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Unformatted text preview: E 45 Laboratory Manual Lab 01 Basics of Mechanical Behavior
1. To understand the what is meant by "mechanical" behavior of materials 2. To measure and compare the hardness (resistance to penetration) of metallic samples 3. To understand the fundamental concepts of mechanical stress and strain 4. To measure and compare the effects of stress concentration on deformation behavior Overview
This experiment offers practical experience with two (2) experimental methods that are very commonly used for rapid and/or elementary assessment of the mechanical properties of materials. They are known as "hardness" texts, and the detailed characteristics of the two methods are highlighted and compared. This laboratory also illustrates a much more complex and critical cause of failure in engineering materials known as "stress concentration." Understanding these elementary concepts and their relationships to the in-service performance of materials is essential for all practicing engineers. Equipment
Hardness Tests 1. Hardness testers: Rockwell™ and Brinell™ 2. Brinell™ microscope with light source 3. Test specimens: mild steel; stainless steel; aluminum; brass; copper (various thicknesses)
Stress Concentration Measurements 5. Opaque rubber test plate with support frame, weight pan, weights 6. Twelve-inch scale graduated in 50ths and 100ths; magnifying glass 7. Photoelastic stress analysis equipment and transparent elastomeric plate specimens Copyright ⓒ 2010, Berkeley Professor Ronald Gronsky Lab 01: Basics of Mechanical Behavior Background
The usefulness of metallic materials in engineering structures often depends upon two mechanical properties: sufficient strength to bear a load, and sufficient ductility to allow the relaxation of stress concentrations through plastic deformation without fracture. Other factors such as dimensional stability, abrasion resistance, corrosion resistance, high impact strength, electrical or thermal conductivity, or others can assume importance for specific applications, but basic mechanical properties are always significant. At ordinary temperatures, metals, alloys, and many other materials will deform when loaded, and for small loads the amount of induced deformation is proportional to the magnitude of the applied load. This strain is "elastic" in nature, that is, the material will return to its original shape when the load is removed. But at high loads, permanent changes may occur in the material. With increasing load, a point is reached beyond which irreversible deformation, called "plastic" deformation, occurs. Most metallic materials are ductile at room temperature, meaning they can be deformed plastically before they fracture. Brittle materials such as glass will fracture, rather than plastically deform, when the elastic limit is exceeded. Metallic materials are useful because they can be fabricated easily into complicated shapes by plastic deformation, but once metallic components have been incorporated into structures, such as wing spars in aircraft, bridge supports, or bio-implants, further plastic deformation under normal service loads could be disastrous. Consequently, engineers must base their designs upon the elastic properties of materials used in service, and such parameters as engineering stress, σ, (the load divided by the area supporting the load), engineering strain, ε, (change in length divided by original length), Young's modulus, E, (ratio of stress to strain), and Poisson's ratio, ν, (the negative value of the ratio of strain in the direction perpendicular to the loading direction to the strain in the loading direction). Hardness can be loosely defined as a material's resistance to deformation, but it specifically refers to a material’s resistance to penetration by an indenter. Hardness is one of the easiest mechanical properties of a material to measure, although it is not the most basic property. For this reason, hardness tests are widely used as a rough guide to the strength of materials. These tests are rapid and often “nondestructive” and, therefore, represent an important means of quality control. Hardness is not a simple property; it depends on a complex set of other material properties. Different kinds of hardness tests assess these various properties differently, so an exact definition of hardness necessarily depends on the testing method used. The different kinds of hardness tests are not equivalent and cannot be directly compared. Therefore the information gained from hardness tests on a given material must be interpreted while considering data obtained from other quantitative measures of mechanical properties, such as the strength. Hardness values from different materials can only be compared when the hardness tests are carried out under identical procedures and conditions. Beyond hardness measurements, this laboratory exercise is also concerned with the elastic behavior of materials and particularly with the effect that a discontinuity, such as a through-hole in a plate, has upon the distribution of strain (and therefore stress) in materials. Metallic specimens are not particularly good for demonstrating