A micro-mechanical model of woven fabric and its application to the analysis of buckling under uniaxial tension. Part 2: buckling analysis

International Journal of Engineering Science 39 (2001) 1±13

www.elsevier.com/locate/ijengsci

A micro-mechanical model of woven fabric and its application to the analysis of buckling under uniaxial tension. Part 2: buckling analysis Y.T. Zhang a,*, Y.B. Fu b a

Department of Mechanics, Tianjin University, Tianjin 300072, People's Republic of China b Department of Mathematics, University of Keele, Sta€ordshire ST5 5BG, UK Received 3 November 1999; accepted 29 November 1999

Abstract With the aid of the micro-mechanical model for woven fabric proposed in Part 1, we analyse the buckling of a fabric sheet when it is subjected to uniaxial tension. To simplify analysis we focus our attention on the case when the inclination angle between the direction of uniaxial tension and that of warp is either zero or 45°. We ®rst show that under the traditional orthotropic continuum model out-of-plane buckling of fabric is not possible whether the inclination angle is zero or 45°. Using our new model, we show that out-of-plane buckling is not possible when the inclination angle is zero but is possible when the inclination angle is 45°. Both results agree with our experimental results. Ó 2000 Elsevier Science Ltd. All rights reserved.

1. Introduction Fabric has two preferred directions: those along the warp and the weft. If a piece of fabric is stretched along the warp (resp. weft) direction, the stretching force is sustained by the ®bres along the warp (resp. weft) direction and ®bres along the weft (resp. warp) direction are stress free. When the same piece of fabric is stretched along other non-preferred directions, out-of-plane buckling can usually be observed. Fig. 1 shows a photograph of a typical buckled con®guration of a piece of fabric which is stretched along a direction making an angle of 45° with the direction of warp (the reader may like to recon®rm this result by stretching his or her trousers). We note that such out-of-plane buckling under stretching does not occur in paper, fabric-reinforced composite

*

Corresponding author. Fax: +86-22-23358329. E-mail address: [email protected] (Y.T. Zhang).

0020-7225/01/$ - see front matter Ó 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 0 2 0 - 7 2 2 5 ( 0 0 ) 0 0 0 1 3 - 6

2

Y.T. Zhang, Y.B. Fu / International Journal of Engineering Science 39 (2001) 1±13

Fig. 1. A typical buckled picture of a sheet of plain fabric which is subjected to a uniaxial tension along a direction making an angle of 45° with respect to the warp direction.

plates or other anisotropic materials. We attribute this di€erence to a unique property of ®bres in fabric, namely that ®bres in fabric are woven and are not ®xed at their intersections so that they can slide and turn almost freely. A close inspection of the shearing behaviour of woven fabric shows that although woven fabric has almost zero shearing sti€ness initially, resistance to shearing is still provided by the squeezing of ®bres along a certain direction. Based on this observation, we proposed in Part 1 of this paper a micro-mechanical model for woven fabric. We anticipate that this new model will be able to explain why out-of-plane buckling can take place when a piece of fabric is stretched along a non-preferred direction. This will be con®rmed by the analysis carried out in the present part. In a typical draping experiment, a piece of fabric is placed on a circular or square table and the properties of the hanging part of the fabric are then visually examined. In the simulation of draping of fabric, the theories of cantilevers, plates, shells, and even rings and particles are applied and the results are mostly numerical, see e.g. [1±3]. A variety of such experimental results can be found in [4]. Whereas much progress has been made on the simulation of fabric draping, see e.g. [1,5±7], literature on the simulation of fabric buckling has been scarce. This is probably because on the one hand, fabric can undergo large deformations and the additional complexity of geometry makes analytical studies of buckling dicult, and on the other hand, there does not exist a satisfactory model which can characterise the mechanical properties of fabric correctly. In Section 2, we formulate the problem and summarize the main governing equations. We consider the problem of a sheet of fabric that is subjected to a simple tension. To simplify analysis we assume that the inclination angle between the direction of tension and that of warp is zero or 45°. In Section 3 we show that, under the traditional orthotropic continuum model, out-of-plane buckling of fabric is not possible whether the inclination angle is zero or 45°. In Section 4, we show using our micro-mechanical model that out-of-plane buckling is not possible when the inclination angle is zero but is possible when the inclination angle is 45°. Both results agree with our experimental results.

