Incompressible Isotropic Material Research Articles

Large elastic deformations of isotropic materials IX. The deformation of thin shells

The theory of the large elastic deformation of incompressible isotropic materials is applied to problems involving thin shells. The inflation of a circular diaphragm of such a material is studied in detail. It is found that the manner in which the extension ratios and curvatures vary in the immediate neighbourhood of the pole of the inflated diaphragm can be determined analytically. However, in order to determine their variation throughout the inflated diaphragm a method of numerical integration has to be employed. Although this is, in principle, valid for any form of the stored-energy function, the calculations are carried through only for the Mooney form. Finally, the problem of the inflation of a spherical balloon, which has already been dealt with by Green & Shield (1950), is discussed in further detail.

Experiments on the Mechanics of Rubber I: Eversion of a Tube

Measurements are made of the changes which take place in the dimensions of rubber tubes when they are turned inside out These are compared with the values predicted from the theory of large elastic deformations of incompressible isotropic materials

Finite extension and torsion of cylinders

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.

Large elastic deformations of isotropic materials VII. Experiments on the deformation of rubber

It is shown in this part how the theory of large elastic deformations of incompressible isotropic materials, developed in previous parts, can be used to interpret the load-deformation curves obtained for certain simple types of deformation of vulcanized rubber test-pieces in terms of a single stored-energy function. The types of experiment described are: (i) the pure homogeneous deformation of a thin sheet of rubber in which the deformation is varied in such a manner that one of the invariants of the strain, I 1 or I 2 , is maintained constant; (ii) pure shear of a thin sheet of rubber (i.e. pure homogeneous deformation in which one of the extension ratios in the plane of the sheet is maintained at unity, while the other is varied); (iii) simultaneous simple extension and pure shear of a thin sheet (i.e. pure homogeneous deformation in which one of the extension ratios in the plane of the sheet is maintained constant at a value less than unity, while the other is varied); (iv) simple extension of a strip of rubber; (v) simple compression (i.e. simple extension in which the extension ratio is less than unity); (vi) simple torsion of a right-circular cylinder; (vii) superposed axial extension and torsion of a right-circular cylindrical rod. It is shown that the load-deformation curves in all these cases can be interpreted on the basis of the theory in terms of a stored-energy function W which is such that δ W /δ I 1 is independent of I 1 and I 2 and the ratio (δ W /δ I 2 ) (δ W /δ I 1 ) is independent of I 1 and falls, as I 2 increases, from about 0*25 at I 2 = 3.

Rheology in the United States

INTEREST in rheology has increased in recent years, and with the relaxation of censorship restrictions on scientific publications several important papers in industrial rheology (ten or fourteen papers) were contained in the programme arranged for the annual meeting of the Society of Rheology held in New York City during October 26-27. Prof. E. C. Bingham found the plotting of fluidities more helpful than the plotting of viscosities when dealing with anomalies of flow, and instanced the linearity of fluidities for many protein solutions. A theoretical discussion on the rate process theory of flow, applied to viscous liquids, by A. Bondi, was illustrated by numerous examples, and an interpretation given of the energy of hole formation in terms of the Lennard-Jones theory of the liquid state. A paper on the theory of strain hardening under combined stresses was given by W. Prager, who was concerned with isotropic incompressible plastic materials. T. Alfrey dealt with stress-strain curves for visco-elastic materials obeying the Boltzmann superposition principle. The industrial papers included the viscosity of oils at high pressure, the effect of plasticizers on the visco-elastic properties of vinyl resins (W. Aiken, T. Alfrey, A. Jansson and H. Mark), the viscosity of cellulose acetate solutions and of dilute solutions of polystyrene. Rotation cylinder viscometers were described, including a cylinder type for measurements during thixotropic recovery for a compounded latex, and a parallel plate type for high polymers. The need for an improved understanding of fundamentals leading to better methods was emphasized.

Strain Hardening Under Combined Stresses

Experimental investigations of the strain hardening of metals under combined stresses are usually conducted so that the directions of the principal stresses as well as the ratios of their magnitudes remain constant during any one test. The paper is concerned with incompressible isotropic materials which are stressed in this manner and deform in accordance with certain postulates. The most general stress-strain relation which can arise under these circumstances is established, and some special cases of this relation are discussed.