By Konstantin A. Lurie
This e-book supplies a mathematical remedy of a singular thought in fabric technology that characterizes the houses of dynamic materials—that is, fabric elements whose homes are variable in house and time. in contrast to traditional composites which are usually present in nature, dynamic fabrics are regularly the goods of contemporary know-how built to keep up the best keep an eye on over dynamic tactics. 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 precise importance is the participation of dynamic fabrics in nearly each optimum fabric layout in dynamics.
The ebook discusses a few basic gains of dynamic fabrics as thermodynamically open platforms; it supplies their sufficient 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 ordinary 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
These observations show that the work of an external force produced over a period is equal to the net increase of the energy of a slow motion. 41). This equation appears to be an Euler equation produced by an eﬀective action density 1 Λ¯ = (ru20t + 2qu0t u0z − pu20z ). 90) 2 As in the beginning of this section, this function generates components of ¯ : an eﬀective energy-momentum tensor W 44 2 An Activated Elastic Bar: Eﬀective Properties ¯ ¯ tt = u0t ∂ Λ − Λ¯ = 1 (ru2 + pu2 ), W 0t 0z ∂u0t 2 ¯ ¯ tz = u0t ∂ Λ = qu2 − pu0t u0z , W 0t ∂u0z ¯ ¯ zt = u0z ∂ Λ = ru0t u0z + qu20z , W ∂u0t ¯ ¯ zz = u0z ∂ Λ − Λ¯ = − 1 (ru2 + pu2 ).
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. For example, if k2 = 10, ρ2 = 9, k1 = ρ1 = 1, then ρ1 ∆k − k1 ∆ρ − m2 ∆k∆ρ = 9 − 8 − 72m2 , and this is ≤ 0 if m2 ≥ 1/72. e. k/ρ increases as we go from material 1 to material 2. Combined with ∆k∆ρ > 0 (irregular case), this means that the increase may be due to that in k and to the less intensive increase (not a decrease) in ρ, or due to the decrease in ρ and the less intensive decrease (not an increase) in k.
Eﬀective parameters K versus P with variable V (case ρ¯ ¯ −k ¯ 1 k ≥ 0). The plots of K versus P with V variable along the curves are given, ¯ ¯ respectively, by Fig. 8 (case ρ¯ ρ1 − k¯ k1 ≥ 0), and Fig. 9 (case ¯ ¯ ρ¯ ρ1 − k¯ k1 ≤ 0). 29): K= k V2− V 2 − k¯ ( ρl¯ ) ( k¯1 ) ¯ 1 ρ , P = 2 ¯ ρ1 ρ2 V − k ρ¯ V2− ¯ 1 ρ ¯ k ρ¯ . 5)); the relevant segments are marked boldface in the figures. e. 50). 29), V 2 − ρ¯ 1¯1 ¯ 1 (k) 2 2 v1 v2 = −¯ a1 a2 ρ . 66) ¯ k 2 ¯ V − k ρ1 Given the observations made earlier in this section, we conclude that v1 and v2 should have opposite signs in a regular case.
An Introduction to the Mathematical Theory of Dynamic Materials by Konstantin A. Lurie