Chart based Intersection approach and results

3.4.4. Chart based Intersection approach and results

In the chart based intersection approach, the potential material sets with the same curvature value for the Vickers and spherical indenters were identify separately through a chart C vs. N for different yield stress Figure 3.10 a b. The property data was then plotted onto another chart for yield stress vs. Work hardening coefficient Figure 3.10 c. The properties were determined at the intersection point between the data for the Vickers and Spherical indenters. As illustrated in Figure 3.10 a, b, c with a known Cv or Cs value, the properties can be determined by drawing a straight line along the curvature value, the intersection point represents data with the same curvature value. These data was then plotted onto the yield stress-work hardening n chart. The intersection point will represent the predicted material properties. The two cases represents two examples for 6 1= 150, n1=0.15 and 6 2= 300, n2=0.28, both are very close to the true material properties. Prediction carry out by determining the value of n work hardening coefficient based on analysis of trend equations at each curvature Through chat fitting and mapping knowable predicted value n1 = 0.174 , n2 =0.209 as described on accuracy study results of inverse FE modeling on Vickers and spherical indentation plotting and mapping data result. Evaluation of accuracy on the approach using Plotting and mapping carry out by taking input data which is characterized by randomize and represent the sample space. The selected input data with 2 variations in the value n, namely n=0.15 and n=0.28 range 0.01 0.30 and 6 = 100, 140, 190 and 300. Accuracy study Plotting and mapping Table.3.4, known that on average value of different predictive of n into n-input as n=0.24 and average accuracy error ∆n n ⁄ is 0.07 both the prediction n on Vickers Indentation and on Spherical Indentation. This shows the selected predictors significantly acceptable within the limit of level confidence less than 0.5 .Accuracy studies results were done for some input value, the results were listed in Table 3.4. The average accuracy, which is much better than the accuracy for the 3D mapping approach. Figure 3.1 Flow chart showing the Research work . Development of FE indentation models of Vickers and Spherical indenters Validation of the FE model and the effect of materials properties on the loading and unloading curves. Theoretical development of the inverse FE modelling program based on indentation curves with different indenter shapes: Curve fitting; 3-D Mapping, Chart method. Evolution of accuracy of each approach with known material properties Tensile shear tests of spot welded joints Impact tests of spot welded joints Development of analysis procedures for drop weight impact testing of spot welded joints Develop a practical approach based conventional hardness testing measurement after unloading Development of FE models of spot welded joints with realistic structures under tensile shear Prediction of the behaviour of spot welded joint with realistic materials properties a b Figure 3.2 a FE model of the Vickers Indentation test; b Typical force- Indentation depth P-h curve during loading and unloading of the Vickers indentation 6 = 100 MPa, n= 0.01 0.00 1.00 2.00 3.00 4.00 5.00 6.00 7.00 0.005 0.01 0.015 P N h mm A B 136 between opposite faces a Effect of the yielding strength b Effect of work hardening coefficient. Figure 3.3 Effect of work hardening and yield strength on the P-h curves of the Vickers indentation. 