On a stress analysis in the infinite elastic body with two neighbouring curved fibres

Composites: Part B 34 (2003) 143–150 www.elsevier.com/locate/compositesb

On a stress analysis in the infinite elastic body with two neighbouring curved fibres Surkay D. Akbarov*, Resat Kosker Department of Mathematical Engineering, Faculty of Chemistry and Metallurgy, Yildiz Technical University, Davutpasa Campus No. 127, Topkapi, 34010 Istanbul, Turkey Received 29 October 2001; accepted 12 August 2002

Abstract In this paper, we attempt to develop a method for the stress analyses in an infinite elastic body containing two neighbouring fibres which are located along two parallel lines, each of them having a periodical curving which is of opposite phase according to the other. The stress distribution is studied when the body is loaded at infinity by uniformly distributed normal forces with intensity p acting in the direction of fibres. The investigation is carried out in the framework of the piecewise homogeneous body model with the use of the three-dimensional geometrically non-linear exact equations of the theory of elasticity. Consequently, in the present investigation the effect of the geometrical non-linearity on the considered stress distribution is also taken into account. The numerous numerical results related to the considered stress distribution and to the influence of the problem parameters on this distribution are given. q 2003 Elsevier Science Ltd. All rights reserved. Keywords: A. Fibres; Geometrical non-linearity

1. Introduction One of the major problems in mechanics of composite materials is a problem associated with the special features of the structure of these materials. Among these specific features of composite material structure, the curving (bending) of reinforcing elements is of considerable importance. As it is well known, these bent, curved structural elements may be the result of specific structural characteristics of composite materials caused by design factors as well as a consequence of technological processes under the action of various factors. The effective practical applications of composite materials, in service conditions, require intensive, systematic investigations to determine the stress deformed state in these materials, taking the curving of reinforcing elements into account. At present, two basic approaches may be distinguished in the study of the mechanics of composite materials with curved structure. In the first, the continuum approaches offer the possibility of taking the influence of reinforcing element curving into account * Corresponding author. Tel.: þ90-212-449-1684; fax: þ 90-212-4491514. E-mail address: [email protected] (S.D. Akbarov).

in calculating the components of the stress –strain state for the areas considerably greater in size than the curving period. These approaches are given in Refs. [1 – 4] and many others. In the second type of approaches, developed considerably later than the first one, methods enabling the influence of reinforcing-element curving to be taken into account in calculating the components in the stress – strain state in areas of size comparable with, or smaller than the curved period, are developed. These approaches were developed in the framework of the continuum theory as well as in the framework of the piecewise homogeneous body model. The review of the related investigations is given in Refs. [5,6]. The consistent consideration of these investigations has been studied in Ref. [7]. It follows from Refs. [5 –7] that, up to now the investigations of the stress – strain state in the unidirectional fibrous composites with curved fibres are carried out for only small concentration of fibres and in this case composite material is modeled as an infinite elastic body containing a single periodical curved fibre. For investigation of such a problem in the framework of the piecewise homogeneous body model with the use of three-dimensional linear theory of elasticity, a method is developed in Ref. [8] and the corresponding numerical

1359-8368/03/$ - see front matter q 2003 Elsevier Science Ltd. All rights reserved. PII: S 1 3 5 9 - 8 3 6 8 ( 0 2 ) 0 0 0 7 7 - X

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S.D. Akbarov, R. Kosker / Composites: Part B 34 (2003) 143–150

results are analysed in Ref. [9]. However, up to now no results related to the cases where the fibre concentration is high in composite have yet been obtained. The investigation on how the stress distribution mentioned earlier is effected by the reciprocal effect between fibres, as volume ratio of fibres gets bigger in composites, is very important. This study, being the first attempt in this field develops a method for solution of these problems and investigates the stress distribution in an infinite elastic body containing two neighbouring fibres which are located along two parallel lines, each having a periodical curving which is of opposite phase according to the other. This stress distribution is studied when the body is loaded at infinity by uniformly distributed normal forces with intensity p acting in the direction of the fibres. The investigation is carried out in the framework of the piecewise homogeneous body model with the use of the three-dimensional geometrical non-linear exact equations of the theory of elasticity. Consequently, in the present investigation the effect of the geometrical nonlinearity on the considered stress distribution is also taken into account. Fig. 1. The geometry of material structure.

