1,720,973 research outputs found
On the asymptotic expansion of the Airy function
No full textSi prova una nuova formula di rappresentazione per la famosa funzione di Airy. Ne viene data applicazione per la determinazione di certi bounds significativi per la funzione stessa
Sharp evaluation of the Oore-Burns integral for cracks subjected to arbitrary normal stress field
No full textThis paper presents a new approximation formula for the Oore-Burns integral related to three-dimensional weight functions. The approach drastically reduces the computational time of the Oore-Burns integral with respect to previous formulations because the mesh over the integration domain can be very coarse without loss of accuracy. This is made possible by analytic evaluation of the coefficient of δ1/2 of the deviation between the integral and its Riemann sum (δ is the size of the mesh over the crack). In the case of a penny-shaped crack, the new equation can be considered as an explicit formulation of the exact weight function of Galin. In order to confirm the accuracy of our new formulation, we consider the case of penny-shaped cracks under different types of mode I loading. Predictions of the stress intensity factor are compared with analytical predictions along the crack, and the new equation appears to be stable with respect to the refinement of the mesh. Furthermore, it is accura...This paper presents a new approximation formula for the Oore-Burns integral related to three-dimensional weight functions. The approach drastically reduces the computational time of the Oore-Burns integral with respect to previous formulations because the mesh over the integration domain can be very coarse without loss of accuracy. This is made possible by analytic evaluation of the coefficient of δ1/2 of the deviation between the integral and its Riemann sum (δ is the size of the mesh over the crack). In the case of a penny-shaped crack, the new equation can be considered as an explicit formulation of the exact weight function of Galin. In order to confirm the accuracy of our new formulation, we consider the case of penny-shaped cracks under different types of mode I loading. Predictions of the stress intensity factor are compared with analytical predictions along the crack, and the new equation appears to be stable with respect to the refinement of the mesh. Furthermore, it is accurate even when the stress field is represented with high-order polynomial terms. Finally, we apply our approximation of the Oore-Burns integral to an elliptical crack with small eccentricity under uniform pressure. Agreement with the Irwin solution is excellent
Stress intensity factors for embedded elliptical cracks in cylindrical and spherical vessels
No full textIn this paper, we give a very accurate approximation of the stress intensity factors of embedded elliptical cracks in cylindrical and spherical vessels. We evaluate the stress intensity factor along the whole crack border; we do so using a polynomial weight function based on a second order approximation of the Oore-Burns integral in terms of the deviation of the contour from a disk. The stress intensity factor is given for uniform internal pressure and is related to the hoop stress. Finally, a comparison with FE stress intensity factor is given
Calcolo dello stress intensity factor di cricche tridimensionali
No full textIl computo dello Stress Intensity Factor (SIF) con il metodo della funzione peso è relegato al calcolo di un integrale che vede come dominio di integrazione la zona interessata dalla cricca e come funzione integranda il prodotto fra il campo di tensione, valutato in assenza di cricca, e la funzione peso che risulta legata alla sola geometria. Nel presente lavoro, utilizzando la funzione peso di Oore-Burns, viene proposta un’equazione in forma chiusa, esatta nell’ambito di una teoria del primo ordine, e valida per il calcolo dello SIF di cricche tridimensionali sollecitate da un campo di tensione uniforme. La soluzione è stata ottenuta considerando una generica omotopia che trasforma la circonferenza di riferimento nella cricca in esame. Successivamente, l’integrale di Oore-Burns è espresso in forma chiusa sfruttando lo sviluppo in serie di Fourier del contorno del difetto. Viene data una espressione esatta dello SIF e valida per una tensione nominale generica di modo I. Come esempio applicativo verrà mostrato il calcolo dello SIF qualora il difetto sia posto in prossimità di un intaglio acuto laddove il campo tensionale mostra una singolarità
An analysis of three-dimensional planar embedded cracks subjected to uniform tensile stress
No full textIn this paper we describe an analytical methodology for calculating Stress Intensity Factors
(SIF) on planar embedded cracks with an arbitrarily shaped front. The approach is based on
a first order expansion of the celebrated integral of Oore–Burns and the actual shapes of
three-dimensional planar flaws are analysed in terms of homotopy transformations of a
reference disk.
