An axisymmetric model for thread forming in polycarbonate and polypropylene screw and boss fasteners.
One of the most popular ways of fastening assemblies together is by means of the boss and screw, an example of which is shown in Fig. 1. The threads of the boss may be pre-tapped, cut by a thread-cutting screw, or formed by a "thread-forming" screw. The second two processes are typically performed as the assembly is made during the first installation of the screw. In the case of thread-forming, the boss material must be sufficiently robust to accommodate the large plastic strains and the significant heat generated by plastic deformation of the boss material and between the surfaces of the boss and screw.
After forming, large residual stresses remain in the boss, particularly where the material has undergone plastic deformation. These large stresses can lead to a number of potential failure modes, including boss cracking from high hoop stresses, environmental stress cracking, reduced fatigue life, and loss of clamp load due to stress relaxation (1, 2), which is a function of the joint and is also a function of time and temperature. Stresses near the yield stress will relax more quickly than those of lower magnitude.
Over the years, screw and boss designs have evolved to give acceptable performance in terms of a strip to drive torque ratio, but retention of clamp load has always been a problem. Clearly, optimized screw and boss designs will be material specific. One trend in automotive applications is replacement of polycarbonate (PC) and ABS by polypropylene (PP). For further optimization of design and process, it would be helpful to have a model that delineates what is happening during the screw driving process and can be used to perform parameter variations. Intuition tells us that if residual stresses can be reduced, clamp load loss would be improved. Further, it may be possible, by design, to minimize the effect of the residual stresses on clamp load loss. In this paper, we present a model of the thread-forming process using PC as a base material for the boss and a steel screw. Results for a typical PP material are also computed for comparison with the base PC material, with emphasis on determining the temperature and stress profiles during the thread-forming process. Subsequent studies will cover stress relaxation and other potential failure modes.
[FIGURE 1 OMITTED]
In order to reduce the complexity of solving a fully three-dimensional formulation for the thread-formation problem, the assumption is made that the thread pitch is reasonably small compared to the screw diameter. This allows one to approximate a segment of the thread during one revolution as an annular ring. The process of visualizing the problem as an axially symmetric one is illustrated in Fig. 2, which depicts unwrapping a helical thread and the subsequent approximation as a series of hoops.
The first step is to visualize one cycle or revolution of a continuous thread. Starting and ending points on a thread cycle are then offset to form a single hoop. For simplicity, no taper was shown on the screw body in Fig. 2. However, a taper on a thread is modeled as a hoop or ring with a major radius that is expanding with time. The expansion rate is dictated by the taper angle on the thread body and the rotation rate, [omega], of the screw. Thus, an individual thread can now be modeled as a series of axially symmetric rings, whose major radius changes with time. Gradients in boss geometry can be accommodated in a similar manner.
[FIGURE 2 OMITTED]
The advancement is dealt with as a discrete series of steps in the axial direction that occurs at the end of each screw revolution. This action, as outlined in Fig. 3, consists of advancing the lead or leading thread in the radial direction, which approximates the increasing diameter of the thread. The thread and screw body undergo heating effects, while the boss body undergoes both plastic deformation and heating effects. Upon completion of the thread-forming process at the end of a single revolution, the threads are advanced by the distance representing one pitch, as shown in Fig. 3, preserving the temperature profile of the thread. Thus, the initial tapered thread or threads would be replaced by subsequent non-thread forming chase threads.
[FIGURE 3 OMITTED]
In this paper, we wish to focus on the heat transfer aspects of this problem, and what follows is a brief outline of the mechanical analysis.
The system described below was solved with DEFORM2D (3), which is a nonlinear quasi-static large-deformation code primarily used in the metal forming industry. The reader is referred to Kobayashi (4) and Simo (5) for the details of the analysis for the elasto-plastic solver. To deal with large deformation, DEFORM2D utilizes an updated Lagrangian formulation (4 Ch 14). Elasto-plasticity uses the radial return algorithm outlined in Simo (5 p 143) with the Green-Naghdi stress-rate formulation (5 p255). The quasi-static velocity field solution is achieved using a Newton-Raphson iteration technique.
