By Konstantin A. Lurie
This ebook offers a mathematical remedy of a unique idea in fabric technological know-how that characterizes the houses of dynamic fabrics - that's, fabric ingredients whose homes are variable in area and time. in contrast to traditional composites which are usually present in nature, dynamic fabrics are typically the goods of contemporary expertise constructed to take care of the best keep watch over over dynamic techniques. those fabrics have various functions comparable to: tunable left-handed dielectrics, optical pumping with high-energy pulse compression, and electromagnetic stealth know-how, to call a number of. Of distinctive importance is the participation of dynamic fabrics in virtually each optimum fabric layout in dynamics. The e-book discusses a few basic positive aspects of dynamic fabrics as thermodynamically open platforms; it provides their enough tensor description within the context of Maxwell's concept of relocating dielectrics and makes a unique emphasis at the theoretical research of spatio-temporal fabric composites (such as laminates and checkerboard structures). a few strange purposes are indexed besides the dialogue of a few general optimization difficulties in space-time through dynamic fabrics.
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Extra info for An Introduction to the Mathematical Theory of Dynamic Materials
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= . 8)) ∆i = V 2 − a2i , i = 1, 2. We shall also use parameters α, β, θ defined by the formulae A α= = C β= a2 ∆ 1 ρ∆ BV =V C C θ= = D 1 ρ∆ k ∆ V2 = k1 k2 1 ∆ 1 ρ∆ = ¯ 1 a2 −1 , V 2 ρ¯ − k¯ = ρ1 ρ2 V V 2 − a2 , V 2 ρ¯ − k¯ V 2 ρ¯ − k¯ 1 k1 k2 ρ1 ρ2 V 2 ¯1 − k ¯ 1 ρ . f. 12)) p= V 2 − θα2 V − θαβ 1 − θβ 2 , q=− , r=− .
31) The subscripts in these formulae denote partial diﬀerentiation over the relevant variables. 31), we obtain the formulae d dt du = u0z + δu1z + δ 2 u2z + . . + δ −1 (u0ξ + δu1ξ + δ 2 u2ξ + . ), dz du = u0t + δu1t + δ 2 u2t + . . − V δ −1 (u0ξ + δu1ξ + δ 2 u2ξ + . ), dt du du du ρ(ξ) = ρ(ξ) − V δ −1 ρ(ξ) dt dt t dt ξ = [ρ(ξ)(u0t + δu1t + δ 2 u2t + . ) − V δ −1 ρ(ξ)(u0ξ + δu1ξ + δ 2 u2ξ + . )]t − V δ −1 [ρ(ξ)(u0t + δu1t + δ 2 u2t + . ) − V δ −1 ρ(ξ)(u0ξ + δu1ξ + δ 2 u2ξ + . )]ξ . 32). 1), we express this one as a power series over δ; we require that the coeﬃcients of powers of δ be set equal to zero.
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.