the elastic behavior page 2 of 10 Lab 01: Basics of Mechanical Behavior of materials because of their high elastic moduli, which cause the response (strain) to an applied stress to be too small to see with the naked eye. It is much easier to observe and measure elastic behavior in an "elastomeric" specimen. There is a significant difference in elastic behavior between a metal and an elastomer. The relationship between stress and strain is linear for metals in the elastic range, but it is nonlinear for elastomers. This difference, however, is more than compensated by the ease with which the relatively large strains can be measured on an elastomeric specimen. The specimens used for this exercise illustrate the critical stress concentrating effect of a hole in a member subjected to load. One of the specimens is a “plate” of opaque black rubber, on which a square grid of lines has been printed so that strain can be seen and measured at various locations where metallic marker pins have been inserted. Strains in both the longitudinal and transverse directions are measured at the top and side edges of the hole, and compared to those along the center line of the specimen, but away from the hole, so that the relationship between stress and strain can be determined. For rubber this will be non-linear; consequently a single elastic modulus is not enough to describe the situation. It will be found that the strains at the edge of the hole are significantly greater than those far away, for the same applied stress. Since stress and strain are related, it should be evident that the presence of a discontinuity, such as a hole, produces a "stress concentration" in the material at the edge of the hole. If stress and strain are linearly related, as they are in metallic materials, the stress concentration is obtained directly from the ratio of the longitudinal strain at the edge of the hole to the longitudinal strain in the uniform section far removed from the hole. The stress concentrating effect of holes, notches, corners, cracks, pits, inclusions, fillets, grooves, threads and other imperfections (sometimes called “stress raisers”) is exceedingly important in practical engineering. Drilling holes in a stressed member for bolting or riveting for example, can lead to local stress concentration exceeding the strength of the material, resulting in unexpected failure. As another example, consider the effect of cracks, even very small ones, upon overall strength. The stress concentration at the crack tip may exceed the elastic strength of the material. Unless this stress (which exists only in a small region at the crack tip) is relieved by plastic yielding, the crack will propagate. The stress concentration factor, k, (the factor by which the stress is increased over what it would be in a section where no cracks or other stress raisers are present) is independent of the material, as long as the material is in its elastic range. It depends on the geometry and on the kind of load (tension, torsion, etc.) that is applied. Photoelastic observations are sometimes used to determine the stress concentration factor, and the points of maximum stress concentration, especially in parts of complex shape. Specimens for the photoelastic apparatus used in this laboratory are plates of a special elastomeric polymer through which visible light (photons) can pass. Plane-polarized light passing through a stressed region is depolarized to different degrees depending on the amount of stress. Since this process is also wavelength dependent, it is possible to display the stress distribution as a multicolored pattern (variable wavelength), produced by additional optical elements. page 3 of 10 Lab 01: Basics of Mechanical Behavior Stress concentration factors are known for many situations. For example, the stress concentration factor (k) for a crack with tip radius r is
c k=2 r where c is the length of the crack. Clearly, this can become very large. The stress concentration factor for a circular hole in a plate loaded in tension is k = 3, independent of the hole radius, provided: (1) the plate is thin compared to the hole diameter; (2) the plate length and breadth are much greater than the hole diameter; and (3) the hole is not located near the edge of the plate. A large number of stress concentration factors have been obtained through mathematical analyses, and through experimental studies, for a variety of shapes and stresses. For additional detail, see for example R.E. Peterson, Stress Concentration Design Factors, Wiley, New York, (1953). Experimental Procedures
Brinell™ Hardness Test In your Brinell hardness tests for this lab, a smooth flat specimen surface is indented with a 10 mm diameter (D) steel ball under a load (P) of 500 kg for soft metals and 3000 kg for the rest. The load is maintained for a standard time, usually 30 seconds. The indenter is then removed and the diameter (d) of the permanent impression is measured by a low-power optical microscope. The hardness values are computed from the mean of two diameter measurements at right angles. The Brinell hardness number (BHN) is obtained by dividing the load by the spherically shaped surface area of the indentation.
