Introduction
Manufacturing processes or production systems are constantly under development, each change is a reaction of the pressures coming from the market growth, the customer, new competitors, the evolution of the technology and new and innovative trends in product and service processes. Furthermore, the majority of commercialized products have a short life cycle in the market, therefore, it is highly recommended to develop adaptable and flexible manufacturing processes in order to satisfy in every moment the existing demand.
The process of Incremental Sheet Forming (ISF) fulfills these requirements [1] and many researchers have focused their interest in the development of this technique. Components obtained by ISF might be functional and have a geometric complexity not achievable by other sheet metal forming processes. The technology is potentially available for any company or institution owning CNC machine tools coupled to CAD/CAM toolpath generators. Furthermore, ISF allows a reduction of the tooling and setup costs compared with the conventional manufacturing process of sheet metal, for instance, deep drawing, in which there is the need to build expensive dies for each type of product.
On the early developments of the technology, metallic materials with good formability at room temperature, such as aluminum alloys (AA1050 or AA3003) or certain steels (DC04, AISI304), were used [2]. However, other metallic materials require an additional heating system in order to improve formability, such as titanium alloys [3] or magnesium alloys [4], have been recently investigated.
Titanium is widely used in aerospace and biomedical sectors, due to its high corrosion resistance, lightweight and good mechanical properties. Fan et al. [5] investigated how to reach the temperature needed to deform TI-6Al-4V by ISF. It was proposed to use an electrical hot incremental forming system that consists of obtaining an electric closed circuit with DC power supply, cables, the tool and the sheet. Due to Joule’s law, when current passes from the tool to the sheet, heat is obtained, therefore, the local temperature increases. This variation of temperature increases the ductility of the material in the contact zone. In a recent work, Ambrogio et al. [6] statistically analyzed the variation of the temperature during the incremental forming of two lightweight alloys, Ti6Al4V and AA5754. They varied the step down and feed rate in order to increase the temperature allowing an improvement of the formability for these materials.
The use of magnesium in aeronautic, automotive, and electronic applications is increasing in the recent years. However, due to its hexagonal-closed-packed (hcp) crystal structure has a very low ductility and formability at room temperature. In Park et al. [4] magnesium sheets were deformed by ISF without using an external heating device, they proposed to increase the local temperature using the rotation of the tool in a CNC milling machine. This process is called RISF (Rotational Incremental Sheet Forming). In this process the sheet is clamped to the framework while the tool rotates to its own axis at high speed, penetrating step by step along the established tool path. This generates an important quantity of heat in the contact zone due to the friction between the tool and the sheet and also due to the energy of plastic deformation. The local heat accelerates the plastic deformation and increases formability.
Several recent publications have shown that the use of thermoplastic materials is becoming one of the research trends in ISF. The first research work was published in 2008 [7], PVC (polyvinylchloride) was formed using the SPIF variant in order to determine the formability limits for a variable wall angle geometry. Since then, many papers have been focused on increasing the knowledge regarding the formability limits and failure modes for other polymers [8], the feasibility evaluation of producing hole-flanged polymer parts by SPIF [9] or the numerical simulation of the process [10]. Bagudanch et al. [11] was the first work in which it was demonstrated the effect that the spindle speed has in SPIF when a polymer material (PVC) is used. The results showed that increasing the spindle speed it was possible to reduce the forming force and to increase formability because of the heat generation due to the friction between the tool and the sheet. In a subsequent work [12], the study was extended for two polymers (PVC and PC -polycarbonate-). Davarpanah et al. had also corroborated the effect of the tool rotation on the forming force for PVC and PLA [13]
Polymers have a drastically different behavior than metals because they are formed by crystalline lattices of atoms being well ordered. The polymer molecules consist of carbon atoms bonded into a long chain. Since the carbon-carbon bond can rotate, the chains can be rearranged into infinite different conformations, mainly depending on the speed of deformation, stress and temperature. At high temperatures, chains can be easily moved with the applied deformation. Below the glass transition temperature, chain mobility drastically decreases and the material virtually vitrifies and becomes more fragile, even though changes in chain conformation are still possible. Therefore, the temperature reached during the deformation plays an important role and causes important variations on the final part results.
The present paper is focused on increasing the knowledge regarding the temperature variation during the SPIF process for three non-biocompatible polymers (PVC, PC and PP -polypropylene-) and two biocompatible polymers (UHMWPE -ultra-high molecular weight polyethylene- and PCL -polycaprolactone-), with the aim of determining the effect of the process parameters as well as providing empirical equations able to predict the maximum temperature of an experimental test using the response surface methodology.