Y.T. Zhang, Y.B. Fu / International Journal of Engineering Science 39 (2001) 1±13

3

2. Governing equations of buckling Consider the fabric sheet shown in Fig. 2. We choose a rectangular coordinate system so that in its undeformed con®guration B0 the sheet is de®ned by ÿ1 < X1 < 1;

jX2 j 6 a;

jX3 j 6 h;

…1†

where the thickness 2h is small compared with the width 2a. We assume that the boundary surfaces X2 ˆ a and X3 ˆ h are traction free and that the application of a uniaxial tension along the X1 -direction carries B0 into another equilibrium con®guration Be . We assume that under the same boundary conditions another non-homogeneously deformed con®guration, called a buckled con®guration and denoted by Bt , also exists. Relative to the same coordinate system, a material particle which has coordinates …XA † in B0 has coordinates …xi † and …~xi † in Be and Bt , respectively. Our ultimate goal is to ®nd the conditions under which such a buckled con®guration can indeed exist. We write ~xi ˆ xi …XA † ‡ ui …xj †;

…2†

where ui …xj † is a small amplitude displacement associated with the deformation Be ! Bt . The  and F and de®ned by deformation gradients arising from B0 ! Be and B0 ! Bt are denoted by F oxi ; FiA ˆ oXA

FiA ˆ

o~xi : oXA

…3†

It is clear from Eq. (2) that FiA ˆ …dij ‡ ui;j †FjA ;

…4†

where here and henceforth a comma indicates di€erentiation with respect to the implied spatial coordinate. Furthermore, the convention whereby upper case indices refer to coordinates in B0 and lower case indices to coordinates in Be will be observed. In the absence of body forces, the equations of equilibrium are given by piA;A ˆ 0;

…5†

where …piA † is the ®rst Piola±Kirchho€ stress.

Fig. 2. The coordinate system for a sheet of plain fabric which is subjected to a uniaxial tension along the X1 -axis.

4

Y.T. Zhang, Y.B. Fu / International Journal of Engineering Science 39 (2001) 1±13

Let N dA denote a vector surface area element in B0 , where N is the unit outward normal to the surface, and n da the corresponding area element in Be . Then, the area elements are connected through Nanson's formula  T n da; N dA ˆ F

…6†

where T signi®es transposition. In the subsequent buckling analysis, it will be assumed that on a boundary where the traction vector pN is prescribed in B0 , this traction will be maintained at this value during the incremental deformation Be ! Bt . Such an assumption is usually referred to as a dead-load traction boundary condition and is represented by iA †NA ˆ 0; …piA ÿ p

…7†

iA is the ®rst Piola±Kirchho€ stress corresponding to the deformation B0 ! Be . where p It is convenient to introduce a tensor function with components vij through iA †FjA ; vij ˆ Jÿ1 …piA ÿ p

…8†

 In terms of this new tensor function, the equilibrium condition (5) and and the where J ˆ det F. dead-load boundary conditions (7) may be written simply as (see e.g. [8]) vij;j ˆ 0;

…9†

vij nj ˆ 0;

…10†

and

respectively. The advantage of working with these new forms of governing equations and boundary conditions is that all quantities are now calculated relative to Be (instead of B0 ). We now proceed to derive the incremental constitutive relations. For the response of fabric, it is customary to use a linear relation between the second Piola±Kirchho€ stress tensor T (ˆ Fÿ1 p) and the Green strain tensor E (ˆ …FT F ÿ I†=2). Relative to the coordinate system OX1 X2 X3 , we have Tij ˆ Lijkl Ekl ;

