Finite elastic deformation of compressible isotropic bodies

1955 ◽  
Vol 227 (1169) ◽  
pp. 271-278 ◽  
Keyword(s):  
Spherical Shell ◽  
Energy Function ◽  
Strain Energy ◽  
Cylindrical Tube ◽  
Strain Energy Function ◽  
Finite Extension ◽  
General Strain ◽  
Circular Cylindrical Tube ◽  
Isotropic Bodies ◽  
Isotropic Spherical Shell

The problem of finite extension, inflation and torsion of an isotropic circular cylindrical tube is considered for compressible bodies, and a formula for the couple required to maintain the torsion is obtained in terms of a general strain-energy function. In addition, the symmetrical inflation of an isotropic spherical shell is examined and a condition, in terms of a general strain-energy function, is obtained when a spherical shell is everted.

Finite deformation of materials exhibiting curvilinear aeolotropy

1955 ◽  
Vol 229 (1176) ◽  
pp. 119-134 ◽  
Keyword(s):  
Spherical Shell ◽  
Strain Energy ◽  
Cylindrical Tube ◽  
Strain Energy Function ◽  
Transversely Isotropic ◽  
Special Cases ◽  
Incompressible Materials ◽  
General Strain ◽  
Limiting Case ◽  
Finite Elastic Deformations

Using tensor notation, a general theory is developed for finite elastic deformations of compressible and incompressible materials which exhibit curvilinear aeolotropy. The theory is formulated for materials which are completely unsymmetrical, orthotropic or transversely isotropic with respect to the curvilinear co-ordinate system which is employed to define the aeolotropy. In applications, attention is confined to cylindrically symmetrical and spherically symmetrical problems, from which emerge as special cases the inflation, extension and torsion of a cylindrical tube, and the inflation of a spherical shell. In addition, the flexure of a cuboid of rectilinearly aeolotropic material is considered as a limiting case of the cylindrically symmetrical problem. The conditions for the tube or spherical shell to be everted, and for the curved faces of the deformed cuboid to be free from applied stress, are obtained in terms of a general strain-energy function in forms which are independent of symmetries in the material.

scholarly journals Dynamics of a viscoelastic spherical shell with a nonconvex strain energy function

1998 ◽  
Vol 56 (2) ◽  
pp. 221-244 ◽  
Author(s):  
Roger Fosdick ◽  
Yohannes Ketema ◽  
Jang-Horng Yu
Keyword(s):  
Spherical Shell ◽  
Energy Function ◽  
Strain Energy ◽  
Strain Energy Function

Finite elastic deformation of incompressible isotropic bodies

1950 ◽  
Vol 202 (1070) ◽  
pp. 407-419 ◽  
Keyword(s):  
Elastic Deformation ◽  
Simple Form ◽  
Energy Function ◽  
Strain Energy ◽  
Strain Energy Function ◽  
General System ◽  
Rotating Cylinder ◽  
The Body ◽  
External Pressures ◽  
Isotropic Bodies

The theory of finite elastic deformation of incompressible isotropic bodies is expressed in a simple form in tensor notation, using a general system of co-ordinates which move with the body as it is deformed. In order to illustrate the advantages of the present methods one problem, previously solved by Rivlin, is re-examined. Two new problems are then solved using a completely general form for the strain energy function, the first problem being that of a rotating cylinder, and the second a uniform spherical shell under symmetrical internal and external pressures.

Some general results in the theory of large elastic deformations

1955 ◽  
Vol 231 (1184) ◽  
pp. 75-90 ◽  
Keyword(s):  
Elastic Body ◽  
Energy Function ◽  
Strain Energy ◽  
General Type ◽  
Cylindrical Tube ◽  
Strain Energy Function ◽  
General Function ◽  
Cylindrical Annulus ◽  
Special Cases ◽  
Wide Range

When the strain-energy function for an elastic body is expressed as a function of the six components of strain, the solution of a given problem for different types of material may assume very different forms. In the present paper, by regarding the strain-energy function as a function of the parameters defining the deformation, results are obtained which are valid for a wide range of materials. The analysis for each problem is performed initially for bodies possessing a suitable type of curvilinear aeolotropy, and results are derived which are independent of symmetries in the elastic material. These results are therefore valid, not only for the general type of material initially considered, but also for isotropic bodies and for materials which are orthotropic or transversely isotropic with respect to the curvilinear co-ordinate system which defines the aeolotropy. Both compressible and incompressible bodies are considered. From this point of view, a general type of cylindrically symmetrical deformation is examined which includes as special cases the problem of flexure, the inflation, extension and torsion of a cylindrical tube, and the shear of a cylindrical annulus. Particular results for these special cases are considered separately, and for the flexure and torsion problems, expressions are found for the resultant forces and couples required to maintain the deformation. A brief analysis is also given for the corresponding types of deformation for a cuboid. In the final section of the paper, a generalized shear problem is considered in which, during deformation, each point of the elastic body moves parallel to a given axis through a distance which is a general function of position in a plane normal to that axis.

