An Introduction to the Mathematical Theory of Dynamic Materials

Lurie, Konstantin A.

doi:10.1007/978-3-319-65346-4

Cited by 40 publications

(64 citation statements)

References 3 publications

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“…The study of space-time materials and, more specifically, of space-time microstructures was initiated in the late nineteen fifties [2][3][4][5][6] and has seen a recent surge of interest [7][8][9][10][11][12][13][14][15][16]: see, for example, the book [17] and the introduction in [1] for a more comprehensive review of the relevant literature, as well as references to ways in which space-time microstructures can be generated. Space-time microstructures represent composites whose material properties vary not just with respect to space, like standard composites, but also with respect to time.…”

Section: Introductionmentioning

confidence: 99%

Field patterns without blow up

Mattei

Milton

2017

New J. Phys.

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Field patterns, first proposed by the authors in Milton and Mattei (2017 Proc. R. Soc. A 473 20160819), are a new type of wave propagating along orderly patterns of characteristic lines which arise in specific space-time microstructures whose geometry in one spatial dimension plus time is somehow commensurate with the slope of the characteristic lines. In particular, in Milton and Mattei (2017 Proc. R. Soc. A 473 20160819) the authors propose two examples of space-time geometries in which field patterns occur: they are two-phase microstructures in which rectangular space-time inclusions of one material are embedded in another material. After a sufficiently long interval of time, field patterns have local periodicity both in time and space.This allows one to focus only on solving the problem on the discrete network on which a field pattern lives and to define a suitable transfer matrix that, given the solution at a certain time, provides the solution after one time period. For the aforementioned microstructures, many of the eigenvalues of this  -symmetric transfer matrix have unit norm and hence the corresponding eigenvectors correspond to propagating modes. However, there are also modes that blow up exponentially with time coupled with modes that decrease exponentially with time. The question arises as to whether there are space-time microstructures such that the transfer matrix only has eigenvalues on the unit circle, so that there are no growing modes (modes that blow-up)? The answer is found here, where we see that certain space-time checkerboards have the property that all the modes are propagating modes, within a certain range of the material parameters. Interestingly, when there is no blow-up, the waves generated by an instantaneous disturbance at a point look like shocks with a wake of oscillatory waves, whose amplitude, very remarkably, does not tend to zero away from the wave front.

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Section: Introductionmentioning

confidence: 99%

Field patterns without blow up

Mattei

Milton

2017

New J. Phys.

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“…This is entirely consistent with the experimental observations in [1] of wave refocusing in the presence of instantaneous time mirrors. Note that the wave field energy is not conserved (and here increases) in such a setting [1] or in general in the presence of time-dependent variations of the underlying medium [17].…”

Section: Introductionmentioning

confidence: 99%

Time-Reversal by Time-Dependent Perturbations

Bal¹,

Fink²,

Pinaud³

2019

SIAM J. Appl. Math.

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We consider the time-reversal of waves in time-dependent media. The constitutive parameters of a wave equation are assumed constant in time during intervals (0, t−τ )∪(t+τ, 2t) for τ t and time-varying during the time interval (t−τ, t+τ ) to model a quasi-instantaneous time mirror. We show that under appropriate hypotheses, a time-reversed signal is generated by such time-dependent fluctuations. At time 2t, the time-reversed signal is an appropriate differentiation of the original initial condition when the time-varying fluctuations are independent of space. In the case of spatially varying fluctuations, we show that the time-reversed signal enjoys stronger refocusing properties when propagation occurs in a highly heterogeneous environment, as in the case of classical time-reversal. This paper offers additional theoretical justifications to the experimental results obtained in [1] and proposes potential extensions. We present numerical simulations that also corroborate and quantify the theoretical predictions. *

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“…There exist two natural ways of changing the material properties in time, thus, leading to two types of space-time microstructure. Using the terminology in the book of Lurie [10] activated materials (the subject matter of this article) are immovable with respect to the laboratory frame and the space-time microstructure is realized by an external mechanism that produces a time switching (either instantaneous or gradual) of the space pattern of the material in a pre-determined manner; in kinetic materials, instead, the space-time microstructure is realized by an actual mechanical motion of the various parts of the composite system with respect to the laboratory frame. In other words, in activated materials the motion (without transfer of matter) is only in terms of the property pattern, whereas in kinetic materials the space-time microstructure is achieved by moving fragments of the material assemblage.…”

Section: Introductionmentioning

confidence: 99%

Field patterns: a new mathematical object

Milton¹,

Mattei²

2017

Proc. R. Soc. A.

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Field patterns occur in space–time microstructures such that a disturbance propagating along a characteristic line does not evolve into a cascade of disturbances, but rather concentrates on a pattern of characteristic lines. This pattern is the field pattern. In one spatial direction plus time, the field patterns occur when the slope of the characteristics is, in a sense, commensurate with the space–time microstructure. Field patterns with different spatial shifts do not generally interact, but rather evolve as if they live in separate dimensions, as many dimensions as the number of field patterns. Alternatively one can view a collection as a multi-component potential, with as many components as the number of field patterns. Presumably, if one added a tiny nonlinear term to the wave equation one would then see interactions between these field patterns in the multi-dimensional space that one can consider them to live, or between the different field components of the multi-component potential if one views them that way. As a result of P T -symmetry many of the complex eigenvalues of an appropriately defined transfer matrix have unit norm and hence the corresponding eigenvectors correspond to propagating modes. There are also modes that blow up exponentially with time.

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An Introduction to the Mathematical Theory of Dynamic Materials

Abstract: The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights.

Cited by 40 publications

References 3 publications

Field patterns without blow up

Field patterns without blow up

Time-Reversal by Time-Dependent Perturbations

Field patterns: a new mathematical object

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