Mathematical analysis of the waves propagation through a rectangular material structure in space–time

Abstract

We consider propagation of waves through a spatio-temporal doubly periodic material structure with rectangular microgeometry in one spatial dimension and time. Both spatial and temporal periods in this dynamic material are assumed to be the same order of magnitude. Mathematically the problem is governed by a standard wave equationWe consider a checkerboard microgeometry where variables cannot be separated. The rectangles in a space-time checkerboard are assumed filled with materials differing in the values of phase velocities k ρ but having equal wave impedance kρ. The difference between dynamic materials and classical static composites is that in the former case the design variables will also be time dependent. Within certain parameter ranges, the formation of distinct and stable limiting characteristic paths, i.e., limit cycles, was observed in [K.A. Lurie, S.L. Weekes, Wave propagation and energy exchange in a spatio-temporal material composite with rectangular microstructure, J. Math. Anal. Appl. 314 (2006) 286-310]; such paths attract neighboring characteristics after a few time periods. The average speed of propagation along the limit cycles remains the same throughout certain ranges of structural parameters, and this was called in [K.A. Lurie, S.L. Weekes, Wave propagation and energy exchange in a spatiotemporal material composite with rectangular microstructure, J. Math. Anal. Appl. 314 (2006) 286-310] a plateau effect. Based on numerical evidence, it was conjectured in [K.A. Lurie, S.L. Weekes, Wave propagation and energy exchange in a spatio-temporal material composite with rectangular microstructure, J. Math. Anal. Appl. 314 (2006) 286-310] that a checkerboard structure is on a plateau if and only if it yields stable limit cycles and that there may be energy concentrations over certain time intervals depending on material parameters. In the present work we give a more detailed analytic characterization of these phenomena and provide a set of sufficient conditions for the energy concentration that was predicted numerically in [K.A. Lurie, S.L. Weekes, Wave propagation and energy exchange in a spatio-temporal material composite with rectangular microstructure, J. Math. Anal. Appl. 314 (2006) 286-310].

The homogenized equations of motion for an activated elastic lamination in plane strain

“…Waves propagating through such structures can experience unique properties such as amplification, de-amplification, screening, etc. [9][10][11]. In fact, the main result of this paper shows the appearance of a Coriolis type 1 force arising from the homogenization of a dynamic elastic laminate.…”

“…Here we used the continuity of the symbols in the square brackets in (7)- (11). We desire to obtain the homogenized equations of motion, so the problem we now face is to determine the expressions for average stress and momentum in the above equation.…”

On Material Optimization in Continuum Dynamics

“…1 we have one stable (boldface curve) limit cycle per period; any two consecutive stable limit cycles are separated by an unstable limit cycle (dashed curve) that repels the neighboring characteristics. The average velocity of a limit cycle is equal to δ/τ [13,14] regardless of the phase velocities a 1 , a 2 . For one period, the energy of the wave increases by the factor (a 2 /a 1 ) 2 because it is pumped twice into the wave; as a result, the energy of the wave grows unlimitedly in time.…”

“…Homogenization in its standard format continues to work for some of the regular situations, particularly for regular laminates of multiple rank [4]. There are, however, regular formations for which this standard technique fails to work; an example is given by a rectangular material checkerboard in space-time [13,14]. Within certain ranges of structural parameters, this construction may support energy (or mass) accumulation in a traveling dynamic disturbance; as a consequence, homogenization becomes inapplicable unless we introduce a bound for the energy (mass) supplied by an external agent.…”

Elements of study on dynamic materials

“…Lurie, in a series of papers [21,22] and a book [2], has advocated the consideration of a checkerboard of space-time variations of material parameters. But he also provided an interesting result obtained by homogenization for long time (and space) propagation where both material coefficients are fast periodic functions of the characteristic right-running variable.…”