Theoretical background for simulation of physical processes in the interfacial layer “solid-liquid”

The paper shows a modification of the quasi-thermodynamic approach to simulation of the interfacial layer from conditions of its mechanical equilibrium. The anisotropy of the interfacial stress tensor is represented as the sum of the ball and deviator parts, where the ball part defines the pressure in the medium and the deviator part forms the components responsible for the liquid surface tension. Obtaining a closed system of equilibrium equations was possible taking into account evaporation from free surface of liquid. The result is a simple expression for determining the thickness of the interfacial layer.

VARYING THE SHAPE OF A SURFACE WHICH IS DISCRETE PRESENTED BY AN IRREGULAR BALANCED GRID

The article discusses a problem, the solution of which is related to the research previously described in previous publications. This paper demonstrates the solution of the problem of forming a discrete frame, in the form of a balanced irregular grid, discretely represented surface. The described problem is solved by one of the methods of discrete modeling, using the static-geometric method of Professor Kovalev S.N. (SGM). The initial conditions for the formation of such irregular balanced discrete networks are the coordinates of the nodes of the reference loop, the topological organization of the grid and the z-coordinate of one of the internal nodes. Note that irregular grids are characterized by different node and cell topologies. This fact can greatly complicate the modeling process, namely, performing the necessary calculations when calculating the coordinates of discrete grid nodes. To facilitate calculations and simplify numbering of discrete grid nodes, it is proposed to use a topological grid scheme based on a regular grid. For regular grids, each node has a specific number, which greatly facilitates the calculation of node coordinates. The operative change in the shape of the grid can be carried out by connecting the classical coordinate calculations of the discrete SGM grid, that is, by solving the system of equilibrium equations of nodes, with an affine transformation, namely the introduction of the scaling factors of coordinates. The disadvantage of this synthesis of the two methods will be the change in the preassigned reference of contour of the mesh, due to the fact that all coordinates of absolutely all grid nodes are multiplied by the corresponding transformation coefficients. To avoid changing the shape of a given reference contour, it is proposed to use a synthesis of three methods in the work, namely SGM, affine coordinate transformation and a method of functional addition of coordinates. This synthesis of methods will maintain the balance of the discrete grid during the modeling process, and will allow you to simply vary (change) the shape of the simulated surface.

BENDING OF AN ELASTIC THREE-LAYER PLATE WITH A HOLE CONNECTED TO THE SOIL FOUNDATION

The relevance of this paper is explained by a demand for the development of mechanical and mathematical models and methods for calculating the stress-strain state of the sandwich structural elements. The statement of the boundary value problem on the deformation of a circular sandwich plate with a central hole, connected to the soil foundation, was given. To describe the kinematics of an asymmetric plate pack, the broken line hypotheses are accepted. In a relatively thick lightweight core, the normal does not change its length, remains rectilinear, but rotates through some additional angle. Tuff, coarse grained soil, granite, and gneiss are accepted as the soil foundation. The bearing reaction is described by the Winkler model. The system of equilibrium equations is obtained by the variational method. Its solution is written in displacements through Kelvin functions. A numerical parametric analysis of displacements and stresses in the plate is carried out, their dependence on the type of soil foundation is shown.

Analysis of force factors of a thin-walled shell metalworks

The method of refining the power factors of a machine-building structure presented in the form of a shell is described. The stability of a thin-walled shell is analyzed using a system of equilibrium equations taking into account changes in its shape. The concept of shell stability is considered for generalized forces. The estimation of the accuracy of the metal structure calculation is justified by solving several problems using the finite element method. The calculation error for various finite elements is determined.

Calculation of the deflection of an arched truss with suspended elements depending on the number of panels

The aim of the work - to propose a scheme and analytical calculation of a statically definable planar truss with a suspended lower belt. Methods. The formula for the dependence of the deflection of the truss under the action of a uniform load on the lower belt on its size and the number of panels is derived in the computer mathematics system Maple. The forces in the rods are found from the solution of the general system of equilibrium equations of all nodes in symbolic form. The deflection is calculated using the Maxwell - Mohr's formula. Generalization of a number of formulas for deflection obtained by increasing the number of panels sequentially to an arbitrary number is performed by double induction using two independent parameters. In this case, special operators of the Maple system are used, allowing for a sequence of coefficients in the desired formula to create and solve recurrent equations that satisfy the elements of the sequences. Results. The obtained solutions have a polynomial form for the number of panels. Curves of deflection dependence on the number of panels are constructed and analyzed. Asymptotic properties of solutions are found in the case of a fixed span length of the structure and a given total load. The proposed scheme is a statically determinate structure with two independent parameters of regularity allows for the finding of a fairly simple analytical solution. The resulting formula is most effective in calculating systems with a large number of elements, where numerical methods tend to accumulate rounding errors.

