Large elastic deformations of isotropic materials VI. Further results in the theory of torsion, shear and flexure

Abstract

The forces necessary to produce certain simple types of deformation in a tube of incompressible, highly elastic material, isotropic in its undeformed state, are discussed. The first type of deformation may be considered to be produced by the following three successive simpler deformations: (i) a uniform simple extension, (ii) a uniform inflation of the tube, in which its length remains constant, and (iii) a uniform simple torsion, in which planes perpendicular to the axis of the tube are rotated in their own plane through an angle proportional to their distance from one end of the tube. Certain special cases of this deformation are considered in greater detail employing a simple stored-energy function of the form W = C$_{1}$(I$_{1}$-3)+C$_{2}$(I$_{2}$-3), where C$_{1}$ and C$_{2}$ are physical constants for the material and I$_{1}$ and I$_{2}$ are the strain invariants. The second type of deformation considered is that in which the simpler deformations (i) and (ii) mentioned above are followed successively by simple shears about the axis of the tube and parallel to it. The forces which must be applied are calculated for the simple form of stored-energy function given above. Finally, the simultaneous simple flexure and uniform extension normal to the plane of flexure of a thick sheet is discussed, and a number of the results obtained in a previous paper (Rivlin 1949 b) are generalized.