Abstract :
In their paper, Carmeliet and de Borst attempt to include the effects of heterogeneity on
material response. The paper recognizes that “damage evolution in quasi-brittle materials
is a complex process in which heterogeneity plays an important role.” For this reason,
stochastic distributions of certain material properties are considered. In particular, the
initial damage threshold level K. is assumed to be a random field with a Gaussian autocorrelation
function. The relevant autocorrelation length 0 is the first length introduced in
their work. As correctly (and obviously) mentioned by the authors, the stochastic approach
does not resolve the issue of change of character in the governing differential equation in
the softening regime. For this reason an additional length scale I is introduced by assigning
non-local properties to the relevant damage variable D. The present Letter to the Editor
addresses (a) important inconsistencies present in the formulation, (b) the physical and
mathematical interpretation of the two length scales introduced, and (c) the importance of
surface effects for the type of problems considered. In this perspective, it is shown herein
that the paper introduces a redundant formulation, physically unreasonable, and pinpoints
towards the wrong direction of research in the subject area.
The length scale present in non-local, gradient and viscous continuum theories
represents, in general, the spatial “range” of significant mechanical interactions among
nearby points. A conjugate length may also be considered, for example by assigning nonlocal
properties to 1 -D, instead of D. In any case, material microstructure is decisive on
the magnitude of that length (if this was not the case, then a universal length scale would
exist). Thus in such theories, the relevant length scale I is directly related to the material
microstructure. Heterogeneity is in all pragmatic terms the realization of (micro)structure.
Thus, the two length scales of Carmeliet and de Borst (1995) should be related to each
other, and we show why and how in the sequence.
For demonstration purposes, we consider a (strictly) uniaxial tension problem. As is
known, i.e. Sluys and de Borst (1994), the non-local strain measure, g or the rate & for a
rate formulation, can be expressed, through series expansion as (the notation is similar to
the one in the paper discussed)
