Discrete and Continuous Dynamical Systems - B
March 2012 , Volume 17 , Issue 2
on analytical and geometrical problems in continuum mechanics
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Human efforts of describing the mechanisms of physical world are a continuous source of mathematical problems. Models are, in fact, representations of classes of phenomena inspired and corroborated by observations intended here as interpretative cataloguing of events. Mathematics is a language having both qualitative and quantitative nature -- the only one with both features -- so it is natural to use it in constructing representations with predictive characteristics of the events in the phenomenological world. In any other human language, in fact, when we refer to quantification, we use concepts coming from mathematics -- the standard use of numbers in counting, for example, the words in this paragraph is a quantitative feature based on a mathematical concept (the one of numbers indeed) that is added to any type of evaluation of the literary/philosophical quality of the phrases themselves. Models appear then as mathematical structures with constituents constrained by the need of having a clear physical meaning. The mathematical questions appearing in models of the physical world have twofold nature: On one side we have technical hitches related, for example, to existence and regularity of solutions to variational minimality requirements and/or to balance equations under boundary and/or initial contitions. On the other side there are foundational problems generated by the quest of the most appropriate form of models. The target is in fact the physical world: appropriateness is then, for a model, the ability to describe the essential structures underlying some classes of phenomena, and their interconnections in non-trivial way, remaining at the same time rather flexible to allow one the possibility of hopefully describing unexpected features that the experimental programs could put in evidence. A model, then, has to be considered not as a manner of justifying the development of a more or less difficult exercise in mathematics, rather it is an occasion of exploring by the tools of a language both qualitative and quantitative, as mathematics is, the intricate and elegant aspects of the physical world. Distinguishing between different possible models of the same class of phenomena is a volatile matter decided by sensibility and culture (the order is not accidental) of the researcher overburden with the judgement.
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In this paper we deal with equations describing fragmentation and coagulation processes with growth or decay, where the latter are modelled by first order transport equations. Our main interest lies in processes with strong fragmentation and thus we carry out the analysis in spaces ensuring that higher moments of the solution exist. We prove that the linear part, consisting of the transport and fragmentation terms, generates a strongly continuous semigroup in such spaces and characterize its generator as the closure of the sum (and in some cases the sum itself) of the operators describing the transport and fragmentation, defined on their natural domains. These results allow us to prove the existence of global in time strict solutions to the full nonlinear fragmentation-coagulation-transport equation.
In this note we take the view that compactness in $L^p$ can be seen quantitatively on a scale of fractional Sobolev type spaces. To accommodate this viewpoint one must work on a scale of spaces, where the degree of differentiability is measured, not by a power function, but by an arbitrary function that decays to zero with its argument. In this context we provide new $L^p$ compactness criteria that were motivated by recent regularity results for minimizers of quasiconvex integrals. We also show how rigidity results for approximate solutions to certain differential inclusions follow from the Riesz--Kolmogorov compactness criteria.
We investigate the asymptotics of obstacle problems for non-local energies in a vector-valued setting. Motivations arise, in particular, in phase field models for ferroelectric materials and variational theories for dislocations.
Many physical phenomena proceed in or on irregular objects which are often modeled by fractal sets. Using the model case of the Sierpinski gasket, the notions of Hausdorff, spectral and walk dimension are introduced in a survey style. These characteristic numbers of the fractal are essential for the Einstein relation, expressing the interaction of geometric, analytic and stochastic aspects of a set.
We discuss the energetic formulation of the Gurtin and Anand model (J. Mech. Phys. Solids, 2005) in strain gradient plasticity, and illustrate the related mathematical analysis concerning the existence of quasi-static evolutions.
We study variational problems modeling the adhesion interaction with a rigid substrate for elastic strings and rods. We produce conditions characterizing bonded and detached states as well as optimality properties with respect to loading and geometry. We show Euler equations for minimizers of the total energy outside self-contact and secondary contact points with the substrate.
