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- ItemOpen AccessA class of surface-independent integrals in three-dimensional elasticity with an application to locating planar cracks
Γενικό Τμήμα (Δημοσ. Π.Π. σε περιοδικά)(Martinus Nijhoff Publishers, 1987-03) Ioakimidis, NikolaosShow more Path-independent integrals have found a wide range of applications in two-dimensional isotropic elasticity particularly in crack problems appearing in fracture mechanics. Similarly, in three-dimensional elasticity, surface-independent integrals have also been studied. Here it is taken into account that every problem in three-dimensional isotropic elasticity can be completely solved on the basis of four (and, quite frequently, only three) harmonic potentials. Two integral identities on a closed surface S inside a three-dimensional isotropic elastic medium, which are direct consequences of the Green identities of real three-dimensional calculus, can be directly used with the aforementioned harmonic potentials of three-dimensional isotropic elasticity. In this way, an infinity of surface-independent integrals in three-dimensional isotropic elasticity can be directly derived. As an application to fracture mechanics a planar crack C lying on the Oxy-plane and under symmetric normal loading conditions is studied with the help of the related Boussinesq–Papkovich harmonic potential G(x, y, z), which is based on the crack opening displacement. With the help of this potential and related surface-independent integrals on an appropriate surface S useful information can be gathered with respect to the planar crack C. For example, for a small crack C its approximate location on the Oxy-plane can be determined. The application of the present results to the penny-shaped crack is trivial. Generalizations of the same approach are also possible.Show more - ItemOpen AccessA note on locating straight-crack tips in finite plane elastic media
Γενικό Τμήμα (Δημοσ. Π.Π. σε περιοδικά)(Martinus Nijhoff Publishers, 1986-09) Ioakimidis, NikolaosShow more The method of complex path-independent integrals is an interesting method and practically useful in some problems of plane isotropic elasticity. Here this method is applied to the problem of locating the tips a and b of an unloaded straight crack [a, b] lying inside a finite plane isotropic elastic medium. This is achieved through the generalization of previous related results concerning two simpler cases of this problem. For this task the two complex potentials Phi(z) and Omega(z) of Kolosov–Muskhelishvili are assumed to be known in advance on a closed contour C surrounding the unloaded straight crack [a, b] with C perhaps coinciding with the boundary of the isotropic elastic medium under consideration. Then by using the Cauchy theorem in complex analysis and four appropriate complex path-independent integrals I0, I1, I2 and I3, which are based on the square of the sum of these two complex potentials, we can easily determine the two crack tips a and b of the present straight crack [a, b] in closed form. This is achieved through the derivation and use of two simple equations including the aforementioned four complex path-independent integrals. The present approach appropriately modified is also directly applicable when the closed contour C surrounds only one crack tip (either b or a) and it leads to the location of this particular crack tip. Naturally, in this computationally much simpler case, a second closed contour C* should be similarly used for the location of the other crack tip (now either a or b). Modifications and generalizations of the present approach based on complex path-independent integrals are also possible.Show more - ItemOpen AccessA numerical replacement of computer algebra methods for the derivation of polynomial equations in mechanics
Γενικό Τμήμα (Δημοσ. Π.Π. σε περιοδικά)(Elsevier Science, 1997-11) Ioakimidis, NikolaosShow more The classical computer algebra methods of Gröbner bases and characteristic sets have been already used for the derivation of polynomial formulae, mainly in classical geometry. Here we derive analogous formulae in applied mechanics by using a completely different approach: the numerical determination of the coefficients in the required polynomial formulae without the use of computer algebra techniques. Two compatibility equations for the stress components in plane elasticity and the equation of caustics in fracture mechanics are used as the vehicle for the illustration of the present elementary numerical approach. Beyond the simplicity of the present method, its powerfulness consists mainly in the fact that the well-known disadvantages of computer algebra methods (in the required computer memory and, mainly, computer time) are now avoided. Generalizations of the present approach follow easily (e.g. from polynomial to differential equations).Show more - ItemOpen AccessA real analytical integral formula for a simple root of a system of two nonlinear equations
