SIMPLE METHODOLOGY FOR CALCULATING THE CONSTANTS OF GAROFALO EQUATION

The examination of the high temperature plastic properties of metallic materials was realised by the torsion plastometer SETARAM or by the compression plastometer DIL805A/D. The brass CuZn30 was used as the test material. Peak stress detection was performed for two independent variables, temperature and velocity of deformation. The set experimental plan forms the test array 54, i.e. five temperatures 650, 700, 750, 800, 850 °C and four strain rates of 0.5, 2.5, 12.5 and 25 s. It was necessary to evaluate measured data and to determine the mathematical model of the peak stress. For this aim, the Garofalo equation was used. This equation contains 4 material constants. The currently method used to determine the material constants takes a relatively long time and requires a number of auxiliary calculations. In this method, nonlinear regression is often used, which requires the initial estimation of parameters. The article presents a mathematical analysis and a simple methodology for calculating the material constants of the Garofalo equation. A general linear regression is used for the calculation that does not require an initial estimate of the material constants. The numerical calculation of material constants is documented on the measured peak stress data.


Introduction
The Garofalo empirical equation [1,2] is used to describe the high temperature deformation, which describes the strain rate in relation to the flow stress and the absolute temperature Extensive analysis of the Garofalo equation ( 1) and the flow of material has been carried out by several authors [3,4,5].Material constants are determined from the measured actual stress values, which are defined by the temperature and the strain rate.[6,7].Different mathematical methods and nonlinear regression [11,12] are used for searching constants n, Q a C [8,9,10].
Calculation of material constant The Eq. ( 1) allows a description of strain stress dependencies for specific thermodynamic conditions of forming.For high strain temperatures and low strain rates, the condition 1 σ α   applies, (the data in row of temperature 850 °C in Table 4), therefore Eq. ( 1) can be reduced to the form [13] , (the data in row of temperature 650 °C in Table 4), that is fulfilled when forming at lower temperatures and higher strain rates, it is possible to reduce Eq. ( 1) to the form [13]   The following procedure is used in [3] to determine the material constant  .The constants n1 and  are calculated from Eq. ( 2) and Eq. ( 3).For this aim, these equations have been transformed to the form of a line equation by using the logarithm Fig. 1 The line directive determines the constant n1 = 5,3420 The data in the row of temperature 850 °C in Table 4 were used to calculate the constant n1 of Eq. ( 5).Similarly, the data in row of temperature 650 °C in constant  of Eq. ( 6) [13,14].Because the data at constant temperature were always used, the first two members on the right of equations ( 5) and ( 6) are also the constants.Eq. ( 5) expresses the dependence σ ln ln    and Eq. ( 6) the dependence σ ln    .The graphical representation of these dependencies is in Fig. 1 and Fig. 2. Obtained values n1 and  allow calculating the material constant  for CuZn30 brass using the Eq. ( 4) where material constant n means gradient of the line (9).For constant temperatures Eq. ( 9) represents a set of lines.The graph of the lines will be view in the coordinates ln ´ln sin(.p)for points with the same temperature [15].Fig. 3 shows five lines for our five temperatures.The gradients of these lines represent the material constant n for a corresponding temperature.The values of this material constant n are calculated in Table 1 for every line.The resulting value of the material constant n for the whole test array is given by the arithmetic mean  Calculation of material constant Q.The material constant Q represents the activation energy of the deformation.To determine the value of this material constant we use Eq. ( 8) [16,17].This method is highly demanding for the data processing time and the determination of the material constant Q [18,19,20].In Eq. ( 8) we introduce transformations means y-intercept of the line, because the strain rate is for the corresponding line constant.Eq. ( 8) takes a form [3,15] const The graph of the line will be shown in the coordinates

 
/T 1000 σ α sinh ln p   in points that have the same strain rate [15].Fig. 4 shows four lines for four used strain rates.The values of gradients ki of every line are shown in   Calculation of material constant C. Material constant C is the last constant which appears in Garofalo equation.Eq. ( 1) can be written in the form .) The left side of the Eq. ( 12) represents Zener  Hollomon parameter Z: From Eq. ( 12) and ( 13) is made next equation ) Fig. 5 Determination of constant C by using Zener-Hollomon parameter [15] Because the activation energy Q has already been calculated, it is possible to calculate Zener  Hollomon parameter from Eq. ( 13).In Eq. ( 14 For calculating the constants of Garofalo equation are used different mathematical methods, most often nonlinear regression.The new solution consists in the use of general linear regression.

