A semi-analytical solution procedure for predicting damage evolution at interfaces

1993 ◽  
Vol 17 (11) ◽  
pp. 807-819 ◽  
Author(s):  
Zhen Chen
Keyword(s):  
Analytical Solution ◽  
Damage Evolution ◽  
Solution Procedure

scholarly journals A Full-Fledged Analytical Solution to the Quantum Harmonic Oscillator for Undergraduate Students of Science and Engineering

Physics ◽  
2020 ◽  
Vol 2 (4) ◽  
pp. 541-570
Author(s):  
Arturo Rodríguez-Gómez ◽  
Ana Laura Pérez-Martínez
Keyword(s):  
Analytical Solution ◽  
Harmonic Oscillator ◽  
Undergraduate Students ◽  
Solution Procedure ◽  
Quantum Harmonic Oscillator ◽  
Web Based ◽  
Chemistry Students ◽  
Physics Students ◽  
Step Procedure ◽  
And Mathematics

The quantum harmonic oscillator is a fundamental piece of physics. In this paper, we present a self-contained full-fledged analytical solution to the quantum harmonic oscillator. To this end, we use an eight-step procedure that only uses standard mathematical tools available in natural science, technology, engineering and mathematics disciplines. This solution is accessible not only for physics students but also for undergraduate engineering and chemistry students. We provide interactive web-based graphs for the reader to observe the shape of the wave functions for an electron and a proton when both are subject to the same potential. Each of the eight steps in our solution procedure is treated as a separate problem in order to allow the reader to quickly consult any step without the need to review the entire article.

An Analytical Solution of the Generalized Phase-Lagging Equation for Ultra-Fast Heat Transfer in One-Dimensional Semi-Infinite Domain

2012 ◽  
Author(s):  
Vladimir Kulish ◽  
Kirill V. Poletkin
Keyword(s):  
Analytical Solution ◽  
Domain Boundary ◽  
Integral Form ◽  
Solution Procedure ◽  
Generalized Derivatives ◽  
Infinite Domain ◽  
One Dimensional ◽  
Confluent Hypergeometric Functions ◽  
The Laplace Transform ◽  
Derivatives Of

The paper presents an integral solution of the generalized one-dimensional phase-lagging heat equation with the convective term. The solution of the problem has been achieved by the use of a novel technique that involves generalized derivatives (in particular, derivatives of non-integer orders). Confluent hypergeometric functions, known as Whittaker’s functions, appear in the course of the solution procedure, upon applying the Laplace transform to the original transport equation. The analytical solution of the problem is written in the integral form and provides a relationship between the local values of the temperature and heat flux. The solution is valid everywhere within the domain, including the domain boundary.

Analytical Solution for Three-Dimensional, Unsteady Heat Conduction in a Multilayer Sphere

2016 ◽  
Vol 138 (10) ◽  
Author(s):  
Suneet Singh ◽  
Prashant K. Jain ◽  
Rizwan-uddin
Keyword(s):  
Heat Conduction ◽  
Analytical Solution ◽  
Coordinate System ◽  
Three Dimensional ◽  
Solution Procedure ◽  
Polar Coordinate System ◽  
Legendre Functions ◽  
Spherical Coordinates ◽  
Azimuthal Direction ◽  
Spherical Coordinate

An analytical solution has been obtained for the transient problem of three-dimensional multilayer heat conduction in a sphere with layers in the radial direction. The solution procedure can be applied to a hollow sphere or a solid sphere composed of several layers of various materials. In general, the separation of variables applied to 3D spherical coordinates has unique characteristics due to the presence of associated Legendre functions as the eigenfunctions. Moreover, an eigenvalue problem in the azimuthal direction also requires solution; again, its properties are unique owing to periodicity in the azimuthal direction. Therefore, extending existing solutions in 2D spherical coordinates to 3D spherical coordinates is not straightforward. In a spherical coordinate system, one can solve a 3D transient multilayer heat conduction problem without the presence of imaginary eigenvalues. A 2D cylindrical polar coordinate system is the only other case in which such multidimensional problems can be solved without the use of imaginary eigenvalues. The absence of imaginary eigenvalues renders the solution methodology significantly more useful for practical applications. The methodology described can be used for all the three types of boundary conditions in the outer and inner surfaces of the sphere. The solution procedure is demonstrated on an illustrative problem for which results are obtained.

