120 J. F. Ganghoffer
which appears as a mixture of boundary and volume terms. Expression 20 is compacted as 21
φ [∂ V
r
x] R
∂
Gx, yd y =
φ []
R
Gx, yd y with self-explanatory notations. The left-hand side represents the average flux of internal en-
tropy through the boundary of due to the normalisation coefficient 1 R
∂
G
i
x, yd y, and the right-hand side is the average internal entropy produced in the volume according to the
normalisation coefficient 1 R
G
i
x, yd y. The concept of representative volume at the material point x, V
r
x, arises from the set of all points z enclosed within the volume delimited by the boundary ∂ Sz in 4.9: equality 21
then defines implicitly the representative volume as the set of points interacting with the point x, such that the internal entropy produced within V
r
x is equilibrated by an equal and opposite flux of internal entropy across the boundary ∂ V
r
x. Thus, we further rewrite 21 a 22
φ [∂ V
r
x] R
γ ∈∂V
r
x
Gx, γ dγ =
φ [V
r
x] R
y ∈V
r
x
Gx, yd y .
Since only the contribution to the dissipation due to non-local variables intervene in 21, equality 22 can be rewritten :
R
γ ∈∂V
r
x
{F
d
γ V
d
} exp −
1 2
h
kx −γ
lx ,γ
i
2
dγ R
γ ∈∂V
r
x
exp −
1 2
h
kx −γ
lx ,γ
i
2
dγ =
23 R
y ∈V
r
x
{F
d
yV
d
} exp −
1 2
h
kx −y
lx ,y
i
2
d y R
y ∈V
r
x
exp −
1 2
h
kx −γ
lx ,γ
i
2
d y with the non-local damage driving force F
d
y defined at the beginning of previous section. The concept of a representative volume is then defined via the internal length lx, di st x, y
which connects the point x - centre of the representative volume - and the point y on its boundary, as the following set of points :
V
r
x : = {y ∈ dist x, y ≤ lx, y}
which is not necessarily a sphere. The internal length is an unknown that is determined from equation 23.
The evaluation of the internal length at each time step is then done in a two-step uncoupled procedure : in the first step, equation 23 is solved, using the value of the damage and rate of
damage at previous time step. In the second step, the local and nonlocal damage variables are updated, according to the return mapping algorithm described in [6].
4. Geometrisation of the interaction
The path selection rule has been obtained under the assumption that the metric g
i j
of this space is given; in fact, the influence function Gx, y is a function of the metric, via the internal length.
New concepts 121
We want to reflect the fact that the forces responsible for the nonlocal interaction can be included into the geometry of the interaction, thus we envisage a situation in which the metric is coupled
to the internal variable distribution. As a matter of simplification, define the coefficients
A
x y
z : =
Gx, zGz, y Cz
, that intervene in equation 20. The coefficient A
x y
depends on the metric, and possibly on the first order spatial gradient of the metric, thus we use the notation A
x y
g
i j
z, g
i j,l
z; the differential element of length dsz involves the metric tensor according to the relation 15.
Latin indices take their values in the set {1, 2, 3}. We now perform the variation in the path
integral 18 with respect to the metric, which gives the variation of the term δ
Z
S[x ,y]
A
x y
g
i j
z, g
i j,l
zdsz
= Z
S[x ,y]
δ A
x y
g
i j
z, g
i j,l
z + A
x y
δ dsz
dsz. We introduce the energy-momentum tensor T
i j
, see [8], defined as T
i j
: =
∂ A
x y
∂ g
i j
− ∂l ∂
A
x y
∂∂ lg
i j
and the Christoffel symbols Ŵ
k i j
: =
1 2
g
km
g
j m,l
+ g
im, j
− g
i j,m
such that the covariant derivative Du
j
of the contravariant vector u : =
d x
j
ds
e
j
, locally tangent to the path, expresses as
Du
j
: = du
j
+ Ŵ
j rt
u
r
u
t
ds. A set of elementary calculation [7] renders the variation
δ Z
S[x ,y]
A
x y
g
i j
z, g
i j,l
zdsz =
24 −
Z
S[x ,y]
T
ik,l
+ δ
il
Du
k
ds A
x y
g
ik
δ x
l
zdsz. Since the variations δx
l
z in 24 are arbitrary, we get the following condition, valid at each point z
∈ S [x, y] : 25
T
ik,l
+ δ
il
Du
k
Ds A
x y
g
i j
= 0, ∀l.
Equation 25 is the sum of one term that contains the energy content of the nonlocal interaction due to the energy-momentum tensor and of a second term that accounts for the geometric part
122 J. F. Ganghoffer
of the interaction via the metric tensor and the covariant derivative of the vector tangent to the path.
The energy-momentum tensor reflects the strength of the nonlocal interaction, and we see from the structure of 25 that the induced curvature also has the meaning of a field strength. The
Christoffel symbols Ŵ
k i j
that intervene in the covariant derivative Du
j
are indeed directly related to the - contracted twice - curvature tensor Ricci tensor, defined as
R
ik
: = Ŵ
l ik,l
− Ŵ
l il,k
+ Ŵ
l ik
Ŵ
m lm
− Ŵ
m il
Ŵ
l km
. The scalar obtained by the contraction g
ik
R
ik
represents the scalar measure of the curvature of space. The higher the strength of the nonlocal interaction, the higher the curvature; this idea is
supported by the well-known case of plasticity within solid materials, where a high density of dislocations at a place curves the space around. Therefore, the physical meaning of relation 25
is that the strength of the interaction is incorporated into the geometry of the space. We follow thereby a trend which is nowadays classical in physics, which started with general relativity the
metric tensor plays there the role of the gravitation potential.
The fact that the nonlocal interaction shall follow certain paths in space can be under- stood from qualitative micromechanical arguments : when these defects are not isotropically
distributed in space, but along certain lines instead, their mutual interaction will follow these lines. As a perspective of development of the present work, we can mention the involvement
of such a formalism to treat the more general case of a tensorial-like internal variable, having plasticity in mind.
Figure 1: Splitting up of all spatial paths from x to y.
New concepts 123
Figure 2: Space slicing. A continuous path from dx to dz can be approximated by a sequence of values dz
1
, dz
2
, . . . , dz
n
. This approximation becomes exact if dz
i −1
, z
i
→ 0.
References
[1] D
E
B
ORST
R., S
LUYS
L. J., M ¨
ULHAUS
H. B.
AND
P
AMIN
J., Fundamental issues in
finite element analyses of localisation of deformation, Eng. Comp. 10 1993, 99–121.
[2] B
AZANT
Z. P.
AND
P
IJAUDIER
-C
ABOT
G., Nonlocal damage theory, J. Eng Mech. 113
