Applications of the finite volumes method for complex flows: From the theory to the practice

In this short paper, we recall some well-known results on hyperbolic systems of conservation laws. We introduce the Godunov finite volume scheme for their approximations. We then present two recent applications to multiphase flows: the computation of a wave breaking and the construction of entropic schemes for phase transition.     

A general method for the determination of appproximations to the spectral distributions from the dynamic response functions

Rheologica Acta - A method, based on the logarithmic differentiation of the dynamic response functions with respect to frequency, is presented which is more general than the Stieltjes transform...     

Gear tooth stress analysis by the complex potentials method

The complex potentials method of plane elasticity theory is applied to spur gear stress analysis. The concept is that the stress field in a gear tooth loaded by a point force can be obtained using the Boussinesq solution and a transformation function mapping a bell-shaped curve approximating the real tooth shape in the z-plane into the semi-infinite ζ-plane. The main features of an original computer code implementing this formulation are presented along with applications.

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Sommario Dopo un'illustrazione di come lo stato di sollectitazione in un dente di una ruota dentata a denti diritti puo' essere ottenuta mediane il metodo delle trasformazioni conformi e dei potenziali complessi della teoria dell'elasticita' piana, viene presentata la struttura di un codice di calcolo originale che implementa questo approccio ed alcune applicazioni.

     

A contour integral method for dynamic stress intensity factors

The contour integral method previously used to determine static stress intensity factors is applied to dynamic crack problems. The required derivatives of the traction in the reference problem are obtained numerically by the displacement discontinuity method. Stress intensity factors are determined by an integral around a contour which contains a crack tip. If the contour is chosen as the outer boundary of the body, the stress intensity factor is obtained from the boundary values of traction and displacement. The advantage of this path-independent integral is that it yields directly both the opening-mode and sliding-mode stress intensity factors for a straight crack. For dynamic problems, Laplace transforms are used and the dynamic stress intensity factors in the time domain are determined by Durbin's inversion method. An indirect boundary element method, incorporating both displacement discontinuity and fictitious load techniques, is used to determine the boundary or contour values of traction and displacement numerically.     

Applications of the dynamic mode decomposition

The decomposition of experimental data into dynamic modes using a data-based algorithm is applied to Schlieren snapshots of a helium jet and to time-resolved PIV-measurements of an unforced and harmonically forced jet. The algorithm relies on the reconstruction of a low-dimensional inter-snapshot map from the available flow field data. The spectral decomposition of this map results in an eigenvalue and eigenvector representation (referred to as dynamic modes) of the underlying fluid behavior contained in the processed flow fields. This dynamic mode decomposition allows the breakdown of a fluid process into dynamically revelant and coherent structures and thus aids in the characterization and quantification of physical mechanisms in fluid flow.     

Analysis of dynamic mixed mode stress fields in bimaterials by the method of caustics

The equations of caustics for dynamically extending curvilinear interface cracks are derived where optical isotropy and anisotropy of the material have been considered, respectively. The equations have been obtained by applying the method of generalized complex potentials. Moreover, an algorithm for the determination of stress intensity factors from experimentally obtained caustics is presented for the case of dynamic crack extension.     

Scattering of flexural waves and dynamic stress concentrations in Mindlin's thick plates with a cutout

Using the complex variable method and conformal mapping, scattering of flexural waves and dynamic stress concentrations in Mindlin's thick plates with a cutout have been studied. The general solution of the stress problem of the thick plate satisfying the boundary conditions on the contour of cutouts is obtained. Applying the orthogonal function expansion technique, the dynamic stress problem can be reduced into the solution of a set of infinite algebraic equations. As examples, numerical results for the dynamic stress concentration factor in Mindlin's plates with a circular, elliptic cutout are graphically presented in sequence. The project supported by the National Natural Science Foundation of China     

Extension of the line element-less method to dynamic problems

Meccanica - The line element-less method is an efficient approach for the approximate solution of the Laplace or biharmonic equation on a general bidimensional domain. Introducing generalized...     

Elastic wave scattering and dynamic stress concentrations in exponential graded materials with two elliptic holes

Based on the elastodynamics, employing complex functions and conformal mapping methods, and local coordinates, the scattering of elastic waves and dynamic stress concentrations in infinite exponential graded materials with two holes are investigated. A general solution of the problem and expression satisfying the given boundary conditions are derived. The problem can be reduced to the solution of an infinite system of algebraic equations. As an example, numerical results of dynamic stress concentration factors for two elliptic holes in exponential graded materials are presented, and the influence of incident wave number and holes spacing on dynamic stress distributions is analyzed.     

