Crack initiation at V-notch tip under in-plane mixed mode loading: a review of the fictitious notch rounding concept Текст научной статьи по специальности «Медицинские технологии»

УДК 539.421

Зарождение трещины у вершины V-образного надреза при нагружении смешанного типа в плоскости: концепция закругления

фиктивного надреза

F. Berto

Падуанский университет, Виченца, 36100, Италия

Концепция закругления фиктивного надреза впервые применена для V-образных надрезов с отверстием у корня в условиях нагружения смешанного типа в плоскости с учетом отклонения растущей трещины от биссектрисы. Радиус фиктивного надреза определяется как функция радиуса реального надреза и угла раскрытия надреза. Ввиду сложности рассматриваемой проблемы применен метод на основе критерия разрушения под действием нормального напряжения. Использование указанного критерия в сочетании с критерием максимального касательного напряжения позволило определить угол распространения трещины. Разработан аналитический метод на основе процедуры Нейбера, в котором значения радиуса реального надреза представлены в виде функции соотношения режимов нагружения и угла раскрытия надреза. Для сравнения фактор нагружения определен численно путем итеративного анализа конечно-элементных моделей.

Ключевые слова: закругление фиктивной трещины, нагружение смешанного типа, V-образный надрез с отверстием у корня, микроструктурное нагружение

Crack initiation at V-notch tip under in-plane mixed mode loading: A review of the fictitious notch rounding concept

F. Berto

University of Padova, Vicenza, 36100, Italy

The fictitious notch rounding concept has been recently applied for the first time to V-notches with root hole subjected to in-plane mixed mode loading. Out-of-bisector crack propagation is taken into account. The fictitious notch radius is determined as a function of the real notch radius (the microstructural support length) and the notch opening angle. Due to the complexity of the problem, a method based on the simple normal stress failure criterion has been used. It is combined with the maximum tangential stress criterion to determine the crack propagation angle. An analytical method based on Neuber's procedure has been developed. The method provides the values of the microstructural support factor as a function of the mode ratio and the notch opening angle. The support factor is considered to be independent of the microstructural support length. Finally, for comparison, the support factor is determined on a purely numerical basis by iterative analysis of finite element models. The present paper is aimed to give a brief overview of the recent findings on this challenging topic making clear the state of the art.

Keywords: fictitious notch rounding, mixed mode loading, V-notches with root hole, microstructural support

1. Introduction

All previous publications on the fictitious notch rounding concept deal with the pure loading modes I, II and III. The difficulty of considering mixed mode loading conditions is due to the fact that the most critical direction, in which cracks might provisionally initiate and propagate, varies as a function of the mode ratio. This direction varies from the notch bisector line in the case of pure mode I load-

ing, to a direction substantially out of the notch bisector line in the case of pure mode II loading. The present work extends the fictitious notch rounding concept to mixed mode I and II loading conditions for the first time and provides a solution to the problem as a function of the mode ratio.

The fictitious notch rounding concept [1, 2] refers to the fact that the theoretical maximum notch stress does not characterize the static strength or fatigue strength of pointed

© Berto F., 2015

or sharply rounded notches. The notch stress averaged over a short radial distance at pointed notches or over a small distance normal to the notch edge at rounded notches (real notch radius p) is the key parameter of the method. The mentioned distance p* is usually called "microstructural support length". In the high-cycle fatigue regime, notch stress averaging should take a path which coincides with the point and direction of fatigue crack initiation and propagation. The basic idea of the fictitious notch rounding concept is to determine the fatigue-effective averaged notch stress directly (i.e. without actual notch stress averaging) by performing the notch stress analysis with a fictitiously enlarged notch radius pf given by

Pf =P + sp*. (1)

By taking advantage of the analytical frame provided by Filippi et al. [3] and Neuber [2], the fictitious notch rounding approach has been applied to V-notches under mode I and mode III loading, respectively [4, 5]. The support factor s was found to be highly dependent on the notch opening angle 2a. A satisfactory agreement was found between the theoretical stress concentration factor Kt (pf) evaluated at the fictitiously rounded notch and the averaging stress concentration factor Kt obtained by integrating the relevant stress over the distance p* in the bisector line of the pointed V-notch.

