Stress tensor

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)