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- Keyword
-
- Blow-up, Euler equations (1)
- Critical buckling load, Hollomon’s law, Axial plastic columns, High strength metals, Work-hardening (1)
- Finite element solution, Ritz-Galerkin method, Nonlinear Euler-Bernoulli beam, Power-law, Work hardening material, Hollomon's equation, Convergence, Error estimate, Hermite elements (1)
- Generalized Proudman-Johnson equation (1)
- Generalized sine, Hamilton system, nonlinear spring, vibration, analytic solution (1)
- Publication
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Articles 1 - 9 of 9
Full-Text Articles in Physical Sciences and Mathematics
Blow-Up Of Solutions To The Generalized Inviscid Proudman-Johnson Equation, Alejandro Sarria
Blow-Up Of Solutions To The Generalized Inviscid Proudman-Johnson Equation, Alejandro Sarria
University of New Orleans Theses and Dissertations
The generalized inviscid Proudman-Johnson equation serves as a model for n-dimensional incompressible Euler flow, gas dynamics, high-frequency waves in shallow waters, and orientation of waves in a massive director field of a nematic liquid crystal. Furthermore, the equation also serves as a tool for studying the role that the natural fluid processes of convection and stretching play in the formation of spontaneous singularities, or of their absence.
In this work, we study blow-up, and blow-up properties, in solutions to the generalized, inviscid Proudman-Johnson equation endowed with periodic or Dirichlet boundary conditions. More particularly,regularity of solutions in an Lp setting will …
Analytic And Finite Element Solutions Of The Power-Law Euler-Bernoulli Beams, Dongming Wei, Yu Liu
Analytic And Finite Element Solutions Of The Power-Law Euler-Bernoulli Beams, Dongming Wei, Yu Liu
Mathematics Faculty Publications
In this paper, we use Hermite cubic finite elements to approximate the solutions
of a nonlinear Euler-Bernoulli beam equation. The equation is derived
from Hollomon’s generalized Hooke’s law for work hardening materials with
the assumptions of the Euler-Bernoulli beam theory. The Ritz-Galerkin finite
element procedure is used to form a finite dimensional nonlinear program
problem, and a nonlinear conjugate gradient scheme is implemented to find
the minimizer of the Lagrangian. Convergence of the finite element approximations
is analyzed and some error estimates are presented. A Matlab finite
element code is developed to provide numerical solutions to the beam equation.
Some …
Influence Of Damping On Hyperbolic Equations With Parabolic Degeneracy, Katarzyna Saxton, Ralph Saxton
Influence Of Damping On Hyperbolic Equations With Parabolic Degeneracy, Katarzyna Saxton, Ralph Saxton
Mathematics Faculty Publications
This paper examines the effect of damping on a nonstrictly hyperbolic 2 x 2 system. It is shown that the growth of singularities is not restricted as in the strictly hyperbolic case where dissipation can be strong enough to preserve the smoothness of solutions globally in time. Here, irrespective of the stabilizing properties of damping, solutions are found to break down in finite time on a line where two eigenvalues coincide in state space.
Critical Buckling Loads Of The Perfect Hollomon’S Power-Law Columns, Dongming Wei, Alejandro Sarria, Mohamed Elgindi
Critical Buckling Loads Of The Perfect Hollomon’S Power-Law Columns, Dongming Wei, Alejandro Sarria, Mohamed Elgindi
Mathematics Faculty Publications
In this work, we present analytic formulas for calculating the critical buckling states of some plastic axial columns of constant cross-sections. The associated critical buckling loads are calculated by Euler-type analytic formulas and the associated deformed shapes are presented in terms of generalized trigonometric functions. The plasticity of the material is defined by the Holloman’s power-law equation. This is an extension of the Euler critical buckling loads of perfect elastic columns to perfect plastic columns. In particular, critical loads for perfect straight plastic columns with circular and rectangular cross-sections are calculated for a list of commonly used metals. Connections and …
On The Global Solvability Of A Class Of Fourth-Order Nonlinear Boundary Value Problems, M.B.M. Elgindi, Dongming Wei
On The Global Solvability Of A Class Of Fourth-Order Nonlinear Boundary Value Problems, M.B.M. Elgindi, Dongming Wei
Mathematics Faculty Publications
In this paper we prove the global solvability of a class of fourth-order nonlinear boundary value problems that govern the deformation of a Hollomon’s power-law plastic beam subject to an axial compression and nonlinear lateral constrains. For certain ranges of the acting axial compression force, the solvability of the equations follows from the monotonicity of the fourth order nonlinear differential operator. Beyond these ranges the monotonicity of the operator is lost. It is shown that, in this case, the global solvability may be generated by the lower order nonlinear terms of the equations for a certain type of constrains.
Travelling Wave Solutions Of Burgers' Equation For Gee-Lyon Fluid Flows, Dongming Wei, Ken Holladay
Travelling Wave Solutions Of Burgers' Equation For Gee-Lyon Fluid Flows, Dongming Wei, Ken Holladay
Mathematics Faculty Publications
In this work we present some analytic and semi-analytic traveling wave solutions of generalized Burger' equation for isothermal unidirectional flow of viscous non-Newtonian fluids obeying Gee-Lyon nonlinear rheological equation. The solution of Burgers' equation for Newtonian flow as a special case. We also derive estimates of shock thickness for non-Newtonian flows.
An H1 Model For Inextensible Strings, Stephen C. Preston, Ralph Saxton
An H1 Model For Inextensible Strings, Stephen C. Preston, Ralph Saxton
Mathematics Faculty Publications
We study geodesics of the H1 Riemannian metric (see article for equation) on the space of inextensible curves (see article for equation). This metric is a regularization of the usual L2 metric on curves, for which the submanifold geometry and geodesic equations have been analyzed already. The H1 geodesic equation represents a limiting case of the Pochhammer-Chree equation from elasticity theory. We show the geodesic equation is C∞ in the Banach topology C1 ([0,1], R2), and thus there is a smooth Riemannian exponential map. Furthermore, if we hold one of the curves fixed, …
Blow-Up Of Solutions To The Generalized Inviscid Proudman-Johnson Equation, Alejandro Sarria, Ralph Saxton
Blow-Up Of Solutions To The Generalized Inviscid Proudman-Johnson Equation, Alejandro Sarria, Ralph Saxton
Mathematics Faculty Publications
For arbitrary values of a parameter --- finite-time blowup of solutions to the generalized, inviscid Proudman Johnson equation is studied via a direct approach which involves the derivation of representation formulae for solutions to the problem.
Some Generalized Trigonometric Sine Functions And Their Applications, Dongming Wei, Yu Liu, Mohamed B. Elgindi
Some Generalized Trigonometric Sine Functions And Their Applications, Dongming Wei, Yu Liu, Mohamed B. Elgindi
Mathematics Faculty Publications
In this paper, it is shown that D. Shelupsky's generalized sine function, and various general sine functions developed by P. Drabek, R. Manasevich and M. Otani, P. Lindqvist, including the generalized Jacobi elliptic sine function of S. Takeuchi can be defined by systems of first order nonlinear ordinary differential equations with initial conditions. The structure of the system of differential equations is shown to be related to the Hamilton System in Lagrangian Mechanics. Numerical solutions of the ODE systems are solved to demonstrate the sine functions graphically. It is also demonstrated that the some of the generalized sine functions can …