On the Theory of the Tension of an Elastic Cylinder

Abstract

In the accepted theory of the stretching of an elastic cylinder by a uniform traction in the direction of its axis, it is assumed that the cylinder which is granted to be elastically isotropic when the applied traction is zero, remains elastically isotropic after the traction is applied. We do not hold this assumption reasonable and we present here a theory of the traction of an elastic cylinder in which allowance is made for the lack of isotropy caused by the applied traction. Since we have very little definitive experimental knowledge of the effect of stress upon the elastic constants of a medium, our theory must be qualitative rather than quantitative. We take the point of view of our previous note1 and so do not attempt to relate the energy of deformation to the (remote) unstressed state but, rather, consider merely the change of energy as we pass from any stressed position of the medium to a neighboring position in which the stress has been changed by an infinitesimal amount.Taking our z-axis parallel to the axis of the cylinder, we assume that the stress is a uniform traction parallel to this axis so that the only nonvanishing component of the stress tensor is T. and we shall denote this simply by T. When T is changed to T + 5T any point (x, y, z) of the cylinder will be displaced to (t, , D) = (x + Ax, y + Sy, z + 8z). We follow the classical theory in assuming that Ax is a function of x alone, that Sy is a function of y alone and that 5z is a function of z alone. This is only a rough approximation to what actually happens, since it implies that straight lines parallel to the axis of the cylinder are deformed into such and that plane cross-sections are deformed into such. As a consequence of this assumption the strain matrix e is diagonal with components en = Jx/boX, ew = 65y/by; enz = a6z/ z (infinitesimals higher than the first order being neglected). We denote by p the density of the medium when under traction T and by 0 the energy of deformation per unit mass when the traction is T + 5T. In order to connect T + ST and e we expand pq as a function of the strain-components as far as terms of the second order. The first order terms in this expansion reduce to Ten since the stress, when 382 PROC. N. A. S.

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