…11†

where Lijkl are the elastic moduli and have the symmetries Lijkl ˆ Ljikl ˆ Lijlk ˆ Ljilk . When referred to the the special coordinate system OX10 X20 X3 where the X10 - and X20 -axes are along the warp and weft directions, respectively, the components Tij0 of T are related to the components Eij0 of E by

Y.T. Zhang, Y.B. Fu / International Journal of Engineering Science 39 (2001) 1±13

8 0 9 T11 > > > > > 0 > > T22 > > > > > > =

2

B11 6 B12 6 6 0 6 33 ˆ 6 0 > 6 0 T23 > > > > > 6 > 0 > > > T13 > > > 4 0 > : 0 ; T12 0

B12 B22 0 0 0 0

0 0 B33 0 0 0

0 0 0 2B44 0 0

0 0 0 0 2B55 0

38 0 9 2B16 > E11 > > > > 0 > > > E 2B26 7 > 22 > > 7> > > < 7 0 = 0 7 E33 ; 7 0 > E23 > 0 7> > > 7> > 0 > 0 5> > E13 > > > : 0 > ; E12 2B66

5

…12†

where from Part 1, 1 B16 ˆ B26 ˆ ÿ mBcc ; 8

3 B66 ˆ Bcc ; 8

0 † and Bcc is the compressive modulus along the direction which is at within which m ˆ sign…E12 equal inclinations with respect to the warp and weft directions. The constitutive relation (12) generalizes the relation (4.7) in Part 1 to three-dimensional deformations. The many zeroes in the sti€ness matrix re¯ect our assumption that the fabric sheet is thin so that the material response in the X10 X20 plane can be assumed to be uncoupled from the response in the other two perpendicular planes. 0 where by Relative to the special coordinate system OX10 X20 X3 , (11) takes the form Tij0 ˆ L0ijkl Ekl 0 comparing with (12) we deduce that the non-zero components of Lijkl are given by

L01111 ˆ B11 ;

L02222 ˆ B22 ;

L01112 ˆ L01121 ˆ B16 ;

L03333 ˆ B33 ;

L02211 ˆ L01122 ˆ B12 ;

L02212 ˆ L02221 ˆ B26 ;

L02323 ˆ L02332 ˆ L03223 ˆ L03232 ˆ B44 ; L03131 ˆ L03113 ˆ L01331 ˆ L03131 ˆ B55 ; L01212 ˆ L01221 ˆ L02112 ˆ L02121 ˆ B66 : If the coordinate system OX1 X2 X3 is obtained by rotating the X10 -axis about the X3 -axis counterclock-wise by an angle of h, then the elastic moduli Lijkl can be calculated according to Lijkl ˆ aim ajn akr als L0mnrs ;

…13†

where the non-zero components of aij are given by a11 ˆ cos h; a12 ˆ ÿa21 ˆ sin h; a22 ˆ cos h, a33 ˆ 1. It turns out that for all the cases considered in the present paper, the constitutive relation relative to the coordinate system OX1 X2 X3 is of the form 8 9 2 9 38 0 0 0 2D16 > E11 > T11 > D11 D12 > > > > > > > > 6 > > > T22 > D21 D22 0 0 0 2D26 7 E22 > > > > > > > > > 7 6 > > > < = 6 0 7 0 D33 0 0 0 7 E33 = 33 6 ˆ6 ; …14† 7 > 6 0 T23 > 0 0 7> E23 > 0 0 2D44 > > > > > > > > 7 6 > > > 4 > > 0 0 0 0 2D55 T > 0 5> E > > > > > > > : 32 > : 32 > ; ; 0 0 0 0 0 2D66 T12 E21

6

Y.T. Zhang, Y.B. Fu / International Journal of Engineering Science 39 (2001) 1±13

where the constants Dij can be related to Bij with the aid of the tensor transformation rule (13). Such relations will be written out for the particular cases to be considered in the following sections. The ®rst Piola±Kirchho€ stress p is related to the second Piola±Kirchho€ stress T by p ˆ FT. On substituting this relation, (4) and (11) into (8) and then linearizing in terms of ui;j , we obtain vij ˆ Ajilk uk;l ;

