Discussion: “A Displacement Equivalent-Based Damage Model for Brittle Materials” (Soh, C. K., Liu, Y., Yang, Y., and Dang, Y., 2003, ASME J. Appl. Mech., 70, pp. 681–695)

Abstract

In the article, the damage discussed is the microcracks existing in any brittle material. It is a time-independent theory. However, the authors seem not to know that I published two books: Rock Rheology by N. Cristescu, 1988, Kluwer Academic, 336 pp. and Time Effects in Rock Mechanics by N. D. Cristescu and U. Hunsche, 1998, Wiley, 342 pp. In these books are chapters on damage: in the first, “Damage and Failure of Rocks” and in the second, “Damage and Creep Failure.” The damage I have considered is based on the same idea: increase or decrease of microcracks, with the distinction that I have considered also the hydrostatic tests, which the authors disregard. For instance, in Fig. 1 (Fig. 4.25) (all figures are from the second book) one can see the initial contribution of the hydrostatic contribution on the alumina powder obtained in a three-axial test apparatus. I have considered the phenomenon to be time-dependent. Thus, if you stop everything at a certain level of stress, the strains are increasing in creep. In addition, the authors consider only elastic properties. I have considered inelastic properties with respect to developing damage. Thus, I have considered damage produced by shear. For instance, on Fig. 2 (Fig. 6.11) is shown, in an octahedral plane, various possible pure three-axial tests. A pure hydrostatic test is shown as (b), (c) is a typical true three-axial test. One can see that initially OA are the microcracks closed during the initial hydrostatic test. They are followed by a continuous increase of the microcracks. Only in the last part one is forming other microcracks and the curve is going down. Finally, curve (d) corresponds to a very high initial hydrostatic tests. When the stress state close to the failure curve, the failure is imminent. Departing from this curve and approaching the incompressible domain would increase the time to failure to infinity. The damage rate is defined by the evolution law ḋ(t)=−Ẇv(t)=−kT1−W(t)H(σ) ∂F∂σ σ−kS ∂S∂σ σ if ∂F∂σ<0 and ∂S∂σ<0.where H(T)=∫0Tσ(t)ε̇VI(t)dt+∫0Tσ′(t):ε̇I′(t)dt measures the irreversibility, H(σ) is the yield function, F(σ) is the viscoplastic potential for transient creep, S(σ) is the potential for the steady-state creep, kTandkS are two viscosity constants, and 〈A〉=A if A>0 and 〈A〉=0 if A≤0.