Difference between revisions of "Hooke law"
From Encyclopedia of Mathematics
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| − | 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 | + | {{TEX|done}} |
| + | 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 | + | 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"> | + | <table> |
| − | + | <TR><TD valign="top">[1]</TD> <TD valign="top"> E.M. Lifshitz, "Theory of elasticity" , Pergamon (1959) (Translated from Russian)</TD></TR> | |
| − | + | <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> | |
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Latest revision as of 20:12, 11 December 2023
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) |
| [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=11627
Hooke law. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Hooke_law&oldid=11627
This article was adapted from an original article by A.B. Ivanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article