# Stress tensor

A tensor defining the distribution of internal stresses in a body under strain. The stress tensor is a symmetric tensor $\sigma_{ij}$ of rank 2. The component $\sigma_{ik}$ is the $i$-th component of the force acting on a unit surface perpendicular to the $x_k$-axis. Thus, acting on a unit area perpendicular to the $x$-axis, one has a normal stress $\sigma_{xx}$ (i.e. a force in the direction of the $x$-axis) and shearing stresses (i.e. forces in the directions of the $y$- and $z$-axes) $\sigma_{yx}$ and $\sigma_{zx}$. 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:

$$\sigma_{ij}'=\frac13\delta_{ij}\sigma_{ll}=p\delta_{ij},$$

with $p$ the hydrostatic pressure.

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

$$\sigma_{ij}''=\sigma_{ij}-\frac13\delta_{ij}\sigma_{ll}=\sigma_{ij}+p\delta_{ij}.$$

#### 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://www.encyclopediaofmath.org/index.php?title=Stress_tensor&oldid=37616