Preface

Polycrystalline structures are conglomerates of a large number of crystals irregularly situated, yet bound to each other strongly enough to behave as a whole. As the size and the shape of these crystals are irregular too, the latter are called grains or crystallites. The boundary surfaces connecting the grains have crystal structures that are not identical to those of the adjacent crystallites. They are distortions, which allow a smooth transition between the structures within the grains in contact. The mosaic of these borderline regions represents an extended block of two dimensional imperfections.

A mechanical loading of these materials leads to deformations. With small loadings, the deformation is elastic, slip and elastic (Young's) modulus being its characteristics. The Young's modulus is an important parameter of the polycrystal materials determining its resistance to deformation.

For higher loadings, the deformation becomes plastic. Theoretical studies, experimental data, as well as practical observations show that this type of deformation involves slippage in the material and active participation of its twodimensional imperfections. With lower temperatures (less than 0.4 Tm for metals and 0.6 Tm for alloys, Tm denoting the melting point) slippage does not occur uniformly, but remains confined to smaller regions, which appear successively. At higher temperatures, the critical break tension drops down. Thus, smaller loadings prove sufficient to bring about deformation effects, such as dislocations slip, twinning, sliding of grain boundaries, etc.

Stress level, stress rate, and temperature are the parameters characterizing the plastic deformation of polycrystalline materials.

In these materials, the macroscopic value of their parameters is a mean value, resulting from taking the average over domains comprising a considerable number of grains with usually different orientations. In this way, they turn out isotropic as compared to the monocrystals, in which sharp anisotropy is observed. In special cases, the orientations of the grains show more or less preferred directions, leading to anisotropy. To the best of my knowledge, one of them – polycrystalline tungsten, is dealt with for the first time in specialized literature by Dr. P. V. Galptshyan in the chapter: "Strength of Polycrystalline Materials".

The book "Polycrystalline Materials" presents theoretical and practical investigations of some polycrystals materials.

Preface IX

damage processes, such as small cracks in the microstructure and fatigue. Damage initialization and evolution are directly influenced by the locally anisotropic behavior of the microstructure, as determined by the combination of random grain shapes and sizes, different crystallographic orientations, inclusions, voids, and other microstructural features. For the bulk of a grain, constitutive models assuming pure anisotropic elasticity, as well as anisotropic elasticity in combination with crystal plasticity have been used. Analytical models for grain geometry, in view of calculating the properties of crystalline aggregates, involve 2D and 3D Voronoi tesselations, whereas the method of X-ray diffraction contrast tomography was utilized to measure

Several cases of 3D Voronoi tesselations, and two cases of stainless steel wire have been treated. Grain boundaries were explicitly modeled, using the cohesive zone approach, with finite elements of zero physical thickness. Initialization and early development of grain boundary damage, with respect to stainless steel, were traced numerically for several constitutive laws. Differences obtained in the results are small when the approach of anisotropic elasticity is compared to combining the latter with crystal plasticity, except for the computation time required- more than two times

**Section II "Methods of Synthesis, Structural Properties Characterization, and Applications of Some Polycrystalline Materials"** includes the following four

"*Nanocrystalline NASICON Materials - Structure and Electrical Properties*" by Dr.

The chapter deals with an important class of solid electrolytes – sodium (NA) super (S)-ionic (I) conductors (CON): Ax By (PO4 )3 , where A denotes an alkali metal ion and B denotes a multivalent metal ion. They are widely tested in energy applications, e.g. electric vehicles, and having the advantage of high ionic conductivity together with the stability of the phosphate units. The authors have investigated the correlation between ionic conduction and phase symmetry for a family of NASICONs, comprising LiTi2(PO4)3 and A3M2(PO4)3 where A = Li, Na and M = Cr, Fe) . Structural characterization was obtained by spectroscopic and diffraction techniques, while mobile ions characterization proceeded through impedance spectra. Application of the

"*Structural Characterization of New Perovskites*" by Dr. A. D. Lozano – Gorrin.

The author discusses some general features of the perovskite-type materials, including relatively new methods of their preparation (solution combustion, sonochemical procedures, microwave assisted synthesis) as well as characterization of their structure and physical properties by a variety of diffraction techniques (X-rays-, electron- and

these properties and make a 3D characterization of the grains.

longer for the second approach.

Lakshmi Vijayan and Prof. G. Govindaraj.

materials studied has been discussed as well.

neutron diffraction).

chapters:

**Section I "Plastic Deformation, Strength and Grain-Scale Approaches to Polycrystals"**  comprises the following three chapters:

"*Scale-Bridging Modelling of Plastic Deformation and Damage Initiation in Polycrystals*" by Dr. Anxin Ma and Prof. Alexander Hartmaier.

