Issue 46

V. Rizov, Frattura ed Integrità Strutturale, 46 (2018) 158-177; DOI: 10.3221/IGF-ESIS.46.16 172 Figure 10 : The strain energy release rate in non-dimensional form plotted against 1 3 / B B s s ratio: curve 1 – for the shaft configuration shown in Fig. 7a (linear-elastic solution), curve 2 – for the shaft configuration shown in Fig. 7a (non-linear solution), curve 3 – for the shaft configuration shown in Fig. 8a (linear-elastic solution), curve 4 – for the shaft configuration shown in Fig. 8a (non-linear solution). Formula (46) is applied to determine * MTL U . For this purpose, * L U and * 0 i L u are replaced with * MTL U and * 0 i MTL u , respectively. Formula (48) is used to obtain the complementary strain energy density, * 0 i MTL u , in the i- th layer of the internal crack arm. For this purpose, * 0 i L u and L  are replaced, respectively, with * 0 i MTL u and  , where  is expressed by (54). * MQ U is obtained by replacing of * L U , 1 n and * 0 i L u , respectively, with * MQ U , 2 n and * 0 i MQ u in (46). * 0 i MQ u is determined by replacing of * 0 i L u and L  , respectively, with * 0 i MQ u and  in (48) and taking into account the fact that the external crack arm is loaded in bending only. Thus, * 0 i MQ u is written as 2 * 0 1 ln ln i i i i i MQ i i i i i i i s s s s u s p p p p p p                         (64) The complementary strain energy cumulated in the un-cracked shaft portion as a result of bending and torsion is calculated by (49). For this purpose, * H U and * 0 i H u are replaced with * MTH U and * 0 i MTH u , respectively. Formula (48) is used to obtain the complementary strain energy density, * 0 i MTH u . For this purpose, * 0 i L u , L  and  are replaced, respectively, with * 0 i MTH u , b  and H  , where H  is found by (38). Finally, * MTL U , * MQ U and * MTH U are added-up and substituted in (44). The result is 1 2 * * * 0 0 0 1 1 1 1 i i i i i i n i n i n MTL MQ MTH i i i b A A A G u dA u dA u dA r                        (65) The integration in (65) should be performed by the MatLab computer program. It should be noted that the strain energy release rate obtained by (65) is exact match of the strain energy release rate calculated by (62), which is a verification of the