Multiscale Modeling of Heterogenous Materials

From Microstructure to Macro-Scale Properties

Oana Cazacu, Oana Cazacu

• Publisher: Wiley-ISTE
• Year: 2013
• ISBN:
  9781118623503
• Edition: 1
• Topic: Technology & Engineering
• Subtopic: Materials Science
• Index: Technology & Engineering

A material's various proprieties is based on itsmicroscopic andnanoscale structures.This book provides an overview of recent advances in computational methods for linking phenomena in systems that span large ranges of time and spatial scales. Particular attention is given to predicting macroscopic properties based on subscale behaviors. Given the book's extensive coverage of multi-scale methods for modeling both metallic and geologic materials, it will be an invaluable reading for graduate students, scientists, and practitioners alike.

Chapter 1

Accounting for Plastic Strain Heterogenities in Modeling Polycrystalline Plasticity: Microstructure-based Multi-laminate Approaches ¹

1.1. Introduction

Models of the mechanical or physical behavior of materials are most efficient when they are microstructure-based. However, reproducing reality is not achievable and energy savings also demand models that do not become highly consumptive of computer space and time. With the goal of bridging scales in models that start from elementary atomistic models to simulate an overall response, the search for compromises between the microstructural descriptions and the resulting simulation accuracy will remain a challenging area for a long time. A smart alternative to running huge “ab initio” computational calculations is to anticipate which microstructural features do really matter at the macro-scale concerned according to the considered situation.

In the specific field of modeling the plastic behavior of heterogenous non-linear metallic materials, effective properties are reasonably approached when simultaneously considering i) a good enough description of the evolving morphology, ii) an appropriate homogenization scheme according to the material morphology and behavior type, iii) a relevant microstructural modeling of intra-crystalline plasticity. In all these domains the last few decades have significantly enriched the available background. With regard to the third point, dislocation dynamic simulations have clarified many features that concern crystal hardening evolutions with strain [DEV 06], the anisotropy of which is one of the most complex questions not yet answered, even for the simplest cubic structures. With regard to the other two domains, enhanced morphological descriptions by integrating n-point statistics with increasing n [TAL 97], as well as improved first-and second-order developments to better describe the non-linearity of the phase plastic behavior laws [PON 98] have enabled remarkable gains in the accuracy of accessible behavior estimates or bounds. The possible further improvements in the global modeling of polycrystalline plasticity discussed here address points that concern in a combined manner the morphology description, the homogenization framework and the behavior approximation in terms of plastic flow criterion. For theoretical details and simulation examples relating to this book, see [FRA 07, FRA 08].

For such aggregates that deform by crystallographic shear mechanisms (i.e. slip, twinning or also transformation plasticity up to a volume change), we first question the granular description that is conventionally used, compared to an alternative description in terms of grain boundary and sub-boundary orientation distribution. Secondly, between the evolution of the sub-boundary spatial arrangement and the shear activity in the material, a link is made that is based on a multi-laminate approach of plastic heterogenities. Such a multi-laminate approach to describe the current morphology of the strained material in turn acts on the homogenization scheme that can be preferentially used. Comparing with the inclusion-based modeling, further advantages are pointed out, as the more natural reference to an equivalent homogenous super-crystal that justifies introducing a single plastic potential for the whole aggregate, or the possibility of accounting for a grain size effect. Section 1.2 illustrates some support to a sub-boundary based morphology description of polycrystal plasticity, section 1.3 introduces the considered multi-laminate representation and section 1.4 summarizes the proposed modeling framework that results.

1.2. Polycrystal morphology in terms of grain and sub-grain boundaries

1.2.1. Some evidence of piece-wise regularity for grain boundaries

When looking at micrographic or nanographic pictures of metallic aggregates, there is plenty of evidence that grains are polyhedral domains whose boundary facets result from the elaboration route. Figure 1.1 shows two examples that concern aggregates of micrometric (left) or nanometric (right) grains. On a topological ground, it is also obvious that if, ideally speaking, all the grains were convex and smooth — i.e. with rounded edges — some complementary matrix phase, of vanishing volume fraction, should be accounted for to ensure matter continuity and compactness. However, part of the grains can even be non-convex polyhedrons, some have sharp edges and the widely used assumption of ellipsoidal grain shape is definitely a rough one. Furthermore, a total space mapping is hardly obtained with a single grain shape, unless with the particular polyhedrons identified in stereology. The related rank-2 correlation functions between pairs of grain centroids (A-A, A-B, A-C, B-B, B-C, etc.) are in general different for shapes that are not congruent and they neither reduce to a single ellipsoidal symmetry nor to a set of different ellipsoidal symmetries, as discussed in [FRA 04a]. These are real limits to inclusion-based aggregate approaches.

Figure 1.1. TEM microstructures of (left) micrometric grains in steel (by courtesy of K. Zhu, LPMTM) and of (right) ultra-finely grained alpha iron (by courtesy of G. Dirras, LPMTM)

An alternative type of morphology description that has emerged is from considering the orientation distribution of the grain boundaries. Not entering into the details here, from the still open analysis domain of boundary networks [SHU 05] or coincident site lattice [COU 05], an essential feature of these aggregate structures is noteworthy: the crystallographic orientations of the boundary facets of such polyhedral grains are mainly oriented parallel to or close to dense planes.

1.2.2. Charact...

Table of contents

• Cover
• Title
• Copyright
• Foreword
• Chapter 1: Accounting for Plastic Strain Heterogenities in Modeling Polycrystalline Plasticity: Microstructure-based Multi-laminate Approaches
• Chapter 2: Discrete Dislocation Dynamics: Principles and Recent Applications
• Chapter 3: Multiscale Modeling of Large Strain Phenomena in Polycrystalline Metals
• Chapter 4: Earth Mantle Rheology Inferred from Homogenization Theories
• Chapter 5: Modeling Plastic Anistropy and Strength Differential Effects in Metallic Materials
• Chapter 6: Shear Bands in Steel Plates under Impact Loading
• Chapter 7: Viscoplastic Modeling of Anisotropic Textured Metals
• Chapter 8: Non-linear Elastic Inhomogenous Materials: Uniform Strain Fields and Exact Relations
• Chapter 9: 3D Continuous and Discrete Modeling of Bifurcations in Geomaterials
• Chapter 10: Non-linear Micro-cracked Geomaterials: Anisotropic Damage and Coupling with Plasticity
• Chapter 11: Bifurcation in Granular Materials: A Multiscale Approach
• Chapter 12: Direct Scale Transition Approach for Highly-filled Viscohyperelastic Particulate Composites: Computational Study
• Chapter 13: A Modified Incremental Homogenization Approach for Non-linear Behaviors of Heterogenous Cohesive Geomaterials
• Chapter 14: Meso- to Macro-scale Probability Aspects for Size Effects and Heterogenous Materials Failure
• Chapter 15: Damage and Permeability in Quasi-brittle Materials: from Diffuse to Localized Properties
• Chapter 16: A Multiscale Modeling of Granular Materials with Surface Energy Forces
• Chapter 17: Length Scales in Mechanics of Granular Solids
• List of Authors
• Index