and justifies the reason why strain loading paths in Figs. 1b and 2b
were assumed as linear.
Moreover, it also follows from Eq. (6) that the critical value
of damage DI
crit associated with stress triaxiality and dilatational
changes in voids defines a straight line with slope ‘−1’ falling from
left to right in close agreement with the FFL (fracture forming line,
Fig. 1b) and the condition of critical reduction of thickness at fracture
ε3f = ln
1 − Rf
, where Rf given by (t0 − tf )/t0 with t0 and tf
being the initialthickness and the thickness atfracture. Conversely,
Eq.(7) allows concluding thatthe critical value of damage DII
crit associated
with in-plane shear and distortional changes in voids defines
a straight line rising from left to right with a slope equal to ‘+1’ in
agreement with the condition of critical distortion f along the SFFL
(shear fracture forming line, Fig. 2b).
In connection to what was said above aboutthe fracture loci, itis
worth noting that if the lower limits of the integrals in Eqs. (6) and
(7) are equal to ε0 rather than zero, corresponding to situations
where there is a threshold strain ε0 below which damage is not
accumulated, the FFL and the SFFL deviate from straight lines and
present ‘upward curvatures’ as it is schematically represented by
the dashed solid curves in Figs. 1b and 2b,