Download - Analysis of freely sliding creeping polycrystals with a Generalized Finite Element Method
8/8/2019 Analysis of freely sliding creeping polycrystals with a Generalized Finite Element Method
http://slidepdf.com/reader/full/analysis-of-freely-sliding-creeping-polycrystals-with-a-generalized-finite 1/23
Analysis of freely sliding creeping polycrystalswith a Generalized Finite Element Method
A. Simonea E. Van der Giessenb C.A. Duartec
a Faculty of Civil Engineering and Geosciences, Delft University of Technology, The Netherlands
b Zernike Institute for Advanced Materials, University of Groningen, The Netherlands
c Department of Civil and Environmental Engineering, University of Illinois at Urbana-Champaign, Illinois, USA
XFEM 2009, September 28-30, 2009, Aachen, Germany 1 / 14
8/8/2019 Analysis of freely sliding creeping polycrystals with a Generalized Finite Element Method
http://slidepdf.com/reader/full/analysis-of-freely-sliding-creeping-polycrystals-with-a-generalized-finite 2/23
Goal... and issues
1 mm
[Ingraffea 2003]
◮
Effect of microstructure on mechanical properties◮ accurate and automated analyses
◮ Consider relevant features at adequate resolution
◮ Meshing issues?
Simone, Van der Giessen & Duarte 2 / 14
8/8/2019 Analysis of freely sliding creeping polycrystals with a Generalized Finite Element Method
http://slidepdf.com/reader/full/analysis-of-freely-sliding-creeping-polycrystals-with-a-generalized-finite 3/23
Classical FEM approach
FEM for
polycrystals
→
polycrystal
topology
◮ User intervention◮ Generation time
◮ Mesh requirement difficult to achieve: accuracy
◮ Daunting task in 3D
Simone, Van der Giessen & Duarte 3 / 14
8/8/2019 Analysis of freely sliding creeping polycrystals with a Generalized Finite Element Method
http://slidepdf.com/reader/full/analysis-of-freely-sliding-creeping-polycrystals-with-a-generalized-finite 4/23
GFEM for polycrystals
background GFEM for
+ =
mesh polycrystals
polycrystal
topology
◮ Grain boundaries cut elements
◮ Grain junctions within elements
[Simone, Duarte & Van der Giessen, IJNME 2006]
Simone, Van der Giessen & Duarte 4 / 14
8/8/2019 Analysis of freely sliding creeping polycrystals with a Generalized Finite Element Method
http://slidepdf.com/reader/full/analysis-of-freely-sliding-creeping-polycrystals-with-a-generalized-finite 5/23
GFEM for polycrystals
background GFEM for
+ =
mesh polycrystals
polycrystal
topology
◮ User time is not wasted on mesh definition
Grain junctions within elements
[Simone, Duarte & Van der Giessen, IJNME 2006]
Simone, Van der Giessen & Duarte 4 / 14
8/8/2019 Analysis of freely sliding creeping polycrystals with a Generalized Finite Element Method
http://slidepdf.com/reader/full/analysis-of-freely-sliding-creeping-polycrystals-with-a-generalized-finite 6/23
Key ingredients
◮ Displacement decomposition:
u = u +
N Gi =1
Hi u i
with
Hi (x ) =
1 if x ∈ Gi
0 otherwise
G3G1
G2
◮ Grains: any constitutive relation
◮ Grain boundaries: any traction-separation law
Akin to XFEM... but easier
[Moës et al. 1999, Daux et al. 2000]
Simone, Van der Giessen & Duarte 5 / 14
8/8/2019 Analysis of freely sliding creeping polycrystals with a Generalized Finite Element Method
http://slidepdf.com/reader/full/analysis-of-freely-sliding-creeping-polycrystals-with-a-generalized-finite 7/23
Keep things simple...
◮ No Level sets
computational geometry tools
◮ No junction enrichments
...but not too simple
◮ Control resolution level → gain accuracy
along grain boundariesat junctions
within grains
Simone, Van der Giessen & Duarte 6 / 14
8/8/2019 Analysis of freely sliding creeping polycrystals with a Generalized Finite Element Method
http://slidepdf.com/reader/full/analysis-of-freely-sliding-creeping-polycrystals-with-a-generalized-finite 8/23
Keep things simple...
