Fatigue failure is a common state of failure in mechanical components that operate under cyclic loading. It is particularly rapid in materials that exhibit brittle behavior, especially for components subjected to bending. Fatigue failure is also more likely to occur in components with geometric discontinuities or defects, such as those resulting from manufacturing or from the migration of inclusions during the elaboration of the materials. These geometric changes can be created intentionally (e.g., holes, grooves, shoulders) or accidentally during service. Shapes that we can consider as geometric discontinuities are notches, whether in blunt-U shape or acute V shape. These shapes are the subject of several studies by researchers in the field of fracture [1–3] and fatigue fracture [4–7] of samples. The presence of these notches in mechanical organs generally induces amplifications of the stresses and becomes a privileged place of stress concentration; several authors are interested in the effects of their presence in mechanical components. A.R. Torabi et al. [1] studied the brittle fracture of Brazilian disks containing central V-notches with end holes (VO-notches) made of polymethylmethacrylate (PMMA) under mode II loading. They used the averaged strain energy density (ASED) criterion, and their contributions demonstrated that the ASED approach works well on specimens containing VO-notches in pure mode II. This approach was used by Lazzarin and Zambardi [8] to predict the static and fatigue behavior of components weakened by sharp reentrant corners, showing that strain energy density (SED) can accurately predict the static and fatigue behavior of heavily notched components. Berto and Lazzarin [9] applied the SED approach to static and fatigue strength assessments of notched and welded structures. These authors indicated that the region around the tip of the notch is controlled by a finite size volume and when sharp V-shaped cracks or notches are considered, the volume becomes a circle or a circular sector, where the radius R0 is the measured size of this circular sector. This parameter R0 has become a propriety of a material, and it depends on a fracture toughness, ultimate tensile strength, and Poisson’s ratio in the case of static loads; it depends on the fatigue strength of the butt ground welded joints and the notch stress intensity factor (NSIF) range in the case of welded joints under high cycle fatigue loading. On the basis of the local energy concept, Aliha et al. [10] used it in their article analyzing cracked specimens (semicircular and triangular crack type) of four different shapes made by PMMA material under pure mode I, pure mode II, and mixed mode I/II. They demonstrated through experimental work that the SED approach can be used as a criterion to predict the behavior of crack specimens under mixed-mode loading.
According to M. Aliha et al. [11], the ASED criterion has been used for evaluating brittle fracture behavior for edge-cracked triangular specimens subjected to symmetric, three-point bend loading, caused by rock materials. It was found that the ASED criterion can successfully predict the fracture loads of investigated marble in the entire range of mode I/II mixities.
Negru et al. [12] used the local SED approach with a theory of critical distance (TCD) for evaluation of brittle fracture of two types of specimens: single edge notched bend (SENB) U-notched specimens and asymmetric semi-circular bend (ASCB) cracked specimens. The specimens were submited to bend loading to predict maximum fracture load for two different polyurethane (PUR) materials. Both criteria have proven their effectiveness in predicting the fracture load. Ayatollahi, Berto, and Lazzarin [13] examined a mixed mode of Brazilian disk samples containing sharp and blunt V-notches made of polycrystalline graphite. The ASEFD is used over a well-defined volume to predict static strength for different values of loading mixtures, V-notch angles, and notch radii [14]. Many researches have focused their studies on mixed mode loading based on the use of the SED criterion to predict the failure of a notched specimen [10]. Salavati et al. [15] and Torabi and Berto [3] employed bainitic functionally graded steels and a generalized ASED criterion to investigate the fracture behavior of brittle and quasibrittle materials, respectively, for two cracked specimens: a cracked Brazilian disk (BD) specimen and a semicircular bend (SCB) specimen under mixed mode I/II loading. The stress intensity factors
This collected research, which was oriented towards the use of the SED approach, has proven that this approach has high reliability in the prediction and evaluation of the parameters governing the behavior of fracture and the fatigue fracture.
On the basis of this research, the objective of this paper is to investigate the effect of notches on fatigue behavior by combining two methods, the extended finite element method (XFEM) and the ASED approach under mixed-mode loading I/II. Experimental tests are conducted on cylindrical specimens notched with two types of notches: blunt U-notches and sharp V-notches made by polymethylmethacrylate (PMMA) material. The obtained results are compared with those obtained experimentally and show good agreement with the numerical results.
The material considered in this study is PMMA, polymethyl methacrylate. It exhibits fragile behavior. PMMA has the following properties:
Experimental rotary bending fatigue tests are performed on the SM1090 machine (see Fig. 1). The recommended specimens are cylindrical in shape (Fig. 2). The specimens are mounted on the rotary bending fatigue machine with one end fixed on the chuck while the other end is subjected to a constant force at a frequency of 50 Hz.
Given the difficulty of machining notches on this type of material, we were able to adopt two different notches for the two configurations of U and V notches. For the sharp V notches, we chose two angles with dimensions of 20° and 140° and for the blunt U notches, we chose two radii of 0.2 mm and 2 mm (see Figs. 2 and 3). The recommended specimen geometries suitable for the rotary bending fatigue machine are cylindrical shapes with the dimensions shown in Figure 3.
