J.P. Parmigiani, M.D. Thouless/J. Mech. Phys. Solids 54(2006)266-287 ""Griffith criterion"for which there is a single value of the critical energy-release rate required for fracture (i.e, Tu=Ti, to one in which fracture occurs only in response to mode-I loading(n>i). The use of Eq (2)in cohesive-zone analyses has been shown to do an excellent job of describing experimental results (Yang and Thouless, 2001; Kafkalidis and Thouless, 2002; Li et al., 2006), and it can mimic mixed-mode fracture criteria for linear-elastic fracture mechanics(LEFM), if the phase angle is defined as y= arctan√乡n/, where y has its usual definition under LEFM conditions of y= arctan(Kn/Kn), and Kn and KI are the nominal mode-II and mode-I stress-intensity factors acting at a crack tip Hutchinson and Suo, 1992) The general forms of the mode-I and mode-lI traction-separation laws used in this study are shown in Fig 3. The mode-I cohesive strength is o, the mode-ll cohesive strength is t, the mode-I toughness is TI, and the mode-lI toughness is Ill. Generalized forms for the traction-separation laws have been used, as the precise shape does not generally have a Mode l Mode ll Fig 3. Schematic illustration of the(a)mode-I, and(b) mode-ll traction-separation laws used for the cohesive. zone model in this paper. Throughout this paper the values of 81/5 and 52/8 were kept at fixed values of 0.01 and 0. 75, respectivel‘‘Griffith criterion’’ for which there is a single value of the critical energy-release rate required for fracture (i.e., GII ¼ GI), to one in which fracture occurs only in response to mode-I loading (GIIbGI). The use of Eq. (2) in cohesive-zone analyses has been shown to do an excellent job of describing experimental results (Yang and Thouless, 2001; Kafkalidis and Thouless, 2002; Li et al., 2006), and it can mimic mixed-mode fracture criteria for linear-elastic fracture mechanics (LEFM), if the phase angle is defined as c ¼ arctan ffiffiffiffiffiffiffiffiffiffiffiffiffiffi GII=GI p , (3) where c has its usual definition under LEFM conditions of c ¼ arctanðKII=KIÞ, and KII and KI are the nominal mode-II and mode-I stress-intensity factors acting at a crack tip (Hutchinson and Suo, 1992). The general forms of the mode-I and mode-II traction-separation laws used in this study are shown in Fig. 3. The mode-I cohesive strength is s^, the mode-II cohesive strength is t^, the mode-I toughness is GI, and the mode-II toughness is GII. Generalized forms for the traction–separation laws have been used, as the precise shape does not generally have a ARTICLE IN PRESS I = d 0 c n ˆ c 0 (a) 1 Mode I ˆ

Γ ∫
2 t c 0 (b) 1 Mode II 2 II = d 0 c Γ ∫ Fig. 3. Schematic illustration of the (a) mode-I, and (b) mode-II traction-separation laws used for the cohesive￾zone model in this paper. Throughout this paper the values of d1=dc and d2=dc were kept at fixed values of 0.01 and 0.75, respectively. 272 J.P. Parmigiani, M.D. Thouless / J. Mech. Phys. Solids 54 (2006) 266–287