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Quantitative error-assessment between upset-test data and a computer-code simulation
Int. J. Mach. "Fools Manufact. Printed in Great Britain
Vol. 31, No. 2, pp.193-201, 1991.
089(I-6955/9153.00 + .00 Pergamon Press plc
QUANTITATIVE ERROR-ASSESSMENT BETWEEN UPSETTEST DATA AND A COMPUTER-CODE SIMULATION J. N. M_mE~trs,* K. P. JErk* and Y. P. Go* (Received 29 May 1990; in final form 15 August 1990) Abstract--A quantitative analysis of axial-force, internal radial displacement and workpiece contour histories
for upset-tests are presented in this paper. Discrete internal deformations are measured by using the technique of "tracer particles". Vertical arrays of tracer particles, aligned in the longitudinal direction of the upset cylinders, permit internal radial deformations to be both visualized and measured as a function of axial diedisplacement. Statistically quantitative axial-force, radial-contour and internal radial deformations vs axial displacement of the die are obtained via two-factor, replicated-pair, tests. These experimental results can be used to verify computer-simulations and provide the necessary information for improving simulations. Also, these results are useful for calibrating, or "back calculating", various input parameters associated with computer-simulations. Lastly, the paper presents a quantitative comparison between the experimental results and a computer-code simulation of the forging force and internal radial displacements.
INTRODUCTION
NUMEROUS investigators have qualitatively predicted force--displacement behaviors, cross-sectional shapes, and grain-size and internal flaw distributions at various stages of the forging process [1-3]. However, quantitative comparisons at explicit workpiece locations are relatively scarce (see review in ref. [4]) and such comparisons yield a lower order of agreement between the predicted and experimental results [4]. Some of the disagreement is due to the inaccuracy (either formulation or input parameters) in the computer-model. Another part of the disagreement is due to statistical variations in the experimental data. Hence the "goodness" of a computer-model can only be determined by quantitative comparison with experimental values which account for variations within and between the samples. However, the authors found no statistical considerations in the investigations reviewed in refs [1-4]. This paper presents experimental data with statistically quantitative (mean, standard deviation and coefficient-of-variation) measures of external forces, external configuration, and internal radial displacements induced during an upset test. This data is useful for evaluating forging simulations since the coefficient-of-variation (100% times the standard deviation/mean value) associated with the data is on the order of only a few per cent. The maximum experimental uncertainty is usually estimated as three times the standard deviation ("3-sigma"), and for engineering purposes, ---5% is considered as highly reproducible. Hence, this paper concentrates on the reproducibility of experimental data measured in an isothermal upset test. Also, the variance F-test (ratio of variances ~arge*t/~ma.est) and Student's t-test (difference in mean-values/[~(1/Nl + l/N2)] 1/2 , ~ = {Nl*~+ N2*~}/ {Nl + N2-2} ) are used to analyse the variation. There are three types of variation: (a) among specimens containing tracer particles, (b) between specimens with and without tracer particles, and (c) between tracer-particles located on different axes within the same sample. All statistical testing is done at the 0.01 level-of-significance. Hence, the probability of rejecting a true hypothesis is one out of 100. The two basic hypotheses are: (1) there is a difference of at least 1.7% between the mean values of any two sets of data, and (2) the standard deviations of the two sets are the same. Hence, if the two sets of data fail the first hypothesis, there is a 99% chance that the reproducibility of the data is within i
*Mechanical Engineering Department, ViUanova University, Villanova, PA 19085, U.S.A. 193
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J.N. MAJERUS et al.
the desired range of 5.1% for 3-sigma. All test values of the variance ratio F and Student's t were taken from ref. [5]. Since these quantitative data should be useful to other investigators, tables containing all the statistical data are given in the Appendix. These tables of data can be used for either calibrating, verify, or improving quantitative computer simulations. Although there are a variety of computer programs that can be used for forging simulations, e.g. ALPID [3], DYNA2D, HEMP, EPIC and other hydrodynamic codes ([6] and [7]), the authors selected ALPID (Version 2.1) for a quantitative comparison. METHODOLOGY USED IN UPSET TESTS
The testing methodology involves three aspects: (a) the basic approach, (b) the usage of internal tracer-particles, and (c) the isothermal upset tests. Each of these three aspects are discussed in the next three sections.
Basic approach The statistical variations, both within and between various forged workpieces, are measured via the usage of "paired conditions" and replicated experiments [5]. Obviously, this entails a large amount of experimental testing and measurements for any forging experiment. Hence, in order to reduce the cost of the forging experiments, one would like to minimize both the required forging force and the cost associated with controlling the temperatures for hot isothermal tests. In order to accomplish this, the authors selected a modelling material to be used for all of the experiments. Although clay and plasticine T M have been widely used [2,3], the authors believed that a metal would have more realistic physical, thermal and mechanical properties. Hence a eutectic tin (63%)-lead(37%) alloy was selected. At room temperature, this alloy is above the recrystallization temperature [8] and hence, a room-temperature test approximates a hot isothermal condition. Also, this material has a well defined solid-liquid transition. This is advantageous since any localized melting induced during a forging process could be detected via cross-sectional micrographic examinations. Old handbook data [9] indicates that this material has a "yield stress" in shear around 40 MPa (at an unspecified strain-rate), and hence forgings can utilize low-tonnage equipment. Also, in order to both accurately measure the axial force and control the velocity of the upper-die, the authors selected a universal testing machine to drive the upper-die or platten. Because proper material characterization is required for any code-simulation, a separate set of tests was conducted to determine a constitutive equation for the tin-lead eutectic modeling material. Also, attempts were made to measure the statistical uncertainties associated with the resultant constitutive equation. The testing procedure and results are given in refs [10-12]. Overall, it appears that the material is an "elastic"/viscoplastic type [13] whose strain-hardening and "apparent" elastic limit are quite sensitive to strain-rate. The viscoplastic behavior in axial compression is given by an equation of the form or = B + Bl*ln(~/t~o) + [B2 + Ba*~/~o)*e, where B, B~, B2 and B3 are material constants, e = effective strain, ~ = effective strain-rate, and eo = nondimensionalizing unit of one second. The elastic strain-limit appears to be very small (0.04--0.08% [12]) and hence elastic recovery should have a negligible influence on the measured results. Because of the strain-rate sensitivity, the upset tests were conducted using a nominal strain rate (1.7% per s) for which the above viscoplastic equation could be reasonably approximated by the standard "power-law" constitutive equation.
Usage of tracer particles Even though properly etched cross-sections can show flow lines, quantitative measurements of internal displacements cannot be obtained. Quantitative internal flow could be measured by means of internal "tracer particles" which are imbedded at discrete locations prior to casting of a sample. Unfortunately, cavity-free samples require a very expensive and time-consuming process.