Γενικό Τμήμα (Δημοσ. Π.Π. σε περιοδικά)(University of Niš, 1986) Ioakimidis, NikolaosShow more A method for the closed-form solution of two real nonlinear algebraic or transcendental equations, f(x,y) = 0, g(x,y) = 0, possessing one simple root (x0, y0) in a finite domain of the Oxy-plane is proposed. This method is based on the Picard method for the calculation of the number of roots of a system of nonlinear equations or, more explicitly, on the classical Gauss (or divergence) theorem in elementary mathematical analysis. The resulting formulae for x0 and y0 contain integrals including the functions f and g and their first partial derivatives. The present results generalize earlier relevant results for a single nonlinear equation.Show more - ItemOpen AccessA theoretical bound for the modulus of the generalized stress intensity factor at an interface crack tip related to the method of caustics
Γενικό Τμήμα (Δημοσ. Π.Π. σε περιοδικά)(Kluwer Academic Publishers, 1989-09) Ioakimidis, NikolaosShow more A theoretical bound is derived for the modulus (or absolute value) of the complex generalized stress intensity factor at a crack tip of an interface crack lying between two isotropic elastic half-planes in plane isotropic elasticity. This bound can be experimentally computed by employing the experimental method of caustics, which is a popular and efficient method for the computation of stress intensity factors at crack tips, but in the present case concerning an interface crack two experiments with appropriately selected overall mechanical–optical constants in the method of caustics are required. The derivation of the formula for the present bound is based on the use of the first complex potential of Kolosov–Muskhelishvili (in its expressions for both half-planes) together with the use of the classical maximum modulus principle in complex analysis. Moreover, the same formula for the bound includes two bielastic constants depending on four elastic constants, more explicitly, on the shear moduli and on the Muskhelishvili constants of the two isotropic elastic half-planes. Evidently, in the special case of just one isotropic elastic medium with a straight crack, the present bound reduces to the simpler bound valid in this special case.Show more - ItemOpen AccessA unified Riemann–Hilbert approach to the analytical determination of zeros of sectionally analytic functions
Γενικό Τμήμα (Δημοσ. Π.Π. σε περιοδικά)(Academic Press, 1988-01) Ioakimidis, NikolaosShow more A general method for the analytical determination (through closed-form integral formulae) of the zeros of sectionally analytic functions in the cut complex plane is proposed. This method is based on the application of the theory of the Riemann–Hilbert boundary value problem for sectionally analytic functions and it is a generalization of the existing relevant methods of Burniston and Siewert and of Anastasselou. These methods result here as special cases of the proposed method. As an application a new formula is derived and numerical results are presented for the root of a classical transcendental equation appearing in the theory of ferromagnetism.Show more - ItemOpen AccessAnnihilation of loading parameters in classical numerical methods with differential equations
Γενικό Τμήμα (Δημοσ. Π.Π. σε περιοδικά)(Pergamon Press (Elsevier Science), 1996-04) Ioakimidis, Nikolaos; Anastasselou, EleniShow more Classical numerical methods for the solution of applied mechanics and engineering problems usually lead to systems of linear algebraic equations. In the case when a parameter appears in the loading conditions, the same parameter appears also in the right-hand side of one or more than one of the aforementioned equations. Here the method of annihilation of this parameter, by applying appropriate linear differential operators to both sides of the linear algebraic equations including the parameter, is used. In this way, the whole problem can be solved by solving the resulting system of ordinary linear differential equations completely numerically, and symbolic computations are avoided. The case of Cauchy-type singular integral equations is used for the illustration of the method and related numerical results are presented. Further possibilities are also discussed in brief. Finally, the automatic derivation of the aforementioned annihilating operators, by using Gröbner bases, is considered.Show more - ItemOpen AccessApplication of Betti’s reciprocal work theorem to the location of cracks in three-dimensional elasticity