1 2 3 4 5 6 7 0.005 0.01 0.015 P h Sy=100, n=0.2 Sy=120, n=0.2 Sy=150, n=0.2 Sy=190, n=0.2 Sy=300, n=0.2 Sy=390, n=0.2 0.000E+00 1.000E+00 2.000E+00 3.000E+00 4.000E+00 5.000E+00 6.000E+00 7.000E+00 0.000E+00 5.000E-03 1.000E-02 P h Sy 300, n=0.01 Sy 300, n=0.1 Sy 300, n=0.2 Sy 300, n=0.3 a FE model of the Spherical b.Close-up view showing the Indentation test the mesh underneath the indenter Figure.3.4a, b FE models of the Spherical Indentation test R=0.5mm. c Figure.3.4.c. Typical force-indentation depth p-h curve during loading and unloading for Spherical Indentation 6 = 500 MPa, n=0.024. 20 40 60 80 100 120 140 0.005 0.01 0.015 0.02 0.025 P N h mm Figure 3.5 a Effect of yield stress 6 as plastic parameter on the p-h curves of the spherical Indentation. Figure 3.5 bEffect of Work hardening coefficient n as a plastic parameter on the P-h curves of the spherical indentation 20 40 60 80 100 0.005 0.01 0.015 0.02 0.025 P N h mm SY=100 n= 0.01 SY=200 n=0.01 SY=300 n=0.01 SY=400 n=0.01 SY=500 n=0.01 0.000E+00 5.000E+00 1.000E+01 1.500E+01 2.000E+01 2.500E+01 3.000E+01 3.500E+01 4.000E+01 4.500E+01 5.000E+01 0.000E+00 5.000E-03 1.000E-02 P h Sy 200, n=0.01 Sy 200, n=0.1 Sy 200, n=0.2 Sy 200, n=0.3 Input experimental Force Vs depth data Simulation Space Figure 3.6. Flow chart showing the inverse FE modelling approach based on the Vickers and Spherical Indentation tests. 50 100 0.01 0.02 0.03 P N h mm 20 40 60 80 100 0.02 0.04 P N h mm 1000 2000 3000 4000 1 2 3 S tr e ss , M p a Strain Three inverse properties prediction methods: • Dimensional analysis approach • 3D mapping approach • Dual indenter Chart based + and n Cv and Cs Figure 3.7 a.Surface mapping linear fitting plot of loading curvature vs material properties. Figure 3.7 b. Typical materials parameter prediction process based on linear 3-D mapping method -3.00E-01 -2.00E-01 -1.00E-01 0.00E+00 1.00E-01 2.00E-01 3.00E-01 4.00E-01 5.00E-01 6.00E-01 7.00E-01 50 100 150 200 250 300 350 n Yield Stress, Mpa Cv 150 Cs 150 Cv 300 Cs 300 0.0 2.0e+5 4.0e+5 6.0e+5 8.0e+5 1.0e+6 1.2e+6 100 120 140 160 180 0.1 0.2 0.3 0.4 C s Sy n 3D Linear mapping 2e+4 4e+4 6e+4 8e+4 1e+5 100 120 140 160 180 0.1 0.2 0.3 0.4 C v SY n 34 Accuracy study Input data Vickers Indentation + Spherical indentation Material properties Value curvature Predicted value Different with input value Accuracy error Predicted value Different with input value Accuracy error n + Cv Cs n ∆n ∆nn n ∆n ∆nn 0.15 100 26008 466275 2.052E-01 5.523E-02 3.682E-01 2.477E-01 9.770E-02 6.513E-01 0.15 140 26008 466275 1.538E-01 3.786E-03 2.524E-02 1.639E-01 1.388E-02 9.253E-02 0.15 190 26008 466275 8.949E-02 -6.051E-02 -4.034E-01 5.911E-02 -9.089E-02 -6.059E-01 0.15 300 26008 466275 -5.198E-02 -2.020E-01 -1.347E+00 -1.714E-01 -3.214E-01 -2.143E+00 0.28 100 61940 