An Integral-Type Model of the Second Kind
Although the foregoing integral model involves a localization limiter in the sense that the solution for the inelastic strain has a finite measure (i. e., the model gives a finite inelastic displacement, ги, and a finite energy dissipation), the localization still occurs over a segment of vanishing size. Planas, Guinea and Elices (1996) have extended the analysis to include a linear term along with the integral term in (13.1.26). They take the integral operator as
T(x) = T [c^(s);x] _ – ує^(х) + ‘ 1 ^ [ a(|s – xj)e^(s) ds (13.1.30)
in which 7 is constant. Obviously, for 7 — 0 we recover the previous model. Considering again a very long bar in which localization takes place in region A far from both ends as sketched in Fig. 13.1.2a, we have Lr — t. Taking the x-origin to lie at the center of the localization zone, as depicted in Pig. 13.1.2a, we can reduce Eqs. (13.1.21)—(13.1.22) to the following Fredholm integral equation of the second kind:
1 4- -v rh/2
~7c^(x)-l————– — / a(|s – xjjs-^s) ds — T/t fora: C [-h/2, h/2]
subjected to the restrictions
Here it is understood that є?(х) = Oforx 0 [—h/2, h/2]. The integral equation (13.1.31) is a Fredholm equation of the second kind that can be solved for a given h by any of several known methods (see, e. g., Mikhlin 1964; Press et al. 1992). The key point here is that h is not known a priori, but that it has to be obtained as part of the solution, because if h is picked at random the solution will fail to satisfy (13.1.32) or (13.1.33), or both.
Planas, Guinea and Elices (1996) investigated the behavior of the problem both theoretically and numerically. On the theoretical side they showed that, for the solutions with a zero-measure to be excluded, 7 must be positive. They also showed that the solution for must be continuous across the interfaces between the softening and unloading regions. On the numerical side, they investigated symmetric modes of localization by discretizing the bar in equal elements of constant and using point collocation at the center of the elements. The integral was evaluated using a single integration point in the center of each element. A certain value of h was initially assumed and the resulting linear system was solved using standard methods (LU decomposition). It was found that if h was too small, condition (13.1.33) was violated, as shown for one particular case in Fig. 13.1.6a by the full lines, while if h was too large, (13.1.32) could not be fulfilled, as shown in the same figure by the dashed line. The solution was
Figure 13.1.6 Uniaxial nonlocal model of the second kind, (a) Solutions of the integral equation (13.1.31) for too small a value of h (full lines) and for too large a value of h (dashed lines), (b) Complete solutions of the problem for various weight functions, (e) Distributions of fracturing strains for various 7. (d) Influence of the factor 7 on the width h of the localization zone. (After Planas, Guinea and lilices 1996.)
found iteratively, first with relatively large elements (£/12 in size) and then for a refined mesh (element size £/100 to £/1000 depending on the cases). The results can be summarized as follows:
1. The distribution of єf is parabolic in shape and is not very sensitive to the shape of the weight function ft, as shown in Fig. 13.1.6b, in which the distributions for the ft-fund ions-in Figs. 13.1.4a, b, and d are compared for 7 — 1. We see that h varies only between 1,3£ ami 1.6£, approximately.
2. The width h of the softening zone is very much influenced by the value of 7, as shown in Fig. 13.1,6c. Indeed, since the exact solution for 7 = 0 is known to be the Dirac 6-function for which h — 0, we must have h -+ 0 for 7 —> 0. For the cases investigated by Planas, Guinea and Flices (1996) the asymptotic relationship is of the power-type: h cc 7m( where m is of the order of 1/3, as shown in Fig. 13.1.6d.