2. Formulation of the problem

follows:

Now, we consider the mathematical formulation of the problem. With the middle line of each fibre, we associate Lagrangian rectilinear Oq x1q x2q x3q and cylindrical Oq rq uq zq system of coordinates (Fig. 1), where q ¼ 1; 2 are related, respectively, to first and second fibre. Between these coordinates we have the following relations (Fig. 2):

rq ¼ R þ

1 X

1k aqk ðuq ; t3 Þ;

zq ¼ t 3 þ

k¼1

nqr ¼ 1 þ

1 X

nqz ¼

1k bqk ðuq ; t3 Þ;

k¼1

1k cqk ðuq ; t3 Þ;

k¼1 1 X

1 X

nq u ¼

1 X

1k dqk ðuq ; t3 Þ;

k¼1

1k fqk ðuq ; t3 Þ

ð3Þ

k¼1

x12 ¼ x22 ;

x13 ¼ x23 ;

r1 eiu1 ¼ R12 þ r2 eiu2 ;

ð1Þ

z1 ¼ z2 ¼ z We assume that the equations of the middle line of each fibre are given as follows:
 2p x13 ; x12 ¼ 0; x11 ¼ L sin ‘

x21 ¼ 2L sin




2p ‘

 x23 ;

ð2Þ

t 3 is a parameter and t3 [ ð21; þ1Þ; the explicit expression of functions aq ðuq ; t3 Þ; …; fq ðuq ; t3 Þ in Eq. (3) are given in Ref. [7]. Throughout the investigations, repeated indices are summed over their ranges; however, underlined repeated indices are not summed. Below the values related to the fibres will be denoted by the upper indices (21) and (22), but those related to the matrix by upper index (1). Thus, within the fibres and infinite matrix in the geometrical

x22 ¼ 0

and the cross-section of each fibre which is perpendicular to the middle line is a circle with constant radius R (Fig. 2) and this is invariant along the entire length of the fibre. Assume that L (curving amplitude of the fibre) is smaller than ‘ (the length period of the curving); we introduce a small parameter 1 ¼ L=‘; ð0 , 1 p 1Þ: If contact surfaces between the fibres and matrix are denoted by S1 and S2, from Eq. (2) and the condition of fibre cross-section, the equations of these surfaces and the components of their normal vectors are derived as

Fig. 2. Chosen coordinates.

S.D. Akbarov, R. Kosker / Composites: Part B 34 (2003) 143–150

non-linear statement and in the cylindrical system of coordinates, we write the governing field equations: 7i ½sðkÞin ðgjn þ 7n uðkÞj Þ ¼ 0; ðkÞ

ðkÞ ðkÞ ðkÞn 21ðkÞ 7m un ; jm ¼ 7j um þ 7m uj þ 7j u

ð4Þ

ðkÞ

ðkÞ ðkÞ ðkÞ n ðkÞ ðkÞ ðkÞ sðkÞ ¼ 1ðkÞ rr þ 1uu þ 1zz ðinÞ ¼ ðl e Þdi þ 2ðm 1ðinÞ Þ; e

It is assumed that on the inter-medium surfaces Sq (Fig. 1), the complete cohesion conditions are satisfied:

sð2qÞin ðgjn þ 7n uð2qÞj ÞlSq nqðjÞ ¼ sð1Þin ðgjn þ 7n uð1Þj ÞlSq nqðjÞ ; ð2qÞ

uj

ð5Þ lSq ¼ uð1Þ j lSq

In the considered case, it is also assumed that the conditions ð1Þ sð1Þ zz ! p; sðijÞ ! 0; rq !1

rq !1

ðijÞ – zz are satisfied. In Eqs. (4) and (5), the conventional tensor notation is used and subindices in parentheses show the physical components of the corresponding tensors. It is known that

sðijÞ ¼ sij Hi Hj ¼ si j

1 ¼ 1ij Hi Hj ; Hi Hj

using Eq. (3) and with some process as detailed in Refs. [7,8], we obtain the contact condition satisfied in rq ¼ R for each approach in Eq. (7). In this case according to Ref. [7] in the field equations related to the zeroth approximation, we neglect the non-linear terms. Further, by direct verification we prove that the field equations related to the first, second and subsequent approximations are the equations of three-dimensional linearised theory of elasticity [10,11]. Assuming that ›uð0Þ ðiÞ =›xðiÞ p 1 ðxðiÞ ¼ r; u; zÞ; we write contact conditions belonging to zeroth and first approximation:

sð2qÞ;0 ¼ sð1Þ;0 uð2qÞ;0 ¼ uð1Þ;0 ðijÞ ðiÞ ðijÞ ; ðiÞ  2q;0   › s ›sðiÞr 2q;0 2q;0 ðiÞr 2q;1 ½sðiÞr 1;1 þ f1q þw1q þgrq ½sðiÞr 1;0 ›r 1;0 ›z 1;0 2q;0

