Instability of nonlinear viscoelastic plates - Semantic Scholar
Applied Mathematics and Computation 162 (2005) 1453–1463
www.elsevier.com/locate/amc
Instability of nonlinear viscoelastic plates Li-Qun Chen a
a,b,*
, Chang-Jun Cheng
a,b
Department of Mechanics, Shanghai University, Suit 1101, Building 2, 555 Hu Tai Road, Shanghai 200436, China b Shanghai Institute of Applied Mathematics and Mechanics, Shanghai 200072, China
Abstract This paper investigates the instability of an isotropic, homogeneous, simply supported rectangular plate subjected to a prescribed periodic in-plane load. The material is assumed to be viscoelastic and obey the Leaderman nonlinear constitutive relation. The equation of motion is derived as a nonlinear integro-partial-differential equation, and is simplified into a nonlinear integro-differential equation by the Galerkin method. The averaging method is developed to establish the condition of instability. Numerical results are presented to compare with the analytical ones. Ó 2004 Elsevier Inc. All rights reserved.
1. Introduction The dynamical stability problems of viscoelastic plates are more complicated than those of elastic plates, since the stress-strain relations of viscoelastic materials often leads to integro-differential equations of motion. In 1990, Aboudi et al. [1] used the concept of the Liapunov exponents to analyze the dynamic stability of viscoelastic homogeneous plates. In 1992, the method was also used by Cederbaum et al. [2] to study the dynamic stability of shear deformable viscoelastic laminated plates. Recently Zhu et al. [3] discussed the dynamical behavior of viscoelastic rectangular plates, and Cheng and Zhang [4] discovered hyperchaotic motion of viscoelastic thin plates with large
*
Corresponding author. E-mail address: [email protected] (L.-Q. Chen).
0096-3003/$ - see front matter Ó 2004 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2004.03.020
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deflection. In addition, Zhang and Cheng [5] studied chaotic behavior of large deflection viscoelastic thin plates in a supersonic flow. In all above mentioned work, the Boltzmann superposition principle was incorporated, enabling the modeling of linear viscoelastic materials. However, it is well known that many viscoelastic materials are not linear hence they should be modeled nonlinearily in order to give an adequate description of a viscoelastic structure behavior. The comparison research [6] has shown that the Leaderman model [7] is one of useful representations, when prediction and simplicity are concerned. In 1994, Touati and Cederbaum [8] employed the Liapunov exponents to study numerically the influence of the various parameters involved on stability of nonlinear viscoelastic plates. In 1995, they [9] further considered the influence of large deflections on the dynamic stability of nonlinear viscoelastic plates. Although the numerical procedure can be used to determine whether the plate is stable for a particular set of parameters, the analytical methods can give more general conclusions. For linear viscoelastic plates, Cederbaum [10] obtained the instability regions analytically via the multiple-scales method. In this paper, the authors derive the integro-partial-differential equation of motion for an isotropic homogeneous, simply supported rectangular plate subjected to a given periodic in-plane excitation, within the Leaderman model of nonlinear materials. The Galerkin method is applied to simplify the equation into a nonlinear integro-differential equation. The averaging method is utilized to obtain a critical value of the load parameter at which instability occurs. Numerical results are presented with comparison to those given by the averaging method.
