CC BY
58
УДК 539.375
A review of the local strain energy density approach applied to V-notches
F. Berto12 and M.R. Ayatollahi2
1 Department of Engineering Design and Materials, Norwegian University of Science and Technology, Trondheim, 7491, Norway 2 Fatigue and Fracture Laboratory, Centre of Excellence in Experimental Solid Mechanic and Dynamics, School of Mechanical Engineering, Iran University of Science and Technology, Narmak, Tehran, 16846, Iran
A synthesis of the work on the strain energy density will be presented. In particular a large bulk of experimental data from static tests of sharp and blunt V-notches and from fatigue tests of welded joints are presented in an unified way by using the mean value of the strain energy density over a given finite-size volume surrounding the highly stressed regions. When the notch is blunt, the control area assumes a crescent shape and R0 is its width as measured along the notch bisector line. In plane problems, when cracks or pointed V-notches are considered, the volume becomes a circle or a circular sector, respectively. The radius R0 depends on material fracture toughness, ultimate tensile strength and Poisson's ratio in the case of static loads; it depends on the fatigue strength ДаА of the butt ground welded joints and the notch stress intensity factor range ДК1 in the case of welded joints under high cycle fatigue loading (with ДаА and ДК1 valid for 5 • 106 cycles). Dealing with welded joints characterized by a plate thickness greater than 6 mm, the final synthesis based on strain energy density summarizes nine hundreds data taken from the literature while a new synthesis from spot-welded joints under tension and shear loading, characterized by a limited thickness of the main plate, is presented here for the first time. Dealing with static tests, about one thousand experimental data as taken from the recent literature are involved in the synthesis. The strong variability of the nondimensional radius R/R0, ranging from about zero to about 1000, makes the check of the approach based on the mean value of the strain energy density severe. In parallel, dealing with welded joints, nine hundred experimental data are here summarized in terms of local strain energy density.
Keywords: strain energy density, control radius, finite size volume, U-notch, V-notch, welded joints
Использование подхода на основе плотности энергии локальной деформации
для V-образных надрезов
F. Berto12, M.R. Ayatollahi2
1 Норвежский университет естественных и технических наук, Тронхейм, 7491, Норвегия 2 Иранский университет науки и технологии, Тегеран, 16846, Иран
В рамках единого подхода на основе усредненной плотности энергии деформации в заданном объеме конечного размера вокруг области с высокими напряжениями обсуждаются экспериментальные данные статических испытаний образцов с острыми и тупыми V-образными надрезами, а также испытаний на усталость сварных швов. В случае тупого надреза исследуемая область принимает изогнутую форму и ее ширина R0 измеряется вдоль линии, перпендикулярной надрезу. В двумерных задачах при рассмотрении трещин или острых V-образных надрезов изучаемый объем принимает форму круга или кругового сектора соответственно. При статическом нагружении радиус R0 зависит от вязкости разрушения материала, предела прочности на растяжение и коэффициента Пуассона. Для сварных швов при многоцикловом усталостном нагружении (при ДаА и ДК1 для 5 • 106 циклов) данная величина зависит от усталостной прочности ДаА стыкового соединения и значений коэффициента интенсивности напряжений в надрезе ДК1. Приведен обзор опубликованных данных по плотности энергии деформации для сварных соединений пластин толщиной более 6 мм, а также недавно полученных данных для точечных сварных соединений пластин ограниченной толщины при растяжении и сдвиге. Помимо этого в статье использованы многочисленные литературные данные статических испытаний. Ввиду большого разброса значений безразмерного радиуса R/R0 в пределах от ~0 до ~1000 выбор критерия на основе усредненной плотности энергии деформации затруднителен. Экспериментальные данные для сварных соединений также обсуждаются с точки зрения плотности энергии локальной деформации.
Ключевые слова: плотность энергии деформации, контрольный радиус, объем конечного размера, U-образный надрез, V-образный надрез, сварной шов
1. Introduction
Dealing with fracture assessment of cracked and notched components a clear distinction should be done between large
© Berto F., Ayatollahi M.R., 2017
and small bodies [1-6]. The design rules applied to large bodies are based on the idea that local inhomogeneities, where material damage starts, can be averaged being large
the volume to surface ratio. In small bodies the high ratio between surface and volume makes not negligible the local discontinuities present in the material and the adoption of a multiscaling and segmentation scheme is the only way to capture what happens at pico-, nano- and microlevels [46]. In this scheme the crack tip has no dimension or mass to speak; it is the sink and source that absorbs and dissipates energy while the stress singularity representation at every level is the most powerful tool to quantify the energy packed by an equivalent crack reflecting both material effect and boundary conditions. This new revolutionary representation implies also a new definition of mass [4, 6]. The distinction between large and small bodies should ever be considered by avoiding to transfer directly the design rules valid for large components to small ones under the hypothesis that all material inhomogeneities can be averaged [1-6].
Keeping in mind the observations above and limiting our considerations to large bodies (i.e. large volume to surface ratio), for which an averaging process is still valid, the paper is addressed to review a volume-based strain energy density approach applied to static and fatigue strength assessments of notched and welded structures [7-14].
The concept of "elementary" volume and "structural support length" was introduced many years ago [15, 16] and it states that not the theoretical maximum notch stress is the static or fatigue strength-effective parameter in the case of pointed or sharp notches, but rather the notch stress averaged over a short distance normal to the notch edge. In high cycle fatigue regime, the integration path should coincide with the early fatigue crack propagation path. A further idea was to determine the fatigue-effective notch stress directly (i.e. without notch stress averaging) by performing the notch stress analysis with a fictitiously enlarged notch radius pf corresponding to the relevant support [15, 16].
