Computation of the plastic hardening curve based on a static tensile test

Numerical calculations, concerning, among other things, metal forming, based on the finite element method, require a description of the relationship between stress and strain in the entire load range. After crossing the yield point, the strain hardening is appearing. This phenomenon can be described by the plastic hardening curve. This paper presents the method of determining the strain hardening curve by means of power function approximation. The load and displacement data recorded in the tensile test using the MTS measurement system were used to determine this curve.

Numerical calculations in engineering tasks, especially in the field of plastic working, require description of the elastic and non-elastic characteristics of the material.While in the case of a linear elastic material it is sufficient to determine two material constants (E and ν), in the plastic domain it is necessary to determine the full relation between stress and strain [4].It can be obtained on the basis of a static tensile testas a result of the description of the curve of plastic strengthening by means of an appropriate approximation.Many ways of describing plasticizing stresses are known.Plastic flow curves can be classified into one of six groups [1]:  group Iincluding functions of the σ = f() type, giving the dependence of stresses on strains; group IIincluding functions of the form σ = f(, ), which additionally take into account the strain speed ε,  group IIIincluding functions of the type σ = f(, , ), which in relation to group II are enriched by the temperature value of the shaping process T;  group IVincluding functions of the form σ = f(, ,  w ), which in relation to group II additionally introduce a parameter describing the internal state of the material σw;  group Vincluding functions of the type σ = f(, , , ), whose variables are: strain , strain speed , temperature T and time t;  group VIincluding the functions of group III, which in the subsequent stages of the test take into account a change in the orientation of the main directions hε; functions take the form σ = f(, , , ℎ ε ).
The curves of groups I-III are most often used to describe the curve of plastic strengthening.
This article presents the method of determining the flow curve, classified in group I, based on the tensile test of steel flat specimen in accordance with PN-EN ISO 6892-1.Experimental tests were carried out using a measuring device consisting of a MTS hydraulic cylinder (equipped with a force sensor and a displacement sensor) and an adapter for tensile test on flat specimen, made according to the author's design (fig.1).The range of loads carried out by the actuator is within ±25 kN.
Two EPSILON extensometers were used to measure longitudinal and transverse deformations.The extensometer measuring longitudinal strain can work in the range of +20 mm/-8 mm, and the extensometer for transverse deformation -in the range of ±5 mm.

Experimental research
The tests were carried out according to the author's tensile test procedure, created in the MTS TestSuit program.Fig. 2 shows a dog bone specimen after an endurance test, on which the extensometers for measurements of longitudinal and transverse deformations were still attached.A crack appears on the damaged specimen, running at an angle of about 45° to the direction of the main stresses, i.e. towards the maximum tangential stresses [2].
As a result of the tests, the dependence of stress on strain over the entire load range was obtained (fig.3).Experimental material parameters are summarized in tab.I.In order to carry out numerical simulation of sheet metal forming processes in FEA-based programs, it is necessary to give the σ-ε dependence in the plastic domain, which entails entering into the calculation program the coordinates of the points from the strengthening curve.This is a very labor-intensive operation, because to obtain a faithful image of dependencies in the non-linear range, a large number of data must be manually entered into the calculation program.An alternative solution is to determine a function that approximates the course of the strengthening curve.In the procedure used in the tensile test all the values needed to determine the function approximating the experimental curve were recorded.
In the plastic domain, the strengthening curve can be described, for example, with the Hollomon fortification rule [3,7], which is a simple power function of the form: where: εreal strain; K, nparameters determined from the power approximation of the test results in the range from the offset yield strength R0,2 to the ultimate tensile strength Rm according to the PN-ISO 10275 standard.
In order to determine the coefficients K and n, the formula (1) is transformed, which consists in logarithmization of pages, which leads to obtaining a linear function in logarithmic coordinates: where: nslope of the linear function, Ky-intercept of the linear function.
The exponent of the strengthening n was determined from the formula: where: lnεj, lnσjvalues calculated on the basis of logarithmized values of strains ε and stresses σ.
The summation in formulas ( 3) and ( 4) takes place in the range from j = 1 to m, i.e. from R0.2 to Rm.On the other hand, the values in formula (3) were determined as follows: where: iaverage values of strains and stresses ranging from R0.2 to Rm.
The factor K is calculated from the dependence: From the formulas (3)-( 5), the stresses σH were determined.The graph (fig.4) shows the experimental curve and the approximation curve according to the Hollomon's rule.At the beginning of the strengthening curve, the stress value σH [5] jumps, and the adjustment to the experimental curve is not satisfactory, which is confirmed by the coefficient of determination R 2 equal to 0.734.
In order to improve the fit of the curves, a modified concept of determining the strengthening curve was used, i.e. the strengthening rule according to Krupkowski [8].
According to it, in the stress formula σK the strain is divided into two values:  K = ( 0 +  p )  (6) where: ε0initial strain, εpplastic strain.
If the value of ε0 in formula ( 6) is to accept strains corresponding to the offset yield point, then the Hollomon formula is obtained [6].
The stress equation formula σK contains three parameters: K, ε0 and n.To find their values, for which the best fit of the Krupkowski curve will occur to the experimental results, the optimization task was defined using the solver built into the Excel spreadsheet.The approximate approximation parameters -K, n and initial strain ε0were used in the optimization task as decision variables.The measure of the fit approximation σK(ε) to the experimental curve σ(ε) is the value calculated as the sum of the squares of the difference between the stress value σK and σ, which is denoted as δ.The objective function in the task is to minimize this value.The parameter has been specified in the range from R02 to Rm:  = ∑ ( K −   )  =1 (7) where: σKstress determined from the Krupkowski formula, σthe value determined on the basis of the force recorded during the tensile test.
As a result of the optimization, modified values of decision variables were obtained, which much better fit the σK approximation curve to the experimental results.
Fig. 5 shows the experimental curve and the approximation curve according to Krupkowski's rule.Thanks to the application of Krupkowski's rule and solving the optimization task, the R 2 coefficient of determination was obtained equal to 1 with the accuracy of three significant digits, which indicates a very good matching of the σK(ε) function to the σ(ε) experimental curve.The δ value of this curve is 5696, which is ten times less than in the case of the Hollomon rule.
Tab. II lists the parameters K, n, ε0, δ and coefficients of determination determined for the described rules of strengthening.

Conclusions
The mechanical properties of the sheet supplied by the manufacturer should be within the limits specified for a given type of steel.Even a slight difference in mechanical properties can affect the production of defective, incorrectly shaped emboss.For this reason, it is very important to carry out the control tests on a regular basis during the punching process, in particular the accurate determination of the curve of material strengthening.
The applied Krupkowski fortification rule very well reflects the experimental results.At the beginning of the plastic hardening curve, a small jump is created, which is acceptable, because the pressing processes run in the central part of this curve.
In summary, it can be concluded that the use of a precise measuring system to determine deformations and real stresses is a proper solution in the case of determining the function describing the curve of plastic strengthening.

Fig. 1 .
Fig. 1.View of the measuring stand

Fig. 4 .
Approximation of the experimental strengthening curve

Fig. 5 .
Approximation of the experimental strengthening curve according to Krupkowski's rule

TABLE I . Material parameters determined in the tensile test
E -Young's modulus, R02offset yield strength, R05 -elastic limit, Rmultimate tensile strength