scholarly journals FEATURES OF STRESSES AT THE APEX OF AN ELASTIC WEDGE, SUPPORTED BY A THIN FLEXIBLE COATING ON THE SIDES

2019 ◽  
Vol 46 (3) ◽  
pp. 59-166
Author(s):  
B. V. Sobol' ◽  
A. N. Soloviev ◽  
M. M. Payzulaev ◽  
E. V. Rashidova ◽  
G. M. Murtazaliev
Keyword(s):  
Boundary Conditions ◽  
Radial Component ◽  
Rigid Base ◽  
Singular Stress ◽  
Special Boundary Condition ◽  
Singular Behaviour ◽  
Positive Roots ◽  
Transcendental Equations ◽  
Stress Behaviour ◽  
Flexible Coating

Objectives To study the problem of determining the degree of stress at the apex of a wedge-shaped area in cases where the sides (or one of them) are covered with a thin flexible coating.Method It is assumed that the coating is not stretchable. On the other side of the wedge-shaped area, the same coating is assumed to be present; it is either fixed, stress-free or in smooth contact with a rigid base. Mathematically, the problem is reduced to the task of determining the roots of characteristic transcendental equations arising from the existence of a nontrivial solution to the system of linear homogeneous equations.Results Values for the specific characteristics of the radial component of a stress tensor are determined for different combinations of boundary conditions and solution angles. In particular, the angles at which the singular behaviour of stresses occurs are determined. The case is considered when a special boundary condition is given on the edge surface, simulating the overlay. Characteristic equations are obtained to determine the index of the degree dependency of the asymptotic solution in its vicinity for four variants of boundary conditions. In two cases, transcendental equations are obtained, which are solved numerically.Conclusion Calculations of the first positive roots of the equations depending on the angle of the edge solution and Poisson's ratio are presented. The values of the angles, at which the singular behaviour of stresses occurs, are determined. In the case of a combination of boundary conditions (III – IV), the singular stress behaviour is observed for the angle ???? = ????/8, while in the case of (III – III) this value is equal to ????/4. 

Analytical-Numerical Determination of Stress Distribution around a Tip of Polygon-Like Inclusion

2016 ◽  
Vol 713 ◽  
pp. 94-98
Author(s):  
Ondřej Krepl ◽  
Jan Klusák ◽  
Tomáš Profant
Keyword(s):  
Stress Distribution ◽  
Stress Field ◽  
Numerical Procedure ◽  
Plane Elasticity ◽  
Necessary Condition ◽  
Singular Stress ◽  
Reliable Assessment ◽  
Generalized Stress Intensity Factors ◽  
Stress Behaviour

A stress distribution in vicinity of a tip of polygon-like inclusion exhibits a singular stress behaviour. Singular stresses at the tip can be a reason for a crack initiation in composite materials. Knowledge of stress field is necessary condition for reliable assessment of such composites. A stress field near the general singular stress concentrator can be analytically described by means of Muskhelishvili plane elasticity based on complex variable functions. Parameters necessary for the description are the exponents of singularity and Generalized Stress Intensity Factors (GSIFs). The stress field in the closest vicinity of the SMI tip is thus characterized by 1 or 2 singular exponents (1 - λ) where, 0<Re (λ)<1, and corresponding GSIFs that follow from numerical solution. In order to describe stress filed further away from the SMI tip, the non-singular exponents for which 1<Re (λ), and factors corresponding to these non-singular exponents have to be taken into account. Analytical-numerical procedure of determination of stress distribution around a tip of sharp material inclusion is presented. Parameters entering to the procedure are varied and tuned. Thus recommendations are stated in order to gain reliable values of stresses and displacements.

