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SD-Lecture17-Stress-Waves-in-Infinite-Media

SD-Lecture17-Stress-Waves-in-Infinite-Media - CEGCEG-5065C...

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CEG CEG-5065C Soil Dynamics 5065C Soil Dynamics Lecture #17 Stress Waves in Infinite Media Luis A. Prieto-Portar 2009
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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
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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 Consider the displacement u in the x-direction, Simplifying, and expanding to all three axes, ° ± ° ± ² ³ ° ± ² ³ ² ³ + - + + - + + - ´ µ · ´ µ ´ µ · · ¸ ¹ ´ µ ¸ ¹ º » º » ¸ ¹ º » = yx zx x x x zx zx yx yx dx dydz dz dxdy dy dxdz x z y u dx dy dz t τ τ σ σ σ τ τ τ τ σ ρ τ σ τ + y x x 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 ρ τ σ τ ρ τ τ σ τ σ ρ
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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 Recall that, and and 2 therefore, = + + = = = + yx zx x yx yx zx zx x x u t x y z G G G u τ τ σ ρ τ γ τ γ σ λε ε τ γ τ γ σ λε ε γ γ ( ) 2 2 2 2 and again recall that, and 2 = + + + = + = + ² ³ = + + + · ¸ ¹ x yx zx yx zx x G G G t x y z v u u w x y z x u v u G G t x y x y ρ λε ε γ γ λε γ γ γ ρ λε ε λε ² ³ + + · ¸ ¹ u w G z z x
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( ) 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 Simplifying, but therefore,simplifying and extending to all three axes, ² ³ = + + + + + + · ∂ ∂ ∂ ∂ ¸ ¹ + + = ∂ ∂ ∂ ∂ u u v w u u u G t x x x y x z x y z u v w x x y x z x u ε ρ λ ρ λ λ λ ε ε 2 2 2 2 2 2 where = + + = + + G G u t x x y z ρ λ ρ λ ρ ( ) ( ) 2 2 2 2 2 2 gradient squared or squared = + + = + + v G G v "del" t y w G G w t z ε λ ε ρ λ
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( ) ( ) ( ) 2 2 2 2 2 2 2 2 2 2 2 2 2 2
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