代写ME 2356-Mechanics of Materials帮做R程序

Department of Mechanical and Industrial Engineering

ME 2356-Mechanics of Materials

Lab 1 Instructions

I. GOALS

The primary goals of this experiment are:

1. To determine the relationship between tensile normal stress and normal strain for various materials

2. To learn techniques for tensile testing on electromechanical testers

3. To analyze stress-strain data to determine mechanical properties including Young’s modulus, yield strength, ultimate tensile strength, and fracture strength

4. To learn about the photoelasticity and stress concentration

II. OVERVIEW OF EXPERIMENT

There are three experiments in this lab.

• The first part includes tension tests of a material on the large Instron mechanical testing machine.

• In the second part you will perform. a manual tensile test to study unloading and reloading of the sample.

• In the third part, you will perform. a photoelasticity test to look at stress concentration on different samples.

III. RELEVANT THEORY

Testing in tension is one of the most common mechanical testing methods, since one test can be used to determine several important mechanical properties. In a tension test, tensile forces are applied to a specimen, and the resulting changes in the specimen’s length are recorded. Standard specimens are of a ‘dogbone’ shape, as shown in Figure 1, with a reduced cross-section in the middle of the sample. This reduced cross-section contains the ‘gauge length’; we will determine strain by measuring the length of this part of the specimen and tracking how it changes during loading. The gauge length is important, as it is necessary to determine the strain from the displacement data.

Figure 1: Typical tensile specimen

The testing machine itself typically measuresload and displacement. However, in order to be able to design with materials, we generally want stress and strain. Stress is given by:

where σ is the normal engineering stress, F is the load, and A0 is the original cross-sectional area of the gauge length.

Engineering strain is given by:

where ε is normal engineering strain, li is the instantaneous length and l0 is the initial gauge length. Strain is unitless, whereas stress is given in MPa. Remember, the machine will provide force data – you will need to convert this to stress during your analysis. Also remember that the machine itself will provide displacement data, and you will need to convert this to strain during your analysis.

The strain data provided by a tensile test machine are often considered to be unreliable. That is because the displacement that the machine reads out reflects the stretching of the sample plus the stretching of the machine’s components. To achieve more reliable data, engineers typically use another means of measuring strain, the extensometer. The extensometer clips onto the sample, extends with the sample (ideally requiring no force for its extension) and reports its elongation. Extensometers typically report their results directly in terms of strain, calculated as described above. However, it is still important to have the raw displacement measurement because the extensometer must be removed well before the test is finished. If the extensometer is still attached when the specimen breaks, it will break, too.

Stress strain curves provide information about a number of properties, as shown in Figure 2. The initial linear portion of the curve is the elastic region. The slope of the curve in this region provides the Young’s modulus, which is a key indicator of the stiffness of a material. The proportional limit is at the of the linear region, where the fitted line to this region deviates from the curve. The elastic region ends at the yield point, which is identified as follows. Draw a line that is parallel to the elastic region but is offset to the right of it by a strain of 0.002. Where this line crosses the stress-strain curve is defined as the yield point. For this reason, it is also referred to as the 0.2% offset yield strength. The maximum stress found during the tensile test is designated the ultimate tensile strength. After this point is exceeded, the gauge section becomes narrower – this is referred to as necking. After necking begins, the sample tends to rapidly progress to failure. The stress at the point of failure is the fracture strength. During the lab you will be using your data to determine these materials properties.

Figure 2: Typical properties determined from a stress strain curve.

All of the values discussed so far have been determined from a plot of engineering stress and strain. However, engineering stress does not consider the changing cross-sectional area of the specimen. True stress does take this into account, and is determined by:

where σT is the true stress and Ai is the instantaneous cross-sectional area. You will be measuring the cross-sectional area of the sample after failure, when the cross section has become thinner due to necking. This will allow you to determine true stress at failure in the necking region.

