Solids & Fluids

Forces which tend to produce an increase in length are called tensile forces. Suppose that a wire or bar or spring is suspended vertically and weights are hung on the end of it: Definitions

Strain is a measure of the deformation produced by an applied stress:  If the force is gradually increased and the corresponding value of extension measured, a graph of stress against strain may be plotted, which has the following ‘typical’ shape: The wire undergoes two types of deformation:
• Between O and B it undergoes elastic deformation. If the stress is reduced in steps, recovery is along BO, i.e. the wire returns to its original length
• B is called the yield point. Beyond B the wire undergoes plastic deformation. If the stress is reduced in steps, recovery is along a line such as CO/, i.e. the wire does not return to its original length, but is permanently stretched
Each point on the graph corresponds to a value of stress and strain. The stress corresponding to the yield point B is called the yield stress. Up to B, the applied stress increases the separation of the atoms in the material, and when the stress is removed, the atoms return to their original separation. When the yield stress is exceeded, a non-recoverable change occurs inside the material due to layers of atoms sliding over each other.

The stress corresponding to point D is called the breaking stress or ultimate tensile stress, and is the maximum that the wire can stand. The corresponding strain is called the breaking strain.

The above graph represents a ductile material, which stretches a lot before breaking (E.g. copper wire).
A brittle material has a short region BD, i.e. it tends to snap suddenly (E.g. glass).

Hooke’s law

Point A in the above graph is called the elastic limit. OA is a straight line passing through the origin, so for this region: This is one form of Hooke’s Law.

(1) can be written as: (1), (2), (3) and (4) are all versions of Hooke’s Law - NB: this law only applies to the linear region OA of the graph.

Note: Since F = k*extension, the bigger the value of k the bigger the force needed for a given extension, so the bigger k, the 'stiffer' a wire, bar or spring is.

Young’s modulus, E

In the linear region OA of the stress-strain graph: Strain has no unit, so E has the same unit as stress, i.e. Nm-2  (Pa).

Example

A 2kg mass is hung from a steel wire of original length 2m and diameter 0.64mm. The extension produced is 0.60mm. Calculate Young’s modulus for steel. (g=10m/s2) Fatigue and creep

A metal which is subject to a large number of cycles of stress may fracture eventually even though the maximum stress may be much less than the ultimate tensile stress. This may occur if tiny cracks gradually get bigger and bigger. This is called fatigue fracture.

Also, a material may continue to deform over a period of time, even if the stress is constant. This is called creep.

A typical creep curve: Strain energy

The work done when a wire is stretched results in energy being stored in it, called strain energy. The above graph of force against extension has the same shape as the corresponding stress against strain graph.  Note: As drawn in the diagram, F/ is not actually constant during the extension. However, though the strip has to be drawn quite wide for clarity, we can imagine it to be as thin as we wish, and the thinner it is, the smaller the change in F/ over the correspondingly small extension.  Note that the above only applies to the linear part of the graph.

Strain energy per unit volume   Example  Calculate:
1. the extension produced
2. the strain energy
3. the strain energy per unit volume   Beams The beam is initially straight. When pushed down in the middle:
• the top surface becomes shorter, in a state of compression
• the bottom surface becomes longer, in a state of tension
Between the compressed and stretched sides there is a region called the neutral plane.

I-shaped steel girders are used in large structures. They are, in effect, beams that have had material taken out of the neutral plane (this saves weight and money). Bridges

(a) Arch bridges Stone is weak in tension but strong in compression. Arch bridges make use of this fact, the stone of which it is built resisting the compression.

(b) Steel girder bridges Intermolecular forces and Hooke's law

The forces between atoms/molecules in a solid must be partly attractive and partly repulsive. We can infer this because if there were no attractive forces, a solid would simply crumble, but if there were only attractive forces, the solid would collapse in on itself, so short-range repulsive forces must exist to stop this.

A graph of the force F between two atoms, versus separation r has the form: The separation r0 is the position at which the force F = 0, so r0 is the equilibrium separation of the atoms. We see that for values of r close to r0 the graph is almost linear. Thus, if a change in r is produced by an applied force, then equal changes in the force will produce equal changes in r. Thus, for a bar/wire, whose atoms have this sort of F-r graph, equal changes in force will produce equal changes in extension, which is what Hooke’s law says.

Elastic moduli

Materials can undergo different types of deformation, and there are three 'elastic moduli', defined according to the type of stress and strain involved. They all have the same basic definition: In each case, the stress is defined such that its unit is the pascal, and strain is defined such that it has no unit. The elastic modulus then has the same unit as stress.

a) Young's modulus, E b) Shear (or rigidity) modulus, G

Consider a block of material with its lower face fixed firmly to a surface and a force then applied tangentially to its upper surface, represented in cross-section by: The force is called a shearing force, and by definition: Note that the twisting of a wire involves shear stress.

c) Bulk modulus, K

Suppose that the external pressure (=force/area) on something increases, and that this produces a decrease in volume.  Note: Solids have values for E, G and K, whereas gases only have values for K.

Crystalline solids

In a crystal the arrangement of atoms, ions or molecules repeats itself many times - we say that there is ‘long range order’.

(Note: an ‘ion’ is an atom that has gained or lost one or more electrons; a ‘molecule’ is two or more atoms joined together)

Many solids have a crystalline structure. The crystalline structure of, say, table salt is evident, but not that of most metals. However, if a steel sample, for example, is polished and then etched (with a dilute acid) its crystalline structure can be seen under a microscope. The steel is seen to consist of many tiny crystals, called grains - it is said to be polycrystalline.

Amorphous solids

These include, for example, wax and glass. An amorphous solid has no definite melting point - as it is heated it becomes progressively more like liquid. Such solids are like a liquid with a very high viscosity. A window several 100 years old may be thicker at the bottom than at the top, due to the glass flowing down - very slowly.

Polymers

These are material made up of  giant chain molecules, each containing many 1000s of atoms.

• Cellulose, rubber, wool, proteins and silk are natural polymers
• Man-made polymers include plastics such as polythene and Perspex, and fibres such as nylon and Terylene
The 'units' that make up a polymer molecule are called 'monomers'. For example: There are a large number of uses of man-made polymers. For example, polythene is commonly used for making carrier bags, because it is light, strong and waterproof.