View source for 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 $\sigma_{ij}$ 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 tensor|deviatoric]] stress tensor:

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

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  L.D. Landau,   E.M. Lifshitz,   "Elasticity theory" , Pergamon  (1959)  (Translated from Russian)</TD></TR></table>



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

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  C. Truesdell,   W. Noll,   "The non-linear field theories of mechanics"  S. Flügge (ed.) , ''Handbuch der Physik'' , '''III/3''' , Springer  (1965)  pp. 1–602</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  C. Truesdell,   R. Toupin,   "The classical field theories"  S. Flügge (ed.) , ''Handbuch der Physik'' , '''III/1''' , Springer  (1960)  pp. 226–793</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  I.S. [I.S. Sokolnikov] Sokolnikoff,   "Mathematical theory of elasticity" , McGraw-Hill  (1956)  (Translated from Russian)</TD></TR></table>