Another material property that can be determined from tensile testing is the ductility, which can be expressed as a reduction in area:

where A0 is the initial cross-sectional area of the sample and Af is the final cross-sectional area of the sample. The reduction in area is measured at the point of fracture.

Finally, the resilience of a material, as measured by the modulus of resilience, can be determined from stress-strain data. Resilience is a measure of the ability of a material to absorb and recover energy in the elastic region. It can be determined by finding the area under the engineering stress-strain curve up to the point of yielding or:

where Ur is the modulus of resilience and εy is the strain at yielding. Assuming the elastic region is linear, this becomes:

where σy is the yield strength.

1. INSTRON – EXPERIMENT 1

There are two distinct methods to measure mechanical properties of a sample rod under tension:

(i) Force-controlled or Fixed-load (F = constant) configuration. This is the simple test you performed in high school to measure spring constant (k) of a rubber band. A rubber band is hung on fixed wall at top. Small weights (F) are added to a pan hanged on the lower end, while the incremental extension (d) is measured using a ruler. A set of data d(F) is plotted to yield a slope of (dF/dd) which is the spring constant. The pseudo elastic modulus is given by

where k = (E A / L) is the spring constant expressed in terms of the elastic modulus, cross-sectional area and original length. This method allows measurement only up to the ultimate strength at A (shown on the graph below) before failure such as necking and fracture occurs. No data can be obtained for necking along curve section AB.

(ii) Displacement-controlled or Fixed-grips (d = constant) configuration. Measurement is performed by universal testing machines (UTM) such as Instron or MTS. Sample rod is firmly held by a lower grip that is stationary and an upper grip that moves at a slow speed. As the sample extends, the associated force is measured by a load cell. The software generates data set of F(d) and the slope of (dF/dd) is the spring constant. This method allows full measurement of curve OAB from initial loading to final failure, including the intermediate necking AB.

Figure 3. Stress-strain curve showing necking region.

We will be using an Instron 5582 testing frame. (shown in Figure 4) for this experiment. This machine has a 100 kN maximum load and can measure in both tension and compression. For small amounts of strain, such as in the elastic region of a tensile stress-strain curve, an additional tool called an extensometer (also described above) is used to provide a more accurate measure of strain. The extensometer is clipped onto the sample to measure these initial strains. The extensometer needs to be removed after the material reaches the yield point, but before the sample breaks. Failure to do this will damage the extensometer. The data is recorded using the Instron Bluehill software.

Figure 4. Instron 5582 Electromechanical Testing Machine.

1.1 INSTRON: PROCEDURE

1. Mount the sample in the Instron machine:

a) Install the specimen into the “jaws” of the Instron machine. Make sure it is lined up with the grooves in the jaws. Hand-tighten the upper jaw first by turning the crank. Slowly move the upper jaw down using the scroll wheel on the key panel and when the sample is all the way down in the lower jaw, hand-tighten the jaw. By placing the sample in the upper jaw first and then the lower jaw, we make sure to minimize sample deformation.

b) Measure the initial length Lo and initial diameter Do of the specimen. (use the distance between the grips as the initial length, Fig 5. right)

Figure 5. Instron components and sample.

c) Connect the extensometer directly to the sample. Hold down the black push button (this aligns a push pin to set the initial length of the extensometer) and slide the clips on to the sample. Ensure that the extensometer is located in the middle of the sample length. Release the push button to retract the alignment pin.

Figure 6. The extensometer.

2. Set up a tensile test in BlueHill software:

a) In the ‘Method’, choose ‘Tension Method’.

b) Set units to ‘SI’.

c) In the ‘Measurements’ tab: Double click on ‘Strain’. This will add ‘Strain 1’ to the list on the right side. This is the strain value from the extensometer.

d) In the ‘Test control’ tab:

i. Under ‘Test’: enter the displacement rate as 10 mm/min.

ii. Under ‘Strain’, make sure that ‘Strain 1’ is chosen as the ‘Primary source’. Select the option to ‘remove extensometer during test’. For measurement choose ‘Strain1’ . Set the value to 2% and for ‘Action during removal’ choose ‘Pause test and continue data capture’.

e) In the ‘Workspace’ tab, under ‘Raw Data’, double click on ‘Strain 1’ to add it to the list on the right.

f) In the ‘Export’ tab: to export the Raw data, under ‘Export 1’, check the ‘Raw Data’ for the ‘Content’ and uncheck the rest.

g) Save and name your method by clicking on ‘Save method’ on the top right (save on Desktop).