A fiber bundle $Y$ having as a prototype fiber a finite-dimensional, differentiable manifold, not embedded in any linear space, and as a base a space-time tube constructed on the $n-$dimensional point space is the geometric environment considered here. After assigning a Lagrangian to its first jet bundle, the notion of defect and the representation of its evolution in this setting is discussed first. Then the geometry is restricted to a base involving fit regions of a three-dimensional Euclidean space, and the balances of actions on defects in the abstract fiber which can be pictured on the basis of $Y$ by a smooth space-type line $l$ are derived in a non-purely conservative setting (a d'Alembert-Lagrange principle is involved). The main result is related to the case involving non-constant peculiar energy along $l$: the covariance of the action balances along $l$, including the configurational ones, is related to the validity of an integral Nöther-like relation which has the meaning of action power relative to the defect motion.
Variational problems for the liquid crystal energy of mappings from three-dimensional domains into the real projective plane are studied. More generally, we study the dipole problem, the relaxed energy, and density properties concerning the conformal $p$-energy of mappings from $n$-dimensional domains that are constrained to take values into the $p$-dimensional real projective space, for any positive integer $p$. Furthermore, a notion of optimally connecting measure for the singular set of such class of maps is given.
A framework for modeling acceleration waves propagation in complex materials is presented. Coupled propagation of elasto-acoustic, microstructural and thermal waves is investigated in the full three dimensional case. The presence of microstructure inside each material element is taken into account without introducing additional hypotheses on the physical nature of the microstructure itself, thus obtaining a general theory that is suitable for the whole class of complex bodies. In particular, jump conditions across the discontinuity interface that identifies the acceleration wave are obtained and the amplitude evolution equation is derived.
In this paper we consider a one-dimensional chain of atoms which interact with their nearest and next-to-nearest neighbours via a Lennard-Jones type potential. We prescribe the positions in the deformed configuration of the first two and the last two atoms of the chain.
We are interested in a good approximation of the discrete energy of this system for a large number of atoms, i.e., in the continuum limit.
We show that the canonical expansion by $\Gamma$-convergence does not provide an accurate approximation of the discrete energy if the boundary conditions for the deformation are close to the threshold between elastic and fracture regimes. This suggests that a uniformly $\Gamma$-equivalent approximation of the energy should be made, as introduced by Braides and Truskinovsky, to overcome the drawback of the lack of accuracy of the standard $\Gamma$-expansion.
In this spirit we provide a uniformly $\Gamma$-equivalent approximation of the discrete energy at first order, which arises as the $\Gamma$-limit of a suitably scaled functional.
In the context of stress theory of the mechanics of continuous media, a generalization of the boundary operator for de Rham currents---the co-divergence operator---is introduced. While the boundary operator of de Rham's theory applies to real valued currents, the co-divergence operator acts on vector valued currents, i.e., functionals dual to differential forms valued in a vector bundle. From the point of view of continuum mechanics, the framework presented here allows for the formulation of the principal notions of continuum mechanics on a manifold that does not have a Riemannian metric or a connection while at the same time allowing irregular bodies and velocity fields.
The main leitmotivs of this paper are the essential nonlinearities, symmetries and the mutual relationships between them. By essential nonlinearities we mean ones which are not interpretable as some extra perturbations imposed on a linear background deciding about the most important qualitative features of discussed phenomena. We also investigate some discrete and continuous systems, roughly speaking with large symmetry groups. And some remarks about the link between two concepts are reviewed. Namely, we advocate the thesis that the most important non-perturbative nonlinearities are those implied by the assumed "large" symmetry groups. It is clear that such a relationship does exist, although there is no complete theory. We compare the mechanism of inducing nonlinearity by symmetry groups of discrete and continuous systems. Many striking and instructive analogies are found, e.g., analogy between analytical mechanics of systems of affine bodies and general relativity, tetrad models of gravitation, and Born-Infeld nonlinearity. Some interesting, a bit surprising problems concerning Noether theorem are discussed, in particular in the context of large symmetry groups.
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