Viscoelastic Behavior of Composite Rotors at Elevated Temperatures

Abstrucr-The composite rotor of a pulsed-power machine is built with radial precompression to enhance mechanical performance; however, the preload might decrease due to the viscoelastic behavior of materials at elevated temperatures. In this investigation, an analytical solution is developed to study the viscoelastic problem of thick-walled cylinders. The analysis accounts for ply-by-ply variations of rotor structural properties, ply orientations, and temperature gradients through the thickness of rotors. Fiber-reinforced composite materials generally illustrate extreme anisotropy in viscoelastic behavior. The viscoelasticity exists mainly in matrix dominant properties, such as transverse and shear, while the fiber dominant properties behave more like elastic mediums. Accordingly, the viscoelastic characteristics of composite cylinders is quite different from those of isotropic cylinders. Currently, finite element packages such as ABAQUS, ANSYS, and DYNA3D are not very suitable for the viscoelastic analysis of composite cylinder because of the lack of anisotropic viscoelastic elements. The prestress in the hoopwound fiber, which generates radial compression in the rotor, might decrease due to Poisson's effect alone from the creep behavior in the transverse properties of composite. The result also shows the effects of layup construction and fiber Orientations on the anisotropic behavior of composite rotors.


I. INTRODUCTION
Composite rotors are currently used as an energy storage device for rotating pulsed-power machines.Prestresses are built in during fabrication of the rotors (cylinders) through a "press-fit'' procedure to enhance the mechanical performance.The rotors are subjected to a radial compression prior to operation.Centrifugal force resulting from the rotation of rotors during operation generates tensile stresses in the radial and circumferential directions.Since the composite rotors are mainly reinforced circumferentially (filament hoop-wound cylinders), the radial tensile is critical to the ultimate performance of the rotors.Accordingly, it is essential to design and build the rotors with radial precompression.However, the polymer matrix composites generally creep in a long period of time, especially at elevated temperatures [ 11.The associated stress relaxation in the composite will result in the loss of the prestresses and potentially lead to the failure of the rotors.The objective of this investigation is to develop an analytical method to study the viscoelastic behavior of the thick-walled composite cylinders.The analysis can be applied to the design of pulsed-power machinery and overwrapped composite gun tubes.
To date, activities in the research of viscoelasticity have mainly concemed isotropic materials including the studies by Muki and Stemberg [2]; Schaperyf31; Williams [41;and Christensen [51.These basic theories of viscoelasticity were then extended to the area of heterogeneous and anisotropic materials for a variety of applications.Hashin [6] used the effective relaxation moduli and creep compliances to define the macroscopic viscoelastic behavior of linear viscoelastic heterogeneous media and its implementation in viscoelastic modeling.The general formulation of linear viscoelastic boundary value problems of composite materials, including the thermal viscoelastic problems for thermorheologically simple materials, and the applications of the correspondence principle were examined by Schapery [7].Rogers and Lee [8] investigated the viscoelastic behavior of an isotropic cylinder.
In the following research, the linear quasi-static viscoelastic behavior of a thick, laminated composite cylinder with an elevated temperature change is studied.The analysis accounts for ply -by-ply variation of properties, temperature changes, and fiber orientations.The thick cylinder is assumed to be in the absence of thermomechanical coupling and to be in the state of generalized plane strain such that all the stress and strain components are independent of the axial coordinate (Tzeng and Chien [9], [lo]).Moreover, due to the nature of axisymmetry, all the stress and strain components are also independent of the circumferential coordinate.The mechanical responses of this thick composite cylinder will, therefore, only have to satisfy the governing equation in the radial direction.
Invoking the Boltzmann superposition integral for the complete spectrum of increments of anisotropic material constants with respect to time, the thermoviscoelastic constitutive relations of the anisotropic composite cylinder can be derived in integral forms.Since the thick composite cylinder is subjected to a constant elevated temperature change, and boundary conditions are all independent of time, formulations of the linear thermal viscoelastic problem can have forms identical to those of the corresponding linear thermoelastic problem by taking advantage of the elastic-viscoelastic correspondence principle.In other words, all of these integral constitutive equations reduce to the algebraic relations, which are very similar to those developed for thermoelastic media when they are Laplace transformed by means of the rule for convolution integrals, The thermoelastic analysis can thus be used to derive the transformed thermal viscoelastic solutions in the frequency domain.Consider a corresponding thermoelastic problem with the msformed displacement components E, 7 , and W , in the axial direction, the circumferential direction, and the radial direction, respectively, in each layer.The misymmetric character of the thick composite cylinder along with the assumption of the state of generalized Plane strain leads to a simplified displacement field, which reflects the circumferential independence and only radial dependence of $,