It is worth mentioning that the notch stress averaging method was originally proposed for brittle fracture problems by Wiegardt [6] and later extended by Weiss [7]. The fictitious notch rounding concept was mathematically formalized by Neuber [2, 8], who provided a general theoretical frame of the method. The application of the concept to strength assessments was demonstrated in Ref. [1]. There is also a correspondence of the concept with the "critical distance approach" proposed and successfully applied by Peterson [9]. It was also applied many years later to notched thin plates subjected to fatigue loading [10, 11] using the characteristic length a0 derived by El-Haddad, Topper and Smith [12] from the conventional endurance limit and the threshold stress intensity factor range.

Referring to Neuber's concept, Radaj [13-16] proposed to apply the fictitious notch rounding method to assess the high-cycle fatigue strength of welded joints (toe or root failures). A worst case assessment for welded low-strength steels, notch radius p = 0 mm, microstructural support length p* = 0.4 mm and support factor s = 2.5, resulted in pf = 1.0 mm. This reference radius was found to be generally applicable to fatigue strength assessments of welded joints in structural steels and aluminum alloys and has become a standardized procedure within the design recommendations of the International Institute of Welding

[17].

A recent work has been devoted to the application of the fictitious notch rounding approach to notches with root hole subjected to pure mode I loading [18, 19]. Some ana-

lytical expressions have been derived for the fictitious notch radius pf and the support factor s taking advantage of some closed form expressions specifically derived for V-notches with root hole [20]. An extension to pure mode III loading has been carried out in [21] on the basis of the same set of equations [20].

The case of in-plane shear loading (mode II) is more complex to analyze than mode I and mode III loading. The reason for the higher complexity is the fact that out-of-bi-sector crack propagation is observed, because the maximum notch stress occurs outside the notch bisector line. The mode II problem was considered from a theoretical point of view resulting in closed-form solutions for elliptical notches and in numerical solutions for keyhole notches [22]. In a recent paper, pointed V-notches subjected to pure mode II loading were investigated [23]. Due to the complexity of the analytical developments, the support factor s was determined numerically using the finite element method. The key problem was the choice of the crack path direction for stress averaging over the microstructural support length. Two criteria available from the literature were used to determine the angle of most probable crack propagation, the maximum tangential stress criterion according to Erdogan-Sih [24] and the minimum strain energy density criterion according to Sih [25].

Taking advantage of the closed form equations provided in Ref. [20] for V-notches with root hole, the fictitious notch rounding concept has been mathematically formalized for this notch shape in the case of in-plane shear loading [26]. Two analytical methods and one numerical method have been proposed in the quoted contribution for determining the fictitious notch radius pf and therefrom the support factor s dependent on the notch opening angle 2a. In all three methods, the crack propagation angle has been determined based on the maximum tangential stress criterion.

A state-of-the art review of the fictitious notch rounding approach comprising also the recent developments just mentioned has been carried out in Refs. [27, 28]. While the application of the fictitious notch rounding approach to the pure loading modes is well developed, in-plane mixed mode loading remains an open problem which is difficult to solve for various reasons. The main difficulty is that the most probable crack propagation angle depends on the mode ratio ranging from pure mode I to pure mode II loading.

Taking advantage of a recent paper [29] the aim of the present paper is to provide a theoretically founded basis for the application of the fictitious notch rounding approach to in-plane mixed mode loading. Using the newly developed analytical frame [20], the values of the support factor s as a function of the mode ratio M and the notch opening angle 2a are obtained. As implied by Eq. (1), the support factor s is considered to be independent of the microstructural support length p*, which is only approximately the case. But all the s values reported in the present contribution are on the safe side when used for strength assessments.

