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1

About one problem of finding equal strength contour inside a viscoelastic rectangle

Kapanadze Giorgi, Kipiani Gela, Rajczyk Marlena

Zeszyty Naukowe Politechniki Częstochowskiej. Budownictwo

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2023

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Z. 29 (179)

41--47

EN

The problem of finding an equal-strength contour inside a viscoelastic rectangle according to the Kelvin-Voigt model is considered. It is assumed that constant normal compressive forces with given principal vectors act on the sides of the rectangle (or the values of constant normal displacements are known), and the inner boundary (the desired equal-strength contour) is free from external forces. The methods of the theory of conformal reflections, Cauchy type integral and boundary value problems of analytic functions are used to study the plate bending problems discussed in the paper. Which in turn are based on the task of constructing a conformally mapping function on a doubly linked circular ring bounded by broken line. The latter is reduced to the Riemann-Hilbert problem for a circular ring based on the solution of which it becomes possible to present the mentioned function in a defective form. It is worth noting that when considering mixed problems of plate bending for doubly connected areas bounded by broken line, it is possible to decompose them into two independent problems, each of which is a Riemann-Hilbert problem.

PL

Rozważono problem znalezienia konturu o stałej wytrzymałości wewnątrz lepko sprężystego prostokąta zgodnie z modelem Kelvina-Voigta. Zakłada się, że na boki prostokąta działają stałe normalne siły ściskające o danych wektorach głównych (lub znane są wartości stałych przemieszczeń normalnych), a wewnętrzna granica (żądany kontur o jednakowej wytrzymałości) jest wolna od sił zewnętrznych. Do badania omawianych w artykule problemów zginania płyt wykorzystano metody teorii odbić konforemnych, zagadnienia całki typu Cauchy’ego i wartości brzegowych funkcji analitycznych, które z kolei opierają się na zadaniu skonstruowania funkcji odwzorowującej konformalnie na podwójnie połączonym pierścieniu kołowym ograniczonym linią przerywaną. To ostatnie sprowadza się do problemu Riemanna-Hilberta dla pierścienia kołowego na podstawie rozwiązania, które możliwe jest jako przedstawienie wspomnianej funkcji w postaci wadliwej. Warto zauważyć, że rozpatrując mieszane problemy zginania blachy dla obszarów podwójnie połączonych ograniczonych linią przerywaną, można je rozłożyć na dwa niezależne problemy, z których każdy jest problemem Riemanna-Hilberta.

2

Control system defined by some integral operator

Majewski M.

Opuscula Mathematica

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2017

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Vol. 37, no. 2

313--325

EN

In the paper we consider a nonlinear control system governed by the Volterra integral operator. Using a version of the global implicit function theorem we prove that the control system under consideration is well-posed and robust, i.e. for any admissible control u there exists a uniquely defined trajectory xu which continuously depends on control u and the operator [formula] is continuously differentiable. The novelty of this paper is, among others, the application of the Bielecki norm in the space of solutions which allows us to weaken standard assumptions.

3

Linear and Non-linear Creep models for a multi-layered concrete composite

Balevicius R., Marciukaitis G.

Archives of Civil and Mechanical Engineering

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2013

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Vol. 13, no. 4

472--490

EN

One- and two-dimensional linear and nonlinear creep models for predicting the time-dependent behavior of a concrete composite under compression are proposed. These models use the analytical and iterative solutions of the Volterra integral equation. The analytical approach is based on the age-adjusted effective modulus method, and the nonlinear technique applies an iterative approach to the system of non-linear equations, implying a generalization of the principle of superposition. Both models are validated in this study. It has been recently found that negative values of the aging coefficient can emerge in early age multi-layered composites when the stress redistribution between the layers is governed by the combination of considerably different creep strains and aging of the layers. In the plane-strain state, the two-dimensional creep analysis of multi-layered composites yields the same vertical stress-time history as that in a one-dimensional case if the Poisson ratios of the layers are equal. This is valid even though the average value of the vertical stress used to calculate the Volterra integral term is dependent on the Poisson ratio of the layers. In particular, the evolution of vertical stress with time is dependent only on the vertical strain and compatibility conditions in a direction parallel to the lamination. A fracture mechanics approach is also introduced to predict the gradual degradation of long-term strength for a multi-layered composite under a sustained compressive load. The results show that the stress redistribution near the crack-tip under the final period of a high level of sustained loading may lead to an additional required compressive stress for complete failure of the composite. Long-term failure primarily begins with the less deformable (stiffer) layers because the more-deformable layers can relieve the initial stresses. Thus, the long-term strength of the composite can exceed its instantaneous strength for early age composites or for composites composed of layers that possess considerably different creep and aging properties.