Non-local damage model with evolving internal length
Gilles Pijaudier-Cabot
1
, Khalil Haidar
1
and Jean-François Dubé
2
1
R&DO, Institut de Recherches en G
!
eenie Civil et M
!
eecanique – GeM/UMR6183, Ecole Centrale de Nantes,
BP 92101, F-44321 Nantes Cedex, France
2
LMGC/UMR5508, Universit
!
ee de Montpellier II, Place Eug
"
eene Bataillon, F-34095 Montpellier Cedex, France
A modified non-local damage model with evolving internal length, inspired from micromechanics, is
developed. It is shown in particular that the non-local influence between two points in the damaged
material depends on the value of damage at each of these points. The resulting weight function is non-
symmetric and truncated. Finite element results and strain localization analysis on a one-dimensional
problem are presented and compared to those of the original non-local damage model. It is shown that in
the course of damage localization, the incremental strain profiles expand according to the modified non-
local model, instead of shrinking according to the original constitutive relation. Comparisons with
experimental data on model materials with controlled porosity are also discussed. Acoustic emission
analyses provide results with which the theoretical model is consistent qualitatively. This model also opens
the path for durability mechanics analyses, where it has been demonstrated that the internal length
in the non-local model should evolve with environmentally induced damage.
KEY WORDS: concrete; damage; cracking; internal length
1. INTRODUCTION
Non-local continuum damage, in an integral or gradient format, is a con sistent general concept
for macroscopic modelling of failure in quasi-brittle materials. The non-local aspect is a
requisite for a realistic description of fracture, including crack inception, crack propagation and
also structural size effect which is a consequence of the exist ence of a finite size fracture process
zone (FPZ).
Among the governing parameters in the non-local damage model, the internal length plays a
pivotal role as it controls the size of the fracture process zone. In most existing proposals, the
internal length is a constant parameter. In fact, one could state that it is already so difficult to
1
measure this parameter with a relatively good accuracy that considering a variation of it may
hardly be en visioned without solid experimental evidences.
There are some theoretical indications, however, which suggest that the internal length should
change in the course of the fracture process. With the help of micromechanics, Bazant [1, 2] has
proposed a description of the interactions between cracks and voids in the course of failure. The
resulting weight function has been derived from statistical analysis by Bazant and Jirasek [3] and
it seems quite natural from these studies that it should evolve during failure. Geers has also
observed that, in a gradient damage model, the internal length should change so that the
continuum-based constitutive relations can describe fracture with a displacement discontinuity
across the crack faces when damage is equal to one [4]. If the introduction of an internal length
in the constitutive relations results from microcracks and their interactions, there is also a simple
reasoning which tells that without micr ocracks the internal length should vanish. In the course
of the degradat ion process, the internal length should evolve.
Recent experiments on model cementitious mate rials with controlled poro sity, i.e. initial
damage, have attempted to capture the evolution of the size of the fracture process zone with
growing damage and growing initial porosity [5, 6]. As we will see in this pa per, acoustic
emission analyses show clearly that the FPZ is widening in the course of its development. An
increase of the width of the FPZ with initial porosity/damage is also observed.
This paper is an attempt to integrate these observations in a modified non-local damage
model. In the first part, an analysis based on micromechanics is presented. It is based on several
simplifying assumptions and provides qualitative arguments about non-locality of damage in
the case where the microcrack density in the material is small. It is shown that the additional
strain induced in the material due to the voids depends on the applie d remote stress, on the sizes
of the voids, and on their distance to the point at which this strain is computed. Experimental
results on model materials with initial porosity are recalled in the second part. Acoustic emission
analyses provide quantitative results on the development of the FPZ, consistent with the
micromechanical analysis and over values of damage that cover a wider range, from diffuse
microcracking to macrocrack propagation. An enhanced non-local damage model is develop ed
in the third part. The non-local influence of one point on another depends on the state of strain
at the first one; it does not exist without damage. The consequences of this enhancement on the
inception of strain localization are considered from theoretical and numerical points of views in
the last section. This non-local damage model should be also capable of capturing the evolution
of the internal length due to the variation of the initial porosity in a heterogeneous material.
2. SIMPLIFIED MICROMECHANICAL MODEL
The objective of the foregoing simplified analysis is to obtain qua litative expressions of non-
local effects induced by the presence of voids in an elastic, homogeneous material. These
expressions deal with the local strains in the material}in between the voids essentially. In the
non-local damage model developed at the end of this paper, we shall consider that it is these
local strains that control damage growth and the non-local effects will be inspired from the
present derivation.
Consider an infinite isotropic two-dimensional solid sub jected to a remote uniform stress field
s
1
: Our purpose is qualitative and we shall consider for more simplicity s
1
as isotropic. Under
this simplification, the considered problem is essentially one dimensional, the strain and stress
2
being volumetric. This solid contains microcracks caused by the loading history that are
distributed in the material arbit rarily. Each crack induces a modification of the local stress and
strain fields. In order to evaluate this perturbation, we assume that the microcracks are far
enough from each other. Under this assumption and as we will see in the next section, the
induced remote stress and strain fields due to a crack are similar to those generated by a circular
void. Therefore, cracks are going to be replaced by circular voids in the foregoing derivations.
