Aalysis of composites with multiple rigid-lie reiforcemets by the BEM Piotr Fedeliski* Departmet of Stregth of Materials ad Computatioal Mechaics, Silesia Uiversity of Techology ul. Koarskiego 18A, 44-100 Gliwice, Polad e-mail: Piotr.Fedeliski@polsl.pl Abstract This paper presets the boudary elemet method (BEM) for aalysis of plates cotaiig multiple rigid-lie reiforcemets. The accuracy of the method is verified by comparig computed stresses i the eighbourhood of a rigid fibre i a ifiite plate subjected to tesio with aalytical solutios. The proposed approach is applied for aalysis of composites reiforced by multiple rigid fibres. The ifluece of the distace betwee the fibres o stress distributio i the matrix ad the effective Youg modulus is aalyzed. Keywords: boudary elemet methods, composites, microstructures, homogeizatio, elasticity 1. Itroductio The aim of usig fibres i composite materials is to icrease stregth ad stiffess of structures. It is kow that these two properties are difficult to improve simultaeously [3]. Stiffess depeds o overall properties of the structure ad stregth o microstructure. If the stiffess of fibers is much greater tha the matrix the the fibers ca be modelled as rigid stiffeers i a deformable matrix. The examples of such atural composites are: silk, acre ad eamel [6]. The aalysis of composites with may fibers requires computatioal methods. Li ad Tig [4] aalyzed a lie iclusio i a aisotropic elastic ifiite plate subjected to uiform loadig at ifiity. They used the Stroh formalism to calculate the displacemet ad stress fields for the rigid ad elastic iclusio. Pigle et al. [6] used the duality priciple to derive rigid lie iclusio solutios from the crack solutios. They computed stress fields aroud a rigid lie iclusio ad derived a compliace cotributio tesor for a sigle ad multiple lie iclusios. Gorbatikh et al. [3] determied the relatio betwee stress itesity factors at the tips of rigid lie iclusios ad effective compliace of the material. Liu et al. [5] applied the fast multipole BEM to aalyze composites reiforced by aotubes. They calculated effective material properties of composites modelled as three-dimesioal structures havig large umber of degrees of freedom. The formulatio ad applicatio of the BEM for a sigle rigid fibre i a fiite plate was preseted by Fedeliski [2]. I the preset work the method is exteded ad applied to solids with multiple rigid-lie reiforcemets. 2. Boudary elemet method for plates with rigid fibres 2.1. Boudary itegral equatios for a plate with fibres Cosider a plate made of homogeous, isotropic ad liearelastic material. The boudary of the plate is deoted by ad its domai by Ω (Fig. 1). The plate is statically loaded alog the exteral boudary by boudary tractios t j ad the domai Ω by body forces f j. The relatio betwee the loadig of the plate ad its displacemets ca be expressed by the Somigliaa idetity [1] cij ( x') u j ( x') + Tij ( x', x) u j( x) d ( x) =, (1) Uij ( x', x) t j ( x) d ( x) + Uij( x', X ) f j( X ) dω( X ) Ω where: x is the collocatio poit, for which the above itegral equatio is applied, x is the boudary poit, X is the domai poit ad c ij is a costat, which depeds o the positio of the poit x, U ij ad T ij are the Kelvi fudametal solutios of elastostatics. I equatios the Eistei summatio covetio is used ad the idices for two-dimesioal problems have values i,j=1,2. Figure 1. Elastic plate - loadig ad displacemets Assume that the plate is reiforced by straight, thi ad rigid fibres, which are perfectly boded to the matrix. If the plate is deformed the forces of iteractio occur alog the lies of attachmet of fibers (Fig. 2). These forces of iteractio ca be treated as particular body forces actig alog the lies i the domai of the body. The boudary itegral equatio (1) for the plate loaded by boudary tractios ad forces of iteractio of fibres has the form [2] *Ackowledgmet: The scietific research is fiaced by the Miistry of Sciece ad Higher Educatio of Polad i years 2010-2012.
