Harmonic Holes and the Inverse Problem

Harmonic Holes and the Inverse Problem

Large stress concentrations at the edges of holes produce serious structural problems. Overstressed areas, no matter how local, often lead to fatigue cracks greatly reducing the life of a structure or even triggering collapse if they are not detected or if they propagate rapidly.

However, two design modifications may be beneficial. The shape of the hole might be changed and/or the hole boundary reinforced by thickening the plate locally or providing a liner of stiffer material. Such an inverse approach, where the geometry of the boundary is left as an unknown, might be called “shape mechanics.” It involves a fundamental reorientation of solid mechan-ics to thinking about design conditions and optimum shapes to control fields rather than accept unsatisfactory fields dictated by prescribed geometry.

The search for optimum structural shapes has traditionally been conducted by engineers using a strategy distinctly different from that of standard math-ematical analysis where the governing equations are solved for a given set of boundary conditions on a body of specified geometry. Instead, the engineer leaves at least some geometric parameters unspecified but imposes extra con-ditions on the stress�'strain field sufficient to then derive a configuration “optimized” to that preselected design criterion.

Linear and shell structures provide a classic example of this alternative design strategy in which the search for optimum shapes to eliminate bending and the resulting effects on building forms is a major historical theme. In startling con-trast, there is essentially no counterpart to the design tradition in elasticity. A 200-year history has produced a rich and elegant mathematical theory in both real and complex representation but, thus far, the inverse approach has generally been bypassed. There are a number of historic and philosophic influences one might postulate to explain, at least in part, this obvious omission but probably the most important factor is the lack of clear design criteria for inverse elasticity that can be translated into precise mathematical statements comparable in power and scope to the no-bending or minimum energy condition in structures.*

Design Condition

The choice of design condition is absolutely crucial to the inverse strategy in that it dictates all that follows. The engineer should logically look for a powerful design condition that can be translated into a precise mathematical statement independent of coordinate systems and incorporating the fundamental proper-ties of the final stress�'strain field everywhere, not just at the unknown boundary.

We have seen that the circular shape for holes or rigid inclusions are opti-mum giving minimum stress concentration, but only for an isotropic field. In this case, not only are the boundary stresses constant and their sum equal to

twice the isotropic stress (i.e., the value of the first invariant) but, quite remarkably, this is also true everywhere in the resulting field as the perturba-tions of the individual stress components damp out. With the circular shape as a model, let us therefore choose for the design criterion the “harmonic field condition” that, when the hole or inclusion is introduced, the first invariant of the stress (or strain) remain unperturbed everywhere in the field. The shape of the new boundary that achieves this is, in turn, called a “harmonic shape.”

This name is used for obvious reasons. If, in any portion of an uncut two-dimensional domain in plane stress or plane strain defined by a known “original” stress function F� o , a new boundary is introduced, such as the hole in Figure 7.6, then the final field, F, is given by

The superscript ^o indicates an original quantity before a harmonic shape is introduced, and the superscript * indicates the change or perturbation of this quantity due to the new harmonic boundary. The final state, being always the sum of these two components, is given no superscript.

Because the “perturbation stress function”, F*, is harmonic, so must all changes in all stress, σ_ij*,  , strain, Eij*, and displacement, uij*, components when the unknown boundary is introduced. For example, the equilibrium equa-tions for the perturbation:

become the Cauchy-Riemann conditions since σ_x* = - σ_y*. Therefore, the normal and shear components of the induced stresses and strains are actually conjugate harmonic functions. Similarly, because there is no change in the first invariant, there can be no elastic rotations and

Since E_x* = - E _y*, u* and -v* must be conjugate harmonic function as well. Thus, the analogy between the harmonic design condition and the mem- brane shapes for arches and shells is complete. For membrane structures the “no bending” condition is actually, of course, a requirement that there be no elastic rotations of any cross-section and the harmonic condition gives a full two-dimensional counterpart imposing “no bending” anywhere in the field when the harmonic shape is introduced.* Thus, as engineers, we might expect, with the benefit of historical perspective, optimum behavior from designs fulfilling the harmonic condition.

Harmonic shapes can be derived in a number of ways. It is, after all, only the perturbation, F*,which is sought and, by imposing the harmonic condi tion, the biharmonic field has been reduced to the simplest elliptic form. It can also be shown that both F* and its normal derivative can be completely specified on the unknown boundary in terms of F� o . Essentially then, the prob-lem becomes directly analogous to the classic question of determining the shape of a free-surface in Laplacian fields and all the experimental, numeri-cal, and mathematical techniques employed in the various disciplines where this problem arises might conceivably be used.*

Since we seek the geometry of an unknown boundary, a complex variable formulation such as presented by Muschelishvili** offers a particularly pow-erful analytic approach. As shown in Figure 7.6 , we can describe an opening of essentially any shape as a function of the complex variable z = x + iy, and it becomes convenient to map from a region containing a circular opening using the inverse transformation z = ω (ζ) . Stresses, strains, and displace-ments for the plane problem without body forces can be expressed in terms of two complex functions  ϕ(ζ) and Ψ(ζ) which, in the standard problems of elasticity, are determined from the boundary condition:

in which σ(without a subscript) equals the value of ζ on the boundary of the unit circle γ. The first of these equations is for the first fundamental problem of elasticity in which F(σ) represents any self-equilibrating boundary loading, while the second is for the second fundamental problem in which g₁ and g₂ are

the known boundary values of any displacement components u and v imposed on γ. The material properties k and 2μ may also be written:

in terms of Young’s modulus, E, and Poisson’s ratio, v.

For standard analysis where geometry is known, the problem becomes the determination of the complex potentials ϕ and Ψ. For the inverse problem on the other hand, the desired shape will be given directly by the unknown mapping function ω(σ), which can be found from the applicable boundary condition only if the unknown perturbation components ϕ* and Ψ* can be eliminated. The har-monic condition allows us to do this since it requires that:

Thus, multiplying each term in Equations (a) and (b) by dσ/(σ- ζ) and inte-grating over the boundary, the basic equations for the form of the harmonic shape (opening or inclusion) are found to be: