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After graduating from Nancy School of Mines, Gilles Canova began his scientific career at McGill University, Montreal, where he obtained his MSc (1980) and PhD (1982) degrees. On returning to France in 1983, he became an Assistant Professor at the University of Metz and obtained his Doctorat d'Etat in 1986. In 1991 he joined the Centre National de la Recherche Scientifique as a Research Director and moved to Grenoble where he was to spend the rest of his scientific career in the GPM2 Laboratory. He was regularly invited as a scientific collaborator or consultant to research centres such as the Los Alamos National Laboratory (USA), Pechiney Research Centre (France), and LEM-ONERA (France). Gilles was about to spend one year in Berkeley at the Lawrence Livermore Laboratory (USA) when he sadly passed away in July 1997.

The activities of Gilles Canova covered a wide range of topics in mechanics and materials science. He was interested in many materials, ranging from metals to rocks, from polymers to biological materials. During his all too brief career he addressed many open questions concerning the mechanical properties of materials, from texture development to dislocation dynamics, from damage and fracture mechanisms to modelling complex topologies in multiphase materials. However, behind this variety, a unifying concept can easily be identified: building bridges between the different scales in materials modelling. In order to perform this ambitious programme, Gilles, who was trained in solid mechanics, had to borrow and apply concepts from materials science. He had to learn about the materials he was modelling, always respectful of the diversity of materials, always searching for the generic concepts. The ultimate goal of Gilles' activity was to find the right model for the right material. In this quest, he managed to involve many collaborations, always eager to share the pleasure of scientific research and discovery, with collaborators and with students.

Besides this general idea, changing scales, which gave to Gilles' scientific career its unity, his work can be described along three lines, which also structure this symposium to honour his memory. These three lines appeared successively in his research, but the new ones did not replace the former topics, and his interests widened and deepened.

The first topic he became interested in was constitutive equations and polycrystal plasticity. This activity filled the first part of his career and led to a long and fruitful collaboration with Fred Kocks, which culminated at the end of his life in co-authoring a chapter in a book on Texture and Anisotropy which is already becoming a classic in the field. During this time, he became interested in topics that were to become leitmotivs in his future scientific career: plastic instabilities, and self-consistent modelling of polyphase materials. He became naturally interested in self-consistent models through texture development modelling. Besides the problem in pure mechanics, which marked the beginning of a long-standing collaboration with his colleagues in Metz, he developed a continuous interest in rocks and geological materials. In his last years, in collaboration with his colleagues in Grenoble, he developed this field of polycrystal plasticity in pioneering publications on damage accumulation and recrystallisation textures. His attitude with respect to models of the mechanics of polyphase materials was typical of his scientific inquisitiveness and honesty: he was eager to identify the value and the limits of the theoretical tools he was developing. In this respect, each new material was a challenge: from the specificity of the materials, he identified the features which would require a new concept. At the very end of his scientific career, his interest in complex materials and topologies was another challenge to be met by the modelling techniques, namely the inclusion of length scales and bicontinuous structures in the framework.

Very rapidly, Gilles felt that an understanding of the length scales occurring in metal plasticity could not come from continuum mechanics alone. Again his contribution was to initiate the now very active field of dislocation simulations. In close collaboration with Ladislas Kubin, he developed the first code to simulate dislocation dynamics in three dimensions, taking into account all the microscopic mechanisms and the interactions between dislocations. This is still the most advanced attempt to bridge the gap between nanometric and macroscopic scales in plasticity. He continuously developed his ideas in this field, toward other crystallographic structures, modelling plastic instabilities. Always eager to compare his theory with experiment, which is the sign of a truly inventive theoretician, he recognised the technical limitations of the simulation in terms of sample size and accessible strains: accordingly his last publications in this field aimed at small-scale structures, including free surfaces, dealing with the indentation problem. His pioneering work in dislocation simulations had a tremendous impact on the physical metallurgy of plasticity, which had traditionally lagged behind the metallurgy of phase transformations as far as computer simulations were concerned. He showed that intensive computer simulations were able to provide efficient tools for what is still the central question of physical metallurgy: to go from the microstructure to the properties.