Fundamentals of critical distance mechanics applied to static failure, state that crack propagation occurs when the normal strain [17] or circumferential stress aee [18] at some critical distance from the crack tip reaches a given critical value. This "point criterion" becomes a "line criterion" in Refs. [19, 20] who dealt with components weakened by sharp V-shaped notches. A stress criterion of brittle failure was proposed based on the assumption that crack initiation or propagation occurs when the mean value of decohesive stress over a specified damage segment d0 reaches a critical value. The length d0 is 2-5 times the grain size and then ranges for most metals from 0.03 to 0.50 mm. The segment d0 was called "elementary increment of the crack length". Dealing with this topic a previous paper [21] was quoted in Refs. [19, 20]. Afterwards, this critical distance-based criterion was extended also to structural elements under multiaxial loading [22, 23] by introducing a nonlocal failure function combining normal and shear stress components, both normalised with respect to the relevant fracture stresses of the material.
Dealing with notched components the idea that a quantity averaged over a finite size volume controls the stress state in the volume by means of a single parameter, the average value of the circumferential aee stress [24].
For many years the strain energy density has been used to formulate failure criteria for materials exhibiting both ductile and brittle behavior. Since Beltrami [25] to nowadays the strain energy density has been found being a powerful tool to assess the static and fatigue behavior of notched and unnotched components in structural engineering. Different strain energy density based approaches were formulated by many researchers.
Dealing here with the strain energy density concept, it is worthwhile contemplating some fundamental contributions [26-34]. The concept of "core region" surrounding the crack tip was proposed in Ref. [26]. The main idea is that the continuum mechanics stops short at a distance from the crack tip, providing the concept of the radius of the core region. The strain energy density factor S was defined as the product of the strain energy density by a critical distance from the point of singularity [27]. Failure was thought of as controlled by a critical value Sc, whereas the direction of crack propagation was determined by imposing a minimum condition on S. The theory was extended to employ the total strain energy density near the notch tip [28], and the point of reference was chosen to be the location on the surface of the notch where the maximum tangential stress occurs. The strain energy density fracture criterion was refined and extensively summarised in Ref. [29]. The material element is always kept at a finite distance from the crack or the notch tip outside the "core region" where the inho-mogeneity of the material due to microcracks, dislocations and grain boundaries precludes an accurate analytical solution. The theory can account for yielding and fracture and is applicable also to ductile materials. Depending on the local stress state, the radius of the core region may or may not coincide with the critical ligament rc that corresponds to the onset of unstable crack extension [29]. The ligament rc depends on the fracture toughness KIC, the yield stress
, the Poisson's ratio v and, finally, on the ratio between dilatational and distortional components of the strain energy density. The direction of amax determines maximum distortion while amin relates to dilatation. Distortion is associated with yielding, dilatation tends to be associated to the creation of free surfaces or fracture and occurs along the line of expected crack extension [29, 30].
A critical value of strain energy density function (dW/dV)c has been extensively used since 1965 [31-34], when first the ratio (dW/dV)c was determined experimentally for various engineering materials by using plain and notched specimens. The deformation energy required for crack initiation in a unit volume of material is called absorbed specific fracture energy and its links with the critical value of Jc and the critical factor Sc were widely discussed. This topic was deeply considered in Refs. [27-29]
where it was showed that (dW/dV)c is equivalent to Sc/r being Sc the critical strain energy density factor and the radius vector r the location of failure. Since distributions of the absorbed specific energy W in notched specimens are not uniform, it was assumed that the specimen cracks as soon as a precise energy amount has been absorbed by the small plastic zone at the root of the notch. If the notch is sufficiently sharp, specific energy due to the elastic deformation is small enough to be neglected as an initial approximation [34]. While measurements of the energy in an infinitely small element are not possible, they can be approximated with sufficient accuracy by calculating the fracture energy over the entire fractured cross section of an unnotched tensile specimen [34]. Notched components loaded under static loads show that the average absorbed specific fracture energy decreases with increasing the notch sharpness, with the absorbed specific fracture energy parameter being plotted as a function of the theoretical stress concentration factor Kh and the temperature [34]. For a common welded structural steel and Kth = 1, the absorbed specific fracture energy value, obtained by tensile tests, is about 1 MJ/m3 while for values of Kth greater than 3 a plateau value is visible [35]. Depending on the considered welded metal, the plateau approximately ranges between 0.15 and 0.35 MJ/m3. These values are not so different from the mean value that characterizes the high cycle fatigue strength of welded joints, Wc = 0.105 MJ/m3 but with reference to a specific control volume [8, 10].
The criterion based on the energy density factor S gave a sound theoretical basis to the experimental findings [3134] and the approach, used in different fields, was strongly supported by a number of researchers [35]. Recently as stated above, the volume energy function has been scaled from macro to micro to take into account the microcracks with a stronger stress singularity [2]. Dealing with pointed V-notches the volume energy density factor S was defined and applied as an extension of the method proposed for the crack case [30]. Potential sites of fracture initiation were assessed and the rate change of volume with surface A V/AA was accurately evaluated by using numerical models showing that the local variation of this parameter should be kept smaller than the global average of A V/AA in the system to assure the reliability of the numerical results. Moreover the critical strain energy density factor Sc was plotted as a function of the notch opening angle both for symmetrical and skew-symmetrical loadings. The fundamental hypothesis was that the location of yield and fracture initiation would coincide with the maximum of the maximum strain energy density function (dW/ dV and maximum of the minimum strain energy density function (dW/ dV )™X with reference to the angular space variable, respectively. The stationary values of dW/dV in a system may have numerous maxima and minima. The pair (dW/ dV)mX and (dW/ dV)™X is unique and corresponds to a specific physical meaning. The peak of (dW/dV)max,
corresponding to a relative maximum, refers to yield because shape alteration with small volume change while the relative minimum corresponding to an increasing volume change is related to fracture. The second hypothesis is that failure by yielding and fracture would occur when (dW/dV)mX and (dW/dV)m£ reach their respective critical values (dW/ dV)p and (dW/ dV)c.
The concept of strain energy density has also been reported in the literature in order to predict the fatigue behavior of notches both under uniaxial and multiaxial stresses [36, 37].
