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.

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.

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.

The Strain-Energy Density of Compressible, Rubber-Like Axishells

1987 ◽  
Vol 54 (2) ◽  
pp. 453-454 ◽  
Author(s):  
J. G. Simmonds
Keyword(s):  
Strain Energy Density ◽  
Strain Energy ◽  
Large Strain ◽  
Axisymmetric Deformation ◽  
Strain Theory ◽  
Three Dimensions ◽  
Compressible Material ◽  
Shells Of Revolution ◽  
First Approximation ◽  
Recent Method

A recent method for deriving, by descent from three dimensions, strain-energy densities in a first-approximation, large-strain theory of incompressible, elastically isotropic shells of revolution undergoing torsionless, axisymmetric deformation (axishells) is adapted to axishells made of compressible material. Application is made to the Blatz-Ko strain-energy densities for a polyurethane (“continuum”) rubber and a foam rubber.

Stress Optical Behavior of Rubber Networks at Large Deformations

1966 ◽  
Vol 39 (5) ◽  
pp. 1436-1450
Author(s):  
K. J. Smith ◽  
D. Puett
Keyword(s):  
Experimental Data ◽  
Mechanical Properties ◽  
Statistical Theory ◽  
Natural Rubber ◽  
Large Deformations ◽  
Strong Support ◽  
Theoretical Development ◽  
Statistical Parameters ◽  
Strain Behavior ◽  
Optical Behavior

Abstract The birefringence of natural rubber networks at large deformations has been investigated experimentally and compared with the simultaneously determined stress—strain behavior. Our data is analyzed using a statistical theory of flexibly jointed chains, derived herein, which is believed to be more significant for the particular range of deformation used than the theories of Treloar and of Kuhn and Grün. In addition, the experimental data of Saunders is commented on in light of our theoretical development. We find that for network extensions exceeding those of the Gaussian region there is little correlation between the observed and theoretical behavior of the stress and birefringence (based upon the theory of flexibly jointed chains) and this lack of agreement is attributed to the fact that the statistical parameters needed for the description of the optical chain properties differ in magnitude from those required for the mechanical properties. Furthermore, by considering the points of incipient crystallization the strain behavior of the stress-optical coefficient is highly indicative of nonGaussian behavior rather than crystallization, and therefore yields strong support for the position that nonGaussian behavior does exist in rubber networks.

Continuum Description of the Time- and Strain-Dependent Poisson’s Ratio of Ligament Under Finite Deformation

2013 ◽  
Author(s):  
Aaron M. Swedberg ◽  
Shawn P. Reese ◽  
Steve A. Maas ◽  
Benjamin J. Ellis ◽  
Jeffrey A. Weiss
Keyword(s):  
Large Scale ◽  
Mechanical Response ◽  
Tensile Deformation ◽  
Large Strain ◽  
Fiber Direction ◽  
Strain Behavior ◽  
Nonlinear Stress ◽  
Poisson's Ratios ◽  
Poisson’S Ratios ◽  
Strain Dependent

Ligament volumetric behavior controls fluid and thus nutrient movement as well as the mechanical response of the tissue to applied loads. The reported Poisson’s ratios for tendon and ligament subjected to tensile deformation loading along the fiber direction are large, ranging from 0.8 ± 0.3 in rat tail tendon fascicles [1] to 2.98 ± 2.59 in bovine flexor tendon [2]. These Poisson’s ratios are indicative of volume loss and thus fluid exudation [3,4]. We have developed micromechanical finite element models that can reproduce both the characteristic nonlinear stress-strain behavior and large, strain-dependent Poisson’s ratios seen in tendons and ligaments [5], but these models are computationally expensive and unfeasible for large scale, whole joint models. The objectives of this research were to develop an anisotropic, continuum based constitutive model for ligaments and tendons that can describe strain-dependent Poisson’s ratios much larger than the isotropic limit of 0.5. Further, we sought to demonstrate the ability of the model to describe experimental data, and to show that the model can be combined with biphasic theory to describe the rate- and time-dependent behavior of ligament and tendon.

The Strength of Thick-Walled Cylinders

1959 ◽  
Vol 81 (2) ◽  
pp. 95-111 ◽  
Author(s):  
B. Crossland ◽  
S. M. Jorgensen ◽  
J. A. Bones
Keyword(s):  
Shear Stress ◽  
Mild Steel ◽  
Close Agreement ◽  
Large Strain ◽  
Tension Test ◽  
Experimental Results ◽  
Maximum Pressure ◽  
Strain Behavior ◽  
Pressure Tests ◽  
Strain Data

Comprehensive pressure tests have been carried out on thick-walled, closed-ended cylinders made from a mild steel and a hardened and tempered steel, the maximum pressure reached being 94,000 lb/in.2 The complete theoretical behavior of the cylinders is computed from shear stress-strain data obtained from torsion tests and is shown to be in very close agreement with the experimental results. In addition, a method is given for deriving the large strain behavior of the cylinders from tension test data. When compared with the experimental results this approach gives larger errors, the theoretical values of pressure being consistently high. Finally, ultimate pressures have been calculated from two empirical expressions.

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