Research Article Finite Element Formulation of Forced Vibration Problem of a Prestretched Plate Resting on a Rigid Foundation

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1 Hindawi Publishing Corporation Journal of Applied Mathematics Volume 7, Article ID 5636, 1 pages doi:1.1155/7/5636 Research Article Finite Element Formulation of Forced Vibration Problem of a Prestretched Plate Resting on a Rigid Foundation M. Eröz and A. Yildiz Received 7 April 7; Revised 1 June 7; Accepted 3 September 7 Recommended by Heinz H. Bauschke The three-dimensional linearized theory of elastodynamics mathematical formulation of the forced vibration of a prestretched plate resting on a rigid half-plane is given. The variational formulation of corresponding boundary-value problem is constructed. The first variational of the functional in the variational statement is equated to zero. In the framework of the virtual work principle, it is proved that appropriate equations and boundary conditions are derived. Using these conditions, finite element formulation of the prestretched plate is done. The numerical results obtained coincide with the ones given by Ufly and in 1963 for the static loading case. Copyright 7 M. Eröz and A. Yildiz. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1. Introduction Elastodynamics problems of prestretched media arise in many areas of applied mathematics, engineering, and natural sciences. Classical linear theory of elastic waves is not sufficient for solving elastodynamics problems involving initially stressed bodies. That is why a general nonlinear theory of elastic waves has been developed since the second half of the th century. An analysis of the studies up to 1986 was made in 1,. Later researches are given in 3. Recent researches involving dynamic stress field in multilayered media with initial stress are given in 4 7. In the present paper, a boundary-value problem of elastodynamics involving initially stressed bodies which has no analytical solution considered and finite element method is utilized to solve the problem numerically. In a study by Akbarov 7, the layers of the slab have infinite length in the radial direction. In a recent paper by Akbarov and Guler 8, the stress field in a half-plane covered by the prestretched layer under the action of arbitrarly linearly located time-harmonic forces is investigated. In 8, the layer considered

2 Journal of Applied Mathematics x P δ(x 1 e iωt q q h a a x 1 Figure.1. The geometry of the strip plate resting on a rigid foundation. is extending to infinity in the x 1 -axis direction. In this study, a prestretched strip plate which has a finite domain both for the x 1 -axis and x -axis is considered. So the method of solution used in 7, 8 is not suitable for the problem at hand.. Formulation of the problem The problem of forced vibration of a prestretched strip plate resting on a rigid foundation is considered. Strip plate and rigid half-plane occupy the regions B = { (x 1,x : a x 1 a, x h } (.1 and {(x 1,x : <x 1 <, <x }, respectively, in Cartesian coordinate system Ox 1 x (see Figure.1. We assume that strip plate is made of linearly elastic material, homogeneous and isotropic. We also assume that, before contact, the plate is stressed from both sides by normal forces having amplitude q. A time-harmonic point-located normal load, P δ(x 1 e iωt, is applied to the upper surface of the plate, where δ(x 1 stands for the Dirac delta function. Following on from the above, it can be assumed that the plane deformation state prevails. According to Guz 1, for the case considered, the equations of motion are σ ij x j q u i x 1 u = ρ i, i = 1,, j = 1,. (. t In (., ρ denotes the density of the material in the natural state, u 1 (x 1,x,t and u (x 1,x,t denote the displacement in the axis x 1 and x, respectively. For an isotropic compressible material, we can write the following mechanical relations: where λ and μ are Láme constants and Here σ ij = λθδ ij με ij, θ = ε 11 ε, (.3 ε ij = 1 ( ui u j. (.4 x j x i ε = { ε 11,ε,ε 1 } T (.5

