Experiment #4

Standard Tension Test

Stephen Mirdo

Performed on October 25, 2010

Report due November 1, 2010

Table of Contents Object ………………………………………..………………………….………….…. p. 1 Theory …………………………………………………………………………..…pp. 1 - 3 Procedure ………………………….…………………………………...……..……..... p. 4 Results ….................................................................................................................. p. 5 - 7 Discussion and Conclusion …………………….......……………………..............…... p. 8 Appendix ……………………………………..…………………..….……..…... pp. 9 - 11

Object The object of this experiment was to establish the stress-strain relation ship of ductile metals and brittle materials.

Theory

The determination of a material’s mechanical properties indicates what type of applications a material can be used for. For example, a material that has a high elastic tendency would be an ideal material to select for the manufacture of a spring, whereas a highly ductile material with a low elastic tendency would be a poor choice. The mechanical properties of a material consist of its yield strength, tensile strength, modulus of elasticity and a host of other properties beyond the scope of this experiment. The term strength is used instead of stress because the strength is a property applied to the test material and stress, σ, is related to the magnitude of the applied load, F, divided by the cross sectional area of the specimen, A.

σ = F/A (Equation 1)

The determination of a material’s tensile properties can be accomplished by means of a uniaxial tension test. A depiction of a tension test apparatus can be seen in Figure 1. A test specimen is locked into the apparatus and a load is applied in either compression or tension. For the purposes of this experiment, a tension test will be the focus. As a tensile force is applied, the material will elongate and eventually fracture. As the material’s dimensions change, observations of the force applied are made and used to determine certain mechanical properties of the material such as the yield strength and the ultimate tensile strength.

Figure 1: Uniaxial tension test apparatus (Source: Fundamentals of Material Science and Engineering: An Integrated Approach, W.D. Callister, Jr and D.G. Rethwish, 2008)

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The yield strength of a material is defined as the maximum stress a specimen can sustain without suffering any plastic, or unrecoverable, deformation. As force is applied to the test specimen, as seen in Figure 1, uniaxial stress occurs in the specimen. The specimen will elongate as the force is applied. However, if the magnitude of the applied force is such that the induced stress in the material does not exceed the material’s proportional limit, once the stress is relieved, the material will regain its original proportions. This implied that the deformation incurred at or below the yield strength of the material results in plastic deformation.

The yield strength can be identified on a stress-strain plot as seen in Figure 2

below. The yield strength, σy, is the point along the stress-strain curve where the relationship between the induced stress and incurred strain ceases to be linear. The expanse between the origin and the yield strength point is the proportional limit of the material. Any further induced stress in the material beyond the yield strength will result in plastic deformation.

Figure 2: Typical stress-strain plot of a ductile material

The ultimate tensile strength of a material is defined as the maximum stress a

specimen can sustain before it begins to fracture. The ultimate tensile strength is located within the plastic deformation region of the stress-strain curve. Prior to this point, indicated as σUT in Figure 2, deformation is uniform in the test specimen. When the ultimate tensile strength has been achieved, necking will occur in the test specimen and any subsequent deformation is confined to this necked region. Necking is defined as a non-uniform reduction in cross sectional area of the test specimen and a visual representation can be seen in Figure 3.

Figure 3: Diagram of the test specimen through tension test. Subset (a) depicts the original proportions of the test specimen, subset (b) depicts elongation that occurs at σy and subset (c) indicates necking that occurs at σUT.

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By employing Hooke’s Law, the modulus of elasticity, E, can also be determined using the stress-strain plot in Figure 2. The slope of the stress-strain curve from the origin to the yield strength is equivalent to the modulus. The modulus of elasticity of a material is defined as a measure of how a material deforms when under stress. This property is indicative of a material’s ability to transmit stress through the test specimen. Hooke’s Law is defined as the proportionality of a load to deflection incurred by the load. For a uniaxial loading scenario, Hooke’s Law is written as:

σ = Eε (Equation 2)

where σ is the uniaxial stress, E is the modulus of elasticity of a material and ε is the uniaxial strain. The uniaxial stress, or in this case, the tensile stress is determined using Equation 1, where F is the applied force and A is the cross-sectional area of the specimen. The strain, ε, is computed by dividing the difference in the gage length of the test specimen (ΔL) due to loading by the original length of the test specimen, L. Strain is usually written as a unitless number, but can be written as a length per length such as in/in.

ε = ΔL / L = (Lf – Li) / L (Equation 3) Rearranging Hooke’s law in Equation 2 to solve for the modulus of elasticity, E, is written as follows:

E = σ / ε (Equation 4) Relating the slope of the stress-strain curve as seen in Figure 2 allows Equation 4 to be written as follows:

E = Δσ / Δε (Equation 5) where Δσ is the difference of stresses at two given points on the stress-strain curve and Δε is a corresponding difference of strains at the defined points.

