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Young's Modulus and Stress Analysis By the end of this practical you should have learned: Understanding that the deformation (strain) of an elastic material is directly proportional to the amount of force applied to it (stress). that Young’s Modulus is a measure of how stiff a material is. that Poisson’s Ratio is a measure of the compressibility of a material in the direction perpendicular to an applied force or stress. Practical skills You will use strain gauges to determine the elastic properties of a material and use a spreadsheet to process data and present your results graphically. Overview The objectives of this practical are 1. to demonstrate Hooke's law; 2. to determine the Young's modulus of the material, and hence identify it; 3. to determine Poisson’s ratio for the material; In 1676, Robert Hooke stated, “The power (by which he meant applied force) of any springy body is in the same proportion with the extension”. This is often restated in a simple form as, “The extension of a springy body is proportional to the load applied, unless the elastic limit is reached”. In engineering terms, this can be stated as a general law of mechanics, “Stress is directly proportional to strain for elastic deformation”. Elastic materials are ones which, if subjected to an applied force or stress, will return to their original shape and size once the stress is removed, provided that the stress is below a characteristic threshold or elastic limit. If this limit is exceeded, the material will undergo plastic deformation, in which a rearrangement of the atomic or molecular structure occurs and the shape and size are changed permanently. Stiffer materials will deform less when subjected to an applied load and Young’s Modulus is a quantity that allows a comparison of the stiffness of materials. A material with a high value of Young’s Modulus will experience less deformation (strain) for a given force (stress) than one with a lower value of Young’s Modulus. When a sample of material is stretched in one direction, it tends to get thinner in the other two directions. Poisson's ratio (ν) is a measure of this tendency. It is defined as the ratio of the strain in the direction of the applied load to the strain normal to the load. For a perfectly incompressible material, the Poisson's ratio would be exactly 0.5. Most practical engineering materials have ν between 0.0 and 0.5. Cork is close to 0.0, most steels are around 0.3, and rubber is almost 0.5. Some materials, mostly polymer foams, have a negative Poisson's ratio; if these materials are stretched in one direction, they become thicker in perpendicular directions. Page 1 of 5 90c52a70-611d-4126-ace1-83b0e6a0f53e.doc7 Experimental details The device used to determine Young’s Modulus and Poisson’s ratio is the electrical resistance strain gauge. This is the most widely used device for measuring elastic strains. It is essentially a strip of metal foil which is firmly glued to the surface where the strain is to be measured, so that when the material is strained, the strain at the surface is fully transmitted to the metal foil. Elastic strain along the length of the strip causes a small change in resistance of the gauge, largely because of the change in length and cross- sectional area of the strip, although there is also a slight change in its resistivity. Small changes in resistance are easy to measure accurately, and so the gauge gives an accurate reading of the small elastic strain along the direction of the strip in the gauge. The change in resistance, and hence the voltage across the strain gauge for a constant current, is proportional to the strain; the gauge manufacturer supplies the value of the constant of proportionality. For the strain gauges and current used here ε = 3.803×10-7 V (a conversion factor provided by the manufacturer), where ε = strain (No units – a dimensionless quantity) and V = voltage across strain gauge (measured in microvolts). The experimental work in this practical is very simple and proper working out of the results will take some time. A results table is provided at the back of this booklet for you to record your results. To make calculations easier, a spreadsheet is available to enter your data and process the results. Alternatively you can use graph paper to plot graphs manually. It will help to arrange members of the group to do specific tasks. Background Information This practical uses a simulation of the simple cantilever bending of the beam to which the strain gauges are attached. The sidearm is not used in this experiment. Identify the 3 strain gauges on the cantilever arm. For this experiment you will use only two gauges; the one that is lined up along the cantilever, the “x” direction and the one lined