TLS - HPC

From Damage to Fracture with level sets
The Thick Level Set model for damage (TLS)
Nicolas Moës
Ecole Centrale de Nantes,
GEM Institute UMR CNRS 6183, France
Contributors
•
•
•
•
•
•
•
Claude Stolz (Director of Research, CNRS)
Paul-Emile Bernard (Post-Doc , FNRS, Belgium)
Nicolas Chevaugeon (Assistant Professor, ECN)
Laurent Stainier (Professor, ECN)
Alexis Salzman (Senior Engineer, ECN)
Fabien Cazes (Post-Doc)
Kevin Moreau (Phd student)
TLS lectures outline
•
Motivations : X-FEM limitation, why local damage
fails, need of a material length, need of a crack
•
•
•
•
TLS key ingredients and theory
•
Perspectives
Application : quasi-static brittle fracture
Application : dynamic brittle fracture
TLS advanced: coupling local and non-local
damage evolution
Motivations
•
X-FEM allows to simulate propagation of existing
cracks
•
We want more : simulate the full scenario : diffuse
damage - localization - crack initiation and growth
•
This is currently not possible routinely from the
industrial point of view
•
Two reasons for this, to our opinion :
•
•
Non-local models are currently too expensive
Even if they were affordable, robust crack
placement is needed (displacement jump) in
damage zones.
Why do local damage fail ?
• Paperboard example on a 1D with a elastodamage model
• It is shown that damage may go to 1 at
some place without any dissipation in the
system.
Limit stress,
the need for a length

c
G  Gc
?
(EGc )
lc ⇠
( c )2

c /3

c
Limit stress (cont’d)

c
G  Gc
?
(EGc )
lc ⇠
( c )2

c /3

c
The need for a better nonlocal damage model
•
Fracture mechanics alone cannot handle crack initiation
and branching
•
Damage mechanics alone without length scale makes no
sense once the stress strain relation reaches a peak.
•
Non-local damage model alone should be OK, but
•
•
•
Non-local models are currently too expensive
Even if they were affordable, robust crack placement is
needed (displacement jump) in damage zones.
The Thick Level Set model for damage is a non-local
damage model solving these two issues
Non-local Damage Models








Integral approach: the damage evolution is governed by a driving force which is nonlocal i.e. it is the average of the local driving force over some region: (Bazant,
Belytschko, Chang 1984, Pijaudier-Cabot and Bazant 1987).
Higher order, kinematically based, gradient approach involving higher order
gradients of the deformation: (Aifantis 1984, Triantafillydis and Aifantis 1986, Schreyer
and Chen 1986) or additional rotational degrees of freedom (Mühlhaus and Vardoulakis
1987).
Higher order, damage based, gradient models: the gradient of the damage is a
variable as well as the damage itself. This leads to a second order operator acting on
the damage: (Fremond and Nedjar 1996, Pijaudier-Cabot and Burlion 1996, Peerlings,
de Borst et al 1996, Lorentz et Andrieux 1999, Nguyen and Andrieux 2005).
Generalized continua, micro-morphic approach Forest (et al.) 2006
Variational approach of fracture: (Francfort and Marigo 1998, Bourdin, Francfort and
Marigo 2000, Bourdin, Franfort and Marigo 2008)
Phase-field approach emanating from the physics community: (Karma, Kessler and
Levine 2001, Hakim and Karma 2005) and more recently revisited by (Miehe,
Welschinger, Hofacker, 2010).
Peridynamics Silling 2000
Comparison papers : Peerlings, Geers et al. 2001, Lorentz et Andrieux 2003
These models apply non-locality everywhere and at all time
These models do not address the crack placement issue
Why is a crack needed?
•
Displacement jump is part of the model:
fragmentation, cutting, hydraulic fracturing,
leakage, ...
•
•
•
•
Avoids element distorsion (ductile fracture)
Allows mesh coarsening away from moving tips
«Better algorithmic convergence».
References on crack placement : Perego et al,
Cesar de sa et al, Feldt-Payet et al, RodriguezFerran et al, Oliver et al...