3. To run the test. Click on home and then ‘Test’. Choose the method you just created, then click ‘OK’.

4. Open ‘System Details’ (can be found on Fig. 7) to add ‘Strain 1’ to the display.

5. To ensure the load cell is balanced and the elongation and strain are zeroed, for each one, click on the ‘Digits’ on the top and click ‘Balance’ or ‘Zero’ (digits shown on Fig 7).

6. Start the test by clicking the ‘Start’ button in the bottom right. The test will pause automatically at 2% strain (when the curve plateaus). Once it pauses, open the Instron door and carefully remove the extensometer. Then continue running the test until the specimen breaks.

7. Before removing the sample’s broken sections from the Instron, close the gap between the broken sections (by manually moving the upper jaw down using the scroll wheel on the key panel) and measure the final length Lf and the final diameter df.

8. To export the data, click on Export (Fig. 7) and choose your sample. Chose “Export to ExportFile1.csv”. Then click “Desktop”. Type your name for the file name and click “Save”.

9. Send the file to you and your groupmates through Canvas Inbox.

Figure 7. Bluehill, Test tab.

1.2 INSTRON: RESULTS, ANALYSIS, AND REPORTING

This section is worth 46 points out of 100.

All the calculations and analyses must be performed in MATLAB. Resources to help you get back up to speed on MATLAB are available on Canvas. The university also has a MATLAB TA who holds office hours to help Northeastern students with MATLAB questions.

1. Write a brief introduction to the lab report (see template) (5 points)

2. Write a brief explanation of the Intron experiment in the Methods section of the template. (5 points) Include:

a. What instruments were used.

b. What sample was tested.

c. What was measured.

d. How the measurements were taken.

e. Show all equations used in your analysis of the data.

3. Report the values of L0, Lf, Do, Df, A0, and Af in a carefully labeled table. (2 points)

4. In Matlab plot the raw force and displacement data for the sample. You should be plotting displacement on the x-axis and force on the y-axis. Label your axes and include the units. (3 points)

5. Convert your experimental data to engineering stress and strain. In Matlab, plot curves for both engineering stress vs. strain and stress vs. extensometer strain on the same graph, labeling each curve. label the axes and include the units. Watch out for units! Check the units in your raw data. Extensometer might be in % strain. (5 pts, including the MATLAB code)

6. Mark the major points and regions on the stress-strain graph. You can use Microsoft Word or Power Point to label the points and regions on your graphs (7 points, 1 point each). This includes:

a. Elastic region.

b. Plastic region.

c. Necking region.

d. Proportional limit.

e. Yield strength.

f. Ultimate strength.

g. 0.2% offset line.

7. Calculate the following mechanical properties, report them in a clearly labeled table: (12 points, 2 points each)

a. Young’s modulus (using extensometer data) by fitting a line to the linear part of the graph.

b. Proportional limit (using crosshead data)

c. Yield strength (0.2% offset)

d. Ultimate tensile strength

e. True stress in the necked region just prior to failure

f. % Reduction in Area (%RA) at failure

8. Discussion: Calculate the elastic energy density of the specimen up to the yield point. Discuss how this quantity is related to the Modulus of Resilience. (3 points)

9. Discussion: Compare the calculated values with reference values from the internet. Name the sources of error and limitations. Discuss reproducibility (if someone else repeated the same experiment, would they get the same results as you? Why or why not?). (4 points)

2 MANUAL TESTER - EXPERIMENT 2

This experiment is to investigate the unloading and reloading behavior. of a material. In this experiment, you will load a sample until plastically deformed, unload it, and then reload it until failure. This will produce a graph similar to the graph in figure 8 (depending on the material used).

