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Комбинированная концепция эквивалентного материала и критерия усредненной плотности энергии деформации для предсказания зарождения трещин в пластинах из алюминиевого сплава с V-образным надрезом при нагружении смешанного типа
A.R. Torabi1, A. Campagnolo2, F. Berto3'4
1 Тегеранский университет, Тегеран, 13741-4395, Иран 2 Падуанский университет, Падуя, 35131, Италия 3 Падуанский университет, Виченца, 36100, Италия 4 Норвежский технологический университет, Тронхейм, 7491, Норвегия
Теоретически и экспериментально исследовано зарождение трещин на границах закругленного V-образного надреза в пластинах из алюминиевого сплава 6061-T6 при нагружении смешанного типа I + II. В экспериментах невооруженным глазом наблюдалось развитие больших пластических деформаций в области вершины надреза в момент зарождения трещины, что свидетельствует о разрушении пластин алюминия в условиях полномасштабной текучести. С использованием комбинированной концепции эквивалентного материала (EMC) и критерия хрупкого разрушения (критерия усредненной плотности энергии деформации (ASED)) получена теоретическая оценка экспериментальных данных по максимальной нагрузке для каждой пластины (несущей способности) без проведения анализа упругопластического разрушения. Показано, что комбинированный критерий EMC-ASED позволяет предсказывать экспериментальные результаты при разных значениях угла и радиуса V-образного надреза.
Ключевые слова: концепция эквивалентного материала, усредненная плотность энергии деформации, несущая способность, V-образный надрез, пластина из алюминия, нагружение смешанного типа
A successful combination of the equivalent material concept and the averaged strain energy density criterion for predicting crack initiation from blunt V-notches in ductile aluminum plates under mixed mode loading
A.R. Torabi1, A. Campagnolo2, and F. Berto3,4
1 Fracture Research Laboratory, Faculty of New Sciences & Technologies, University of Tehran, Tehran, 14395-1561, Iran 2 Department of Industrial Engineering, University of Padova, Padova, 35131, Italy 3 Department of Management and Engineering, University of Padova, Vicenza, 36100, Italy
4 Department of Engineering Design and Materials, Norwegian University of Science and Technology, Trondheim, 7491, Norway
Crack initiation from blunt V-notch borders in ductile Al 6061-T6 plates is investigated experimentally and theoretically under mixed mode I/II loading. Experimental observations with naked eye during loading indicated large plastic deformations around the notch tip at the onset of crack initiation, demonstrating large-scale yielding failure regime for the aluminum plates. To theoretically predict the experimentally obtained value of the maximum load that each plate could sustain, i.e. the load-carrying capacity, without performing elastic-plastic failure analyses, the equivalent material concept (EMC) is combined with a well-known brittle fracture criterion, namely the averaged strain energy density (ASED) criterion. It is shown that the combined EMC-ASED criterion could successfully predict the experimental results for various V-notch angles and radii.
Keywords: equivalent material concept, averaged strain energy density, load-carrying capacity, V-notch, aluminum plate, mixed mode loading
1. Introduction
Due to high ductility and relatively low density, alumi-
num alloys are extensively used in many engineering struc-
tures, particularly aerospace structures. There are many
structures in which notches of various shapes, e.g. blunt V-notches, are employed for specific design purposes. For instance, one can refer to a typical rectangular access panel in aerostructures containing four round corners. Each cor-
© Torabi A.R., Campagnolo A., Berto F., 2016
ner is, in fact, a blunt V-notch with 90° opening angle. A notch concentrates stresses around its tip and increases the risk of initiating crack(s) from the tip due to stress concentration. Such crack(s) may grow during service and lead to final fracture of the notched member. In the example of the access panel mentioned above, several cases of crack initiation from the panel corners have been observed by the first author in civil aircrafts made mainly of aluminum alloys. Although design of notched aluminum components are usually performed such that no yielding takes place in the component (even at the notch neighborhood), some excess loads may be applied to the aluminum component from unexpected external sources, leading to localized plastic deformations at the notch vicinity. If the magnitude of such loads is larger than a specific value, crack initiation may take place from the notch border, resulting in dramatic decrease of the load that the notched component could sustain. Therefore, the load-carrying capacity (LCC) of notched aluminum components should essentially be examined theoretically and experimentally in order to prevent catastrophic failures.
