Closed-form solution of Cattaneo equation including volumetric source in relation to laser short-pulse heating

2011 ◽  
Vol 89 (7) ◽  
pp. 761-767 ◽  
Author(s):  
H. Al-Qahtani ◽  
B.S. Yilbas
Keyword(s):  
Stress Field ◽  
Closed Form ◽  
Short Pulse ◽  
Closed Form Solution ◽  
Transformation Method ◽  
Substrate Material ◽  
Form Solution ◽  
Cooling Period ◽  
Wave Nature ◽  
Heating Model

The wave nature of the heating model is considered, incorporating the Cattaneo equation with the presence of a volumetric heat source. The volumetric heat generation resembles the step input laser short-pulse intensity. The governing of the heat equation is solved analytically using the Laplace transformation method. The stress field generated due to thermal contraction and expansion of the substrate material is formulated and the closed-form solution is presented. It is found that the wave nature of the heating is dominant during the period of the irradiated short-pulse; however, in the late cooling period, the wave nature of heating is replaced by diffusional heat conduction, governed by Fourier’s law. The stress field during the heating cycle is compressive and becomes tensile in the cooling cycle.

Formulation of laser-induced thermal stresses: Stress boundary at the surface

2003 ◽  
Vol 217 (4) ◽  
pp. 423-434 ◽  
Author(s):  
B S Yilbas ◽  
N Al-Aqeeli
Keyword(s):  
Stress Field ◽  
Closed Form ◽  
Stress Wave ◽  
Laser Heating ◽  
Thermal Stresses ◽  
High Stress ◽  
Closed Form Solution ◽  
Substrate Material ◽  
Solid Bulk ◽  
Stress Boundary

Laser heating of solids results in a high-temperature gradient inside the substrate material. This, in turn, results in high stress levels in the region irradiated by a laser beam. In the present study, laser heating of solid substrate is formulated and a closed-form solution for the stress field inside the substrate material is obtained. Time exponentially decaying laser pulse intensity is employed in the analysis. In order to account for the recoil pressure effect on the resulting stress field, the stress boundary at the solid surface is employed in the analysis. The Laplace transformation method is used when deriving the closed-form solutions. It is found that a stress wave propagates into the solid bulk with a wave speed c1. The amplitude of the stress wave reduces as the distance from the substrate increases towards the solid bulk. The occurrence of peak stress inside the substrate material differes for the stress-free boundary and the stress boundary at the free surface cases.

New Forward and Inverse Solutions for Wet Fins Generalized Profiles With All Nonlinear Phenomena

2020 ◽  
Vol 143 (2) ◽  
Author(s):  
Ranjan Das ◽  
Balaram Kundu
Keyword(s):  
Closed Form ◽  
Radiation Effects ◽  
Transmission Rate ◽  
Closed Form Solution ◽  
Maximum Error ◽  
Transformation Method ◽  
Form Solution ◽  
Nonlinear Phenomena ◽  
Maximum Heat ◽  
Practical Applications

Abstract This study establishes forward closed-form and inverse analyses of wet fins of various profiles involving all modes of heat transfer. Existing limitations in the literature are addressed here by choosing the appropriate nonlinear variation of thermal conductivity and radiation effects. The error between linear and nonlinear methodologies is found to be within 60%. Furthermore, the maximum error between the closed-form solution based on the differential transformation method (DTM), and the numerical solution is observed as 0.5%. After necessary validations, optimization of various fin profiles is carried out by the maximization of the net fin heat transmission rate under a defined fin volume and thermogeometrical constraints. For the optimum criterion, the suitability of the artificial bee colony (ABC)-based metaheuristic technique is established. The identification of thermogeometrical parameters is realized by analyzing combinations obtained from 100 runs of ABC and the decision-making criterion is adopted on the basis of the maximum thermal performance. Among the studied profiles, concave parabolic geometry yields the maximum heat transport rate, which is followed by triangular, convex, and rectangular geometries for the same fin volume. The present combination of DTM and ABC techniques is proposed to be useful in practical applications toward design and the selection of evaporator fins for air-conditioning and refrigeration appliances operating under wet conditions in a more accurate and optimized manner.

Full spectrum of nonlinear Jaynes–Cummings and anti-Jaynes–Cummings models

2017 ◽  
Vol 15 (08) ◽  
pp. 1740008
Author(s):  
Francesco A. Raffa ◽  
Mario Rasetti
Keyword(s):  
Closed Form ◽  
Unitary Transformation ◽  
Energy Levels ◽  
Closed Form Solution ◽  
Transformation Method ◽  
Form Solution ◽  
Full Spectrum ◽  
Nonlinear Generalization ◽  
Jc Model

The unitary transformation method is utilized for the closed form solution of the nonlinear generalization of the Jaynes–Cummings (JC) model, which includes arbitrary multiphoton and intensity-dependent interactions between radiation and matter. Specifically, the unitary transformation leading to the diagonalization of the relevant Hamiltonian is established and energy levels and eigenstates are derived. It is also shown that the results for the JC model can be extended to the anti-Jaynes–Cummings (AJC) model through the map between the two models.

scholarly journals Closed-Form Solution Procedure for Simulating Debonding in FRP Strips Glued to a Generic Substrate Material

Fibers ◽  
2021 ◽  
Vol 9 (4) ◽  
pp. 22
Author(s):  
Enzo Martinelli
Keyword(s):  
Closed Form ◽  
Full Range ◽  
Solution Process ◽  
Closed Form Solution ◽  
Solution Procedure ◽  
Substrate Material ◽  
Form Solution ◽  
Numerical Code ◽  
Mathematical Expressions ◽  
Frp Strips

The present paper proposes a useful closed-form solution for a wide class of mechanical problems, among which one of the most relevant and debated is the deboning process of Fiber-Reinforced Polymer (FRP) strips glued to generic materials and possibly intended as a mode-II fracture process. Specifically, after outlining well-known equations, a novel piecewise analytical formulation based on a cascading solution process is proposed with the aim of keeping the mathematical expressions of the relevant mechanical quantities as simple as possible. Although other analytical solutions and numerical procedures are already available in the literature, the present one is capable of handling the softening or snap-back response deriving from the full-range simulation of the depending process with no need for complex numerical techniques. This is obtained by considering the slip at the free end of the strip as the main displacement control parameter. After some comparisons between the proposed closed-form solution and experimental results available in the literature, some mechanical considerations are highlighted by elaborating on the results of a parametric study considering the variation of the main geometric and mechanical quantities. The numerical code implemented as part of the present study is available to readers in Open Access.

On a Closed-Form Solution of Prandtl's System of Equations

2013 ◽  
Vol 40 (2) ◽  
pp. 106-114
Author(s):  
J. Venetis ◽  
Aimilios (Preferred name Emilios) Sideridis
Keyword(s):  
Closed Form ◽  
Closed Form Solution ◽  
Form Solution ◽  
System Of Equations
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