Mathematical Models Applied to Wear Estimation

Generally, where there is a complex behaviour such as wear, it is desirable to find a mathematical description to aid analysis. This is the case for the abrasion tester device developed in this study. The calculation included each reading to be as precise as possible allowing the standard deviation (SD) of the samples to be calculated by having at least three readings for the cutting sample and wheel at each specific time.

Most experimental studies express their data distribution by using either the standard deviation (SD) or the standard error of the mean (SE). There can be confusion between the SD and SE in research. Nevertheless, they are statistically different and each of them has its own calculation process and meaning. SD refers to how close the mean is to the actual sample data (it is a dispersion of data in a normal distribution), while SE statistically extracts this sampling distribution (it is the SD of the theoretical distribution of the sample means), therefore, the SE is a result of SD per square root of the number of data points [224]. The standard deviation is calculated by applying equation (7-5) [224]. In this work,

156 Microsoft Excel software was used to find the standard deviation and to plot results to represent wear curves of cut-off wheel and samples against cutting time.

𝑺𝑫𝒆𝒗. = √∑(𝒙−𝒙`)_𝒏−𝟏 ………Equation 7-5

At this point the scatter plot can be used to estimate wear properties of the testing samples, using the plotted curves with error bars (see Figure (7-9) for example), to find wear properties of samples.

In many experiments regression analysis is a useful tool to understand the behaviour of data and the trend (here this is the volumetric wear (W)). Where there is a linear relationship between two parameters plotted against each other, the regression equation can be used to estimate the relationship between the dependent variable and the independent one [225] (W/t in the current study).

Here, we have assumed that the curve is divided into three stages, each of near-linear form, by which the Linear Least Squares Method equation (7-6) can be applied for data fitting.

Y = b X + a ………. Equation 7-6

where: Y represents dependent variable (volumetric lose in mm3) X represents independent variable (time in Sec)

b is slope of line , b= ∑ 𝑿𝒀− 𝟏 𝒏∑ 𝑿 ∑ 𝒀 ∑[𝑿𝟐_]−𝟏 𝒏 (∑ 𝑿)𝟐 ……….Equation 7-7 a is y intercept (that fixed value of Y when X=0), a=

∑ 𝒀− 𝒃 ∑ 𝑿

𝒏 …Equation 7-8

n is a number of data points

An important assumption in linear regression is that the relationship is indeed linear, that the slope does not change. The assumption of linearity is used when it can at least offer a measure of the trend [225]. Another important factor therefore when using the linearity assumption is to estimate how close it is to reality; this factor is called a Coefficient of determination r2

(XY) or (R2) [226], a value between 0-1 (close to 1 means the regression

linearity is high, and vice versa if it is close to zero). This factor can be found by equation (7-9).

R2 = 1- 𝑆𝑆𝑒𝑟𝑟

𝑆𝑆𝑡𝑜𝑡 ………Equation 7-9

157 SSerr=∑(𝑌𝑖− 𝑌∗)2

The linear regression model is used for prediction of Y with given X (if they are independent). When using it, it is good practice to state the R2 of the equation [226], [227]. Many computer programs can be used to do these calculations, particularly for a large amount of data. One of the most widely available is Microsoft Excel which is used here. An example result, for the first stage of wheel wear in the cutting of Ni, Figure (7-9), is shown below.

The first step is filling first two columns in Table (7-1) from raw data (data readings from test), then calculating the next two columns and finding the summation of all values. The next step is substituting these values in to equations (7-7 and 7-8) to find the slope and intercept of the least square fitting line (b and a) as follows:

b= 260.3975− 1 5 ×5× 180.165 7.5− 1₅ ×25 = 32.092 a= 180.165−32.092 × 5 5 = 3.9394

Now the linear regression equation is ready substituting a and b in Equation 7-6

to be Y= 32.092 X + 3.9394 ………....Equation 7-10

at this point we can find R2 when using Sum. values of SSerr and SStot from Table (7-1) as

that shown with green highlighted in this table.

Table (7-1): Values used in least square regression method analysis and correlation.

X Y X*Y X^2 Y^2 Y*(bX+a) SStot SSerr

0 0 0 0 0 3.94 1298.38 15.52 0.5 23.53 11.77 0.25 553.66 19.99 156.33 12.56 1 35.06 35.06 1.00 1229.20 36.03 0.95 0.94 1.5 59.16 88.73 2.25 3499.31 52.08 534.62 50.09 2 62.42 124.84 4.00 3896.26 68.12 696.27 32.53 Sum 180.17 260.40 7.50 9178.44 2686.55 111.65

Mean 36.03 SSerr/ Stot 0.0416

R2 1-0.0416

158

Chapter 8 Abrasion Wear Assessment of AMDCs (Cutting

Test)

This chapter will focus on the primary abrasive wear test, which assesses the resistance of a material to a process sometimes named cutting wear [134]. A test is developed and described which can be used to assess the performance of the MMDCs produced here under conditions of abrasive cutting.

To verify the test and be sure the observed wear in the cutting blade was caused by composite effects only we have also cut one of the empty mild steel moulds (without filling by composite) showing no noticeable wear on the cutting blade.

Displaying the tested wear properties of all samples (Al, Sn and epoxy matrices) processed by molten metal infiltration in histogram plots together, allows comparison of the wear behaviour, Figure (8-1) a and b for samples and wheels, respectively and Figure (8-2) for relative wear of composites.

Through this study the effect on abrasive wear of a number of material parameters is assessed, including the effects of diamond particle size and its surface condition. These are explored for the fixed condition of aluminium as the matrix and processing by the gas infiltration process. After it is found which particle size is the best, the result will be compared with aluminium reinforced by the same size reinforcement processed by other manufacturing methods; conventional powder metallurgy (PM) and spark plasma sintering (SPS). Then ceramic particles (alumina and silicon carbide) are substituted for the reinforcement using the same way of manufacturing (GI).

In the next chapter, other matrices are used with different production methods applied to study their impact on wear behaviour in MMDCs. This chapter continues with composites manufactured by melting matrices.

159

Figure (8-1): Summary of abrasive wear of composites a-samples and b-wheels used to cut composites.

a

160

Figure (8-2): Summary of relative wear of composites.

8.1 Effects of Different Diamond Particle Sizes and Surface