2. The simplified mathematical model of nonlinear viscoelastic plates Consider the motion of an isotropic, homogeneous, simply supported viscoelastic plate subjected to prescribed in-plane loads. Geometrical relations are supposed to be linear, but the transverse projections of in-plane loads are concerned. The dynamical equation of the plate is € ¼ 0; Mxx;xx þ 2Mxy;xy þ Myy:yy þ Nx W;xx þ Ny W;yy þ 2Nxy W;xy þ qhW
ð1Þ
where Mxx , Mxy , and Myy are the stress couples, Nx and Ny are the in-plane loads in the x and y directions, Nxy is the in-plane shear force, W is the deflection in the transverse direction, q and h are respectively the material density and the plate thickness. For nonlinear viscoelastic materials, the constitutive relations are given by [7]
L.-Q. Chen, C.-J. Cheng / Appl. Math. Comput. 162 (2005) 1453–1463
rðtÞ ¼ Qð0Þg½eðtÞ þ
Z
1455
t
Q_ ðt sÞg½eðsÞ ds;
ð2Þ
0þ
where T
rðtÞ ¼ ð rxx ðtÞ ryy ðtÞ rxy ðtÞ Þ ; eðtÞ ¼ ð exx ðtÞ 0 1 exx ðtÞ þ bxx e2xx ðtÞ þ cxx e3xx ðtÞ B C gðeðtÞÞ ¼ @ eyy ðtÞ þ byy e2yy ðtÞ þ cyy e3yy ðtÞ A exy ðtÞ þ bxy e2xy ðtÞ þ cxy e3xy ðtÞ
eyy ðtÞ exy ðtÞ Þ
T
ð3Þ
in which bxx , byy , bxy , cxx , cyy , and cxy are constants. For the state of plane stress of isotropic viscoelastic plates, and in the case that the Poisson ratio m is a constant 2
Q11 ðtÞ Q12 ðtÞ QðtÞ ¼ 4 Q21 ðtÞ Q22 ðtÞ 0 0
3 0 0 5; Q66 ðtÞ
ð4Þ
where Q11 ðtÞ ¼ Q22 ðtÞ ¼ Q66 ðtÞ ¼
E ðt Þ ; 1 m2
Q12 ðtÞ ¼ Q21 ðtÞ ¼
mEðtÞ ; 1m
E ðt Þ 2ð1 þ mÞ
ð5Þ
in which EðtÞ is a time-dependent relaxation function, and Eð0Þ is the initial Young modulus of the material. For a homogeneous thin plate, the strain-displacement relations and the stress couples are respectively given by exx ¼ zW;xx ; eyy ¼ zW;yy ; exy ¼ 2zW;xy ; Z h=2 Z h=2 zrxx dz; Myy ¼ zryy dz; Mxx ¼ h=2
ð6Þ Mxy ¼
h=2
Z
h=2
zrxy dz: h=2
ð7Þ Substituting Eq. (6) into Eq. (2) and using Eq. (4), one obtains from Eq. (7)
Mxx ¼ F1 W;xx þ mW;yy þ cxx F2 fW;xx g þ cyy mF2 W;xy ;
Myy ¼ F1 mW;xx þ W;yy þ cxx mF2 fW;xx g þ cyy F2 W;xy ;
Mxy ¼ F1 ð1 mÞW;xy þ 4cxx ð1 mÞF2 W;xy ;
ð8Þ
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L.-Q. Chen, C.-J. Cheng / Appl. Math. Comput. 162 (2005) 1453–1463
where for any function f ðx; y; tÞ the functional F1 and F2 are defined as Z t I _ ðt sÞf ð x; y; sÞ ds ; E E ð 0 Þf ð x; y; t Þ þ F1 f f ð x; y; tÞg ¼ 1 m2 0þ Z t J 3 _ ðt sÞf 3 ð x; y; sÞ ds F2 f f ð x; y; tÞg ¼ E E ð 0 Þf ð x; y; t Þ þ 1 m2 0þ
ð9Þ
in which Z
h=2
h3 I¼ z dz ¼ ; 12 h=2 2
J¼
Z
h=2
z4 dz ¼
h=2
h5 : 80
ð10Þ
Substituting of Eq. (8) into Eq. (1) leads to
n
F1 W;xxxx þ 2W;xxyy þ W;yyyy þ cxx ½F2 fW;xx g;xx þ cyy F2 W;yy ;yy
o þ cyy m F2 W;yy ;xx þ cxx m½F2 fW;xx g;yy þ 8cxy ð1 mÞ F2 W;xy ;xy € ¼ 0: þ Nx W;xx þ Ny W;yy þ 2N;xy W;xy þ qhW
ð11Þ
Eq. (11) is the mathematical model of nonlinear viscoelastic plates. If the separation of variables method can be employed, let W ð x; y; tÞ ¼ qðtÞUð x; y Þ:
ð12Þ
Substituting Eq. (12) into Eq. (11) gives h i I J 3 þ 2U F U þ U ð t Þ þ cxx ðU;xx Þ ;xxxx ;xxyy ;yyyy 1 2 2 ;xx 1m 1m h h h i 3 i 3 i 3 þ cyy U;yy þ cyy m U;yy þ cxx m ðU;xx Þ ;yy ;xx ;yy h 3 i þ 8cxy ð1 mÞ U;xy F2 ðtÞ þ Nx U;xx qðtÞ þ Ny U;yy qðtÞ ;xy
qðtÞ ¼ 0; þ 2Nxy U;xy qðtÞ þ qhU€ where F1 ðtÞ ¼ Eð0ÞqðtÞ þ
Z
t
E_ ðt sÞqðsÞ ds; 0
3
F2 ðtÞ ¼ Eð0Þq ðtÞ þ
ð13Þ
Z
ð14Þ
t
E_ ðt sÞq3 ðsÞ ds: 0
For a simply supported square plate with the side length l, applying the Galerkin method to Eq. (13) and neglecting higher modes, one can suppose px py sin : ð15Þ Uð x; y Þ ¼ sin l l Substituting Eq. (15) into Eq. (13) and integrating the resulting equation respect to x and y on ½0; l, together with Eqs. (10) and (14), one derives
L.-Q. Chen, C.-J. Cheng / Appl. Math. Comput. 162 (2005) 1453–1463
Nx þ Ny þ Nxy 2 € qþx 1 q þ kq3 N Z t Z t 2 _ Dðt sÞqðsÞ ds k D_ ðt sÞq3 ðsÞ ds; ¼ x 0þ
1457
ð16Þ
0þ
where p 4 h2 E ð 0Þ p 2 h3 E ð 0Þ E ðt Þ ; ; N¼ ; DðtÞ ¼ 2 4 2 2 3ð1 m Þql 3ð1 m Þl E ð 0Þ 9p8 h4 Eð0Þ k¼ cxx ð1 þ mÞ þ cyy ð1 þ mÞ þ 8cxy ð1 mÞ : 2 8 1280ð1 m Þql x2 ¼
ð17Þ
Eq. (16) is the simplified mathematical model of nonlinear viscoelastic plates.
