Solids

• Graph for a spring: It obeys Hooke's law F = kx where k is the stiffness constant. For springs this works in compression

  
• Graph for a rubber band: It does not obey Hooke's law and has a non-linear relationship.

  
• Force-extension graph for a metal wire: Metal wire is stretched beyond its limit of proportionality. When the load is removed, the extension is decreased, and the unloading line is parallel to the loading line (as the stiffness constant is the same) the forces between the atoms are the same as during loading. This is because it is stretched beyond its elastic limit it is deformed plastically - permanently stretched, so the unloading line does not go through the origin. The area between the two lines is work done to permanently deform the wire.

  
• The equation for work done is Energy (J) = Force (N) x Distance (m)
• In a spring graph to find the energy stored from the graph is through finding the area under the graph
• To find Elastic Potential Energy or Elastic Strain Energy: E = ½ FΔx and F = kΔx. So E = ½ k Δx Δx = ½ k (Δx)2
• Graph for rubber band: It does not obey Hooke's law. The difference is due to energy being lost as heat. It also undergoes hysteresis, which is the difference between the strain energy required to generate a given stress in a material and the material's elastic energy at that stress.

  
• Hooke's law states up to a given load, the extension of a spring is directly proportional to the force applied to the spring and is given by F = kx, where k is the stiffness of the object. The graph initially has a linear region when the spring is still coiled. As the spring loses its 'springiness', the extension increases disproportionately. k is the spring constant
• Most materials behave like springs as the bonds between atoms and molecules are stretched when they are loaded. By studying force-extension graphs of materials, you can see if the material obeys Hooke's law
• It is not always possible to use spring setup as metals will often require very large forces to produce measurable extensions, so different specialised equipment needed
• A material is said to be elastic if it regains its original dimensions when the deforming force is removed
• The atoms in a solid are held together by bonds. These behave like springs between the particles as the wire is stretched, the atomic separation increases. In the elastic region, the atoms return to their original positions when the deforming force is removed
• Elastic limit is when beyond that point the wire stops becoming elastic. This is where the wire has passed the point of reversibility and will undergo permanent deformation
• A material is plastic if it retains its shape after a deforming force is removed
• To obtain a force-extension graph for rubber can be done in the same way as the equipment for spring. The rubber band stretches very easily at first, reaching a length of three or four times its original value. It then becomes very stiff and difficult to stretch as it approaches its breaking point
• Natural rubber is a polymer. It contains long chains of atoms that are normally tangled in a disordered fashion. Relatively small forces are needed to 'untangle' these molecules so a large extension is produced for small loads. When the chains are fully extended, additional forces need to stretch the bonds between the atoms, so much smaller extensions are produced for a given load; the rubber becomes stiffer. Band returns to its original length if the force is removed at any stage prior to breaking as is elastic
• When a stress is applied to a material, the strain is the effect of that stress
• Stress is usually written as stress = force/cross-sectional area (σ = F / A). Has the units pascal
• Strain is written as strain = extension/original length (ε = Δl/l) and because it is a ratio of two lengths it has no unit.
• Young modulus (E) is a property of materials that undergo tensile or compressive stress
• The Young Modulus is calculated by stress/strain (E = σ/ε = Fl/AΔl) and is measured in pascals
• The importance of a stress and strain as opposed to a force-extension is that they are properties of the material: a stress-strain graph is always the same for a given material but for force-extension it depends on the dimensions of the sample used
• The shape of stress-strain graphs is much the same as force-extension but for large extensions, a reduction in cross-sectional area will result in an increase in the stress for a particular load. The main advantage is that the information gained from the graph relates to the properties of the material used and not just those of the particular sample tested
• O-A represents the Hooke's law region. Strain is proportional to stress up to this point. The Young Modulus of copper can be found directly by taking the gradient of the graph in this section. B is the elastic limit. If the stress is removed below this value, the wire returns to its original state. The stress at C is termed the yield stress. For stresses greater than this, copper will become ductile and deform plastically. D is the maximum stress that the copper can endure. It is called the ultimate tensile strength (UTS).

  
• E is the breaking point. There may be an increase in stress at this point due to a narrowing of the wire at the position on the wire where it breaks, which reduces the area at that point