Methodology
Geometry and materials
The test geometry is a pyramidal frustum with circular generatrix, for which the wall angle varies at each depth increment. The length of the edges of the pyramid is 105mm, the initial wall angle 45º and the generatrix radius 80mm (Figure 1a).
Tests with five polymers were performed, PVC, PC, PP, PCL and UHMWPE. The sheet thickness was 2 mm.
Experimental setup
The ISF tests were performed using a Kondia®HS1000 3-axis milling machine. The details of the clamping system and the setup (Figure 1b) are widely described in previous works [12].
A thermographic camera IRBIS ImageIR®3300 (Figure 1c) has been used to measure the material temperature values during the SPIF manufacturing process, at a frequency of 1 image/10 seconds. The temperature distribution on the sheet during the entire experimental tests was registered (Figure 1d). This information was processed in order to obtain the maximum temperature (Tmax=y) of each test for the analyzed materials.
Design of experiments
Box-Behnken designs (BBD) [14] are a type of efficient three-level designs for fitting second order response surfaces. The design for four factors consists of 27 experimental runs (Table 1) that can be split into three blocks with one center point at each block, being the considered parameters and levels: tool diameter (Dt: 6/10/14mm), spindle speed (S: Free/1000/2000rpm), feed rate (F: 1500/2250/3000mm/min) and step down (Δz: 0.2/0.35/0.5mm).
Model selection procedure
The procedure used to select a model is based on standard methodology [15]. The first step was to start by estimating the full model with all first order (FO) terms, two-way interaction (TWI) and pure quadratic (PQ) terms. Then, the non-significant terms (one at each step) were sequentially removed based on the tests on individual regression coefficients and groups of coefficients. Each model is analyzed in terms of the fit statistics: R2, R2-adjusted, R2-predicted and RMSE. The model shows a good fit if the R2 statistics are high and the RMSE is low.
For each model the test for significance of regression (p-value associated to Model in the ANOVA table) was observed, where a p-value<α (being α the significance level) indicates that the regression is significant. It has also been examined the lack of fit test that can be computed on BBD because it includes true replicates on the center point. It is an indicator of the tentative model satisfactorily describing the data when its p-value is high.
The last step before validating the model is the adequacy checking. It is important to examine that the fitted model offers an adequate approximation to the real system and to corroborate that the last squares regression assumptions are met, that is, that residuals are normal i.i.d. N(0,σ2). The first can be checked by plotting observed values (y) vs. fitted values (y ̂) and the second by looking at different model diagnostics graphs. In the present paper, the residuals vs y ̂ graph to check for independency and homoscedasticity and the normal probability plot to check normality, were used. Conclusions on graphical analysis can be confirmed with tests such as Shapiro-Wilk test or the test on correlation coefficient.
Note that coded variables (low level=-1, medium level=0, and high level=1) were used in order to be able to compare the size of the coefficients and that the significance level in all cases is α=0.05.
Results and discussion
The maximum temperatures obtained in all the experimental tests are summarized in Table 1. The statistical models proposed for each material are shown in the following subsections.
PVC
The model selected for explaining the Tmax for PVC is the one showed on Table 2 that can be written as:
y ̂=46.99+9.77·S-0.29·F+4.48·S^2+6.61·F^2 E(1)
The model depends significantly on S and S2 as well as F2 being the most influential factor S (higher estimate and lower variability). The model is significant and has a non-significant lack of fit. The residuals are normally distributed (Shapiro-Wilk normality test p-value=0.9283), independent and homoscedastic.
The response surface and contour plot are shown in Table 2. There is a stationary point at S=0 and F=0. Any increment or decrement on F will have an effect on increasing Tmax. The same happens when S increases, although its effect is acuter when moving from 1000rpm to 2000rpm.
PC
The model selected for explaining the Tmax for PC is the one showed on Table 2 that can be written as:
y ̂=48.97-1.40·Dt+10.90·S-2.01·F+0.55·∆z+7.30·F^2+8.74·〖∆z〗^2 E(2)
The model that better accomplishes the regression assumptions is the one that includes Dt even if this term is not individually significant. However note that the test on the group of first order terms is significant.
Models without Dt showed a significant lack of fit test (p-value<α). The low p-value of the lack of fit test is explained because the 3 replicates on the center point have very low variability (model-independent estimator of σ2), while the variability of the errors from the model (model-dependent estimator of σ2) is very high. In the selected model the p-value>α, then there is no evidence that the model does not fit well the data. Residuals are normal and i.i.d. (Shapiro-Wilk normality test p-value=0.9283).
