

Seminar on the History of Mathematics
September 5, 2019 18:00, St. Peterburg






The history of the application of Volterra integral equations at Leningrad State University
I. I. Demidova^{} 
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Abstract:
The history of the application of Volterra integral equations to describe the behavior of real polymeric materials in nonisothermal conditions and to solve the problems of thermoviscoelasticity and photothermoviscoelasticity is analyzed.To the 105th anniversary of Yu.N. Rabotnov, L.M. Kachanov, E.M. Polishchuk, and the 90th anniversary of I.I. Bugakov.
The famous Italian mathematician and physicist Vito Volterra (18601940) was a corresponding member of the Physics and Mathematics Department of the St. Petersburg Academy of Sciences (1908), an honorary member of the Academy of Sciences of the USSR (1926). His research were in the field of partial differential equations, elasticity theory, integral and integrodifferential equations of functional analysis. He proposed a special type of integral equations, which was called the Volterra integral equation. The scientist suggested using such equations in demography, in insurance mathematics, and in the study of viscoelastic materials. He proposed a special type of integral equations, which was called the Volterra integral equation. The scientist suggested using such equations in demography, in insurance mathematics, and in the study of viscoelastic materials. Since in the 60s polymeric materials began to be used extensively in engineering, the question arose of the strength and reliability of structures made of such materials. L. M. Kachanov (1914–1993), the permanent head of the department of the theory of elasticity of matte fur at Leningrad State University from 1956 to 1977, made a significant contribution to the theory of plasticity, creep, and fracture mechanics. L.M. Kachanov drew attention to the possibility of describing the mechanical properties and the functioning of structures based on the Volterra equations. He proposed a thesis topic to the graduate student I.I. Bugakov (19291989). To use the Volterra equations, it was first necessary to check the possibility of describing the mechanical properties of the material under different thermalforce loads and obtain the necessary functions corresponding to the core of the integral. Since at the department of the theory of elasticity, I.I. Bugakov mastered the polarizationoptical method for studying stresses, not only mechanical, but also optical studies of the properties of polymeric materials began. Solving problems simultaneously using numerical and experimental methods allows us to estimate the correctness of applying the required equations when solving problems of thermoviscoelasticity. After conducting creep experiments at room temperature, I.I. Bugakov showed that the results of studies on celluloid are well described not only at the loading stage, but also during unloading and cyclic loading. Further experiments were carried out on creep on epoxy resins under different loads and temperatures. Generalized creep curves were constructed, both for the mechanical properties of the material, and for optical. Since it was impossible to describe generalized curves with simple analytic functions [4], after consulting with Professor S.G. Mikhlin (19081990) and Senior researcher V.Ya. Rivkind (1940–1996), it was decided to numerically solve the tasks. The calculations were carried out on the computers M222 on the basis of the computer center of the Faculty of Mathematics and Mechanics of Leningrad State University. The computation programs were compiled and debugged by the graduate student GF. Lobanova and engineer A.A. Utkin. Temperature problems of viscoelasticity under the action of homogeneous and inhomogeneous stationary and nonstationary temperature fields, as well as problems for composite bodies, were solved. The use of Volterra integral equations made it possible to establish relations that greatly simplified the solution of the problems of thermoviscoel and photothermoviscoelasticity. The research results were also used to assess the functioning of bioconstructions. Note that theoretical and experimental studies were continued for materials with nonlinear properties, taking into account the dependence of reduced time on stresses [3, 6]. Research materials published in monographs [24] and the work of colleagues.
References
1. Polishchuk Å.M. Vito Volterra. Leningrad: Nauka. 1977. 114 p.
2. Bugakov I.I. Polzuchest' polimernyh materialov. (Creep of polymeric materials). Moscow: Nauka 1973. 288 p.
3. Bugakov I.I. Fotopolzuchest'. (Photocreep). Moscow: Nauka.1991. 165 p.
4. Bugakov I.I., Demidova I.I. Metod fototermovyazkouprugosti. (Photothermoviscoelasticity method). SPb., 1993. 166 p.
5. Åkel'chik V.S., Demidova I.I. Ob opisanii reologii polimerov s pomoshch'yu summ drobnoeksponencial'nyh funkcij (On the description of the rheology of polymers using sum of fractional exponential functions ) // Issledovaniya po uprugosti i plastichnosti. LGU, 1978. ¹12. p. 2530.
6. Fedorovskij G.D. Deformirovanie reologicheski slozhnyh polimernyh sred (Deformation of rheologically complex polymer media): avtoreferat diss. kandidata fizikomatematicheskih nauk. SanktPeterburg, 1998. 15 p.

