7. Stress and Strain

Describe what happens on the atomic level when a wire is stretched within the elastic region

- atoms in lattice stay in position relative to each other - force applied pulls them away from each other - at first, extension is proportional to force applied - as more force applied, extension no longer proportional - atoms return to original position when force is removed - energy required to deform material is regained

The maximum kinetic energy of an aeroplane propelled by a stretched rubber band will less than the total energy stored in the elastic band prior to release. With reference to the graph of force vs extension, explain why.

- not all elastic potential energy is transferred to KE of the plane. - some is lost to heat energy when band is released - area under loading line is greater than area under unload line - area between two lines represents the energy converted to heat energy

A copper wire is stretched by applying ever increasing loads to it. It is extended beyond its elastic limit. What would the line on the graph of force - extension look like if the load were gradually removed and the extension plotted against force?

- originate from last reading - be straight and parallel to the Hooke's law region - meet the x axis at a greater value than zero (it has undergone permanent deformation so is now longer than before)

Describe what happens on the atomic level when a further deforming force is applied to a wire which is already at the yield point.

- sample undergoes permanent deformation - large strain relative to stress (not proportional) - bonds break at a dislocation in the crystal structure - crystal plane shifts one position relative to the neighbouring plane with no restorative forces - further force may cause whole plains of atoms to slide past each other, greatly increasing strain in the wire - sample does not regain orginal shape when deforming force is removed - energy required to deform material is not regained when force removed

Describe what happens to a polymer when you apply a stretching force to it

- stretches easily as the chain molecules untangle - then stiffens when they are aligned

After removing the load which has caused permanent deformation of a copper wire, what happens if you gradually apply the load again?

- the wire regains its original stiffness when removed - it then follows the same loading curve as the first time

Describe the energy transfers taking place when the force on a rubber band is increased and then decreased. (2)

--Force increased-- (1) energy transferred to elastic potential energy (and some heat) (2) elastic potential energy is reduced and some is converted to heat (the EPE is used in restoring the elastic band to the original dimensions)

- what is the hysteresis loop for a stress-strain graph? - what does it represent?

-the area between the line for loading and the line for unloading - the energy per unit volume transferred to internal energy during the load-unload cylce

What does each region and point represent?

0-A = Hooke's law region Strain is proportional to stress Young's modulus = gradient A = limit of proportionality B = elastic limit - if the stress is below this value, the wire returns to original state C = yield stress - material plastic for stresses greater than this D = UTS (ultimate tensile strength) or strength of material E = breaking point - stress may increase as wire narrows

Describe the changes in the molecular structure of a rubber band during one loading - unloading cycle (4 steps)

1) - band is initially slightly stiff until the weak cross-links between the tangled chains are broken. 2) - chains are then uncoiled, giving a large increase in strain for a little extra stress until the molecules become aligned. 3) - band now becomes stiff as the strong covalent bonds between the atoms are stretched. 4) - On releasing the stress, the chains recoil until the initial amorphous state is regained. Think about the image of a rubber under a microscope! (google it)

What is the effective spring constant of springs 1 and 2 in series?

1/K = 1/K1 + 1/K2 (the opposite to what happens with resistors!)

How can you use a graph of stress against strain to determine the Young's modulus of a material?

Calculate the gradient of the Hooke's law region, ie. the part that is a straight line.

What do the regions under the line represent?

Down for answer . . A = energy stored as elastic potential energy B = work done in plastically deforming the wire (energy lost in thermal energy ) A + B = work done in breaking the wire

give the formula for Hooke's law

F = k∆x k is a constant and is the "stiffness" of the spring x is the extension of the spring (distance stretched)

Turn over for question!

Hooke's law (or stress strain) obeyed up to point A (1) A is limit of proportionality (1) elastic limit between A and region B (1) region C shows plastic behaviour or wire is ductile (1) region B to C wire will not regain original length (1) beyond region C necking occurs (and wire breaks) (1)

What do Hooke's law and the Young Modulus tell us?

Hooke's law gives the spring constant for the *particular device* (ie, spring, wire) Young Modulus gives the ratio of stress to strain for a *material*

What is the effective spring constant of springs 1 and 2 in parallel?

K = K1 + K 2

define ultimate tensile strength (or simply strength)

Not necessarily the breaking point, but wire has greater strain for much less stress

What is the unit of stress?

Pa

Define hysteresis in rubber

The difference between the amount of energy absorbed when a rubber is stretched and the amount of energy released when the rubber is relaxed.

show that the units of the Young's modulus are Pa

Young's modulus is stress / strain stress = F/A strain = ∆L/L ∴ E = (F/A) / (∆L/L) rearrange to . . . E = FL/ A∆L E = Nm / m²×m E = Nm / m³ E = Nm⁻² Nm⁻² = Pa (as required)

define breaking point

a this value of stress, material breaks at a weak point (a property of an individual sample rather than a material)

define yield point

a value of stress beyond which a material will behave plastically, undergoing permanent deformation (Point at which) for little increase in stress, there will be a large increase in strain

How can you compare the strength of two materials by looking at their stress/ strain graphs?

compare the heights of the UTS. The higher the UTS, the stronger the material

how do you calculate energy density? (2 ways)

for a material that obeys Hooke's law - Energy density is the area under the line of a stress-strain graph - or . . ½ stress x strain

define stress

force per unit (cross-sectional) area (like pressure but for solids, cross-sectional if a wire)

What property of lead makes it malleable but not ductile?

it can be hammered into sheets, but it is too dense to hold its own weight when stretched into thin wires

What is the unit of strain?

no units! (extension / original length = m/m units cancel out)

How would a value for the total *elastic* energy stored in a stretched wire be found?

plot a graph of force against extension for various forces. Energy = area under the line ½ force x extension (area of a triangle)

define Young's modulus

ratio of tensile stress to tensile strain

What does a small area under the line of a stress - strain graph indicate? (right up until fracture)

that the material has low energy density and is brittle

define strain

the extension of a body divided by original length. (extension/ original length)

define elastic limit

the maximum amount that a material can be stretched (by a force) and still return to its original length when the force is removed

Elastoplastic region

the range of values of stress between the elastic limit and the yield point in which the material will behave partially elastically and partially in a plastic manner

define hooke's law region

the range of values of stress for which the strain is proportional to the stress it is subjected to (Young's Modulus can be found by taking the gradient of the straight line in this region of a graph of stress vs strain)

What does the hysteresis loop for a force extension graph represent? (load - unload return to origin)

the total energy transferred to internal energy during the load-unload cylce

What does the hysteresis loop for a force extension graph represent? (unload doesn't return to origin)

the total energy transferred to internal energy during the load-unload cylce AND energy needed to permanently deform the material

Define elastic limit

the value of stress below which a material will return to its original state if deforming force is removed

Define yield stress

the value of stress beyond which a material becomes plastic and undergoes permanent deformation

define energy density

the work done in stretching a specimen per unit volume of the sample

what property is associated with the ability to absorb large amounts of energy per unit volume before fracturing?

toughness

State Hooke's law (in words)

up to the limit of proportionality, the extension (or compression) of a spring is directly proportional to the force applied to the it

what is the order of magnitude of the Young's Modulus of metals, like copper or steel?

×10¹¹

What is the equation of strain?

ε = ∆x/x strain = extension / original length

what is the equation of stress

σ = F / A (stress = Force divided by cross sectional area)