BMES 345 CH03 Problem Set 20220927

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Drexel University *

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Mechanical Engineering

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Apr 3, 2024

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BMES 345 PROBLEM SET – CHAPTER 3. LINEAR ELASTICITY Contents Problem 3.1: Linear Elasticity ....................................................................................................................................... 3 Problem 3.2: 1-D Hooke’s Law ...................................................................................................................................... 4 Problem 3.3: Linear Elasticity ....................................................................................................................................... 5 Problem 3.4: 1-D Hooke’s Law ...................................................................................................................................... 6 Problem 3.5: Linear Elasticity ....................................................................................................................................... 7 Problem 3.6: Linear Elasticity ....................................................................................................................................... 8 Problem 3.7: 3-D Hooke’s Law ...................................................................................................................................... 9 Problem 3.8: 3-D Hooke’s Law .................................................................................................................................... 10 Problem 3.9: 3-D Hooke’s Law .................................................................................................................................... 11 Problem 3.10: 3-D Hooke’s Law .................................................................................................................................. 12 Problem 3.11: Linear Elasticity ................................................................................................................................... 13 Problem 3.12: 3-D Hooke’s Law .................................................................................................................................. 14 Problem 3.13: Hooke’s Law in Shear ......................................................................................................................... 15 Problem 3.14: 3-D Hooke’s Law .................................................................................................................................. 16 Problem 3.15: 3-D Hooke’s Law .................................................................................................................................. 17 Problem 3.16: 1-D Hooke’s Law .................................................................................................................................. 18 [Solution] Problem 3.1 ................................................................................................................................................... 19 [Solution] Problem 3.2 ................................................................................................................................................... 20 [Solution] Problem 3.3 ................................................................................................................................................... 21 [Solution] Problem 3.4 ................................................................................................................................................... 22 [Solution] Problem 3.5 ................................................................................................................................................... 23 [Solution] Problem 3.6 ................................................................................................................................................... 25 [Solution] Problem 3.7 ................................................................................................................................................... 26 [Solution] Problem 3.8 ................................................................................................................................................... 28 [Solution] Problem 3.9 ................................................................................................................................................... 30 [Solution] Problem 3.10 ................................................................................................................................................ 33 [Solution] Problem 3.11 ................................................................................................................................................ 35 [Solution] Problem 3.12 ................................................................................................................................................ 36 [Solution] Problem 3.13 ................................................................................................................................................ 38 [Solution] Problem 3.14 ................................................................................................................................................ 40 1
[Solution] Problem 3.15 ................................................................................................................................................ 41 [Solution] Problem 3.16 ................................................................................................................................................ 43 2
Problem 3.1: Linear Elasticity Consider 4 bars of different dimensions, made of the same homogeneous, isotropic linear elastic material (note that the pictures below represent the undeformed dimensions of the bars): Each bar is subjected to the same tensile axial normal strain . Answer the following questions: a) Which bar(s) will experience the highest tensile normal force? b) Which bar(s) will experience the largest change in length? c) Which bar(s) will experience the largest tensile normal stress? d) Which bar(s) will experience the largest lateral normal strain? 3
Problem 3.2: 1-D Hooke’s Law The following data were collected from compression testing of cortical bone. The cortical bone sample was a cylinder, with a height of 2 mm and a diameter of 6 mm. If the elastic modulus of this cortical bone sample is E = 19 GPa, fill in all of the blanks of the following table: Sample Height (mm) Strain Force (N) Stress (MPa) 2.00 0.000 0.0 0.0 1.99 0.010 13430.3 950.0 4
Problem 3.3: Linear Elasticity A sample of cardiac muscle is subjected to tensile testing. Based on this testing, the properties of the muscle (assuming it is a linear elastic material) are determined to be E = 33 kPa and ν = 0.35. Using this information, complete the missing entries in the table: Sample Length (mm) Sample Width (mm) Force (N) Stress (kPa) Strain 6.00 2.0000 0 0 0 6.05 1.9942 0.00055 0.275 6.10 1.9883 0.00110 0.0167 6.15 1.9825 0.825 0.0250 6.20 0.00220 1.100 0.0333 1.9708 0.00275 1.375 0.0417 5
Problem 3.4: 1-D Hooke’s Law Consider a portion of the femoral artery under tension: The femoral artery segment is subjected to tensile loads F = 0.17 N at each end of the vessel, and the resulting elongation of the vessel is δ = 1 mm. a) What is the elastic modulus of the femoral artery? b) If the tensile load is increased to F = 0.55 N, what is the resulting elongation? 6
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