The load-carrying capacity of notched aluminum components has been investigated mainly during the past four years. The first work in this subject belongs to Madrazo etal. [1]. They studied theoretically and experimentally the failure of the notched compact-tension specimens made of Al 7075-T651 under pure mode I loading conditions. Using the well-known theory of critical distances applied to the linear elastic stress distributions around the notch, they could predict the experimental load-carrying capacities successfully [1]. By means of a series of mode I fracture tests on the single-edge-notched-bend specimens, Vratnica et al. [2] measured the load-carrying capacities of commercial aluminum alloy weakened by U-shaped notches of different radii. A fracture criterion was also utilized to predict the experimentally obtained load-carrying capacities [2]. Despite considerable plastic deformations around the notches at the onset of crack initiation from the notch tip, the linear elastic stress field around the notch was utilized in Refs. [1, 2] for the theoretical failure assessments. While the results reported in [1, 2] are undoubtedly valuable, disregarding the plastic zone effects on failure strength of the aluminum specimens can be recognized as a weak point for the two researches.
With the aim to consider the effects of plastic deformations on static failure of notched aluminum components and
to avoid using elastic-plastic fracture criteria in theoretical failure assessments, Torabi and co-researchers have recently joint the equivalent material concept (EMC) [3-6] to some well-known fracture criteria in the context of the linear elastic notch fracture mechanics (LENFM) [7-10]. Torabi and Alaei [8] have investigated the crack initiation from the round border of blunt V-notches in Al 7075-T6 thin sheets under pure mode I loading. The experimental observations and the elastic-plastic finite element analyses have indicated that the V-notched Al 7075-T6 thin sheets fail by moderate-scale yielding regime [8]. To predict the experimentally recorded LCC values for the tested specimens, they have combined the equivalent material concept with two brittle fracture criteria, namely the point-stress and mean-stress criteria, and successful results have been achieved, especially for the relatively small and medium notch radii [8]. Similar work has been successfully performed in [9] on ductile failure of U-notched Al 7075-T6 and Al 6061-T6 plates subjected to pure tension. It has been reported in [9] that the Al 7075-T6 plates fail by moderate-scale yielding regime, while the Al 6061-T6 plates by large-scale yielding regime. Two works have also been performed in the same field under mixed mode I/II loading conditions. In [7, 10], the crack initiation has been analyzed experimentally and theoretically in U-notched Al 6061-T6 plates and in blunt V-notched Al 7075-T6 ones, respectively under mixed mode loading. One of the two main results reported in [7, 10] is that both plates fail by large-scale yielding regime, and the crack growth and final fracture in Al 6061-T6 plates are stable, while those in Al 7075-T6 are unstable. The other result is that the well-known stress-based brittle fracture criteria, namely the maximum tangential stress and mean-stress criteria, could well be combined with equivalent material concept to predict the LCC of U-notched Al 6061-T6 and V-notched Al 7075-T6 plates [7, 10].
According to equivalent material concept, a real ductile material is equated with a virtual brittle material, called the equivalent material, exhibiting linear elastic stress-strain behavior till final fracture [7-10]. Hence, equivalent material concept can be combined with different fracture criteria in the context of the linear elastic notch fracture mechanics for predicting crack initiation from notch borders in ductile materials without requiring time-consuming and complex elastic-plastic analyses. One of the most famous and powerful fracture criteria in the LENFM domain is the averaged strain energy density (ASED) criterion [11-13].
