Large elastic deformations of isotropic materials. V. The problem of flexure

Abstract

A cuboid of highly elastic incompressible material, whose stored-energy function W is a function of the strain invariants, has its edges parallel to the axes x, y and z of a rectangular Cartesian co-ordinate system. It can be bent so that: (i) every plane, initially normal to the x-axis, becomes part of the curved surface of a cylinder whose axis is the z-axis; (ii) every plane, initially normal to the y-axis, becomes a plane containing the z-axis; (iii) there is no displacement parallel to the z-axis. It is found that such a state of flexure can be maintained by the application of surface tractions only, and these are calculated explicitly in terms of the derivatives of W with respect to the strain invariants. The surface tractions are normal to the surfaces on which they act, in their deformed state. Those acting on the surfaces initially normal to the x-axis are uniform over each of these surfaces. The assumption is then made that the stored-energy function W has the form, originally suggested by Mooney (1940), for rubber, W = C$_{1}$($\lambda _{1}^{2}$+$\lambda _{2}^{2}$+$\lambda _{3}^{2}$-3)+C$_{2}$($\lambda _{2}^{2}\lambda _{3}^{2}$+$\lambda _{3}^{2}\lambda _{1}^{2}$+$\lambda _{1}^{2}\lambda _{2}^{2}$-3), where C$_{1}$ and C$_{2}$ are physical constants for the material and $\lambda _{1}$, $\lambda _{2}$, $\lambda _{3}$ are the principal extension ratios. For this case-and therefore for the incompressible neo-Hookean material (Rivlin 1948a, b, c), which is obtained from this by putting C$_{2}$ = 0- it is found that the flexure can be maintained without the application of surface tractions to the curved surface, provided that 2(a$_{1}$ - a$_{2}$) (r$_{1}$r$_{2}$)$^{\frac{1}{2}}$ = r$_{1}^{2}$ - r$_{2}^{2}$, where (a$_{1}$ - a$_{2}$) is the initial dimension of the cuboid, parallel to the x-axis, and r$_{1}$ and r$_{2}$ are the radii of the curved surfaces. When this condition is satisfied, the system of surface tractions applied to a boundary initially normal to the y-axis is equivalent to a couple M, proportional to (C$_{1}$ + C$_{2}$). It is also found that the surface tractions applied to a boundary normal to the z-axis has a resultant F$_{2}$ proportional to (C$_{1}$ - C$_{2}$).