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Introduction
Published in William G. Pariseau, Notes on Geoplasticity, 2019
Theories in the mechanics of solids, as a practical matter, are simplified descriptions of actual material behavior. They are continuum models. Plasticity theories describe the response of model solids strained beyond an elastic limit. Prior to reaching this limit, the response is purely elastic in accordance with Hooke’s law. As in any continuum theory, motion of the material must be consistent with physical laws and kinematics, often expressed in terms of stress equations of motion and strain-displacement relationships. A plasticity theory is completed when the elastic limit and the stress-strain relationship beyond the elastic limit are specified. Stress at the elastic limit may be defined as material strength. At the elastic limit, the material is at yield. The constitutive equations (material laws) are then Hooke’s law, a strength criterion, and an elastic-plastic stress-strain relationship. Generally, the resulting system of nonlinear partial differential equations leads to boundary value problems that are too complicated for much analytical progress and for solution to problems of engineering interest. An exception occurs in plane analyses and to a much lesser extent in axial symmetry. In this regard, plane strain analyses are of particular interest and form the focus of much that follows in the form of limiting equilibrium analyses.
Nanosystems, Quantum Mechanics, and Mathematical Models
Published in Sergey Edward Lyshevski, Mems and Nems, 2018
The total energy E(t, r1, r2,…,rN−1, rN) must be found using the nucleus-nucleus Coulomb repulsion energy as well as the electron energy. It is very difficult, or impossible, to solve analytically or numerically the nonlinear partial differential equation (6.2.1). Taking into account only the Coulomb force (electrons and nuclei are assumed to interact due to the Coulomb force only), the Hartree approximation is applied. In particular, the N-electron wavefunction Ψ(t, r1, r2,…,rN−1, rN) is expressed as a product of N one-electron wavefunctions as
Quantum Mechanics and Its Applications
Published in Sergey Edward Lyshevski, Nano- and Micro-Electromechanical Systems, 2018
It is very difficult, or impossible, to solve analytically or numerically the nonlinear partial differential Equation 7.1. Taking into account only the Coulomb force (electrons and nuclei are assumed to interact due to the Coulomb force only), the Hartree approximation is applied. In particular, the N-electron wave function Ψ(t, r1,r1,…rN−1,rN)is expressed as a product of N one-electron wave functions as Ψ(t,r1,r2,⋅⋅⋅,rN−1,rN)=ψ1(t,r1)ψ2(t,r2)⋅⋅⋅ψN−1(t,rN−1)ψN(t,rN)
New conservation laws and exact solutions of coupled Burgers' equation
Published in Waves in Random and Complex Media, 2021
Arzu Akbulut, Mir Sajjad Hashemi, Hadi Rezazadeh
Nonlinear partial differential equations are used describing important models in mathematical physics and engineering. The obtaining exact solutions of nonlinear partial differential equations has been one of the most important area of researchers [1].
Inverse method identification of thermophysical properties based on solotone effect analysis for discontinuous Sturm–Liouville systems
Published in Inverse Problems in Science and Engineering, 2019
The commercial mathematical package Maple was used to implement the solotone analysis method. Working precision was fixed to 20 decimal places by setting the Digits environment variable (Digits:=20). The numerical root finder NextRoot, which is part of the RootFinding package, was used to locate roots of each eigencondition. NextRoot finds the next zero of a function along the real line and can be used to iterate through the roots of a function. Occasionally NextRoot failed to converge to a root, a problem attributed to the number of guard digits being used. This issue was overcome by increasing guardDigits to two (the default value being one). An alternative root finding approach would be to search for roots in a user-defined window. In this way, successive eigenvalues are obtained by marching forward in steps. This is the method used in [3], and implemented using Mathematica. However, with this method, the window size needs to be kept very small to avoid missing an eigenvalue. In certain scenarios, it may be possible to compute lower and upper bounds for each eigenvalue due to the presence of asymptotes within the eigencondition, this is the case in [4] where a 2D Cartesian multilayer problem is discussed. The lower and upper bounds are then used to direct the numerical search. However, even when such eigenvalue bounds exist, it might be prudent to employ additional methods to generate the eigenvalue spectra. This would allow the root finding methodology and software used to be checked for consistency. One possibility would be to use a variational approach, where the eigenvalues of certain partial differential equations are equal to the minima of associated integrals [5, Chapter VI]. In addition, variational methods may be modified to allow for discontinuities in the coefficients of the partial differential equations [6]. An interesting development is the variational iteration method [7] which can provide approximate solutions to nonlinear partial differential equations without linearization or small perturbations. This method has been applied to linear Sturm–Liouville problems [8] and used to obtain approximate solutions to inverse problems involving diffusion equations [9]. The variational iteration method has also been applied to nonlinear oscillators with discontinuities [10]. An alternative method to numerically compute the eigenvalues would be to use the cardinal sine function and related sampling theorems. These methods uses the sinc function to find numerical solutions, and such methods manage discontinuities better than procedures employing polynomials. For example, in [11] the sinc method is used to compute the eigenvalues of discontinuous Sturm–Liouville problems where eigenparameters appear within the boundary conditions.