The analytical procedure given by Neuber [2] for determining the factor s is applied in edge-normal directions outside the bisector line. This is an extension of what has been done by the authors [4, 5] in the cases of pure mode I, mode II and mode III loading. By this method, V-notches with root hole, both in the real and in the fictitious configuration, are considered. Pointed notches are the limit case obtained when the real notch root radius tends to zero. The proposed method requires a numerical solution of the rather complex governing equation to determine the values of s, but easy-to-survey tables and diagrams present these values as a function of the mode ratio M for various notch opening angles 2a in order to provide a simple tool for the engineering application of the fictitious notch rounding approach in the case of mixed mode loading. Finally, for verification of the method, the factor s is also determined on a purely numerical basis by iteration of finite element models.

2. Analytical frame for V-notches with root hole subjected to mixed mode loading conditions

Using the normal stress criterion in combination with the maximum tangential stress criterion (for the crack propa-

m

Fig. 1. Fictitious notch rounding concept applied to mixed mode loading (mode I + II): real root hole notch with stress averaged over p* in direction of crack propagation (a) and substitute root hole notch with fictitious notch radius pf producing amax = a (b)

gation angle), an analytical method has been developed for evaluating the fictitious notch radius pf and the support factor s therefrom as a function of the mode ratio M defined below. The method refers to Fig. 1 where the real root radius p is substituted by the fictitious notch radius pf.

The V-notch with root hole subjected to mode I loading is considered. Taking the relevant boundary conditions into account, the stress components for the symmetric mode result from Ref. [20]:

K

GOÛ =

Ip

A -1

v5n (1+xi)+9i( y) {

cos(i -À1) ex

(1+Ài)ii( e)

2 Xi

vO

+ m ( Y)cos(1 + À1) e

+V12 ( e)x ii( e)

i+(i-M

/p\ 2 V1

vrv

/ \2 Xi p1

+ (2 + X,)

2tf.i+1)

(2)

KI

X,-i

G... =

V2n (1 + Xi) + qh(Y) (3-x,) -v n(e )

cos(1 -Xj) ex

/ \2 X / p \ 1

-TVi2 (e )x ii (e )

2 1+1

+ q>j ( Y)cos(1 + X1) e

(3+X.)

V 2X1

-1 - (2 + Xi)

KIp

^ 2(X. +1)

(3)

Tre V2n (1+x.)+ffl.(Y)

V2X1

(1 -X.)n( e)

+ ( Y)sin(1 + Xj)e

sin(i -X1) ex

+V12 ( e) x i2 ( e )

1+1

(1+Xi)

V2X1

+1 - (2 + Xi)

^ X 2(X]+1)

(4)

The parameter \ is the Williams mode I eigenvalue [30] which is dependent on the notch opening angle 2a.

The generalised mode I notch stress intensity factor K1p can be expressed as follows:

V^r1^1 aee (r,0) [(1 + Xi) + Y) ]

Kt„ =■

\2 X

1 gi + g 2

+ g 3

/p\ 2 Xi +1

+g 4

/p\2 Xi + 2

(5)

When the notch radius p tends to zero, Klp tends to the mode I notch stress intensity factor Kl defined according

to Gross and Mendelson [31]:

Ki =>/2^ lim^ ae r1-\ (6)

Considering the V-notch with root hole under mode II loading, the stress components for the antisymmetric mode result also from Ref. [20]:

aee = -

K iip r X 2%

V2n (1 -X,)+^2(y)'

x<!sin(1 -X2)e

-v 22(e)xx22 (e^)

(X, +1)-y 2i (e)| r

2X,

2X, +1

2X,

1 + (1 -X,)| p| + (2+ X2)|p

+ % (y)sin(1 + A2) ex

2(X2 +1)

Ki

= -

iip

X, -1

(1-X,)y)'

x^sin(1 -X2)e

22 (e)xx22 (e)

-^2( y) sin(1+x2) ex

\ 2X 2

1 -(3 + X2)| P| + (2 + X2)|P

2X2

(3-X2)21 (e)^r I +

2X, +1 "

2( X 2 +1)

K

Tre =

iip

X, -1

(1 -X,)y)

x<!cos(1 -X2)e

-v 2,(e)x21(e)

(1 -X,) -i^21 (e)| r

2X,

2X, +1

2X,

1 + (1 + X,)| P| - (2+ X,)|p

(7)

(8)

+ ^2 (y) cos(1+X,) ex

2( X2 +1) ^

(9)

The parameter X2 is Williams mode II eigenvalue [30], which is dependent on the notch opening angle 2a.

