Ch.0. Group Work Units. Continuum Mechanics Course (MMC) - ETSECCPB - UPC
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1 Ch.0. Group Work Unts Contnuum Mechancs Course (MMC) - ETSECCPB - UPC
2 Unt 1 2 Prove that the followng expresson holds true: ee 6 k k
3 Unt 1 - Soluton 3 ee e k e 1 + e k e 2 + e k e 3 + k k e121e121 + e122e122 + e123e e131e131 + e132e132 + e133e e211e211 + e212e212 + e213e e221e221 + e222e222 + e223e e231e231 + e232e232 + e233e e311e311 + e312e312 + e313e e e + e e + e e e e + e e + e e
4 Unt 2 4 Prove the followng property of the tensor product s true: u ( v w) ( u v) w
5 Unt 2 - Soluton 5 u ( v w) ( u v) w c 2 nd order vector tensor (matrx) 1 st order tensor (vector) scalar vector 1 st order tensor (vector) ( ) [ ] ( ) ( vw ) c u v w u v w u u vw k k k k k k-component of vector c ( u v) w v [ w] c u u vw k k k k k-component of vector c
6 Unt 3 Prove de followng propertes of the scalar or dot product uv vu u0 0 ( α β ) α( ) β( ) u v+ w u v + u w uu > 0 uu 0 u 0 u 0 u v 0, u 0, v 0 u v Lnear operator 6
7 7 Unt 3 - Soluton
8 Unt 4 8 When does the relaton nt Tn hold true?
9 Unt 4 - Soluton 9 n T T n c vector 2 nd order tensor vector c nt c Tn c nt k k c k Tkn nt k c k c k f Tk Tk
10 Unt 4 - Soluton 10 nt Tn COMPACT NOTATION nt k Tkn k { 1, 2, 3} INDEX NOTATION T T [ n] [ T] [ T][ n] T c c c MATRIX NOTATION
11 Unt 4 - Soluton 11 T T [ n] [ T] [ T][ n] T c c c MATRIX NOTATION T11 T12 T13 T11 T12 T13 n1 n1 n2 n 3 T21 T22 T 23 T21 T22 T 23 n 2 T T T T T T n [,, ] c1 c1 c2 c3 c 2 c 3 [ ]
12 Unt 5 12 Prove that: ( T ) A: B Tr A B A B Tr ( AB )
13 Unt 5 - Soluton 13 ( T ) T T A B A B A [ B] AB AB c Tr k kk k k k k c ( AB) [ AB] [ A] [ B] AB AB Tr kk k k k k k
14 Unt 6 Prove the followng propertes of the open product: ( u v) ( v u) ( u v) w u ( v w) u( v w) ( v w) u u ( αv+ βw) αu v+ βu w u ( v w) ( u v) w ( u v) w w( u v) Lnear operator 14
15 15 Unt 6 - Soluton
16 Unt 7 Prove the followng propertes of the dot product: 1 A A A 1 ( ) ( ) ( ) A B+ C AB + AC A BC AB C ABC AB BA 16
17 17 Unt 7 - Soluton
18 Unt 8 Prove the followng propertes: ( T ) ( T ) ( T) ( T) AB : Tr A B Tr B A Tr AB Tr BA BA : 1: A TrA A: 1 ( ) ( T ) ( T ) A: BC B A: C AC : B A: ( u v) u ( A v) ( u v) : ( w x) ( u w) ( v x) REMARK A: B C: B A C 18
19 19 Unt 8 - Soluton
20 20 Unt 8 - Soluton
21 Unt 9 21 Prove that det A e k AA A. k 1 2 3
22 Unt 9 - Soluton 22 det A det A A A A A A A A A A A A + A A A + A A A A A A A A A A A A
23 Unt 9 - Soluton 23 e AA 1 2 A 3 e111a11a21a31 + e112 A11A21A32 + e113a11a21a e121a11a22 A31 + e122 A11A22 A32 + e123a11a22 A e131a11a23a31 + e132 A11A23A32 + e133a11a23a A A A + e A A A + e A A A + k k k 1 k 2 k e221a12 A22 A31 + e222 A12 A22 A32 + e223a12 A22 A e231a12 A23A31 + e232 A12 A23A32 + e233a12 A23A e311a13a21a31 + e312 A13A21A32 + e313a13a21a e A A A + e A A A + e A A A e A A A + e A A A + e A A A A11A22 A33 + A12 A23A31 + A13A21A32 A13A22 A31 A12 A21A33 A11A23A32
