Some Alternative Forms and Properties of the Nonlocal Model
The solution of (13.3.9) can be written as:
Д5(,)(х) – Д50ї(х) – [ К(х,£)ТїйЩ$)сІУ(£) (13.3.11)
in which function ЛГ(х,£) is the resolvent of the kernel Л(х, £). (This resolvent could be calculated numerically in advance of the nonlocal finite element analysis, but it would not allow a simple physical interpretation and a closed-form expression.) With the notation
-v-v – Ко (13.3.12)
where Sци = Kronecker delta, Eq. (13.3.8) can be transformed to
The macro-continuum counterpart of this discrete matrix relation is
[ Ф(x,£)AS(l>(£)dK(£) = AS0>(x)’ (13.3.14)
= / ДS<»(£)a(x,0dK(0
which represents an integral equation of the first kind for the. unknown function Л5’0(£). Obviously,
Ф(х,£) = <5(х-£) – Л(х,£) (13.3.15)
where S(x — £) = Dirac delta function in two or three dimensions; indeed, substitution of this expression into Eq. (13.3.14) yields Eq. (13.3.9).
Defining the inverse square matrix:
[Вци] = 1V]"‘ (13.3.16)
we may write the solution of the equation system (13.3.13) as
ASjP = £ BIAS’S0 = J2cnX^sx), ^a-£ (13.3.17)
v А у
with а„л = a(x„, £л)- Т^е macro-continuum counterpart of the last equation is
AS(1)(x) = / B(x,$)AS<0(OdV(0= f С(х,0А5(І)(0^(0 (13.3.18)
where B(xM,£„) – €(BliU)/Vc and C(x,£) = fy B(x,£)a(£,x)dV(£). The kernel B(x,£) represents the resolvent of the kernel Ф(х, £) of (13.3.14). Furthermore,
B(x, d = S(x-()~K(x,() (13.3.19)
because substitution of this equation into Eq. (13.3.18) furnishes Eq. (13.3.11). With (13.3.18) we have reduced the nonlocal formulation to a similar form as (13.3.3) for the previous nonlocal damage formulations (Pijaudier-Cabot and Bazant 1987; Bazant and Pijaudier-Cabot 1989; Bazant and Ozbolt 1990, 1992). However, the presence of the Dirac delta function in the last equation makes Eq. (13.3.18) inconvenient for computations. Aside from that, it seems inconvenient to calculate in finite element codes function B(x, £). Another difference is that the weight function (i. e., the kernel) is anisotropic (and, in the present simplification, associated solely with the principal inelastic stresses).
Note also that if we set Л(х, £) — 0, the present model would become identical to the aforementioned previous nonlocal damage model. But this would not be realistic. The directional and tensorial interactions characterized by Л(х, £) appear to be essential.
Because the nonlocal integral in (13.3.21) is additive to the local stress AS, the present nonlocal model can be imagined as an overlay of two solids that are forced to have equal displacements at all points: (i) the given solid with all the damage due to cracks, but local behavior (no crack interactions); and (ii) an overlaid solid that describes only crack interactions. The nonlocal stress AS represents the sum of the stresses from both solids. This is the stress that is to be used in formulating the differential equilibrium equations for the solid.
For the sake of simplicity, we have so far assumed that the influence of point £ on point x depends only on the orientation of the maximum principal inelastic stress at £. Since at £ there might be cracks normal to all the three principal stresses (denoted now by superscripts г — 1,2,3 in parentheses), it might be more realistic to consider that each of them separately influences point x. In that case, Eqs. (13.3.8) and (13.3.9) can be generalized as follows:
Д5(і)(х) ■ f ^AM(x, Z)ASU)(Z)dV(Z) = ASW(x) (z-1,2,3) (13.3.21)
Similar generalizations can be made in the subsequent equations, too. Note that when the body is infinite, all the present summations or integrations are assumed to follow a special path labeled by ©, which will be defined in the next section.
The heterogeneity of the material, such as the aggregate in concrete, is not specifically taken into account in our equations. Although the heterogeneity obviously must influence the nonlocal properties (e. g., Pijaudier-Cabot and Bazant 1991), this influence is probably secondary to that of microcracking. The reason is that the prepeak (hardening) inelastic behavior, in which microcracking is much less pronounced than after the peak while the heterogeneity is the same, can be adequately described by a local continuum. The main effect of heterogeneity (such as the aggregates in concrete, or grains in ceramics) is indirect; it determines the spacing, orientations, and configurations of the microcracks.