Large Deformations of a Rotating Solid Cylinder for Non-Gaussian Isotropic, Incompressible Hyperelastic Materials

2000 ◽  
Vol 68 (1) ◽  
pp. 115-117 ◽  
Author(s):  
C. O. Horgan ◽  
G. Saccomandi
Keyword(s):  
Angular Velocity ◽  
Strain Energy ◽  
Large Deformations ◽  
Rotating Cylinder ◽  
Solid Cylinder ◽  
Hyperelastic Materials ◽  
Steady Rotation ◽  
Non Gaussian ◽  
Limiting Value ◽  
Small Angular Velocity

The purpose of this research is to investigate the steady rotation of a solid cylinder for a class of strain-energy densities that are able to describe hardening phenomena in rubber. It is well known that use of the classic neo-Hookean strain energy gives rise to physically unrealistic response in this problem. In particular, solutions exist only for a sufficiently small angular velocity. As the velocity approaches this limiting value, the analysis predicts that the rotating cylinder collapses to a disk. It is shown here that this nonphysical behavior does not occur when generalized neo-Hookean models, which exhibit hardening at large deformations, are used.

Failure Prediction of Unidirectional Composites Undergoing Large Deformations1

2015 ◽  
Vol 82 (7) ◽  
Author(s):  
Jacob Aboudi ◽  
Konstantin Y. Volokh
Keyword(s):  
Strain Energy ◽  
Large Deformations ◽  
Static Stability ◽  
Energy Functions ◽  
Micromechanical Analysis ◽  
Hyperelastic Materials ◽  
Failure Envelopes ◽  
Special Cases ◽  
Enhanced Strain ◽  
Strain Energy Functions

In previous publications, strain-energy functions with limiters have been introduced for the prediction of onset of failure in monolithic isotropic hyperelastic materials. In the present investigation, such enhanced strain-energy functions whose ability to accumulate energy is limited have been incorporated with a finite strain micromechanical analysis. As a result, macroscopic constitutive equations have been established which are capable to predict the onset of loss of static stability in a hyperelastic phase of composite materials undergoing large deformations. The details of the micromechanical analysis, based on a tangential formulation, for composites with periodic microstructure are presented. The derived micromechanical analysis includes the capability to model a possible imperfect bonding between the composite’s constituents and to provide the field distribution in the composite. The micromechanical method is verified by comparison with analytical and finite difference solutions for porous hyperelastic materials that are valid in some special cases. Results are given for a rubberlike matrix characterized by softening hyperelasticity, reinforced by unidirectional nylon fibers. The response of the composite to various types of loadings is presented up to the onset of loss of static stability at a location within the hyperelastic rubber constituent, and initial failure envelopes are shown.

Stability of a Rigid Body With an Oscillating Particle: An Application of MACSYMA

1985 ◽  
Vol 52 (3) ◽  
pp. 686-692 ◽  
Author(s):  
L. A. Month ◽  
R. H. Rand
Keyword(s):  
Angular Velocity ◽  
Rigid Body ◽  
Classical Problem ◽  
Moment Of Inertia ◽  
Model Parameters ◽  
Stability Chart ◽  
The Stability ◽  
Transition Curves ◽  
Minimum Moment ◽  
Small Angular Velocity

This problem is a generalization of the classical problem of the stability of a spinning rigid body. We obtain the stability chart by using: (i) the computer algebra system MACSYMA in conjunction with a perturbation method, and (ii) numerical integration based on Floquet theory. We show that the form of the stability chart is different for each of the three cases in which the spin axis is the minimum, maximum, or middle principal moment of inertia axis. In particular, a rotation with arbitrarily small angular velocity about the maximum moment of inertia axis can be made unstable by appropriately choosing the model parameters. In contrast, a rotation about the minimum moment of inertia axis is always stable for a sufficiently small angular velocity. The MACSYMA program, which we used to obtain the transition curves, is included in the Appendix.

