129 research outputs found
Dispersion relations and wave operators in self-similar quasicontinuous linear chains
We construct self-similar functions and linear operators to deduce a self-similar variant of the Laplacian operator and of the D'Alembertian wave operator. The exigence of self-similarity as a symmetry property requires the introduction of nonlocal particle-particle interactions. We derive a self-similar linear wave operator describing the dynamics of a quasicontinuous linear chain of infinite length with a spatially self-similar distribution of nonlocal interparticle springs. The self-similarity of the nonlocal harmonic particle-particle interactions results in a dispersion relation of the form of a Weierstrass-Mandelbrot function that exhibits self-similar and fractal features. We also derive a continuum approximation, which relates the self-similar Laplacian to fractional integrals, and yields in the low-frequency regime a power-law frequency-dependence of the oscillator density
Mass transport in morphogenetic processes: a second gradient theory for volumetric growth and material remodeling
International audienceIn this work, we derive a novel thermo-mechanical theory for growth and remodeling of biological materials in morphogenetic processes. This second gradient hyperelastic theory is the first attempt to describe both volumetric growth and mass transport phenomena in a single-phase continuum model, where both stress- and shape-dependent growth regulations can be investigated. The diffusion of biochemical species (e.g. morphogens, growth factors, migration signals) inside the material is driven by configurational forces, enforced in the balance equations and in the set of constitutive relations. Mass transport is found to depend both on first- and on second-order material connections, possibly withstanding a chemotactic behavior with respect to diffusing molecules. We find that the driving forces of mass diffusion can be written in terms of covariant material derivatives reflecting, in a purely geometrical manner, the presence of a (first-order) torsion and a (second-order) curvature. Thermodynamical arguments show that the Eshelby stress and hyperstress tensors drive the rearrangement of the first- and second-order material inhomogeneities, respectively. In particular, an evolution law is proposed for the first-order transplant, extending a well-known result for inelastic materials. Moreover, we define the first stress-driven evolution law of the second-order transplant in function of the completely material Eshelby hyperstress
On Love-type waves in a finitely deformed magnetoelastic layered half-space
In this paper, the propagation of Love-type waves in a homogeneously and finitely deformed layered half-space of an incompressible non-conducting magnetoelastic material in the presence of an initial uniform magnetic field is analyzed. The equations and boundary conditions governing linearized incremental motions superimposed on an underlying deformation and magnetic field for a magnetoelastic material are summarized and then specialized to a form appropriate for the study of Love-type waves in a layered half-space. The wave propagation problem is then analyzed for different directions of the initial magnetic field for two different magnetoelastic energy functions, which are generalizations of the standard neo-Hookean and Mooney–Rivlin elasticity models. The resulting wave speed characteristics in general depend significantly on the initial magnetic field as well as on the initial finite deformation, and the results are illustrated graphically for different combinations of these parameters. In the absence of a layer, shear horizontal surface waves do not exist in a purely elastic material, but the presence of a magnetic field normal to the sagittal plane makes such waves possible, these being analogous to Bleustein–Gulyaev waves in piezoelectric materials. Such waves are discussed briefly at the end of the paper
Formation of soliton complexes in dispersive systems
The concept of soliton complex in a nonlinear dispersive medium is formulated. It is shown that interacting identical topological solitons in the
medium can form bound soliton complexes which move without radiation.
This phenomenon is considered to be universal and applicable to various physical systems. The soliton complex and its “excited” states are described analytically and numerically as solutions of nonlinear dispersive
equations with the fourth and higher order spatial or mixed derivatives.