2q;0

þ guq ½sðiÞu 1;0 þ gzq ½sðiÞz 1;0 ¼ 0     ›uðiÞ 2q;0 ›uðiÞ 2q;0 2q;1 þw1q ¼0 ½uðiÞ 1;1 þ f1q ›r 1;0 ›z 1;0 ð8Þ where ðijÞ ¼ rr; r u; rz; ðiÞ ¼ r; u; z and ð2qÞ;s ½w2q;s 2 wð1Þ;s ; 1;s ¼ w

1 ; H i Hj ð6Þ

1ðijÞ ¼ 1ij

145

uðiÞ ¼ ui Hi ¼ ui

1 Hi

w1q ¼ grq ¼

2Rd0q ðt3 Þcosuq ;

dq ðt3 Þ R

2

d00q ðt3 ÞR

gzq ¼ 2d0q ðt3 Þcosuq ;

f1q ¼ dq ðt3 Þcosuq ; ! cosuq ;

guq ¼

dq ðt3 Þ R

sinuq ;

dq ðt3 Þ ¼ ‘ sinð2pt3 =‘Þ

where ðijÞ ¼ rr; uu; zz; r u; rz; zu; ðiÞ ¼ r; u; z: Here and in the previous equations, the contravariant (covariant) components of corresponding tensors or vectors are indicated by upper (lower) indices. Moreover, in Eq. (6), the Lame´’s coefficients are denoted by Hi and the covariant components of the unit normal vector to the surface Sq are denoted by nqðjÞ : Writing the expression of the Lame´’s coefficients in the cylindrical coordinate system and rearranging we can obtain the expression of Eqs. (4) and (5) in the cylindrical system of coordinates. Thus, with the above-stated, the formulation of the considered problem is exhausted.

Similar contact conditions are obtained for the subsequent approximations. Now, we determine the unknown values belonging to the zeroth and first approximations. Assume that the materials of each fibre are the same and Poisson coefficient of this material nð21Þ ¼ nð22Þ (nð2qÞ denotes Poisson coefficient of qth fibre) is equal to Poisson coefficient of matrix material denoted by n (1). Thus, for zeroth approach we obtain:

3. Method of solution

sð1Þ;0 ¼ p; zz

For investigation of this problem. we use the boundary shape perturbation method developed in Refs. [7,8] according to which the unknown values are presented in series form in 1: ðmÞ ðmÞ {sðmÞ ðijÞ ; 1ðijÞ ; uðiÞ }

¼

1 X

1

k

ðmÞ;k ðmÞ;k {sðmÞ;k ðijÞ ; 1ðijÞ ; uðiÞ };

k¼1

ð9Þ

sð21Þ;0 ¼ sð22Þ;0 ¼ zz zz

ð21Þ;0 1zz ¼ 1ð22Þ;0 ¼ 1ð1Þ;0 ¼ zz zz

Eð2Þ p; Eð1Þ

p ; Eð1Þ

uzð21Þ;0 ¼ uð22Þ;0 ¼ uð1Þ;0 ¼ 1ð1Þ;0 z z zz z;

ð10Þ z ¼ z1 ¼ z2

ð2qÞ;0 sðijÞ ¼ sð1Þ;0 ðijÞ ¼ 0; ðijÞ ¼ rr; uu; r u; uz; rz

ð7Þ

ðijÞ ¼ rr; uu; zz; r u; rz; uz; ðiÞ ¼ r; u; z From Eq. (4), we obtain equations set for each approximation in Eq. (7). Substituting the expression (7) in Eq. (5),

The assumption of equality Poisson coefficients of fibre and matrix materials have not a considerable effect on numerical results, as is known from Ref. [7]. This assumption is accepted just to simplify the solution procedure. Now we consider the equations for the values of the first

146

S.D. Akbarov, R. Kosker / Composites: Part B 34 (2003) 143–150

approximations. These are

given, for example, in Ref. [10]. In Eq. (15) the functions of C, x satisfy the following equations: ! 2 2 › D1 þ j1 2 c ¼ 0; ›z ð16Þ ! ! 2 2 2 › 2 › D 1 þ j2 2 D 1 þ j3 2 x ¼ 0 ›z ›z

›sðkÞ;1 1 ›sðkÞ;1 ›sðkÞ;1 1 rr rz ru þ þ ðsðkÞ;1 þ 2 sðkÞ;1 uu Þ r ›u r rr ›r ›z ðkÞ;1

ðkÞ;0

þ szz

›2 ur ›z 2

¼ 0;