This chapter reviews the current state of modelling the phenomenon of plastic deformation in its various aspects. The authors' analysis involves the macro-, meso-, micro-, and atomic scales using the finite element method, representative volume element approaches, the dislocation dynamics method, and molecular dynamics simulation. Steels have been used to generate realistic microstructures for all multiphase polycrystalline materials studied. Possible approaches in order to bridge the different length scales have been discussed and successful multiscale modelling applications reported. On the basis of recent constitutive models provided by continuum mechanics, the authors have elaborated a number of numerical procedures aiming at the integration of results obtained for several length scales. In this way, they were able to build representative volume element (RVE) models for the mechanical behavior of materials, which are heterogeneous, with respect to the nature of grains and phases they are made of. Once a RVE for a given microstructure is constructed and the critical deformation and damage mechanisms are included into the constitutive relations, this RVE can be applied to calculate stress-strain curves and other mechanical data. The advantage of this approach is that the effects of grain size and strength of the grain boundaries on the macroscopic mechanical response of a material can be predicted.

#### "*Strength of a Polycrystalline Material*" by Prof. P. A.Galptshyan

It is shown that due to the greater concentration of stresses in it, the polycrystalline material has a strength less than that of a monocrystal made of the same substance. Hence, in order to enhance its strength, one has to reduce the stresses in the material. A remarkable case is polycrystalline tungsten, whose elastic anisotropy factor proves zero. This kind of tungsten is known to be a most durable substance, more so than the tungsten monocrystals and even the diamond. For example, the ultimate strength under tension of unanealed wires of polycrystalline tungsten is in the range of 1800 MPa to 4150 Mpa, depending on the diameter of the wire. For diamond monocrystals, it equals 1800 MPa at 200 C. It is worth noting that a correspondence has been found for polycrystalline metals between their ultimate strength and their modulus of elasticity: the two parameters are changing in the same direction.

"*Grain-Scale Modelling Approaches for Polycrystalline Aggregates*" by Dr. I. Simonovski and Dr. I. Cizelj

It has been shown that polycrystalline aggregates their microstructure, which plays an important role in the evolution of stresses and strains, and in the development of damage processes, such as small cracks in the microstructure and fatigue. Damage initialization and evolution are directly influenced by the locally anisotropic behavior of the microstructure, as determined by the combination of random grain shapes and sizes, different crystallographic orientations, inclusions, voids, and other microstructural features. For the bulk of a grain, constitutive models assuming pure anisotropic elasticity, as well as anisotropic elasticity in combination with crystal plasticity have been used. Analytical models for grain geometry, in view of calculating the properties of crystalline aggregates, involve 2D and 3D Voronoi tesselations, whereas the method of X-ray diffraction contrast tomography was utilized to measure these properties and make a 3D characterization of the grains.

VIII Preface

of some polycrystals materials.

material can be predicted.

and Dr. I. Cizelj

"*Strength of a Polycrystalline Material*" by Prof. P. A.Galptshyan

elasticity: the two parameters are changing in the same direction.

comprises the following three chapters:

Dr. Anxin Ma and Prof. Alexander Hartmaier.

The book "Polycrystalline Materials" presents theoretical and practical investigations

**Section I "Plastic Deformation, Strength and Grain-Scale Approaches to Polycrystals"** 

"*Scale-Bridging Modelling of Plastic Deformation and Damage Initiation in Polycrystals*" by

This chapter reviews the current state of modelling the phenomenon of plastic deformation in its various aspects. The authors' analysis involves the macro-, meso-, micro-, and atomic scales using the finite element method, representative volume element approaches, the dislocation dynamics method, and molecular dynamics simulation. Steels have been used to generate realistic microstructures for all multiphase polycrystalline materials studied. Possible approaches in order to bridge the different length scales have been discussed and successful multiscale modelling applications reported. On the basis of recent constitutive models provided by continuum mechanics, the authors have elaborated a number of numerical procedures aiming at the integration of results obtained for several length scales. In this way, they were able to build representative volume element (RVE) models for the mechanical behavior of materials, which are heterogeneous, with respect to the nature of grains and phases they are made of. Once a RVE for a given microstructure is constructed and the critical deformation and damage mechanisms are included into the constitutive relations, this RVE can be applied to calculate stress-strain curves and other mechanical data. The advantage of this approach is that the effects of grain size and strength of the grain boundaries on the macroscopic mechanical response of a

