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Анализ упругопластического разрушения пластин с надрезами из алюминиевого сплава Al 7075-T6 на основе локальной энергии и концепции эквивалентного материала
A.R. Torabi1, F. Berto2, A. Campagnolo3
1 Тегеранский университет, Тегеран, 13741-4395, Иран 2 Падуанский университет, Виченца, 36100, Италия 3 Падуанский университет, Падуя, 35131, Италия
В статье проведен анализ разрушения тонких пластин алюминиевого сплава Al 7075-T6 с закругленными V-образными надрезами в условиях растяжения. Испытания на разрушение проводились на прямоугольных пластинах с отверстием в форме ромба с закругленными боковыми углами. Экспериментальные наблюдения показали, что зона пластичности зарождается у вершины надреза и распространяется по мере монотонного увеличения растягивающей нагрузки, в результате чего происходит мгновенное разрушение с большим раскрытием вершины надреза. Высокие значения пластических деформаций в области вершины надреза, а также наклонные плоскости разрушения в исследуемых образцах свидетельствуют о вязком разрушении пластин из сплава Al 7075-T6 с V-образным надрезом. В результате экспериментов выявлено, что несущая способность пластин с надрезом соответствует времени начала роста трещины от вершины надреза. Выполнено теоретическое предсказание экспериментальных результатов на основе концепции эквивалентного материала с использованием известного критерия хрупкого разрушения — критерия усредненной плотности энергии деформации. Без проведения конечно-элементного анализа упругопластической деформации показано, что рассмотрение усредненной плотности энергии деформации в сочетании с концепцией эквивалентного материала позволяет эффективно предсказывать несущую способность пластин из сплава Al 7075-T6 с надрезом, разрушающихся в режиме среднемасштабной текучести.
Ключевые слова: концепция эквивалентного материала, плотность энергии деформации, V-образный надрез, упругопласти-ческое разрушение, алюминиевый сплав Al 7075-T6
Elastic-plastic fracture analysis of notched Al 7075-T6 plates by means of the local energy combined with the equivalent material concept
A.R. Torabi1, F. Berto2, and A. Campagnolo3
1 Fracture Research Laboratory, Faculty of New Science and Technologies, University of Tehran, Tehran, 13741-4395, Iran 2 Department of Management and Engineering, University of Padova, Vicenza, 36100, Italy 3 Department of Industrial Engineering, University of Padova, Padova, 35131, Italy
The main goal of the present research is to analyze tensile fracture in Al 7075-T6 thin plates weakened by blunt V-notches. For this purpose, first, 27 fracture tests are carried out on rectangular plates containing a central rhombic hole with two blunt V-shaped corners horizontally located. The experimental observations indicated that a plastic region initiates from the notch tip and grows as the tensile load monotonically increases, and finally, fracture happens suddenly with a significant opening of the notch tip. By showing significant plastic deformations around the notch tip and also inclined fracture planes, the specimens after fracture confirm well the ductile rupture in V-notched Al 7075-T6 plates. As the main experimental result, the load-carrying capacity of the notched plates corresponding to the onset of crack initiation from the notch tip is recorded. To theoretically predict the experimental results, the equivalent material concept is utilized together with the well-known brittle fracture criterion, namely the averaged strain energy density criterion. Without requiring elastic-plastic finite element analysis, it is shown that the combination of the averaged strain energy density and equivalent material concept is successful in predicting the load-carrying capacity of the V-notched Al 7075-T6 plates that fail by moderate-scale yielding regime.
Keywords: equivalent material concept, strain energy density, V-notch, elastic-plastic fracture, Al 7075-T6
Nomenclature
ASED — averaged strain energy density in the control volume near the notch tip; E — elastic modulus; EMC — equivalent material concept; K — strain-hardening coefficient; Kc — fracture toughness of material; KIc — plane-strain fracture toughness of material; n — strain-hardening exponent; SED — strain energy density;
e* — strain at crack initiation for the equivalent material; ep — true plastic strain;
eu — engineering plastic strain at maximum load; eu ^ — true plastic strain at maximum load; ep — true plastic strain at yield point; ey — elastic strain at yield point; p — notch radius; a — true stress;
af — tensile strength of the equivalent material; ay — yield strength.
