Contact problems for an inhomogeneous elastic wedge with variable Poisson’s ratio

Plane contact problems of the elasticity theory are investigated for a wedge when Poisson’s ratio is an arbitrary smooth function with respect to the angular coordinate while shear modulus is constant. For this case Young’s modulus is also variable with respect to the angular coordinate. A finite contact domain is given on one wedge face, it does not include the wedge apex, while the other wedge face is rigidly fixed (problem A) or stress-free (problem B). To reduce the problems to integral equations with respect to the contact pressure, we use the general Freiberger type representation for the solution of elastic equilibrium equations written in polar coordinates with variable Poisson’s ratio. Exact solutions of auxiliary problems are constructed with the help of Mellin integral transforms. The regular asymptotic method employed is effective for contact domains relatively distant from the wedge apex. It is shown that logarithmic terms appear in the asymptotic solutions for the inhomogeneous material which are missing in the well-known asymptotics for the homogeneous one. In contact problem C which is corresponding to problem A, the friction and roughness are taken into account in the contact region. The roughness of the wedge surface is simulated by a Winkler type coating. The collocation method is used for solving integral equations of the second kind. Unlike problem A, in problem C the contact pressure does not have square root singularities at end-points where it takes finite values. Calculations are made for the cases when Poisson’s ratio and Young’s modulus increase or decrease from the surface of the wedge.

Calculation of the tense-deformed state of two ortogonal attended plates at the regional terms of symmetry through the matrices of Grina type

Purpose. Calculate the tense-deformed state of two ortogonal attended plates through special the built matrices of Grina type. Research methods. Bases of theory of laminas, apparatus of trigonometric rows of Fourier, methods: border-component tasks, variation of arbitrary permanent, matrices of Grina type. Results. A task of elastic elastic equilibrium of plate pairs connected at right angle was considered. On parallel edges of component body to connection rib special edge conditions – conditions of symmetry – were chosen. From the physical point of view the probed body can be the model of lateral walls of parallelepiped. It is provided the special terms of symmetry on both edges of component body, which are parallel to the rib of connection of plates. The method of calculation allows to calculate the tense-deformed state of spatial construction consisting of two plates in edge conditions of arbitrary surface loading. The results of calculation (as lines of level) of basic characteristics of static deformation of considered rectangular connection of two plates are given. Scientific novelty. The method of calculation of pair of plates, united at right angles was improved at the regional terms of symmetry, with subsequent graphic illustration of achived results. Practical value. The task probed in-process designs the phenomena which take place, at deformation of elements of vulcanization equipment. Achived results allow to find pequliarities of elements work of construction of complext structure and to promote its efficiency by optimization of component parameters parts.

The Solution of Boundary Value Problems of Various Types with Consideration of Volume Forces for Anisotropic Bodies of Revolution

In this work, we studied the axisymmetric elastic equilibrium of transversely isotropic bodies of revolution, which are simultaneously under the influence of surface and volume forces. The construction of the stress-strain state is carried out by means of the boundary state method. The method is based on the concepts of internal and boundary states conjugated by an isomorphism. The bases of state spaces are formed, orthonormalized, and the desired state is expanded in a series of elements of the orthonormal basis. The Fourier coefficients, which are quadratures, are calculated. In this work, we propose a method for forming bases of spaces of internal and boundary states, assigning a scalar product and forming a system of equations that allows one to determine the elastic state of anisotropic bodies. The peculiarity of the solution is that the obtained stresses simultaneously satisfy the conditions both on the boundary of the body and inside the region (volume forces), and they are not a simple superposition of elastic fields. Methods are presented for solving the first and second main problems of mechanics, the contact problem without friction and the main mixed problem of the elasticity theory for transversely isotropic finite solids of revolution that are simultaneously under the influence of volume forces. The given forces are distributed axisymmetrically with respect to the geometric axis of rotation. The solution of the first main problem for a non-canonical body of revolution is given, an analysis of accuracy is carried out and a graphic illustration of the result is given

SIMULATION OF THE MACHINED SURFACE AFTER END MILLING WITH SELF-OSCILLATIONS

Thin-walled parts are widely used in the aviation industry. It is mainly carried out with end mills and is accompanied by self-oscillation during rough milling. They negatively affect the quality of the machined surface. Therefore, it is important to model it taking into account the dynamics of the milling process to predict the accuracy. In the early works of the authors, the mechanism of the profile forming of the machined surface was determined. In this case, the identity of the shape of the cutting surface and the oscillogram of part’s oscillations during milling is taken as a basis. The first wave of self-oscillations takes part in the shaping of the machined surface during cut-up milling with self-oscillation, and during cut-down milling - the last wave. The change in the distances of the cut depressions to the position of the elastic equilibrium of the part is periodically repeated from the maximum value to the minimum. Based on this, when modeling the waviness pitch of the machined surface after cut-up milling, it is necessary to know the feed rate and how many cuts were made by the tool from the largest to the smallest depression. When modeling the machined surface after cut-down milling, you need to know the length of the cutting surface. It is calculated based on cutting speed and cutting time. The formula for determining the waviness pitch after cut-down milling is derived taking into account the tool feed. The waviness height of the machined surface after cut-up and cut-down milling is determined as the difference between the largest and smallest depressions. To determine the size of the pitch and the height of the waviness, formulas are derived for converting electrical and time values of oscillograms into linear ones. These formulas also allow you to determine areas of the oscillogram with oscillations of the part during cutting and the resulting surface areas on the profilogram. The methods for modeling machined surfaces were tested after cut-up and cut-down milling with self-oscillation. In this case, the pitch and height of the waviness on the profilograms were compared with those calculated from the results of measurements of the oscillograms. Based on their analysis, refined formulas for calculating the waviness height have been derived. The error between the measurements of the waviness pitch and height and the calculated values is within 6%.

