Installation — business terrible  1 part
September 8th, 2015
In a series of papers, Bazant and PijaudierCabot developed an isotropic nonlocal damage model, whose uniaxial version was thoroughly investigated (Bazant and PijaudierCabot 1988; PijaudierCabot and Bazant 1987, 1988). The underlying local damage model is similar to the damage models analyzed in
Sections 8.4.1 and 8.4.4, which can be rewritten as
ct – (1 — Q)Es (13.1.34)
fi is the damage, a nondecreasing variable that is made nonlocal using adequate flow rules. Bazant and PijaudierCabot used two different sets of flow rules which are equivalent in the local version but lead to slightly different models in the nonlocal version. For the underlying local model they assume, based on energy considerations, that the driving force for the growth of damage is the damage energy release rate У, defined as
dR = 1 FV2 8Q 2
in which U — (1/2)<t£ — (1 /2) (1 – Q)e2 is the elastic energy density. Once the driving force is defined, the evolution of Q is assumed to be described by a unique function of the maximum driving force ever experienced by the material:
n = F(Y) with Y — max(F) (13.1.36)
where F(Y) is a monotonically increasing function of Y. Because F(Y) is monotonic, it turns out that. F[max(y)] = гшх[Р(У)], and thus, on purely nonlocal grounds, the foregoing growth rule is strictly equivalent to writing
fi — max (из) with из — F(Y) (13.1.37)
Although these two formulations are equivalent in the local framework, they lead to two different nonlocal models according to whether the nonlocal averaging is applied to Y in (13.1.36) or to из in (13.1.37). In the first case PijaudierCabot and Bazant (1987) introduced the nonlocal variable Y as
Y(x) — — f a(.s – xj) y(s) ds (13.1.38)
i’r JL
and then modified (13.1.36) to read
f2 = F(y) with У = max(y) (13.1.39)
They called this the energy averaging approach because of the meaning of У. In their second formulation (Bazant and PijaudierCabot 1988) they averaged the intermediate variable w in (13.1.37) (which they called the damage averaging approach). The new nonlocal variable из is defined as
Zo(x) — — I a(s — x) U3(s) ds with из = F(Y) (13.1.40)
‘‘r J L
and then the evolution of Q is defined as
Q — 57 with 57 = max (to) (13.1.41)
Recently, Jirasek (1996) showed that averaging of different variables yields models with very different postpeak responses, and suggested that averaging of the inelastic strain or damage seems to be most realistic.
PijaudierCabot and Bazant (1987, 1988) used the energy average model to investigate dynamic strain localization in a bar subjected to two shock waves traveling from both ends and converging in the center of the bar. The analyses confirmed that the nonlocal formulation does prevent zero measure fracture modes. Furthermore, the computations were shown to be meshobjective. In a further work, Bazant and PijaudierCabot (1988) analyzed the static localization in a bar subjected to tension. Although the complete analysis is globally nonlinear, it is incrementally linear, and the incremental formulation for the initiation of localization takes a form similar to that described in the preceding paragraph, namely, that of a Fredholm equation of the second kind subjected to certain restrictions. To see this, consider the damage average formulation (13.1.41), and assume that the bar is homogeneously deformed up to a point
on the softening branch. We want to analyze the initiation of localization, and so we consider the rate (or incremental) equation derived by differentiating (13.1.34) with respect to time:
E(l – fio)c’o — Ee0Q = it (= const.) (13.1.42)
Here we have set fi = Qq and є — £o – These are the values reached prior to localization, which are, by hypothesis, uniform. Note also that equilibrium requires it to be uniform. Inserting (13.1.40) into (13.1.41) and differentiating, we get
in which F'(Y) = dF(Y)/dY and (•) are the Macauley brackets, equal to its argument if it is positive, or zero if it is negative; the second equality holds because F'(Y) and є are positive (remember that F(Y) is monotonically increasing). Therefore, (13.1.42) can be rewritten as follows:
E( — Clo)i(x) — 2FF'(Y0)Yo J a(s — x) i(s) ds^ = it (= const.) (13.1.44)
This is an integral equation of the second kind in є which, however, is not linear because of the presence of the Macauley brackets. Bazant and PijaudierCabot solved this integral equation numerically for various weight functions and characteristic lengths. They found the typical strainrate profiles shown in Fig. 13.1.7a. They also found that the size h of the localized z. onc was proportional to i, and that the convergence with mesh refinement was fully satisfactory, as shown in Fig. 13.1.7b.
The results for the strainrate profiles are remarkably similar to those found in the previous paragraph for the socalled integral model of the second kind, and Planas, Guinea and Flices (1996) examined whether the two problems were related. It turned out that they were: it suffices to write the problem in terms of an inelastic strain rate єf defined as
(,зл’45)
Solving this equation for є and inserting the result into (13.1.44), the integral equation is transformed into
^ ‘ …………………………….. ~ ‘ .46)
Now, to analyze this problem we can split the bar, as before, into region A in which localization occurs (and hence the expression into angle brackets is positive) and region В in which unloading occurs and
thus £•? — 0 and the expression in brackets is negative. Taking further into account that the stress evolves along the softening branch for which a — — dj < 0, the foregoing equation can be split into two:
ІО" 1 
for x С A 
(13.1.47) 

E{ 1 – 
По) 

И 
for х Є В 
(13.1.48) 

E( 1 
По) 
For very long bars, this system reduces to (13.1.31)(13.1.33) if we introduce the following correspondences
This result, combined with the analysis in the previous paragraph, shows that the relationship between the characteristic length £ and the extent of the localization zone h depends on the characteristics of the softening function at the point where the localization occurs. This was pointed out by Baz. ant and PijaudierCabot (1988), and can now be quantitatively assessed using the plot in Fig. 13.1.6d and the expression for 7 from the preceding formula.
13.1 Consider the bellshaped averaging function defined in (13.1.4), restricted to two dimensions. Determine po so that the value of A,. — J a(s – x)d/l(s) coincide with that for a uniform distribution over a circle of diameter l. [Hint: use polar coordinates to carry out the integral and get po = /3/2.]
13.2 Consider the bellshaped averaging function defined in (13.1.4), restricted to one dimension. Determine po so that the value of Lr – JL a(s – x)ds coincide with that for a uniform distribution over a segment of length l.
13.3 Consider a nonlocal model with a uniform weight function and its high gradient harmonic approximation. Determine the relationship between l and A.
13.4 Consider a nonlocal model with the parabolic weight function defined in Fig. 13.1.4b, and its high gradient harmonic approximation. Determine the relationship between l and A.
13.5 Show that the energy and damage averaging in (13.1.39) and (13.1.41) are exactly equivalent if F(Y) is linear in Y.
13.6 Show that along the softening branch 2F’(Y)Y – (1 – П) > 0 if no localization occurs, and that the denominators in (13.1.49) are always positive.