রূপান্তর সূত্র
সমজাতীয় সমদৈশিক রৈখিক স্থিতিস্থাপক পদার্থগুলোর স্থিতিস্থাপক বৈশিষ্ট্য রয়েছে, যা এর মধ্যে যে কোনও দুটি মডিউল দ্বারা স্বতন্ত্রভাবে নির্ধারিত হয়। ত্রিমাত্রিক উপাদান (ছকের প্রথম অংশ) এবং দ্বিমাত্রিক উপাদান (ছকের দ্বিতীয় অংশ) উভয়ের জন্যই দেওয়া এই সূত্রগুলো অনুসারে স্থিতিস্থাপক মডিউলের অন্য যে কোনোটি গণনা করা যেতে পারে।
ত্রিমাত্রিক সূত্র
K
=
{\displaystyle K=\,}
E
=
{\displaystyle E=\,}
λ
=
{\displaystyle \lambda =\,}
G
=
{\displaystyle G=\,}
ν
=
{\displaystyle \nu =\,}
M
=
{\displaystyle M=\,}
টীকা
(
K
,
E
)
{\displaystyle (K,\,E)}
3
K
(
3
K
−
E
)
9
K
−
E
{\displaystyle {\tfrac {3K(3K-E)}{9K-E}}}
3
K
E
9
K
−
E
{\displaystyle {\tfrac {3KE}{9K-E}}}
3
K
−
E
6
K
{\displaystyle {\tfrac {3K-E}{6K}}}
3
K
(
3
K
+
E
)
9
K
−
E
{\displaystyle {\tfrac {3K(3K+E)}{9K-E}}}
(
K
,
λ
)
{\displaystyle (K,\,\lambda )}
9
K
(
K
−
λ
)
3
K
−
λ
{\displaystyle {\tfrac {9K(K-\lambda )}{3K-\lambda }}}
3
(
K
−
λ
)
2
{\displaystyle {\tfrac {3(K-\lambda )}{2}}}
λ
3
K
−
λ
{\displaystyle {\tfrac {\lambda }{3K-\lambda }}}
3
K
−
2
λ
{\displaystyle 3K-2\lambda \,}
(
K
,
G
)
{\displaystyle (K,\,G)}
9
K
G
3
K
+
G
{\displaystyle {\tfrac {9KG}{3K+G}}}
K
−
2
G
3
{\displaystyle K-{\tfrac {2G}{3}}}
3
K
−
2
G
2
(
3
K
+
G
)
{\displaystyle {\tfrac {3K-2G}{2(3K+G)}}}
K
+
4
G
3
{\displaystyle K+{\tfrac {4G}{3}}}
(
K
,
ν
)
{\displaystyle (K,\,\nu )}
3
K
(
1
−
2
ν
)
{\displaystyle 3K(1-2\nu )\,}
3
K
ν
1
+
ν
{\displaystyle {\tfrac {3K\nu }{1+\nu }}}
3
K
(
1
−
2
ν
)
2
(
1
+
ν
)
{\displaystyle {\tfrac {3K(1-2\nu )}{2(1+\nu )}}}
3
K
(
1
−
ν
)
1
+
ν
{\displaystyle {\tfrac {3K(1-\nu )}{1+\nu }}}
(
K
,
M
)
{\displaystyle (K,\,M)}
9
K
(
M
−
K
)
3
K
+
M
{\displaystyle {\tfrac {9K(M-K)}{3K+M}}}
3
K
−
M
2
{\displaystyle {\tfrac {3K-M}{2}}}
3
(
M
−
K
)
4
{\displaystyle {\tfrac {3(M-K)}{4}}}
3
K
−
M
3
K
+
M
{\displaystyle {\tfrac {3K-M}{3K+M}}}
(
E
,
λ
)
{\displaystyle (E,\,\lambda )}
E
+
3
λ
+
R
6
{\displaystyle {\tfrac {E+3\lambda +R}{6}}}
E
−
3
λ
+
R
4
{\displaystyle {\tfrac {E-3\lambda +R}{4}}}
2
λ
E
+
λ
+
R
{\displaystyle {\tfrac {2\lambda }{E+\lambda +R}}}
E
−
λ
+
R
2
{\displaystyle {\tfrac {E-\lambda +R}{2}}}
R
=
E
2
+
9
λ
2
+
2
E
λ
