# Deformation tensor

A tensor describing the locations of the points of a body after deformation with respect to their location before deformation. It is a symmetric tensor of the second rank,

(*) |

where are the Cartesian rectangular coordinates of a point in the body prior to deformation and are the coordinates of the displacement vector . In the theory of elasticity the deformation tensor is decomposed into two constituent tensors:

The tensor describes a spatial deformation and is known as the spherical deformation tensor:

The tensor describes solely the change in form, and the sum of its diagonal elements is equal to zero:

The tensor is known as the deviator of the deformation tensor.

In the case of a small deformation, second-order magnitudes are neglected, and the deformation tensor (*) is defined by the expression:

In spherical coordinates the linearized deformation tensor (*) assumes the form:

In cylindrical coordinates it has the form

#### References

[1] | L.D. Landau, E.M. Lifshits, "Theory of elasticity" , Pergamon (1959) (Translated from Russian) |

[2] | , [Soviet] Physical Encyclopedic Dictionary , 1 , Moscow (1960) pp. 553 (In Russian) |

#### Comments

Strictly speaking the of (*) do not form a tensor in the mathematical sense; they do not transform as tensors. The "tensor" is also known as the strain tensor or the rate-of-strain tensor.

Let and denote line elements (infinitesimal distances) before and after deformation. Then , so that describes the change in an element of length as the body is deformed.

#### References

[a1] | A.C. Eringen, "Mechanics of continua" , Pergamon (1967) |

**How to Cite This Entry:**

Deformation tensor.

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Deformation_tensor&oldid=14896