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Corresponding author. The evolution law of the model consists of two maincomponents: a nonlinear kinematic hardening component whichdescribes the translation of the yield surface in stress spacethrough the back stress a, and an isotropic hardening componentdescribing the change of the equivalent stress defining the size ofthe yield surface, as a function of plastic deformation. The isotropiccomponent of the model is defined as follows [6]:s0¼ s0 þ Qð1 exp beÞ (1)where s0 is the yield stress at zero plastic strain, Q is themaximumchange in the size of the yield surface, and b is the rate atwhich thesize of the yield surface changes as plastic strain develops.In the Abaqus software, the hardening laws for each back stressare calculated using the following formula:ak ¼ Cks0ðs aÞe gkake (2)where Ck and gk are thematerial parameters for each back stress ak.The overall back stress is computed from the relation:a ¼X Nk¼1ak (3)where N is the number of back stresses.As presented, the kinematic hardening component is based onthe two major parameters Ck (the initial kinematic hardeningmodule) and gk (rate at which the kinematic hardening modulesdecrease with increasing deformation). These parameters arecalibrated based on half-cycle test data (uniaxial tension orcompression) or can be obtained from the stabilised cycle test(when the strain–stress curve no longer changes its shape fromonecycle to the next). These approaches are usually used in industry.However, parameters can also be specified directly in the FEmodel.Descriptions of these approaches are given below:- The first approach is usually adequate when the simulationinvolves only a few cycles of loading. For each data point (si, ei)the value of the overall back stress, which is the sum of all backstresses at each point, is obtained from the test data as adifference between the current value of stress and the size of theyield surface. Integration of the back stress evolution law over ahalf cycle yields the formulas that are used to calculate Ck and gk.- The second approach is used when experimental data fromsymmetric strain cycles are available. For each pair (si, ei) theoverall back stresses are obtained from the test data: ai = si ss,where ss =(s1 + sn)/2 is the stabilised size of the yield surface.Integration of the back stress evolution laws over this uniaxialstrain cycle, with an exact match for the first data pair (s1, 0),provides the expressions that are used to calculate Ck and gk.- The third method also requires cyclic test data, as well asadditional calculations (e.g., application of the inverse method)[7,8]. A schematic of the approach is presented in Fig. 2. Theoptimum parameters are identified by searching for theminimum of the objective function, which is defined as a squareroot error between the measured and calculated loads: where Fmij and Fcij are themeasured and calculated loads, Nps is thenumber of loadmeasurements in the test, p is the vector of processparameters (strain rates, temperatures), x is the vector ofcoefficients in the combined hardening model.The identification process is performed as follows: first, initialmodel parameters are selected, and the direct problem is solvedusing Abaqus Standard code. Next, calculated load vs. displace-ment data are compared with the experimental data. Then, theoptimization module is used to minimise the goal function withrespect to x [7], and a newset of parameters is obtained. Thewholeprocedure is repeated until the stop criterion is reached.In the first two described cases, strain-stress data from halfcycle and stabilised cycle tests, respectively, are introduced intothe FE code in a tabular form. Additionally in these twosimulations, the effect of a different number of back stresses(between 1 and 5) on the accuracy of results can be analysed. In thethird case, material parameters are specified directly in the FEmodel. Comparison of the modelling results obtained using thementioned approaches is shown in Fig. 3. Additionally, an isotropicmodel was used as a reference to highlight the differences in theobtained results.The best results were achieved when the combined hardeningmodel with the parameters specified directly is used (Fig. 3c). Thedisadvantage of this approach is that it requires a series ofadditional FE calculations and optimisation trials at the modelidentification stage according to the inverse approach presented inFig. 2. Results from Fig. 3a and b also reveal that by increasing thenumber of back stresses to three, additional improvement isobtained. However, further increase of the number of back stressesdoes not cause any further changes in the results. Based on this, itcan be stated that in order to obtain a high level of accuracy duringFE simulation an inverse analysismethod should be applied duringthe model identification stage.2.3. Selection of the testing equipmentThe selection of appropriate testing conditions and equipmentis the second goal of the research. A Gleeble systemwas consideredfirst. It provides a wide range of possibilities in defining thethermo-mechanical cycles but is mainly used to evaluate materialresponse under large deformation degrees.The smallest amplitude that provided reliable results duringcyclic T/C tests was identified as 0.25 mm. The applied thermalcycle was heating to 1000 8C, holding for 600 s and cooling to thedeformation temperature 650 8C. Obtained results are used duringidentification of the material model based on inverse analysis, asseen in Fig. 4. The agreement between experimental and numericalresponses is very good. However, during the levelling operation,material is usually subjected to lowcyclic deformation. That iswhya Zwick tension/compression testing machine which has thecapability to obtain realistic results at lower strains is also used inthe research. An amplitude that replicates levelling during cyclic T/C was identified as 0.125 mm. Different thermal cycles wereapplied: the sample was directly heated to the deformation temperature 650 8C.
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