Nonlinear Periodic Oscillations of Micro-Voids in a Class of Incompressible Hyper-Elastic Spheres

2011 ◽  
Vol 52-54 ◽  
pp. 220-225
Author(s):  
Miao Miao Cai ◽  
Jia Na Meng ◽  
Xue Gang Yuan ◽  
Da Tian Niu
Keyword(s):  
Differential Equation ◽  
Oscillation Amplitude ◽  
Tensile Load ◽  
Periodic Oscillation ◽  
Transversely Isotropic ◽  
Material Model ◽  
Symmetric Motion ◽  
Order Ordinary Differential Equation ◽  
Periodic Oscillations ◽  
Hyper Elastic

The problem of radially symmetric motion is examined for a pre-existing micro-void in the interior of a sphere under a suddenly applied outer surface tensile load, where the sphere is composed of a homogeneous incompressible hyper-elastic material. Through qualitatively analyzing the second-order ordinary differential equation that describes the motion of the pre-existing micro-void with time, some interesting conclusions are proposed. For any given values of surface tensile loads, it is proved that the motion of the pre-existing micro-void with time presents a nonlinear periodic oscillation, however, in certain cases, the oscillation amplitude increases discontinuously with the increasing values of surface tensile loads. Finally, based on the known transversely isotropic incompressible Gent-Thomas material model as an example, numerical simulations are carried out.

scholarly journals Exact Solutions and Conserved Vectors of the Two-Dimensional Generalized Shallow Water Wave Equation

Mathematics ◽  
2021 ◽  
Vol 9 (12) ◽  
pp. 1439
Author(s):  
Chaudry Masood Khalique ◽  
Karabo Plaatjie
Keyword(s):  
Differential Equation ◽  
Wave Equation ◽  
Shallow Water ◽  
Water Wave ◽  
Elliptic Integral ◽  
Momentum Conservation ◽  
Two Dimensional ◽  
Order Ordinary Differential Equation ◽  
Closed Form Solutions ◽  
Shallow Water Wave

In this article, we investigate a two-dimensional generalized shallow water wave equation. Lie symmetries of the equation are computed first and then used to perform symmetry reductions. By utilizing the three translation symmetries of the equation, a fourth-order ordinary differential equation is obtained and solved in terms of an incomplete elliptic integral. Moreover, with the aid of Kudryashov’s approach, more closed-form solutions are constructed. In addition, energy and linear momentum conservation laws for the underlying equation are computed by engaging the multiplier approach as well as Noether’s theorem.

scholarly journals Variation of Passive Biomechanical Properties of the Small Intestine along Its Length: Microstructure-Based Characterization

2021 ◽  
Vol 8 (3) ◽  
pp. 32
Author(s):  
Dimitrios P. Sokolis
Keyword(s):  
Small Intestine ◽  
Orientation Angle ◽  
Axial Direction ◽  
Material Model ◽  
Biomechanical Properties ◽  
Intestinal Wall ◽  
Model Parameters ◽  
Material Models ◽  
Hyper Elastic ◽  
Rat Tissue

Multiaxial testing of the small intestinal wall is critical for understanding its biomechanical properties and defining material models, but limited data and material models are available. The aim of the present study was to develop a microstructure-based material model for the small intestine and test whether there was a significant variation in the passive biomechanical properties along the length of the organ. Rat tissue was cut into eight segments that underwent inflation/extension testing, and their nonlinearly hyper-elastic and anisotropic response was characterized by a fiber-reinforced model. Extensive parametric analysis showed a non-significant contribution to the model of the isotropic matrix and circumferential-fiber family, leading also to severe over-parameterization. Such issues were not apparent with the reduced neo-Hookean and (axial and diagonal)-fiber family model, that provided equally accurate fitting results. Absence from the model of either the axial or diagonal-fiber families led to ill representations of the force- and pressure-diameter data, respectively. The primary direction of anisotropy, designated by the estimated orientation angle of diagonal-fiber families, was about 35° to the axial direction, corroborating prior microscopic observations of submucosal collagen-fiber orientation. The estimated model parameters varied across and within the duodenum, jejunum, and ileum, corroborating histologically assessed segmental differences in layer thicknesses.