© Torabi A.R., Berto F., Campagnolo A., 2016
1. Introduction
The aluminum alloy Al 7075 with various heat treatments is widely utilized in aerostructures. Its most traditional versions are Al 7075-T6 and Al 7075-T651 having high yield and ultimate tensile strength, high fatigue strength, good fracture toughness etc. Notches of different features are extensively employed in aerostructures for joining two or more structural components, e.g. O-notches in riveted joints. Moreover, bolts and screws made of aluminum alloys, particularly Al 7075, are traditionally used to join two or more plates and thin sheets in aerostructures. The threads of such bolts are normally V-shaped which can be considered as regularly oriented V-notches. Such threads mainly sustain tensile stresses and hence, it is essential to use appropriate bolts capable of carrying the tensile loads with a high level of safety. Due to the stress concentration around the V-notch, it is prone to crack initiation and therefore, determination of the load-carrying capacity of V-notches is necessary. Although the design of notched components in aerostructures is always performed in a way the stresses remain below the yield strength of material, it is possible in some cases that some excessive undesired loads are applied to the notched member and, as a result, a region around the notch experiences plastic deformations which can lead to fracture of the notched member. Thus, the fracture behavior of notched components made of aluminum alloys should be studied in the presence of considerable plastic deformations around the notch and the load-carrying capacity of such components should carefully be determined both experimentally and theoretically.
As is well known, the fracture evaluation of ductile components with elastic-plastic behavior in the presence of sharp cracks is usually performed by using the elastic-plastic fracture mechanics [1, 2]. All common failure criteria in the context of elastic-plastic fracture mechanics utilized for predicting crack growth in ductile materials, e.g. the critical J-integral, the crack tip opening displacement, the crack tip opening angle etc. need numerical elastic-plastic analyses which are more complex and time-consuming with respect to the linear-elastic ones. Although the level of stress gradient for blunt V-notches is significantly lower than that for sharp cracks, the stress gradient for blunt V-notches, especially for those having small notch tip radii, is also high enough to make the notched component vulnerable to crack initiation. Therefore, it is necessary to accurately predict the onset of crack initiation from the notch border in the
presence of significant plastic deformations around the notch by means of appropriate failure criteria in the field of the notch fracture mechanics.
Susmel and Taylor [3] were probably the first ones who employed the notch fracture mechanics principles for predicting elastic-plastic failure of notches. They applied the theory of critical distances [4, 5] to the linear-elastic stress distributions around notches of different features and predicted the tensile load-carrying capacity of notched plates made of very ductile commercial steel [3]. Although the large-scale yielding failure regime was recognized in their experiments at the onset of crack initiation from the notch tip, they demonstrated that the theory of critical distances in the linear-elastic regime is surprisingly successful [3]. It has been stated in Ref. [6] that the success of the linear elastic theory of critical distances reported in Ref. [3] is probably due to the close values of the yield and ultimate strength of the steel studied. With the aim to overcome the restrictions of the valuable analysis reported by Susmel and Taylor [3], the equivalent material concept, by which a ductile material having valid fracture toughness KIc value is equated with a virtual brittle material having the same elastic modulus and fracture toughness but different tensile strength, was proposed by Torabi [6]. By using the equivalent material concept in conjunction with the point-stress and mean-stress brittle fracture criteria, the experimental elastic-plastic results reported in Ref. [3] were well predicted in Refs. [6-8]. The accuracy of the proposed approach was also reverified successfully in Ref. [9] by means of some experimental results obtained from tensile fracture tests on ductile steel bolts containing V-shaped threads. Since the equivalent material concept acts as a bridge between linear-elastic and elastic-plastic analyses, it is expected that it could fundamentally be joined by different brittle fracture criteria, e.g. the stress-based maximum tangential stress and mean-stress [10-22], the averaged strain energy density [23-41], the cohesive zone model [42-45] and the finite fracture mechanics [46-51] etc.
