Stress concentration factors: determination of ranges of values by the method of quantifier elimination

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Ioakimidis, Nikolaos
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The interesting and modern method of quantifier elimination in computer algebra was already applied to a very large number of problems including, e.g., ranges of functions and several applied mechanics problems. Here this method is applied to the determination of ranges of stress concentration factors. The classical related handbook is Peterson's Stress Concentration Factors. Here three classical elasticity problems with stress concentration factors included in this handbook are studied in detail with respect to their ranges of values. These problems concern (i) a tension strip with two opposite semi-circular edge notches, (ii) a tension strip with a circular hole and (iii) an infinite medium with a rectangular hole with rounded corners. In these problems, the ranges of the approximate stress concentration factors on physically appropriate intervals for their variables are determined. Beyond the very simple case of stress concentration factors with no parameters (only their variables), the more interesting cases (i) of stress concentration factors with parameters (here dimensionless length parameters) and (ii) of parametric intervals, i.e. intervals with symbols as their endpoints, are also studied. In all cases, the efficient implementation of quantifier elimination in the popular computer algebra system Mathematica was used for the derivation of the ranges of the corresponding stress concentration factors frequently in parametric forms. Naturally, such a range can directly provide the related numerical range for specific value(s) of the parameter(s) involved. Extensions of the approach, e.g. to stress intensity factors in crack problems, are also possible.
Intervals, Functions, Ranges, Stress concentration factors, Elasticity, Strips, Edge notches, Circular holes, Rectangular holes, Upper bounds, Lower bounds, Minimization, Maximization, Symbolic computations, Quantifiers, Universal quantifier, Quantified formulae, Quantifier elimination, Quantifier-free formulae, Computer algebra systems, Mathematica