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- ItemOpen AccessAn application of Ben-Haim's info-gap decision theory (IGDT) to Todinov's method of algebraic inequalities by employing the method of quantifier elimination
Γενικό Τμήμα (Τεχνικές Αναφορές)(Κανένας, 2022-10-10) Ioakimidis, Nikolaos; Ιωακειμίδης, ΝικόλαοςProblems under uncertainty conditions can be studied by using the very interesting and popular Ben-Haim's info-gap (or information-gap) decision theory (IGDT). On the other hand, recently, Todinov proposed an interesting and efficient method based on algebraic inequalities for the reduction of risk and uncertainty as well as for the generation of new knowledge and the optimization of systems and processes. One of the main problems where Todinov applied his new method is the problem concerning the equivalent resistances of n resistors in an electrical circuit connected both in series and in parallel. Here we consider the same problem, but now with the related algebraic inequality used as the performance requirement in Ben-Haim's IGDT. The methodology used here is based on the computational method of quantifier elimination. This method constitutes a very interesting approach for the transformation of quantified formulae to logically equivalent formulae, but now free from the quantifiers and the quantified variables. The same method is implemented in some computer algebra systems including Mathematica, which is used here. The problems studied here and related to the equivalent resistances of two or three resistors concern (i) two resistors with one horizon of uncertainty including the cases of parametric nominal value(s) of one resistance or both resistances here by using a fractional-error uncertainty model in Ben-Haim's IGDT, (ii) two resistors again, but with two horizons of uncertainty, (iii) three resistors with one horizon of uncertainty and (iv) two resistors again, but with the use of an ellipsoidal uncertainty model. The use of negated existentially quantified formulae instead of universally quantified formulae is also studied.
- ItemOpen AccessGeneralized interval-based polynomial approximations to functions in applied mechanics by using the method of quantifier elimination
Γενικό Τμήμα (Τεχνικές Αναφορές)(Αυτο-έκδοση, ) Ioakimidis, Nikolaos; Ιωακειμίδης, ΝικόλαοςThe method of quantifier elimination constitutes an interesting computational approach in computer algebra already implemented in few computer algebra systems. In applied mechanics, this method was already used for the determination of ranges of functions. Here the application of the same method, quantifier elimination, is generalized to the determination of generalized interval-based polynomial approximations to functions again in applied mechanics. The main idea behind the present application is the use of linear interval enclosures for the approximation to functions and, more generally, the use of parameterized solutions to parametric interval systems of linear algebraic equations. This idea is mainly due to Lubomir V. Kolev. Here the present method is at first applied to two simple examples concerning (i) a rational function and (ii) the exponential function with their variables lying in intervals. Next, the same method is also applied to functions in applied-mechanics problems with variables also lying in intervals: (i) the problem of a beam on a Winkler elastic foundation with related function the dimensionless deflection of the beam, (ii) the problem of free vibrations of an oscillator with critical damping with related function the dimensionless displacement of the oscillator and (iii) the problem of a seven-member truss with related functions the nodal displacements. In this application, the stiffness of a bar is an uncertain, interval variable and, moreover, the classical perturbation method is also used. From the present results it is concluded that the method of quantifier elimination constitutes a useful tool for the derivation of simple parameterized interval-based polynomial approximations to functions in applied mechanics.
- ItemOpen AccessDetection of loading/geometrical singularities in isotropic elastic media with the use of Gröbner bases
Γενικό Τμήμα (Τεχνικές Αναφορές)Ioakimidis, Nikolaos; Ιωακειμίδης, ΝικόλαοςThe method of Gröbner bases in computer algebra is applied to the detection and location of loading or geometrical singularities inside two- or three-dimensional isotropic elastic media. The approach consists in using available experimental data concerning the stress/strain components away from the singularity in order to decide whether these components verify the relations corresponding to the particular singularity or not. These conditions can frequently be obtained by the method of Gröbner bases and the related Buchberger algorithm. In the case of an affirmative conclusion, we can further proceed to the location of the singularity. Alternatively, the present approach yields compatibility equations for the stress/strain components in elastic media with loading/geometrical singularities. These equations are displayed in detail for some common stress fields corresponding to concentrated forces as well as to a crack tip.