It should be remembered that in referring to small-scale yielding, a method based on the averaged of the stress and strain product within the elastic-plastic domain around the notch was extended to cyclic loading of notched components [38]. In particular in Ref. [39] it was proposed a fatigue master life curve based on the use of the plastic strain energy per cycle as evaluated from the cyclic hysteresis loop and the positive part of the elastic strain energy density. The two views, cyclic hysteresis loop concept evaluating the plastic energy for tensile specimens [38, 39] and the criterion evaluating the local accumulated strain energy density near the crack tip [27], although formally different, are strictly connected and both tied to the concept of absorbed specific fracture energy.
The averaged strain energy density criterion, proposed in Refs. [7-14, 40], states that brittle failure occurs when the mean value of the strain energy density over a control volume (which becomes an area in two dimensional cases) is equal to a critical energy Wc. The strain energy density approach is based both on a precise definition of the control volume and the fact that the critical energy does not depend on the notch sharpness. Such a method was formalized and applied first to sharp, zero radius, V-notches and later extended to blunt U- and V-notches under mode I loading [11] and successfully applied to welded joints [10]. The control radius R0 of the volume, over which the energy has to be averaged, depends on the ultimate tensile strength, the fracture toughness and Poisson's ratio in the case of static loads, whereas it depends on the unnotched specimen fatigue limit, the threshold stress intensity factor range and the Poisson's ratio under high cycle fatigue loads. The approach was successfully used under both static and fatigue loading conditions to assess the strength of notched and welded structures subjected to predominant mode I and also to mixed mode loading [7-14]. The extension of the strain energy density approach to ductile fracture is possible, with a major problem being the definition of the control volume and the influence of the dilatational and distor-tional components of the strain energy density. Recently, the effect of plasticity in terms of strain energy density over a given control volume has been considered by the present authors, showing different behaviours under tension and torsion loading, as well as under small and large scale yielding [14].
Several criteria have been proposed to predict fracture loads of components with notches, subjected to mode I loading [19, 20, 41-52]. Recently, fracture loads of notched specimens (sharp and blunted U- and V-notches) loaded under mode I have been successfully predicted, using a criterion based on the cohesive zone model [53-56], and in parallel by applying the local strain energy density [7-14]. The problem of brittle failure from blunted notches loaded under mixed mode is more complex than in mode I loading and experimental data, particularly for notches with a non-negligible radius, is scarce. The main aim of some recent papers was to generalize the previous results valid for components with blunted notches loaded under mode I, to notched components loaded under mixed mode [57-60]. This generalization is based on the hypothesis that fracture mainly depends on the local mode I and on the maximum value of the principal stress or the strain energy density. The proposal of mode I dominance for cracked plates was suggested first in Ref. [61 ] when dealing with cracked plates under plane loading and transverse shear, where the crack grows in the direction almost perpendicular to the maximum tangential stress in radial direction from its tip. Two different methods are used to verify such a hypothesis: the cohesive zone model and the model based on the strain energy density over a control volume [57-59]. Both methods allow to evaluate the critical load under different mixed mode conditions when the material behaviour can be assumed as linear elastic. Dealing with the strain energy density approach it is worth noting that the case of pure compression or combined compression and shear, for example, would require a reformulation for the control radius of the volume R0 and should also take into account the variability of the critical strain energy density Wc with respect to the case of uniaxial tension loads. To the best of the author's knowledge, the first contribution that modifies the total strain energy density criterion (Beltrami hypothesis) to account for the different strength properties exhibited by many materials under pure tension and pure compression uniaxial tests was dated 1926 [62].
Dealing with both notched and welded components and summarizing the most recent experimental results reported in the literature, the main aim of the present contribution is to present a complete review of the analytical frame of the volume-based strain energy density approach together with a final synthesis of more than 1900 experimental data from static and fatigue tests. Very different materials have been considered with a control radius R0 ranging from 0.4 to 500.0 |m.
2. Some expressions for strain energy density in the control volume
With the aim of clarifying the base of the final synthesis carried out in this paper, this section summarizes the analytical frame of strain energy density approach.
Fig. 1. Notch geometry and coordinate system
2.1. Stress distributions due to U- and V-notches
With reference to the polar coordinate system shown in Fig. 1, with the origin located at point O, mode I stress distribution ahead of a V-notch tip is given by the following expressions [63]:
~ ^ -1 Gj = air 1
fj (e, a) +
vr°y
gj (e, a)
(1)
where > |1 and the parameter a1 can be expressed either via the notch stress intensity factor K1 in the case of a sharp, zero radius, V-notch or by means of the elastic maximum notch stress a tip in the case of blunt V-notches. In Eq. (1) r0 is the distance evaluated on the notch bisector line between the V-notch tip and the origin of the local coordinate system; r0 depends both on the notch root radius R and the opening angle 2a (Fig. 2), according to the expression r0 = R[(n-2 a)/(2 n-2a)]. The angular functions fj and gij are given in Ref. [63]:
fee
frr
fre
1
1 + + Xb1(1 -V
(i+À1)cos(1 -à1) e (33)cos(1 -à 1) e (1 -À1)sin(1 -à1) e
cos (1+à1) e
-cos (1+à1) e
sin(1+à1) e
(2)
4( q -1)[1 +À1 +Xb1(1 '(1+^1)cos[(1 -^1) e]
Xd <(331)c°s[(1 -^1)e]
(1 - ^1) sin [(1 -m) e] ' cos [(1e] P +Xq |-cos[(1en . (3)
J sin[(1e] J^
The eigenfunctions f depend only on Williams' eigenvalue, À1 which controls the sharp solution for zero notch radius [64]. The eigenfunctions g j mainly depend on eigen-
Fig. 2. Critical volume (area) for sharp V-notch (a), crack (b) and blunt V-notch (c) under mode I loading. Distance r0 = R (n - 2a )/(2rc - 2a)
value but are not independent from A^ Since ^ <A^ the contribution of ^-based terms in Eq. (1) rapidly decreases with the increase of the distance from the notch tip. All parameters in Eqs. (2), (3) have closed form expressions but here, for the sake of brevity, only some values for the most common angles are reported in Table 1 [63].