The last topic of Gilles' activity, and the last item of this symposium, was complex materials. It encompasses both polymers, which are often polyphase materials, and materials with a complex two-phase topology. His interest in polymers paralleled his interest in rocks, as these materials are especially challenging for modellers due to the drastic changes in properties close to the glass transition temperature. Again, Gilles collaborated with people from the field of polymer science, in order to learn about the specificity of the material he wanted to understand. The question of percolating reinforcements that resist any modelling attempt using a standard mean field approach was also a challenging one. This open question led Gilles to explore the behaviour of cellular structures, the mechanics of Cosserat media. This was in its infancy, but already very promising, when death brutally cut short Gilles' activity.

Sadly enough, one of the last of Gilles' publications is entitled `Matériaux àstructure renforçante percolante: quel type d'approche?' (`Materials with percolating reinforcement: what type of model?'). This sounds like a summary of Gilles' scientific career: seeking what in the material really needs a new modelling technique, and developing the most appropriate tool to understand the mechanical behaviour of this specific material, which can be further applied to a whole new class of materials. Over the few years that Gilles spent doing research, all his collaborators and students have shared his enthusiasm for understanding and modelling materials, have been grateful for the opportunity to have worked with him, and now mourn the sadness of his untimely departure. It was impossible to work with Gilles without becoming his friend, and almost part of his family.

The Symposium on Multi-scale Modelling of Mechanical Properties of Materials, published in this issue of Modelling and Simulation in Materials Science and Engineering, was held in memory of Gilles, in Autrans, France, during June 1998. The atmosphere was not meant to be sombre: we wanted this meeting to be as Gilles liked scientific discussion: open and lively. The three topics which filled Gilles' career are the titles of the three days, and the theme of the discussions. But the real unifying idea remains as a motto for the materials scientist and the engineer: to understand the mechanical behaviour of materials at all scales.

That is the meaning of this memorial symposium: it is the wish of his former collaborators and students to share with others the wealth of his influence, the creativity of his ideas, and to travel further on the path he has opened up, and on which we have been fortunate enough to be in his friendly company.

Y Brechet, J Y Cavaille, R Dendievel, M Dupeux, P Duval, L Flandin, L Guetaz, L Kubin, F Louchet, A Molinari and E Rauch

Organizing Committee of the Gilles Canova Memorial Symposium

This paper discusses the differences between the Taylor model and visco-plastic self-consistent (VPSC) models in describing plastic flow of polycrystals. The aspects covered are: invention of a length scale, consideration of the morphology of the microstructure, tuning of modelling parameters and the general applicability of the two different approaches.

Torsion testing is useful but cumbersome: useful as a technique to determine large-strain plastic behaviour in a plane-strain mode; cumbersome because the testing of a solid rod provides reliable data in only the simplest of circumstances, for example, when there is no strain hardening or rate sensitivity. The testing of short thin-walled tubes is widely regarded as the best current technique to determine general constitutive behaviour; a drawback is the requirement for large specimen blanks (say, 3 cm cube) and the complex machining procedure. Gilles Canova proposed an alternative - the testing of a series of solid rods of differing diameters, proving the principle by using 10 such rods for one test. We have undertaken a similar series of tests on just four rods plus one short tube for comparison. We found the results from the two types of torsion test in excellent agreement; however, it was not a critical test, inasmuch as the rate sensitivity of the pure copper used was too small. Had it been greater (as, for example, in aluminium and at higher temperature), the evaluation of perhaps six rods would have provided the constitutive response with regard to both hardening and rate sensitivity at the same time - which would require two short-tube tests (plus duplications). The main drawback of the multiple-rod test is that it requires considerable numerical effort in the evaluation. A closer integration with modelling of the constitutive behaviour would be helpful.

The overall response of heterogeneous viscoplastic materials is obtained using self-consistent schemes with the local response governed by convex flow potentials. The tangent model, originally proposed by Molinari et al is reviewed and extended and a discussion of the nonlinear inclusion problem is presented. A new general formulation of the self-consistent scheme is proposed and algorithms of calculations are suggested.

The viscoplastic self-consistent (VPSC) large strain polycrystal plasticity theory proved to be very useful for modelling the deformation in Earth materials. In contrast to most metals, rocks are composed of low-symmetry minerals with few slip systems. Also, most minerals have a high strain-rate sensitivity. Consequently, different orientations deform at different rates, contrary to the assumptions of the traditional Taylor model. The self-consistent method has been applied to predict textures and microstructures in many mineral systems and some highlights will be reviewed.

