DSpace at VNU: Bending analysis of three-phase polymer composite plates reinforced by glass fibers and titanium oxide particles

Computational Materials Science 49 (2010) S194–S198 Contents lists available at ScienceDirect Computational Materials Science journal homepage: www.elsevier.com/locate/commatsci Bending analysis of three-phase polymer composite plates reinforced by glass fibers and titanium oxide particles Nguyen Dinh Duc a,*, Dinh Khac Minh b a b University of Engineering and Technology, Vietnam National University, Hanoi, Viet Nam Shipbuilding Science and Technology Institute, Viet Nam a r t i c l e i n f o Article history: Received October 2009 Received in revised form 12 April 2010 Accepted 12 April 2010 Available online 10 May 2010 Keywords: Three-phase composite Bending analysis Plate Shear deformation a b s t r a c t This paper presents a method to determine bending deflection of three-phase polymer composite plates consisting of reinforced glass fibers and titanium oxide (TiO2) particles This method analyzes bending with taking into account two important effects which are interaction between titanium oxide particles and polymer matrix and shear deformation Mechanical properties of the three-phase composite plates are clearly related to mechanical properties and volume fractions of constituent materials These explicit expressions play an important role in design of composite materials as well as optimal behavior of threephase composite plates Ó 2010 Elsevier B.V All rights reserved Introduction The composite material consists of two or more constituent materials Three-phase polymer composite which consists of polymer matrix, reinforced fiber and spherical particles are used to increase the mechanical and physical performance of material There have been some calculations determining the material constants for three-phase composite, which is mainly calculated by the numerical methods or only for YoungÕs modulus [1,2] Although some researchers have treated problem of bending analysis of composite rectangular plates [3,4], the bending analysis for three-phase composite plate includes polymer matrix, fibers and particles, to the authorsÕ best knowledge, has not been considered In this article, bending deflection of three-phase composite plate consists of polymer matrix, reinforced glass fibers and titanium oxide (TiO2) particles is analyzed A composite rectangular plate of length a, the width b, thickness h, referred as a thin sheet and built from three-phase composite consisting of the polymer matrix, reinforced glass fibers and titanium oxide particles, is considered (Fig 1) The purpose of present study is to analyze bending of mentioned three-phase composite plates taking into account effects of each phase (fiber and particle) and shear deformation (e44, e55 – 0, ignoring the Kirchhoff–Love hypothesis) on bending behavior of plates The relationship between deformation and stress of plate in this case are determined as [4,5]: r11 ¼ A11 e11 ỵ A12 e22 ; r22 ẳ A22 e22 þ A12 e11 ; r66 ¼ A66 e66 ; r44 ¼ A44 e44 ; r55 ¼ A55 e55 ; ð1Þ where E11 E22 ; A22 ¼ ; À m12 m23 À m12 m23 E11 m23 E22 m12 ¼ ¼ ; A66 ¼ G12 ; À m12 m23 À m12 m23 A11 ¼ A12 Eij and mij are the YoungÕs module and Poisson ratios of anisotropic elastic materials [6] The coefficients A44 and A55 will be concerned in the below mention Governing equations for bending analysis of composite plates by the first order shear deformation theory Transverse shear stress components are defined as [5]: r44 ¼ f zịwx; yị; r55 ẳ f zịux; yị where f zị ẳ 0927-0256/$ - see front matter ể 2010 Elsevier B.V All rights reserved doi:10.1016/j.commatsci.2010.04.016   2 z À h 2 and w(x, y), u(x, y) denotes the rotations of the midplane on the (x, y) plane From Eqs (1) and (2) we have e44 ¼ * Corresponding author E-mail address: ducnd@vnu.edu.vn (N Dinh Duc) 2ị 1 r44 ẳ f zị wx; yị; A44 A44 e55 ẳ 1 r55 ẳ f zị ux; yÞ A55 A55 ð3Þ S195 N Dinh Duc, D.K Minh / Computational Materials Science 49 (2010) S194–S198 Considering two first terms on the right side of Eq (8), which accounts for the interaction between matrix and particles, the moduli of the effective matrix material are defined as [8]: G ¼ Gm À nc ð7 À 5mm ÞH ; ỵ nc 10mm ịH K ẳ Km þ 4nc Gm Lð3K m ÞÀ1 ð9Þ À 4nc Gm L3K m ị1 where Lẳ Fig Conguration and coordinate of a three-phase composite plate E¼ The strain–displacement relations are given by: @u @v @u @ v ; e22 ẳ ỵ ; e66 ẳ ; @x @y @x @y @w @w ỵ wx; yị; e55 ẳ ỵ ux; yị: ¼ @y @x e11 ¼ ð4Þ e44 ð5Þ where u, v are displacement components in the x, y directions, respectively, and w is transverse displacement known as deflection of the plate Equilibrium equations of the plate in terms of three variables w, w, and u are defined as: @u z > ỵ @w ẳ I2 hị > > @x @y > > h > > > Dx @ w3 ỵ D1 ỵ 2Dx;y ị @ w2 À 123 I1 Dy > > A55 @x @x@y h > > > i < D1 ỵDxy ị @ u ỵ A44 @x@y ỵ I2 u ẳ > > h > > 3 Dy > > Dy @@yw3 ỵ D1 ỵ 2Dx;y ị @x@ 2w@y 12 > I1 A > 44 h > > i > > > D ỵD ị : ỵ xy @ w ỵ I2 w ẳ A55 @x@y 0:5h h ; I1 hị ẳ zI0 zịdz ¼ À 120 À0:5h Dx ¼ h A11 ; 12 D1 ¼ h A12 ; 12 @2 u @x2 D xy ỵ A55 ( @2 u @y2 E22 ẳ 6ị @ w @y2 ỵ Dxy @ w A44 @x2 m21 Z m23 0:5h I2 hị ẳ f zịdz ẳ h ; 12 0:5h h A22 ; 12 Dxy ¼ h A66 12 ð7Þ Eq (6) is a set of basic equations for determining deflection of the three-phase composite plates with shear deformation is considered The coefficients A11, A22, A12, A66, A44, A55 are used to determine coefficients Dij in Eq (7) Also, Aij quantities can be calculated from the elastic coefficients of the composite material 3K À 2G 6K À 2G In this work, the elastic moduli of the three-phase composite are estimated based on the two-phase composite material theoretical models Firstly the elastic moduli of two-phase composite which consists of particles and original polymer matrix is calculates [7–9] This two-phase material is assumed to be a ‘‘effective” matrix material And then a new two-phase composite consists of ‘‘effective” matrix material and reinforced fiber is considered [8] Refs [8,10–16] have been proposed such approach and solve a series of issues for composite materials In this work, stress components in the composite plate are defined as follows [8]: ð8Þ ð10Þ ; m221 E11 þ Þ 8Gna ð1 À na Þðma À m na ỵ xna ỵ na ịxa 1ị GGa ; "  1ị ỵ va 1ịv  ỵ 2na ị GG 21 na ịv a   na ỵ na ịva 1ị GG na ỵ v a #)1 v na ị ỵ ỵ na vị GGa ; ỵ2 v ỵ na ỵ na ị GGa  ỵ 1ịm  ma ịna v  ẳm ;  na ỵ vna ỵ na ịva 1ị GGa "  ị GG  ỵ ỵ na v na ịv E22 m221 E22 a ẳ ỵ E11 8G v ỵ na ỵ na ị GGa #  1ị ỵ va 1ịv  ỵ 2na ị GG 21 na ịv a ;  na ỵ na ịva 1ị GG na ỵ v a G12 ẳ G 8G na ỵ na ị GGa na ỵ ỵ na ị GGa ; G23 ẳ G 11ị v ỵ na ỵ na ị GGa ;  ỵ ỵ v  na Þ GG ð1 À na Þv a where Ga ¼ m ¼ Determining the elastic coefficients of the three-phase polymer composite material rij ẳ roij ỵ rij ỵ r ij ỵ 3K ỵ G m ¼ ; Gm =Gc À ; 10mm ỵ 5mm ị GGmc E11 ẳ na Ea ỵ na ịE ỵ Dy ¼ 9KG H¼ E, m, G, K, n are the YoungÕs modulus, Poisson ratio, shear modulus, bulk modulus, and the volume ratio of the component materials in composite, respectively The subscript m, a, c indicates the polymer material, fiber, and particle, respectively The YoungÕs modulus of the polymer composite reinforced titanium oxide particles is presented in [9] The YoungÕs modulus in the Eqs (9), (10) is consistent with the developed expression and results in [7,9] Elastic moduli of the three-phase composite material is then calculated based on the effective matrix material and fiber with six elastic constants [17] It is given by: where Z Kc À Km ; K