Spatial dynamical behavior of narrow composite beams

Applied Mathematics and Computation 216 (2010) 3627–3633

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Spatial dynamical behavior of narrow composite beams Li Zhang *, Shu Tang Liu College of Control Science and Engineering, Shandong University, Jinan 250061, China

a r t i c l e

i n f o

Keywords: Spatial chaos Nonlinear forced term Composite Bifurcation Discrete

a b s t r a c t In this article, we study the nonlinear dynamical behavior of composite beams with nonlinear forced term. We obtain a fourth order partial differential equation (PDE) from the motion law of composite beams and discretize the PDE system. We analyze the corresponding discrete system and obtain the nonlinear spatial behavior such as chaotic and bifurcation behavior of interlaminar displacement. Crown Copyright Ó 2010 Published by Elsevier Inc. All rights reserved.

1. Introduction Multilayered composites are of great use in the design of modern structures and many advanced engineering applications. Their dynamical behavior is presently of interest and important because of improved precision and reliability in the applications in severe operating environments. Therefore, it is of great concern to analyze the laminated composite in many advanced engineering problems. In recent years, numerous scholars have focussed on dynamical behavior in composite beams [1–14]. When predicting possible behavior of composite beams, nonlinear phenomenon exists in every physical system and in many cases. Nevertheless, the characteristic of nonlinear systems is more difficult to predict. Thus, a theory that can predict the nonlinear dynamical behavior of composite beams with nonlinear factor is worth studying. However, there is a little research on spatial dynamical behavior of composite materials with nonlinear forced term. The purpose of the paper is to obtain the spatial chaotic and bifurcation behavior of composite beams. The structure of the paper is as follows. In Section 1, a stress–displacement analysis of composite beams with nonlinear forced term is carried out. A fourth order partial differential equation (PDE) of displacement for composite beams with nonlinear forced term is introduced. Using the classical 13 point difference method, we discretize the PDE system in Section 2. The spatial chaotic and bifurcation behavior of the interlaminar displacement are predicted in Section 3. Section 4 ends the paper with the concluding remarks. 2. Spatial dynamics of interlaminar displacement of composite beam In this article, all the fundamental elasticity relationships between the components of stress, strain and displacement fields are explicitly maintained throughout the elastic continuum and rx, ry, rz, syx, sxz, syz stand for stresses, ex, ey, ez, cyx, czy, czx represent strains and u, v, w mean displacements permanently. We consider a composite, narrow beam made up of a number of homogeneous isotropic and/or orthotropic linear elastic laminae of the uniform thickness shown in Fig. 1, in which the length of this model is expressed as L; the uniform thickness of the material as h0; the total thickness of this composite beam model as h. The top surface of the beam has a transversely distributed load. Under such a condition, the beam domain is in the 2D state of plane stress. * Corresponding author. E-mail address: [email protected] (L. Zhang). 0096-3003/$ - see front matter Crown Copyright Ó 2010 Published by Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2010.05.010

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Fig. 1. Configuration of the composite, narrow beam.

In the 2D beam domain, we get known

rz ¼ szx ¼ syz ¼ 0; ez ¼ @w=@z ¼ 0;

czx ¼ @u=@z þ @w=@x ¼ 0; cyz ¼ @w=@y þ @ v =@z ¼ 0: The 2D differential equations of equilibrium are

@ rx =@x þ @ sxy =@y þ Bx ¼ 0; @ ry =@y þ @ sxy =@x þ By ¼ 0; where Bx, By are the body forces per unit volume in the x and y directions, respectively. However, in the real environment, the dynamic phenomenon is complex under plane stress conditions. So, thinking over all kinds of noise in the real physical world, we consider the system as below:

@ rx =@x þ @ sxy =@y þ g ¼ 0; @ ry =@y þ @ sxy =@x þ g ¼ 0; where g is a nonlinear function about stresses rx, ry, sxy. According to the relationships between the elastic displacement and strain:

ex ¼ @u=@x; ey ¼ @ v =@y;

cyx ¼ @u=@y þ @ v =@x; cxy ¼ @u=@y þ @ v =@x;

and the relationships between strain and stress:

rx ¼ kh þ 2lex ; ry ¼ kh þ 2ley ; cyx ¼ lcxy ; where h = ex + ey = @u/@x + @v/@ y, k and l are the Lamé constants, we get the equilibrium equation of displacement:

ðk þ lÞð@=@x; @=@yÞh þ lr2 ðu; v Þ þ ðg; gÞ ¼ 0;

ð1Þ

where

r2 ¼ @ 2 =@x2 þ @ 2 =@y2 : Using Laplace operator r2 to act on the system (1), we have the fourth order partial differential equation

r4 ðu; v Þ ¼ ðk þ lÞ=lÞð@=@xÞ; ð@=@yÞr2 h  1=lr2 ðg; gÞ ¼ 0:

ð2Þ

For easily considered, we suppose that

ðk þ lÞ=lð@=@x; @=@yÞr2 h  1=lr2 ðg; gÞ ¼ ðf ðuÞ; f ðv ÞÞ;

ð3Þ

where f is a nonlinear function. From (3) and (2), the following system can be derived

r4 ðu; v Þ ¼ ðf ðuÞ; f ðv ÞÞ:

ð4Þ

It is enough to deliberate system (4) if we only consider the dynamical behavior of one of the displacements u or v. Without loss of generality, the following system

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1

C(m,n)

0.5

0

−0.5

−1 80 60 40 20 0

n

0

10

20

40

30

60

50

70

m

1

C(m,n)

0.5

0

−0.5

−1 80 31

60 30.5

40

30

20

n

29.5 0

29

m

Fig. 2. The spatial chaotic behavior of interlaminar displacement and its section.

r4 u ¼ f ðuÞ

ð5Þ

is discussed, where f is a nonlinear function. The corresponding discrete system of (5) is derived by applying the standard 5-point Laplace’s operator twice as follows:

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Fig. 3. The spatial bifurcation behavior of interlaminar displacement and its section..

L. Zhang, S. Tang Liu / Applied Mathematics and Computation 216 (2010) 3627–3633

Fig. 4. The spatial Lyapunov exponent of system (10) and its section.

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20um;n þ umþ2;n þ um;nþ2 þ um2;n þ um;n2  8umþ1;n  8um1;n  8um;nþ1  8um;n1 þ 2umþ1;nþ1 þ 2umþ1;n1 þ 2um1;nþ1 þ 2um1;n1 ¼ f ð20um;n Þ;

ð6Þ

where m, n 2 Nr = {r, r + 1, r + 2, . . .jr 6 0, r 2 Z}. 3. Spatial chaotic and bifurcation behaviors of interlaminar displacement For nonlinear system(6), the forced term f can be assumed to be a continuous function without loss of generality. From Weierstrass approximation theorem[15], f can be always approximated by a polynomial without loss of generality. Then we can choose f(x) = a0 + a1x + a2x2. From (6), we get the following system:

20um;n þ umþ2;n þ um;nþ2 þ um2;n þ um;n2  8umþ1;n  8um1;n  8um;nþ1  8um;n1 þ 2umþ1;nþ1 þ 2umþ1;n1 þ 2um1;nþ1 þ 2um1;n1 ¼ a0 þ eð20um;n Þ þ a2 ð20um;n Þ:

ð7Þ

In this paper, let

C m;n ðrÞ ¼ umþ2r;n þ um;nþ2r þ um2r;n þ um;n2r  8umþr;n  8umr;n  8um;nþr  8um;nr þ 2umþr;nþr þ 2umþr;nr þ 2umr;nþr þ 2umr;nr ; r = 0, 1, 2, . . .. Then we obtain

8 > < C m;n ð1Þ ¼ umþ2;n þ um;nþ2 þ um2;n þ um;n2 8umþ1;n  8um1;n  8um;nþ1  8um;n1 þ 2umþ1;nþ1 þ 2umþ1;n1 þ 2um1;nþ1 þ 2um1;n1 ; > : C m;n ð0Þ ¼ 20um;n : Consequently, system (7) reduces to

C m;n ð1Þ ¼ C m;n ð0Þ þ a0 þ eC m;n ð0Þ þ a2 C m;n ð0Þ2 :

ð8Þ

For system (8), let

(

y ¼ x þ ðe þ 1Þ=2a2 Þ;

l ¼ ð1 þ eÞ2 =4  ð1 þ eÞ=2  a0 a2 :

ð9Þ

then system (8) can be reduced to the following system:

C m;n ð1Þ ¼ 1  lC m;n ð0Þ:

ð10Þ

2

For (10), when l = (1 + e) /4  (1 + e)/2  a0a2 > 1.55, the system (10) is spatial chaotic [16]. For example, if a0 = 0.5, a2 = 0.16, e = 3, then l = 1.92 > 1.55 and the system (10) reduces to Cm,n(1) = 1  1.92Cm,n(0). Fig. 2(a) and (b) show the corresponding spatial chaotic behavior. In addition, since l = (1 + e)2/4  (1 + e)/2  a0a2 is a variable real parameter, when e is changed, the spatial bifurcation of interlaminar displacement can occur as shown in Fig. 3(a) and (b). Using the definition of spatial Lyapunov exponent [16] for system (10), we can get the spatial Lyapunov exponent as shown in Fig. 4(a) and (b). 4. Conclusion In this paper, we induced a fourth order partial equation from the analysis of the interlaminar displacements in composite, narrow beam. The spatial chaos and bifurcation behavior of interlaminar displacement were studied. The analysis of nonlinear dynamic behaviors of interlaminar displacement provided us with more information to understand and solve the nonlinear dynamics of delamination in composite structures. Acknowledgements This work has been supported in part by the National Natural Science Foundation of China (Grant No. 60874009 and No. 10971120) and the Foundation for the Author of National Excellent Doctoral Dissertation of China (Grant No. 200444) for financial support. References [1] M.T. Ahmadian, R.A. Jafari-Talookolaei, E. Esmailzadeh, Dynamics of a laminated composite beam on Pasternak-viscoelastic foundation subjected to a moving oscillator, J. Vib. Control. 14 (2008) 807–830. [2] M.A. Benatta, A. Tounsi, I. Mechab, M.B. Bouiadjra, Mathematical solution for bending of short hybrid composite beams with variable fibers spacing, Appl. Math. Comput. 212 (2009) 337–348.

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