Approximate Analysis of Penetration of a Cylindrical Roller Into a Thin Elastic Coating

1990 ◽  
Vol 112 (3) ◽  
pp. 460-468 ◽  
Author(s):  
Tsung-Ju Gwo ◽  
Thomas J. Lardner
Keyword(s):  
Approximate Solution ◽  
Closed Form ◽  
Approximate Analytical Solution ◽  
Experimental Studies ◽  
Preliminary Design ◽  
Two Dimensional ◽  
Rigid Substrate ◽  
Closed Form Solutions ◽  
Finite Element Calculations ◽  
Cylindrical Roller

An approximate analytical solution to the problem of two-dimensional indentation of a frictionless cylinder into a thin elastic coating bonded to a rigid substrate has been obtained using the approach introduced by Matthewson for axisymmetric indentation. We show by comparing the results of the approximate solution to the exact solutions and to finite element calculations that the approximate solution is accurate for a/h> 2. The advantage of this approach is that the results are expressed in closed form and the accuracy of the approximate solution improves with increasing values of a/h. For a/h>2, for a given load, the theory overestimates the value of a/h compared to the exact solution by less than 10 percent. In many experimental studies and in preliminary design, it is convenient to have closed-form solutions exhibiting the dependence of the parameters.

On the Approximate Solution of Viscous-Flow Problems

1963 ◽  
Vol 30 (2) ◽  
pp. 263-268 ◽  
Author(s):  
J. A. Schetz
Keyword(s):  
Approximate Solution ◽  
Exact Solutions ◽  
Viscous Flow ◽  
Closed Form ◽  
New Method ◽  
Two Dimensional ◽  
General Technique ◽  
Closed Form Solutions ◽  
Flow Problems

The need for a general technique for the approximate solution of viscous-flow problems is discussed. Existing methods are considered and a new method is presented which results in simple closed-form solutions. The accuracy of the method is demonstrated by comparisons with the results of known exact solutions, and finally the general technique is employed to determine a new solution for the fully expanded two-dimensional laminar nozzle problem.

Two-dimensional planar plumes: non-Boussinesq effects

2014 ◽  
Vol 750 ◽  
pp. 245-258 ◽  
Author(s):  
T. S. van den Bremer ◽  
G. R. Hunt
Keyword(s):  
Closed Form ◽  
Uniform Density ◽  
Two Dimensional ◽  
Boussinesq Model ◽  
Axisymmetric Case ◽  
Closed Form Solutions ◽  
Plume Model ◽  
Entrainment Velocity ◽  
Planar Area ◽  
Pure Plume

AbstractIn an accompanying paper (van den Bremer & Hunt, J. Fluid Mech., vol. 750, 2014, pp. 210–244) closed-form solutions, describing the behaviour of two-dimensional planar turbulent rising plumes from horizontal planar area and line sources in unconfined quiescent environments of uniform density, that are universally applicable to Boussinesq and non-Boussinesq plumes, are proposed. This universality relies on an entrainment velocity unmodified by non-Boussinesq effects, an assumption that is derived in the literature based on similarity arguments and is, in fact, in contradiction with the axisymmetric case, in which entrainment is modified by non-Boussinesq effects. Exploring these solutions, we show that a non-Boussinesq plume model predicts exactly the same behaviour with height for a pure plume as would a Boussinesq model, whereas the effects on forced and lazy plumes are opposing. Non-intuitively, the non-Boussinesq model predicts larger fluxes of volume and mass for lazy plumes, but smaller fluxes for forced plumes at any given height compared to the Boussinesq model. This raises significant questions regarding the validity of the unmodified entrainment model for planar non-Boussinesq plumes based on similarity arguments and calls for detailed experiments to resolve this debate.

Gust Loading on a Thin Aerofoil

1971 ◽  
Vol 22 (3) ◽  
pp. 301-310 ◽  
Author(s):  
B. D. Mugridge
Keyword(s):  
Approximate Solution ◽  
Closed Form ◽  
Incompressible Flow ◽  
Velocity Component ◽  
Closed Form Expression ◽  
Sound Power ◽  
Normal Velocity ◽  
Two Dimensional ◽  
Form Expression ◽  
Final Expression

SummaryA closed-form expression is derived which gives an approximate solution to the lift generated on a two-dimensional thin aerofoil in incompressible flow with a normal velocity component of the form exp [i(ωt–xx+yy)]. The inaccuracy of the solution when compared with other published work is compensated by the simplicity of the final expression, particularly if the result is required for the calculation of the sound power radiated by an aerofoil in a turbulent flow.

scholarly journals Two-Dimensional Electrostatic Problem in a Plane with Earthed Elliptic Cavity due to One or Two Collinear Charged Electrostatic Strips

2007 ◽  
Vol 2007 ◽  
pp. 1-9 ◽  
Author(s):  
B. M. Singh ◽  
J. G. Rokne ◽  
R. S. Dhaliwal
Keyword(s):  
Charge Density ◽  
Closed Form ◽  
Integral Equations ◽  
Integral Transform ◽  
Weight Functions ◽  
Two Dimensional ◽  
Electrostatic Problem ◽  
Closed Form Solutions ◽  
Triple Integral Equations ◽  
Elliptic Cavity

A two-dimensional electrostatic problem in a plane with earthed elliptic cavity due to one or two charged electrostatic strips is considered. Using the integral transform technique, each problem is reduced to the solution of triple integral equations with sine kernels and weight functions. Closed-form solutions of the set of triple integral equations are obtained. Also closed-form expressions are obtained for charge density of the strips. Finally, the numerical results for the charge density are given in the form of tables.

On the Displacement of a Two-Dimensional Eshelby Inclusion of Elliptic Cylindrical Shape

2017 ◽  
Vol 84 (7) ◽  
Author(s):  
Xiaoqing Jin ◽  
Xiangning Zhang ◽  
Pu Li ◽  
Zheng Xu ◽  
Yumei Hu ◽  
...  
Keyword(s):  
Closed Form ◽  
Three Dimensional ◽  
Companion Paper ◽  
Stress And Strain ◽  
Cylindrical Shape ◽  
Two Dimensional ◽  
Elasticity Solution ◽  
Strain Fields ◽  
Closed Form Solutions ◽  
Eshelby Inclusion

In a companion paper, we have obtained the closed-form solutions to the stress and strain fields of a two-dimensional Eshelby inclusion. The current work is concerned with the complementary formulation of the displacement. All the formulae are derived in explicit closed-form, based on the degenerate case of a three-dimensional (3D) ellipsoidal inclusion. A benchmark example is provided to validate the present analytical solutions. In conjunction with our previous study, a complete elasticity solution to the classical elliptic cylindrical inclusion is hence documented in Cartesian coordinates for the convenience of engineering applications.

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