Determination of stress intensity factors by the dynamic method of caustics for optically isotropic materials

Summary The method of caustics is an optical method for experimental determination of stress intensity factors at crack tips. The paper generalizes the method of caustics to dynamic situations and the dynamic stress intensity factor at the tip of a running crack in an optically isotropic material is determined. Higher order terms of the Westergaard type stress functions are included and their effect on the shape and extension of the highly constrained zone surrounding a crack tip is discussed. Analytical equations for the caustic are presented. For the singular solution it is found that dynamic K-values associated with larger shadow spots are lower than their static counterparts. Higher order terms induce a generalized evaluation formula for the stress intensity factor where powers of the order n + 5/2 (n = 0, ...) of the caustic diameter are present. The effect of superposition of dynamic and higher order term corrections on the K-value is discussed. The dynamic correction implies that the K(c)-characteristic (c ... crack velocity) is to be modified towards lower values of K. This correction is negligible for small and moderate crack velocities justifying the use of static equations for practical purposes. The K-values for crack branching, however, turn out to be smaller than assumed hitherto, a fact which is of particular interest in connection with SEN-type fracture test specimens.

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Übersicht Die schattenoptische Methode — auch Methode der Kaustik genannt — ist ein wichtiges Verfahren zur experimentellen Bestimmung des Spannungsintensitätsfaktors K an einer Rißspitze. Die vorliegende Arbeit verallgemeinert die schattenoptische Methode auf dynamische Rißausbreitungsvorgänge in optisch isotropen Materialien. Die Bestimmung des dynamischen Spannungsintensitätsfaktors an der Rißspitze schnellaufender Risse aus der Geometrie des Schattenflecks wird aufgezeigt und eine dynamische Korrekturfunktion angegeben.Glieder höherer Ordnung in den Westergaardschen Spannungsfunktionen werden beibehalten und deren Einfluß auf die Gestalt und Ausdehnung der Zone großer Verzerrungen um die Rißspitze sowie die Form und Größe der Kaustik werden untersucht. Bei Berücksichtigung des alleinigen singulären Anteiles des Spannungsfeldes um die Rißspitze zeigen die Rechnungen, daß die mit laufenden Rissen verbundenen K-Werte kleiner als die entsprechenden statischen sind, der Unterschied für praktische Zwecke jedoch erst bei Rißgeschwindigkeiten in der Umgebung der Verzweigungsgeschwindigkeit berücksichtigt werden muß.Die Glieder höherer Ordnung beeinflussen die Struktur der Beziehung zwischen Spannungsintensitätsfaktor und Kaustikdurchmesser. Die Auswerteformel, die im (statischen und dynamischen) singulären Fall eine D ^5/2- Abhängigkeit des Spannungsintensitätsfaktors vom Kaustikdurchmesser zeigt, weist bei Berücksichtigung höherer Terme Abhängigkeiten der höheren Ordnungen n + 5/2(n = 1, ...). auf. Die Korrekturfaktoren für die K-Werte, die im singulären Fall nur von der Rißgeschwindigkeit abhängig sind, werden explizite Funktionen des Kaustikdurchmessers. Das Zusammenwirken von dynamischen Effekten und Einflüssen zufolge des nichtsingulären Spannungsfeldanteiles kann bei bestimmten Arten der Probenbelastung und Probengeometrie zu einer erheblichen Korrektur des K-Wertes führen.Im experimentell gewonnenen Schaubild, das die Rißgeschwindigkeit in Abhängigkeit des Spannungsintensitätsfaktors darstellt, verursacht die dynamische Korrektur eine Verschiebung der Kurve zu etwas niedrigeren K-Werten. Dies bedeutet, daß der K-Wert bei der Rißverzweigung kleiner ist als bisher angenommen; eine Tatsache, die besonders in Verbindung mit der SEN-Bruchprobe von Interesse ist.

     

A general method for obtaining shear stress and normal stress functions from parallel disk rheometry data

The problems of converting the torque and normal force versus rim shear rate data generated by parallel disk rheometers into shear stress and normal stress difference as functions of shear rate are formulated as two independent integral equations of the first kind. Tikhonov regularization is used to obtain approximate solutions of these equations. This way of handling parallel disk rheometer data has the advantage that it is independent of the rheological constitutive equation and noise amplification is kept under control by the user-specified parameter in Tikhonov regularization. If the fluid under test exhibits a yield stress, Tikhonov regularization computation will simultaneously give an estimate of the yield stress. The performance of this method is demonstrated by applying it to a number of data sets taken from the published literature and to laboratory measurements conducted specifically for this investigation.     

