???????? ??????? ???????

??? ??	continuum physics (en) ? ????????
???????	continuum (en)

????? - ????? ????? - ????? ???? ??????

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?????? ???????? ??????? ??????? ?? ??????? ?????????? ?????? ?????? ???????? ???????? ?? ?? ???? ???????? ???? ???????? ???. ?? ?????? ?? ??? ??????? ?????????? ????????? ??? ????? ?????? ??? ??????? ???????? ???????? ?? ????? ?????? ?? ???? ?????????? ?????? ????????.

????? ????? ?????? [ ??? ]

????? ??????? ??????? ?????? ???????? ???????? ? ?? ?????? ???? ????? ??????. ??? ??????? ???????? ?????? ???? ?????????. ????? ???? ????? ????? ???????? ????? ??????? ???? ?????? ???? ????? ?? ?? ?????? ?? ????? ????? ???? ???? (?????) ????? ??????? ???? ?????? ?? ?????? ???????. ????? ?????? ?? ??? ???? ?????? ???????? ?????? ?????? ????? ?? ???????? ?????? ?????? ???.

???? ?????? ?? ?????? ?????? ????? ?????? ???????? ???????? ????? ??? ?? ???? ????? ??????? ????? ????? ??? ?? ?? ???? ????? ?????? ??????? ??????? ?????? ????????. ????? ???? ?????/?????? ????? ?????? ????? ??? ?????? ????? ???????? ????????? ??????? ?????? ??? ??? ???-?????. ???? ??? ????? ???? ??? ??? ???? ????? ????????? ????? ????? ??????? ???? ????????? ????????? (???????? ?????? ???? ?????? ??????/??? ?????? ?? ???????? ????????) ???????? ??????? ????? ????? ????????? ???????? ?????????? ?????? ????????. ^[2]

??? ??? ???? ?????? ??????? ?? ????? ???? ????? ??? ???? ?? ??? ???? ?? ??? ?????? ?????? ????????? ???? ?????? ?????? ????????? ????? ???? ????? ??? ?????? ?????? (?????) ???????. ???? ?????? ????? ?? ???? ?????? ???????????????? ?????? ??????? ????????. ???? ??????? ?????? ?????? ???????? ??????? ?????? ???????? ???????? ??????? ??????? ??????????? ?????????. ???? ????? ?????? ?????? ???????? ??? ???? ????? ?? ???? ???????? ????????: ????? ???? ????????? ????????? ??????? ???????? (??????? ?????? ?? ?? ?????????).

?? ??????? ??? ??? ???????? ??????? ??? ??????? ?????? ???? ???? ???? ????? ??????????? ????.

???? ????? ??? ???? ??? ????? ?????? [ ??? ]

????? ???? ??? ???????? ??? ???? ???? ????? ???? ???????. ???? ????? ????? ??? ???? ???? ?????????? ???? ????????? ????? ?????? ?? ????? ?????? ???? ???????? ?? ???? ?????? ??????? ????? ?????? ????????. ???? ??? ????? ?????? ?? ??? ?????? ?? ????? ??????-??????? ?????? ?? ????? ??????? ??????? ???? ???.

???? ???????? ???? ?? ???: ${\displaystyle x}$ ???? ??????? (??????????) ??? ?????? ?????? ?????? ${\displaystyle t}$ ?? ????? (????????)? ρ(x,t) ????? ???????? ??? ?????? ????? (?????? ?????/?? ?? ??????)? ?u(x,t) ???? ?????? (?????? ???????) ???? ???????? ≪???≫ ???? ${\displaystyle x}$.

?????? ??????? ????? ????? ?? ???????? [ ??? ]

?? ???? ???????? ?? ????? ??? ?????. ????? ?? ?????? ?? ????????: ?? ??????? ??????? ??????? ?? ??? ???????? ?? ?????? ${\displaystyle x=a(t)}$ ??? ??????? ??????? ???????? ???? ???????? ?? ?????? ${\displaystyle x=b(t)}$. ????? ???????? ???????? ?? ??? ????????: ${\displaystyle N=\int _{a(t)}^{b(t)}\rho (x,t)\,dx}$. ??? ?? ???????? ?????? (??? ??? ???? ?????? ???????? ?? ????? ?? ?? ?????? ??????? ???? ???????? ?? ???????? ??? ?????.) ${\displaystyle dN/dt=0}$. ???? ?????? ????? ????? ??????? ???? ???:

${\displaystyle {\begin{aligned}{}{\frac {dN}{dt}}&={\frac {d}{dt}}\int _{a(t)}^{b(t)}\rho (x,t)\,dx\\&=\int _{a}^{b}{\frac {\partial \rho }{\partial t}}\,dx+\rho (b,t){\frac {db}{dt}}-\rho (a,t){\frac {da}{dt}}\\&=\int _{a}^{b}{\frac {\partial \rho }{\partial t}}\,dx+\rho (b,t)u(b,t)-\rho (a,t)u(a,t)\\&=\int _{a}^{b}\left[{\frac {\partial \rho }{\partial t}}+{\frac {\partial }{\partial x}}(\rho u)\right]dx\end{aligned}}}$

??? ??????? ????? ????? ??? ?????????? ?? ??? ???????? ${\displaystyle [a,b]}$. ??????? ??????? ???? ???? ?? ???? ???? ??????? ??????? ??? ?? ???????? ?? ??? ??? ???? ??????? ??????? ????? ???? ?? ???? ??????? ${\displaystyle x}$. ???????? ???? ?? ???????? ?????? ???????? ????????? ??????? ??? ?????? ?? ??????? ??????:

${\displaystyle {\frac {\partial \rho }{\partial t}}+{\frac {\partial }{\partial x}}(\rho u)=0}$

??? ??????? ??? ?????? ?????.

?? ????? ??? ???????? ????????? ??????? ??? ???? ??? ???????? ????? ?? ???? ????? ??? ??????? ???????? ?????? ??????? ?????????? ????????? ?????? ??????? ????? ??????? ??? ??? ???.

???????? ???? ??????? [ ??? ]

???????? ????????? ??????? ??????? ?? ?????? ????? ????????? ??? ????? ????? ???? ?????? ????? ???? ?????. ????? ???? ????? ?????? ????? ?? ???????? ??????? ??????? ????? ????? ?????????. ?? ???? ???? ??? ???????? ?? ???????? ???? ?? ???????? ????? ???? ????? ????? ??? ???????? ${\displaystyle u=V(\rho )}$ ???? ???? ?? ???? ????????? ${\displaystyle V}$ ?? ???? ?????? ???????. ??? ???? ??????: ???? ??????? ?? ??? ?????? ??? ???? ?????? ??? ???? ????? (???????? ???? ???????? ????????) ?? ???? ${\displaystyle u=V(\rho )=27.5\ln(142/\rho )}$ (??/?? ??????? ??????? ??????/??).

??????? ???????? ?????? ????? ??????? ?? ???? ???? ??? ???????? ???? ??? ????????? ????????? ???????:

${\displaystyle {\frac {\partial \rho }{\partial t}}+{\frac {\partial }{\partial x}}[\rho V(\rho )]=0}$

???? ????? ???????? ${\displaystyle \rho (x,t)}$ ??? ?????? ??????.

????????? ??????? [ ??? ]

?????? ??? ?????? ${\displaystyle \mathbf {u} }$ ??? ???? ??? ??? ?? ??? ??? ?? ?????? ????? ?????? ?? ?????? ????? ?? ?? ???? ?? ???? ????? ???? ?? ???? ????????? ??????? ???? ???? ??????? ?????? ????? ???? ????? ??? ??????? ????? ?????. ??? ????? ??? ????????? ??? ?? ???? ???? ???? ?????? ?????? ?????? ?? ?? ???? ${\displaystyle \mathbf {x} }$ ?? ????? ${\displaystyle \mathbf {\Omega } }$ ??? ???? ??????? (?? 1 ??? 3) ??? ?? ????????? ??????? (??? ?? ????? ????). ???? ??? ?????? ?:

• ????? ?????? ${\displaystyle \mathbf {\epsilon } }$ ??? ????? ?? ?????? 3*3 ?? ???? ???? ?????.
• ????? ????? ${\displaystyle \mathbf {\sigma } }$ ??? ???? ??? ????? ????? ?????? ????? ?? ?????? 3*3 ?? ???? ???? ????? ??????? 2*2 ?? ???? ????? ???? ?? ??????? ??? ????? ??????.

??? ??? ?????? ????? ????????? ??????? ??????? ???????: ^[3]
${\displaystyle div(\mathbf {\sigma } (\mathbf {u} ))+\mathbf {f} =\rho (\mathbf {x} ){\frac {d^{2}\mathbf {u} }{d\mathbf {x} ^{2}}}}$
?????:

• ${\displaystyle \rho }$ ??????? ???????
• ${\displaystyle div}$ ??????: ${\displaystyle \mathrm {div} {\vec {.}}={\frac {\partial {._{x}}}{\partial {x}}}+{\frac {\partial {._{y}}}{\partial {y}}}+{\frac {\partial {._{z}}}{\partial {z}}}}$
• ${\displaystyle {\frac {d^{2}\mathbf {u} }{d\mathbf {x} ^{2}}}}$ ???????

??????? [ ??? ]

?? ????? ??? ?????? ??: ???????? ??????? ???????

• ????? ????????
• ????? ????
• ????? ?????
• ????? ????? ?????????