For this rate-based formulation, the plastic strain-rate, [dot.[epsilon].sub.ij], is a measure of deformation and is given by,
[dot.[epsilon].sub.ij] = [1/2]([[partial derivative][u.sub.i]/[partial derivative][x.sub.j]] + [[partial derivative][u.sub.j]/[partial derivative][x.sub.i]]) (1)
where [u.sub.i] are the velocity components. The flow stress is considered to be a function of temperature, T, and strain rate, [dot.[epsilon]], as dictated by the Eyring model (6, 7),
[[sigma].sub.y] = aT [In(2c [dot.[epsilon]]) + [[DELTA]E/RT]], (2)
where R = 8.314 J/mole [degrees]C. Again, for a complete description of mechanical formulation see DEFORM2D (3).
An energy balance for both the deformable and rigid bodies, that is, the plastic boss and screw, is expressed by the parabolic equation.
[rho][c.sub.p][[partial derivative]T/[partial derivative]t] = [nabla] * k[nabla]T + [kappa]([sigma]:[dot.[epsilon]]) (3)
where [c.sub.p] = [c.sub.p](T) and k = k(T) are the heat capacity and the thermal conductivity of the material. The final term in Eq 3 represents a heat production term resulting from plastic work in deformable bodies, which formally couples the mechanical and thermal analyses.
On surfaces where the screw comes in contact with the deformable boss, heat is generated through plastic work as well as interfacial friction heating. With this in mind, the heat generated at the interface between the rigid screw and the plastic boss can be expressed as,
[q.sub.f] = [f.sub.s][omega]r (4)
where, [omega]r, is the rotational speed of the screw relative to the boss. The friction shear stress, [f.sub.s], is taken to be either [f.sub.s] = f [[sigma].sub.n] or [f.sub.s] = f [k.sub.s], with [[sigma].sub.n] being the interface pressure and [k.sub.s], the shear strength of the deformable material. For interfaces under large plastic deformation, the latter is the appropriate form for the friction shear stress, [f.sub.s], (4).
At this point, the heat generated in the contact region due to friction must be distributed to the two contacting materials forming the interface. The boundary conditions for the materials assumes that the heat will be partitioned using the thermal conductivity of the materials at the interface, [k.sub.1], and, [k.sub.2], as a lever rule, as discussed in Rabinowicz (8). Thus, the flux boundary condition for the materials on either side of the interface are,
- [k.sub.1] [nabla]T * n = f[k.sub.s][omega]r [[k.sub.1]/[k.sub.1] + [k.sub.2]] (5)
- [k.sub.2] [nabla]T * n = f[k.sub.s][omega]r [[k.sub.2]/[k.sub.1] + [k.sub.2]] (6)
Elsewhere, the boss and screw are in contact with air, thus, a heat tranfer coefficient boundary condition,
k[nabla]T * n = h(T - [T.sub.[infinity]]) (7)
was applied to Eq 3.
Based on Eqs 1-7, a number of material parameters are required for both boss materials (PC and PP) and for the steel screw. Required physical properties for the steel screw are taken to be be k = 43 W / mK and [c.sub.p] = 3850 kJ/[m.sup.3] K independent of temperature. It is also assumed that the steel screw does not deform.
For the polymer materials, an empirical relationship exists for the thermal conductivity, which is valid for amorphus polymers like PC (9).
k(T) = k([T.sub.g])(T/[T.sub.g])[.sup.0.22] for T [less than or equal to] [T.sub.g]
k(T) = k([T.sub.g])[1.2 - 0.2(T/[T.sub.g])] for T > [T.sub.g] (8)
where [T.sub.g] is the glass transition temperature. Though semicrystalline, we have found this form of the above empirical equation to fit available conductivity data for PP. The parameter value used in the study are k([T.sub.g]) = 0.235 W / m[degrees]K. [T.sub.g] = 418[degrees]K for PC and k ([T.sub.g]) = 0.120 W / m[degrees]K, [T.sub.g] = 263[degrees]K for PP. Over the temperature range of interest, PC has almost twice the thermal conductivity as PP. The values for both plastics are more than two orders of magnitude lower than that for steel.
Polymers typically have a discontinuous change in heat capacity at the glass transition temperature (9); however, this can be ignored in these calculations since the PC and PP materials will be below and above their glass transition temperature, respectively. Given this condition, the following correlation were used for heat capacities,
[rho][c.sub.p](T) = 3.86T + 84.4 for PC, (9)
[rho][c.sub.p](T) = 5.78T for PP, (10)
where [rho][c.sub.p] has units of (kJ/[M.sup.3][degrees]K) and T is expressed in ([degrees]K). The data used for fitting the thermal properties can be found in Schramm et al. (10) for PC and Brandrup et al. (11) for PP. Over the temperature range of interest, PP has a significantly higher heat capacity than PC.
