Caustics,pseudocaustics and the related illuminated and dark regions with the computational method of quantifier elimination |
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| Institution: | 1. Centre National d’Études Spatiales, 18 Avenue Édouard Belin, 31401 Toulouse Cedex 9, France;2. Laboratoire MIPS, Université de Haute-Alsace, 12 rue des frères Lumière, 68093 Mulhouse, France;3. Département Mécanique Appliquée, Institut FEMTO-ST, 24 chemin de l’Épitaphe, 25000 Besançon, France;1. Centro de Investigaciones en Óptica, A.C., Loma del Bosque No. 115, León Gto, Mexico;2. Centro Nacional de Metrología, Querétaro 76246, México;3. Brookhaven National Laboratory – NSLS II 50 Rutherford Dr. Upton, 11973-5000 New York, USA;1. Luleå University of Technology, Department of Engineering Sciences and Mathematics, Luleå, S-971 87 Sweden;2. University of Mosul, College of Engineering, Department of Mechanical Engineering, Mosul, Iraq;1. School of Electronic Information Engineering, Tianjin University, Tianjin, 300072 PR China;2. State Key Laboratory of Structural Analysis for Industrial Equipment, Dalian University of Technology, Dalian, 116024 PR China |
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| Abstract: | The method of caustics is a powerful experimental method in elasticity and particularly in fracture mechanics for crack problems. The related method of pseudocaustics is also of interest. Here we apply the computational method of quantifier elimination implemented in the computer algebra system Mathematica in order to determine (i) the non-parametric equation and two properties of the caustic at a crack tip and especially (ii) the illuminated and the dark regions related to caustics and pseudocaustics in plane elasticity and plate problems. The present computations concern: (i) The derivation of the non-parametric equation of the classical caustic about a crack tip through the elimination of the parameter involved (here the polar angle) as well as two geometrical properties of this caustic. (ii) The derivation of the inequalities defining the illuminated region on the screen in the problem of an elastic half-plane loaded normally by a concentrated load with the boundary of this illuminated region related to some extent to the caustic formed. (iii) Similarly for the problem of a clamped circular plate under a uniform loading with respect to the caustic and the pseudocaustic formed. (iv) Analogously for the problem of an equilateral triangular plate loaded by uniformly distributed moments along its whole boundary, which defines the related pseudocaustic. (v) The determination of quantities of interest in mechanics from the obtained caustics or pseudocaustics. The kind of computations in the applications (ii) to (iv), i.e. the derivation of inequalities defining the illuminated region on the screen, seems to be completely new independently of the use here of the method of quantifier elimination. Additional applications are also possible, but some of them require the expansion of the present somewhat limited power of the quantifier elimination algorithms in Mathematica. This is expected to take place in the future. |
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| Keywords: | Caustics Pseudocaustics Illuminated and dark regions Cracks Plates Elasticity |
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