Error-assessment
195
Therefore, rather than casting special workpieces, the authors chose to modify a basic workpiece. The basic workpiece was chosen to be a longitudinal section taken from a single bar. This circular bar was extruded to a diameter of 25.4 mm (with a maximum variation of -+0.03 mm along the length). Note that this extrusion causes the equilibrium eutectic microstructure to be replaced by a metastable fine-grain microstructure. This workpiece was modified by drilling a series of axial holes using a twist drill whose diameter is only 0.42 mm. Note that the hole diameter corresponds to only 3.3% of the bar radius, and the cross-sectional area of four holes is only 0.11% of the bar's cross-sectional area. The axial holes were precisely located using a CNC machine with a precision of 0.02 mm. After drilling through the workpiece, the hole locations on the bottom side were measured. When extreme care was taken, the bottom-hole radial distances match within -+ 0.03 mm of the top distances and the corresponding workpiece was accepted. These holes were then filled with a pure-copper (99%) powder of approximately uniform size (50-100 micron in "diameter"). The filling process involved consecutively packing a small amount of powder into the hole using a drill-rod until the hole was filled. This vertical column of powder now becomes the "tracer particles". Note that previous investigators [14] obtained quantitative internal deformations by using wires in the holes, and transversely slicing the samples after forging to observe the ends of the wires. Isothermal upset tests
Using the eutectic alloy, five samples were subjected to upset testing at room temperature under the same axial displacement rate (0.212 mm/s). All samples were to be compressed to the same 60% reduction in height. Each sample was a cylinder with a diameter of 25.40 mm, with a maximum variation of +0.006 mm along the length. The ends of each cylinder were polished to insure parallel surfaces, and the lengths ranged from 12.70 to 12.80 mm among the five samples. Three of the five samples contained an axial array of the tracer particles, and are denoted as TPA, TPB and TPC. Two of the samples (D and E) contained no tracer particles. The axial array consisted of five different locations: one at the center, and four located circumferential at qb = 0, ~r/2, qr, and 3~r/2, respectively. These circumferential locations had the same radial distance of 10.16 mm to the center-line of the tracer particles. Tables 1 and 2 lists the dimensions measured for each sample after the upset test. Although the displacement of the upper die was fixed for 60% reduction in height for all samples, excess lubricant must first be "squeezed out" prior to setting the limitswitch in the testing machine. Failure to do this caused sample TBA to have a reduction in height of only 53%. Height measurements are the average of six readings taken at different radial locations. The maximum difference was only -+0.03 mm. This uniformity in the workpiece height confirms the parallelism of the upper and lower plattens. Note that the out-of-roundness of the top and bottom surfaces after upsetting is very small, with the largest coefficient of variation being less than 0.25%.
TABLE 1. DIMENSIONS
Sample TPA TPB TPC
D E
Dia.* at AA
Dia. at BB
Heightt
Tracer particles
37.49 41.17 40.54 40.59 39.98
37.47 40.51 40.51 40.64 39.88
5.97 (53.0%) 5.11 (59.8%) 5.18 (59.2%) 5.06 (60.2%) 5.26 (58.6%)
Yes
*Measurements taken at mid-height. tPercentages are reduction-in-height. gT# 31:2-0
(mm) MEASURED AFTER UPSET TESTING
Yes Yes
No No
J.N. MAJERUSet al.
196
TABLE 2. MEAN RADIUS* (R), STANDARD DEVIATIONt SD AND COEFFICIENT-OF-VARIATIONC . O . V . ~ OF THE TOP AND Bo'vrOM SURFACES AFTER UPSETTING
Sample TPA TPC D E
Top (R)
Top SD
C.O.V.
Bottom (R)
Bottom SD
C.O.V.
18.47 19.38 20.00 19.69
0.025 0.025 0.035 0.014
0.14 0.13 0.18 0.07
18.51 20.10 20.02 19.68
0.021 0.046 0.014 0.035
0.11 0.23 0.07 0.18
*All values are in mm. tEstimated using [Y(Ri ~Per cent.
(R))2/(N
-
1)] '/2,
N = 6-8.
The diameters associated with two different axis are also listed in Table 1. A x i s - A A corresponds to a diametrical line t h r o u g h tracer particles located at y = 0 and ~r, whereas axis BB is the o r t h o g o n a l line associated with y = ~r/2 and 3~r/2. The orientation of the axes was always the same with respect to the u p p e r platten. A t a reduction in height of about 60%, the difference in m e a n diameter between the "with and without tracer particles" samples is only 0.41 m m and the Student t-test shows that this difference is not significant. Internal flow lines associated with diametrical cross-sections taken from sample T P A are shown in Fig. 1. N o t e that the center section of the sample is not shown in the micrographs in this figure. The micrographs in Fig. l(a) are along a diametral axis oriented at 45 ° with respect to A x i s - A A and contained no tracer particles. Although
the flow lines do not appear at the exact same radial locations to the right and left of the centerline, the radial development of the flow-line contours and spacings is symmetrical about the centerline. The microgrpahs in Fig. l(b) are along A x i s - A A containing the tracer particles. Micrographic examination of the flow lines in Fig. l(b) shows some CENTERLINE
Stile
(al °m
5 . 0 8 mm
(b)
FIG. 1. Micrographs showing flow-lines along diametral axes with (b) and without (a) tracer particles after workpiece was reduced in height by 53% (etchent: 2 ml HCL + 100 ml alcohol).
Error-assessment
197
disturbance adjacent to the hole containing the tracer particles. However, the hole containing the tracer particles has a contour quite close to the flow line at a similar radial location in Fig. l(a). Furthermore, the radial development of the flow-line contours and spacing appears to be the same along both diametral axes. Micrograph comparisons were also made of sections along diametral Axis-AA of samples with about 60% reduction in height (TPB and TPC versus D and E). Although there was some difference in asymmetry about mid-height, the radial development of the flow-line contours and spacings was similar in all cross-sections. These micrographic observations imply that the measured radial displacements of the cavity containing the tracer particles is a good measure of the internal radial displacements of the modeling material. The statistical values (mean or average, standard deviation and coefficient-ofvariation) associated with all five samples are presented in tables in the Appendix. The upset-force data (Table A1) indicates that the reproducibility between samples is excellent, with a three-sigma coefficient-of-variation ranging from 0.9 to 5.4%. These tests showed no statistical difference between either the mean values or the standard deviations for the samples with and without tracer particles. Quantitative data was obtained from enlarged photographs of the micrographical cross-sections. The upset height of each sample was divided into ten equal intervals. The vertical and radial uncertainty is estimated to be about ±0.05 mm. The radial location of the centerline of the tracer particles was determined at each interval of height. This determination used the average value of the measured distances to the outer and inner edge of the hole. This technique compensates for any skewing of the cross-section. Tables A2 and A3 present the quantitative data associated with the tracer particles.
