An Integral-Type Model of the First Kind

A simple integral functional was investigated by Planas, Elices and Guinea (1993, 1994). It leads to an integral equation of the first kind. Its solution, surprisingly, can be obtained in a closed form, and it turns out to be a cohesive crack.

In this model, the expression for the nonlocal variable is

T(x) — T (s);xl ее —- f a(|s — x|) £^(s) ds (13.1.26)

Or Jl

in which the weight function a is assumed to be smooth and to have a maximum only at the center. This function is normalized so thata(O) = 1, and Lr is given by the uniaxial version of (13.1.2), which ensues by replacing the volume integral by a simple integral and Vr by Lr. For very long bars (L extending from —oo to +oo) LT = t = characteristic length. Examples of such weight functions are given in Fig. 13.1.4.

Planas, Elices and Guinea (1993, 1994) showed that when a very long bar is considered and the foregoing expression for the functional is substituted into Eqs. (13.1.21)—(13.1.22), the resulting problem accepts a solution consisting of a Dirac’s 6-function:

є1 — w8(x) , with Vi = Тлі (13.1.27)

where we assume the origin of coordinates to coincide with the spike location; w is the displacement jump associated with the 6-function, i. e., the crack opening. Since a — ф{ХА), the foregoing result indicates that the solution of this nonlocal model is physically equivalent to a cohesive crack model with a softening function

cr = f(w) = ф(у) (13.1.28)

Note also the remarkable similitude of the foregoing result and Eq. (8.3.2) for the crack band model.

That the foregoing expression is indeed a solution is easily shown by substituting (13.1.27) into (13.1.26) and performing the integration; the result is

ї(і) = Та o(ij (13.1.29)

which shows that, since a(0) = 1, T = at the origin where єf > 0, and T < everywhere else, as required. Fig. 13.1.5b shows the distribution for the nonlocal variable T; Figs. 13.1,3c—d display the distributions for the fracturing strain and displacement.

Certainly, however, this is not the only solution, at least on pure mathematical grounds. First, the location of the 6-spike is arbitrary. Second, an array of any number of 6-functions is also possible, which is equivalent to having multiple cohesive cracks. However, the principle of minimum second-order work indicates, similar to the localization analysis in Chapter 8, that only one crack will occur in reality. Planas, Elices and Guinea (1994) further showed that if the weight function satisfies very mild conditions, solutions with bounded strains distributed over a finite support are not possible. Therefore, it appears that the single 6-spike is the solution of this simple nonlocal model. This provides theoretical support for the cohesive crack models.

Г7; A__

Подпись: Figure 13.1.5 Nonlocal integral model of the first kind, (a) Bar subjected to tension, with a localized zone reduced to point A and unloading zones B. Distributions for Ї (b), ES (C), and inelastic displacement u! (d). (Adapted from Planas, Klices and Guinea 1993.)

(а) в I ________ в