{\displaystyle R={\sqrt {E^{2}+9\lambda ^{2}+2E\lambda }}}
(
E
,
G
)
{\displaystyle (E,\,G)}
E
G
3
(
3
G
−
E
)
{\displaystyle {\tfrac {EG}{3(3G-E)}}}
G
(
E
−
2
G
)
3
G
−
E
{\displaystyle {\tfrac {G(E-2G)}{3G-E}}}
E
2
G
−
1
{\displaystyle {\tfrac {E}{2G}}-1}
G
(
4
G
−
E
)
3
G
−
E
{\displaystyle {\tfrac {G(4G-E)}{3G-E}}}
(
E
,
ν
)
{\displaystyle (E,\,\nu )}
E
3
(
1
−
2
ν
)
{\displaystyle {\tfrac {E}{3(1-2\nu )}}}
E
ν
(
1
+
ν
)
(
1
−
2
ν
)
{\displaystyle {\tfrac {E\nu }{(1+\nu )(1-2\nu )}}}
E
2
(
1
+
ν
)
{\displaystyle {\tfrac {E}{2(1+\nu )}}}
E
(
1
−
ν
)
(
1
+
ν
)
(
1
−
2
ν
)
{\displaystyle {\tfrac {E(1-\nu )}{(1+\nu )(1-2\nu )}}}
(
E
,
M
)
{\displaystyle (E,\,M)}
3
M
−
E
+
S
6
{\displaystyle {\tfrac {3M-E+S}{6}}}
M
−
E
+
S
4
{\displaystyle {\tfrac {M-E+S}{4}}}
3
M
+
E
−
S
8
{\displaystyle {\tfrac {3M+E-S}{8}}}
E
−
M
+
S
4
M
{\displaystyle {\tfrac {E-M+S}{4M}}}
S
=
±
E
2
+
9
M
2
−
10
E
M
{\displaystyle S=\pm {\sqrt {E^{2}+9M^{2}-10EM}}}
এখানে দুটি বৈধ সমাধান রয়েছে।
ν
≥
0
{\displaystyle \nu \geq 0}
বিয়োগ চিহ্ন বাড়লে
ν
≤
0
{\displaystyle \nu \leq 0}
(
λ
,
G
)
{\displaystyle (\lambda ,\,G)}
λ
+
2
G
3
{\displaystyle \lambda +{\tfrac {2G}{3}}}
G
(
3
λ
+
2
G
)
λ
+
G
{\displaystyle {\tfrac {G(3\lambda +2G)}{\lambda +G}}}
λ
2
(
λ
+
G
)
{\displaystyle {\tfrac {\lambda }{2(\lambda +G)}}}
λ
+
2
G
{\displaystyle \lambda +2G\,}
(
λ
,
ν
)
{\displaystyle (\lambda ,\,\nu )}
λ
(
1
+
ν
)
3
ν
{\displaystyle {\tfrac {\lambda (1+\nu )}{3\nu }}}
λ
(
1
+
ν
)
(
1
−
2
ν
)
ν
{\displaystyle {\tfrac {\lambda (1+\nu )(1-2\nu )}{\nu }}}
λ
(
1
−
2
ν
)
2
ν
{\displaystyle {\tfrac {\lambda (1-2\nu )}{2\nu }}}
λ
(
1
−
ν
)
ν
{\displaystyle {\tfrac {\lambda (1-\nu )}{\nu }}}
ব্যবহার করা যাবে না যখন
ν
=
0
⇔
λ
=
0
{\displaystyle \nu =0\Leftrightarrow \lambda =0}
(
λ
,
M
)
{\displaystyle (\lambda ,\,M)}
M
+
2
λ
3
{\displaystyle {\tfrac {M+2\lambda }{3}}}
(
M
−
λ
)
(
M
+
2
λ
)
M
+
λ
{\displaystyle {\tfrac {(M-\lambda )(M+2\lambda )}{M+\lambda }}}
M
−
λ
2
{\displaystyle {\tfrac {M-\lambda }{2}}}
λ
M
+
λ
{\displaystyle {\tfrac {\lambda }{M+\lambda }}}
(
G
,
ν
)
{\displaystyle (G,\,\nu )}
2
G
(
1
+
ν
)
3
(
1
−
2
ν
)
{\displaystyle {\tfrac {2G(1+\nu )}{3(1-2\nu )}}}