QUANTITATIVE COMPARISON WITH A COMPUTER SIMULATION
The computer-code studied in this comparison is the ALPID program which was developed for the U.S. AFWAF by Altan and Oh. This program is used by a number of industries and a commercial version is available from Structural Dynamics Research Corp. and runs under their IDEAS T M package. However, all simulations done by the authors utilized the stand-alone program (Version 2.1), and hence certain special features were not available. The preliminary computer model used an initial square-grid with a spacing of 1.27 mm, i.e. 100 elements to represent one-half of the workpiece. In order to estimate the influence of element size upon the results, a second set of calculations was done using a 200-element model. In this model, twice as many elements were taken in the vertical direction. The model used a standard ALPID "power law" constitutive-equation, i.e. ~r = crf + A ~'~ e n, where o- = effective stress, e = effective strain, ~ = effective strain-rate, crf = the flow stress, and A, m, n are the constitutive parameters. Reference [12] contains the explicit derivation of the experimental parameters for several tin-lead eutectic alloys. Reference [12] shows that the alloy used in the current tests can either soften or harden with respect to strain-rate, and that the standard "power form" does not fit the data with a high degree of accuracy. The best fit for axial compression is achieved using an equation of the form tr = B + Bl,ln(~/~o) + [B2 + B3*~/~o]*C. Since our version of ALPID would not permit this constitutive form, we used a regression analysis to obtain the best "power form" parameters for a fixed strain-rate of 1.7%/s. This rate is the nominal value predicted by ALPID within the material during the upset test. The best "power form" parameters were m = 0.00, n = 1.00, A(~) = 4.07 MPa, crf(~) = 38.6 MPa. This form, when applied to our data, has some statistical uncertainty associated with it [12]. Assuming that the various sources of uncertainty are independent, the propagation-of-error [5] dictates that the total 3-sigma uncertainty is ± 18.0%. Assuming that the predicted forging force is linearly related to the effective stress, the estimated error on the ALPID force predictions corresponds to this total uncertainty.
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J.N. MAJEXUS et al.
Figure 2 shows the predicted force compared with the highest and lowest sets of experimental data. In order to predict deformation, the code needs the constant shear-factor friction (interface friction factor usually denoted as m) f associated with a particular die/lubricant/workpiece material. There are no tabulated values of f for the tin-lead euctectic material, and computer-simulations could be applied to ring test data to "back calculate" some average value of f. However, as stated in ref. [13], "the difference in flow behaviour is attributed to the estimated value of the die/workpiece friction used to generate the ALPID simulations". Therefore, to minimize uncertainties in the simulation due to f values, the authors used the mean radius for the top and bottom surfaces to "back calculate" the corresponding values of f. Hence, the friction factors f on the top and bottom platten were varied until the predicted top and bottom radius matched (---0.02 mm) the experimental values. The top and bottom f values were both 0.12 for sample TPA and the corresponding internal radial displacements are compared in Fig. 3. Similar reductions gave ftop = 0.12, fbottom = 0.11 for sample TPB, whereas ftop = 0.13 and fbottom = 0.11 for sample TPC. Figure 4 shows the corresponding predicted internal radial displacements compared with the data for sample TPB. s0000
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FIo. 3. Comparison of internal radial displacements for sample TPA (53% reduction-in-height) with ALPID
predictions, with the -*3-sigma error-bar for total uncertainty (-.0.47 mm) of the experimental data.
Error-assessment
199
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RADIAL LOCATION (ram) OF LINE INITIALLY AT 10.18 m m
F[~. 4. Comparison of internal radial displacements for sample TPB (60% reduction-in-height) with ALPID predictions, with the ±3-sigma error-bar for total uncertainty (±0.56 mm) of the experimental data.
The total uncertainty shown in Fig. 3 is associated with three uncertainties of the radial deformation: (1) uncertainty in the pairs of data (+-0.44 ram), (2) uncertainty in the initial location (---0.08 mm), and (3) uncertainty in the cross-sectional measurement (+-0.15 mm). The propagation-of-error yields a total 3-sigma uncertainty of ---0.47 mm. Similarly, uncertainty in the pairs of data (-+0.53 mm) for sample TPB yields a total 3-sigma uncertainty of +-0.56 mm shown in Fig. 4. The ALPID predictions shown in both figures are for the 200-element model. The predicted radial flow for the 200-element model was slightly larger than that of the 100-element model, with the average increase being less than 0.03 ram. This indicates that the modeling error due to the selected grid is quite small. SUMMARY
Based upon upset test results from five samples, it appears that the methodology yields quantitative data that are quite reproducible. The forging forces exhibited very small coefficients of variation (the corresponding 3-sigma ranges from 0.9 to 5%) at reductions in height greater than about 8%. The external shape of the upset samples showed some circumferential difference in the diameters, but the per cent differences were quite small (maximum difference was 1.6%). Furthermore, the difference in diameter showed no consistent pattern with respect to the circumferential direction. Also random radial asymmetry of internal deformations (coefficients of variation ranged from 0.2 to 1.7% depending upon axial location and test) was also observed. Hence, it was concluded that more meaningful internal deformations are obtained by averaging over orthogonal pairs, i.e. four radial locations of the tracer particles. The statistical tests (variance F and Student's t) showed that the majority of the data passed the hypothesis on the standard deviation. Other than a few comparisons at a specific height, most data failed the hypothesis on the mean values. Hence, the tracer particles allow one to accurately (i.e. coefficients of variation less than 1.7%) determine the internal radial displacements for the modeling material. The tracer particles also permit the calculation of mean strains within certain radial regions of the workpiece. At a 60% reduction-in-height (samples TPB, TPC), the average values of both the tracer-particle location (15.54 mm) and the outside radius (20.37 mm) were measured at mid-height. Hence, at mid-height, the inner radial region (initially 64% of the volume) experienced a mean radial strain of 53% in tension. Conversely, the outer radial region displayed a mean radial strain of 48% in compression.
200
J.N. MAJERUSet al.
