By Konstantin A. Lurie
This publication provides a mathematical remedy of a unique notion in fabric technological know-how that characterizes the houses of dynamic materials—that is, fabric ingredients whose houses 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 constructed to keep up the best keep watch over over dynamic approaches. those fabrics have varied purposes: tunable left-handed dielectrics, optical pumping with high-energy pulse compression, and electromagnetic stealth know-how, to call a number of. Of certain importance is the participation of dynamic fabrics in virtually each optimum fabric layout in dynamics.
The booklet discusses a few normal positive aspects of dynamic fabrics as thermodynamically open platforms; it supplies their sufficient tensor description within the context of Maxwell’s thought 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 besides the dialogue of a few normal optimization difficulties in space-time through dynamic materials.
This booklet is meant for utilized mathematicians drawn to optimum difficulties of fabric layout for platforms ruled through hyperbolic differential equations. it is going to even be necessary for researchers within the box of clever metamaterials and their functions to optimum fabric layout in dynamics.
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Extra info for An Introduction to the Mathematical Theory of Dynamic Materials
20) is not equal to V unless V = 0. 9). 1) in which the coeﬃcients ρ and k are defined as fast periodic functions of the argument ξ = (z − V t)/δ, δ → 0. The period in ξ is taken equal to 1. 1) represented in the form of a power series over δ: u = u0 (z, t, ξ) + δu1 (z, t, ξ) + δ 2 u2 (z, t, ξ) + . . 30) where ui , i = 0, 1, 2, . . are assumed 1-periodic in ξ. 1) should be recalculated by the rule of diﬀerentiating composite functions: d F (z, t, ξ) = Fz + δ −1 Fξ , dz d F (z, t, ξ) = Ft − V δ −1 Fξ .
2) that depart from the interface. On the interface V = 0, we have two characteristics, with the slopes ±a1 , at the side occupied by material 1, and two characteristics with the slopes ±a2 , at the opposite side occupied by material 2; here, ai , i = 1, 2, denotes the phase velocity ki /ρi of waves in material i; we assume below, without sacrificing generality, that a2 > a1 . Clearly, two out of the four characteristics, specifically, those with slopes a2 and −a1 , depart from the interface. e.
We first eliminate the τ -variable by applying the Laplace transform: ∞ e−sτ u(ζ, τ )dτ. 8)). Assume that ζ ≥ 0, and that material 1 occupies the intervals (n − m1 )δ ≤ ζ ≤ nδ, n = 0, 1, 2, . . 43) while material 2 is concentrated within supplementary intervals nδ ≤ ζ ≤ (n + m2 )δ, n = 0, 1, . . 44) Here m1 and m2 denote, as before, the volume fractions of materials 1 and 2 in the laminate; clearly, m1 + m2 = 1. 45) with P (µ1 , ζ), . . , Q(µ2 , ζ) being δ-periodic in ζ. 45), A1 and A2 denote the coeﬃcients to be determined by the boundary conditions, and µ1 , µ2 represent the Floquet characteristic exponents given by the formula (see Appendix 1) µ1,2 δ = V (θ1 /a1 + θ2 /a2 ) ± χ.