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Stress tensor

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A tensor defining the distribution of internal stresses in a body under strain. The stress tensor is a symmetric tensor of rank 2. The component is the -th component of the force acting on a unit surface perpendicular to the -axis. Thus, acting on a unit area perpendicular to the -axis, one has a normal stress (i.e. a force in the direction of the -axis) and shearing stresses (i.e. forces in the directions of the - and -axes) and . The stressed state defined by the components of the stress tensor can be resolved into two stressed states. The first is that characterized by the isotropic stress tensor:

with the hydrostatic pressure.

The second stressed state is characterized by the components of the deviatoric stress tensor:

References

[1] L.D. Landau, E.M. Lifshitz, "Elasticity theory" , Pergamon (1959) (Translated from Russian)


Comments

In general continuum theory, the stress tensor is not necessarily symmetric. The symmetry is lost when coupled stresses are present (polar materials).

References

[a1] C. Truesdell, W. Noll, "The non-linear field theories of mechanics" S. Flügge (ed.) , Handbuch der Physik , III/3 , Springer (1965) pp. 1–602
[a2] C. Truesdell, R. Toupin, "The classical field theories" S. Flügge (ed.) , Handbuch der Physik , III/1 , Springer (1960) pp. 226–793
[a3] I.S. [I.S. Sokolnikov] Sokolnikoff, "Mathematical theory of elasticity" , McGraw-Hill (1956) (Translated from Russian)
How to Cite This Entry:
Stress tensor. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Stress_tensor&oldid=33051
This article was adapted from an original article by A.B. Ivanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article