Random point fields with Markovian refinements and the geometry of fractally disordered media

We give a general construction of the probability measure for describing stochastic fractals that model fractally disordered media. For these stochastic fractals, we introduce the notion of a metrically homogeneous fractal Hansdorff-Karathéodory measure of a nonrandom type. We select a classF[q] of random point fields with Markovian refinements for which we explicitly construct the probability distribution. We prove that under rather weak conditions, the fractal dimension D for random fields of this class is a self-averaging quantity and a fractal measure of a nonrandom type (the Hausdorff D-measure) can be defined on these fractals with probability 1. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 124, No. 3, pp. 490–505, September, 2000.     

Towards thermomechanics of fractal media

Hans Ziegler’s thermomechanics [1,2,3], established half a century ago, is extended to fractal media on the basis of a recently introduced continuum mechanics due to Tarasov [14,15]. Employing the concept of internal (kinematic) variables and internal stresses, as well as the quasiconservative and dissipative stresses, a field form of the second law of thermodynamics is derived. In contradistinction to the conventional Clausius–Duhem inequality, it involves generalized rates of strain and internal variables. Upon introducing a dissipation function and postulating the thermodynamic orthogonality on any lengthscale, constitutive laws of elastic-dissipative fractal media naturally involving generalized derivatives of strain and stress can then be derived. This is illustrated on a model viscoelastic material. Also generalized to fractal bodies is the Hill condition necessary for homogenization of their constitutive responses.     

Towards thermomechanics of fractal media

Hans Ziegler’s thermomechanics [1,2,3], established half a century ago, is extended to fractal media on the basis of a recently introduced continuum mechanics due to Tarasov [14,15]. Employing the concept of internal (kinematic) variables and internal stresses, as well as the quasiconservative and dissipative stresses, a field form of the second law of thermodynamics is derived. In contradistinction to the conventional Clausius–Duhem inequality, it involves generalized rates of strain and internal variables. Upon introducing a dissipation function and postulating the thermodynamic orthogonality on any lengthscale, constitutive laws of elastic-dissipative fractal media naturally involving generalized derivatives of strain and stress can then be derived. This is illustrated on a model viscoelastic material. Also generalized to fractal bodies is the Hill condition necessary for homogenization of their constitutive responses.     

Fractal measures and polynomial sampling: I.F.S.–Gaussian integration

Measures generated by Iterated Function Systems can be used in place of atomic measures in Gaussian integration. A stable algorithm for the numerical solution of the related approximation problem – an inverse problem in fractal construction – is proposed. Dedicated to Walter Gautschi.     

Patch size effects at anisotropic reinforcement corners

The structural situation of corners of a laminate reinforcement patch can cause singularities in the mechanical in-plane fields due to the geometry and different material properties in the reinforced and nonreinforced domains, respectively. In the present study, an asymptotic analysis of the cross-sectional force fields near the corners of a laminate reinforcement patch is performed. Using the complex potential method based on Lekhnitskii’s approach, the mechanical in-plane fields at a two-dimensionally modeled interface corner can be determined in a closed-form manner. Various configurations of interface corners are examined, and their effect on the singular characteristics of the cross-sectional force field is studied. In particular, the size effect of the reinforcement patch is considered. Russian translation published in Mekhanika Kompozitnykh Materialov, Vol. 43, No. 3, pp. 319–338, May–June, 2007.     

Fractal approach to the intephase in filled polymers

A fractal model of a filled polymer is proposed. The model allows us to estimate quantitatively the volume of an interphase, to take into account geometrical factors influencing the character and the interphase adhesion level, and to predict the mechanical strength of interfacial bonds. This makes it possible to calculate the size of the critical structural defect and the mechanical properties of filled polymers, which are in agreement with the experimental data. Translated from Mekhanika Kompozitnykh Materialov, Vol. 36, No. 1, pp. 3–32, January–February, 2000.     

Thermoelastoplastic deformation of complexly reinforced shells

The problem on the elastoplastic deformation of reinforced shells of variable thickness under thermal and force loadings is formulated. A qualitative analysis of the problem is carried out and its linearization is indicated. Calculations of isotropic and metal composite cylindrical shells have shown that the load-carrying capacity of shell structures under elastoplastic deformations is several times (sometimes by an order of magnitude) higher than under purely elastic ones; the heating of shells with certain patterns of reinforcement sharply reduces their resistance to elastic deformations, but only slightly affects their resistance to elastoplastic ones; not always does the reinforcement in the directions of principal stresses and strains provide the greatest load-carrying capacity of a shell; there are reinforcement schemes that ensure practically the same resistance of shells at different types of their fastening. __________ Translated from Mekhanika Kompozitnykh Materialov, Vol. 42, No. 6, pp. 707–728, November–December, 2006.     

