scholarly journals INTERFERENCE OF CLOSABLE CRACKS AND NARROW SLITS IN AN ELASTIC PLATE UNDER BENDING

2020 ◽  
Vol 14 (2) ◽  
pp. 51-68
Author(s):  
Taras M. Dalyak ◽  
Ivan P. Shatskyi
Keyword(s):  
Numerical Solutions ◽  
Infinite Plate ◽  
Singular Integral Equations ◽  
Normal Displacement ◽  
Biharmonic Equations ◽  
Rectilinear Crack ◽  
Free Surfaces ◽  
Dimensional Statement ◽  
Negative Jump ◽  
Classical Hypothesis

The problem of bending of an infinite plate containing an array of trough closable cracks and narrow slits is considered in a two-dimensional statement. A crack is treated as a mathematical cut, the edges of which are able to contact along the line on the plate outside. A slit is referred to as a cut with contact stress-free surfaces and the negative jump of normal displacement can occur on this cut. The crack closure caused by bending deformation was studied based on the classical hypothesis of direct normal and previously developed model of the contact of edges along the line. A new boundary problem for a couple of biharmonic equations of plane stress and plate bending with interconnected boundary conditions in the form of inequalities on the cuts is formulated. The method of singular integral equations was applied in order to develop approximate analytical and numerical solutions to the problem. The forces and moments intensity factors near the peaks of defects and contact reaction on the closed edges of the cracks are calculated. A detailed analysis was carried out for parallel rectilinear crack and slit, depending on their relative location. Presented results demonstrate qualitative differences in the stress concentration near the defects of different nature.

Rough Contacts Modeled by Nonlinear Coatings

2007 ◽  
Author(s):  
I. I. Kudish
Keyword(s):  
Contact Problem ◽  
Numerical Solutions ◽  
Asymptotic Methods ◽  
Experimental Studies ◽  
Nonlinear Function ◽  
Normal Displacement ◽  
Experimental Fact ◽  
Axially Symmetric ◽  
Elastic Solids ◽  
Local Pressure

A number of experimental studies [1–3] revealed that the normal displacement in a contact of rough surfaces due to asperities presence is a nonlinear function of local pressure and it can be approximated by a power function of pressure. Originally, a linear mathematical model accounting for surface roughness of elastic solids in contact was introduced by I. Shtaerman [4]. He assumed that the effect of asperities present in a contact of elastic solids can be essentially replaced by the presence of a thin coating simulated by an additional normal displacement of solids’ surfaces proportional to a local pressure. Later, a similar but nonlinear problem formulation that accounted for the above mentioned experimental fact was proposed by L. Galin. In a series of papers this problem was studied by numerical and asymptotic methods [5–9]. The present paper has a dual purpose: to analyze the problem analytically and to provide some asymptotic and numerical solutions. The results presented below provide an overview of the results obtained on the topic and published by the author earlier in the journals hardly accessible to the international tribological community (such as Russian and mathematical journals) and, therefore, mostly unknown by tribologists. A number of recent publications on contacts of rough elastic solids supports the view that these results are still of value to the specialists involved in nanotribology. The existence and uniqueness of a solution of a contact problem for elastic bodies with rough (coated) surfaces is established based on the variational inequalities approach. Four different equivalent formulations of the problem including three variational ones were considered. A comparative analysis of solutions of the contact problem for different values of initial parameters (such as the indenter shape, parameters characterizing roughness, elastic parameters of the substrate material) is done with the help of calculus of variations and the Zaremba-Giraud principle of maximum for harmonic functions [10,11]. The results include the relations between the pressure and displacement distributions for rough and smooth solids as well as the relationships for solutions of the problems for rough solids with fixed and free contact boundaries. For plane and axially symmetric cases some asymptotic and numerical solutions are presented.

scholarly journals An Analytical Study of Adversely Affecting Radiation and Temperature Parameters on a Magnetohydrodynamic Elasto-viscous Fluid

2019 ◽  
Vol 24 (1) ◽  
pp. 31
Author(s):  
Nazish Shahid
Keyword(s):  
Viscous Fluid ◽  
Analytical Study ◽  
Numerical Solutions ◽  
Variable Temperature ◽  
Infinite Plate ◽  
Flow Speed ◽  
Inverse Laplace Transform ◽  
Diffusion Effects ◽  
Radiative Diffusion ◽  
Fluid Parameters

