Investigation of the stability problems of elastic bodies using the method of mathematical theory of elasticity

1981 ◽  
Vol 2 (1) ◽  
pp. 51-78 ◽  
Author(s):  
Wang Zhen-ming
Keyword(s):  
Mathematical Theory ◽  
Theory Of Elasticity ◽  
Stability Problems ◽  
Elastic Bodies ◽  
The Stability

Mathematical Theory of Transversally Isotropic Shells of Arbitrary Thickness at Static Load

2019 ◽  
Vol 968 ◽  
pp. 496-510
Author(s):  
Anatoly Grigorievich Zelensky
Keyword(s):  
Mathematical Theory ◽  
Three Dimensional ◽  
Nonlinear Problems ◽  
Theory Of Elasticity ◽  
Elastic Plates ◽  
Dimensional Problem ◽  
Boundary Problems ◽  
Complex Boundary ◽  
Plates And Shells ◽  
Basic Equations

Classical and non-classical refined theories of plates and shells, based on various hypotheses [1-7], for a wide class of boundary problems, can not describe with sufficient accuracy the SSS of plates and shells. These are boundary problems in which the plates and shells undergo local and burst loads, have openings, sharp changes in mechanical and geometric parameters (MGP). The problem also applies to such elements of constructions that have a considerable thickness or large gradient of SSS variations. The above theories in such cases yield results that can differ significantly from those obtained in a three-dimensional formulation. According to the logic in such theories, the accuracy of solving boundary problems is limited by accepted hypotheses and it is impossible to improve the accuracy in principle. SSS components are usually depicted in the form of a small number of members. The systems of differential equations (DE) obtained here have basically a low order. On the other hand, the solution of boundary value problems for non-thin elastic plates and shells in a three-dimensional formulation [8] is associated with great mathematical difficulties. Only in limited cases, the three-dimensional problem of the theory of elasticity for plates and shells provides an opportunity to find an analytical solution. The complexity of the solution in the exact three-dimensional formulation is greatly enhanced if complex boundary conditions or physically nonlinear problems are considered. Theories in which hypotheses are not used, and SSS components are depicted in the form of infinite series in transverse coordinates, will be called mathematical. The approximation of the SSS component can be adopted in the form of various lines [9-16], and the construction of a three-dimensional problem to two-dimensional can be accomplished by various methods: projective [9, 14, 16], variational [12, 13, 15, 17]. The effectiveness and accuracy of one or another variant of mathematical theory (MT) depends on the complex methodology for obtaining the basic equations.

SUBSTANTIATION FOR SIMPLIFICATION OF CALCULATION MODELS FOR ASSESSMENT OF THE STABILITY OF ROOF ROCKS

2021 ◽  
pp. 25-36
Author(s):  
Oleksandr Ahafonov ◽  
◽  
Daria Chepiga ◽  
Anton Polozhiy ◽  
Iryna Bessarab ◽  
...  
Keyword(s):  
Differential Equations ◽  
Coal Seam ◽  
Cross Section ◽  
Three Dimensional ◽  
Theory Of Elasticity ◽  
Unit Width ◽  
Distributed Load ◽  
The Stability ◽  
Roof Rocks ◽  
Calculation Models

Purpose. Substantiation of expediency and admissibility of use of the simplified calculation models of a coal seam roof for an estimation of its stability under the action of external loadings. Methods. To achieve this purpose, the studies have been performed using the basic principles of the theory of elasticity and bending of plates, in which the coal seam roof is represented as a model of a rectangular plate or a beam with a symmetrical cross-section with different support conditions. Results. To substantiate and select methods for studying the bending deformations of the roof in the coal massif containing the maingates, the three-dimensional base plate model and the beam model are compared, taking into account the kinematic boundary conditions and the influence of external distributed load. Using the theory of plate bending, the equations for determining the deflections of the coal seam roof in three-dimensional basic models under certain assumptions have a large dimension. After the conditional division of the plate into beams of unit width and symmetrical section, when describing the normal deflections of the middle surface of the studied models, the transition from the partial derivative equation to the usual differential equations is carried out. In this case, the studies of bending deformations of roof rocks are reduced to solving a flat problem in the cross-section of the beam. A comparison of solutions obtained by the methods of the three-dimensional theory of elasticity and strength of materials was performed. For a beam with a symmetrical section, the deflection lies in a plane whose angle of inclination coincides with the direction of the applied load. The calculations did not take into account the difference between the intensity of the surface load applied to the beam. Differences in determining the magnitude of the deflections of the roof in the model of the plate concerning the model of the beam reach 5%, which is acceptable for mining problems. Scientific novelty. To study the bending deformations and determine the magnitude of the roof deflection in models under external uniform distributed load, placed within the simulated plate, a strip of unit width was selected, which has a symmetrical cross-section and is a characteristic component of the plate structure and it is considered as a separate load-bearing element with supports, the cross-sections of this element is remained flat when bending. The deflection of such a linear element is described by the differential equations of the bent axis of the beam without taking into account the integral stiffness of the model, and the vector of its complete displacement coincides with the vector of the force line. Practical significance. In the laboratory, to study the bending deformations and their impact on the stability of the coal seam roof under external loads, it is advisable to use a model of a single width beam with a symmetrical section with supports, the type of which is determined by rock pressure control and secondary support of the maingate at the extraction layout of the coal mine.

