Stress Calculations on Multiply Connected Domains

Abstract

The outstanding problem of finding a simple Muskhelishvili-type integral equation for stress problems on multiply connected domains is solved. Complex potentials are represented in a way which allows for the incorporation of cracks and inclusions. Several numerical examples demonstrate the generality and extreme stability of the approach. The stress field is resolved with a relative error less than ½¼ ½¼ on a large, yet simply reproducible, setup with a loaded square plate containing 4096 holes and cracks. Comparison with previous results in the literature indicates that general purpose finite element software may perform better than many special purpose codes.

Stress computations on perforated polygonal domains

“…Preconditioners are needed. We instead use a recently derived [29] nonsingular Fredholm second kind equation which is free from this problem. Furthermore, the unknown quantity VðzÞ in the integral equation is the limit of an analytic function.…”

“…This quantity is easy to compute since VðzÞ is minus the outside limit of FðzÞ on the holes [29]. With external uniform unit traction applied in the y-direction we get…”

“…In fact, in Ref. [29] we showed that for simple setups, involving a few holes and cracks in finite domains, these commercial packages often perform better than many specialized codes. Similar observations made by others has lead to a belief in the solid mechanics community that the finite element method will be the only numerical method in the future.…”

“…The paper is organized as follows: Section 2 discusses advantages with and shortcomings of integral equation methods. Section 3 reviews a recently derived and very efficient equation for multiply connected 2D domains [29]. Sections 4 and 5 contain the new material.…”

“…New placement of discretization points increases the stability. Higher order quadrature increases the convergence rate, compared to the earlier implementation [28,29]. Section 5 gives several numerical examples.…”