Under the plane strain conditions, the eigenfunctions fj and gj satisfy the following expressions:
fzz (6) = v(/ee (8)+frr (6)),
gzz (6) = v( gee (6) + grr (6)), 4
whereas fzz (6) = gzz (6) = 0 under plane stress conditions.
2.2. Strain energy density approach
The strain energy density approach is based on the idea that under tensile stresses failure occurs when W = Wc, where the critical value Wc obviously varies from material to material. If the material behavior is ideally brittle, then Wc can be evaluated by using simply the conventional ultimate tensile strength a t, so that Wc =o?/(2 E).
Often unnotched specimens exhibit a nonlinear behavior whereas the behavior of notched specimens remains linear. Under these circumstances the stress at should be substituted by "the maximum normal stress existing at the edge at the moment preceding the cracking", as underlined in Ref. [20] where it is also recommended to use tensile specimens with semicircular notches.
In plane problems, the control volume becomes a circle or a circular sector with a radius R0 in the case of cracks or
pointed V-notches in mode I or mixed, I + II, mode loading (Fig. 2, a, b). Under plane strain conditions, a useful expression for R0 has been provided considering the crack case [40]:
Ro =
(1 + v)(5 -8 v)
4 k
K
\2
IC
(5)
If the critical value of the notch stress intensity factor is determined by means of specimens with 2a ^ 0, the critical radius can be estimated by means of the expression:
-,1/(2 -2 AT)
, (6)
R =
I1K1C
4À1( k -a) EWc
IC-
when 2a = 0, K1C equals the fracture toughness K
In the case of blunt notches, the area assumes a crescent shape, with R0 being its maximum width as measured along the notch bisector line (Fig. 2, c) [11]. Under mixed-mode loading, the control area is no longer centred with respect to the notch bisector, but rigidly rotated with respect to it and centred on the point where the maximum principal stress reaches its maximum value [57, 58]. This rotation is shown in Fig. 3 where the control area is drawn for a U-shaped notch both under mode I loading (Fig. 3, a) and mixed-mode loading (Fig. 3, b).
The parameter a1 of Eq. (1) can be linked to the mode I notch stress intensity factor by means of the simple expression
a =
V2K
(7)
Parameters for stress distributions and local strain energy [63]
Table 1
2a, rad
q
Ml
x bl
x cl
Xdi
cb1
F(2a)
0
2.0000
0.5000
-0.5000
1.0000
4.0000
0.0000
1.000
0.7850
n/ 6
1.8333
0.5014
-0.4561
1.0707
3.7907
0.0632
1.034
0.6917
n/ 4
1.7500
0.5050
-0.4319
1.1656
3.5721
0.0828
1.014
0.6692
n/ 3
1.6667
0.5122
-0.4057
1.3123
3.2832
0.0960
0.970
0.6620
n/ 2 2n/ 3 3n/ 4 5n/ 6
1.5000 1.3334 1.2500 1.1667
0.5448 0.6157 0.6736 0.7520
-0.3449 -0.2678 -0.2198 -0.1624
1.8414 3.0027 4.1530 6.3617
2.5057 1.5150 0.9933 0.5137
0.1046 0.0871 0.0673 0.0413
0.810 0.570 0.432 0.288
0.7049 0.8779 1.0717 1.4417
where K1 assumes the following form according to the definition given in Ref. [66]
K = lim [g0 (r, 8 = 0)] r1^1. (8)
r
In the presence of a notch root radius equal to zero, the distance r0 is also zero, and all ^-related terms in Eq. (1) disappear. It is possible to determine the total strain energy over the area of radius R0 and then the mean value of the elastic strain energy density referred to the area The final relationship is
/ x2
Wi =
Il
4 EÀ1(n-a)
Ki
(9)
where À1 is Williams' eigenvalue and K1 the mode I notch stress intensity factor. The parameter I1 is different under plane stress and plane strain conditions and is given in Table 2 for different values of Poisson's ratio v [59]. Equation (9) was extended to pointed V-notches in mixed, I + II, mode [7] as well as to cases where mode I loads where combined with mode III loads [9]. While describing Table 2 it is important to underline the influence of Poisson's ratio on the I1 values in the case of sharp notches. For a notch opening angle smaller than 60°, I1 varies strongly from v = 0.1 to v = 0.4. This fact confirms the important
and not negligible effect of the Poisson's ratio while discussing sharp or quasi-sharp notches in agreement with the effect of this parameter on the kind of singularities, weak or strong, highlighted in Ref. [1] in the case of a constrained micronotch at the front of a free edge macrocrack.