The Effect of the Fuel Injector Internal Geometry Upon the Primary Zone Aerodynamics

1998 ◽  
Author(s):  
R. A. Hicks ◽  
M. Whiteman ◽  
C. W. Wilson
Keyword(s):  
Boundary Conditions ◽  
Fluid Dynamic ◽  
Radial Component ◽  
Fuel Injector ◽  
Air Blast ◽  
Fuel Injectors ◽  
Cfd Models ◽  
Primary Zone ◽  
Flow Through ◽  
Semi Empirical

One of the major aims of research in gas turbine combustor systems is the minimisation of non-desirable emissions. The primary method of reducing pollutants such as soot and NOx has been to run the combustion primary zone lean. Unfortunately, this causes problems when the combustor is run under idle and relight conditions as the primary zone air fuel ratio (AFR) can exceed the flammability limit. Altering this AFR directly affects the primary zone aerodynamics through changes in the spray profile. One method of determining the influence of changes in AFR upon the primary zone is to use Computational Fluid Dynamic (CFD) models. However, to model the flow through an air-blast fuel injector and accurately predict the resulting primary zone aerodynamics requires hundreds of thousands, if not millions, of cells. Therefore, with current computer capabilities simplifications need to be made. One simplification is to model the primary zone as a 2-D case. This reduces the number of cells to a computationally solvable level. However, by reducing the problem to 2-D the ability to accurately model air-blast fuel injectors is lost as they are intrinsically 3-D devices. Therefore, it is necessary to define boundary conditions for the fuel injector. Currently, due to difficulties in obtaining experimental measurements inside a air-blast fuel injector, these boundary conditions are often derived using semi-empirical methods. This paper presents and compares two such models; the model proposed by Crocker et al. in 1996 and one developed at DERA specifically for modelling air-blast fuel injectors. The work also highlights the importance of the often neglected radial component upon the primary zone aerodynamics.

scholarly journals Experimental study of the impact response of geocells as components of rockfall protection embankments

2009 ◽  
Vol 9 (2) ◽  
pp. 459-467 ◽  
Author(s):  
S. Lambert ◽  
P. Gotteland ◽  
F. Nicot
Keyword(s):  
Experimental Study ◽  
Numerical Model ◽  
Boundary Conditions ◽  
Composite Structures ◽  
Impact Force ◽  
Impact Response ◽  
Rigid Base ◽  
Transmitted Force ◽  
The Impact ◽  
Rockfall Protection

Abstract. Rockfall protection embankments are ground levees designed to stop falling boulders. This paper investigates the behaviour of geocells to be used as components of these structures. Geocells, or cellular confinement systems, are composite structures associating a manufactured envelope with a granular geomaterial. Single cubic geocells were subjected to the impact resulting from dropping a spherical boulder. The geocells were filled with fine or coarse materials and different boundary conditions were applied on the lateral faces. The response is analysed in terms of the impact force and the force transmitted by the geocell to its rigid base. The influence on the geocell response of both the fill material and the cell boundary conditions is analysed. The aim was to identify the conditions resulting in greatest reduction of the transmitted force and also to provide data for the validation of a specific numerical model.

Series Solution and Stress Analysis of Rectangular Plate With Mixed Boundary Conditions

2005 ◽  
Author(s):  
Jiemin Liu ◽  
Jintang Liu ◽  
Toshiyuki Sawa
Keyword(s):  
Stress Concentration ◽  
Boundary Conditions ◽  
Rectangular Plate ◽  
Mixed Boundary ◽  
Mixed Boundary Conditions ◽  
Size Factor ◽  
Singular Stress ◽  
Feature Size ◽  
Concentrated Force ◽  
Displacement Boundary

Stress functions expressed from Fourier series, suitable for arbitrary stress boundary conditions, were derived using method of variable separation. General displacement expressions containing the displacement of rigid body were also derived. A method of solving mixed boundary problems (in which external forces acting at a part of the whole boundaries are known and displacements at the rest boundaries are known) was presented. As an example, a rectangular plate, one side of which was fixed and objective side was subjected to a concentrated force, was analyzed. In addition, characteristics of stress distributions in the regions of stress concentration were questioned. It was found from the presented results of calculation that describing stress concentration with the singular stress at a point was unworkable. Describing stress concentration with the average stress in the feature size instead of the singular stress at a point was operative and reflected objectively practical stress and displacement boundary conditions. The concept of feature-size-factor was introduced.

scholarly journals Weber-Type Integral Transform Connected with Robin-Type Boundary Conditions

Mathematics ◽  
2020 ◽  
Vol 8 (8) ◽  
pp. 1335
Author(s):  
Thanaa Elnaqeeb ◽  
Nehad Ali Shah ◽  
Dumitru Vieru
Keyword(s):  
Boundary Conditions ◽  
Integral Transform ◽  
Transcendental Equation ◽  
Micro Channels ◽  
Type Integral ◽  
Positive Roots ◽  
Weber Type ◽  
Type Boundary ◽  
Mathcad Software ◽  
Transport Problems

