Ensemble averaging stress–strain fields in polycrystalline aggregates with a constrained surface microstructure – Part 1: anisotropic elastic behaviour

TLDR: In this paper, the effect of 3D grain morphology on the deformation at a free surface in polycrystalline aggregates is investigated by means of a large-scale finite element and statistical approach.

Abstract: The effect of three-dimensional (3D) grain morphology on the deformation at a free surface in polycrystalline aggregates is investigated by means of a large-scale finite element and statistical approach. For a given two-dimensional surface at z = 0 containing 39 grains with given crystal orientations, 17 random 3D polycrystalline aggregates are constructed having different 3D grain shapes and orientations except at z = 0, based on an original 3D image analysis procedure. They are subjected to overall tensile loading conditions. The resulting stress–strain fields at the free surface z = 0 are analyzed. Ensemble average and variance maps of the stress field at the observed surface are computed. In the case of an anisotropic elastic behaviour of the grains, fluctuations ranging between 5% and 60% are found in the equivalent stress level at a given material point of the observed surface from one realization of the microstructure to another. These results have important implications in the way of comparing fin...

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Ensemble averaging stress-strain elds in polycrystalline
aggregates with a constrained surface
microstructure-Part 1: Computational tools and
application to anisotropic elastic behaviour
Asmahana Zeghadi, Franck N’Guyen, Samuel Forest, Anne-Françoise
Gourgues, Olivier Bouaziz
To cite this version:
Asmahana Zeghadi, Franck N’Guyen, Samuel Forest, Anne-Françoise Gourgues, Olivier Bouaziz.
Ensemble averaging stress-strain elds in polycrystalline aggregates with a constrained surface
microstructure-Part 1: Computational tools and application to anisotropic elastic behaviour. Philo-
sophical Magazine, Taylor & Francis, 2007, 87, pp.1401-1424. �10.1080/14786430601009509�. �hal-
00513781�

For Peer Review Only
Ensemble averaging stress-strain fields in polycrystalline
aggregates with a constrained surface microstructure-Part
1:
Computational tools and application to anisotropic elastic
behaviour
Journal:
Philosophical Magazine & Philosophical Magazine Letters
Manuscript ID:
TPHM-06-Apr-0094.R1
Journal Selection:
Philosophical Magazine
Date Submitted by the
Author:
21-Jun-2006
Complete List of Authors:
Forest, Samuel; Ecole National Supérieure des Mines de
Paris/CNRS, Centre des Matériaux UMR 7633
Zeghadi, Asmahana; EDF
Nguyen, Franck; Ecole des Mines de Paris
Gourgues, Anne-Francoise; Ecole des Mines de Paris
Bouaziz, Olivier; Arcelor Research
Keywords:
anisotropic elasticity, polycrystalline metals
Keywords (user supplied):
ensemble average, finite element
Note: The following files were submitted by the author for peer review, but cannot be converted
to PDF. You must view these files (e.g. movies) online.
zeghadi-part1-final.tex
zeghadi-part1-final.bbl
http://mc.manuscriptcentral.com/pm-pml
Philosophical Magazine & Philosophical Magazine Letters

