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Deformation tensor

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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