Cátedra de Mecánica del Continuo – FI-UNER Cursado: Primer cuatrimestre de 2008 Material de Estudio Tema III: El tensor de tensiones. Parte I Introducción ρ ! " ρ = # + $ # # & % ! ' ! & ' ' $! + % () ) ! * '# + # - ' , % ' ( , + '# . ' " * ( * # & El vector de tensiones y el “principio de tensión” de Cauchy % ' $ &/ $ & # ! # ' ' 0 *# ' !$ 1 ! # + & + # # 3 + ! ! # &- # + + % & # & # . & ρ ' " 0 # & + &- ! 4 ρ = + + # 5# + 2 2 2 ( 2 2 () µ . - +" 67& 8 # ) ' [ρ ]= # * ρ & + + ' # ' 2 + 2 2 2 2 ' ! $ & ( ! ( [ρ / &: ]= ! ! 2 ρ + + + →< 1 [ + <= ( & () &9 4 5! % <= + 2 + ; − 2 ! ! ] − 2 = − ' + # ! =− / # $ & > − 4 ! 5 ' # $ ' !# 1 " ! * ! # # 3 # & # $ ! 1 # ' + 1 3 & 1 + + + ! & El Tensor de Tensiones ? # +" ! + # 0 ( &@ # # + ! + & & ' + +" . & 2! & [ρ ]= ρ + + − − 2 2 + + − A " 2 ! . 2 " & - # ' + ' ( ( ! B + & &C A ! $! ρ / = ρ ( − ⋅ ) − 2 ( ) ⋅ − ( ⋅ ) + D →< 1 $ " + & ; ( <=− ( = # ⋅ )− 2 ⋅ )+ 2 ( ( ⋅ )− ( ⋅ )+ ( →< )+ ⋅ E ) ⋅ & ( = [ ⋅[ ⋅ )+ ( + + = ⋅ = 3 ] 0 ⋅ )+ ! ( ] ⋅ ) < % 0 & + = # #" # # & # # # & & : % < = ⋅ [ ]= ⋅( # )= ⋅( # # # ) 2 = ⋅ $ ( - ' 2 # # +" # = ⋅( = " # # # )= ( ⋅ ) # ; # &- # ' # + ' #1 & ; +" 1 + & $ " ! " ! & $ " 1 ! +1 = # & " ' + & # 0 $ # $ 3 & + & ' . & $ + % > + # & . & $ =σ # +" =τ # % ' ! 0 ! ≠ # & . Simetría del Tensor de Tensiones @ ' % & # 0+ . & ;& > ×ρ ×ρ = + × = " ( F "! " F ) "" ) () ; = ) ) 2 ( () ( F !! ! F ! F " !" ) () . # F > ) / ! ; + % + + % ( &- , [ ρ( / ! × ]= ) @ ×ρ + →< 1 > + % <= % & × →< ! × & A A& A <= ⋅ % × →< = % & % D ( <= 1 →< + − !" 2 +" "& G+ !" −" ! 2 ! + − ; ! ] × [ = % →< # + 2 ( ; + − " " 2 − " 2 2 2 = ] D 3 ( ( ! + + − "" 2 − −" " −" 3!# # . & 2 [ # & 8 " ⋅ →< + = − "" > 2 " 2 + E 2 − 2 " > →< 1 " ! + <= 2 − " " 2 + 2 − " 2 2< " + " H ' !" & 2 # 2; A !" = "! ! = ! + ' # 22 2 2 # # = ' 2; # " # # 2 " & = & = ! + & > ' # & D Tensiones principales y círculo de Mohr Definiciones ! + ⋅ (λ # λ ! 2= !# ( λ ! # ( && 2=+ & # ! " ! & ! " ! ( & ! & G+ & 2= # ( ⋅ ( ⋅ (λ - ' 2> + & $ + 1 )λ # (< 2A +" λ ) ' λ2 F '2 λ ) ' ( < ' '2 ' 2D ' ' ( '2 ( B2 2E ## ) # # < ' ( E ! # # & Tensiones normales máximas y mínimas @ $ # & ' ! ! 