…15†

where Ajilk ˆ Jÿ1 Fja Flb Labce Fic Fke ‡ Jÿ1 Fia Fkb Tab djl ;

…16†

the Tab being the components of T in Be . Relation (16) can also be obtained by combining Ogden's [9] Eqs. (6.1.22) and (6.1.29). In the following analysis, we shall look for buckling solutions in which u1  0 and u2 ; u3 are independent of x1 . It then follows that v21  0, v31  0 and other stress components of v are all independent of x1 . The equilibrium equation (9) then reduces to v22;2 ‡ v23;3 ˆ 0;

v32;2 ‡ v33;3 ˆ 0;

…17†

and from (10) the boundary conditions are v23 ˆ v33 ˆ 0

on x3 ˆ h;

…18†

v22 ˆ v32 ˆ 0

on x2 ˆ a:

…19†

and

We shall see later that it will not be possible to satisfy both (18) and (19) point-wise simultaneously. We shall seek a solution that satis®es (18) pointwise, and in place of (19), we shall satisfy a relaxed condition that the resultants of surface tractions on x2 ˆ a are zero, that is Z

h ÿh

Z v22 dx3 ˆ

h ÿh

v32 dx3 ˆ 0:

…20†

We now show that these conditions in fact follow automatically from (17) and (18). To this end, we ®rst note that by integrating (17) from ÿh to h and making use of (18), we have d dx2

Z

h

d v22 dx3 ˆ dx2 ÿh

Z

h ÿh

v32 dx3 ˆ 0:

…21†

For the type of buckling solutions to be discussed later, see (23), v22 and v32 in general have the form v22 ˆ f1 …x3 † cos…kx2 † ‡ f2 …x3 † sin…kx2 †;

v32 ˆ f3 …x3 † cos…kx2 † ‡ f4 …x3 † sin…kx2 †;

…22†

Y.T. Zhang, Y.B. Fu / International Journal of Engineering Science 39 (2001) 1±13

7

where k is the mode number and f1 ; f2 ; f3 ; f4 are known functions. It follows by substituting (22) into (21) that these functions must necessarily satisfy Z

h

ÿh

fj …x3 † dx3 ˆ 0;

j ˆ 1; 2; 3; 4;

from which then follows (20). 3. Buckling analysis under traditional continuum model We now consider the case when the fabric sheet is subjected to uniaxial tension along the x1 direction and the inclination angle between the x1 -axis and the warp direction is either zero or 45°. When the inclination angle is zero, the x1 - and x2 -axes are clearly principal axes of stretch. When the inclination angle is 45°, we showed in Part 1 that under the additional assumption B11 ˆ B22 , the x1 - and x2 -axes are also principal axes of stretch. Thus in both cases, we have  ˆ diag ‰k1 ; k2 ; 1Š, where k1 and k2 are principal stretches. F Under the traditional continuum model, the only non-zero component of Tab in Eq. (16) is T11 ˆ 2Q, where 2Q is just the p in Part 1. When the X1 -axis coincides with the warp direction we have D11 ˆ B11 ;

D12 ˆ B12 ;

D21 ˆ B12 ;

D22 ˆ B22 ;

D66 ˆ B66 ;

D16 ˆ 0;

D26 ˆ 0:

When the inclination angle is 45°, to simplify analysis we assume B11 ˆ B22 . We then have D11 ˆ …B11 ‡ B12 ‡ 2B66 †=2; D44 ˆ B44 ;

D55 ˆ B55 ;

D12 ˆ …B11 ‡ B12 ÿ 2B66 †=2;

D66 ˆ B11 ÿ B12 ;

D16 ˆ 0;

D21 ˆ D12 ;

D22 ˆ D11 ;

D26 ˆ 0:

In fact, all of the results in this section are expressed in terms of Dij and our ®nal conclusions are independent of the above relations between Dij and Bij . For the eigenvalue problem (17)±(19), we look for a buckling solution of the form u1 ˆ 0;

uj ˆ Hj …kx3 †eikx2 ‡ C:C:;

j ˆ 2; 3;