Heat conduction of shells reinforced by fibers with constant and variable cross sections

We propose a model for heat conduction of a spatially reinforced medium and present its generalization to the case of a polyreinforced layer. We consider the heat-conduction equations for fibrous shells and construct a procedure for reduction of a three-dimensional problem of heat conduction to a two-dimensional one. Analytic solutions of a stationary problem of heat conduction are found for thin conic shells of revolution for various structures of reinforcement, and a graphical comparison of the corresponding results is performed. We study one of the approaches to rational reinforcement of thin shells, according to which the thermal “transparency” of a shell in the transverse direction is taken as a criterion of rational design. Institute of Mathematics, Ukrainian Academy of Sciences, Kiev. Translated from Matematychni Metody ta Fizyko-Mekhanichni Polya, Vol. 41, No. 2, pp. 132–150, April–June, 1998.     

Stability analysis of CFRP-wrapped concrete columns strengthened with external longitudinal CFRP sheets

The external confinement by CFRP wrappings is a very efficient method to increase the load-carrying capacity of round concrete columns. Nevertheless, the serviceability of such columns under loads exceeding the strength of unconfined concrete is limited by different factors. One of them is the reduced stability of the columns due to the significantly reduced tangent elastic modulus inactive loading. To increase the critical load of buckling instability of concrete columns, an additional longitudinal composite reinforcement can be used. In this paper, the stability and strength of concrete columns confined by circumferential wrappings and strengthened with a longitudinal external CFRP reinforcement are studied. Plain and confined columns of length 300 and 1500 mm were tested. Theoretical predictions show that the additional longitudinal reinforcement is efficient in improving the stability of confined columns in the region of moderate slenderness. The prediction for the ultimate strength and stability of the columns coincides rather well with experimental results. __________ Translated from Mekhanika Kompozitnykh Materialov, Vol. 44, No. 3, pp. 295–308, May–June, 2005.     

Nonstandard analysis,parastatistics, and fractals

To estimate divergent integrals, it is convenient, on one hand, to use ideas of nonstandard analysis and, on the other hand, to approximate the integral with a special lattice model that can be interpreted as space quantization. We apply these methods in the case of noninteger (fractal) and negative (hole) dimensions and present some refined formulas, in particular, for the spectrum of flicker noise. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 153, No. 2, pp. 262–270, November, 2007.     

Truncated fractal basin boundaries in the pendulum with nonperiodic forcing

Summary It is well known that oscillators such as the pendulum can have fractal basin boundaries when they are periodically forced with the consequence that the long term behavior of the system may be unpredictable. In engineering and physical applications, the forcing is often nonperiodic and eventually decays to zero, and simulation of the pendulum with decaying forcing (M. Varghese, J. S. Thorp, Physical Review Letters, vol. 60, no. 8, pp. 665–668, Feb. 1988) exhibits truncated fractal basin boundaries which also limit the system predictability. We develop a coordinate change for the pendulum with decaying forcing that allows us to apply standard qualitative methods to study the basin boundaries. We prove that the basin boundaries cannot be fractal and show by example how the extreme stretching and folding leading to a truncated fractal basin boundary may arise.     

Scaling of mathematical fractals and box-counting quasi-measure

The same term, ‘fractals’ incorporates two rather different meanings and it is convenient to split the term into physical or empirical fractals and mathematical ones. The former term is used when one considers real world or numerically simulated objects exhibiting a particular kind of scaling that is the so-called fractal behaviour, in a bounded range of scales between upper and lower cutoffs. The latter term means sets having non-integer fractal dimensions. Mathematical fractals are often used as models for physical fractal objects. Scaling of mathematical fractals is considered using the Barenblatt–Borodich approach that refers physical quantities to a unit of the fractal measure of the set. To give a rigorous treatment of the fractal measure notion and to develop the approach, the concepts of upper and lower box-counting quasi-measures are presented. Scaling properties of the quasi-measures are studied. As examples of possible applications of the approach, scaling properties of the problems of fractal cracking and adsorption of various substances to fractal rough surfaces are discussed.     

Simplicity and superrigidity of twin building lattices

Kac–Moody groups over finite fields are finitely generated groups. Most of them can naturally be viewed as irreducible lattices in products of two closed automorphism groups of non-positively curved twinned buildings: those are the most important (but not the only) examples of twin building lattices. We prove that these lattices are simple if the corresponding buildings are irreducible and not of affine type (i.e. they are not Bruhat–Tits buildings). Many of them are finitely presented and enjoy property (T). Our arguments explain geometrically why simplicity fails to hold only for affine Kac–Moody groups. Moreover we prove that a nontrivial continuous homomorphism from a completed Kac–Moody group is always proper. We also show that Kac–Moody lattices fulfill conditions implying strong superrigidity properties for isometric actions on non-positively curved metric spaces. Most results apply to the general class of twin building lattices. Dedicated to Jacques Tits with our admiration     

Finite-element modeling of the interaction of reinforcement with concrete matrix

A finite-element model of a reinforced concrete beam with rebars modeled by a 3-D deformable body has been developed. An analysis of the stress-strain state of the beam allowed us to determine the stress distribution on cross sections of the rebars and the location of zones with cracks in concrete. It is found that the break of bond between the reinforcement and concrete goes outside the areas of intensely cracked concrete matrix. __________ Translated from Mekhanika Kompozitnykh Materialov, Vol. 44, No. 3, pp. 309–316, May–June, 2008.     

