A Simple Family of Nonlocal Models

A set of nonlocal models with a common underlying local formulation can be formulated with relative ease, as done by Planas, Klices and Guinea (1993, 1994). These models have the advantage of decoupling the material nonlinearity, involved in the softening curve, from the nonlinearity introduced by the localization. To be specific, w’c select the uniaxial softening model with strength degradation described in Section 8.4.2, for which

(13.1.14)

Подпись:(13.1.15)

(13.1.16)

where in the last condition the equality holds whenever and are both increasing.

We know from the analysis in Section 8.3 that this model leads to strain localization into a zone of measure zero. To avoid this, various nonlocal modifications are possible. The simplest is probably to modify only the last equation (13.1.16) and let it depend on a nonlocal variable П. Since the last equation defines the evolution of strength, we can call this type of model a nonlocal strength model.

There are various ways to include nonlocality in the strength equation. A simple one is to modify

(13.1.16) to read

Подпись: (13.1.17)< <KT)

T is a nonlocal variable defined from the fracturing strain distribution as

T(x) = F[ef(s)x] (13.1.18)

in which T [e^(s); x denotes a spatial operator relating the distribution of inelastic strains to the nonlocal variable. This operator can, in principle, be of the differential or integral type, or of other types.

To see how the general equations are obtained, consider a very long bar (i. e., neglect the end effects) and assume that, upon reaching the peak, strain localization occurs within zone A while over the remainder of the bar unloading takes place (Fig. 13.1.2a). This means that > 0, and a ■– ф(T) for x Є A, and = 0 and a < ф{T) for x С B. Equilibrium further requires that a constant along the bar. Therefore, if we assume, as usual, that after peak the function ф(T) is monotonically decreasing as depicted in Fig. 13.1.2b, the foregoing conditions can be rewritten as

ef >0аікІТ = Тл forze A (13.1.19)

0 and T < Тл for ж Є В (13.1.20)

in which Тл is the constant value that the nonlocal variable assumes in the softening zone. Given Тл, the stress is obviously obtained as a — ф(Тл).

Substituting now T from (13.1.18) in the two last equations, the problem is reduced to the functional equation

Подпись: (13.1.21)

A Simple Family of Nonlocal Models

T [e^(.s); x = Тл for x and s Є A

subjected to the restriction

T [e^(s); x <Yл for x Є В and s Є A (13.1.22)

The solution of this equation yields the distribution of for each Тл, which is the basic problem to solve. Note that appropriate jump conditions at the interface between zones A and В may be necessary to complete the solution. They depend on the type of operator envisaged. Note also that the zone A over which localization takes place is not known a priori, and so it must ensue as a part of the solution. This means that, even if the nonlocal operator T is linear, the overall problem is not.

We tum next to the analysis of three types of operators and their properties. They are all linear operators, and so the localization problem in Eqs. (13.1.21)—(13.1.22) is quasi-linear. In this way, the material nonlinearity, included in the softening function, is decoupled from the localization problem, which sheds light on the mathematical aspects of the problem.

__ __ i-h = -H

(a) i_ В ; Л ; I! J

r x

A Simple Family of Nonlocal Models

Figure 13.1.3 Nonlocal gradient model with harmonic operator, (a) Bar subjected to tension, with a localized zone A and unloading /.ones B. Distributions for T (b), e* (c) and inelastic displacement xJ (d). (Adapted from Planas, Elices and Guinea 1993.)