Constitutive Equations for Hyperelastic Materials Based on the Upper Triangular Decomposition of the Deformation Gradient

2018 ◽  
Vol 24 (6) ◽  
pp. 1785-1799 ◽  
Author(s):  
Y. Q. Li ◽  
X.-L. Gao
Keyword(s):  
Energy Density ◽  
Constitutive Equations ◽  
Strain Energy Density ◽  
Strain Energy ◽  
Deformation Gradient ◽  
Density Functions ◽  
Triangular Decomposition ◽  
Deformation Gradient Tensor ◽  
Hyperelastic Materials ◽  
Distortion Tensor

The upper triangular decomposition has recently been proposed to multiplicatively decompose the deformation gradient tensor into a product of a rotation tensor and an upper triangular tensor called the distortion tensor, whose six components can be directly related to pure stretch and simple shear deformations, which are physically measurable. In the current paper, constitutive equations for hyperelastic materials are derived using strain energy density functions in terms of the distortion tensor, which satisfy the principle of material frame indifference and the first and second laws of thermodynamics. Being expressed directly as derivatives of the strain energy density function with respect to the components of the distortion tensor, the Cauchy stress components have simpler expressions than those based on the invariants of the right Cauchy-Green deformation tensor. To illustrate the new constitutive equations, strain energy density functions in terms of the distortion tensor are provided for unconstrained and incompressible isotropic materials, incompressible transversely isotropic composite materials, and incompressible orthotropic composite materials with two families of fibers. For each type of material, example problems are solved using the newly proposed constitutive equations and strain energy density functions, both in terms of the distortion tensor. The solutions of these problems are found to be the same as those obtained by applying the polar decomposition-based invariants approach, thereby validating and supporting the newly developed, alternative method based on the upper triangular decomposition of the deformation gradient tensor.

Slow steady rotation of axially symmetric bodies in a viscous fluid

1961 ◽  
Vol 10 (1) ◽  
pp. 17-24 ◽  
Author(s):  
R. P. Kanwal
Keyword(s):  
Angular Velocity ◽  
Flow Field ◽  
Viscous Fluid ◽  
Stokes Flow ◽  
Flow Problem ◽  
Analytic Solutions ◽  
The Body ◽  
Axially Symmetric ◽  
Steady Rotation ◽  
Axis Of Symmetry

The Stokes flow problem is considered here for the case in which an axially symmetric body is uniformly rotating about its axis of symmetry. Analytic solutions are presented for the heretofore unsolved cases of a spindle, a torus, a lens, and various special configurations of a lens. Formulas are derived for the angular velocity of the flow field and for the couple experienced by the body in each case.

scholarly journals Generalized shear deformations for isotropic incompressible hyperelastic materials

1977 ◽  
Vol 20 (2) ◽  
pp. 129-141 ◽  
Author(s):  
James M. Hill
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Energy Function ◽  
Strain Energy ◽  
Strain Energy Function ◽  
Shear Deformations ◽  
Partial Differential ◽  
Hyperelastic Materials ◽  
Restricted Form ◽  
Consistent Solution

AbstractFor isotropic incompressible hyperelastic materials the single function characterizing generalized shear deformations or as they are sometimes called anti-plane strain deformations must satisfy two distinct partial differential equations. Knowles [5] has recently given a necessary and sufficient condition for the strain–energy function of the material which if satisfied ensures that the two equations have consistent solutions. It is shown here for the general material not satisfying Knowles' criterion that the only possible consistent solution of the two partial differential equations are those which are already known to exist for all strain–energy functions. More general types of generalized shear deformations for such meterials are shown to exist only for special or restricted form ot the strain-energy function. In derving these results we also obtain an alternative derivation of Knowles' criterion.

scholarly journals Large elastic deformations of isotropic materials. III. Some simple problems in cyclindrical polar co-ordinates

1948 ◽  
Vol 240 (823) ◽  
pp. 509-525 ◽  
Keyword(s):  
Large Deformations ◽  
Equations Of Motion ◽  
Compressive Force ◽  
Cylindrical Tube ◽  
Cylindrical Symmetry ◽  
Simple Extension ◽  
Solid Cylinder ◽  
Normal Surface ◽  
Incompressibility Condition ◽  
Special Cases

Expressions for the components of strain and the incompressibility condition, for large deformations, are obtained in a cylindrical polar co-ordinate system. The stress-strain relations, equations of motion and boundary conditions for an incompressible, neo-Hookean material, in such a coordinate system, are also obtained and specialized to the case of cylindrical symmetry. These results are applied to the special cases of the simple torsion of a solid cylinder and of a hollow, cylindrical tube and to their combined simple extension and simple torsion. In the case of a solid cylinder, it is found that a state of simple torsion can be maintained by surface tractions applied to the ends of the cylinder only, and these consist of a torsional couple together with a compressive force. The necessary torsional couple is proportional to the amount of torsion and the compressive force to the square of the torsion. In the case of a hollow, cylindrical tube, it is again necessary to exert a torsional couple, proportional to the torsion, and a compressive force, proportional to the square of the torsion, on the plane ends, but it is also necessary to exert a normal surface traction, acting in a positive radial direction, on one or other of the curved surfaces of the tube and proportional to the square of the torsion.