BH N = P √ πD 2 D− D 2 − d2 In Brinell hardness measurements the indentation mark is quite large, and may cause damage to finished products, making this a "destructive" evaluation method in most cases. For all Brinell tests, specimens must be flat and securely supported. The specimen must be thick enough so that no bulge appears on the opposite face during penetration by the indenter and should preferably be ten (10) times as great in thickness as the depth of the impression. Impressions should not be made within two-and-one-half (2.5) diameters of the specimen edge and should be at least five (5) diameters from other test impressions. The 500 kg load should be applied for at least thirty (30) seconds and the 3000 kg load for at least fifteen (15) seconds.
Rockwell™ Hardness Test The Rockwell test is more rapid and leaves a smaller and less conspicuous indentation on the specimen than does the Brinell test. In the Rockwell test the hardness value is an arbitrary number that is inversely related to the depth of the indentation which results from plastic deformation of the material being tested. The test surface should be flat and free from scale, oxide films, pits and foreign material that may affect the results. A pitted surface may give erratic readings, owing to some indentations being near the edge of a depression. This permits a free flow of metal around page 4 of 10 Lab 01: Basics of Mechanical Behavior the indenting tool and results in a low reading. Oiled surfaces generally give slightly lower readings than dry ones because of the reduced friction under the indenter. The bottom surface should be free from scale, dirt, or other foreign material that might crush or flow under the test pressure and so affect the results. To minimize the effects of surface irregularities, the initial setting (“SET”) of the scale is made after a minor load (10 kg) is applied to preset the indenter into the specimen. The scale reads from 0 to 100 in units of 0.002 mm and graduated in a direction such that the greater the penetration, the lower the reading. The load and the design of the penetrator depend on the kind of material under study. For hard materials a diamond (“brale”) indenter and a 150 kg major load are ordinarily used, which is called the Rockwell C scale. For softer material a 1/16” steel ball penetrator and 100 kg load are commonly used for the Rockwell B scale. A number of other scales are used for specific purposes. For testing surface hardened materials or thin samples a superficial hardness tester uses units of 0.001 mm. The initial setting (labeled “SET”) of the dial is at “0” on the black scale for the brale indenter, and at “30” on the red scale for the ball indenter. If a 1/16” ball indenter is used with samples harder than Rockwell B 100, they become flattened and give incorrect readings. If samples of hardness lower than RB = 0 are tested with the 1/16" ball and 100 kg load, the chuck may contact the sample. The specimen must be carefully supported in the Rockwell test because any unrecovered vertical motion will change the apparent Rockwell hardness by one unit for every 0.002 mm displacement.
Hardness Testing Methods Specimens of several metals and alloys will be tested on both Brinell™ and Rockwell™ Hardness testers. Brinell Hardness tests will be performed on specimens of mild steel and stainless steel (3000 kg load) and on specimens of aluminum, brass, and copper (500 kg load). The Brinell Hardness numbers will be read from a table for 500 kg or 3000 kg loads, or will be calculated from the formula given above. Rockwell Hardness tests will be performed on mild steel and stainless steel specimens as well as on aluminum, brass, pure copper, copper alloy 230, copper alloy 157, and copper alloy 066 specimens, all using the Rockwell A scale, with a 60 kg load. When measuring Rockwell hardness in this lab you should record three (3) readings at different spots on the specimen, in order to arrive at an average readings for a hardness value. Successive readings should be taken at least 1/8” apart. The average Rockwell A readings will then be converted to Brinell hardness numbers using conversion charts posted in the laboratory.