The PC used in this study was Lexan LS2 (GE Plastics) and the PP used as a basis for comparison was Pro-fax SD-242 (Basell) impact-modified polypropylene. Moduli and Eyring parameters were determined by tensile tests of (127 X 12.7 X 3.18 mm) rectangular strips using an Instron 5565 hydraulic testing machine. The moduli of PC and PP are assumed to be a function of temperature, i.e., E = E(T), which are plotted in Fig. 4, while Poisson's ratio was taken to be a constant values of [v.sub.pc] = 0.35 and [v.sub.pp] = 0.38. PP has a lower modulus than PC and PP's modulus drops more rapidly with temperature than does PC's modulus. The parameters for the Eyring model, given in Eq 2, were taken to be [a.sub.pc] = 4.08 X [10.sup.-3] N/[m.sup.2][degrees]K, [c.sub.pc] = 1.0 X [10.sup.-31] s, and [DELTA][E.sub.pc] = 3.16 X [10.sup.5] J/mole, as reported by Bauwens and Crowet (12) for PC and [a.sub.pp] = 6.00 X [10.sup.-3] N/[m.sup.2][degrees]K, [c.sub.pp] = 7.0 X [10.sup.-12] s, and [DELTA][E.sub.pp] = 9.83 X [10.sup.4] J/mole for PP as determined from our data. The friction factor for polycarbonate was taken from Yamaguchi (13). The variation of friction with surface pressure for PC was [f.sub.pc] = 0.3 below 80 MPa, [f.sub.pc] = 0.7 above 150 MPa, with a linear ramp between the two surface pressures, while the PP material had a constant factor of [f.sub.pp] = 0.3.
[FIGURE 4 OMITTED]
The initial mesh for the deformable boss and rigid bodies representing the threads are shown in Fig. 5. Typical element and mesh densities for the rigid bodies are 200 elements with 170 nodes with the deformable bodies having 8422 elements with 8788 nodes. The mesh density in the regions where thread forming will take place are specified to be approximately 400 times the average element density in the deformable body. The time step used through this study was [DELTA]t = 0.0022 s, which corresponds to a screw speed of [omega] = 340 rpm with each revolution containing 80 increments. The initial temperature for the entire system was set to T = 20[degrees]C.
[FIGURE 5 OMITTED]
As a numerical experiment, the influence of friction heating versus heating due to plastic work was investigated for the polycarbonate boss/screw configuration by performing simulations with and without interfacial friction. f. The results are shown in Fig. 6 and Fig. 7 just after the final thread has been formed at t = 1.056 s.
For the lowest thread (or the lead tapped thread) on the right side of the plot, the plastic heating zone is clearly evident just ahead of the thread in the interior of the boss. The resulting maximum temperature rise above the initial temperature of 20[degrees]C is 15[degrees]C, which is the result of plastic work, which is represented as the term. [sigma]:[dot.[epsilon]] in Eq 3, with some degree of cooling due to heat transfer to the screw thread and body. This is contrasted by the plot on the left in Fig. 6, which includes interfacial friction. The temperatures are at a much higher level with a maximum value of 123[degrees]C above the initial temperature. This indicates that friction is a larger contributor to heating for polycarbonate and, in fact, though plastic work is present, a separate heating zone is not distinguishable. Given the critical role of friction in determining the boss temperature, it is clear that any material, be it lubricant or contaminant, that changes the friction properties could have a dramatic impact on boss behavior.
[FIGURE 6 OMITTED]
One interesting observation is that the maximum temperature with friction heating included occurs at the first chase thread and not the first leading tapered thread. The leading thread has the largest contact time and pressure with the boss; thus, it would receive the largest amount of friction heating, when compared to other threads. However, the leading thread is also moving on to new, cool material at discrete intervals, when forming the next thread; thus, a cooling effect occurs. For upper or trailing threads (or chase threads), there is generally less time in contact with the boss, as well as lower pressures, resulting in less relative heating by friction. The lack of friction, coupled with the fact that screw threads entering the boss are cool, causes the trailing threads to be cooler. Thus, the maximum temperature is predicted to occur somewhere between the leading and trailing threads, though this has not been experimentally confirmed.