Causality in physical processes

A mathematical model is presented that describes how the diverse phenomena in nature can arise from a common foundation rooted in an objective framework of relational causality. Key equations of conventional physical theories are derived anew. Rigorous derivations are shown for such fundamental relationships as the Schrödinger equation, Bohr’s formula for the hydrogen spectrum, Newton’s gravitational law, Coulomb’s electrostatic law, Compton shift formula, and other principal equations of electromagnetism, atomic theory, optics, and thermodynamics. Nuclear forces that bind nucleons and pry them apart are shown to arise from a single universal electrostatic force that also gives rise to the Coulomb force. Dark matter is shown to be an ever-present material content that subsists in the fabric of space everywhere. The precise value of Coulomb’s constant is derived on purely theoretical grounds, while hitherto unknown values are predicted for the electric dipole moments of electrons and protons. The first of three components of the core hypothesis of this model is that there exists a primordial seed particle of which all that lies in the universe is ultimately composed, and that it is defined by Planck’s constant. Space is modeled as a lattice of contiguous cells that are composed entirely of these particles, and are inherently elastic. The second component is that the universe evolves temporally through a recursive process, emulating a system of cellular automata. The third component stipulates that any compression produced in a cell by its ambient conditions generates potential energy according to classical elasticity. Thus, every cell seeks to minimize its stored potential energy at each step in its temporal evolution, in pursuit of elastic equilibrium. This core hypothesis in respect to the physical character of space, time, and energy is shown to lead logically to a universal dynamic that gives rise to the physical effects observed in nature at every scale.

Isostasy with Love: I Elastic equilibrium

Summary Isostasy explains why observed gravity anomalies are generally much weaker than what is expected from topography alone, and why planetary crusts can support high topography without breaking up. On Earth, it is used to subtract from gravity anomalies the contribution of nearly compensated surface topography. On icy moons and dwarf planets, it constrains the compensation depth which is identified with the thickness of the rigid layer above a soft layer or a global subsurface ocean. Classical isostasy, however, is not self-consistent, neglects internal stresses and geoid contributions to topographical support, and yields ambiguous predictions of geoid anomalies. Isostasy should instead be defined either by minimizing deviatoric elastic stresses within the silicate crust or icy shell, or by studying the dynamic response of the body in the long-time limit. In this paper, I implement the first option by formulating Airy isostatic equilibrium as the linear response of an elastic shell to a combination of surface and internal loads. Isostatic ratios are defined in terms of deviatoric Love numbers which quantify deviations with respect to a fluid state. The Love number approach separates the physics of isostasy from the technicalities of elastic-gravitational spherical deformations, and provides flexibility in the choice of the interior structure. Since elastic isostasy is invariant under a global rescaling of the shell shear modulus, it can be defined in the fluid shell limit, which is simpler and reveals the deep connection with the asymptotic state of dynamic isostasy. If the shell is homogeneous, minimum stress isostasy is dual to a variant of elastic isostasy called zero deflection isostasy, which is less physical but simpler to compute. Each isostatic model is combined with general boundary conditions applied at the surface and bottom of the shell, resulting in one-parameter isostatic families. At long wavelength, the thin shell limit is a good approximation, in which case the influence of boundary conditions disappears as all isostatic families members yield the same isostatic ratios. At short wavelength, topography is supported by shallow stresses so that Airy isostasy becomes similar to either pure top loading or pure bottom loading. The isostatic ratios of incompressible bodies with three homogeneous layers are given in analytical form in the text and in complementary software.

Design Contributions to the Elaboration of New Modeling Schemes for the Buckling Assessment of Hydraulic Actuators

Hydraulic cylinders represent the main actuating/positioning element for standalone lifting equipment or equipment for various transport platforms. This type of actuator represents a structural component responsible for the operational safety of the equipment it serves. One of the most common and dangerous reasons concerning the end of life for this equipment is the buckling or loss of stability of the elastic equilibrium shape. This article aims to compare the classical approach of the problem in accordance with the strength of materials theory in relation to the numeric algorithms used in the applications for the analysis of structure behavior and the algorithms that are based on the finite element method. The subject of study is a hydraulic cylinder that is installed in a self-lifting platform and because of the manifestation of the phenomenon under analysis, it has led to a technical accident. For this purpose, an estimation of the value for the buckling critical load of the cylinder assembly was carried out.

Certain cases of elastic equilibrium of a composed wedge with an interface crack

Problems of interface cracks starting from the common corner points of pairs of perfectly glued wedges of different isotropic elastic materials are addressed. It is demonstrated that for a few particular configurations and a restrictive condition imposed on values of elastic constants (corresponding to vanishing of the second Dundurs parameter), the problem of elastic equilibrium may be solved by Khrapkov’s method. These configurations are: (i) the wedges forming a half-plane; (ii) the wedges forming a plane; (iii) one of the wedges being a half-plane. In all cases, the external boundaries are supposed to be free of stresses. By applying Mellin’s transform for all three configurations the problem has been reduced to vector Riemann’s problem, and the matrix coefficient has been factorized for the case of the mentioned restrictive condition. The first configuration, i.e. the problem of an inclined edge crack located along the boundary separating two wedges of different elastic isotropic materials forming a half-plane is considered in more detail. The solution has been obtained for both uniform (corresponding to remote loading) and non-uniform (loading applied at the crack faces) problems. Numerical results are presented and compared with the available results obtained by other authors for particular cases. The obtained solutions appear especially valuable for analysing extreme cases of parameters.