3345 チュートリアル 1 HP テンソル代数 テンソル解析 - - 連続体力学の数理的基礎 - 第 4 講テンソル解析 - テンソル場の微積分 - 登坂宣好 第 4 講概要 2, 3 1 筆者紹介 1971 Engineering Science gradient divergence rota
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1 3345 チュートリアル 1 HP テンソル代数 テンソル解析 - - 連続体力学の数理的基礎 - 第 4 講テンソル解析 - テンソル場の微積分 - 登坂宣好 第 4 講概要 2, 3 1 筆者紹介 1971 Engineering cience gradient divergence rotation nabla ol.20, No
2 1 [1,2] [3] 2 [4] 3 E P 3 A 3 (P 3, 3 E ) A3 O 3 E G = {g 1, g 2, g 3 } (O, G) P P = O + x = O + x i g i (1) P x 3 E O P P = x = x i g i A 3 f : A 3 R v : A 3 3 E T : A 3 L(E 3) f(p ) = f(x 1, x 2, x 3 )1 (2) v(p ) = v i (P )g i = v i (x 1, x 2, x 3 )g i (3) T (P ) = T i j(p )(g i g j ) = T i j(x 1, x 2, x 3 )(g i g j ) (4) (2) 1 R 3 4,5 3.1 a B (a B)[u] := (a B[u]) ( u E ) (5) (a B) u (a B[u]) E (a B[u]) = (B[u] a) B a (B a)[u] := (B[u] a) (6) (a B) = (B a) (7) B I (a I)[u] = (a I[u]) = (a u) (I a)[u] = (I[u] a) = (u a) (27-2) ol.20, No
3 B B = (b c) 3.2 (a (b c)) = ((a c) b) (8) (a B)[u] := (a B[u]) = ((a B[g j ]) g j )[u] (9) (B a)[u] := (B[u] a) = ((B[g j ] a) g j )[u] (10) (a B), (B a) E (a B) = (B a) (11) B I (a I)[u] = (a I[u]) = (a u) (I a)[u] = (I[u] a) = (u a) B = (b c) 3.3 (a (b c)) = ((a b) c) (12) (a B)[u] : = (a B[u]) = ((a B[g i ]) g i )[u] (13) (B a)[u] := B(a u) (14) (a B[u]) (B[u] a) (a I)[u] = (a I[u]) = (a u) (I a)[u] = I(a u) B = (b c) (a (b c)) = ((a b) c) (a b c) (15) (directional derivative) Φ Φ (D u Φ) P := lim h 0 Φ(P + hu) Φ(P ) h x k Φ(P )uk (16) Φ u P P u (D u Φ) P = (dφ) P [u] (17) (dφ) P Φ P 4.2 f f(p + hu) f(p ) (D u f) P := lim f(x k + hu k ) f(x k ) = lim x k f(p )uk = x k f(p )(gk u) = ( x k gk f(p ) u) (18) σ k f P (27-3) ol.20, No
4 x k f(p )uk = x k f(p )(σk [u]) = ( x k f(p )σk )[u] (df) P [u] (19) (18) (D u f) P = (df) P [u] = ( x k gk f(p ) u) (20) df f(p ) P grad f(p ) := x k gk f(p ) (21) grad f(p ) 4.3 (16) v(p + hu) v(p ) (D u v) P := lim v i (P + hu k ) v i (P ) = lim g i x k vi (P )u k g i x k vi (P )g i (g k u) (22) (D u v) P = ( x k vi (P )(g i g k ))[u] = (dv) P [u] v P (dv) P = x k vi (P )(g i g k ) (23) ((dv) a ) P = x k vi (P )(g k g i ) (24) v div v(p ) := T r [(dv) P ] (25) = (g m (dv) P [g m ]) x k vk (P ) = (g k v(p )) xk (26) rot v(p ) := (((dv) a ) P [g m ] g m ) (27) x k vi (P )(g k g i ) = (g k v(p )) xk x k ϵkjl g ij v i (P )g l (28) grad v(p ) := ((dv) a ) P (29) x k vi (P )(g k g i ) = (g k v(p )) (30) xk [5] (30) grad v(p ) = (dv) P ((dv) a ) P 6.1 v (dv) P W (P ) := (dv) P ((dv) a ) P u (44) W (P )[u] = x k vi (P )(((g i g k ) (g k g i ))[u]) x k (v(p )(gk u) g k (v(p ) u)) x k (gk v(p )) u = (rot v(p )) u (31) (dv) P [6] 4.4 (27-4) ol.20, No
5 T T (P ) = T i j (P )(g i g j ) T (P + hu) T (P ) (D u T ) P := lim T i j = lim (xk + hu k ) T i j (xk ) (g i g j ) x k T i j(p ) u k (g i g j ) x k T i j(p )(g i g j )(g k u) (32) u (D u T ) P x k T i j(p )((g i g j ) g k )[u] = (dt ) P [u] (33) (dt ) P := x k T i j(p )((g i g j ) g k ) x k T i j(p )(g i (g j g k )) (34) (dt a ) P = x k T i j(p )((g j g i ) g k ) x k T i j(p )(g j (g i g k )) (35) (a) grad T (P ) := ((dt a ) a ) P = ( x k T i j(p )((g j g i ) g k )) a x k (gk (T i j(p )(g j g i ))) x k (gk T a (P )) (36) (b) div T (P ) := ((dt ) P [g m ])[g m ] x k T i j(p )g jk g i x k T i j(p )g i (g j g k ) x k T i j(p )(g i g j )[g k ] x k T (P )[gk ] (37) (c) rot T (P ) := (g n g m )((dt a ) P [g m g n ]) x k ϵkjn T i j(p )(g n g i ) x k T i j(p )((g k g j ) g i ) x k ((gk g j ) (T (P )[g j ])) x k (((gk g j ) g j )T a (P )) x k (gk I)T a (P ) (38) 4.5 (nabla) [7] := g k x k (39) x k E grad f(p ) = f(p ) (40) grad v(p ) = ( v(p )) (41) grad T (P ) = ( T a (P )) (42) div v(p ) = ( v(p )) (43) div T (P ) = T (P )[ ] (44) rot v(p ) = ( v(p )) rot T (P ) = ( I)T a (P ) (45) = ( T a (P )) 4.6 (46) 3.4 v(p ) = T a (P )[u] (47) (27-5) ol.20, No
