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(3) where δmax = elongation at maximum load (miniature specimen) εmax = strain at maximum load (standard uniaxial test) leq = equivalent gage length of miniature specimen true strain (εt)max = ln(1+εmax) (4) hence the value of true strain at the maximum load can be determined from experimental load-elongation curve. This is the extra advantage of using dumb-bell shaped miniature specimen. This is because miniature specimen is also in tension mode in the same fashion as the` standard specimen. This type of relationship is difficult to formulate for miniature shear punch tests. c) The miniature load-elongation curve is pided into n linear segments. For each point of load-elongation curve, corresponding strain is determined by piding the elongation by equivalent gage length. From these strain values, true strains are determined. Elastic strains are subtracted from true strain values in order to get true plastic strains. d) The second data point in the uniaxial true stress – true plastic strain curve (σt2, εt2) can be obtained by performing the finite element analysis by giving the inputs as load P2, elastic modulus E, the yield stress σt1 and corresponding true plastic strain εt1 which is zero, the true plastic strain εt2 and corresponding assumed value of σt2. The assumed value of σt2 may be taken a little higher than the σt1 to start iterations. Now the elongation δp given by finite element analysis is compared against δ2 which is experimental value of elongation at load P2. If 2 p δ δ α − ≥ then σt2 is modified as follows. 2 δ ⎝⎠ ⎣⎦mWhere (σt2)m+1 = true stress at load P2 in (m+1)th iteration (σt2)m = true stress at load P2 in mth iteration β = Convergence control parameter ( 01 β < ≤ ) Iterations may be stopped when 2 p δ δ α − ≤ . At the final iteration (σt2, εt2) is the required second data point in the uniaxial true stress-true plastic strain curve. e) Similarly, the third data point in the true stress – true strain curve is obtained by considering the load P3, elastic modulus E, the previous points on the true stress-true strain curve i.e. (σt1, εt1), (σt2, εt2), the true strain (εt3) as obtained from the miniature test and the assumed value of stress σt3. Now the inverse FE analysis is carried out by iterating on σt3 till the complete match is obtained between the load-elongation curves of the miniature test and the inverse FE. At the final iteration the assumed stress value σt3 and the true strain εt3 is the required third data point on the uniaxial true stress-true strain curve. Likewise rest of the data points on the true stress –true strain curve are obtained by proceeding in the same manner. f) The material properties of the unknown material are determined in terms of Young’s modulus, yield stress and data points on uniaxial true stress-true plastic strain curve. Once the material properties are known, the direct finite element analysis can simulate miniature specimen test in much better way and can do in-depth analysis of miniature specimen test. 4. Results The inverse finite element algorithm developed in the present study for the analysis of dumb-bell shaped miniature specimen under in-plane tensile load was implemented with the help of the ABAQUS code. The miniature test load-elongation curves in combination with the inverse finite element method may provide best estimates of conventionally measured tensile properties for different materials. Figure 3 shows that the load-elongation curves as obtained from the inverse finite element procedure are matching completely against the one obtained from the miniature test on the specimen from the AR66 alloy for a unique combination of constitutive parameters. Table 2 shows the comparison of elastic modulus value obtained from the inverse finite element model and from the standard test. The starting value of Young’s modulus is taken as 70 GPa. Table 2. Comparison of elastic modulus (E) from standard test and inverse FE for AR66 alloy Method (E) in GPa Uniaxial test 70.65 Inverse FEM 71.00 The developed procedure requires repeated finite element analysis as different data points on the true stress-true strain curve are obtained by considering the different segments of miniature test load-elongation curve. The procedure employed here is the one that consists simply of matching the load–elongation curve obtained from the inverse finite element analysis to the experimental curve up to the segment under consideration. When a complete match is found between the load-elongation curves of the miniature test and the one from the inverse finite element method, it gives the constitutive behaviour of the unknown material. Figure 6 shows the comparison of true stress–true strain curve obtained from the standard tensile test with that obtained by using inverse finite element procedure for the miniature specimen from used in the present investigation. It is observed from the figure that the results are encouraging and the predicted true stress-true strain diagram corroborates well with the standard test curve. 4004505005506006507000 0.03 0.06 0.09 0.12True plastic strainTrue stress in MPaUniaxial testInverse FEM Fig. 6. Comparison of true stress–true plastic strain by inverse FE and uniaxial test for the AR66 alloy. 5. Conclusions The determination of constitutive tensile behavior of any unknown material is made possible using the inverse finite element procedure as outlined above. The elastic modulus of unknown material is determined with the help of the initial slope of the miniature specimen’s load-elongation curve and repeated inverse finite element analysis. The results from the inverse finite element are comparing well with the standard test results. The approach seems to have potential to predict the mechanical properties of the material in the in-service structures, which could be used in remaining life estimation. The proposed dumb-bell shaped miniature specimen and the designed loading geometry in the present investigation have some special advantages in finite element modeling.
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