In this research, the elastic-plastic fracture behavior of V-notched Al 7075-T6 thin plates is investigated both experimentally and theoretically under mode I loading conditions. In the experimental part, 27 fracture tests are performed on rectangular thin plates weakened by central V-notches of various notch angles and different notch radii which are horizontally located. The experimental results are mainly the load-carrying capacity of the notched
Table 1
Chemical composition of Al 7075-T6 Element Si Fe Cu Mn Mg Zn Ni Cr Pb Sn Ti
Weight, % 0.06 0.32 1.72 0.03 2.44 4.63 0.004 0.2 0.002 0.001 0.037 Element B Cd Bi Ca P Sb V Zr Co Li AI
Weight, % 0.001 0.001 0 0.001 0.001 0.001 0.007 0.017 0.003 0.001 90.5
Table 2
Mechanical properties of Al 7075-T6
Material property Value
Elastic modulus E, GPa 71
Poisson's ratio 0.33
Tensile yield strength, MPa 521
Ultimate tensile strength, MPa 583
Elongation at break, % 5.8
Engineering strain at maximum load 0.047
True fracture stress, MPa 610
Fracture toughness K c, MPa • m^2 50
Strain-hardening coefficient, MPa 698
Strain-hardening exponent 0.046
Al 7075-T6 plates and the macroscopic analysis of the fracture planes. Such results indicate significant plastic deformations around the notch at the onset of crack initiation from the notch tip and the elastic-plastic finite element analyses suggest a moderate-scale yielding failure regime for the plates. For the first time, the equivalent material concept is combined with the averaged strain energy density criterion to predict the experimentally obtained load-carrying capacity of the V-notched Al 7075-T6 plates without elastic-plastic analyses. It is shown that the experimental results could be predicted well by means of the combined EMC-averaged strain energy density criterion.
2. Experimental program
2.1. Material
The material studied is the aluminum alloy Al 7075-T6 with the chemical composition and mechanical properties presented in Tables 1 and 2, respectively. The main standard tests performed for determining the mechanical properties are the tensile tests, the Poisson's ratio tests and the fracture toughness tests according to ASTM E8 [52], ASTM E132-04 [53] and ASTM B646-12 [54], respectively.
The engineering and true stress-strain curves for the tested Al 7075-T6 are depicted in Fig. 1.
2.2. Test specimen
The test specimen is a thin rectangular plate containing a central rhombic hole with four V-shaped corners; two of which are horizontally located and subjected to the opening mode (i.e. the pure mode I loading) as a result of the remote tension. The V-notched specimen is schematically represented in Fig. 2 together with its geometric parameters.
The parameters 2a, p, 2a, L, W, and P in Fig. 2 are the notch angle, the notch radius, twice the notch length (i.e. the total slit length), the specimen length, the specimen width and the remotely applied tensile load, respectively. The
Strain
Fig. 1. The stress-strain curves for the tested Al 7075-T6
values of the parameters considered in the fracture tests are as follows: 2a = 3060°, and 90p = 1, 2 and 4 mm, 2a = = 25 mm, L = 160 mm and W = 50 mm. For all specimens, the thickness is equal to 2 mm. Taking into account the three values for each of the notch angle and notch radius, nine different notch geometries are examined. Three tests are carried out for each of the notch geometries in order to check the repeatability of the experiments. All in all, 27 fracture experiments are conducted in this investigation.
For fabricating the test samples, an Al 7075-T6 plate of 2 mm thick is first provided. Then, each specimen is sketched and the corresponding electronic file is given to a high-precision 2D CNC water jet cutting machine. The test samples are ultimately cut from the aluminum plate. The fracture tests are performed under the displacement-control conditions. The test speed is set to be equal to 2 mm/min (the strain rate is approximately equal to 0.0002) providing quasi-static loading conditions.
2.3. Experimental results
Figure 3 shows some of the V-notched Al 7075-T6 plates before, during and after the fracture tests. In particular Fig. 3, b depicts the set up during the test and Fig. 3, c shows the broken specimens after the tests.