The generalised mode II notch stress intensity factor KIIp can be expressed as follows [20]:

V^r1^2 Tre (r, 0)

Kiip =

1 + m P

2X 2

+ a, I ^

2 X 2 +1

+ A, I ^

2 X 2 +2

(10)

When the notch root radius tends to zero, KIIp tends to the mode II notch stress intensity factor KII defined according to the following expression:

KII = V2^ limr^0 Tr0 r1"*2. (11)

It is important to underline that the property of KIIp to converge to the stress intensity factor of the pointed V-notch, KII, when the notch root radius tends to zero, is a characteristic of the set of equations provided in Ref. [20] for V-notches with root hole. Other sets of equations [32, 33] applicable to rounded V-notches of different shape do not have this property as discussed in Ref. [34]. In that paper it was also documented the stable trend of the generalized notch stress intensity factors given in Eqs. (6) and (12) as a function of the distance r. The problem of the oscillating trend of KIp and KIIp described in [33, 34] for blunt V-notches with flanks tangent to the notch root radius is overcome by employing the set of equations reported in [20]. Fictitious notch rounding is considered for V-notches with root hole subjected to mode I and mode II loading based on Eqs. (2) and (7). Based on the maximum tangential stress criterion, the crack propagation angle 60 is obtained from the following condition

dae/ d9 = 0. (12)

The hoop stress field CTee depends on the position variables r and e in such a way the condition dae /de = 0 does not determine the angle of crack propagation unless a value of r is specified. The angle e0 increases by increasing the distance r due to the higher contribution of mode II with respect to mode I loading at greater distances from the notch tip. In this paper the values of p = 0 (worst case configuration) and r = 0.005 mm (early crack propagation) have been set to evaluate the crack initiation angle e0.

It will be shown in the following that the specific value of r allows us to match the values of s obtained for pure mode I [19] and pure mode II [26]. The normal stress criterion is now introduced for the averaged stress o and the maximum tangential stress criterion for the crack propagation angle e0. Based on Eqs. (2) and (7) the maximum theoretical notch stress ctth (r, e0) is obtained and the averaged stress in the direction of e0 is determined:

1 p+p*

c = — J cth(r>eo)dr-

(13)

The determinate integral in Eq. (13) has first been solved in its indeterminate form separated into mode I and mode II components (termed ii1 and ii2), and only then in its determinate form (termed di1 and di2).

The determinate integrals referring to mode I and mode II, respectively, result in the following form:

dii(p, p*, e0) = iii( p+p*, e0) - iii(p, e0), (14)

di2(p, p*, e0) = ii2(p + p*, e0)-ii2(p, e0). (15) For the sake of brevity of presentation and due to the length of the final expressions, the explicit forms of di1 and di2 are omitted here. The limit value of the average stress for p* ^ 0 (and p = pf) is:

limp*^o di1(p. p* 9o) = pf1 1Kip x

p=pf

x{4cos [(1 + f 1) 0o ] q^ + cos [(1 - f 1) 0o ] x

x(1 + f 1 + V 12(6o) X11(9)) +'V11(9o ))}x x(V2n (1+f1 +91))-1,

limp*^ 0 di2( p. p*. 0o) = pf2 -1 Klip x

p=pf

(16)

x{4 sin[(1 + f 2 )0o ]^2 + sin[(1 - f 2 )0o ] x x(1 + f 2 - V22 (9o )?C22 (9o) - 1 (0o ))}x x(>/2n(f 2 -^2 -1))-1 (17)

