Issue 48

J. Christopher et alii, Frattura ed Integrità Strutturale, 48 (2019) 554-562; DOI: 10.3221/IGF-ESIS.48.53 558 M ETHODOLOGY FOR PARAMETRIC OPTIMIZATION he physical constants associated with these models have been determined by the minimization of errors between experimental and predicted relaxation stress vs. hold time data for two different strain hold levels of 1.3 and 2.5% at 873 K for E911 steel. In order to obtain the unknown constants such as s and  in Model-I, least-square optimisation based on Levenberg–Marquardt algorithm has been used. Tab. 1 shows the optimised constants associated with the Model-I for E911at 873 K for the strain holds of 1.3 and 2.5%. An iterative procedure has been invoked to obtain the constants associated with the Model-II. As a first step in the iteration, random initial value of parameters within the bounds has been seeded for numerical integration. The coupled differential equations (i.e. Eqns. (3), (8) and (9)) defining the of relaxation stress ( r in C       ), inelastic strain and internal stress with time have been numerically integrated by the fourth-order Runge-Kutta method. Following numerical integration, the least-square error value has been estimated and then the parameter values are adjusted using interior-point algorithm for obtaining low least-square error value [9]. The fitting procedure has been repeated for several iterations to reach the optimised parameters. The optimised constants associated with Model-II for E911at 873 K for the strain holds of 1.3 and 2.5% have been presented in Tab. 2. For the numerical integration, the values of constants such as M = 3; b = 0.268 nm;  = 64420 MPa; k = 1.38  10  23 J/K;  D = 1  10 13 s  1 and R= 8.314 J mol  1 K  1 have been considered. The mobile dislocation density values of 1  10 13 m  2 have been chosen for E911 steel [9]. The activation energy value of 285 kJ mol  1 has been fixed for E911 steel [11]. For Model-II, the bounds for the physical constants are fixed based on the following consideration. The upper and lower limit values of  ro were chosen close to the initial relaxation stress. Hence, the bounds are fixed between 275 to 350 MPa. Since initial internal stress (  io ) should be less than the initial relaxation stress, the bounds for  io are fixed between 200 to 275 MPa. The value of 'm' is allowed to vary between 0 and 1 based on the literature values [10]. Once the three unknown independent parameters (  ro,  io and m) are optimized, the dependent parameter h has been calculated using 0 0 i m r h    . Parameter  r0 , MPa s, MPa  , s  1 Hold strain levels 1.3 % 311.55 20.0 0.34 2.5 % 327.23 20.8 0.50 Table 1 : Optimised parameters associated with the Model-I for E911 steel. Parameter  r0 , MPa  i0 , MPa m h, MPa 1.3 % 309.55 227.08 0.812 2.15 2.5 % 324.68 238.97 0.809 2.22 Table 2 : Optimised parameters associated with the Model-II for E911 steel. R ESULTS AND DISCUSSION he experimental as well as predicted relaxation stress (  r )-time (t) data have been shown in Fig. 4 as double logarithmic plots for the strain holds of 1.3 and 2.5 % at 873 K. A marginal decrease in relaxation stress with time for the initial hold durations followed by rapid linear decrease in stress values at longer durations has been T T