Generalized plane stress in a semi-infinite strip under arbitrary end-load

1964 ◽  
Vol 281 (1385) ◽  
pp. 184-206 ◽  
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Boundary Conditions ◽  
Fourth Order ◽  
Plane Stress ◽  
Stress Function ◽  
Infinite Strip ◽  
Order Ordinary Differential Equation ◽  
Orthogonal Functions

The problem involves the determination of a biharmonic generalized plane-stress function satisfying certain boundary conditions. We expand the stress function in a series of non-orthogonal eigenfunctions. Each of these is expanded in a series of orthogonal functions which satisfy a certain fourth-order ordinary differential equation and the boundary conditions implied by the fact that the sides are stress-free. By this method the coefficients involved in the biharmonic stress function corresponding to any arbitrary combination of stress on the end can be obtained directly from two numerical matrices published here The method is illustrated by four examples which cast light on the application of St Venant’s principle to the strip. In a further paper by one of the authors, the method will be applied to the problem of the finite rectangle.

Dynamic Stability of an Axially Moving Yarn with Time-Dependent Tension

2014 ◽  
Vol 900 ◽  
pp. 753-756 ◽  
Author(s):  
You Guo Li
Keyword(s):  
Differential Equation ◽  
Homotopy Perturbation Method ◽  
Time Dependent ◽  
Second Law ◽  
Order Ordinary Differential Equation ◽  
Vibration Equation ◽  
Transversal Vibration ◽  
First Order ◽  
Homotopy Perturbation ◽  
Axially Moving

In this paper the nonlinear transversal vibration of axially moving yarn with time-dependent tension is investigated. Yarn material is modeled as Kelvin element. A partial differential equation governing the transversal vibration is derived from Newtons second law. Galerkin method is used to truncate the governing nonlinear differential equation, and thus first-order ordinary differential equation is obtained. The periodic vibration equation and the natural frequency of moving yarn are received by applying homotopy perturbation method. As a result, the condition which should be avoided in the weaving process for resonance is obtained.

EXISTENCE THEOREMS OF POSITIVE SOLUTIONS FOR FOURTH-ORDER FOUR-POINT BOUNDARY VALUE PROBLEMS

2004 ◽  
Vol 02 (01) ◽  
pp. 71-85 ◽  
Author(s):  
YUJI LIU ◽  
WEIGAO GE
Keyword(s):  
Differential Equation ◽  
Boundary Value Problems ◽  
Positive Solutions ◽  
Fourth Order ◽  
Existence Theorems ◽  
Boundary Value ◽  
Growth Conditions ◽  
Point Boundary ◽  
Order Ordinary Differential Equation ◽  
Order Four

In this paper, we study four-point boundary value problems for a fourth-order ordinary differential equation of the form [Formula: see text] with one of the following boundary conditions: [Formula: see text] or [Formula: see text] Growth conditions on f which guarantee existence of at least three positive solutions for the problems (E)–(B1) and (E)–(B2) are imposed.

scholarly journals On the Existence of Eigenmodes of Linear Quasi-Periodic Differential Equations and their Relation to the MHD Continuum

1982 ◽  
Vol 37 (8) ◽  
pp. 830-839 ◽  
Author(s):  
A. Salat
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Differential Equations ◽  
Infinite Sequence ◽  
Periodic Coefficient ◽  
Second Order ◽  
Order Ordinary Differential Equation ◽  
Magnetic Surfaces ◽  
Toroidal Geometry

The existence of quasi-periodic eigensolutions of a linear second order ordinary differential equation with quasi-periodic coefficient f{ω1t, ω2t) is investigated numerically and graphically. For sufficiently incommensurate frequencies ω1, ω2, a doubly indexed infinite sequence of eigenvalues and eigenmodes is obtained.The equation considered is a model for the magneto-hydrodynamic “continuum” in general toroidal geometry. The result suggests that continuum modes exist at least on sufficiently ir-rational magnetic surfaces

DERIVATION OF 2-POINT ZERO STABLENUMERICAL ALGORITHM OF BLOCK BACKWARD DIFFERENTIATION FORMULA FOR SOLVING FIRST ORDER ORDINARY DIFFERENTIAL EQUATIONS

2021 ◽  
Vol 5 (2) ◽  
pp. 579-583
Author(s):  
Muhammad Abdullahi ◽  
Bashir Sule ◽  
Mustapha Isyaku
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Taylor Series ◽  
Taylor Series Expansion ◽  
Order Ordinary Differential Equation ◽  
Differentiation Formula ◽  
First Order ◽  
Backward Differentiation Formula

This paper is aimed at deriving a 2-point zero stable numerical algorithm of block backward differentiation formula using Taylor series expansion, for solving first order ordinary differential equation. The order and zero stability of the method are investigated and the derived method is found to be zero stable and of order 3. Hence, the method is suitable for solving first order ordinary differential equation. Implementation of the method has been considered

Sign in / Sign up

Export Citation Format

Share Document