Starting at the surface of the Earth, ice deforms mainly by basal slip. Texture patterns observed in experiments and in the large polar ice sheets are well predicted with the VPSC model. In sediments, concentrations of salt (halite) deform by buoyant upwelling into salt domes. When VPSC was applied to halite, entirely different textures were predicted than those with the Taylor model, in better accordance with low temperature deformation experiments where {110}10 is the prevalent slip system. Calcite has been an excellent example to illustrate how textures measured in natural rocks can be used to infer the deformation history in the Earth's crust. In calcite, VPSC automatically simulates the effects of `curling' in axial compression, producing plane strain deformation at the microscopic scale. Many minerals are recrystallized. VPSC has been used as the basis of a model for dynamic recrystallization which balances nucleation of highly deformed grains and growth of less deformed grains. Applying it to quartz made it possible to explain textures in naturally deformed quartzites, particularly those deformed in simple shear. The upper mantle of the Earth is largely composed of olivine and deforms in large convection cells that extend over thousands of kilometres. Polycrystal plasticity predicts a highly heterogeneous texture evolution along streamlines with strong development of preferred orientation. Since single crystals of olivine are elastically anisotropic, oriented polycrystals also display anisotropy. Predicted anisotropies of seismic wave velocities of 5-10% in the model mantle agree well with those observed by seismologists. Finally the still highly enigmatic inner core is composed of solid -iron (hexagonal close packed) and seismologists have observed that wave velocities are slightly higher parallel to the Earth's axis than in the equatorial plane. Again, VPSC simulations suggest that this anisotropy in the centre of the Earth could be due to deformation during convection.

The examples illustrate, on a grand scale, that VPSC has not only helped us to unravel the deformation history of the Earth but also contributes towards to a better understanding of the deformation behaviour of complex and anisotropic materials.

Viscoplastic self-consistent polycrystal models have been successful in addressing and explaining features of plastic deformation which cannot be treated with the Taylor condition of isostrain. In particular, these models have been applied to the simulation of plastic deformation and texture development in materials with hexagonal, trigonal, orthorhombic and triclinic symmetry.

An important assumption required to solve the equilibrium equation within self-consistent formulations is that the strain-rate varies linearly with the stress in the homogeneous effective medium surrounding the inclusion. The characteristic of such a linear relation has been a matter of debate and two extreme cases can be identified: the tangent and the secant approaches. The secant approach has associated with it a stiffer inclusion-matrix interaction than the tangent approach and is closer to the Taylor approach.

In this work we perform a systematic study of the implications of both assumptions on the response of cubic and hexagonal materials (texture development, system activity, stress and strain-rate deviations). In addition, we argue that the strength of the matrix-inclusion interaction should not be constant but should depend on the capability of each orientation to accommodate the particular deformation mode imposed externally. As a consequence, we propose a relative directional compliance (RDC) criterion for defining a variable interaction between grain and matrix depending on their relative compliances and compare the predictions of the RDC approach with the predictions of the secant and the tangent schemes.

A viscoplastic self-consistent (VPSC) multiscale model for predicting the plastic behaviour of materials with lamellar microstructures is presented. In this approach, three different scales are defined: the microscopic, at single crystal (single lamella) scale; the mesoscopic, at the scale of a single lamellar colony and the macroscopic, at the lamellar aggregate level. This formulation is compared with the two-site VPSC model and both are applied to the prediction of the plastic anisotropy of an as-cast sample of a lamellar -TiAl based alloy (Ti-48 at%Al-2 at%Cr) tested under compression.

A brief summary of the talks presented during the second day of the Symposium on Multiscale Modelling of Mechanical Properties of Materials held in Autrans (France) on 13-15 June, 1998 to commemorate Gilles Canova is given, along with an account of discussions and the author's views.