Table 1
Chemical composition of Al 6061-T6 resulted from a metallographic test Element Si Fe Cu Mn Mg Zn Ni Cr Pb Sn Ti
Weight, % 0.610 0.480 0.170 0.050 0.860 0.020 0.003 0.200 0.001 0.001 0.080 Element B Cd Bi Ca P Sb V Zr Co Li Al
Weight, % 0.0040 0.0010 0.0000 0.0010 0.0020 0.0080 0.0250 0.0002 0.0030 0.0010 97.6000
Table 2
Mechanical properties of Al 6061-T6 resulted from the standard tests
Material property Value
Elastic modulus E, GPa 67
Poisson's ratio 0.33
Tensile yield strength, MPa 276
Ultimate tensile strength, MPa 292
Elongation at break, % 11
Engineering strain at maximum load 0.034
True fracture stress, MPa 299
Fracture toughness Kc, MPa • m1/2 38
Strain-hardening coefficient, MPa 314
Strain-hardening exponent 0.021
The averaged strain energy density criterion has been frequently utilized for successfully predicting brittle fracture in several test specimens weakened by different notch features, e.g. V- and U-notches, made of various brittle materials, e.g. PMMA, graphite, etc., and under different loading conditions, e.g. mode I [14-17], mode II [18-20], mode III [21, 22], mixed modes I/II [23-32] and I/III [33, 34], etc.
In the present research, first, the crack initiation and propagation behaviors of Al 6061-T6 plates weakened by blunt V-notches are experimentally investigated under mixed mode I/II loading conditions, and the LCC of the plates are measured for various V-notch geometries. The experimental observations with naked eye during loading and after the final rupture, and the numerical elastic-plastic stress analyses indicate that the entire notched Al 6061-T6 plates fail by large-scale yielding regime. Then, the equivalent material concept is combined with the well-known averaged strain energy density criterion to theoretically predict the experimentally obtained LCCs. It is shown that the EMC-ASED criterion could successfully predict the onset of crack initiation from the blunt V-notch borders for different notch angles and radii. By means of such a combined criterion, it could be possible to predict ductile crack initiation in V-notched plates without using time-consuming and complex elastic-plastic failure criteria.
2. Experiments
2.1. Material
The ductile aluminum alloy Al 6061-T6 is considered for conducting the failure experiments. This aluminum alloy is widely utilized in engineering structures, e.g. aero-structures. The chemical composition and some of the mechanical properties of Al 6061-T6 are presented in Tables 1 and 2, respectively. The data of Table 1 is obtained
Fig. 1. The engineering and true stress-strain curves of Al 6061-T6
from a simple metallographic test and that of Table 2 from the standard mechanical tests, such as the tensile, Poisson's ratio, and fracture toughness tests according to the ASTM E8 [35], ASTM E132-04 [36] and ASTM B646-12 [37], respectively.
The engineering and true stress-strain curves of the tested Al 6061-T6 are depicted in Fig. 1. As seen in the engineering curve, the material reaches its ultimate point at a strain value of about 0.03 and the specimen cross-section begins decreasing till final rupture. The relatively large strain interval between the peak and final rupture points (about 0.08) suggests that a stable crack propagation behavior is expected for such a ductile material.
2.2. Test specimen
To perform the failure experiments on round-tip V-notches, a rectangular plate containing a central rhombic slot having four blunt V-shaped corners is considered. The test specimen is schematically shown in Fig. 2 with the geo-
W
Fig. 2. The blunt V-notched test specimen for mixed mode failure experiments
Fig. 3. Some of the V-notched Al 6061-T6 plates before (a), during (b) and after fracture (c)
metric parameters included. As seen in Fig. 2, the loading mode, e.g. mode I and mixed mode I/II etc., on the blunt V-notch is controlled by the rotation angle P, which is the angle between the notch bisector line and the horizontal axis. For P = 0, the bisector line of the two main V-corners lies on the horizontal axis and the V-notches experience pure mode I loading conditions. In order to produce mixed mode I/II loading conditions at the notch neighborhood, the slot is rotated counterclockwise by the angle p. The contribution of mode II loading enhances as the value of P increases. Therefore, for various P values, different mode mixity ratios are achieved.
