قانون هوک - ویکی‌پدیا، دانشنامهٔ آزاد

${\displaystyle \mathbf {F} =-k\mathbf {x} \ }$

${\displaystyle \sigma =E\varepsilon }$

یا

${\displaystyle \Delta L={\frac {F}{EA}}L={\frac {\sigma }{E}}L.}$

${\displaystyle \tau =G\gamma }$

γ = tan(θ) ≈ θ

${\displaystyle F=-kx\,}$

${\displaystyle PE={1 \over 2}kx^{2}}$

${\displaystyle \omega ={\sqrt {k \over m}}}$

${\displaystyle f={1 \over 2\pi }{\sqrt {k \over m}}.}$

${\displaystyle {\boldsymbol {\sigma }}={\mathsf {c}}:{\boldsymbol {\epsilon }}~.}$

${\displaystyle \sigma _{ij}=c_{ijk\ell }~\epsilon _{k\ell }}$

${\displaystyle {\boldsymbol {\epsilon }}={\mathsf {s}}:{\boldsymbol {\sigma }}\qquad {\rm {or}}\qquad \epsilon _{ij}=s_{ijk\ell }~\sigma _{k\ell }~.}$

${\displaystyle \varepsilon _{ij}=\left({\tfrac {1}{3}}\varepsilon _{kk}\delta _{ij}\right)+\left(\varepsilon _{ij}-{\tfrac {1}{3}}\varepsilon _{kk}\delta _{ij}\right)}$

${\displaystyle {\boldsymbol {\varepsilon }}=\mathrm {vol} ({\boldsymbol {\varepsilon }})+\mathrm {dev} ({\boldsymbol {\varepsilon }})~;~~\mathrm {vol} ({\boldsymbol {\varepsilon }}):={\tfrac {1}{3}}~\mathrm {tr} ({\boldsymbol {\varepsilon }})~\mathbf {I} ~;~~\mathrm {dev} ({\boldsymbol {\varepsilon }}):={\boldsymbol {\varepsilon }}-\mathrm {vol} ({\boldsymbol {\varepsilon }})}$

${\displaystyle \sigma _{ij}=3K\left({\tfrac {1}{3}}\varepsilon _{kk}\delta _{ij}\right)+2G\left(\varepsilon _{ij}-{\tfrac {1}{3}}\varepsilon _{kk}\delta _{ij}\right)\,~;~~{\boldsymbol {\sigma }}=3K~\mathrm {vol} ({\boldsymbol {\varepsilon }})+2G~\mathrm {dev} ({\boldsymbol {\varepsilon }})}$

${\displaystyle {\boldsymbol {\sigma }}=\lambda ~\mathrm {tr} ({\boldsymbol {\varepsilon }})~\mathbf {I} +2\mu ~{\boldsymbol {\varepsilon }}={\mathsf {c}}:{\boldsymbol {\varepsilon }}~;~~{\mathsf {c}}=\lambda ~\mathbf {I} \otimes \mathbf {I} +2\mu ~{\mathsf {I}}}$

${\displaystyle \sigma _{ij}=\lambda ~\varepsilon _{kk}~\delta _{ij}+2\mu ~\varepsilon _{ij}=c_{ijk\ell }~\varepsilon _{k\ell }~;~~c_{ijk\ell }=\lambda ~\delta _{ij}~\delta _{k\ell }+\mu ~(\delta _{ik}~\delta _{j\ell }+\delta _{i\ell }~\delta _{jk})}$

${\displaystyle {\boldsymbol {\varepsilon }}={\tfrac {1}{2\mu }}~{\boldsymbol {\sigma }}-{\tfrac {\lambda }{2\mu (3\lambda +2\mu )}}~\mathrm {tr} ({\boldsymbol {\sigma }})~\mathbf {I} ={\tfrac {1}{2G}}~{\boldsymbol {\sigma }}+\left({\tfrac {1}{9K}}-{\tfrac {1}{6G}}\right)~\mathrm {tr} ({\boldsymbol {\sigma }})~\mathbf {I} }$

${\displaystyle {\mathsf {s}}=-{\tfrac {\lambda }{2\mu (3\lambda +2\mu )}}~\mathbf {I} \otimes \mathbf {I} +{\tfrac {1}{2\mu }}~{\mathsf {I}}=\left({\tfrac {1}{9K}}-{\tfrac {1}{6G}}\right)~\mathbf {I} \otimes \mathbf {I} +{\tfrac {1}{2G}}~{\mathsf {I}}}$

${\displaystyle {\boldsymbol {\varepsilon }}={\tfrac {1}{E}}~{\boldsymbol {\sigma }}-{\tfrac {\nu }{E}}\left[\mathrm {tr} ({\boldsymbol {\sigma }})~\mathbf {I} -{\boldsymbol {\sigma }}\right]}$