…23†

p where i ˆ ÿ1, k is the mode number and C.C. denotes the complex conjugate of the preceding term. On substituting (23) into (17), we obtain two second-order di€erential equations for H2 and H3 . Solving these equations yields ÿiH2 …kx3 † ˆ C2 sinh…p1 kx3 † ‡ C4 sinh…p2 kx3 † ‡ C1 cosh…p1 kx3 † ‡ C3 cosh…p2 kx3 †;

…24†

H3 …kx3 † ˆ q1 ‰C2 cosh…p1 kx3 † ‡ C1 sinh…p1 kx3 †Š ‡ q2 ‰C4 cosh…p2 kx3 † ‡ C3 sinh…p2 kx3 †Š;

…25†

8

Y.T. Zhang, Y.B. Fu / International Journal of Engineering Science 39 (2001) 1±13

where Cj are disposable constants and D22 k22 ÿ D44 pj2 qj ˆ ; D44 pj

j ˆ 1; 2:

…26†

The p1 and p2 are two of the four roots of the equation ÿ
ÿ
D22 k22 ÿ D44 p2 ÿ D44 k22 ‡ D33 p2 ÿ D244 k22 p2 ˆ 0

…27†

that are given by q p p1 ˆ a 1 ‡ g;

…28†

q p p2 ˆ a 1 ÿ g;

where

r D22 ; a ˆ k2 2D44

gˆ1ÿ4

D244 : D22 D33

…29†

Since the above general solution contains only four disposable constants, we can in general satisfy only four boundary conditions exactly. On substituting the above solution for u2 and u3 into the four boundary conditions (18), we obtain b1 …h† ˆ 0;

b1 …ÿh† ˆ 0;

b2 …h† ˆ 0;

b2 …ÿh† ˆ 0;

…30†

where the two functions b1 and b2 are de®ned by b1 …h† ˆ C2 D22 k22 p2 cosh…hkp1 † ‡ C4 D22 k22 p1 cosh…hkp2 † ‡ C1 D22 k22 p2 sinh…hkp1 † ‡ C3 D22 k22 p1 sinh…hkp2 †;

…31†

b2 …h† ˆ C2 …D22 k22 ÿ D44 p12 † sinh…hkp1 † ‡ C4 …D22 k22 ÿ D44 p22 † sinh…hkp2 † ‡ C1 …D22 k22 ÿ D44 p12 †p2 cosh…hkp1 † ‡ C3 …D22 k22 ÿ D44 p22 †p1 cosh…hkp2 †:

…32†

The following identities may be used to simplify the above expression of b2 …h†: D22 k22 ÿ D44 p22 ˆ D44 a2 …1 ‡

p g†;

D22 k22 ÿ D44 p12 ˆ D44 a2 …1 ÿ

p g†:

The four boundary conditions (30) are clearly equivalent to the two sets of relations b1 …h† ‡ b1 …ÿh† ˆ 0;

b2 …h† ÿ b2 …ÿh† ˆ 0;

…33†

which are two linear equations for C2 and C4 , and b1 …h† ÿ b1 …ÿh† ˆ 0;

b2 …h† ‡ b2 …ÿh† ˆ 0;

…34†

Y.T. Zhang, Y.B. Fu / International Journal of Engineering Science 39 (2001) 1±13

