The determination of principal stresses from photoelastic data

Current developments in the automation of photoelastic analysis have enabled the fast and accurate collection of the isochromatic and isoclinic parametersl from photoelastic specimens. Since the isochromatic parameter yields the difference in the principal stresses then a suitable procedure has to be used for their separation. This paper provides a survey of the stress separation techniques available with a view to incorporating the most suitable method into an automated full field polariscope.     

On the photoelastic determination of complex stress intensity factors

An improvement of the one-parameter extrapolation method of photoelastic determination of complex (mixed-mode) stress intensity factors at straight or curvilinear crack tips in a plane isotropic elastic medium due to Smith et al. [12, 13] can be achieved by measuring the absolute value of such a factor on the isochromatic fringes along properly selected polar directions and not at the maxima of the isochromatic fringes. In this way, the unknown value of the constant term of the stress field near the crack tip is taken into account. It is seen that it is always possible to find at least one appropriate polar direction to measure the absolute value of the stress intensity factor. In the case of Mode I stress intensity factors, these polar angles are = ± 120° and not = ± 90° as generally considered previously. Some numerical results are also presented in this special case and show the efficiency of the present method.     

A simple method for the photoelastic determination of mode I stress intensity factors

A new simple method for the photoelastic determination of Mode I stress intensity factors from isochromatics is proposed. This method takes into account the fact that a considerable part of the error committed in the photoelastic determination of Mode I stress intensity factors K_I at crack tips, based on experimentally obtained isochromatic fringe patterns, is due to ignoring the non-singular part of the stress field near the crack tips for the evaluation of these factors. This error can, in most cases, be minimized by an appropriate selection of the polar direction from the crack tip on which the experimental measurements for the subsequent evaluation of the stress intensity factors K_I are made. The suitable polar direction for determining K_I depends in general on the distance of the point where measurements on the isochromatics are made from the crack tip. The method was applied to the problem of a simple crack inside an infinite medium under uniaxial and biaxial loading. A comparison of the present method whith the employed analogous methods shows the superiority of the proposed method.     

A remark on the numerical solution of singular integral equations and the determination of stress-intensity factors

Summary As is well-known, an efficient numerical technique for the solution of Cauchy-type singular integral equations along an open interval consists in approximating the integrals by using appropriate numerical integration rules and appropriately selected collocation points. Without any alterations in this technique, it is proposed that the estimation of the unknown function of the integral equation is further achieved by using the Hermite interpolation formula instead of the Lagrange interpolation formula. Alternatively, the unknown function can be estimated from the error term of the numerical integration rule used for Cauchy-type integrals. Both these techniques permit a significant increase in the accuracy of the numerical results obtained with an insignificant increase in the additional computations required and no change in the system of linear equations solved. Finally, the Gauss-Chebyshev method is considered in its original and modified form and applied to two crack problems in plane isotropic elasticity. The numerical results obtained illustrate the powerfulness of the method.     

Notch effects on photoelastic determination of mixed-mode stress intensity factors

An investigation into the notch effect on the photoelastic determination of the mixed mode stress intensity factors is presented. This accomplished by comparing the isochromatic loops generated for a crack and an ellipse in an infinite plate. The generated mathematical loops are based on the exact solutions. A method of analysis is proposed and used to correct the distorted maximum angle $\text{θ}{}_{\mathrm{m}}$ and maximum shear stress $\text{τ}{}_{\max }$ due to the notch effect. The results obtained from the photoelastic measurements based on the proposed method compare favourably with those by an earlier investigation.     

A photoelastic determination of Ki stress intensity factors

The quantitative relation between the exact solution of the stress field at the vicinity of a crack tip derived fron Westergaard's formulation and the well-known Irwin singular solution was established and results obtained were correlated with photoelastic data for the study of the reigon near the crack tip. The maximum shear stress distribution expressed by the isochromatic pattern for the exact and the singular solution were calculated respectively for uniaxial and biaxial tension. The region where accurate measurements in the isochromatic pattern are possible to evaluate the stress intensity factor to any desired decree of accuracy was established and the extrapolation law for the analysis of the region near the crack tip from data obtained fron the far-field of isochromatics was demonstrated. Experimental evidence corroborated this technique. The method was compared with other already existing experimental methods for the determination of K_I.     

