Degenerate Scale Problem for a Hypocycloid Hole in an Infinite Plate in Plane Elasticity

2016 ◽  
Vol 32 (4) ◽  
pp. N7-N10
Author(s):  
Y.-Z. Chen
Keyword(s):  
Boundary Condition ◽  
Conformal Mapping ◽  
Closed Form ◽  
Infinite Plate ◽  
Closed Form Solution ◽  
Plane Elasticity ◽  
Form Solution ◽  
Scale Problem ◽  
Exterior Region ◽  
Degenerate Scale

AbstractBased on the conformal mapping, this paper provides a closed form solution for the degenerate scale of the hypocycloid hole in plane elasticity. In the derivation, we assume the vanishing displacements along the boundary in the degenerate scale problem. Some functions in the boundary condition are decomposed into three parts with particular behavior. Even the displacements are vanishing along the boundary of an exterior region, the displacements and stresses are not equal to zero in the exterior region. This is a particular feature in the degenerate scale problem.

Closed form Solution for Degenerate Scale Problem of Joukowcki Airfoil Configuration in Antiplane Elasticity

2013 ◽  
Vol 29 (4) ◽  
pp. N21-N23 ◽  
Author(s):  
Y. Z. Chen
Keyword(s):  
Conformal Mapping ◽  
Closed Form ◽  
Closed Form Solution ◽  
Mapping Function ◽  
Form Solution ◽  
Scale Problem ◽  
Thin Airfoil ◽  
Degenerate Scale ◽  
Antiplane Elasticity

ABSTRACTThis paper provides a closed form solution for degenerate scale problem of the Joukowcki thin airfoil configuration in antiplane elasticity. The solution depends on a conformal mapping function for the Joukowcki thin airfoil configuration.

Thermal and Fluid Dynamic Model for Capillary-Driven Heat Pipe With Closed-Form Solution

2017 ◽  
Author(s):  
Jesse Maxwell
Keyword(s):  
Boundary Condition ◽  
Heat Flux ◽  
Closed Form ◽  
Heat Pipe ◽  
Closed Form Solution ◽  
Constant Heat Flux ◽  
Form Solution ◽  
Constant Heat ◽  
Micro Channels ◽  
Flow Through

A model is derived for the steady state performance of capillary-driven heat pipes on the basis treating fluid flow through miniature- and micro-channels and applied as bulk properties to a large aspect ratio quasi-one-dimensional two-phase system. Surface tension provides the driving force based on an equivalent bulk capillary radius while laminar flow through micro-channels and the vapor core are treated. Heat conduction is accounted for radially while isothermal advection is treated along the axis. A closed-form solution is derived for a steady state heat pipe with a constant heat flux boundary condition on the evaporator as well as a constant heat flux or a convective boundary condition along the condenser. Two solution methods are proposed, and the result is compared to empirical data for a copper-water heat pipe. The components of the closed-form solution are discussed as contributors to driving or frictional forces, and the existence of an optimal pore radius is demonstrated.

scholarly journals Buckling Behavior of Nanobeams Placed in Electromagnetic Field Using Shifted Chebyshev Polynomials-Based Rayleigh-Ritz Method

Nanomaterials ◽  
2019 ◽  
Vol 9 (9) ◽  
pp. 1326 ◽  
Author(s):  
Subrat Kumar Jena ◽  
Snehashish Chakraverty ◽  
Francesco Tornabene
Keyword(s):  
Boundary Condition ◽  
Closed Form ◽  
Chebyshev Polynomials ◽  
Ritz Method ◽  
Closed Form Solution ◽  
Form Solution ◽  
Buckling Load ◽  
Mode Shapes ◽  
Critical Buckling Load ◽  
Buckling Behavior

In the present investigation, the buckling behavior of Euler–Bernoulli nanobeam, which is placed in an electro-magnetic field, is investigated in the framework of Eringen’s nonlocal theory. Critical buckling load for all the classical boundary conditions such as “Pined–Pined (P-P), Clamped–Pined (C-P), Clamped–Clamped (C-C), and Clamped-Free (C-F)” are obtained using shifted Chebyshev polynomials-based Rayleigh-Ritz method. The main advantage of the shifted Chebyshev polynomials is that it does not make the system ill-conditioning with the higher number of terms in the approximation due to the orthogonality of the functions. Validation and convergence studies of the model have been carried out for different cases. Also, a closed-form solution has been obtained for the “Pined–Pined (P-P)” boundary condition using Navier’s technique, and the numerical results obtained for the “Pined–Pined (P-P)” boundary condition are validated with a closed-form solution. Further, the effects of various scaling parameters on the critical buckling load have been explored, and new results are presented as Figures and Tables. Finally, buckling mode shapes are also plotted to show the sensitiveness of the critical buckling load.

