a result of our consideration of the coupled bending-extension problem, a complete system of relations to obtain the analytical solution for the case of concentrated forces and moments is deduced. Namely, it is given by the potentials (15) and Eqs. (8)±(13), (17) and (18). As shown, for the case of a concentrated moment with respect to the transversal axis, we need additional information about the self-equilibrated components of loading (double forces and double moments), as well as about continuity of normal de¯ection. Without such information, the solution which determines the response of laminate is not unique. It is also shown, that the properties of the in®nite straight dislocation in the transversal direction principally differ from the classical pure in-plane problem, [13]. The explanation is in the use of averaged 2D theory of laminate in this analysis. The obtained system of solutions can be used as kernels of boundary integral equations (similarly to [10, 11], for example), as well as to simplify the boundary value problem with respect to complex potentials in the multiply-connected region. References

1. Muskhelishwili, N.I.: Some basic problems of the mathematical theory of elasticity. Groningen: Noordhoff 1953 2. England, A.H.: Complex variables methods in elasticity. London: Wiley-Interscience 1971 3. Stroh, A.N.: Steady state problem in anisotropic elasticity. J. Math. Phys. 41 (1962) 77±103 4. Lekhnitskii, S.G.: Anisotropic plates. New York: Gordon & Breach 1968 5. Ting, T.C.T.: Anisotropic elasticity. Theory and Applications. Oxford: Oxford Univ. Press 1996 6. Jones, R.M.: Mechanics of composite materials. New York: McGraw-Hill 1975 7. Becker, W.: Concentrated forces and moments on laminates with bending extension coupling. Composite Struct. 30 (1995) 1±11 8. Zakharov, D.D.: Formulation of boundary-value problems of statics for thin elastic asymmetricallylaminated anisotropic plates and solution using functions of a complex variable. J. Appl. Math. Mech. (PMM) 59 (1995) 615±623 9. Zakharov, D.D.: Green's tensor and the boundary integral equations for thin multi-layer asymmetric anisotropic plates. J. Appl. Math. Mech. (PMM) 61 (1997) 483±492 10. Totenham, H.: The boundary element method for plates and shells. In: Banerjee, P.K.; Butter®eld, R. (eds.) Development in boundary element method, pp. 173±205, Appl. Sci. Publ. 1979 11. Kupradze, V.D.: Potential methods in the theory of elasticity. Moscow: Fizmatgiz 1963 (in Russian) 12. Stroh, A.N.: Dislocations and cracks in anisotropic elasticity. Phil. Mag. 3 (1958) 625±646 13. Steeds, J.W.: Introduction to anisotropic elasticity theory of dislocations. Oxford: Clarendon Press 1973

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