Comparison of the ALPID simulations with the test data showed that all trends were correctly predicted. The asymmetries of deformation could be simulated by changing the die friction factor m by 10-20%. The predicted forging-force exhibited the observed nonlinear behavior with respect to axial displacement. Also, the nonlinear internal deformation behavior was predicted. Quantitatively, the ALPID-predicted forging force was consistently higher than the average data, with the difference ranging from 5.8 to 15%. At 53% reduction-in-height, the predicted radial deformation was consistently lower than the average data, with the difference ranging from 2.7% to 4.9%. Conversely, at about 60% reduction-in-height, the predicted deformation was higher than the average data in 9 out of 11 locations. The difference between the predicted and experimental results ranged from -0.51 to 4.51%, with the average being 2.56%. The authors believe that these force and deformation accuracies are well within the desired range for engineering design. However, there appear to be some consistent bias in the predicted behavior. Considering that the statistical uncertainties are both positive and negative, one would expect the experimental data to oscillate in some fashion around the predicted values. Consequently, the predicted behavior should oscillate between the bounds of the total statistical uncertainty. However, the predicted values are consistently close to one of the bounds of each set of data. Furthermore, there are explicit differences in the predicted trends. The predicted forging force increases at a faster rate with axial deformation than the experimental data. Furthermore, even though the top and bottom radial deformations were matched via "back calculating" the values of m of each sample, the internal displacements were predicted to have more curvature than was observed with respect to the axial height. Also the internal displacements were under predicted at the 53% reduction in height, whereas they were over predicted at the higher 60% reduction in height. Acknowledgements--This work is part of a study conducted under grant No. MSM-87-13554 from the National Science Foundation. We wish to thank the U.S. AFWAL for granting us permission to use the ALPID computer program and the Structural Dynamics Research Corp. for providing us with the object-code for Version 2.1 of the program. We also thank the DuPont Corp. for an educational grant which supplied some of the supplemental equipment. REFERENCES [1] N. REBELO and S. KOBAYASm,Int. J. Mech. Sci. 22, 699-705, 707-718 (1980). [2] T. G. BYRER(Ed.), Forging Handbook, Sections 2 and 3. American Soc. of Metals, Metals Park, Ohio (1985). [3] T. ALTAN, S. OH and H. GEGEL, Metal bbrming: Fundamentals and Applications, Chapters 12-14 and 20. American Soc. of Metals, Metals Park, Ohio (1983). [4] H. L. GEGEL,J. C. MALAS, S. M. DORAIVELUand V. A. SHENDE,Computer-aided process design for bulk forming. In Metals Handbook, 9th edn, Vol. 14. American Soc. of Metals, Metals Park, Ohio (1988). [5] J. MANDEL, The Statistical Analysis of Experimental Data. Dover Publications, New York (1984). [6] G. L. GOUDREAUand J. O. HALLQUIST,Recent developments in large scale finite element Lagrangian hydrocode technology, J. Comput. Meth. appl. Mech. Engng, 33, 725-757 (1982). [7] P. C. CHou and L. Wu, A dynamic relaxation finite element method for metal forming processes, Int. J. Mech. Sci. 28, 231-250 (1986). [8] W. B. HAMPSHIRE,Metals Handbook, 9th edn, Vol. 4, p. 740. ASM, Metals Park, Ohio (1981). [9] W. R. LEWIS, Notes on Soldering. Tin Research Institute, Ohio (1948). [10] J. N. MAJERUS,K. P. JEN and Y. P. Gu, Quantitative data for assessing the accuracy of computersimulations to forging processes, Proc. ASM's Conf. on Near Net Shape Manufacturing, pp. 147-157. ASM International (1988). [11] K. P. JEN, J. N. MAJERUSand Y. P. Gu, A video technique for obtaining corrected true-compressive stress, SEM's Exp. Tech., 13, 11-14 (1989). [12] K. P. JEN, J. N. MAJERUSand Y. P. Gu, True stress-strain equations for eutectic tin-lead solders, submitted to Exp. Mech. [13] N. CRISTESCU,Dynamic Plasticity, Chapter 10. North-Holland, Amsterdam (1967). [14] V. K. JAIN, R. SRINIVASAN,L. E. WATSONand H. L. GEGEL, Determination of strain in large plastic deformation. Proc. ll6th TMS Annual Meeting, Denver, Colorado, The Metallurgical Society (1987).
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APPENDIX
Tables of statistical data obtained from three samples (TPA, TPB and TPC) containing tracer particles and two samples (D and E) without tracer particles TABLE A1.
STATISTICAL FORCE DATA ASSOCIATED WITH THE FIVE UPSET TESTS
Axial disp.
Mean forcet
Standard deviationS;
(ram)
(N)
(N)
1.27 2.03 2.79 3.56 4,32 5.08 5.84 6.35
With TP*
No TP
With TP*
No TP
20760 22360 24020 26150 28540 31460 35080 37990
21130 22620 24400 26480 28810 31800 35500 38450
160 100 210 250 330 420 560 670
60 90 90 140 210 320 380 470
Coef. of variation (%) With TP* No TP 0.8 0.5 0.9 1.0 1.2 1.4 1.6 1.8
0.3 0.4 0.4 0.5 0.7 1.0 1.1 1.2
*TP refers to the tracer particles. tUncertainty of load reading is -+45 N. :~Estimated using [~(F~ - (F))21(N - 1)] 1/2, N = 3 for TP, N = 2 for No TP.
TABLE A2.
STATISTICAL DATA FOR THE CENTERLINE LOCATION OF TRACER PARTICLES ASSOCIATED WITH SAMPLES TPA, TPB, AND T P C AT MAXIMUM REDUCTION IN HEIGHT*
Axial location (mm) TPA TPB TPC
0.60 1.19 1.79 2.39 2,99 3.58 4.18 4.78 5.37 5.97
Bottom 0.51 1.02 1.52 2.03 2.54 3.07 3.58 4.09 4.60 5.11
0.51 1.04 1.53 2.08 2.59 3.10 3.63 4.14 4.67 5.18
Mean radial location (ram) TPA TPB TPC
Standard dev.t (ram) TPA TPB TPC
14.96 15.02 15.11 15.20 15.28 15.35 15.31 15.23 15.15 14.99 14.84
0.05 0.06 0.09 0.12 0.15 0.16 0.19 0.22 0.20 0.20 0.19
15.77 15.80 15.80 15.85 15.83 15.76 15.58 15.32 15.07 14.92 14.82
16.68 16.71 16.68 16.61 16.52 16.29 15.90 15.49 15.22 14.99 14.84
0.27 0.27 0.27 0.20 0.19 0.15 0.25 0.22 0.03 0.04 0.02
0.04 0.08 0.09 0.19 0.22 0.17 0.07 0.07 0.15 0.13 0.13
Coef. of variation (%) TPA TPB TPC 0.3 0.4 0.6 0.8 0.8 1.0 1.2 1.4 1.3 1.3 1.3
1.7 1.0 1.0 1.3 1.2 1.0 1.6 1.4 0.4 0.3 0.2
0.2 0.4 0.5 1.1 1.2 1.0 0.4 0.4 1.0 0.9 0.9
*See Table 1 for specific values. tEstimated using [~(Ri - (R))2/(N - 1)] 1'2, N = 6 or 8 radial readings.
TABLE A3.
MEAN RADIAL LOCATION* AND STANDARD DEVIATIONSt S D FOR TRACER PARTICLES ON AXES AND B B IN SAMPLES TPA AND T P B AT MAXIMUM REDUCTION IN HEIGHT~
AA
Axial location TPA TPB
Axis A A TPA TPB
Axis A A SD TPA TPB
Axis BB (R) TPA TPB
Axis BB SD TPA TPB
Bottom 0.60 0.51 1.19 1.02 1.79 1.52 2.39 2.03 2.99 2.54 3.58 3.07 4.18 3.58 4.78 4.09 5.37 4.60 5.97 5.11
14.99 15.05 15.15 15.29 15.39 15.47 15.44 15.39 15.30 15.11 14.95
0.07 0.09 0.13 0.11 0.11 0.11 0.11 0.07 0.02 0.07 0.16
14.94 15.00 15.06 15.13 15.16 15.24 15.18 15.06 14.99 14.86 14.73
0.00 0.02 0.04 0.05 0.07 0.11 0.16 0.18 0.14 0.22 0.18
15.55 15.57 15.57 15.67 15.67 15.61 15.41 15.20 15.02 14.92 14.85
0.07 0.07 0.07 0.04 0.00 0.02 0.02 0.02 0.02 0.02 0.02
*All dimensions are in mm. tEstimated using [E(Ri - (R))2/(N - I)] '/2, N = 4 radial readings. ~tSee Table 1 for the specific values.
16,00 16.03 16,03 16,02 15,99 15,91 15,75 15.43 15.11 14.92 14.83
0.04 0.04 0.04 0.02 0.05 0.16 0.29 0.31 0.18 0.09 0.04
Vol. 31, No. 2, pp.193-201, 1991.