Stress Concentration Analysis The first sample for the stress-concentration portion of the lab is a rubber plate with a through-hole “stress-raiser,” shown in Fig. 1. The sample is loaded by placing weights on a tray suspended from the bottom of the plate. Small metal pins embedded in the rubber plate act as landmarks used to measure changes in dimensions along both horizontal and vertical directions, from which the strain is calculated. You should record measurements x1 and y1, the horizontal and vertical distances between pins that are away from the hole (location 1 in Fig. 1, the “nominal values in the absence of any “stress concentration”), as well as measurements x2 and y2, the horizontal and vertical distances be- page 5 of 10 Lab 01: Basics of Mechanical Behavior tween pins that are nearest to the hole (location 2 in Fig. 1, where stress concentration may exert an effect). Please remember that your first set of measurements should be recorded with no load. 1 Figure 1 Schematic of rubber “plate” with through-hole “stressraiser” to be used in experiments on stress concentration. Measurements of the distances between two sets of pins, one (location 1) far enough away from the hole to be representative of the “nominal” behavior of the plate, and the other (location 2) near the hole, are compared to quantify the stress “concentration” introduced by the throughhole. 2 You will also need to measure the cross sectional area A1, the nominal cross sectional area of the test sheet (width times thickness) and cross sectional area A2, the local cross sectional area of the text sheet minus the largest area removed by the presence of the hole. These values will be needed to calculate stresses; in region 2 the same load is carried by a reduced cross-sectional area of the sample, so the stress (load/area) is “concentrated.” The second sample in this lab exercise on stress concentration is an array of three photoelastic plates, one without stress-raisers, and two with through-holes of different sizes and shapes, all contained within a special viewing chamber providing polarized illumination, shown in Fig. 2. The plates exhibit color changes under load, offering a direct mapping of the “stress-concentrating” effect of the holes. The plate with no holes provides calibration of the color page 6 of 10 Lab 01: Basics of Mechanical Behavior changes. You should make careful observations of these visible changes in the color pattern of the photoelastic plates. Both the color and the shape of these patterns change with applied stress. It is the stress DISTRIBUTION that is most evident in experiments of this type. Figure 2 Photograph of the photoelastic plate samples used to visualize stress-concentration in the vicinity of stressraisers. The plate on the right is intact and offers a color calibration under various loads. Below this image frame are weight trays attached to each sample. DataSheet 1 — Brinell Hardness Material Load P (kg) Indentation diameter d (mm) Load ÷ Projected Area Load ÷ Indentation Area The first three columns of this DataSheet 1 (only the first few rows are shown here -- there should be as many rows as needed to present all of your data) will be completed in lab, and all data should be carefully entered into your lab notebook, including the diameter (D) of the penetrator ball. Afterwards you should set up a spreadsheet with at least the five (5) columns shown here, and program the appropriate functions to arrive at the data requested in the last two columns. This data sheet must be submitted with your formal lab report. Please note all relevant parameters in DataSheet 1. The diameter of penetrator ball is D, expressed in mm. The diameter of the indentation made in the plane of the sample surface is d, also expressed in mm. From the measured value of d, two area terms are developed. One is a simple “projected” area (Ap), which ignores the fact that the indentation has a spherical shape, and calculates the area projected onto the top surface of the sample, or
2 d Ap = π 2
page 7 of 10 Lab 01: Basics of Mechanical Behavior The other is the actual indentation area (Ai) with “hemispherical-cap” shape, given by the expression
Ai = πD D − D 2 − d2 2 These are the equations that should be programmed into your spreadsheet to complete DataSheet 1.
DataSheet 2 — Rockwell Hardness Material Trial # 1 2 3 Average 1 2 3 Average Scale Major Load Minor Load Hardness Conversion Data recorded in the lab should be retrieved from your lab notebook to construct DataSheet 2 as a spreadsheet with at least the seven (7) columns shown here, and as many rows as needed to present three (3) trials and averages for all samples studied. Note that the last column is the topic of Question 1, and asks for the conversion of the Rockwell hardness number measured in your experiments to an “equivalent” Brinell hardness number. No calculations are required; conversion tables are found in the laboratory.