◮ No Level sets
computational geometry tools
◮ No junction enrichments
...but not too simple
◮ Control resolution level → gain accuracy
along grain boundariesat junctions
within grains
Simone, Van der Giessen & Duarte 6 / 14
8/8/2019 Analysis of freely sliding creeping polycrystals with a Generalized Finite Element Method
http://slidepdf.com/reader/full/analysis-of-freely-sliding-creeping-polycrystals-with-a-generalized-finite 9/23
Mesh refinement: a necessary companion
◮ Very simple local mesh refinement strategy
split cut elements...as if there were no discontinuities!
preserve aspect ratio[Rivara 1984]
Simone, Van der Giessen & Duarte 7 / 14
8/8/2019 Analysis of freely sliding creeping polycrystals with a Generalized Finite Element Method
http://slidepdf.com/reader/full/analysis-of-freely-sliding-creeping-polycrystals-with-a-generalized-finite 10/23
Mesh refinement: a necessary companion
◮ Very simple local mesh refinement strategy
split cut elements...as if there were no discontinuities!
preserve aspect ratio[Rivara 1984]
Simone, Van der Giessen & Duarte 7 / 14
8/8/2019 Analysis of freely sliding creeping polycrystals with a Generalized Finite Element Method
http://slidepdf.com/reader/full/analysis-of-freely-sliding-creeping-polycrystals-with-a-generalized-finite 11/23
Mesh refinement: a necessary companion
◮ Very simple local mesh refinement strategy◮ split cut elements...as if there were no discontinuities!◮ relax meshing requirements & preserve aspect ratio
[Rivara 1984]
Simone, Van der Giessen & Duarte 7 / 14
8/8/2019 Analysis of freely sliding creeping polycrystals with a Generalized Finite Element Method
http://slidepdf.com/reader/full/analysis-of-freely-sliding-creeping-polycrystals-with-a-generalized-finite 12/23
Mesh refinement: a necessary companion
◮ Higher accuracy? Use hp -adaptivity![Duarte, Reno & Simone, IJNME 2007]
split cut elements...as if there were no discontinuities![Rivara 1984]
Simone, Van der Giessen & Duarte 7 / 14
8/8/2019 Analysis of freely sliding creeping polycrystals with a Generalized Finite Element Method
http://slidepdf.com/reader/full/analysis-of-freely-sliding-creeping-polycrystals-with-a-generalized-finite 13/23
Creeping Polycrystals
Simone, Van der Giessen & Duarte 8 / 14
8/8/2019 Analysis of freely sliding creeping polycrystals with a Generalized Finite Element Method
http://slidepdf.com/reader/full/analysis-of-freely-sliding-creeping-polycrystals-with-a-generalized-finite 14/23
Grain boundary sliding
[Langdon 2006]
◮ Important contribution to anelasticity at high temperatures
◮ May account for 10% to 65% of total creep strain
[Ghahremani 1980]
Simone, Van der Giessen & Duarte 9 / 14
S h f
8/8/2019 Analysis of freely sliding creeping polycrystals with a Generalized Finite Element Method
http://slidepdf.com/reader/full/analysis-of-freely-sliding-creeping-polycrystals-with-a-generalized-finite 15/23
Stress enhancement factor
◮ GB sliding accelerates creep rate◮ Homogeneous material:
εc e (t ) = ε0
σe (t )
σ0
n
◮ Overall response of polycrystal with GB sliding:
˙εc e (t ) = ε0
f σe (t )
σ0
n
◮ Stress enhancement factor f > 1
[Crossman & Ashby 1975]
Simone, Van der Giessen & Duarte 10 / 14
St h t f t
8/8/2019 Analysis of freely sliding creeping polycrystals with a Generalized Finite Element Method
http://slidepdf.com/reader/full/analysis-of-freely-sliding-creeping-polycrystals-with-a-generalized-finite 16/23
Stress enhancement factor
◮ Creep exponent n = 1 → viscoelasticity
◮ Periodic cell in uniaxial tension:
f =
˙ε11
ε11
f is the ratio of relaxed to unrelaxed strain rate
[Onck & Van der Giessen 1997]
◮ f = 1.158 for regular hexagonal grains
[Ghahremani 1980]
Simone, Van der Giessen & Duarte 10 / 14
R d t b ti
8/8/2019 Analysis of freely sliding creeping polycrystals with a Generalized Finite Element Method