- ItemOpen AccessA new semi-Gaussian quadrature rule for finite-part integrals in crack problems with a second-order singularity
Γενικό Τμήμα (Τεχνικές Αναφορές)Ioakimidis, Nikolaos; Ιωακειμίδης, ΝικόλαοςFinite-part (hypersingular or Hadamard-type) integrals appear, naturally, in two-dimensional crack and additional problems in three-dimensional elasticity. Their computation along the radial direction leads to a one-dimensional finite-part integral on the interval [0,1] with a second-order singularity at the left end, x=0, of this interval. Kutt's Gauss–Jacobi-equivalent quadrature rule (on [0,1]) is the natural approach to the computation of this finite-part integral, but the appearance of two complex conjugate nodes (and analogous weights) outside the integration interval is a physically serious disadvantage to its use. Although the subtraction of the singularity is completely possible, here a new, semi-Gaussian quadrature rule is suggested for n = 4, 5, . . . , 10 nodes, where no node lies outside the integration interval [0,1] (exactly as in classical Gaussian quadrature rules) and, moreover, no derivative of the integrand appears in the approximation to the integral on [0,1]. The polynomial accuracy of the present quadrature rule for finite-part integrals is very high and equal to 2n-3. The proposed quadrature rule seems to be a reasonable choice (as an alternative both to Kutt's rule and to the subtraction of the singularity approach) during the evaluation of two-dimensional finite-part integrals in crack and related problems. The computation of the nodes and the weights (for n = 4, 5, . . . , 10 nodes) is described in brief and numerical results for both of these quantities are displayed together with experimental numerical results, which illustrate both the accuracy and the rapid convergence of the present quadrature rule.
- ItemOpen AccessUncertainty intervals/regions for the stress intensity factors at crack tips under uncertain loading by using the ellipsoidal model and numerical integration
Γενικό Τμήμα (Τεχνικές Αναφορές)(Αυτο-έκδοση, ) Ioakimidis, Nikolaos; Ιωακειμίδης, ΝικόλαοςQuantifier 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.
- ItemOpen AccessInterval computations in the formulae for the stress intensity factors at crack tips using the method of quantifier elimination
Γενικό Τμήμα (Τεχνικές Αναφορές)Ioakimidis, Nikolaos; Ιωακειμίδης, ΝικόλαοςThe concept of the stress intensity factor at a crack tip is extremely well known and it plays a very important role in fracture mechanics. On the other hand, uncertainty is often present in engineering problems mainly because of measurement errors and it is frequently represented with the help of interval variables. Here we consider the case of formulae for the computation of stress intensity factors at crack tips with one or more than one variable in such a formula being an interval variable. In this case, we compute the related intervals for the stress intensity factors, which, naturally, are also interval variables. This computation is based on the related existentially quantified formulae and it is made with the help of the interesting computational method of quantifier elimination as this method is efficiently implemented in the computer algebra system Mathematica. More explicitly, here the following four classical crack problems are studied: (i) the problem of a straight crack in an infinite plane isotropic elastic medium under a tensile loading at infinity normal to the crack, (ii) the related problem of a slant straight crack with respect to the loading at infinity, (iii) the problem of a crack in a similar medium now under an exponential normal loading on its edges and (iv) the problem of a periodic array of collinear straight cracks again in an infinite plane isotropic elastic medium under a tensile loading at infinity normal to the cracks. In the third and the fourth problems, approximate formulae for the stress intensity factors are used. The present results permit the efficient evaluation of the intervals (the ranges) for the stress intensity factors at crack tips when interval variables instead of crisp (deterministic) variables are present in the related formulae without any overestimation of the intervals for the stress intensity factors. Naturally, the present method is also applicable to more difficult crack problems provided, of course, that the total number of variables in the existentially quantified formulae used for quantifier elimination is small (generally up to five or six variables); otherwise, quantifier elimination may fail to yield a QFF (quantifier-free formula) at least in a reasonable time interval. The present results constitute one more application of quantifier elimination and interval analysis to applied mechanics, here to fracture mechanics.