${\displaystyle {\begin{aligned}\varepsilon _{11}&={\tfrac {1}{E}}\left[\sigma _{11}-\nu (\sigma _{22}+\sigma _{33})\right]\\\varepsilon _{22}&={\tfrac {1}{E}}\left[\sigma _{22}-\nu (\sigma _{11}+\sigma _{33})\right]\\\varepsilon _{33}&={\tfrac {1}{E}}\left[\sigma _{33}-\nu (\sigma _{11}+\sigma _{22})\right]\\\varepsilon _{12}&={\tfrac {1}{2G}}~\sigma _{12}~;~~\varepsilon _{13}={\tfrac {1}{2G}}~\sigma _{13}~;~~\varepsilon _{23}={\tfrac {1}{2G}}~\sigma _{23}\end{aligned}}}$

${\displaystyle {\begin{bmatrix}\varepsilon _{11}\\\varepsilon _{22}\\\varepsilon _{33}\\2\varepsilon _{23}\\2\varepsilon _{31}\\2\varepsilon _{12}\end{bmatrix}}={\begin{bmatrix}\varepsilon _{11}\\\varepsilon _{22}\\\varepsilon _{33}\\\gamma _{23}\\\gamma _{31}\\\gamma _{12}\end{bmatrix}}={\cfrac {1}{E}}{\begin{bmatrix}1&-\nu &-\nu &0&0&0\\-\nu &1&-\nu &0&0&0\\-\nu &-\nu &1&0&0&0\\0&0&0&2(1+\nu )&0&0\\0&0&0&0&2(1+\nu )&0\\0&0&0&0&0&2(1+\nu )\end{bmatrix}}{\begin{bmatrix}\sigma _{11}\\\sigma _{22}\\\sigma _{33}\\\sigma _{23}\\\sigma _{31}\\\sigma _{12}\end{bmatrix}}}$

${\displaystyle {\begin{bmatrix}\sigma _{11}\\\sigma _{22}\\\sigma _{33}\\\sigma _{23}\\\sigma _{31}\\\sigma _{12}\end{bmatrix}}={\cfrac {E}{(1+\nu )(1-2\nu )}}{\begin{bmatrix}1-\nu &\nu &\nu &0&0&0\\\nu &1-\nu &\nu &0&0&0\\\nu &\nu &1-\nu &0&0&0\\0&0&0&(1-2\nu )/2&0&0\\0&0&0&0&(1-2\nu )/2&0\\0&0&0&0&0&(1-2\nu )/2\end{bmatrix}}{\begin{bmatrix}\varepsilon _{11}\\\varepsilon _{22}\\\varepsilon _{33}\\2\varepsilon _{23}\\2\varepsilon _{31}\\2\varepsilon _{12}\end{bmatrix}}}$

${\displaystyle {\begin{bmatrix}\sigma _{11}\\\sigma _{22}\\\sigma _{33}\\\sigma _{23}\\\sigma _{31}\\\sigma _{12}\end{bmatrix}}={\begin{bmatrix}2\mu +\lambda &\lambda &\lambda &0&0&0\\\lambda &2\mu +\lambda &\lambda &0&0&0\\\lambda &\lambda &2\mu +\lambda &0&0&0\\0&0&0&\mu &0&0\\0&0&0&0&\mu &0\\0&0&0&0&0&\mu \end{bmatrix}}{\begin{bmatrix}\varepsilon _{11}\\\varepsilon _{22}\\\varepsilon _{33}\\2\varepsilon _{23}\\2\varepsilon _{31}\\2\varepsilon _{12}\end{bmatrix}}}$

${\displaystyle {\begin{bmatrix}\varepsilon _{11}\\\varepsilon _{22}\\2\varepsilon _{12}\end{bmatrix}}={\cfrac {1}{E}}{\begin{bmatrix}1&-\nu &0\\-\nu &1&0\\0&0&2(1+\nu )\end{bmatrix}}{\begin{bmatrix}\sigma _{11}\\\sigma _{22}\\\sigma _{12}\end{bmatrix}}}$

${\displaystyle {\begin{bmatrix}\sigma _{11}\\\sigma _{22}\\\sigma _{12}\end{bmatrix}}={\cfrac {E}{1-\nu ^{2}}}{\begin{bmatrix}1&\nu &0\\\nu &1&0\\0&0&{\cfrac {1-\nu }{2}}\end{bmatrix}}{\begin{bmatrix}\varepsilon _{11}\\\varepsilon _{22}\\2\varepsilon _{12}\end{bmatrix}}}$

${\displaystyle \sigma _{ij}={\cfrac {\partial U}{\partial \epsilon _{ij}}}\quad \implies \quad c_{ijk\ell }={\cfrac {\partial ^{2}U}{\partial \epsilon _{ij}\partial \epsilon _{k\ell }}}~.}$