A new Weber-type integral transform and its inverse are defined for the representation of a function f(r,t), (r,t)∈[R,1]×[0,∞) that satisfies the Dirichlet and Robin-type boundary conditions f(R,t)=f1(t), f(1,t)−α∂f(r,t)∂r|r=1=f2(t), respectively. The orthogonality relations of the transform kernel are derived by using the properties of Bessel functions. The new Weber integral transform of some particular functions is determined. The integral transform defined in the present paper is a suitable tool for determining analytical solutions of transport problems with sliding phenomena that often occur in flows through micro channels, pipes or blood vessels. The heat conduction in an annular domain with Robin-type boundary conditions is studied. The subroutine “root(⋅)” of the Mathcad software is used to determine the positive roots of the transcendental equation involved in the definition of the new integral transform.

The effects of capillarity on free-streamline separation

1975 ◽  
Vol 70 (2) ◽  
pp. 333-352 ◽  
Author(s):  
R. C. Ackerberg
Keyword(s):  
Surface Tension ◽  
Boundary Conditions ◽  
Pressure Gradient ◽  
Classical Solution ◽  
Capillary Wave ◽  
Small Region ◽  
Linear Boundary ◽  
Small Surface ◽  
First Approximation ◽  
Singular Behaviour

The effect of a small surface-tension coefficient on the classical theory of free-streamline separation from a sharp trailing edge is studied. The classical solution fails in a small region surrounding the edge, where it predicts singular behaviour, and an inner solution, satisfying linear boundary conditions, is required to obtain a uniformly valid first approximation. The solution valid near the edge removes the curvature and pressure-gradient singularities of the classical solution and predicts a standing capillary wave along the free streamline.

On the Formulation of Nonreflecting Boundary Conditions for Turbomachinery Configurations: Part I — Theory and Implementation

2020 ◽  
Author(s):  
Christian Frey ◽  
Daniel Schlüß ◽  
Nina Wolfrum ◽  
Patrick Bechlars ◽  
Maximilian Beck
Keyword(s):  
Boundary Condition ◽  
Boundary Conditions ◽  
Frequency Domain ◽  
Harmonic Balance ◽  
Radial Component ◽  
Rotational Surface ◽  
Nonreflecting Boundary Conditions ◽  
Meridional Velocity ◽  
The Impact ◽  
Rotational Surfaces

Abstract With unsteady flow simulations of industrial turbomachinery configurations becoming more and more affordable there is a growing need for accurate inlet and outlet boundary conditions as numerical reflections alone can lead to incorrect trends in engine efficiency, noise and aeroelastic analysis parameters. This is the first of two papers on the formulation of unsteady boundary conditions which have been implemented for both time-domain and frequency-domain solvers. Giles’ original idea for steady solvers to formulate the boundary condition in terms of characteristics generalizes to frequency-domain solvers. The boundary condition drives the value of the incoming characteristics to ideal values that are computed using the modal decomposition of linearized 2D Euler flows. The present paper explains how to generalize 2D nonreflecting boundary conditions to real 3D annular domains by applying them in certain conical rotational surfaces. For a flow with zero radial component and an annular boundary that is perpendicular to the machine axis, these surfaces are the cylindrical streamsurfaces. For more general flows and geometries, however, there is no natural choice for the rotational surfaces. In this paper, two choices are discussed: the surfaces that are generated by the boundary normals and those that are defined by the circumferentially averaged meridional velocity. The impact of the boundary condition on the stability of the harmonic-balance solver is analyzed by studying the pseudo-time evolution of certain energy integrals. For a model problem which consists of a small disturbance of an inviscid flow, the increase or decrease of this energy integral is shown to be directly related to the normal characteristic variables along the boundary. This shows that the actual boundary condition should be formulated as a control problem for the normal characteristics. Moreover, the application of the harmonic balance solver to a simple duct configuration with prescribed disturbances demonstrates that using the characteristics based on the meridional velocity may prevent the solver from converging. In contrast, the 2D theory can be formulated in a different surface without impairing the robustness of the overall approach. These findings are illustrated by a simple test case. The impact of the choice of the rotational surface for the 2D theory is studied for various duct segments and a low-pressure turbine configuration in the second paper. There it is shown that applying the 2D theory to the meridional-velocity surfaces may be advantageous in that it leads to more accurate results.

