Rotating Variable-Thickness Inhomogeneous Cylinders : Part II — Viscoelastic Solutions and Applications

Analytical solutions for the rotating variable-thickness inhomogeneous, orthotropic, hollow cylinders under plane strain assumption are developed in Part I of this paper. The extensions of these solutions to the viscoelastic case are discussed here. The method of effective moduli and Illyushin's approximation method are used for this purpose. The rotating fiber-reinforced viscoelastic homogeneous isotropic hollow cylinders with uniform thickness are obtained as special cases of the studied problem. Numerical application examples are given for the dimensionless displacement of and stresses in the different cylinders. The influences of time, constitutive parameter and elastic properties on the stresses and displacement are investigated.


Introduction
In recent years the subject of viscoelasticity has received considerable attention from analysts and experimentalists.The stress state of a viscoelastic hollow cylinder with the help of internal pressure and temperature field is analyzed in the literature [1,2].A modified numerical method is introduced by Ting and Tuan [3] to study the effect of cyclic internal pressure on the stress and temperature distributions in a viscoelastic cylinder.Talybly [4] has investigated the state of stress and strain for a viscoelastic hollow cylinder fastened to an elastic shell under nonisothermal dynamic loading.Feng et al. [5] have obtained the solution for finite deformations of a viscoelastic solid cylinder subjected to extension and torsion.The thermomechanical behavior of a viscoelastic finite circular cylinder under axial harmonic deformations is presented by Karnaukhov and Senchenkov [6].
The determination of stress and displacement fields is an important problem in design of engineering structures using fiber-reinforced composite materials.The analytical solution for the rotating fiber-reinforced viscoelastic cylinders becomes very complex when the thickness along the radius of the cylinder is variable, even for simple cases.Methods for solving quasi-static viscoelastic problems in composite structures have been developed by a number of authors [7][8][9].Allam and Appleby [10] have used the realization method of elastic solutions to solve the problem of bending of a viscoelastic plate reinforced by unidirectionally elastic fibers.In other work [11], they have used the method of effective moduli to determine the stress concentrations around a circular hole or circular inclusion in a fiber-reinforced viscoelastic plate under uniform shear.Allam and Zenkour [12] have used the small parameter method as well as the method of effective moduli for the bending response of a fiberreinforced viscoelastic arched bridge model with quadratic thickness variation and subjected to uniform loading.In [13], they have also obtained the stresses around filled and unfilled circular holes in a fiber-reinforced viscoelastic plate under bending.The same author [14] have developed closed form solutions for the rotating fiberreinforced viscoelastic solid and annular disks with variable thickness by applying the generalization of Illyushin's approximation method.In addition, Allam et al. [15] have determined the stress concentrations around a triangular hole in a fiber-reinforced viscoelastic composite plate under uniform tension or pure bending.Also, Zenkour et al. [16] have presented the elastic and viscoelastic solutions to rotating functionally graded hollow and solid cylinders.
In the present paper, the rotating fiber-reinforced viscoelastic hollow cylinder is analytically studied.The thickness of the cylinder and the elastic properties are taken to be functions in the radial coordinate.The governing second-order differential equation is derived and solved with the aid of some hypergeometric functions.The displacement and stresses for rotating fiberreinforced viscoelastic inhomogeneous orthotropic hollow cylinder with variable thickness and density subjected to various boundary conditions are obtained.Special cases of the studied problem are established and numerical results are presented in graphical forms.