Γενικό Τμήμα (Δημοσ. Π.Π. σε περιοδικά)(Kluwer Academic Publishers, 1990-04) Ioakimidis, NikolaosShow more In this note, the solution of the problem of location of a crack C of arbitrary shape inside a finite three-dimensional isotropic elastic medium D is proposed through a combination of experimental and numerical methods and on the basis of the classical Betti’s reciprocal work theorem. More explicitly, the position, shape and orientation of the crack C can be characterized by a set of n unknown quantities a_j. Next, for the aforementioned medium D under consideration both the tractions t_i0 (generally known in advance) and the displacements u_i0 on the whole boundary S of D can become available by using one of the methods of experimental stress analysis. Next, a number m of loadings t_ik is assumed applied on the surface S of D and the corresponding displacements u_ik on the same surface S are computed by using an appropriate numerical method such as here the boundary element method. Under these circumstances and by appropriately using Betti’s reciprocal work theorem we can derive a system of m nonlinear algebraic equations with respect to the n unknown quantities a_j (evidently with m greater than or equal to n). The approximate solution of this system permits the computation of the unknown quantities a_j and, therefore, the determination of the position, shape and orientation of the crack C inside D. Generalizations of the present experimental–computational approach to more complicated crack problems, problems of arrays of cracks, holes, inclusions, etc. inside D are also easily possible.Show more - ItemOpen AccessApplication of computer-generated finite-difference equations to decision and inverse problems in elasticity
Γενικό Τμήμα (Δημοσ. Π.Π. σε περιοδικά)(Pergamon Press (Elsevier Science), 1998-09) Ioakimidis, NikolaosShow more Differential equations constitute a common tool for the solution of classical elasticity problems. Here we consider the possibility of using finite-difference equations in similar problems on the basis of experimental measurements of an elastic quantity (displacement, strain or stress component) at a small number of equidistant points. These equations can be derived by computer algebra techniques (Gröbner bases, characteristic sets and resultants) and they can be used for arriving at “decisions” whether a “singularity” (concentrated force, hole, inclusion, crack, etc.) exists or not and, further, for the determination of characteristic quantities concerning this singularity (such as the position of the centre and the radius of a circular hole or inclusion, the magnitude and the direction of a concentrated force or the intensity of the tensile loading, its direction at infinity, etc.). Elementary problems as well as the problems of a concentrated force, a crack and a circular hole in an infinite plane isotropic elastic medium (the latter under a tensile loading at infinity) illustrate the present approach. Furthermore, the problem of torsion of an elastic bar with an elliptical cross-section is used for the construction of a finite-difference equation in two dimensions. Futher possibilities for this kind of decision and inverse problems in elasticity are also suggested in brief. The Buchberger algorithm for the derivation of Gröbner bases (as this is implemented in the computer algebra system Maple V) has been used here as the tool for the derivation of the present finite-difference equations.Show more - ItemOpen AccessApplication of Gröbner bases to problems of movement of a particle
Γενικό Τμήμα (Δημοσ. Π.Π. σε περιοδικά)(Pergamon Press, 1994-02) Ioakimidis, Nikolaos; Anastasselou, EleniShow more The classical method of Gröbner bases for multivariate polynomials in computer algebra and the related Buchberger's algorithm and its modifications for the computation of such bases are applied to some elementary problems of kinematics as well as to the classical Kepler–Newton problem in celestial mechanics, where, beyond the variables in the polynomials, the differential operator D appears as well. The popular computer algebra system Maple V and the related standard package were used for this purpose and several possibilities of using Gröbner bases for the proof and/or the derivation of formulae in mechanics are illustrated. The present results generalize well-known results for the proof/derivation of geometric theorems by using classical Gröbner bases and related techniques and they illustrate the power of commercial computer algebra systems in the aforementioned tasks in kinematics. Modifications and generalizations of the present approach are also possible.Show more - ItemOpen AccessApplication of MATHEMATICA to the direct semi-numerical solution of finite element problems
Γενικό Τμήμα (Δημοσ. Π.Π. σε περιοδικά)(Pergamon Press, 1992-12) Ioakimidis, NikolaosShow more Several numerical methods, such as the finite element method, reduce applied mechanics and additional engineering problems to systems of linear algebraic equations. It has been already suggested that the inclusion of a symbolic parameter in the corresponding numerical results leads to a generality and a wide applicability of these results. Here we suggest the direct solution of these equations by using the popular computer algebra system MATHEMATICA. Assuming the results expressed in a Taylor–Maclaurin series form with respect to the selected symbolic parameter, the whole problem is reduced to the solution of an appropriate number of systems of purely numerical linear equations. This can be achieved either inside MATHEMATICA or by using efficient external numerical routines. As an application the above modification of the finite element method was used in the classical problem of a tapered elastic beam. The obtained semi-numerical results by the finite element method were seen to be in agreement with the available theoretical results. Further possibilities are also suggested in brief.Show more - ItemOpen AccessApplication of MATHEMATICA to the direct solution of torsion problems by the energy method