Transient thermal stress-intensity factors for short edge cracks with equal depth of crack tips

Stress intensity factors were determined for very short edge cracks with different inclinations but equal depths of crack tips from the heated edge of a plate under transient thermal stresses. The critical orientation of edge cracks was found experimentally.     

Elastodynamic stress-intensity factors for a semi-infinite crack under 3-D combined mode loading

The dynamic stress intensity factor histories for a half plane crack in an otherwise unbounded elastic body are analyzed. The crack is subjected to a traction distribution consisting of two pairs of suddenly-applied shear point loads, at a distance L away from the crack tip. The exact expression for the combined mode stress intensity factors as the function of time and position along the crack edge is obtained. The method of solution is based on the direct application of integral transforms together with the Wiener-Hopf technique and the Cagniard-de Hoop method, which were previously believed to be inappropriate. Some features of solutions are discussed and the results are displayed in several figures.     

Influence of geometric non-linearity on stress-intensity factors

The influence of geometric non-linearity on the stress-intensity factor for a centrally cracked plate subjected to uniformly distributed load is studied. The bending and stretching stress-intensity factors have been derived by strain energy release rate technique. It is found that the bending stress-intensity factor varies in a non-linear fashion as load increases for large deflections of the plate and the resulting in-plane stretching of the plate introduces a stretching stress-intensity factor.     

Analytical solution for bonded wedges under thermal stresses

After a change in temperature, high stresses leading to destruction may occur in bonded dissimilar materials near the point of the interface line intersection with the edge. In terms of linear elasticity, these high stresses are described by the singular terms of the stress field expansion at the corner point. In the present paper, the explicit representation of the singular terms and exact values of the stress intensity factors in the case of infinite wedge-shaped joint geometry are obtained by the Mellin transform technique. Systematic comparison with the FEM results for samples of finite size has shown that the values of stress intensity factors are in good agreement if the singularity is not too strong (the singularity orders _k<0.2). With the stronger singularity, the analytical solution is in qualitative agreement with the FEM one, such that it can be used for fast parametrical study of finite samples as well.     

Dynamic stress-intensity factors for finite strip problems

Two simple opening-mode finite strip problems are discussed. The discussion is limited to a dynamic steady-state solution within the realms of the classical theory of elasticity. It is shown that by using Fourier transform methods, the problems are reduced to equations of the Wiener-Hopf type. The complete stress and displacement distributions are difficult to obtain, and no attempt is made to arrive at this. By application of the asymptotic properties of the Fourier transform the stress-intensity factor is, instead, derived.

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Zusammenfassung Zwei einfache Probleme über den Öffnungsverlauf eines Defektes in einem Blech endlicher Abmessungen werden besprochen.Die Erörterung beschränkt sich auf eine augenblickliche dynamische Lösung auf dem Gebiete der klassischen Elästizitätstheorie. Man zeigt, daß mit Hilfe der Fourier-Transformation das Problem in die Lösung von Gleichungen des Typs Wiener-Hopf überführt werden kann. Die Bestimmung der vollständigen Verteilung der Spannungen und Verschiebungen ist jedoch schwierig und ein Versuch these Frage zu lösen wurde auch nicht unternommen. Es wurde hingegen unter Anwendung der asymptotischen Eigenschaften der Fourier-Transformation der Spannungsintensitätsffaktor abgeleitet.

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Résumé On discute deux problèmes simples de modes d'ouverturre d'un défaut dans une bande de dimensions finies.On se limite dans la discussion à une solution dynamique instantanée dans le domaine de la théorie classique de l'élasticité. On montre que, en utilisant la méthode des transformations de Fourier, les problèmes se réduisent à des équations du type Wiener-Hopf. Il est difficile de déterminer complètement les distributions des contraintes et des déplacements. Aussi ne tente-t-on pas d'y arriver. Par contre, il est possible de déduire le facteur d'intensité de contraintes en appliquant les propriétés asymptotiques de la transformée de Fourier.

     

Calculation of stress-intensity factors of cracks in nozzles

A three-dimensional finite element analysis of the stress-intensity factor for two different crack depths in nozzle of a nuclear reactor pressure vessel is presented.The stress-intensity factors are calculated using first a three-dimensional elastic crack-tip singularity element and secondly a modified compliance method.The results of the two methods are compared and show good agreement.

