Issue 46

V. Rizov, Frattura ed Integrità Strutturale, 46 (2018) 158-177; DOI: 10.3221/IGF-ESIS.46.16 165 Figure 4 : Non-linear    diagram. The strain energy cumulated in half of the shaft as a result of the torsion is obtained as T TL TH U U U   (35) where TL U and TH U are the strain energies in the internal crack arm and the un-cracked shaft portion, respectively. The strain energy in the internal crack arm is written as 1 0 1 i i i n TL TL i A U a u dA      (36) where 0 i TL u is the strain energy density in the i -th layer as a result of the torsion. In principle, the strain energy density is equal to the area, OPQ , enclosed by stress-strain curve in Fig. 4. Thus, formula (18) can be used to obtain 0 i TL u . For this purpose, 0 i FL u , L  , i s and i p are replaced, respectively, with 0 i TL u ,  , i f and i g , where  is expressed by (28). The strain energy cumulated in the un-cracked shaft portion as a result of the torsion is expressed as   0 1 i i i n HL TH i A U l a u dA      (37) where the strain energy density in the i -th layer, 0 i TH u , is obtained by (18). For this purpose, 0 i FL u , L  , i s and i p are replaced, respectively, with 0 i TH u , H  , i f and i g . Here, the distribution of the shear strains is written as H q r R    (38) By substituting of (22), (35), (36) and (37) in (21), one obtains 1 0 0 1 1 1 i i i i i n i n q m III TL TH i i b b b A A T G u dA u dA r r R r                               (39) The total strain energy release rate, G , is written as II III G G G   (40)