The equation a(p, p*) = limp* ^o(a), according to the procedure given by Neuber [2], results in the following equation with di1(p, p*, 0o) and di2(p, p*, 0o) according to Eqs. (14), (15):

di1(p, p*, 0o) - di2(p, p*, 0o) = pf1 -1 Kip x x{4cos [(1 + f1 )0o + cos [(1 -f1 )0o ]x x(1 + f 1 + V 12(0o)X1() + Xj11(0o ))}x

(1 + f +qh))-1 -pf 2 -1 Kii p x x{4sin[(1 + f2 )0o ]^2 + sin [(1 -f2 )0o ]

x (1 + f 2 - V22 (0o )X22 () - ■V>21 (0o ))}x

(+f 2 -92 -1)-1. (18) Solving Eq. (18), with K ip = K i and K iip = K ii, it is possible to determine the fictitious radius pf and therefrom the support factor s by using the following expression derived from Eq. (1):

s = pf-p (19)

p* '

The factor s under mixed mode loading results as a function of the mode I and mode II stress intensity factors and of the crack propagation angle 0o. Equation (18) can be solved in general terms for any ratio Kiip /Kip of the notch stress intensity factors, the crack propagation angle 0o being dependent on this ratio. The mode ratio M is defined as follows based on the ratio X of the notch stress intensity factors K iip and K ip:

M =— arctan X, n

X = Kiip jf2-f X = jf .

K

(20) (21)

ip

Pure mode I loading results in M = 0 and pure mode II loading in M = 1. The dimension of the notch stress intensity factors being dependent on the relevant eigenvalues f 2 or f 1, which are not identical in general, a length parameter jf is introduced in Eq. (21) with the condition Jf = 1.

In the case of pointed notches, p = 0, Eq. (18) can be applied by replacing Kip by Ki and Kiip by Kii while

0o remains the angle of crack propagation. Pointed notches are relevant in worst case considerations of strength assessments. The following expression is valid for X, updating Eq. (21):

X = Kii jf 2 -f1 X=KTf .

(22)

The simplified hypothesis, Ki = Kip and Kii = Kiip, which has been inherently assumed for the analytical calculations, avoids an iterative procedure for the determination of s and then of pf. This is the hypothesis implicitly used by Neuber without the explicit introduction of the stress intensity factors [2], but is approximately true only when the fictitiously enlarged notch root radius is sufficiently small. In the reality, for large pf, Kip and Kiip do not match K i and Kii [19, 34].

3. Evaluation of the support factor under mixed mode loading conditions

By considering Eq. (18) and solving it for different values of the mode ratio M, it is possible to obtain the support factor s as a function of the mode ratio M. To evaluate the values of s, the condition p ~ 0 approximating pointed notches has been used. This choice was made for two specific reasons. The first reason is that pointed notches are the most critical ones in strength assessments. In many practical cases, the real notch radius is not equal to zero, but it is very small and it can be approximated in the safe direction by the worst case condition p = 0. The second reason is that by considering pointed notches, it is possible to define the mode ratio M uniquely.

The mode ratio M is evaluated here by using K ip = K i and K iip = K ii. Equation (12) has been employed here to determine the crack propagation angle 0o by applying the maximum tangential stress criterion to pointed V-notches.

Due to the fact that Eq. (18) has been expressed in terms of the real notch radius p, this radius has to remain finite. In this contribution, p = 0.001 mm has been introduced in order to model quasi-pointed V-notches. The values of s have been evaluated by solving Eq. (18) numerically for different values of the mode ratio M and keeping constant p* = 0.1 mm together with p = 0.001 mm. The data are plot-

Fig. 2. Support factor s as a function of the mode ratio M for different values of the notch opening angle 2a

Fig. 3. Geometry and dimensions of the double V-notched quadratic plate specimen considered in the finite element analysis; remote loading by prescribed nominal stress Gn, dimensions w = 100 mm and 2a = 14.14 mm; real pointed versus fictitiously rounded (root hole) V-notch

ted in Fig. 2. The support factor s rises with the mode ratio M varying from M = 0 (pure mode I) to M = 1 (pure mode II). Whereas the rise is rather weak for the keyhole (2a = 0), it is rather strong for larger notch opening angles (2a = 60°). It has to be noted that large values of s mean large fictitious notch radii pf, i.e. uncritical strength conditions.