${\displaystyle [{\boldsymbol {\sigma }}]={\begin{bmatrix}\sigma _{11}\\\sigma _{22}\\\sigma _{33}\\\sigma _{23}\\\sigma _{31}\\\sigma _{12}\end{bmatrix}}\equiv {\begin{bmatrix}\sigma _{1}\\\sigma _{2}\\\sigma _{3}\\\sigma _{4}\\\sigma _{5}\\\sigma _{6}\end{bmatrix}}~;~~[{\boldsymbol {\epsilon }}]={\begin{bmatrix}\epsilon _{11}\\\epsilon _{22}\\\epsilon _{33}\\2\epsilon _{23}\\2\epsilon _{31}\\2\epsilon _{12}\end{bmatrix}}\equiv {\begin{bmatrix}\epsilon _{1}\\\epsilon _{2}\\\epsilon _{3}\\\epsilon _{4}\\\epsilon _{5}\\\epsilon _{6}\end{bmatrix}}}$

${\displaystyle [{\mathsf {C}}]={\begin{bmatrix}c_{1111}&c_{1122}&c_{1133}&c_{1123}&c_{1131}&c_{1112}\\c_{2211}&c_{2222}&c_{2233}&c_{2223}&c_{2231}&c_{2212}\\c_{3311}&c_{3322}&c_{3333}&c_{3323}&c_{3331}&c_{3312}\\c_{2311}&c_{2322}&c_{2333}&c_{2323}&c_{2331}&c_{2312}\\c_{3111}&c_{3122}&c_{3133}&c_{3123}&c_{3131}&c_{3112}\\c_{1211}&c_{1222}&c_{1233}&c_{1223}&c_{1231}&c_{1212}\end{bmatrix}}\equiv {\begin{bmatrix}C_{11}&C_{12}&C_{13}&C_{14}&C_{15}&C_{16}\\C_{12}&C_{22}&C_{23}&C_{24}&C_{25}&C_{26}\\C_{13}&C_{23}&C_{33}&C_{34}&C_{35}&C_{36}\\C_{14}&C_{24}&C_{34}&C_{44}&C_{45}&C_{46}\\C_{15}&C_{25}&C_{35}&C_{45}&C_{55}&C_{56}\\C_{16}&C_{26}&C_{36}&C_{46}&C_{56}&C_{66}\end{bmatrix}}}$

${\displaystyle [{\boldsymbol {\sigma }}]=[{\mathsf {C}}][{\boldsymbol {\epsilon }}]\qquad {\text{or}}\qquad \sigma _{i}=C_{ij}\epsilon _{j}~.}$

${\displaystyle [{\mathsf {S}}]={\begin{bmatrix}s_{1111}&s_{1122}&s_{1133}&2s_{1123}&2s_{1131}&2s_{1112}\\s_{2211}&s_{2222}&s_{2233}&2s_{2223}&2s_{2231}&2s_{2212}\\s_{3311}&s_{3322}&s_{3333}&2s_{3323}&2s_{3331}&2s_{3312}\\2s_{2311}&2s_{2322}&2s_{2333}&4s_{2323}&4s_{2331}&4s_{2312}\\2s_{3111}&2s_{3122}&2s_{3133}&4s_{3123}&4s_{3131}&4s_{3112}\\2s_{1211}&2s_{1222}&2s_{1233}&4s_{1223}&4s_{1231}&4s_{1212}\end{bmatrix}}\equiv {\begin{bmatrix}S_{11}&S_{12}&S_{13}&S_{14}&S_{15}&S_{16}\\S_{12}&S_{22}&S_{23}&S_{24}&S_{25}&S_{26}\\S_{13}&S_{23}&S_{33}&S_{34}&S_{35}&S_{36}\\S_{14}&S_{24}&S_{34}&S_{44}&S_{45}&S_{46}\\S_{15}&S_{25}&S_{35}&S_{45}&S_{55}&S_{56}\\S_{16}&S_{26}&S_{36}&S_{46}&S_{56}&S_{66}\end{bmatrix}}}$

${\displaystyle c_{pqrs}=l_{pi}~l_{qj}~l_{rk}~l_{s\ell }~c_{ijk\ell }}$

${\displaystyle [\mathbf {e} _{i}']=[L][\mathbf {e} _{i}]}$

${\displaystyle C_{ij}~\epsilon _{i}~\epsilon _{j}=C_{ij}'~\epsilon '_{i}~\epsilon '_{j}~.}$

${\displaystyle C_{ij}=C'_{ij}\quad \implies \quad C_{ij}~(\epsilon _{i}~\epsilon _{j}-\epsilon '_{i}~\epsilon '_{j})=0~.}$

