Elements of study on dynamic materials

Rousseau, Martine; Maugin, Gérard A.; Berezovski, Mihhail

doi:10.1007/s00419-010-0461-4

Cited by 12 publications

(6 citation statements)

References 20 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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“…Finally, it may be noted that the results of the paper can be used when constructing dynamic materials [36,37], and, in particular, dynamic metamaterials [38][39][40][41]. In this case a dedicated controllable artificial device [40,[42][43][44][45] can play a role of a spring with negative stiffness.…”

Section: Discussionmentioning

confidence: 89%

Non-stationary localized oscillations of an infinite Bernoulli-Euler beam lying on the Winkler foundation with a point elastic inhomogeneity of time-varying stiffness

Shishkina

Гаврилов

Mochalova

2019

Journal of Sound and Vibration

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We consider non-stationary localized oscillations of an infinite Bernoulli-Euler beam. The beam lies on the Winkler foundation with a point inhomogeneity (a concentrated spring with negative time-varying stiffness). In such a system with constant parameters (the spring stiffness), under certain conditions a trapped mode of oscillation exists and is unique. Therefore, applying a non-stationary external excitation to this system can lead to the emergence of the beam oscillations localized near the inhomogeneity. We provide an analytical description of non-stationary localized oscillations in the system with time-varying properties using the asymptotic procedure based on successive application of two asymptotic methods, namely the method of stationary phase and the method of multiple scales. The obtained analytical results were verified by independent numerical calculations. The applicability of the analytical formulas was demonstrated for various types of an external excitation and laws governing the varying stiffness. In particular, we have shown that in the case when the trapped mode frequency approaches zero, localized low-frequency oscillations with increasing amplitude precede the localized beam buckling. The dependence of the ampli- tude of such oscillations on its frequency is more complicated in comparison with the case of a one degree of freedom system with time-varying stiffness.

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“…It is worth noting that another consequence of the exponentially accumulation of energy in macroscopic systems is the thermodynamic non-equilibrium of such materials which, on the contrary, seem to be thermodynamically open systems: only when analyzed together with the surrounding environment can they be considered as thermodynamically closed systems. We should point out that the crossing of pure temporal interfaces, in correspondence to which the properties of the material change instantaneously, leads to very interesting phenomena, such as the Doppler effect analyzed, for instance, by Rousseau et al [30].…”

Section: Introductionmentioning

confidence: 93%

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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Elements of study on dynamic materials

Cited by 12 publications

References 20 publications

Field patterns without blow up

Field patterns without blow up

Non-stationary localized oscillations of an infinite Bernoulli-Euler beam lying on the Winkler foundation with a point elastic inhomogeneity of time-varying stiffness

Field patterns: a new mathematical object

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