The assessment of the risk of failure for a structure subjected to fatigue is based on failure criteria that designers consider in their designs. To predict the fatigue behavior of the mixed mode surrounding the tip notch, an energy criterion was formalized by Sih [17], which gave a theoretical basis to Gillemot’s experiment [17, 18]. The energy criterion was strongly supported by several researchers, and it offered an alternative approach that could be used to characterize the level of stress at the tip notch without recourse to mesh fineness around the tip notch. He postulated that the failure occurs when the quantity of energy reaches a critical value of Sc, while the direction of crack propagation was determined by imposing a minimum condition on S [17, 19, 20].
The basic idea of the strain energy density criterion consisted of considering a finite size volume around the notch over which the strain energy is averaged [8, 21]. According to this criterion, fracture occurs when the variation of local energy over the structural volume Δ
Using the Williams’ series expansions [23, 24], the stress relationships can be expressed nearness of the notch tip as follow:
Substituting the stress components Eq. (4) into Eq. (5), and integrating the strain energy density over a control area on a boundary defined by a distance of the radius
The elastic strain energy density per cycle,
For fatigue problems [9–11], the combined mode I and II can easily be evaluated by means of the SED averaged over a control volume, considered as a material property and assumed to be a circular sector of radius
El Haddad, Topper, and Smith [33] provide a parameter to link the stress intensity threshold term Δ
In a highly stressed region, all stress and strain components are correlated to mode I and mode II (NSIF). The variation of total strain energy density averaged Δ
In the case of blunt notches, the control volume is assumed to have crescent shape, with
Dealing with blunt notches under fatigue loading, the following expression can be used to evaluate the strain energy for notches in mode I,
To account for the shear contribution, Lazzarin et al. [40] dealt with a torsion problem in which they developed an expression for the strain energy density over volume control for the U-notch and the blunt V-notch. from this relation we used it in the shear action that contributes to the mode –II of sliding mode, this formula is expressed as follows [22]:
This study is divided into two parts: the first is the experimental part, realized on the rotary bending fatigue machine, which provided us with results from which we deduced curves of the average total local strain energy as a function of the number of cycles to failure (W–N
Under the conditions of plane strain and by using the extended finite element method, the specimen is converted into discrete Q4 elements with a total number of 3224 elements (Fig. 8).
The presence of a geometric discontinuity defined by a notch in the specimen undergoing fatigue loading leads to crack initiation after a certain number of loading cycles, followed by its propagation. Numerically, the XFEM is used to represent the discontinuities independent of the mesh. The discontinuities can be modeled by enriching all discontinuous elements using enrichment functions that satisfy the discontinuous behavior and adding additional nodal degrees of freedom. It enables us to control the initiation and propagation of fatigue cracks without the need to remesh the structure.
Crack initiation is localized at the point where the principal stress criterion
Based on the local approach of strain energy density which consists in the first step of defining the volume control zone where it rules the elaboration of the zone damage and then leads to the initiation and propagation of cracks. This zone is located around the notch tip and includes a quantity of elastic deformation energy. If this energy reaches the value of the critical energy defined by Wc, the structure undergoes a crack initiation and rupture. A calculation is made of the total local energy, which takes into account two quantities of energy linking to the bending and shearing energies Δ
Figure 10 shows the distribution of the ASED normalized by the critical energy in dimensionless units as a function of the fatigue life for the two types of notches, V and U. The results obtained are compared with those from the experiment. At low fatigue cycles, the energy values reach maximum values for both types of notches. The trend follows a decay towards a stabilization of the energy at a high life cycle. In terms of amplitude, the local energy of the sharp notch is higher than the energy values of the blunt notch, and this is due to the attenuation of the stress concentrations at the notch level. The presence of notches in a structure always leads to the amplification of energy concentrations and the reduction of the structure’s lifetime. The introduction of the fatigue behavior curve of the smooth specimen (lack of notches) in Figure 10 (blunt U notch) shows that the lifetime of these specimens is longer (more cycles) than the specimens with notches and with lower energy quantities. This implies that a high concentration of energy is present in specimens with notches.
Figure 11 shows approximations of the mean local energy SED by power equations of the experimental values for the different notches. Table 2 gives the approximation equations of all the notches with correlations coefficients. For a sharp V-notch, the critical volume becomes a circular sector of radius Rc centered at the notch tip (Fig. 6). For a blunt notch under mixed-mode loading, the critical volume is no longer centered at the notch tip, but rather at the point where the principal stress reaches its maximum value along the border of the notch (Fig. 7).
