By Freeadman M.H., Teichner P.
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Additional info for 4-Manifold topology II: Dwyers filtration and surgery kernels
4, ineqs. 5) become satisfied by a due choice of V ; bearing this in mind, we shall now calculate the eﬀective parameters of an activated elastic bar. 1 Longitudinal vibrations of activated elastic bar Fig. 2. An immovable interface: V = 0. Fig. 3. A moving interface: | V |< a1 . 21 22 2 An Activated Elastic Bar: Eﬀective Properties Fig. 4. A moving interface: a1 < V < a2 . Fig. 5. A moving interface: −a2 < V < −a1 . 1 Longitudinal vibrations of activated elastic bar Fig. 6. A moving interface: | V |> a2 .
91) may be interpreted as an eﬀective The quantity W energy density measured in the laboratory frame. 96), this density takes on the form ¯ tt = 1 ρ(1 − V P )2 − kP 2 u20t + 1 k(1 + Q)2 − ρV 2 Q2 u20z . f. 86)) 1 ρ(1 − V P )2 + kP 2 u20t − ρV Q(1 − V P ) − kP (1 + Q) u0t u0z 2 1 + k(1 + Q)2 + ρV 2 Q2 u20z . e. for a static laminate. This diﬀerence is non-zero because of a temporal activation. We therefore expect that if we go to a co-moving coordinate frame τ, η in which an interface between layers in an activated composite remains immovable, then the ¯ τ τ and Tτ τ evaluated for this system, may vanish.
9) bearing in mind that derivatives uτ , vτ are continuous across the interfaces ζ=const immovable in a new frame; for this reason, they remain unaﬀected by averaging: uτ = uτ , vτ = vτ . Preserving symbols uζ , uτ , vζ , vτ for the averaged quantities uζ , uτ , vζ , vτ , we arrive at the system uζ = BV uτ − Cvτ , vζ = −Duτ + BV vτ , where we introduced notation B= 1 ∆ , C= 1 ρ∆ , D= k ∆ . 7); after some calculation, we arrive at the system vt = puz − qut , vz = quz + rut . 12) . 13) with the symbol A defined as A = BV 2 − 1 = a2 V − a2 2 A direct calculation shows that ¯ 1 a21 a22 V2 −1 , ∆1 ∆2 a2 1 B= (V 2 − a2 ), ∆1 ∆2 1 ¯ C= (V 2 ρ¯ − k), ρ1 ρ2 ∆ 1 ∆ 2 ¯ ¯1 1 k1 k2 − V2 D= ∆1 ∆2 k ρ A= .