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BIOE 2350: Biomechanics 1 Homework Lab 3: Stress-Strain Due Sunday, November 20th Include a well-commented m-file Part I. Stress-Strain Curve A tension test of a metal rod has been completed on an Instron testing machine. Diameter of the sample was 8.94 mm ; the length was 102.62 mm . The load (N) displacement ( mm ) data for the sample is given in the posted spreadsheet. 1. Plot the stress/strain curve . Label the axes with units. (10 points) Be mindful of units. You may want to convert the diameter, length, and displacement column from mm to m so stress is in Pascals (Pa) 2. Find the yield stress . (10 points) Calculate the slope of the linear part of the σ - ε curve ( Hint : use either polyfit or fitlm in Matlab, see HMWL 1). Because some metals do not have a pronounced yield point, the “0.2% offset method” is used. A line is constructed parallel to the initial portion of the stress-strain curve but offset by 0.2% from the origin. The 0.2% offset yield strength is the stress at which the constructed line intersects the stress-strain curve as shown: stress (Pa)

BIOE 2350: Biomechanics 2 Plot the 0.2% offset line. Find the intersect point. It is not trivial to do this automatically in Matlab. It is fine if you want to do it by clicking on the graph to manually obtain the intersection point. Label the yield stress on the graph and indicate its value.

BIOE 2350: Biomechanics 3 3. Calculate and report Young’s modulus . (5 points) Use the first 80% of the linear region, i.e. up to 80% of the yield stress. Remember: slope of linear region = Young’s modulus. 2.158862130678830e+10 4. Calculate and report the fracture toughness, i.e. the area under the curve. (5 points) Hint : use function Q = trapz (X,Y) in Matlab. Units will be the same units as stress. 2.331569155950089e+07 stress (Pa)

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BIOE 2350: Biomechanics 4 Part II. Finite element model Create a finite element model of the cylinder. Apply the elastic modulus you found above. Apply a load within the elastic region. 1. Show contour plots for stress and strain in the x-direction (along the length of the beam) . (6 pts) Make sure scale bar is included in screenshots. a. Comment on the contour plots. Is the value what you would calculate theoretically? (4 pts) The contour plots clearly show the most stress and strain around the non-fixed parts. This is because of the sides distance from the actual force, and the fixture

BIOE 2350: Biomechanics 5 being non moveable. Yes, for stress the order of magnitude, 8, is the same in both our calculations and in the simulation. Here are some questions to get you thinking (i.e. not required) if you’re stuck on how to answer this question: - Is the color throughout the beam uniform or distributed? Does this make sense? - Can this result be justified by the stress and strain equations? Hint: think about which of the variables are constants. 1. Show the contour plot for displacement in the x-direction and determine the displacement of the end. (6 pts) (i.e. What is the displacement value at the end of the beam? Use the scale bar.)

BIOE 2350: Biomechanics 6 a. Compare this value to the value from the experiments at the same load. Are they the same? Discuss (4 pts) The value of 5.526e-1 does roughly match the value at 5 Kn of force (extension of .346). Despite minor differences, the similarity in the order of magnitude leads me to believe that these are relatively similar. The young’s modulus used in the simulation was an approximation as well, this could contribute to minor differences. Some different variables could include an imperfect application of force in the experiment, difference in metallic composition in the experiment, and overall human error. Hint: Think about the stress/strain equations. Which variables are different between the experiment and the simulation? Is the Young’s modulus used in the simulation an approximation or the exact value? Other things to try (if you want to play with FEA, but not required): 1. Increase the modulus and determine if the displacement decreases. 2. Try another load and determine how it maps to your experimental curve. Are the displacement from the two loads multiples of each other (i.e. if you apply 2x the load, do you get 2x the displacement)? 3. Figure out how to apply a displacement and calculate the reaction force load. 4. Determine if the geometry of your sample affects the results. Create a square beam with the same cross-sectional area. close all ;clearvars;clc; data = xlsread( 'LoadDisplacementData-1.xlsx' ); %seperate data, remove hidden middle column extension = data(:,1)/1000; load = data(:,3); initialL = 102.62/1000; %constant radius = 8.94/2000; %constant strain = extension/initialL; %strain calculation stress = load/(pi*radius^2); %stress calculation c = plot(strain, stress); %stress vs strain curve hold on youngsmod = (stress(300) - stress(1))/(strain(300)-strain(1)); %calculating tangent line x = .001:.0001:.015; y = youngsmod*(x/1.02); plot(x,y) %laying line on top of stress strain curve xlabel( 'strain m/m' ) ylabel( 'stress (Pa)' ) Fractough = trapz(strain, stress) %facture toughness

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BIOE 2350: Biomechanics 7

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