scholarly journals An Approximate Solution for Flow between Two Disks Rotating about Distinct Axes at Different Speeds

2007 ◽  
Vol 2007 ◽  
pp. 1-16 ◽  
Author(s):  
H. Volkan Ersoy
Keyword(s):  
Reynolds Number ◽  
Approximate Solution ◽  
Angular Velocity ◽  
Analytical Solution ◽  
Viscous Fluid ◽  
Approximate Analytical Solution ◽  
Velocity Difference ◽  
Small Angular Velocity

The flow of a linearly viscous fluid between two disks rotating about two distinct vertical axes is studied. An approximate analytical solution is obtained by taking into account the case of rotation with a small angular velocity difference. It is shown how the velocity components depend on the position, the Reynolds number, the eccentricity, the ratio of angular speeds of the disks, and the parameters satisfying the conditionsu=0andν=0in midplane.

Development of Hyperelastic Model for Weldlines Containing Natural Rubber Molded Part

2013 ◽  
Vol 747 ◽  
pp. 631-634
Author(s):  
Watcharapong Chookaew ◽  
Jirachai Mingbunjurdsuk ◽  
Pairote Jittham ◽  
Somjate Patcharaphun
Keyword(s):  
Energy Density ◽  
Strain Energy Density ◽  
Strain Energy ◽  
Mathematical Formulation ◽  
Constitutive Models ◽  
Hyperelastic Materials ◽  
Design Engineers ◽  
Strain Energy Density Function ◽  
Molded Part ◽  
Rubber Part

Several constitutive models of non-linear large elastic deformation based on strain-energy-density functions have been developed for hyperelastic materials. These models, coupled with the Finite Element Method (FEM), can effectively utilized by design engineers to analyze and design elastomeric products operating under the deformation states. However, due to the complexities of the mathematical formulation which can only obtained at the moderate strain and the assumption of material used for the analysis. Therefore it is formidable task for design engineer to make use of these constitutive relationships. In the present work, the strain-energy-density function of weldline containing rubber part was constructed by using the Neural Network (NN) model. The analytical results were compared to those obtained by Neo-Hookean, Mooney-Rivlin, Ogden models. Good agreement between developed NN model and the existing experimental data was found, especially at very low strain and at very high strain.

Finite Bending and Straightening of Hyperelastic Materials: Analytical Solution and FEM

2019 ◽  
Vol 11 (09) ◽  
pp. 1950084 ◽  
Author(s):  
Sara Sheikhi ◽  
Mohammad Shojaeifard ◽  
Mostafa Baghani
Keyword(s):  
Finite Element ◽  
Strain Energy ◽  
Strain Energy Function ◽  
Optimization Procedure ◽  
Element Analysis ◽  
Energy Functions ◽  
Hyperelastic Materials ◽  
Finite Bending ◽  
Strain Energy Functions ◽  
Analytical Approaches

In this research, an incompressible, isotropic, nonlinear elastic rectangular block and a circular cylindrical sector are studied under bending and straightening moments, respectively. Analytical approaches are presented on implementing of the left Cauchy–Green tensor and Cauchy stresses. In addition, finite element analysis of both problems is carried out using UHYPER user-defined subroutine in ABAQUS to verify the analytical methods. Four different invariant-based strain energy functions, including neo-Hookean, Mooney–Rivlin, Arruda–Boyce, and recently proposed polynomial Exp-Exp models, are examined, and the results are compared. Material parameters of silicon rubber for the strain energy functions are identified by applying an optimization procedure. Finite element method results confirmed the analytical approach with great compatibility. Results showed that the length of the unbent beam does not affect the stress. Likewise, the initial angle of curved structure does not affect the unbending moment and stresses. Moreover, the Exp-Exp model had a slightly different result rather than other strain energies, which means that this model is more conservative than its counterparts. Furthermore, the Exp-Exp strain energy function is calibrated for tissue-like phantom and is compared with experimental data.

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