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For Peer Review Only
Ensemble averaging stress–strain fields in
polycrystalline aggregates with a constrained
surface microstructure–Part 1: Anisotropic
elastic behaviour
A. Zeghadi
a
, F. N’Guyen
a
, S. Forest
a,∗
, A.–F. Gourgues
a
,
O. Bouaziz
b
,
a
Centre des Mat´eriaux / Mines Paris, Paristech, CNRS UMR 7633, B.P. 87,
91003 Evry Cedex, France
b
ARCELOR Research, Voie Romaine, B.P. 30320,
57283 Maizi`eres–l`es–Metz, France
Abstract
The effect of three–dimensional grain morphology on the deformation at a free
surface in polycrystalline aggregates is investigated by means of a large sc ale finite
element and statistical approach. For a given 2D surface at z = 0 containing 39
grains with given crystal orientations, 17 random 3D polycrystalline aggregates are
constructed having different 3D grain shapes and orientations except at z = 0,
based on an original 3D image analysis procedure. They are subjected to overall
tensile loading conditions. The resulting stress–strain fields at the free surface
z = 0 are analysed. Ensemble average and variance maps of the stress field at the
observed surface are computed. In the case of an anisotropic elastic behaviour of
the grains, fluctuations ranging between 5% and 60% are found in the equivalent
stress le vel at a given material point of the observed surface from one realization
of the microstructure to another. These results have important implications in the
way of comparing finite element simulations and surface strain field measurements
in metal polycrystals.
Key words: Polycrystal, Anisotropic elasticity, Ensemble average, Finite element,
Copper
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For Peer Review Only
1 Introduction
The development of field measurement methods in the past 15 years provides us with a precise knowledge
of the heterogeneity of strains, stresses (via the determination of elastic strains) and lattice rotations
in metallic polycrystals (Becker, 1991; Allais et al., 1994; Ziegenbein et al., 1998). The deformation
of grids deposited on the surface specimen, interpreted by means of image correlation analysis, X–
ray microdiffraction, and Electron Back–Scatter Diffraction (EBSD) are respectively used to evaluate
the fields of total strain, elastic strain and lattice orientation at various stages of the deformation of
polycrystals, with a pixel resolution below the micron scale (Mohamed et al., 1997; Delaire et al.,
2000; Eberl et al., 2002; Parisot et al., 2001; Mary et al., 2005). The observation of detailed strain
fields inside the individual grains for a large number of grains reveals the extreme heterogeneity of
deformation due to crystal slip processes. The development of bands of intense plastic deformation
crossing several grains is often reported, in addition to the strong intragranular heterogeneities induced
by strain incompatibilities at grain boundaries. These now standard experimental techniques can be
applied at the free surface of a polycrystal subjected to various loading conditions such as tension and
shear. The analysis of grid deformation usually provides the in–plane components of the displacement
field, from which in–plane components of the strain tensor can be evaluated. EBSD analysis gives the
full 3D orientation map of all surface grains. Micro–diffraction analyses using for instance synchrotron
radiation provide 2D maps of the 3D elastic strain tensor field for material points at the surface of
polycrystals (Eberl et al., 2002).
Such field measurements can be used to validate the theoretical framework of continuum crystal plas-
ticity settled in (Mandel, 1971; Asaro, 1983a). The results of finite element simulations based on the
numerical integration of the constitutive equations for single crystals, can be compared to the corre-
sponding experimental information (Teodosiu et al., 1993; Delaire et al., 2000; Eberl et al., 2002; Erieau
and Rey, 2004; Cheong and Busso, 2005). Continuum crystal plasticity takes the crystallographic na-
ture of plastic slip into account via a set of slip variables γ
s
associated with each slip system s. The first
finite element simulations of the elastic-viscoplastic behavior of polycrystals go back to (Miyamoto,
1972; Asaro, 1983b; Harren and Asaro, 1989; Havlicek et al., 1990; Teodosiu et al., 1993) in which
∗
Corresponding author. Tel.: +33-1-60-76-30-51; Fax: +33-1-60-76-31-50
Email address: samuel.forest@ensmp.fr (S. Forest).
2
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Frequently asked questions

Q1. What contributions have the authors mentioned in the paper "Ensemble averaging stress-strain fields in polycrystalline aggregates with a constrained surface microstructure-part 1: computational tools and application to anisotropic elastic behaviour" ?

In this paper, the effect of grain morphology on the deformation at a free surface in polycrystalline aggregates is investigated by means of a large scale finite element and statistical approach. 

Q2. What is the usual strategy for a direct comparison between strain field measurements and finite element results?

The usual strategy for a direct comparison between strain field measurements and finite element results consists in producing a finite element mesh of the surface morphology of the observed grains and subjecting it to boundary conditions as close as possible to the experimental ones. 

Q3. What is the effect of the constrained displacement on the microstructure?

The constrained displacement associated with the changes of the underlying microstructure leads to marked strain incompatibilities and strongly different local stress levels from one realization to another. 

Q4. What is the definition of continuum crystal plasticity?

Continuum crystal plasticity takes the crystallographic nature of plastic slip into account via a set of slip variables γs associated with each slip system s. 

Q5. What is the main problem in assessing the quality of the simulation results compared with the available?

The main problem in assessing the quality of the simulation results compared to the available experimental data such as strain and lattice rotation fields lies in the fact that the usually unknown actual 3D morphology of the grains can significantly affect the strain and lattice rotation fields at the surface. 

Q6. How is the growth rate of the grain determined?

In the case of expanding grains, the growth rate chosen initially is kept constant for the subsequent layers until the depth z = d0/2 is reached in average. 

Q7. how many grains are required to reach the stress–strain field at the constrained free surface?

The number of grains within the thickness must be large enough for the stress–strain field at the constrained free surface to be practically unaffected by a further increase of the thickness. 

Q8. What is the process of convexifying grains?

The grains are replaced by their convex envelop; this convexifying procedure is run until all the grains in the volume are convex; this may require several iterations; if the procedure does not converge, the volume is excluded. 

Q9. What is the quality of the description of the variables close to grain boundaries?

the quality of the description of the variables close to grain boundaries can be restored if the number of elements per grain is sufficiently high (Barbe et al., 2001b; Diard et al., 2005).