BMES 345 CH03 Problem Set 20220927
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BMES 345 PROBLEM SET – CHAPTER 3. LINEAR ELASTICITY
Contents
Problem 3.1: Linear Elasticity
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3
Problem 3.2: 1-D Hooke’s Law
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4
Problem 3.3: Linear Elasticity
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5
Problem 3.4: 1-D Hooke’s Law
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6
Problem 3.5: Linear Elasticity
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7
Problem 3.6: Linear Elasticity
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8
Problem 3.7: 3-D Hooke’s Law
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9
Problem 3.8: 3-D Hooke’s Law
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10
Problem 3.9: 3-D Hooke’s Law
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11
Problem 3.10: 3-D Hooke’s Law
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12
Problem 3.11: Linear Elasticity
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13
Problem 3.12: 3-D Hooke’s Law
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14
Problem 3.13: Hooke’s Law in Shear
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15
Problem 3.14: 3-D Hooke’s Law
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16
Problem 3.15: 3-D Hooke’s Law
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17
Problem 3.16: 1-D Hooke’s Law
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18
[Solution] Problem 3.1
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19
[Solution] Problem 3.2
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20
[Solution] Problem 3.3
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21
[Solution] Problem 3.4
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22
[Solution] Problem 3.5
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23
[Solution] Problem 3.6
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25
[Solution] Problem 3.7
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26
[Solution] Problem 3.8
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28
[Solution] Problem 3.9
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30
[Solution] Problem 3.10
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33
[Solution] Problem 3.11
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35
[Solution] Problem 3.12
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36
[Solution] Problem 3.13
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38
[Solution] Problem 3.14
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40
1
[Solution] Problem 3.15
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41
[Solution] Problem 3.16
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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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Related Questions
10. A standard steel specimen of 0.504" diameter elongated 0.0125" in an 8" gage length during a test where it was subjected
to a tensile force of 6249 lb. If the specimen's gage length was measured to be 8.0025" after the test was over, what was the
permanent plastic deformation? What is the elastic deformation and what is the modulus of elasticity of that metal?
r PL
AE
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No wrong answer please , i could downvote
The piece of suture is tested for its stress relaxation properties after cutting 3 cm long sample with a diameter of 1mm. The initial force recorded after stretching 0.1 cm between grips was 5 Newtons. Assume the suture material behave as if it has one relaxation time. The gage length was 1 cm.
a. Calculate the initial stress.
b. Calculate the initial strain.
c. Calculate the modulus of elasticity of the suture if the initial stretching can be considered as linear and elastic.
d. Calculate the relaxation time if the force recorded after 10 hours is 4 Newtons.
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4. What useful informations are obtained from the tensile test of a ductile material?
5. What do you mean by factor of safety?
6. List the important factors that influence the magnitude of factor of safety.
7. What is meant by working stress and how it is calculated from the ultimate stress or yield stress of a
material? What will be the factor of safety in each case for different types of loading?
8. Describe the procedure for finding out the stresses in a composite bar.
9. Explain the difference between linear and lateral strain.
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help
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1. An arbitrary material has an elastic modulus of 118 GPa. How much tensile force was applied to a
100 mm-long cylindrical bar that was originally 10 mm in diameter to reduce the area by 0.1%?
The material has a Poisson's ratio of 0.29.
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ASAP
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MECHANICAL PROPERTIES OF AGRICULTURAL AND BIOLOGICAL MATERIALS
Discuss the following mechanical properties of durable and perishable crops:Grains:a. Elasticityb. Hardnessc. Strengthd. Other properties
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In the First project: you have been asked to perform tensile testing for four different materialsand analyse the results and work on some NDT process selection:a. For the results shown in Table 1 of the tensile testing that you have performed, find thefollowing, if you know that the original length of specimen is 20.8 mm and the original diameteris 6.4 mm. Fill the calculated results in the summary table below (Table 1):1. Plot the engineering stress versus engineering strain for each material and L-D Diagram.2. Compute the modulus of elasticity, E in GPa.3. Determine the yield strength at a strain offset of 0.002.4. Determine the tensile strength in MPa.5. What is the approximate %El ductility, measured by percent elongation?6. Compute the modulus of resilience.7. Determine the fracture stress in MPa.8. Compute the final area (Af) in mm2.
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Is the bulk modulus used to measure the stiffness of a volume of material?
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Define the given terms related to mechanical properties of materials.
1. elastic material
2. Apparent Viscosity
3. Bend test
4. Bingham Plastic
5. Dilatant
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Question 1
You are working on a design team at a small orthopaedic firm. You have been asked to select a cobalt-
chrome-molybdenum (CoCr) material that will not experience plastic deformation under a specific mechanical test, as follows...
A tensile stress is applied along the long axis of a solid cylindrical rod that has a diameter of 10 mm. An applied load of some
magnitude F produces a 7x10³ mm change in diameter (see figure below, original shape is blue, elongated shape is unshaded).
Q1H: Provide a brief rationale based on calculations used to support your answer. That is, how would you explain the behavior of
the "new alloy" material to your design team?