As depicted in Fig. 2, the notch angle, the notch radius, twice the notch length (i.e. the slot length), the specimen length, the specimen width, the notch rotation angle, and the remotely applied tensile load are denoted by the parameters 2a, p, 2a, L, W, P, and P, respectively. The values of these parameters are considered in the tests to be as follows: 2a = 30°, 60°, and 90°; p = 1, 2, and 4 mm; 2a = = 25 mm; L = 160 mm; W = 50 mm. The values of P for the notch angles of 30°, 60° and 90° are also considered to be equal to (0°, 30° and 60°), (0° and 30°) and (0° and 30°), respectively. Thickness of the whole specimens is constant and equal to 4 mm. Each test is repeated three times, and sixty three failure tests are conducted in this experimentation. The specimens are fabricated by means of a two-dimensional CNC water jet cutting machine and the monotonic tests are carried out under the strain-controlled conditions with a stain rate of about 2.7 • 10
4 c-1
2.3. Experimental results
During the failure tests, it is observed that a plastic region nucleates from the notch border and grows as the test continues. At the onset of crack initiation from the notch border for the entire specimens, a large plastic zone is observed with naked eye around the V-notch, demonstrating the large-scale yielding failure regime. Such large plastic deformations are clearly seen in Fig. 3, c. The initiated crack is also observed to propagate relatively slowly till the final rupture. Figure 4 shows a sample load-displacement curve recorded from testing a V-notched Al 6061-T6 specimen loaded under mixed mode conditions.
This figure confirms the experimental observations in a graphical form. In Fig. 4, the large nonlinear portion before the peak point proves the existence of large amount of plastic deformations around the V-notch at the crack initiation instance. Meanwhile, no sudden load drop from the peak to zero demonstrates that growth of the crack emanating from the V-notch border in Al 6061-T6 specimens takes place in a stable manner.
Table 3 presents the experimentally obtained load-carrying capacity of the V-notched Al 6061-T6 plates for different notch geometries and loading conditions. Each failure load in the three repeated tests is denoted by Pi (i = = 1, 2, 3) and Pav denotes the average of the three failure loads.
In order to numerically confirm the experimental observations regarding the large-scale yielding failure regime for the V-notched Al 6061-T6 plates, a sample elastic-plastic finite element stress-strain analysis is performed for a plate with 2a = 60 °, p = 2 mm and P = 30 ° to determine the size of the plastic zone around the notch border at crack initiation instance. The finite element model is created in the commercial software ABAQUS and the mean experimental failure load presented in Table 3 (i.e. Pav = 34 921 N) is applied to the finite element model. To introduce the material to the finite element software, two approaches generally exist. The first one is to present the stress-strain be-
Fig. 4. A sample load-displacement curve recorded from testing a V-notched Al 6061-T6 specimen loaded under the mixed mode conditions
Table 3
The experimentally obtained load-carrying capacity of the V-notched Al 6061-T6 plates for different notch geometries and loading conditions
2a p, mm ß p, N P2, N P3, N P , N av '
30° 1 0° 29 997 30 350 30 567 30 304
2 0° 31 913 31 884 31 674 31 824
4 0° 31 652 31 615 31 862 31 710
60° 1 0° 31 876 31 685 31 325 31 629
2 0° 31 711 31 538 31 668 31 639
4 0° 31 946 31 869 32 119 31 978
90° 1 0° 30 114 30 626 29 334 30 025
2 0° 31 485 31 468 31 501 31 485
4 0° 31 450 31 398 31 516 31 455
30° 1 30° 33 433 32 641 34 054 33 376
2 30° 35 158 35 239 35 150 35 182
4 30° 34 275 34 573 34 326 34 391
60° 1 30° 35 500 35 515 35 154 35 390
2 30° 34 845 34 974 34 944 34 921
4 30° 34 764 34 738 34 227 34 576
90° 1 30° 36 383 36 618 35 956 36 319