Specifications for a new model
• Avoid ad hoc crack placement
• Avoid cohesive model
• Avoid using non-local everywhere and at all
times (we want a localization limiter and
not introduce a length everywhere at any
time)
The issue is not (only) numerical, it is also theoretical.
Need for a new model.
Fracture Mechanics
Energy
State Law
Dissipation
Z
W (u, a)
@W
@a
G=
Gȧ
Evolution law ȧ =
@
Damage Mechanics
w(u, d)
⌦
Y =
Z
Y d˙
@w
@d
⌦
⇤
(G)
@G
d˙ =
@
⇤
(Y )
@Y
d˙
Initiation ?
ȧ
Crack placement ?
The Thick Level Set Model (TLS)
Local and non-local damage zones
⌦
+
Non-Local
Local
d = d( ), kr k = 1
⌦
d = d( ), kr k < 1
= lc
⌦
=0
⌦+
c
Figure 2:
⌦c
c
Local (⌦ ) and localization (⌦+ ) domains as well as cracks faces ( c ). Inside ⌦+ , we
k = 1. It can be noted that r (and thus rd) is discontinuous along the dashed line
he so-called skeleton of the distance function.
Ri
Ri + L r
Ri + l
Local and non-local
Figure 3: Notations
for the pull out test.
Damage
,
|r | = 1
|r | < 1
0
|r | = 1
⌦+
R
r
⌦
˙
˙
D ˙ [ ] = 0 =) [ ] = r · [r ]
Sketch of the variation of
dening.
along the radius when the damage mo
Fracture
TLS Damage
Damage
W (u, a)
Z
Z
@W
@a
G=
ȧ =
⌦
⇤
(G)
@G
@
g=
Z
Gȧ
@
w(u, d( ))
R
w
⌦
@
g˙
0
˙=
@
⇤
(g)
@g
w(u, d)
⌦
Y =
Z
@w
@d
Y d˙
⌦
@
˙
d=
⇤
(Y )
@Y
Thermodynamics: The Free Energy
• Free energy and local state laws
In the implementation, a more complex free energy is used to take
into account disymetric behavior in tension and compression
• Global potential energy
Thermodynamics: The Free Energy
•
There exist a configurational force associated to the front movement
Appendix A for details:
¯˙
d (s, t) =
R l(s,t)
0
d˙ (z, t) J (z, t) dz
R l(s,t)
0
J (z, t) dz
e 2. A damage band, the curvilinear coordinate system attached to it, the length l(s, t) (the upper b
e interval we integrate over in (21) and (22)) and the local and non-local energy release rate Y an
e non-local fields appear to be some average of their local counterpart in the direction o
set function gradient. Note that definition of the non-local energy release rate introduc
ional weighting function through the first derivative of the damage profile. An evolution m
Thermodynamics: Dissipation
• The dissipation is given by
=
•
Z
Y d˙ d⌦
⌦oc
We observe a duality between the configurational
force g and the front velocity vn and also a duality
between the non-local Y and the non-local d˙
Rl
0
Y d ( )(1
0
Y = Rl
0 ( )(1
d
0
⇢0 ) d
⇢0 ) d
Rl
˙
d(1
0
˙
d = Rl
(1
0
⇢0 )
d
)
d
⇢0
The driving force for damage is local when there is not
yet any damage -> no wrong location of damage initiation
Similarities and differences with the nonlocal integral approach
Pijaudier-Cabot, Bazant 1987
In the TLS model, the length over which averaging is performed
in not constant but evolving in time starting from zero.
Also, crack placement is automatic.
Evolution laws
• The local evolution law is reused with the
non-local quantities
d˙ =
• The
@
⇤
(Y )
@Y
!
d˙ =
@
⇤
(Y )
@Y
update of the front is based on the
following sequence
!