On Some Classes of Correct Problems in the Membrane Theory of Convex Shells

On the basis of the theory of the modified Riemann-Hilbert problem for generalized analytic functions, a geometric description is given of a fairly wide family of correct by I. N. Vekua of boundary value problems of the membrane theory of convex hulls with a piecewise smooth boundary. Solutions to the corresponding Riemann-Hilbert problem for an elliptic system of equilibrium equations are found in the classes of N.I. Muskhelishvili and realize a state of tense equilibrium under the condition of stress concentration in corner points. An effective formula is given for calculating the index of the boundary condition, which allows us to formulate the results in a visible form. Families of shells are found for which the solvability picture of the main boundary-value problem coincides with the solvability picture of the Vekua problem for shells with a smooth border.

ANALYTICAL CALCULATION OF DEFLECTION OF A MULTI-LATTICE TRUSS WITH AN ARBITRARY NUMBER OF PANELS

The scheme of a planar externally statically indeterminate truss with four supports is proposed. In analytical form, for several types of loads, the problem of forces in the rods and deflectionof the structure is solved, depending on the number of panels, the size and intensity of the load. The solution uses the Maple computer mathematics system. The deflectionat Midspan is determined using Maxwell – Mohr's formula, the forces in the rods – the method of cutting out nodes from the system of equilibrium equations for all nodes, which includes four reactions of the supports. By induction, a series of solutions for trusses with a consistently increasing number of panels is generalized to an arbitrary number of panels. For the elements of the sequences of coefficientare developed and are solved by homogeneous linear recurrence equations. The resulting formulas for the deflectio of the structure under various loads have the form of polynomials in the number of panels. A linear asymptotic solution for the number of panels is found. The kinematic degeneration of the structure and the distribution of node speeds corresponding to this case were found. The dependences of the reaction of supports and forces in the most compressed and stretched rods on the number of panels are determined.

Modeling of stress-strain state of piping systems with erosion and corrosion wear

The problems of modeling the stress-deformed state of erosion or corrosion-worn rectilinear sections and the ball-shaped bends of pipeline systems are proposed to solve in a cylindrical coordinate system. For this purpose, formulas of Christophell type II, non-zero components of the strain tensor and a system of equilibrium equations in the framework of linear torsional theory are given. The system of equilibrium equations is reduced to one equation, which is the basic equation of the Lame’s problem. Formulas for the calculation of ring stresses that occur in the wall of erosion or corrosion worn rectilinear sections, and the removal of pipelines from the action of internal pressure are derived. The influence of the change in the wall thickness of the pipeline bends in the place of their erosion or corrosion wear on the amount of ring stresses is determined.

The Replication Hypothesis along the Take-Off Run and a System of Equilibrium Equations at the Lift-Off of a Protobird

An extant bird resorts to flapping and running along its take-off run to generate lift and thrust in order to reach the minimum required wing velocity speed required for lift-off. This paper introduces the replication hypothesis that posits that the variation of lift relative to the thrust generated by the flapping wings of an extant bird, along its take-off run, replicates the variation of lift relative to the thrust by the flapping wings of a protobird as it evolves towards sustained flight. The replication hypothesis combines experimental data from extant birds with evidence from the paleontological record of protobirds to come up with a physics-based model of its evolution towards sustained flight while scaling down the time span from millions of years to a few seconds. A second hypothesis states that the vertical and horizontal forces acting on a protobird when it first encounters lift-off are in equilibrium as the protobird exerts its maximum available power for flapping, equaling its lift with its weight, and its thrust with its drag.

Form-Finding of Tensegrity Model with Triangular Cells

Tensegrity structures is a light-weight structure compared to concrete structures that are heavy and rigid in shape. The studies on form-finding for tensegrity configuration are still ongoing and have been extensively conducted. Additionally, many proposed tensegrity structures have not been built for real applications. This study aims to determine potential self-equilibrated configurations of three-stage Class I tensegrity model assemblage with triangular cells, which may be applied as deployable towers. The form-finding methodology involves phases in establishment of desired form and formulation for the self-equilibrated state. The system of equilibrium equations was solved by Moore-Penrose generalized inverse method. A range of twist angles 10o – 50o for triangular cells was investigated in the form-finding process. It was found that the form-finding method via changing of twist angles has successfully search self-equilibrated tensegrity models.