Approximation method for the problems of mechanics of inhomogeneous hereditarily elastic bodies

We consider a boundary-value problem of mechanics of inhomogeneous hereditarily elastic bodies formulated as a linear equation with an operator of fractional integration, partial derivatives with respect to time and spatial variables, and polynomial-type coefficients of one of the variables. An approximate solution of this problem is constructed according to Dzyadyk's a-method combined with the use of the Laplace transformation. It is proved that the errors of the approximation of the required function and its derivatives decrease in geometric progression.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 46, No. 9, pp. 1234–1245, September, 1994.     

Isomorphic properties of intersection bodies

We study isomorphic properties of two generalizations of intersection bodies - the class of k-intersection bodies in Rn and the class of generalized k-intersection bodies in Rn. In particular, we show that all convex bodies can be in a certain sense approximated by intersection bodies, namely, if K is any symmetric convex body in Rn and 1≤k≤n−1 then the outer volume ratio distance from K to the class can be estimated by

     

L_p-mixed intersection bodies

In this paper the author first introduce a new concept of Lp-dual mixed volumes of star bodies which extends the classical dual mixed volumes. Moreover, we extend the notions of Lp- intersection body to Lp-mixed intersection body. Inequalities for Lp-dual mixed volumes of Lp-mixed intersection bodies are established and the results established here provide new estimates for these type of inequalities.     

On intersection and mixed intersection bodies

Duals of the basic projection and mixed projection inequalities are established for intersection and mixed intersection bodies.      

On polars of mixed projection bodies

Recently, Lutwak established general Minkowski inequality, Brunn-Minkowski inequality and Aleksandrov-Fenchel inequality for mixed projection bodies. In this paper, following Lutwak, we established their polar forms. As applications, we prove some interrelated results.     

Brunn-Minkowski inequality for mixed intersection bodies

Dual of the Brunn-Minkowski inequality for mixed projection bodies are established for mixed intersection bodies.     

Inequalities for dual quermassintegrals of mixed intersection bodies

In this paper, we first introduce a new concept ofdual quermassintegral sum function of two star bodies and establish Minkowski’s type inequality for dual quermassintegral sum of mixed intersection bodies, which is a general form of the Minkowski inequality for mixed intersection bodies. Then, we give the Aleksandrov-Fenchel inequality and the Brunn-Minkowski inequality for mixed intersection bodies and some related results. Our results present, for intersection bodies, all dual inequalities for Lutwak’s mixed prosection bodies inequalities.     

On dual quermassintegrals of mixed intersection bodies

In this paper, it is shown that a family of inequalities involving mixed intersection bodies holds. The Busemann intersection inequality is the first of this family. All of the members of this family are strengthened versions of classical inequalities between pairs of dual quermassintegrals of a star body.     

On asymmetry of some convex bodies

We determine Minkowski's measure of asymmetry for convex hulls of a point and some sets for which the asymmetry is known. Some properties of the asymmetry measure are found which may indicate some interesting properties of convex bodies. Received March 1, 2001, and in revised form June 29, 2001. Online publication December 17, 2001.     

Quantized mechanics of affinely rigid bodies

In this paper, we develop the main ideas of the quantized version of affinely rigid (homogeneously deformable) motion. We base our consideration on the usual Schrödinger formulation of quantum mechanics in the configuration manifold, which is given, in our case, by the affine group or equivalently by the semi‐direct product of the linear group and the space of translations , where n equals the dimension of the “physical space.” In particular, we discuss the problem of dynamical invariance of the kinetic energy under the action of the whole affine group, not only under the isometry subgroup. Technically, the treatment is based on the 2‐polar decomposition of the matrix of the internal configuration and on the Peter‐Weyl theory of generalized Fourier series on Lie groups. One can hope that our results may be applied in quantum problems of elastic media and microstructured continua.     

Load-displacement relations for elastic bodies

Zusammenfassung Es wird gezeigt, dass bei statischen Eigenwertproblemen der infinitesimalen Elastizitätstheorie die an einem Teil der Oberfläche, an welchem die Verschiebungen vorgeschrieben sind, angreifende Dyname in den vorgeschriebenen Grössen ausgedrückt werden kann. Das Verfahren verlangt die Kenntnis der Oberflächenverschiebungen und Oberflächenspannungen unter gewissen behelfsmässig definierten Gleichgewichtszuständen. Es wird auf einen Zylinder mit vorgeschriebenen Endverschiebungen und auf den Halbraum unter einem reibungsfreien starren Stempel angewandt.     