2 30° 35 434 35 434 34 838 35 235
4 30° 34 694 34 886 34 444 34 675
30° 1 60° 43 623 44 476 44 248 44 116
2 60° 44 358 44 285 44 182 44 275
4 60° 40 794 41 798 41 585 41 392
havior of material in the plastic zone by some well-known expressions, like the power-law and the Ramberg-Osgood expressions, and the second one is to give the true stressstrain curve of material (see Fig. 1) to the software point-by-point. Since the second approach is based on the experimental data exactly resulted from the standard tensile tests, its accuracy is expected to be better than the accuracy of the first approach. Therefore, the second approach is utilized for introducing the material. Figure 5 represents the Von Mises stress distribution around the notch border at the onset of crack initiation. As seen in Fig. 5, the plastic zone is recognized by a butterfly wing area. It is evident that a large portion of the ligament experiences plastic deformations at crack initiation instance, demonstrating well the large-scale yielding failure regime for the V-notched Al 6061-T6 plate.
As seen in Figs. 3-5, the V-notched Al 6061-T6 plates fail by large-scale yielding regime. To predict the experimental LCCs presented in Table 3, some failure criteria in the context of the elastic-plastic notch fracture mechanics, e.g. the notch tip opening displacement and the critical
J-integral, could commonly be used. To avoid using such criteria, which are rather time-consuming and complicated and usually need numerical elastic-plastic (e.g. the finite element) stress-strain analyses, the equivalent material concept is briefly described in forthcoming sections and utilized in conjunction with the well-known brittle fracture criterion, namely the averaged strain energy density criterion. It is revealed that by means of such a combined criterion, the LCC of notched ductile plates can be predicted by only linear elastic analyses.
3. The equivalent material concept
In order to avoid using elastic-plastic criteria in failure prediction of notched ductile components, a new concept, called the equivalent material concept, has been recently proposed by Torabi [3]. Based on equivalent material concept, a ductile material having elastic-plastic behavior is equated with a virtual brittle material exhibiting linear elastic behavior till final fracture. This equality has been performed in [3] to make a justification for using LENFM-based fracture criteria to predict the crack initiation from the notch borders encountering considerable plastic deformations at failure. Despite details of the equivalent material concept has been given in [3-10], it is briefly described here for making the readers more convenient.
The equivalent material concept has two basic assumptions as (i) the ductile material has a valid Abased fracture toughness (i.e. KIc or Kc) and the ductile material and equivalent brittle material have the same values of fracture toughness and elastic modulus, but various values of the tensile strength, and (ii) both the real ductile and virtual brittle materials absorb the same amount of tensile strain energy density for the crack initiation to occur. If tensile strength of the equivalent material is determined, the LENFM-based fracture criteria can simply be used for predicting crack initiation in notched ductile components.
The tensile strength of the equivalent material has been reported in the literature by a closed-form expression. It is [7-10]
Mises
(Avg: 75%)
302.10
241.90
— 181.70
— 121.40
61.19 0.96
Plastic zone
Fig. 5. A von Mises stress distribution around a round-tip V-notch border in Al 6061-T6 plate (the stress values are in MPa). The plastic zone is recognized by a butterfly wing area
Fig. 6. Control volume (area) for sharp V-notch (a), sharp crack (b) and blunt V-notch (c) under mode I loading. Distance r0 =p(n-2a)/(2n-2a)
a* =
2 2 EK r n+i
ay + ~T I8u,true '
n +1
-(0.002)n+1].
