Deformation of Solids

Deformation: tensile and compressive forces, load and extension

Deformation is a change in the shape or size of an object, and it happens because forces act on it. A tensile force is a pair of forces that pull outwards along the object, stretching it. A compressive force is a pair of forces that push inwards along it, squashing it. Everything in this topic is one dimensional: a wire, a rod, a spring or a column is stretched or squashed along its length, and nothing is twisted, bent or sheared.

The load is the force applied to the object. A load is usually hung on a wire, so the load is the weight of the masses added to the hanger, in newtons. The extension is the increase in length, so extension = new length - original length. A compression is a decrease in length, and the size of that decrease is also measured in metres. Always quote the original length with the extension, because the same force stretches a long sample further than a short one.

The behaviour of a solid comes from the forces between its atoms. Each atom sits at an equilibrium separation from its neighbours, held by forces that behave like very stiff springs. A small tensile force pulls the atoms slightly further apart against those forces, and when the force is removed the atoms spring back to their original positions. Push the material far enough, and whole layers of atoms slide past one another into new positions; the change can no longer be undone, and the material is left permanently deformed.

Keep two things apart, because examiners test both. The extension is not the same as the length of the wire, and the load is a force in newtons, not a mass in kilograms. If a 0.50 kg mass hangs from a wire, the load is the weight of that mass: 0.50 x 9.81 = 4.9 N.

• Tensile forces pull an object apart and produce an extension; compressive forces push it together and produce a compression.
• Load is the force applied, measured in newtons; extension is the increase in length, measured in metres.
• Extension = new length - original length, and the original length is always needed as well.
• The load a hanging mass produces is its weight, so load = mass x 9.81.

Hooke's law and the spring constant

Hooke's law states that the extension of an object is directly proportional to the force applied to it, provided that the limit of proportionality is not exceeded. Doubling the force doubles the extension, and trebling it trebles the extension, so a graph of force against extension is a straight line through the origin. In symbols, F = kx, where F is the load, x is the extension and k is the spring constant of the object.

Rearranged, the spring constant is k = F / x, and its unit is newtons per metre, N m-1. The spring constant measures how stiff the particular object is: a stiff spring has a large k, so a large force produces only a small extension, while a soft spring has a small k. The value of k belongs to the object, not to the material, so a long wire and a short wire of the same steel have different spring constants.

On a graph of force against extension the spring constant is the gradient of the straight-line section. Beyond the limit of proportionality the line curves over, so the force is no longer proportional to the extension and the graph is no longer a straight line through the origin. For example, a spring with k = 250 N m-1 extends by 0.040 m when a force of F = kx = 250 x 0.040 = 10 N is applied.

Notice that the limit of proportionality is a statement about proportionality, not about permanent damage. It is the last point at which force and extension are still proportional. The point at which the material stops returning to its original length comes slightly later, and is called the elastic limit.

• Hooke's law: the extension is directly proportional to the applied force, up to the limit of proportionality.
• F = kx, with F in newtons, x in metres and k in N m-1.
• The spring constant is the gradient of the straight-line section of a force-extension graph.
• A stiffer spring has a larger spring constant, so the same force gives a smaller extension.

Stress, strain and the Young modulus

Force and extension describe one particular sample of a material, so they are not much use for comparing materials. A long wire extends more than a short one under the same load, and a thick wire extends less than a thin one. What is needed are two quantities that describe the material itself and do not depend on the size or shape of the sample.

Stress is the force per unit cross-sectional area, stress = F / A, where A is the area of the cross-section perpendicular to the force. Its unit is N m-2, which is the pascal, Pa. Strain is the extension per unit original length, strain = x / L. Length divided by length gives no unit at all, so strain is a plain number and has no unit.

The Young modulus E of a material is the ratio of the stress to the strain: E = stress / strain = (F L) / (A x). Because stress has the unit Pa and strain has no unit, the Young modulus is also measured in Pa. A large Young modulus means a stiff material, one that needs a large stress to produce even a small strain. The Young modulus only has a single, fixed value while the material is obeying Hooke's law, because only then are stress and strain proportional to each other.

A worked check. A copper wire of cross-sectional area 2.0 x 10^-7 m2 carries a load of 26 N while its length of 3.00 m increases by 3.0 mm. The stress is 26 / (2.0 x 10^-7) = 1.3 x 10^8 Pa and the strain is (3.0 x 10^-3) / 3.00 = 1.0 x 10^-3, so E = 1.3 x 10^8 / 1.0 x 10^-3 = 1.3 x 10^11 Pa, which is the textbook value for copper.

Determining the Young modulus of a metal wire

Two long wires of the same material are hung side by side from the same rigid support, so that they are close together and can be compared. One is the test wire, with a hanger of slotted masses on the end; the other is the reference wire, which carries a fixed load to take up any slack. A main scale is fixed to the bench and a vernier scale, or a fiducial mark, is attached to the test wire so that its extension can be read accurately.

The original length L is measured with a metre rule from the support to the fiducial mark. The diameter is measured with a micrometer screw gauge, at several places along the wire and in two perpendicular directions at each place, and a mean diameter is used to find the cross-sectional area from A = pi d^2 / 4. Masses are then added one at a time, and each time the wire is allowed to settle before the extension is read from the vernier scale. After the largest load the masses are removed one at a time and the readings are checked, to make sure the wire has not been permanently stretched.