An investigation of how the velocity of elasto-viscous fluid past an infinite plate, with slip and variable temperature, is influenced by combined thermal-radiative diffusion effects has been carried out. The study of dynamics of a flow model leads to the generation of characteristic fluid parameters ( G r , G m , M, F, S c and P r ). The interaction of these parameters with elasto-viscous parameter K ′ is probed to describe how certain parametric range and conditions could be pre-decided to enhance the flow speed past a channel. In particular, the flow dynamics’ alteration in correspondence to the slip parameter’s choice, along with temperature provision to the boundary in temporal pattern, is determined through uniquely calculated exact expressions of velocity, temperature and mass concentration of the fluid. The complex multi-parametric model has been analytically solved using the Laplace and Inverse Laplace transform. Through study of calculated exact expressions, an identification of variables, adversely (M, F, S c and P r ) and favourably ( G r and G m ) affecting the flow speed and temperature has been made. The accuracy of our results have also been tested by computing matching numerical solutions and by graphical reasoning. The verification of existing results of Newtonian fluid with varying boundary condition of velocity and temperature has also been completed, affirming the veracity of present results.

scholarly journals Numerical Solution of Non-Newtonian Fluids Flow Past an Accelerated Vertical Infinite Plate in the Presence of Free Convection Currents

2013 ◽  
Vol 18 (3) ◽  
pp. 761-777
Author(s):  
M. Patel ◽  
M.G. Timol
Keyword(s):  
Fluid Flow ◽  
Free Convection ◽  
Prandtl Number ◽  
Numerical Solution ◽  
Numerical Solutions ◽  
Infinite Plate ◽  
Similarity Analysis ◽  
Convection Current ◽  
Linear Ordinary Differential Equations ◽  
Flow Past

Abstract A similarity analysis of non-Newtonian fluid flow past an accelerated vertical infinite plate in the presence of free convection current is carried out. A group theoretic generalized dimensional analysis is employed to achieve the governing non-linear ordinary differential equations in the most general form. Numerical solutions of these equations are given with the plot of their velocity profiles with the effects of Pr-Prandtl number and Gr-Grashof number

Stress Distributions for a Quarter Plane Containing an Arbitrarily Oriented Crack

1983 ◽  
Vol 50 (1) ◽  
pp. 43-49 ◽  
Author(s):  
L. M. Keer ◽  
J. C. Lee ◽  
T. Mura
Keyword(s):  
Stress Intensity ◽  
Singular Integral ◽  
Numerical Solutions ◽  
Singular Integral Equations ◽  
Integral Transforms ◽  
Stress Distributions ◽  
Intensity Factors ◽  
Quarter Plane ◽  
Dislocation Densities ◽  
Arbitrarily Oriented Crack

A solution for an elastic quarter plane containing an arbitrarily oriented crack is presented. The problem is formulated by means of Mellin integral transforms and reduced to a system of two coupled singular integral equations where the unknown quantities are the dislocation densities that characterize the crack. Numerical solutions are investigated for various orientations of the cracks. In each case the stress intensity factors are computed for the different parameters.

Fractional calculus and Sinc methods

2011 ◽  
Vol 14 (4) ◽  
Author(s):  
Gerd Baumann ◽  
Frank Stenger
Keyword(s):  
Differential Equations ◽  
Fractional Calculus ◽  
Integral Equations ◽  
Fractional Derivatives ◽  
Numerical Solutions ◽  
Singular Integral Equations ◽  
Fractional Integrals ◽  
Sinc Methods ◽  
Weakly Singular Integral Equations ◽  
Fractional Differential

AbstractFractional integrals, fractional derivatives, fractional integral equations, and fractional differential equations are numerically solved by Sinc methods. Sinc methods are able to deal with singularities of the weakly singular integral equations of Riemann-Liouville and Caputo type. The convergence of the numerical method is numerically examined and shows exponential behavior. Different examples are used to demonstrate the effective derivation of numerical solutions for different types of fractional differential and integral equations, linear and non-linear ones. Equations of mixed ordinary and fractional derivatives, integro-differential equations are solved using Sinc methods. We demonstrate that the numerical calculation needed in fractional calculus can be gained with high accuracy using Sinc methods.

A Slip Model for Finite-Element Plasticity

1978 ◽  
Vol 45 (3) ◽  
pp. 527-532 ◽  
Author(s):  
H. M. van Rij ◽  
P. G. Hodge
Keyword(s):  
Finite Element ◽  
Numerical Solutions ◽  
Virtual Work ◽  
Tangential Velocity ◽  
Element Model ◽  
Displacement Component ◽  
Tangential Component ◽  
Normal Displacement ◽  
Perfectly Plastic ◽  
Dimensional Finite Element

Conventional finite-element models are based on displacement or velocity fields which are at least continuous. However, it is known that perfectly plastic materials may exhibit discontinuities of the tangential velocity component along certain lines. In this investigation, a two-dimensional finite-element model is proposed which will allow for such discontinuities. A regular pattern of triangular elements is assumed. The elements are assumed to be rigid, and across the line separating any two adjoining elements the normal displacement component is continuous, but a discontinuity may exist in the tangential component. The defining equations—compatibility, equilibrium, and constitutive—are developed with the aid of the Principle of Virtual Work. Prandtl’s punch problem for contained flow is solved under plane strain conditions. Comparison is made with existing analytical and other numerical solutions, in order to evaluate the merits of allowing for discontinuities.

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