( ) ( ) ( ) ( ) ( ) λ = 1 + t t. θ = t ε t. Continuum Mechanics. Chapter 1. Description of Motion dt t. Chapter 2. Deformation and Strain

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Download "( ) ( ) ( ) ( ) ( ) λ = 1 + t t. θ = t ε t. Continuum Mechanics. Chapter 1. Description of Motion dt t. Chapter 2. Deformation and Strain"

Ζοροβάβελ Κυπραίος
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1 Continm Mechanics. Official Fom Chapte. Desciption of Motion χ (,) t χ (,) t (,) t χ (,) t t Chapte. Defomation an Stain s S X E X e i ij j i ij j F X X U F J T T T U U i j Uk U k E ( F F ) ( J J J J) Eij X j Xi Xi X j T T T i j k k e ( F F ) ( j j j j) eij j i i j T s i j ( J J ) ij j i Ω3 T a i j Ω ( J J ) Ω ij Ω3 j i Ω λ T E T λ t t s λ S λ λ X E XX t e t e λ cos cos Θ t t sin ( ) T E T T E T T E T ( ) t e t t e t t e t () () Θ XY π E XY acsin E E XX π e acsin e e γ YY

2 Continm Mechanics. F Q U V Q T U F F V FF T F Ω Q Ω U Vt F t V e F V ( T ) V F a F A F s sm() l a w skew() l ij w ij i j j i i j j i w ω w w 3 3 F F F l F s F T F l E F F V V ( a) a ( ( ) l ) ( ) Chapte 3. Compatibilit Eqations S S S S S S ( ) S,,,, mjq ni ij, q ij kl kl ij ik jl jl ik

3 Continm Mechanics. Consieing e e e3 e ef e e e e e e 3 e e e, e e e e e e an ef 3 : ef Chapte 4. Stess b a, ± τ s s s s s cos τ sin τ in τ co t V Chapte 5. Conseation an Balance Eqations Vt µ µ V µ V Vt ( μ) V μn S V t A V V V V ψ ψ V V ψ S n V V V ψ ψv k V ψ n S ja n S k A ja V b V t S Vt V V V V b V t S Vt V V V 3

4 Continm Mechanics. b t : Pe V S V V V V Vt V V Q V S e qn V V : q s V V S V V s n V ( i) s local s ( i) s con s s : ψ s s : ψ s Chapte 6. Elasticit λ T µ β l δ µ β δ ( ) ν ν T α E E ij ll ij ij ij ν E δ αδ ( ) m ν E ij ll ij ij ij Ke G ij ij λ ν E µ E ( ν)( ν) ( ν) ( 3 ) λ ν ( λ µ ) E µ λ µ λ µ β E ( ν) α E K 3 ( ν) G β 3 λ µ α K µ G 3λ µ 3 λ µ µ b t * λ ( n ) µ ( ) n t en Γ E ( ν ) ˆ * * t t ( β ) n βˆ β ( β ) 4

5 Continm Mechanics. Chapte 7. Plane Linea Elasticit Plane stain: ν ν E ( ν ) ν C ( ν)( ν) ν ν ( ν ) ν ( ) Plane stess: ν E C ν ν ν ν ( ) ν tg t ' ± τ τ, ± t cnt tg α α α tg α tg ( ) tg t τ τ ' ± Chapte 8. Plasticit I J ij ij kk I oct m t oct J 3 3 ' 3J ep H E E E H 3 J e e 3 s s3 s s3 sinϕ c cosϕ tn c ntg 3 α J β with m sin 6c cos α ; β 3 3 sin 3 3 sin ( ) ( ) 5

6 Continm Mechanics. Chapte 9. Constittie Eqations in Flis p λ T µ p l µ ij ij ll ij ij C λ µ I C ijkl lδijδ kl µ ( δikδ jl δilδ jk ) K p p λ µ p 3 WR pt ( ) K W T µ : D Chapte. Fli Mechanics p p p ω P ω p ( p) t p ( p) p ( λ µ ) ( ) µ b WR WD ( k ) p cnt g g Chapte. Vaiational Pinciples ( ; ) F E Ω T Γ Ω Γ Γ * s W ; b a V t Γ s: V V V Γs Γ * [ : ] V b a V t Γ V Γ V * : V b V t S V V V t 6

7 Continm Mechanics. 7 Infinitesimal Stain Tenso Clinical Cooinates Spheical Cooinates cot sin sin cot sin sin cos ),, ( sin cos,, sin sin cos ϕ ϕ ϕ

8 Continm Mechanics. Defomation Rate Tenso Clinical Cooinates é ù ê ë úû æ ö ç - çè ø æ ö ç çè ø æ ö ç çè ø cos (,, ) sin Spheical Cooinates cot sin sin cot sin sin cos (,, ) sin sin cos 8

9 Continm Mechanics. Clinical Cooinates Naie Eqations e G ω ω t e ω ω λ G G G b t ( λ G) G b e G G ω ( λ G) ( ω ) b t whee ω ω Ω Ω cos (,, ) sin ( ) ω Ω e ( ) Spheical Cooinates e G G ω λ ω b sin sin t ( G) ( sin ) ( λ G) e G ω G ( ω sin) b sin sin t ( λ ) G e G G ω ( ω) b sin t whee ω ( sin ) Ω sin sin ( ) ω Ω sin ω Ω ( ) e sin sin sin sin cos (,, ) sin sin cos 9

10 Continm Mechanics. Stess Tenso fo Newtonian Flis Incompessible fli Catesian Cooinates µ p µ p µ p τ τ τ τ τ τ µ µ µ Clinical Cooinates µ p τ τ µ µ p τ τ µ µ p τ τ µ ( ) Spheical Cooinates µ p µ p cotg s µ p sin τ τ µ sin τ τ µ sin sin τ τ µ sin ( ) ( sin ) sin sin

11 Continm Mechanics. Continit Eqations Catesian Cooinates Clinical Cooinates Spheical Cooinates ( ) ( ) ( ) t t ( ) ( ) ( ) ( ) ( sin) ( ) t sin sin Catesian Cooinates Naie - Stokes Eqations Incompessible fli; an µ constants Component p µ b t Component p µ b t Component p µ b t

12 Continm Mechanics. Clinical Cooinates Component p µ ( ) b t Component p µ ( ) b t Component p b µ t Spheical Cooinates Component p µ ( ) sin ( sin) sin sin sin sin b t sin Component p cotg µ ( sin) sin sin sin cotg b t sin Component p cotg µ ( sin) sin sin sin sin sin b ϕ cotg t sin

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