The calculation of the perturbation stresses and strains between the voids uses the superposition
theorem. It follows the method proposed by Kachanov [7], although we will deal here with a
much simpler, not to say simplistic, pro blem. Note that cracks or voids are far from each others
and that the perturbation stresses are computed far from them. We look at a material that has
been slightly damaged only. The following derivation cannot be applied to cases where the
material contains a large number of microcracks, or close to material failure when damage
exhibits a localized pattern.
2.1. Interaction between two voids in an elastic material
Consider the medium described in Figure 1. The diameters of the two circular inclusions,
denoted as S
1
and S
2
; are a
1
and a
2
; respectively. We may decompose this problem into two
S
1
S
2
r
a
1
a
2
S
1
S
2
P
2
=
-
P
1
=
-
Sub-problem I
Sub-problem II
σ
∞
σ
∞
σ
∞
σ
∞
σ
∞
σ
∞
=
+
Figure 1. Superposition scheme for an isotropic solid with a two circular voids.
3
sub-problems:
*
Sub-problem I: The solid is considered without any inclusion. It is subjected to the remote
tensile volumetric stress s
1
:
*
Sub-problem II: Inside each inclusion, a distribution of normal pressures P
1
¼ P
2
¼s
1
is
applied.
Superposition of these two sub-problems provides the distribution of stress inside the medium
containing voids.
Consider now sub-problem II. Again, we can apply the principle of superposition (Figure 2)
in order to compute the interaction stress fields due to the presence of each inclusion, and
subsequently the distribution of internal normal pressure inside the voids: in sub-problem II-1
(Figure 2), the inclusion S
1
is considered alone. It is loaded by an unknown normal pressure
%
PP
1
;
which causes on the imaginary location of S
2
a normal pressure p
21
: We have a similar situation
for sub-problem II-2. In order to evaluate
%
PP
1
and
%
PP
2
; we may again use the superposition
principle and write
P
1
¼ P
1
þ p
12
P
2
¼ p
21
þ P
2
ð1Þ
Note that in an exact derivation,
%
PP
1
; p
21
;
%
PP
2
; p
21
should not be normal pressures only. Tangential
distributed forces should also be present. These terms are neglected in our equations, but a
complete derivation, that includes tangential terms, would not change the conclusions at whi ch
we will arrive as far as the mathematical form of the leadin g terms e ntering in the expression of
the pressures are concerned. It would be, however, slightly more complex.
The interaction pressures have the following expressions:
p
12
/P
2
a
2
2
r
2
ð2aÞ
S
1
S
2
S
1
S
2
p
21
1
P
S
1
S
2
p
12
2
P
Sub-problem II-1
Sub-problem II-2
=
+
Figure 2. Superposition scheme in subproblem II.
4
and
p
21
/P
1
a
2
1
r
2
ð2bÞ
where r is the distance between the centres of the two voids. It is the perturbation stress field due
to a circular inclusion loaded by an internal pressure obtained from the well-known Eshelby
solution (see e.g. Ref erence [8]). It is also similar to the long-range asymptotic crack influence
function derived by Bazant [2]. According to our assumptions, the first Equation (2a) holds far
from void 2 and the second holds far from void 1. These expressions are singular r goes to zero,
but they are not correct in such a case since we deal with a diffuse void distribution only.
We can recast Equation (1) into an algebraic system format:
P
1
P
2
"#
¼
1 l
2
12
l
2
21
1
"#
P
1
P
2
"#
ð3Þ
with the notations l
12
/ a
2
=r and l
21
/ a
1
=r: These interaction terms, also called influence
coefficients, are assumed to be constant over each void location. Since P
1
¼ P
2
¼s
1
the
solution of this system of two equations is
P
1
¼s
1
1 þ l
2
12
1 l
2
21
l
2
12
and
P
2
¼s
1
1 þ l
2
21
1 l
2
21
l
2
12
ð4Þ
Since the voids (or cracks) are far away from each other, r4a
1
; a
2
and they are assumed to be
constant over the void inner surfaces. Equation (4) may be further simplified:
P
1
s
1
ð1 þ l
2
12
Þ and P
2
s
1
ð1 þ l
2
21
Þ
ð5Þ
If the tangential forces had been considered, they would be proportional to l
2
12
and l
2
21
on ea ch
void surfaces, respectively, which indicates that the normal and tangential terms that reflect the
influence of interaction are of a similar format, with different pre-factors (angular functions) in
the influence coefficients.
2.2. St ress and strain fields in an elastic material with voids
The extension to the problem of n circular interacting voids S
i
ði ¼ 1 nÞ of diameter a
i
;
arbitrarily distributed in an elastic matrix follows exactly the same technique, with the same
simplifying assumptions (Figure 3). The remote traction s
1
is transformed into distributed
normal forces P
i
¼ðs
1
:n
i
Þ acting inside each void S
i
; where n
i
is the outward normal vector to
the inner contour G
i
of each void S
i
: The effect of the pressure inside the voids S
j
on void S
i
is
written as the sum of the p
ij
ði=jÞ; where p
ij
represents the normal pressure at the imaginary
location of S
i
produced by S
j
:
p
ij
/P
j
a
2
j
r
2
ij
ð6Þ
r
ij
being the distance between the centres of voids i and j: Again, we replace the problem of n
interacting voids by the superposition of n problems in which each void is considered alone in
5