925826 5.749E-01 2.949E-01 1.053E+00 6.391E-01 3.591E-01 1.283E+00 0.28 140 61940 925826 5.235E-01 2.435E-01 8.695E-01 5.553E-01 2.753E-01 9.833E-01 0.28 190 61940 925826 4.592E-01 1.792E-01 6.398E-01 4.506E-01 1.706E-01 6.091E-01 0.28 300 61940 925826 3.177E-01 3.770E-02 1.346E-01 2.201E-01 -5.994E-02 -2.141E-01 Table 3.1. Accuracy study results of inverse FE modelling on Vickers and Spherical indentation by 3D Linier Predicted chart mapping And Plotting data result from Cv and Cs chart to used predicted materials Figure 3.8 a.Surface mapping nonlinear fitting plot of loading curvature vs material properties. Figure 3.8 b Typical materials parameter prediction process based on nonlinear 3-D mapping method. -6.00E-01 -4.00E-01 -2.00E-01 0.00E+00 2.00E-01 4.00E-01 6.00E-01 8.00E-01 100 200 300 400 n Yield Stress, Mpa Cv 150 Cv 300 Cs 150 Cs 300 Cs 180 2e+4 4e+4 6e+4 8e+4 1e+5 100 120 140 160 180 0.1 0.2 0.3 0.4 C v Sy n 0.0 2.0e+5 4.0e+5 6.0e+5 8.0e+5 1.0e+6 1.2e+6 120 140 160 180 0.1 0.2 0.3 0.4 C s Sy n 3D Non linear chart mapping 36 Table 3.2. Accuracy study results of inverse FE modelling on Vickers and Spherical indentation by 3D Non Linier Parabolic Predicted chart mapping Accuracy study Input data Vickers Indentation- Spherical indentation Material properties Value curvature Predicted value Different with input value Accuracy error Predicted value Different with input value Accuracy error Sy n + Cv Cs n ∆n ∆nn n ∆n ∆nn 150 0.15 100 26008 466275 0.230977 8.098E-02 5.398E-01 0.270383 1.204E-01 8.026E-01 150 0.15 140 26008 466275 0.166904 1.690E-02 1.127E-01 0.175556 2.556E-02 1.704E-01 150 0.15 190 26008 466275 0.077722 -7.228E-02 -4.819E-01 0.034696 -1.153E-01 -7.687E-01 150 0.15 300 26008 466275 - - - - - - 300 0.28 100 61940 925826 0.490031 2.100E-01 7.501E-01 0.562677 2.827E-01 1.010E+00 300 0.28 140 61940 925826 0.454171 1.742E-01 6.220E-01 0.505893 2.259E-01 8.068E-01 300 0.28 190 61940 925826 0.412688 1.327E-01 4.739E-01 0.43741 1.574E-01 5.622E-01 300 0.28 300 61940 925826 0.33883 5.883E-02 2.101E-01 0.301386 2.139E-02 7.638E-02 180 0.25 100 61940 925826 0.354054 1.041E-01 4.162E-01 0.399732 1.497E-01 5.989E-01 180 0.25 140 61940 925826 0.30762 5.762E-02 2.305E-01 0.326902 7.690E-02 3.076E-01 180 0.25 190 61940 925826 0.251198 1.198E-03 4.792E-03 0.233276 -1.672E-02 -6.689E-02 180 0.25 300 61940 925826 0.138376 -1.116E-01 -4.465E-01 0.001922 -2.481E-01 -9.923E-01 Figure 3.9a Effects of Yield strength 6 on the value loading curvature Cv and work hardening coefficient n on Vickers Indentation. Figure 3.9 b Effects of Yield strength 6 on the value loading curvature Cs and work hardening coefficient n on Spherical Indentation y = 374.1e 3.196x 200 400 600 800 1000 1200 1400 1600 0.000 0.100 0.200 0.300 0.400 0.500 C v s y 3 4 n Sy=100 Sy=120 Sy=150 Sy=190 Sy=300 Prediction Cv Expon. Prediction Cv y = 8011.e 1.983x 5000 10000 15000 20000 25000 30000 35000 0.2 0.4 0.6 0.8 C s S y 3 4 n Sy=100 Sy=120 Sy=140 Sy=190 Sy=300 prediction Cs Expon. prediction Cs Figure 3.9c. Typical materials parameter prediction process based on the intersection between properties line for the Vickers and Spherical indentation 6 = 120 Mpa, n=0.2. 