2.3. Strain energy density for blunt V-notches under mode I loading
In the presence of rounded V-notches it is possible to link the parameter a1 of Eq. (1) to the maximum principal stress present at the notch tip:
a1 =
a tip ro
1-X1
1+ C01
(10)
where ( is the parameter already listed in Table 1. By using the elastic maximum notch stress, it is possible to determine the total strain energy over the area ^ and then the mean value of the strain energy density. When the area embraces the semicircular edge of the notch (and not its rectilinear flanks), the mean value of strain energy density can be expressed in the following form [11]:
W1 = F (2a) H
2a,R '
R
V J
tip E
(11)
where F(2a) depend on previously defined parameters
Table 2
Parameter I for sharp, zero radius, V-notches (plane strain) [7, 59]
2 a y/rc, rad À1 I1
v = 0.10 v = 0.15 v = 0.20 v = 0.25 v = 0.30 v = 0.35 v = 0.40
0° 1 0.5000 1.1550 1.0925 1.0200 0.9375 0.8450 0.7425 0.6300
15° 23/24 0.5002 1.1497 1.0880 1.0162 0.9346 0.8431 0.7416 0.6303
30° 11/12 0.5014 1.1335 1.0738 1.0044 0.9254 0.8366 0.7382 0.6301
45° 7/8 0.5050 1.1063 1.0499 0.9841 0.9090 0.8247 0.7311 0.6282
60° 5/6 0.5122 1.0678 1.0156 0.9547 0.8850 0.8066 0.7194 0.6235
90° 3/4 0.5445 0.9582 0.9173 0.8690 0.8134 0.7504 0.6801 0.6024
120° 2/3 0.6157 0.8137 0.7859 0.7524 0.7134 0.6687 0.6184 0.5624
135° 5/8 0.6736 0.7343 0.7129 0.6867 0.6558 0.6201 0.5796 0.5344
150° 7/12 0.7520 0.6536 0.6380 0.6186 0.5952 0.5678 0.5366 0.5013
Table 3
Values of the function H for blunted V-shaped notches [11]
Table 4
2a, rad R0IR H
v = 0.30 v = 0.35 v = 0.20
0.01 0.5638 0.5432 0.5194
0.05 0.5086 0.4884 0.4652
0 0.10 0.4518 0.4322 0.4099
0.30 0.3069 0.2902 0.2713
0.50 0.2276 0.2135 0.1976
1.00 0.1314 0.1217 0.1110
0.01 0.6395 0.6162 0.5894
0.05 0.5760 0.5537 0.5280
n/ 6 0.10 0.30 0.5107 0.3439 0.4894 0.3264 0.4651 0.3066
0.50 0.2531 0.2386 0.2223
1.00 0.1428 0.1333 0.1226
0.01 0.6678 0.6436 0.6157
0.05 0.5998 0.5769 0.5506
n 3 0.10 0.5302 0.5087 0.4842
0.30 0.3543 0.3372 0.3179
0.50 0.2597 0.2457 0.2301
1.00 0.1435 0.1349 0.1252
0.01 0.6290 0.6063 0.5801
0.05 0.5627 0.5415 0.5172
n/ 2 0.10 0.30 0.4955 0.3296 0.4759 0.3144 0.4535 0.2972
0.50 0.2361 0.2246 0.2115
1.00 0.1328 0.1256 0.1174
0.01 0.5017 0.4836 0.4628
0.05 0.4465 0.4298 0.4106
2n/ 3 0.10 0.3920 0.3767 0.3591
0.30 0.2578 0.2467 0.2339
0.50 0.1851 0.1769 0.1676
1.00 0.1135 0.1079 0.1015
0.01 0.4114 0.3966 0.3795
0.05 0.3652 0.3516 0.3359
3n/ 4 0.10 0.3206 0.3082 0.2938
0.30 0.2082 0.1997 0.1900
0.50 0.1572 0.1504 0.1427
1.00 0.1037 0.0988 0.0932
F (2a) =
(
-1
\2(1-^)
q
V2K
1+ Ô) 1
(12)
and is reported in the last column of Table 1. H is summarized in Tables 3 and 4 as a function of opening angles and Poisson's ratios. By simply using the definition of the
H values for U-notched specimens [11]
R0IR H
v = 0.10 v = 0.15 v = 0.20 v = 0.25 v = 0.30
0.0005 0.6294 0.6215 0.6104 0.5960 0.5785
0.0010 0.6286 0.6207 0.6095 0.5952 0.5777
0.0050 0.6225 0.6145 0.6033 0.5889 0.5714
0.0100 0.6149 0.6068 0.5956 0.5813 0.5638
0.0500 0.5599 0.5515 0.5401 0.5258 0.5086
0.1000 0.5028 0.4942 0.4828 0.4687 0.4518
0.3000 0.3528 0.3445 0.3341 0.3216 0.3069
0.5000 0.2672 0.2599 0.2508 0.2401 0.2276
1.0000 0.1590 0.1537 0.1473 0.1399 0.1314
mode I notch stress intensity factor for blunt V-notches [67] a simple relationship between atip and KRj can be obtained as follows:
/ 1 \1-Aj KV =V2^| i-1 R '
1 + (B1
= V F (2a) o tIp R1^1. (13)
Then it is possible to rewrite Eq. (11) in a more compact form:
W1 = H\ 2a, R° 1 \ R
( K VI)2
1
R
2(1-^) 0
(14)
Equation (14) will be used to summarise all results from blunt notches (U- and V-notches) subjected to mode I loading.
2.4. Strain energy density for blunt notches under mixed mode loading
Under mixed mode loading the problem becomes more complex than under mode I loading, mainly because the maximum elastic stress is out of the notch bisector line and its position varies as a function of mode I to mode II stress distributions (see Fig. 3). The problem was widely discussed considering different combination of mode mixity [57, 58].
The expression for U-notches under mixed mode is analogous to that valid for notches in mode I:
W(e) = H * f 2a, 1 (15)
[ R J 4 E
where amax is the maximum value of the principal stress along the notch edge and H depends again on the normalized radius R/R0, Poisson's ratio v and the loading conditions. For different configurations of mode mixity, the function H, analytically obtained under mode I loading, was shown to be very close to H . This idea of equivalent local mode I was discussed in previous works [57, 58]. Equation (15) will be used here to summarise all experi-
Fig. 4. Strain energy density curves for a U-shaped notch under mode I loading
mental data from U-notches under mixed mode conditions. When the notch opening angle is different from zero, the strain energy density should be directly evaluated from finite element models being ^ X2 in that case.
In order to better justify the choice of the control area Q, consider Fig. 4 where a number of curves with different strain energy density values are plotted for a U-shaped notch (2a = 0). The centre of the curved lines, as obtained from a finite element analysis, is approximately located at a distance r0 = R/2 with respect to the notch tip that is in correspondence of the origin of the local coordinate system. An increase of the radius R2, due to an increase of R0, results in a reduction of the strain energy density level. Consider now a blunt V-notch with 2a = 135° (Fig. 5). The centre of the strain energy density curves is seen to be much closer to the notch tip than in the previous case. A distance r0 = R/ 5, in agreement with the theoretical model seems to be a good approximation. An increase of R2 (due to an increase of R0, r0 being constant) allows us to reach the curves with minor strain energy density. The greater R0, the smaller the sensitivity to the notch is. When R = 0, the radius R2 coincides with R0 and becomes independent of the notch opening angle.
