118 J. F. Ganghoffer
which is nothing else than the influence coefficient Gz
1
, z when z
1
is replaced by z
2
and ǫ by 2ǫ defining l0 :
= 1. The integration over z
1
has been performed over the whole space of reels, i.e. over ]
−∞, +∞[, whereas the solid occupies a finite volume in space : it is thought however that this is a valid approximation when the length of the interactions is much smaller
than the global dimension of the body. Repeating this process then leads to 16
Gx, z =
1 √
2 lim inf
ǫ →0
lnǫ ln
− 1ǫ exp
− k
2lnǫ
2
x − z
2
. The continuous case in 16 is recovered when taking the limit of the discrete influence coeffi-
cient in the following way : the length of the segments tends to zero, thus n → ∞, ǫ → 0,
nǫ → dist x, z, and the internal length becomes lnǫ → ldx, z: we obtain the final
expression 17
Gx, z =
1 √
2 exp
− k
2ldi st x, z
2
x − z
2
. When a more complete expansion of the function Bx, z is retained, we believe that a closed
form of the kernel Gx, y is much more difficult to derive. The nonlocal damage is further defined as
dx : =
1 R
Gx, yd y Z
Gx, ydyd y using the kernel determined in 12.
The differences with the more traditional approaches, e.g. the nonlocal damage model in [2] are: the internal length is a function of both the material point considered, and on the dis-
tance between the points x and y, whereas the traditional models assume it is a fixed, uniform quantity. In 17, lx, y is not defined, thus a complementary rule is still needed; this rule will
be elaborated in the next section. Note furthermore that in 13, the normalisation condition
∀y ∈ , dy = 1 ⇒ ∀x ∈ , dx = 1 is satisfied.
The path integral formulation can further be interpreted in the following way : consider a path S joining x and z, and make a partition of the space in all sets of possible such paths.
Then, for a fixed point y, the integral 14 can be formally expressed as a sum over all possible paths of the damage rate convected along each path with an amplitude equal to the influence
coefficient K x, z. The set of paths that effectively contribute to the damage rate at a given point shall be selected from a thermodynamic criterion elaborated in the next section.
3. Selection rule for the path
We first rewrite formally the dissipation as an integral involving a product of - local - thermody- namic forces F
i
and associated fluxes V
i
in the more condensed form 18
φ =
1 R
G
i
x, yd y Z
F
i
y.V
i
yG
i
x, yd y in which the summation is intended over the index i . A specific kernel G
i
x, y is associated to each dissipative mechanism, and it has the form established in the previous section. Considering
New concepts 119
nonlocal damage coupled to elasticity, the fluxes are the rate of irreversible strain and the rate of damage with a minus sign, and the associated thermodynamic forces are the local stress and the
damage driving force, respectively : according to 17, we can identify the force and the kernel associated to the irreversible strain as
F
σ
y : = σ y;
G
σ
x, y : = δx − y
and the force conjugated to the damage rate splits into a local contribution F
l d
y : = W
e
y ;
G
F
l d
: = δx − y,
and a nonlocal contribution F
d
y : = β1 − dy,
in which the kernel Gx, y has the form defined in the previous section. We then perform the variation of the integral in 18, considering that x is a fixed point in space :
δφ =
1 R
G
i
x, yd y Z
δ {F
i
yV
i
yG
i
x, y } dy −
19
− δ
R
G
i
x, yd y R
G
i
x, yd y
2
Z
{F
i
yV
i
yG
i
x, y } dy.
We first work out the term R
δ {F
i
yV
i
yG
i
x, y } dy , which is rewritten as the path integral
Z
δ {F
i
yV
i
yG
i
x, y } dy =
Z
F
i
yV
i
yδ X
z ∈S[x,y]
Z
S[x ,y]
G
i
x − z
G
i
z, y C
i
z dsz
d y
where the summation defined by the symbol P
is performed over all paths S [x, y] that connect points x and y. The variation in 18, 19 is performed with the quantities at the extremal points
x and y considered as fixed. Furthermore, in a first step, we consider the metric of space as a given quantity, at each point along any path.
As a matter of simplification, we set up Sz : = S [x, y]; the stationarity condition of the
internal entropy condition, δ
d S
i
dt
= 0 , is finally expressed into the condensed form [7] Z
G
i
x, yd y Z
y ∈∂
F
i
yV
i
yd y X
Sz
Z
γ ∈∂ Sz
G
i
x, γ G
i
γ , y
C
i
γ dγ
20 −
Z
y ∈∂
G
i
x, yd y Z
{F
i
yV
i
yG
i
x, y } dy = 0
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