3 M. Eröz and A. Yildiz 3 denotes the deformation tensor and σ = { σ 11,σ,σ 1 } T (.6 denotes the stress tensor. In (.3, δ ij denotes the Kronecker delta, i j, δ ij = 1, i = j. (.7 In the case considered the relations λ = Eν (1 ν(1 ν, μ = E (1 ν = G (.8 are valid, where E denotes the elasticity moduli and ν denotes the Possion ratio. We assume that the following boundary conditions exist: ( q u 1 x 1 σ 11 u x= 1 =, u =, x= ( =, q u σ 1 =, x 1=±a x 1 x 1=±a σ x=h 1 =, σ = P x=h δ ( x 1 e iωt. (.9 Since the applied point-located load is time-harmonic, all the dependent variables are also harmonic and can be represented as { ui,σ ij,ε ij } = {ûi, σ ij, ε ij } e iωt, (.1 where the superposed caret denotes the amplitude of the corresponding quantity. Hereafter the carets will be omitted. Using (.3 and(.4 in(., we have the linearized equations of motion in terms of displacement as follows: (λ μ q u 1 x 1 (μ q u x 1 μ u 1 x (λ μ u x (λ μ u x 1 x = ρ ω u 1, (λ μ u 1 x 1 x = ρ ω u. ( Finite element formulation We have the dimensionless coordinate system by the following transformation: x 1 = x 1 h, x = x h. (3.1

4 4 Journal of Applied Mathematics By multiplying both sides of the equations with h after substituting (3.1 in(., we have h σ 11 h σ 1 q u 1 x 1 x x 1 = ρ ω h u 1, h σ 1 h σ q u x 1 x x 1 = ρ ω h u. (3. Under the coordinate transformations (3.1, boundary conditions (.9willbeasfollows: u 1 x= =, u x= =, ( q u ( 1 σ 11 =, q u σ 1 =, x 1 x 1=±a/h x (3.3 1 x 1=±a/h σ 1 x=1 =, σ x=1 = P δ ( h x 1 e iωt. We first multiply (3. by the test functions v 1 = v 1 ( x 1, x andv = v ( x 1, x, respectively, and then add the resultant equations side by side. After integrating the equation over the domain we get = B = {( x 1, x : x1 a/h, x 1 }, (3.4 h σ 11 v 1 h σ 1 v h σ 1 v 1 h σ x 1 x 1 x ρ ω h ( u 1 v 1 u v d x1 d x. v q u 1 x x 1 v 1 q u x 1 v d x 1 d x Applying integration by parts to (3.5 weget hσ 11 v 1 cos ( n, x 1 hσ 1 v cos ( n, x 1 hσ 1 v 1 cos ( n, x hσ v cos ( n, x B q u 1 v 1 cos ( u n, x 1 q v cos ( n, x 1 ds x 1 x 1 v hσ 1 v 11 hσ v 1 hσ 1 v 1 hσ q u 1 v 1 q u v d x 1 d x x 1 x 1 x x x 1 x 1 x 1 x 1 = ρ ω h ( u 1 v 1 u v d x1 d x, (3.5 (3.6 where B denotes the boundary of the domain B. If we collect the domain integrals in (3.6togetheranddefine T ij = σ ij σ u j in, σ 11 x = q, σ in = forin 11, (3.7 n

5 M. Eröz and A. Yildiz 5 B 3 B 1 x! n cos(! n,x = 1 B 1! n cos(! n,x 1 = 1 cos(! n,x 1 = 1 B cos(! 4 n,x = 1 x 1 a/h a/h Figure 3.1. The form of the boundary B. we get = ht ij v j B d x 1 d x ρ x ω h u i v i i hσ 11 v 1 cos ( n, x 1 hσ 1 v cos ( n, x 1 hσ 1 v 1 cos ( n, x hσ v cos ( u n, x q 1 v 1 cos ( u n, x 1 q v cos ( n, x 1 ds. x 1 x 1 (3.8 The integral over the boundary B in (3.8 can be calculated as follows: let the boundary B be in the form given in Figure 3.1. According to Figure 3.1 boundary B can be written as B 1 B B 3 B 4.Therighthand side of (3.8 canbewrittenasfollows: B { cos ( n, x 1 hσ 11 v 1 hσ 1 v q u 1 v 1 q u v cos ( } n, x hσ 1 v 1 hσ v ds. x 1 x 1 (3.9 We have the integrals hσ 11 v 1 hσ 1 v q u 1 v 1 q u v d x, x 1 x 1 1 hσ 1 v 1 hσ v d x1, ( 1 hσ 11 v 1 hσ 1 v q u 1 x 1 v 1 q u x 1 v ( 1 hσ 1 v 1 hσ v d x1, d x, (3.1