3. Condition of the instability Assume that the relaxation function in Eq. (5) is given by EðtÞ ¼ A þ Begt :
ð18Þ
The in-plane loads in the x and y directions are both simple harmonic excitations Nx ¼ Ny ¼ Nxy ¼ fN cos Xt:
ð19Þ
Inserting Eqs. (18) and (19) into Eq. (16) yields € q þ x2 ð1 4f cos XtÞq þ kq3 Z t Z t gt 2 gs gs 3 ¼ gbe x e qðsÞ ds þ k e q ðsÞ ds ; 0
ð20Þ
0
where a¼
A ; AþB
b¼
B : AþB
ð21Þ
Next, we will use the idea of the averaging method to analyze the long-time dynamical behavior governed by Eq. (20). Assume that the solution of Eq. (18) have the form q ¼ q cos u;
u ¼ xt þ w;
q_ ¼ xq sin u;
ð22Þ
where w varies slowly with time t. Hence the solution satisfies the condition _ sin u ¼ 0: q_ cos u wq
ð23Þ
Substituting Eq. (22) into Eq. (20) and considering only the case X ¼ x like the previous studies [8,10], one obtains
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q_ sin u qw_ cos u ¼ 4f xq cos 2ðu wÞ cos 2u þ
k 3 q cos3 u x
1 Gðq; u; w; tÞ; x
ð24Þ
where Gðq; u; w; tÞ ¼ gbq 4x2 þ 3kq2 ðg cos w þ x sin wÞ=4 g2 þ x2 kq2 ðg cos 3w þ 3x sin 3wÞ=4 g2 þ 9x2 ð1 egt Þ:
ð25Þ
In evaluating the integrals, the slow variable w is treated as a constant. For the long-time dynamical behavior, the term egt that decreases rapidly can be neglected. Therefore Gðq; u; w; tÞ in Eq. (24) can be replaced by ð4x2 þ 3kq2 Þðg cos w þ x sin wÞ H ðq; u; wÞ ¼ gbq 4ð g2 þ x 2 Þ 2 kq ðg cos 3w þ 3x sin 3wÞ þ : ð26Þ 4ðg2 þ 9x2 Þ Solving Eqs. (23) and (24), one gets k 1 q_ ¼ 4f xq cos 2ðu wÞ cos 2u q3 cos3 u þ H ðq; u; wÞ sin u; x x k 1 qw_ ¼ 4f xq cos 2ðu wÞ cos 2u q3 cos3 u þ H ðq; u; wÞ cos u: x x ð27Þ Averaging Eq. (27) over ½0; 2p respect to / yields the averaged system Z 1 2p k 3 1 3 q_ ¼ 4f xqcos2ðu wÞcos2u q cos u þ H ðq;u;wÞ sinudu; 2p 0 x x Z 2p 1 k 1 4f xqcos2ðu wÞcos2u q3 cos3 u þ H ðq;u;wÞ cosudu: qw_ ¼ 2p 0 x x ð28Þ Inserting Eq. (26) into Eq. (27) and evaluating the integrals, one obtains bgx2 3bkg 2 _q ¼ q f x sin 2w þ q ; 8ð g2 þ x 2 Þ 2ð g2 þ x 2 Þ bxg2 3 bg2 _ C q2 : qw ¼ q f x cos 2w þ þ k 1 8 2ðg2 þ x2 Þ xðg2 þ x2 Þ ð29Þ The steady-state motion of Eq. (20) corresponds to the equilibrium states of Eq. (29) given by
L.-Q. Chen, C.-J. Cheng / Appl. Math. Comput. 162 (2005) 1453–1463
1459
bgx2 3bkg q2 ¼ 0; q f x sin 2w þ 8ðg2 þ x2 Þ 2ð g2 þ x 2 Þ bxg2 3 bg2 C q2 ¼ 0: q f x cos 2w þ þ k 1 8 2ðg2 þ x2 Þ xðg2 þ x2 Þ ð30Þ For the trivial case of q ¼ 0, Eq. (29) deduces to bgx2 3bkg 2 q : q_ ¼ q f x sin 2w þ 8ðg2 þ x2 Þ 2ð g2 þ x 2 Þ
ð31Þ
Based on the linearization analysis, the solution q ¼ 0 is stable if f < fc ¼
bgx : 2ðg2 þ x2 Þ
ð32Þ
If f > fc , q ¼ 0 may become unstable and the nontrivial solution corresponding to the limit circle may occur.