- ItemOpen AccessA modification of the generalized airfoil equation and the corresponding numerical methods
Γενικό Τμήμα (Τεχνικές Αναφορές)Ioakimidis, Nikolaos; Ιωακειμίδης, ΝικόλαοςThe two-dimensional problem of steady, inviscid, irrotational, subsonic flow around a straight or curvilinear thin airfoil or an array of such airfoils inside a wind tunnel is generally reduced to a one-dimensional Cauchy type real or complex singular integral equation called generalized airfoil equation. Here a new form of this equation is suggested (with no change in the unknown function) with index equal to 1 instead of 0. The new equation is supplemented by an integral condition assuring the uniqueness of its solution. This modification of the generalized airfoil equation permits the application of the theoretical results and the algorithms for the numerical solution of Cauchy type singular integral equations with index equal to 1 (mainly appearing in crack problems in the theory of plane elasticity) to the generalized airfoil equation and it establishes the relationship between crack and airfoil problems. Moreover, it permits the utilization of the classical Chebyshev polynomials instead of the airfoil polynomials. Three applications are also made and numerical results are presented.
- ItemOpen AccessIntervals for the resultants of interval forces with existentially and/or universally quantified formulae with the help of the method of quantifier elimination
Γενικό Τμήμα (Τεχνικές Αναφορές)Ioakimidis, Nikolaos; Ιωακειμίδης, ΝικόλαοςThe problem of the computation of the interval of the resultant of collinear uncertain forces represented by intervals without overestimation has been recently studied in two papers (i) by Elishakoff, Gabriele and Wang (2016) and (ii) by Popova (2017). In the first paper, a modification of classical interval arithmetic is proposed whereas the methodology proposed in the second paper is based on the algebraic extension of classical interval arithmetic. Here the general case of the computation of the interval of this resultant is studied in detail on the basis of the use of quantified formulae including the existential and/or the universal quantifiers with respect to the interval forces. Many quantified formulae are possible in a resultant problem and the method of quantifier elimination in its implementation in the computer algebra system Mathematica is used for the derivation of the related quantifier-free formulae. After the illustration of the present approach in the elementary subtraction problem, which is well known for the overestimation phenomenon, the same approach is illustrated in problems (originally studied in the above papers) concerning the resultants of two, three and four collinear forces with different directions as well as in the problem of three collinear forces acting on a box. Symbolic intervals with parameters one or two of the forces are also computed. The case of the resultant of many collinear interval forces is also successfully studied. The conclusion drawn is that several overestimation-free, exact intervals can be computed for the resultant of interval forces (frequently including a degenerate interval: sharp resultant) and the derived interval (if it exists) strongly depends on the quantifiers used for the interval forces.
- ItemOpen AccessHypersingularities and cracks in plane and three-dimensional elasticity
Γενικό Τμήμα (Τεχνικές Αναφορές)Ioakimidis, Nikolaos; Ιωακειμίδης, ΝικόλαοςCrack problems in plane elasticity are often interpreted as edge dislocation arrays. Singular stress fields, such as those due to concentrated forces, dislocations and centres of rotation and dilatation, often prove useful in the interpretation and/or the solution of elasticity problems. Here a new kind of singular stress fields called hypersingularities, since these fields are straightforwardly related to hyperintegrals in mathematics, is introduced. The stress components of hypersingularities are seen to tend to infinity more rapidly than in other singular stress fields. The hypersingularities considered here are related to simple crack problems in plane and three-dimensional isotropic elasticity. Generalizations and further applications of the present results are quite possible.
- ItemOpen AccessA real closed-form integral formula for roots of nonlinear equations
Γενικό Τμήμα (Τεχνικές Αναφορές)Ioakimidis, Nikolaos; Ιωακειμίδης, ΝικόλαοςAn analytical integral formula for a single simple root of a nonlinear equation in a finite interval is proposed. The derivation of this formula is based on Picard's method for the number of roots of nonlinear equations and on Green's theorem. The present formula is particularly useful for the location of roots of nonanalytic functions. A comparison between the present formula and a related classical formula based on the theory of analytic functions is also made.