Rotation of Viscoelastic Cylinders
According to the elastic solution given in part I, we can use the method of effective moduli and Illyushin's approximation method to solve the rotation problem of variable thickness and density viscoelastic hollow cylinder reinforced with unidirectionally elastic fibers.
For an orthotropic cylinder, the compliance parameters ij  can be expressed in terms of the engineering characteristics as [17]: in which = 1 2 , r r zr rz z z r z rz where i E are Young's moduli and ij  are Poisson's rations which are related by the reciprocal relations: Now, consider a hollow cylinder made of a composite material composed of two components.A viscoelastic material as a first component, reinforced by unidirectional elastic fibers as a second component.The first of these components plays the role of filler and may posses the properties of a linear viscoelastic material, and it is described by the modulus f E and Poisson's ratio f  .
The other component will be serve as the reinforcement and is an elastic material with modulus of elasticity E and Poisson's ratio  .
Under the above considerations and using the method of effective moduli [14,18], Young's moduli and Poisson's ratios, with , are given by [19]: where  is the volume fraction of fiber reinforcement.
Thus, it is obvious that the reciprocal relations given in Equation ( 3) are fulfilled.Note that, the viscoelastic modulus f E is given by: where K is the coefficient of volume compression (the bulk modulus) and it is assumed to be not relaxed, i.e. = K const., and  is the dimensionless kernel of relaxation function which is related to the corresponding Poisson's ratio by the formula: Substituting from Equations ( 5) and ( 6) into Equation (4) yields where = / p K E is the constitutive parameter.With the help of Equations ( 1) and ( 8), one can rewrite the solutions given in Part I of this paper; see Equations ( 20) and ( 23)-(25); in the form: where It is to be noted that, in elastic composites, the radial displacement and stresses are functions of  and r while in viscoelastic composites they are operator functions of the time t and r .According to Illyushin's approximation method [11,19,20], the function u can be represented in the form:  are some known kernels, constructed on the base of the kernel  and may be chosen in the form: ,( = 1, 2) given in Equation ( 9).The coefficients ( ) i A r are determined from the system of algebraic equations where   Now, let us consider the relaxation function in an exponential form   thus, we get ) Equation ( 12) for a viscoelastic composite may be recorded to obtain explicit formula for the radial displacement as function of r and time t in the form:  16) and (18).Using the same technique once again to obtain the radial, circumferential and axial stresses for the rotating fiber-reinforced viscoelastic hollow cylinder with variable thickness and density by replacing only ( , ) and making the suitable changes in this case.

Applications
In this section, some numerical examples for the rotating fiber-reinforced viscoelastic inhomogeneous variablethickness cylinder will be introduced.The results of the present problem will be given for three sets of geometric parameters k and n for the thickness profile.The numerical applications will be carried out for the radial displacement and stresses that being reported herein are in the following dimensionless forms: The effect of the elastic properties of the cylinder, constitutive and time parameters on the dimensionless radial displacement and stresses will be shown.The calculations will be carried out for the following values of parameters: .Also, the coefficient  is still unknown and the time parameter is given in terms of it.The distributions of the dimensionless stresses and displacement through the radial direction of the rotating fiber-reinforced viscoelastic inhomogeneous variablethickness cylinder are plotted in Figures 1-3 according to the FF, CC, FC and CF boundary conditions, respectively.For all hollow cylinders, the dimensionless radial displacement r u is the largest in the same position for small k , i.e. = 0.6 k .For FF and CF hollow cylinders, the dimensionless stresses are the largest for small n .The minimum values of the dimen- sionless radial stress r  at the outer surface of the CC and FC hollow cylinders are larger for = 0.6 k .Also, other parameters are taken (except otherwise stated) as: = 0.2, = 2.5, = 0

Figure 6 .
Figure 6.Distribution of dimensionless stresses and displacement through the radial direction of a FC variablethickness viscoelastic hollow cylinder.viscoelastic inhomogeneous cylinder subjected to various boundary conditions with different values of the parameter m .The stresses and displacement for = 1 m are the smallest when compared to the results for = 0 m

Figure 7 .Figure 8 9 .Figure 8 .
Figure 7. Distribution of dimensionless stresses and displacement through the radial direction of a CF variablethickness viscoelastic hollow cylinder.and 1  .For FF and FC hollow cylinders, the dimensionless radial displacement r u has changed concavity.The dimensionless radial stress r  increases firstly to get its maximum value then it decreases again at the

Figure 9 .
Figure 9.The effect of time parameter  on (a) r u , (b) r  , (c)   and (d) z  of a variable-thickness viscoelastic hollow