In the presence of combined tension and shear loads, the point of the maximum principal stress does not coincide with the notch tip as stated above. The control volume moves as shown in Fig. 6: it rotates with respect to the centre of the notch root, according to an angle that depends on K^n to K^j ratio, where KR^ and KR^ are the mode II and mode I notch stress intensity factors for blunt U- and V-notches. However, in general, a major difficulty in applying the present approach to mixed-mode problems is the definition of the control volume, due to the differences between the critical value Wc under tension and compression tests exhibited by many materials. A possible strategy might be that of modifying Eq. (14) by adding an energetic term proportional to the mean stress, in order to take
Fig. 5. Strain energy density curves for a V-shaped notch with 2a = 135° under mode I loading
into account the sign of the stresses. The new expression should be compared with a critical value equal to Wc* = /(2E), where P gives the compression strength to tensile strength ratio.
2.5. Some advantages of the strain energy density
As opposed to the direct evaluation of the notch stress intensity factors, which needs very refined meshes, the mean value of the elastic strain energy density on the control volume can be determined with high accuracy by using coarse meshes [68]. Very refined meshes are necessary to directly determined the notch stress intensity factors from the local stress distributions. Refined meshes are not necessary when the aim of the finite element analysis is to determine the mean value of the local strain energy density on a control volume surrounding the points of stress singularity. The strain energy density in fact can be derived directly from nodal displacements, so that also coarse meshes are able to give sufficiently accurate values for it. Some recent contributions document the weak variability of the strain energy density as determined from very refined meshes and coarse
Fig. 6. Strain energy density curves for a U-shaped notch under shear loading
Fig. 7. Geometrical parameters for transverse nonload carrying fillet welded joints and modulus used at the weld toe for the strain energy density evaluation
meshes, considering some typical welded joint geometries and provide a theoretical justification to the weak dependence exhibited by the mean value of the local strain energy density when evaluated over a control volume centred at the weld root or the weld toe. On the contrary singular stress distributions are strongly mesh dependent. The notch stress intensity factors can be estimated from the local strain energy density value of pointed V-notches in plates subjected to mode I, mode II or a mixed mode loading. Taking advantage of some closed-form relationships linking the local stress distributions ahead of the notch to the maximum elastic stresses at the notch tip the coarse mesh strain energy density based procedure is used to estimate the relevant theoretical stress concentration factor Kt for blunt notches considering, in particular, a circular hole and a U-shaped notch, the former in mode I loading, the latter also in mixed I + II mode [68]. Figure 7 shows for a particular geometry analyzed in Ref. [68] the typical coarse mesh used to evaluate the strain energy density at the weld toe. Inside the control volume eight elements were used and the entire model of the welded joint contained forty elements being negligible the difference in terms of strain energy density with the results obtained from finer meshes used for the notch stress intensity factors evaluation.
Other important advantages can be achieved by using the strain energy density approach. The most important are as follows: (i) it permits consideration of the scale effect which is fully included in the notch stress intensity factor approach; (ii) it permits consideration of the contribution of different modes; (iii) it permits consideration of the cycle nominal load ratio; (iv) it overcomes the complex problem tied to the different notch stress intensity factor units of measure in the case of different notch opening angles (i.e. crack initiation at the toe (2a = 135°) or root (2a = 0o) in a welded joint); (v) it overcomes the complex problem of multiple crack initiation and their interaction on different planes; (vi) it directly takes into account the T-stress and this aspect becomes fundamental when thin structures are analysed [69]; (vii) it directly includes three-dimensional effects and out-of-plane singularities not assessed by Williams' theory [70-74].
3. Synthesis based on strain energy density in a control volume
The mean value of the strain energy density in a circular sector of radius R0 located at the fatigue crack initiation sites has been used to summarize fatigue strength data from steel welded joints of complex geometry (Fig. 4).
Local strain energy density AW averaged in a finite size volume surrounding weld toes and weld roots is a scalar quantity which can be given as a function of mode I-II notch stress intensity factors in plane problems [8] and mode I-II—III notch stress intensity factors in three dimensional problems [9]. The evaluation of the local strain energy density needs precise information about the control volume size. From a theoretical point of view the material properties in the vicinity of the weld toes and the weld roots depend on a number of parameters as residual stresses and distortions, heterogeneous metallurgical microstructures, weld thermal cycles, heat source characteristics, load histories and so on. To device a model capable of predicting R0 and fatigue life of welded components on the basis of all these parameters is really a task too complex. Thus, the spirit of the approach is to give a simplified method able to summarise the fatigue life of components only on the basis of geometrical information, treating all the other effects only in statistical terms, with reference to a well-defined group of welded materials and, for the time being, to arc welding processes.
In a plane problem all stress and strain components in the highly stressed region are correlated to mode I and mode II notch stress intensity factors. Under a plane strain hypothesis, the strain energy included in a semicircular sector shown in Fig. 2 is [7, 13]
aw=\ei
AK
N
Ro
l-x,
+eL
AK
N
Ro
1-x„
+
AK
N
Ro
l-X,
k (16)
where R0 is the radius of the semicircular sector and ex, e2 and e3 are functions that depend on the opening angle 2a and Poisson's ratio v (see Table 5).
