Notch stress state

In order to start the multiscale modeling, internal state variables were adopted to reflect void/crack nucleation, void growth, and void coalescence from the casting microstructural features (porosity and particles) under different temperatures, strain rates, and deformation paths [115, 116, 221, 283]. Furthermore, internal state variables were used to reflect the dislocation density evolution that affects the work hardening rate and, thus, stress state under different temperatures and strain rates [25, 283-285]. In order to determine the pertinent effects of the microstructural features to be admitted into the internal state variable theory, several different length scale analyses were performed. Once the pertinent microstructural features were determined and included in the macroscale internal state variable model, notch tests [216, 286] and control arm tests were performed to validate the model s precision. After the validation process, optimization studies were performed to reduce the weight of the control arm [287-289]. 

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... 

Correlation of results from one test to another for a given material becomes difficult because of different stress states of the specimen and the associated strain rates in different tests. In the tensile-impact test, the stress state is uniaxial and it measures the tensile property at a high strain rate. In Izod and Charpy tests, the presence of notch gives a triaxial state of stress. The falling-... 

Studies of the dynamic crack propagation behaviour of commercial RTPMMA materials [52,53] showed that crack propagation occurred by a stick-slip process. The toughening was attributed to crack-tip blunting by shear yielding. Unfortunately, the use of thin bhmtly-notched specimens means that the stress state in the samples cannot be specified with certain. ... 

Notched laminate analysis predicts the stress state around the hole and evaluates the criticality of these stresses. Stress analyses for notched anisotropic laminates are extensions to the analyses of notched isotropic plates. Specific failure criteria have been... 

Crazing requires the presence of dUatational component in the stress tensor and may be inhibited by hydrostatic pressure. On the other hand, it is enhanced by the presence of triaxial tensile stress (Kinloch and Young 1983). Unfortunately, such a stress state exists ahead of large flaws or notches in relative thick specimens (plane-strain conditions). Therefore, the presence of sharp cracks, notches, or defects in thick specimens wUl favor craze initiation leading to brittle fracture, which is opposite to a bulk shear yielding mechanism that leads usually to ductile behavior. 

The results presented in Fig. 11.21 are very meaningful as they clarify some of the key issues concerning the contribution of void formation to toughness in polymer blends. It appears that the cavitation is extremely important in notched specimens because it allows the blend to yield under plane-strain conditions at stiU moderate stresses due to increased sensitivity to the mean stress. It implies that this modification of yielding does not result from eliminating geometrical constraints and converting a state of plane-strain to plane-stress state, as it has been frequently postulated in the past (Bucknall and Paul 2009). 

Many materials that give low Izod impact values on notched specimens fail in a ductile manner when tested in sheet form. Notched impact strength and the impact properties of sheet are not correlated because the stress states and the material responses differ. 

This must not be confused with the stress state near notches, addressed in chapter 4. If notches are present, the stress state is usually (almost) purely elastic, but here the material yields over the whole cross section. The stress states in the two cases are completely different. 

Loading of the material in a triaxial stress state that keeps Mohr s circle small and shifts it to the right in the direction of the cleavage strength. Such a stress state can be found in the notched bar impact bending test [42]. In components, changes in cross section and notches cause such... 

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. 

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. 

The maximum external force during the test is larger in the notched than in the un-notched case. This is again due to the triaxial stress state and can be explained in the same way as for the onset of yielding above. It has to be kept in mind that the diameter of the un-notched specimen is the same as that of the notched specimen at the notch position. Thus, it should not be assumed that a specimen can be made stronger by notching it. 

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