up laterally across the cantilever, the “y” direction. These are represented in the diagram, above, by the strain gauges on the left and right respectively. Measurements are taken using a single meter, which is attached in turn to different strain gauges on the top and bottom of the beam. You will use four of these. TOP – denotes the gauges measuring tension on the upper surface. BOTTOM – denotes the gauges measuring compression on the lower surface. ALONG – denotes the strain in the x direction, along the beam. LATERAL – denotes the strain in the y direction, across the beam. Method – Measurements 1. Select the strain gauge labeled TOP, ALONG – x direction in tension 2. Take a reading with no load. There may be some drift in this value. Wait until a steady reading is obtained, or estimate the average value. 3. Suspend weights from point “A”, and apply successively larger loads to the end of the beam. Record the strain gauge outputs (in μV) produced in each case. It may be useful to remove some weights from time to time to check that you get the same (or almost the same) readings. 4. Repeat the method selecting the following strain gauges: TOP, LATERAL – y direction in tension. BOTTOM, ALONG – x direction in compression. BOTTOM, LATERAL – y direction in compression. Page 2 of 5 90c52a70-611d-4126-ace1-83b0e6a0f53e.doc7 Treatment of Results To verify Hooke’s Law for this material Plot suitable graphs of the gauge readings as a function of the applied. The linearity of the graphs will demonstrate the validity of Hooke's law. To determine Young’s Modulus for this material Convert the strain gauge readings to strains εx and εy, (Use the conversion factor provided by the manufacturer) and the loads to Newtons (1 kg = 9.81N). From the εx and εy readings (no units), the loads applied (in Newtons) and the dimensions of the beam (measure these yourself – each apparatus is different), calculate the Young's modulus E and Poisson's ratio v of the beam. Use the gradients of the stress/strain lines, rather than individual readings. NOTE: The LINEST function in Excel is useful here; or you could plot the graphs on graph paper. Make an estimate of the accuracy of your values. The theory for this part of the practical is given in appendix 1. Formulae to use: For Young' s Modulus 6 Fl E x wh 2 6l gradient.wh 2 For Poisson' s Ratio... y x Where E = Young’s Modulus F = Force (N) l = Distance between the strain gauge and the eye holding the weights (m) εx = Strain in the x direction (no units) εy = Strain in the y direction (no units) w = Width of the beam (m) h = Thickness of the beam (m) ν = Poisson’s ratio Material Young’s Modulus (Pa) Poisson’s Ratio Magnesium 45 x 109 0.29 Aluminium 69 x 109 0.35 Brass 103 x 109 0.34 Titanium 105 x 109 0.32 Steel 200 x 109 0.29 Tungsten 400 x 109 0.28 What material do you think the beam is made of? Page 3 of 5 90c52a70-611d-4126-ace1-83b0e6a0f53e.doc7 Appendix: Elasticity theory Cantilever beam theory Due to the applied force F, a couple of moment F l (the bending moment) acts on the beam cross section at the gauge position. The bending stress σx varies linearly from a maximum σmax at the top surface to a minimum - σmax (compressive) at the bottom surface. Hence at a height z from the centre line of the beam, the stress σ is: This stress acts on an area = w.dz at a distance from the centre line (moment arm) = z. The total moments from all of these stresses internal to the beam must balance the moment F l applied externally. Page 4 of 5 90c52a70-611d-4126-ace1-83b0e6a0f53e.doc7 Results Tables – Use the Interactive Young’s Modulus simulation to collect data. Enter your readings in these tables, then transfer them to the Excel Spreadsheet Dimensions of material tested 1 (cm) 2 (cm) 3 (cm) Distance of weights to strain gauge = l Width of the beam = w Thickness of the beam = h Direct Readings from Strain Gauges (microvolts) TOP BOTTOM Load / g ALONG (x) LATERAL (y) ALONG (x) LATERAL (y) 0 100 150 200 250 300 350 400 450 500 To convert μV to measurements of elastic strain (ε), these readings are multiplied by 3.803 x 10-7 NOTE BENE Enter your data in the corresponding blue columns in the Excel spreadsheet. As time is very short, much of the routine calculation is done for you in the spreadsheet – this includes the plotting of the graphs. You will, however, need to print your graphs. Explain from the shape of the graphs whether your material demonstrates the validity of Hooke’s Law. You then need to calculate Young’s Modulus and Poisson’s ratio for your material, using the formulae shown on page 3, and by comparing your value to those in the table on page 3 state which material you have been using. A website you may find useful :- http://schools.matter.org.uk/ Page 5 of 5 90c52a70-611d-4126-ace1-83b0e6a0f53e.doc7