›sðkÞ;1 ›sðkÞ;1 1 ›sðkÞ;1 2 uz ru uu þ þ sðkÞ;1 þ r ›u r ru ›r ›z

ð11Þ

ðkÞ;1

ðkÞ;0

þ szz

›2 uu ›z 2

¼ 0;

ðkÞ;1 ›sðkÞ;1 1 ›suz ›sðkÞ;1 1 rz zz þ þ sðkÞ;1 þ r ›u r rz ›r ›z

Taking the expression of the contact conditions into account, the solutions to Eq. (16) are found as follows:

ðkÞ;1

ðkÞ;0

þ szz

›2 uz ›z 2

¼ 0;

cð2qÞ ¼ a sinaz ðkÞ;1

ðkÞ ðkÞ;1 n sðkÞ;1 Þdi þ 2ðmðkÞ 1ðinÞ Þ ðinÞ ¼ ðl e

e

ðkÞ;1

xð2qÞ ¼ cosaz

1ðkÞ;1 rr ¼ 1ðkÞ;1 zz ¼ 1ðkÞ;1 uz ¼

›r ›uðkÞ;1 z ›z

; 1ðkÞ;1 uu ¼ ;

1ðkÞ;1 ru ¼

›uðkÞ;1 u r ›u 1 2

þ

uðkÞ;1 r

›uðkÞ;1 r r ›u

r þ

›r

2

uðkÞ;1 u r

! ;

cð1Þ ¼ a sinaz ð13Þ

! 1 ›uðkÞ;1 ›uðkÞ;1 z u þ ; 2 ›z r ›u

1 1ðkÞ;1 zr ¼ 2

›uðkÞ;1 z

þ

›uðkÞ;1 r

Cnð2qÞ In ðj1ð2qÞ arq Þexpðinuq Þ

1 X

½Anð2qÞ In ðjð2qÞ 2 ar q Þ

n¼21 ð2qÞ þBn In ðj3ð2qÞ arq Þexpðinuq Þ

;

›uðkÞ;1 u

1 X n¼21

ð12Þ

ðkÞ;1 ðkÞ;1 ¼ 1ðkÞ;1 rr þ 1uu þ 1zz ;

›uðkÞ;1 r

In Eq. (16) ji ði ¼ 1;2;3Þ are constants and determined by the following expressions: sffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m þ s0zz m þ s0zz l þ2m þ s0zz j1 ¼ ; j2 ¼ ; j3 ¼ ð17Þ m m l þ2m

2 1 X X

ð18Þ

Cnð1Þq Kn ðjð1Þ 1 arq Þexpðinuq Þ

q¼1 n¼21

xð1Þ ¼ cosaz

2 1 X X

ð1Þ ½Að1Þq n Kn ðj2 arq Þ

q¼1 n¼21 ð1Þ þBð1Þq n Kn ðj3 arq Þexpðinuq Þ

!

ð19Þ

From Eqs. (8) –(10) for these approximations, we obtain the following contact conditions:

where a ¼ 2p=‘ and In ðxÞ; Kn ðxÞ are Bessel functions of a purely imaginary argument and Macdonald functions, in turn. Moreover, the unknowns Cnð2qÞ ;…;Bð2qÞ are the n complex constant and satisfy the relations:

2q;1 ½srr 2q;1 1;1 ¼ 0; ½sru 1;1 ¼ 0;

ð2qÞ ð2qÞ ð2qÞ ð2qÞ ð2qÞ Að2qÞ n ¼ A2n ; Bn ¼ B2n ; Cn ¼ C2n ;

›r

›z

2q;1 ð1Þ;0 ð2Þ;0 ½srz 2q;1 1;1 ¼ 2pðszz 2 szz Þcosat3 cosu; ½ur 1;1 ¼ 0;

ð14Þ

ð2qÞ ð2qÞ Im Að2qÞ 0 ¼ Im B0 ¼ Im C0 ¼ 0

2q;1 ½uu 2q;1 1;1 ¼ 0; ½uz 1;1 ¼ 0

ð1Þq ð1Þq ð1Þq ð1Þq ð1Þq Að1Þq n ¼ A2n ; Bn ¼ B2n ; Cn ¼ C2n ;

By direct verification, we prove that Eqs. (11) – (14) are the equations of the three-dimensional linearised theory of stability [10,11]. Therefore, we can use the Guz’s representations in the cylindrical system of coordinates.