Experimental Material and Methods
The CuZn30 brass was used for high temperature plastic deformation tests [15].This brass belongs to the alpha brasses, which have good cold formability.It is the brass used in the munitions industry.The test material was in the form of bars with a diameter  12 mm.The chemical composition of this brass is given in Table 3 and corresponds to the German standard DIN 17 660 w.n.2.0265.Courses of stress in dependence on deformation were obtained on a torsion plastometer SETARAM.The dimensions of the test sample are given in Fig. 6.In order to use the Garofalo equation for the evaluation of the tests, it is necessary to follow the elaborated test plan.The test plan is represented by array 5t4´ (5 temperatures  4 strain rates).Every element of the array signifies one measurement at defined temperature and deformation rate.
Variables t1 to t5 represent temperatures 850, 800, 750, 700 and 650 °C.Variables ´1 to ´4 mean strain rates of 0.5, 2.5, 12.5 and 25 s -1 .The matrix of tests represents 54=20 measurements.Measured values of peak stress resulting from courses of stress in dependence on deformation in accordance with the test plan are shown in Table 4.

Results and Discussion
New calculation methodology of material constant .In Eq. ( 5) and Eq. ( 6) the constants n1 and  mean the gradient of line.To calculate material constant  it is not necessary to obtain where s represents the number of strain rates at which the strain stress was measured (in Table 4 s=4).Numeric value of material constant  is determined by Eq. ( 4), using values calculated from Eq. ( 16) and Eq.(17).New calculation methodology of material constant n, Q a C. Calculation procedure of material constants n, Q and C is as follows, use logarithm on Eq. ( 1) and adjust it Introduce next transformations in Eq. ( 18) ) ) and constants in Eq. ( 18) replace with substitutions ) ) then Eq. ( 18) will take shape Unknown constants b0, b1 and b2 in Eq. ( 25) will be obtain by general linear regression, because functions f1 and f2 are known, defined by Eq. ( 20) and Eq. ( 21).The determinant equations have the following form where p represents total number of measurements (in Table 4 p=20).Eq. (26) to Eq. (28) are a system of three linear equations of three unknowns b0, b1 and b2.After solving the system of equations, material constants will be calculated by Eq. (22) to Eq. ( 24) Example of calculation of material constants: n, Q and C. For calculating the particular sums in Eq. ( 26) to Eq. (28) we use data from Values of peak stress will be determine from the Garofalo equation ( 1) that should be adjust for direct calculation of stress Based on the specified material constants n, C, Q, and , the values of stress p are calculated by Eq. ( 35).The rate of correlation between the measured and calculated values of peak stress gives the correlation coefficient (R 2 =0,97998).The comparison of the calculated material constants of the Garofalo equation, according to the old and new methodology is given in Table 5.The use of results obtained by plastometer in metal forming processes is documented by several publications [22,23].A special application in the rolling process is given in the publication [24].

Conclusions
The results of determination of material constants , n, Q and C obtained by new methodology are in good agreement with the old methodology.The main benefit of the new method of calculating the material constants for the Garofalo equation lies in its simplicity and the speed of the calculation.Material constants are calculated directly as their final values  The new methodology for calculating the constants of the Garofalo equation does not require an initial estimate of constants as nonlinear regression  The calculation of material constants does not require complicated software as for nonlinear regression  The new methodology does not require to calculate line equations, see on Fig. 3 to Fig. 5  The new methodology allows the calculation of constants in a fully automatic mode  The Garofalo equation was linearized using logarithm what allows to calculate its constants through general linear regression  The new methodology is suitable for the evaluation of the measured data by the torsion plastometer SETARAM and also by the compression plastometer DIL805A/D.

Fig. 2
Fig. 2 The line directive determines the constant  = 0,0495 Calculation of material constant n.If we use the logarithm operation on the Eq.(1), we get linearized form of this equation   p σ α sinh ln n RT Q C ln ln        (8.)In Eq. (8) we use transformations   ln y and

Fig. 4
Fig.4 Line plots at different strain rate determining material constant Q[15]

. 5 )
material constant C signifies y-intercept y0 of the line.Eq. (15) represents one line for whole set of measured values.The graph of the line will be shown in the coordinates , from which it results DOI 10.12776/ams.v23i4.1017p-ISSN 1335-1532 e

Fig. 6
Fig. 6 Test specimen for hot torsion tests

complete regression equations according to Fig. 1 a. 2 .
FigIt is sufficient to calculate only the gradients of lines that represent constants n1 a .According to the theory of mathematical statistics, we define them as follows

Table 2 .
The resulting value of the constant k for the whole test array is given by the arithmetic mean

Table 4
Peak stress of CuZn30 brass (MPa)

Table 4 .
The numerical form of these equations is as follows