As it can clearly be seen in Fig. 3, c, the V-notch seriously experiences permanent opening for all of the notch angles and in some cases, the profile of the notch round border considerably changes (see the mid specimen), demonstrating the existence of moderate or large-scale yielding conditions around the notch at failure. During the tensile tests, it is observed that a plastic region nucleates from the notch tip and propagates around the notch and finally, fracture happens abruptly so that the onset of crack nucle-
L
Fig. 2. The V-notched specimen together with its geometric parameters
Fig. 3. The V-notched Al 7075-T6 plates before (a), during (b) and after the fracture tests (c)
ation from the notch tip could not be captured by naked eye. The experimental observations also indicated that significant plastic deformations around the notch at failure do not guarantee the stable crack propagation and rupture in ductile components. The crack initiation and propagation behaviors of a ductile material can be interpreted by using typical engineering stress-strain curve. Despite the strain to failure which is normally large for typical ductile materials, the strain at peak (i.e. at the ultimate strength) is a key parameter in understanding the material cracking behavior. A small strain interval between the ultimate and the final rupture points (like for the present Al 7075-T6 alloy tested herein; see Fig. 1) means that crack initiates in the material by great plastic deformations and the material fractures rapidly or abruptly by unstable crack growth. Conversely, if the strain at the ultimate point covers a small portion of the total strain to rupture, crack initiates rapidly with relatively small plastic deformations and propagates slowly by large plastic deformations till the final rupture.
Figure 4 represents a sample load-displacement curve for a V-notched Al 7075-T6 plate. From Fig. 4, a clear but relatively small non-linear portion can be seen in the curve between the end of the linear zone and the peak point suggesting considerable plastic deformations around the notch tip at crack initiation instance. It is essential to notice that
Fig. 4. A sample load-displacement curve for a V-notched Al 7075-T6 plate
such amount of plastic deformations could not be resulted from a small-scale yielding regime, since in the small-scale yielding failure regime, no clear non-linear portion is normally realized in the load-displacement curve.
Table 3 summarizes the experimentally obtained critical loads of the V-notched Al 7075-T6 specimens for different notch angles and radii; all of them are recorded at the onset of the abrupt fracture. In Table 3, P (i = 1, 2, 3) and Pav denote each failure load in the repeated experiments and the average of the three failure loads, respectively.
In the next section, the equivalent material concept, which equates a ductile material with a virtual brittle material, is briefly described. By means of the equivalent material concept, it may be possible to use brittle fracture criteria for predicting elastic-plastic fracture in notched members.
3. A brief description of the equivalent material concept
From the viewpoint of the strain energy density, the equivalent material concept equates a ductile material having valid fracture toughness (KIc or Kc) value with a virtual brittle material having the same elastic modulus and
Table 3
The experimentally obtained critical loads of the V-notched Al 7075-T6 specimens for different notch angles and radii
2a
30°
60°
90°
p, mm
1
Pi, N
27 554
27 826
28 290
27 550
27 922
27 874
26 012
28 337
27 900
P2, N
27 337
28 271
28 661
27 918
28 496
28 304
23 776
27 616
28 098
P3, N
27 870
28 146
28 529
27 874
27 547
28 702
25 328
26 358
28 113
P , N
27 587
28 081
28 493
27 781
27 988
28 293
25 039
27 437
28 037
Strain s Sf
Fig. 5. A sample stress-strain curve for a typical ductile material
fracture toughness, but various tensile strength [6-8]. The tensile strength of the equivalent material is determined by considering the same values of the strain energy density needed by the real ductile and virtual brittle materials for the crack initiation to occur. A sample stress-strain curve for a typical ductile material is depicted in Fig. 5.
In the plastic region, the power-law stress-strain relationship can be written as
a = Ken, (!)
where a, ep, K and n are the true stress, the true plastic strain, the strain-hardening coefficient, and the strain-hardening exponent, respectively. The total strain energy density is
composed of the elastic and plastic components as follows:
1
(SED)tot= (SED)e + (SED)p = - ayey + J ad£p, (2)
2 eP
where a y, e y and e y are the yield strength, the elastic strain at yield point and the true plastic strain at yield point, respectively. Substituting ey = ayjE and Eq. (1) into Eq. (2), it results (E is the Young's modulus):
-2 e p
(3)
(SED)tot ^ + ) K e nd ep.
