MAAE2202_Lab A

Carleton University Laboratory Report Course #: MAAE 2202 Lab #: A Lab Section #: Load Deformation Behaviour for Engineering Materials Summary

The purpose of the lab was to obtain the stress – strain relationship from tensile force testing for aluminum 2011 series and brass 360 series. Using the TECQUIPMENT Universal Testing Machine SM100, an increasing tensile force was applied to each of the specimens and measured using the software. By attaching an extensometer, the change in length of the specimens due to the tensile force was measured precisely. Using these values as well as the measured cross-sectional area, the stresses and strains on the specimens were calculated. With this a stress – strain curve was plotted, and the elastic modulus was determined by the slope of the linear portion of the curve. The elastic modulus for aluminum was calculated to be 65 GPa and brass 81 GPa. The yield stress was determined by offsetting the line for Young’s modulus by 0.2% strain and finding the point in which it intercepted with the original curve. The yield stress for aluminum was calculated to be 316 MPa and brass 261 MPa. Nomenclature Symbol Parameter Unit F Tensile Load kN ΔL Extension mm L original Original Length mm ε Strain Unitless σ Stress GPa σ y Yield Stress MPa E Elastic/Young’s Modulus GPa A Cross-Sectional Area mm 2 d Diameter mm %error Error Percentage % Table 1: Nomenclature used for this lab report. Theory and Analysis Strain is the ratio of extension to original length of a component, and can be represented by equation 1: ε = ∆ L L original Stress is the force per unit area within a component, with area being the cross-sectional area. Stress can be represented by equation 2: σ = F A = 4 F π d 2 Young’s modulus of a material is a property that describes how easily it can stretch and deform. It is the ratio between stress and strain, and can be represented by equation 3: E = σ ε = ∆σ ∆ε Since Young’s modulus is the gradient between stress and strain, the slope of the line on a stress-strain graph will be the elastic modulus. P a g e 2 | 12

Load, F Gauge Extension, ΔL Strain, ε Stress, σ kN mm unitless GPa 0.0 0.00 0 0 0.2 0.00 0.00006 0.002640688 0.7 0.00 0.00018 0.009242409 2.2 0.00 0.00032 0.029047571 2.8 0.00 0.00042 0.036969636 3.6 0.04 0.00064 0.047532389 4.5 0.08 0.00084 0.059415486 5.9 0.09 0.001 0.077900304 6.6 0.10 0.00116 0.087142713 7.3 0.12 0.0013 0.096385122 8.6 0.14 0.0015 0.113549596 9.3 0.17 0.00164 0.122792005 9.9 0.19 0.0018 0.13071407 10.5 0.21 0.00196 0.138636135 11.8 0.24 0.0021 0.155800609 12.4 0.27 0.00224 0.163722674 12.9 0.28 0.0024 0.170324394 13.8 0.32 0.00262 0.182207492 15.0 0.34 0.00274 0.198051621 15.7 0.37 0.00292 0.20729403 16.4 0.39 0.0031 0.216536439 17.1 0.43 0.00334 0.225778848 18.4 0.46 0.00348 0.242943322 19.2 0.50 0.00376 0.253506075 19.8 0.53 0.00404 0.26142814 20.4 0.55 0.0042 0.269350205 21.5 0.59 0.00444 0.283873991 22.1 0.64 0.00484 0.291796055 22.6 0.69 0.00512 0.298397776 23.1 0.74 0.00566 0.304999497 23.8 0.84 0.00666 0.314241906 24.1 0.94 0.0079 0.318202938 24.3 1.04 0.00948 0.320843627 24.5 1.12 0.01052 0.323484315 24.8 1.20 0.01118 0.327445347 24.8 1.28 0.01144 0.327445347 24.9 1.32 0.0116 0.328765691 25.0 1.36 0.0118 0.330086036 25.2 1.45 0.01184 0.332726724 P a g e 4 | 12

Table 2: Calculated stress and strain values from applied tensile force, change in length of extensometer, original length of extensometer, and cross-sectional area of the aluminum 2011 specimen. Load, F Gauge Extension, ΔL Strain, ε Stress, σ kN mm unitless GPa 0.0 0.000 0 0 0.1 0.000 0 0.001331203 0.4 0.004 0.00008 0.005324814 1.3 0.008 0.00016 0.017305644 1.6 0.011 0.00022 0.021299255 1.9 0.014 0.00028 0.025292865 2.2 0.016 0.00032 0.029286475 3.1 0.019 0.00038 0.041267306 3.6 0.026 0.00052 0.047923323 4.2 0.031 0.00062 0.055910543 4.9 0.038 0.00076 0.065228967 6.5 0.049 0.00098 0.086528222 7.0 0.054 0.00108 0.093184239 7.8 0.063 0.00126 0.103833866 8.4 0.070 0.0014 0.111821086 9.5 0.077 0.00154 0.126464324 10.0 0.085 0.0017 0.133120341 10.4 0.093 0.00186 0.138445154 11.4 0.107 0.00214 0.151757188 13.1 0.121 0.00242 0.174387646 13.8 0.136 0.00272 0.18370607 14.3 0.145 0.0029 0.190362087 14.9 0.155 0.0031 0.198349308 15.7 0.161 0.00322 0.208998935 16.1 0.175 0.0035 0.214323749 16.6 0.186 0.00372 0.220979766 17.1 0.198 0.00396 0.227635783 18.0 0.212 0.00424 0.239616613 18.7 0.230 0.0046 0.248935037 19.3 0.253 0.00506 0.256922258 19.7 0.264 0.00528 0.262247071 20.1 0.275 0.0055 0.267571885 20.5 0.297 0.00594 0.272896699 20.7 0.314 0.00628 0.275559105 20.9 0.333 0.00666 0.278221512 21.3 0.357 0.00714 0.283546326 21.4 0.388 0.00776 0.284877529 P a g e 5 | 12