The solution is proposed in terms of Fourier series and the first order approximation of
the coefficients is given independently from the homotopy transformations.
The comparison with numerical results, taken from scientific literature, indicates that
the proposed equation is very accurate when the flaw presents a small deviation from
the circular shape. Finally, the closed form solution is used to predict the SIF of many types
of convex and non-convex planar flaws present in engineering components such as welded
structures or casting components
New weight functions and second order approximation of the Oore-Burns integral for elliptical cracks subject to arbitrary normal stress field
No full textIn this paper, by means of a specific coordinate transformation, the singularity of the weight function is overcome. A strong advantage is obtained for a penny-shaped crack. In this case, a new exact formulation is given and a new alternative non-singular integral is proposed in terms of trigonometric functions. The new approach gives a remarkable streamlining of the Galin’s function with the advantage of reducing the complexity of the double integral. Furthermore, we give a second order analytical approximation of Oore-Burns integral with respect to deviations from the disk. This approach drastically simplify the computational procedure without loss of accuracy
Stress intensity factors in three-dimensional planar cracks subjected to uniform tensile stress
No full textIn this paper we present a closed-form solution for mode I Stress Intensity Factors (SIF) in three-dimensional planar flaws based on homotopy transformations of a disk. The SIF is given for each point of the crack border under the hypothesis of an isolated crack under tensile loading. The solution is proposed in terms of the Fourier series and the first order approximation of the coefficients is given using the explicit form. The results indicate that the proposed equation is very accurate when the flaw is a small deviation from a circle. Our solution is used to predict the SIF of many types of planar flaws and the results are compared with numerical predictions taken from the literature
First order Oore-Burns integral for nearly circular cracks under uniform tensile loading
No full textIn this paper we describe an analytical methodology for calculating Stress Intensity Factors (SIF) on planar embedded cracks with an arbitrarily shaped front. The approach is based on a first order expansion of the celebrated integral of Oore-Burns and the actual shapes of three-dimensional planar flaws are analysed in terms of homotopy transformations of a reference disk.
The solution is proposed in terms of Fourier series and the first order approximation of the coefficients is given independently from the homotopy transformations.
The comparison with numerical results, taken from scientific literature, indicates that the proposed equation is very accurate when the flaw presents a small deviation from the circular shape. Finally, the closed form solution is used to predict the SIF of many types of convex and non-convex planar flaws present in engineering components such as welded structures or casting components
Analytical evaluation of J-Integral for an elliptical notch
No full textIn the present paper the J-integral for an elliptical notch has been analytically calculated by using the stress distribution for the case of an isolated ellipse under remote tensile loading. The material is thought of as obeying a purely linear elastic law and the difference between the J-integral of a crack and the equivalent ellipse is discussed. For instance it is analytically verified that when the
notch tip radius tends toward zero the well known J-integral formula for a crack is obtained. Finally, as an engineering application, an accurate formula is given for the evaluation of the Notch Stress Intensity Factors of a crack from the peak stress of an equivalent ellipse
Analytical evaluation of J-integral for elliptical and parabolic notches under mode I and mode II loading
No full textIn the present work the J-integral (indicated
here as JVρ because two parallel flanks are not
present) was calculated by using, along the free border,
the exact analytical stress distribution for the ellipse and
the asymptotic one for parabolic notches. The material
was assumed as homogeneous isotropic and linear elastic.
First, for an ellipse under remote tensile loading, the
expression of JVρ has been analytically calculated on
the basis of Inglis’ equations. The equations have been
used to prove that, in terms of J-integral, the crack is the
limit case of an equivalent elliptic notch. Furthermore,
by distinguishing the symmetric and skew-symmetric
terms, the well-known Stress Intensity Factors (SIF) of
mode I and II for a crack in a wide plate under tension
are obtained by adding a limiting condition. Second, by
means of Creager–Paris’ equations, JVρ has been analytically
calculated for a parabolic notch of assigned
tip notch radius ρ. The asymptotic value of JVρ and the
relationship between the peak stress and the relative
SIF are the same as the ellipse. Finally, as an engineering
application, we provide an accurate formula for the
evaluation of the Notch Stress Intensity Factors of a
crack, mainly subjected to tensile stress, from the peak
stress of the equivalent ellipse under the same loading
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