6 6 Journal of Applied Mathematics for the boundaries { B 1 = ( x1, x : x1 = a } { h, x a 1, B = ( x1, x : h x 1 a } h, x = 1, { B 3 = ( x1, x : x1 = a } { h, x a 1, B 4 = ( x1, x : h x a } h, x =, (3.11 respectively. Using the conditions (.9in(3.1 we get the integral term Consequently, (3.8canbewrittenas or hσ x=1 ν x=1 d x 1. (3.1 v a/h j ht ij ρ x ω h u i v i d x 1 d x = hσ x=1 ν x=1 d x 1 (3.13 i v hσ 1 v 11 hσ v 1 hσ 1 v 1 hσ q u 1 v 1 q u v x 1 x 1 x x x 1 x 1 x 1 x 1 ρ ω h ( u 1 v 1 u v d x 1 d x = hσ x=1 ν x=1 d x 1. (3.14 The mechanical relations (.3 and(.4 under the transformation (3.1 can be written explicity as follows: σ 11 = (λ μ 1 u 1 λ 1 u, h x 1 h x Afterusingboundarycondition σ = λ 1 u 1 h σ 1 = μ 1 h (λ μ 1 x 1 h ( u1 u. x x 1 u x, (3.15 σ x=1 = P δ ( h x 1 e iωt (3.16 and the property the right-hand side of the (3.13canbewrittenas δ ( a(x = 1 δ(x, (3.17 a (x P δ ( x 1 ν x=1 d x 1. (3.18

7 M. Eröz and A. Yildiz 7 After substituting relations (3.15in(3.13 and using (3.18 as right-hand side we get { (λμq u 1 λ u x 1 { λ u 1 (λ μ u x 1 x = P δ ( x 1 ν x=1 d x 1. } { v1 μ u 1 (μq u } { v u1 μ x x 1 x x 1 x 1 } v ρ x ω h ( u 1 v 1 u v d x 1 d x x u x 1 } v1 x (3.19 By dividing both sides of (3.19toLáme constant μ, weget {( q μ λ μ u1 λ } { ( } { u v1 u1 q x 1 μ x x 1 x μ 1 u v u1 u } v1 x 1 x 1 x x 1 x { ( } λ u 1 λ μ x 1 μ u v ρ ω h ( u1 v 1 u v d x 1 d x x x μ P = μ δ( x 1 ν x=1 d x 1. (3. By (3., we get bilinear form a(u,v and linear form l(v. Introducing the distortion wave velocity μ c = (3.1 ρ and the dimensionless frequency Ω = ωh, (3. c we obtain J(u = (1/a(u,u l(u, the total energy functional, where u = u(u 1,u as follows: J ( u 1,u ( ( u1 u = 1 q μ ( ( λ μ u1 x 1 x 1 x 1 ( u x { u1 u } λ u 1 x x 1 μ x 1 u x Ω ( u 1u P d x 1 d x μ δ( x 1 u x=1 d x 1. (3.3 As known from calculus of variation 9, by equating the first variational of the total energy functional, J(u given in(3.3, to zero one must derive(.11 and boundary conditions (3.3. In order to achieve this result we use δj ( u 1,u =. (3.4