4. Numerical results In order to obtain a model easy to compute numerically, one usually differentiates Eq. (18) via the Leibnitz rule, and yields a three-order ordinary differential equation. However the operation is quite complicated. Here the auxiliary variable is introduced as Z t gt x ¼ gbe egs x2 qðsÞ þ kq3 ðsÞ ds ð33Þ 0
then x_ ¼ gx þ gb x2 q þ kq3 :
ð34Þ
Inserting Eq. (33) into Eq. (20) yields € q þ x2 ð1 4f cos XtÞq þ kq3 þ x ¼ 0:
ð35Þ
In the present paper, the numerical results herein are obtained by using, X ¼ 2, a ¼ 0:1 and b ¼ 0:9. Fix g ¼ 0:01 and k ¼ 0:01. In this case, from Eq. (35), we have fc ¼ 0:00449955. Now we consider the long-time dynamical behavior of Eq. (33) for some values of f . For a small excitation f < fc the system is asymptotically stable with response tending to zero. The case that f ¼ 0:0021 is depicted in Fig. 1. However the response approaches to zero significantly slowly if f is close to fc . The case that f ¼ 0:004 is shown in Fig. 2. For a excitation f > fc instability occurs in the system and its dynamical behavior
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L.-Q. Chen, C.-J. Cheng / Appl. Math. Comput. 162 (2005) 1453–1463
Fig. 1. The response for f ¼ 0:0021.
Fig. 2. The response for f ¼ 0:004.
asymptotically tends a periodic motion. If f is slightly bigger than fc the response approaches the stable limit cycle very slowly. The case that f ¼ 0:0045 is depicted in Fig. 3. It is also noted that the amplitude of the response increases with the excitation parameter f . The case that f ¼ 0:009 is shown in Fig. 4. Fix k ¼ 0:1 and f ¼ 0:025. Then from Eq. (32) s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi# " b b g > gc ¼ 1 : ð36Þ 4f 4f In this case, gc ¼ 0:0557. The case that g ¼ 0:07 is depicted in Fig. 5. However the response approaches to zero slowly if a is very near gc . The case that g ¼ 0:06 is shown in Fig. 6. For gc but still adequate big, the system asymptotically tends a periodic motion. The response in the case that g ¼ 0:05 are depicted in Fig. 7. It is noted that the amplitude of the response decreases with g. The results in the case that g ¼ 0:01 are shown in Fig. 8.
L.-Q. Chen, C.-J. Cheng / Appl. Math. Comput. 162 (2005) 1453–1463
Fig. 3. The response for f ¼ 0:0045.
Fig. 4. The response for f ¼ 0:009.
Fig. 5. The response for g ¼ 0:07.
1461
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L.-Q. Chen, C.-J. Cheng / Appl. Math. Comput. 162 (2005) 1453–1463
Fig. 6. The response for g ¼ 0:06.
Fig. 7. The response for g ¼ 0:05.
Fig. 8. The response for g ¼ 0:01.
L.-Q. Chen, C.-J. Cheng / Appl. Math. Comput. 162 (2005) 1453–1463
1463
5. Conclusions A nonlinear integro-partial-differential equation (12) that describes the motion of an isotropic homogeneous thin simply supported viscoelastic plate subjected to given in-plane loads is established. The plate is made of a nonlinear viscoelastic material that is assumed to obey the Leaderman constitutive relation (2). The Galerkin method is used to simplify Eq. (12) into a integrodifferential equation (14). Under the assumption that the relaxation function is given by Eq. (16), we employed the idea of the averaging method to analyze the dynamical behavior of the plate. Based on the averaged system (27), the critical value of the load parameter (30) at which instability appears is presented. Numerical results of the ordinary differential equations (34) and (35) indicate that the system successively appears asymptotical equilibrium state and periodic motion with the increase of the excitation parameter f and with decrease of the material coefficient g. The analytical critical value of the excitation parameter fc and the material coefficient gc are confirmed by the numerical simulations. Acknowledgements The work was supported by the National Natural Science Foundation of China (Project No. 10172056). References [1] J. Aboudi, G. Cederbaum, I. Elishakoff, Stability of viscoelastic plates by Lyapunov exponent, J. Sound Vib. 139 (1990) 459–468. [2] G. Cederbaum, J. Aboudi, I. Elishakoff, Dynamic stability of viscoelastic composite plates via the Lyapunov exponent, Int. J. Solids Struct. 28 (1991) 317–327. [3] Y.-Y. Zhu, N.-H. Zhang, F. Miura, Dynamical behavior of viscoelastic rectangular plates, in: W.-Z. Chien (Ed.), Proc. 3rd Inter. Conf. Nonlinear Mech., Shanghai UP, 1998, pp. 445–450. [4] C.-J. Cheng, N.-H. Zhang, Chaos and hyperchaos motion of viscoelastic rectangular plates under a transverse periodic load, Acta Mech. Sin. 30 (1998) 690–699 (in Chinese). [5] N.-H. Zhang, C.-J. Cheng, Chaos behavior of viscoelastic plates in supersonic flow, in: W.-Z. Chien (Ed.), Proc. 3rd Inter. Conf. Nonlinear Mech., Shanghai UP, 1998, pp. 432–436. [6] J. Smart, J.G. Williams, A comparison of single integral non-linear viscoelasticity theories, J. Mech. Phys. Solids 20 (1972) 313–324. [7] H. Leaderman, Large longitudinal retarded elastic deformation of rubberlike network polymers, Polym. Trans. Soc. Rheol. 6 (1962) 361–382. [8] D. Touati, G. Cederbaum, Dynamic stability of nonlinear viscoelastic plate, Int. J. Solids Struct. 31 (1994) 2367–2376. [9] D. Touati, G. Cederbaum, Influence of large deflections on the dynamic stability of nonlinear viscoelastic plates, Acta Mech. 113 (1995) 215–231. [10] G. Cederbaum, Parametric excitation of viscoelastic plates, Mech. Struct. Mech. 59 (1992) 37–51.
