The generalized plane deformation analysis presented by Hyer [1] is the basis of this work. However, instead of formulating a problem in terms of a scaling variable, ζ, such that ζ = Ωr. This scaling is done to avoid numerical problems, which is explained later. The formulation is summarized in the post-consolidation section. The theory has been enhanced significantly to model the effects of winding tension in the insitu-process, and the redistribution of stresses resulting from the removal of the mandrel at any point during the process. Both the post- and insitu-consolidation models incorporate the temperature dependence of the thermoelastic response of the material. The model is then used to investigate the interaction between the cylindrical part and the tool (mandrel), and also the effect of winding tension on the residual stresses in the cylinder. It is hoped that this methodology would serve as valuable tools for the evaluation of the design and performance of both post- and insitu- consolidated laminated cylinders under various combination of loading conditions.

Theoretical Development of Post-Consolidation Model

It is desired to determine the response of a laminated cylinder subjected to both mechanical and thermal loading. The geometry is defined in the cylindrical x-θ-r coordinate system as shown in Figure 1. Since only axi-symmetric loading is considered, the stresses, strains and displacements are independent of the circumferential coordinate, θ. Attention is focused on the response of the cylinder away from the ends and therefore the stresses and strains are independent of the axial coordinate, x.

Figure 1. Definition of the Coordinate axes.

The cylinder is composted of N layers, which the innermost layer being referred to as the first layer. The inner radius of the cylinder is ri and the outer radius is ro. The interface between the kth and the (k+1)th is denoted as rk+1. The hydrostatic pressure acting on the outer radius is Po while the hydrostatic pressure acting on the inner radius is Pi. The cylinder is subjected to an axial force Faxial and a torsional load τo. Each layer may also be subjected to a thermal loading ΔT(k), where the superscript (k) denotes the kth layer.

The response of the cylinder is determined from the interaction of individual layers that comprise the thickness. The response of each layer is written in terms of constants obtained from the integration of the elasticity equations for that layer. In general, the set of constants vary from layer to layer. These constants are determined by enforcing the traction boundary conditions at the inner and outer radii, by enforcing continuity of displacements and tractions at the interfaces between layers, and by enforcing equilibrium of forces on the cylinder.

The constitutive behavior of each layer is expressed in the x-θ-r coordinate system as:

               [1]

The terms are the elastic constants and the αj terms are the coefficients of thermal expansion of the kth layer in the global x-θ-r system, which are obtained by rotating the lamina properties in the principle 1-2-3 material system by the winding angle ϕ. This rotational transformation of the material properties is accomplished using the transformation rules found in any standard textbook on composites [4]. The two transformation systems are defined in Figure 1. As mentioned earlier, the problem is formulated in terms of the scaled variable, ζ. Based on the axisymmetric and plane deformation assumptions, the components of strain for each layer can be expressed in terms of the axial displacement, u(x,ζ), the circumferential displacement, v(x,ζ) and the radial displacement w(ζ) as follows:

;        ;        ;        ;        ;                [2]

As shown by Hyer [1], the displacements for the kth layer can be obtained as:

               [3]

               [4]

From Equations (2), (3) and (4), γθr = γxr = 0, and therefore, from Equation (1), τθr=τxr = 0. The differential equation governing the radial displacement, w is then obtained as follows:

               [5]

The solution to this equation is:

       [6]

where

               [7]

and

               [8]

The above equations are valid if . For a material that is transversely isotropic in the 2-3 plane (;;;), the governing differential equation for the radial displacement simplifies to:

               [9]

whose solution is:

               [10]

A1 and A2 vary between layers, while εo and γo have been shown to be constant for all layers. Therefore, for a laminated cylinder consisting of N layers, 2N+2 constants need to be determined. These constants are determined by enforcing the traction boundary conditions at the inner and outer radii, by enforcing continuity of displacements and tractions at the interfaces between layers, and by enforcing equilibrium of forces on the cylinder. The applicable boundary conditions at the inner and outer radii of the cylinder are

                               [11]

The internal pressure Pi is positive in the direction of the outward normal, while the external pressure Po is positive in the direction of the inward normal. Continuity of displacements at the (N+1) layer interface results in:

       

                       k=1,2,….,(N-1)        [12]

Application of the condition of equilibrium of forces in the axial direction results in:

               [13]