089(I-6955/9153.00 + .00 Pergamon Press plc
QUANTITATIVE ERROR-ASSESSMENT BETWEEN UPSETTEST DATA AND A COMPUTER-CODE SIMULATION J. N. M_mE~trs,* K. P. JErk* and Y. P. Go* (Received 29 May 1990; in final form 15 August 1990) Abstract--A quantitative analysis of axial-force, internal radial displacement and workpiece contour histories
for upset-tests are presented in this paper. Discrete internal deformations are measured by using the technique of "tracer particles". Vertical arrays of tracer particles, aligned in the longitudinal direction of the upset cylinders, permit internal radial deformations to be both visualized and measured as a function of axial diedisplacement. Statistically quantitative axial-force, radial-contour and internal radial deformations vs axial displacement of the die are obtained via two-factor, replicated-pair, tests. These experimental results can be used to verify computer-simulations and provide the necessary information for improving simulations. Also, these results are useful for calibrating, or "back calculating", various input parameters associated with computer-simulations. Lastly, the paper presents a quantitative comparison between the experimental results and a computer-code simulation of the forging force and internal radial displacements.
INTRODUCTION
NUMEROUS investigators have qualitatively predicted force--displacement behaviors, cross-sectional shapes, and grain-size and internal flaw distributions at various stages of the forging process [1-3]. However, quantitative comparisons at explicit workpiece locations are relatively scarce (see review in ref. [4]) and such comparisons yield a lower order of agreement between the predicted and experimental results [4]. Some of the disagreement is due to the inaccuracy (either formulation or input parameters) in the computer-model. Another part of the disagreement is due to statistical variations in the experimental data. Hence the "goodness" of a computer-model can only be determined by quantitative comparison with experimental values which account for variations within and between the samples. However, the authors found no statistical considerations in the investigations reviewed in refs [1-4]. This paper presents experimental data with statistically quantitative (mean, standard deviation and coefficient-of-variation) measures of external forces, external configuration, and internal radial displacements induced during an upset test. This data is useful for evaluating forging simulations since the coefficient-of-variation (100% times the standard deviation/mean value) associated with the data is on the order of only a few per cent. The maximum experimental uncertainty is usually estimated as three times the standard deviation ("3-sigma"), and for engineering purposes, ---5% is considered as highly reproducible. Hence, this paper concentrates on the reproducibility of experimental data measured in an isothermal upset test. Also, the variance F-test (ratio of variances ~arge*t/~ma.est) and Student's t-test (difference in mean-values/[~(1/Nl + l/N2)] 1/2 , ~ = {Nl*~+ N2*~}/ {Nl + N2-2} ) are used to analyse the variation. There are three types of variation: (a) among specimens containing tracer particles, (b) between specimens with and without tracer particles, and (c) between tracer-particles located on different axes within the same sample. All statistical testing is done at the 0.01 level-of-significance. Hence, the probability of rejecting a true hypothesis is one out of 100. The two basic hypotheses are: (1) there is a difference of at least 1.7% between the mean values of any two sets of data, and (2) the standard deviations of the two sets are the same. Hence, if the two sets of data fail the first hypothesis, there is a 99% chance that the reproducibility of the data is within i
*Mechanical Engineering Department, ViUanova University, Villanova, PA 19085, U.S.A. 193
194
J.N. MAJERUS et al.
the desired range of 5.1% for 3-sigma. All test values of the variance ratio F and Student's t were taken from ref. [5]. Since these quantitative data should be useful to other investigators, tables containing all the statistical data are given in the Appendix. These tables of data can be used for either calibrating, verify, or improving quantitative computer simulations. Although there are a variety of computer programs that can be used for forging simulations, e.g. ALPID [3], DYNA2D, HEMP, EPIC and other hydrodynamic codes ([6] and [7]), the authors selected ALPID (Version 2.1) for a quantitative comparison. METHODOLOGY USED IN UPSET TESTS
The testing methodology involves three aspects: (a) the basic approach, (b) the usage of internal tracer-particles, and (c) the isothermal upset tests. Each of these three aspects are discussed in the next three sections.
Basic approach The statistical variations, both within and between various forged workpieces, are measured via the usage of "paired conditions" and replicated experiments [5]. Obviously, this entails a large amount of experimental testing and measurements for any forging experiment. Hence, in order to reduce the cost of the forging experiments, one would like to minimize both the required forging force and the cost associated with controlling the temperatures for hot isothermal tests. In order to accomplish this, the authors selected a modelling material to be used for all of the experiments. Although clay and plasticine T M have been widely used [2,3], the authors believed that a metal would have more realistic physical, thermal and mechanical properties. Hence a eutectic tin (63%)-lead(37%) alloy was selected. At room temperature, this alloy is above the recrystallization temperature [8] and hence, a room-temperature test approximates a hot isothermal condition. Also, this material has a well defined solid-liquid transition. This is advantageous since any localized melting induced during a forging process could be detected via cross-sectional micrographic examinations. Old handbook data [9] indicates that this material has a "yield stress" in shear around 40 MPa (at an unspecified strain-rate), and hence forgings can utilize low-tonnage equipment. Also, in order to both accurately measure the axial force and control the velocity of the upper-die, the authors selected a universal testing machine to drive the upper-die or platten. Because proper material characterization is required for any code-simulation, a separate set of tests was conducted to determine a constitutive equation for the tin-lead eutectic modeling material. Also, attempts were made to measure the statistical uncertainties associated with the resultant constitutive equation. The testing procedure and results are given in refs [10-12]. Overall, it appears that the material is an "elastic"/viscoplastic type [13] whose strain-hardening and "apparent" elastic limit are quite sensitive to strain-rate. The viscoplastic behavior in axial compression is given by an equation of the form or = B + Bl*ln(~/t~o) + [B2 + Ba*~/~o)*e, where B, B~, B2 and B3 are material constants, e = effective strain, ~ = effective strain-rate, and eo = nondimensionalizing unit of one second. The elastic strain-limit appears to be very small (0.04--0.08% [12]) and hence elastic recovery should have a negligible influence on the measured results. Because of the strain-rate sensitivity, the upset tests were conducted using a nominal strain rate (1.7% per s) for which the above viscoplastic equation could be reasonably approximated by the standard "power-law" constitutive equation.
Usage of tracer particles Even though properly etched cross-sections can show flow lines, quantitative measurements of internal displacements cannot be obtained. Quantitative internal flow could be measured by means of internal "tracer particles" which are imbedded at discrete locations prior to casting of a sample. Unfortunately, cavity-free samples require a very expensive and time-consuming process.