It is shown that due to the greater concentration of stresses in it, the polycrystalline material has a strength less than that of a monocrystal made of the same substance. Hence, in order to enhance its strength, one has to reduce the stresses in the material. A remarkable case is polycrystalline tungsten, whose elastic anisotropy factor proves zero. This kind of tungsten is known to be a most durable substance, more so than the tungsten monocrystals and even the diamond. For example, the ultimate strength under tension of unanealed wires of polycrystalline tungsten is in the range of 1800 MPa to 4150 Mpa, depending on the diameter of the wire. For diamond monocrystals, it equals 1800 MPa at 200 C. It is worth noting that a correspondence has been found for polycrystalline metals between their ultimate strength and their modulus of

"*Grain-Scale Modelling Approaches for Polycrystalline Aggregates*" by Dr. I. Simonovski

It has been shown that polycrystalline aggregates their microstructure, which plays an important role in the evolution of stresses and strains, and in the development of Several cases of 3D Voronoi tesselations, and two cases of stainless steel wire have been treated. Grain boundaries were explicitly modeled, using the cohesive zone approach, with finite elements of zero physical thickness. Initialization and early development of grain boundary damage, with respect to stainless steel, were traced numerically for several constitutive laws. Differences obtained in the results are small when the approach of anisotropic elasticity is compared to combining the latter with crystal plasticity, except for the computation time required- more than two times longer for the second approach.

**Section II "Methods of Synthesis, Structural Properties Characterization, and Applications of Some Polycrystalline Materials"** includes the following four chapters:

"*Nanocrystalline NASICON Materials - Structure and Electrical Properties*" by Dr. Lakshmi Vijayan and Prof. G. Govindaraj.

The chapter deals with an important class of solid electrolytes – sodium (NA) super (S)-ionic (I) conductors (CON): Ax By (PO4 )3 , where A denotes an alkali metal ion and B denotes a multivalent metal ion. They are widely tested in energy applications, e.g. electric vehicles, and having the advantage of high ionic conductivity together with the stability of the phosphate units. The authors have investigated the correlation between ionic conduction and phase symmetry for a family of NASICONs, comprising LiTi2(PO4)3 and A3M2(PO4)3 where A = Li, Na and M = Cr, Fe) . Structural characterization was obtained by spectroscopic and diffraction techniques, while mobile ions characterization proceeded through impedance spectra. Application of the materials studied has been discussed as well.

"*Structural Characterization of New Perovskites*" by Dr. A. D. Lozano – Gorrin.

The author discusses some general features of the perovskite-type materials, including relatively new methods of their preparation (solution combustion, sonochemical procedures, microwave assisted synthesis) as well as characterization of their structure and physical properties by a variety of diffraction techniques (X-rays-, electron- and neutron diffraction).

#### X Preface

"*Controlled Crystallization of Isotactic Polypropylene Based on Alpha/Beta Compounded Nucleating agents - From Theory to Practice*" by Prof. Zhong Xin and Dr.Yaoqi Shi.

Isotactic polypropylene (iPP), one of the most important thermoplastic polymers, exhibits very interesting polymorphic behavior. Its different crystalline forms have different optical and mechanical properties. In this respect, alpha/beta compounded nucleating agents for polypropylene attract more and more attention.

Three kinds of well studied alpha/beta compounded NAs (phosphate/amide, sorbitol/amide, and phosphate/carboxylate) have been reviewed by discussing their influence on the crystallization kinetics, crystallization morphologies, and mechanical properties of iPP. The results show that these three NAs are able to not only increase the crystallization temperature of iPP, but also to shorten its crystallization half-time. Consequently, they are able to considerably reduce the molding cycle time. It has also been found that the type of nucleation of the polymer could be changed, while the geometry of its crystal growth remains the same.

### "*Influence of Irradiation on Mechanical Properties of Materials*" by Prof. V. I. Krasilnikov.

This chapter discusses substantial changes in the mechanical properties of materials, radiation embrittlement, and hardening being two of its most common and important effects. Both of them depend on the temperature of the irradiated material.

The author has proposed a model, based on the interaction of vacancies with interstitial barriers in order, to explain and investigate the saturation of the dependence of yield strength on radiation dosage. In the framework of this model, equations describing the evolution of barrier densities, as well as yield strength have been obtained in analytical form. It has been shown that with increasing the intensity of the barrier recombination processes, the yield strength of the irradiated material decreases, the dependence being nonlinear. In the case of radiation hardening, the model proves valid for both low and large doses.

