Elastic-Softening Model with Strength Degradation

Consider now the same basic parallel crack model of Fig. 8.4.2a, but assume that upon unloading the cracks cannot close, as depicted in Fig. 8.4.2c. This is, of course, a tremendous simplification, but frictional grain interlock, as well as debris and surface mismatch, can prevent cracks to a large extent from closing in materials such as concrete. Obviously, the resulting model is of a plastic type with softening such that ef = ep where sp is the plastic strain.

Now we need only to specify that, since we do not consider compression, ef = if at all times and that the monotonic curve cannot be exceeded. As before, we thus have

a – ф(і{) < 0 (8.4.6)

as the plastic criterion. Note that in this case we cannot use the criterion in terms of єЛ Note also that it may seem that making a distinction between ef ands^ is superfluous. However, for the three-dimensional case the distinction will be essential because ef is a second-order tensor, while if will remain a scalar (known as the equivalent uniaxial inelastic strain).