Admissibility of Uniform Inelastic Stress Fields

In the previous nonlocal formulations, the requirement that a field of uniform inelastic stress and damage must represent at least one possible solution led to the aforementioned normalizing condition for the weight function a. Similarly, we must now require that the homogeneous stress field — ДS’O)

satisfy (13.3.8) and (13.3.9) identically. This yields the conditions that the integral of A(x, £) or the sum of Л,,,, over an infinite body vanish. However, the asymptotic behavior of Л(х, £) forr —> oc which will be discussed later causes this integral or sum to be divergent. Therefore, the conditions must be imposed in a special form—the integral in polar coordinates is required to vanish only for a special path, labeled by ©, in which the angular integration is completed before the limit r —> oo is calculated, that is,

Подпись: A(x,£)dV(€) — lim

Подпись: [° A(x,S)dV(t) J v Admissibility of Uniform Inelastic Stress Fields

A(x, £)rd$ ] dr — 0 (for 2D)

г. ф are polar coordinates; г, в, ф are spherical coordinates. Furthermore, again labeling by © a similar summation path (or sequence) over all the cracks v in an infinite body, the following discrete condition needs to also be imposed;

О

(13.3.23)

I/

This condition applies only to an array of infinitely many microcracks that are, on the macroscale, perfectly random and distributed statistically uniformly over an infinite body (or are periodic). By the same reasoning, for an infinite body we must also have

Подпись: (13.3.24)Подпись: (13.3.25)K(x,£)dK(£):= 0

/°Ф(х, О^Й) – Г B(x, S)dV(S) = [° C(x, OdV(Z) = 1; Jv Jv JV

and in the discrete form

(13.3.26)

For integration paths in which the radial integration up to r –> cp is carried out before the angular integration, the foregoing integrals and sums are divergent.