Issue 47

P. Foti et alii, Frattura ed Integrità Strutturale, 47 (2019) 104-125; DOI: 10.3221/IGF-ESIS.47.09 108         2 2 2 2 1 2 2 1 (1) (1) (1) (1) (1) (1) (1) (1) (1) (1) 1 1 1 , 2 2(1 ) 2 rr zz rr zz zz rr r W r r K E                                            (9)         2 2 2 2 2 2 2 1 (2) (2) (2) (2) (2) (2) (2) (2) (2) (2) 2 2 1 , 2 2(1 ) 2 rr zz rr zz zz rr r W r r K E                                            (10)     1 2 2 (1) (2) (1) (2) (1) (2) 12 1 2 (1) (2) (1) (2) (1) (2) (2) (1) (2) (1) (2) (1) (1) (2) 1 , [ 2(1 ) rr rr zz zz rr zz zz rr rr zz zz rr r r W r r K K E                                                                       (11) In order to evaluate the averaged value of the strain energy density, a sector-shaped cylinder of radius 0 R along the notch tip line, called ‘control volume’, is considered. For more consideration about the shape of the control volume, we remand to ref [25–27]. The control volume radius 0 R depends only on the material and its value decreases with increasing brittleness. In plane problems, both in mode I and mixed mode (I+II) loading, the control volume becomes a circle or a circular sector with radius 0 R respectively in the case of cracks and pointed V-notches, as shown in Fig. 3. Figure 3 : Control volume (area) for: a) sharp V-notch; b) crack. The radius 0 R can be estimated for the crack case under plane strain and plane stress conditions as follows [28–30]: 2 0 (1 )(5 8 ) 4 IC t K R plane strain              (12) 2 0 (5 3 ) 4 C t K R plane stress            (13) While in the case of a pointed V-notch the critical radius can be assessed by [8]: 1 1 1 1 2 2(1 ) 2 2(1 ) 1 1 1 1 0 1 1 4 ( ) 2 ( ) C C C t I K K I R EW                                     (14) The values of  and 1 I depending on  and on the stresses field are reported in Tab. 1. In the case of mode I loading 1 C K equals the fracture toughness IC K .