[Proc. 15th Semi-Annual Eastern Photoelasticity Conf. pp. 2331, 1942]
Formulation of the Problem
In this note a method is developed which permits the approximate solution of the following class of boundary value problems of the theory of elasticity. A body, bounded by one or more cylindrical surfaces with parallel generators and by two parallel planes perpendicular to the axes of the cylindrical surfaces, is acted upon by tractions applied to the cylindrical portion of the boundary, the boundary tractions being parallel to the bounding planes and moreover non-varying in the direction of the axes of the cylinders.
Mathematically the problem consists in finding solutions of the following system of nine equations:
for the nine unknowns Vx,Vy,Vz,Wxy,Wxz,Wyz, u, v, w, subject to the following boundary conditions
In these conditions 2h is the distance between the bounding planes, gi(x, y) = 0 the equation of the ith cylindrical bounding surface, s the arc length along the circumference of the cylinders and px and py the x and y-components of the boundary tractions.
The two-dimensional theory of plane stress can be described as a method of approximately solving the system of Eqs. (1)(4), based on the following
assumptions:
It is known that these assumptions are in general consistent only with the system of differential equations (1)(3) when the value of Poisson's ratio, Q, is zero.
Otherwise it is necessary to disregard two of the shear stress strain relations, Eqs. (2f) and (2g), since it is impossible to satisfy them for a state of stress obtained from Eqs. (1), (2a)(2d) and (4)(6).
The usefulness of the approximate theory lies in the fact that experimental evidence and mechanical intuition indicate that in many cases the results of this approximate theory are very close to the corresponding exact results. However, the same evidence and intuition indicate also that the results of the approximate theory may be seriously in error in such regions of the body where appreciable changes in stress occur over distances which are of the same order of magnitude as the thickness 2h of the body. Such changes occur for instance in the edge zone of holes (cut-outs) when the diameter of the holes is of the order of magnitude of the thickness 2h.
It is for a quantitative analysis of such effects that it is desirable to calculate three-dimensional corrections for the two-dimensional theory of plane stress.
Since it is believed that an exact solution of the three-dimensional problem presents very great analytical difficulties, an approximate method for its solution is here developed which reduces the three-dimensional problem mathematically to one in two-dimensions while retaining the three-dimensional mechanical characteristics of the solution.
The method employed here is an appropriate application of the Principle of Least Work.
Method of Solution
The approximate analysis of the three-dimensional effect may be based on replacing the assumptions, Eqs. (6), of two-dimensional stress by the following assumptions:
while the assumptions, Eqs. (5), of plane stress are replaced by
In Eqs. (7) and (8) the functions S, T, s and t depend on x and y only while the functions g, which are to be determined suitably, depend on z only.
The first set of conditions here imposed on the stresses of Eqs. (7) and (8) is that they have to satisfy the equilibrium conditions (1) and the boundary conditions (3) and (4).
From the equilibrium conditions (1) follows that g1 and g2 depend on g and that txz, tyz and sz depend on sx, sy and sz in the following way,
(where dashes denote differentiation with respect to z) and
The boundary conditions (3) are satisfied by prescribing
and the boundary conditions (4) imply the following set of five conditions
In view of the special form of the expressions (7) and (8) it is still not in general possible to satisfy all the stress strain relations exactly. Since, however, Eqs. (7) and (8) are more general expressions than Eqs. (5) and (6) imply, it will be possible to satisfy the stress strain relations more nearly than is done in the two-dimensional theory of plane stress.
The way in which this closer approximation is obtained is the following. Use is made of the basic minimum principle for the stresses to determine the functions of Eqs.
(7)(10) by an application of the direct methods of the calculus of variations. The minimum principle can be stated in the following form: ''Among all possible states of stress which satisfy the equilibrium conditions in the interior of the body and the stress boundary conditions on the surface of the body, the correct state of stress makes the difference of the strain energy and of the work of the unprescribed boundary stresses a minimum." In the case that all boundary conditions are stress boundary conditions the strain energy itself is a minimum for the correct state of stress. The more general minimum principle permits, however, extension of the present results to problems in which displacement as well as stress boundary conditions are prescribed.
On the basis of the equilibrium state of stress (7) and (8) a "best" approximation will here be obtained by determining the arbitrary functions sx, sy and txy such that the strain energy of the body becomes as small as is possible with expressions for the stresses of the form Eqs. (7) and (8).
For the function g(z) which determines the shape of the stress curves over the thickness 2h of the body the following assumption is made
This expression for g satisfies the surface conditions (11).
It will be shown that application of the minimum energy principle leads to a system of simultaneous partial differential equations for a stress function I from which the average stresses Sx, Sy and T are derived and for the stress corrections sx, sy and txy. The advantage which this system of equations possesses compared with the basic system of Eqs. (1) and (2) is that the number of independent variables is
reduced from three to two and that it is of a form which permits an explicit solution of the boundary value problem for the cases that the solid is bounded by two parallel planes (problem of the infinite strip) or by two concentric circular cylinders (problem of the annulus). The latter case includes the problem of a circular hole in an infinite sheet which seems to be of the greatest practical interest among those which may be analyzed by the method of this paper.