AN INNOVATIVE SOLUTION IN CLOSED FORM AND NUMERICAL ANALYSIS FOR DISSIMILAR ELLIPTICAL INCLUSION IN PLANE ELASTICITY

2014 ◽  
Vol 06 (06) ◽  
pp. 1450080 ◽  
Author(s):  
Y. Z. CHEN
Keyword(s):  
Closed Form ◽  
Closed Form Solution ◽  
Plane Elasticity ◽  
Form Solution ◽  
Complex Potentials ◽  
Infinite Matrix ◽  
Elliptical Inclusion ◽  
The Matrix ◽  
Innovative Solution ◽  
Remote Loading

This paper provides a closed form solution for dissimilar elliptical inclusion in plane elasticity. A dissimilar elliptical inclusion is embedded in the infinite matrix with different elastic properties. The infinite matrix is applied by the constant remote loading. Complex variable method is used and two sets of the complex potentials are assumed in the analysis. One set is used for the matrix portion, and other for the inclusion portion. Catching the idea from the eigenstrain problem, we can assume the stresses in the inclusion to be constant. From the continuity conditions for stresses and displacements along the interface, we can get the two sets of the complex potentials in a closed form. In the analysis, an adequate form of the complex potential defined in the elliptical inclusion portion is analyzed in detail.

Exact solution for time exponentially varying pulsed laser heating: Convective boundary condition case

2001 ◽  
Vol 215 (5) ◽  
pp. 591-606 ◽  
Author(s):  
M Kalyon ◽  
B S Yilbas
Keyword(s):  
Boundary Condition ◽  
Exact Solution ◽  
Closed Form ◽  
Temperature Rise ◽  
Laser Heating ◽  
Closed Form Solution ◽  
Convective Boundary Condition ◽  
Form Solution ◽  
Heating Process ◽  
Convective Boundary

Laser heating offers considerable advantages over conventional methods. The closed-form solution for the temperature rise in the substrate during the laser heating process gives insight into the physical phenomena involving during the heating process and the material response to a laser heating pulse. In the present study, the exact solution for the temperature rise due to a time exponentially varying pulse and convective boundary condition at the surface is obtained. The closed-form solution to the solutions available in the literature for a step input intensity pulse with a convective boundary condition at the surface as well as a time exponentially varying pulse with a non-convective boundary condition at the surface is deduced. A Laplace transformation method is used in the analysis. In order to account for a pulse resembling a typical laser pulse, an intensity function resulting in exponentially increasing and decaying intensity distribution is employed in the source term in the governing transport equation. The effects of the pulse parameters β′, β′/γ′ and Biot number Bi on the resulting temperature profiles are presented and the material response to a pulse profile resembling a typical actual laser pulse is discussed. It is found that the closed-form solution obtained in the present study becomes identical with those presented in the previous studies for different pulse and boundary conditions. Moreover, the coupling effect of pulse parameter β and Bi is significant for the temperature rise at the surface.

A SIMPLE MODEL FOR FRACTURE IN A LAMINATED STRIP AND AN APPROXIMATE SOLUTION IN CLOSED FORM

1994 ◽  
Vol 05 (02) ◽  
pp. 219-221
Author(s):  
T.Y. Fan ◽  
M. Maier ◽  
S. Kerth
Keyword(s):  
Simple Model ◽  
Closed Form ◽  
Closed Form Solution ◽  
Plane Elasticity ◽  
Form Solution ◽  
Mapping Technique ◽  
Nonlinear Behaviour ◽  
Complex Variable Function ◽  
Variable Function ◽  
Material Nonlinear

This paper presents a simple model on the fracture of a laminated strip. Based on the method of complex variable function for solving plane elasticity and the confomal mapping technique, a closed form solution for the simplified model is constructed. The most important physical quantity for fracture analysis—the stress intensity factor—includes analytically the effects of structure geometry and the material constants. The analysis is very easily extended to solve the problem of material nonlinear behaviour and the problem of any 2n+1 layers with n+1 different materials. These may provide useful insight into the field of interlaminar fracture of composite materials.

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