Ductility is another important mechanical property and can be expressed as a percent elongation or percent area reduction. The specimen under tension will elongate and naturally have a longer final length. The percent elongation is calculated as follows:

%EL = [(Lf – Li) / Li] * 100 (Equation 6)

As the specimen elongates during the tension test, it will experience a reduction in cross-sectional area. The percentage of reduction relates to the ductility of the material. The greater the value of percent reduction in area, %RA, a material has, the greater the ductile property of the material. The percent reduction in area can be calculated with the following equation:

%RA = [(Af – Ai) / Ai] * 100 (Equation 7)

where Af is the final cross-sectional area measured after the fracture of the specimen and Ai is the initial cross-sectional area measured before the tension test.

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Procedure Equipment: Tension Test Apparatus Aluminum 2017 Test Specimen Steel C1045 Test Specimen Polymethyl Methacrylate (PMMA) Test Specimen Experiment:

1) Measure and mark the initial length and diameter of the gage section of the test specimen. If the specimen is not circular in cross-section, measure the thickness and width of the mater for computation of the cross-sectional area.

2) Mount the specimen into the grips of the tension testing apparatus. Ensure

that the threads of the specimen are screwed far enough into the mount. If using a specimen that is not of circular cross-section, ensure that the specimen is loaded completely vertical.

3) Zero all load ranges using the range selector and appropriate zero adjustor. 4) Close the unload valve.

5) Apply the load slowly and record the yield load.

6) Continue loading the test specimen and record the ultimate load.

7) Remove the broken test specimen from the machine. 8) Record the final gage section length and the neck diameter.

9) Repeat steps 1 – 8 for each test specimen.

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Results

Table 1: Initial and final dimension of the Aluminum 2017 test specimen Initial Diameter (in) 0.505 Diameter at Fracture (in) 0.370 Initial Gage Length (in) 2.000 Gage Length at Fracture (in) 2.580

Stress-Strain Plot of Aluminum 2017

0

10000

20000

30000

40000

50000

60000

70000

80000

0 0.05 0.1 0.15 0.2 0.25

Strain (in/in)

Stre

ss (p

si)

Figure 4: Stress-strain plot of Aluminum 2017 from which the elastic modulus was

calculated and the yield strength and ultimate tensile strength was determined. Table 2: Calculated modulus of elasticity as well as determined yield and ultimate yield strengths with applicable percent error.

Experimental Theoretical % Error Modulus of Elasticity E (ksi) 9873.7 10500 6.0% Yield Strength (psi) 48900 40000 22.3% Ultimate Tensile Strength (psi) 69800 62000 12.6% % Elongation at Fracture 23.8% N/A N/A % Reduction in area at Fracture 46.3% N/A N/A

The elastic modulus for the aluminum specimen was determined by using

Equation 5, employing Hooke’s Law. The data points selected for this calculation were within the linear portion of the stress strain curve. The yield strength was determined by examining where the stress-strain curve ceased to be linear. The tensile strength was determined by examining the peak of the stress strain curve. The reduction in area was calculated by using Equation 7 and percent elongation was determined by using Equation 6.

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Table 3: Initial and final dimension of the Steel C1045 test specimen Initial Diameter (in) 0.505 Diameter at Fracture (in) 0.370 Initial Gage Length (in) 2.000 Gage Length at Fracture (in) 2.580

Stress-Strain Plot of Steel C1045

0

20000

40000

60000

80000

100000

120000

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2

Strain (in/in)

Stre

ss (p

si)

Figure 5: Stress-strain plot of Steel C1045 from which the elastic modulus was calculated and the yield strength and ultimate tensile strength was determined.

Table 4: Calculated modulus of elasticity as well as determined yield and ultimate yield strengths with applicable percent error.

Experimental Theoretical % Error Modulus of Elasticity E (ksi) 26002 29000 10.3% Yield Strength (psi) 84700 84800 0.1% Ultimate Tensile Strength (psi) 113300 95000 19.3% % Elongation at Fracture 17.90% N/A N/A % Reduction in Area at Fracture 37.3% N/A N/A

The elastic modulus for the steel specimen was determined by using Equation 5,

employing Hooke’s Law. The data points selected for this calculation were within the linear portion of the stress strain curve. The yield strength was determined by examining where the stress-strain curve ceased to be linear. The tensile strength was determined by examining the peak of the stress strain curve. The reduction in area was calculated by using Equation 7 and percent elongation was determined by using Equation 6.

