• Corpus ID: 219305655

A unifying perspective on linear continuum equations prevalent in science. Part III: Canonical forms for dynamic equations with moduli that may, or may not, vary with time

@article{Milton2020AUP,
  title={A unifying perspective on linear continuum equations prevalent in science. Part III: Canonical forms for dynamic equations with moduli that may, or may not, vary with time},
  author={Graeme W. Milton},
  journal={arXiv: Mathematical Physics},
  year={2020}
}
  • G. Milton
  • Published 3 June 2020
  • Mathematics
  • arXiv: Mathematical Physics
Enlarging on Parts I and II we write more equations in the desired format of the extended abstract theory of composites. We focus on a multitude of full dynamic equations, including equations where the medium is moving or otherwise changing in time. The motivation is that results and methods in the theory of composites then extend to these equations. 
A unifying perspective on linear continuum equations prevalent in science. Part IV: Canonical forms for equations involving higher order gradients
Enlarging on Parts I, II, and III we write more equations in the desired format of the extended abstract theory of composites. We focus on a multitude of equations involving higher order derivatives.
A unifying perspective on linear continuum equations prevalent in science. Part I: Canonical forms for static, steady, and quasistatic equations
Following some past advances, we reformulate a large class of linear continuum science equations in the format of the extended abstract theory of composites so that we can apply this theory to better
A unifying perspective on linear continuum equations prevalent in physics. Part II: Canonical forms for time-harmonic equations
Following some past advances, we reformulate a large class of linear continuum science equations in the format of the extended abstract theory of composites so that we can apply this theory to better
A unifying perspective on linear continuum equations prevalent in science. Part VI: rapidly converging series expansions for their solution
We obtain rapidly convergent series expansions of resolvents of operators taking the form ${\bf A}=\Gamma_1{\bf B}\Gamma_1$ where $\Gamma_1({\bf k})$ is a projection that acts locally in Fourier
A unifying perspective on linear continuum equations prevalent in physics. Part V: resolvents; bounds on their spectrum; and their Stieltjes integral representations when the operator is not selfadjoint
We consider resolvents of operators taking the form ${\bf A}=\Gamma_1{\bf B}\Gamma_1$ where $\Gamma_1({\bf k})$ is a projection that acts locally in Fourier space and ${\bf B}({\bf x})$ is an
An extremal problem arising in the dynamics of two-phase materials that directly reveals information about the internal geometry
Author(s): Mattei, Ornella; Milton, Graeme W; Putinar, Mihai | Abstract: In two phase materials, each phase having a non-local response in time, it has been found that for appropriate driving fields
A unifying perspective on linear continuum equations prevalent in physics. Part VII: Boundary value and scattering problems
We consider simply connected bodies or regions of finite extent in space or space-time and write conservation laws associated with the equations in Parts I-IV. We review earlier work where, for
Some open problems in the theory of composites
  • G. Milton
  • Engineering
    Philosophical Transactions of the Royal Society A
  • 2021
A selection of open problems in the theory of composites is presented. Particular attention is drawn to the question of whether two-dimensional, two-phase composites with general geometries have the

References

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A unifying perspective on linear continuum equations prevalent in physics. Part II: Canonical forms for time-harmonic equations
Following some past advances, we reformulate a large class of linear continuum science equations in the format of the extended abstract theory of composites so that we can apply this theory to better
A unifying perspective on linear continuum equations prevalent in physics. Part I: Canonical forms for static and quasistatic equations
Following some past advances, we reformulate a large class of linear continuum physical equations in the format of the extended abstract theory of composites so that we can apply this theory to
On cloaking for elasticity and physical equations with a transformation invariant form
In this paper, we investigate how the form of the conventional elastodynamic equations changes under curvilinear transformations. The equations get mapped to a more general form in which the density
The Theory of Composites
Einstein, have contributed to the theory of composite materials. Mathematically, it is the study of partial differential equations with rapid oscillations in their coefficients. Although extensively
An Introduction to the Mathematical Theory of Dynamic Materials
A General Concept of Dynamic Materials.- An Activated Elastic Bar: Effective Properties.- Dynamic Materials in Electrodynamics of Moving Dielectrics.- G-closures of a Set of Isotropic Dielectrics
EFFECTIVE PROPERTIES OF SMART ELASTIC LAMINATES AND THE SCREENING PHENOMENON
Field patterns: a new mathematical object
  • G. Milton, Ornella Mattei
  • Physics
    Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
  • 2017
TLDR
As a result of PT-symmetry many of the complex eigenvalues of an appropriately defined transfer matrix have unit norm and hence the corresponding eigenvectors correspond to propagating modes.
Sharp inequalities that generalize the divergence theorem: an extension of the notion of quasi-convexity
  • G. Milton
  • Mathematics
    Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
  • 2013
Subject to suitable boundary conditions being imposed, sharp inequalities are obtained on integrals over a region Ω of certain special quadratic functions f(E), where E(x) derives from a potential
Dynamic Green’s functions in anisotropic piezoelectric, thermoelastic and poroelastic solids
  • A. Norris
  • Engineering
    Proceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences
  • 1994
A procedure is described to generate fundamental solutions or Green’s functions for time harmonic point forces and sources. The linearity of the field equations permits the Green’s function to be
Field patterns without blow up
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
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