Stress notch root

The criteria (Eqs. 11 and 12) are similar and are derived from studies on materials that are elastic at initiation of crazing, while more ductile materials like polycarbonate show a more pronounced sensitivity to the hydrostatic tension. This has been found experimentally by Ishikawa and coworkers [1, 27] for notched specimens of polycarbonate. Crazing appears ahead of the notch root, at the intersection of well-developed shear bands. From a slip fine field analysis, the tip of the plastic zone corresponds to the location of the maximum hydrostatic stress. This has been confirmed by Lai and Van der Giessen [8] with a more realistic material constitutive law. Therefore, Ishikawa and coworkers [1,27] suggested the use of a criterion for initiation based on a critical hydrostatic stress. Such a stress state condition can be expressed by Eq. 11 with erg = 0 and I r = B°/A°. Thus, the criterion (Eq. 11) can be considered general enough to describe craze initiation in many glassy polymers. For the case of polycarbonate, a similar criterion is proposed in [28] as... 

Figure 6. Diagrams of planar shape and thickness profile of full grown craze and shear zone beyond root of notch in BPA polycarbonate specimen. Shear bands initiate from positions of maximum shear stress at root. Arrows indicate craze growth directions.

Figure 7, Temperature dependence of failure stresses in Instron three-point bend tests on Vs inch notched Izod bars cut from (a) extruded polycarbonate sheet and (b) compression molded block polymer B. Crosshead rate = 0,02 inch /min. Span = 2 inches, o = net section stress = force/net cross-section at notch root, O, Craze initiation , ductile failure X, brittle failure ...

The mechanism for craze nucleation and growth describai here is essentially possible in semicrystalline polymers since the criterion is only related with a stress field due to plastic constraint. Therefore, the size and geometry of a local plastic zone at the notch root is responsible for the formation of crazes (sometimes named internal crazes by the authors). 

For a brittle glassy polymer like PMMA, the results for both sharp and blunted cracks are reproducible and show negligible scattering. The sharp cracks give a lower estimation of the toughness but this is not only related to a notch root effect. In the case of a blunt crack with notch radius of 250 micrometers, stress induced birefringence related to plasticity is observed... 

Analysis of the data of the fractured specimens reveals that a fracture crack propagated from the points where additional stress concentrations were present. This confirms the assumption that the fracture of a loaded ceramic specimen starts from a small crack ahead of a machined notch root. It is beheved that is influenced more by the sharpness of the notch root, rather than by its shape. The data in Table 2.1 are from three to four-point flexure tests performed on several monolithic ceramics. Table 2.2 shows the Kic values at RT and high temperatures attained by the SEVNB method for these notched specimens. 

This test method parallels the use of F 519, specimen Type le in specimen design and concept. The steel specimen has been modified to have a lower hardness of 51 1 HRC, and the notch root radius is 20-mil instead of 9-mil in order to reduce the stress concentration. The modifications were made with the intent of making the specimen less sensitive because the application was lower hardness fasteners (33-44 HRC) for automotive applications as compared to the higher hardness (45-52 HRC) steel used in aerospace apphcations. 

Figure 4.1 shows sketches of the stress trajectories near differently shaped notches. If we look at the stress trajectories at a cross section at the notch root, we see that they are not evenly distributed, but become more narrow at the notch root. Thus, there is a local stress concentration, with a maximum stress Crnax in the notch root as shown in figure 4.2. The shape and size of the... 

Fig- 4.1. Stress trajectories in notched components. The stress trajectories are aligned with the maximum principal stress, their density is a measure of the stress level. At the notch root, there is a stress concentration in both geometries... 

In the centre of the specimen at the position of the notch, the stress is smaller than the net-section stress (Tnss- The reason for this is that the total force transferred through the cross section does not change and that the stress at the notch root is larger than (Tnss-... 