Chebyshev analysis of stress concentrations

A specialized method of Chebyshev polynomial curve fitting is developed for stress-concentration analysis. The method is shown to be both economical and highly accurate.     

Applications of the integral equation method to delay differential equations

In this paper, we compare two approaches for determining the amplitude equations; namely, the integral equation method and the method of multiple scales. To describe and compare the methods, we consider three examples: the parametric resonance of a Van der Pol oscillator under state feedback control with a time delay, the primary resonance of a harmonically forced Duffing oscillator under state feedback control with a time delay, and the primary resonance together with 1:1 internal resonance of a two degree-of-freedom model. Using the integral equation method and the method of multiple scales, the amplitude equations are obtained. The stability of the periodic solution is examined by using the Floquet theorem together with the Routh–Hurwitz criterion (without time delay) and the Nyquist criterion (with time delay). By comparison with the solution obtained by the numerical integration, we find that the accuracy of the integral equation method is much better.     

Calculation of stresses in a multiply-connected anisotropic plate using stress functions of multiple complex variables

On the basis of anisotropic mathematical elasticity, using multiple conformal representations, the stress functions of multiple complex variables for an infinite multiply-connected anisotropic plate are derived. The functions are developed in Fourier series on unit circles, and the unknown coefficients of the functions are determined by undetermined coefficients method. Then the stresses in the plate can be calculated. A plate containing multiple elliptical holes or cracks is discussed, and the corresponding FORTRAN77 program is developed. Five examples are given. The results show that this method is very effective and convenient. The project supported by Aeronautical Science Foundation of China     

An approximation method to relate the linear viscoelastic functions

A method based on rational approximations is presented to interpolate the data from sinusoidal experiments in linear viscoelasticity. Bounds to the corresponding dynamical function and a discrete approximation to the spectrum are established. From this approximation the related viscoelastic functions can be computed. The method is checked by considering two theoretical models of physical interest and a satisfactory accuracy is achieved.     

Extended Kantorovich method for local stresses in composite laminates upon polynomial stress functions

The extended Kantorovich method is employed to study the local stress concentrations at the vicinity of free edges in symmetrically layered composite laminates subjected to uniaxial tensile load upon polynomial stress functions. The stress fields are initially assumed by means of the Lekhnitskii stress functions under the plane strain state. Applying the principle of complementary virtual work, the coupled ordinary differential equations are obtained in which the solutions can be obtained by solving a generalized eigenvalue problem. Then an iterative procedure is estab-lished to achieve convergent stress distributions. It should be noted that the stress function based extended Kantorovich method can satisfy both the traction-free and free edge stress boundary conditions during the iterative processes. The stress components near the free edges and in the interior regions are calculated and compared with those obtained results by finite element method (FEM). The convergent stresses have good agreements with those results obtained by three dimensional (3D) FEM. For generality, various layup configurations are considered for the numerical analysis. The results show that the proposed polynomial stress function based extended Kan-torovich method is accurate and efficient in predicting the local stresses in composite laminates and computationally much more efficient than the 3D FEM.     

Material functions generated by the complex viscosity

Based on the complex viscosity model various steady-state and transient material functions have been completed. The model is investigated in terms of a corotational frame reference. Also, BKZ-type integral constitutive equations have been studied. Some relations between material functions have been derived. C ^–1 Finger tensor - F[], (F ^–1[]) Fourier (inverse) transform - rate of deformation tensor in corotating frame - h(I, II) Wagner's damping function - J (x) Bessel function - m parameter inh (I, II) - m(s) memory function - m _k, n_k integers (powers in complex viscosity model) - P principal value of the integral - parameter in the complex viscosity model - rate of deformation tensor - shear rates - [], [] incomplete gamma function - (a) gamma function - steady-shear viscosity - ^* complex viscosity - , real and imaginary parts of ^* - ₀ zero shear viscosity - ⁺, ₁ ⁺ stress growth functions - ^–, ₁ ^- stress relaxation functions - (s) relaxation modulus - ₁(s) primary normal-stress coefficient - ø(a, b; z) degenerate hypergeometric function - ₁, ₂ time constants (parameters of ^*) - frequency - extra stress tensor