Uniqueness theorem

# Uniqueness theorem - Uniqueness theorem Let us consider now...

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Unformatted text preview: Uniqueness theorem Let us consider now whether our equations can have more than one correct solution corresponding to given surface and body forces. The equations of equilibrium expressed in terms of stresses have been established as , ij i j f σ + = (a) In the expanded form, one has xy 0 x xy x xz x y yz y yz xz z z X x y z X y z X x y z τ σ τ τ σ τ τ τ σ ∂ ∂ ∂ + + + = ∂ ∂ ∂ ∂ ∂ ∂ + + + = ∂ ∂ ∂ ∂ ∂ ∂ + + + = ∂ ∂ ∂ (2-12) The kinematic relationships are ( 29 , , 1 2 ij i j j i u u ε = + (b) In the expanded form, one has xy , , x y u v u v x y y x ε ε γ ∂ ∂ ∂ ∂ = = = + ∂ ∂ ∂ ∂ (2-2) and xz yz , , z w w u w v z x z y z ε γ γ ∂ ∂ ∂ ∂ ∂ = = + = + ∂ ∂ ∂ ∂ ∂ (2-3) With the compatibility constrains of , , , , ij kl kl ij ik jl jl ik ε ε ε ε + = + (c) Again, in the expanded form 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 2 1 2 1 2 y xy x y yz z x xz z yz xy x xz y yz xy xz yz xy xz z y x x y z y y z x z x z y z x x y z x z y x y z x y z x y z ε γ ε ε γ ε ε γ ε γ γ ε γ ε γ γ γ γ γ γ ε ∂ ∂ ∂ + = ∂ ∂ ∂ ∂ ∂ ∂ ∂ + = ∂ ∂ ∂ ∂ ∂ ∂ ∂ + = ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ =- + + ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ =- + ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ = +- ∂ ∂ ∂ ∂ ∂ ∂ (2-13) and the constitutive law is given by ( 29 1 1 ij ij ij kk E ε μ σ μδ σ = +- (d) In the expanded form ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 11 11 11 11 22 33 1 1 1 1 1 1 xx xx xx yy zz xx yy zz E E E ε μ σ μ δ σ σ σ ε μ σ μ σ σ σ σ μ σ...
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Uniqueness theorem - Uniqueness theorem Let us consider now...

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