**Dr.
Ricardo A. Lebensohn**

**1) Second-order homogenization of viscoplastic
polycrystals:** The computation of
large-strain mechanical behavior and texture evolution of viscoplastic (VP)
polycrystals using self-consistent (SC) models is nowadays a standard approach
in the materials science community. For this, several classical SC
approximations for non-linear materials are available (e.g. the
"secant", "tangent" and "affine" formulations),
all of them based on linearization schemes at the local level that use
information on mechanical field averages only, disregarding higher-order
statistical information (i.e. average field fluctuations) in the grains.
However, the above assumption may be questionable for single-phase aggregates
of highly anisotropic grains or for two-phase polycrystals, where a strong
directionality and large variations in local properties are to be expected. The
non-dependence with higher-order statistical moments is particularly critical
for the treatment of those highly-contrasted materials, since such information
is essential to capture (in an average sense) the effect of the strong
deformation gradients that are likely to develop inside grains which are highly
anisotropic, or adjacent to another phase. Therefore, in order to overcome the
above limitations we have implemented inside the code VPSC [1] a rigorous
non-linear homogenization scheme [2] that takes into account information on the
average field fluctuations at grain level [3].

**2) Homogenization of the dilatational viscoplastic
behavior of anisotropic porous polycrystals**: The originally incompressible VPSC formulation [4] has been recently
extended to deal with porous polycrystals taking into account: a) the
dilatational deformation associated with void growth [5], and b) second-order
statistical information in terms of average field fluctuations inside the
constituent grains. Such extended model allows us to account for porosity
evolution in voided polycrystals, while preserving the anisotropy and rate
sensitivity capabilities of the VPSC formulation. This extended VPSC model has
been applied to address the problem of coupling between texture, plastic
anisotropy, void shape, triaxiality, rate sensitivity and porosity evolution
[6,7].

**3)
Simulation of forming of anisotropic materials by direct coupling between SC
polycrystal models and Finite Element (FE) codes:** Simulating the forming of anisotropic materials,
such as zirconium, requires a description of the anisotropy of the aggregate
and the single crystal, and also of its evolution with deformation (due to
texture development and hardening by slip and twinning). As a consequence,
using polycrystal constitutive laws inside FE codes represents a considerable
improvement over using empirical constitutive laws, since the former provides a
physically-based description of anisotropy and its evolution. We have developed
a polycrystal constitutive description for pure clock-rolled Zr deforming under
quasi-static conditions at room and liquid nitrogen temperatures, using tensile
and compressive experimental data to adjust the constitutive parameters of a SC
polycrystal model [8]. This model was in turn implemented in an explicit FE
code, assuming a full polycrystal at the position of each integration point.
This methodology was successfully applied to simulate the inhomogeneous
three-dimensional deformation of zirconium bars deforming under four-point bend
conditions.

**4) ****Macroscopic
anisotropic yield functions and FE analysis to account for the effect of
twinning on anisotropy and work-hardening:** We have recently combined physically-based and
experimentally-adjusted models for description of the twinning process and the
interaction between twinning and slip systems in zirconium [8] and a
macroscopic yield criterion that accounts for the effect of twinning on
yielding [9] under quasiestatic (elastoplastic) [10] and dynamic
(elasto-viscoplastic) [11] loading conditions, using a multiscale methodology
based on: texture measurements, uniaxial mechanical tests, SC polycrystal
models, interpolation techniques at macroscopic level, and FE analysis. We are
also in the process of applying the above methodology to titanium [12].

**5)
Full-field formulation based on Fast Fourier Transforms for the calculation of
the micromechanical behavior of plastically deformed 3-D polycrystals:** This formulation (originally developed as a fast
algorithm to compute the linear and nonlinear response of composites using as
input a digital image of their microstructures [13]) has been adapted by us to
treat 3-D polycrystals deforming by dislocation glide [14]. The
FFT-based model provides an exact solution of the governing equations, has
better performance than a Finite Element calculation for the same purpose and
resolution, and can use voxelized microstructure data as direct input. Among
the applications of this formulation we can mention: a) validation of
predictions of statistical models on effective properties, field fluctuations,
global and local texture evolution and hardening in plastically deformed cubic
and hexagonal materials (e.g. [15,16]); b) direct comparison with intragranular
orientation maps and micromechanical fields measured by imaging techniques
(OIM, DIC, etc) in plastically deformed Cu polycrystals [17]; c) study of the
effect of microstructure on strain localization [18]; d) prediction of subgrain
structure formation, GND densities and implementation of length-scale-sensitive
hardening laws [19].