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Table 5: Initial and final dimension of the Polymethyl Methacrylate test specimen Initial Area of Cross-section (in^2) 0.03236 Cross-sectional Area at Fracture (in^2) Unknown Initial Gage Length (in) 1.000 Gage Length at Fracture (in) 1.047

Figure 6: Stress-strain plot of Polymethyl Methacrylate (PMMA) from which the elastic modulus was calculated using a secant line due to the material’s nonlinear elastic region of its stress-strain curve. Table 6: Calculated modulus of elasticity as well as determined yield and ultimate yield strengths with applicable percent error. Experimental Theoretical % Error Modulus of Elasticity E (ksi) 115.2 479 76.0% Yield Strength (psi) ? 40000 N/A Ultimate Tensile Strength (psi) 4879 10400 53.1% % Elongation at Fracture 4.7% 2% - 5% N/A % Reduction in area at Fracture Unknown N/A N/A

Some materials, such as the polymer PMMA, do not exhibit a linear stress-strain

curve beneath the material’s proportional limit. In lieu of calculating the slope of the actual curve, the slope of a secant line from the origin to an arbitrary point on the curve was used. The calculated elastic modulus is not in agreement with accepted values. The yield strength of the material is indeterminable using this experimental method. The ultimate tensile strength was determined by the stress at fracture. The elongation was calculated using Equation 6. The reduction in area is unknown as the final dimensions were not given.

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Discussion & Conclusion

From the results of the percent reduction in area calculations, a comparison of the ductility of the two metal test specimens, aluminum 2017 and steel C1045 can be made. It was noted that the %RA of the aluminum, 46.3%, had a higher value than that of the steel, 37.3%. This implies that steel would be better suited to an application where the strength of the material is an asset and aluminum would be a good choice for an application where a more ductile material is needed.

The steel specimen has an ultimate tensile strength value of 113,300 psi, which is

greater than that of aluminums, 69,800 psi. This implies that the steel can sustain higher stresses before necking and inevitably fracturing. Therefore, it can be concluded that the steel is a tougher material than the aluminum.

Another method of determining the strength of a material using the stress-strain

curve employs calculus. Develop an equation for the stress-strain curve and integrate it with respect to the strain axis. The integrand will yield the area under the stress-strain curve and its value can be interpreted as the strength of the material.

The PMMA test specimen used in this experiment is considered to be a brittle

material. The material fractured at a stress concentration that was far less than the values yielded by the steel or aluminum. A percent reduction in area calculation, used to determine the ductility of the material, could not be carried out as the final cross-sectional area of the specimen was not recorded. A percent elongation calculation, however, can also be used to determine a material’s ductility. The calculation indicated that at 4.7% elongation before fracture, PMMA is not a very ductile material. It is better suited for applications where a lightweight, yet rigid material is sought.

Error is present in this experiment. The PMMA test specimen did not appear to

be loaded completely vertically in the test apparatus. This discrepancy may have caused the skewed values for elastic modulus and tensile strength. Another source of error was that the testing apparatus for the ductile and brittle materials were not zeroed out before the test began. The applied force before testing began may have skewed the actual values. Error is also present in that the extensometer used in the aluminum specimen’s tension test may have slipped during the experiment. This would cause the strain values used in calculation to be inaccurate.

Improvements to this experiment are few and far between. The tension test is an

excellent way to gather certain mechanical properties of materials. However, the rate at which the load is applied to the test specimen is “tweakable.” If the load is applied at a slow rate, such as 100 lbf/min, the mechanical properties of the material can be more accurately obtained. The speed of loading can influence how a material will behave. Another improvement to this experiment would be to know the treatments, such as coldworking, a material specimen received before undergoing a tension test to demonstrate how said treatments influence material properties.

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Appendix Data Usage Sample calculation of elastic modulus for the aluminum 2017 test specimen:

(32700 psi – 2080 psi) / (0.00493499 in/in - 0.001833813 in/in) = 9873.7 ksi

Sample calculation of percent area reduction for steel C1045 test specimen:

| (π/4) * [(0.505 in)2 – (0.400 in)2] | / [(π/4) * j33] x 100 = 37.3% Sample calculation for percent elongation at fracture for PMMA test specimen:

|1.047 in – 1.00 in| / 1.00 in x 100 = 4.7%

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Bibliography

Mechanics of Materials, 2nd Edition Timothy A. Philpot (2011)

Fundamentals of Material Science and Engineering: An Integrated Approach

W.D. Callister, Jr and D.G. Rethwish (2008)

Materials Laboratory Manual, Fall 2010 University of Memphis, Department of Mechanical Engineering

MatWeb Material Property Data

http://www.matweb.com