c þ 4G3m 0:5Ea ; ð1 þ ma Þ 3K À 2G 6K À 2G ; Gc ¼ 0:5Ec ; ð1 ỵ mc ị Eẳ 9KG 3K ỵ G ; : x ẳ 4m 12ị Some numerical results and discussions To illustrate proposed approach, a three-phase composite square plate (a = b) consists of polymer matrix, particles reinforced, and fibers reinforced along the x axis is considered Table Characteristics of investigated three-phase composite Material component Epoxy Glass fiber Titanium oxide particle YoungÕs modulus Poisson ratio Symbol Value (GPa) Symbol Value Em Ea Ec 2.75 72.38 147 mm ma mc 0.35 0.2 0.2 S196 N Dinh Duc, D.K Minh / Computational Materials Science 49 (2010) S194–S198 Table The deflection w1 (na, nc) na 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 nc 0.1 0.15 0.2 0.25 0.3 0.35 0.4 3.2572 3.1616 3.0665 2.9715 2.8757 2.7788 2.6802 2.5795 2.4762 2.3698 2.2600 2.4845 2.4240 2.3646 2.3058 2.2469 2.1874 2.1269 2.0647 2.0006 1.9339 1.8640 2.0219 1.9778 1.9352 1.8935 1.8523 1.8110 1.7693 1.7267 1.6827 1.6370 1.5888 1.7101 1.6755 1.6425 1.6106 1.5794 1.5486 1.5176 1.4863 1.4542 1.4208 1.3858 1.4833 1.4550 1.4283 1.4028 1.3781 1.3539 1.3299 1.3058 1.2812 1.2559 1.2293 1.3095 1.2858 1.2637 1.2427 1.2225 1.2030 1.1837 1.1646 1.1452 1.1254 1.1047 1.1710 1.1510 1.1323 1.1148 1.0980 1.0819 1.0662 1.0507 1.0351 1.0193 1.0028 Fig h/a ratio when na = versus volume ratio nc of particle component To satisfy the simply supported boundary condition, the following approximate functions are assumed w ¼ A sin Fig The relationship between the deflection w1 and na, nc The shear deformation is considered in this calculation, therefore, a plate with thickness is 15 times smaller than itÕs width and length (h/a = h/b = 1/15) is considered The plate is assumed to be simply supported on the all edges and subjected to a transverse load distributed on the surface as q ¼ q0 sin px a sin py : a A11 ¼ A22 ¼ E11 ¼ E a sin py a ; u ¼ B cos px a sin py a ; w ¼ C sin px a cos   w ẳ w0 ỵ k ; where k ẳ  2 h ; G ð1 À mÞ a p2 E q0 a4 px py sin ; sin a a 4Dp4 ð16Þ "    2 # q0 a4 px py p2 E h : sin 1ỵ sin w ẳ w0 ỵ k ẳ a a G0 ð1 À mÞ a 4Dp4 ð17Þ A44 ¼ A55 ¼ G23 ¼ G0 E11 E A66 ¼ G12 ẳ ẳ ẳG 21 ỵ m12 ị 21 ỵ mị 13ị w ẳ 0jxẳ0;xẳa ; w ẳ 0; My ¼ 0; u ¼ 0jy¼0;y¼b : The maximum bending deflection occurs at the center of plate x = y = a/2 and is given by:   3q0 að1 À m212 ị p2 E11 n2 ỵ E11 n3 p4 G23 ð1 À m12 Þ   3q a m12 ị p2 ỵ m12 ị 3q a ẳ 04 w1 ; ỵ ẳ 04 E11 n3 5G23 n p p wMax ¼ The simply supported boundary conditions are defined as Mx ¼ 0; : a 15ị A12 ẳ E11 m12 ẳ Em w ¼ 0; py The bending deflection of square plate can be calculated from Eq (6): w0 ¼ Three-phase composite material is a transversely isotropic medium with Oxy We have following relations px ð14Þ ð18Þ Table h/a ratio when na = nc h/a C1 C2 0.1 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.0726 0.1027 0.0732 0.1035 0.0737 0.1043 0.0743 0.1051 0.075 0.1061 0.0757 0.1071 0.0765 0.1082 0.0774 0.1095 0.0785 0.111 0.0797 0.1127 0.0812 0.1148 S197 N Dinh Duc, D.K Minh / Computational Materials Science 49 (2010) S194–S198 Table h/a ratio when na = 0.2 nc h/a C1 C2 0.1 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.0369 0.0521 0.0386 0.0546 0.0404 0.0572 0.0423 0.0599 0.0443 0.0626 0.0464 0.0656 0.0485 0.0687 0.0509 0.0719 0.0533 0.0754 0.056 0.0792 0.0589 0.0833 where For the first case C1, Eq (16) shows that the difference (w–w0)/ w0 < 5% when k 0.25 The h/a ratio is then given by:   ð1 À m212 Þ p2 ỵ m12 ị w1 ẳ ỵ ; E11 n3 5G23 n ð19Þ h a and sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0:25G0 ð1 À mÞ ð1 À mÞG0 1 m12 ịG23 ẳ ẳ : p2 E E E11 2p 2p The h/a ratio for the second case C2 (k 0.5) is given by: n ¼ h=a; m12 E11 ¼ m21 : E22 In this work, a three-phase composite material with characteristics in Table is