www.elsevier.com/locate/amc
Instability of nonlinear viscoelastic plates Li-Qun Chen a
a,b,*
, Chang-Jun Cheng
a,b
Department of Mechanics, Shanghai University, Suit 1101, Building 2, 555 Hu Tai Road, Shanghai 200436, China b Shanghai Institute of Applied Mathematics and Mechanics, Shanghai 200072, China
Abstract This paper investigates the instability of an isotropic, homogeneous, simply supported rectangular plate subjected to a prescribed periodic in-plane load. The material is assumed to be viscoelastic and obey the Leaderman nonlinear constitutive relation. The equation of motion is derived as a nonlinear integro-partial-differential equation, and is simplified into a nonlinear integro-differential equation by the Galerkin method. The averaging method is developed to establish the condition of instability. Numerical results are presented to compare with the analytical ones. Ó 2004 Elsevier Inc. All rights reserved.
1. Introduction The dynamical stability problems of viscoelastic plates are more complicated than those of elastic plates, since the stress-strain relations of viscoelastic materials often leads to integro-differential equations of motion. In 1990, Aboudi et al. [1] used the concept of the Liapunov exponents to analyze the dynamic stability of viscoelastic homogeneous plates. In 1992, the method was also used by Cederbaum et al. [2] to study the dynamic stability of shear deformable viscoelastic laminated plates. Recently Zhu et al. [3] discussed the dynamical behavior of viscoelastic rectangular plates, and Cheng and Zhang [4] discovered hyperchaotic motion of viscoelastic thin plates with large
*
Corresponding author. E-mail address: [email protected] (L.-Q. Chen).
0096-3003/$ - see front matter Ó 2004 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2004.03.020
1454
L.-Q. Chen, C.-J. Cheng / Appl. Math. Comput. 162 (2005) 1453–1463
deflection. In addition, Zhang and Cheng [5] studied chaotic behavior of large deflection viscoelastic thin plates in a supersonic flow. In all above mentioned work, the Boltzmann superposition principle was incorporated, enabling the modeling of linear viscoelastic materials. However, it is well known that many viscoelastic materials are not linear hence they should be modeled nonlinearily in order to give an adequate description of a viscoelastic structure behavior. The comparison research [6] has shown that the Leaderman model [7] is one of useful representations, when prediction and simplicity are concerned. In 1994, Touati and Cederbaum [8] employed the Liapunov exponents to study numerically the influence of the various parameters involved on stability of nonlinear viscoelastic plates. In 1995, they [9] further considered the influence of large deflections on the dynamic stability of nonlinear viscoelastic plates. Although the numerical procedure can be used to determine whether the plate is stable for a particular set of parameters, the analytical methods can give more general conclusions. For linear viscoelastic plates, Cederbaum [10] obtained the instability regions analytically via the multiple-scales method. In this paper, the authors derive the integro-partial-differential equation of motion for an isotropic homogeneous, simply supported rectangular plate subjected to a given periodic in-plane excitation, within the Leaderman model of nonlinear materials. The Galerkin method is applied to simplify the equation into a nonlinear integro-differential equation. The averaging method is utilized to obtain a critical value of the load parameter at which instability occurs. Numerical results are presented with comparison to those given by the averaging method.