A Note on the Finite Elastic Inflation of a Thin Spherical Shell

1972 ◽  
Vol 1 (3) ◽  
pp. 158-160
Author(s):  
J. B. Haddow ◽  
M. G. Faulkner
Keyword(s):  
Spherical Shell ◽  
Numerical Method ◽  
Energy Function ◽  
Strain Energy ◽  
Strain Energy Function ◽  
Algebraic Equations ◽  
Programmable Calculator ◽  
Isotropic Materials ◽  
Nonlinear Algebraic Equations ◽  
Thin Spherical Shell

The finite elastic inflation of a thin spherical shell is considered for compressible isotropic materials. A numerical method is used which allows the inflation problem to be done for any strain energy function. This method reduces to the solution of two nonlinear algebraic equations which can be solved on a desk-type programmable calculator.

Finite extension and torsion of cylinders

1951 ◽  
Vol 244 (876) ◽  
pp. 47-86 ◽  
Keyword(s):  
Hydrostatic Pressure ◽  
Cross Section ◽  
Cross Sections ◽  
Complex Variable ◽  
Energy Function ◽  
Strain Energy ◽  
Strain Energy Function ◽  
Finite Extension ◽  
Order Effects

The theory of finite elastic deformations of an isotropic body, in which a completely general strainenergy function is used, is applied to the problem of a small twist superposed upon a finite extension of a cylinder which has a constant cross-section. The law which relates the force necessary to produce the large extension, with the torsional modulus for the small torsion superposed on that extension, is given by a simple general formula. When the material is incompressible the corresponding law is independent of the particular form of the strain-energy function which applies to the material. When the cylinder is not a circular cylinder or a circular cylindrical tube the twisting couple vanishes for a certain value of the extension ratio, this value being independent of the particular form of the strain-energy function when the material is incompressible. The problems of a small twist superposed upon a hydrostatic pressure, or upon a combined hydrostatic pressure and tension, are also solved. Attention is then confined to isotropic incompressible rubber-like materials using a strain-energy function suggested by Mooney, and the second-order effects which accompany the torsion of cylinders of constant cross-sections are examined. The problem is reduced to the determination of two functions of a complex variable which are regular in the cross-section of the cylinders and which satisfy a suitable boundary condition on the boundary of the cross-section. The solution is expressed as an integral equation and applications are made to cylinders with various cross-sections. This theory is then generalized to include the second-order effects in torsion superposed upon a finite extension of the cylinders. Complex variables are used throughout this part of the paper, and the problem is reduced to the determination of four canonical functions of a complex variable, these functions being the solutions of certain integral equations. An explicit solution is given for an elliptical cylinder but without using the integral equations.

Mechanical power and hyperelasticity

2017 ◽  
Author(s):  
David J. Steigmann
Keyword(s):  
Energy Function ◽  
Strain Energy ◽  
Strain Energy Function ◽  
Mechanical Power ◽  
Elastic Materials ◽  
Cyclic Motion

This chapter covers the notion of hyperelasticity—the concept that stress is derived from a strain—energy function–by invoking an analogy between elastic materials and springs. Alternatively, it can be derived by invoking a work inequality; the notion that work is required to effect a cyclic motion of the material.

scholarly journals Modelling the Inflation and Elastic Instabilities of Rubber-Like Spherical and Cylindrical Shells Using a New Generalised Neo-Hookean Strain Energy Function

2021 ◽  
Author(s):  
Afshin Anssari-Benam ◽  
Andrea Bucchi ◽  
Giuseppe Saccomandi
Keyword(s):  
Experimental Data ◽  
Energy Function ◽  
Limit Point ◽  
Strain Energy ◽  
Cylindrical Shells ◽  
Strain Energy Function ◽  
Model Parameters ◽  
Wide Range ◽  
Cylindrical Tubes ◽  
Gent Model

AbstractThe application of a newly proposed generalised neo-Hookean strain energy function to the inflation of incompressible rubber-like spherical and cylindrical shells is demonstrated in this paper. The pressure ($P$ P ) – inflation ($\lambda $ λ or $v$ v ) relationships are derived and presented for four shells: thin- and thick-walled spherical balloons, and thin- and thick-walled cylindrical tubes. Characteristics of the inflation curves predicted by the model for the four considered shells are analysed and the critical values of the model parameters for exhibiting the limit-point instability are established. The application of the model to extant experimental datasets procured from studies across 19th to 21st century will be demonstrated, showing favourable agreement between the model and the experimental data. The capability of the model to capture the two characteristic instability phenomena in the inflation of rubber-like materials, namely the limit-point and inflation-jump instabilities, will be made evident from both the theoretical analysis and curve-fitting approaches presented in this study. A comparison with the predictions of the Gent model for the considered data is also demonstrated and is shown that our presented model provides improved fits. Given the simplicity of the model, its ability to fit a wide range of experimental data and capture both limit-point and inflation-jump instabilities, we propose the application of our model to the inflation of rubber-like materials.

Sign in / Sign up

Export Citation Format

Share Document