One process variable to be considered is changes in the screw speed [OMEGA], the effects of which are shown in Fig. 8. This figure compares the temperature distribution at the point where the last thread is formed for screw speeds of [omega] = 340 rpm and [omega] = 170 rpm. which corresponds to process times of t = 1.056s and t = 2.112 s, respectively. The visible temperature drop is on the order of 20[degrees]C: that is, the maximum temperature for the screw drops from 147[degrees]C to 126[degrees]C, while the polycarbonate boss drops from 140[degrees]C to 120[degrees]C. One might expect a greater temperature drop for the slower screw speed due to a reduction by half in the flux of heat given in Eqs 5 and 6. However, the total energy transfer will be roughly the same because the contact time will double for the lower screw speed. The actual reduction in temperature is a result of the longer time for heat to be conducted away along the screw body and, to a lesser extent, in the polycarbonate boss. This would allow one to conclude that screw speed has only a secondary influence on the internal temperature achieved in the boss.
[FIGURE 7 OMITTED]
[FIGURE 8 OMITTED]
In addition to temperature profiles, it is possible to determine the resulting stress state in the boss at the present operating conditions. The axial component of stress, [[sigma].sub.z], as shown in Fig. 9, is found to be compressive in between successively formed threads. This may be somewhat counterintuitive, as the thread-forming process generally is perceived as pushing outwards from the center of the boss. This would lead one to believe that the hoop stress, [[sigma].sub.[theta]], which is in tension, would control the resulting stress state. However, the material for the geometry under consideration tends to get "pinched" between successive wedge-shaped threads, as shown in Fig. 9. This leads to the overall compressive nature of the stress in the regions between threads, which could be important for materials where Bauschinger effects are significant, such as filled polymers.
The final part of the thread-forming process is the stripping stage, during which the screw head engages the top of the boss. This is modeled in a discrete system by having relative motion between the screw head and the body of the screw. Figure 10 illustrates the large stresses, shown as effective stress, that are developed between the screw head and the boss. Shown is a high level of stripping in which the threads tend to pull out of the boss body in the axial direction and the polymer material separates from the low face of each thread. Owing to bulk elastic deformation of the boss in the axial direction, the lower faces of the upper threads are shown to have separated from the boss to a greater extent than the lower ones. Once the polycarbonate material undergoes the large plastic deformation associated with the stripping process, the assumed incompressible material tends to flow towards the center of the screw on the upper face of each thread, as shown on the right in Fig. 10. A maximum thread temperature of 147.8[degrees]C occurs during the stripping process. This maximum temperature is in good agreement with infrared measurements, which yielded a maximum temperature of 150[degrees]C.
Another quantity that can be determined experimentally is the torque exerted on a tool while driving a screw into the boss. To acquire this data, a fixture was built to secure the boss shown in Fig. 1, and the screw was driven into position using a fixed rpm. A load transducer was used to monitor the torque in real time. The resulting torque-time curve for a fixed speed of 340 rpm is shown in Fig. 11. The experimental curve exhibits a rapid rise in torque, followed by a plateau region. This initial rise is caused by the high loading on the initial tapered thread, which is cool. As the initial thread progresses into the boss, it experiences similar loading because the polymer is always cool. However, the subsequent chase threads enter the boss and cool the material that was heated by the tapered thread. This action results in a thermal contraction of the material near the tip of the chase thread, which has the effect of reducing the normal load. Thus, one should not expect each subsequent chase thread to contribute as much to the torque time curve as the initial tapered thread. An additional factor that contributes a reduced loading on the chase threads is the bulk elastic response of the boss as lead threads force the boss to expand circumferentially.
[FIGURE 9 OMITTED]
[FIGURE 10 OMITTED]
The predicted torque curve based on the finite-element analysis, shown as the dotted line in Fig. 11. compares reasonably well considering that there were no adjustable parameters. The effect of the discretized motion into the boss is apparent by the cyclic nature generated during the formation of the six-thread section, as compared to the formation of one helical thread. One feature that is not captured is the slight fall-off of the experimental data, which might be explained by relaxation of the polymer. In order to predict such behavior, a viscoelastic formulation of the problem would be needed.
[FIGURE 11 OMITTED]
Figure 12 shows the development of hoop strain, [[epsilon].sub.[theta]], near the upper outer edge of the boss. High maximum values could be used as an indication of potential cracking and part failure. The strain levels are shown to rise quickly as the initial threads are formed at the top of the boss, followed by a gradual reduction. The reduction in hoop strain is due to thermal contraction of the material that was formerly heated during the thread forming process. The maximum value is close to 0.5%, which is consistent with our experimental observations.