24 Unt 10 Prove that c a b b a
25 Unt 10 - Soluton
26 Unt Gven the vector determne ( ) xxxˆ xxˆ ˆ x1 3 v v x e + e + e v, v, v.
27 Unt 11 - Soluton 27 ( ) xxxˆ xxˆ ˆ x1 3 v v x e + e + e Dvergence: [ v] xxx xx x 1 v v x v v v v v + + xx x1 x x1 x2 x3
28 Unt 11 - Soluton 28 Dvergence: v v x In matrx notaton: [ v] xxx xx x 1 T xxx symb symb symb T v [ ] [ v],, xx 1 2 x1 x2 x 3 x symb ( xxx 1 2 3) ( xx 1 2) x1 xxx + xx + x + + xx + x x x x x x x
29 Unt 11 - Soluton 29 Rotaton: [ v] e k v x In ndex notaton: k [ v] xxx xx x 1 v [ v] k ek x v v v v v v e + e + e + e + e + e x1 x1 x2 x2 x3 x3
30 Unt 11 - Soluton 30 Rotaton: v [ ] 2 v3 v1 v e 12 + e 13 + e21 + x x x In matrx notaton v v v + e + e + e x2 x3 x3 [ v] xxx xx x 1 v3 v2 e123 + e132 x2 x v [ ] 3 v1 v xx e + e x1 x x 2 xx 1 3 v2 v 1 e312 + e321 x1 x2 1 1 In compact notaton: ( xx 1) ˆ ( x xx ) v e + eˆ
31 Unt 11 - Soluton 31 Rotaton: v Calculated drectly n matrx notaton: v v x ˆ ˆ ˆ 1 e1 e2 e3 v1 symb v v 2 det x 2 x1 x2 x 3 v 3 v1 v2 v3 x3 v3 v 2 v1 v 3 v2 v 1 eˆ ˆ ˆ 1+ e2 + e3 x2 x3 x3 x1 x1 x2 + eˆ ( xx 1) eˆ ( x xx ) xxx xx x
32 Unt 11 - Soluton 32 Gradent: v v v v x v In matrx notaton x 1 xx 2 3 x2 1 symb symb symb T v v v xxx 1 2 3, xx 1 2, x 1 xx 1 3 x1 0 x 2 xx In compact notaton: x [ ] [ ] [ ] [ ] [ ][ ] [ ] 3 1 v xxx v xx eˆ eˆ + xeˆ eˆ + eˆ eˆ + xx eˆ eˆ + xeˆ eˆ + xx eˆ eˆ xx x
33 Unt 12 Establsh the followng denttes nvolvng a smooth scalar feld and a smooth vector feld v, ( ) φv φ v+ v φ ( ) φv φ v + φ v MMC - ETSECCPB - UPC 10/2/16
34 Unt 12 - Soluton ( ) φv φ v+ v φ [ v] v v φ φ φ ( φv) v + φ φ + v x x x x x ( ) φ v+ v φ φv φ v + φ v [ φv] φ v ( φ v ) v + φ φ + φ x x x v v [ ] [ ] MMC - ETSECCPB - UPC 10/2/16
35 Unt 13 Establsh the followng denttes nvolvng the smooth scalar felds and ψ, smooth vector felds u and v, and a smooth second order tensor feld A, φψ φ ψ + φ ψ ( ) ( ) ( ) φa φ A+ φ A ( ) A v A v+ A: v ( ) ( ) u v u v+ u v ( ) ( ) ( ) u v u v u v ( ) ( ) φ MMC - ETSECCPB - UPC 10/2/16
