Neuber’s rule

If the nominal stress range is indicated by AS, then the nominal strain Ae = AS/E. Then according to Neuber s rule ... 

Petrequin, Roche and Tortel derived a plastic strain magnification factor (similar to the ASME Code Kg factor) for notches using Neuber s rule and cyclic stress-strain curves.This factor is calculated as a function of elastic strain range. 

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. 

If we assume that the net-section stress apss is smaller than the yield strength, we can use Hooke s law Snss = o nss/E to derive Neuber s rule... 

Neuber s rule can be derived as an approximation. In the case of a parabolic notch and under the assumption of a certain simple stress-strain curve, a formula can be derived that simplifies to Neuber s rule in the case of a very sharp notch. For large notch radii, Neuber s rule is a conservative approximation i.e., it overestimates stresses and strains [106]. This conservative property of the rule is valid for most other notch geometries as well [59]. 

Fig. 4.6. Determination of

An important step in local strain fatigue analysis under irregular variation of load with time, as schematized in Fig. 6.16a, is the knowledge of the local notch strain-stress history, as shown in Fig. 6.16c. This, in turn, necessitates the knowledge of the cyclic stress-strain characteristic of the material (see Sect. 1.3.1). Advanced elastic-plastic analysis technique such as finite element computer code is needed or it can be used a simplified Neuber s rule as described in the next section. 

Equation (6.29) is known as Neuber s rule. Recalling the definition of... 

Note that for any given value of stress level cr the hrst member of Eq. (6.31) is a constant, also known as Neuber s constant, so that (6.31) is the equation of a hyperbola. The intersection of the Neuber s hyperbola with the stress-strain curve of the material yields the solution of the elastic-plastic problem. This is schematically shown in Eig. 6.21 where the coordinates of point A = (s.cr) are the notch tip strain and stress, respectively. Moving from monotonic loading to fatigue cycles Neuber s rule applies substituting the theoretical stress concentration factor ki with the notch factor fey (see Sect. 7.1), as suggested by Wetzel [31] in 1968 and Topper et al. [32] in 1969, obtaining what is known as the Neuber modihed equation... 

Fig. 6.21 Graphic representation of Neuber s rule. The working point A is the intersection of Neuber s hyperbola with the stress-strain curve...

Nevertheless, the most significant deviations pertain the field of nominal stresses higher than the yield strength and that of low values of the strain hardening exponent n. For nominal stress lower than yield strength, ffr/ffy < 1, discrepancies are rather contained. The issue of Neuber s rule application to fatigue has involved many researchers and study [31, 32, 38-43] with questions... 

Fig. 6.27 Comparison between concentration factors k and k obtained with FE calculations and Neuber s rule, respectively, vs. the non-dimensional ratio of nominal stress (T to yield strength (jy [38]...

Fig. 6.28 Deviations from Neuber s rule found by Huang in a plate with a central hole for various applied nominal stresses and strain hardening exponents n (modified from [39])...

Fig. 6.29 Elastic strain calculated either with Neuber s rule using the theoretical stress concentration factor and a triaxial cyclic curve or with notch factor kf and uniaxial cyclic characteristic of the material (modified from [42])...

A series of 17 smooth specimens of carbon steel type A 106 B, 6.35 mm (0.25") diameter and 15.24 mm (0.6") length, has been tested in push-and-pull traction (R = —1) at 288 °C under strain controlled conditions. Results are shown in Fig. 6.42. A further testing with notched specimens having k, — 3, root radius p = 0.28 mm and a diameter D — 8 mm with the reduced notch section having the same diameter d — 6.35 mm as smooth specimens, has yielded results listed in Table 6.14. Analyze the results using Manson-Coffin S-N curve and Neuber s rule. 

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