C l l ( T ( t ) , i ) = C,, (To,h(t)) and L(t) = aT ( T ( f ) ) '
Here, To is the base temperature, and aT is the temperature shift factor.pij(T,r) is given by Bij(T,r) = C,"(T,t) .aw,.It is often desirable to use the inverse form of the consatutive relation (l), where vi (T, t) is the tensor product of the creep compli- kt ance Aij (T,t) and the thermal creep coefficient Since the elevated temperature change AT is constant above some reference value in time, the relaxation moduli and creep compliances are evaluated at that reference temperature regardless of whether or not the material is thermorheologically simple by employing the temperature shift-factor.
The Laplace transform of a functionfft) is defined as where s is the Laplace transform variable.Applying (3) with the convolution rule to (1) and ( 2) reduces the integral constitutive equations to the following algebraic relations: Substituting these transformed displacement components into the strain-displacement relations and invoking the compatibility conditions, one can derive explicit forms of U , and 7. Since each layer of the thick, laminated cylinder is cylindrically monoclinic in reference to the global coordinates, there is no coupling between transverse shears and other deformations.Accordingly, the vanishing shear traction boundary conditions and interface continuity conditions generate zero out-of-plane shear tractions and shear strains for each layer.Moreover, owing to the absence of torsional deformation, the transformed displacement components and v ,become where the constant quantity?has the physical interpretation of transformed axial strain of a layer.In fact, E O, according to the present formulation, also represents the transformed axial strain of the entire composite cy1 . The calculation of 2 requires the knowledge of end boundary conditions and will be given later.Likewise, solving for W(r) requires the information of transformed strain components, the constitutive The previously transformed displacement field gives the Furthermore, from the previous discussions, it can be shown that two of the three equilibrium equations are satisfied automatically.The only nontrivial equilibrium equation is the one in the radial direction:  where ri and ro are inner and outer radii, respectively, of the kth layer.The continuity conditions at each interface between two adjacent layers require continuous radial traction and continuous radial displacement at any instant as shown in Fig. 2. Thus, when written in the transformed form, they become and where k = 1 , -e , N -1; and subscripts i and o denote inner and outer surfaces, respectively.
Accordingly, the formulation accounts for ply-by-ply variations of material properties and temperature change.The matrix form numerical solution procedure with parallel computing techniques resolved the complexity and timeconsuming calculation procedures in Laplace transform of a multilayered composite cylinder [lo].

RESULTS AND DISCUSSION
The time-dependent thermal viscoelastic behavior of a 100-layer AS43502 graphite epoxy composite cylinder subjected to a temperature increase AT = 150" C is examined.The compliance in the fiber direction, S,, = 5.9 .lO-*/psi, and Poisson's ratios, v12 = ~1 3 = 0.3, ~2 3 = 0.36, are assumed to be time independent.The composite's thermal expansion coefficients in three principal directions are all = -0.5 .lo4 /"C and a;?.r, = a3, = 40.0-C , where the negative value indicates shrinkage with temperature increase.
Figs. 3 and 4 show radial displacement and radial stress profiles across the thickness of the cylinder at three instants, t = loT3 min, I = IO4.' min, and t = 1OI2 min, respectively.The radial tractions and displacements satisfied the continuity itions at the interfaces of layers at all instants.The "saw" shaped radial stress distribution is resulted from the discontinuity of material properties and various fiber orientations.The radial displacement, w(t), reaches a steady state at t = 1OI2 min due to a long-term creep behavior.The radial displace- The hoop stress, om(t) through the thickness of the cylinder are illustrated at three instants in Figs.5(A), S(B), and 5(C), respectively.There exist two distinct values (discontinuity) of o,(t) across each interface of two adjacent layers due to the various fiber orientations through the thickness.There is a trend showing the creep behavior of the hoop stress profiles from t = min to t = 10l2 min.The hoop stresses in 60" and 90" layers show a fairly steep gradient across the cylinder thickness initially.However, the gradient gradually disappears as time approaches to infinity.Likewise, Figs.@A), 6(B), and 6(C) present the axial stress, -- However, the axial stresses in 30" and 60" layers show an increase outwardly from the inside of the cylinder in the early stage.Figs.

