Stress and Strain: Young's Modulus

Question 1

what is the simple relationship between stress and strain

Answer 1

• when stress is applied to a material

- the strain is the effect of that stress

Question 2

what is the equation for stress

Answer 2

o(sigma) = F / A

Question 3

what are the units of stress and why

Answer 3

• pascals (Pa or just Nm-2)

- as pressure = F / A

Question 4

what is the equation for strain

Answer 4

• e(emf sign) = delta l / l

- extension / original length

Question 5

what are the units of strain and why

Answer 5

• doesnt have any
• as the units for extension and length cancel out
• making it just a ratio

Question 6

what is the equation for the young modulus

Answer 6

• young modulus = stress / stain
• E = o / e
• E = Fl / A delta l

Question 7

what are the units of the young modulus

Answer 7

pascals

Question 8

what is the main difference between the variables stress and strain, and force and extension

Answer 8

• stress and strain are properties of the material itself
• so a graph of it would always look the same for a material
• whereas force-extension graphs depend on the dimensions of the sample used

Question 9

what are the values of the young modulus in terms of their size usually like and why

Answer 9

• very large, up to hundreds of GPa
• because the cross sectional area of the wire is usually very small (1mm2 = 1x10-6 m2)
• so stress is very large

Question 10

for a stress over strain graph, what is the first landmark of the line (A)

Answer 10

• the point where the wire stops obeying hookes law

- up to this point the line has been linear

Question 11

what can be calculated in this section of the line

Answer 11

• the young modulus of the material

- by calculating the gradient

Question 12

what is the second landmark of the line (B)

Answer 12

• the elastic limit

- before this point the wire would return to its original state if the stress were removed

Question 13

what is the third landmark (C)

Answer 13

• the yield stress has been reached
• where stresses grater than this will cause the wire to become ductile
• meaning it would deform plastically

Question 14

what is the fourth landmark (D)

Answer 14

• the ultimate tensile strength (UTS)

- the maximum stress the wire can endure

Question 15

what is the fifth landmark (E)

Answer 15

the breaking point

Question 16

what is the general shape of the stress over strain graph

Answer 16

• similar to force over extension graphs
• the line begins linearly with a high positive gradient
• then begins to curve downwards
• the line continues with a slight negative gradient
• then very slightly curves upwards right before the line breaks (breaking point)

Question 17

why does the line curve slightly upwards just before it breaks

Answer 17

• the point where the wire is breaking would ‘stretch’ and become thinner
• as stress = F / A, decreasing the cross sectional area at that point increases stress

Question 18

what is the young modulus actually measuring

Answer 18

the stiffness of a material

Question 19

how would you find the young modulus from a compressive stress over stain graph (the material has been compressed)

Answer 19

by calculating the area under the line

Question 20

what is energy density

Answer 20

• the work done in stretching an object

- per unit volume of it

Question 21

what worded equation could be derived from that

Answer 21

energy density = work done / volume

Question 22

for a wire that obeys hookes law (at least up until it doesnt), how does that worded equation turn into 1/2 * stress * strain

Answer 22

• work done / volume = W / Al
• W / Al = F delta l / Al
• because of the triangle the line would form on the graph, 1/2 * F delta l / Al
• = 1/2 * F / A * delta l / l
• = 1/2 * stress * strain

Question 23

therefore what the real equation for energy density

Answer 23

energy density = (stress x strain) / 2

Question 24

what is the energy density of a material directly measuring

Answer 24

that materials toughness

Question 25

what is weird about rubbers stress over strain graph when loading and unloading it are both plotted on the same graph

Answer 25

• the lines dont coincide (they arent on top of each other) like you would think they would be
• the shapes of the lines are similar
• but the unloading line is under the loading line all the way
• but they start and end at the same places

Question 26

what does the area under the loading line give you

Answer 26

the work done per unit volume on the band as it stretches

Question 27

what does the area under the unloading line give you

Answer 27

the work done per unit volume of the band by the band as it relaxes

Question 28

what is this difference in work done per unit volumes on and by the band called

Answer 28

hysteresis

Question 29

what is the hysteresis loop

Answer 29

the area in between the loading and unloading line

Question 30

what does the hysteresis loop on a stress over strain graph represent

Answer 30

• the energy per unit volume transferred to internal energy

- during the load-unload cycle

Question 31

what does the hysteresis loop on a force over extension graph represent

Answer 31

the total energy transferred to internal energy for each cycle

Question 32

therefore, why does a rubber band become warm when stretched multiple times in a row

Answer 32

• because each time a loading and unloading cycle is completed
• some energy would remain in the internal energy stores of the rubber
• this energy stockpiles until some of it is dissipated as thermal energy we can feel

Question 33

what does rolling resistance refer to in vehicles

Answer 33

The amount of kinetic energy transferred by a moving vehicle occurring during hysteresis loops in tyres