Prevalent Mode II Fracture Investigation of VO-Notched Specimens Made of Tungsten–Copper Functionally Graded Materials

In this research, the prevalent mode fracture of V-notches with end holes (VO-notch) in tungsten–copper functionally graded material (W–Cu FGM) was studied experimentally and theoretically. W–Cu FGM specimens were made by powder metallurgy technique in the experimental part. Several fracture tests were done on VO-notched W–Cu FGM specimens under prevalent mode loading for different notch tip radii and notch depths. To predict the fracture loads of VO-notched FGM specimens under prevalent mode loading, the averaged strain energy density over a well-defined control volume was applied in the theoretical section. Moreover, the effect of notch tip radius, notch depth and notch opening angle on the fracture loads was investigated. This study shows that strain energy density theory works well on VO-notched FGM specimens under prevalent mode loading.


INTRODUCTION
V-, U-, VO-and keyhole-notches are different types of notches which exist in structural elements. These notches decrease the strength of structures [1] and brittle fracture investigation of them is an important issue which has been studied by many researchers. In order to evaluate the critical fracture load of notched components, several fracture criteria have been suggested in the literature. The volume based on the strain energy (SED) criterion is one of the most eminent criteria that reminds the Sih's criterion [2]. According to the strain energy density, when the averaged strain energy density over a well-defined control volume reaches its critical value, the static fracture happens [3,4]. One of the most important advantages of this criterion is that the averaged value of strain energy density over the control volume can be computed with high accuracy by finite element approach using the low number of elements [5,6]. Other applications of this criterion are considering the T-stress and three-dimensional effects, assessing uniaxial and multiaxial fatigue failure of welded joins and notched members [7,8] and also investigating different loading modes. This criterion has been applied to fatigue and fracture investigation of different types of homogeneous materials. But, it is still needed to work more on the fracture behavior of advanced materials. Recently, this theory has been utilized for fracture assessment of additively manufactured materials [9].
One of the main groups of advanced material is functionally graded material (FGM), in which the spatial chemical properties change gradually to achieve the specific features. Various techniques such as the bulk (particulate processing), preform processing, layer processing and melt processing can be employed to fabricate these materials. Three types of functionally graded materials which have been fabricated and investigated in the recent years, are well-known metal-ceramic FGMs, functionally graded steels (FGSs) and tungsten-copper FGMs (W-Cu FGM) [10].
Metal-ceramic functionally graded materials are usually applied to increase the properties of thermalbarrier systems. Functionally graded steels are one of functionally graded materials which have elastic-plastic behavior. Recently, these materials have been fabricated from austenitic stainless steel and carbon steel using electroslag refining (ESR) method [11,12]. In order to obtain composites with several layers consisting of ferrite, austenite, bainite and martensite, the thickness of the primary ferritic and austenitic steel electrodes should be selected as appropriate.
The tensile strength and microhardness profile of austenitic graded steel was modeled by Nazari et al. [13] based on the strain gradient plasticity theory. In other works, the flow stress of dual layer austeniticmartensitic, bainitic and martensitic FGSs under hot deformation loading was investigated [14,15]. Nazari et al. [16,17] studied the impact energy of FGS with crack divider configuration using the strain gradient plasticity theory and modified stress-strain curve data. Salavati et al. [18] developed a new analytical expression for the relationship between the Charpy impact energy and notch tip position for FGSs. The plates fabricated from austentic-martensitic FGS weakened by U-notch were investigated by Barati et al. [19]. Salavati and Mohammadi [20] investigated the ductile fracture of bainitic FGS and performed fracture tests on U-notched specimens under mode I loading. They employed the strain energy density with equivalent material concept to predict the ductile fracture of bainitic FGS.
Tungsten-copper (W-Cu) functionally graded materials are widely used as ultra-high voltage electric contact materials and heat-sink materials [21]. Due to large differences in melting point between these two metals, it is difficult to fabricate W-Cu functionally graded material. Therefore, some production approaches such as spark plasma sintering [22], microwave sintering [23], laser sintering [24], one-step resistance sintering [25] have been suggested in the literature. Fracture behavior of W-Cu functionally graded material weakened by different kind of notches has been studied in [26][27][28][29].
In this research, the fracture of W-Cu FGM specimens containing VO-notch under prevalent mode II is investigated both numerically and experimentally. The averaged strain energy density theory is used to predict the fracture loads of VO-notched specimens which show a good agreement with the experimental results. Moreover, the effects of notch parameters such as notch tip radius, notch depth and notch opening angle on the fracture load was studied.

