FE solution depends on the mesh

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? FE solution depends on the mesh
the ABC of modeling localized deformation (in 14
slides)
where is the problem?
a simple case plane strain compression test
this is clearly inacceptable!
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the ABC of modeling localized deformation
where is the problem?
another, even simpler case simple shear
x1
L
x2
stress and strain
force and velocity
s12 t e12 g/2
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the ABC of modeling localized deformation
solving the rate problem
balance equations
history of B.C.
V(t) T(t)
constitutive model
brittle
ductile
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the ABC of modeling localized deformation
elementary solution for the rate problem
homogeneous solution
or
or
with
another possibility heterogeneous solution
for the (elastically) unloading part
for the (plastically) loading part
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the ABC of modeling localized deformation
rate problem
Initial Boundary Value problem
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the ABC of modeling localized deformation
two extreme solutions
homogeneous (no localization)
Kotronis et al. (2008), Acta Geotechnica
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the ABC of modeling localized deformation
a classical continuum model is just not
equipped with such
information need of an internal
length i.e., need to incorporate (in a way or
another) some information from the micro
scale in the description of the continuum (macro)
model
continua with microstructure
Cosserat, micro-polar, second
gradient,
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continua with microstructure
materials with microstructure (Mindlin, 64)
a constrained class of materials with micro
structure
the Virtual Work Equation reads (Germain 1973)
double stress
double force
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how to put all this in a FE context
Virtual Work Equation for second gradient models
can of course be used in a FE code ? but, in this
case well need C1 functions for displacements,
since VWE contains second spatial
derivatives of displacement ! ? alternative to
write in weak form, by introducing
Lagrange multipliers (Chambon et al. 2001)
? three fields ui, vij, lij
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FE implementation
u displacement v displacement gradient l
Lagrange multipliers
u1 u2
Q8
v11 v12 v21 v22
Q4
l11 l12 l21 l22
Q1
36 DOF by element !
implemented in the FE code Lagamine (ULg)
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examples of FE results objectivity?
plane strain compression
Gauss points in the plastic regime
FE results are objective, i.e., do not depend on
the FE mesh
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examples of FE results uniqueness?
non-homogeneous solutions obtained by a
directional search algorithm after the threshold
of bifurcation has been satisfied (starting from
the homogeneous solution). The solutions
presented (after 10 of total softening, for the
most part) produce 1, 2 or 3 bands, some bands
can be deactivated during the loading
these are all objective solutions of the same
problem (similar to what is observed from actual
experiments)
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examples of FE results complex (and realistic)
a more interesting problem (borehole drilling)
once again, there is more than one solution
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concluding message for the modeling
? onset and progression (in time and space) of
localization ? overall behavior in the presence
of multiple regions of localized damage,
possibly interacting with each other ?
hydro-thermo-mechanical coupling modeling
localized damage requires advanced, non
conventional models