${\displaystyle {\begin{bmatrix}\sigma _{1}\\\sigma _{2}\\\sigma _{3}\\\sigma _{4}\\\sigma _{5}\\\sigma _{6}\end{bmatrix}}={\begin{bmatrix}C_{11}&C_{12}&C_{13}&0&0&0\\C_{12}&C_{22}&C_{23}&0&0&0\\C_{13}&C_{23}&C_{33}&0&0&0\\0&0&0&C_{44}&0&0\\0&0&0&0&C_{55}&0\\0&0&0&0&0&C_{66}\end{bmatrix}}{\begin{bmatrix}\epsilon _{1}\\\epsilon _{2}\\\epsilon _{3}\\\epsilon _{4}\\\epsilon _{5}\\\epsilon _{6}\end{bmatrix}}}$

${\displaystyle {\begin{bmatrix}\epsilon _{\rm {xx}}\\\epsilon _{\rm {yy}}\\\epsilon _{\rm {zz}}\\2\epsilon _{\rm {yz}}\\2\epsilon _{\rm {zx}}\\2\epsilon _{\rm {xy}}\end{bmatrix}}={\begin{bmatrix}{\tfrac {1}{E_{\rm {x}}}}&-{\tfrac {\nu _{\rm {xy}}}{E_{\rm {x}}}}&-{\tfrac {\nu _{\rm {xz}}}{E_{\rm {x}}}}&0&0&0\\-{\tfrac {\nu _{\rm {yx}}}{E_{\rm {y}}}}&{\tfrac {1}{E_{\rm {y}}}}&-{\tfrac {\nu _{\rm {yz}}}{E_{\rm {y}}}}&0&0&0\\-{\tfrac {\nu _{\rm {zx}}}{E_{\rm {z}}}}&-{\tfrac {\nu _{\rm {zy}}}{E_{\rm {z}}}}&{\tfrac {1}{E_{\rm {z}}}}&0&0&0\\0&0&0&{\tfrac {1}{G_{\rm {yz}}}}&0&0\\0&0&0&0&{\tfrac {1}{G_{\rm {zx}}}}&0\\0&0&0&0&0&{\tfrac {1}{G_{\rm {xy}}}}\\\end{bmatrix}}{\begin{bmatrix}\sigma _{\rm {xx}}\\\sigma _{\rm {yy}}\\\sigma _{\rm {zz}}\\\sigma _{\rm {yz}}\\\sigma _{\rm {zx}}\\\sigma _{\rm {xy}}\end{bmatrix}}}$

${\displaystyle {\begin{bmatrix}\varepsilon _{\rm {xx}}\\\varepsilon _{\rm {yy}}\\2\varepsilon _{\rm {xy}}\end{bmatrix}}={\begin{bmatrix}{\frac {1}{E_{\rm {x}}}}&-{\frac {\nu _{\rm {xy}}}{E_{\rm {x}}}}&0\\-{\frac {\nu _{\rm {yx}}}{E_{\rm {y}}}}&{\frac {1}{E_{\rm {y}}}}&0\\0&0&{\frac {1}{G_{\rm {xy}}}}\end{bmatrix}}{\begin{bmatrix}\sigma _{\rm {xx}}\\\sigma _{\rm {yy}}\\\sigma _{\rm {xy}}\end{bmatrix}}\,.}$

${\displaystyle {\begin{bmatrix}\sigma _{\rm {xx}}\\\sigma _{\rm {yy}}\\\sigma _{\rm {xy}}\end{bmatrix}}={\cfrac {1}{1-\nu _{\rm {xy}}\nu _{\rm {yx}}}}{\begin{bmatrix}E_{\rm {x}}&\nu _{\rm {xy}}E_{\rm {y}}&0\\\nu _{\rm {yx}}E_{\rm {x}}&E_{y}&0\\0&0&G_{\rm {xy}}(1-\nu _{\rm {xy}}\nu _{\rm {yx}})\end{bmatrix}}{\begin{bmatrix}\varepsilon _{\rm {xx}}\\\varepsilon _{\rm {yy}}\\2\varepsilon _{\rm {xy}}\end{bmatrix}}\,.}$

${\displaystyle {\begin{bmatrix}\sigma _{1}\\\sigma _{2}\\\sigma _{3}\\\sigma _{4}\\\sigma _{5}\\\sigma _{6}\end{bmatrix}}={\begin{bmatrix}C_{11}&C_{12}&C_{13}&0&0&0\\C_{12}&C_{11}&C_{13}&0&0&0\\C_{13}&C_{13}&C_{33}&0&0&0\\0&0&0&C_{44}&0&0\\0&0&0&0&C_{44}&0\\0&0&0&0&0&{\tfrac {1}{2}}(C_{11}-C_{12})\end{bmatrix}}{\begin{bmatrix}\epsilon _{1}\\\epsilon _{2}\\\epsilon _{3}\\\epsilon _{4}\\\epsilon _{5}\\\epsilon _{6}\end{bmatrix}}}$