2. The simplified mathematical model of nonlinear viscoelastic plates Consider the motion of an isotropic, homogeneous, simply supported viscoelastic plate subjected to prescribed in-plane loads. Geometrical relations are supposed to be linear, but the transverse projections of in-plane loads are concerned. The dynamical equation of the plate is € ¼ 0; Mxx;xx þ 2Mxy;xy þ Myy:yy þ Nx W;xx þ Ny W;yy þ 2Nxy W;xy þ qhW
ð1Þ
where Mxx , Mxy , and Myy are the stress couples, Nx and Ny are the in-plane loads in the x and y directions, Nxy is the in-plane shear force, W is the deflection in the transverse direction, q and h are respectively the material density and the plate thickness. For nonlinear viscoelastic materials, the constitutive relations are given by [7]
L.-Q. Chen, C.-J. Cheng / Appl. Math. Comput. 162 (2005) 1453–1463
rðtÞ ¼ Qð0Þg½eðtÞ þ
Z
1455
t
Q_ ðt sÞg½eðsÞ ds;
ð2Þ
0þ
where T
rðtÞ ¼ ð rxx ðtÞ ryy ðtÞ rxy ðtÞ Þ ; eðtÞ ¼ ð exx ðtÞ 0 1 exx ðtÞ þ bxx e2xx ðtÞ þ cxx e3xx ðtÞ B C gðeðtÞÞ ¼ @ eyy ðtÞ þ byy e2yy ðtÞ þ cyy e3yy ðtÞ A exy ðtÞ þ bxy e2xy ðtÞ þ cxy e3xy ðtÞ
eyy ðtÞ exy ðtÞ Þ
T
ð3Þ
in which bxx , byy , bxy , cxx , cyy , and cxy are constants. For the state of plane stress of isotropic viscoelastic plates, and in the case that the Poisson ratio m is a constant 2
Q11 ðtÞ Q12 ðtÞ QðtÞ ¼ 4 Q21 ðtÞ Q22 ðtÞ 0 0
3 0 0 5; Q66 ðtÞ
ð4Þ
where Q11 ðtÞ ¼ Q22 ðtÞ ¼ Q66 ðtÞ ¼
E ðt Þ ; 1 m2
Q12 ðtÞ ¼ Q21 ðtÞ ¼
mEðtÞ ; 1m
E ðt Þ 2ð1 þ mÞ
ð5Þ
in which EðtÞ is a time-dependent relaxation function, and Eð0Þ is the initial Young modulus of the material. For a homogeneous thin plate, the strain-displacement relations and the stress couples are respectively given by exx ¼ zW;xx ; eyy ¼ zW;yy ; exy ¼ 2zW;xy ; Z h=2 Z h=2 zrxx dz; Myy ¼ zryy dz; Mxx ¼ h=2
ð6Þ Mxy ¼
h=2
Z
h=2
zrxy dz: h=2
ð7Þ Substituting Eq. (6) into Eq. (2) and using Eq. (4), one obtains from Eq. (7)
Mxx ¼ F1 W;xx þ mW;yy þ cxx F2 fW;xx g þ cyy mF2 W;xy ;
Myy ¼ F1 mW;xx þ W;yy þ cxx mF2 fW;xx g þ cyy F2 W;xy ;
Mxy ¼ F1 ð1 mÞW;xy þ 4cxx ð1 mÞF2 W;xy ;
ð8Þ
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L.-Q. Chen, C.-J. Cheng / Appl. Math. Comput. 162 (2005) 1453–1463
where for any function f ðx; y; tÞ the functional F1 and F2 are defined as Z t I _ ðt sÞf ð x; y; sÞ ds ; E E ð 0 Þf ð x; y; t Þ þ F1 f f ð x; y; tÞg ¼ 1 m2 0þ Z t J 3 _ ðt sÞf 3 ð x; y; sÞ ds F2 f f ð x; y; tÞg ¼ E E ð 0 Þf ð x; y; t Þ þ 1 m2 0þ
ð9Þ
in which Z
h=2
h3 I¼ z dz ¼ ; 12 h=2 2
J¼
Z
h=2
z4 dz ¼
h=2
h5 : 80
ð10Þ
Substituting of Eq. (8) into Eq. (1) leads to
n
F1 W;xxxx þ 2W;xxyy þ W;yyyy þ cxx ½F2 fW;xx g;xx þ cyy F2 W;yy ;yy
o þ cyy m F2 W;yy ;xx þ cxx m½F2 fW;xx g;yy þ 8cxy ð1 mÞ F2 W;xy ;xy € ¼ 0: þ Nx W;xx þ Ny W;yy þ 2N;xy W;xy þ qhW
ð11Þ
Eq. (11) is the mathematical model of nonlinear viscoelastic plates. If the separation of variables method can be employed, let W ð x; y; tÞ ¼ qðtÞUð x; y Þ:
ð12Þ
Substituting Eq. (12) into Eq. (11) gives h i I J 3 þ 2U F U þ U ð t Þ þ cxx ðU;xx Þ ;xxxx ;xxyy ;yyyy 1 2 2 ;xx 1m 1m h h h i 3 i 3 i 3 þ cyy U;yy þ cyy m U;yy þ cxx m ðU;xx Þ ;yy ;xx ;yy h 3 i þ 8cxy ð1 mÞ U;xy F2 ðtÞ þ Nx U;xx qðtÞ þ Ny U;yy qðtÞ ;xy
qðtÞ ¼ 0; þ 2Nxy U;xy qðtÞ þ qhU€ where F1 ðtÞ ¼ Eð0ÞqðtÞ þ
Z
t
E_ ðt sÞqðsÞ ds; 0
3
F2 ðtÞ ¼ Eð0Þq ðtÞ þ
ð13Þ
Z
ð14Þ
t
E_ ðt sÞq3 ðsÞ ds: 0
For a simply supported square plate with the side length l, applying the Galerkin method to Eq. (13) and neglecting higher modes, one can suppose px py sin : ð15Þ Uð x; y Þ ¼ sin l l Substituting Eq. (15) into Eq. (13) and integrating the resulting equation respect to x and y on ½0; l, together with Eqs. (10) and (14), one derives
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Nx þ Ny þ Nxy 2 € qþx 1 q þ kq3 N Z t Z t 2 _ Dðt sÞqðsÞ ds k D_ ðt sÞq3 ðsÞ ds; ¼ x 0þ
1457
ð16Þ
0þ
where p 4 h2 E ð 0Þ p 2 h3 E ð 0Þ E ðt Þ ; ; N¼ ; DðtÞ ¼ 2 4 2 2 3ð1 m Þql 3ð1 m Þl E ð 0Þ 9p8 h4 Eð0Þ k¼ cxx ð1 þ mÞ þ cyy ð1 þ mÞ þ 8cxy ð1 mÞ : 2 8 1280ð1 m Þql x2 ¼
ð17Þ
Eq. (16) is the simplified mathematical model of nonlinear viscoelastic plates.