Mesoscale simulations have recently been developed in order to better understand the collective behaviour of dislocations and their effects on the mechanical response. Those simulations deal with dislocations discretized into segments which are allowed to move in a three-dimensional (3D) discrete network. This network is a sublattice of the original crystalline lattice network. The minimum distance between two points is defined by the annihilation distance for two edge dislocations, i.e. the minimum distance for which two edge dislocations can coexist without instantaneous collapse. The elastic theory can still be applied in the simulated volume, since the minimum distance is large compared to the dislocation core radius within which nonlinear expressions should be taken into account in the dislocation-dislocation interaction. This property allows us to use the superposition principle to enforce boundary conditions on the simulation box. This paper details the rigorous boundary conditions applied when the simulation box is supposed to be either a bulk crystal, a free standing film or a finite crystal submitted to a complex loading.

One avenue being pursued in the development of dislocation-based models of plasticity is the explicit simulation of the dynamics of dislocations, based on the recognition that such dislocations are the carriers of plasticity. The construction of models of dislocation dynamics requires insights into both the nucleation and interaction of dislocations, many of the details of which fall outside the domain of validity of the linear theory of elasticity. The aim of the present paper is to show how preliminary steps have been made to elucidate the mechanisms of dislocation nucleation and interaction, and to illustrate how such information can be imported into explicit models of dislocation dynamics. This effort reflects, in part, the research program of Gilles Canova, to whom the present volume is dedicated and to whom the authors dedicate this paper.

Three-dimensional discrete dislocation dynamics (DDD) has been used successfully to investigate different aspects of plasticity. An investigation of thin-film plasticity with the help of these DDD schemes requires detailed modelling of the interfaces and surfaces of the film. One possible method is to compensate for the normal stresses that a dislocation population exerts on a surface by appropriate point loads. This traction compensation method is extended to a free-standing film, where the interaction of the two opposing free surfaces must be taken into account. The influence of the free surfaces on the operation of a Frank-Read source is investigated.

For the investigation of the dynamic properties of dislocation systems different algorithms suitable to run on parallel computers are reviewed. In the first part of the paper parallel methods of the numerical integration of the equations of motion of dislocations are outlined. In the second part the concept of stochastic dislocation dynamics is described. Results of numerical investigations on dislocation patterning obtained on different parallel computers are also presented.

This paper aims to provide an overview of observed situations in heterogeneous materials, both in terms of the geometry of the multiphase structure (topology, shapes and scales) and in terms of the mechanical contrast between the constituents. An overview of some tools available in micromechanics will be given in order to answer the following two questions: What is the best model class for a given material? What is the best model material to test a mechanical theory? Some remarks which probably require new conceptual approaches will be outlined in the conclusion.

The stress-strain behaviour of polyethylene, determined from mechanical tests at constant true strain rate, is characterized by high strain hardening in uniaxial tension and strain softening in simple shear. The strong influence of the deformation path is correlated with the different crystalline textures developed. This behaviour is analysed on a theoretical basis in terms of the specific response of a polycrystalline viscoplastic aggregate and of a rubber-like hyperelastic network. For the 100% crystalline model (by Canova), the tensile hardening is overestimated with respect to experiment, but the shear softening is correctly predicted. In contrast, for the 100% amorphous model (by van der Giessen), the tensile hardening is smaller, but the shear response cannot reproduce the softening. Therefore, the discussion is focused on a composite model which mixes the constitutive equations of the crystalline phase and of the amorphous phase through the self-consistent scheme of Hervéand Zaoui. Originally established for elastic multiphase materials, this model is applied here to the nonlinear case of polyethylene by means of an incremental tangent computation. The stress-strain curves predicted by this approach can be fitted with very good precision to the experimental curves in tension and shear with reasonable values of the microstructural parameters.

Classical homogenization methods do not account for size effects in the effective properties of polycrystals: they do not predict the influence of grain size nor the width of shear bands in localization phenomena. The polycrystal can be regarded as an example of a heterogeneous Cosserat material. A method is proposed to estimate the effective properties of aggregates of Cosserat media. A validation of the methodology is given in the case of linear Cosserat elasticity. Some steps towards an extension of the self-consistent scheme to Cosserat materials are then presented.