Y
Y d ( )(1
⇢0 )
Y
!
local - >non-local
Rl
0
Y = Rl
0
0 ( )(1
d
0
⇢0 )
d
d
d˙
!
d˙
non-local -> local
0 ˙ 0
0 ˙
˙
d = d d/d = d
Fracture
W (u, a)
@W
@a
G=
ȧ =
Z
w(u, d( ))
⌦
Y =< Y >
Z
Gȧ
@
TLS Damage
⇤
(G)
@G
Y d˙ d⌦
⌦
d˙ =
@
⇤
(Y )
@Y
Damage
Z
w(u, d)
⌦
Y =
Z
@w
@d
Y d˙
⌦
@
˙
d=
⇤
(Y )
@Y
Numerical aspects
•
•
Staggered explicit approach :
•
For known damage, elastic fields are computed (nonlinear problem in
general) giving Y and Y
•
•
•
•
•
Computation of the local and non-local update of damage
Localization front between the local and non-local zone is moved
(In quasi-static analysis : Load is computed (dissipation control))
Crack location is updated, iso-lc
New front is inserted (if norm of grad phi above 1)
Technical aspects
•
•
•
•
Computation of the average Y
X-FEM Ramped Heaviside enrichment
Double cuts on elements
Adaptive octree grid
Example : Brittle Time-Independent model with no
hardening
Local
d˙ =
@
⇤
(Y )
,
@Y
⇤
(Y ) = I (Y
Yc )
Non-Local version
d˙ =
@
⇤
(Y )
@Y
Y  Yc ,
d˙
0,
(Y
Yc )d˙ = 0
Variational Principle for Y
Z
Z
0
⌦oc
d
YY⇤+@
lc
@ Y@
⌦oc
⇤
d⌦ +
@ Y ⇤ d⌦ =
Z
Z
µ
⇤
Z
⌦oc
d0
Y Y ⇤ d⌦
lc
8Y ⇤ 2 Y
8
d =0
o
µ⇤ d = 0
⇤
2L
8µ⇤ 2 M
o
Y
Y
Moes et al. IJNME 2011
Brazilian Test
Figure 20: Domain definition and boundary conditions for the brazilian test.
Figure 20: Domain definition and boundary conditions for the brazilian test.
Figure 21: The solid black line corresponds to the automatic initiation procedure, based on local criterion
Y = Yc . For the dashed blue line, a single centered damage has been forced and no other damage initation
was authorized, as often computed in the literature. A severe snap-back is observed in both computations
Figure 21: The solid
blackfor
linethe
corresponds
the automatic
initiation
procedure, under
based on
criterion
on the load-displacement
curves
braziliantotest.
The vertical
displacement
thelocal
load
U and theFigure 22: Norm of the displacement field for the brazilian test. The bold black line is the iso-zero delimiting
Ramped Heaviside enrichment
Iso-0
Iso-lc
No Enrichment
Enrichment
There may be more than one enrichment
needed on the support (two needed below)
Three independent pieces
on the support
Iso-lc
Double cut of the cracked
elements on iso-lc
Displacement
Double Cut Algorithm
• The information given by the level set (scalar
distance function if not enough).
• Cuts should be allowed even if positive values
• Vector distance is needed
+
+
+
+
+
+
-
+
Figure 2: Two edges with similar vector distance information but with dif-
Figure 1: Two edges for which the level set is positive atferent
bothcutting
vertices.
scenario. Thanks to the level set sign, the two cases may be
+
Figure 2: Two edges with similar vector distance information but with different cutting scenario. Thanks to the level set sign, the two cases may be
distinguished.
Double cut algorithm
x2
c2
•
c1
New edge cut algorithm
based on level set sign
and vector distance
function
d1
x1
-/0
+
•
New element cut
algorithm
d2
-/0
Figure 3: An edge for which both vertices are outisde the fully damaged
zone (the level set is strictly positive). This case leads to 0 or 2 cuts on the
edge. The cuts may be located at the same location. In the picture above
two cuts are created.
-/0
-/0 +
+
+
4
+
+
+
+
+
+
+
+
+
+
3D cases (no degenerate)
All 3D cases (with degen.)