Material Properties
In this paper, W-Cu FGM specimens consist of 3 regions, WBA tungsten-based alloy (WBA) region, copper region, and the graded region which joints the WBA region to copper region (Fig. 1). Copper and tungsten-based alloy are two extremes of the functionally graded material. Generally, these specimens contain six layers that there is a continuously change from a tungsten-based alloy with the chemical composition of W-Ni-Mn-Cu respectively as 90-4-3.33-2.67 wt % to pure copper. Therefore, 4 innermost layers (layers 2 to 5) are considered as graded region. Table 1 presents the chemical composition of the layers. It is to be mentioned that, the fabrication of W-Cu FGM is hard, because there is the large difference between the melting points of two metals. Several fabrication techniques have been suggested by some researchers for W-Cu FGM. The procedure of fabrication has been provided in details in [30]. In this research, in order to produce a six-layered W-Cu FGM specimens, the powder metallurgy method is employed. In this method, the thicknesses of regions are obtained by metallography technique. According to this, the averaged thickness of WBA region, the graded region, and copper region are considered to be 0.95, 2.3, and 0.75 mm, respectively. This new material has been made by the cooperation of Shahid Bahonar University of Kerman, Amirkabir University of Technology and Norwegian University of Science and Technology [30].  The power law function is used to express the mechanical properties of the graded region such as the elasticity modulus, Poisson's ratio, ultimate tensile stress and fracture toughness K Ic . Therefore, this function can be written as  where E is the elasticity modulus,  denotes the Poisson's ratio,  ut is the ultimate tensile stress, K Ic is the fracture toughness, n, n 1 , n 2 are the power law exponents, Z indicates the thickness coordinate of the graded region -h/2 < Z < h/2, and h = 2.3 mm is the thickness of the graded region. Table 2 shows the boundary condition properties of Eqs. (1)-(4) and the mechanical properties of FGM layers [30].

Experimental Tests
In order to do the experimental tests, the VO-notched beams are considered. The geometry of the specimens and dimensions of the layers is given in Fig. 1. Thickness, width, and length of the specimens are considered to be equal to 2, 4, and 18 mm, respectively, which are in agreement with ASTM E1820. The span length between two supports is set to be equal 8 mm. The VO-notch is drawn from WBA side of the specimens as oblique in Fig. 2. To perform the experimental tests, two notch tip radii  of 0.2, 0.4 mm and three notch depth a of 1.1, 1.2, 1.4 mm are considered. Moreover, the notch opening angles 2 = 60 is considered for all specimens. Two specimens have been fabricated for each geometrical feature with 6 VOnotched beam specimens.     Fig. 3. A specimen loaddisplacement curves is shown in Fig. 4. Table 3 shows the fracture loads of each test.

FRACTURE OF VO-NOTCH UNDER THE PREVALENT MODE II IN FUNCTIONALLY GRADED MATERIAL USING THE AVERAGED STRAIN ENERGY DENSITY CRITERION
The brittle fracture happens when the value of averaged strain density (SED) over a specific control volume reaches to a critical value W c [3,4]. W c is a material-dependent value which is independent of notch geometry. For brittle materials, W c can be written as the following relation [4]: The critical radius R c is measured along the notch bisector line and is determined as [31]: The outer radius of the control volume is considered as R c + r 0 that illustrated in Fig. 5a for mode I and Fig. 5b for mode II loading. The value of r 0 can be obtained as [32]: It should be noted that in functionally graded material R c changes point by point due to a continuous change in the material properties. In other words, the outer boundary of the control volume has no longer a circular arc shape. Several researches have used this issue on the fracture of notched specimens [30]. For specimens fabricated of functionally graded material with VO-notch under the prevalent mode, in which the material varied in the x direction, the outer boundary of control volume as depicted in Fig. 6 can be obtained using the following equations:  where x and y are the coordinates of a point on the outer boundary,  denotes the corresponding angle to that point as shown in Fig. 6, a is notch depth, R c (x) shows the critical length as a function of x coordinate, and r 0 can be obtained by Eq. (4).

