A research on the stress-strain behaviour of structures made of filled shells

Introduction. The study of the stress-strain state of structures, made of filled shells, remains relevant in terms of interaction between principal structural elements (the shell, the filler, the bed), identification of the optimal combination of shell/filler characteristics, and conditions of their contact. The article addresses the findings of the research on a structural model of a thin cylindrical shell with a filler. Materials and methods. The problem of determining an effective ratio of basic dimensions of a structure is solved in the context of the maximally uniform distribution of forces in a shell with regard for the accepted constraints concerning the conditions of loading, fixing, and describing interaction between the model elements. In addition, the ratio of indicators of mechanical properties of the filler material and the shell is taken into account. The efficiency criterion is determined as a result of evaluating the stress state of a structure, at which forces inside the shell are distributed most evenly and the values of the radial forces are close to the forces directed along the generatrix of a shell. Results. The range of the effective ratio of the main dimensions of a shell is identified with regard for the ratio of the values of the indicator of the stress-strain behaviour of the shell and the filler, which is identified for larger groups of the internal filler, distinguished by the value of indicators of deformation characteristics. The co-authors have identified the ranges of ratios of geometric, rigidity and mechanical parameters of the system, that allow the structure to be attributed to the category of filled shells whose analysis can be performed with the help of applicable provisions of computational modeling. Conclusions. The results of the study allow for the selection of structural parameters based on the pre-set conditions and the criterion of uniformity of distribution of forces in a shell. The study also enables to identify the ratios of the above geometric, rigidity and mechanical parameters at which the structure should be attributed to the category of filled shells or “shells that have a filler” under the pre-set design conditions.

Dynamics of hyperbolic correspondences

This paper establishes the geometric rigidity of certain holomorphic correspondences in the family $(w-c)^q=z^p$ , whose post-critical set is finite in any bounded domain of $\mathbb {C}$ . In spite of being rigid on the sphere, such correspondences are J-stable by means of holomorphic motions when viewed as maps of $\mathbb {C}^2$ . The key idea is the association of a conformal iterated function system to the return branches near the critical point, giving a global description of the post-critical set and proving the hyperbolicity of these correspondences.

Sharp estimates of the geometric rigidity on the first Heisenberg group

We prove quantitative stability of isometries on the first Heisenberg group with sub-Riemannian geometry: every (1 + )-quasi-isometry of the John domain of the Heisenberg group is close to some isometry with order of closeness in the uniform norm and with the order of closeness+ in the Sobolev norm. An example demonstrating the asymptotic sharpness of the results is given.

Manhattan curves for hyperbolic surfaces with cusps

In this paper, we study an interesting curve, the so-called Manhattan curve, associated with a pair of boundary-preserving Fuchsian representations of a (non-compact) surface; in particular, representations corresponding to Riemann surfaces with cusps. Using thermodynamic formalism (for countable state Markov shifts), we prove the analyticity of the Manhattan curve. Moreover, we derive several dynamical and geometric rigidity results, which generalize results of Burger [Intersection, the Manhattan curve, and Patterson–Sullivan theory in rank 2. Int. Math. Res. Not.1993(7) (1993), 217–225] and Sharp [The Manhattan curve and the correlation of length spectra on hyperbolic surfaces. Math. Z.228(4) (1998), 745–750] for convex cocompact Fuchsian representations.