Stress and Strain: Young's Modulus Flashcards
what is the simple relationship between stress and strain
- when stress is applied to a material
- the strain is the effect of that stress
what is the equation for stress
o(sigma) = F / A
what are the units of stress and why
- pascals (Pa or just Nm-2)
- as pressure = F / A
what is the equation for strain
- e(emf sign) = delta l / l
- extension / original length
what are the units of strain and why
- doesnt have any
- as the units for extension and length cancel out
- making it just a ratio
what is the equation for the young modulus
- young modulus = stress / stain
- E = o / e
- E = Fl / A delta l
what are the units of the young modulus
pascals
what is the main difference between the variables stress and strain, and force and extension
- 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
what are the values of the young modulus in terms of their size usually like and why
- 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
for a stress over strain graph, what is the first landmark of the line (A)
- the point where the wire stops obeying hookes law
- up to this point the line has been linear
what can be calculated in this section of the line
- the young modulus of the material
- by calculating the gradient
what is the second landmark of the line (B)
- the elastic limit
- before this point the wire would return to its original state if the stress were removed
what is the third landmark (C)
- the yield stress has been reached
- where stresses grater than this will cause the wire to become ductile
- meaning it would deform plastically
what is the fourth landmark (D)
- the ultimate tensile strength (UTS)
- the maximum stress the wire can endure
what is the fifth landmark (E)
the breaking point
what is the general shape of the stress over strain graph
- 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)
why does the line curve slightly upwards just before it breaks
- 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
what is the young modulus actually measuring
the stiffness of a material
how would you find the young modulus from a compressive stress over stain graph (the material has been compressed)
by calculating the area under the line
what is energy density
- the work done in stretching an object
- per unit volume of it
what worded equation could be derived from that
energy density = work done / volume
for a wire that obeys hookes law (at least up until it doesnt), how does that worded equation turn into 1/2 * stress * strain
- 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
therefore what the real equation for energy density
energy density = (stress x strain) / 2
what is the energy density of a material directly measuring
that materials toughness
what is weird about rubbers stress over strain graph when loading and unloading it are both plotted on the same graph
- 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
what does the area under the loading line give you
the work done per unit volume on the band as it stretches
what does the area under the unloading line give you
the work done per unit volume of the band by the band as it relaxes
what is this difference in work done per unit volumes on and by the band called
hysteresis
what is the hysteresis loop
the area in between the loading and unloading line
what does the hysteresis loop on a stress over strain graph represent
- the energy per unit volume transferred to internal energy
- during the load-unload cycle
what does the hysteresis loop on a force over extension graph represent
the total energy transferred to internal energy for each cycle
therefore, why does a rubber band become warm when stretched multiple times in a row
- 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
what does rolling resistance refer to in vehicles
The amount of kinetic energy transferred by a moving vehicle occurring during hysteresis loops in tyres