8 8 Journal of Applied Mathematics In (3.4 we equate the integrands involving δu 1 and δu to zero. Consequently, we obtain u 1 ρ ω = q μ u 1 x 1 λ ( u λ μ x 1 x μ u 1 u 1 x 1 x u x 1 x, u ρ ω = q u μ x1 u 1 u x 1 x x1 λ ( u 1 λ μ x 1 x μ u x, ( q u 1 σ 11 =, σ 1 x=1 =, x 1 x 1=±a/h ( q u σ 1 =, σ x=1 = P δ( x 1, x 1 x 1=±a/h (3.5 (3.6 as boundary conditions. It is worthy of noting that we get (3.5 by using functional (3.3 which is obtained by(3.. Therefore, if we get the total energy functional using (3.19, we obtain (. the linearized equations of motion. The total energy functional, J(u, will be minimized using Rayleigh-Ritz method. Firstly, we divide the domain B into finitely many B i subdomains. We utilize displacement-based finite element method, so the functions to be sought in each finite element will be displacements. Thus, in the eth finite element, we get u (e 1 = k=1 u (e = k=1 M a k N k (r,s, M b k N k (r,s. (3.7 In (3.7, M denotes the nodes in eth finite element. The shape functions N k (r,s, defined on the unit square 1,1 1,1, are N 1 (r,s = 1 ( r r ( s s N (r,s = 1 ( r r ( s s N 3 (r,s = 1 ( r r ( s s, N 4 (r,s = 1 ( r r ( s s 4 N 5 (r,s = 1 ( r 1 ( s s N 6 (r,s = 1 ( r r ( s 1, N 7 (r,s = 1 ( r 1 ( s s N 8 (r,s = 1 ( r r ( s 1 N 9 (r,s = ( r 1 ( s 1 (3.8 (see Figure 3.. Substituting (3.7in(3.3 we get the linear algebraic system Au = f, (3.9 according to Rayleigh-Ritz method. In (3.9, matrix A denotes the stiffness matrix of the system and is given by A (e A (e 11 A (e 1 = A (e 1 A (e (3.3

9 M. Eröz and A. Yildiz 9 4 s r 1 5 Figure 3.. The order of the nodes of a finite element defined on 1,1 1,1. on the eth finite element. In (3.3 wehave A (e 1 11 = A (e 1 A (e 1 A (e = 1 = 1 = 1 Nij N ( kl λ x x μ Nij N kl x 1 x 1 Nij N kl λ N ij x 1 x μ N kl x x 1 d x 1 d x, Ω N ij N kl d x 1 d x, Nij N kl λ N ij N kl d x 1 d x, x x 1 μ x 1 x Nij N ( kl λ x 1 x 1 μ Nij N kl Ω N ij N kl d x 1 d x. x x (3.31 In (3.31, the subscripts i, j,k, andl should take the values i, j, k, l = 1,...,M. In(3.9 the vector u is as follows: u = { a k bk } T, (3.3 and the components of the vector u gives values of the displacements at the nodes in the directions x 1 and x.thevector f in (3.9isgivenby f = { P μ N ij x1= x =1 By using the displacements obtained from solving (3.9 in we obtain the stresses. In (3.34, matrix D is given by } T. (3.33 σ = DBu, (3.34 D = 1 ν ν ν 1 ν, ( ν 1 ν