Mechanical properties of PMMA
Tensile strength (MPa) | Flexural strength (MPa) | Modulus of elasticity (MPa) | Density (kg/m3) | Elongation (%) | Poisson’s rate | Fracture Toughness (MPa.√m) |
---|---|---|---|---|---|---|
70.5 | 110 | 3000 | 1190 | 6 | 0.3 | 1.863 |
Local energy modeling near notch
Experimental | ||
---|---|---|
Notch type | Correlation coefficients | Experimental: Local energy near notch |
U-notch radius 0.2 mm | R2 = 0.9881 | |
V-notch angle 20° | R2 = 0.9591 | |
V-notch angle 140° | R2 = 0.9656 | |
U-notch radius 2 mm | R2 = 0.8875 | |
U-notch radius 0.2 mm | R2 = 0.967 | |
V-notch angle 20° | R2 = 0.857 | |
V-notch angle 140° | R2 = 0.890 | |
U-notch radius 2 mm | R2 = 0.8305 |
It was fundamentally assumed that the crescentshaped volume rotates rigidly under mixed mode, with no change in shape and size (Fig. 7) [3]. The results obtained are in good agreement with the experimental results (Fig. 10).
Figure 12 shows the area where the first maximum principal stress (
Another interest in this study is to analyze the variation of cyclic loading on the evolution of the average strain density energy under the effect of the presence of notches. Figures 13 and 14 show the distribution of the average strain energy normalized by critical energy via fatigue life: the high loads develop high-amplitude average SEDs. For low angles in V notches, the intensity of the average SED is greater than for more open angles. The lifetime is reduced compared to the more open angles. The values obtained by numerical calculation are in good approximation compared with those of the experimental ones.
For U-shaped notches with a significant radius, the high loading leads to high mean SED values compared to low-radius notches and a longer lifetime compared to sharp notches. Sharp V-notches develop higher mean SED values than blunt notches. This suggests that the loading amplitude plays an influential role on the ASED and the fatigue life of structures subjected to bending fatigue.
Several simulations were performed for a sharp notch V140° at different mesh element numbers. Figure 15 shows convergence to the reference result from a mesh element number of 3224 elements. As the mesh element number increases, the results tend to deviate from the reference value.
A statistical calculation was performed to compare the experimental and numerical results. The results of the calculation showed a good correlation between the two sets of results. The standard deviation of the experimental test results was slightly lower than that of the simulation test results. The minimum and maximum values of the two data sets were also very close (see Table 3).
Correlations of experimental and numerical results
Experimental test | |||||||
---|---|---|---|---|---|---|---|
Notch type | Number of test repetitions | Min. number of cycles | Max. number of cycles | Standard deviation of cycles | Minimum energy | Maximum energy | Standard deviation of energy |
U-notch radius 0.2 mm | 4 | 22640 | 201,865 | 82391.597 | 0.293 | 0.507 | 0.094 |
V-notch angle 20° | 4 | 521,218 | 151,883 | 75622.87 | 0.229 | 0.461 | 0.107 |
V-notch angle 140° | 4 | 11032 | 311,170 | 129198.70 | 0.166 | 0.504 | 0.143 |
U-notch radius 2 mm | 5 | 25,239 | 341,478 | 125086.76 | 0.112 | 0.283 | 0.074 |
U-notch radius 0.2 mm | 4 | 15254 | 326,645 | 146,519.222 | 0.255 | 0.438 | 0.080 |
V-notch angle 20° | 5 | 150.43 | 151,883 | 67,693.93 | 0.229 | 0.609 | 0.141 |
V-notch angle 140° | 4 | 3890.97 | 85,337 | 42857.97 | 0.253 | 0.538 | 0.139 |
U-notch radius 2 mm | 5 | 22938.2 | 442,920 | 169,677.83 | 0.124 | 0.313 | 0.081 |
The combination of two numerical tools, the extended finite element method (XFEM) and the local strain energy density approach, allowed us to analyze the influence of the presence of notches on the fatigue behavior of structures subjected to rotating bending.
The mixed loading mode appropriate for this type of solicitation allowed us to consider the combination of two modes, I and II. Under these conditions, we used two methods: experimental and numerical. The projection of the experimental work onto the numerical led to the ability to project the degree of fatigue of rotary bending into a twodimensional bending specimen.
The study carried out identified the following results:
A high concentration of local energy is around the notch tip, The value of the average strain energy density acting on the control volume zone differs from one notch to another. For the sharp notch the control volume is located in a circular area in the center, which is located at the notch tip and the maximum of the principal stress is at the notch tip where the crack initiation zone is. In the case of blunt notches, the control volume is assumed to have a crescent shape, its maximum width is measured along the bisector of the notch, In mixed-mode case, the maximum principal stress occurs at a point on the edge of the notch, outside the bisector. The control volume is no longer centered on the bisector of the notch but turned rigidly with respect to it and located far from the point of the notch tip at the beginning of the notch lip end.
In components weakened by a notch subjected to rotational bending fatigue, the control volume follows the crack tip as it propagates, so that the fracture process zone begins and mode I remains the dominant mode. The local energy magnitude for a blunt notch is lower than that for a sharp notch, and the lifetime is longer for blunt notches. Therefore, the lifetime of components is affected by the presence of notches. The current work showed good concordance between experimental results and numerical results.