Stress is force per area; strain is extension divided by original length. Within the linear elastic region the stress-strain slope is Young modulus. Releasing the load recovers the original length in this model.
A long thin wire stretches more than a short thick wire under the same load. Modulus describes the material; the wire stiffness also depends on its dimensions.
An ordinary stretched wire contracts sideways. Poisson ratio compares this negative lateral strain with the positive longitudinal strain. At small strain, the volume change is approximately (1−2ν)ε.
The force rises from zero as a wire extends or a beam bends, so stored work is half final force times deflection. Beam geometry enters through its cross-section second moment: greater depth sharply reduces bending.