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- ItemOpen AccessTwo elementary analytical formulae for roots of nonlinear equations
Γενικό Τμήμα (Δημοσ. Π.Π. σε περιοδικά)(Gordon and Breach, Science Publishers, 1985) Ioakimidis, NikolaosShow more Two new simple closed-form integral formulae for the evaluation of one simple root of a nonlinear algebraic or transcendental equation in a finite real interval are proposed. The construction of these formulae is based simply on the classical method of integration by parts. Although the integrand in the first formula is discontinuous, the second formula has a continuous integrand. Numerical results are also presented.Show more - ItemOpen AccessFurther possibilities of application of the method of caustics
Γενικό Τμήμα (Δημοσ. Π.Π. σε περιοδικά)(Martinus Nijhoff Publishers, 1985-09) Ioakimidis, NikolaosShow more The classical method of caustics in fracture mechanics constitutes a particularly efficient means of experimental determination of stress intensity factors at crack tips. In this note, the same method, the method of caustics, is suggested to be used for locating points or regions of singular behavior of the stress and strain fields inside a simple elastic medium. The problem of the determination of the magnitude P and the position of a concentrated load of direction normal to the surface of an isotropic elastic half-space, but applied at an interior point of the half-space (more explicitly, at a depth h from its surface) is considered as the vehicle for the illustration of the present approach. By using the classical elasticity-based solution of this problem and the two equations of caustics (for its initial curve and the caustic itself) we reduce this problem to a system of two non-linear algebraic equations. The numerical solution of this system permits us to derive both the magnitude P and the position h (the depth of the point of application) of the concentrated load. More complicated related problems of possible applicability of the method of caustics such as the problem of location of geometrical discontinuities (for example cracks, small cavities and inclusions) inside a three-dimensional isotropic elastic medium (non-destructive testing of the elastic medium), which may cause its fracture, are also mentioned in brief. These problems can be studied by the same approach, the method of caustics, and, finally, be solved through the solution of an appropriate resulting system of non-linear algebraic equations.Show more - ItemOpen AccessOn the inversion of the first equation of caustics
Γενικό Τμήμα (Δημοσ. Π.Π. σε περιοδικά)(Martinus Nijhoff Publishers, 1985-09) Ioakimidis, NikolaosShow more The first equation of caustics, which is a complex equation, establishes the correspondence between a point on a specimen with this point defined by a complex number z (with z := x+iy on the specimen) and the corresponding point on the screen with this point defined by a complex number W (with W := X+iY on the screen). This equation appears in the optical method caustics in experimental mechanics, but here considered only when this method is applied to classical plane isotropic elasticity problems. In this short note, the inversion of this equation, the first equation of caustics, is studied in brief. This is achieved through the simultaneous consideration both of the initial equation, the first equation of caustics, and of its complex conjugate equation and, next, through the elimination of the conjugate bar(z) (with bar(z) := x-iy) of the complex number z (with z := x+iy) between these two complex equations, which is very easily possible. Then there results a single complex equation with respect to the complex number z. This final equation can be solved (i) either numerically by using an appropriate numerical method or (ii) in closed form now by using an appropriate integral formula for the analytical solution of an equation of a single complex variable z in the complex plane. The case of more than one solution of the present equation, the first equation of caustics, is also mentioned in brief. This case happens when more than one point on the specimen correspond to the same point on the screen. The present brief results extend previous related results concerning the second equation of caustics.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 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 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 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 AccessOn the practical application of the method of complex path-independent integrals to problems of fracture mechanics
Γενικό Τμήμα (Δημοσ. Π.Π. σε περιοδικά)(Kluwer Academic Publishers, 1988-03) Ioakimidis, Nikolaos; Ioakimidis, NikolaosShow more Real and complex path-independent integrals have been already repeatedly used for the determination of quantities of interest in fracture mechanics (mainly stress intensity factors at crack tips), but they have also been used for the location of crack tips in plane isotropic elastic media. Here by appropriately using both experimental optical methods of pseudocaustics and of caustics for a plane isotropic elastic medium it is observed that it is possible to experimentally measure the second derivative Phi’(z) = phi’’(z) (with z = x+iy) of the first complex potential phi(z) of Kolosov–Muskhelishvili. Next, by additionally experimentally measuring an auxiliary complex function S(z, z bar) (either by appropriately using the method of caustics or, alternatively, the method of photoelasticity) we can compute the first derivative Psi(z) = psi’(z) of the second complex potential psi(z) of the same authors. In this way, a variety of complex path-independent integrals based on Phi’(z), Psi(z) and, possibly, additional analytic functions can be computed on a closed contour C of the isotropic elastic medium under consideration and these computations permit, for example, the computation of stress intensity factors at crack tips.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 AccessThe hypersingular integrodifferential equation of a straight crack along the interface of two bonded isotropic elastic half-planes
