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

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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



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



Classical FEM approach

FEM for

polycrystals

→

polycrystal

topology

◮ User intervention◮ Generation time

◮ Mesh requirement difficult to achieve: accuracy

◮ Daunting task in 3D




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]




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]




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]




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




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]





◮ Very simple local mesh refinement strategy

split cut elements...as if there were no discontinuities!

preserve aspect ratio[Rivara 1984]





◮ Very simple local mesh refinement strategy◮ split cut elements...as if there were no discontinuities!◮ relax meshing requirements & preserve aspect ratio

[Rivara 1984]





◮ Higher accuracy? Use hp -adaptivity![Duarte, Reno & Simone, IJNME 2007]

split cut elements...as if there were no discontinuities![Rivara 1984]




Creeping Polycrystals




Grain boundary sliding

[Langdon 2006]

◮ Important contribution to anelasticity at high temperatures

◮ May account for 10% to 65% of total creep strain

[Ghahremani 1980]


S h f



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]


St h t f t



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]


R d t b ti



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


R d t b ti



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


Stress enhancement factor f (viscoelasticity n 1)



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






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






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


Final remarks



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


Minisymposium at ECCM 2010 in Paris



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


analysis of freely sliding creeping polycrystals with a generalized finite element method

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