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Résumé On présente une analyse par éléments finis à trois dimensions du facteur d'intensité d'entaille correspondant à deux différentes profondeurs de fissuration dans un piètement d'une cuve de réacteur nucléaire.Les facteurs d'intensité des contraintes sont calculés en utilisant en premier lieu des éléments tridimensionnels décroissant la singularité élastique à la pointe de la fissure, et en second lieu une méthode de compliance modifiée.Les résultats des deux méthodes sont comparés et on constate leur bon accord.

     

Non-destructive identification of residual stresses in steel under thermal loadings

We discuss computational aspects of the inverse and ill-posed problem of identifying residual stresses in steel structures under thermal loading. This corresponds to an inverse source problem in linear thermo-elasticity. The studies aim in investigating whether thermal loadings for the excitation of structures are sufficient in order to detect reliably inherent residual stresses. These stresses may result from the construction process or later thermal or mechanical treatment of the structure-like welding. By answering the raised question positively, our method provides an important basis for successful thermal straightenings. The quality of the solution of the inverse problem depends on a series of parameters, like material parameters, noise in the measurements, and the experimental setup. We numerically study the effects of these parameters and quantify the uncertainties in the results of the inverse problems by means of Sobol indices.     

Plane thermal stresses around holes under steady distribution of temperature

Summary A method for determining the plane thermal stress distribution in a multiply connected region under steady distribution of temperature is presented. The analysis is based on the complex variable approach and permits, if the thermal field is known, the simple determination of theKolosoff functions. The method is illustrated using two examples.

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Zusammenfassung Es wird eine Methode zur Bestimmung des ebenen Wärmespannungszustandes in einem mehrfach zusammenhängenden Bereich unter stationärer Temperaturverteilung angegeben. Die Analyse benützt die komplexe Darstellung und erlaubt die einfache Bestimmung derKolosoff-Funktionen, wenn das Temperaturfeld bekannt ist. Die Methode wird anhand von zwei Beispielen illustriert.

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Notation c constant which take valuesc=E/(1–) for plane strain and - c E for plane stress - Re{} real part of - u, v components of displacement vector - T temperature - z complex variable (x+i y) - coefficient of thermal expansion - complex variable in a mapped plane - boundary value ofz or - Kolosoffs constant which takes values =3–4 for plane strain and =(3–)/(1+) for plane stress - shear modulus of elasticity - Poissons ratio - _xx,_yy,_xy components of stress tensor - () mapping function     

Mode II stress-intensity factors for cracked wood beams

The mode II (forward shear) fracture of western hemlock wood beams is studied. Results of a finite element analysis are presented showing the relationship between the mode II stress-intensity factor and the geometry of the wood beam. Two types of loading are considered: a concentrated load and a uniformly distributed load. The critical value for the stress-intensity factor, $\text{K}{}_{\mathrm{IIC}}$, is obtained by experiments on a small end-cracked beam. This critical value, and the results of the finite element analysis, are used to predict ultimate loads for end-cracked, large-size beams cut from dimension lumber. Excellent agreement is shown between theoretical predictions and experimental results on the larger beams. Species considered for the experimental program was western hemlock.     

An improved compounding method for calculating stress-intensity factors

The compounding methods for calculating stress-intensity factors for complex geometrical configurations are reexamined. It is shown that techniques which were developed specifically for problems involving localized loads, e.g. a pin-loaded hole with a crack at its edge, can also be used when the loading is remote from the crack. When these techniques are used for cracks at unloaded holes, and for cracks in stiffened sheets, the compounding equations are as simple as those derived previously. However, fewer additional calculations are now needed and numerical tests show that there is no loss of accuracy. These advantages make the new procedures preferable to those used in the earlier compounding method.     

Verification of stress-intensity factors for various middle-crack tension test specimens

Recently, the stress-intensity factor equation used in the ASTM Standard Test Method for Measurement of Fatigue Crack Growth Rates (E-647) for the middle-crack tension M(T) specimens (friction-gripped or pin-loaded) has been questioned due to the influence of the specimen “height” specified in the standard. A boundary-element code has been used to calculate the stress-intensity factors for a wide range of crack-length-to-width ratios and various height-to-width ratios for M(T) specimens under remote uniform stress, remote uniform displacement, or pin-loaded holes. Comparisons are made with some of the well-known stress-intensity factor solutions and equations in the literature. Recommended specimen heights and stress-intensity factor equations have been made.