The factors that mainly contribute to explain Tmax are S, F2 and Δz2. The most influential factor is S because estimates of F2 and Δz2 are lower and moreover have larger variability (they are less precise). The response surface plots are shown in Table 2. The surface corresponding to S-F is similar to S-Δz (estimates of F2 and Δz2 does not greatly differ) and again similar to the surface obtained in the PVC model for Tmax.
The surface obtained by plotting F-Δz shows a stationary point (minimum response) at F=0.138 and Δz=-0.032 which corresponds to F=2353mm/min and Δz=0.345mm in original units.
PP
The model selected for explaining Tmax for PP is the one showed on Table 3 that can be written as:
y ̂=70.50+6.37·Dt+26.030·S+5.54·∆z E(3)
Dt and Δz were kept in the model because with those terms the regression assumptions were better accomplished. However, the test on the group of first order terms is significant and the lack of fit test shows that the model fits well the data. Residuals are normal and i.i.d. (Shapiro-Wilk normality test p-value=0.7045).
The most significant factor is S (higher estimate) while Dt and Δz have a lower and similar effect on Tmax. The response surface plot of S-Dt is shown on Table 2 (the surface plot of S-Δz is very similar). The maximum value of Tmax is obtained when S, Dt and Δz are at their maximum values, and the minimum at the opposite.
UHMWPE
The model selected for explaining Tmax for UHMWPE is the one showed on Table 3 that can be written as:
y ̂=71.92+3.67·Dt+13.73·S+2.67·F+5.57·Dt·S+6.85·S·F-7.92·S^2 E(4)
The contribution of the first order terms to the model is significant. It has been decided to keep in the model the interactions Dt·S and S·F even if they are not individually significant because they improve the model fit and adequacy.
It has been checked that the residuals of this model are normally distributed (Shapiro-Wilk normality test p-value=0.9836), independent and homoscedastic. There are no influential observations.
Tmax on UHMWPE is mainly explained by S and its quadratic effect S2 (greater coefficient estimates). However, the quadratic term of S has a negative estimate. When Dt=S=F=0 the average Tmax is 71.92ºC. If S increases to 1 (2000rpm) the average Tmax is 77.73ºC (95%CI=[71.41, 84.05]) while when S reduces to -1 (Free spindle speed) the average Tmax is 50.26ºC (95%CI=[43.95, 56.59]), that is, the effect on increasing/reducing S is not symmetric on Tmax.
Table 3 shows the response surface plots for the selected model. The minimum values on Tmax are found when S=-1 and Dt and F at 0 or 1, while maximum Tmax is found at S=1 and again Dt and F at 0 or 1.
PCL
A preliminary exploratory analysis of Tmax shows a right skewed distribution (non-normal pattern) and a clear atypically high temperature corresponding to experiment 15 (74.38ºC). The material melted under those conditions and before the experiment could be stopped the temperature raised very quickly (melting temperature of PCL is 60ºC).
After trying different models, it was decided to keep the value of 74.38ºC in experiment 15 and to transform the response variable by the use of the natural log. This transformation improves the fit and the adequacy of the model. Note that all tested models have the same significant terms: Intercept, Dt, S and S2.
The parameter estimates of the retained log model as well as the standard error and the test on individual regression coefficients are shown in Table 3.
And the model is of the form:
ln〖y ̂=3.69+0.07·Dt+0.22·S+0.1·S^2 〗 E(5)
Or equivalently:
y ̂=e^(3.69+0.07·Dt+0.22·S+0.1·S^2 ) E(6)
Table 3 shows the ANOVA table including information about the model and the residuals. Both the first order terms as well as the quadratic term significantly contribute to the model. The lack of fit test with its high p-value shows that the model adequately describes the data.
The graphical model adequacy checking (upper Figure 2) shows homoscedasticity and independence of residuals (r_(Resid,y ̂ )≈0) with a high residual on run 15 (already detected as an extreme observation on the exploratory analysis). The normality of residuals is accomplished, except for run 15. In order to verify whether this observation is an influential one, two extra model diagnostics graphs were plotted (lower graphs on Figure 2). The square root of standardized residuals should lie in the interval [0,√3] and run 15 is a clear outlier. However, its Cook’s distance is less than 1, which is the usually considered limit for influential points, which have to be avoided because greatly influence the fitted model.
To check the influence of run 15, the model was re-estimated by changing the response value of run 15 to 60ºC and it was confirmed that the estimates of the parameters do not greatly differ. The final model to explain Tmax on PCL is the one shown in Equation 5.