A graph of load against extension is plotted. Because F = kx, the gradient of the straight line is the spring constant k of the wire, and the Young modulus is then found from E = k L / A. Alternatively the stress F / A and the strain x / L are calculated for every load, a graph of stress against strain is plotted, and the gradient of that graph is the Young modulus directly.

The wire must be long and thin. A long wire gives a large enough extension to measure accurately, and a thin wire gives a large enough stress for a modest load. Keep the loads below the limit of proportionality, or the graph will curve and every value calculated from it will be wrong. Watch for parallax when reading the scale, and make sure the wire has no kinks in it.

• Measure the original length with a metre rule and the diameter with a micrometer screw gauge, as a mean of several readings.
• Add masses one at a time, let the wire settle, and read the extension from the vernier scale or fiducial mark.
• Plot load against extension and use E = kL / A, where k is the gradient of the graph.
• Check the readings again as the load is removed, to prove the wire was not permanently stretched.

Elastic and plastic deformation and the elastic limit

Elastic deformation is deformation in which the material returns to its original shape and length when the forces are removed. Plastic deformation is deformation in which the material does not return to its original shape and length, so it is left permanently deformed. The elastic limit is the point beyond which the material, once the force is removed, is no longer the same shape and size as it was before.

The two limits are not the same thing, and this catches students out. The limit of proportionality is where the graph stops being a straight line through the origin, so force and extension are no longer proportional. The elastic limit is where the material stops returning to its original length. For a metal wire the limit of proportionality is reached slightly before the elastic limit, so there is a short range of loads in which the wire still returns to its original length but Hooke's law no longer applies.

When a metal wire is loaded beyond its elastic limit and then released, it is left longer and slightly thinner. The atoms have been made to slide past one another into new equilibrium positions, and when the load is taken off they stay there. A small amount of energy has also been used to do this, which is why a permanently bent paper clip feels warm at the bend.

Rubber behaves differently again. It is elastic over very large strains, stretching to several times its original length and returning afterwards, but its force-extension graph is curved rather than a straight line, so rubber does not obey Hooke's law and has no sharp elastic limit. Its loading and unloading curves are not even the same, which is why a rubber band that has been stretched feels warmer when it is allowed to contract.

• Elastic deformation: the material returns to its original shape and length when the forces are removed.
• Plastic deformation: the material is left permanently deformed after the forces are removed.
• Elastic limit: the load beyond which the material no longer returns to its original length.
• Limit of proportionality: the load beyond which force is no longer proportional to extension; for a metal it comes slightly before the elastic limit.

Work done and elastic potential energy

Stretching a material transfers energy to it, and that energy is stored as elastic potential energy as long as the deformation is elastic. The energy transferred is the work done by the stretching force, and on a graph of force against extension it is equal to the area under the graph, because work done = force x distance moved in the direction of the force, and the area of a force-extension graph is a force multiplied by a distance.

Within the limit of proportionality the graph is a straight line from the origin, so the area under it is the area of a triangle: elastic potential energy Ep = 1/2 x base x height = 1/2 F x. Substituting F = kx from Hooke's law gives the second form, Ep = 1/2 k x^2. Both expressions give the same answer; use 1/2 Fx when you know the force, and 1/2 kx^2 when you know the spring constant.

Using the spring from the earlier block, k = 250 N m-1 stretched by x = 0.040 m stores Ep = 1/2 k x^2 = 0.5 x 250 x (0.040)^2 = 0.20 J. The same answer comes from the force, since F = kx = 10 N, so Ep = 1/2 Fx = 0.5 x 10 x 0.040 = 0.20 J.

Two cautions. First, the formulae 1/2 Fx and 1/2 kx^2 only work while the material is obeying Hooke's law. Beyond the limit of proportionality the graph is a curve, and the energy is found by counting the squares under that curve instead. Second, if the material is deformed plastically the energy is not stored and recoverable: it ends up in the thermal store of the material instead, and the wire stays stretched.

• The area under a force-extension graph is the work done in stretching the material.
• Within the limit of proportionality the graph is a straight line through the origin, so the area is the triangle 1/2 Fx.
• Ep = 1/2 Fx = 1/2 kx^2, with Ep in joules, F in newtons, x in metres and k in N m-1.
• Beyond the limit of proportionality, count squares under the curve; the formulae no longer apply.

Force-extension graphs and stress-strain graphs compared

A force-extension graph describes one particular sample. Take two wires of the same steel, one long and thin and the other short and thick, and hang the same loads on both: the long thin wire stretches much further, so its graph is much less steep. The gradient of a force-extension graph is the spring constant k = F / x of that sample, and k depends on the original length and the cross-sectional area as well as on the material.

A stress-strain graph removes those dimensions. Stress is force divided by area and strain is extension divided by original length, so both quantities are the same for the long thin wire and for the short thick wire at the same stage of loading. The gradient of a stress-strain graph is therefore the Young modulus, which is a property of the material alone: both steel wires give the same straight line, with a gradient of about 2.0 x 10^11 Pa.

The shapes of the two graphs are also useful for comparing materials. A metal wire gives a straight line through the origin, followed by a gentle curve, and eventually the material crosses its elastic limit and breaks. A rubber band gives a curve that is not a straight line at all: the initial stiffness is low, the stress rises steeply as the rubber approaches its limit, and the loading and unloading curves do not lie on top of one another. The area under the force-extension graph is the energy stored in that sample, so a graph that is steeper, or that encloses a smaller area, stores less energy for the same force.