Consider an uniaxial model in which a certain scalar variable —for example, the uniaxial strain— is assumed to be nonlocal, as given by the second of (13.1.5), and assume further that the function a is a rectangular or bell-shaped function as sketched in Fig. 13.1.Id. If we further assume that the bar is
very long compared to l, so that the averaging integral would extend from — oo to +co, then, setting и — s — x, we may rewrite the expression for the nonlocal variable as
a(juj) є(х + и) du
If the local variable є is assumed to be smooth and varying slowly over a segment centered at x in which a is different from zero, we can approximate ф + u) by a truncated Taylor power series expansion about point x. Thus, we get the following expansion:
Ф) = є(х) + —(х)Єіи
in which Ці are the dimensionless moments of the weight function, defined as
Since a is even, the odd moments are equal to zero, and only the even moments need to be retained in the foregoing expansion. In cases in which the local variable (є in this example) varies slowly over the length і (and thus can be approximated by an arc of second-degree parabola), a two-term expansion is a good approximation of the nonlocal variable. Therefore, setting /ф2/2 — (A/27t)2, wc get
as proposed by Bazant (1984b, Eqs. 44, 55, 64,70 and 73).
We thus see that, under certain smoothness conditions, the nonlocal integral operator can be approximated by a differential operator involving even-order gradients. For the second-order case, the differential operator reduces to the harmonic operator in (13.1.13).
The harmonic operator as well as fourth-order differential operators have been proposed to describe materials with softening. They have the advantage of leading to differential equations which are easier to treat analytically and numerically than the integral equations posed by the full nonlocal approach (following Bazant 1984b, such models, called second gradient models, have also been introduced by Miihlhaus and Aifantis 1991, de Borst and Miihlhaus 1992, and others).
It is easy to show that the gradient approaches also display multiple solutions if the nonlocal variable can take arbitrary positive or negative values. We thus turn attention to the nonlocal models (including their gradient approximations) based on assuming that the nonlocal variable is one irreversible (nondecreasing) internal variable.