All 3D cases
(with degen. and permutation)
Implementation geometrical 3D test
Example of cutting
Adaptive Octree Grid
Fine grid everywhere
Fine grid only where needed
Implementation on the way
Defect evolving in shear loading
3D Bending of a reinforced
beam (without double cut)
3D example
three point bending with double cut
Displacement field
Deformed shape
Undeformed shape
Note that the displacement field is clearly discontinuous
and gives crack opening
Emmental
3D corner
Other movies, please
d
conditions:
thea constitutive
Dirichlet boundary
conditiondenoted
ud andby
thefunctionals
Neumann Fboundary
condition
n.
We
provide
model,
formally
=
F
("
(u) , d)
placement
u
,
the
initial
velocity
u̇
and
the
initial
damage
d
.
We
also
impose
two
0
0domain frontier such that @⌦
0
ctively
on
parts
⌦
and
⌦
of
the
[
@⌦
=
@⌦
and
D " is the
N small strain tensor. We impose three initial
D
N
F
("
(u)
,
d)
,
where
conditions:
the
d
conditions:
the
Dirichlet
boundary
condition
u
and
the
Neumann
boundary
condition
d boundary value problem:
⌦
=
;
.
It
leads
to
the
study
of
the
following
initial
N
placement
u̇0 domain
and the frontier
initial damage
d0 .@⌦
We also
impose
two
0 , the
ctively
on u
parts
⌦Dinitial
and velocity
⌦N of the
such that
[
@⌦
=
@⌦
and
D
N
the Dirichlet
boundary
condition initial
ud andboundary
the Neumann
boundary condition
⌦conditions:
=
;
.
It
leads
to
the
study
of
the
following
value
problem:
N
⇢ü = r ·
ctively on parts ⌦D and ⌦N of the domain frontier such that @⌦D [ @⌦N = @⌦ and
>
=
in ⌦ ⇥value
T problem:
(1)
⌦N = ;. It leads
to
the
of the following initial boundary
⇢ü = rstudy
·
= F> (" (u) , d)
=
in ⌦ ⇥ T
(1)
⇢ü = r ·
d˙ =
=F
F>d ("
(u)
,, d)
("
(u)
d)
=
in ⌦ ⇥ T
(1)
˙=
d
= FFd ("
("(u)
(u),,d)
d)
conditions:
d˙ = Fd (" (u) , d)
conditions:
u = u0
onditions:
u̇ = u̇0
in ⌦ ⇥ {0}
(2)
u = u0
d = d0
= uu̇00
in ⌦ ⇥ {0}
(2)
uu̇ =
= u̇d00
in ⌦ ⇥ {0}
(2)
dary conditions:u̇d =
d = d0
dary conditions:u = ud
on @⌦D ⇥ T
(3)
dary conditions:· n = T d
on @⌦N ⇥ T
u = ud
on @⌦D ⇥ T
(3)
· un =
=tensor
on@⌦
@⌦
is the Cauchy stress
The
aims
ofTT
the TLS is to provide
uTdd and n is the outward normal. on
(3)a
DN⇥⇥
ve model that ·isnconsistent
with regard to the localization
model
= Td
onproblem.
@⌦N ⇥ TThe constitutive
Cauchy
stress
tensor
and potential
n is the outward
The aims
of the ⇤TLS
to provide
ys the
defining
both
a free
energy
' and anormal.
dissipation
potential
andisapplying
ona
Application: Explicit Dynamics
⇤
Ȳ =
c
1
1
exp
a
1
(92)
⌧c
Yuc (x, t)
a
Y
are the displacement field
(x, t). We have to
+ and the level set c function
+
two
partial
equations:
orderinitiates
balance
momentum
where
Yc is differential
the critical energy
release ratethe
oversecond
which damage
and of
propagates,
⌧c is theequation
characteristic
and a is an homogeneous
parameter
that
influence
the
rate at order
which the
damage central
rder
level settime
propagation
equation
(26).
