Volume 8, Issue 6, 01 June 1937
Index of content:
8(1937); http://dx.doi.org/10.1063/1.1710316View Description Hide Description
In this paper equations are set up which describe the behavior of a solid under the action of combined stresses when a steady state of creep is assumed to exist. A material is said to be in a steady state of creep if it yields with a constant speed under a given constant stress. In an ordinary creep test the steady state exists when the deformation has reached the point where the ``creep curve'' becomes a straight line. In mathematical language this can be expressed by saying that the stress is a function of the rate of strain and is independent of the amount of strain. Cases are worked out in which this stress vs. strain rate relation is a simple power function. Various types of flow including purely plastic flow and viscous flow can be discussed by changing the value of the exponent in the power function relation. Several specific examples which include the case of the thick walled cylinder with closed ends loaded by internal pressure and the case of a disk having a circular hole and loaded in its plane are worked out. Equations are developed which show the steady state stresses in such cases. Expressions for the stress concentration factors in creep are also obtained.
8(1937); http://dx.doi.org/10.1063/1.1710317View Description Hide Description
8(1937); http://dx.doi.org/10.1063/1.1710318View Description Hide Description
Two methods are given for treating a new type of potential problem arising in a recent study of the author on the encroachment of water into an oil sand. The problem is that of finding the potential distributions in two regions of different ``constants'' (``conductivities''), separated by a moveable surface of unknown shape, this motion and shape to be determined by the conditions that each point of the interface should have a velocity proportional to the vector gradient of the potential at the point, and that the area swept out by the moving surface shall assume the ``constant'' appropriate to that of the encroaching side of the interface. The first method is a perturbation method, valid when the difference between the ``constants'' is small, and reduces the problem of determining the shape and motion of the interface and the potential distributions simultaneously, to a series of problems in which potential distributions are to be found on the sides of known interfaces, and the motion of the surface is to be found in a known (time varying) potential field. This same type of reduction is effected also in the second method, which is one of direct successive approximations, but which may be used for any values of the potential ``constants.'' The perturbation theory is carried through explicitly to the third order for the case of a one dimensional system, and agrees to that order with the exact results which were derived for this case in a previous paper. The method of successive approximations is carried out for this same case through the second approximation for a ratio of the ``conductivity'' of the interior side of the interface to that on the exterior side equal to 5, and through the third approximation for the ratio 1/5. In the former, the calculations give a rapid monotonic convergence to the correct solution, and in the latter the convergence is oscillatory, and only slightly less rapid. These behaviors are shown to be generally true for the linear problem, in the course of the construction of an analytic convergence proof for the one dimensional problem. The convergence proof of the method of successive approximations is also given for the case of radial symmetry.
8(1937); http://dx.doi.org/10.1063/1.1710319View Description Hide Description
The problem of the flow of heat through a long tunnel wall in an infinite solid has been treated and the results for the rate of flow and total heat flow have been expressed in a convenient form for numerical computations. This general problem has become of interest in connection with the cooling of deep mines.