Issue 33

F. Berto et alii, Frattura ed Integrità Strutturale, 33 (2015) 229-237; DOI: 10.3221/IGF-ESIS.33.29 233 the notch tip. Being the radius at the notch tip very small (  less than 0.1 mm), the Mode 1 and Mode 3 notch stress intensity factors K 1 and K 3 can be used to re-analyse the fatigue strength data related to V-notched specimens in terms of the SED. These field parameters were calculated by means of linear elastic FE analysis considering a sharp V-notch with  = 0, see Fig. 2. In particular, considering a cylindrical coordinate system (r, θ, z) centered at the notch apex, where r is the radial coordinate,  is the angle between a particular point and the notch bisector line while z is the longitudinal axis of the specimens, the Mode 1 and Mode 3 Notch Stress Intensity Factors (NSIFs) can be defined according to the following expressions: 1 1 1 0 2 lim ( , 0) r K r r           (3) 3 1 3 0 2 lim ( , 0) z r K r r           (4) In the case of a V-notch opening angle equal to 90 degrees, the eigenvalues  1 and  3 are equal to 0.545 and 0.667 respectively. On the other hand in conditions of linear elasticity the NSIFs can be linked to the nominal stress components according to the following expression: 1 1 1 1 nom K k d       (5a) 3 1 3 3 nom K k d       (5b) where d is the notch depth (d = 6.0 mm) while k 1 and k 3 are non-dimensional factors derived from FE analysis. They simply represent the shape factors, in analogy with the representation of Linear Elastic Fracture Mechanics. The harmonic element solid plane 83 of the Ansys  code was used in the finite element analysis. Taking advantage of the geometric and loading symmetry, it was possible to model only one quarter of the specimen. FE models gave k 1 = 1.000 and k 3 = 1.154. The stress field is controlled by the first singular term (NSIF) up to a distance from the notch tip about equal to 1.0 mm. Substituting the notch depth of the specimens examined here, d = 6 mm, in Eqs. (5a) and (5b), one can obtain: 1 2.260 nom K     (in MPa ·mm 0.445 ) (6a) 3 2.096 nom K     (in MPa ·mm 0.333 ) (6b) Taking into account the range of the nominal stresses at N A = 2·10 6 cycles relating to V-notched specimens tested under pure tension and pure torsion with a nominal load ratio R = -1 (Tab. 1) and substituting them into the Eq. (6a) and (6b), it can be obtained: 0.445 1 2.260 200 452 A K MPa mm      (7a) 0.333 3 2.096 580 1216 A K MPa mm      (7b) In the case of a component weakened by a sharp V-notch and in conditions of linear elasticity, the SED averaged over a control volume, which embraces the notch tip, can be calculated by means of the following expression:     1 3 2 2 3 1 1 3 2 1 2 1 1c 3c 1 K K W e e E R R                    (8) where  K 1 and  K 3 represent the Mode I and Mode III NSIF ranges, R 1c and R 3c are the radii of the control volume related to Mode I and Mode III loadings while e 1 and e 3 are two parameters that summarize the dependence from the V- notch geometry. These parameters are directly linked to the integrals of the angular functions over the control volume and they can be determined a priori by means of closed-form expressions once known the V-notch opening angle. Since the specimens examined here are characterized by a notch opening angle 2  equal to 90 degrees, e 1 and e 3 are equal 0.146 and 0.310 respectively, with the Poisson’s ratio  = 0.3. The use of refined meshes in the close neighborhood of the stress singularity is necessary in the calculation of NSIFs. On the other hand the SED averaged over a control volume is insensitive to the mesh refinement, it can be accurately evaluated also by means of coarse meshes because it directly depends on nodal displacements. Some of the advantages linked to the use of the averaged SED are described in details in Ref. [30].