2
G
(
1
+
ν
)
{\displaystyle 2G(1+\nu )\,}
2
G
ν
1
−
2
ν
{\displaystyle {\tfrac {2G\nu }{1-2\nu }}}
2
G
(
1
−
ν
)
1
−
2
ν
{\displaystyle {\tfrac {2G(1-\nu )}{1-2\nu }}}
(
G
,
M
)
{\displaystyle (G,\,M)}
M
−
4
G
3
{\displaystyle M-{\tfrac {4G}{3}}}
G
(
3
M
−
4
G
)
M
−
G
{\displaystyle {\tfrac {G(3M-4G)}{M-G}}}
M
−
2
G
{\displaystyle M-2G\,}
M
−
2
G
2
M
−
2
G
{\displaystyle {\tfrac {M-2G}{2M-2G}}}
(
ν
,
M
)
{\displaystyle (\nu ,\,M)}
M
(
1
+
ν
)
3
(
1
−
ν
)
{\displaystyle {\tfrac {M(1+\nu )}{3(1-\nu )}}}
M
(
1
+
ν
)
(
1
−
2
ν
)
1
−
ν
{\displaystyle {\tfrac {M(1+\nu )(1-2\nu )}{1-\nu }}}
M
ν
1
−
ν
{\displaystyle {\tfrac {M\nu }{1-\nu }}}
M
(
1
−
2
ν
)
2
(
1
−
ν
)
{\displaystyle {\tfrac {M(1-2\nu )}{2(1-\nu )}}}
দ্বিমাত্রিক সূত্র
K
2
D
=
{\displaystyle K_{\mathrm {2D} }=\,}
E
2
D
=
{\displaystyle E_{\mathrm {2D} }=\,}
λ
2
D
=
{\displaystyle \lambda _{\mathrm {2D} }=\,}
G
2
D
=
{\displaystyle G_{\mathrm {2D} }=\,}
ν
2
D
=
{\displaystyle \nu _{\mathrm {2D} }=\,}
M
2
D
=
{\displaystyle M_{\mathrm {2D} }=\,}
টীকা
(
K
2
D
,
E
2
D
)
{\displaystyle (K_{\mathrm {2D} },\,E_{\mathrm {2D} })}
2
K
2
D
(
2
K
2
D
−
E
2
D
)
4
K
2
D
−
E
2
D
{\displaystyle {\tfrac {2K_{\mathrm {2D} }(2K_{\mathrm {2D} }-E_{\mathrm {2D} })}{4K_{\mathrm {2D} }-E_{\mathrm {2D} }}}}
K
2
D
E
2
D
4
K
2
D
−
E
2
D
{\displaystyle {\tfrac {K_{\mathrm {2D} }E_{\mathrm {2D} }}{4K_{\mathrm {2D} }-E_{\mathrm {2D} }}}}
2
K
2
D
−
E
2
D
2
K
2
D
{\displaystyle {\tfrac {2K_{\mathrm {2D} }-E_{\mathrm {2D} }}{2K_{\mathrm {2D} }}}}
4
K
2
D
2
4
K
2
D
−
E
2
D
{\displaystyle {\tfrac {4K_{\mathrm {2D} }^{2}}{4K_{\mathrm {2D} }-E_{\mathrm {2D} }}}}
(
K
2
D
,
λ
2
D
)
{\displaystyle (K_{\mathrm {2D} },\,\lambda _{\mathrm {2D} })}
4
K
2
D
(
K
2
D
−
λ
2
D
)
2
K
2
D
−
λ
2
D
{\displaystyle {\tfrac {4K_{\mathrm {2D} }(K_{\mathrm {2D} }-\lambda _{\mathrm {2D} })}{2K_{\mathrm {2D} }-\lambda _{\mathrm {2D} }}}}
K
2
D
−
λ
2
D
{\displaystyle K_{\mathrm {2D} }-\lambda _{\mathrm {2D} }}
λ
2
D
2
K
2
D
−
λ
2
D
{\displaystyle {\tfrac {\lambda _{\mathrm {2D} }}{2K_{\mathrm {2D} }-\lambda _{\mathrm {2D} }}}}
2
K
2
D
−
λ
2
D
{\displaystyle 2K_{\mathrm {2D} }-\lambda _{\mathrm {2D} }}