Γενικό Τμήμα (Δημοσ. Π.Π. σε περιοδικά)(Pergamon Press, 1992-05) Ioakimidis, NikolaosShow more The computer algebra system MATHEMATICA is used for the solution of classical torsion problems in the theory of elasticity together with the energy or variational method (or even, equivalently, the Ritz method). The cases of an elliptic, a triangular and a rectangular cross-section are studied. Although the first two cases lead easily to exact results, more important is the third case, where SAN (semi-analytical/numerical) approximate results are obtained. These results incorporate the dimensions of the rectangular cross-section as well as all additional symbolic quantities. The same results are also seen to converge to the corresponding numerical results for particular numerical values of the dimensions of the cross-section. Generalizations of the present results are also briefly proposed.Show more - ItemOpen AccessApplication of MATHEMATICA to the iterative SAN solution of singular integral equations appearing in crack problems
Γενικό Τμήμα (Δημοσ. Π.Π. σε περιοδικά)(Elsevier Science Publishers, 1992) Ioakimidis, NikolaosShow more The modern computer algebra system MATHEMATICA is applied to the iterative solution of Cauchy-type singular integral equations of the first kind appearing in crack and related problems of plane and antiplane elasticity. This application is made directly without transforming the singular integral equation to a Fredholm integral equation of the second kind. The semi-analytical/numerical (SAN) results by this approach are expressed in appropriate Taylor–Maclaurin series of the parameter used. A related MATHEMATICA package is presented in detail as well as SAN applications to two problems of periodic arrays of cracks in plane isotropic elasticity. The obtained SAN results show the efficiency and convergence of the proposed approach.Show more - ItemOpen AccessApplication of quadrature rules to the determination of plane equipotential lines and other curves defined by harmonic functions
Γενικό Τμήμα (Δημοσ. Π.Π. σε περιοδικά)(Elsevier Science Publishing, 1988-08) Ioakimidis, NikolaosShow more Many important kinds of curves (like equipotentials) in plane problems of physics and engineering are determined by an equation of the form u(x,y) = C, where u(x,y) is a harmonic function and C a constant. Here a solution of the previous equation is suggested (on the basis of previous analogous results for analytic functions). This solution contains a parameter varying along the curve under consideration and requires the use of convergent quadrature rules for the numerical evaluation of the integral appearing in it. An application to a simple problem of potential theory (e.g. heat transfer) is also made and points of lines of heat flow are determined by the present method. Finally, a generalization of the present results to the complicated equation of the theory of caustics in dynamic plane fracture mechanics (where u(x,y) is not a harmonic function any more) is made.Show more - ItemOpen AccessApplication of quantifier elimination to a simple elastic beam finite element below a straight rigid obstacle
Γενικό Τμήμα (Δημοσ. Π.Π. σε περιοδικά)(Pergamon Press (Elsevier Science), 1995-05) Ioakimidis, NikolaosShow more The method of quantifier elimination constitutes an interesting algebraic/applied logic computational technique concerning the derivation of formulae free from the existential and/or the universal quantifiers and the related quantified variables on the basis of original formulae generally including equalities and/or inequalities. The derived formulae are called quantifier-free formulae (QFFs). Here we consider the application of this method to a simple beam finite element of an isotropic elastic beam. This beam element is assumed lying below a straight rigid obstacle at a distance equal to d0. Hence, the deflection v(x) of the beam should continuously exceed –d0 (v(x) > –d0). The deflections and the rotations (almost equivalently the slopes) at the two tips of this beam finite element are assumed already determined by the classical finite element method. In fact, here we have a cubic polynomial for the deflection v(x) of the present beam finite element and we wish that v(x) > –d0 on the whole beam finite element. This inequality constraint (explicitly appearing in the related universally quantified formula) is transformed to an equivalent quantifier-free formula (QFF) here by using the method of quantifier elimination. The present QFF was based on the results for the continuous positivity of the cubic polynomial recently derived and communicated to the author by Collins. The present beam problem constitutes a direct link between algebraic/computational techniques for quantifier elimination and applied/computational mechanics.Show more - ItemOpen AccessApplication of quantifier elimination to inverse buckling problems