Resonant frequencies for a system of time-harmonic elastic wave

In this work, we consider the propagation of elastic waves outside a (compact) obstacle with smooth boundary. Existence, uniqueness results for the solution are established in a simple way. We study the meromorphic continuation to the whole complex plane of the solution. A proof alternative of the existence of resonant frequencies is given.     

Supporting and separating subsets for invex bodies

Invex bodies represent an important class of bodies which are considered as a generalization of convex bodies.In this article,the author studies the supporting for this class of bodies as well as the separating subsets of two bodies.     

Junction problem for elastic and rigid inclusions in elastic bodies

An equilibrium problem for an elastic body is considered. It is assumed that the body has a thin elastic inclusion and a thin rigid inclusion. We analyze a junction problem assuming that the inclusions have a joint point. Different equivalent problem formulations are discussed, and existence of solutions is proved. A set of junction conditions is found. We investigate a convergence to infinity and to zero of a rigidity parameter of the elastic inclusion. A delamination of the elastic inclusion is also investigated. In this case, inequality‐type boundary conditions are imposed at the crack faces to prevent a mutual penetration between crack faces. Copyright © 2015 John Wiley & Sons, Ltd.     

An extension of a result by Lonke to intersection bodies

In this paper we prove that intersection bodies cannot be direct sums using Fourier analytic techniques. This extends a result by Lonke. We also prove a necessary regularity condition and a convexity condition for a body of revolution to be an intersection body of a star body.     

Pointwise estimates for marginals of convex bodies

We prove a pointwise version of the multi-dimensional central limit theorem for convex bodies. Namely, let μ be an isotropic, log-concave probability measure on Rn. For a typical subspace E⊂Rn of dimension nc, consider the probability density of the projection of μ onto E. We show that the ratio between this probability density and the standard Gaussian density in E is very close to 1 in large parts of E. Here c>0 is a universal constant. This complements a recent result by the second named author, where the total variation metric between the densities was considered.     

Saint-venant principle for paraboloidal elastic bodies

We study the asymptotic behavior of solutions to the linear problem of elasticity in a domain Ω with paraboloidal exit at infinity. Properties of solutions and the condition of the existence of solutions depend on a parameter γ∈[0,1] characterizing the velocity of extending the paraboloid (a cylinder and a cone correspond to the cases γ=0¤γ=1 respectively). Asymptotic formulas are deduced for displacement fields generating forces and moments “applied at infinity”. The Saint-Venant principle is verified for “oblong” bodies such as paraboloids, cylinders, and narrow cones. The following question turns out to be a key one: What rigid displacements belong to the energy space obtained by completion of $C_0^\infty (\bar \Omega )^3 $ by the energy norm? The dimension d_γ of the lineal R_γ of rigid energy displacements is computed (in this case, d₀=6, d₁=0, and the function γ?d_γ has jumps at the points γ=1/4, 1/2, 3/4). We also clarify the reasons why it is necessary to distinguish the notions “energy solution” and “solution with finite energy”. We also discuss the phenomenon of a boundary layer that appears near the endpoints of spindle-like rods and is described by energy solutions in paraboloids. As is shown, in order to have the well-posed formulation of the boundary conditions in one-dimensional models of such rods, it is necessary to use the weakened Saint-Venant principle, i.e., replace R₀ with R_γ: for γ>1/4. If we apply the strong principle, we arrive at an overdetermined limit one-dimensional problem. Bibliography: 71 titles.     

Some optimization problems for elastic bodies

Two optimization problems for finite deformations of an elastic body are formulated. In the first problem, the motion that gives minimum value (maximum value) to a certain functional is chosen from a single family of controllable motions. The necessary conditions for optimality are stated for one specific problem of this type. In the second problem, the minimizing (maximizing) motion is chosen from the class of all possible motions connecting two configurations of the body. The necessary conditions for the optimality of singular solutions are obtained for one specific example of this type.This work was supported by the Pokrajinska Zajednica za Naucni Rad, Novi Sad, Yugoslavia.