(
K
2
D
,
G
2
D
)
{\displaystyle (K_{\mathrm {2D} },\,G_{\mathrm {2D} })}
4
K
2
D
G
2
D
K
2
D
+
G
2
D
{\displaystyle {\tfrac {4K_{\mathrm {2D} }G_{\mathrm {2D} }}{K_{\mathrm {2D} }+G_{\mathrm {2D} }}}}
K
2
D
−
G
2
D
{\displaystyle K_{\mathrm {2D} }-G_{\mathrm {2D} }}
K
2
D
−
G
2
D
K
2
D
+
G
2
D
{\displaystyle {\tfrac {K_{\mathrm {2D} }-G_{\mathrm {2D} }}{K_{\mathrm {2D} }+G_{\mathrm {2D} }}}}
K
2
D
+
G
2
D
{\displaystyle K_{\mathrm {2D} }+G_{\mathrm {2D} }}
(
K
2
D
,
ν
2
D
)
{\displaystyle (K_{\mathrm {2D} },\,\nu _{\mathrm {2D} })}
2
K
2
D
(
1
−
ν
2
D
)
{\displaystyle 2K_{\mathrm {2D} }(1-\nu _{\mathrm {2D} })\,}
2
K
2
D
ν
2
D
1
+
ν
2
D
{\displaystyle {\tfrac {2K_{\mathrm {2D} }\nu _{\mathrm {2D} }}{1+\nu _{\mathrm {2D} }}}}
K
2
D
(
1
−
ν
2
D
)
1
+
ν
2
D
{\displaystyle {\tfrac {K_{\mathrm {2D} }(1-\nu _{\mathrm {2D} })}{1+\nu _{\mathrm {2D} }}}}
2
K
2
D
1
+
ν
2
D
{\displaystyle {\tfrac {2K_{\mathrm {2D} }}{1+\nu _{\mathrm {2D} }}}}
(
E
2
D
,
G
2
D
)
{\displaystyle (E_{\mathrm {2D} },\,G_{\mathrm {2D} })}
E
2
D
G
2
D
4
G
2
D
−
E
2
D
{\displaystyle {\tfrac {E_{\mathrm {2D} }G_{\mathrm {2D} }}{4G_{\mathrm {2D} }-E_{\mathrm {2D} }}}}
2
G
2
D
(
E
2
D
−
2
G
2
D
)
4
G
2
D
−
E
2
D
{\displaystyle {\tfrac {2G_{\mathrm {2D} }(E_{\mathrm {2D} }-2G_{\mathrm {2D} })}{4G_{\mathrm {2D} }-E_{\mathrm {2D} }}}}
E
2
D
2
G
2
D
−
1
{\displaystyle {\tfrac {E_{\mathrm {2D} }}{2G_{\mathrm {2D} }}}-1}
4
G
2
D
2
4
G
2
D
−
E
2
D
{\displaystyle {\tfrac {4G_{\mathrm {2D} }^{2}}{4G_{\mathrm {2D} }-E_{\mathrm {2D} }}}}
(
E
2
D
,
ν
2
D
)
{\displaystyle (E_{\mathrm {2D} },\,\nu _{\mathrm {2D} })}
E
2
D
2
(
1
−
ν
2
D
)
{\displaystyle {\tfrac {E_{\mathrm {2D} }}{2(1-\nu _{\mathrm {2D} })}}}
E
2
D
ν
2
D
(
1
+
ν
2
D
)
(
1
−
ν
2
D
)
{\displaystyle {\tfrac {E_{\mathrm {2D} }\nu _{\mathrm {2D} }}{(1+\nu _{\mathrm {2D} })(1-\nu _{\mathrm {2D} })}}}
E
2
D
2
(
1
+
ν
2
D
)
{\displaystyle {\tfrac {E_{\mathrm {2D} }}{2(1+\nu _{\mathrm {2D} })}}}
E
2
D
(
1
+
ν
2
D
)
(
1
−
ν
2
D
)
{\displaystyle {\tfrac {E_{\mathrm {2D} }}{(1+\nu _{\mathrm {2D} })(1-\nu _{\mathrm {2D} })}}}
(
λ
2
D
,
G
2
D
)
