Models with Nonlocal Strain
The simplest model imaginable is the nonlocal version of the classical linear elastic model. Its uniaxial version simply reads
a(x) — Ііє, є — l’a(s – 2|) £(s) ds (13.1.5)
where a and є are the uniaxial stress and strain. Consider, for the sake of simplicity, an infinitely long bar subjected to uniaxial stress a, and assume that the averaging rule is rectangular, such that a(r) — 1
for |r| < 5/2 and zero otherwise. The equilibrium of the bar requires a to be constant, and so vve need to solve the equation
— / e(s) ds — a (= const.) (13.1.6)
f Jx – tp-
This is an integral equation that accepts as a trivial solution є = а/Е. However, the solution is not unique. Indeed, we can write the general solution as є = a/E + є* and substitute into the foregoing equation to find that the condition to be satisfied by the unknown function є* is
This equation simply states that the mean of the function over any segment of length і is zero. There are infinitely many solutions of this equation since any harmonic function whose wavelength is a submultiple of l satisfies this condition, i. e., any function of the type
is a solution whatever the constants A and В and the nonzero integer n.
It may be shown that many bell-shaped curves also lead to multiple solutions. To avoid this problem, the averaging function a must have the property that its Fourier transform is positive for any wave number (see Bazant and Cedolin 1991, Sec. 13.10). One particular possibility is to lake a weight function with a Dirac 5-spike at its center. Then the nonlocal elastic equation (13.1.5) can be rewritten as
c = 715 є + (1 — y)Eє, є = ~ I as(|s — xl) e(s) ds (13.1.9)
in which the first term comes from the spike, and 0 < 7 < 1; 7 is a constant that measures the relative weight of the spike, and as is the smooth part of the weight function. It is obvious that for 7 = 1 the response is purely local (in which case the elastic solution is unique), while for 7 = 0 the response is purely nonlocal and displays the aforementioned multiple solutions. If 7 is selected large enough, then the multiple solutions can be avoided.
This kind of approach, with intermediate values of 7, can be interpreted as a parallel coupling of a local elastic model with a nonlocal model, in which 7 has the meaning of the volume fraction of local medium. Such a model may also be regarded as a nonlocal continuum model overlaid by a local elastic continuum. The overlay by an ordinary elastic continuum (called the imbricate continuum) was introduced to stabilize the solutions for softening nonlocal continua (Bazant, Belytschko and Chang 1984; Bazant 1986c). However, such an overlay prevents strain-softening from reducing the stress to zero. Other later formulations were proposed, such that this artificial expedient could be avoided.
Before proceeding to other models, we may note that in the foregoing simple analysis the multiplicity of solutions arises because the strain can accept alternating solutions. This is so because the strain can, in principle, take any value, positive or negative. However, if nonlinear ever-increasing variables (such as cumulated plastic work or damage) were to appear in nonlocal equations similar to (13.1.7), then no arbitrary solution could exist, because the average of a nonnegative variable cannot be zero unless the variable vanishes everywhere. This is at the root of the most recent nonlocal models in which the stress and strain are considered to be local, while some nondeereasing internal variable is taken as nonlocal. We will examine a number of models of this kind in the sequel. But before doing so, it is useful to generalize the nonlocal idea of averaging to other kinds of operators, in particular, differential. operators which lead to the so called gradient models.