There are several standard models for how materials respond to stress:

- Elastic -- a material has a rest shape and its shape departs away from the rest shape due to stress. The amount of departure from rest shape is called strain, the departure itself is called deformation. The resistance to deformation is called Young's modulus. A spring obeying Hooke's law is a one-dimensional linear version of a general elastic body.
- Viscous -- a material has no rest shape, but its velocity depends on the applied forces. A dashpot (a shock absorber) is a one-dimensional version of a viscous material.
- Viscoelastic -- a material that is elastic, but also has damping.
- Plastic -- a material that, when the stress exceeds a threshold, changes its rest shape in response. The material commonly known as "plastic" is named after this property.

Elasticity extensively uses tensors to describe stresses, strains, and the relationship between them.

Typically, elasticity uses linear models to relate stresses and strains (see Linear elasticity). However, true materials exhibit non-linear behavior.

- L.D. Landau, E.M. Lifshitz,
*Course of Theoretical Physics: Theory of Elasticity*Butterworth-Heinemann, ISBN 075062633X - J.E. Marsden, T.J. Hughes,
*Mathematical Foundations of Elasticity*, Dover, ISBN 0486678652 - P.C. Chou, N. J. Pagano,
*Elasticity: Tensor, Dyadic, and Engineering Approaches*, Dover, ISBN 0486669580 - R.W. Ogden,
*Non-linear Elastic Deformation*, Dover, ISBN 0486696480