Issue 38

M. Lutovinov et alii, Frattura ed Integrità Strutturale, 38 (2016) 237-243; DOI: 10.3221/IGF-ESIS.38.32 239 2 2 2 1 1 2 2         eq (1) where  eq is equivalent stress. In all cases, estimations started from the yielding point. In case of proportional loading it was only natural since the modification of inputs was performed so that the first non-zero input values corresponded to the yielding state. In non- proportional load cases, it was checked whether the value of equivalent stress was higher than yield strength. If that condition was not met, the solutions for the current step were equal to the elastic solution. The solutions to the sets of equations provided by each method were obtained using MATLAB function fsolve . The results of estimations were compared with the elastic–plastic FE solutions presented in the original articles. The only exception was Reinhardt’s method, the results of which were compared with Moftakhar’s method. In case of non-proportional loading, besides the multilinear approximation of cyclic stress–strain curve, the material model described by the Ramberg–Osgood expression was used to compare the quality of estimates. After the verification of the implementation of the methods, the finite element analyses were carried out. The aim of those analyses was to provide inputs that were obtained on FE models with a mesh denser than used to be common, when the evaluated methods were first presented. To get a model with a representative fine mesh, sensitivity analyses were carried out on FE models with an elastic material model and different mesh densities. The appropriate model was then the model for which changes in values of equivalent stress were insignificant (less than 1% for cases of proportional loading, and less than 4% for cases of non-proportional loading). For proportional loading, simulations on two types of specimens were performed. The first one was a solid bar with a circumferential groove loaded simultaneously with tension and torsion. Another specimen was a hollow tube with a circumferential groove loaded by internal pressure and tension. Both types of specimens and loads corresponded to those presented in [3]. For non-proportional loading, a round bar with a circumferential groove loaded by torsion in the first step and by increasing tension and constant torsion in the second step, was used. This type of model is presented in [6]. After results from FE simulations were obtained, they were used as inputs for estimations in order to assess and compare prediction quality of selected methods. R ESULTS OF VERIFICATION OF METHODS IMPLEMENTATION he same results as presented in [3] were obtained for Moftakhar’s method using 4 increments in elastic–plastic region and bilinear approximation as the material model. Same settings were used for Reinhardt’s method and it led to the same results as in case of Moftakhar’s method. The implementation of Hoffmann–Seeger’s method was verified using the results from [5]. The same results as in [6] were obtained for Singh’s method using 28 increments per loading step and 6 linear approximation of the cyclic stress–strain curve. The attempt to get the same results as in [7] for Buczynski’s method was not successful. The fsolve function failed to solve the set of equations defined by the method. Numerical instability was observed for both definitions of the material model, with the Ramberg–Osgood expression or with the linear approximation. It was also observed that changing the settings of TolFun parameter, which defines termination tolerance on the fsolve function value [8], affects this behaviour. The smallest instability is observed in the case of 6 piecewise linear material model when estimations start from plasticized state with settings of TolFun to 1e-5. C OMPARISON WITH USE OF MODERN FE MODELS Proportional loading n Fig. 1, the estimates of stress and strain components are presented. The stress estimates are very similar to those presented in [3]. There is a slight difference in minimum principal stress component for the bar specimen at the end of load sequence, where the prediction is most accurate as the elastic– plastic result is bounded by two Moftakhar’s or two Reinhardt’s estimates. There is also a slight difference in case of the hollow tube, where the estimate according to Moftakhar’s method with the ESED rule is closer to the elastic–plastic results from the FE simulation. T I