Finite Axisymmetric Deformation of Rubber-Like Shells of Revolution

1988 ◽  
Vol 55 (2) ◽  
pp. 332-340
Author(s):  
W. J. Keppel ◽  
D. A. DaDeppo
Keyword(s):  
Differential Equations ◽  
Axisymmetric Deformation ◽  
Rate Equations ◽  
Shells Of Revolution ◽  
Tangent Stiffness Matrix ◽  
Strain Energy Density Function ◽  
History Of ◽  
Generalized Newton ◽  
Raphson Iteration ◽  
Incremental Loading

The finite axisymmetric deformation of a thin shell of revolution is treated in this analysis. The governing differential equations are given for a hyperelastic shell material with the Mooney-Rivlin strain-energy-density function. These equations are solved numerically using a 4th-order Runge-Kutta integration method. A generalized Newton-Raphson iteration procedure is used to systematically improve trial solutions of the differential equations. The governing differential equations are differentiated with respect to a set of generalized coordinates to derive associated rate equations. The rate equations are solved numerically to generate the tangent stiffness matrix which is used to determine the load deformation history of the shell with incremental loading. Numerical examples are presented to illustrate the major characteristics of nonlinear shell behavior.

On a Nonlinear Theory for Muscle Shells: Part I—Theoretical Development

1991 ◽  
Vol 113 (1) ◽  
pp. 56-62 ◽  
Author(s):  
L. A. Taber
Keyword(s):  
Muscle Activation ◽  
Large Strain ◽  
Axisymmetric Deformation ◽  
Theoretical Development ◽  
Nonlinear Material ◽  
Shells Of Revolution ◽  
Strain Behavior ◽  
Strain Energy Density Function ◽  
Lines Of Curvature ◽  
Transverse Shear Strains

This paper presents a theory for studies of the large-strain behavior of biological shells composed of layers of incompressible, orthotropic tissue, possibly muscle, of arbitrary orientation. The intrinsic equations of the laminated-shell theory, expressed in lines-of-curvature coordinates, account for large membrane [O(1)] and moderately large bending and transverse shear strains [O(0.3)], nonlinear material properties, and transverse normal stress and strain. An expansion is derived for a general two-dimensional strain-energy density function, which includes residual stress and muscle activation through a shifting zero-stress configuration. Strain-displacement relations are given for the special case of axisymmetric deformation of shells of revolution with torsion.

scholarly journals The Ermakov equation: A commentary

2008 ◽  
Vol 2 (2) ◽  
pp. 146-157 ◽  
Author(s):  
P.G.L. Leach ◽  
S.K. Andriopoulos
Keyword(s):  
Nonlinear Systems ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
English Translation ◽  
Singularity Theory ◽  
Discrete Math ◽  
Previous Article ◽  
Short History ◽  
History Of ◽  
Modern Context

We present a short history of the Ermakov equation with an emphasis on its discovery by thewest and the subsequent boost to research into invariants for nonlinear systems although recognizing some of the significant developments in the east. We present the modern context of the Ermakov equation in the algebraic and singularity theory of ordinary differential equations and applications to more divers fields. The reader is referred to the previous article (Appl. Anal. Discrete math., 2 (2008), 123-145) for an english translation of Ermakov's original paper.

The Axisymmetric Deformation of Linearly and Nonlinearly Elastic Spinning Tubes Under End Thrusts and Torques

1998 ◽  
Vol 65 (1) ◽  
pp. 99-106
Author(s):  
T. J. McDevitt ◽  
J. G. Simmonds
Keyword(s):  
Anisotropic Material ◽  
Large Strain ◽  
Axisymmetric Deformation ◽  
Large Strains ◽  
Shells Of Revolution ◽  
Large Rotation ◽  
Elastic Tubes ◽  
Axisymmetric Bending ◽  
Combined Parameters ◽  
Nonlinearly Elastic

We consider the steady-state deformations of elastic tubes spinning steadily and attached in various ways to rigid end plates to which end thrusts and torques are applied. We assume that the tubes are made of homogeneous linearly or nonlinearly anisotropic material and use Simmonds” (1996) simplified dynamic displacement-rotation equations for shells of revolution undergoing large-strain large-rotation axisymmetric bending and torsion. To exploit analytical methods, we confine attention to the nonlinear theory of membranes undergoing small or large strains and the theory of strongly anisotropic tubes suffering small strains. Of particular interest are the boundary layers that appear at each end of the tube, their membrane and bending components, and the penetration of these layers into the tube which, for certain anisotropic materials, may be considerably different from isotropic materials. Remarkably, we find that the behavior of a tube made of a linearly elastic, anisotropic material (having nine elastic parameters) can be described, to a first approximation, by just two combined parameters. The results of the present paper lay the necessary groundwork for a subsequent analysis of the whirling of spinning elastic tubes under end thrusts and torques.

New Integration Techniques for Chemical Kinetic Rate Equations: Part II—Accuracy Comparison

1986 ◽  
Vol 108 (2) ◽  
pp. 348-353 ◽  
Author(s):  
K. Radhakrishnan
Keyword(s):  
Differential Equations ◽  
Time Derivative ◽  
Iterative Solution ◽  
Chemical Kinetic ◽  
Conservation Equation ◽  
General Purpose ◽  
Rate Equations ◽  
Arbitrary System ◽  
Kinetic Rate ◽  
Integration Techniques

A comparison of the accuracy of several techniques recently developed for solving stiff differential equations is presented. The techniques examined include two general-purpose codes EPISODE and LSODE developed for an arbitrary system of ordinary differential equations, and three specialized codes CHEMEQ, CREK1D, and GCKP84 developed specifically to solve chemical kinetic rate equations. The accuracy comparisons are made by applying these solution procedures to two practical combustion kinetics problems. Both problems describe adiabatic, homogeneous, gas-phase chemical reactions at constant pressure, and include all three combustion regimes: induction, heat release, and equilibration. The comparisons show that LSODE is the most efficient code—in the sense that it requires the least computational work to attain a specified accuracy level—currently available for chemical kinetic rate equations. An important finding is that an iterative solution of the algebraic enthalpy conservation equation for the temperature can be more accurate and efficient than computing the temperature by integrating its time derivative.

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1980 ◽  
Vol 64 (430) ◽  
pp. 304-306
Keyword(s):  
Differential Equations ◽  
Statistical Mechanics ◽  
Numerical Solution ◽  
Primary Schools ◽  
History Of Mathematics ◽  
Penguin Book ◽  
Cambridge University ◽  
Princeton University ◽  
Calculus Students ◽  
History Of

Axisymmetric deformation of flexible nonhomogeneous shells of revolution

1982 ◽  
Vol 18 (4) ◽  
pp. 314-318
Author(s):  
V. I. Klimanov ◽  
V. V. Chupin
Keyword(s):  
Axisymmetric Deformation ◽  
Shells Of Revolution
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