L05_State_of_Stress

L05_State_of_Stress - CE507 Lecture 5 5.1 Definition of...

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CE507 Lecture 5 Tensor, State of Stress 5.1 Definition of Tensor Recall from Class 2 (5.1) [ ] [ ][ ][ ] ' T Aa A a = [ ] a : direction cosine matrix [ ] A : linear transformation where (5.2) [ ]{ } { } A xy = ±± in 1 st coordinate system [ ]{ } { } '' ' Ax y = in 2 nd coordinate system In index notation, (5.3) ii j j y = kk l l y ′′ = where (5.4) i i y ay ′ = ll j j jl j l x ax x =⇒ = so (5.5) i ik i i j j y ay aAx == = ki ij lj l aAax = kl l or (5.6) kl ki lj ij a A ′ = Tensor Definitions: (5.7) Order 0 (scalar) α ′ = in any direction Order 1 (vector) j j x ′ = transformation of coordinates Order 2 (matrix) kl ki lj ij ta a t ′ = (9 terms) Order 3 lmn li mj nk ijk a a t = (27 terms) Order 4 mnpq mi nj pk ql ijkl a a a t = (81 terms) 5.1.1 Isotropic Tensors : Tensors that remain invariant (the same) under transformation. Q : Is the only tensor of order 1 (vector) that is isotropic is the zero vector? 1 x 2 x 3 x 3 ' x 2 ' x 1 ' x
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Example of isotropic 2 nd order tensor: [ ] I The identity matrix is isotropic: (5.8) [] [] [] II I TT aa a a == = or (5.9) ij ik jl kl ik jk ij δ δδ = From the cubic polynomial 32 12 3 0 I λλ λ −+ = 123 ,, I are invariant (same in all coordinate system) (5.10) 1 kk I A = kk ki kj ij ij ij ii kk Aa a A AAA = = (5.11) 2 21 1 () 2 ij ji I IA A =− need only to show that ij ji A A is invariant (5.12) ij ji AA ′′ = ( ) ik jl kl jm in mn aaA a aA = ik in jl jm kl mn aaaa AA = kn lm kl mn where kn ik in = and lm jl jm = = kn kl lm mn = nl ln = ij ji (5.13) 31 2 3 det ijk ijk I A A ε need to first show that ijk ε is invariant ???HW???
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M Δ ± F Δ ± ˆ n B 1 x 2 x 3 x 1 T ± 2 T ± 3 T ± S Δ T ± ˆ n o 1 x 2 x 3 x 33 τ 32 31 22 23 21 11 12 13 5.2 State of Stress Given a body B. Consider an element with area S Δ On or inside the surface, with normal ˆ n , force ˆ F Δ and moment ˆ M Δ on it. Assume that (5.14) 0 lim 0 s Md M Sd S Δ == Δ + ±± zero couple in the limit (per unit area) Define traction, (5.15) 0 lim s Fd F T S Δ Δ + ± force / area Special Case For a given coordinate system 123 x xx , consider a cubic element whose forces are parallel to the axes.
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This note was uploaded on 02/28/2008 for the course CE 507 taught by Professor Lee during the Fall '07 term at USC.

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L05_State_of_Stress - CE507 Lecture 5 5.1 Definition of...

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