- ItemOpen AccessThe energy method in problems of buckling of bars with quantifier elimination(Elsevier Science and The Institution of Structural Engineers, 2018-02)The classical energy method for the approximate determination of critical buckling loads of bars is revisited. This method is based on the stability condition of the bar and on the appropriate selection of an approximation to the deflection of the bar. Moreover, it is frequently related to the Rayleigh quotient or to the Timoshenko quotient for the determination of the critical buckling load. Here we will use again the energy method for the determination of critical buckling loads of bars but now on the basis of a new computational approach. This new approach consists in using the modern computational method of quantifier elimination efficiently implemented in the computer algebra system Mathematica instead of partial differentiations when we use the stability condition of the bar or essentially equivalently when we minimize the Rayleigh quotient or the Timoshenko quotient. This approach, which avoids partial differentiations, is also more rigorous than the classical approach based on partial derivatives because it does not require the use of the conditions for a minimum based on second partial derivatives, which are generally ignored in practice. Moreover, it is very simple to use inside the powerful computational environment offered by Mathematica. The present approach is illustrated in several buckling problems of bars including parametric buckling problems. Buckling problems of bars with two internal unilateral constraints, where the classical energy method is difficult to apply, are also studied. Even in this rather difficult application the critical buckling load is directly determined with a sufficient accuracy.
- ItemOpen AccessSupplementing the numerical solution of singular/hypersingular integral equations/inequalities with parametric inequality constraints with applications to crack problems(Institute of Fundamental Technological Research of the Polish Academy of Sciences, 2017)Singular and hypersingular integral equations appear frequently in engineering problems. The approximate solution of these equations by using various numerical methods is well known. Here we consider the case where these equations are supplemented by inequality constraints mainly parametric inequality constraints, but also the case of singular/hypersingular integral inequalities. The approach used here is simply to employ the computational method of quantifier elimination efficiently implemented in the computer algebra system Mathematica and derive the related set of necessary and sufficient conditions for the validity of the singular/hypersingular integral equation/inequality together with the related inequality constraints. The present approach is applied to singular integral equations/inequalities in the problem of periodic arrays of straight cracks under loading- and fracture-related inequality constraints by using the Lobatto–Chebyshev method. It is also applied to the hypersingular integral equation/inequality of the problem of a single straight crack under a parametric loading by using the collocation and Galerkin methods and parametric inequality constraints.
- ItemOpen AccessApplication of quantifier elimination to mixed-mode fracture criteria in crack problems(Springer-Verlag Germany, 2017-10)Several criteria for mixed-mode fracture in crack problems are based on the maximum of a quantity quite frequently related to stress components. This quantity should not reach a critical value. Computationally, this approach requires the use of the first and the second derivatives of the above quantity although frequently the use of the second derivative is omitted because of the necessary complicated computations. Therefore, mathematically, the determination of the maximum of the quantity of interest is not assured when the classical approach is used without the second derivative. Here a completely different and more rigorous approach is proposed. The present approach is based on symbolic computations and makes use of modern quantifier elimination algorithms implemented in the computer algebra system Mathematica. The maximum tangential stress criterion, the generalized maximum tangential stress criterion (with a T-stress term), the T-criterion and the modified maximum energy release rate criterion are used for the illustration of the present new approach in the mode I/II case. Beyond the conditions of fracture initiation, the determination of the fracture angle is also studied. The mode I/III case is also considered in brief. The present approach completely avoids differentiations, similarly the necessity of a distinction between maxima and minima, always leads to a global (absolute) and not to a local (relative) maximum and frequently to closed-form formulae and automatically makes a distinction of cases in the final formula whenever this is necessary. Moreover, its use is easy and direct and the maximum of the quantity of interest is always assured.
- ItemOpen AccessApplication of quantifier elimination to inverse buckling problems(Springer-Verlag Austria, 2017-10)The inverse buckling problem for a column is the problem where both the loading and the buckling mode are defined in advance (the latter generally in a polynomial form) and the flexural rigidity of the column is sought in a similar form with the help of the related ordinary differential equation. This problem was proposed and studied in many buckling problems by Elishakoff and his collaborators. A serious difficulty in its solution is that the resulting flexural rigidity should be positive along the column. Here in order to check this positivity the modern computational method of quantifier elimination is proposed and used inside the computational environment offered by the computer algebra system Mathematica and mainly based on the Collins cylindrical algebraic decomposition algorithm. At first, the simple inverse buckling problem of an inhomogeneous column under a concentrated load is studied with respect to the aforementioned positivity requirement. Next, the much more difficult problem concerning a variable distributed loading is also studied both in the case of one parameter and in the case of two parameters in this loading. Parametric rational and trigonometric forms of the flexural rigidity are also studied. Naturally, the resulting conditions for the positivity of the flexural rigidity are rather simple for one loading parameter, but they may become sufficiently complicated for two loading parameters. The present computational approach constitutes a simple, efficient and mathematically rigorous way for the derivation of positivity conditions for the flexural rigidity of a column in a variety of inverse buckling problems.
- ItemOpen AccessCaustics, pseudocaustics and the related illuminated and dark regions with the computational method of quantifier elimination(Elsevier, 2017-01)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.