3. Condition of the instability Assume that the relaxation function in Eq. (5) is given by EðtÞ ¼ A þ Begt :
ð18Þ
The in-plane loads in the x and y directions are both simple harmonic excitations Nx ¼ Ny ¼ Nxy ¼ fN cos Xt:
ð19Þ
Inserting Eqs. (18) and (19) into Eq. (16) yields € q þ x2 ð1 4f cos XtÞq þ kq3 Z t Z t gt 2 gs gs 3 ¼ gbe x e qðsÞ ds þ k e q ðsÞ ds ; 0
ð20Þ
0
where a¼
A ; AþB
b¼
B : AþB
ð21Þ
Next, we will use the idea of the averaging method to analyze the long-time dynamical behavior governed by Eq. (20). Assume that the solution of Eq. (18) have the form q ¼ q cos u;
u ¼ xt þ w;
q_ ¼ xq sin u;
ð22Þ
where w varies slowly with time t. Hence the solution satisfies the condition _ sin u ¼ 0: q_ cos u wq
ð23Þ
Substituting Eq. (22) into Eq. (20) and considering only the case X ¼ x like the previous studies [8,10], one obtains
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q_ sin u qw_ cos u ¼ 4f xq cos 2ðu wÞ cos 2u þ
k 3 q cos3 u x
1 Gðq; u; w; tÞ; x
ð24Þ
where Gðq; u; w; tÞ ¼ gbq 4x2 þ 3kq2 ðg cos w þ x sin wÞ=4 g2 þ x2 kq2 ðg cos 3w þ 3x sin 3wÞ=4 g2 þ 9x2 ð1 egt Þ:
ð25Þ
In evaluating the integrals, the slow variable w is treated as a constant. For the long-time dynamical behavior, the term egt that decreases rapidly can be neglected. Therefore Gðq; u; w; tÞ in Eq. (24) can be replaced by ð4x2 þ 3kq2 Þðg cos w þ x sin wÞ H ðq; u; wÞ ¼ gbq 4ð g2 þ x 2 Þ 2 kq ðg cos 3w þ 3x sin 3wÞ þ : ð26Þ 4ðg2 þ 9x2 Þ Solving Eqs. (23) and (24), one gets k 1 q_ ¼ 4f xq cos 2ðu wÞ cos 2u q3 cos3 u þ H ðq; u; wÞ sin u; x x k 1 qw_ ¼ 4f xq cos 2ðu wÞ cos 2u q3 cos3 u þ H ðq; u; wÞ cos u: x x ð27Þ Averaging Eq. (27) over ½0; 2p respect to / yields the averaged system Z 1 2p k 3 1 3 q_ ¼ 4f xqcos2ðu wÞcos2u q cos u þ H ðq;u;wÞ sinudu; 2p 0 x x Z 2p 1 k 1 4f xqcos2ðu wÞcos2u q3 cos3 u þ H ðq;u;wÞ cosudu: qw_ ¼ 2p 0 x x ð28Þ Inserting Eq. (26) into Eq. (27) and evaluating the integrals, one obtains bgx2 3bkg 2 _q ¼ q f x sin 2w þ q ; 8ð g2 þ x 2 Þ 2ð g2 þ x 2 Þ bxg2 3 bg2 _ C q2 : qw ¼ q f x cos 2w þ þ k 1 8 2ðg2 þ x2 Þ xðg2 þ x2 Þ ð29Þ The steady-state motion of Eq. (20) corresponds to the equilibrium states of Eq. (29) given by
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bgx2 3bkg q2 ¼ 0; q f x sin 2w þ 8ðg2 þ x2 Þ 2ð g2 þ x 2 Þ bxg2 3 bg2 C q2 ¼ 0: q f x cos 2w þ þ k 1 8 2ðg2 þ x2 Þ xðg2 þ x2 Þ ð30Þ For the trivial case of q ¼ 0, Eq. (29) deduces to bgx2 3bkg 2 q : q_ ¼ q f x sin 2w þ 8ðg2 þ x2 Þ 2ð g2 þ x 2 Þ
ð31Þ
Based on the linearization analysis, the solution q ¼ 0 is stable if f < fc ¼
bgx : 2ðg2 þ x2 Þ
ð32Þ
If f > fc , q ¼ 0 may become unstable and the nontrivial solution corresponding to the limit circle may occur.