Local Analytical Solution of One Dimension Consolidation Equation

2011 ◽  
Vol 243-249 ◽  
pp. 3113-3116
Author(s):  
Bo Feng ◽  
Run Tao Zhan ◽  
Feng Zhou
Keyword(s):  
Differential Equation ◽  
Analytical Solution ◽  
Pore Water ◽  
Pore Water Pressure ◽  
Water Pressure ◽  
Simple Relation ◽  
Solution Procedure ◽  
Rate Of Change ◽  
One Dimension ◽  
Fractional Differential

One dimension consolidation equation can be transformed into a fractional differential equation by Laplace transform. The transformed equation can leads to a simple relation between pore water pressure and its time revolution. When local rate of change of the pore water pressure is determined, the local pore water pressure can be obtained without having to solve the consolidation equation within the entire domain. The simplicity of the solution procedure is highlighted considering by a example..

scholarly journals A new analytical solution procedure for nonlinear integral equations

2012 ◽  
Vol 55 (7-8) ◽  
pp. 1892-1897 ◽  
Author(s):  
Majid Khan ◽  
Muhammad Asif Gondal ◽  
Sunil Kumar
Keyword(s):  
Analytical Solution ◽  
Integral Equations ◽  
Solution Procedure ◽  
Nonlinear Integral Equations

Cavity Expansion in Soil of Finite Radial Extent

2014 ◽  
Vol 553 ◽  
pp. 464-469
Author(s):  
Mohammad Pournaghiazar ◽  
Adrian R. Russell ◽  
Nasser Khalili
Keyword(s):  
Analytical Solution ◽  
Quartz Sand ◽  
Stress Condition ◽  
Constitutive Models ◽  
Solution Procedure ◽  
Cavity Expansion ◽  
Finite Boundary ◽  
Spherical Cavities ◽  
Radial Extent ◽  
Zero Radius

The problem of drained cavity expansion in a soil of finite radial extent is investigated. Spherical cavities expanded from zero radius subjected to a constant stress condition at the finite boundary are considered. The new analytical solution procedure presented enables more advanced constitutive models to be implemented than possible than when using other solution procedures. Cavity expansion results generated for a Sydney quartz sand highlight substantial differences between cavity limit pressures for boundaries of finite and infinite radial extent.

Non-dimensional thermoelastic model of a compound annular cylinder in a stress-free state with internal heat generation

2020 ◽  
pp. 095440622097369
Author(s):  
Srisharan G Govindarajan ◽  
Gary L Solbrekken
Keyword(s):  
Analytical Solution ◽  
Outer Cylinder ◽  
Internal Heat ◽  
Solution Procedure ◽  
Internal Heating ◽  
Interfacial Separation ◽  
Physical Insight ◽  
Stress Free State ◽  
Dimensional Variable ◽  
Insight Into

A non-dimensional, axisymmetric, thermal-stress model for a three-layer cylinder with internal heating has been developed. Such a geometry is encountered in an annular target for isotope production. The middle cylinder is the heat generating source with an assumed thermal expansion coefficient that is smaller than that of the other two cylinders enclosing it. Hence, the development of the solution is based on the assumption that interfacial separation occurs at the interface of the middle cylinder and the outer cylinder, while contact is reinforced between the middle cylinder and the inner cylinder. The commercial finite element code Abaqus FEA is used to obtain a numerical model which is validated using the developed analytical solution. The non-dimensional analytical solution has been presented in a simplified, generalized form, and can be applied to either of the cylinders by adjusting a few parameters. The non-dimensional variable groupings allow physical insight into how the stresses and temperature distributions evolve. A detailed solution procedure along with a discussion of the results has been provided.

An Analytical Solution of the Generalized Phase-Lagging Equation for Ultra-Fast Heat Transfer in One-Dimensional Semi-Infinite Domain

2012 ◽  
Author(s):  
Vladimir Kulish ◽  
Kirill V. Poletkin
Keyword(s):  
Analytical Solution ◽  
Domain Boundary ◽  
Integral Form ◽  
Solution Procedure ◽  
Generalized Derivatives ◽  
Infinite Domain ◽  
One Dimensional ◽  
Novel Technique ◽  
The Laplace Transform ◽  
Derivatives Of

The paper presents an integral solution of the generalized one-dimensional phase-lagging heat equation with the convective term. The solution of the problem has been achieved by the use of a novel technique that involves generalized derivatives (in particular, derivatives of non-integer orders). Confluent hyper-geometric functions, known as Whittaker’s functions, appear in the course of the solution procedure, upon applying the Laplace transform to the original transport equation. The analytical solution of the problem is written in the integral form and provides a relationship between the local values of the temperature and heat flux. The solution is valid everywhere within the domain, including the domain boundary.

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