In the context of notches, the stress concentration is always related to the net-section stress stress concentration at the notch root has to be compared to the nominal stress far away from the notch or, more precisely, (Tnooi infinitely far away from the notch), the increase in stress due to the reduced cross section and the stress concentration at the notch root have to be multiplied. 

To design notched components, knowledge of Ki is required. Therefore, empirical formulae have been determined that can be used to calculate for different geometries and load cases. They are collected in tables e. g., Peterson s Stress Concentration Factors [109] or Dubbel [18]. One example, a shaft with a circumferential notch under tensile load, is shown in figure 4.3. The dimensions in the figure are the outer diameter D, the diameter at the notch root d, the notch depth t (with 2t = D — d), and the notch radius q. 

If the available materials to construct the shaft are a ceramic with i2m = 400 MPa or the aluminium alloy AlSilMgMn with Rpo.2 = 202 MPa and Rm = 237 MPa, we can expect the ceramic to fail because the stress at the notch root is much larger than the tensile strength. For the aluminium alloy, the tensile strength is also exceeded, and we thus might expect its failure as well. However, the calculation is not valid in the case of a ductile material, for equation (4.1) is valid only for a linear-elastic material, whereas the alloy AlSi 1 MgMn yields at Rpo.2 = 202 MPa. This increases the strain at the notch root and reduces the stress concentration. The actual stress at the notch root cannot be calculated with the tools introduced so far. In the next section, we will discuss Neuber s rule that allows to estimate the stresses. 

In the previous section, we defined the stress concentration factor Kt (equation (4.1)) for linear-elastic materials. As the example at the end of the previous section shows, it cannot be used directly for the case of ductile materials, for yielding at the notch root reduces the stresses. In this section, we discuss how the influence of a notch can be taken into account even in ductile materials. 

If cTmax exceeds the yield strength of the material, the material yields at the notch root and Hooke s law is no longer valid. As shown in figure 4.4, this increases e ax compared to the linear-elastic case. The maximum stress <7max, on the other hand, is reduced due to local unloading. Therefore, Kt,o- < Kt,e holds. The numerical values of Kt,o- and Kt.e are still unknown, though. 

Strictly speaking, equivalent stresses (for example, the von Mises equivalent stress) should be used to calculate stresses and strains due to the multiaxial stress state. Furthermore, the equation iLt.cr = Kt,s is only approximately valid in the elastic region because of the transversal contraction caused by the radial and circumferential stresses. For engineering purposes, a uniaxial calculation is sufficient, especiaffy so if we consider the scatter in the material parameters. The multiaxiality of the stress state at the notch root is discussed in section 4.3. 

Fig. 4.5. Qualitative dependence of the stress concentration factors on the load. Sp is the strain at yielding (stress Rp) in the notch root...

If we approximate the stress state at the notch root as uniaxial, the material state must lie on the stress-strain curve measured in tensile tests. This provides another relation between Cmax and max, which are therefore uniquely determined. Graphically, equation (4.5) corresponds to a hyperbola in the a-e space of the stress-strain diagram, since the right side is constant for a given load case. The stresses and strains at the notch root can be found as the intersection of the hyperbola and the stress-strain curve as shown in figure 4.6. 

In reality, the stress state is biaxial at the notch root (the radial stress at the surface is zero), so that there is no difference to the uniaxial case if the Tresca yield criterion is used. If the von Mises yield criterion is used, there is a slight difference which is neglected here. 

As explained in chapter 4, notches cause a stress concentration in a component. Thus, it should be expected that notches also affect the fatigue strength of a component. The stress concentration at the notch root is again described by the stress concentration factor Kt according to equation (4.1) ... 

Whether a growing crack can be stopped in this way depends on how rapidly the stress decreases at the notch root. To quantify this decrease, we define the relative stress gradient at the notch root... 

As we will see below, this is not true an5rmore as soon as linear elastic fracture mechanics can be applied. In this case, the increase of the stress intensity factor due to the growing crack is larger than the decrease of the stress in the notch root, see figure 10.38. 

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