Cátedra de Mecánica del Continuo – FI-UNER Cursado: Primer cuatrimestre de 2008 Material de Estudio Tema II: Algebra Tensorial y Derivada Material Notación indicial y convención de suma ! = = " ! % ! # & $ ! ' ( % % $ " & ! ! = = ) ( % & " % * & + , ( % " " ) " + ! " $ . & / ! = ! 0 = & 1 ( 1 " / ( & $ 1 % % " / & + & = + 2 & = ( 3 & 4 % +"! " / & ' Tensores # 3 ! 4 . " & 5 6 α α # $ 7 α 8 ! - 9 # 9 : & ; ) $ ' : ( $ ;⋅ : ;⋅ , $ Productos entre vectores y tensores ( $ ! ⋅ = ! ⋅ # = = = ! ! = = ! ! = = # = ( ) ⋅ 0 #δ ! = # = δ δ ≠ 6 < ) ⋅ (# ⋅ =( . ( = # = #δ = # ) ⋅ = ( & ) 1 4 > =( ⋅ = ( )⋅ ) ⋅ 2 = δ = ( % % 4 )(# =( = ' # / ( & 5 : δ! δ! ; ; ! - δ! ! δ! ; ' ? ! ! : 5 @ 6 !⋅ ? : $ !⋅ : & 3 & " & ) ) ; ! δ! !⋅ " " !⋅ ' ' & / ' / & ! + & " 4 [ ] = 4 [ # ( )](' ) = # ' [ ]( ) ! ! ! ! ! = # '! δ ! = #' [ ] ( ) : =( ⋅ & ; ) / = ! ! ! ! ! ! ⋅( = [ ] 4 =( ⋅ : A: : ; ; ; ! ) ) = ) ] )] , ) 0 ⋅( ! :δ ! ;⋅ B⋅ ! ) : !⋅ ;⋅ !⋅ ⋅ !⋅ ⋅ 1 !⋅ ! / ! 5 [ # ( )](' = # ' [ ]( = # ' ( )⋅ [ = (' ) ⋅ [ # ( [ ] . ) 1 : !⋅ ;⋅ C 2 ? ? " % & ⋅ ⋅ 3 ⋅ ⋅ !6 Componentes de un tensor @ 4 / & D ' ' ( $ ( $ $! & ! ! $ . > ) ( = ) ( ) = ! = $ & " $ [ = $ > !! ]( [ =$ ! =$ ! =$ ! ( =$ ! δ! ⋅( ! ) ]( ! ) ! ! ) ⋅ ! ) !) =$ $ & ! ⋅ E $ / $ ⋅ $ $ !, & / + & ⋅ F !, !0 !0 $ / / * ⋅ ⋅ ⋅ $ Algunos tensores especiales E !1 > ⋅ ⋅ )! )! ! !4 % % & " )! ! ( !2 ! !3 & )! δ! δ ? E &? / / $ ( ' )6 ( )⋅ = ( , ⋅ $ ' )⋅ = ( $$ )⋅ (# ) = ! ! $$# ) ) , ( )⋅ =( ⋅ ( )⋅ = # $! ! D )⋅ ( ) = ) $ # ) ) $$ = $ )! $ $ )) Operadores diferenciales espaciales G * ∇φ = ∂φ ∂φ ∂φ + + ∂ ∂ ∂ = ∇⋅ = = ! = ∂φ ∂ )0 & # * ∂ ∂ ∂ ∂ + + = ∂ ∂ ∂ ∂ )1 & ∇ ), ∂φ ∂ & φ - ∂H∂ φ ∂φ ∂φ ∂φ + + ∂ ∂ ∂ ! ∇φ = ∇ $$ = $ $ 4 F φ? 7 + 8 φ? % ∆φ = ∇ φ = ∇ ⋅ (∇φ ) =∇⋅ ∂φ ∂φ ∂φ + + ∂ ∂ ∂ = ∂φ ∂φ ∂φ + + ∂ ∂ ∂ = ∂φ ∂ ∂ )2 & ∆ = ∇ = ∇ ⋅∇ = ∂ ∂ ⋅ ∂ ∂ = ∂ ∂ ∂ ∂ ( ⋅ )= ∂ ∂ ∂ ∂ δ = ∂ ∂ ∂ ∂ )3 0 G& + ∂ H∂ ∂ % ∂ H∂ & (% / " ' . > ∂ ∂ ∇ = ∇⋅ ∆ = ∇ ⋅ (∇ )= = ∂ ∂ ∂ ∂ ⋅ ($ ! ∂ ∂ ⋅ ! ( )= ∂ ( ) = ∂$ ⋅( ! ∂ ! ! ) = ,6 ∂ ∂ ∂ ! ⋅ ) = ∂$ ! ∂ ∂ ∂ ! ! δ ! = = ∂$ ! ∂ ∂ ! δ ∂ ∂ , ! ! = ∂ ! ∂ ∂ ! , Conceptos de cinemática y movimiento en el continuo. Descripción material y espacial % & @ " $ . 5 " ( " & I & % " $ & $ & / D + & " * F & ( " & " " & 1 " ( & & " 6 & 5 " & . ,! + * . ,! + . * & 6 ,) # 6 * @ 6 ,, $ $ & $ ϕ > J & / " ϕ ( " & ( +* * > ϕ =ϕ J ,0 & % E & & , ϕ = ϕK ,1 ϕ - $ ϕ 5 & & * & + & $ " & " & G ,, E & H & $ & & ' Derivada material y derivada total. Aceleración de una partícula % H & > 2 D & J # ∂ H∂ . + G ( 5 & & ( $ # & H& F 4 ( / # H ( " ( & + / D ' " + E " L ( % " & ,2 + - ,2 " + = @ + ∂ ∂ ,3 " 4 " . ,3 ϕ ϕ ,0 $% $ 4 & & &ϕ ∂ϕ = & ∂ φ ' ,1 . & φ # + $ % & 06 & ( & ∂φ ∂φ = ∂ ∂ . % ( + $ φ > & & 0 0 " φ φ ' & ( " & φ 3 φ + φ " φ: ; + " > ∂φ ∂ ∂φ ∂ G ≠ = ∂φ ∂φ + ∂ ∂ = ∇φ ⋅ ≠ ∂ ∂ ∂ ∂ 0 = ⋅ / & % & &φ ∂φ = + ⋅ ∇φ & ∂ ( ( 0! / # 4 % & - # - ' 4 " φ 4 4 " % & φ . % $ & = ∂φ + ∂ ⋅ ∇φ 0) # 0) & . 0) . & $ $ 4 $ & . = C % ' 0 ! = = = % + & & ∂#! ∂ = 0, 0, ∂#! ∂# + ! ∂ ∂ ] $ & ∂ = & ∂ & ∂#! ∂# +# ! ∂ ∂ ∂ + [∇ ∂ " & ≠ ∂ ∂ 00 ≠ = ∂ + ⋅ [∇ ∂ 00 ' ∇ ]= ∂ ∂ + ( ⋅ ∇) / & C 6 Descripción cinemática del flujo. Camino de partícula, línea de corriente y línea de traza ( $ . & $ $ % 5 + # ( & /% ( " + % / " & % " # + " / + ( + ' + ' ' & % E & / $ " . $ & " $ = = E # = 01 = # 02 # $ & " " $ & 03 5 " $ " L @ + > + & 16 + " 1 / E" " 4 6 1 6 " / & + " 6 = 6 $ ( 6 ) 1!