http://slidepdf.com/reader/full/analysis-of-freely-sliding-creeping-polycrystals-with-a-generalized-finite 17/23
Random perturbations
r
αr θ
α ∈ [0; 1]
θ ∈ [0;2π] rad
periodic cell, uniaxial tension
s
◮ new junction position defined by angle θ and offset αr
ρ = 1KAG
K k =1
L(k )2 1 + sin
2ψ(k )
ρ = 0.289
Simone, Van der Giessen & Duarte 11 / 14
R d t b ti
8/8/2019 Analysis of freely sliding creeping polycrystals with a Generalized Finite Element Method
http://slidepdf.com/reader/full/analysis-of-freely-sliding-creeping-polycrystals-with-a-generalized-finite 18/23
Random perturbations
periodic cell, uniaxial tension
L(k )ψ(k )
n (k )
facet k
L(k )
◮ new junction position defined by angle θ and offset αr
◮ ρ = 1KAG
K k =1
L(k )2 1 + sin
2ψ(k )
ρ = 0.289
Simone, Van der Giessen & Duarte 11 / 14
Stress enhancement factor f (viscoelasticity n 1)
8/8/2019 Analysis of freely sliding creeping polycrystals with a Generalized Finite Element Method
http://slidepdf.com/reader/full/analysis-of-freely-sliding-creeping-polycrystals-with-a-generalized-finite 19/23
Stress enhancement factor f (viscoelasticity, n = 1)
r = 48%s r = 42%s r = 36%s r = 30%s r = 24%s r = 18%s r = 12%s
r = 6%s r = 0
ρ
f
0.440.420.40.380.360.340.320.30.28
1.4
1.35
1.3
1.25
1.2
1.15
1.1
◮ Each point relates to a polycrystal
◮ 1001 random realizations of the same simulation
no user intervention
Simone, Van der Giessen & Duarte 12 / 14
Stress enhancement factor f (viscoelasticity n 1)
8/8/2019 Analysis of freely sliding creeping polycrystals with a Generalized Finite Element Method
http://slidepdf.com/reader/full/analysis-of-freely-sliding-creeping-polycrystals-with-a-generalized-finite 20/23
Stress enhancement factor f (viscoelasticity, n = 1)
r = 48%s r = 42%s r = 36%s r = 30%s r = 24%s r = 18%s r = 12%s
r = 6%s r = 0
ρ
f
0.440.420.40.380.360.340.320.30.28
1.4
1.35
1.3
1.25
1.2
1.15
1.1
◮ Regular hexagonal grains: 15.8% increase creep rate
◮ Randomness increases f up to 16.6% (f : 1.158 → 1.35)
no user intervention
Simone, Van der Giessen & Duarte 12 / 14
Stress enhancement factor f (viscoelasticity n 1)
8/8/2019 Analysis of freely sliding creeping polycrystals with a Generalized Finite Element Method
http://slidepdf.com/reader/full/analysis-of-freely-sliding-creeping-polycrystals-with-a-generalized-finite 21/23
Stress enhancement factor f (viscoelasticity, n = 1)
r = 48%s
r = 42%s
r = 36%s
r = 30%s
r = 24%s
r = 18%s
r = 12%s
r = 6%s
r = 0
ρ
f
0.440.420.40.380.360.340.320.30.28
1.4
1.35
1.3
1.25
1.2
1.15
1.1
gfem reference
[Onck & Van der Giessen 1997]
◮ Differences with reference trend (red line)
◮ Higher dispersion◮ Values above and below reference value
f = 1.158 for r = 0
Simone, Van der Giessen & Duarte 12 / 14
Final remarks
8/8/2019 Analysis of freely sliding creeping polycrystals with a Generalized Finite Element Method
http://slidepdf.com/reader/full/analysis-of-freely-sliding-creeping-polycrystals-with-a-generalized-finite 22/23
Final remarks
◮ No need for mesh generators◮ background mesh◮ polycrystalline topology
◮
No need for user intervention◮ accurate and automated analyses
◮ Random micro-structures can be easily addressed
Simone, Van der Giessen & Duarte 13 / 14
Minisymposium at ECCM 2010 in Paris
8/8/2019 Analysis of freely sliding creeping polycrystals with a Generalized Finite Element Method
http://slidepdf.com/reader/full/analysis-of-freely-sliding-creeping-polycrystals-with-a-generalized-finite 23/23
Minisymposium at ECCM 2010 in Paris
Generalized/extended FEM, meshless and related approaches
C.A. Duarte
University of Illinois at Urbana-Champaign
USA
J.S. Chen
University of California, Los Angeles
USA
M.A. Schweitzer
Rheinische Friedrich-Wilhelms Universitaet Bonn
Germany
A. Simone
Delft University of Technology
The Netherlands
Simone, Van der Giessen & Duarte 14 / 14