We
apply
the
second
accurate
rate tends toward ⌧1 . Operators h·i+ and h·i respectively stand for the positive and negative part of
ntegration
scheme to the first one and the first order accurate forward Euler method
a quantity:
e, leading to:
Explicit time integration
c
1. for a scalar:
k+1
u
k+1
k
=u +
=
k
2. for a 2nd order tensor:1
ük+1 =
u̇
k+1
⇢
+
r·F
1
= u̇ +
2
k
k hxi+1=
tu̇ +
tF
1
t2 2(xü+k |x|)
2 1
hxi = k+1
(x |x|)k
2
" u
,
" uk+1
,d
3
h"i+ =
k
X
(93)
(28)
(94)
(29)
(30)
k+1
h"i i+ v i ⌦ v i
i=1 k+1
t ü +
ü
3
X
h"i =
h"i i v i ⌦ v i
(95)
(31)
(96)
straightforward
way tomodel
discretize ini=1time the TLS approach. Both schemes are
Physical
where
"i are if
eigenvalues
and v iare
are eigenvectors
of the tensor.
main
stable
time steps
upper bounded
by Ita reads:
critical value tc . Note that the
k
is fully explicit,
model
itself
is
computed
from
. Knowing
= (1since
d) the
htr"ievolution
I
+
2µ
h"i
+
(1
hd)
h
tr
"i
I
+
2µ
h"i
(97)
+
+
✓ k
◆
k+1
k+1
t field u
and
set
function
,
we
deduce
the
strain
field
"
from
1the level
1
2
2
Y
=
h
tr
"i
+
µ
h"i
:
h"i
+
h
h
tr
"i
+
µ
h"i
:
h"i
(98)
+
+
+
k
2
2 energy release rate Y k+1 by applying
age field d from (9)
and
derive
the
local
✓
✓ ⌧
◆◆
¯˙k+1
1
Ȳ k+1
˙
d = non-local
1 exp fields
a
(99) deduce
e then compute
Ȳ 1 from (21) and d
from (24) and
⌧c
Yc
+
k+1
+k
˙
ction rate field
from (26) only in domain ⌦ . We then extend the level set
where (99) is a delay-damage evolution model [4].
Kalthoff experiment
Kaltho↵ and Winkler experiments.
Figure: Geometry for the Kaltho↵ and Winkler plate.
EXPLICIT DYNAMICS WITH A NON-LOCAL DAMAGE MODEL USING THE TLS APPROACH
damage
0
0.5
damage
Y
1
Z
X
0
(a) t = 30.09 µs
0.5
(c) t = 65.26 µs
Y
1
damage
Y
1
Z
X
0
0.5
X
Y
1
(d) t = 85.59 µs
Figure 11. Damage fields for Kalthoff and Winkler experiments.
3,000
Z
(b) t = 45.72 µs
damage
0
0.5
Z
X
19
Single-edge notched tension test
Square in traction, branching.
Figure: Geometry for the single-edge notched tension test.
damage
0
0.5
damage
Y
1
Z
X
0
(a) t = 0.79 µs
0.5
1
Z
X
(b) t = 1.19 µs
damage
0
0.5
Y
damage
Y
1
Z
X
0
0.5
Y
1
Z
X
Ri
Ri + L r
Ri + l
TLS advanced :
Coupling
local
and
non-local
damage
Figure 3: Notations for the pull out test.
,
|r | = 1
|r | < 1
0
|r | = 1
⌦+
R
r
⌦
˙
˙
D ˙ [ ] = 0 =) [ ] = r · [r ]
Sketch of the variation of
dening.
along the radius when the damage mo
boundary
is the iso-contour
= lc .lc .In
terms ofthe
thezone
surrogate
va
is relation
introduces
a length scale
Finding
d = 1,
0
f
(d)
=
d
(
(d))
(1)
reads
patial
damage
distribution
satisfies
at
all
time
nce the level set is not strictly limited
to lc but
may
go beyond
TLS
is a fmodel
which
the gradient
is where
bounded
(d)1 is
by operation.