Γενικό Τμήμα (Δημοσ. Π.Π. σε περιοδικά)(Springer-Verlag Austria, 2017-10) Ioakimidis, NikolaosShow more 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.Show more - ItemOpen AccessApplication of quantifier elimination to mixed-mode fracture criteria in crack problems
Γενικό Τμήμα (Δημοσ. Π.Π. σε περιοδικά)(Springer-Verlag Germany, 2017-10) Ioakimidis, NikolaosShow more 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.Show more - ItemOpen AccessApplication of the Green and the Rayleigh–Green reciprocal identities to path-independent integrals in two- and three-dimensional elasticity
Γενικό Τμήμα (Δημοσ. Π.Π. σε περιοδικά)(Springer-Verlag, 1993-03) Ioakimidis, Nikolaos; Anastasselou, EleniShow more An elementary but quite general method for the construction of path-independent integrals in plane and three-dimensional elasticity is suggested. This approach consists simply in using the classical Green formula in its reciprocal form for harmonic functions and, further, the more general Rayleigh–Green formula also in its reciprocal form, but for biharmonic functions. A large number of harmonic and biharmonic functions appears in a natural way in the theory of elasticity. Therefore, the construction of path-independent integrals (or, probably better, surface-independent integrals in the three-dimensional case) becomes really a trivial task. An application to the determination of stress intensity factors at crack tips is considered in detail and only the sum of the principal stress components is used in the path-independent integral. Further applications of the method are easily possible.Show more - ItemOpen AccessAutomatic derivation of positivity conditions inside boundary elements with the help of the REDLOG computer logic package
Γενικό Τμήμα (Δημοσ. Π.Π. σε περιοδικά)(Elsevier Science, 1999-12) Ioakimidis, NikolaosShow more The very well known shape functions are used in classical boundary element analysis for the construction of the polynomial interpolation function p for the approximation to the unknown field quantity u. In this note, we show the usefulness of the recent REDLOG Reduce computer logic package of Dolzmann and Sturm for the construction of positivity and analogous parametric feasibility conditions, based on the nodal values of the polynomial p, inside the whole boundary element. A simple one-dimensional case, based on the classical quadratic element, and a more difficult two-dimensional case, based on a quadratic triangular element, are used for the illustration of the approach, whereas several related conclusions are also drawn. The present results are applicable to cases where the positivity or the negativity or just the boundedness of the unknown quantity is required (such as the case of the pressure distribution in contact problems and the opening displacement in crack problems) and significantly extend the already available recent related computational quantifier elimination research results.Show more - ItemOpen AccessBeams on tensionless elastic foundation: approximate quantifier elimination with Chebyshev series
Γενικό Τμήμα (Δημοσ. Π.Π. σε περιοδικά)(John Wiley & Sons, 1996-02) Ioakimidis, NikolaosShow more The well-known Sturm’s theorem (based on Sturm’s sequences) for the determination of the number of distinct real zeros of polynomials in a finite or infinite real interval has been already used in elementary quantifier elimination problems including applied mechanics and elasticity problems. Here it is further suggested that this theorem can also be used for quantifier elimination, but in more complicated problems where the functions involved are not simply polynomials, but they may contain arbitrary transcendental functions. In this case, it is suggested that the related transcendental equations/inequalities can be numerically approximated by polynomial equations/inequalities with the help of Chebyshev series expansions in numerical analysis. The classical problem of a straight isotropic elastic beam on a tensionless elastic foundation, where the deflection function (incorporating both the exponential function and trigonometric functions) should be continuously positive (this giving rise to a quantifier elimination problem along the length of the beam) is used as an appropriate vehicle for the illustration of the present mixed (symbolic–numerical) approach. Two such elementary beam problems are considered in some detail (with the help of the Maple V computer algebra system) and the related simple quantifier-free formulae are established and seen to coincide with those already available in the literature for the same beam problems. More complicated problems, probably necessitating the use of more advanced computer algebra techniques (together with Sturm’s theorem), such as the Collins well-known and powerful cylindrical algebraic decomposition method for quantifier elimination, can also easily be employed in the present approximate (because of the use of Chebyshev series expansions) symbolic–numerical computational environment.Show more