Multiple Linear Model to Study Mechanical Properties of Bronze Alloys Using MATLAB Language

This work include prepare five alloys of bronze (Cu-5wt.%Sn, Cu-10wt.%Sn, Cu-15wt.%Sn, Cu-20wt.%Sn, Cu-25wt.%Sn).These alloys were then subjected to tensile tests to determine the tensile strength (MPa) and elongation(%) for each alloy. Tensile tests were performed according to ASTM-E8.MATLAB language was used to determine the mathematical model for tensile strength and elongation with respect to alloy composition.MATLAB results showed that there is a multiple linear model between tensile results and alloy composition.


Introduction
Bronze are generally considered alloys of copper containing tin and can certainly contain other elements [1].Chemical, civil, mechanical, materials, aerospace, and biomedical engineers need to predict the tensile strength of metal parts as a function of their alloy composition [2].A model is a representation of something constructed and used for a particular purpose.We use models constantly in all walks of life because they present a simplified view of the world which highlights the parts which interest us [3,4].In this study we conclude the multiple linear model which interest us in engineering metallurgy.
The multiple linear model can be described as follows: Suppose that y is a linear function of the two or more variables x,, x 2 ,....For example: To find the coefficient values of a 0 , a b and ai to fit a set of data (y, x h x 2 ) in the least squares sense, we can make use of then fact that the left-division method for solving linear equations uses the least square method when the equation set is overdetermined.To use this method, let n be the number of data points and the linear equation in matrix form as follows:  Using MATLAB Language where x 1i, x 2i , and y 1i are the data, i=l......,n.The solusion for the coefficient is given by a=X/y [5,6].

Experimental Procedure
The material provided for this study was in the form of commercially produced copper and tin.These materials was subjected to casting to prepare five-alloys of bronze as shown in Table (1).

Results and Discussions
The tensile strength and elongation data for alls are reported in Table ( A correlation between mechanical properties and alloy composition can be discussed as follows:

Tensile Strength and Alloy Composition
The tension strength (y) required to break a bronze bar is a function of the percentages X] and X2 of each of two alloying elements present in the metal.The Table (2) gives the pertinent data and the linear model obtaining from these data using MATLAB language is (y= 0+ 7.04 xi+ 5 x 2 ).The script file is shown in Figure (2).The vector y p is the vector of tensile strength values predicted by the model.The scalar Max_Percent_Error is the maximum percent error in the five predications.
Figure (3) and Figure( 4) illustrate \ the tensile strength as a function of the f percentage X) and X2 of each of two alloying elements present in the bronze.;

Elongation and Alloy Composition
In the same way the correlation of t the alloys is represented by (y) as as function to the percentage xi and x 2 off each of two alloying elements present inf the bronze.The linear model is (y=0 +| 1.455 X, + 0.555 x 2 ) and the script fiji' is shown in Figure (5) /doi.org/10.30684/etj.29.6.4University of Technology-Iraq, Baghdad, Iraq/2412-0758 This is an open access article under the CC BY 4.0 license http://creativecommons.org/licenses/by/4.0Eng.&Tech.Vol.26,No.6,2008Multiple Linear Model to Study Mechanical Properties of Bronze Alloys

Figure( 6
) ancj Figure(7) show the accorelation ol elongation as a function of percentage Xi and x 2 of each of alloying elements present in the bronz Conclusions 1.A multiple linear model of tensjj strength of bronze alloys is: Tensile-Strength (MPa)= 7.04(wt.%Sn)+5(wt.%Cu).2. Tensile strength change linearly i function to the percentages of Sn Cu. 3. A multiple linear model of elongatj of bronze alloys is: Elongation (%)= 0+ ]A55(vd'J & +0.555(wt.%Cu).4. Elongation change linearly function to the percentages of Sn| Cu.