6 grad v(p ) = grad (T a (P )[u]) = ( v(p )) = ( (T a (P )[u])) = (T (P )[ ] u) = ( T a (P ))[u] = (grad T (P ))[u] (48) div v(p ) = div (T a (P )[u]) = ( v(p )) = ( (T a (P )[u])) = ( T a (P ))[u] = (T (P )[ ] u) = (div T (P ) u) (49) rot v(p ) = rot (T a (P )[u]) = ( v(p )) = ( (T a (P )[u])) = ( T a (P ))[u] = (rot T (P ))[u] (50) u T grad T (P ) := ( T a (P )) div T (P ) := ( T a (P )) rot T (P ) := ( T a (P )) div T (P ) div T (P ) ( T a (P )) E u div v(p ) E E E ( T a (P ))( E) = T (P )[ ]( E ) (49) 4.7 P g[f(p )] := grad f(p) = f(p ) = ( 1)[f(P )] g := ( 1) (51) g[v(p )] := grad v(p ) = ( v(p )) = ( I)[v(P )] g := ( I) (52) d[v(p )] := div v(p ) = ( v(p )) = ( I)[v(P )] d := ( I) (53) r[v(p )] := rot v(p ) = ( v(p )) = ( I)[v(P )] r := ( I) (54) G[T (P )] := grad T (P ) = ( T a (P )) = ( Υ)[T (P )] G := ( Υ) (55) D[T (P )] := div T (P ) = ( T a (P )) = ( Υ)[T (P )] D := ( Υ) (56) R[T (P )] := rot T (P ) = ( T a (P )) = ( Υ)[T (P )] R := ( Υ) (57) I Υ (27-6) ol.20, No
7 5 5.1 A 3 P A 3 D (l, m) l f P T l (P ) : i f f(l) = l l (l, m)g i T m (P ) := i f f(l) = m m (l, m)g i d(p ) := T l (P ) T m (P ) da ( da := dl dm) (58) n(p ) := T l(p ) T m (P ) T l (P ) T m (P ) (59) (T l (P ), T m (P ), n(p )) (58) d(p ) := n(p ) d = (T l (P ) T m (P )) da (60) A 3 P (30) d (P ) := [(dx 1 )g 1 (dx 2 )g 2 (dx 3 )g 3 ] 5.2 = [g 1 g 2 g 3 ] dx 1 dx 2 dx 3 = g dv (dv := dx 1 dx 2 dx 3 ) (61) A 3 Φ ; x = f(u) D Φd := Φ(P )d(p ) = Φ(f(u)) T l (u) T m (u) da D (62) (60) D A 3 Φ Φ d := Φ(P ) d (P ) = Φ(P ) g dv (63) b a d dx f(x)dx = [f(x)]b a (divergence theorem) A 3 f(p ) f(p ) d(p ) = f(p )n(p ) d(p ) = grad f(p ) d (P ) = f(p ) d (P ) (64) (27-7) ol.20, No
8 6.2 (a) (d(p ) v(p )) = (n(p ) v(p ))d(p ) = div v(p )d (P ) = ( v(p ))d (P ) (b) (d(p ) v(p )) = = = (c) (d(p ) v(p )) = = = 6.3 (65) (n(p ) v(p ))d(p ) rot v(p )d (P ) ( v(p ))d (P ) (66) (n(p ) v(p ))d(p ) grad v(p )d (P ) ( v(p ))d (P ) (67) (a) T (P )[d(p )] = T (P )[n(p )]d(p ) = (div T (P ))d (P ) = T (P )[ ]d (P ) (68) (d(p ) T a (P )) = (n(p ) T a (P ))d(p ) = ( T a (P ))d (P ) = (( Υ)T (P ))d (P ) = div T (P )d (P ) (69) (b) (d(p ) g j ) g j )T a (P ) = (d(p ) T a (P )) = (((n(p ) g j ) g j )T a (P ))d(p ) = (n(p ) T a (P ))d(p ) = ((n(p ) Υ)T a (P ))d(p ) = rot T (P )d (P ) = ((( g j ) g j )T a (P ))d (P ) = (( Υ)T (P ))d (P ) (70) (c) (d T a (P ))d(p ) = (n T a (P ))d(p ) = ( T a (P ))d (P ) = (( Υ)T (P ))d (P ) = grad T (P )d (P ) (71) (69)(71) Υ 4.7 T a (P ) = Υ[T (P )] (64) (65)(67) (68)(71) [3] (27-8) ol.20, No
9 [1] ol. 18, (2013) [2] ol. 20, (2015) [3] (2015) [4] (1987) [5] Gurtin, M. E.: An Introduction to Continuum Mechanics, Academic Press, (2003) [6] (2006) [7] Green, A. E. and W. Zerna : Theoretical Elasticity (Theoretical Continuum Mechanics), Oxford Univ. Press, (1954) [8] Gurtin, M. E. : The Linear Theory of Elasticity ( in Encyclopedia of Physics, ol. Ia/2), pringer- erlag, (1972) (isotropic function) (27-9) ol.20, No
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