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A tensor defining the distribution of internal stresses in a body under strain. The stress tensor is a symmetric tensor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090460/s0904601.png" /> of rank 2. The component <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090460/s0904602.png" /> is the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090460/s0904603.png" />-th component of the force acting on a unit surface perpendicular to the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090460/s0904604.png" />-axis. Thus, acting on a unit area perpendicular to the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090460/s0904605.png" />-axis, one has a normal stress <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090460/s0904606.png" /> (i.e. a force in the direction of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090460/s0904607.png" />-axis) and shearing stresses (i.e. forces in the directions of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090460/s0904608.png" />- and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090460/s0904609.png" />-axes) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090460/s09046010.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090460/s09046011.png" />. 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:
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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:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090460/s09046012.png" /></td> </tr></table>
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$$\sigma_{ij}'=\frac13\delta_{ij}\sigma_{ll}=p\delta_{ij},$$
  
with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090460/s09046013.png" /> the hydrostatic pressure.
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with $p$ the hydrostatic pressure.
  
The second stressed state is characterized by the components of the deviatoric stress tensor:
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The second stressed state is characterized by the components of the [[Deviatoric tensor|deviatoric]] stress tensor:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090460/s09046014.png" /></td> </tr></table>
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$$\sigma_{ij}''=\sigma_{ij}-\frac13\delta_{ij}\sigma_{ll}=\sigma_{ij}+p\delta_{ij}.$$
  
 
====References====
 
====References====

Latest revision as of 19:53, 22 January 2016

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://encyclopediaofmath.org/index.php?title=Stress_tensor&oldid=11269
This article was adapted from an original article by A.B. Ivanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article