Γενικό Τμήμα (Δημοσ. Π.Π. σε περιοδικά)(Kluwer Academic Publishers, 1988-12) Ioakimidis, NikolaosShow more The use of hypersingular integral equations (based on finite-part integrals) constitutes an interesting approach in elasticity and, particularly, in crack problems in fracture mechanics. In this note, this approach is generalized to the problem of a straight crack (analogously for a system of collinear straight cracks) along the interface of two bonded isotropic elastic half-planes. The loading of this interface straight crack consists both of a normal loading and of a shear loading (complex loading on the crack), which are assumed to be the same on both edges of the crack. In the present generalization, there results a hypersingular integrodifferential equation with unknown function the complex crack opening displacement. The construction of this equation is based on the use of the two complex potentials of Kolosov–Muskhelishvili for crack problems in plane isotropic elasticity as well as on the use of the Plemelj formulae for an appropriately defined sectionally analytic function. No condition of single-valuedness of displacements is required contrary to what happens in the case of Cauchy-type singular integral equations for ordinary straight crack problems. Furthermore, in the special case of a single isotropic elastic plane (homogeneous isotropic elastic medium), the present hypersingular integrodifferential equation reduces to the well-known hypersingular integral equation valid for ordinary straight crack problems. Next, in the case of boundaries and/or additional cracks, an additional ordinary integral term (now with a regular kernel, a Fredholm kernel in this term) appears in the present hypersingular integrodifferential equation. Finally, for the numerical solution of this equation the collocation method, which is based on the expansion of the unknown function (here the complex crack opening displacement) into a series of polynomials together with the appropriate weight function and the determination of the unknown coefficients in this series through the use of appropriately selected collocation points along the interface crack seems to be an excellent computational approach.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 AccessInteraction of a moment with a crack tip for the determination of weight functions
Γενικό Τμήμα (Δημοσ. Π.Π. σε περιοδικά)(Pergamon Press, 1990) Ioakimidis, NikolaosShow more A new approach is used for the determination of weight functions and the computation of mode I stress intensity factors at crack tips in plane isotropic elasticity problems. This approach consists in assuming a moment (couple of forces) acting on the crack edges near the crack tip and, next, applying Betti’s reciprocal work theorem. In this way, the advantages of the weight function method (over the Green’s function method) are preserved and, simultaneously, an interesting physical interpretation is given to this method. The problem of the simple straight crack is used for the illustration of the present approach. Generalizations follow without difficulty.Show more - ItemOpen AccessHypersingular Cauchy-type integrals in crack problems with hypersingular stress fields
Γενικό Τμήμα (Δημοσ. Π.Π. σε περιοδικά)(Kluwer Academic Publishers, 1990-02) Ioakimidis, NikolaosShow more Cauchy-type integrals with –1/2 power singularities at the endpoints a and b of the integration interval [a, b] proved to be extremely useful in crack problems in fracture mechanics for straight cracks [a, b] in a plane elastic medium and, particularly, for the computation of the stress intensity factors at the two crack tips t = a and t = b. Here hypersingular Cauchy-type integrals (interpreted in the sense of finite-part integrals) with –3/2 strong power singularities at the same endpoints a and b (the crack tips for a crack problem) are introduced in the complex plane z = x+iy with z a point outside the crack [a, b]. It is proved that near these endpoints t = a and t = b there exist (i) a strong, –3/2, singularity, (ii) a less strong, –2/2, singularity and (iii) an ordinary inverse-square-root weak, –1/2, singularity. The hypersingular Cauchy-type integrals introduced here are appropriate for crack problems with hypersingular stress fields near their tips with a –3/2 order of singularity for the stress components and a –1/2 order of singularity for the displacement components. Hypersingular stress fields near crack tips were introduced and extensively used by Bueckner and Stern for the computation of stress intensity factors at crack tips based on the classical Betti’s reciprocal work theorem.Show more - ItemOpen AccessTwo-dimensional principal value hypersingular integrals for crack problems in three-dimensional elasticity
