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Difference between revisions of "Hooke law"

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$$u_{ik}=\frac{1}{9K}\delta_{ik}\sigma_{ll}+\frac{1}{2\mu}\left(\sigma_{ik}-\frac13\delta_{ik}\sigma_{ll}\right),$$
 
$$u_{ik}=\frac{1}{9K}\delta_{ik}\sigma_{ll}+\frac{1}{2\mu}\left(\sigma_{ik}-\frac13\delta_{ik}\sigma_{ll}\right),$$
  
where $\delta$ is the Kronecker symbol, $K$ is the modulus of compression and $\mu$ is the shear modulus. See [[Elasticity, mathematical theory of|Elasticity, mathematical theory of]].
+
where $\delta$ is the [[Kronecker symbol]], $K$ is the modulus of compression and $\mu$ is the shear modulus. See [[Elasticity, mathematical theory of]].
  
 
In its simplest form the law was experimentally established by R. Hooke in 1660.
 
In its simplest form the law was experimentally established by R. Hooke in 1660.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  E.M. Lifshitz,  "Theory of elasticity" , Pergamon  (1959)  (Translated from Russian)</TD></TR></table>
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<table>
 +
<TR><TD valign="top">[1]</TD> <TD valign="top">  E.M. Lifshitz,  "Theory of elasticity" , Pergamon  (1959)  (Translated from Russian)</TD></TR>
 +
</table>
  
  
Line 17: Line 19:
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  I.S. [I.S. Sokolnikov] Sokolnikoff,  "Mathematical theory of elasticity" , McGraw-Hill  (1956)  (Translated from Russian)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  S.P. Timoshenko,  J.N. Goodier,  "Theory of elasticity" , McGraw-Hill  (1970)</TD></TR></table>
+
<table>
 +
<TR><TD valign="top">[a1]</TD> <TD valign="top">  I.S. [I.S. Sokolnikov] Sokolnikoff,  "Mathematical theory of elasticity" , McGraw-Hill  (1956)  (Translated from Russian)</TD></TR>
 +
<TR><TD valign="top">[a2]</TD> <TD valign="top">  S.P. Timoshenko,  J.N. Goodier,  "Theory of elasticity" , McGraw-Hill  (1970)</TD></TR>
 +
</table>

Revision as of 19:19, 13 January 2016

A law describing the relation between stress and deformation in an elastic body, within a certain range. It states that a small deformation is proportional to the forces applied to the body, i.e. the deformation tensor $u_{ik}$ is a linear function of the stress tensor $\sigma_{jk}$:

$$u_{ik}=\frac{1}{9K}\delta_{ik}\sigma_{ll}+\frac{1}{2\mu}\left(\sigma_{ik}-\frac13\delta_{ik}\sigma_{ll}\right),$$

where $\delta$ is the Kronecker symbol, $K$ is the modulus of compression and $\mu$ is the shear modulus. See Elasticity, mathematical theory of.

In its simplest form the law was experimentally established by R. Hooke in 1660.

References

[1] E.M. Lifshitz, "Theory of elasticity" , Pergamon (1959) (Translated from Russian)


Comments

References

[a1] I.S. [I.S. Sokolnikov] Sokolnikoff, "Mathematical theory of elasticity" , McGraw-Hill (1956) (Translated from Russian)
[a2] S.P. Timoshenko, J.N. Goodier, "Theory of elasticity" , McGraw-Hill (1970)
How to Cite This Entry:
Hooke law. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Hooke_law&oldid=37520
This article was adapted from an original article by A.B. Ivanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article