cannotprovided
go beyond
is given
aintedious
This is
the level set
e
surrogate
variable
,
condition
(1)
may
be
rewritten
as
comes into play. Variable d is expressed in terms of a level set as d
For instance,
kr k if01d is linear with respect to , the gradient of dama
f (d)
= d ( introduces
(d))
Figureby1. aThis
relation
a length scale lc . Finding the zone d =
constant
krdk  f (d) on ⌦
k  1provided
well-posed
since the level set is not strictly limited to lc but may go bey
f (d) is given by
1(1), may
For
instance,
if d isvariable
linear with
respect to
the gradient
of damage
the
use
of
the
surrogate
,
condition
be
rewritten
as
d = to
/lc , 2 [0, 1] =) krdk 
d( ) Identical
lc
by
a
constant
0
here8⌦ isf (d)
the
of interest. The choice of th
= ddomain
( (d))
<kr k  1
0 1 ), the bound depends on t
whereas
for
more
complex
function
d(
n
(1)
is
related
to
d(
)
by
f
(d)
=
d
( (d)) (the
prime
indicati
n what
follows.
As
damage
evolves,
one
eventually
d
=
/l
,
2
[0,
1]
=)
krdk

c
:
d=
d( )
For
instance,
if
is
with
respect
to
the
gradient
of dama
ford the
profile
shown
in
1, we
have
d with respect to
). linear
The
function
d(lcFigure
) is, called
the damage
one for
which
d = 1. However, finding the iso-con
by
a
constant
is the
main
ingredient
of
the
TLS.
Equation
(2)
above
indicate
p
0
for more
complex
d(
),=the
depends
onind
the
2d (bound
where whereas
fExamples
(d) din=(1)
is
related
to
d(
)
by
f
(d)
(d))
(the
prime
2function
1 the(1zone
( /lwhere
krdk

1 is dactive (we nam
c )) =)
nce
function
in
the
constraint
For instance,
the profile
in Figure
1 lc d(1, )we
derivative
of d withfor
respect
to ).shown
The function
is have
called the dam
d = /lc , 2 [0, 1] =) krdk 
alization
zone).
The
evolution
of
a
distance
function
has
been
an
l
c
unction and is the main ingredient of the TLS. p
Equation (2) above ind
2obtain
For
a
general
power
law
with
n
1,
we
2
g is
algorithm
proposed
[11].
In the
localization
the evolu
d=
1 (1 in
(in/l
=)
krdk

1 dzone,
a distance
function
the
where
the constraint
is active
(we
c ))zone
lc the bound depends on t
whereas
for
more
complex
function
d(
),
al,
indeed
zone
the localization zone). The nevolution of a distance
function
has bee
n
1 1/n
d = 1 for
(1 the
( /l
krdk

(1 1,d)we have
c )) =)
For
instance,
profile
shown
in
Figure
and updating
algorithm
proposed
lclocalization zone, the e
For a general
power
law withinn[11].1,Inwethe
obtain
b
a
c
+
⌦
kr (x)k < 1 ) Local constitutive
model ⌦at
x
d
e
⌦
+ =7[b,
Page
of d],
22 ⌦ = [a, b] [ [d, e],
3 Distribution ofconstitutive
on a 1D domain:model
⌦ = [a, e],at
⌦
kr (x)k = 1 )Figure
Non-Local
x
Slopes at 45 degrees on ⌦+ indicate that behaves as a distance function (kr k = 1)
kr (x)k > 1
kr k < 1 on ⌦ . Point c is the skeleton of the distance function.
forbidden
we have novelty
assumed Yof
(if not it compared
needs to be averaged
by form
c uniform
The first condition is where
the major
this paper,
to previous
˙
Finally, we write the relation giving ˙ in terms of d:
n the TLS. At any time t, the domain may thus be decomposed into three
Z
Z
+
verlapping zones : a locald˙zone
⌦
,
a
non-local
zone
⌦
and a fully damaged
0
= d0 ˙ , d0 2 A :
d0 a d! =
d a d!, 8a 2 A
⌦
⌦
x
⌦c
+
b
a
⌦⌦ =
+
d sectione we illustrate the average formula on the 1D example
Tocend this
in
+ Figure 3. Averages are given by
⌦c [ ⌦ ⌦+[ ⌦
⌦
Rc 0
+ = [b, d],
( e],)ldx
⌦ of =
{x
2
⌦
<
(x)
<
}
Distribution
on a 1D
domain:
⌦ := kr
[a, e], ⌦(x)k
⌦ 1,= [a,
b]yd
[ [d,
=
{b, d}.