ð1Þq ð1Þq Im Að1Þq 0 ¼ Im B0 ¼ Im C0 ¼ 0

1 › ›2 › 1 ›2 c2 x; uu ¼ 2 c 2 x; r ›u ›r r ›r ›z ›r ›z ! 2 21 0 › u3 ¼ ðl þ mÞ ðl þ2mÞD1 þðm þ szz Þ 2 x; ›z

ur ¼

D1 ¼

›2 1 › 1 ›2 þ 2 2 þ 2 r ›r r ›u ›r

ð15Þ

ð20Þ

Now we attempt to satisfy the contact condition (14). For this purpose, we must represent the expressions (18) and (19) in the qth ðq ¼ 1;2Þ cylindrical coordinate system to satisfy the contact conditions on the qth fibre – matrix interface Sq : The expressions (18) are already presented in the qth cylindrical system of coordinates. To make these operations for the expressions (19), we use the summation theorem [12] for the Kn ðxÞ function, which can be written for the case at hand as follows: rm expðium Þ ¼ Rmn expðiwmn Þþrn expðiun Þ

S.D. Akbarov, R. Kosker / Composites: Part B 34 (2003) 143–150

Kn ðcrn Þexpðinun Þ ¼

1 X

ð21Þn Ik ðcrm ÞKn2n ðcRmn Þ

k¼21

£exp½iðn 2nÞwmn expðikum Þ mn ¼ 12;21; m;n ¼ 1;2; rm , Rmn ; R12 ¼ R21 ;

w12 ¼ 0; w21 ¼ p

ð21Þ

Using Eqs. (18) –(21) we obtain from Eq. (14) an infinite system of algebraic equations with respect to the unknown constants (Eq. (20)). Introducing the notation ð1Þq ð1Þq Cnð1Þq Kn ðjð1Þ 1 kÞ ¼ yn1 þizn1 ; ð1Þq ð1Þq ð1Þ Að1Þq n Kn ðj2 kÞ ¼ zn2 þiyn2

ð22Þ

ð1Þq ð1Þq ð2Þq ð2Þq ð1Þ ð2Þ ð2Þq Bð1Þq n Kn ðj3 kÞ ¼ zn3 þiyn3 ; Cn In ðj1 kÞ ¼ yn1 þizn1 ð2Þq ð2Þq ð2Þq ð2Þq ð2Þ ð2Þ Að2Þq Bð2Þq n In ðj2 kÞ ¼ zn2 þiyn2 ; n In ðj3 kÞ ¼ zn3 þiyn3

ðkÞq

ðkÞq

z

y

n1

n1







ðkÞq

ðkÞq ðkÞq Zn ¼ zn2 ; Yn ¼ yðkÞq

;



n2

ðkÞq

ðkÞq

z

y

n3 n3 ð1Þq ð2Þq ð2Þq Dð1Þq nv ¼ kdrs ðn;vÞk; Dn ¼ kdrs ðnÞk ð1Þq Fnv ¼ kfrsð1Þq ðn;vÞk; Fnð2Þq ¼ kfrsð2Þq ðnÞk; q ¼ 1;2; r;s ¼ 1;2;3;

For numerical investigations the infinite system of algebraic Eq. (25) must be approximated by a finite system. To validate such a replacement it must be shown that the determinant of these infinite system of equations must be of normal type [13]; this holds if we can prove the convergence of the series M¼

1 X 1 X

Znð1Þ1 þ

ð1Þ2 ð2Þ1 Dð1Þ2 þDð2Þ1 nv Zv n Zn ¼ 0;

v¼0

Znð1Þ2 þ

1 X

ð23Þ ð1Þ1 ð2Þ2 Dð1Þ1 þDð2Þ2 nv Zv n Zn ¼ 0

v¼0

Ynð1Þ1 þ

1 X

ð1Þ2 ð1Þ2 ðnÞ1;0 Fnv Yv þFnð2Þ1 Ynð2Þ1 ¼ 2pd3n ðsð1Þ;0 Þ zz 2 szz

v¼0

Ynð1Þ2 þ

1 X

ð1Þ1 ð1Þ1 ðnÞ1;0 Fnv Yv þFnð2Þ2 Ynð2Þ2 ¼ 22pd3n ðsð1Þ;0 Þ zz 2 szz

v¼0

ð24Þ where n ¼ 0;1;2;…;1; d33 ¼ 1; d3n ¼ 0; if n – 3: ðkÞq Note that the component of the matrices DðkÞq nv ; Fnv ; ð2Þq ð2Þq Dn ; Fn are obtained from the formulae (12) – (15), (19) – (21) and since it is cumbersome we omit here the detailed expressions. It follows from Eq. (23) that ZnðkÞq ¼ 0; k ¼ 1; 2; q ¼ 1; 2: Moreover, it follows from the symmetry with respect to the plane mm1 (Fig. 1) that YnðkÞ1 ¼ 2YnðkÞ2 : Taking this into account, from Eq. (24) we obtain Ynð1Þ1 2

