The Stress Distribution in the Composite Materials with Locally Curved Fibers

Transcription

1 Journal of Conteporary Applied Matheatics V. 7, No, 207, June IN The tress Distribution in the Coposite Materials with Locally Curved Fibers Hubet Aliyev Abstract. Nowadays coposite aterials are widely used in industry as they can withstand high loads. In this paper, in the case of the piecewise-hoogeneous body odel with the use of the three -diensional linearized theory of elastic stability the proble of stress distribution in the coposite aterials with curved layers is investigated. The case in which layers are antiphased locally curved is considered. It is also assued that external copressive forces act at innity in the direction along the ber. Key Words and Phrases: stability, coposite aterials, ber, atrix, ller, echanics, defored solid, stress, noral vector, anisotropic layers. 200 Matheatics ubject Classications: 90C0, 65K05, 90C5, 49M7, 5A8 Investigations of the stability, stress strain state of coposite aterials have been carried out in any works [-5]. Due to wide usage of coposite aterials in industry, stress strain distribution, stability of coposite aterials are very iportant probles. Guz, Ilyushin, Akbarov ade great contribution to the theory of coposite aterials. In this paper, coposite aterial with innite nuber of non-intersecting bers is considered. It is assued that atrix and ller are anisotropic. This paper also investigates the inuence of antiphased piecewise-hoogeneous body odel using the three-diensional linearized theory of elastic stability proble of stress distribution in coposite aterials with curved bers. Eleents of atrix and ller will be denoted by and 2 respectively. To each ber the Cartesian coordinate syste O k x k xk 2 xk k =, 2; =, 2,,...is assigned. uppose that bers lie in x 2 x2 2 x2, and width of each ller is a constant. It is assued that the external copressive forces act at innity in the direction along the ber. For each ber equation of equilibriu, generalized Hook's law and Cauchy relations are given in the for σ k x k ε k rs = 2 j = 0 σ k u k r x k s = C k rs εk rs ; uk s x k ; i, j, r, s =, 2 r 6 c 20 JCAM All rights reserved.

2 The tress Distribution in the Coposite... 7 In generally accepted notations are used. If we denote upper liit of k th ber by, the lower liit by, condition of full contact can be written as σ σ u i n, = σ 2 n, j j ; n, j = σ 2 n, j ; 2 u i = u 2 ; = u 2 ; i, j, r, s =, 2, Here n,± j noral vectors to the surface ±. Assue that equation of the iddle line of the 2 ller is given in the for x 2 2 = F x 2 = ε f x 2 in ε [, 0 is a sall diensionless constant. Using assuption that width of llers are constant and we obtain the equation for ± : where T t = [ x 2± = t H 2 df t dt T t dft dt x 2± 2 = F t ± H 2 T t 4 ] 2 2?? Fro 4 after soe algebra we derive equations for the noral vectors to the surfaces in the for where = dx2± 2 V ± t ; n,± n,± dt V ± t = dx 2± 2 dt 2 = dx2± dt V ± t ; 5 dx 2± 2 t dt In 4-6 t is the paraeter and < t < ; x 2±,x2± 2 -coordinates of the ± ; H 2 half of the width of th ber. Quantities that express stress strain state of any -th ber will be searched in the for

3 8 Hubet Aliyev σ k = q=0 ε q σ k,q ; ε k = u k = q=0 q=0 ε q ε k,q ; ε q u k,q 7 Expression for x 2± i and n± i also written as a series in ter of εand expression of each approxiation of 7 expanded in placecitytaylor series, and fro 2 we obtain a necessary relation for each approxiation. Generalized Hook's law for this three diensional case produces [ ] ε k = E k ν k 2 Ek E k σ k _ ν k 2 νk νk 2 σ k 22 ; ε k 2 = 2G k 2 [ ε k 22 = E k ν k 2 νk νk 2 σk Ek E k 2 σ k 2 8 Ek 2 E k ] ν k 2 2 σ k 22 in 8 used notation fro [6]. The values of zero's approxiation correspond to the stress strain distribution in the coposite aterial with horizontal non curved bers and balances with force p. = p γ γ 2 σ,0 σ 2,0 = E 2 E E 2 E u k,0 = E k u k,0 2 = E k ν k ν ν 2 2 E E ν ν 2 2 E 2 E 2 2 E k ν k 2 νk νk 2 C k = const; γ k = E k 2 E E 2 E 2 E 2 ; σ,0 ; 9 σ k,0 x k ; σ k,0 x k 2 C k ; H k H H 2