- ItemOpen AccessA quadrature method for the numerical solution of two real nonlinear equations in two unknowns
Γενικό Τμήμα (Τεχνικές Αναφορές)Ioakimidis, Nikolaos; Ιωακειμίδης, ΝικόλαοςA quadrature method for the numerical solution of a system of two real nonlinear equations in two unknowns in a region of the xy-plane, based on the use of two appropriate numerical integration rules for ordinary integrals in this region, is proposed. The main advantage of the method, beyond its originality and peculiarity, is the fact that no initial approximation to the sought solution is required. Three numerical applications, where the Gauss– and Lobatto–Chebyshev quadrature rules have been used, show the efficiency and the convergence of the proposed method. A generalization to the case of four real nonlinear equations in four unknowns (by using quaternions) is also suggested in brief.
- ItemOpen AccessRobust reliability under uncertainty conditions by using modified info-gap models with two to four horizons of uncertainty and quantifier elimination
Γενικό Τμήμα (Τεχνικές Αναφορές)(Κανένας, ) Ioakimidis, Nikolaos; Ιωακειμίδης, ΝικόλαοςQuantifier elimination for real variables constitutes an interesting computational tool with efficient implementations in some popular computer algebra systems and many applications in several disciplines. On the other hand, many practical problems concern situations under uncertainty, where uncertainty intervals and, more generally, reliability regions of uncertain quantities have to be computed. Here the interest is in the popular Ben-Haim's IGDT (info-gap or information-gap decision theory) for problems under severe uncertainty based on info-gap models, where quantifier elimination already proved to constitute a possible tool for the computation of the related reliability regions and robustness functions. Here Ben-Haim's IGDT is considered again, but now in a modified form, where more than one horizon of uncertainty is present (here two, three or four). More explicitly, here each uncertain quantity is assumed to have its own horizon of uncertainty contrary to the usual case in the IGDT, where only one horizon of uncertainty is present in the related info-gap model. Six applications are presented showing the usefulness of the present computational approach. These applications (mainly based on fractional-error info-gap models) concern (i) a linear system, (ii) a sum, (iii) the area of a rectangle, (iv) the volume of a rectangular cuboid, (v) the buckling load of a fixed-free column and (vi) the von Mises yield criterion in two-dimensional elasticity. Beyond the uncertain quantities (here two, three or four) one, two or three parameters may also be present and appear in the derived QFFs (quantifier-free formulae). Of course, it is noted that quantifier elimination generally has a doubly-exponential computational complexity and this restricts its applicability to problems with a small total number of variables (quantified and free).
- ItemOpen AccessDirect solution of plane elasticity problems by using the Muskhelishvili functional equation and computer algebra software
Γενικό Τμήμα (Τεχνικές Αναφορές)Ioakimidis, Nikolaos; Ιωακειμίδης, ΝικόλαοςThe classical Muskhelishvili functional equation for the solution of plane isotropic elasticity problems is revisited. The first complex potential φ(z) of Kolosov–Muskhelishvili is assumed to have the approximate form of a polynomial with unknown coefficients inside the elastic medium. Then the boundary conditions permit the expression of the boundary values for the second complex potential ψ(z) of Kolosov–Muskhelishvili in terms of the same coefficients. This potential, ψ(z), is then obtained by using the classical Cauchy integral formula in complex analysis. Finally, the unknown coefficients are determined by using an appropriate number of collocation points for ψ(z) outside the elastic medium. Both cases of the first and the second fundamental problems are considered. The present method was implemented and applied to the problem of an elliptical elastic medium by using the computer algebra system Mathematica and its modern and powerful language. Generalizations of the present results are also possible and suggested in brief. It is hoped that the present approach will be considered as an alternative to the classical methods of solution of plane isotropic elasticity problems like the finite and boundary element methods.