2E ey
Thus
(SED)fot =
aE+"KT [(ep)n+1 -(8p)"+1:
2E n +1
(4)
If ey is considered to be equal to 0.002 (obtained from 0.2% offset yield strength), then
2K
(5)
(SED)tot + K [e"+1 -(0.002)"+1].
2E n +1
For calculating the total strain energy density associated with the onset of crack initiation, which is equal to the area under the a-e curve from beginning to the peak (see Fig. 5), one should substitute ep in Eq. (5) with eutrue, i.e. the true plastic strain at the ultimate point, which could easily be computed by using the expression eut = ln(1 + eu),
where eu is the engineering plastic strain at the ultimate point. Therefore
(SED)necking + -+: [eu+rue -(0-002)
2E n +1
n+1-
(6)
Figure 6 shows a typical stress-strain curve for the equivalent brittle material. In Fig. 6, the parameters e* and a* are the strain at fracture and the tensile strength of the equivalent brittle material, respectively. The strain energy density absorbed by the equivalent material till the crack initiation is
(SED)em
. (a*)2
2E
(7)
According to the equivalent material concept, the strain energy density values presented in Eqs. (6) and (7) should
be identical. Thus
2
kr
(8)
f = aE+^[eu+1ue - (0.002)n+1;
2E 2E n +1
The tensile strength of the equivalent brittle material a* can finally be extracted as follows:
af = Ja?, + 2JK [e^ - (0.002)
n +1
, n +1-
(9)
The parameter a* presented in Eq. (9) can be used together with the material fracture toughness in different brittle fracture criteria for predicting the load-carrying capacity of notched engineering components made of ductile materials with elastic-plastic behavior. Considering the values presented in Table 2 for Al 7075-T6, the value of the parameter a* can be calculated from Eq. (9) to be equal to about 1845 MPa.
In the forthcoming sections, the equivalent material concept is utilized in conjunction with the averaged strain energy density approach, which is basically a brittle fracture
Strain 8 sj
Fig. 6. A typical stress-strain curve for the equivalent material
Fig. 7. Control volume (area) for sharp crack (a), sharp (b) and blunt (c) V-notches under mode I loading. Distance r0 = p( n - 2a)/(2n - 2 a)
criterion, to predict the experimentally obtained critical loads presented in Table 3.
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 [23, 24], 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 [55] 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 in Ref. [55] is a point-wise criterion, while the averaged strain energy density approach as presented in Refs. [23, 24] suggests that brittle fracture takes place when the strain energy density averaged over a known control volume, W, 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 Klc 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 [23] and later extended to blunt U- and V-notches [24, 41]. Some recent developments and applications are summarized in Refs. [24, 41, 56], with some considerations also to three-dimensional effects [57-60], which have been widely discussed in Ref. [61].
For sharp cracks, the control volume is a circle of radius Rc centered at the crack tip (Fig. 7, a). Under planestrain conditions, the critical length Rc can be evaluated by the following expression [62]:
\2
where Klc 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. 7, b), while for a blunt V-notch under mode I loading, the volume assumes the crescent shape shown in Fig. 7, 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. 7, c). Such a distance depends on the V-notch opening angle 2a, according to the expression r0 = p( n- 2a )/(2tc- 2a) [24].
