Appendix A
Figure A1: Work of separation (𝜙 𝜙⁄ 𝑛)as a function of normal (∆𝑛⁄ ) and tangential (∆𝛿𝑛 𝑡⁄ ) components of 𝛿𝑡 the interface separation vector for the MP model with q=0.43 and r=0: (a) m=0 (XN model); (b) m = 1; (c)
m=5. Dotted line indicates a path comprising of mixed-mode separation followed by normal separation.
MP model potential surfaces for the case of 𝑞 < 1 (work of mode II separation less than work of mode I separation) are presented in Figure A1. Specifically, it is assumed that 𝑞 = 0.43, as is commonly implemented for the XN model. The case of 𝑚 = 0 (the XN model) is presented in Figure A1(a). Once again a separation path is illustrated in which a mixed-mode separation is followed by a normal separation. As detailed in Section 2.1, residual normal tractions must be overcome during the second (normal) phase of the separation despite the preceding full mixed-mode separation. As illustrated in Figure A1(b) and (c) for 𝑚 = 1 and 𝑚 = 5 respectively, the zone in which residual normal tractions are computed is reduced as the parameter 𝑚 is increased. Hence, for 𝑚 = 5 no residual normal tractions are computed during the second (normal) phase of the separation. Once again it is worth mentioning that the MP model is identical to the XN model in pure mode I and mode II separation.
Figure A2: (a) Tangential traction (𝑇𝑡⁄𝜏𝑚𝑎𝑥) as a function of tangential displacement (∆𝑡⁄ ) for q=0.43 and 𝛿𝑡
r=0 during a mixed-mode separation where 𝑇𝑎𝑛−1(∆𝑛⁄∆𝑡) = 20𝑜. (b) Normal traction (𝑇𝑛⁄𝜎𝑚𝑎𝑥) as a function of normal displacement (∆𝑛⁄ ) representing a normal separation subsequent to the mixed-mode separation 𝛿𝑛 shown in (a). Curves are shown for the XN model and the MP model (m=1, 2 and 5).
Figure A2(a) shows, for 𝑞 = 0.43, the tangential traction-separation curves for mixed-mode separation (20o to the mode II axis) for the XN model and for the MP model with 𝑚 =1, 2 and 5. For all cases, the tangential tractions reduce to zero at a tangential separation of ∆𝑡⁄𝛿𝑡= 3. However, it should be noted that an increase in the parameter 𝑚 leads to an increase in the computed peak tangential traction beyond the peak mode II traction (𝜏𝑚𝑎𝑥). For the case of 𝑚 = 5, a peak tangential traction of 𝑇𝑡⁄𝜏𝑚𝑎𝑥 ≈ 1.7 is computed during the first (mixed-mode) phase of the deformation. The reduction of the residual normal traction zone for high values of 𝑚 results in a strong influence of the normal work of separation ∅𝑛 on mixed-mode separations, even when the mode angle is tending towards mode II. Figure A2 (b) shows the normal traction-separation curves during the second (normal) phase of the separation path when tangential separation is held constant at a value of ∆𝑡⁄𝛿𝑡 = 5. No residual tractions are computed for the MP model for 𝑚 = 5, as the mixed-mode path followed during the first phase of the deformation extends beyond the residual normal traction zone of the potential surface. However, as 𝑚 is reduced, the magnitude and region of residual normal tractions is increased. For 𝑚 = 1 a residual normal traction of 𝑇𝑛⁄𝜎𝑚𝑎𝑥 = 0.3 is computed at ∆𝑛⁄𝛿𝑛 ≈ 2. When ∆𝑛⁄𝛿𝑛 ≥ 4, computed residual normal tractions are not significant. However, for the XN model (𝑚 = 0), residual normal tractions are still evident when the normal separation is increased to ∆𝑛⁄𝛿𝑛 = 6.
Appendix B
Positive instantaneous dissipation is computed for all paths shown in Figure 13. Figure B1 shows the plot of 𝑑𝜙𝑖 corresponding to Figure 16(a) (XN model) and Figure 16(d) (NP2 model), demonstrating positive dissipation throughout. Dissipation for the MP model is also shown.
Figure B1: Normalised instantaneous dissipation (𝑑𝜙𝑖⁄ ) during constant traction controlled mode mixity 𝜓 𝜙𝑛
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