4. Numerical results In order to obtain a model easy to compute numerically, one usually differentiates Eq. (18) via the Leibnitz rule, and yields a three-order ordinary differential equation. However the operation is quite complicated. Here the auxiliary variable is introduced as Z t gt x ¼ gbe egs x2 qðsÞ þ kq3 ðsÞ ds ð33Þ 0
then x_ ¼ gx þ gb x2 q þ kq3 :
ð34Þ
Inserting Eq. (33) into Eq. (20) yields € q þ x2 ð1 4f cos XtÞq þ kq3 þ x ¼ 0:
ð35Þ
In the present paper, the numerical results herein are obtained by using, X ¼ 2, a ¼ 0:1 and b ¼ 0:9. Fix g ¼ 0:01 and k ¼ 0:01. In this case, from Eq. (35), we have fc ¼ 0:00449955. Now we consider the long-time dynamical behavior of Eq. (33) for some values of f . For a small excitation f < fc the system is asymptotically stable with response tending to zero. The case that f ¼ 0:0021 is depicted in Fig. 1. However the response approaches to zero significantly slowly if f is close to fc . The case that f ¼ 0:004 is shown in Fig. 2. For a excitation f > fc instability occurs in the system and its dynamical behavior
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Fig. 1. The response for f ¼ 0:0021.
Fig. 2. The response for f ¼ 0:004.
asymptotically tends a periodic motion. If f is slightly bigger than fc the response approaches the stable limit cycle very slowly. The case that f ¼ 0:0045 is depicted in Fig. 3. It is also noted that the amplitude of the response increases with the excitation parameter f . The case that f ¼ 0:009 is shown in Fig. 4. Fix k ¼ 0:1 and f ¼ 0:025. Then from Eq. (32) s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi# " b b g > gc ¼ 1 : ð36Þ 4f 4f In this case, gc ¼ 0:0557. The case that g ¼ 0:07 is depicted in Fig. 5. However the response approaches to zero slowly if a is very near gc . The case that g ¼ 0:06 is shown in Fig. 6. For gc but still adequate big, the system asymptotically tends a periodic motion. The response in the case that g ¼ 0:05 are depicted in Fig. 7. It is noted that the amplitude of the response decreases with g. The results in the case that g ¼ 0:01 are shown in Fig. 8.
L.-Q. Chen, C.-J. Cheng / Appl. Math. Comput. 162 (2005) 1453–1463
Fig. 3. The response for f ¼ 0:0045.
Fig. 4. The response for f ¼ 0:009.
Fig. 5. The response for g ¼ 0:07.
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Fig. 6. The response for g ¼ 0:06.
Fig. 7. The response for g ¼ 0:05.
Fig. 8. The response for g ¼ 0:01.
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5. Conclusions A nonlinear integro-partial-differential equation (12) that describes the motion of an isotropic homogeneous thin simply supported viscoelastic plate subjected to given in-plane loads is established. The plate is made of a nonlinear viscoelastic material that is assumed to obey the Leaderman constitutive relation (2). The Galerkin method is used to simplify Eq. (12) into a integrodifferential equation (14). Under the assumption that the relaxation function is given by Eq. (16), we employed the idea of the averaging method to analyze the dynamical behavior of the plate. Based on the averaged system (27), the critical value of the load parameter (30) at which instability appears is presented. Numerical results of the ordinary differential equations (34) and (35) indicate that the system successively appears asymptotical equilibrium state and periodic motion with the increase of the excitation parameter f and with decrease of the material coefficient g. The analytical critical value of the excitation parameter fc and the material coefficient gc are confirmed by the numerical simulations. Acknowledgements The work was supported by the National Natural Science Foundation of China (Project No. 10172056). References [1] J. Aboudi, G. Cederbaum, I. Elishakoff, Stability of viscoelastic plates by Lyapunov exponent, J. Sound Vib. 139 (1990) 459–468. [2] G. Cederbaum, J. Aboudi, I. Elishakoff, Dynamic stability of viscoelastic composite plates via the Lyapunov exponent, Int. J. Solids Struct. 28 (1991) 317–327. [3] Y.-Y. Zhu, N.-H. Zhang, F. Miura, Dynamical behavior of viscoelastic rectangular plates, in: W.-Z. Chien (Ed.), Proc. 3rd Inter. Conf. Nonlinear Mech., Shanghai UP, 1998, pp. 445–450. [4] C.-J. Cheng, N.-H. Zhang, Chaos and hyperchaos motion of viscoelastic rectangular plates under a transverse periodic load, Acta Mech. Sin. 30 (1998) 690–699 (in Chinese). [5] N.-H. Zhang, C.-J. Cheng, Chaos behavior of viscoelastic plates in supersonic flow, in: W.-Z. Chien (Ed.), Proc. 3rd Inter. Conf. Nonlinear Mech., Shanghai UP, 1998, pp. 432–436. [6] J. Smart, J.G. Williams, A comparison of single integral non-linear viscoelasticity theories, J. Mech. Phys. Solids 20 (1972) 313–324. [7] H. Leaderman, Large longitudinal retarded elastic deformation of rubberlike network polymers, Polym. Trans. Soc. Rheol. 6 (1962) 361–382. [8] D. Touati, G. Cederbaum, Dynamic stability of nonlinear viscoelastic plate, Int. J. Solids Struct. 31 (1994) 2367–2376. [9] D. Touati, G. Cederbaum, Influence of large deflections on the dynamic stability of nonlinear viscoelastic plates, Acta Mech. 113 (1995) 215–231. [10] G. Cederbaum, Parametric excitation of viscoelastic plates, Mech. Struct. Mech. 59 (1992) 37–51.