Figure 8. Stress-strain curve with unloading and reloading.

2.1 MANUAL TESTER: PROCEDURE

All specimens have a nominal diameter of 3.33 mm and a nominal working length of 3.2 cm.

1. Mount a sample in the Materials Testing Machine:

a. Place the sample (short threaded side down) through the top hole and screw it into the bottom hole. Screw in the sample until the top edge of the last thread is flush with the top of the load cell (Fig 9. Red arrows). The longer threaded section of the sample should go up through the hole in the center of the load bar. Turn the crank counter-clockwise to flush the surfaces shown by blue arrows in Fig 9.

b. Check that the height of the load bar is correct; the bottom edge of the longerthreaded section should be flush with the bottom of the load bar.

c. While holding the sample so that it does not turn, screw the sample nut onto the longer threaded section on the top until the sample is held “almost tightly”. Do not overtighten.

d. Attach the safety shields.

Figure 9. Mounted sample.

2. The PASCO Capstone software has been prepared for you.

3. Set a Pre-Load

a. Click Record to begin collecting data and rotate the crank clockwise until the force is up to 50-100 N.

b. Click Stop and DO NOT change the crank position. Since the Materials Testing Machine is set to automatically zero itself the next time you start recording data, this puts a pre- load of 50 N to 100 N on the sample, which results in better data.

4. Initial load cycle (you will pass the yield point) :

a. Click Record to begin collecting your real data.

b. Slowly rotate the crank clockwise to load the specimen in tension. Watch the software’s real-time graph of force and position. Continue increasing the load until afterthe specimen has completely passed the yield point (i.e. until the force-position curve has completely flattened out).

c. Once you’re convincingly past the yield point, stop turning the crank. Do NOT stop the data recording.

d. Unload cycle: Slowly rotate the crank counterclockwise to remove the load from the specimen. Watch the force-position graph as you unload the specimen. Continue unloading until the force returns to about 100N. Do NOT stop the data recording.

5. Second load cycle (load to failure):

a. Slowly rotate the crank clockwise to load the specimen in tension. Continue loading until the specimen breaks.

b. Click Stop to end the data recording.

c. Export the force and position data of only the last run on the computer desktop as a .csv file with a name that identifies who you are and what type of material was tested.

6. Export the data:

a. Click on “file” then click “Export Data”.

b. Click “Deselect all” and then in the lefthand menu select “Position” Run #2 and “Force” Run #2

c. Click export to file and save the data as a .csv file with your name and the material type (steel, brass, Al, etc.)

d. Go to Chrome, send the file to you and your teammate through Canvas.

7. Remove the specimen and put it into the recycling basket before you leave.

2.1 MANUAL TESTER: RESULTS, ANALYSIS, AND REPORTING

This section is worth 32 points out of 100.

All the calculations and analyses must be performed in MATLAB.

1. Write a brief explanation of the manual tester experiment in the Methods section of the template. (5 points) Include:

a. What instruments were used.

b. What sample was tested.

c. What was measured.

d. How the measurements were taken, etc.

e. Show all equations used in your analysis of the data.

2. In Matlab plot the raw force (load) and displacement data for the sample. You should be plotting displacement on the x-axis and force on the y-axis. Label your axes and include units. (5 points)

3. Convert your experimental data to engineering stress and strain. In Matlab, plot curves for stress vs. strain. label the axes and include the units. (5 pts, including the MATLAB code)

a. Note that: All specimens have a nominal diameter of 3.33 mm and a nominal working length of 3.2 cm.

4. Label the following on your stress-strain plot. You can use word or Power Point to Label (4 points):

a. Initial loading Young’s modulus by fitting a line to the linear part of the graph.

b. Reloading Young’s modulus.

c. Yield strength (0.2% offset)

d. Ultimate tensile strength.

5. Calculate the following mechanical properties and report them in a clearly labeled table (8 points):

e. Initial loading Young’s modulus by fitting a line to the linear part of the graph.

f. Reloading Young’s modulus.

g. Yield strength (0.2% offset)

h. Ultimate tensile strength.

6. Discussion: Compare the calculated values with reference values from the internet. Name the sources of error and limitations. Discuss reproducibility (if someone else repeated the same experiment, would they get the same results as you? Why or why not?). (5 points)

3 POLARISCOPE – EXPERIMENT 3

When you performed the mechanical tests above, you used specimens that had a uniform. cross-section along the entire length being measured. The cross-section of the specimens increased gradually from the area being studied to the larger area that was held in the grips. These constant cross-sections and gradual changes in area are important because they prevent stress concentrations from forming. You may have already discussed stress concentrations in class, and you may discuss them more in the future. Whether you have or not, this portion of the lab will give you the opportunity to learn about how they work firsthand and to see how practicing engineers can evaluate them experimentally.

Stress concentration occurs at a point in a specimen where the loading cross-section changes suddenly. The change in the cross-section could be a sudden change in the width of the cross-section, a hole punched out of the middle of the specimen, etc. Whenever there is a stress concentration, the stress is locally higher than the average value on the cross-section.

Photoelasticity describes changes in the optical properties of a material under mechanical deformation. It is often used to experimentally determine the stress distribution in a material, where it gives a picture of stress distributions around discontinuities in materials (i.e. at stress concentrations). Photoelastic experiments (also informally referred to as photoelasticity) are an important tool for determining critical stress points in a material (i.e., the points that experience the highest stress) and are used for determining stress concentration in irregular geometries.

A polariscope is used to visualize stress distributions via photoelasticity. The polariscope consists of two linear polarizers and a light source. First the light is passed through the first polarizer which converts the light into plane polarized light. The apparatus is set up in such a way that this plane polarized light then passes through the stressed specimen. This light then follows, at each point of the specimen, the direction of principal stress at that point. The light is then made to pass through the analyzer, and we finally get the fringe pattern.

Finite element analysis can also be used to represent the stress distribution in a sample as a color distribution. Figure 10 below shows a comparison of photoelasticity vs. finite element analysis.

Figure 10. Photoelasticity vs. Finite Element Analysis

3.1 POLARISCOPE: PROCEDURE

In this part, two samples will be used: a sample with a hole and a dogbone. The procedure is the same for both samples.

1. Lower the loading screw until it barely touches the sample. Do not load the sample just yet.

2. Take a picture of the unloaded sample through the polarizer.

3. Load the sample until you see a slight change in the color over the sample. Take another picture in this state (low compression).

4. Turn the crank for another half-round and take a picture (high compression).

Figure 11. Polariscope setup

3.2 POLARISCOPE: RESULTS, ANALYSIS, AND REPORTING

This section is worth 22 points out of 100

For sample 1 and sample 2: (6 photos in total, 3 photos for each sample)

1. Photos of the unloaded samples (2 photos, 4 points)

2. Photos of the samples under load when the slight change in color appears. Mark the region where high stress starts to appear. You can use Word to mark the area where stress starts to appear. (4 points)

3. Photos of the high compression (after you turned the crank another half round). Mark the regions of high stress in the photo. Can you word to mark this. (4 points)

4. Discussion: Compare the result (i.e. the location where the highest stresses are observed) with the stress concentrations that you expect under each loading condition. (5 pts)

5. Discussion: Discuss possible applications of this method in industry. (5 pts)

REPORTING

Use the “report template” posted on the web page to prepare your report. All graphs should have clearly labeled axes including appropriate units, and captions that indicate the equipment the data was gathered on and the sample name.

It is required that you use Matlab for all data figures and analysis (when specified). Copy your code in the appendix.