1 X

ð1Þ2 ð1Þ1 Fnv Yv þ Fnð2Þ1 Ynð2Þ1 ¼ 2pd3n ðsð1Þ;0 2 sð2Þ;0 zz zz Þ

v¼0

ð25Þ

ð26Þ

For investigation of the series (26), we use the following asymptotic estimates of the functions In ðxÞ and Kn ðxÞ :
 1 lxl n ; c1 ¼ const; In ðxÞ , c1 n! 2 ð27Þ
n 2 Kn ðxÞ < c2 ðn 2 1Þ! ; c2 ¼ const lxl These hold for large n and fixed x. Furthermore, we use the following inequality R=ðR12 2 2LÞ . R=R12 ;

R12 =R . 2

ð28Þ

which means that the fibres do not touch each other. Thus, taking Eqs. (27) and (28) into account and analysing the ð1Þ2 expressions of Fnv we obtain the following estimate for the series (26) M , c3

1 X

ð1Þ2 lFnv l

n¼0 v¼0

k ¼ 2pR=‘ we can unite this infinite set of equations in the following form

147

1 X

nc4 ðr 2 1Þ2n ;c3 ;c4 ¼ const;

r ¼ R12 =R

ð29Þ

n¼0

As the series on the right-hand side converges, so does Eq. (26). Thus the determinant of the infinite system of Eq. (25) is normal and the infinite systems can be replaced by the finite system for numerical purposes. The required number of equations in this finite system must be determined from the convergence numerical results. This concludes the discussion of the first approximatons. Subsequent approximations can be found likewise.

4. Numerical results According to the mechanical consideration, for the considered case as a result of the interaction between the fibres, the self-equilibrium shear stresses which arise as a result of the fibre curving on the interfaces must decrease and normal stress (denoted as snn ) must increase. This situation is also proven by the obtained numerical results which are not shown here. Therefore, here we consider only the numerical results related to the self-equilibrium normal stress snn : Note that the stress snn has a great significance under the estimation of the adhesion strength of the considered composite material. Moreover, note that the present investigation and the investigation carried out in Ref. [9] and others listed in Refs. [5,6] show that the stress snn has its maximum at the point N1 on the surface S1 or at

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S.D. Akbarov, R. Kosker / Composites: Part B 34 (2003) 143–150

Table 1 The values of snn =lpl for various k and p=Eð1Þ with Eð2Þ =Eð1Þ ¼ 50; R12 =R ¼ 2:5

k

p=Eð1Þ Tension

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Compression

5 £ 1025

5 £ 1024

5 £ 1023

5 £ 1022

25 £ 1025

25 £ 1024

25 £ 1023

25 £ 1022

0.8050 1.5421 2.1265 2.4602 2.4941 2.2792 1.9332 1.5646 1.2347 0.9642

0.8050 1.5415 2.1246 2.4566 2.4892 2.2737 1.9281 1.5602 1.2312 0.9615

0.8041 1.5352 2.1062 2.4218 2.4408 2.2204 1.8781 1.5179 1.1974 0.9351

0.7957 1.4756 1.9377 2.1191 2.0397 1.7929 1.4848 1.1873 0.9327 0.7278

20.8050 21.5422 22.1269 22.4610 22.4952 22.2804 21.9343 21.5655 21.2355 20.9648

20.8051 21.5429 22.1288 22.4645 22.5002 22.2859 21.9395 21.5699 21.2390 20.9675

20.8060 21.5491 22.1476 22.5005 22.5508 22.3423 21.9926 21.6150 21.2750 20.9955

20.8146 21.6149 22.3548 22.9251 23.1900 23.0928 22.7226 22.2410 21.7744 21.3804

the point N2 on the surface S2. The points N1 and N2 correspond, respectively, to the case gt3 ¼ p=2; u1 ¼ 0 (for N1) and u2 ¼ 0 (for N2) (Fig. 1). Taking the above stated and the symmetry of the material structure with respect to the plane mm1 (Fig. 1) into account, we consider below only the values of the snn obtained at point N1 on the interface S1. In this study, numerical results are obtained in the framework of the zeroth and first approximations. Therefore in the quantitative sense, these results can be acceptable for the small degree of the curving, i.e. for the small values of the parameter 1. Therefore, we assume that 1 ¼ 0.015 and all numerical results which are discussed below are obtained for this case. However, in the qualitative sense these results have also a great significance for moderately greater values of 1. In these cases, the accuracy of the numerical results can be improved by determination of the second and subsequent terms of the series (7). All numerical results which will be discussed below have been obtained by the use of PC and corresponding algorithm and programming are developed and composed by the authors. Table 1 shows the values of snn =lpl for various k ¼ 2pR=‘ and p=Eð1Þ under, Eð2Þ =Eð1Þ ¼ 50; R12 =R ¼ 2:5: Note that the results corresponding to the case p=Eð1Þ ¼ 5 £ 1025 can be taken as the results obtained in the geometrical linear