{\displaystyle (\lambda _{\mathrm {2D} },\,G_{\mathrm {2D} })}
λ
2
D
+
G
2
D
{\displaystyle \lambda _{\mathrm {2D} }+G_{\mathrm {2D} }}
4
G
2
D
(
λ
2
D
+
G
2
D
)
λ
2
D
+
2
G
2
D
{\displaystyle {\tfrac {4G_{\mathrm {2D} }(\lambda _{\mathrm {2D} }+G_{\mathrm {2D} })}{\lambda _{\mathrm {2D} }+2G_{\mathrm {2D} }}}}
λ
2
D
λ
2
D
+
2
G
2
D
{\displaystyle {\tfrac {\lambda _{\mathrm {2D} }}{\lambda _{\mathrm {2D} }+2G_{\mathrm {2D} }}}}
λ
2
D
+
2
G
2
D
{\displaystyle \lambda _{\mathrm {2D} }+2G_{\mathrm {2D} }\,}
(
λ
2
D
,
ν
2
D
)
{\displaystyle (\lambda _{\mathrm {2D} },\,\nu _{\mathrm {2D} })}
λ
2
D
(
1
+
ν
2
D
)
2
ν
2
D
{\displaystyle {\tfrac {\lambda _{\mathrm {2D} }(1+\nu _{\mathrm {2D} })}{2\nu _{\mathrm {2D} }}}}
λ
2
D
(
1
+
ν
2
D
)
(
1
−
ν
2
D
)
ν
2
D
{\displaystyle {\tfrac {\lambda _{\mathrm {2D} }(1+\nu _{\mathrm {2D} })(1-\nu _{\mathrm {2D} })}{\nu _{\mathrm {2D} }}}}
λ
2
D
(
1
−
ν
2
D
)
2
ν
2
D
{\displaystyle {\tfrac {\lambda _{\mathrm {2D} }(1-\nu _{\mathrm {2D} })}{2\nu _{\mathrm {2D} }}}}
λ
2
D
ν
2
D
{\displaystyle {\tfrac {\lambda _{\mathrm {2D} }}{\nu _{\mathrm {2D} }}}}
ব্যবহার করা যাবে না যখন
ν
2
D
=
0
⇔
λ
2
D
=
0
{\displaystyle \nu _{\mathrm {2D} }=0\Leftrightarrow \lambda _{\mathrm {2D} }=0}
(
G
2
D
,
ν
2
D
)
{\displaystyle (G_{\mathrm {2D} },\,\nu _{\mathrm {2D} })}
G
2
D
(
1
+
ν
2
D
)
1
−
ν
2
D
{\displaystyle {\tfrac {G_{\mathrm {2D} }(1+\nu _{\mathrm {2D} })}{1-\nu _{\mathrm {2D} }}}}
2
G
2
D
(
1
+
ν
2
D
)
{\displaystyle 2G_{\mathrm {2D} }(1+\nu _{\mathrm {2D} })\,}
2
G
2
D
ν
2
D
1
−
ν
2
D
{\displaystyle {\tfrac {2G_{\mathrm {2D} }\nu _{\mathrm {2D} }}{1-\nu _{\mathrm {2D} }}}}
2
G
2
D
1
−
ν
2
D
{\displaystyle {\tfrac {2G_{\mathrm {2D} }}{1-\nu _{\mathrm {2D} }}}}
(
G
2
D
,
M
2
D
)
{\displaystyle (G_{\mathrm {2D} },\,M_{\mathrm {2D} })}
M
2
D
−
G
2
D
{\displaystyle M_{\mathrm {2D} }-G_{\mathrm {2D} }}
4
G
2
D
(
M
2
D
−
G
2
D
)
M
2
D
{\displaystyle {\tfrac {4G_{\mathrm {2D} }(M_{\mathrm {2D} }-G_{\mathrm {2D} })}{M_{\mathrm {2D} }}}}
M
2
D
−
2
G
2
D
{\displaystyle M_{\mathrm {2D} }-2G_{\mathrm {2D} }\,}
M
2
D
−
2
G
2
D
M
2
D
{\displaystyle {\tfrac {M_{\mathrm {2D} }-2G_{\mathrm {2D} }}{M_{\mathrm {2D} }}}}