Error-assessment
195
Therefore, rather than casting special workpieces, the authors chose to modify a basic workpiece. The basic workpiece was chosen to be a longitudinal section taken from a single bar. This circular bar was extruded to a diameter of 25.4 mm (with a maximum variation of -+0.03 mm along the length). Note that this extrusion causes the equilibrium eutectic microstructure to be replaced by a metastable fine-grain microstructure. This workpiece was modified by drilling a series of axial holes using a twist drill whose diameter is only 0.42 mm. Note that the hole diameter corresponds to only 3.3% of the bar radius, and the cross-sectional area of four holes is only 0.11% of the bar's cross-sectional area. The axial holes were precisely located using a CNC machine with a precision of 0.02 mm. After drilling through the workpiece, the hole locations on the bottom side were measured. When extreme care was taken, the bottom-hole radial distances match within -+ 0.03 mm of the top distances and the corresponding workpiece was accepted. These holes were then filled with a pure-copper (99%) powder of approximately uniform size (50-100 micron in "diameter"). The filling process involved consecutively packing a small amount of powder into the hole using a drill-rod until the hole was filled. This vertical column of powder now becomes the "tracer particles". Note that previous investigators [14] obtained quantitative internal deformations by using wires in the holes, and transversely slicing the samples after forging to observe the ends of the wires. Isothermal upset tests
Using the eutectic alloy, five samples were subjected to upset testing at room temperature under the same axial displacement rate (0.212 mm/s). All samples were to be compressed to the same 60% reduction in height. Each sample was a cylinder with a diameter of 25.40 mm, with a maximum variation of +0.006 mm along the length. The ends of each cylinder were polished to insure parallel surfaces, and the lengths ranged from 12.70 to 12.80 mm among the five samples. Three of the five samples contained an axial array of the tracer particles, and are denoted as TPA, TPB and TPC. Two of the samples (D and E) contained no tracer particles. The axial array consisted of five different locations: one at the center, and four located circumferential at qb = 0, ~r/2, qr, and 3~r/2, respectively. These circumferential locations had the same radial distance of 10.16 mm to the center-line of the tracer particles. Tables 1 and 2 lists the dimensions measured for each sample after the upset test. Although the displacement of the upper die was fixed for 60% reduction in height for all samples, excess lubricant must first be "squeezed out" prior to setting the limitswitch in the testing machine. Failure to do this caused sample TBA to have a reduction in height of only 53%. Height measurements are the average of six readings taken at different radial locations. The maximum difference was only -+0.03 mm. This uniformity in the workpiece height confirms the parallelism of the upper and lower plattens. Note that the out-of-roundness of the top and bottom surfaces after upsetting is very small, with the largest coefficient of variation being less than 0.25%.
TABLE 1. DIMENSIONS
Sample TPA TPB TPC
D E
Dia.* at AA
Dia. at BB
Heightt
Tracer particles
37.49 41.17 40.54 40.59 39.98
37.47 40.51 40.51 40.64 39.88
5.97 (53.0%) 5.11 (59.8%) 5.18 (59.2%) 5.06 (60.2%) 5.26 (58.6%)
Yes
*Measurements taken at mid-height. tPercentages are reduction-in-height. gT# 31:2-0
(mm) MEASURED AFTER UPSET TESTING
Yes Yes
No No
J.N. MAJERUSet al.
196
TABLE 2. MEAN RADIUS* (R), STANDARD DEVIATIONt SD AND COEFFICIENT-OF-VARIATIONC . O . V . ~ OF THE TOP AND Bo'vrOM SURFACES AFTER UPSETTING
Sample TPA TPC D E
Top (R)
Top SD
C.O.V.
Bottom (R)
Bottom SD
C.O.V.
18.47 19.38 20.00 19.69
0.025 0.025 0.035 0.014
0.14 0.13 0.18 0.07
18.51 20.10 20.02 19.68
0.021 0.046 0.014 0.035
0.11 0.23 0.07 0.18
*All values are in mm. tEstimated using [Y(Ri ~Per cent.
(R))2/(N
-
1)] '/2,
N = 6-8.
The diameters associated with two different axis are also listed in Table 1. A x i s - A A corresponds to a diametrical line t h r o u g h tracer particles located at y = 0 and ~r, whereas axis BB is the o r t h o g o n a l line associated with y = ~r/2 and 3~r/2. The orientation of the axes was always the same with respect to the u p p e r platten. A t a reduction in height of about 60%, the difference in m e a n diameter between the "with and without tracer particles" samples is only 0.41 m m and the Student t-test shows that this difference is not significant. Internal flow lines associated with diametrical cross-sections taken from sample T P A are shown in Fig. 1. N o t e that the center section of the sample is not shown in the micrographs in this figure. The micrographs in Fig. l(a) are along a diametral axis oriented at 45 ° with respect to A x i s - A A and contained no tracer particles. Although
the flow lines do not appear at the exact same radial locations to the right and left of the centerline, the radial development of the flow-line contours and spacings is symmetrical about the centerline. The microgrpahs in Fig. l(b) are along A x i s - A A containing the tracer particles. Micrographic examination of the flow lines in Fig. l(b) shows some CENTERLINE
Stile
(al °m
5 . 0 8 mm
(b)
FIG. 1. Micrographs showing flow-lines along diametral axes with (b) and without (a) tracer particles after workpiece was reduced in height by 53% (etchent: 2 ml HCL + 100 ml alcohol).
Error-assessment
197
disturbance adjacent to the hole containing the tracer particles. However, the hole containing the tracer particles has a contour quite close to the flow line at a similar radial location in Fig. l(a). Furthermore, the radial development of the flow-line contours and spacing appears to be the same along both diametral axes. Micrograph comparisons were also made of sections along diametral Axis-AA of samples with about 60% reduction in height (TPB and TPC versus D and E). Although there was some difference in asymmetry about mid-height, the radial development of the flow-line contours and spacings was similar in all cross-sections. These micrographic observations imply that the measured radial displacements of the cavity containing the tracer particles is a good measure of the internal radial displacements of the modeling material. The statistical values (mean or average, standard deviation and coefficient-ofvariation) associated with all five samples are presented in tables in the Appendix. The upset-force data (Table A1) indicates that the reproducibility between samples is excellent, with a three-sigma coefficient-of-variation ranging from 0.9 to 5.4%. These tests showed no statistical difference between either the mean values or the standard deviations for the samples with and without tracer particles. Quantitative data was obtained from enlarged photographs of the micrographical cross-sections. The upset height of each sample was divided into ten equal intervals. The vertical and radial uncertainty is estimated to be about ±0.05 mm. The radial location of the centerline of the tracer particles was determined at each interval of height. This determination used the average value of the measured distances to the outer and inner edge of the hole. This technique compensates for any skewing of the cross-section. Tables A2 and A3 present the quantitative data associated with the tracer particles.