0.000000E+00 5.000000E+01 1.000000E+02 1.500000E+02 2.000000E+02 2.500000E+02 3.000000E+02 0.1 0.2 0.3 0.4 0.5 0.6 Y ie ld t re ss M p a Work Hardening coefficient n Syn,Cv Syn,Cs 1.200E+02 39 Input data Accuracy study Material properties Value of Curvature Predicted values Differences with input value Accuracy error σ y n Cs Cv σ y n ∆ σ y ∆ n ∆ σ y σ y ∆ nn 100 0.1 303841.62 15725.77 102 0.09 -2 0.010 -0.020 0.10000 100 0.2 387159.42 22631.67 105 0.19 -5 0.010 -0.050 0.05000 100 0.3 495483.94 32535.09 111 0.29 -11 0.010 -0.110 0.03333 200 0.1 5.260E05 2.771E+04 202 0.09 -2 0.010 -0.010 0.10000 200 0.2 6.297E05 3.767E+04 195 0.21 5 -0.010 0.025 -0.05000 200 0.3 7.547E05 5.061E+04 195 0.31 5 -0.010 0.025 -0.03333 290 0.1 6.892E05 3.734E+04 275 0.12 15 -0.020 0.052 -0.20000 290 0.2 8.039E05 4.885E+04 265 0.22 25 -0.020 0.086 -0.10000 290 0.3 9.365E05 6.349E+04 255 0.31 35 -0.010 0.121 -0.03333 Table.3.3. Accuracy study results of inverse FE modelling on Vickers and Spherical indentation Figure 3.10 a Typical curves of the Cv vs. n for Vickers indentation and the interception line to identified the material parameters with identical Cv value. b Figure.3.10 b Typical curves of the Cs vs. n for Spherical indentation and the interception line to identified the material parameters with identical Cs value. y = 14834e 3.353x y = 11002e 3.608x y = 19655e 3.061x y = 29685e 2.614x 10000 20000 30000 40000 50000 60000 70000 0.1 0.2 0.3 0.4 C v n Sy=140 Sy=100 sy=190 Sy=300 Expon. Sy=140 Expon. Sy=100 Expon. sy=190 Expon. Sy=300 26008 y = 23802e 2.447x R² = 1 y = 42235e 1.818x R² = 0.999 y = 32353e 2.098x R² = 1 y = 61251e 1.448x R² = 0.999 0.000000E+00 2.000000E+05 4.000000E+05 6.000000E+05 8.000000E+05 1.000000E+06 1.200000E+06 1.400000E+06 0.2 0.4 0.6 C s n sy=100 sy=190 sy=140 sy=300 Expon. sy=100 Expon. sy=190 Expon. sy=140 Expon. sy=300 466275 Figure 3.10 c. Typical materials parameter prediction process based on dual indenter chart method. -0.3 -0.2 -0.1 0.1 0.2 0.3 0.4 0.5 0.6 50 100 150 200 250 300 350 n Yield Stress, Mpa Cv150, 0.15 Cs150, 0.15 Cv300, 0.28 Cs300, 0.28 42 Table.3.4. Accuracy study results of inverse FE modelling on Vickers and Spherical indentation by Plotting and mapping data result from Cv and Cs chart to used predicted materials Accuracy study Input data Vickers Indentation Spherical indentation Material properties Value curvature Predicted value Different with input value Accuracy error Predicted value Different with input value Accuracy error n + Cv Cs n ∆n ∆nn n ∆n ∆nn 0.15 100 26008 466275 0.23831 8.832E-02 5.888E-01 0.274451 1.245E-01 8.297E-01 0.15 140 26008 466275 0.167607 1.761E-02 1.174E-01 0.174204 2.420E-02 1.614E-01 0.15 190 26008 466275 0.091527 -5847E-02 -3.898E-01 0.083707 -6.629E-02 -44.420E-01 0.15 300 26008 466275 -0.05067 -2.007E-01 -1.338E+00 -0.18814 -3.381E-01 -2.254E+00 0.28 100 61940 925826 0.478695 1.987E+01 7.096E-01 0.554414 2.744E-01 9.801E-01 0.28 140 61940 925826 0.42664 1.466E-01 5.237E-01 0.50114 2.211E-01 7.898E-01 0.28 190 61940 925826 0.375109 9.511E-02 3.397E-01 0.45000 1.700E-01 6.071E-01 0.28 300 61940 925826 0.28181 1.810E-03 6.465E-01 0.284901 4.901E-03 1.750E-02

4. Mechanical testing and FE Modelling of spot