**6) Other projects**:

a) Simulation of** **texture
development and grain refinement during equal channel angular extrusion (ECAE)
of fcc materials [20].

b) Prediction of intracrystalline fields in deformed ice and its
implications in the deformation-recrystallization behavior of polar ice sheets
[21].

c) Second-order micromechanical modeling of rocks of the Earth's
interior and implementation of a multiscale simulation of the regional upper
mantle flow beneath a mid-ocean ridge [22].

d) In the context of an European project [23], we'll soon begin the interface of the FFT-based
approach and the Elle platform [24]. Elle is a code for the simulation
of micro-process coupling in geological materials via an iterative modification
of the microstructure by a whole range of processes (grain boundary migration,
diffusion, sub-grain evolution). The Elle and FFT-based approaches share a
similar numerical structure and the incorporation of the latter in Elle will
provide a fully anisotropic crystal plasticity capability, an essential feature
to model plastic deformation and its interaction with other micro-process of
low-symmetry geological materials.

e) SC modeling of
semi-crystalline polymers for the study their constitutive behavior, texture
and morphology evolution during large plastic deformation [25].

f) Micromechanical study
the role played by dislocation glide and grain boundary accommodation in the
determination of the plastic behavior of nanostructured materials [26].

g) Based on a recent
mesoscale description of martensitic materials [27] using a relaxation
technique by coarse-graining over a martensitic microstructure and Fast Fourier
Transforms, we have proposed a theoretical, numerical and experimental study to
understand the interplay between the slip and transformation-induced
deformation in Ni-Ti shape-memory alloys [28].

h) In the context of the
Generation IV Reactor Materials Working Group of the Advanced Fuel Cycle
Initiative (AFCI) Program, we are involved in the development of improved
models for the creep response of irradiated structural materials by incorporating irradiation hardening in polycrystal
models of austenitic and ferritic-martensitic steels [29].

**References:**

[1] The code VPSC (developed and maintained by R.A. Lebensohn and C.N Tome) is a multipurpose
polycrystal plasticity research code, based on the knowledge of the mechanisms
of slip and twinning that are active in single crystals of arbitrary symmetry.
VPSC can be used to predict the effective stress-strain response, texture
evolution, anisotropy, etc., and it is presently used as a predictive tool for
metallic and geological material systems, for parameter identification,
interpretation of experimental results and multiscale calculations, in academic
and industrial applications, etc., by numerous research groups worldwide (at
present, the VPSC code distribution list has more than registered 80 users).

[2] P.
Ponte Castaneda: "Second-order homogenization estimates for nonlinear
composites incorporating field fluctuations. I. Theory". J. Mech. Phys. Solids 50, 737, 2002.

[3] R.A. Lebensohn, C.N. Tome and P. Ponte Castaneda:
"Improving the self-consistent predictions of texture development of
polycyrystals incorporating intragranular field fluctuations". Materials
Science Forum, 495-497, 955, 2005.

[4] R.A. Lebensohn and C.N. Tome: "A selfconsistent
approach for the simulation of plastic deformation and texture development of
polycrystals: application to zirconium alloys" Acta Mater 41, 2611, 1993.

[5] R.A. Lebensohn, C.N Tome and P.J. Maudlin: "A
selfconsistent formulation for the prediction of the anisotropic behavior of
viscoplastic polycrystals with voids". J Mech. Phys. Solids 52, 249, 2004.

[6] R.A. Lebensohn: "Anisotropic modelling of the plastic
deformation of damaged polycrystalline materials". TCG-I Meeting,

[7] R.A. Lebensohn, C.N. Tome and P. Ponte Castaneda, in
preparation.

[8] C.N. Tome, P.J. Maudlin, R.A. Lebensohn and G.C.
Kaschner: "Mechanical response of zirconium. Part I: Derivation of a
polycrystal constitutive law and finite element analysis". Acta Mater. 49,
3085, 2001.

[9] O. Cazacu, B.
Plunkett, F. Barlat: "Orthotropic yield criterion for hexagonal closed
packed metals". Int. J. Plasticity 22, 1171, 2006.