- ItemOpen AccessSymbolic computations for the approximate solution of singular integral equations: application to a crack problem
Γενικό Τμήμα (Τεχνικές Αναφορές)Ioakimidis, Nikolaos; Ιωακειμίδης, ΝικόλαοςWe propose the application of symbolic SAN (semi-analytical–numerical) computations to the numerical solution of SIEs (singular integral equations), which are the BIEs (boundary integral equations) for crack problems in plane and antiplane, isotropic and anisotropic elasticity. The case of a periodic array of collinear cracks (with a variable distance of the cracks) together with the modified Gauss–Chebyshev method (also based on the natural interpolation/extrapolation formula) for the numerical solution of SIEs are used for the illustration of the proposed approach. The obtained SAN results are seen to be very good approximations of the analytical exact results even for a very small number of nodes in the modified Gauss–Chebyshev method. The computer algebra system Derive has been used for the derivation of the present SAN results.
- ItemOpen AccessA hybrid method for the solution of problems of bending of thin plates
Γενικό Τμήμα (Τεχνικές Αναφορές)Ioakimidis, Nikolaos; Ιωακειμίδης, ΝικόλαοςAn extremely elementary hybrid method for the solution of the problem of thin plates (either finite or infinite with holes) under the action of bending moments and normal forces along the boundary of the plate is proposed. This method is based on the optical method of studying the deformed shape of the boundary of the plate (method of reflected light or `pseudocaustics') and, subsequently, on the application of elementary analytical techniques from the theory of complex variables. All quantities of interest (deflections, displacements, bending and torsional moments and shear stresses) are determined completely in the whole plate by the present hybrid method.
- ItemOpen AccessQuantifier-elimination-based interval computations in beam problems studied by using the approximate methods of finite differences and of finite elements
Γενικό Τμήμα (Τεχνικές Αναφορές)(Αυτο-έκδοση, ) Ioakimidis, Nikolaos; Ιωακειμίδης, ΝικόλαοςThe rather recent interesting computational method of quantifier elimination already implemented in four computer algebra systems has been already used in many problems of engineering interest including several problems of applied and computational mechanics. Among the previous applications of interest here is mainly the problem of a beam with parametric inequality constraints and under the presence of a loading parameter. This problem was solved by the popular methods (i) of finite differences and (ii) of finite elements in combination with the method of quantifier elimination. Here the same approach is generalized to the case where the loading parameter belongs to an interval. The methods (i) of finite differences and (ii) of finite elements are used again (leading to parametric systems of linear equations) with the computation of the approximate intervals concerning (i) the dimensionless deflection, (ii) the rotation and (iii) the dimensionless bending moment on the whole beam computed on the basis of their values at the nodes used on the beam. In the application of the finite difference method both (i) the purely existential case and (ii) a mixed universal–existential case are considered evidently with respect to the interval loading parameter. The REDLOG computer logic package of the REDUCE computer algebra system has been used again in the present interval computations and the excellent convergence of the obtained approximate intervals computed with the finite difference method is observed. In the purely existential case, up to 3072 intervals on the beam have been successfully used and this is an extremely satisfactory situation in quantifier elimination because it concerns a total number of 3076 variables.
- ItemOpen AccessSharp bounds based on quantifier elimination in truss and other applied mechanics problems with uncertain, interval forces/loads and other parameters
Γενικό Τμήμα (Τεχνικές Αναφορές)Ioakimidis, Nikolaos; Ιωακειμίδης, ΝικόλαοςThe computational method of quantifier elimination in computer algebra for real numbers related to the elimination of the universal quantifier "for all" and/or the existential quantifier "exists" in quantified formulae has already been efficiently implemented in few computer algebra systems and also applied to several interesting problems of mathematics, physics and engineering. Here this method is applied to some simple problems of applied mechanics including truss problems in structural mechanics in the case of uncertain, interval parameters related to the applied loads and/or to the parameters of the structure. Therefore, the present results are related to classical interval analysis and they permit the determination of sharp bounds for the mechanical quantities of interest such as resultants of forces, reactions and displacements in truss problems. The implementation of quantifier elimination in the powerful and user-friendly computer algebra system Mathematica has been selected as the most efficient and appropriate tool for the present computational tasks and it offers one more computational possibility in simple applied mechanics problems under uncertainty that is described by interval parameters. The five applied mechanics problems studied in detail here concern (i) the resultant of three forces acting on a box, (ii) the resultant of four forces acting on a particle, (iii) a block resting and sliding on a horizontal plane, (iv) a three-member truss and, finally, (v) a six-member truss. All these problems were already proposed and solved with interval parameters by other researchers. The present results are in agreement with the already available related results and, moreover, they always provide sharp bounds for the interval quantities of interest.
