SD-Lecture06-Infinite-Media

SD-Lecture06-Infinite-Media - Soil Dynamics Lecture 06...

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Unformatted text preview: Soil Dynamics Lecture 06 Stress Waves in Infinite Media © Luis A. Prieto-Portar, August 2006. The equation of motion of a stress wave in an elastic medium. Consider an element of an elastic medium, as shown below, with all the possible stresses on each of its six faces. x The equation of motion can be found through a summation of the forces along all three axes, and using Newton’s second law ( F = ma ), ( )( )( ) 2 2 yx zx x x x zx zx yx yx y x Consider the displacement u in t he x direction, dx dydz dz dxdy dy dxdz x z y u dx dy dz t Simplifying,and expanding to all three axes, x τ τ σ σ σ τ τ τ τ σ σ τ τ τ τ σ σ τ τ τ τ σ σ τ τ τ τ ρ τ σ − ∂ ∂ ∂ + − + + − + + − ∂ ∂ ∂ ∂ = ∂ ∂ ∂ + ∂ 2 2 2 2 2 2 x zx xy y zy yz xz z u y z t v x y z t w x y z t τ ρ τ σ τ τ σ τ τ σ τ τ σ τ ρ τ τ σ τ σ τ σ τ σ ρ ∂ ∂ + = ∂ ∂ ∂ ∂ ∂ ∂ ∂ + + = ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ + + = ∂ ∂ ∂ ∂ Compression stress waves (P-waves, or Primary waves or Dilatational waves). The stress wave of motion in the x-direction was developed on the previous slide, ( ) ( ) ( ) ( ) 2 2 2 2 2 2 2 2 2 yx zx x yx yx zx zx x x x yx zx yx zx x u t x y z Re call that, G and G and G therefore, u G G G t x y z and again recall that, v u u w and x y z x u v u G G t x y x y τ τ σ ρ τ γ τ γ σ λ ε ε τ γ τ γ σ λ ε ε τ γ τ γ σ λ ε ε τ γ τ γ σ λ ε ε ρ λ ε ε γ γ ρ λ ε ε γ γ ρ λ ε ε γ γ ρ λ ε ε γ γ γ γ γ γ γ γ γ γ ρ λ ε ε ρ λ ε ε ρ λ ε ε ρ λ ε ε ∂ ∂ ∂ ∂ = + + ∂ ∂ ∂ ∂ = = = + ∂ ∂ ∂ ∂ = + + + ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ = + = + ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ = + + + ∂ ∂ ∂ ∂ ∂ u w G z z x ∂ ∂ ∂ + + ∂ ∂ ∂ ( ) 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 Simplifying, u u v w u u u G t x x x y x z x y z but u v w x x y x z x therefore,simplifying and extending to all three axes,...
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SD-Lecture06-Infinite-Media - Soil Dynamics Lecture 06...

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