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

Sanguinet, William

doi:10.1002/zamm.201000208

Cited by 3 publications

(3 citation statements)

References 8 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…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.

View full text Add to dashboard Cite

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.

show abstract

Section: Introductionmentioning

confidence: 99%

Field patterns without blow up

Mattei

Milton

2017

New J. Phys.

View full text Add to dashboard Cite

show abstract

“…The extension of the theory to space-time microstructures with properties varying in a two-dimensional space and in time reveals new aspects totally absent in the one-dimensional case. For instance, in the work of Sanguinet [31], the homogenization of an elastic laminate in plane strain (with both inertial and elastic properties varying in time and space) leads to two new additional forces, one of which is of the Coriolis-type. This force, arising from the dynamics and the plane strain hypothesis, vanishes when the model is restricted to the one-dimensional case.…”

Section: Introductionmentioning

confidence: 99%

Field patterns: a new mathematical object

Milton¹,

Mattei²

2017

Proc. R. Soc. A.

View full text Add to dashboard Cite

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.

show abstract

“…These ranges do not include the purely uniform material. In elastodynamics, the concept of dynamic materials has been previously studied in references [2,7].…”

mentioning

confidence: 99%

Propagation of dilatation and shear waves through a dynamic checkerboard material geometry in 1D space + time

Sanguinet

Lurie

2013

Z Angew Math Mech

Self Cite

View full text Add to dashboard Cite

We study the propagation of dilatational and shear waves through an isotropic elastic material having the dilatation and shear moduli variable in space and time. Two isotropic materials alternate occupying rectangular cells in 1D space + time producing a double periodic checkerboard material assembly. The materials are assumed to differ in their wave velocities (dilatational and shear) and to have pairwise equal values of wave impedances for each type of wave. We show, however, that, for both types of waves traveling normally to spatial interfaces between the materials, the average velocity of propagation is the same for certain ranges of material and structural parameters. Also, the energy is accumulated in those waves, and such accumulation occurs in very narrow pulses. This is unlike the wave propagation in a uniform static material, where both types of waves propagate at different speeds. The coincidence of average speeds of propagation appears to be due to the checkerboard material geometry. It creates the "plateau effect" within the ranges of material and structural parameters mentioned above [3,4,6]. These ranges do not include the purely uniform material. In elastodynamics, the concept of dynamic materials has been previously studied in references [2,7].

show abstract

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

Cited by 3 publications

References 8 publications

Field patterns without blow up

Field patterns without blow up

Field patterns: a new mathematical object

Propagation of dilatation and shear waves through a dynamic checkerboard material geometry in 1D space + time

Contact Info

Product

Resources

About