Deformation (mechanics)

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 (mechanics)

From Wikipedia, the free encyclopedia

Deformation in continuum

mechanics is the transformation

of a body from a reference

configuration to a current

configuration.[1] A

configuration is a set

containing the positions of all

particles of the body.

A deformation may be caused by

external loads,[2] 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.

Contents

1 Strain

1.1 Strain measures

1.1.1 Engineering strain

1.1.2 Stretch ratio

1.1.3 True strain

1.1.4 Green strain

1.1.5 Almansi strain

1.2 Normal strain

1.3 Shear strain

1.4 Metric tensor

2 Description of deformation

2.1 Affine deformation

2.2 Rigid body motion

3 Displacement

3.1 Displacement gradient tensor

4 Examples of deformations

4.1 Plane deformation

4.1.1 Isochoric plane deformation

4.1.2 Simple shear

5 See also

6 References

7 Further reading

Strain

A strain is a normalized measure of deformation representing the displacement between particles in the body relative to a reference length.

A general deformation of a body can be expressed in the form where is the reference position of material points in the body. Such a measure does not distinguish between rigid body motions (translations and rotations) and changes in shape (and size) of the body. A deformation has units of length.

We could, for example, define strain to be

where is the identity tensor. Hence strains are dimensionless and are usually expressed as a decimal fraction, a percentage or in parts-per notation. Strains measure how much a given deformation differs locally from a rigid-body deformation.[3]

A strain is in general a tensor quantity. Physical insight into strains can be gained by observing that a given strain can be decomposed into normal and shear components. The amount of stretch or compression along material line elements or fibers is the