QUANTITATIVE COMPARISON WITH A COMPUTER SIMULATION
The computer-code studied in this comparison is the ALPID program which was developed for the U.S. AFWAF by Altan and Oh. This program is used by a number of industries and a commercial version is available from Structural Dynamics Research Corp. and runs under their IDEAS T M package. However, all simulations done by the authors utilized the stand-alone program (Version 2.1), and hence certain special features were not available. The preliminary computer model used an initial square-grid with a spacing of 1.27 mm, i.e. 100 elements to represent one-half of the workpiece. In order to estimate the influence of element size upon the results, a second set of calculations was done using a 200-element model. In this model, twice as many elements were taken in the vertical direction. The model used a standard ALPID "power law" constitutive-equation, i.e. ~r = crf + A ~'~ e n, where o- = effective stress, e = effective strain, ~ = effective strain-rate, crf = the flow stress, and A, m, n are the constitutive parameters. Reference [12] contains the explicit derivation of the experimental parameters for several tin-lead eutectic alloys. Reference [12] shows that the alloy used in the current tests can either soften or harden with respect to strain-rate, and that the standard "power form" does not fit the data with a high degree of accuracy. The best fit for axial compression is achieved using an equation of the form tr = B + Bl,ln(~/~o) + [B2 + B3*~/~o]*C. Since our version of ALPID would not permit this constitutive form, we used a regression analysis to obtain the best "power form" parameters for a fixed strain-rate of 1.7%/s. This rate is the nominal value predicted by ALPID within the material during the upset test. The best "power form" parameters were m = 0.00, n = 1.00, A(~) = 4.07 MPa, crf(~) = 38.6 MPa. This form, when applied to our data, has some statistical uncertainty associated with it [12]. Assuming that the various sources of uncertainty are independent, the propagation-of-error [5] dictates that the total 3-sigma uncertainty is ± 18.0%. Assuming that the predicted forging force is linearly related to the effective stress, the estimated error on the ALPID force predictions corresponds to this total uncertainty.
198
J.N. MAJEXUS et al.
Figure 2 shows the predicted force compared with the highest and lowest sets of experimental data. In order to predict deformation, the code needs the constant shear-factor friction (interface friction factor usually denoted as m) f associated with a particular die/lubricant/workpiece material. There are no tabulated values of f for the tin-lead euctectic material, and computer-simulations could be applied to ring test data to "back calculate" some average value of f. However, as stated in ref. [13], "the difference in flow behaviour is attributed to the estimated value of the die/workpiece friction used to generate the ALPID simulations". Therefore, to minimize uncertainties in the simulation due to f values, the authors used the mean radius for the top and bottom surfaces to "back calculate" the corresponding values of f. Hence, the friction factors f on the top and bottom platten were varied until the predicted top and bottom radius matched (---0.02 mm) the experimental values. The top and bottom f values were both 0.12 for sample TPA and the corresponding internal radial displacements are compared in Fig. 3. Similar reductions gave ftop = 0.12, fbottom = 0.11 for sample TPB, whereas ftop = 0.13 and fbottom = 0.11 for sample TPC. Figure 4 shows the corresponding predicted internal radial displacements compared with the data for sample TPB. s0000
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FIG. 2. Comparison of experimental forging force with ALPID predictions, with the -+3-sigma ALPID error-
bar based upon total uncertainty (-+18.0%) of the constitutive equation.
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RADIAL LOCATION (ram) OF LiNE I N I T I A L L Y A T 1 0 . 1 e rnm
FIo. 3. Comparison of internal radial displacements for sample TPA (53% reduction-in-height) with ALPID
predictions, with the -*3-sigma error-bar for total uncertainty (-.0.47 mm) of the experimental data.
Error-assessment
199
6,0' top die
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RADIAL LOCATION (ram) OF LINE INITIALLY AT 10.18 m m
F[~. 4. Comparison of internal radial displacements for sample TPB (60% reduction-in-height) with ALPID predictions, with the ±3-sigma error-bar for total uncertainty (±0.56 mm) of the experimental data.
The total uncertainty shown in Fig. 3 is associated with three uncertainties of the radial deformation: (1) uncertainty in the pairs of data (+-0.44 ram), (2) uncertainty in the initial location (---0.08 mm), and (3) uncertainty in the cross-sectional measurement (+-0.15 mm). The propagation-of-error yields a total 3-sigma uncertainty of ---0.47 mm. Similarly, uncertainty in the pairs of data (-+0.53 mm) for sample TPB yields a total 3-sigma uncertainty of +-0.56 mm shown in Fig. 4. The ALPID predictions shown in both figures are for the 200-element model. The predicted radial flow for the 200-element model was slightly larger than that of the 100-element model, with the average increase being less than 0.03 ram. This indicates that the modeling error due to the selected grid is quite small. SUMMARY
Based upon upset test results from five samples, it appears that the methodology yields quantitative data that are quite reproducible. The forging forces exhibited very small coefficients of variation (the corresponding 3-sigma ranges from 0.9 to 5%) at reductions in height greater than about 8%. The external shape of the upset samples showed some circumferential difference in the diameters, but the per cent differences were quite small (maximum difference was 1.6%). Furthermore, the difference in diameter showed no consistent pattern with respect to the circumferential direction. Also random radial asymmetry of internal deformations (coefficients of variation ranged from 0.2 to 1.7% depending upon axial location and test) was also observed. Hence, it was concluded that more meaningful internal deformations are obtained by averaging over orthogonal pairs, i.e. four radial locations of the tracer particles. The statistical tests (variance F and Student's t) showed that the majority of the data passed the hypothesis on the standard deviation. Other than a few comparisons at a specific height, most data failed the hypothesis on the mean values. Hence, the tracer particles allow one to accurately (i.e. coefficients of variation less than 1.7%) determine the internal radial displacements for the modeling material. The tracer particles also permit the calculation of mean strains within certain radial regions of the workpiece. At a 60% reduction-in-height (samples TPB, TPC), the average values of both the tracer-particle location (15.54 mm) and the outside radius (20.37 mm) were measured at mid-height. Hence, at mid-height, the inner radial region (initially 64% of the volume) experienced a mean radial strain of 53% in tension. Conversely, the outer radial region displayed a mean radial strain of 48% in compression.
200
J.N. MAJERUSet al.
