Approximate laplace transform inversions in viscoelastic stress analysis

An investigation was made of approximate methods for inverting Laplace transforms that occur in viscoelastic stress analysis when use is made of the elastic-viscoelastic analogy. Alfrey's and ter Haar's methods and Schapery's direct method were examined and shown to be special cases of a general inversion formula due to Widder. Schapery's least squares method and several techniques based on orthogonal function theory were also examined. Viscoelastic solutions to two problems involving deformations and stresses in solid propellant rocket motors under axial and transverse acceleration loads were obtained by use of several of the methods discussed. The problems were typical of the type where the associated elastic solution is known only numerically. The use of the orthogonal polynomial methods is explained in detail, and their limitations discussed. From the investigation described, it was concluded that Schapery's direct method and ter Haar's method generally give good results when applicable. Widder's general inversion formula, which includes Alfrey's method as a special case, is not useable for the type problems of interest here. Although the orthogonal polynomial methods possess characteristics that make them especially suited to the type problems considered, their use appears limited by severe computational difficulties. Schapery's least squares method gives good results to most problems of interest.