scholarly journals A General Theorem on the Stability of a Class of Functional Equations Including Quartic-Cubic-Quadratic-Additive Equations

Mathematics ◽  
2018 ◽  
Vol 6 (12) ◽  
pp. 282
Author(s):  
Yang-Hi Lee ◽  
Soon-Mo Jung
Keyword(s):  
Functional Equations ◽  
Direct Method ◽  
General Theorem ◽  
Stability Problems ◽  
General Stability ◽  
Stability Theorems ◽  
Additive Type ◽  
The Stability

We prove general stability theorems for n-dimensional quartic-cubic-quadratic-additive type functional equations of the form by applying the direct method. These stability theorems can save us the trouble of proving the stability of relevant solutions repeatedly appearing in the stability problems for various functional equations.

scholarly journals Insights into the dynamic trajectories of protein filament division revealed by numerical investigation into the mathematical model of pure fragmentation

2021 ◽  
Vol 17 (9) ◽  
pp. e1008964
Author(s):  
Magali Tournus ◽  
Miguel Escobedo ◽  
Wei-Feng Xue ◽  
Marie Doumic
Keyword(s):  
Mathematical Theory ◽  
Length Distribution ◽  
Numerical Approach ◽  
Distribution Profile ◽  
Fragmentation Dynamics ◽  
Mathematical Equations ◽  
Fragmentation Reactions ◽  
The Stability ◽  
Fragmentation Equation ◽  
The Mathematical Model

The dynamics by which polymeric protein filaments divide in the presence of negligible growth, for example due to the depletion of free monomeric precursors, can be described by the universal mathematical equations of ‘pure fragmentation’. The rates of fragmentation reactions reflect the stability of the protein filaments towards breakage, which is of importance in biology and biomedicine for instance in governing the creation of amyloid seeds and the propagation of prions. Here, we devised from mathematical theory inversion formulae to recover the division rates and division kernel information from time dependent experimental measurements of filament size distribution. The numerical approach to systematically analyze the behaviour of pure fragmentation trajectories was also developed. We illustrate how these formulae can be used, provide some insights on their robustness, and show how they inform the design of experiments to measure fibril fragmentation dynamics. These advances are made possible by our central theoretical result on how the length distribution profile of the solution to the pure fragmentation equation aligns with a steady distribution profile for large times.

Lamb’s problem: a brief history

2019 ◽  
Vol 25 (3) ◽  
pp. 501-514
Author(s):  
Mohamad Emami ◽  
Morteza Eskandari-Ghadi
Keyword(s):  
Special Interest ◽  
Mathematical Theory ◽  
Three Dimensional ◽  
Elastic Solid ◽  
Theory Of Elasticity ◽  
Scientific Investigation ◽  
Lamb’S Problem ◽  
Solution Methods ◽  
Dimensional Version ◽  
History Of

In this review note, a historical scientific investigation is presented for Lamb’s problem in the mathematical theory of elasticity. This problem first appeared in 1904 in the pioneering paper of Professor Sir Horace Lamb (Lamb, H. On the propagation of tremors over the surface of an elastic solid. Philos Trans R Soc Lon 1904; 203: 1–42). Of special interest here are the analytical studies of the three-dimensional version of Lamb’s problem, which consists of a semi-infinite, homogeneous, isotropic elastic solid that is set in motion by the exertion of a dynamical point force applied suddenly on the surface of the domain. The objective of this paper is to offer a comprehensive introduction to Lamb’s problem for the reader, along with discussing its mathematical complexities. An account is given of the history of this ever-significant problem from its earlier stages to the more recent investigations via outlining and discussing different rigorous approaches and methods of solution that have been hitherto suggested. The limitations of different methods, if they exist, are also discussed. Eventually, various solution methods are compared considering their nature, advantages, and restrictions.

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