The material parameter R0 can be estimated by using the fatigue strength Aga of the butt ground welded joints
Table 5
Values of the parameters in Eq. (16), valid for the Beltrami hypothesys and Poisson's ratio v = 0.3
2 a, rad
Y, rad
x,
Plane strain
e2
Axisymmetric
e3
0.5000
0.5000
0.5000
0.13449
0.34139
0.41380
n/12
23n/24
0.5002
0.5453
0.5217
0.13996
0.30588
0.39659
n 6
1 in/12
0.5014
0.5982
0.5455
0.14485
0.27297
0.37929
n 3
5n/ 6
0.5122
0.7309
0.6000
0.15038
0.21530
0.34484
n 2
3n/ 4
0.5445
0.9085
0.6667
0.14623
0.16793
0.31034
2n/ 3
2n/ 3
0.6157
1.1489
0.7500
0.12964
0.12922
0.27587
3n/ 4
5n/ 8
0.6736
1.3021
0.8000
0.11721
0.11250
0.25863
(in order to quantify the influence of the welding process, in the absence of any stress concentration effect) and the notch stress intensity factor based fatigue strength of welded joints having a V-notch angle at the weld toe constant and large enough to ensure the non singularity of mode II stress distributions.
A convenient expression is [7]:
1
> (17)
R =
V^AKi
Aa,
where both A1 and e1 depend on the V-notch angle. Equation (17) makes it possible to estimate the R0 value as soon as AkN and AaA are known. At NA = 5 -106 cycles and in the presence of a nominal load ratio R equal to zero a mean value Ak/A equal to 211 MPa • mm0326 was found reanalyzing experimental results taking from the literature [10]. For butt ground welds made of ferritic steels a mean value AaA = 155 MPa (at NA = 5 • 106 cycles, with R = 0) was found [75]. Then, by introducing the above mentioned value into Eq. (17), one obtains for steel welded joints with failures from the weld toe R0 = 0.28 mm.
It is interesting to learn that, for welded joints made of structural steels, different expressions for AKth taken from the literature were reported in Ref. [76], from which AKth = = 180 MPa • mm05 (5.7 MPa • m05). In the case 2a = 0 and fatigue crack initiation at the weld root Eq. (17) gives R0 = = 0.36 mm, by neglecting the mode II contribution and using e1 = 0.133, Eq. (7), Ak1A = 180 MPa- mm0-5, and, once again, Aa A = 155 MPa. This means that the choice to use a critical radius equal to 0.28 mm both for toe and root failures is a sound engineering approximation.
By modeling the weld toe regions as sharp V-notches and using the local strain energy, more than 300 fatigue strength data from welded joints with weld toe failure were analyzed and the first theoretical scatter band in terms of strain energy density was obtained [8]. The geometry exhibited a strong variability of the main plate thickness (from 6 to 100 mm), the transverse plate (from 3 to 200 mm) and
the bead flank (from 110° to 150°). The synthesis of all those data is shown in Fig. 8, where the number of cycles to failure is given as a function of A W1 (the mode II stress distribution being nonsingular for all those geometries). The figure includes data obtained both under tension and bending loads, as well as from "as-welded" and "stress-relieved" joints. The scatter index TW, related to the two curves with probabilities of survival Ps = 2.3% and 97.7%, is 3.3, to be compared with the variation of the strain energy density range, from about 4.0 to about 0.1 MJ/m3. TW = 3.3 becomes equal to 1.50 when reconverted to an equivalent local stress range with probabilities of survival Ps = 10% and 90% (Ta =V33/1.21 = 1.5). The scatterband proposed was latter applied in [10] to a larger bulk of experimental data, which included also fatigue failures from the weld root.
Fig. 8. Strain energy-based scatter band summarizing more than 300 fatigue strength data from steel welded joints with weld toe failure. The main plate thickness ranged from 6 to 100 mm. R— nominal load ratio, 2a—V-notch angle at the weld toe [8]
X
X
n
Fig. 9. Fatigue strength of welded joints as a function of the averaged local strain energy density; R is the nominal load ratio
A final synthesis based on 900 experimental data is shown in Fig. 9 where some recent results from butt welded joints, three-dimensional models and hollow section joints have been included. A good agreement is found, giving a sound, robust basis to the approach when the welded plate thickness is equal to or greater than 6 mm.
A new synthesis of data from single spot-welded joints (see Fig. 10) characterized by thin plates (0.65 mm < t < < 1.75 mm) is presented here for the first time. The control radius of the three-dimensional volume around the slit tip and its depth are equal to 0.28 mm. The value of AW at 2-106 cycles is higher than that reported in Fig. 9 and referred to main plate thicknesses greater than 6 mm (AW = = 0.18 N - mm/mm3 for single spot-welded joints against AW = 0.105 N - mm/mm3 in Fig. 9). Future developments will be presented and widely discussed by the present authors in forthcoming publications.
Dealing with static loading, the local strain energy density values are normalized to the critical strain energy density values (as determined from unnotched specimens) and plotted as a function of the R/R0 ratio. The data related to
Fig. 10. Synthesis of data from spot-welded joints under tension and shear loading. Strain energy density values have been determined by means of three-dimensional models
the experimental program of PMMA tested at -60°C [5558] are summarized together with other data taken from a database due related to PMMA tested at room temperature [44, 53, 54]. Dealing with cracked and V-sharp specimens under mixed mode I + II loading recent data taken from the literature are also considered in the present synthesis [77, 78]. In particular data summarized in Ref. [77] are from sharp V-specimens made of an acrylic resin and tested under mixed mode loading whereas in Ref. [78] data are from diagonally loaded square cracked plate specimens made of PMMA. Dealing with the acrylic resin, V-notched plates presented different opening angles (30°, 60° and 90°), and were also characterised by a notch axis differently inclined with respect to the minor side of the specimens (0°, 5°, 15° and 30°) [77]. The notch root radius was nominally equal to zero, whereas the R0 was about equal to 0.07 mm (ultimate tensile strength g t = 72.57 MPa, fracture toughness = 37.1 MPa - mm0-5, the Young's modulus E = 3230 MPa and Poisson's ratio v = 0.3). In the presence of tensile loads, the geometries assured a strong variability of the mode I and mode II notch stress intensity factors. However all the specimens were subjected to prevalent mode I. A new test configuration for mixed mode fracture was proposed in Ref. [78] and a complete set of experimental data from diagonally loaded square cracked plate specimens made of PMMA ranging from pure mode I to pure mode II was provided.
