By Konstantin A. Lurie

This booklet provides a mathematical therapy of a singular idea in fabric technology that characterizes the homes of dynamic fabrics - that's, fabric components whose homes are variable in house and time. not like traditional composites which are frequently present in nature, dynamic fabrics are more often than not the goods of recent expertise built to take care of the simplest keep watch over over dynamic strategies. those fabrics have different functions comparable to: tunable left-handed dielectrics, optical pumping with high-energy pulse compression, and electromagnetic stealth know-how, to call a couple of. Of distinct value is the participation of dynamic fabrics in virtually each optimum fabric layout in dynamics. The ebook discusses a few normal positive factors of dynamic fabrics as thermodynamically open platforms; it offers their enough tensor description within the context of Maxwell's idea of relocating dielectrics and makes a distinct emphasis at the theoretical research of spatio-temporal fabric composites (such as laminates and checkerboard structures). a few strange purposes are indexed in addition to the dialogue of a few usual optimization difficulties in space-time through dynamic fabrics.

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Additional resources for An Introduction to the Mathematical Theory of Dynamic Materials

Example text

In a laboratory frame (z, t), the eﬀective velocities take the values (V ∓ a1 )(V ∓ a2 ) a1 a2 ∓ V a +V = . ±¯ a−V ±¯ a−V Particularly, for V = 0 we obtain vst = ± a−1 whereas for V = ∞ −1 , vtemp = ± a . 70) for vst and vtemp when we apply them to the case γ1 = γ2 . 1) governing the wave propagation through an immovable elastic bar represents an Euler equation generated by the action density 2 2 ∂u ∂u 1 1 − k . 74) ∂z ∂ ∂u ∂t ∂z ∂t Wtt = Wtz Wzt Wzz = ∂u ∂Λ 1 −Λ=− ρ ∂z ∂ ∂u 2 ∂z ∂u ∂t 2 1 − k 2 ∂u ∂z 2 − the momentum flux density.

75) we have the rate of increase DW of the Dt energy of a unit segment of the bar; this rate is calculated as the sum of the ∂Wtz tt that is brought into a unit segment local change ∂W ∂t and the energy ∂z tt is equal to the through its endpoints per unit time. 77) − − 2 ∂t ∂t ∂t ∂z produced, per unit time, by an external agent against the variable property pattern. 76). In this section, we shall see in detail how the energy-momentum balance manifests itself through homogenization. 76). 31). 76) to the following form (Wtt )t + (Wtz )z − V δ −1 (Wtt )ξ + δ −1 (Wtz )ξ 1 = V δ −1 ρξ (ut − V δ −1 uξ )2 − kξ (uz + δ −1 uξ )2 , 2 (Wzt )t + (Wzz )z − V δ −1 (Wzt )ξ + δ −1 (Wzz )ξ 1 = − δ −1 [ρξ (ut − V δ −1 uξ )2 − kξ (uz + δ −1 uξ )2 ].

Eﬀective parameters K versus P with variable V (case ρ 1 ρ −k the signs of v1 and v2 are the same if V 2 is taken within the interval    1 , a21  for the slow range, ¯ ρ¯ k1 1 k ≤ 0). 67) and within the interval a22 , k¯ ¯ 1 ρ for the fast range. 59) that such intervals may exist in the irregular case. For each of them, the homogenized waves propagate in the same direction relative to a laboratory frame; this direction may be switched to opposite as we go from V to −V . We thus arrive at what will be termed coordinated wave propagation.