Exceptional Modules for Tubular Canonical Algebras

Let Λ be a tubular canonical algebra of quiver type over a field. We show that each exceptional Λ-module can be exhibited by matrices involving as coefficients 0, 1 and –1 if Λ is of type (3,3,3), (2,4,4) or (2,3,6) and by matrices involving as coefficients 0, 1, –1, λ, –λ and λ–1 if Λ is of type (2,2,2,2) and defined by a parameter λ. Presented by Claus M. Ringel.     

Structure of nonlinear elastic relationships for the highly anisotropic layer of a nonthin shell

Laminated nonthin shells made of nonlinearly elastic fiber composites are considered. The composite material is assumed to be transversely isotropic in planes perpendicular to reinforcement. The asymptotic method and the condition of material stability are applied to analyze the structure of constitutive relations. To introduce a small parameter, the high stiffness in the reinforcement direction of the fiber composite is used. This allows us to obtain simplified constitutive relations containing functions with one or two arguments instead of five as in the initial general case. Kazan State Architectural Building Academy, Tatarstan, Russia. Translated from Mekhanika Kompozitnykh Materialov, Vol. 35, No. 5, pp. 615–628, September–October, 1999.     

Clustering in an ideal gas in nanostructures as a Bose-condensation-type phenomenon in an asymptotically probabilistically quantized space

We describe a space quantization model that allows treating the phenomenon of particle clustering in an ideal gas in nanostructures with a fractal dimension. Dedicated to Vasilii Sergeevich Vladimirov __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 157, No. 3, pp. 468–469, December, 2008.     

Equal-stressed reinforcement of metal-composite plates with fibers of constant cross section in steady-state creep

The problem on equal-stressed reinforcement of metal-composite plates with fibers of constant cross section, loaded in their plane and operating under the conditions of steady-state creep, is formulated. A qualitative analysis of the corresponding system of resolving equations and boundary conditions is performed. An important case of the problem is considered, where both the reinforcing fibers and the matrix are equally stressed. A method for numerically solving this problem is developed. Particular analytical and approximate solutions are discussed, with the example of which changes in the reinforcement structure in relation to the stress level in the reinforcing fibers are clarified. It is shown that nonunique solutions to such problems can exist. __________ Translated from Mekhanika Kompozitnykh Materialov, Vol. 44, No. 1, pp. 11–34, January–February, 2008.     

A population dynamics model with strong maternal care

A two-sex age-structured nondispersing population dynamics deterministic model is presented taking into account strong maternal and weak paternal care of offspring. The model includes a weighted harmonic-mean type pair formation function and neglects the spatial dispersal and separation of pairs. It is assumed that each sex has pre-reproductive and reproductive age intervals. All adult individuals are divided into single males, single females, permanent pairs, and female-widows taking care of their offsprings after the death of their partners. All pairs are of two types: pairs without offspring under parental care at the given time and pairs taking child care. All individuals of pre-reproductive age are divided into young and juvenile groups. The young offspring are assumed to be under parental or maternal (after the death of their father) care. Juveniles can live without parental or maternal care but they cannot reproduce offsprings. It is assumed that births can only occur from couples. The model consists of nine integro-PDEs subject to the conditions of integral type. A class of separable solutions is studied, and a system for macro-moments evolving in time is derived in the case of age-independent vital ones. __________ Translated from Lietuvos Matematikos Rinkinys, Vol. 46, No. 2, pp. 215–255, April–June, 2006.     

New Approximations and Tests of Linear Fluctuation-Response for Chaotic Nonlinear Forced-Dissipative Dynamical Systems

We develop and test two novel computational approaches for predicting the mean linear response of a chaotic dynamical system to small change in external forcing via the fluctuation–dissipation theorem. Unlike the earlier work in developing fluctuation–dissipation theorem-type computational strategies for chaotic nonlinear systems with forcing and dissipation, the new methods are based on the theory of Sinai–Ruelle–Bowen probability measures, which commonly describe the equilibrium state of such dynamical systems. The new methods take into account the fact that the dynamics of chaotic nonlinear forced-dissipative systems often reside on chaotic fractal attractors, where the classical quasi-Gaussian formula of the fluctuation–dissipation theorem often fails to produce satisfactory response prediction, especially in dynamical regimes with weak and moderate degrees of chaos. A simple new low-dimensional chaotic nonlinear forced-dissipative model is used to study the response of both linear and nonlinear functions to small external forcing in a range of dynamical regimes with an adjustable degree of chaos. We demonstrate that the two new methods are remarkably superior to the classical fluctuation–dissipation formula with quasi-Gaussian approximation in weakly and moderately chaotic dynamical regimes, for both linear and nonlinear response functions. One straightforward algorithm gives excellent results for short-time response while the other algorithm, based on systematic rational approximation, improves the intermediate and long time response predictions.