9

which are two linear equations for C1 and C3 . Eqs. (33) have non-trivial solutions for C2 and C4 only if the determinant of their coecient matrix vanishes, i.e. p p …35† p2 …1 ‡ g† cosh…khp1 † sinh…khp2 † ÿ p1 …1 ÿ g† cosh…khp2 † sinh…khp1 † ˆ 0: When this relation is satis®ed, Eqs. (34) can only admit the trivial solution for C1 and C3 . The corresponding u2 is an odd function of x3 and we refer to the corresponding buckling solution as a ¯exural mode. Similarly, Eqs. (34) have non-trivial solutions for C1 and C3 only if p p …36† p2 …1 ‡ g† cosh…khp2 † sinh…khp1 † ÿ p1 …1 ÿ g† cosh…khp1 † sinh…khp2 † ˆ 0; and we refer to the corresponding buckling solution as an extensional mode. We now show that for any kh 6ˆ 0, there exist no values of Dij and k2 that can satisfy either Eqs. (35) or (36). We distinguish two cases: g > 0 in which case p1 and p2 are both real, and g < 0 in which case p1 and p2 are complex conjugates of each other. We ®rst consider (35). When p1 and p2 are real numbers, we can write p p p1 ˆ a 1 ‡ b; p2 ˆ a 1 ÿ b; p where b ˆ g, and it is clear that p1 > p2 . Eq. (35) then becomes p2 …1 ‡ b† tanh…p2 kh† ˆ p1 …1 ÿ b† tanh…p1 kh†:

…37†

De®ning f …kh† ˆ p2 …1 ‡ b† tanh…p2 kh† ÿ p1 …1 ÿ b† tanh…p1 kh†, we have df p2 …1 ‡ b† p2 …1 ÿ b† ˆ 2 2 ÿ 1 2 : d…kh† cosh …p2 kh† cosh …p1 kh† It follows from p22 …1 ‡ b† ˆ a2 …1 ÿ b2 † ˆ p12 …1 ÿ b† and p2 < p1 that f 0 …kh† > 0 and hence that f …kh† is an increasing function of kh. But since f …0† ˆ 0, we have f …kh† > f …0† ˆ 0. Thus f …kh† is always positive and there exist no values of Dij and k2 that can satisfy Eq. (35). When p1 and p2 are complex conjugates of each other, we can write p1 ˆ r ‡ is;

p2 ˆ r ÿ is;

p where r ˆ a…1 ‡ b2 †1=4 cos…h=2†; s ˆ a…1 ‡ b2 †1=4 sin…h=2†; b ˆ jgj, and h is the principal value of the argument of 1 ‡ ib, i.e. tan h ˆ b. It is clear that h=2 6 p=4 and r P s. Eq. (35) becomes …br ÿ s† sinh…2rkh† ˆ …r ‡ bs† sin…2skh†:

…38†

De®ning g…kh† ˆ …br ÿ s† sinh…2rkh† ÿ …r ‡ bs† sin…2skh†, we have dg ˆ 2r…br ÿ s† cosh…2rkh† ÿ 2s…r ‡ bs† cos…2skh†: d…kh† With the aid of the identity   ÿ

1=2 h h h 1 ÿ 2 1=2 2 cos ‡ b sin tan h ˆ r…br ÿ s†; s…r ‡ bs† ˆ a 1 ‡ b sin ˆ a2 1 ‡ b2 2 2 2 2

…39†

10

Y.T. Zhang, Y.B. Fu / International Journal of Engineering Science 39 (2001) 1±13

we deduce that dg=d…kh† > 0. Since g…0† ˆ 0, g…kh† is always positive and so again there exist no values of Dij and k2 that can satisfy Eq. (35). Next we consider the bifurcation condition (36) for the extensional mode. When p1 and p2 are complex conjugates of each other, Eq. (36) becomes …br ÿ s† sinh…2rkh† ˆ ÿ…r ‡ bs† sin…2skh†:

…40†

Using the same argument as that for (38), we may show that there exist no values of Dij and k2 that can satisfy Eq. (40). Finally, when p1 and p2 are real numbers, (36) becomes p2 …1 ‡ b† tanh…p1 kh† ˆ p1 …1 ÿ b† tanh…p2 kh†;

…41†

i.e. p p 1 ‡ b…sinh…p1 ‡ p2 † ÿ sinh…p2 ÿ p1 †† ˆ 1 ÿ b…sinh…p1 ‡ p2 † ÿ sinh…p1 ÿ p2 ††: Let lˆ

p p 1 ‡ b ‡ 1 ÿ b;

mˆ

p p 1 ‡ b ÿ 1 ÿ b:

Eq. (41) becomes m sinh…alkh† ˆ ÿl sinh…amkh†: It is clear that y ˆ m sinh…alkh† and y ˆ ÿl sinh…amkh† have no intersections except at kh ˆ 0. We conclude that there exist no values of Dij and k2 that can satisfy Eq. (36). 4. Buckling analysis under modi®ed model When the direction of tension coincides with that of the warp the stress components Tab are given by Eq. (5.4) in Part 1, i.e. Tab ˆ 0 except T11 ˆ 2Q, where Q corresponds to p=2 in Part 1 and is related to k2 by (5.5) in Part 1. The Dij are related to Bij by D11 ˆ B11 ;

D12 ˆ B12 ;

D21 ˆ B12 ;

D22 ˆ B22 ;

D66 ˆ 3Bcc =8;

D16 ˆ ÿBcc =8;

D26 ˆ ÿBcc =8: Following the same procedure as that used in the previous section, we may again show that buckling is not possible in this case. We next consider the case when the x1 -axis (which de®nes the direction of tension) is obtained by rotating the warp direction counter-clock-wise by 45°. To simplify analysis, we assume that B11 ˆ B22 , B12 ˆ 0. According to our analysis in Part 1, the Tab in Eq. (16) are such that Tab ˆ 0 except T11 ˆ 2Q, T22 ˆ ÿQ. The relationship between Dij and Bij is given by equation (5.8) in Part 1 and we have D16 ˆ D26 ˆ 0.

Y.T. Zhang, Y.B. Fu / International Journal of Engineering Science 39 (2001) 1±13

11

We look for the same form of buckling solution as Eq. (23) in the previous section. The solutions for H2 and H3 are again given by (24) and (25), but now qj …j ˆ 1; 2† are de®ned by D22 k22 ÿ D44 pj2 ÿ Q : D44 pj

qj ˆ

…42†

The p1 ; p2 are two of the four roots of
ÿ
ÿ D22 k22 ÿ D44 p2 ÿ Q ÿ D44 k22 ‡ D33 p2 ‡ k22 Q ÿ D244 k22 p2 ˆ 0 that are given by p p1 ˆ m ‡ n;

p2 ˆ

…43†

p m ÿ n;

…44†

where mˆ rˆ

ÿ
1 D22 D33 k22 ÿ D33 Q ÿ D44 k22 Q ; 2D33 D44

ÿ

D22 D33 k22

ÿ D33 Q ÿ


2 D44 k22 Q

ÿ

nˆ

4D33 D44 k22

ÿ

p r ; 2D33 D44

D22 D44 k22

ÿ D44 Q ÿ

D22 k22 Q


‡ Q2 :

…45†

The boundary conditions are again given by (30) but now the two functions b1 and b2 are de®ned by ÿ
ÿ
b1 …h† ˆ C2 D22 k22 ÿ Q p2 cosh…hkp1 † ‡ C4 D22 k22 ÿ Q p1 cosh…hkp2 † ÿ
ÿ
‡ C1 D22 k22 ÿ Q p2 sinh…hkp1 † ‡ C3 D22 k22 ÿ Q p1 sinh…hkp2 †;

…46†

ÿ
ÿ
b2 …h† ˆ C2 D22 k22 ÿ Q ÿ D44 p12 sinh…hkp1 † ‡ C4 D22 k22 ÿ Q ÿ D44 p22 sinh…hkp2 † ÿ
ÿ
‡ C1 D22 k22 ÿ Q ÿ D44 p12 p2 cosh…hkp1 † ‡ C3 D22 k22 ÿ Q ÿ D44 p22 p1 cosh…hkp2 †:

…47†

Following the same procedure as in the previous section, we ®nd that the buckling conditions for the ¯exural and extensional modes are given by ÿ
ÿ
D22 k22 ÿ Q ÿ D44 p22 D22 k22 ÿ Q p2 cosh…khp1 † sinh…khp2 † ÿ
ÿ
ÿ D22 k22 ÿ Q ÿ D44 p12 D22 k22 ÿ Q p1 cosh…khp2 † sinh…khp1 † ˆ 0;