[FIGURE 12 OMITTED]
Figure 13 shows the effective stress and temperature fields for the comparative case of forming threads in polycarbonate versus polypropylene at a point where the first four threads have engaged the body of the boss. One quick observation is the relative difference between the maximum stress levels for PC (63 MPa) and PP (44 MPa) are much closer than expected, given their bulk mechanical properties. The reason can be that the PC temperature levels are higher, which causes a greater reduction in its mechanical properties relative to PP. Compared to PC, the high stress (relative to its yield stress) in the PP material may result in rapid stress relaxation, which would lead to potential issues with clamp load loss.
[FIGURE 13 OMITTED]
The source for the large amount of heat generated for PC is internal heating and interfacial friction. Given that both the PC and PP simulations would have similar strain rates, the mechanical properties of PC would result in the generation of more internal heating. However, as discussed earlier, friction heating is the biggest contributor to adding energy to the boss/screw system. There are several contributing factors that lead to much lower temperature increases for PP relative to PF. First, the PP/steel interface has a friction factor that is close to half that of PC/steel. Second, PP has a lower thermal conductivity of PP than PC resulting in more of the heat going into the steel where it is carried away. Finally, PP has a higher heat capacity than PC resulting in a lower temperature rise for a given amount of heat generation. It is these differences that account for the higher temperatures in PC and the resulting reduction of mechanical properties.
Figures 14 and 15 show the loading profiles for the screw head after it comes into contact with the boss. Simulations for both PC and PP are plotted, as well as the corresponding PP FEA data for the various indicated points in the forming process. The point of inflection in the loading curve occurs when stripping starts; that is, the lower face of the thread separates from the polymer, as shown. Also, it can be seen that the curves reach a maximum load resulting from plastic deformation or yielding of the polymer in the system. Clearly visible in the superimposed contour plots is the rise in temperature on the top face of the thread due to the large shearing and plastic work in that region.
[FIGURE 14 OMITTED]
[FIGURE 15 OMITTED]
[FIGURE 15 OMITTED]
As would be expected, the load on the screw head for the PC boss (26kN) is higher than the PP boss (13kN) at the same stage in thread forming process because of higher mechanical properties. However, these loading levels are below that which can be calculated assuming yielding takes place at the processing temperature. The lower loading levels occur because the material yields at the higher temperatures seen in the shear zone mention above. Thus, the reduction of mechanical properties in the region adjacent to the thread surfaces limits the load on the screw head. Also, friction modifiers would have little effect, since the heat generated in this region is mainly due to plastic deformation. This is shown by the fact that the material region above the thread that eventually goes through large plastic deformation has a temperature range of 22[degrees]C to 24[degrees]C before stripping and a range of 32[degrees]C to 39[degrees]C near the end of the stripping calculations. Given an initial temperature of 20[degrees]C, more of the temperature rise in PP is due to plastic deformation.
A framework describing a numerical model for the thread-forming process has been outlined in this report. Though the work is preliminary, promising simulation results were obtained, indicating the potential for future work in the "upfront" design stage. Generally, the concept estimating torque-time curves seems to be viable, which is a requirement for specifying strip torque to loading ratios for a specific boss/screw material.
The selection of material models for thread forming in polymer materials is also a critical issue. Though polymers typically display some type of stress relaxation, it is apparent that this is not critical and results from the very short time scales involved during the initial forming process are reasonable, though some improvement may be gained by a more complex constitutive model. The internal temperatures of the polycarbonate boss were estimated, resulting in a strong correlation with measured maximum temperatures. Also, screw speed was shown to have only a secondary effect on the ultimate temperature rise in the boss, while changes in the friction factor were shown to have a significant effect. Though the torque magnitudes shown in Fig. 11 are reasonable, we believe that refinement of the friction model should improve the correlation of predicted torque time behavior.
Future endeavors should focus on incorporating the second phase of the boss/screw assembly; namely, the stress relaxation phase. Given the impractical nature of forming a fully viscoelastic model with locally large stains and stresses, the proposed elasto-plastic model could be used to initialize a creep model representing the second phase.
The authors would like to thank Joon Park of Visteon and Chris Fisher of Scientific Forming Technologies Corporation for their input regarding technical issues.
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KEVIN R. J. ELLWOOD*, DON FESKO, and DAVID R. BAUER
Ford Research Laboratory
Materials Science Department
Dearborn, MI 48121-2153
*To whom correspondence should be addressed.
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|Author:||Ellwood, Kevin R.J.; Fesko, Don; Bauer, David R.|
|Publication:||Polymer Engineering and Science|
|Date:||Aug 1, 2004|
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