(1)
Equation (1) has been obtained by assuming that the ductile material obeys the power-law stress-strain relationship in the plastic region. In Eq. (1), the parameters a*, ay, E, K, n, and 8 utrue denote the tensile strength of the equivalent material, yield strength of the ductile material, elastic modulus, strain-hardening coefficient, strain-hardening exponent, and the true plastic strain at the ultimate point, respectively. The value of 8u,true can easily be computed by means of Eq. (2) in which 8u is the engineering plastic strain at the ultimate point
8 u,true = ln(1 + 8 u). (2)
To know more about the equivalent material concept, the readers are encouraged to study [3-10]. Now, a* and KIc (or Kc) values can be used in various brittle fracture criteria for predicting the LCC of notched ductile members encountering significant plastic deformations at the notch vicinity. Using Eq. (1) and the data summarized in Table 2, the value of a* is computed for Al 6061-T6 to be equal to about 1066 MPa.
In forthcoming sections, the experimentally recorded LCCs of the V-notched Al 6061-T6 specimens summarized in Table 3 are predicted using the equivalent material concept together with the well-known averaged strain energy density fracture criterion.
4. A brief description of the averaged strain energy density criterion
The most important point for designers is certainly the existence of appropriate failure models to predict the load-carrying capacity of components weakened by notches. With the aim to provide such models, a strain energy density based criterion has been proposed by Lazzarin and co-authors [11, 13], by which the experimental fracture loads of notched specimens can be estimated very well.
The strain energy density factor S was defined for sharp cracks by Sih [38] as the product of the strain energy density by a specified critical distance measured from the crack tip. Fracture was thought of as controlled by a critical value
Sc, whereas the crack growth direction was determined by imposing a minimum condition on the factor S.
The method proposed by Sih is a point-wise criterion while the averaged strain energy density approach as presented in [11] suggests that brittle fracture takes place when the strain energy density averaged over a known control volume is equal to a critical value Wc. This value varies from material to material but it is independent of the notch geometry. The control volume is thought of as dependent on the ultimate tensile strength and on the fracture toughness KIc in the case of brittle or quasi-brittle materials subjected to static and monotonic loads. Such a method was formalized and applied first to sharp V-notches under mode I and mixed mode I/II loadings [11] and later extended to blunt U and V-notches [25, 26, 29-31]. Some recent developments and applications are reported in [13, 20].
For sharp cracks, the control volume is a circle of radius Rc centered at the crack tip (Fig. 6, b). Under planestrain conditions, the critical length Rc can be evaluated by the following expression [39]:
R =
(1 + v)(5 - 8v)
4n
K
2
(3)
In Eq. (3), KIc is the fracture toughness, v is the Poisson's ratio and au is the ultimate tensile strength of material.
For a sharp V-notch, the control volume becomes a circular sector of radius Rc centered at the notch tip (Fig. 6, a) while for a blunt V-notch under mode I loading, the volume assumes the crescent shape shown in Fig. 6, c, where Rc is the depth measured along the notch bisector line. The outer radius of the crescent shape is equal to Rc + r0, being r0 the distance between the notch tip and the origin of the local coordinate system (see Fig. 6, c). Such a distance depends on the V-notch opening angle 2 a, according to the expression r0 =p( n-2a)/(2 n-2a).
Dealing with blunt U- and V-notches the concept of equivalent local mode I, although not exact in principle, can be seen as an accurate engineering approximation [25, 26, 29-31]. In particular the averaged strain energy density was generalized from mode I to mixed mode I/II, under the hypothesis of an equivalent local mode I along the normal line to the notch edge, at the point where the principal stress
Fig. 7. Control volume (area) under in-plane mixed mode loading
reaches its maximum value (see Fig. 7). The approach was used to assess rupture loads of U-notched components made of PMMA and tested at -60oC under mixed mode loading [25, 26, 29-31].
