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## Numerical simulation of superplastic bulge forming test

Superplastic blow forming is a technology of shell parts production. The development of these processes requires computer simulation which cannot be realized without accurate parameters of the applied material. This characterization can be based on results of free bulging tests. Characterization techniques utilize the models of the dome growth during the bulging test. This study is devoted to the assessing of friction coefficient effect on the linear behavior of normalized thickness - normalized height relation.

This study proposes a method for determining the material constitutive equations and optimal forming conditions on the basis of free bulging tests. The blow-forming tests were carried out at the temperature of 420 °C using aluminum alloy (AMg-6) sheets of a 1 mm thickness. Each test was performed at constant pressure. For each fixed value of the pressure, a series of experiments was carried out with different forming times to obtain evolutions of dome height H and thickness s. These data were processed by the proposed method to obtain the flow stress dependence on the effective strain rate. The constitutive equations were obtained by least squares minimization of deviations between the experimental variations of H and s and ones predicted by a simplified engineering model. On the basis of the obtained data, the optimum strain rate for AMg-6 processing was determined as one corresponding to the maximum strain rate sensitivity.

Computer simulations are fast growing approach for doing research in sciences. It is auxiliary to experimental and analytical research. The main goal of the conference is in the development of methods and algorithms which take into account trends in the hardware development, and which may help to intensive research. Conference should play role of the venue were senior scientists and students may have opportunity to speak each other and exchange ideas and views on the developments in the area of high-performance computing in most sciences.

This paper presents the research of the flow characteristics of the Ti-6V-4Al alloy in wide ranges of temperature (725 ‑ 950 °C) and strain rate (10-5 ‑ 10-2 s-1). The material processing maps were constructed based on the basis of dynamic materials model (DMM) developed by Prassad and modified by Narayana Murty. For the construction of such maps the data of the material’s flow stress at different temperatures and strain rates is necessary. To obtain such data the stepped tensile tests which allow get the stress - strain rate dependence at a given temperature are ideal. The experiments conducted consist of the tensile-testing of samples’ series at various temperatures with stepped change of the deformation speed. By the results of these tests the constitutive equations, which describe relationship between stress and strain rate for each temperature were obtained. The data was analyzed in terms of the two different approaches proposed by Prassad and Narayana Murty to assess the impact of deformation conditions on the formability and flow stability of the material. Based on these approaches, the processing maps which allow identifying the conditions of the Ti-6V-4Al alloy superplasticity were constructed.

We consider the time-dependent 1D Schrödinger equation on the half-axis with variable coefficients becoming constant for large x. We study a two-level symmetric in time (i.e. the Crank-Nicolson) and any order finite element in space numerical method to solve it. The method is coupled to an approximate transparent boundary condition (TBC). We prove uniform in time stability with respect to initial data and a free term in two norms, under suitable conditions on an operator in the approximate TBC. We also consider the corresponding method on an infinite mesh on the half-axis. We derive explicitly the discrete TBC allowing us to restrict the latter method to a finite mesh. The operator in the discrete TBC is a discrete convolution in time; in turn its kernel is a multiple discrete convolution. The stability conditions are justified for it. The accomplished computations confirm that high order finite elements coupled to the discrete TBC are effective even in the case of highly oscillating solutions and discontinuous potentials.

The study is carried out by the first author within The National Research University Higher School of Economics' Academic Fund Program in 2012-2013, research grant No. 11-01-0051.

Computer simulation of mechanical testing is used primarily for the correct interpretation of their results and is particularly relevant in cases, when the properties of the material during deformation are essentially nonlinear. For example: when we study mechanical properties of materials with high rate sensitivity. First of all it is superplastic titanium alloys. Superplastic materials exhibit the ability to severe plastic deformation without discontinuities if forming occurs in a narrow range of strain rates, specific to each alloy and temperature-dependent. In the study of superplastic materials, it’s necessary to maintain a constant rate of deformation of the sample. This is achieved by conducting an experiment with a special program loading, crosshead speed at which change during the experiment.

This aim of this paper is the interpretation of the results of mechanical testing of materials to determine their properties under hot deformation. As an example, a simulation of rod stretching in superplasticity mode was considered. Comparing obtained data with the analytical solution was conducted.

The volume contains articles of scientific staff and faculty of the Department of Computer Science and Applied Mathematics and Scientific-Educational Center of computer modeling of unique buildings and complexes of Moscow State University of Civil Engineering (National Research University), devoted to actual problems of applied mathematics and computational mechanics.

The purpose of this study is to find out the characteristics of hot forming of Ti-6Al-4V titanium alloy in order to determine the conditions of its superplastic behavior. The experiments were performed in two stages: the stepped tensile-tests series (temperature range 700 – 925 °С) and the constant strain rate tensile-test series (temperature range 775 – 925 °С). By the results of stepped tensile tests the constitutive equations which describe relationship between stress and strain rate for each temperature were constructed. On the base of obtained data, the temperature and strain-rate ranges which ensure the realization of superplasticity at forming of Ti-6Al-4V alloy as well as optimal strain rates which corresponds to the maximum value of strain rate sensitivity exponent were determined. In was shown that at low temperatures (700 – 775C) the Ti-6Al-4V alloy shows all signs of superplasticity, however at these temperatures the optimal strain rates are too slow for industrial technological procedures. The dependence between optimum strain rate and reciprocal temperature appears to be well fitted by exponential low. At the second stage of the experimental research, the tensile-tests with a constant, optimum for each temperature strain-rate were carried in order, to estimate the real initial flow stress and the character of strain hardening of the material during the deformation with optimum strain rate. In was found that flow stress values obtained by stepped tensile tests matches the values form constant-strain-rate tests with effective strain value equal to 0,2 and the strain hardening during the deformation with optimal strain rates is significant.

We consider certain spaces of functions on the circle, which naturally appear in harmonic analysis, and superposition operators on these spaces. We study the following question: which functions have the property that each their superposition with a homeomorphism of the circle belongs to a given space? We also study the multidimensional case.

We consider the spaces of functions on the m-dimensional torus, whose Fourier transform is p -summable. We obtain estimates for the norms of the exponential functions deformed by a C1 -smooth phase. The results generalize to the multidimensional case the one-dimensional results obtained by the author earlier in “Quantitative estimates in the Beurling—Helson theorem”, Sbornik: Mathematics, 201:12 (2010), 1811 – 1836.

We consider the spaces of function on the circle whose Fourier transform is p-summable. We obtain estimates for the norms of exponential functions deformed by a C1 -smooth phase.

Let G be a semisimple algebraic group whose decomposition into the product of simple components does not contain simple groups of type A, and P⊆G be a parabolic subgroup. Extending the results of Popov [7], we enumerate all triples (G, P, n) such that (a) there exists an open G-orbit on the multiple flag variety G/P × G/P × . . . × G/P (n factors), (b) the number of G-orbits on the multiple flag variety is finite.

I give the explicit formula for the (set-theoretical) system of Resultants of m+1 homogeneous polynomials in n+1 variables