Figure 5: Plot of relationship between stress and strain for brass 360 using extensometer. As was for the aluminum, the orange section of the curve was used for the slope. Young’s modulus can be seen as the slope of the offset orange line, 80.924 GPa. The intercept of this 0.2% offset line and the stress – strain curve is the yield stress, 0.261 GPa or 261 MPa. Requirement C Aluminum 2011 Calculated Value Published Value Young’s Modulus, E (GPa) 65.239 70.3 Yield Stress, σ y (MPa) 316 296 Table 4: Comparing calculated values to MatWeb’s published values for Aluminum 2011. The calculated elastic modulus is very close to the published value, having a error percentage of only 7.2%. The Yield Stress was also very close to the published yield tensile strength, having an error percentage of 6.7%. Brass 360 Calculated Value Published Value Young’s Modulus E (GPa) 80.924 97.0 Yield Stress, σ y (MPa) 261 124-310 Table 5: Comparing calculated values to CCT’s published values for Brass 360. The calculated elastic modulus for brass was much less accurate than the aluminum when comparing to the published values, with an error percentage of 16.6%. The calculated yield stress fell in between the range of the published values. P a g e 7 | 12

When analyzing the stress – strain curves for the two specimens, the aluminum had a much clearer and longer linear portion. This made the trend line much more accurate due to the greater number of data points used for the slope. The curve for the brass specimen almost looks like a logarithmic curve with little to no linear section. This made identifying the elastic section difficult and led to greater inaccuracies in the calculations. Requirement D As was calculated and observed from the published values, the elastic modulus for the brass alloy was greater than the aluminum alloy. Young’s modulus of elasticity is essential to predicting behaviour of metals. A higher value means the material is more rigid; therefore, the brass 360 alloy is stiffer than aluminum 2011 alloy. In engineering context, this could be useful in determining which material to use for a building, bridge, engine, or any project requiring physical materials. Of course, other properties of the materials would be analyzed as well. Like Young’s modulus, yield stress is a property that would be considered when determining what metals to use for a project. The calculated value for yield stress of brass was lower than the calculated value for aluminum. Yield Stress is used in engineering to determine the maximum allowable load in mechanical components. In a project where components would need to be stiffer and have greater loads applied, the brass alloy would be a better fit than aluminum. Requirement E Figure 6: Plot of relationship between stress and strain for brass 360 using LVDT. P a g e 8 | 12

References MAAE 2202 - Lab Manual pages 3-10 (MAAE 2202A/B Brightspace course page, Administration tab) “Aluminum 2011-T3.” MatWeb , https://www.matweb.com/search/datasheet_print.aspx? matguid=8c05024423d64aaab0148295c5a57067. “ Brass Data Sheet.” CCT Precision Machining , https://www.cctprecision.com/materials/brass-data- sheet/. Appendices Appendix A: Specimen Drawing Figure 8: Engineering Drawing of tensile specimen used in the experiment, taken from the lab manual. Appendix B: Calculations Calculation for strain ε = ∆ L L original Example calculation, with the original length being the length of the extensometer, 50 mm P a g e 10 | 12

ε = 1.71 mm 50 mm = 0.0342 Calculation for stress σ = F A = 4 F π d 2 Example calculation for the aluminum which had a measured diameter of 9.82 mm σ = 4 ∙ 24.5 kN π ( 9.82 mm ) 2 = 0.3235 kN m m 2 = 0.3235 GPa Calculation for Young’s Modulus E = σ ε Since the graph was set up in a way that the y axis was stress, σ and the x axis was strain, ε, the slope of the line (y/x) or (σ/ε) was E, Young’s Modulus Calculation for error percentage %error = | x calc − x pub x pub | × 100% Example calculation using Young’s modulus for aluminum %error = | 65.239 − 70.3 70.3 | × 100% = 7.2% Appendix C: Data Acquisition Software Figure 11: Information provided by the data acquisition software. P a g e 11 | 12