1 " # ! ' # λ ≥ λ2 ≥ λ & ⋅ ( ⋅ ⋅ ! ⋅λ ( (λ ⋅ ( ⋅ ⋅ ⋅λ ( (λ & 2 $ (α Fβ 2 " Fγ α2 F β2 F γ2 ( # I I( ⋅ ( ⋅ ⋅ ( α ⋅ ( α Fβ Fβ 2 Fγ 2 Fγ ⋅ ⋅α ⋅ αλ Fβ F β λ2 2 2 Fγ Fγλ ⋅ ( α2 λ F β2 λ2 F γ2 λ + ⋅ ( λ ( λ α2 F β2 F γ2 ≥ α2 λ F β2 λ2 F γ2 λ ( ⋅ ; ⋅ ( λ ( λ α2 F β2 F γ2 ≤ α2 λ F β2 λ2 F γ2 λ ( ⋅ & = # # ' 1 ! ! # & Tensiones tangenciales máximas # ' ' ! + 1 & # 2 ! &9 ( ! < ( ⋅ ( ⋅ λ ( F 2 λ2 2 λ F & > ' λ ⋅ ( ( F 2 2 ( , λ2 λ F ⋅ 2 2 λ2 F 2 F 2 2 F λ& A ' ' 2 2 2 ( λ 2F + 2 2 I2 ) (I λ2 2 F 2 λ 2 2 λ F 2 ) # 2 (∂ λ2 F 2 λ 2& D + 1 # ! B∂ 2 2 2 ' 2 2 9 λ F 2 F∂ B∂ 2 2 2 F∂ B∂ 2 ( <& E J 1 ∂ 2 # B∂ ( <& 2 F 2 2 2 F $ $ ' ( ;< & & + E F & 2 ( K∂ 2 ; 2 B∂ 2 F E + )η L F K∂ 2 ; η + & ( <& B∂ 2 ! )η 2L 2 F K∂ 2 B∂ )η (η & L (< ;2 2 ∂ 2 B∂ (η ∂ 2 B∂ 2 (η 2 ! ∂ 2 B∂ ; & ; 3 & ;< !η& 2 ! + ( < <) (< <) ( (< < ) 2 ± 2 < ) ( 2 <± 2 ) (< ) 2 ± 2 & ;; + 1 &: $ $ & + % & 0 % ;; & D + 2 ( B; λ ) λ2 2 ;= 2 ( B; λ ) λ 2 ;> 2 ( B; λ2 ) λ 2 ;A + & ;; & + + # ' % ! ' I I ( B2 Kλ( ) λ L λ( ( ' λ ! λ ( 1 λ & 1 & ( ;D λ( ! λ ' ! # ! & & ;; # ;= ;A ' ' + ! 1 & Círculo de Mohr. 1 +3 & 9 # + ' G ' & / # M # $ " 1 # &9 ' # 3 ! 0 & 9 σ ≥ σ2 ≥ σ &C $& # " ! !2 ! ! $ + ' # 2 . & =& / & ' # =< ;E σ2 2 2 )σ σ α α σ !2 ! ! . : # # % % α! + 2 F σ2 2 2 F (< =< α ( σ 9 ! !2 + σ ( = ! α F σ2 2 αF (< = ! . & ! !!2 =& & = σ α) α) α(< =2 σ2 α) αF α(< = + & =; ! σ ) ) α(< =; σ2 ) F B α(< == == σ ) σ2 ) α F B α ( 2B αF B α (< 2α 2α σ ) σ2 B2 ( 9 + + & =; ! σ F σ2 ) 2 B α) ! α(2 2α B == 2α % & α (< =D 2α σ ) σ2 B2 =E + =A ! + & 1 . & !( 1 () φ >< ) () φ > & =A ! =E ' ' & πB; ;= ! α # α ( < πB2 < σ ! %# " !) $ # = =E ! ( >& α ( πB; =A ! # ) % =A + B α) F ) σ F σ2 B2 ( 1 => E< σ2 # % ' ( σ ) σ2 B2 ( σ F σ2 B2& ; σ −σ 2 2 2α σ +σ 2 σ2 σ 2 . > 3 !2 σ −σ σ −σ 2 2 2 σ 2 −σ 2 σ σ2 σ 3! 3! . ' ' & !2 A 1 # " 1 $ . & A& = + ' + ' ! # . & . 1 A& $ + + # • ' ' + 2 # ;=N • ' & ! 1 + ! -$ & # 1 $ +" % λ =σ ' ! # " ! λ2 = σ 2 !! # =σ! ! =τ ! . & 1 σ !σ2 D& + ! ' . . & + σ τ ! # $ # # E& +3 1 3 & ! # σ =& + = σ2 =& − ' ! !! ! - =σ & # σ! τ! 3 ' = σ ! +σ 2 σ ! +σ 2 σ ! −σ + 2 + τ !2 2 σ ! −σ − >2 2 2 )# τ = + τ !2 1 σ −σ2 2 = σ! −σ 2 ' 2 + τ !2 >2 > / $ # + " # ! !! =σ! =σ ! =τ ! & σ *! *! ! σ! . D *! σ σ & σ2 σ! . E A