By Konstantin A. Lurie
This ebook provides a mathematical remedy of a singular thought in fabric technology that characterizes the homes of dynamic materials—that is, fabric elements whose houses are variable in area and time. not like traditional composites which are usually present in nature, dynamic fabrics are often the goods of recent expertise built to keep up the best regulate over dynamic procedures. those fabrics have different functions: tunable left-handed dielectrics, optical pumping with high-energy pulse compression, and electromagnetic stealth expertise, to call a couple of. Of designated value is the participation of dynamic fabrics in virtually each optimum fabric layout in dynamics.
The booklet discusses a few normal good points of dynamic fabrics as thermodynamically open platforms; it provides their enough tensor description within the context of Maxwell’s conception 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 functions are indexed besides the dialogue of a few regular optimization difficulties in space-time through dynamic materials.
This e-book is meant for utilized mathematicians attracted to optimum difficulties of fabric layout for platforms ruled by way of hyperbolic differential equations. it is going to even be invaluable for researchers within the box of clever metamaterials and their functions to optimum fabric layout in dynamics.
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Additional resources for An Introduction to the Mathematical Theory of Dynamic Materials
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. 74). 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.
A matrix microstructure in space-time violating ineqs. 5). 5). 6) this co-moving frame is travelling with velocity V in the positive z-direction. 8) we rewrite this as uζ = V 1 k V uτ − vτ , vζ = − uτ + vτ . 1, indicate that parameters ρ, k depend on the argument z − V t = ζ; we shall assume that these parameters are periodic functions, with a unit period, of the fast variable ξ = ζ/δ, δ → 0. 9) will now be averaged over the unit period in ξ. Introduce the symbol · = m1 (·)1 +m2 (·)2 for the arithmetic mean of (·), with materials 1 and 2 represented in a unit period at the volume fractions m1 , m2 ≥ 0 (m1 + m2 = 1).
2 = (u0t − V u1ξ )2 + 2δ(u0t − V u1ξ )(u1t − V u2ξ ) + . . , (uz + δ −1 uξ )2 = (u0z + u1ξ )2 + 2δ(u0z + u1ξ )(u1z + u2ξ ) + . . 78), the latter equation includes terms of order δ −1 , δ 0 , δ, etc. 78), should be set equal to each other. We are particularly interested in the coeﬃcients of δ 0 because they carry information about the energy flows as we pass to the limit δ → 0. 84) = V ρξ (u0t − V u1ξ )2 − V kξ (u0z + u1ξ )2 , 2 2 −V whereas the balance of δ 0 -terms is expressed by ∂ 1 ∂ ρ(u0t − V u1ξ )2 + k(u0z + u1ξ )2 − [k(u0t − V u1ξ )(u0z + u1ξ )] 2 ∂t ∂z ∂ −V [ρ(u0t − V u1ξ )(u1t − V u2ξ ) + k(u0z + u1ξ )(u1z + u2ξ )] ∂ξ ∂ − [k(u0t − V u1ξ )(u1z + u2ξ ) + k(u0z + u1ξ )(u1t − V u2ξ )] ∂ξ = V [ρξ (u0t − V u1ξ )(u1t − V u2ξ ) − kξ (u0z + u1ξ )(u1z + u2ξ )] .
An Introduction to the Mathematical Theory of Dynamic Materials by Konstantin A. Lurie