[10] B. Plunkett, R.A. Lebensohn, O. Cazacu^{ }and
F. Barlat: "Evolving yield function of hexagonal materials taking into
account texture development and anisotropic hardening". Acta Mater 54,
4159, 2006.

[11] B. Plunkett, O.
Cazacu, R.A. Lebensohn, and F. Barlat: "Strain-rate effects on the texture
evolution of high-symmetry and low-symmetry metals: modeling and validation
using the

[12] R.A. Lebensohn (PI) et al.:
"Characterization of hcp Materials". LANL's WFO Program, sponsored by
Eglin Air Force Research Laboratory, 2005-2007.

[13] H.
Moulinec and P. Suquet: "A fast numerical method for computing the linear
and nonlinear properties of composites". C.R. Acad. Sci. Paris II 318,
1417, 1994.

[14] R.A. Lebensohn: "N-site modelling of a 3D
viscoplastic polycrystal using Fast Fourier Transform". Acta Mater. 49,
2723, 2001.

[15] R.A. Lebensohn, Y. Liu and P. Ponte Castaneda: "On
the accuracy of the self-consistent approximation for polycrystals: comparison
with full-field numerical simulations". Acta Mater. 52, 5347, 2004.

[16] O. Castelnau, R. Brenner and R.A. Lebensohn: "The
effect of strain heterogeneity on the work-hardening of polycrystals predicted
by mean-field approaches". Acta Mater. 54, 2745, 2006.

[17] R.A. Lebensohn, O. Castelnau, R.
Brenner: "Full-field modelling and experimental validation of subgrain
texture and microstrucure evolution of polycrystalline copper". 9^{th}
Int. Conf. on Numerical Method in Industrial Forming Process (NUMIFORM 2007),

[18] A. Rollett, S. Lee, S. Sintay,
R.A. Lebensohn: "Exploration of the effect of polycrystal microstructure
on strain localization with a Fourier Transform viscoplastic model". 2007
TMS Annual Meeting & Exhibition,

[19] R.A. Lebensohn: "A
formulation based on Fast Fourier Transforms for the calculation of the
micromechanical behavior of plastically deformed 3-D polycrystals". 3rd International Conference on
Multiscale Materials Modeling,

[20] I.J. Beyerlein, R.A. Lebensohn and C.N.
Tome: "Modeling of texture and microstructural evolution in the equal
channel angular process". Mat. Sci.

[21] R.A. Lebensohn, M. Montagnat and P. Duval:
"Modeling the viscoplastic behavior and calculation of intracrystalline
fields in columnar ice polycrystals", in preparation.

[22] O. Castelnau, D. Blackman,
R.A. Lebensohn: "Modelisation micromecanique des roches du manteau
terrestre: Influence sur l'ecoulement regional sous une dorsale
oceanique". Demande de mise a disposition, Olivier Castelnau, CNRS,
2006-2007.

[23] S. Piazolo
(PI) et al.: "Subgrain
structure development in rocks and metals". 2006-2008 Collaborative Research Project, EUROCORES Programme, European
Science Foundation.

[24] M.W.
Jessell, P.D Bons, L. Evans., T. Barr and K. Stuwe: "Elle: a micro-process
approach to the simulation of microstructures". Computers and Geosciences
27, 17, 2001.

[25] S.
Nikolov, R.A. Lebensohn and D. Raabe: "Self consistent modeling of large
plastic deformation, texture and morphology evolution in semi-crystalline
polymers". J. Mech. Phys. Solids 54, 1350, 2006.

[26] R.A.
Lebensohn, E.M Bringa and A. Caro: "A viscoplastic micromechanical model
for the yield strength of nanocrystalline materials", Acta Mater, in
press.

[27] K.
Bhattacharya and P. Suquet: "A model problem concerning recoverable
strains of shape-memory polycrystals". Proc. Royal Soc. London A 461,
2797, 2005.

[28] K.
Bhattacharya (PI), R. Lebensohn and G. Ravichandran: "Interplay between
slip and transformation induced deformation in polycrystalline solids". US
Department of Energy, Basic Energy Sciences (BES) proposal, 2006.

[29] C.
Deo, C.N. Tome, R.A. Lebensohn and S. Maloy: "Modeling and simulation of
irradiation hardening in structural ferritic steels for advanced nuclear
reactors". Journal of Nuclear Materials, submitted.