We present experimental results for texture evolution in NiAl polycrystals under compression at elevated temperatures of initially extruded samples. Based on these experimental observations and on existing experimental results, we discuss the operating slip systems in NiAl and address the lack of five independent slip systems in this ordered intermetallic material. We also propose the use of crystal plasticity modelling to simulate texture evolution in NiAl. We use the projection model of Parks and Ahzi which accounts explicitly for the missing degrees of kinematic freedom due to the activation of only three independent slip systems in NiAl crystals. Different sets of slip systems were used in this model and in the Taylor model to simulate texture evolution under uniaxial tension of initially isotropic NiAl polycrystals, and uniaxial compression of initially textured polycrystals. The contribution of each set of slip systems to texturing is discussed. Predicted results are compared with experimental observations. We conclude that the use of the projection model with the three independent slip systems, experimentally observed to operate at elevated temperatures, leads to texture predictions in good agreement with experiments.

In order to furnish some input data to Monte Carlo codes developed for the simulation of static recrystallization in low carbon steels, two polycrystalline models are used in conjunction with four different hardening laws to estimate numerically the stored energy within individual grains, which is due to the increase in dislocation density during rolling. Three quantities are calculated as a function of final orientation, which are believed to be good estimates of this energy: these are the average dislocation density (linked to the square of an average reference shear stress), the total plastic work and the final plastic work rate. It is thus found that the three selected parameters present the same variation trends for a given model, whatever the hardening law. However, the Taylor and VPSC models lead to opposite conclusions: at the end of the simulated rolling process, the (respectively ) orientations are the hardest (respectively softest) with the Taylor model and the softest (respectively hardest) with the VPSC one; thus, the present data cannot be used in the present state to perform recrystallization simulations but may be used to validate the different polycrystalline models, since they are more sensitive to the interaction law than the texture evolution or macroscopic response.

This work deals with the AC electrical properties measurements under large strain of composites strongly contrasted in the viewpoint of their electrical and mechanical properties. First, the possibility to model a strongly heterogeneous material (from the viewpoint of electrical properties) using an improvement of a RC-type model is explored; this model relates the R and C values to the geometrical characteristics of the network of reinforcing fibres. Secondly, a powerful method is developed to characterize the filler rearrangement by measuring the electrical properties. It is concluded that the evolution of the real part of the conductivity is related to the damage of the percolating network, whereas the imaginary part of the conductivity reflects the global rearrangement of the filler in the matrix under large strain.

The paper is concerned with the problem of constructing equivalent stress-equivalent strain curves at large strains. For the equivalent strain, the average accumulated crystallographic shear is used, while for the equivalent stress, the resolved shear stress is employed. The latter is obtained from the work conjugacy condition. In such a construction of the hardening curve, the Taylor factor appears to be the major factor that can be calculated from polycrystal deformation texture models. In this paper, the viscoplastic Taylor and self-consistent approaches are employed to calculate the Taylor factors. The self-consistent model was calibrated on the torsion texture development which is the most sensitive to the polycrystal model parameters at large strains. The obtained Taylor factors show important variations in torsion, compression and rolling. They have been used to convert experimentally measured work hardening data on copper into resolved shear stress-resolved shear strain curves. The effect of the Taylor factor on the absolute hardening rate was found to be significant at a large strain range of deformation. The simulation textures were markedly different from the measured textures at very large strains where both polycrystal texture deformation models fail to predict the correct texture evolution. For this reason, the textures were measured at increasing strains at 11 points in rolling, at 12 points in compression and at four points in torsion from where the Taylor factors were calculated by both of the models in order to construct the equivalent stress-equivalent strain curves.

In body centred cubic (bcc) crystals at low temperatures, the thermally activated motion of screw dislocations by the kink-pair mechanism governs the yield properties and also affects the strain hardening properties. In this work, the average strength of dislocation junctions is derived and numerically estimated in the case of Nb and Ta crystals. This allows us to extend an existing simulation of dislocation dynamics in bcc crystals to the case of the motion of a screw dislocation line through a random distribution of forest obstacles. Numerical results are presented in the case of Ta crystals and at two temperatures, 160 K and 215 K. They are complemented by a simple model that applies quite generally to bcc metals at low temperatures. It is shown that forest hardening is made up of two contributions, a free-length effect that depends on the length of the mobile screw segments and whose dependence on forest obstacle density is logarithmic and a line tension effect linearly proportional to the obstacle density. As a result of the thermally activated character of screw dislocation mobility, the relative weight of the two contributions to forest hardening depends on the temperature and strain rate.