## ::Deformation (mechanics)

### ::concepts

Mathbf::partial    Strain::length    Material::bmatrix    Begin::gamma    Books::strain''    Which::alpha

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The deformation of a thin straight rod into a closed loop. The length of the rod remains almost unchanged during the deformation, which indicates that the strain is small. In this particular case of bending, displacements associated with rigid translations and rotations of material elements in the rod are much greater than displacements associated with straining.

Deformation in continuum mechanics is the transformation of a body from a reference configuration to a current configuration.<ref name=Truesdell>Truesdell, C. and Noll, W., (2004), The non-linear field theories of mechanics: Third edition, Springer, p. 48.</ref> A configuration is a set containing the positions of all particles of the body.

A deformation may be caused by external loads,<ref name=wu>H.-C. Wu, Continuum Mechanics and Plasticity, CRC Press (2005), ISBN 1-58488-363-4</ref> body forces (such as gravity or electromagnetic forces), or changes in temperature, moisture content, or chemical reactions, etc.

Strain is a description of deformation in terms of relative displacement of particles in the body that excludes rigid-body motions. Different equivalent choices may be made for the expression of a strain field depending on whether it is defined with respect to the initial or the final configuration of the body and on whether the metric tensor or its dual is considered.

In a continuous body, a deformation field results from a stress field induced by applied forces or is due to changes in the temperature field inside the body. The relation between stresses and induced strains is expressed by constitutive equations, e.g., Hooke's law for linear elastic materials. Deformations which are recovered after the stress field has been removed are called elastic deformations. In this case, the continuum completely recovers its original configuration. On the other hand, irreversible deformations remain even after stresses have been removed. One type of irreversible deformation is plastic deformation, which occurs in material bodies after stresses have attained a certain threshold value known as the elastic limit or yield stress, and are the result of slip, or dislocation mechanisms at the atomic level. Another type of irreversible deformation is viscous deformation, which is the irreversible part of viscoelastic deformation.

In the case of elastic deformations, the response function linking strain to the deforming stress is the compliance tensor of the material.

Deformation (mechanics) sections