Γενικό Τμήμα (Δημοσ. Π.Π. σε περιοδικά)(Springer-Verlag, 1990-03) Ioakimidis, NikolaosShow more A principal value definition of the basic hypersingular integral in the fundamental integral equation for two-dimensional cracks in three-dimensional isotropic elasticity is proposed. As is the case with the corresponding definitions of Cauchy-type one-dimensional and two-dimensional principal value singular integrals as well as Mangler-type one-dimensional principal value hypersingular integrals, the present definition is based on the special consideration of an appropriate region around the singular point. The cases of circular, square and equilateral triangular regions are considered in some detail.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 AccessSymbolic computations: A powerful method for the solution of crack problems in fracture mechanics
Γενικό Τμήμα (Δημοσ. Π.Π. σε περιοδικά)(Kluwer Academic Publishers, 1990-06) Ioakimidis, NikolaosShow more Computer algebra (or symbolic computations or manipulations) is a powerful tool available to mathematicians, engineers and scientists since more than twenty years through the use of appropriate computer software packages frequently called computer algebra systems such as MACSYMA, DERIVE, REDUCE and MATHEMATICA. Here reference is made to several practical applications of symbolic computations to problems of science and engineering including, of course, applied mechanics. On the other hand, symbolic computations can be incorporated into classical numerical methods simply through the use of symbolic parameters in these methods. This permits the derivation of SAN (Semi-Analytical–Numerical or Symbolic–Analytical–Numerical) results, which are of general validity (because of the appearance of parameters in them) and can be employed as analytical formulas, e.g. differentiated. Here this SAN approach is applied to the simple problem of a single straight crack [–1, 1] in plane isotropic elasticity under a parametric exponential pressure distribution p(x) = exp(c x), where c is a parameter. By using the related integral formula for the computation of the stress intensity factor k at the right crack tip x = 1 and the popular Gauss–Chebyshev quadrature rule with n = 1, 2, 3 and 4 nodes we derive the related approximate formulas for this stress intensity factor, which, clearly, include the exponential function as well as the parameter c in it. Next, these formulas are expanded in Maclaurin series, which have polynomial forms with variable the parameter c and permit us to observe their very satisfactory convergence. Further development and use of SAN approaches in crack problems of fracture mechanics is expected in the future.Show more - ItemOpen AccessConstruction of the equation of caustics in dynamic plane elasticity problems with the help of REDUCE
Γενικό Τμήμα (Δημοσ. Π.Π. σε περιοδικά)(Pergamon Press, 1991) Ioakimidis, NikolaosShow more The method of caustics has become a very efficient tool in crack, hole and many additional plane elasticity problems. Unfortunately, the fundamental equation of the caustics frequently requires complicated algebraic computations including that of a Jacobian determinant. Here we show that computer algebra software can prove very efficient in these computations using as a vehicle for this illustration the already known fundamental equation of caustics in dynamic plane elasticity (both for crack problems in fracture mechanics as well as for hole and additional problems). We have used the programming capabilities of REDUCE, a very popular computer algebra system, for our algebraic computations. Moreover, we illustrate the “learning” abilities of REDUCE especially for the derivation of the complex form of this equation. The case of static plane elasticity results simply as a special case of dynamic plane elasticity. Additional possibilities are suggested in brief.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 AccessLocating branch points of sectionally analytic functions by using contour integrals and numerical integration rules
Γενικό Τμήμα (Δημοσ. Π.Π. σε περιοδικά)(Gordon and Breach Science Publishers, 1992) Ioakimidis, NikolaosShow more The classical method for locating zeros (and/or poles) of analytic (and/or meromorphic) functions in the complex plane on the basis of appropriate complex contour integrals is shown to be applicable (after an appropriate modification) to the location of branch points of sectionally analytic functions. The classical trapezoidal quadrature rule is the standard rule used during the necessary numerical integrations. Numerical experiments illustrate the efficiency of the proposed method. Generalizations of the method are also suggested.Show more - ItemOpen AccessComputation of the orders of singularity of sectionally analytic functions
Γενικό Τμήμα (Δημοσ. Π.Π. σε περιοδικά)(Elsevier Science Publishing, 1992-03) Ioakimidis, NikolaosShow more In several practical problems of science and engineering the solutions are expressed with the help of a sectionally analytic function with an open arc of discontinuity. A simple method is proposed here, on the basis of numerical integration rules, for the computation of the orders of singularity of this function at the tips of its arc of discontinuity. The values of these orders are of significant practical importance. Just the values of the sectionally analytic function (and, if possible, of its first derivative as well) on a closed contour surrounding the arc of discontinuity are required. Complex contour integrals are used for this evaluation. Numerical results are also displayed and they illustrate the validity and the rapid convergence of the proposed computational technique. Generalizations of the approach are easily possible.Show more