Faxial represents the total axial force resulting from the applied traction Fa in the axial direction, as well as the axial forces due to the internal and external hydrostatic pressures on the end fittings of a closed-ended cylinder. For the laminated cylinder under consideration, Equation (13) can be written in terms of ζ as follows:

               [14]

If the cylinder is open-ended, then the axial forces due to the hydrostatic pressures are absent and only the first term on the right hand side of Equation (14) is retained. The integral condition representing the applied torsion, τo, can be written as:

               [15]

or in terms of ζ as:

               [16]

Equations (11), (12), (14), and (16) provide the necessary 2N+2 conditions required to determine the N*(A1)’s and N*(A2)’s and εo and γo terms, and therefore the displacements, strains and stresses in each layer. A flowchart of the program methodology is shown in Figure 2. 

Theoretical Development for the Insitu-Consolidation Model

In the insitu-consolidation process, the cylinder wall thickness is incrementally increased layer by layer. The theoretical basis is the same as in post-consolidation but the mode has been modified to incorporate the effects of winding tension and also the redistribution of stresses upon mandrel removal. Ghasemi-Nejhad [5] used a finite element approach to model the insitu-consolidation of a filament wound thermoplastic cylinder with a localized heat source. His analysis showed that the resulting stress distribution for insitu-consolidation was axi-symmetric and consistent with the assumptions made in this analysis.

Figure 2.        Flow Chart for the post-consolidation model. 

In this ‘onion-skin’ model, each layer is added on sequentially, and the incremental strains and stresses are due to the addition of a layer are calculated. The layer that is being wound on is subject to a winding tension, and is at the processing temperature, Tp. This layer is then allowed to cool to the mandrel preheat temperature Tm, before the next layer is added. It is assumed that the temperature of all the other plies is constant at the mandrel preheat temperature. In this manner, the cylinder geometry is built up incrementally, as are the residual strain and stress profiles. The incorporation of winding tension into the analysis is discussed in a subsequent section.

The wound cylinder, now uniformly at the mandrel preheat temperature, can then be cooled (or heated) to the operating temperature, Top. In this step, the temperature change occurs uniformly in all the layers (at this point, the transient heat transfer solution is not applicable to the manufacturing process, but is used for stress analysis during firing)

At this point in the analysis, if the mandrel is present, it may be removed. The incremental strains and stresses introduced into the composite cylinder on removal of the mandrel are computed by the mandrel removal analysis. The details of this analysis are presented in a later section.

Finally the response of the cylinder to the mechanical loads is determined. The final states of strain and stress are determined by summing up the incremental contributions from all of the above processes. A flow chart of this process is shown in Figure 3.

Figure 3.        Flow Chart for the insitu-consolidation model.

Stresses Due to Thermal Loads

In this analysis, it is assumed that the resin has a negligible modulus above Tg, and has its room temperature modulus below Tg. The instantaneous thermoelastic properties of the composite are calculated using continuous fiber micromechanics. Since the properties of the material are assumed to be temperature dependent as described above, the thermal stresses need to be calculated in an incremental fashion, as shown in Figure 4. At each increment in time, the instantaneous temperature and the ΔT corresponding to that time increment is first calculated. Based on the temperature, the instantaneous resin modulus is determined. Continuous fiber micromechanics is then used to determine the effective thermoelastic properties of the composite lamina in the 1-2-3 lamina reference frame, from which the effective stiffness constants and the coefficients of thermal expansion in the global system are determined through the appropriate rotational transformations by the winding angle ϕ. The incremental stresses and strains corresponding to the increment in temperature are calculated as before.

It should be noted that above Tg, the composite stiffness in the fiber direction may be several orders of magnitude higher than the composite stiffness in the transverse direction. Depending on the orientation of the ply, the parameter λ(k) defined in Equation (7) may be of the order of 1000. If the problem had been formulated in terms of r instead of ζ , the term rλ+1 may cause a numerical overflow for r>1, while   r-λ+1 may cause a numerical underflow for r<1. The exact value of rλ+1 or r-λ+1 which will cause the overflow will depend on the computer system in use. This problem is overcome by scaling the variable r such that the new variable ζ does not cause this overflow.

Figure 4, Flow chart for the thermal stress calculation.

Stresses Due to Winding Tension

Cirino [6] included the effects of winding tension in the plane-stress analysis of filament wound rings. His analysis was, however, limited only to hoop wound rings. The analysis presented herein is applicable to winding tension at any winding angle.

