Uncertainty intervals/regions for the stress intensity factors at crack tips under uncertain loading by using the ellipsoidal model and numerical integration

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Ioakimidis, Nikolaos
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Quantifier elimination constitutes an interesting computational approach in computer algebra already successfully applied to several disciplines. Here we apply this approach to crack problems in fracture mechanics with respect to the two stress intensity factors at the crack tips, but under uncertainty conditions as far as the loading of the crack(s) is concerned. At first, a single straight crack loaded by two uncertain concentrated normal loads satisfying an ellipsoidal inequality constraint is studied. Next, the more interesting case of an uncertain distributed normal load on the crack(s) is also considered in the problems of (i) a single straight crack, (ii) a periodic array of collinear cracks and (iii) a periodic array of parallel cracks. In these problems, the inequality constraint satisfied by the loading is assumed to have a quadratic (`energy'-type) integral form. Beyond quantifier elimination the computational approach consists in using either (i) the closed-form formulae for the stress intensity factors (for a single crack) or (ii) the method of Cauchy-type singular integral equations and, next, the quadrature method for their numerical solution, more explicitly, the Lobatto–Chebyshev method (for all three aforementioned crack problems). Moreover, for the integral inequality constraint the Gauss–Chebyshev quadrature rule is used. By performing quantifier elimination to the relevant existentially quantified formulae and computing the related QFFs (quantifier-free formulae), we were able to derive both (i) the uncertainty intervals (or uncertainty ranges) for the stress intensity factors and (ii) the related uncertainty regions. These results show the uncertainty propagation from the loading of the crack(s) to the resulting stress intensity factors.
Convex models, Ellipsoidal model, Inequality constraints, Uncertainty, Uncertainty intervals, Uncertainty regions, Uncertainty propagation, Cracks, Collinear/parallel cracks, Stress intensity factors, Fracture mechanics, Singular integral equations, Lobatto–Chebyshev method, Numerical integration, Quantifier elimination, Quantifier-free formulae, Mathematica