DataSheet 3 — Stress Concentration Load,P y1 y2 x1 x2 (lbs) (in) (in) (in) (in) Nominal stress (psi) Local stress (psi) Nominal y strain (in/in) Local y strain (in/in) Nominal x strain (in/in) Local x strain (in/in) DataSheet 3 records the total applied load (P) in units of pounds added to the weight tray suspended from the opaque rubber plate sample, and the measured distances between location pins on the plate. Please remember that the first set of measurements should be made for P = 0. When retrieved from your lab notebook, these data are entered into a spreadsheet programmed to generate additional columns, such as those listed here. Again this is a minimum number page 8 of 10 Lab 01: Basics of Mechanical Behavior of columns. Additional intermediate columns can be added to facilitate calculations or presentation of the data. The number of rows will depend upon the number of experimental runs made during the lab exercises. Recall that stress is defined as load divided by the cross sectional area over which the load is borne. In these calculations the full cross sectional area A1 and reduced cross sectional area A2 are taken as constant, the values obtained under a no-load (P = 0) condition. Nominal stress is the quotient of load and A1, while local stress is the quotient of load and A2. Also, strain is defined as a change in length per unit (initial) length. The “nominal y strain” is the strain in the y-direction that is measured far away from the stress-raiser, while the “local y strain” is that measured in the vicinity of the stress-raiser. In all calculations of strain, the initial length is the dimension along the x direction or the y direction under no load (P = 0). Consequently only the numerators change in the last four columns, not the denominators. Lab Report
In your report, please be sure to include all three (3) datasheets, complete with analyses and calculations, along with answers to the following questions.
Question 1 Using conversion charts provided in the laboratory, convert average Rockwell A numbers to equivalent Brinell numbers. Can you suggest a reason for the poor agreement between converted and measured Brinell numbers in some cases?
Question 2 What if the Brinell hardness number were defined by a much simpler function, namely, the load (P) divided by the projected area Ap of the impression, defined above? How does this number change with a change in applied load for the materials you tested in this lab? Would this quantity (P/Ap) be as good a hardness index as the BHN (= P/Ai)? Explain.
Question 3 A hardness test can be used to give a rough estimate of a material's strength. A tensile test, in which a specimen is strained to complete failure, is obviously a much more accurate test of strength. Why would hardness tests ever be used instead of tensile tests? Give three reasons. Which test would you use in engineering design? Which test would you use for quality control? Explain.
Question 4 Using your data from the opaque rubber plate sample, plot longitudinal and transverse strain versus nominal applied stress (load divided by cross sectional area) for positions 1 and 2. Note that this amounts to a plot of gross section stress against the y1 and x1 strains as well as the net section stress against the y2 and x2 strains (4 plots total). Plot stress on the ordinate (y-axis) and the strain on the abscissa (x-axis). Remember that these are the applied stresses and may not be equal to the local stresses. What position in the opaque rubber plate sample that you tested is subjected to the highest tensile stress? Note that this will be the position of highest tensile strain for the same applied load. What
page 9 of 10 Lab 01: Basics of Mechanical Behavior position in the plate sample is subjected to the highest compressive stress? As above, this will be the position of highest compressive strain for the same applied stress.
Question 5 What is the ratio between the highest tensile stress and the average longitudinal tensile stress (i.e., the stress concentration factor)? Stress concentration can be easily obtained from your plots using a vertical line connecting two data sets for locations 1 and 2.
Question 6 Using your data from the transparent photoelastic plate sample, rank the relative stress concentration at all four (4) non-circular openings from highest to lowest. Explain your results. Why should the shape of a “stress-raiser” matter?
Question 7 When a uniaxial stress σy is applied to a material (Fig. 3), there are resulting elastic strains (εy and εx) along both the y and x directions that are proportional to the applied stress, where E is Young's Modulus and ν is Poisson's Ratio.
εy = 1 σy E −ν σy E
y εx = y Figure 3 Schematic of strain resulting from a uniaxial stress applied along the ydirection as illustrated for a plate with a through-hole. The x-direction is horizontal. x x Figure 4 Schematics of solid plate with throughhole subjected to biaxial state of stress y y Write down the expressions for εy and εx when the same part is subject to a biaxial stress state, that is, both σx and σy, as illustrated in Fig. 4. Hint: the principle of “superposition” applies here. Begin by writing the equations for the strains when only σx is applied, add the equations for the strains when only σy is applied (above), and simplify. page 10 of 10 ...
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This note was uploaded on 11/19/2010 for the course E 45 taught by Professor Gronsky during the Fall '08 term at University of California, Berkeley.
- Fall '08