Elasticity

Elasticity is the ability to recover original shape and size after a load is removed.

Plastic deformation leaves a permanent change after unloading. A perfectly plastic body is an ideal model of this behaviour.

Real materials can show different behaviour at different strain levels. One familiar label never describes every possible loading condition.

Stress is measured in pascals. Strain is a ratio, so it has no unit.

Under equal tensile forces, a thinner wire has greater stress because its cross-sectional area is smaller.

This calculation gives strain only. It does not prove that the band follows Hooke’s law at 10% strain.

Hooke’s law applies while stress remains proportional to strain. This is the graph’s initial straight region.

The proportional limit ends that straight-line relationship. It does not always end elastic recovery.

The elastic limit comes later for many materials. Beyond it, unloading leaves permanent strain.

The straight region’s slope is Young’s modulus. A steeper slope means less strain for the same stress.

Young’s modulus compares longitudinal stress with longitudinal strain in the proportional region.

Steel has a higher Young’s modulus than rubber. Under equal stress, steel develops less strain within the relevant elastic range.

A perfectly rigid body would have infinite Young’s modulus in the ideal model. Real materials are not perfectly rigid.

Bulk modulus measures resistance to volume change under uniform pressure.

The minus sign reflects decreasing volume under increasing pressure. Compressibility is the reciprocal of bulk modulus.

Gases are generally much easier to compress than liquids and solids. Liquids still are not absolutely incompressible.

A Hookean spring has restoring force F = −kx. The minus sign shows that the restoring force opposes displacement.

For applied-force magnitude against extension, the graph’s slope is k. Its unit is newtons per metre.

A spring balance uses calibrated extension to infer force. This method works within the spring’s proportional range.

A slowly stretched Hookean spring stores elastic energy relative to its undeformed state.

Without energy losses, this equals the triangular area under the applied-force-versus-extension graph.

Elastic-property data help engineers estimate how much a structural material will deform under a load.

Stress identifies the force intensity. A suitable modulus connects that stress with the expected deformation.

Elastic energy also explains motion after release. Stored energy can become kinetic energy when a spring moves.

The word elastic does not simply mean “stretches far.” The loading type, modulus, and limits all matter.