cij ( x') u j ( x') + Tij ( x', x) u j( x) d ( x) = N, (2) Uij ( x', x) t j( x) d ( x) + Uij ( x', X ) t j ( X ) d( X ) = 1 where: N is the umber of fibres, is the lie of attachmet ad t j are the forces of iteractio. The cosidered structure ad therefore each fibre is i equilibrium. The forces actig o each fibre (Fig. 4) should satisfied the followig equilibrium equatios t1 ( x ) d ( x ) = 0, (5) t2 ( x ) d ( x ) = 0, (6) [ t1 ( x)si α + t2 ( x)cos α ] r( x) d ( x) = 0, (7) The last equatio (7) is the equatio of momets of forces with respect of the fibre tip x o. Figure 2. Elastic plate with fibres 2.2. Displacemets ad equatios of equilibrium for stiff fibres Figure 3. Displacemets of the rigid fibre 2.3. Numerical implemetatio of the method The boudary of the plate ad the fibres are divided ito boudary elemets (Fig. 5). I the developed computer code 3- ode quadratic boudary elemets are used. Alog the exteral boudary of the plate the variatios of coordiates, displacemets ad tractios, ad alog the fibres the variatios of iteractio forces are iterpolated. The boudary itegral equatios (2) are used for odes alog the exteral boudary ad the fibres. The displacemets of fibre odes ca be expressed by the displacemets of fibre tips ad their agles of rotatio, by usig Eqs (3) ad (4). These equatios ca writte i the followig matrix form u= Iu f, (8) where the matrix u cotais the compoets of displacemets of fibre odes, the matrix I depeds o the positio of odes ad the matrix u f cotais compoets of displacemets of fibre tips ad their agles of rotatio. Figure 4. Forces actig o the fibre Deformatios of the matrix ifluece displacemets of fibres. The displacemet of a arbitrary poit x of the fibre ca be expressed by the displacemets of the fibre tip x o ad the agle of rotatio of the fibre ϕ (Fig. 3). For small agles of rotatio of fibres, the compoets of displacemets of a arbitrary poit of the fibre are expressed i the form u1( x) = u1( xo) ϕr( x)siα, (3) u2( x) = u2( xo ) + ϕr( x)cosα, (4) Figure 5. Discretizatio of the matrix ad fibres by quadratic boudary elemets The equilibrium equatios for fibres (5), (6) ad (7) ca be writte i the matrix form Et f = 0, (9) where the matrix E depeds o the positio of fibre odes ad the matrix t f cotais odal values of compoets of tractios i fibres. The matrix E is obtaied by itegratio of expressios i Eqs (5), (6) i (7), by assumig quadratic variatios of forces alog the fibres. Because the equilibrium equatios have very simple forms, the itegrals are computed aalytically. where: α is the iitial agle betwee the fibre ad the axis x 1 of the global coordiate system, r is the distace betwee the poit x ad the fibre tip x o.
Boudary itegral equatios (2), supplied with Eqs (8) ad (9) ca be writte i the matrix form H 0 Gee G ee ef ue te H fe I = G fe G ff, (10) u 0 0 f t 0 f E where the submatrices with the idex e are related to the exteral boudary ad the submatrices deoted by the idex f are related to the fibres. The submatrices H ad G deped o boudary itegrals of fudametal solutios, shape fuctios ad are itegrated umerically by usig the Gauss quadrature. Next the system of algebraic equatios is rearraged. The ukow quatities are o the left side of the equatio ad the kow quatities o the right side of the equatio. The first modificatio refers to the ukow iteractio forces t f Hee Gef 0 u e G ee H fe G ff I t f = G fe [ te]. (11) 0 0 0 E u f I the fial modificatio the kow ad ukow boudary coditios are rearraged. The modified matrix equatio is solved ad the ukow displacemets ad tractios alog the exteral boudary ad the displacemets ad iteractio forces for fibres are obtaied. 3. Numerical examples To demostrate the accuracy of the method ad possible applicatios three umerical examples are solved. 3.1. Rigid fibre i a ifiite plate A ifiite plate with a rigid fibre of legth 2l is subjected to the parallel loadig q 1 or to the perpedicular loadig q 2, as show i Fig. 6. The structure is modelled as a fiite square plate with a rigid fibre ad the dimesios of the plate are 10 times larger tha the fibre. The material of the plate has the Poisso ratio ν=0.25 ad is i plae stress state. The rigid fibre is divided ito 20 boudary elemets ad the square plate ito 160 boudary elemets. Stresses are computed at 245 iteral poits i the eighbourhood of the fibre. This field is marked as a grey square i Fig. 6. The cotour plot of ormalized stresses σ 11 /q 1 for the parallel loadig q 1 ad ormalized stresses σ 22 /q 2 for the perpedicular loadig q 2 are show i Fig. 7. Figure 7: Normal stresses i the viciity of the rigid fibre: parallel loadig ormalized stress σ 11 /q 1, perpedicular loadig ormalized stress σ 22 /q 2 For the parallel loadig stresses σ 11 have the smallest values alog the fibre ad the largest values alog the extesio of the fibre. For the perpedicular loadig the stresses σ 22 are more uiformly distributed i the aalyzed field. The maximum values of stresses exceed the applied tesio by about 2%. The distributio of stresses agree very well with stresses computed aalytically ad preseted by Pigle et al. [6]. 