# Solid mechanics

**Solid mechanics** is the branch of mechanics, physics, and mathematics that concerns the behavior of solid matter under external actions (e.g., external forces, temperature changes, applied displacements, etc.). It is part of a broader study known as continuum mechanics. One of the most common practical applications of Solid Mechanics is the Euler-Bernoulli beam equation. Solid mechanics extensively uses tensors to describe stresses, strains, and the relationship between them.

## Response Models

There are three models that describe how a solid responds to an applied stress:

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 deformation, the proportion of deformation to original size is called strain. If the applied stress is sufficiently low (or the imposed strain is small enough), almost all solid materials behave in such a way that the strain is directly proportional to the stress; the coefficient of the proportion is called the modulus of elasticity or Young's modulus. This region of deformation is known as the linearly elastic region.

Typically, solid mechanics uses linear models to relate stresses and strains (see linear elasticity). However, real materials often exhibit non-linear behavior.

- Elastically – Linearly elastic materials can be described by the linear elasticity equations such as Hooke's law: when the applied stress is removed, the materials returns to its previous state.
- Viscoelastically – These are materials that behave elastically, but also has damping: when the stress is applied and removed, work has to be done against the damping effects and is converted in heat within the material resulting in a a hysteresis loop in the stress–strain curve.
- Plastically – Materials that behave elastically when the applied stress is less than a yielding value. When the stress is greater than the yield stress, the material behaves plastically and does not return to its previous state.

## See also

- Strength of materials - Specific definitions and the relationships between stress and strain.
- Applied mechanics
- Viscosity
- Thermoplasticity
- Materials science
- Chord modulus

## References

- L.D. Landau, E.M. Lifshitz,
*Course of Theoretical Physics: Theory of Elasticity*Butterworth-Heinemann, ISBN 0-7506-2633-X - J.E. Marsden, T.J. Hughes,
*Mathematical Foundations of Elasticity*, Dover, ISBN 0-486-67865-2 - P.C. Chou, N. J. Pagano,
*Elasticity: Tensor, Dyadic, and Engineering Approaches*, Dover, ISBN 0-486-66958-0 - R.W. Ogden,
*Non-linear Elastic Deformation*, Dover, ISBN 0-486-69648-0 - S. Timoshenko and J.N. Goodier," Theory of elasticity", 3d ed., New York, McGraw-Hill, 1970.
- A.I. Lurie, "Theory of Elasticity", Springer, 1999.
- L.B. Freund, "Dynamic Fracture Mechanics", Cambridge University Press, 1990.
- R. Hill, "The Mathematical Theory of Plasticity", Oxford University, 1950.
- J. Lubliner, "Plasticity Theory", Macmillan Publishing Company, 1990.

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