The only influential factors on Tmax are Dt, S and S2. The response surface plot of Dt and S is shown in Table 3 (Tmax on original scale). The maximum Tmax is obtained with a combination of Dt and S at their high values while the minimum is obtained at the opposite. The plot shows no stationary point but a falling ridge on the direction (-1,-1) that is coherent with the analyzed factors.
Summary
In this last section, the proposed models for each material are compared according to not only the statistical results but also considering the material properties and from a technological point of view. The coefficients (estimates and standard deviation) of the selected models for the five materials are rewritten in Table 4 to ease comparison. Note that the model for PCL describes log(Tmax).
As stated before, Box-Behnken designs allow estimation of interactions and second order coefficients with a low number of experiments. However, the standard error of the coefficients is high, that is, the precision of the estimates is poor. For example, the estimate of S in the model for PP has a standard deviation of 3.41, which means that the true value of that coefficient (95% confidence) is in the interval [18.98;33.08]. This is the value that Tmax is increased (decreased) when factor S is moved to its high (low) value. In the case of the PCL model, the estimate of Dt is 0.07 and in original scale it indicates that Tmax is multiplied (divided) by e0.22 when S is moved to its high (low) value.
The Tmax in the center point (all factors to 0 level) is similar in PVC(46.99) and PC(48.97); it is lower in PCL(e3.69=40.04); and it is higher in PP(70.50) and UHMWPE(71.92). Dt helps to explain Tmax only for the PCL material. On the other hand, the factor S is significant and positive in all models, and in three of them it has a quadratic effect. In all cases, an increment (decrement) of S factor increases (decreases) Tmax. Two-way interactions have been considered only for the UHMWPE model (Dt·S,S·F), even if their joint test for significance shows a p-value=0.0529 slightly higher than α. They have been kept in the model assuming a higher probability of incorrectly considering them significant. More experiments should be run in order to verify the significance of those interactions.
The temperature evolution during several tests for UHMWPE is represented in Figure 3 and Figure 4. In Figure 3a it can be observed that there is an increase of the temperature with the increase of the tool diameter. This variation becomes more important when the experiment approaches the end of the process. Similar results have been obtained for the rest of the materials. This fact occurs because with higher tool diameters the contact zone between the tool and the sheet is higher and it is more difficult to dissipate the heat generated due to the friction.
As it has been mentioned in previous sections of the paper, the spindle speed has an important role in the temperature variation during the experimental test for all the analyzed materials. An increase of the spindle speed causes an important increase of the heat generated due to friction between the sheet and the tool. In the case of UHMWPE the variation can be around 35-40ºC (Figure 3b) considering the extreme values of the spindle speed (Free and 2000rpm).
Regarding the influence of the feed rate, it can be observed from Figure 4a that for UHMWPE it is not very important although its presence improves the statistical model. Decreasing the feed rate obviously leads to an increase of the forming time, but the maximum temperature reached in the experiments is almost the same that the one obtained using higher feed rates. Similar results are found in the case of the step down (Figure 4b), the maximum temperature is the same independently of the step down level and the only difference is the duration of the test.
Conclusions
In this work SPIF experimental tests using five different polymers have been done following a Box-Behnken design of experiments for four factors. The variation of the process parameters causes a change on the friction conditions which lead to a different amount of heat generation during the test, therefore, the polymer material might change its mechanical behavior.
The maximum temperature achieved in the experiments has been statistically analyzed and empirical models for each material have been obtained. These models can be useful to be able to control the temperature, which in turn, will modify the mechanical properties. Therefore, the material formability could be increased and guarantee that the level of forming forces will not exceed the machine operating limits. Moreover, it will be possible to predict which will be the increase of the temperature under new process conditions, determining, for example, whether the material will overpass the glass transition or melting temperatures.
Acknowledgements
This research has received funding from the Spanish Ministry of Education (DPI2012-36042), the Spanish Ministry of Economy and Competitiveness (MTM2012-33236) and the Catalan Agency for Management of University and Research Grants (2014 SGR551). The first author gratefully acknowledges the support provided by grant FPU12/05402 (Spanish Ministry of Education). The authors would like to express their gratitude to Instituto Tecnológico y de Estudios Superiores de Monterrey (ITESM) and Centro de Investigación de Química Aplicada (CIQA) for supplying PCL sheets. Finally, thank Jordi Canal, Marc López and Gerard Gutiérrez for their collaboration during the experimental campaign.