…48†

ÿ
ÿ
D22 k22 ÿ Q ÿ D44 p22 D22 k22 ÿ Q p2 cosh…khp2 † sinh…khp1 † ÿ
ÿ
ÿ D22 k22 ÿ Q ÿ D44 p12 D22 k22 ÿ Q p1 cosh…khp1 † sinh…khp2 † ˆ 0;

…49†

and

12

Y.T. Zhang, Y.B. Fu / International Journal of Engineering Science 39 (2001) 1±13

respectively. Out-of-plane buckling is possible only if the buckling conditions (48) and (49) have non-trivial solutions for kh 6ˆ 0. We note that similar bifurcation conditions to (48) and (49) have been derived for pre-stressed elastic plates, see e.g. [10,11] and the references therein. As an illustration, we assume that B33 ˆ B11 ;

B44 ˆ B11 =2;

B55 ˆ B11 =2:

The buckling conditions (48) and (49) then depend only on kh, k2 and a non-dimensional material parameter g de®ned by g ˆ B11 =Bcc . The corresponding stretching force (per unit area) p can be calculated using equation (5.11) in Part 1, that is p=Bcc ˆ g…1 ÿ k22 †=…2g ÿ 1†. In Fig. 3, we have shown solutions of (48) and (49), as curves of p=Bcc against kh, for g ˆ 2:0. Similar curves can be obtained for other values of g. Buckling with respect to the ¯exural (resp. extensional) modes cannot take place in the region bounded by the horizontal axis and the solid (resp. dotted) curve in Fig. 3. Thus it is clear that as the stretching force is increased gradually, buckling ®rst takes place in the form of a ¯exural mode. This agrees with what we observed in various experiments which we have conducted on woven fabric. Since for ¯exural modes the buckling stretching force is an increasing function of kh, the critical stretching force corresponds to the minimum value of k that is possible. Although our boundary conditions (18) and (20) do not induce any constraint on the

Fig. 3. The bifurcation curves for g ˆ 2:0.

Y.T. Zhang, Y.B. Fu / International Journal of Engineering Science 39 (2001) 1±13

13

value of k, a minimum of k can be obtained in a number of ways. For instance, we may require that u2 ˆ 0 at x2 ˆ a. From (23) and (24), this is satis®ed only if sin ka ˆ 0 which yields a minimum of p=a for k. The corresponding critical stretching force can then be obtained from Fig. 3. This critical value can in principle be compared with that determined from experiments. We do not present results from such comparisons since our aim here is to demonstrate that buckling is indeed possible using our modi®ed constitutive model. Acknowledgements This work was carried out when the ®rst author was visiting Keele University. This work was supported by National Natural Science Foundation of China (No. 19772032). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11]

D.M. Stump, W.B. Fraser, Textile Research Journal 66 (1996) 506. D.E. Breen, D.H. House, M.J. Wozny, Textile Research Journal 64 (1994) 663. B. Eberhardt, A. Weber, W. Strasser, IEEE Computer Graphics and Applications 16 (1996) 52. J. Amirbayat, J.W.S. Hearle, Journal of The Textile Institute 80 (1989) 51. J.H. Kim, Fabric mechanics analysis using large deformation orthotropic shell theory, Ph.D. Thesis, Department of Mechanical and Aerospace Engineering, North Carolina State University, 1991. T.J. Kang, W.R. Yu, Journal of the Textile Institute 86 (1995) 635. B.J. Chen, M. Govindaraj, Textile Research Journal 65 (1995) 324. Y.B. Fu, R.W. Ogden, Continuum Mechanics and Thermodynamics 11 (1999) 141. R.W. Ogden, Non-linear Elastic Deformations, Ellis Horwood, New York, 1984. D.G. Roxburgh, R.W. Ogden, International Journal of Engineering Science 32 (1994) 427. G.A. Rogerson, Y.B. Fu, Acta Mechanica 111 (1995) 59.