By Konstantin A. Lurie
This ebook supplies a mathematical therapy of a unique proposal in fabric technology that characterizes the houses of dynamic materials—that is, fabric components whose houses are variable in area and time. not like traditional composites which are usually present in nature, dynamic fabrics are usually the goods of contemporary expertise built to take care of the best regulate over dynamic approaches. those fabrics have various functions: tunable left-handed dielectrics, optical pumping with high-energy pulse compression, and electromagnetic stealth expertise, to call a number of. Of designated importance is the participation of dynamic fabrics in virtually each optimum fabric layout in dynamics.
The e-book 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 unique emphasis at the theoretical research of spatio-temporal fabric composites (such as laminates and checkerboard structures). a few strange functions are indexed in addition to the dialogue of a few usual 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 by way of hyperbolic differential equations. it's going to even be valuable for researchers within the box of shrewdpermanent metamaterials and their functions to optimum fabric layout in dynamics.
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Extra resources for An Introduction to the Mathematical Theory of Dynamic Materials
37). 35) over the period 1 in ξ and by using the 1-periodicity of u1 and u2 , we arrive at the relation: ( ρ u0t )t − V Vρ V −k 2ρ − ( k u0z )z − B A ρ u0t + 2 k− u0z C V ρ−k C t B A k Vk ρ− u0t + 2 k− V 2ρ − k C V ρ−k C ρ− = 0. 2 as m1 (·)1 + m2 (·)2 . 11). Now it is easy to see that S(0) = T (0) = S(1) = T (1) = 0 and, as a consequence, X(0) = X(1) = Y (0) = Y (1) = 0. 9) may be viewed as a linear system with coeﬃcients that are periodic in ζ with period δ. 3 will follow from this solution in a low frequency asymptotic limit.
K2 /ρ2 > k1 /ρ1 . 52) 34 2 An Activated Elastic Bar: Eﬀective Properties a21 ≤ a21 k¯ ≤ a22 , ρ¯ ¯ 1 ρ ≤ ≤ a22 . 53) Also, 1 ≤ a22 , ¯ 1 k ρ¯ ¯ 1 ρ a21 ≤ k¯ . 56) both of the latter possibilities will be termed irregular. 54) will be complemented by the following: a21 ≤ 1 ρ¯ ¯ 1 k , k¯ ¯ 1 ρ ≤ a22 . 58) ¯ 1 ρ¯ k and the range for which a22 ≤ k¯ ¯ 1 ρ . 61) here we applied notation ∆(·) = (·)2 − (·)1 . It is clear that the diﬀerence ¯ 1/¯ ρ k1 − a21 is positive in the regular case when ∆k > 0, ∆ρ < 0; however, in irregular case, when the signs of ∆k and ∆ρ are the same, this diﬀerence may become negative.
59) will hold once m2 falls into the range (1/72, 71/72). 59) to hold, it is necessary that ∆k and ∆ρ should both be non-zero. 59). 25) for K and P . 64) V 2 > a22 . 15) shows that θ ≥ 0 for both ranges once k, ρ are of the same sign for all participating materials. 65). 59)), this interval may come to existence in the irregular case. 36 2 An Activated Elastic Bar: Eﬀective Properties K V2=k(ρ–1) V2=∞ k k–1 –1 V2=0 V =1/ρ(k 2 k–1 2 V =k/ρ –1 ρ (ρ )–k(k ) –1 –1 V2=α12 ρ(ρ–1)–1 V2=(ρ–1)/(k–1) –1–1 0 2 ) –1 ρ 2 V =k/ρ P ρ 2 V =α2 V2=k(ρ–1) ¯ 1 ρ Fig.
An Introduction to the Mathematical Theory of Dynamic Materials by Konstantin A. Lurie