According to the coordinate system shown in Fig. 7, the stress component g00 normalized to its maximum value occurring along the notch edge has been found equal to the mode I theoretical solution reported in [40]. This observation leads to the conclusion that under mixed mode loading the line normal to the notch edge and starting from the point of maximum principal stress behaves as a virtual bisector
line under pure mode I, confirming the applicability of the equivalent local mode I concept.
5. Application of equivalent material concept in combination with averaged strain energy density criterion
The averaged strain energy density approach is applied here considering the material properties of the equivalent material. The critical strain energy density Wc EMC is evaluated by using the following equations and considering a* = = 1066 MPa:
a*2
Wc,EMC = — • (4)
The critical strain energy density results to be equal to 8.48 MJ/m3. The control radius Rc is evaluated by using Eq. (3) with au = a* = 1066 MPa. It is found Rc = 0.317 mm. The averaged strain energy density occurring inside the control volume embracing the edges of V-notches (W) is calculated numerically by using the finite element code ANSYS. For each notch geometry, a finite element model is created by accurately defining the control volume, where the strain energy density should be averaged. According to the procedure described in the previous section, the control
Table 4
Critical load predicted by means of averaged strain energy density criterion in combination with equivalent material concept
2a P p, mm P N P2, N P3, N Pas ED 'N A! A 2 A3
30° 0° 1 29 997 30 350 30 567 31 711 0.95 0.96 0.96
0° 2 31 913 31 884 31 674 35 951 0.89 0.89 0.88
0° 4 31 652 31 615 31 862 42 518 0.74 0.74 0.75
60° 0° 1 31 876 31 685 31 325 29 943 1.06 1.06 1.05
0° 2 31 711 31 538 31 668 34 349 0.92 0.92 0.92
0° 4 31 946 31 869 32119 41 092 0.78 0.78 0.78
90° 0° 1 30114 30 626 29 334 28 559 1.05 1.07 1.03
0° 2 31 485 31 468 31 501 32 465 0.97 0.97 0.97
0° 4 31 450 31 398 31 516 39 009 0.81 0.80 0.81
30° 30° 1 33 433 32 641 34 054 35 232 0.95 0.93 0.97
30° 2 35 158 35 239 35 150 39 086 0.90 0.90 0.90
30° 4 34 275 34 573 34 326 45 352 0.76 0.76 0.76
60° 30° 1 35 500 35 515 35 154 33 171 1.07 1.07 1.06
30° 2 34 845 34 974 34 944 39 315 0.89 0.89 0.89
30° 4 34 764 34 738 34 227 45 277 0.77 0.77 0.76
90° 30° 1 36 383 36 618 35 956 36 319 1.00 1.01 0.99
30° 2 35 434 35 434 34 838 42 689 0.83 0.83 0.82
30° 4 34 694 34 886 34 444 48 565 0.71 0.72 0.71
30° 60° 1 43 623 44 476 44 248 55 727 0.78 0.80 0.79
60° 2 44 358 44 285 44 182 57 701 0.77 0.77 0.77
60° 4 40 794 41 798 41 585 58 600 0.70 0.71 0.71
Fig. 8. Synthesis of fracture data in terms of normalized averaged strain energy density
volume is centered at the point of maximum principal stress along the curvilinear edge of the notch. All finite element analyses are performed by using eight-node 2D finite elements (PLANE 183) under plane strain conditions.
6. Results and discussion
Table 4 summarizes the outlines of the experimental, numerical and theoretical findings for V-notched specimens with three different notch radii (p = 1, 2, 4 mm) analyzed by means of the averaged strain energy density approach. In particular, Table 4 reports the experimental loads to failure P for all notch radii p compared with the theoretical values PASED based on the averaged strain energy density evaluation. PASED is the theoretical load obtained by keeping constant the critical strain energy density Wc EMC equal to 8.48 MJ/m3.
The last columns of Table 4 present the deviations between the values of the experimental failure loads and the theoretical ones evaluated by means of averaged strain energy density criterion. A is defined as the ratio between the experimental load and the theoretical one for each case.