investigated According to Eqs (11), (13), and (16), bending deflection of the three-phase composite plate versus variation of the volume fraction of fiber phase na, and particle phase nc when accounting the shear deformation is presented in Table and Fig Table and Fig shows that the deflection is decreased when increasing volume ratio of either fiber or particle component The deflection changes due to the variation of fiber volume ratio larger than the particle volume ratio The condition for considering the shear deformation In this section, the difference between w (accounted shear deformation) and w0 (without shear deformation) is considered This difference is depended on the factor k (see Eq (16)) Factor k is investigated in the first case (C1) of (w–w0)/w0 < 5%, and the second one (C2) of (w–w0)/w0 < 10% Eq (16) shows that the factor k is depended on h/a ratio, na and nc Fig h/a ratio when na = 0.2 versus volume ratio nc of particle component h a sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0:5G0 ð1 À mÞ ð1 À m12 ÞG23 ¼ : p2 E p 2E11 The ratio h/a will be investigated in several specific composites when changing volume ratios of the component materials Table and Fig shows the h/a ratio when na = (two-phase composite of matrix and particles without fiber component) It shows that the h/a value is not much changed when changing volume ratio of particle component in the two-phase composite Therefore, the h/a ratio is main factor when plate bending analysis Table and Fig shows the h/a ratio when na = 0.2 (three-phase composite with 20% reinforced fiber) with volume ratio nc changes from zero to 50% The thin plate (h/a 1/20) theory can be used when nc smaller than 0.35, the thick plate theory with accounting of shear deformation has to be accounted when nc larger than 0.35 for the first case C1 The thin plate theory cannot be used for the case C2 Table and Fig shows the h/a ratio when nc = 0.2 (three-phase composite with 20% reinforced particle) with volume ratio na changes from zero to 50% It shows that the shear deformation decreases when increasing the volume ratio of fiber component Fig h/a ratio when nc = 0.2 versus volume ratio na of fiber component Table h/a ratio when nc = 0.2 na h/a C1 C2 0.1 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.075 0.1061 0.0584 0.0826 0.0509 0.072 0.0468 0.0662 0.0443 0.0626 0.0427 0.0605 0.0418 0.0592 0.0414 0.0586 0.0414 0.0585 0.0416 0.0589 0.0422 0.0597 S198 N Dinh Duc, D.K Minh / Computational Materials Science 49 (2010) S194–S198 The thin plate theory can be used for the case C1 when volume ratio of fiber component na larger than 0.15 Conclusion A bending analysis of three-phase composite plate with accounting shear deformation and particle–matrix interaction is proposed This analysis shows that the deflection is decreased when increasing volume ratio of either fiber or particle component The deflection changes due to the variation of fiber volume ratio larger than the particle volume ratio Bending deflection and elastic moduli of three-phase composite are explicitly expressed on component material mechanical properties and geometry of plate Therefore, the composite material characteristic and also structure design can be effectively optimized Acknowledgment This work has been supported by ‘‘Vietnam National University– Hanoi key science research project”, coded QGTÐ 09.01 References [1] B Jaing, R.C Batra, Intel Mater Syst Struct 12 (2001) 165–182 [2] J.C Afonso, G Ranalli, Compos Sci Technol 65 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Reddy, Mechanics of Laminated Composite Plates and Shells: Theory and Analysis, CRC Press, 2004 [5] A.K Malmeicter, V.P Tamuz, G.A Teterc, Strength of Composite Materials, ‘‘Zinatie”, Riga, 1980