This investigation deals with the response of the semi-infinite, linear elastic, homogeneous, isotropic plate in plane strain, subject to symmetric normal loads acting, in the absence of shear stress, on its edge. Near- and far-field approximations are found, the far-field approximations giving the __integral of the Airy integral for both problems. In the present work, this technique is used on non-mixed problems of the stress type; is employed to obtain long-term information for two problems involving symmetric end loads acting, in the absence of shear stress, on the edge of the semi-infinite plate.
The far-field approximations lead to integrals of the Airy integral for both problems—the same function that arises in mixed problems (see [I]). To these are added, for the excitation of the plate which is symmetrical with respect to the x-axis, the criteria of symmetry~. Applying boundary conditions further reduces the number of boundary unknowns to be found to two.
Now R(s, p) equals zero is a general form of the Ray Leigh-Lamb frequency equation for symmetric waves in an infinite elastic plate. As Miklowitz notes in [ 1 ], R(s, p) thus has an infinite series of zeros in each quadrant of the complex s-plane, and in particular there exists s = s. Accordingly, a residual evaluation of the inner integral in (1. 8) would lead to exponentially unbounded waves if x - oo, a violation of (1. 4.
1. 24), the boundedness condition, corresponds to equation {23) of [I] and is the required system of integral equations for completing the evaluation· of the boundary unknowns.
PROBLEM A: UNIFORM LOAD
The long-time behavior of the waveguides we are currently studying can be deduced by approximating u(s, y, p) and ~(s, y, p) of (1.17) for small p after the unknowns have been set of the edges. It follows that the determination, via the boundary condition, of the edge unknowns satisfying (2.1) for small p will provide the long-term formal solution to the present problem. For elementary theory, the equation of motion is derived directly by assuming that the stress is uniform over any cross section of the plate perpendicular to the direction of propagation of the long waves (x direction).
It should be noted that (2. 2) cannot in general be realized as a consistent specialization of the equations of motion and the. Introducing the edge-strain estimates of the elementary theory, given in (2.5), into the first of the boundary conditions in (2.1), reveals that these elementary theory quantities produce the normal. The first term on the right-hand side of the above is zero by virtue of the continuity and periodicity of f(y); the second term is o(l /n), for n - oo, as a consequence of the Riemann-Lebesgue lemma.
So we have an example of the advertised coefficient decay. stress acting on the end of the plate in problem A. Therefore, the additional contributions to the edge stresses are for problem. As a result of the angle condition, this is a continuous definition of these functions of y on ] -oo, oo. Note that (2.9) automatically establishes the validity of the differentiation of the series per term.
Note also that, due to the large n behavior assured for a~(p), b'n (p), the coefficients in (2. 9) must satisfy the order conditions. Now add the contributions of the elementary theory given in (2.5) (yielding an integration of the last result. Now, on the premise that our elementary theory will in fact describe the nature of the dominant time variation in the very long run we need ord {u(p)} ~ ord [.!.2}, ord{a (p)}~ord{.
Considering the leading terms of the real and imaginary parts of the quantities in (2. 21) one sees that. Because of the uniformity of the asymptotics on p in s and y, for this case, (2. 30) can be differentiated with respect to x and y to produce the near-field, long-time strains. Both u(p) components in the above have poles in the right half, s -plane at s = p/c • These poles are not admissible in vie\lt/ of the.
On the contrary, we have shown that the elementary solution, and the elementary solution alone, is obtained from the first long approximation of the exact theory for problem A. To conclude the inversion of the small p, formal solution for problem A, we consider the third s range, namely; p little, s-o.
PROBLEM B: LINE-LOAD
In view of this, the boundary conditions for problem B are - which can be expressed via (1. 2) by. As in Section 2, we now need to postulate forms for the boundary unknowns to open up the boundary condition' (1. 24). The associated boundary values for this problem will contribute to the regular parts of the edge.
Task B3 will provide the dominant, long-term time dependence of the edge sets, thereby completing the selection of the representations for the regular parts of the edge unknowns. To obtain this additional information about the well-known Flamant problem, we solve problem Bl as the limit, f j - 0, of the sequence of kernel problems denoted by o(y, fj) in (3.2). Nevertheless, since providing values for integration of (1. 24) is the ultimate purpose of the treatment, existence in such a sense is sufficient.
0 in produces displacements and stresses consistent with the usual forms, valid away from the origin, for the Flamant problem. Indeed, using (3. 3) in conjunction with standard results for the Flamant problem and determining the derivative of the signum function from. It should be noted that currently we are just posting the forms for the singular parts of the edge unknowns.
Accordingly, there are only reasonable guesses as to what these expressions might be, based on the thesis that the long-time singular nature in the near field of a problem involving exclusively outward-propagating perturbations is the same as the singular behavior of the corresponding elastostatic problem . If this turns out not to be the case for problem B, the convergence of the Fourier series. Depending on the validity of the thesis that the forms v will describe all singular contributions to the marginal unknowns, the terms in (3.11) will actually be correct and consequently a (p), b (p ) will be subject to large n, order.
Now, from the premise that problem B3 will indeed describe the nature of the dominant time variation 1n the very long -time, we require that u(p), a (p), b (p) obey the same small p order . For n ~ 7 these values actually decrease faster than 1 /n3• Such numerical convergence supports our thesis that the long-time, near-field singular nature of Problem B is the same as the singular behavior of the corresponding elastostatic problem. The displacement amounts thus calculated agree to within 1% of the finite element values for all y.
By looking at our shapes for the boundary unknowns for problem B, (3. 12), and doing the simple inversion on p, we can define. Benthem, 'A Laplace transform method for the solution of semi-infinite and finite strip problems in stress analysis.
FINITE-ELEMENT ANALYSIS
1\ CTx=Ux Uxy= 0
APPENDIX 2. NUMERICAL RESULTS FOR THE SINE INTEGRAL
To avoid this difficulty, Newton 1's method is repeated until the roots are accurate to 10 decimal places, and a check of this accuracy is given by substituting the values of the roots back into sin z. The e given in the table is the maximum of the e and e values mentioned.