It is clearly seen in Table 4 that the great majority of the predictions are well inside the scatter of ±20% with some of the results inside the scatter ±10%. A synthesis in terms of the square root value of the local strain energy averaged over the control volume of radius Rc (W), normalized with respect to the critical energy of the material Wc EMC as a function of the notch angle is shown in Fig. 8. The plotted parameter is proportional to the fracture load. The goal is to study the influence of the notch geometry on the fracture prediction on the basis of the averaged strain energy density. From the graphical point of view, it is obvious that the great majority of values fall inside a scatter ranging from 0.8 to 1.2 with some data inside a scatter ranging from 0.9 to 1.1. The synthesis confirms also the choice of the control volume which seems to be suitable to characterize the material behavior under mixed mode I/II loading. The scatter of the experimental data presented here is in very good agreement with the recent database in terms of averaged strain energy density reported in a recent review of the approach and dealing with brittle and quasi-brittle failure [13].
As is well-known, the standard stress-strain curve of aluminum alloys is usually modeled by the elastic-perfectly plastic behavior for metal forming purposes. This is because most of aluminum alloys show negligible strain hardening in the plastic zone of the curve. While the elastic-perfectly plastic model is useful in metal forming (since only the energy absorbed by the material in the plastic region is the important parameter), it is not a good model in the fracture mechanics context, because it is not able to interpret the crack initiation, crack growth and the final rupture in aluminum members. As an example, let us consider the failure of Al 6061-T6 plates investigated in this research and that of Al 7075-T6 plates reported in [7-10]. Both aluminum alloys have the tensile stress-strain curves very close to the elastic-perfectly plastic curve. However, the notched Al 6061-T6 plates exhibit stable crack propagation and final rupture, while the notched Al 7075-T6 plates show unstable crack growth and abrupt fracture. This is because, in the standard tensile stress-strain curve, the ultimate point of Al 6061-T6 takes place at a rather low strain value, while that of Al 7075-T6 happens at a strain value very close to the strain to failure. As a result, the real engineering stressstrain curve of aluminum alloys must be utilized for predicting the fracture behavior of such metallic alloys, especially in the presence of stress concentrators like notches.
The authors believe that the combined EMC-ASED criterion has an important limitation which should be highlighted herein. It is that the EMC-ASED criterion is only capable of predicting the LCC of ductile V-notched components under mixed mode I/II loading conditions, and it is not able to predict the extension path of the crack(s) emanating from the V-notch border in ductile components or to predict the fracture plane. This is because ductile and brittle materials have basically various fracture behaviors.
7. Conclusions
This manuscript is aimed to analyze the in-plane mixed mode fracture in Al 6061-T6 thin plates weakened by blunt V-notches. Some fracture tests were carried out on rectangular plates weakened by central blunt V-shaped notches. The experimental observations indicated large plastic deformations around the notch tip at the onset of crack initiation, demonstrating large-scale yielding failure regime for the aluminum plates. The loads corresponding to the onset of crack initiation from the notch tip were recorded. To theoretically predict the experimental results, the equivalent material concept was employed together with the averaged strain energy density over a material-dependent control volume. Without requiring time-consuming and complex elastic-plastic finite element analyses, it was shown that the combination of the averaged strain energy density approach and the equivalent material concept can successfully predict the load-carrying capacity of the V-notched Al 6061-T6 plates characterized by large plastic deformations ahead of the notch tip.
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Поступила в редакцию 07.04.2016 г.
Сведения об авторах
Ali Reza Torabi, PhD, Assist. Prof., University of Tehran, Iran, a_torabi@ut.ac.ir Alberto Campagnolo, PhD Stud., University of Padova, Italy, campagnolo@gest.unipd.it Filippo Berto, Prof., University of Padova, Italy, berto@gest.unipd.it