c
b
˙
R c k = 1) whereas
c] : y function
= (kr
,
d(x)
=
45 degrees on ⌦+ indicate that behavesOn
as a[b,
distance
0
d ( ) dx
1 on ⌦ . Point
+ c is the skeleton of the distance function.
b
⌦
= {x 2 ⌦ : kr (x)k = 1, R(x)
< lc }
d
0
yd ( ) dx
c
˙
On [c, d] : y = R d
, d(x)
=
0 ( ) dx
⌦c = {x 2 ⌦ : (x) lc }
d
c
have assumed Yc uniform (if not it needs to be averaged by formula (25)).
˙
we write the relation giving ˙ in terms of d:
We define also
the boundary
Z
Z
c
Rc
˙ dx
d
Rb c
dx
b
Rd
d˙ dx
c
Rd
dx
c
of the fully damaged zone and the interf
l.
Fiber pull-out 1D test
Page 11 of 22
T, U
ri
rl
re
r
Figure 4 Pull-out of an infinite fiber of radius ri from a tube of radius re . Radius rl indicates the
(evolving) extent of the non-local damage zone.
τ/τc
1.4
1.2
1
0.8
0.6
0.4
0.2
0
0
1
2
3 4 5
γ/(τc/µ)
6
7
8
Figure 5 Local constitutive model: stress versus rising strain (case dc = 0.5) .
gimes may be solved analytically whereas the two last one may not. We how-
1
φ/lc
0.8
0.6
0.4
0.2
0
1
1.05 1.1 1.15 1.2 1.25 1.3 1.35 1.4
r/ri
1
0.8
d
0.6
0.4
0.2
0
1
1.05 1.1 1.15 1.2 1.25 1.3 1.35 1.4
r/ri
T
1.6
OA: elastic
B C
1.4 AB: loc. dam.
1.2 BC: loc./non-loc.
1 CD: non-loc.
A
0.8
0.6
0.4
0.2
D
0o
0 0.2 0.4 0.6 0.8 1 1.2 1.4
U
Figure 6 Force-displacement curve in the case n=2. The elastic part of the solution is on part
[OA] of the curve. The part [AB] corresponds to the development of local damage whereas part
[BC] corresponds to both local and non-local damage development. Finally, part [CD] is governed
only by non-local damage. Letters A, B and C are located at T = 1, T = 1.351 and
1.4
1.2
1
T
0.8
0.6
0.4
0.2
0
0 0.2 0.4 0.6 0.8 1 1.2 1.4
U
Figure 8 Force-displacement curves for n=1 (small dots), n=2 (solid line), n=3 (big dots).
Strengths of the TLS (I)
•
•
Cracks appear automatically from damage growth
•
Displacement jump may be used : hydraulic fracture, leakage,
friction, fragmentation, ...
•
•
Avoids mesh distortion in large strain like ductile fracture
Allows mesh coarsening with X-FEM discontinuity enrichment
A cohesive model is not needed in the transition
damage/fracture since crack appear if and only if
d=1
•
No need for an additional constitutive law. The damage model
is enough, the non-trivial transition from damage to cohesive
is avoided
•
The cracks unveils automatically in fully damage zone
Strengths (II)
•
TLS lies between damage and fracture : moving of a geometrical
object (level set) and use of damage models. It is not a crack
regularization approach.
•
•
•
It takes into account material size effect
•
•
Preservation of the duality in the non-local quantities
It avoids incorrect damage initiation.
Non-local formulation to avoid spurious localization is only used
where needed. Thus, 99% of the domain is local damage where
smooth damage may develop. Dramatic reduction of computer
time, even further reduced by mesh coarsening.
All 2D examples shown in this presentation and our papers run in
less than 15 minutes on a single processor. 3D timing is on the
way.
Other/Coming work
• Shear bands in quasi-brittle media
• Ductile failure
• Fragmentation in ductile and quasi-brittle
media
• Structure comminution
Acknowledgements:
References
IJNME, 2011, 36:358-380
CMAME,2012, 233-236:11-27
IJF, 2012, 174:43-60