Comparison of the ALPID simulations with the test data showed that all trends were correctly predicted. The asymmetries of deformation could be simulated by changing the die friction factor m by 10-20%. The predicted forging-force exhibited the observed nonlinear behavior with respect to axial displacement. Also, the nonlinear internal deformation behavior was predicted. Quantitatively, the ALPID-predicted forging force was consistently higher than the average data, with the difference ranging from 5.8 to 15%. At 53% reduction-in-height, the predicted radial deformation was consistently lower than the average data, with the difference ranging from 2.7% to 4.9%. Conversely, at about 60% reduction-in-height, the predicted deformation was higher than the average data in 9 out of 11 locations. The difference between the predicted and experimental results ranged from -0.51 to 4.51%, with the average being 2.56%. The authors believe that these force and deformation accuracies are well within the desired range for engineering design. However, there appear to be some consistent bias in the predicted behavior. Considering that the statistical uncertainties are both positive and negative, one would expect the experimental data to oscillate in some fashion around the predicted values. Consequently, the predicted behavior should oscillate between the bounds of the total statistical uncertainty. However, the predicted values are consistently close to one of the bounds of each set of data. Furthermore, there are explicit differences in the predicted trends. The predicted forging force increases at a faster rate with axial deformation than the experimental data. Furthermore, even though the top and bottom radial deformations were matched via "back calculating" the values of m of each sample, the internal displacements were predicted to have more curvature than was observed with respect to the axial height. Also the internal displacements were under predicted at the 53% reduction in height, whereas they were over predicted at the higher 60% reduction in height. Acknowledgements--This work is part of a study conducted under grant No. MSM-87-13554 from the National Science Foundation. We wish to thank the U.S. AFWAL for granting us permission to use the ALPID computer program and the Structural Dynamics Research Corp. for providing us with the object-code for Version 2.1 of the program. We also thank the DuPont Corp. for an educational grant which supplied some of the supplemental equipment. REFERENCES [1] N. REBELO and S. KOBAYASm,Int. J. Mech. Sci. 22, 699-705, 707-718 (1980). [2] T. G. BYRER(Ed.), Forging Handbook, Sections 2 and 3. American Soc. of Metals, Metals Park, Ohio (1985). [3] T. ALTAN, S. OH and H. GEGEL, Metal bbrming: Fundamentals and Applications, Chapters 12-14 and 20. American Soc. of Metals, Metals Park, Ohio (1983). [4] H. L. GEGEL,J. C. MALAS, S. M. DORAIVELUand V. A. SHENDE,Computer-aided process design for bulk forming. In Metals Handbook, 9th edn, Vol. 14. American Soc. of Metals, Metals Park, Ohio (1988). [5] J. MANDEL, The Statistical Analysis of Experimental Data. Dover Publications, New York (1984). [6] G. L. GOUDREAUand J. O. HALLQUIST,Recent developments in large scale finite element Lagrangian hydrocode technology, J. Comput. Meth. appl. Mech. Engng, 33, 725-757 (1982). [7] P. C. CHou and L. Wu, A dynamic relaxation finite element method for metal forming processes, Int. J. Mech. Sci. 28, 231-250 (1986). [8] W. B. HAMPSHIRE,Metals Handbook, 9th edn, Vol. 4, p. 740. ASM, Metals Park, Ohio (1981). [9] W. R. LEWIS, Notes on Soldering. Tin Research Institute, Ohio (1948). [10] J. N. MAJERUS,K. P. JEN and Y. P. Gu, Quantitative data for assessing the accuracy of computersimulations to forging processes, Proc. ASM's Conf. on Near Net Shape Manufacturing, pp. 147-157. ASM International (1988). [11] K. P. JEN, J. N. MAJERUSand Y. P. Gu, A video technique for obtaining corrected true-compressive stress, SEM's Exp. Tech., 13, 11-14 (1989). [12] K. P. JEN, J. N. MAJERUSand Y. P. Gu, True stress-strain equations for eutectic tin-lead solders, submitted to Exp. Mech. [13] N. CRISTESCU,Dynamic Plasticity, Chapter 10. North-Holland, Amsterdam (1967). [14] V. K. JAIN, R. SRINIVASAN,L. E. WATSONand H. L. GEGEL, Determination of strain in large plastic deformation. Proc. ll6th TMS Annual Meeting, Denver, Colorado, The Metallurgical Society (1987).
Error-assessment
201
APPENDIX
Tables of statistical data obtained from three samples (TPA, TPB and TPC) containing tracer particles and two samples (D and E) without tracer particles TABLE A1.
STATISTICAL FORCE DATA ASSOCIATED WITH THE FIVE UPSET TESTS
Axial disp.
Mean forcet
Standard deviationS;
(ram)
(N)
(N)
1.27 2.03 2.79 3.56 4,32 5.08 5.84 6.35
With TP*
No TP
With TP*
No TP
20760 22360 24020 26150 28540 31460 35080 37990
21130 22620 24400 26480 28810 31800 35500 38450
160 100 210 250 330 420 560 670
60 90 90 140 210 320 380 470
Coef. of variation (%) With TP* No TP 0.8 0.5 0.9 1.0 1.2 1.4 1.6 1.8
0.3 0.4 0.4 0.5 0.7 1.0 1.1 1.2
*TP refers to the tracer particles. tUncertainty of load reading is -+45 N. :~Estimated using [~(F~ - (F))21(N - 1)] 1/2, N = 3 for TP, N = 2 for No TP.
TABLE A2.
STATISTICAL DATA FOR THE CENTERLINE LOCATION OF TRACER PARTICLES ASSOCIATED WITH SAMPLES TPA, TPB, AND T P C AT MAXIMUM REDUCTION IN HEIGHT*
Axial location (mm) TPA TPB TPC
0.60 1.19 1.79 2.39 2,99 3.58 4.18 4.78 5.37 5.97
Bottom 0.51 1.02 1.52 2.03 2.54 3.07 3.58 4.09 4.60 5.11
0.51 1.04 1.53 2.08 2.59 3.10 3.63 4.14 4.67 5.18
Mean radial location (ram) TPA TPB TPC
Standard dev.t (ram) TPA TPB TPC
14.96 15.02 15.11 15.20 15.28 15.35 15.31 15.23 15.15 14.99 14.84
0.05 0.06 0.09 0.12 0.15 0.16 0.19 0.22 0.20 0.20 0.19
15.77 15.80 15.80 15.85 15.83 15.76 15.58 15.32 15.07 14.92 14.82
16.68 16.71 16.68 16.61 16.52 16.29 15.90 15.49 15.22 14.99 14.84
0.27 0.27 0.27 0.20 0.19 0.15 0.25 0.22 0.03 0.04 0.02
0.04 0.08 0.09 0.19 0.22 0.17 0.07 0.07 0.15 0.13 0.13
Coef. of variation (%) TPA TPB TPC 0.3 0.4 0.6 0.8 0.8 1.0 1.2 1.4 1.3 1.3 1.3
1.7 1.0 1.0 1.3 1.2 1.0 1.6 1.4 0.4 0.3 0.2
0.2 0.4 0.5 1.1 1.2 1.0 0.4 0.4 1.0 0.9 0.9
*See Table 1 for specific values. tEstimated using [~(Ri - (R))2/(N - 1)] 1'2, N = 6 or 8 radial readings.
TABLE A3.
MEAN RADIAL LOCATION*
AA
Axial location TPA TPB
Axis A A
Axis A A SD TPA TPB
Axis BB (R) TPA TPB
Axis BB SD TPA TPB
Bottom 0.60 0.51 1.19 1.02 1.79 1.52 2.39 2.03 2.99 2.54 3.58 3.07 4.18 3.58 4.78 4.09 5.37 4.60 5.97 5.11
14.99 15.05 15.15 15.29 15.39 15.47 15.44 15.39 15.30 15.11 14.95
0.07 0.09 0.13 0.11 0.11 0.11 0.11 0.07 0.02 0.07 0.16
14.94 15.00 15.06 15.13 15.16 15.24 15.18 15.06 14.99 14.86 14.73
0.00 0.02 0.04 0.05 0.07 0.11 0.16 0.18 0.14 0.22 0.18
15.55 15.57 15.57 15.67 15.67 15.61 15.41 15.20 15.02 14.92 14.85
0.07 0.07 0.07 0.04 0.00 0.02 0.02 0.02 0.02 0.02 0.02
*All dimensions are in mm. tEstimated using [E(Ri - (R))2/(N - I)] '/2, N = 4 radial readings. ~tSee Table 1 for the specific values.
16,00 16.03 16,03 16,02 15,99 15,91 15,75 15.43 15.11 14.92 14.83
0.04 0.04 0.04 0.02 0.05 0.16 0.29 0.31 0.18 0.09 0.04