In metals, the atomic lattice changes size and shape when forces are applied (energy is added to the system).
This is known as perfect elasticity, in which a given object will return to its original shape no matter how strongly it is deformed.
The material's elastic limit or yield strength is the maximum stress that can arise before the onset of plastic deformation.
[2] The curve is generally nonlinear, but it can (by use of a Taylor series) be approximated as linear for sufficiently small deformations (in which higher-order terms are negligible).
[3] For rubber-like materials such as elastomers, the slope of the stress–strain curve increases with stress, meaning that rubbers progressively become more difficult to stretch, while for most metals, the gradient decreases at very high stresses, meaning that they progressively become easier to stretch.
In response to a small, rapidly applied and removed strain, these fluids may deform and then return to their original shape.
For most commonly used engineering materials, the elastic modulus is on the scale of gigapascals (GPa, 109 Pa).
A geometry-dependent version of the idea[a] was first formulated by Robert Hooke in 1675 as a Latin anagram, "ceiiinosssttuv".
He published the answer in 1678: "Ut tensio, sic vis" meaning "As the extension, so the force",[5][6] a linear relationship commonly referred to as Hooke's law.
[7] Although the general proportionality constant between stress and strain in three dimensions is a 4th-order tensor called stiffness, systems that exhibit symmetry, such as a one-dimensional rod, can often be reduced to applications of Hooke's law.
A material is said to be Cauchy-elastic if the Cauchy stress tensor σ is a function of the deformation gradient F alone: It is generally incorrect to state that Cauchy stress is a function of merely a strain tensor, as such a model lacks crucial information about material rotation needed to produce correct results for an anisotropic medium subjected to vertical extension in comparison to the same extension applied horizontally and then subjected to a 90-degree rotation; both these deformations have the same spatial strain tensors yet must produce different values of the Cauchy stress tensor.
By also requiring satisfaction of material objectivity, the energy potential may be alternatively regarded as a function of the Cauchy-Green deformation tensor (
), in which case the hyperelastic model may be written alternatively as Linear elasticity is used widely in the design and analysis of structures such as beams, plates and shells, and sandwich composites.
Hyperelasticity is primarily used to determine the response of elastomer-based objects such as gaskets and of biological materials such as soft tissues and cell membranes.
More specifically, the fraction of pores, their distribution at different sizes and the nature of the fluid with which they are filled give rise to different elastic behaviours in solids.
Microscopically, the stress–strain relationship of materials is in general governed by the Helmholtz free energy, a thermodynamic quantity.
As such, microscopic factors affecting the free energy, such as the equilibrium distance between molecules, can affect the elasticity of materials: for instance, in inorganic materials, as the equilibrium distance between molecules at 0 K increases, the bulk modulus decreases.