36 Unt 13- Soluton ( φψ ) ( φ ) ψ + φ ( ψ ) ( ) ( ) φψ φ ψ φψ ψ + φ [ φ] ψ + φ[ ψ] x x x ( ) ( φψ ) φ( ψ) + φa φ A+ φ A [ ] φ A φ A A φ ( φa) A + φ φ + A x x x x x [ A] [ ] [ A] [ A A] φ + φ φ + φ MMC - ETSECCPB - UPC 10/2/16
37 Unt 13- Soluton ( A v) ( A) v+ A: v ( A v ) A v ( A v) v + A x x x [ ] [ ] [ ] [ ] ( ) A v + A v A v+ A: v ( u v) ( u) v+ u v ( u v ) u v ( u v) v + u x x x ( )[ ] [ ] [ ] ( ) u v + u v u v+ u v MMC - ETSECCPB - UPC 10/2/16
38 Unt 13- Soluton ( ) ( ) u v u v u v ( ε v ) ( v ) ku k u k u vk ( u v) εk εk vk + εku x x x x u vk εk vk u ε k [ v] [ u] [ u] [ v] k k x x ( ) u v u v [ v] ε k v x k MMC - ETSECCPB - UPC 10/2/16
39 Unt 14 Establsh the followng denttes nvolvng the smooth scalar feld φ and the smooth vector felds u and v, ( u v) ( u) v ( v) u ( φv) ( φ) v φ( v) ( u v) ( u ) v v( u) u( v) u ( v) MMC - ETSECCPB - UPC 10/2/16
40 Unt 14- Soluton ( u v) ( u) v+ ( v) u ( u v ) u v ( u v) v + u [ u] [ v] + [ u] [ v] x x x ( ) ( ) u v+ v u ( φv) ( φ) v φ( v) ( φvk ) ( v) + φ vk φ ε k ε k v k + φε k x x x v ε k φ k + φε k φ + φ x k [ ] v ( ) v ( v) MMC - ETSECCPB - UPC 10/2/16
41 Unt 14- Soluton εkε pqk δpδ q δqδ p ( u v) ( u ) v v( u) + u( v) u ( v) ( u v) [ u v] ( ε u v ) k klm l m ε k εk x x ul vm εkεlmk vm + εkεlmkul x x ul vm ( δδ l m δmδ l ) vm + ( δδ l m δmδ l ) ul x x u u v v v v + u u x x x x [ u] [ v] ( u)[ v] [ u] ( v) [ u] [ v] + + MMC - ETSECCPB - UPC 10/2/16
42 Unt 14- Soluton ( u v) ( u ) v v( u) + u( v) u ( v) ( u v)... (contnued) [ u] [ v] ( u)[ v] [ u] ( v) [ u] [ v] [ u ] [ v] ( u)[ v] [ u] ( v) [ u] [ v] ( u ) v v( u) u( v) u ( v) + ( u ) v v( u) u( v) u ( v) + MMC - ETSECCPB - UPC 10/2/16
43 Unt where Use the Generalzed Dvergence Theorem to show that x S x n ds s the poston vector of Vδ n n V A n ds A V dv
44 Unt 15 - Soluton 44 x n ds S x n ds S Applyng the Generalzed Dvergence Theorem: S x n ds Vδ V x nds x V dv Applyng the defnton of gradent of a vector: [ x ] x [ ] x x x x
45 Unt 15 - Soluton 45 The Generalzed Dvergence Theorem n ndex notaton: Then, S x n ds V x x dv S x n ds Vδ x x n ds dv δ dv δ V S V V x
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