- ItemOpen AccessSymbolic intervals for the unknown quantities in simple applied mechanics problems with the computational method of quantifier elimination
Γενικό Τμήμα (Τεχνικές Αναφορές)Ioakimidis, Nikolaos; Ιωακειμίδης, ΝικόλαοςQuantifier elimination constitutes an interesting computational method in computer algebra for real polynomial and rational functions permitting the elimination of the universal quantifier "for all'' and/or the existential quantifier "exists'' in quantified formulae. This method was successfully applied to several applied mechanics problems during the last twenty-five years. On the other hand, interval analysis was also successfully employed in a very large number of applied mechanics problems under uncertainty conditions mainly during the same period permitting the determination of intervals for quantities of interest. Recently, the efficient implementation of quantifier elimination in the computer algebra system Mathematica was used in some applied mechanics problems for the computation of such intervals mainly for forces and for displacements in simple truss problems. On the other hand, because computer algebra systems are mainly used for symbolic computations, it seems natural to extend the application of quantifier elimination to the determination of intervals for quantities of mechanical interest from numerical intervals to symbolic intervals. This seems to be a simple task and, hence, it can be successfully applied to simple applied mechanics problems. Here a classical problem of a six-member truss is used for the illustration of this approach and sharp, exact symbolic intervals including one or two parameters of the problem (one or two cross-sectional parameters and/or a loading parameter) are derived of course based on numerical intervals again for one or two interval parameters. Next, the case of symbolic intervals for one or two interval parameters is also considered. In both cases, the derivation of symbolic intervals significantly extends the derivation of numerical intervals as far as the generality of the computed intervals is concerned.
- ItemOpen AccessREDLOG-aided derivation of feasibility conditions in applied mechanics and engineering problems under simple inequality constraints
Γενικό Τμήμα (Τεχνικές Αναφορές)Ioakimidis, Nikolaos; Ιωακειμίδης, ΝικόλαοςProblems involving symbolic computations and quantified variables in parametric inequality constraints appear quite naturally and frequently in applied mechanics and engineering. In this technical report, we illustrate the use of REDLOG, a recent logic package of the well-known Reduce computer algebra system, for the solution of several such problems, i.e. for the derivation of simultaneously necessary and sufficient parametric feasibility conditions (free from the quantified variables) so that our parametric inequality constraints can be completely satisfied. The present applications concern some simple problems from the theory of plates, heat transfer, elasticity and strength of materials, whereas an extremely large number of additional related problems appearing in engineering practice can also be solved with the help of REDLOG.
- ItemOpen AccessElementary quantifier-free formulae in boundary elements
Γενικό Τμήμα (Τεχνικές Αναφορές)Ioakimidis, Nikolaos; Ιωακειμίδης, ΝικόλαοςThe use of elementary algebraic quantifier elimination techniques is suggested during the numerical solution of elasticity problems by boundary element methods in problems where the derived solutions need to be verified as far as the related physical constraints (in inequality forms) are concerned. The cases of crack problems, where the crack opening displacement must be non-negative, and of contact problems, where the pressure distribution between the bodies in contact must also be non-negative, constitute two such classical elementary examples where the derived ordinary numerical solutions need to be verified with respect to the aforementioned elementary constraints or simply rejected. As a simple such application, elementary quantifier-free formulae are derived (by using Sturm's theorem) for simple univariate polynomials of the first and of the second degree which should remain positive along a finite interval. These formulae are directly applicable to the aforementioned two elasticity problems.