The layer being wound is subject to a winding tension, , in the fiber direction. This winding tension gives rise to strains in the 1-2-3 principle lamina system, which are given by:

               [17]

The above strains are transformed from the principle lamina system into the global cylindrical system using the rotational transformation matrices to yield the strains [εw] due to winding tension.

As before, the response of the cylinder is determined from the response of a single layer. The response of each layer is written in terms of constants obtained from the integration of the elasticity equations for that layer. In general, these set of constants vary from layer to layer. These constants are determined by enforcing the traction boundary conditions at the inner and outer radii, by enforcing continuity of displacements and tractions at the interfaces between layers, and by enforcing equilibrium of forces on the cylinder.

To incorporate the effect of winding tension, the constitutive equation for each layer in the x-q-r coordinate system is modified as follows:

               [18]

The strain displacement relations given by Equations (2)-(8) remain unchanged except for equation (8) which is written as:

               [19]

As before, the governing differential equation for the radial displacement assuming transverse isotropy in the 2-3 plane simplifies to:

               [20]

whose solution is:

               [21]

It may be noted that εw can be considered as a stress free strain due to the applied winding tension σw, in much the same context as αΔT is the stress free strain due to a temperature change ΔT. All of the applied winding tension may not be preserved upon laydown [7,8]. The amount of winding tension preserved is dependent on the relative stiffness of the substrate and the layer. The more compliant the substrate is relative to the layer, the lesser will be the extent to which the winding tension is preserved in that layer. As before, the 2N+2 constants, and hence, the incremental stresses due to the winding tension are determined.

Mandrel Removal Calculations

The filament winding operation is usually done over a metallic mandrel. The mandrel can be incorporated into the above analysis by considering it to be just another layer with the isotropic properties of the metal. It is now desired to determine the effect of mandrel removal on the stress state in the composite cylinder.

Let σx(ζ), σθ(ζ), σr(ζ) and τxθ(ζ) be the stress distributions in the filament wound composite cylinder with the mandrel still in place. It is desired to determine the incremental stresses, Δσx(ζ),Δσθ(ζ), Δσr(ζ) and Δτxθ(ζ) introduced into the composite cylinder upon removal of the mandrel. The state of stress in the cylinder after mandrel removal will then be [σx(ζ)+Δσx(ζ)],[σθ(ζ)+Δσθ(ζ)],[σr(ζ)+Δσr(ζ)] and [τxθ(ζ)+Δτxθ(ζ)]. The removal of the mandrel introduces incremental deformations in the cylinder given by:

               [22]

               [23]

       [24]

Note that since ΔT(k) = 0 for the mandrel removal operation, (k) = 0 for all layers. The (2N+2) constants are determined as before, by the enforcement of the traction boundary conditions at the inner and outer radii, by the enforcement of continuity of displacements and tractions at the interfaces between layers, and by enforcing equilibrium of forces on the cylinder. The boundary condition at the outer radius is now written as:

       (since Po =, for mandrel removal)        [25]

The boundary condition at the inner radius of the composite cylinder is:

        (since Pi=0 for mandrel removal)        [26]

or:

               [27]

Note here at , since it is the radial stress component between the mandrel and the first composite layer. Comparing the above to Equation 911), may be regarded as a fictitious internal pressure, analogous to Pi. Continuity of displacements at the (N-1) layer interfaces results in:

       

               k=1,2,….,(N-1)        [27]

Application of the condition of equilibrium of forces in the axial direction results in:

         (since Faxial = 0 for mandrel removal)        [28]

or

               [29]

Note that the integral on the right hand side is evaluated only over the composite layers, the mandrel being excluded. The right hand side of Equation (29) may be regarded as a fictitious axial load, analogous to Faxial in Equation (13). Similarly, the integral condition for torsion can be written as:

          (since τo = 0 for mandrel removal)        [30]

or

               [31]

The right side of Equation (31) may be regarded as a fictitious torsional load, analogous to τo in Equation (16). From Equations (25) thru (31), the (2N+2) unknowns, and hence the Δσ contribution from mandrel removal can be determined. Thus, incorporation of the mandrel removal into the analysis in this manner automatically ensures the satisfaction of the displacement and traction boundary conditions, the continuity of displacement and tractions at the interfaces between layers and the equilibrium of forces in the cylinder. The flow chart of mandrel removal methodology is presented in Figure (5).

Figure 5.        Flow Chart for the mandrel removal procedure.

.