4. Validation of the proposed fictitious notch rounding approach for mixed mode loading conditions

The fictitious notch rounding approach applied to inplane mixed mode loading is validated by comparisons based on the finite element method considering inclined V-notches with end holes (notch radius pf = sp*) characterized by different notch opening angles 2a. Geometry and dimensions of the double V-notched plate are shown in Fig. 3 together with the applied remote boundary conditions. The values of the notch stress intensity factors K I and K jj found by the finite element analysis for the corresponding pointed V-notches (p = 0) are presented in Table 1 for different notch opening angles 2a and notch inclination

angles p. Table 1 also summarizes the mode ratio M, the crack propagation angle 0O and the support factor s determined by solving Eq. (18) for the specific parameters.

The following parameters are considered in the numerical investigation by the finite element analysis:

- the notch opening angle 2a = 0°, 30°, 45° and 60°,

- the notch inclination angle P = 15°, 30°, 45° and 60° corresponding to a value of M varying between 0.16 and 0.73,

- the microstructural support length p* varying between 0.05 and 0.3 mm,

- the notch depth 2a = 10 V2 mm combined with the plate width w =100 mm.

The averaged stress has been obtained by numerical integration over the length p* of the hoop stress component (ath = 000) along the direction 0O evaluated by using Eq. (12)

_ 1 p*

a = -tJ ath (r,00)dr-

P* 0

(23)

The stress concentration factor characterising the averaged notch stress over p* in pointed V-notches is defined in Eq. (24) with reference to the nominal stress a n

1 *

Kt = — = —— J a th

(24)

CTn p on0

The stress concentration factor Kt (pf) of the fictitiously rounded notch is defined as follows:

Kt (pf):

:(p*, s)

CT n

(25)

(26)

The relative deviation can be defined as follows:

A = Kt (pf) - Kt Kt (pf)

5. Comparison with finite element results

The maximum stress CTmax (p*, s) has been determined in this section directly by means of the finite element analysis modeling the plate shown in Fig. 3 with pf = sp*, while CT values are those evaluated analytically. The finite element analysis was carried out by using ANSYS (release 13.0). In total 2000 models have been analyzed, each model charac-

Table 1

Notch stress intensity factors KI and KII and crack propagation angle 00 for different mode ratios M resulting in support factors s

2a P KI? MPa • mm1-X Kn,MPa • mm1-X X 2 00 X M s

15° 472 128 0.550 0.660 -13.49 0.272 0.169 2.56

45° 30° 376 222 0.550 0.660 -25.73 0.592 0.340 3.16

45° 244 257 0.550 0.660 -36.52 1.053 0.516 4.24

60° 112 223 0.550 0.660 -46.70 1.993 0.704 6.08

Table 2

Fictitious notch rounding results for mixed mode loading conditions; normal stress failure criterion combined with the maximum tangential stress criterion for 60, different values of the microstructural support length p* and the mode ratio M

2a ß X M 00 s p*, mm pf, mm K, Eq. (24) K (Pf), FEM A, %

45° 15° 0.05 0.128 16.90 16.93 0.17

0.272 0.169 -13.50° 2.56 0.10 0.256 12.06 12.57 4.07

0.20 0.511 8.61 9.24 6.86

0.30 0.767 7.07 7.81 9.47

30° 0.05 0.158 14.62 14.73 0.75

0.592 0.340 -25.73° 3.16 0.10 0.316 10.58 10.92 3.11

0.20 0.631 7.67 8.18 6.28

0.30 0.947 6.36 6.97 8.79

45° 0.05 0.212 11.08 11.19 0.99

1.053 0.516 -36.52° 4.24 0.10 0.424 8.18 8.56 4.48

0.20 0.848 6.05 6.65 9.03

FEM—finite element method

terized by the specific values of 2a, P and pf. The results from some models exhibiting relative deviations less than or equal to 10% are listed in Table 2 as example. The errors are due not only to the adopted simplified hypothesis, KI = KIp and Kjj = KIIp, but also to the fact that the maximum stress component amax takes place outside the notch bisector, creating an additional stress concentration effect linked to the nominal stress component parallel to the notch inclination angle P.

