By Konstantin A. Lurie

This publication supplies a mathematical remedy of a singular idea in fabric technological know-how 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 quite often the goods of contemporary know-how constructed to take care of the simplest keep an eye on over dynamic procedures. those fabrics have assorted functions resembling: tunable left-handed dielectrics, optical pumping with high-energy pulse compression, and electromagnetic stealth expertise, to call a couple of. Of specific value is the participation of dynamic fabrics in nearly each optimum fabric layout in dynamics. The booklet discusses a few normal positive aspects of dynamic fabrics as thermodynamically open structures; it offers their enough tensor description within the context of Maxwell's concept of relocating dielectrics and makes a different emphasis at the theoretical research of spatio-temporal fabric composites (such as laminates and checkerboard structures). a few strange functions are indexed besides the dialogue of a few general optimization difficulties in space-time through dynamic fabrics.

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**Example text**

Homogenization: Averaging Processes in Periodic Media - Mathematical Problem in the Mechanics of Composite Materials. 1 Preliminary remarks The analysis given in Chapter 2 may be treated as introductory. We applied the model of a thin elastic bar to illustrate the concept of activated dynamic material. Within this concept, we defined the eﬀective material parameters of a spatio-temporal composite, specifically, a laminate in one spatial dimension and time. Such parameters emerge as we introduce a proper coordinate frame in which the homogenized system is reduced to a canonical form, with diagonal matrix of the eﬀective material constants.

55). 50) for the eﬀective velocities may then be illustrated by the following elementary argument. e. a transmitted wave travelling in material 2. 6) where the interfaces stay immovable. An elementary calculation now specifies the average velocity of waves passing through a unit period in ξ in this frame: m1 ±a1 −V 1 + m2 ±a2 −V = (V ∓ a1 )(V ∓ a2 ) . 6). 72) are identical. 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 .

50) in a low frequency limit ω → 0. In this capacity, they represent the group velocities of the low frequency waves propagating through an activated dynamic lamination. 17), ¯1 2 . 19), ¯1 2 . 54) now show that always 2 vst ≤ a22 , 2 vtemp ≥ a21 . 59) show that it is possible that 2 2 ≤ a21 , vtemp ≥ a22 . vst Combining these inequalities, we conclude that, in a regular case, both vst and vtemp fall into the interval (a1 , a2 ), whereas in an irregular case, they may fall outside this interval: vst may become less than a1 , and vtemp - greater than a2 .