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 equiva-
«of
Rc =
(1 + v)(5 - 8v)
4n
KT,
(10)
Fig. 8. Example of the fine mesh employed in the control volume (a); the same results can be achieved by using a coarse mesh shown in (b)
Table 4
Critical loads predicted by means of the averaged strain energy density criterion in combination with equivalent material concept
p, mm 2a Pi, N P2, N P3, N PASED, N Ai A 2 A3
1 27554 27337 27 870 23 653 1.16 1.16 1.18
2 30° 27 826 28271 28146 28496 0.98 0.99 0.99
4 28290 28661 28 529 35019 0.81 0.82 0.81
1 27550 27918 27 874 22534 1.22 1.24 1.24
2 60° 27922 28496 27547 27411 1.02 1.04 1.00
4 27 874 28304 28702 33 929 0.82 0.83 0.85
1 26012 23 776 25 328 21565 1.21 1.10 1.17
2 90° 28337 27616 26358 26073 1.09 1.06 1.01
4 27900 28098 28113 32283 0.86 0.87 0.87
lent material. The critical strain energy density Wc EMC is evaluated by using Eq. (7) and considering a* = 1845 MPa. The critical strain energy density results to be equal to 23.97 MJ/m3. The control radius Rc is evaluated by using Eq. (10) with au = a* = 1845 MPa. It is found Rc = = 0.18 mm. The averaged strain energy density, W, occurring inside the control volume embracing the edges of V-notches is calculated numerically by using the finite element code ANSYS. For each geometry, a finite element model is created, which requires an accurate definition of the control volume, where the strain energy density should be averaged (see Fig. 7, c). The linear elastic finite element analyses are performed by using 2D eight-node solid elements (PLANE 183) under plane-strain conditions. A detail of the adopted finite element mesh is reported in Fig. 8. In particular, typical fine and coarse finite element meshes are shown in Fig. 8, a and b, respectively. As is well known from Refs. [56, 63, 64], the averaged strain energy density can be correctly evaluated also with very coarse finite element meshes and for this reason all finite element models are performed here using the mesh shown in Fig. 8, b.
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 a constant averaged strain energy density equal to 23.97 MJ/m3 over the control volume.
The last columns of the table present the deviations between the values of the experimental failure loads and the theoretical ones evaluated by means of the averaged strain energy density criterion. The deviation 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 almost all 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 energy averaged over the control volume (of radius Rc), normalized with respect to the critical energy of the material as a function of the notch angle is shown in Fig. 9. The plotted parameter is proportional to the fracture load. The goal is to study the influence of the notch geometry on the fracture predictions based on the averaged strain energy density. Also from the graphical point of view, it is obvious that almost all values fall inside a scatter ranging from 0.8 to 1.2 with the majority of the data inside 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 pure mode I 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, which deals with brittle and quasi-brittle failures [24].
6. Plastic zone size
Referring to Sect. 2, the experimental observations indicated that the Al 7075-T6 plates fail by crack initiation
^c,EMC ~ ^c,EMC = = 23.97 MJ/m3 0.18 mm
+20% +10% A
ö / 0
n. © R
0 «) u
^ 0
-20% \j -10%
0.0 h-1-1-1-1-
0° 20° 40° 60° 80° 100° Notch opening angle 2a
Fig. 9. Synthesis of fracture data in terms of normalized averaged strain energy density
Fig. 10. A sample plastic region around the V-notch tip at failure instance corresponding to the specimen with 2a = 30° and p =1 mm
from the notch tip under moderate-scale or large-scale yielding conditions. In order to accurately determine the type of elastic-plastic failure regime, i.e. moderate-scale yielding or large-scale yielding, the tensile test of the aluminum plates is simulated and the elastic-plastic finite element analyses are conducted for the entire notch geometries tested. Note that in the finite element analyses, the average experimental failure load is applied to the specimen. A sample plastic region around the V-notch tip is depicted in Fig. 10, corresponds to the specimen characterized by 2a = 30° and p = = 1 mm at failure instance.
According to Fig. 10, about 3.12 mm ahead of the notch tip (one fourth of the ligament) experiences plastic deformations. For the other notch geometries, similar results are achieved. It is found from the elastic-plastic finite element analyses that the percentage of the ligament occupied by plastic deformations at the onset of crack initiation varies between 25% and 30%, depending upon the notch geometry. Therefore, the finite element analyses prove the moderate-scale yielding failure regime for the tested Al 7075-T6 plates weakened by blunt V-notches.
7. Conclusions
The present research is aimed to analyze the tensile fracture in Al 7075-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 a non negligible plasticity ahead of the notch tip. 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 measured in a control volume. Without requiring 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 7075-T6 plates characterized by significant local plasticity ahead of the notch tip.
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Поступила в редакцию
__27.10.2015 г.
Сведения об авторах
Ali Reza Torabi, Assist. Prof., University of Tehran, Iran, [email protected] Filippo Berto, Prof., University of Padova, Italy, [email protected]
Alberto Campagnolo, PhD Student, University of Padova, Italy, [email protected]