Another model quantitively describing the dependence of the yield strength of irradiated materials on their temperature has also been introduced and applied. The results show its usefulness in dealing with the processes of plastic deformation under irradiation. Some implications about materials used in the construction of nuclear reactors have been discussed.

The research has been carried out to increase the lifetime of III and IV generation reactors and practical ITER-materials.

> **Prof. D.Sc.Eng. Zachari Zachariev** Institute of Polymers Bulgarian Academy of Sciences, Sofia Bulgaria

X Preface

"*Controlled Crystallization of Isotactic Polypropylene Based on Alpha/Beta Compounded Nucleating agents - From Theory to Practice*" by Prof. Zhong Xin and Dr.Yaoqi Shi.

Isotactic polypropylene (iPP), one of the most important thermoplastic polymers, exhibits very interesting polymorphic behavior. Its different crystalline forms have different optical and mechanical properties. In this respect, alpha/beta compounded

Three kinds of well studied alpha/beta compounded NAs (phosphate/amide, sorbitol/amide, and phosphate/carboxylate) have been reviewed by discussing their influence on the crystallization kinetics, crystallization morphologies, and mechanical properties of iPP. The results show that these three NAs are able to not only increase the crystallization temperature of iPP, but also to shorten its crystallization half-time. Consequently, they are able to considerably reduce the molding cycle time. It has also been found that the type of nucleation of the polymer could be changed, while the

"*Influence of Irradiation on Mechanical Properties of Materials*" by Prof. V. I. Krasilnikov.

effects. Both of them depend on the temperature of the irradiated material.

This chapter discusses substantial changes in the mechanical properties of materials, radiation embrittlement, and hardening being two of its most common and important

The author has proposed a model, based on the interaction of vacancies with interstitial barriers in order, to explain and investigate the saturation of the dependence of yield strength on radiation dosage. In the framework of this model, equations describing the evolution of barrier densities, as well as yield strength have been obtained in analytical form. It has been shown that with increasing the intensity of the barrier recombination processes, the yield strength of the irradiated material decreases, the dependence being nonlinear. In the case of radiation hardening, the

Another model quantitively describing the dependence of the yield strength of irradiated materials on their temperature has also been introduced and applied. The results show its usefulness in dealing with the processes of plastic deformation under irradiation. Some implications about materials used in the construction of nuclear

The research has been carried out to increase the lifetime of III and IV generation

**Prof. D.Sc.Eng. Zachari Zachariev**

Bulgarian Academy of Sciences, Sofia

Institute of Polymers

Bulgaria

nucleating agents for polypropylene attract more and more attention.

geometry of its crystal growth remains the same.

model proves valid for both low and large doses.

reactors have been discussed.

reactors and practical ITER-materials.

**Part 1** 

**Plastic Deformation, Strength and Grain –** 

**Scale Approaches to Polycrystals** 

**Part 1** 

**Plastic Deformation, Strength and Grain – Scale Approaches to Polycrystals** 

**0**

**1**

*Germany*

**Scale Bridging Modeling of Plastic Deformation**

*Interdisciplinary Centre for Advanced Materials Simulation, Ruhr-University Bochum*

Plastic deformation of polycrystalline materials includes dislocation slip, twinning, grain boundary sliding and eigenstrain produced by phase transformations and diffusion. These mechanisms are often alternative and competing in different loading conditions described by stress level, strain rate and temperature. Modelling of plasticity in polycrystalline materials has a clear multiscale character, such that plastic deformation has been widely studied on the macro-scale by the finite element methods, on the meso-scale by representative volume element approaches, on the micro-scale by dislocation dynamics methods and on the atomic scale by molecular dynamics simulations. Advancement and further improvement of the reliability of macro-scale constitutive models is expected to originate from developments at microstructural or even smaller length scales by transfering the observed mechanisms to the macro-scale in a suited manner. Currently efficient modelling tools have been developed for different length scales and there still exists a challenge in passing relevant information between models on different scales. This chapter aims at overviewing the current stage of modelling tools at different length scales, discussing the possible approaches to bridge

different length scales, and reporting successful multiscale modelling applications.

Fig. 1. Multiphase polycrystalline RVE (right) with 90% matrix and 10% precipitate. The grain size has a normal distribution (middle) and the [111] polfigure (left) shows a random

**1. Introduction**

texture.

**and Damage Initiation in Polycrystals**

Anxin Ma and Alexander Hartmaier