New interesting data from pointed V-notches samples made of PMMA and tested at low temperature under prevalent mode II loading are available in Refs. [60, 79]. The problem of brittle failure from blunted notches loaded under mixed mode is more complex than in mode I loading and experimental data, particularly for notches with a non-negligible radius, is scarce. The experimental programme has been performed with V-notched specimens, varying the notch inclination, the notch angle and the boundary conditions to obtain different mode (I + II) mixity. The new re-
(W/Wc)
1/2
1.6
1.2
0.80.4-
0.0
PMMA, mixed mode V-notches and cracks Scatterl (W\!WXqX = 0.95) with^c \(0.64) (0.85) (0.27) (0.52)/ band for V-notches, , = 0.045 mm (0.80) L15
^^ / j Y /
R0 = 0.045 mm °'85 o x > 0.95 O 0.70 < X < 0.90 x^0 = o.l mm, cracks [78] A 0.27 <%< 0.64 Room temperature
0
1000 2000 3000 4000 5000
Fig. 11. Synthesis of all experimental data from pointed V-notched plates based on two different values of the control volume radius; comparison with recent results from cracked plates tested at room temperature [60]
(W/Wc)m
1.4-
—
1.0-
0.6-
T
R = 0*- 0.1 1.0 10 100 R/Rq
o Pointed V-notches, mixed mode I + II [60] + Pointed V-notches, mixed mode I + II,
R0 = 0.07 mm [77] x Cracks, R0 = 0.1 mm [78] ♦ PMMA, -60°, U-notch, 0 < R < 4 mm,
R0 = 0.035 mm, mixed mode I + II [57, 58] □ PMMA, -60°, U- and V-notch, 0.05 < R < 4 mm, 0° < 2a < 150°, Rq = 0.035 mm, mode I [55, 56] o PMMA, room temperature, V-notch, R = 0,
15° < 2a < 150°, R0 = 0.011 mm, mode I [53, 54] A PMMA, room temperature, U-notch, 0.1 <R < 4 mm, R0 = 0.011 mm, model [44]
Fig. 12. Static failure data in terms of normalized strain energy density
search is focused on sharp notches and the notch root radius R is always less than 0.1 mm, ranging between 20 and 72 ^m. To achieve different mixed mode loading and to analyze the field of prevalent mode II, two types of V-notched specimens were studied; beams with vertical notches, with notch angles a = 30o, 60o and 90o, and beams with tilted notches at 45o, with notch angles 30o, 60o and 90o. Samples are loaded by using a three point bending configuration. The position of the loading point is modified such as the span length to vary the mode I contribution respect to mode II contribution. Dealing with the new results summarized in Ref. [60], the ratio x = W1/Wtot which gives the mixity degree in terms of local strain energy density has been analyzed (see Fig. 11). The mean value of the ratio s¡WI Wc to which the critical loads are linked, is found to be about equal to 1.0 but values greater than 1.1 characterize specimens with x ~ 1.00 (dominant mode I), x = = 0.27 (dominant mode II) and 0.52 < x < 0.64 (mode II comparable with mode I) independently of the mode mixity being this effect tied to the intrinsic scatter of the experimental data.
The final synthesis has been carried out by normalizing the local strain energy density to the critical strain energy density values (as determined from unnotched, plain specimens) and plotting this nondimensional parameter as a function of the R/R0 ratio. A scatterband is obtained whose mean value does not depend on R/R0, whereas the ratio between the upper and the lower limits are found to be about equal to 1.3/0.8 = 1.6 (Fig. 12). The strong variability of the non-dimensional radius RR0 (notch root radius to control volume radius ratio, ranging here from about zero to about 500) makes stringent the check of the ap-
Fig. 13. Synthesis of data taken from the literature. Different materials are summarised, among the others AISI O1 and durallu-minium
proach based on the local strain energy density. The complete scatterband presented here for the first time (Fig. 13) has been obtained by updating the database containing failure data from 20 different ceramics, 4 PVC foams and some metallic materials [59] with the new results reported in Refs. [60, 77, 78].
4. Conclusions
For many years the strain energy density has been used to formulate failure criteria for materials exhibiting both ductile and brittle behavior. The strain energy density is the most fundamental quantity in mechanics being all physical quantities expressible in terms of it. From pico- to macroscopic scale the energy absorption and dissipation can explain the most complex phenomena tied to fracture initiation and propagation.
Keeping in mind that the design rules valid for large bodies (i.e. high volume to surface ratio) can not be directly translated and applied to small bodies where local inhomogeneities play a fundamental role for the material damage initiation and propagation and being also aware of the recent contributions and efforts to develop a multiscaling and segmentation scheme able to capture the complex phenomena that happen at every level from pico to macro, the main purpose of the paper is to present a review of the approach based on the mean value of the local strain energy density.
Dealing with static loading the approach is applied here to different materials and geometries both, under mode I and mixed mode (I + II) loading. About one thousand experimental data, taken from the recent literature, are involved in the synthesis. They were from U- and V-notched specimens made of very different materials. A scatter band is proposed by using as a synthesis parameter the value of the local energy averaged over control volume (of radius R0), normalized by the critical energy of the material. Such a
normalized energy is plotted as a function of the notch radius to critical radius ratio R/R0.
The strain energy density in a circular sector of radius R0 located at the crack initiation sites has successfully been used to summarize also about nine hundred data from fatigue failures of welded joints.
Under the hypothesis that all material inhomogeneities can be averaged, that ceases to be valid at pico- and micro-levels but at the same time is the basis of the volume-based theories applied to structural components, the strain energy density approach is shown to be a powerful tool both for static and fatigue strength assessment of notched and welded structures.
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Поступила в редакцию 29.08.2016 г.
Сведения об авторах
Filippo Berto, PhD, Prof., NTNU, Norway, filippo.berto@ntnu.no
Majid R. Ayatollahi, PhD, Prof., Director, Iran University of Science and Technology, m.ayat@iust.ac.ir