ASSESSMENT OF THE GENERALIZED NOTCH STRESS INTENSITY FACTORS, MODE MIXITY AND CRACK INITIATION ANGLE
Zappalorto and Lazzarin [33] introduced the generalized notch stress intensity factors for the VOnotch as the following relation: All the constants of Eqs. (6) have been shown in [13].
To obtain the stress components   and  r along the notch bisector, finite element models are applied. They are calculated at a distance r from the origin of the local coordinate system placed at the center of the hole. The values of the parameters g i , h i , and  i have been presented in [33].
In order to remove the weak dependence on the notch tip distance, the mean values of the generalized notch stress intensity factors can be obtained as follow [34]: where parameter  has been considered equal to 0.2 in this research according to [34]. The mode mixity has been calculated using the following relation [35]: The point on the notch edge can be considered as the crack initiation point. On this point, the principal stress reaches to the maximum value. The crack initiation angle is defined as the angle between the notch bisector line and the line passing through crack initiation point and center of the hole that is shown by angle  (Fig. 6).
In order to evaluate the generalized notch stress intensity factors, mode mixity and crack initiation angles for different geometries of VO-notch, finite element analyses are used [30]. For this purpose, an eight-node element is considered. Table 4 presents the mean value of the generalized notch stress intensity factors and mode mixity . In the theoretical section of this paper, the commercial finite element ABAQUS 6.11 software is used to calculate the averaged value of strain energy density over the control volume by some finite element analyses (Fig. 7). All these analyses have been performed under plane strain conditions and linear elastic hypothesis. Moreover, for each geometry two models were created. The first model was applied to determine the point where the maximum principal stress was located; the second one was used to obtain the averaged strain energy density over the well-defined control volume. Figure 8 shows the maximum principal stress and SED contour lines for the configuration with  = 0.4 mm and a = 1.1 mm. It should be noted that, both the averaged value of strain energy density over the control volume and also the fracture load are not sensitive to the increasing of the number of elements [28]. Therefore, the coarse mesh with approximately 100 elements is applied to assess the fracture loads ( Fig. 9).
If the arbitrary load is applied, the fracture load can be resulted as the following relation: where F app is the applied load, F th is the theoretical fracture load, SED indicates the averaged value of strain energy density over the control volume, and W c denotes the critical value of strain energy density corresponding to the notch tip.

COMPARISON BETWEEN THE EXPERIMENTAL AND NUMERICAL RESULTS
The average values of experimental and theoretical fracture loads (F exp and F th , respectively) based on the averaged strain energy density criterion are provided in Table 5, where the experimental results are compared with the theoretical results. There is a good agreement between the results. It should be noted that, although the experimental tests did not repeat in most cases, the theoretical solution well estimates the average value of experimental tests and their trend.

FRACTURE LOADS BASED ON THE NOTCH PARAMETERS
The effect of notch geometrical parameters such as notch radius , notch depth a, and notch opening angle 2 on the fracture load are studied. In order to eva-  luate the effect of notch geometries on the fracture load, different geometries can be proposed in the numerical analyses. Table 6 shows the complete results.  In this table, F cr FGM fracture load for the FGM material and F cr Hom shows the critical fracture load for the specimen made of the same homogeneous material corresponding to the layer including the notch tip in FGM material. Figures 10 and 11 show the variation of the critical fracture load versus the notch depth for different notch radii for two notch opening angles, respectively. As both figures show, the critical fracture load decreases by increasing the notch depths.

CONCLUSION
In this paper, the fracture load of VO-notch specimens fabricated of W-Cu FGM under the prevalent mode II was predicted using averaged strain energy density criterion. A six-layered W-Cu FGM was successfully fabricated by powder metallurgy technique. A new set of experimental data on fracture of VO-notched W-Cu FGM under prevalent mode II was presented in this work for the first time. The outer boundary of the control volume was determined numerically and it was shown that the shape of control volume is appropriate for fracture assessment of notched specimens made of functionally graded materials.
The effect of notch parameters (notch tip radius, notch depth, and notch opening angle) on the critical fracture load was studied. The results show that the critical fracture load decreases by increasing the notch depths for different notch tip radii in a constant notch opening angle. The average deviation between the experimental and theoretical results is about 2% which shows a good agreement.