10 1 Journal of Applied Mathematics u x 1 /h elements 3 elements 4 elements 8 elements Figure 4.1 and matrix B is given by B = N 1 r N 9 r N 1 s N 9 s N 1 N 9 s s N 1 N 9 r r ( Numerical results In order to see the validity of the algorithm and programmes, the case where Ω = and q = is selected and the number of finite elements in the direction of x 1 -axis is increased. The results obtained in the case considered approach the corresponding ones in the static loading case which are given in Uflyand 1. Let η denote the dimensionless parameter characterizing the initial stress in the strip plate and is given by η = q/μ. Consider the distribution of stresses and displacements in the interface plane (where x /h = when η =. It is seen that the problem is an axisymmetric problem with respect to x 1 = plane. The numerical results in Figures 4.1, 4.,and4.3 are obtained under the following assumptions: Ω =, q =, ν =.33, and h/a =.. In Figures 4.1 and 4., the distribution of the displacements u 1 and u in the direction of x 1 -axis is given. Increasing the number of finite elements in the direction of x 1 -axis, it is seen that the displacements are approaching to each other asymptotically. In Figure 4.3 the stress distribution in the direction of x 1 -axis is given. The results obtained here coincide with the ones given in 1. It is seen that the values of σ decrease towards the sides of the strip plate where x 1 =±a. In Figure 4.4 the stress distribution in the direction of x 1 -axis for various Ω values is given. It is seen that increasing the dimensonless frequency Ω increases the stress σ.

11 M. Eröz and A. Yildiz 11 u x 1 /h.8 elements 3 elements 4 elements 8 elements Figure 4. σ h/p x 1 /h elements 3 elements Figure elements 8 elements However, there are such points on the interface plane along the x 1 -axis at which the dimensonless frequency Ω does not affect the values of the stress σ. 5. Conclusions In this paper, mathematical formulation of forced vibration of a prestretched strip plate resting on a rigid foundation is given in the framework of the three-dimensional linearized theory of elastodynamics. A numerical algorithm is developed for both the static and dynamic loading cases. The numerical results are presented for the distribution of displacements and stresses. These numerical results indicate the validity of the formulation. It is seen that there are some points on the interface plane at which the dimensionless frequency does not affect the stress σ.

12 1 Journal of Applied Mathematics σ Λ h/p 1 1 x 1 /h Ω = Ω =.3 Figure 4.4 Ω =.5 Ω =.8 References 1 A.N.Guz,Elastic Waves in a Body with Initial Stresses, I.General Theory, Naukova Dumka, Kiev, Russia, A.N.Guz,Elastic Waves in a Body with Initial Stresses, II. Propagation Laws, Naukova Dumka, Kiev, Russia, A. N. Guz, Elastic waves in bodies with initial (Residual stresses, International Applied Mechanics, vol. 38, no. 1, pp ,. 4 V. B. Rudnitsky and N. N. Dikbtaryaruk, Inintially stressed elastic strip strengthenedby elastic stringers, International Applied Mechanics, vol. 38, no. 11, pp ,. 5 S. V. Matniak, On Rolling rigid cylinder along a band with initial stresses, International Applied Mechanics, vol. 39, no. 3, pp , 3. 6 S. D. Akbarov, A. D. Zamanov, and T. R. Suleimanov, Forced vibration of a prestretched twolayer slab on a rigid foundation, Mechanics of Composite Materials, vol. 41, no. 3, pp. 9 4, 5. 7 S. D. Akbarov, On the dynamical axisymmetric stress field in a finite pre-stretched bilayered slab resting on a rigid foundation, Journal of Sound and Vibration, vol. 94, no. 1-, pp. 1 37, 6. 8 S. D. Akbarov and C. Guler, On the stress field in a half-plane covered by the pre-stretched layer under the action of arbitrary linearly located time-harmonic forces, Applied Mathematical Modelling, vol. 31, no. 11, pp , 7. 9 A. Cherkaev and E. Cherkaev, Calculus of Variations and Applications, Lecture Notes, 3. 1 S. I. Uflyand, Integral Transformations in the Theory of Elastcity, Nauka, Moscow, Russia, M. Eröz: Mathematics Department, Faculty of Science and Letters, Sakarya University, Esentepe Campus, Sakarya, Turkey address: mustafaeroz@gmail.com A. Yildiz: Mathematics Department, Faculty of Science and Letters, Sakarya University, Esentepe Campus, Sakarya, Turkey address: yildiz@sakarya.edu.tr