statement. It follows from Table 1 that as a result of the geometrical non-linearity under tension of the considered material (i.e. under p=Eð1Þ . 0), the values of snn decrease monotonically with p=Eð1Þ : However, under compression of this material (i.e. under p=Eð1Þ , 0), the absolute values of snn increase monotonically with lpl=Eð1Þ : Moreover, Table 1 shows that the dependence between snn =lpl and k is nonmonotonic. It should be noted that such character of this dependence is observed for other values of the problem parameters Eð2Þ =Eð1Þ ; R12 =R: The non-monotonic character of the dependence between snn =lpl and k had also been observed for the single curved fibre in the infinite matrix and corresponding results have been detailed in Refs. [7,9]. Note that under the consideration of compression we assume that lpl=Eð1Þ , lpcr l=Eð1Þ ; where pcr is the critical values of the compressed force intensity p obtained for the stability loss problem of the two fibres in an infinite matrix. The investigation of this stability loss problem and the values of lpcr =Eð1Þ are given in Refs. [14,15]. In Tables 2 –4 the values of snn =lpl are given for various p=Eð1Þ ; R12 =R under Eð2Þ =Eð1Þ ¼ 20; 50 and 100, respectively, in the case where 2pR=‘ ¼ 0:4: It follows from these tables that in the case where R12 =R ! 1 and p=Eð1Þ ¼ 5 £ 1025 the values of snn =lpl approach the corresponding ones obtained in Ref. [9] in which the considered problem had been

Table 2 The values of snn =lpl for various p=Eð1Þ and R12 =R with Eð2Þ =Eð1Þ ¼ 20; 2pR=‘ ¼ 0:4 p=Eð1Þ

5 £ 1025 5 £ 1024 5 £ 1023 5 £ 1022 25 £ 1025 25 £ 1024 25 £ 1023 25 £ 1022

R12 =R 1

10

5

4

3

2.5

0.2582 [7,9] 0.2578 0.2542 0.2240 20.2583 [7,9] 20.2586 20.2624 20.3086

0.2693 0.2689 0.2651 0.2327 20.2694 20.2698 20.2738 20.3229

0.3645 0.3640 0.3590 0.3161 20.3646 20.3651 20.3702 20.4310

0.4466 0.4460 0.4405 0.3919 20.4467 20.4473 20.4530 20.5187

0.6510 0.6504 0.6441 0.5872 20.6512 20.6518 20.6582 20.7295

0.9362 0.9355 0.9288 0.8664 20.9363 20.9370 20.9438 21.0173

S.D. Akbarov, R. Kosker / Composites: Part B 34 (2003) 143–150

149

Table 3 The values of snn =lpl for various p=Eð1Þ and R12 =R with Eð2Þ =Eð1Þ ¼ 50; 2pR=‘ ¼ 0:4 R12 =R

p=Eð1Þ

5 £ 1025 5 £ 1024 5 £ 1023 5 £ 1022 25 £ 1025 25 £ 1024 25 £ 1023 25 £ 1022

1

10

5

4

3

2.5

0.5774 [7,9] 0.5757 0.5590 0.4363 20.5778 [7,9] 20.5796 20.5977 20.8856

0.6069 0.6050 0.5874 0.4570 20.6073 20.6091 20.6283 20.9288

0.8585 0.8561 0.8330 0.6560 20.8590 20.8614 20.8862 21.2452

1.0785 1.0757 1.0494 0.8422 21.0791 21.0818 21.1098 21.4956

1.6403 1.6371 1.6057 1.3456 21.6410 21.6442 21.6771 22.0930

2.4602 2.4566 2.4218 2.1191 22.4610 22.4645 22.5005 22.9251

Table 4 The values of snn =lpl for various p=Eð1Þ and R12 =R with Eð2Þ =Eð1Þ ¼ 100; 2pR=‘ ¼ 0:4 R12 =R

