Elasticity and mechanical properties of materials
Material-property questions connect forces and deformations to measurable geometry. Separate the behavior of an entire specimen from a property used to characterize its material. A longer wire extends more than a shorter wire of the same material and cross-sectional area under the same tensile force, so force divided by extension is not by itself a geometry-independent material constant.
Engineering tensile stress is force divided by the original cross-sectional area. Engineering tensile strain is extension divided by the original length. Stress is measured in pascals; strain is a dimensionless ratio. Within a specified linear elastic range, Young's modulus is the ratio of tensile stress to tensile strain. Combining these definitions gives extension = force times original length divided by area times Young's modulus. This relation requires the stated tensile geometry and linear model; it must not be applied through plastic deformation merely because the arithmetic is convenient.
For a hypothetical specimen, take length 2.0 m, cross-sectional area 1.0 square millimetre and a supplied Young's modulus of 200 gigapascals. A 100 N tensile load gives stress 100 megapascals. The strain is 0.00050 and extension is 0.0010 m, or 1.0 mm. Converting area is essential: one square millimetre is one millionth of a square metre. These values are supplied exercise data, not allowable loads or a specification for any real material or installation.
Hooke's spring law relates restoring force to displacement from equilibrium as F = -kx. Its minus sign indicates direction. For a slowly loaded linear spring, the elastic energy stored is one half kx squared, equivalently one half force times extension. More generally, the area under a force-extension loading curve gives work done, with force and extension expressed in compatible units. If loading and unloading curves differ, their enclosed area represents energy dissipated in a cycle; it should not be counted as recoverable elastic energy.
Read graph axes before interpreting slopes. The gradient of a force-extension graph is spring stiffness, whereas the gradient of a stress-strain graph in its linear range is Young's modulus. The elastic limit concerns recovery after unloading; the proportional limit concerns a straight-line relation. They need not coincide. Yielding and fracture are different events. A physical experiment should record its loading range and measurement uncertainty instead of extending a fitted line beyond the observed material behavior.