Determination of the stress concentration in the corner point of the wedge-shaped region reinforced by a more rigid thin coating

2020 ◽  
Vol 26 (1) ◽  
pp. 80-89
Author(s):  
AN Soloviev ◽  
BV Sobol ◽  
EV Rashidova ◽  
AI Novikova
Keyword(s):  
Boundary Conditions ◽  
Corner Point ◽  
Theory Of Elasticity ◽  
The Other ◽  
Thin Coating ◽  
Various Boundary Conditions ◽  
Singular Behaviour ◽  
Isotropic Theory ◽  
Made In

We analysed the problem of determining the exponents in the asymptotic solution of the isotropic theory of elasticity problem at the top of the wedge-shaped region where its sides (or one of them) are supported by a thin coating and lean without friction on the rigid bases. On the other side of the wedge-shaped region, it is assumed that there are various boundary conditions, including when there is a thin coating. Mathematically, the problem reduces to the problem of determining the roots of transcendental characteristic equations arising from the condition for the existence of a nontrivial solution of a system of the linear homogeneous equations. The characteristics of the stress tensor components have been determined for the various combinations of boundary conditions and physical and geometric parameters. The qualitative conclusions are made. In particular, we have established the combinations of the values of these parameters at which the singular behaviour of stresses arises.

scholarly journals An Extended Kantorovich Method Used for Static Solution of an Arbitrarily Edge Supported Sandwich Cylindrical Shell Panel

2019 ◽  
Vol 8 (6) ◽  
pp. 4184-4193
Keyword(s):  
Cylindrical Shell ◽  
Differential Equations ◽  
Boundary Conditions ◽  
Laminated Composite ◽  
Static Solution ◽  
Radial Component ◽  
Kantorovich Method ◽  
Power Series Method ◽  
Arbitrary Boundary ◽  
Shell Panel

Exact solution of complex problems like composite shells with arbitrarily supported boundary conditions through analytical three-dimensional (3-D) approach is mathematically challenging. In the present work an analytical 3-D elasticity solution for the static bending problem of a laminated composite cylindrical shell panel having any arbitrary boundary conditions is proposed. The governing Partial Differential Equations (PDE) problems are obtained by the application of the Ressiner-type mixed variational principle in cylindrical coordinate system. The extended Kantrovich method [10] is applied to solve these equations by reducing them to Ordinary Differential Equations (ODE). Further, the set of ODEs corresponding to the radial component & the circumferential components are solved utilizing modified power series method & Pagano’s approach respectively. Through numerical studies of sandwich shell panels it is shown that this method accurately predicts the deflections, stresses, boundary effects and interfacial disruptions being generated of laminate scheme, material property variations and configuration of the shell panel. Crucially, this is achieved with just two or three terms and few iterations, hence attributes faster computation as compared to other numerical techniques.

scholarly journals CHARGED POLYTROPIC STARS AND A GENERALIZATION OF LANE–EMDEN EQUATION

2004 ◽  
Vol 13 (07) ◽  
pp. 1441-1445 ◽  
Author(s):  
RODRIGO PICANÇO ◽  
MANOEL MALHEIRO ◽  
SUBHARTHI RAY
Keyword(s):  
Equation Of State ◽  
Differential Equations ◽  
Electric Field ◽  
Boundary Conditions ◽  
Radial Component ◽  
Spherically Symmetric ◽  
Field Equations ◽  
Emden Equation ◽  
Polytropic Equation ◽  
Structure Equations

In this paper we discuss charged stars with polytropic equation of state, where we derive an equation analogous to the Lane–Endem equation. We assume that these stars are spherically symmetric, and the electric field have only the radial component. First we review the field equations for such stars and then we proceed with the analog of the Lane–Emden equation for a polytropic Newtonian fluid and their relativistic equivalent (Tooper, 1964).1 These kind of equations are very interesting because they transform all the structure equations of the stars in a group of differential equations which are much more simple to solve than the source equations. These equations can be solved numerically for some boundary conditions and for some initial parameters. For this we assume that the pressure caused by the electric field obeys a polytropic equation of state too.

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