p=Eð1Þ

5 £ 1025 5 £ 1024 5 £ 1023 5 £ 1022 25 £ 1025 25 £ 1024 25 £ 1023 25 £ 1022

1

10

5

4

3

2.5

0.9450 [7,9] 0.9405 0.8973 0.6214 20.9461 [7,9] 20.9507 20.9997 22.1536

0.9984 0.9935 0.9478 0.6542 20.9995 21.0043 21.0561 22.2483

1.4610 1.4545 1.3922 0.9752 21.4625 21.4691 21.5386 22.9245

1.8769 1.8692 1.7957 1.2871 21.8786 21.8864 21.9676 23.4447

2.9772 2.9674 2.8732 2.1768 22.9794 22.9892 23.0911 24.6723

4.6526 4.6411 4.5295 3.6455 24.6551 24.6667 24.7851 26.3951

with approaching the fibres each other (i.e. with decreasing R12 =R) the stretching (compression) increases monotonically, therefore the values of snn =lpl also increase monotonically. Table 5 shows the numerical convergence of the values of snn =lpl obtained for various p=Eð1Þ under various numbers of the linear equations in the framework of which this investigation is carried out. It is assumed that Eð2Þ =Eð1Þ ¼ 50; k ¼ 0:3; R12 =R ¼ 2:5: The comparison of the numerical results given in Table 5 and obtained in various number of the linear equations shows that the used solution method is highly effective in the convergence sense.

investigated for a single fibre in the geometrical linear statement, i.e. with the use of the linear theory of elasticity. The comparison of the results attained for various Eð2Þ =Eð1Þ show that the absolute values of snn =lpl increase monotonically with Eð2Þ =Eð1Þ : Moreover, these tables show that the absolute values of snn =lpl increase monotonically with decreasing R12 =R: This fact can be explained as follows. Under tension (compression) of the considered body the fibre at the selected point ðgt3 ¼ p=2; u1 ¼ 0Þ; of the interface stretches (compresses) the matrix material. As a result of this stretching (compression), the self-balanced normal stress arises. It is evident that in the considered case

Table 5 The numerical convergence of the values of snn =p with the various numbers of the linear equations ðEð2Þ =Eð1Þ ¼ 50; 2pR=‘ ¼ 0:3; R12 =R ¼ 2:5Þ p=Eð1Þ

5 £ 1025 5 £ 1024 5 £ 1023 5 £ 1022 25 £ 1025 25 £ 1024 25 £ 1023 25 £ 1022

Number of the equations 16

28

40

52

64

76

94

118

1.4391 1.4378 1.4246 1.3045 21.4394 21.4407 21.4542 21.6040

1.9337 1.9320 1.9153 1.7617 21.9341 21.9358 21.9529 22.1418

2.0761 2.0743 2.0563 1.8918 22.0765 22.0783 22.0967 22.2989

2.1135 2.1117 2.0934 1.9259 22.1139 22.1158 22.1345 22.3404

2.1232 2.1213 2.1029 1.9347 22.1236 22.1254 22.1442 22.3511

2.1256 2.1238 2.1054 1.9369 22.1260 22.1279 22.1467 22.3538

2.1264 2.1245 2.1061 1.9376 22.1268 22.1287 22.1475 22.3546

2.1265 2.1246 2.1062 1.9377 22.1269 22.1288 22.1476 22.3547

150

S.D. Akbarov, R. Kosker / Composites: Part B 34 (2003) 143–150

5. Conclusion In the present paper, in the framework of the piecewise homogeneous body model with the use of the threedimensional geometrically non-linear exact equations of the theory of elasticity, the method for the determination of the stress – strain state in the unidirected fibrous composites with periodically curved fibres is developed for the cases where the interaction between the fibres is taken into account. All investigations are made for the infinite elastic body containing neighbouring two periodically curved neighbouring fibres which are located along two parallel lines. It is assumed that the curving of the fibres is of opposite phase according to each other. It is also assumed that the middle line of the fibres is on the same plane and the uniformly distributed normal forces act at infinity in the direction of the fibres. The numerical results, related to the self-balanced normal stress which acts on the interface and arises as a result of the fibres curving, are given. Here, these results are illustrated for the point at which the normal stress has its maximum. From the analyses of the results are derived the following conclusions for the considered fibres location: † The value of the self-balanced normal stress increases monotonically with the fibres approaching each other. † As a result of the geometrical non-linearity, the absolute value of the normal stress increases in compression but decreases in tension. Obtained numerical results agree with the well-known mechanical consideration and in the particular cases coincide with the corresponding known results. The approach proposed in the present paper, with concrete problems as an example, after some obvious changes and developments, can successfully be applied for investigations the numerous other problems on the determination of the stress – strain state in the fibrous composite materials

with curved fibres under high concentration of fibres in these composites.

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