The dispersive sine-Gordon, double and triple sine-Gordon, and piecewise
models are studied in detail. Mechanisms and conditions of the formation
of soliton complexes, and peculiarities of their stationary dynamics are investigated. A phenomenological approach to the description of the complexes and the classification of all the possible complex states are proposed. Some examples of physical systems, where the phenomenon can
be experimentally observed, are briefly discussed.Формулюється концепція солітонних комплексів в нелінійному дисперсійному середовищі. Показано, що взаємодіючі тотожні топологічні солітони у такому середовищі здатні утворювати солітонні комплекси, що рухаються без випромінювання. Вважається, що це явище є універсальним і зустрічається у різних фізичних системах. Солітонні комплекси і їхні збуджені стани описуються аналітично і чисельно як розв’язки нелінійних дисперсійних рівнянь з четвертою і
вищими просторовимі і змішаними похідними. Ретельно вивчаються дисперсійні системи синус-Гордон, подвійний і потрійний синус-Гордон, а також кусково-лінійна модель. Досліджуються механізми і
умови формування солітонних комплексів і особливості їхньої стаціонарної динаміки. Запропоновано феноменологічний підхід до описання комплексів і класифікацію усіх можливих їхніх станів. Стисло
розглянуті кілька прикладів фізичних систем, в яких це явище може
експериментально спостерігатись
Entropy Identity and Material-Independent Equilibrium Conditions in Relativistic Thermodynamics
On the basis of the balance equations for energy-momentum, spin, particle and
entropy density, an approach is considered which represents a comparatively
general framework for special- and general-relativistic continuum
thermodynamics. In the first part of the paper, a general entropy density
4-vector, containing particle, energy-momentum, and spin density contributions,
is introduced which makes it possible, firstly, to judge special assumptions
for the entropy density 4-vector made by other authors with respect to their
generality and validity and, secondly, to determine entropy supply and entropy
production. Using this entropy density 4-vector, in the second part,
material-independent equilibrium conditions are discussed. While in literature,
at least if one works in the theory of irreversible thermodynamics assuming a
Riemann space-time structure, generally thermodynamic equilibrium is determined
by introducing a variety of conditions by hand, the present approach proceeds
as follows: For a comparatively wide class of space-time geometries the
necessary equilibrium conditions of vanishing entropy supply and entropy
production are exploited and, afterwards, supplementary conditions are assumed
which are motivated by the requirement that thermodynamic equilibrium
quantities have to be determined uniquely.Comment: Research Paper, 30 page
Optical metrics and birefringence of anisotropic media
The material tensor of linear response in electrodynamics is constructed out
of products of two symmetric second rank tensor fields which in the
approximation of geometrical optics and for uniaxial symmetry reduce to
"optical" metrics, describing the phenomenon of birefringence. This
representation is interpreted in the context of an underlying internal
geometrical structure according to which the symmetric tensor fields are
vectorial elements of an associated two-dimensional space.Comment: 24 pages, accepted for publication in GR
Cartan's spiral staircase in physics and, in particular, in the gauge theory of dislocations
In 1922, Cartan introduced in differential geometry, besides the Riemannian
curvature, the new concept of torsion. He visualized a homogeneous and
isotropic distribution of torsion in three dimensions (3d) by the "helical
staircase", which he constructed by starting from a 3d Euclidean space and by
defining a new connection via helical motions. We describe this geometric
procedure in detail and define the corresponding connection and the torsion.
The interdisciplinary nature of this subject is already evident from Cartan's
discussion, since he argued - but never proved - that the helical staircase
should correspond to a continuum with constant pressure and constant internal
torque. We discuss where in physics the helical staircase is realized: (i) In
the continuum mechanics of Cosserat media, (ii) in (fairly speculative) 3d
theories of gravity, namely a) in 3d Einstein-Cartan gravity - this is Cartan's
case of constant pressure and constant intrinsic torque - and b) in 3d Poincare
gauge theory with the Mielke-Baekler Lagrangian, and, eventually, (iii) in the
gauge field theory of dislocations of Lazar et al., as we prove for the first
time by arranging a suitable distribution of screw dislocations. Our main
emphasis is on the discussion of dislocation field theory.Comment: 31 pages, 8 figure
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