IV. CONCLUSIONS
An analysis has been developed to study the thermal viscoelastic behavior of thick-walled laminated composite cylinders.The relaxation of thermal stresses in a cylinder subjected to a uniform temperature change are properly predicted to illustrate the capability of the analysis.The developed analysis is very useful to study the viscoelastic behavior of cylinders under thermal loads and residual stresses due to manufacturing processes.Because the viscoelastic properties exist only in matrix dominant properties, such as transverse and shear properties, the viscoelastic behavior of anisotropic laminated cylinders is quite different from those of isotropic cylinders.The stress relaxation in hoop-wound fiber layers results mainly from Poisson's effects of creep in the transverse direction since the properties are assumed to be elastic in the fiber direction.The stresses will not diminish completely due to elastic properties in the fiber direction.However, the reduction and redistribution of hoop stresses will cause a decrease of the radial compression in the cylinders in a long period.As the radial displacement approaches to a uniform state through the thickness, the radial stress will diminish completely depending upon the material properties and ~emperat~es.

2 Fig. 1 .
Manuscript received April 8. 19%.I. Tzeng, e-mail jtzeng@arl.mil,fax 410-278-6135: and phone 410-278-6805.U.S. Government Work Not Protected by U.S. Copyright 11. GENERAL FORMULATION IN LAPLACE SPACE Consider a filament-wound axisymmetric thick composite cylinder consisting of N layers with the axial coordinate z, the radial coordinate r, and the circumferential coordinate 8, as shown in Fig. 1.This composite cylinder has the inner radius a, the outer radius b, and the length L. The Boltzmann superposition integral of stress oii fi, j = 1, 2, 3) and strain cii (i, j = 1, 2, 3) relation for an isothermal viscoelastic problem with a constant temperature increase AT and the thermal expansion coefficient aw is aE kl( Cylindrical coordinate system (r,B,z) of an axisymmetric thick, lamin8ted composite cylinder containing N layers with the inner radius a, the outer radius b, and the length L. drpij(T, 'IAT7 (l) 7%where C' ;' ( T , t ) is the relaxation modulus dependent on temperature T and time t, kl-' Furthermore, it can be shown that hy] = [Cij] .

-Fig. 2 .
Fig. 2. Nonnal traction continuity and normal displacement continuity at the interfaces between two adjacent layers.The composite cylinder has an inner radius a = 3.5 in, an outer radius b = 4.1 in, and a thickness of each layer 25

7 (A), 7 (
B), and 7(C) are the in-plane shear stresses cr&) through the cylinder thickness at the three specified instants, respectively.The in-plane shear stresses of 00 and 90" layers vanishes in the entire time history due to generalized plane strain assumption.A drastic change occurs in the 30" and 60" layers resulting from the combined effects of Poisson's ratio and creep characteristics of materials.

Fig. 7 .
Fig. 7. Inplane shear profiles across the thickness of the cylinder at three different instants: (A) t = min.and (C) t = 10" min.min, (€3) t = and A2 are coefficients to be determined from boundary and continuity conditions.Finally, it is understood that the initial condition of the original thermoviscoelastic problem is displacement-free state of rest.The boundary conditions are of free rractions and, hence, of free transformed tractions on both inner and outer circular surfaces:On both end surfaces, stress resultants are zero: , and (8) into (9), the transformed stress components Grr and gee are obtained in terms of the transformed radial displacement i3.Incorporating the resulting i f , and See functions with (10) gives a nonhomogex 2 = -.Solving (11) for S yields where and 2,