3.2. Two rigid fibres i a ifiite plate Two parallel rigid fibres i a ifiite plate are subjected to tesio q 1 (Fig. 8). The legth of the fibres is 2l ad the distace betwee the fibres is 2d 2. The same material properties ad discretizatio are assumed as i the previous example 3.1. The ifluece of the distace betwee the fibres o stress distributio is studied. The ormalized stresses σ 11 /q 1, i the marked field show i Fig. 8, for the distaces d 2 /l=0.5 ad d 2 /l=1.0 are preseted i Fig. 9. For the cosidered distaces the iteractio of fibres has small ifluece o stress distributio. Figure 6: Rigid fibre i a ifiite plate dimesios ad loadig Figure 8. Two rigid fibres i a ifiite plate dimesios ad loadig
Figure 11. Rigid fibres i a rectagular plate iitial shape (dashed lie) ad deformed shape (cotiuous lie) Figure 9. Ifluece of the distace betwee two fibres o the stress distributio σ 11 /q 1 : d 2 /l=0.5, d 2 /l=1.0 3.3. Rigid fibres i a rectagular plate A rectagular plate of legth 2b ad height 2h cotais 13 rigid fibres of legth 2l, as show i Fig. 10. The horizotal distace betwee cetres of eighbourig fibres is d 1 ad the vertical distace is d 2. The ratios of dimesios are: b/l=5, h/l=4, d 1 /l=3 ad d 2 /l=1.6. The material of the plate has the Poisso ratio ν=0.3 ad is i plae strai state. The plate is subjected to the horizotal loadig q 1. Each rigid fibre is divided ito 8 boudary elemets ad the exteral boudary ito 72 boudary elemets. The iitial shape ad the deformed shape of the plate are show i Fig. 11. The stress distributios i the matrix, i the marked field i Fig. 10, are show i Fig. 12. The relative effective Youg modulus is computed as E c /E m =1.345, where E c ad E m are the Youg modulus of the composite ad the matrix, respectively. The effective Poisso ratio of the composite is the same as for the matrix. c) Figure 12. Stress distributios i the matrix: σ 11 /q 1, σ 22 /q 1, c) σ 12 /q 1 Figure 10: Rigid fibres i a rectagular plate dimesios ad loadig
The ifluece of the horizotal ad vertical distace betwee the fibres o the effective Youg modulus is aalyzed. The dimesios of fibres are costat ad the dimesios of the plate are chaged proportioally to the distace betwee the fibres. The depedece of the relative Youg modulus o the horizotal ad vertical distace is preseted i Fig. 13. If the horizotal or vertical distace are smaller tha half of the legth of the fibre the stiffess of the composite sigificatly icreases. For the icreasig distace betwee the fibres the stiffess of the composite teds to the stiffess of the matrix. Refereces [1] Becker A.A., The boudary elemet method i egieerig. A complete course. McGraw-Hill Book Compay, Lodo, 1992. [2] Fedeliski, P., Boudary elemet method for aalysis of elastic structures with rigid fibers, Eg. Modell., 32, pp. 135-142, 2006 (i Polish). [3] Gorbatikh, L., Lomov S.V. ad Verpoest, I., Relatio betwee elastic properties ad stress itesity factors for composites with rigid-lie reiforcemets, It. J. Fract., 161, pp. 205-212, 2010. [4] Li, Q. ad Tig, T.C.T., Lie iclusios i aisotropic elastic solids, Tras. ASME, J. Appl. Mech., 56, pp. 556-563, 1989. [5] Liu, Y., Nishimura, N. ad Otai, Y., Large-scale modelig of carbo-aotube composites by a fast multipole boudary elemet method, Comp. Mater. Sc., 34, pp. 173-187, 2005. [6] Pigle, P., Sherwood, J. ad Gorbatikh, L., Properties of rigid-lie iclusios as buildig blocks of aturally occurrig composites, Composites Sc. Techol., 68, pp. 2267-2272, 2008. Figure 13. Ifluece of the distace betwee the fibres o the relative effective Youg modulus E c /E m : horizotal distace d 1 /2l, vertical distace d 2 /2l 4. Coclusios The boudary elemet method is a very efficiet method for modellig elastic composites with rigid iclusios, particularly fibres. The modellig of composites is sigificatly simplified i compariso to domai methods, for example the fiite elemet method, because odes are situated oly alog the exteral boudary ad fibers. Positios, shapes ad dimesios of fibers ad their umber ca be very simply modified. The compariso of stresses for the fibre i the ifiite plate obtaied by the preset method ad the aalytical method shows that the proposed approach gives very accurate results. For distaces betwee the fibres smaller tha the half legth of fibres a ifluece of iteractio of fibres o stress distributio i the matrix ad sigificat icrease of stiffess ca be observed.