Finally it is worth mentioning that another consistent approach for mixed-mode loading is based on the strain energy density averaged on a defined control volume [3540]. This approach has been successfully used to assess the static behaviour of notched plates made of brittle material as well as the high-cycle fatigue strength of welded joints. Different version of the approach have been successfully used in [41-47] with some extensions to three-dimensional problems [48, 49]. For completeness it is also worth of mentioning some very recent and successful applications of the fictitious notch rounding approach [50-55] and two useful contributions describing useful concepts for applications related to notches [56, 57]. Some recent developments of strain energy density approach refer to fatigue behaviour of notched components at high temperature [58-62]

In the presence of pointed notches subjected to mixed mode I + II loading conditions, the control volume (a semicircular sector under plane strain or plane stress conditions) is introduced as constant, at first for mode I loading conditions. In the case of blunt U- or V-notches, the crescent shape control volume is rigidly rotated with respect to the notch bisector line and centred in the point of maximum tangential stress (or maximum strain energy density) on the notch edge resulting in an "equivalent" mode I loading condition. One of the main advantages of the average strain

energy density approach is that it can be applied using coarse meshes.

6. Conclusions

Based on fictitious notch rounding concept used in combination with the normal stress criterion for the averaged notch stress and the maximum tangential stress criterion for the crack propagation angle, the support factor has been analytically and numerically determined for V-notches with root hole subjected to in-plane mixed mode loading. A suitable definition and quantification of the mode ratio for pointed V-notches has been found. Taking advantage of a recently conceived analytical frame for V-notches with root hole, the original Neuber procedure for determining the fictitious notch radius and the support factor has been applied to out-of-bisector crack propagation, the propagation direction being determined by the maximum tangential stress criterion as a function of the mode ratio and the notch opening angle. Different values of this length, of the notch depth and of the notch opening angle have been considered as well as different mode ratios. The obtained values of the support factor are well suited for engineering usage in structural strength assessments. The relative deviations have been found variable from case to case. A large set of cases have been found to be characterized by relative deviations less than 10%, directly using the original Neuber's procedure.

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Сведения об авторе

Filippo Berto, Prof., University of Padova, Italy, berto@gest.unipd.it

57. Negru R., Muntean S., Pasca N., Marsavina L. Failure assessment of the shaft of a pumped storage unit // Fatigue Fract. Eng. Mater. Struct. -2014. - V. 37. - P. 807-820.

58. Gallo P., Berto F., Lazzarin P. High temperature fatigue tests of notched specimens made of titanium grade 2 // Theor. Appl. Fract. Mech. - 2015. - V. 76. - P. 27-34.

59. Gallo P., Berto F., Lazzarin P., Luisetto P. High temperature fatigue tests of Cu-Be and 40CrMoV13.9 alloys // Proc. Mater. Sci. - 2014. -V. 3. - P. 27-32.

60. Berto F., Lazzarin P., Gallo P. High-temperature fatigue strength of a copper-cobalt-beryllium alloy // J. Strain Anal. Eng. Des. - 2014. -V. 49. - P. 244-256.

61. Berto F., Gallo P., Lazzarin P. High temperature fatigue tests of un-notched and notched specimens made of 40CrMoV13.9 steel // Mater. Des. - 2014. - V. 63. - P. 609-619.

62. Berto F., Gallo P., Lazzarin P. High temperature fatigue tests of a CuBe alloy and synthesis in terms of linear elastic strain energy density // Key Eng. Mater. - 2014. - V. 627. - P. 77-80.

Поступила в редакцию 06.07.2015 г.