Introducing Other Inelastic Effects

In the foregoing we have adopted the simplest approach and assumed that the inelastic behavior is com­pletely due to cracking. This allows building models with a minimum of information. Indeed, for the pure damage or pure strength-degradation models all that is needed is the function ф{є^).deduced from a tensile test. For the mixed model, a further function relating ep to is required.

However, this simple approach neglects other sources of inelastic behavior that may take place in the bulk material between the cracks, such as plastic-type strains, creep (viscoelasticity or viscoplasticity), or shrinkage. A simple, yet effective way of modeling more complex behaviors is to relax the assumption of elastic material between cracks implicit in (8.4.1) and allow the bulk to suffer inelastic strains, too. This can be conveniently sketched as a scries coupling of the cracking strain, that we represent by a fracturing element in Fig. 8.4.6, with a bulk element that can be purely elastic as in Fig. 8.4.6a, or include inelastic strains as in 8.4.6b, which are represented by a black-box where we can introduce the desired inelastic

Подпись: Figure 8.4.6 (a) Elastic-fracturing scries coupling, (b) Elastic-bulk-inelastic-fracturing model. (c) Stress-strain curve split in which all the prepeak inelasticity is confined to the bulk, and all the postpeak softening is confined into the fracturing element.

behavior. For example, Bazant and Chern (1985a) used a fracturing element coupled with a viscoelastic element and a shrinkage element. This means that the inelastic strain is now split into an inelastic strain associated with bulk behavior єгЬ and an inelastic strain associated to fracturing є?, that is,

£ ^ ^ + eib + Ef (8.4.15)


Focusing on time-independent models, the bulk inelasticity and the fracturing strain can each be modeled as done in the foregoing analysis in which a single inelasticity mechanism was assumed. Of course, more information is required to model the behavior. In particular, one function фь{єгЬ) is required to describe the growth of the inelastic bulk strain, in addition to the function ф/(єf) that describes the evolution of the fracturing strain.

Introducing Other Inelastic Effects Introducing Other Inelastic Effects Подпись: (8.4.16)

Obviously, the experimental determination of the two functions is very difficult. One particular sim­plifying hypothesis may help to get an easy-to-handle model. It consists in assuming that all the prepeak inelasticity in tension is due to the bulk inelasticity. Then the stress-strain curve in the softening branch may be split as shown in Fig. 8.4.6c, so that the fracturing part would be the only part that has to be scaled according to the size of the element. So, Fq. (8.3.20) is reduced to the simple form

However, this is a split for mathematical convenience only, since most material scientists will agree that cracking (fracturing) starts before the peak. Nevertheless, since the amount of cracking prior to the peak is only a small fraction of the total, the (id hoc split can be justified on practical grounds.

Due to the enormous variety of combinations that arise as soon as one combines the two inelastic strain mechanisms, the following analysis will be restricted only to the fracturing mechanism. The other mechanism can be added as convenient based on classical inelasticity models without softening.


8.18 Consider the uniaxial constitutive equation a — Еєе~Ьє and assume that it unloads to the origin. Determine the evolution law for the damage parameter u. [Answer: и = і — e“be.]

8.19 Consider the uniaxial elastic-softening model defined by an exponential softening curve о =

for monotonic straining, and assume that the unloading is to the origin, (a) Show that the full stress-strain curve can be written as є =*- (1 /Е + C*)o in which is a function of є* — max(e^), and determine this function, (b) Determine the fracturing work supply per unit volume of material 7′, defined as the external work supply density when the fracturing strain increases up toe^ and then the stress is fully released. [Answer: 7^ = f’teo[ 1 — (1 + є*/2єо) exp(—є/єо)-3 (c) Determine 7f, the fracture work per unit volume for complete rupture.

8.20 For the model in the previous exercise, (a) determine the rate of the fracturing work density; (b) show that 7^ — a2Cf/2; (c) generalize the result to any model that unloads to the origin.

Подпись: (c)

Подпись: Figure 8.5.1 (a) Idealized crack band, (b) Detail of crack displacements, (c) Base vectors.
Introducing Other Inelastic Effects

(a) (b)


8.21 Write tbe rate of fracturing work (as defined in the previous two exercises) for a model with stiffness degradation as a function of the damage parameter u.