Table of properties: Assume Poisson's ratio is 0.3 for all materials
Process
Elastic Modulus (GPa)
Material
CoCr F75
As cast/Annealed
210
CoCr F90
Hot forged
New alloy
Z
X
↑ F
df
O
210
110
Yield Strength (MPa)
450-517
900-1200
600
Tensile Strength (MPa)
655-890
1400-1600
700
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Question 1
You are working on a design team at a small orthopaedic firm. You have been asked to select a cobalt-
chrome-molybdenum (CoCr) material that will not experience plastic deformation under a specific mechanical test, as follows...
A tensile stress is applied along the long axis of a solid cylindrical rod that has a diameter of 10 mm. An applied load of some
magnitude F produces a 7x10³ mm change in diameter (see figure below, original shape is blue, elongated shape is unshaded).
Q1C-D: Using the table of material properties below, calculate the magnitude of stress (o) and applied load (F) required to produce
the 7x10-³ mm change in diameter for rods fabricated from F75 CoCr alloy (as cast) and F90 CoCr alloy (hot forged) materials.
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Question 1
You are working on a design team at a small orthopaedic firm. You have been asked to select a cobalt-
chrome-molybdenum (CoCr) material that will not experience plastic deformation under a specific mechanical test, as follows...
A tensile stress is applied along the long axis of a solid cylindrical rod that has a diameter of 10 mm. An applied load of some
magnitude F produces a 7x10-³ mm change in diameter (see figure below, original shape is blue, elongated shape is unshaded).
Q1E: Of those two materials (F75 CoCr alloy (as cast) and F90 CoCr alloy (hot forged)), which materials would you select to assure
that the deformation is entirely elastic (No yield!)?
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Question 1
You are working on a design team at a small orthopaedic firm. You have been asked to select a cobalt-
chrome-molybdenum (CoCr) material that will not experience plastic deformation under a specific mechanical test, as follows...
A tensile stress is applied along the long axis of a solid cylindrical rod that has a diameter of 10 mm. An applied load of some
magnitude F produces a 7x10-³ mm change in diameter (see figure below, original shape is blue, elongated shape is unshaded).
Q1G: If your design required using the new material to create a wire, what is the largest diameter that would lead to ductile
behavior while still avoiding plastic deformation when exposed to the above loading conditions?
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Stuck need help!
Problem is attached. please view attachment before answering.
Really struggling with this concept.
Please show all work so I can better understand !
Thank you so much.
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Pls. Answer thank you?
If the moduli of rigidity and elasticity of a material are 15,000 ksi and 40,000 ksi, respectively, its Poisson’s ratio is
__________?
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Please solve only problem 3.
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need help...thnks.....
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Please make brief and clear definitions of the terms below.
1.Stress, Strain2.Elastic Deformation3.Plastic Deformation4. Young Modulus5. Yield Strenght
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Please help. Do not just guess. Prove your answer. I will give a good feedback. thank you :)
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Where Expressions from the linear theory of elasticity are in use?
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1) Draw (using a normal graph paper) a conventional
stress-strain diagram for ANY metallic material (e.g. steel,
aluminium, copper, brass, iron, tungsten). The diagram
should be as accurate as possible using a suitable scale
(e.g. 1cm: 10 N).
2) Calculate the Modulus of Elasticity, Modulus of
Toughness and Modulus of Resilience for the material
from the stress-strain diagram. Show your calculations in
detail on a separate A4 piece of paper.
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How this questions solve??
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What is the measure of the stiffness of a volume of material?
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9) An aircraft component is made of an aluminum alloy that has a fracture toughness of 60MPavm. When a crack of 2.5mm exists in the
component, the material fractures when a stress of 440MPA is reached. If a crack of 5.5mm exists in the material, what stress would need to
be reached to cause failure?
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Discuss the following mechanical properties of durable and perishable crops:Fruits and vegetables:a. Compressive strengthb. Impact resistancec. Vibrations resistanced. Other properties
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Given your understanding of what initiates and controls failure in materials, which of the following will increase the failure strength or lifetime
of a test piece or component and why?
a. Polishing to reduce surface defects
b. Decreasing the temperature below the brittle-ductile transition temperature, to make it harder
C. Increasing its volume, to give a larger cross sectional area
d. Increasing the grain size so there are less grain boundaries to initiate failure
e. Decreasing the difference between the maximum and minimum stress values, as this effects the stress concentration factor
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Differentiate stress and strain of materials?
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Answer part A
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!A tension test performed on a metal specimen to fracture produced the stress-
strain relationship shown in Figure Pi.14. Graphically determine the following
(show units and all work):
a. Modulus of elasticity within the linear portion.
b. Yield stress at an offset strain of 0.002 m/m.
c. Yield stress at an extension strain of 0.005 m/m.
d. Secant modulus at a stress of 525 MPa.
e. Tangent modulus at a stress of 525 MPa.
750
600
450
300
150
0.002
0.004
0.006
0.008
0.01
0.012
Strain, m/m
FIGURE P1.14
Stress, MPa
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