4 The tress Distribution in the Coposite... 9 The value of the rst, second and the following approxiations correspond to the stress distribution with curved bers. Using 9 and we expand values of each approxiations to the placecitytaylor series around t, ±H k, after soe algebra, fro 2 we obtain necessary relations for each approxiation. Then, applying Fourier transfor to the syste of dierential equations and evaluations we obtain non-hoogenous syste of linear equations with respect to unknown coecients. In this article, stress distribution in coposite aterials with innite nuber of antiphased and locally curved bers is investigated. ince bers are located periodically with the period of 4 H 2 H, fro the coposite aterials we select only four bers, denoted as, 2, 2 and 2 2. Equation of the iddle of the surface of 2 is taken in the for x 2 2 = F x 2 = ε f x 2 = A exp x2 /L2 0 and equation of the iddle of the surface 2 2 in the for x 2 22 = F 2x 2 2 = ε f 2x 2 2 = A exp x2 2 /L2 In 0 and A and L introduced to describe local characteristics of the bers. And new diensionless paraeter ε = A/Lis introduced. Nuerical results and discussions Let us introduce γ = H 2 /L, that represents the concentration of layers. We will use coonly accepted notation: σ nnσ nn-is the stress in the direction of the noral vector nto the surface σ ττ σ ττ -is the stress in the direction of the tangent vector τ to the surface σ nτ is the stress in tangent direction to the surface η 2 is the concentration of the ller in coposite aterial The concrete nuerical investigations were carried out in the case, when the aterials of the atrix and ller are hoogeneous and anisotropic with elastic characteristics E Young's odulus and νpoisson coecient. Before nuerical analysis it is necessary to note that, the values related to the will be denoted by the upper indices and the values related to -by the upper indices -. For nuerical analysis it is assued that ν 2 = ν 2 = ν = 0.; E 2 /E = 50and ε = 0.05 σ nn σ ττ, σnn σ ττ -are stress coponents in the direction of τ n on the surfaces and respectively, σ nτ -stress coponent in tangent direction of. Let us analyze the ipact of E2, E2 E G 2 2 to the stress distribution of the stresses listed above, Table, 2, and, are calculated for the case γ = 0. and η 2 = 0.5; 0.2; 0.respectively. In these tables values of σ nn/σ,0, σ nn/σ,0, σ ττ /σ,0 are calculated at t /L = 0.8, the value of σ nτ /σ,0 at t /L =.6.

5 40 Hubet Aliyev Analysis of the results shows that, with decreasing E2 and increasing E2 the value of E G 2 2 σ nn/σ,0 is decreasing. Ipact of E2 to the σ nn/σ,0 is ore signicant than ipact of G 2 2

6 The tress Distribution in the Coposite... 4 E 2. This conclusion is valid for allη 2. For all η 2 increasing E2 reduces the absolute value E G 2 2 of σnn/σ,0. Increasing the value of E2 increases, decreasing of E2 decreases the absolute G 2 2 E value of σ nτ /σ,0. Increasing E2 and decreasing E2 increases the value of σ G 2 2 E ττ /σ,0 and σττ /σ,0. Ipact of E2 to σ G 2 ττ /σ,0 and σττ /σ,0 is ore signicant than E2. 2 E In this paper proble of stress distribution in the coposite aterials with curved layers was investigated. Piecewise hoogenous body odel was considered, and by the use of three-diensional linearized theory, stress distributions in the coposite aterials with antiphased locally curved layers, when external forces acted at innity considered. Analyzing results show that obtained results agree with the results found in the previous investigations. References [] Akbarov,. D. 20, tability Loss and Buckling Delaination: Three-Diensional Linearized Approach for Elastic and Viscoelastic Coposites. pringer, Heidelberg, tatenew York. [2] Akbarov,.D. and Yahnioglu, N. 20, Buckling delaination of a sandwich platestrip with piezoelectric face and elastic core layers, Appl. Math. Model. 7, [] Akbarov,.D. and Rzayev, O.G. 2002, On the buckling of the elastic and viscoelastic coposite circular thick plate with a penny-shaped crack, Eur. J Mech. A olid. 22, [4].D. Akbarov, A.N. Guz tatics of lainated and brous coposites with curved structures. Appl. Mech. Rev., vol.. 45 N2, February 992. p [5] D.H. Li, X. Wang, " A odied Fletcher-RGuz, A.N Elastic waves in bodies with initial residual stresses. A.C.K., Kiev. [6] Guz, A.N. Elastic waves in bodies with initial residual stresses. A.C.K., Kiev, [7] Lexniskiy.Q. Teoriya upruqosti anizotropnogo tela, Nauka, 977. Hubet Aliyev Baku Engineering University E-ail:hualiyev@beu.edu.az Received 4 March 207 Accepted 2 May 207